Chapter 1
OGCMs and MRI.COM
This chapter outlines the ocean general circulation models (OGCM) and the status of MRI.COM.
1.1
What do OGCMs cover?
OGCMs are supposed to simulate relatively largescale phenomena such as globalscale thermohaline circulations, basinscale winddriven circulations, and mesoscale eddies (Figure 1.1). Smallscale processes that are either unresolved or neglected might be incorporated in some forms of subgrid scale (SGS) parameterizations. In the future, the coverage of the OGCMs will be extended to directly resolve the smaller spacescale and the shorter timescale phenomena such as tides and vertical convection, but they are not the primary targets of the current OGCMs. The current OGCMs could cover phenomena from thermohaline circulations to mesoscale eddies. However, it is almost impossible to conduct a simulation long enough to achieve a quasisteady state of a thermohaline circulation with a horizontal resolution (∼ several km) that is sufficiently high to resolve mesoscale eddies, even with the present computation resources. For these reasons, the standard practice in the ocean model community is to use a low horizontal resolution (a few hundred kilometers) model to study global thermohaline circulations and to use a limiteddomain model to study an eddying ocean. Some research projects seek to conduct a severalhundredyear integration of a high resolution model that resolves mesoscale eddies using enormous resources (e.g., the Earth Simulator), but such a resource is not available to everyone.
1.2
Classification of OGCMs
Most OGCMs used by ocean research scientists and by operational centers for forecasting climate and oceanic states numerically solve almost the same set of equations for the Boussinesq and hydrostatic ocean. The fundamental equations consist of the momentum equation for continuous fluid, the advectiondiffusion equation for temperature and salinity, the equation of state of sea water, and the mass conservation equation, collectively called primitive equations (Chapter 2). If necessary, equations for additional processes such as surface mixedlayer physics, sea ice, and bottom boundary layer physics are added. Most OGCMs that attempt to simulate realistic oceanic states adopt the finite difference method to discretize the equations. The spectral approach widely used by an atmospheric model would have difficulty treating lands that completely block ocean circulation in the zonal direction, and thus this approach is not usually adopted in generalpurpose models. Ocean models are classified by how they discretize the vertical direction. The choice of the vertical coordinate leads to fundamental differences among the models. There are three classes: zcoordinate models or zmodels adopt depth as the vertical coordinate, σ coordinate or terrain following models adopt fractional depth between the sea surface and the sea floor as the vertical coordinate, and ρ coordinate or isopycnal models adopt isentropic surfaces (isopotential density surfaces) as the vertical coordinate. Each class has its advantages and disadvantages and recent efforts are directed toward adopting generalized vertical coordinates, i.e., remedying each model’s disadvantages by using advantages of other classes. Readers are referred to the book by Griffies (2004) for more –
1
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Chapter 1
OGCMs and MRI.COM
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Figure 1.1. Various oceanic phenomena in terms of their space and time scales and coverage of the ocean general circulation model. The figure for oceanic phenomena is adopted from von Storch and Zwiers (2001). general discussion about OGCMs.
1.2.1 Zcoordinate models (zmodels) The first zcoordinate general circulation model was developed by Dr. Kirk Bryan and his colleagues at the Geophysical Fluid Dynamics Laboratory (GFDL) in the 1960’s. This model is sometimes referred to as the BryanCoxSemtner model or GFDL model. The zmodels utilize the character of the ocean that the local pressure is expressed as a function of depth by zeroorder approximation, which makes implementating the equation of state straightforward. Implementation of bottom topography and drawing of results are also straightforward. The models of this class are most widely used in the community because of their versatility. Such models were first used as components of coupled atmosphereocean models. The descendant of the GFDL model is called the Modular Ocean Model (MOM; Griffies et al., 2004) and is the most widely used. Most climate centers participating in climate model intercomparison projects use the zcoordinate models; MRI.COM also adopts zcoordinates. In Japan, the Center for Climate System Research Ocean Component Model (COCO) at the University of Tokyo (Hasumi, 2006) and the Research Institute of Applied Mechanics Ocean Model (RIAMOM) are also in this class. The major disadvantages of this class of models are as follows. • The vertical resolution in shallow seas and near the sea floor tends to be low and the processes that arise near the coast and the sea floor tend to be poorly reproduced. • Numerical inaccuracy in the tracer transport algorithm immediately leads to spurious diapycnal mixing of – 2 –
1.3. About MRI.COM the transported properties, while the diapycnal mixing is supposed to be very small in the ideal ocean interior. The first disadvantage is expected to be remedied by the σ models and the second by the ρ models. However, zmodel’s disadvantages are not completely overcome; these substitutes have their own difficulties.
1.2.2 Sigmacoordinate models (σ models) The first sigmacoordinate model was developed by Dr. George Mellor and his colleagues at Princeton University. Since the number of vertical grid points is invariable throughout the model domain, σ models are widely used for coastal ocean simulations. Two major σ models are widely used in the community: The Princeton Ocean Model (POM; Mellor, 2004) and the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams, 2003; 2005). The major disadvantages of this class of models are as follows. • An accurate representation of the horizontal pressure gradient is difficult near steeply sloping bottom topography. • The lateral mixing along the same vertical layer near the continental slope region might lead to mixing of the shoreward light water and the seaward dense water. These problems might prohibit using σ models in longterm integrations of the global ocean.
1.2.3
Isopycnalcoordinate models (ρ models)
The first isopycnalcoordinate model was developed by Dr. Rainer Bleck at the University of Miami. The development of this class of models is based on the fact that sea water moves along isopycnal surfaces in the interior. Thus, the character of a water mass is well maintained in the ocean interior. Since many theoretical studies of physical oceanography use an isopycnalcoordinate framework, the ρ models have the great advantage of providing good correspondence between theory and numerical models. A major ρ model widely used in the community is the Miami Isopycnic Coordinate Ocean Model (MICOM; Bleck and Boudra, 1986) developed at the University of Miami. The major disadvantages of this class of models are as follows. • Implementation of surface mixed layer models into a ρ model is in itself inappropriate. • Since the density levels are prescribed, this class of models might not be appropriate for studying a drastic climate change that could lead to great variations in density of major water masses. The Hybrid Coordinate Ocean Model (HYCOM; Bleck et al., 2002) has been developed in an effort to remedy some of these disadvantages.
1.3 About MRI.COM MRI.COM is a zcoordinate model. The horizontal grid arrangement is Arakawa’s Bgrid (Arakawa, 1972). Coast lines are defined by the periphery of the grid cell centered by the velocity points, i.e., the lines connecting the tracer points. This arrangement is suitable for the discrete expressions for the side boundary conditions for velocity, and transport through a narrow passage can be achieved with a single grid cell. –
3
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Chapter 1
OGCMs and MRI.COM
The program structure of MRI.COM is presented along with the typical computational cost of each process in Figure 1.2. subroutine name [program ﬁle name] exchange with atmosphere
initialization chap 16
cost 2% 11%
[stmdlp.F90, rdjobp.F90, rdbndt.F90 glatlon.F90, rdinit.F90, calcoe.F90, etc.]
surface ﬂuxes
bulk formula sea ice
chap 8 chap 9
mkﬂux [mkﬂux.F90] bulk [bulk.F90] simain [ice_main_cat.F90]
1%
continuity equation (determine vertical transport)
chap 3
cont [cont.F90]
19%
equation of motion (baroclinic component)
chap 5
clinic [clinic.F90]
8%
equation of motion (barotropic component)
chap 4
surfce [surfce_ctl.F90] [surfce_integ.F90]
34%
advection of tracers
chap 6 chap 13
tracer [tracer.F90] [trcadv.F90, som_adv.F90]
17%
diﬀusion of tracers (isopycnal and diapycnal)
chap 6 chap 12
trcimp [trcimp.F90] ipcoef [ipcoef.F90] ipycmix[ipycmix.F90]
1.5%
calculation of density stability check and adjustment
chap 2 chap 6
difajs [difajs.F90] cnvajs [cnvajs.F90] [stable.F90, stbden.F90]
1%
preparation for the next time step
5%
mixed layer model determine vertical mixing coeﬀ.
chap 7
mysl25 [my25.F90] nkoblm [nkoblm.F90]
0.5%
output to ﬁles with speciﬁed intervals
chap 16
writdt [writdt.F90]
rewrit [rewrit.F90] sfc_rewrit [surface_ctl.F90]
termination
Figure 1.2. Program Structure of MRI.COM
Though the program code should ideally use MKS units, MRI.COM basically uses cgs units for historical reasons. The sea ice model, the mixedlayer model, and some surface bulk formulae are coded in MKS units and converted into cgs units before their outputs are used by the main part. Details are described in each chapter.
– 4 –
1.4. Future of OGCMs and MRI.COM
1.4
Future of OGCMs and MRI.COM
As OGCMs acquire scientific and numerical integrity, they expand the area of their usage and begin to fulfill social as well as scientific needs. The developer of an OGCM thus has the responsibility to ensure its scientific and numerical integrity and to acknowledge its limitations. Feedback from users of various fields and continuous efforts to overcome limitations will certainly improve the OGCMs. The advance of OGCMs has kept pace with that of supercomputers. The mainstream of super computing is shifting from vector computation to massively parallel computation with distributed memory. To rewrite the current vectorfriendly program codes to adapt to this shift is immediately needed for some OGCMs. With increasing computing power, ever higher resolution simulations will be achieved. The result will be more strongly affected by how subgrid scale processes are parameterized and thus subgrid scale schemes should be selected carefully. To continue to stand as a multipurpose model, an OGCM should be easily coupled with other component models and data assimilation schemes. Having an interface to universal couplers and an adjoint code should be mandatory for such a multipurpose OGCM. These are the main subjects for developing MRI.COM in the coming years.
References Arakawa, A., 1972: Design of the UCLA general circulation model, Numerical Simulation Weather and Climate, Tech. Rep. No. 7, Dep. of Meteorology, University of California, Los Angeles, 116 pp. Bleck, R., and D. B. Boudra, 1986: Winddriven spin up in eddyresolving ocean models formulated in isopycnic and isobaric coordinates, J. Geophys. Res., 91, 76117621. Bleck, R., G. Halliwell, A. Wallcraft, S. Carroll, K. Kelly, K. Rushing, 2002: Hybrid Coordinate Ocean Model (HYCOM) User’s Manual, available online at http://hycom.rsmas.miami.edu/hycommodel/documentation.html. Mellor, G. L., 2004: Users guide for a threedimensional, primitive equation, numerical ocean model, Prog. in Atmos. and Ocean. Sci, Princeton University, 53pp., available online at http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/PubOnLine/POL.html. Griffies, S. M., 2004: Fundamentals of ocean climate models, Princeton University Press, 518pp. Griffies, S. M., M. J. Harrison, R. C. Pacanowski, and A. Rosati, 2004: A Technical Guide to MOM4, GFDL Ocean Group Technical Report No. 5, NOAA/Geophysical Fluid Dynamics Laboratory, Version prepared on March 3, 2004, available online at http://www.gfdl.noaa.gov. Hasumi, H., 2006: CCSR Ocean Component Model (COCO) version 4.0, CCSR report, 25, 103pp. Shchepetkin, A. F., and J. C. McWilliams, 2003: A method for computing horizontal pressuregradient force in an oceanic model with a nonaligned vertical coordinate, J. Geophys. Res., 108, 30903124. Shchepetkin, A. F., and J. C. McWilliams, 2005: The Regional Ocean Modeling System: A splitexplicit, freesurface, topographyfollowingcoordinate oceanic model, Ocean Modell., 9, 347404. von Storch, H., and F. Zwiers, 2004: Statistical Analysis in climate research, Cambridge university press, 484pp.
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5
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Part I
Configuration
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7
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Chapter 2
Governing Equations
In this chapter, the governing equations for the general ocean circulation are formulated. These equations are usually called primitive equations. The numerical methods to solve these equations are detailed in later chapters.
2.1 2.1.1
Formulation Coordinate System
The governing equations of an ocean general circulation model need to be formulated on a sphere. Originally, spherical coordinates were adopted, and the equations are discretized on a geographical (latitudelongitudedepth) grid. A problem arises for a global model because the North Pole is singular in spherical coordinates. Since the zonal grid widths within five latitudinal degrees from the Pole become less than a tenth of those in middle to low latitudes, a short time step is required owing to the limitation of the CourantFriedrichsLewy (CFL) condition.∗ This could hinder longterm integration. One simple means to remove this North Pole singularity is to shift both poles to land. In this case, one could use the spherical coordinate model without major modification by just adjusting the Coriolis parameter. Unfortunately, there are not many pairs of points on land that are symmetric about the Earth’s center.† One would like to choose a pair that provides the largest zonal grid size for the oceanic grid point having the smallest zonal grid size. The land mass around the shifted pole should be as large as possible or the oceanic grid point nearest to the shifted pole should be as far from the pole as possible. However, even the most suitable pair with poles on Greenland and Antarctica (near the Ross Sea) places the nearest oceanic grid point to the shifted pole only five degrees away. One might also be concerned that the Equator is not represented as a line on the shifted grid arrangement.‡ To resolve these issues, the model equations are represented on generalized orthogonal coordinates instead of spherical coordinates. Users may choose the coordinate system most suitable to their purposes. For example, the resolution of a target region can be intentionally enhanced by placing a pole of the distorted grid near the target region. Of course, a regional model without the North Pole could be arranged on geographical coordinates since the spherical coordinates are one form of the generalized orthogonal coordinates. In our model, the model equations are formulated on generalized orthogonal coordinates. Chapter 14 summarizes the concepts and calculus related to the generalized orthogonal coordinates.
2.1.2
Momentum Equation
Most ocean general circulation models use the Boussinesq approximation, where the density of sea water is replaced by a reference density (ρ0 ) for all terms except for the buoyancy term. On a general orthogonal coordinate system ((μ , ψ , r)) whose unit vectors are eμ , eψ , and er , the momentum equation for velocity v = ueμ + veψ + wer , ∗ The
time step, Δt, needs to satisfy vΔt/Δx ≤ 1, where v is velocity and Δx is the grid width. and Antarctica, China and Argentina, Kalimantan and Columbia. ‡ If the grid size is fine enough, the Kelvin wave in the shiftedpole model will propagate along the Equator as the theory suggests. † Greenland
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9
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Chapter 2
Governing Equations
where u = hμ μ˙ , v = hψ ψ˙ , w = hr r˙, is represented by ∂ hμ ∂ hψ ∂ hμ ∂u ∂ hr 1 ∂P v w − u− v − w + Vμ , (2.1) + v · ∇u + fψ w − f v = − u− ∂t ρ0 hμ ∂ μ hμ hψ ∂ ψ ∂μ hr hμ ∂r ∂μ ∂ hψ ∂ hψ ∂ hμ ∂v ∂ hr 1 ∂P w u − w − v− u + Vψ , (2.2) + v · ∇v + f u − f μ w = − v− ∂t ρ0 hψ ∂ ψ hψ hr ∂ r ∂ψ hμ hψ ∂ μ ∂ψ ∂ hμ ∂w ρ ∂ hr ∂ hr ∂ hr v 1 ∂P u w− w− + v · ∇w + f μ v − fψ u = − −g − u − v + Vr , (2.3) ∂t ρ0 h r ∂ r ρ0 hr hμ ∂ μ ∂r hψ hr ∂ ψ ∂r where P is pressure, V is viscosity, and g is the acceleration due to gravity. The radial distance from the Earth’s center is represented by r and the gravity is in the negative direction of r. The Coriolis force is represented by 2Ω × v = (2Ωψ w − 2Ωr v)eμ + (2Ωr u − 2Ωμ w)eψ + (2Ωμ v − 2Ωψ u)er ,
(2.4)
where Ω = Ωμ eμ + Ωψ eψ + Ωr er is the rotation vector of the Earth. We designate f μ = 2Ωμ , fψ = 2Ωψ , and f = fr = 2Ωr . The above expression is the most general form under the Boussinesq approximation. This form is used to formulate a nonhydrostatic model and a quasihydrostatic model. Further approximations are adopted for the standard model calculations. Since the vertical scale of motion of a water particle is far smaller than the Earth’s radius (a), the radial distance r as a scalar quantity is replaced by the Earth’s radius a. The new vertical coordinate (z) is the distance (positive upward) from the geoid (sea surface height in the state of rest) and ∂ /∂ r is replaced by ∂ /∂ z. To conserve angular momentum under the above approximations, we neglect the metric terms that involve w for the horizontal components, all the metric terms for the vertical components, and the Coriolis terms that do not involve f (Phillips, 1966). Hydrostatic approximation is adopted for the vertical momentum equation: 0=−
∂P − ρ g. ∂z
The resultant momentum equation described below is discretized by the default version of MRI.COM: ∂ hμ ∂ hψ ∂u v 1 ∂P − A (u) − u− v + Vu , − fv = − ∂t ρ0 h μ ∂ μ hμ hψ ∂ ψ ∂μ ∂ hψ ∂ hμ ∂v 1 ∂P u − A (v) − v− u + Vv . + fu = − ∂t ρ0 h ψ ∂ ψ hμ hψ ∂ μ ∂ψ The term A in (2.6) and (2.7) represents advection. For a scalar quantity α , A is defined by ∂ (hψ uα ) ∂ (hμ vα ) ∂ (wα ) 1 A (α ) = + + . hμ hψ ∂μ ∂ψ ∂z
(2.5)
(2.6) (2.7)
(2.8)
The viscosity term is represented by V , which is calculated separately in the lateral and vertical directions. For lateral viscosity, a harmonic (default) or biharmonic scheme is used. The harmonic scheme is used for vertical viscosity. The specific form for harmonic viscosity is as follows: 1 ∂ ∂ ∂u 1 ∂ 2 , hψ νH DT + 2 h2μ νH DS + νV 2 hψ hμ ∂ μ hμ hψ ∂ ψ ∂z ∂z 1 ∂ ∂ ∂v 1 ∂ 2 Vv = 2 hψ νH DS − 2 h2μ νH DT + νV , hψ hμ ∂ μ hμ hψ ∂ ψ ∂z ∂z
Vu =
– 10 –
(2.9) (2.10)
2.1. Formulation where νH is the horizontal viscosity coefficient, νV is the vertical viscosity coefficient, DT is the horizontal tension and DS is the horizontal shear: ∂ u ∂ v − hμ , (2.11) DT = hψ hμ ∂ μ hψ hψ ∂ ψ h μ ∂ u ∂ v DS = hμ + hψ . (2.12) hψ ∂ ψ hμ hμ ∂ μ hψ The above representation of the viscous term was derived by Bryan (1969) and is consistent with Smagorinsky (1963). Note that in the latter the coefficients also varies according to DT and DS (cf. Section 5.2.1). If we take hμ = 1, hψ = 1, the coordinate system is Cartesian. In this case, the viscosity term is reduced to the Laplacian form if the viscosity coefficient is a constant. When the biharmonic scheme is selected for lateral viscosity, the above harmonic operation is repeated twice. The friction terms Vu and Vv obtained by the first procedure are signreversed and substituted for u and v in (2.11) and (2.12) in the second procedure.
2.1.3
Continuity equation
Mass conservation is represented by the continuity equation for an incompressible fluid: A (1) = 0.
2.1.4
(2.13)
Temperature and salinity equation
The advectiondiffusion equation is solved for temperature (potential temperature) and salinity:
∂T (2.14) = −A (T ) + D(T ) + ST , ∂t ∂S (2.15) = −A (S) + D(S) + SS , ∂t where D is the diffusion operator and Sτ represents internal sources and sinks for tracer τ caused by nudging, convective adjustment (Section 6.4), short wave absorption (Section 8.3.1), etc. There are several forms for the diffusion operator. By default, the diffusion operator mixes a tracer in each direction of the model coordinates with the harmonic scheme. For horizontal diffusion, the biharmonic scheme can be used instead of the harmonic scheme. Using (14.22) and (14.23), the harmonictype diffusivity is represented as follows: hψ κ H ∂ T h μ κH ∂ T ∂ ∂ ∂ ∂T 1 (2.16) + + κV D(T ) = hμ hψ ∂ μ hμ ∂ μ ∂ψ hψ ∂ ψ ∂z ∂z ∂ hψ FμT ∂ hμ FψT ∂ FT 1 + z , = + (2.17) hμ hψ ∂μ ∂ψ ∂z hψ κ H ∂ S h μ κH ∂ S ∂ ∂ ∂ ∂S 1 (2.18) + + κV D(S) = hμ hψ ∂ μ hμ ∂ μ ∂ψ hψ ∂ ψ ∂z ∂z ∂ hψ FμS ∂ hμ FψS ∂ FS 1 + z , = + (2.19) hμ hψ ∂μ ∂ψ ∂z where κH and κV are the horizontal and vertical diffusion coefficients and the diffusive fluxes are represented by F. When the biharmonictype is selected for horizontal diffusion, the above Laplacian operation is repeated twice (the diffusion coefficient has a negative value). – 11 –
Chapter 2
Governing Equations
In the real ocean, transport and mixing would occur along neutral (isopycnal) surfaces. Thus horizontal mixing along the constant depth surface is generally inappropriate since neutral surfaces are generally slanting relative to the constant depth surface. To represent excursion and mixing of water masses along neutral surfaces, neutral physics schemes are developed and widely used in place of the horizontal diffusion scheme presented above. When the neutral physics schemes are selected, the advectiondiffusion equation for any scalar quantity (τ ) is expressed as follows (Gent and McWilliams, 1990):
∂ ∂
Dτ τ ∇H · (−κT ∇H ρ /ρz ) = D(τ ) + Sτ , (2.20) + ∇H · τ (κT ∇H ρ /ρz ) + Dt ∂z ∂z where the first term on the r.h.s. is the isopycnal diffusion, whose form is given by D(τ ) = ∇ · (κI K∇τ ), where
⎛
ρy2 + ρz2 1 ⎜ K= 2 ⎝ −ρx ρy ρx + ρy2 + ρz2 −ρx ρz
⎞ −ρx ρz ⎟ −ρy ρz ⎠ ρx2 + ρy2
−ρx ρy ρx2 + ρz2 ⎛
−ρy ρz
(2.21)
(2.22)
⎞ −(ρx /ρz )(ρy /ρz ) −ρx /ρz 1 ⎜ ⎟ = 1 + (ρx /ρz )2 −ρy /ρz ⎝ −(ρx /ρz )(ρy /ρz ) ⎠, 1 + (ρx /ρz )2 + (ρy /ρz )2 2 2 −ρx /ρz −ρy /ρz (ρx /ρz ) + (ρy /ρz ) 1 + (ρy /ρz )2
(2.23)
(Redi, 1982). In the above, the Cartesian notation is used for simplicity. The subscript x represents ∂ /(hμ ∂ μ ) and y represents ∂ /(hψ ∂ ψ ). The isopycnal diffusion coefficient is represented by κI . Diapycnal diffusion is not considered here. The second and third terms on the l.h.s. of (2.20) have the form of advection terms with a transport velocity vector (uT , vT , wT ): ∂ 1 ∂ρ ∂ρ , (2.24) uT ≡ κT / ∂z hμ ∂ μ ∂ z ∂ 1 ∂ρ ∂ρ vT ≡ κT / , (2.25) ∂z hψ ∂ ψ ∂ z ∂ hμ ∂ ρ ∂ ρ 1 ∂ hψ ∂ ρ ∂ ρ wT ≡ − κT / κT / + , (2.26) hμ hψ ∂ μ hμ ∂ μ ∂ z ∂ψ hψ ∂ ψ ∂ z (Gent and McWilliams, 1990). This velocity can be understood as the advection caused by the thickness diffusion of an isopycnal layer, where κT is thickness diffusivity. Note that these could be rewritten as G (τ ) = ∇ · (κT A∇τ ) with
⎛ ⎜ A=⎝
0
0
0 ρx / ρz
0 ρy /ρz
⎞ −ρx /ρz ⎟ −ρy /ρz ⎠ .
(2.27)
(2.28)
0
Comparing with (2.23), we notice that the isopycnal diffusion and the thickness diffusion terms are combined to yield a concise form (Griffies, 1998) and (2.20) can be rewritten as: Dτ = ∇ · {(κI K − κT A)∇τ } + Sτ . Dt
–
12
–
(2.29)
2.1. Formulation
2.1.5
Equation of state of sea water
The in situ density of sea water is a function of (potential) temperature, salinity, and pressure:
ρ = ρ (T, S, P).
(2.30)
The equation of state is usually given as a polynomial fit to the available measurements. A detailed description of this will be presented later (Section 2.2.5).
2.1.6 Boundary conditions a. Momentum equation Upper surface (z = 0): At the sea surface, momentum flux enters the ocean in the form of wind stress (or stress from sea ice where sea ice exists):
νV
(τx , τy ) ∂ (u, v) = . ∂ z z=0 ρ0
(2.31)
Note that z is defined positive upward (the surface wind stress is positive into the ocean). In the model algorithm, this is expressed as a body force to the first level velocity, u v , Fsurf )= (Fsurf
(τx , τy ) , ρ0 Δz 1
(2.32)
2
where Δz 1 is the thickness of the first layer, and τx and τy are zonal and meridional stress, respectively. 2 Surface fresh water flux is assumed to have zero velocity. Bottom (z = −H): Bottom friction (τxb in zonal and τyb in meridional direction) exists at the sea floor. For the bottom level velocity (u, v), the stress caused by bottom friction is proportional to the magnitude of the velocity and has an angle (θ0 + π ) to the velocity vector (Weatherly, 1972): (τxb , τyb ) = −Cbtm ρ0 u2 + v2 (u cos θ0 − v sin θ0 , v cos θ0 + u sin θ0 ), (2.33) where Cbtm is a dimensionless drag coefficient with a value 1.225 × 10−3 and θ0 = ±10◦ . The sign of θ0 is positive in the northern hemisphere and negative in the southern hemisphere. Side wall: No slip condition is applied (u = v = 0).
b. Temperature and Salinity Upper surface (z = 0): At the sea surface, heat and fresh water are exchanged with air and sea ice. Salt is also exchanged below the sea ice. All these exchanges are expressed as surface fluxes and become surface boundary conditions.
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13
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Chapter 2
Governing Equations
The surface boundary conditions for temperature and salinity are expressed as follows: ∂ T κV = FzT , (2.34) ∂ z z=0 ∂ S κV = FzS , (2.35) ∂ z z=0 where flux is defined positive downward (positive into the ocean). Fresh water flux is explicitly incorporated into the sea level equations ((2.44) and (2.66)). It could also exert a force on temperature and salinity. The water transported through the airsea interface (WAO ) and ice bottom (WIbot ) is assumed to have the first level temperature (T1 ). The water transported from the ice surface (WIsurf > 0) is assumed to have the freezing point temperature (Tfreeze ). The water exchanged with ice is assumed to have low salinity (SI = 4.0 [psu]). Note that the freezing point temperature (Tfreeze ) is given by mSI , where m = −0.0543 [K / psu]. The expression for the surface fluxes is given by Q + (WAO +WIbot ) · T1 +WIsurf · Tfreeze , ρ0 C p
FzT
=
FzS
= (WIbot +WIsurf ) · SI ,
(2.36) (2.37)
where Q is heat flux, and Cp is the specific heat of sea water. To avoid an unexpected rising or falling trend of sea level during a longterm integration, fresh water flux might be converted to salt flux. In this case, the surface flux is given by Q −WIsurf · (T1 − Tfreeze ), ρ0 C p
FzT
=
FzS
= −WAO · S1 − (WIbot +WIsurf ) · (S1 − SI ),
(2.38) (2.39)
where S1 is the first level salinity (see Chapter 8 for derivation). Note that fresh water flux should not be added to the sea level in this case. In a longterm run driven by surface flux, temperature and salinity might exhibit unexpected drift. To avoid this, surface temperature and salinity might be restored to observational or climatological values (T ∗ , S∗ ): FzT FzS
1 = − (T − T ∗ )Δz 1 , 2 γt 1 = − (S − S∗ )Δz 1 , 2 γs
(2.40) (2.41)
where Δz 1 is the surface layer thickness, and γt and γs are restoring time for temperature and salinity in units of 2
seconds. Bottom (z = −H): At the sea floor, the adiabatic boundary condition is applied:
∂T ∂S = 0, = 0. ∂z ∂z
(2.42)
Side wall: For any tracer, the adiabatic condition is applied at the side wall:
∂T ∂S = 0, = 0, (2.43) ∂n ∂n where n denotes the direction perpendicular to the wall. River discharge is expressed as a part of the surface fresh water flux. –
14
–
2.1. Formulation c. Continuity equation Upper surface (z = 0): At the sea surface, vertical velocity is generated because a water parcel moves following the freely moving sea surface: ∂η dη 1 ∂η 1 ∂η w= +v − (P − E + R), (2.44) − (P − E + R) = +u dt ∂t hμ ∂ μ hψ ∂ ψ where P is precipitation, E is evaporation, and R is river discharge. Bottom (z = −H): At the sea floor, vertical velocity is generated because the water parcel moves following the bottom topography: 1 ∂H 1 ∂H w=− u . +v hμ ∂ μ hψ ∂ ψ
(2.45)
d. Mixing at the surface boundary layer Near the sea surface, strong vertical mixing could occur for stably stratified situations because of active turbulence. These processes occur on a small scale (< several meters) and cannot be resolved in a large scale model with typical horizontal and vertical resolutions. These processes are parameterized as enhanced vertical viscosity and diffusivity. The user may specify either a high vertical viscosity and diffusivity a priori or use a surface boundary layer model. MRI.COM supports three surface boundary layer models: Mellor and Blumberg (2004), Noh and Kim (1999), and Large et al. (1994). In the surface boundary layer models, vertical viscosity and diffusivity change in time and are calculated every time step. See Chapter 7 for details.
2.1.7 Acceleration method It usually takes several thousand years before the global thermohaline circulation reaches a steady state under (quasi)steady forcing. The limiting factor for the time step is the phase speed of external gravity waves (∼ 200 [m/s]). A severalthousandyear integration will not be a workable exercise as long as we are restricted by this criteria in determining the time step. Bryan (1984) proposed a method to accelerate the ocean’s convergence to equilibrium. In this method, the phase speed of waves is reduced by modifying the governing equations, and a thermally balanced state is quickly reached by reducing the specific heat. Specifically, they are achieved by multiplying a constant to the tendency terms (α for momentum and γ for tracers) to increase inertia and to reduce specific heat. When a steady state is reached in these equations, we expect that the same balance as the undistorted equations will be obtained, because the only difference between the distorted and undistorted equations are tendency terms, which are expected to be zero in the steady state. The modified momentum equation is given by
∂ hμ ∂ hψ u− v + Vu , ∂ψ ∂μ ∂ hμ ∂ hψ ∂v u 1 ∂P α − A (v) + u− v + Vv . + fu = − ∂t ρ0 h ψ ∂ ψ hμ hψ ∂ ψ ∂μ
∂u v 1 ∂P α − A (u) − − fv = − ∂t ρ0 h μ ∂ μ hμ hψ
–
15
–
(2.46)
(2.47)
Chapter 2
Governing Equations
The modified temperature and salinity equations are given by
∂T ∂t ∂S γ ∂t
γ
= −A (T ) + D(T ) + ST ,
(2.48)
= −A (S) + D(S) + SS .
(2.49)
These modifications are equivalent to changing time to t = t/α and the BruntVaisala frequency to N 2 = N 2 α /γ . In this case, the equivalent depth for the nth mode of the vertical mode decomposition becomes Hn = Hn /α . The dispersion relation for the free inertiagravity waves becomes:
ω2 =
f 2 gHn 2 2 + (k + l ). α2 α
(2.50)
Since the angular frequency ω is inversely proportional to α 1/2 , the phase speed becomes low for large α . The model can be run with a long time step. The dispersion relation for Rossby waves becomes:
f 2 −1 ω = −β k α (k2 + l 2 ) + . gHn
(2.51)
Again, a large α yields a low phase speed. In standard practice, a value from several tens to a few hundreds is used as α , a value of one is used near the sea surface, and a value about a tenth is used near the bottom as γ . It should be noted that when α is too large, the model field is prone to baroclinic instability. Since this should not occur in nature, an integration of the model should be performed carefully by checking outputs during the integration.
2.2 Numerical Methods 2.2.1 Discretization To solve the primitive equations formulated in the previous section, the equations are projected on a discrete lattice and then advanced for a discrete time interval using solution algorithms. Since MRI.COM basically adopts zcoordinate, a fixed Eulerian lattice is arranged.§ A detailed description of the grid arrangement is given in Chapter 3. The equations are then volume integrated over a discrete model grid cell. This approach is called a finite volume approach or sometimes a flux form expression. The finite volume approach realizes a smooth transition from zcoordinates in the interior to σ coordinates near the sea surface as detailed in Chapter 4. A vertically integrated expression for the primitive equations is useful for describing the solution procedure. These are called semidiscrete equations. The body force and the metric term will be simply multiplied by the grid width. The material derivative and the subgrid scale flux need some attention. The material derivative of any quantity α , ∂ (hψ uα ) ∂ (hμ vα ) ∂α ∂ (wα ) Dα 1 + + = + Dt ∂t hμ hψ ∂μ ∂ψ ∂z
(2.52)
§ But note that the upper several levels are allowed to move with the undulating sea surface like the σ coordinate models. See Chapter 4 for details.
–
16
–
2.2. Numerical Methods is vertically integrated over a (k + 12 )th grid cell bounded by zk and zk+1 to give zk zk zk ∂ (hψ uα ) ∂ (hμ vα ) ∂α 1 ∂ (wα ) dz + + dz + dz ∂μ ∂ψ ∂z zk+1 ∂ t zk+1 hμ hψ zk+1 ∂ zk ∂ zk ∂ zk 1 α dz + hψ uα dz + hμ vα dz = ∂ t zk+1 hμ hψ ∂ μ zk+1 ∂ ψ zk+1 ∂z ) ∂ z ) ∂ z v(z u(z k k k k k + − w(zk ) α (zk ) − + ∂t hμ ∂ μ hψ ∂ ψ ∂z u(zk+1 ) ∂ zk+1 v(zk+1 ) ∂ zk+1 k+1 + + − w(zk+1 ) α (zk+1 ). + ∂t hμ ∂μ hψ ∂ψ The first line on the r.h.s. is expressed in a semidiscrete form as ∂ ∂ ∂ 1 Δzα , hψ Δzuα hμ Δzvα + + ∂t hμ hψ ∂ μ ∂ψ k+ 12 k+ 12 k+ 12
(2.53)
(2.54)
where any quantity is assumed to have a uniform distribution within a grid cell. For an interior grid cell, the last two lines reduce to the difference between vertical advective fluxes, w(zk )α (zk ) − w(zk+1 )α (zk+1 ).
(2.55)
For the sea surface (k = 0; z0 = η ) and the bottom (k = N; zN = −H), kinematic conditions (2.44) and (2.45) are used to give −(P − E + R)α (0) − w(z1 )α (z1 ) (2.56) at the surface and wzN−1 α (zN−1 ) at the bottom. Similarly, the vertical integral of the divergence of the subgrid scale fluxes gives zk zk ∂ (hψ Fμ ) ∂ (hμ Fψ ) 1 ∂ Fz dz + + dz ∂μ ∂ψ zk+1 hμ hψ zk+1 ∂ z ∂ ∂ 1 Δz hψ Fμ Δz h = + F μ ψ hμ hψ ∂ μ ∂ψ k+ 12 k+ 12 F (z) ∂ z F (z) ∂ z F (z) ∂ z F (z) ∂ z μ ψ μ ψ + − Fz (z) + + − Fz (z) . − hμ ∂ μ hψ ∂ ψ hμ ∂ μ hψ ∂ ψ k k+1
(2.57)
(2.58)
The quantity α = (P − E + R)α (0) + Fsurf
F (z ) ∂ z Fψ (z0 ) ∂ z0 μ 0 0 + − Fz (z0 ) (≡ −Fzα 0 ) hμ ∂ μ hψ ∂ ψ
(2.59)
taken from (2.56) and (2.58) could be regarded as a surface forcing term and corresponds to the surface flux (positive downward) given in the previous section. The first term on the r.h.s. of (2.59) is the tracer transport by the fresh water flux, and the second term is the microstructure flux calculated by subgrid scale (SGS) parameterizaα = −F α ). tions such as bulk formula. We generalize this flux as a vertical flux (i.e., Fsurf z 0
2.2.2 Momentum equation By default, the momentum equation with hydrostatic and Boussinesq approximation is solved. Equations are (2.6) and (2.7). To integrate these equations in time, the instantaneous vector field and pressure should be known. – 17 –
Chapter 2
Governing Equations
The vector field of the previous time level is used. The pressure field might be obtained by integrating the hydrostatic equation vertically: P(μ , ψ , z,t) = Ps (μ , ψ ,t) + g
0 z
ρ (μ , ψ , z )dz .
(2.60)
This equation indicates that the surface pressure Ps (μ , ψ ,t) should be known. There is no problem if the surface height η is known (Ps (μ , ψ ,t) = ρ0 gη ). To obtain the surface height, we should solve vertically integrated equations of motion. Since the external gravity waves caused by the rise and fall of the sea level have high phase speeds, a short time step is required to satisfy the CFL condition. However, when a target phenomenon has a longer time scale, the external gravity waves are usually not important. One might want to separate or remove this wave since its phase speed is two orders of magnitude greater than that of other waves. Historically, external gravity waves are removed from the model by prohibiting the vertical movement of the sea surface (rigidlid approximation). In this case, the vertically integrated equations lead to the vorticity equation in a form of the Poisson equation and are solved iteratively. The surface pressure is diagnostically obtained as pressure to push the lid. When the sea surface is allowed to move vertically, the problem of external gravity waves can be resolved by separating the barotropic mode from the baroclinic mode. We can achieve a long time step for the baroclinic mode by reflecting a temporally averaged state of the barotropic mode that is subcycled with a short time step. Since the free surface option is more suitable for parallel computation, only the free surface option is now officially supported by MRI.COM.
a. Barotropic mode If we put U=
η −H
N
∑ uk− 1 Δzk− 1 ,
udz =
k=1
2
2
V=
η −H
vdz =
N
∑ vk− 12 Δzk− 12 ,
(2.61)
k=1
then the vertically summed semidiscrete momentum equations are
∂U g(η + H) ∂ η + X, − fV = − ∂t hμ ∂μ ∂V g(η + H) ∂ η +Y, + fU = − ∂t hψ ∂ψ where X
=
(2.62) (2.63)
N v ∂h ∂ hψ μ − (Δz(u, v)u) u − v Δzk− 1 ∑ ∑ k− 12 2 ∂ψ ∂μ k− 12 k=1 k=1 hμ hψ N
N 1 1 0 u u −∑ gρμ dz Δzk− 1 + ∑ (ΔzVHu )k− 1 + Fsurf Δz 1 + Fbottom ΔzN− 1 2 2 2 2 ρ h z μ 0 k=1 k=1 k− 1
−∇H ·
N
2
( ≡
N
∑ Fμ ),
k=1
Y
(2.64)
N u ∂h ∂ hψ μ + u − v Δzk− 1 (Δz(u, v)v) 1 ∑ ∑ k− 2 2 ∂ψ ∂μ k− 12 k=1 hμ hψ k=1 0 N
N 1 1 v v −∑ gρψ dz Δzk− 1 + ∑ (ΔzVHv )k− 1 + Fsurf Δz 1 + Fbottom ΔzN− 1 2 2 2 2 ρ h z ψ 0 1 k=1 k=1 k−
= −∇H ·
N
2
( ≡
N
∑ Fψ ).
(2.65)
k=1
– 18 –
2.2. Numerical Methods The vertically integrated continuity equation is given by ∂η 1 ∂ (hψ U) ∂ (hμ V ) = (P − E + R). + + ∂t hμ hψ ∂μ ∂ψ
(2.66)
We solve these equations for U, V , and η . See Chapter 4 for details.
b. Baroclinic mode To solve the baroclinic mode, we could use Ps (μ , ψ ,t) that can be diagnostically obtained using barotropic equations (2.62), (2.63), and (2.66). However, we could avoid the procedure of obtaining Ps (μ , ψ ,t) by using the method described below. Velocity is decomposed into a barotropic component and a baroclinic component as follows: u
= uˆ + u, ¯
(2.67)
v
= vˆ + v, ¯
(2.68)
where u¯ and v¯ are barotropic components and uˆ and vˆ are baroclinic components. We consider updating a new velocity (u , v ) using a momentum equation where the surface pressure gradient term is dropped:
uk− 1 Δz 2
(new) (old) − uk− 1 Δz 1 k− 12 k− 2 2
Δt (new) (old) vk− 1 Δz 1 − vk− 1 Δz 1 k− 2
2
where
1 1 0 ρ0 hμ zk− 1
= − f uΔzk− 1 + Fψ ,
(2.70)
2
v ∂h ∂ hψ μ u − v Δzk− 1 2 hμ hψ ∂ ψ ∂μ k− 12 k− 12 gρμ dz Δzk− 1 + (ΔzVHu )k− 1 − Fzu k + Fzu k+1 ,
Fμ = −∇H · Δz(u, v)u −
(2.69)
2
k− 2
2
Δt
= f vΔzk− 1 + Fμ ,
−
2
2
u ∂h ∂ hψ μ Fψ = −∇H · Δz(u, v)v + u− v Δzk− 1 2 hμ hψ ∂ ψ ∂μ k− 12 k− 12
1 1 0 − gρψ dz Δzk− 1 + (ΔzVHv )k− 1 − Fzv k + Fzv k+1 . 2 2 ρ0 hψ zk− 1 2
(2.71)
(2.72)
2
Summing over the whole water column gives (new)
(old)
∑Nk=1 (uk− 1 Δzk− 1 ) − ∑Nk=1 (uk− 1 Δzk− 1 ) 2
2
2
2
Δt
(old)
∑Nk=1 (vk− 1 Δzk− 1 ) − ∑Nk=1 (vk− 1 Δzk− 1 ) 2
2
2
N
(old)
2
Δt
N
∑ vk− 12 Δzk− 12 + ∑ Fμ ,
k=1
(new)
=f
= −f
N
(2.73)
k=1
(old)
N
∑ uk− 12 Δzk− 12 + ∑ Fψ .
k=1
(2.74)
k=1
From (2.62), (2.63), (2.64), (2.65), we have N
∑ Fμ =
(new)
(new)
2
2
k=1 N
∑ Fψ =
k=1
(old)
∑Nk=1 (uk− 1 Δzk− 1 ) − ∑Nk=1 (uk− 1 Δzk− 1 ) (new)
2
Δt
2
(new) 2
N
(old)
∑ vk− 12 Δzk− 12 +
k=1 (old)
∑Nk=1 (vk− 1 Δzk− 1 ) − ∑Nk=1 (vk− 1 Δzk− 1 ) 2
−f
2
2
Δt
+f
N
k=1
–
19
–
(old)
∑ uk− 12 Δzk− 12 +
η (old) + H ∂ Ps , ρ0 h μ ∂ μ
(2.75)
η (old) + H ∂ Ps . ρ0 h ψ ∂ ψ
(2.76)
Chapter 2
Governing Equations
From the above equations, we have (new)
(new)
∑Nk=1 (uk− 1 − uk− 1 )Δzk− 1 2
2
Δt
(new)
2
Δt
η (old) + H ∂ Ps , ρ0 h μ ∂ μ
(2.77)
=
η (old) + H ∂ Ps . ρ0 h ψ ∂ ψ
(2.78)
(new)
∑Nk=1 (vk− 1 − vk− 1 )Δzk− 1 2
=
2
2
Thus we obtain (new) k− 12
(u
z
(new) k− 12
(v
+ uk− 1 − u 2
z
z
k− 12
Δt z
+ vk− 1 − v 2
)Δz
k− 12
(new) (old) − uk− 1 Δz 1 k− 12 k− 2 2
(new)
= f vΔzk− 1 2
)Δz
Δt
(new) (old) − vk− 1 Δz 1 k− 12 k− 2 2
η (old) + H Δzk− 12 ∂ Ps − (new) + Fμ η + H ρ0 h μ ∂ μ
(2.79)
(new)
= − f uΔzk− 1 2
η (old) + H Δzk− 12 ∂ Ps − (new) + Fψ , + H ρ0 hψ ∂ ψ η
(2.80)
z
where (...) denotes the thickness weighted vertical average. z z z z Since we could regard u(new) + u − u¯ = u and v(new) + v − v¯ = v as the real updated velocity, the baroclinic component is expressed as uˆ = u − u¯ and vˆ = v − v¯ . To summarize, we first solve for (u , v ) using (2.69) and (2.70), and then compute the baroclinic component by ¯ uˆ = u − u¯ and vˆ = v − v¯ . The absolute velocity is obtained by u = uˆ + u¯ and v = vˆ + v.
2.2.3 Continuity equation The vertical component of velocity is obtained by vertically integrating the continuity equation from top to bottom. By using a flux form, the surface boundary condition (2.44) could be naturally included. The vertical integration for the kth vertical level is performed as follows: ∂ (hψ Δzk− 1 uk− 1 ) ∂ (hμ Δzk− 1 vk− 1 ) 1 2 2 2 2 wk = wk−1 + , (2.81) + hμ hψ ∂μ ∂ψ where Δzk− 1 is the width of the (k − 12 )th layer with Δz 1 = Δz 1 const + η and 2
2
w0 =
2.2.4
2
∂η 1 ∂ (hψ U) ∂ (hμ V ) . + − (P − E + R) = − ∂t hψ hμ ∂μ ∂ψ
(2.82)
Temperature and salinity equation
We solve for potential temperature instead of in situ temperature, because we want to express vertical mixing as a simple mixing of water masses. If we use in situ temperature, the temperature change caused by the change of pressure accompanied by vertical excursion of a water parcel should be calculated before mixing.
– 20 –
2.2. Numerical Methods a. A semidiscrete expression The equation for potential temperature and salinity is an advectiondiffusion equation (2.14) and (2.15) (or (2.20)). Its semidiscrete expression is as follows: ∂ − (wT )k−1 + (wT )k (Tk− 1 Δzk− 1 ) = − ∇H · Δzhψ uT, Δzhμ vT 2 2 ∂t k− 1 2 − ∇H · Δzhψ FμT , Δzhμ FψT − FzT k−1 + FzT k + ST Δzk− 1 , 1
(2.83)
∂ − (wS)k−1 + (wS)k (Sk− 1 Δzk− 1 ) = − ∇H · Δzhψ uS, Δzhμ vS 2 2 ∂t k− 1 2 − ∇H · Δzhψ FμS , Δzhμ FψS − FzS k−1 + FzS k + SS Δzk− 1 . 1
(2.84)
k− 2
2
k− 2
2
Several options for discretizing each term on the r.h.s. are detailed in Chapter 6.
b. Treating the unstably stratified layer Since the hydrostatic approximation is used, an unstable stratification should be removed somehow. Generally, we assume that vertical convection occurs instantaneously to remove unstable stratification. We call this convective adjustment, which is explained in Section 6.4. One might also choose to mix tracers by setting the local vertical diffusion coefficient to a large value such as 10000 [cm2 s−1 ] where stratification is unstable. In this case, the tracer equation should be solved using the partial implicit method, which is described in Section 12.5.
2.2.5
Equation of state
The in situ density is needed to calculate the pressure gradient term in the momentum equation. As indicated in (2.30), the equation of state is a function of pressure, temperature and salinity. Here we present the specific form of the equation of state.
a. Basics of the equation of state The standard equation of state provided by UNESCO (1981) is a function of in situ temperature, salinity, and pressure. Note that in situ temperature is used, not potential temperature. Density (ρw ) of pure water (S = 0) under sea level pressure is given as a function of temperature (T ):
ρw (T ) = 999.842594 + 6.793952 × 10−2 T − 9.095290 × 10−3 T 2
(2.85)
+ 1.001685 × 10−4 T 3 − 1.120083 × 10−6 T 4 + 6.536332 × 10−9 T 5 . Density at the sea surface (ρ0 = ρ (T, S, 0)) is expressed using sea surface temperature and salinity:
ρ0 = ρw
(2.86)
+ (0.824493 − 4.0899 × 10−3 T + 7.6438 × 10−5 T 2 − 8.2467 × 10−7 T 3 + 5.3875 × 10−9 T 4 )S 3
+ (−5.72466 × 10−3 + 1.0227 × 10−4 T − 1.6546 × 10−6 T 2 ) S 2 + 4.8314 × 10−4 S2 .
–
21
–
Chapter 2
Governing Equations
Density in the interior is calculated using the secant bulk modulus K(S, T, P). The pure water value Kw is given by Kw = 19652.21 + 1.484206 × 102 T − 2.327105T 2 −2 3
(2.87)
−5 4
+ 1.360477 × 10 T − 5.155288 × 10 T . The value at the sea surface is given by K0 = Kw + (54.6746 − 0.603459T + 1.09987 × 10−2 T 2 − 6.1670 × 10−5 T 3 )S + (7.944 × 10
−2
−2
−4 2
(2.88)
3 2
+ 1.6483 × 10 T − 5.3009 × 10 T ) S ,
and the value at pressure P is given by K = K0
(2.89) −3
−4 2
−7 3
+ P (3.239908 + 1.43713 × 10 T + 1.16092 × 10 T − 5.77905 × 10 T ) + P (2.2838 × 10−3 − 1.0981 × 10−5 T − 1.6078 × 10−6 T 2 ) S 3
+ P (1.91075 × 10−4 ) S 2 + P2 (8.50935 × 10−5 − 6.12293 × 10−6 T + 5.2787 × 10−8 T 2 ) + P2 (−9.9348 × 10−7 + 2.0816 × 10−8 T + 9.1697 × 10−10 T 2 ) S. Density is computed using the following equations,
ρ = ρ0 /(1 − P/K)
(2.90)
σ = ρ − 1000.0.
(2.91)
and Since potential temperature (θ ) is the prognostic variable, an equation of state should be given as a function of potential temperature, salinity, and pressure. To do this, potential temperature should be converted to in situ temperature. The conversion equation is obtained as follows using the adiabatic lapse rate Γ(T, S, P): T (θ0 , S, P) = θ0 +
P P0
Γ(T, S, P )dP .
(2.92)
A polynomial for the adiabatic lapse rate Γ(T, S, P) is given by UNESCO: Γ(T, S, P) = a0 + a1 T + a2 T 2 + a3 T 3 + (b0 + b1 T )(S − 35) + {c0 + c1 T + c2 T 2 + c3 T 3 + (d0 + d1 T )(S − 35)} P + (e0 + e1 T + e2 T 2 ) P2 ,
–
22
–
(2.93)
2.2. Numerical Methods where a0 = +3.5803 × 10−5 , −6
c2 = +8.7330 × 10−12 , −14
a1 = +8.5258 × 10 ,
c3 = −5.4481 × 10
a2 = −6.8360 × 10−8 ,
d0 = −1.1351 × 10−10 ,
a3 = −6.6228 × 10−10 ,
d1 = +2.7759 × 10−12 ,
b0 = +1.8932 × 10−6 ,
e0 = −4.6206 × 10−13 ,
b1 = −4.2393 × 10−8 ,
e1 = +1.8676 × 10−14 ,
c0 = +1.8741 × 10−8 ,
e2 = −2.1687 × 10−16 ,
(2.94)
,
c1 = −6.7795 × 10−10 . With the converted in situ temperature, the equation of state is used to calculate density using salinity and pressure.
b. An equation of state used by MRI.COM An equation of state of MRI.COM is in the same polynomial form as UNESCO, but has a modified set of parameters. The parameters are determined by the least square fit for a realistic range of potential temperature and salinity. We follow the method of Ishizaki (1994), but prescribe a different range (−2 ≤ θ ≤ 40 [◦ C], 0 ≤ S ≤ 42 [psu], and 0 ≤ P ≤ 1000 [bar]). We first calculate density at the sea surface (potential density or σθ ) using equations (2.85) and (2.86) without any modification. The pressure dependent part, or specific volume K(θ , S, P) is given by K(θ , S, P) = e1 (P) + e2 (P)θ + e3 (P)θ 2 + e4 (P)θ 3 + e5 (P)θ 4
(2.95)
+ S( f1 (P) + f2 (P)θ + f3 (P)θ + f4 (P)θ ) 2
3
3
+ S 2 ( f5 (P) + f6 (P)θ + f7 (P)θ 2 ), where e1 (P) = ec1 + (gc1 + hc1 P)P,
f1 (P) = fc1 + (gc5 + hc4 P)P,
e2 (P) = ec2 + (gc2 + hc2 P)P,
f2 (P) = fc2 + (gc6 + hc5 P)P,
e3 (P) = ec3 + (gc3 + hc3 P)P,
f3 (P) = fc3 + (gc7 + hc6 P)P,
e4 (P) = ec4 + gc4 P, e5 (P) = ec5 ,
(2.96)
f4 (P) = fc4 , f5 (P) = fc5 + gc8 P, f6 (P) = fc6 , f7 (P) = fc7 .
The set of coefficients in the above equation is computed using a least square fit as follows. Given 43×43×101 combinations of the above range of potential temperature, salinity, and pressure, in situ temperature is first computed using (2.92). Density is then calculated by the UNESCO equations using in situ temperature and salinity. The above coefficients are determined using these data of density, potential temperature, salinity, and pressure by the least square method. They are given as follows. – 23 –
Chapter 2
Governing Equations ec1 ec2 ec3
19659.35 144.5863 −1.722523
fc1 fc2 fc3
52.85624 −3.128126 × 10−1 6.456036 × 10−3
ec4 ec5
1.019238 × 10−2 −4.768276 × 10−5
fc4 fc5
−5.370396 × 10−5 3.884013 × 10−1
fc6 fc7
9.116446 × 10−3 −4.628163−4
gc1 gc2
3.185918 2.189412 × 10−2
hc1 hc2
2.111102 × 10−4 −1.196438 × 10−5
gc3 gc4
−2.823685 × 10−4 1.715739 × 10−6
hc3 hc4
1.364330 × 10−7 −2.048755 × 10−6
gc5 gc6 gc7
6.703377 × 10−3 −1.839953 × 10−4 1.912264 × 10−7
hc5 hc6
6.375979 × 10−8 5.240967 × 10−10
gc8
1.477291 × 10−4
2.3 Appendix 2.3.1 Physical constants Here we list basic physical constants used for MRI.COM. These are defined in param.F90. Physical constants used to calculate surface fluxes and sea ice processes are listed in Chapters 8 and 9, respectively. value
variable name in MRI.COM
radius of the Earth acceleration due to gravity
6375.0 × 105 cm
RADIUS
980.1 cm2 · s−1
angular velocity of the Earth’s rotation
π /43082.0 radian · s−1 273.16 K 1.036 g · cm−3 3.99 × 107 erg · g−1 · K−1 (1.0 erg · g−1 · K−1 = 1.0 × 10−4 J · kg−1 · K−1 )
GRAV OMEGA
the absolute temperature of 0 ◦ C the average density of sea water the specific heat of sea water
TAB RO CP
References Bryan, K., 1969: A numerical method for the study of the circulation of the world ocean, J. Comput. Phys., 4, 347376. Bryan, K., 1984: Accelerating the convergence to equilibrium of oceanclimate models, J. Phys. Oceanogr., 14, 666673. Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20, 150155. Griffies, S. M., 1998: The GentMcWilliams skew flux, J. Phys. Oceanogr., 28, 831841. – 24 –
2.3. Appendix Ishizaki, H., 1994: A Simulation of the abyssal circulation in the North Pacific Ocean. Part II: Theoretical Rationale, J. Phys. Oceanogr., 24, 19411954. Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization, Rev. Geophys., 32, 363403. Mellor, G. L., and A. Blumberg, 2004: Wave breaking and ocean surface layer thermal response, J. Phys. Oceanogr., 34, 693698. Noh, Y., and H.J. Kim, 1999: Simulations of temperature and turbulence structure of the oceanic boundary layer with the improved nearsurface process, J. Geophys. Res., 104, 15,62115,634. Phillips, N., 1966: The equation of motion for a shallow rotating atmosphere and the “Traditional approximation”, J. Atmos. Sci., 23, 626–628. Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation, J. Phys. Oceanogr., 12, 1154–1158. Smagorinsky, J., 1963: General circulation experiments with the primitive equations: I. The basic experiment, Mon. Weather Rev., 91, 99–164. UNESCO, 1981: Tenth report of the Joint Panel on Oceanographic Tables and Standards, Sidney, B. C., September 1980, Unesco Technical papers in marine science, 36, 25pp. Weatherly, G. L., 1972: A study of the bottom boundary layer of the Florida current, J. Phys. Oceanogr., 2, 54–72.
–
25
–
Chapter 3
Spatial grid arrangement and definition of continuity equation
The model ocean domain is defined as a threedimensional aggregate of rectangular grid cells limited by latitudinal circles, longitudinal lines, and horizontal surfaces with fixed depths. Just above the bottom, vertical thickness of the cell can be locally changed. The horizontal and vertical lengths of the cells are regarded as the horizontal and vertical grid sizes, respectively. In MRI.COM, the grid size can be varied spatially in each direction (variable grid size).
3.1
Horizontal grid arrangement
Figure 3.1(a) illustrates the horizontal grid arrangement. Horizontal components of velocity and bottom depth are defined at the center of the grid cell (×), and tracers such as temperature and salinity, density, and seasurface height (SSH) are defined at the four corners of the cell (◦). Hereafter, for simplicity, the velocity point is referred to as the Upoint; the grid cell centered on the Upoint, the Ucell (Figure 3.1(a)); the tracer point as the Tpoint; and the grid cell centered on the Tpoint, the Tcell (Figure 3.3). The Tcells are staggered from the Ucells by a half grid size and consists of partial cells along the coast lines (Figure 3.3). The coast lines are defined by the periphery of the Ucells, i.e., the lines connecting the Tpoints. This type of horizontal grid arrangement is called Arakawa’s Bgrid arrangement (Arakawa, 1972). Although Arakawa’s Bgrid arrangement is also used in MOM (NOAAGFDL, USA) and COCO (CCSR, U. Tokyo, Japan), the primary cell is the Tcell in those models and the coast lines are defined by the lines connecting the Upoints (Figure 3.1(b)). In the case of the variable grid size the Tpoints are defined just at the centers of the Tcells as seen in Figure 3.1(c), but the Upoints are not at the centers of the Ucells. The Upoints are arranged so that the Ucell boundary stands at the mid point between two neighboring Upoints.
3.2
Vertical grid arrangement
A variable grid size is usually used for the vertical grid arrangement, i.e., fine near the surface and coarse at depth. As illustrated in Figure 3.2(a), tracers (◦) and velocity (×) are defined at just the middepth level of the cell, and the vertical mass fluxes W (, 2) are defined at the boundary of the cell. There are two kinds of W , one for the Tcell (W T ; ) and another for the Ucell (W U ; 2), W U being obtained by an averaging operation on W T . Their horizontal locations are the Tpoints and the Upoints depicted in Figure 3.1(a). In order to express the gentle bottom slope as smoothly as possible, the thickness of the deepest Ucell at each horizontal location is variable, with a limitation that it must exceed 10 percent of the nominal thickness of the layer to avoid violating the vertical CFL condition (Figure 3.2(b)). Otherwise, as presented in Figure 3.2(c), the gentle bottom slope is expressed by wide, flat bottoms and cliffs here and there with height of vertical grid size Δz. In –
27
–
Chapter 3
Spatial grid arrangement and definition of continuity equation
Figure 3.1. Horizontal grid arrangement. (a) MRI.COM (◦ :T, S, η , ×:u, v, H), (b) MOM and COCO (◦ :T, S, η , H, ×:u, v), (c) Variable grid size in MRI.COM these regions, the vertical velocity is concentrated at the cliffs, resulting in relatively strong fictitious horizontal currents there.
3.3
Indices and symbols
The conventions for indexing and the definitions of symbols used in finite difference expressions of the equations throughout this document are given here. The distance corresponding to an increment of Δμ in the zonal direction of the generalized orthogonal coordinate is expressed as follows: Δx ≡ hμ Δμ ,
(3.1)
where hμ is the scaling factor. The meridional distance is defined similarly: Δy ≡ hψ Δψ .
(3.2)
The vertical distance is expressed by Δz. For a discretized grid cell, the horizontal area is expressed by ΔS and the volume is expressed by ΔV . The subscript indices expressing the horizontal grid position in the finite difference expression of the equations are usually integers for the Tpoints, i.e., (i, j) and are increased by a half for the Upoints (i + 12 , j + 12 ). In some cases vice versa, with a notice. In the vertical subscript index for the finite difference expression, the upper level of a grid cell, where the vertical mass flux is defined, is numbered as k (k = 0 being the sea surface), the levels of the Tpoint and Upoint are numbered as k + 12 (Figure 3.2(a)). In some cases, the Tpoint and Upoint levels may be numbered as k, with a notice.
– 28 –
3.4. Continuity equation (a) k1 k1/2 k
○䇭T,S ×䇭U,V T 䂦䇭W U □䇭W
k+1/2 k+1 k+3/2
(b)
(c)
Figure 3.2. Vertical grid arrangement. (a)Vertical grid arrangement. (b) Smooth bottom slope. (c) Stairlike bottom slope.
3.4
Continuity equation
The mass (volume) fluxes, which are fundamental for estimating the advection of momentum and tracers, are calculated based on the finite difference expression of the continuity equation. The finite difference expression of the continuity equation differs for the Tcell and Ucell (Figure 3.3). In MRI.COM, the mass continuity for the Tcell is fundamental and that for the Ucell is derived from the former by an averaging operation. By this, we can avoid spurious vertical mass fluxes for the Ucell continuity, which appear when the Ucell continuity is calculated independently of the Tcell continuity, with the largest error magnitude increasing as the grid size decreases (Webb, 1995). The finite difference expression of the continuity equation for the Tcell is given as follows, based on the mass fluxes passing through each side of the grid cell (Figure 3.3):
MCi,T j
T T T T T T ≡ Ui− 1 , j −Ui+ 1 , j +Vi, j− 1 −Vi, j+ 1 +Wi, j,k+1 −Wi, j,k
= 0,
2
2
2
2
(3.3)
where T ∗ T ∗ Ui+ 1 , j = ui+ 1 , j Δyi+ 1 , j Δz, Vi, j+ 1 = vi, j+ 1 Δxi, j+ 1 Δz, 2
2
2
u∗i+ 1 , j
=
v∗i, j+ 1 2
=
2
2
2
1 (u 1 1 + ui+ 1 , j− 1 ), 2 2 2 i+ 2 , j+ 2 1 (v 1 1 + vi− 1 , j+ 1 ). 2 2 2 i+ 2 , j+ 2
2
(3.4)
(3.5)
The finite difference analog of the continuity for the partial Tcell along the coastline (Figure 3.3(b)) is defined as follows: 1 1 (3.6) Vi,Tj+ 1 = v∗i, j+ 1 Δxi, j+ 1 Δz, Vi,Tj− 1 = v∗i, j− 1 Δxi, j− 1 Δz, 2 2 2 2 2 2 2 2 –
29
–
Chapter 3
Spatial grid arrangement and definition of continuity equation (a)
(b)
i, j+1 i, j+1 v*i, j+1/2
i+1, j+1 Ucell
v*i, j+1/2
vm i, j
vmi, j u*i1/2, j
i+1/2, j+1/2
u*i+1/2, j i, j umi, j
i1, j Tcell
i+1, j
i, j
u*i+1/2, j
v*i, j1/2
v*i, j1/2
i, j1
(c)
i+1/2, j+3/2
(d)
i+1, j+1
i, j+1 v*i, j+1/2
v*i, j+1/2
i1/2, j+1/2
Ucell i+3/2, j+1/2
vmi, j u*i1/2, j i, j
u*i+1/2, j
vmi, j
u*i1/2, j
umi, j
u*i+1/2, j
i, j umi, j
i1, j
v*i, j1/2
Tcell
i+1, j
v*i,j1/2 i+1/2, j1/2
i, j1
Figure 3.3. Horizontal arrangement of variables for the continuity equation. (a) Relationship between Tcell and Ucell (standard form). (b),(c) Relationship between Tcell and Ucell near the coast. (d) Diagonal square grid cell and mass fluxes. v∗i, j+ 1 = vi+ 1 , j+ 1 , 2
2
v∗i, j− 1 = vi+ 1 , j− 1 ,
2
2
2
u∗i− 1 , j = 0.
2
(3.7)
2
For the corner part of land as shown in Figure 3.3(c), it is given as follows: 1 ∗ 1 T Ui− ui− 1 , j Δyi− 1 , j Δz, Vi,Tj− 1 = v∗i, j− 1 Δxi, j− 1 Δz, 1,j = 2 2 2 2 2 2 2 2
(3.8)
u∗i− 1 , j = ui− 1 , j+ 1 , v∗i, j− 1 = vi+ 1 , j− 1 . 2
2
2
2
2
(3.9)
2
The boundary condition for Wi,Tj is as follows: Wi,Tj = Ui− 1 , j Δyi− 1 , j −Ui+ 1 , j Δyi+ 1 , j + Vi, j− 1 Δxi, j− 1 −Vi, j+ 1 Δxi, j+ 1 /ΔSi, j 2
at the surface (where U
2
2
= ∑Nk=1 uk− 1 Δzk− 1 2 2
2
and V =
2
2
∑Nk=1 vk− 1 Δzk− 1 ), 2 2
2
2
and
Wi,Tj = 0 at the bottom. On the other hand, the finite difference expression of the continuity equation for the Ucell (i + 12 , j + 12 ) is defined as follows (Figure 3.3(a, b, and c)): U MCi+ 1 , j+ 1 ,k+ 1 2 2 2
≡
MCi,T j,k+ 1 2
Ni, j,k+ 1
= 0,
2
+
T MCi+1, j,k+ 1 2
Ni+1, j,k+ 1 2
+
MCi,T j+1,k+ 1 2
Ni, j+1,k+ 1 2
+
T MCi+1, j+1,k+ 1 2
Ni+1, j+1,k+ 1 2
(3.10) – 30 –
3.4. Continuity equation where Ni, j,k+ 1 is the number of sea grid cells around the Tpoint (i, j) in the (k + 12 )th layer. Usually, N = 4 2
for Tcells away from land (Figure 3.3(a)), but N < 4 for the partial Tcells along coast lines (Figure 3.3(b,c)). This equation means that the mass convergence in a Ucell consists of the sum of the contributions from four surrounding Tcells. The standard form of the mass continuity, which applies for Ucells (i + 12 , j + 12 ) away from coast lines is as follows: U MCi+ 1 , j+ 1 2
2
1 T T T (MCi,T j + MCi+1, j + MCi, j+1 + MCi+1, j+1 ) 4 = 0. ≡
(3.11)
This can be rewritten as 1 U 1 U 1 U U U (Ui, j + Ui,Uj+1 ) − (Ui+1, j + Ui+1, j+1 ) + (Vi, j + Vi+1, j ) 2 2 2 1 U U U − (Vi,Uj+1 + Vi+1, j+1 ) = Wi+ 1 , j+ 1 ,k −Wi+ 1 , j+ 1 ,k+1 . 2 2 2 2 2 Terms Ui,Uj , Vi,Uj , and W U 1
i+ 2 , j+ 12 ,k
(3.12)
are defined as follows: Ui,Uj = umi, j Δyi, j Δz, Vi,Uj = vmi, j Δxi, j Δz,
umi, j
=
vmi, j
=
1 ∗ (u 1 + u∗i− 1 , j ), 2 i+ 2 , j 2 1 ∗ ∗ (v 1 + vi, j− 1 ), 2 i, j+ 2 2
(3.13)
(3.14)
1 T U T T T (W + Wi+1, (3.15) Wi+ 1 , j+ 1 ,k = j,k + Wi, j+1,k + Wi+1, j+1,k ). 4 i, j,k 2 2 Thus, for the standard form of the Ucell continuity, the following equations are derived from (3.4), (3.13), and (3.14): Ui,Uj
=
Vi,Uj
=
1 T T (U 1 + Ui− 1 , j ), 2 i+ 2 , j 2 1 T T (V 1 + Vi, j− 1 ). 2 i, j+ 2 2
(3.16)
All the above relationships hold for the cases with variable grid sizes (Figure 3.1(c)). The l.h.s. of the standard form of the continuity equation (3.12) expresses the convergence of mass fluxes along the horizontal coordinate axes and it is completed as far as the continuity equation is concerned. However, when the mass continuity is used to calculate the momentum advection, the l.h.s. of (3.12) is rewritten as follows to express the convergence of the diagonal mass fluxes to the coordinate axes, and is used together with its original form (3.12): 1 U 1 U 1 U U U (Ui, j + Vi,Uj ) − (Ui+1, j+1 + Vi+1, j+1 ) + (Ui, j+1 −Vi, j+1 ) 2 2 2 1 U U U U − (Ui+1, j −Vi+1, j ) = Wi+ 1 , j+ 1 ,k −Wi+ 1 , j+ 1 ,k+1 . 2 2 2 2 2
(3.17)
Let us explain the meaning taking the first term on the l.h.s. of (3.17) Ui,Uj + Vi,Uj =
1 T T T (Ui− 1 , j + Vi,Tj− 1 ) + (Ui+ 1 , j + Vi, j+ 1 ) 2 2 2 2 2 –
31
–
(3.18)
Chapter 3
Spatial grid arrangement and definition of continuity equation
T as an example, where (3.16) is used. If the flow is horizontally nondivergent, the horizontal mass fluxes Ui− 1,j 2
and Vi,Tj− 1 in the first term on the r.h.s. are expressed by streamfunction at two pairs of Upoints, (i − 12 , j + 12 ) and 2
(i − 12 , j − 12 ), and (i − 12 , j − 12 ) and (i + 12 , j − 12 ), respectively. Then, their sum corresponds to the net mass flux crossing the diagonal section connecting the two Upoints (i − 12 , j + 12 ) and (i + 12 , j − 12 ) (Figure 3.3(d)). The second term on the r.h.s. expresses the same quantity, though the route is different. Thus, multiplying by a factor of two, the l.h.s. of (3.17) means the horizontal mass convergence in the diagonal square defined by four Upoints (i − 12 , j + 12 ), (i + 12 , j − 12 ), (i + 32 , j + 12 ), and (i + 12 , j + 32 ). Multiplying by a factor of 12 , the l.h.s. of (3.17) itself means the horizontal mass convergence in the Ucell (i + 12 , j + 12 ), whose area is a half of that of the diagonal square. Taking the l.h.s. of (3.12) as Ai+ 1 , j+ 1 and that of (3.17) as Bi+ 1 , j+ 1 , the standard form of the continuity equation 2 2 2 2 for the Ucell used for the calculation of the momentum advection is generally expressed as: U U α Ai+ 1 , j+ 1 + β Bi+ 1 , j+ 1 = Wi+ 1 , j+ 1 ,k −Wi+ 1 , j+ 1 ,k+1 ,
(3.19)
α + β = 1.
(3.20)
2
2
2
2
2
2
2
2
where As shown later in Chapter 5, (α , β ) = (2/3, 1/3) for the generalized Arakawa scheme and (α , β ) = (1/2, 1/2) for the standardized form derived from the continuity equation generalized for arbitrary bottom topography. What Webb (1995) proposed corresponds to (α , β ) = (1, 0).
3.5 Calculation of area 3.5.1 General orthogonal coordinates When equations are solved in MRI.COM, the temporal variations of physical quantities are calculated as a budget of their fluxes through the boundaries of the Ucells or Tcells. In those situations, it is necessary to know the area and volume of the grid cells. These are numerically calculated as follows. The longitude and latitude (λ , ψ ) of grid points on the sphere are defined by user as a function of the model coordinate (μ , ψ , a)
λ = λ (μ , ψ ), φ = φ (μ , ψ ). For example, the distance from a Tpoint (μ (i), ψ ( j)) to a point a half grid size to the east (μ (i + 12 ), ψ ( j)) (variable name in the model: dx bl) is approximated numerically as follows taking μ1 = μ (i), μ2 = μ (i + 12 ), and
ψ1 = ψ ( j): M
∑L
λ μ1 + (m − 1)δ μ , ψ1 , φ μ1 + (m − 1)δ μ , ψ1 , λ μ1 + mδ μ , ψ1 , φ μ1 + mδ μ , ψ1 .
(3.21)
m=1
Here, L[λ1 , φ1 , λ2 , φ2 ] is the distance between the two points (λ1 , φ1 ) and (λ2 , φ2 ) on the sphere along a great circle and δ μ = (μ2 − μ1 )/M (divided by M ∼ 20 between μ1 and μ2 ). Similarly, a quarter grid area (a bl) surrounded by four points (μ (i), ψ ( j)), (μ (i + 12 ), ψ ( j)), (μ (i + 12 ), ψ ( j + 1 1 1 2 )), and ( μ (i), ψ ( j + 2 )) is, taking ψ2 = ψ ( j + 2 ) and δ ψ = (ψ2 − ψ1 )/N (divided by N ∼ 20 between ψ1 and
– 32 –
3.5. Calculation of area
ψ2 ), calculated as:
1 λ μ1 + (m − 1)δ μ , ψ1 + (n − )δ ψ , 2 n=1 m=1 1 λ μ1 + mδ μ , ψ1 + (n − )δ ψ , 2 ×
1 L λ μ1 + (m − )δ μ , ψ1 + (n − 1)δ ψ , 2 1 λ μ1 + (m − )δ μ , ψ1 + nδ ψ , 2 N
1 φ μ1 + (m − 1)δ μ , ψ1 + (n − )δ ψ , 2 1 φ μ1 + mδ μ , ψ1 + (n − )δ ψ 2
M
∑ ∑L
1 φ μ1 + (m − )δ μ , ψ1 + (n − 1)δ ψ , 2 1 φ μ1 + (m − )δ μ , ψ1 + nδ ψ . 2
(3.22)
As depicted in Figure 3.4, a bli, j is the area of the lower left quarter of the central Upoint. Those for the lower right a bri, j , upper left a tli, j , and upper right a tri, j quarters are obtained similarly. The unit area centered on Upoint (areaui, j ) is then expressed as: areaui, j = a bli, j + a bri, j + a tli, j + a tri, j ,
(3.23)
and the area centered on Tpoint (areati, j ) as areati, j = a bli, j + a bri−1, j + a tli, j−1 + a tri−1, j−1 .
(3.24)
Following the conventions for indexing introduced in Section 3.3, the above equations are expressed in later chapters as follows: areaui+ 1 , j+ 1 = a bli+ 1 , j+ 1 + a bri+ 1 , j+ 1 + a tli+ 1 , j+ 1 + a tri+ 1 , j+ 1 ,
(3.25)
areati, j = a bli+ 1 , j+ 1 + a bri− 1 , j+ 1 + a tli+ 1 , j− 1 + a tri− 1 , j− 1 .
(3.26)
2
2
2 2
3.5.2
2
2
2
2
2 2
2 2
2 2
2 2
2 2
Geographic coordinate
Let us examine the situation of Tcell quarterly divided (Figure 3.4). The area of the northeastern quarter (anhft, the same as that of the northwestern quarter) is obtained by the latitudinal integration of the thick line in Figure 3.4, where Δφ = φ ( j + 12 ) − φ ( j − 12 ). Using the latitude of Tpoint φ ( j), the zonal width of the grid unit for Tpoints Δλ = λ (i + 12 ) − λ (i − 12 ), and the Earth’s radius a, the length of the thick line along the latitude circle (Δs) is expressed as: Δs = a
Δλ cos φ . 2
(3.27)
Integrating this in the latitudinal direction, we obtain the following. anhfti, j
= =
= = =
φ + Δφ 2
Δφ
a2 Δλ φ + 2 a2 Δλ Δsad φ = cos φ d φ = 2 2 φ φ Δφ Δφ a2 Δλ cos φ + sin 4 4 φ Δφ Δ Δφ sin a2 Δλ cos φ cos − sin φ sin 4 4 4 φ φ φ Δ Δ Δ a2 Δλ cos φ cos 1 − tan φ tan sin 4 4 4 2 a Δφ Δφ 1 − tan φ tan . Δλ cos φ sin 2 2 4 – 33 –
Δφ sin φ + − sin φ 2
(3.28)
Chapter 3
Spatial grid arrangement and definition of continuity equation
(a)
dy_tli, j dx_tli, j
dx_tri1, j dy_bri1, j
dy_bli, j
dy_tri, j
Ui, j
dx_tri, j
dy_bri, j
Ti, j dx_bri1, j dy_tri1, j1
dx_bri, j
dx_bli, j dy_tli, j1
dy_tri, j1
dx_tli, j1
dx_tri1, j1
(b) anhfti, j
a_tri, j
a_tli, j
Ui, j anhfti1, j
a_bli, j
Ti, j
a_bri, j
Ӡs
ashfti, j
ashfti1, j
Figure 3.4. Variables that define a grid unit. (a) distance. (b) area. Grid indices (i, j) follow array indices in program codes.
– 34 –
3.5. Calculation of area Similarly, the area of the southeastern quarter of the Tcell (variable name in the model: ashft, the same as that of the southwestern quarter) is expressed as: ashfti, j =
a2 Δφ Δλ cos φ sin 2 2
1 + tan φ tan
Δφ 4
.
(3.29)
At the north and the south poles, where φ = ±90◦ , we obtain the following by changing (3.28) and (3.29) to the following forms. a2 Δφ Δφ Δλ sin cos φ − sin φ tan (3.30) anhfti, j = 2 2 4 a2 Δφ Δφ ashfti, j = Δλ sin cos φ + sin φ tan . (3.31) 2 2 4 At the north pole: anhfti, j
= 0
(3.32)
a2
ashfti, j
=
anhfti, j
=
2
Δλ sin
Δφ Δφ tan . 2 4
(3.33)
At the south pole:
ashfti, j
a2 Δφ Δφ Δλ sin tan 2 2 4 = 0.
(3.34) (3.35)
In our model a bli, j = anhfti, j ,
a bri, j = anhfti+1, j ,
a tli, j = ashfti, j+1 , a tri, j = ashfti+1, j+1 , and the areas of the grid cells centered on the Upoints and Tpoints are calculated by (3.23) and (3.24).
References Arakawa, A., 1972: Design of the UCLA general circulation model, Numerical Simulation Weather and Climate, Tech. Rep. No. 7, Dep. of Meteorology, University of California, Los Angeles, 116 pp. Webb, D. J., 1995: The vertical advection of momentum in BryanCoxSemtner ocean general circulation models, J. Phys. Oceanogr., 25, 3186–3195.
–
35
–
Part II
Main Processes
–
37
–
Chapter 4
Equations of motion (barotropic component)
Historically, iterative methods have been used to solve the barotropic part of the momentum equations by applying the rigidlid approximation. The number of iterations to get convergence of the solution is of order N, where N is the larger number of grid points in the horizontal direction. Thus the number of iterations increases as the number of grid points increases. This means that the iterative process could occupy a large part of the total CPU time. This is a severe burden and should be remedied. An alternative is to replace the rigid lid with a free surface. The number of short barotropic time steps in each long baroclinic time step is roughly the ratio of linear wave speeds, around 70 to 100, which becomes smaller than N when the number of grid points of the model is large.∗ In addition, this method is more suitable for parallel computation than the iterative methods. Thus, for fineresolution models, the freesurface formulations have numerous advantages over the iterative methods. The freesurface formulation has a problem when it is used with a mixedlayer model. To appropriately resolve the surface mixed layer, the uppermost layer should be less than a few meters thick. However, this freesurface model does not work when the sea surface is below the bottom of the uppermost layer and the thickness of this layer vanishes. This occurs in world ocean models, because the difference between the maximum and minimum of the sea surface height becomes several meters.† To remedy this problem, the σ coordinates are introduced for the freesurface formulation near the sea surface. In this method, the thicknesses of the several upper layers (z ≤ −HB ) vary as the sea surface height does (Figure 4.1). These layers are referred to as the σ layers. The σ coordinates are written as
σ≡
z−η . HB + η
(4.1)
This extension of the freesurface formulation is described in Section 4.4. The freesurface formulation was adopted in the BryanCoxSemtner numerical ocean general circulation models by Killworth et al (1991). The σ coordinates were adopted in the freesurface formulation by Hasumi (2006). In this chapter, we explain the freesurface formulation of MRI.COM adopting these two methods.
∗A
North Pacific Model with 1/4◦ × 1/6◦ resolution has 742 grid points in the zonal direction. example, the maximum and minimum seasurface heights in a 1◦ × 1◦ world ocean model are about 1 [m] (subtropical gyres) and 2 [m] (the Ross Sea). † For
–
39
–
Chapter 4
Equations of motion (barotropic component)
z=η
z = HB
Figure 4.1. Schematic of the near surface σ layers.
4.1 Governing equations As described in Chapter 2, the prognostic variables in the freesurface model are the surface elevation (η ) and the vertically integrated velocity (U and V ). The momentum equation is given as
∂U − fV ∂t ∂V + fU ∂t
g(η + H) ∂ η + X, hμ ∂μ g(η + H) ∂ η +Y, = − hψ ∂ψ = −
where X
= −∇H ·
N
∑
k=1
Δz(u, v)u
k− 12
(4.2) (4.3)
v ∂h ∂ hψ μ −∑ u− v Δzk− 1 2 ∂ψ ∂μ k− 12 k=1 hμ hψ N
1 1 0 N u u gρμ dz Δzk− 1 + ∑ (ΔzVHu )k− 1 + Fsurf Δz 1 + Fbottom ΔzN − 1 2 2 2 2 k=1 ρ0 hμ zk− 1 k=1 2 N u ∂h N ∂ hψ μ −∇H · ∑ Δz(u, v)v k− 1 + ∑ u− v Δzk− 1 2 2 ∂ψ ∂μ k− 12 k=1 k=1 hμ hψ N
N 1 1 0 v v −∑ gρψ dz Δzk− 1 + ∑ (ΔzVHv )k− 1 + Fsurf Δz 1 + Fbottom ΔzN − 1 . 2 2 2 2 k=1 ρ0 hψ zk− 1 k=1 N
−∑
Y
=
(4.4)
(4.5)
2
The continuity equation is
∂η 1 + ∂t hμ hψ
∂ (hψ U) ∂ (hμ V ) + ∂μ ∂ψ
= (P − E + R),
(4.6)
where P is precipitation (positive downward), E is evaporation (positive upward) and R is the river discharge rate (positive into the ocean). Figure 4.2 illustrates the grid arrangement of the freesurface formulation. The variable η is defined at Tpoints, and the variables U and V are defined at Upoints. Forcing terms (X and Y ) in Eqs. (4.2) and (4.3) are calculated in the subroutine for the baroclinic component and defined at Upoints.
4.2 Time integration Figure 4.3 presents schematics of the time integration of the barotropic modes in the freesurface formulation. When the time integration of the baroclinic mode is performed from step n (t = tn ) to step n + 1 (t = tn+1 , Δt = – 40 –
4.2. Time integration Δx η i1, j+1
η i, j+1
η i+1, j+1
U i1/2, j+1/2
U i+1/2, j+1/2
η i, j
η i1, j
Δy
η i+1, j
U i+1/2, j1/2
U i1/2, j1/2
η i1, j1
η i+1, j1
η i, j1
Figure 4.2. Grid arrangement of the freesurface formulation tn+1 − tn ), the corresponding time integration of the barotropic mode is carried out from step n to step n + 2 using the barotropic time interval Δttr and the vertically integrated values (X,Y ) at t = tn calculated in the program that solves the baroclinic mode. The timeaveraged value of the barotropic mode over the two baroclinic time intervals between t = tn and t = tn+2 is used to calculate the total velocity at t = tn+1 . The “Euler forwardbackward” scheme is a stable and economical numerical scheme for linear gravity wave equations without advection terms (Meisinger and Arakawa, 1976), and this is adopted for the governing equations in the freesurface formulation of MRI.COM. This scheme is more stable than the leapfrog scheme. The time step can be doubled for the linear gravity wave equations. In this numerical scheme, either the continuity equation or the momentum equation is calculated first, and then the estimated values are used for calculating the remaining equations. In the procedure of MRI.COM, the surface elevation is first calculated using the continuity equation; the calculated surface elevation is then used to calculate the pressure gradient terms of the momentum equations. Killworth et al. (1991) recommended using the Euler backward (Matsuno) scheme for the free surface model except for the tidal problem. The Eulerbackward scheme damps higher modes and is more stable. However, the computer burden increases considerably because this scheme calculates the equations twice for one time step. In MRI.COM, stable solutions are efficiently obtained by using this Euler forwardbackward scheme because the time filter is applied for the barotropic mode.‡ The finitedifference expression of the continuity equation (Eq. 4.6) is
(ηi, j − ηi, j ) Δttr
+
1 ψ μ (δμ hψ U )i, j + (δψ hμ V )i, j = (P − E + R)i, j , (hμ hψ )i, j
(4.7)
where the subscripts are labeled on the basis of Tgrid points. The variables ηi, j and wi, j are located at the Tgrid points, and the variables Ui+ 1 , j+ 1 and Vi+ 1 , j+ 1 are located at Ugrid points. (They are located at (i + 12 , j + 12 ) on 2 2 2 2 the basis of Tgrid points; see Figure 3.2). The finitedifference operator is defined as follows:
δ μ Ai ≡ μ
Ai ≡
‡ An
Ai+ 1 − Ai− 1 2
Δ μi
2
2
Ai+ 1 + Ai− 1 2
, δμ Ai+ 1 ≡
2
2
Ai+1 − Ai , Δμi+ 1 2
μ
, Ai+ 1 ≡ 2
Ai+1 + Ai . 2
option (FSEB) in MRI.COM uses the Euler backward scheme for the barotropic equations.
–
41
–
(4.8)
Chapter 4
Equations of motion (barotropic component)
The same applies to ψ . In the program codes, the above variables are multiplied by the area of each model cell at the Tgrid point (ΔST ):
(ηi, j −ηi, j ) · ΔST i, j (4.9) ψ ψ μ μ = Δttr · (P − E + R)i, j · ΔST i, j − Δyi+ 1 , jU i+ 1 , j − Δyi− 1 , jU i− 1 , j − Δxi, j+ 1 V i, j+ 1 − Δxi, j− 1 V i, j− 1 . 2
2
2
2
2
2
2
2
Each operator is defined the same way as the previous one. This equation is used to obtain the new surface elevation, ηi, j . After obtaining ηi, j , the momentum equations, Eqs. (4.2) and (4.3), are solved. A longer time step can be used when the semiimplicit scheme is applied for the Coriolis terms in the momentum equations. Their finitedifference forms are
(Ui+ 1 , j+ 1 −Ui+ 1 , j+ 1 ) 2
2
2
2
−
Δttr
f (Vi+ 1 , j+ 1 + Vi+ 1 , j+ 1 ) 2
2
2
2
2
=−
g(Hi+ 1 , j+ 1 + η 2
2
μ ,ψ
i+ 12 , j+ 12 )
(hμ )i+ 1 , j+ 1 2
ψ
δμ η i+ 1 , j+ 1 + Xi+ 1 , j+ 1 2
2
2
2
2
(4.10)
(Vi+ 1 , j+ 1 −Vi+ 1 , j+ 1 ) 2
2
2
2
Δttr
f (Ui+ 1 , j+ 1 + Ui+ 1 , j+ 1 )
+
2
2
2
2
2
=−
g(Hi+ 1 , j+ 1 + η 2
2
μ ,ψ
i+ 12 , j+ 12 )
(hψ )i+ 1 , j+ 1 2
μ
δψ η i+ 1 , j+ 1 + Yi+ 1 , j+ 1 . 2
2
2
2
2
(4.11)
Next, we solve these equations for Ui+ 1 , j+ 1 and Vi+ 1 , j+ 1 . Let the r.h.s. of the above equations be GX and GY . 2 2 2 2 Multiplying both sides by Δttr , we have
f Δttr (Vi+ 1 , j+ 1 + Vi+ 1 , j+ 1 ) = Δttr GXi+ 1 , j+ 1 , 2 2 2 2 2 2 2 f Δttr −Vi+ 1 , j+ 1 ) + (Ui+ 1 , j+ 1 + Ui+ 1 , j+ 1 ) = Δttr GYi+ 1 , j+ 1 , 2 2 2 2 2 2 2 2 2
(Ui+ 1 , j+ 1 −Ui+ 1 , j+ 1 ) − 2
2
(Vi+ 1 , j+ 1 2
2
2
2
(4.12) (4.13)
leading to
f Δttr V 1 1 2 i+ 2 , j+ 2 f Δttr + Ui+ 1 , j+ 1 2 2 2
Ui+ 1 , j+ 1 − 2
2
Vi+ 1 , j+ 1 2
2
f Δttr V 1 1 + Δttr GXi+ 1 , j+ 1 , 2 2 2 i+ 2 , j+ 2 f Δttr − Ui+ 1 , j+ 1 + Δttr GYi+ 1 , j+ 1 . 2 2 2 2 2
= Ui+ 1 , j+ 1 +
(4.14)
= Vi+ 1 , j+ 1
(4.15)
2
2
2
2
Letting the r.h.s. of the above equations be RX and RY , we could simplify the equations as follows:
Ui+ 1 , j+ 1 2
2
Vi+ 1 , j+ 1 2
2
= =
f Δttr 2 f Δttr 1+ , RXi+ 1 , j+ 1 + RYi+ 1 , j+ 1 2 2 2 2 2 2 f Δttr 2 f Δttr 1+ . RYi+ 1 , j+ 1 − RXi+ 1 , j+ 1 2 2 2 2 2 2
(4.16) (4.17)
We obtain η , U, and V at t = tn+1 by averaging them between [tn ,tn+2 ]. We can see from Figure 4.3 that there are 2 × Δt/Δttr barotropic time steps between the two baroclinic time intervals [tn ,tn+2 ]. Thus the value at t = tn+1 is the average of 2 × Δt/Δttr + 1 barotropic steps centered at t = tn+1 . It is not always necessary to calculate the governing equations for the barotropic modes until t = tn+2 to obtain the value at t = tn+1 . We can use the value at t = tn+1 without any time averaging when they are calculated until t = tn+1 . This method requires less computational cost than the above method. We apply the timefilter method because we empirically know that this method is more computationally stable. The degree of the timefilter procedure can be specified by the namelist parameter ntflt, which specifies the additional time steps to be calculated past the time t = tn+1 . When ntflt=Δt/Δttr , the averaged barotropic mode – 42 –
4.3. Prognostics of physical properties in the uppermost layer at t = tn+1 is evaluated by calculating the additional Δt/Δttr barotropic modes, and then averaging 2 × Δt/Δttr + 1 barotropic steps. This is the default setting and is applied when ntflt=1 or namelist ntflt is not described in the input namelist files. If ntflt= 0, timefiltering is not applied. If ntflt satisfies 0 < ntflt < Δt/Δttr , then the averaged barotropic mode is evaluated by the average of 2 × ntflt+1 steps centered at t = tn+1 . We usually recommend the default setting.
n+1
n
baroclinic
n+2
Δt
depthintegrated properties
averaged
barotropic m1
Δttr
m
m+1
Figure 4.3. Schematic figure of the time integration of the barotropic mode and its timefiltering procedure Subroutine SURFCE diagnoses the volume of the uppermost layer and the mass flux due to the surface elevations using the prognostic variables η , U, and V . The mass flux due to the surface elevations (labeled ws in the program codes) is defined at Tpoints and is obtained by vertically integrating the continuity equation without precipitation and evaporation, wsi, j
= wi, j · ΔST i, j ψ ψ μ μ = −Δt · Δyi+ 1 , jU i+ 1 , j − Δyi− 1 , jU i− 1 , j + Δxi, j+ 1 V i, j+ 1 − Δxi, j− 1 V i, j− 1 . 2
2
2
2
2
2
2
(4.18)
2
The relationship between the continuity equation and the vertical velocity at the surface is explained later.
4.3 4.3.1
Prognostics of physical properties in the uppermost layer Standard scheme
The flux form is used in MRI.COM for calculating the advection and diffusion (viscosity) of momentum, temperature, and salinity. The prognostic variables at the next time step are obtained as the convergence of these fluxes divided by the corresponding volume. In the freesurface formulation, the volume in the uppermost layer changes with time. This requires many careful and special procedures for calculating the prognostic variables in the uppermost layer. In MRI.COM, the prognostic variables themselves and (the prognostic variables) × (their volume) are saved for the prognostic procedures, which are labeled as uv1a, vv1a, and trcv1a in the program codes. In the prognostic procedures of subroutines clinic and tracer for momentums and tracers in the uppermost layer, their values times their volume at the new time step are temporarily stored in uv1a, vv1a, and trcv1a. The prognostic value at the new time step is obtained by dividing them by the volume of the new time step, volt1, which is calculated at the end of the subroutine surfce. In this procedure, the global integrals of the prognostic variables are conserved. However, they are not conserved locally (see the next subsection). The following is the calculation of temperature (T ) in the uppermost layer when the leapfrog time is used. trcv1b = ΔVt old · Told ↓ –
43
–
Chapter 4
Equations of motion (barotropic component) trcv1a = trcv1b + Convergence of the fluxes ↓
Tnew = trcv1a/ΔVt new ↓
trcv1 = trcv1a , trcv1b = trcv1 In MRI.COM, we also use the flux form for body forcing such as windforcing and restoration of temperature and salinity to the prescribed values in the uppermost layer. The body force for the uppermost layer becomes ∂ u ∂ v 1 (τμ , τψ ) = ... + , , ∂ t ∂ t k= 12 ρ0 Δz 1 + η
(4.19)
2
where (τμ , τψ ) are the wind stress at the surface (momentum flux), and Δz 1 is the standard thickness of the upper2 most layer. Thus, the uppermost layer is more accelerated when this layer is thinner than the standard value. When the restoring condition is applied at the surface, the corresponding temperature and salinity fluxes are FzT ∂ T 1 = ... + , (4.20) FzT = − (T − T ∗ )Δz 1 , 2 γt ∂ t k= 1 Δz 1 + η 2 2 FzS ∂ S 1 FzS = − (S − S∗ )Δz 1 , = ... + . (4.21) 2 γs ∂ t k= 1 Δz 1 + η 2
2
Thus, the temperature and salinity are more strongly restored to the prescribed values when this layer is thinner than the standard value.
4.3.2
Locally conserved scheme (option FSMOM)
For the standard freesurface formulations stated above, the relation between the surface elevation, convergence of the barotropic flow, and P − E + R (precipitation minus evaporation) is
η n+1 − η n−1 + ∇ ·U n = P − E + R. 2Δt
(4.22)
Therefore, the mass is not locally conserved even if the r.h.s. (P − E + R) is zero. However, the mass is globally conserved owing to the flux form. Thus, starting from the uniform salinity distribution, we have small deviation from it even if P − E + R is not applied.§ This inconsistency occurs because η n+1 is evaluated as the average of the barotropic steps from tn to tn+2 , while U n is evaluated as the average of the barotropic steps from tn−1 to tn+1 . To remedy this problem, the following procedure can be used in MRI.COM as option FSMOM. In this option,
ηˆ n+1
is calculated in addition to η n+1 so that
ηˆ n+1 − η n−1 + ∇ ·U n = P − E + R 2Δt
(4.23)
is satisfied. The temperature and salinity in the uppermost layer are obtained based on ηˆ n+1 , and local conservation is maintained. However, global conservation is not maintained in this case. Thus, we do not usually use this option.
§ If the times of the barotropic and baroclinic steps are the same, and timefiltering for the freesurface is not applied (ntflt=0), then the uniform salinity distribution is maintained. This is not practical for simulations but is useful for checking program codes.
– 44 –
4.4. Introduction of σ coordinates near the sea surface
4.4 Introduction of σ coordinates near the sea surface This section introduces the σ coordinates in the upper layers as an extension of the freesurface formulation.¶ Introducing the σ coordinates does not require significant modification in the program codes because the barotropic equations are calculated based on the flux form. In MRI.COM, the σ coordinates are automatically introduced when option FREESURFACE is used. The number of the σ layer, ksgm, must be specified in the model configuration file configure.in and should be less than or equal to the total number of vertical grid levels km. The bottom topography should not appear for 1 ≤ k ≤ ksgm.
4.4.1
Formulation of σ layer model
There is an arbitrariness in the definition of the σ coordinates. We follow the notation of the Princeton Ocean Model (POM; Mellor 2004). Let the surface elevation z = η (μ , ψ ,t). A vertical coordinate (σ coordinate) that is normalized by the thickness between z = −H (< 0) and z = η (μ , ψ ,t) is defined as
σ=
z−η H +η
(−1 < σ < 0).
(4.24)
See Figure 4.1. Thus, the values of σ range from σ = 0 at z = η to σ = −1 at z = −H. Hereinafter, we define the thickness of the σ layer as D, i.e., D ≡ H + η . The quantity Dd σ = (η + H )d σ corresponds to the real vertical grid size dzu in the program code of MRI.COM. (Note that dzu varies with time.) For the general σ coordinates, the depth (H) is also a function of horizontal position, but here H is assumed to be constant. Next we derive the governing equations in the σ coordinates. The equations expressed in terms of the original Cartesian coordinates z(μ ∗ , ψ ∗ , z∗ ,t ∗ ) are transformed into sigma coordinates σ (μ , ψ , σ ,t), where
μ = μ ∗, ψ = ψ ∗, σ =
z∗ − η , t = t ∗. H +η
(4.25)
The partial derivatives transform according to
∂ α∗ ∂ μ∗ ∂ α∗ ∂ ψ∗ ∂ α∗ ∂ z∗ ∂ α∗ ∂ t∗
= = = =
∂α 1+σ ∂η ∂α − , ∂μ D ∂μ ∂σ ∂α 1+σ ∂η ∂α − , ∂ψ D ∂ψ ∂σ 1 ∂α , D ∂σ ∂α 1+σ ∂α ∂η − . ∂t D ∂σ ∂t
(4.26) (4.27) (4.28) (4.29)
The material derivatives transform according to ∗ dα ∗ ∂ α ∗ u∗ ∂ α ∗ v∗ ∂ α ∗ ∗ ∂α = + + w + ∗ ∗ ∗ ∗ dt ∗ ∂t hμ ∂ μ hψ ∂ ψ ∂z ∗ ∂α u ∂α v ∂α w ∂α 1+σ ∂α u ∂η v ∂η ∂η . = + + + − + + ∂t hμ ∂ μ hψ ∂ ψ D ∂σ D ∂ σ hμ ∂ μ hψ ∂ ψ ∂t
Here, we define ω as
ω≡
¶ The
w∗ 1 + σ dσ − = dt ∗ D D
v ∂η ∂η u ∂η + + h μ ∂ μ hψ ∂ ψ ∂t
introduction of the σ layer, and the expression of equations are based on Hasumi (2006).
–
45
–
(4.30)
.
(4.31)
Chapter 4
Equations of motion (barotropic component)
The above equation then becomes
∂α u ∂α v ∂α ∂α dα dα ∗ = + +ω ≡ + . dt ∗ ∂t hμ ∂ μ hψ ∂ ψ ∂σ dt
(4.32)
Thus, they take nearly the same form as in zcoordinates. In the program code, we do not directly calculate ω , but diagnose Dω and Dd σ from the equation
ω
∂α ∂α = Dω . ∂σ D∂ σ
(4.33)
They are stored in the model code as variables wlwl and dzu. (Their dimensions are the same as those in the zcoordinates, and their magnitudes are comparable.)
4.4.2
Governing equations in the σ coordinates
a. Continuity equations The continuity equation in the σ layer is ∂ (hψ u) ∂ (hμ v) 1 ∂ω 1 u ∂η v ∂η ∂η + + + + + = 0. hμ hψ ∂μ ∂ψ ∂ σ D hμ ∂ μ hψ ∂ ψ ∂t
(4.34)
In the flux form, this becomes simpler.
∂η 1 + ∂t hμ hψ
∂ (hψ uD) ∂ (hμ vD) ∂ω +D + = 0. ∂μ ∂ψ ∂σ
(4.35)
In the program code, this equation is used to diagnose Dω , which is used as vertical velocity w (wlwl) in the σ layer.
b. Advection terms Advection terms in the flux form A (α D) are as follows: ∂ (hψ uα D) ∂ (hμ vα D) 1 ∂ (ωα ) A (α D) = + . +D h μ hψ ∂μ ∂ψ ∂σ In the program code, the convergence of the advective flux per unit area is calculated as follows: ∂ (hψ uα Dd σ ) ∂ (hμ vα Dd σ ) 1 A (α Dd σ ) = + hμ hψ ∂μ ∂ψ
(4.36)
(4.37)
+ (α Dω )upper − (α Dω )lower . c. Pressure gradient term The hydrostatic equation is
1 ∂p = −ρ g. D ∂σ When the atmospheric pressure at the surface is neglected, the pressure in the σ coordinates is p(σ ) = g
0 σ
Dρ d σ = gD –
46
–
0 σ
ρ dσ .
(4.38)
(4.39)
4.4. Introduction of σ coordinates near the sea surface After some manipulations, the following expression for the pressure gradient in the σ coordinates is obtained: 0 ∂ p∗ ∂ p 1+σ ∂η ∂ p ∂ρ σ ∂η ∂ρ ∂η − = gD − , ∗ = dσ + ρ g ∂μ ∂μ D ∂μ ∂σ ∂ μ D ∂ μ ∂μ ∂ σ σ 0 ∂ρ ∂ p∗ ∂ p 1+σ ∂η ∂ p σ ∂η ∂ρ ∂η dσ + ρ g − = gD − . ∗ = ∂ψ ∂ψ D ∂ψ ∂σ ∂ψ D ∂ψ ∂σ ∂ψ σ
(4.40)
The first term on the r.h.s. corresponds to the baroclinic part of the pressure gradient. The second term on the r.h.s. is the barotropic part of the pressure gradient. Density ρ in the last term on the r.h.s. is replaced by the reference density ρ0 .
d. Time integration of velocities and tracers Horizontal viscosity and diffusion are assumed to be parallel to the σ layer in the σ layer model. In MRI.COM, the volume integral of a tracer with variable name trcl, trcv1 (≡ trcl ∗ tvol1), is stored to calculate trcl in the next step. Introduction of the σ layer corresponds to the direct extension of trcv1 in the uppermost layer to all σ layers. The following equation expresses the tracer equations per unit area: ∂ (hψ uT D) ∂ (hμ vT D) ∂ ∂ 1 − + [ω T D] + DD (T ). [T D] = − (4.41) ∂t hμ hψ ∂μ ∂ψ ∂σ
4.4.3 Redistribution of tracers among the σ layers The volume change due to freshwater flux from the atmosphere is distributed among all the σ layers, so that the ratio of the thicknesses for each σ layer remains unchanged. However, in reality freshwater flux like rain should affect only the uppermost layer. In addition, the uppermost layer gets excess heat flux because the heat flux due to this freshwater flux is only applied to the uppermost layer. In order to fix this problem, tracers are redistributed among the σ layers at the beginning of the subroutine tracer. See the schematic diagram in Figure 4.4. Let us consider the kth layer in the σ layer (1 ≤ k ≤ ksigma ). Let surface height, temperature, and salinity before this redistribution be η n+1(old) , T n+1(old) , and Sn+1(old) . The surface height due to the freshwater flux is
η n+1 = η n+1(old) + δ η ,
(4.42)
where δ η = 2Δt (P − E + R) during the leapfrog time integration. The following thicknesses are redistributed at the lower boundary and the upper boundary of the kth layer. k−1
δ zTk = 1 − ∑ Δσl δ η (1 < k ≤ ksigma ), and δ z1T = 0,
δ zBk
=
l=1 k
(4.43)
1 − ∑ Δσl δ η (1 ≤ k < ksigma ), and δ zBksigma = 0,
(4.44)
l=1
where zBk , and zTk are the redistributed thickness at the lower end and the upper end of the kth layer, respectively (δ zTk = δ zBk−1 ). The redistribution of tracer (S) among the σ coordinate layer depends on the sign of δ η : When δ η > 0 (Figure 4.4a), n+1(old)
Skn+1 (η n+1 + H )Δσk = Sk
n+1(old)
(η n+1(old) + H )Δσk − Sk – 47 –
n+1(old)
δ zBk + Sk−1
δ zTk .
(4.45)
Chapter 4
Equations of motion (barotropic component)
When δ η < 0 (Figure 4.4b), n+1(old)
Skn+1 (η n+1 + H )Δσk = Sk
n+1(old)
(η n+1(old) + H )Δσk − Sk+1
n+1(old)
δ zBk + Sk
δ zTk .
(4.46)
In the program, trcv1b is substituted with trcv1a at the beginning of the subroutine tracer, then trcv1a is modified using either (4.45) or (4.46). (b)
(a)
Sk1
Sk1 Sk1
Sk1 T
T
δzk
δzk Sk
Sk Sk
Sk B k
B k
δz
δz Sk+1
Sk+1
Sk+1
Sk+1
Figure 4.4. Schematic diagram of the redistribution of tracer among the σ coordinate layers. When fresh water is (a) added, and (b) removed from the sea surface.
References Hasumi, H., 2006: CCSR Ocean Component Model (COCO) version 4.0, CCSR report, 25, 103pp. Killworth, P. D., D. Stainforth, D. J. Webb, and S. M. Paterson, 1991: The Development of a FreeSurface BryanCoxSemtner Ocean Model, J. Phys. Oceanogr., 31, 1333–1348. Meisinger, F., and A. Arakawa 1976: Numerical methods used in atmospheric models, GARP Publications Series, 17, 65pp. Mellor, G. L., 2004: Users guide for a threedimensional, primitive equation, numerical ocean model, Prog. in Atmos. and Ocean. Sci, Princeton University, 53pp., available online at http://www.aos.princeton.edu/WWWPUBLIC/htdocs.pom/PubOnLine/POL.html.
–
48
–
Chapter 5
Equations of motion (baroclinic component)
This chapter explains the advection terms (Section 5.1) and the viscosity terms (Section 5.2) of momentum. One of the unique characteristics of MRI.COM’s momentum advection terms is that there are exchanges of momentum between Ucells that share only a corner without a common side boundary. This scheme enables the flow field around and over the bottom topography to be naturally expressed. Furthermore, quasienstrophies, (∂ u/∂ y)2 and (∂ v/∂ x)2 , for the Ucells away from land are conserved in calculating the momentum advection for horizontally nondivergent flows. The description of momentum advection in Section 5.1 is based on Ishizaki and Motoi (1999). The discrete expressions for the viscosity terms in the momentum equations are based on the generalized orthogonal coordinates. A harmonic operator is used as the default assuming a noslip condition on the landsea boundaries. A biharmonic operator (option VISBIHARM) and a parameterization of viscosity as a function of the velocity gradients (option SMAGOR) could also be used.
5.1 Advection terms Chapter 3 demonstrated that the mass fluxes used for calculating momentum advection are those for the mass continuity of the Ucell and are obtained by an averaging operation (3.10) of those for the Tcell mass continuity (3.3) to (3.9). This is the preliminaries for constructing the general mass flux form over an arbitrary bottom and coastal topography. Its vertical part can express diagonally upward mass fluxes over bottom relief and its horizontal part can express horizontally diagonal mass fluxes along coast lines (Ishizaki and Motoi, 1999). Here we explain how to obtain the mass fluxes to be used in the momentum advection and how to get the finite difference expression of the advection terms. The horizontal subscript indices of variables are integers for the Tpoint (i, j), and therefore, (i + 12 , j + 12 ) for the Upoint. In the vertical direction, integer k is used for the level of the vertical mass fluxes and the level for the T and Upoints a half vertical grid size lower is expressed by k + 12 (Figure 3.2(a)).
5.1.1
Vertical mass fluxes and its momentum advection
According to the definition (3.3) and (3.10) in Chapter 3, the vertical mass flux at the upper surface, level k, of U the Ucell (i + 12 . j + 12 , k + 12 ), W i+ 1 , j+ 1 ,k , is defined by surrounding W T as 2
U
W i+ 1 , j+ 1 ,k = 2
2
2
Wi,Tj,k Ni, j,k+ 1 2
+
T Wi+1, j,k
Ni+1, j,k+ 1
+
2
Wi,Tj+1,k Ni, j+1,k+ 1 2
+
T Wi+1, j+1,k
Ni+1, j+1,k+ 1
where Ni, j,k+ 1 is the number of sea grid cells around the Tpoint Ti, j in layer k + 1/2. 2
–
49
–
2
,
(5.1)
Chapter 5
Equations of motion (baroclinic component)
On the other hand, the vertical mass flux at the bottom surface, level k, of the Ucell (i + 12 , j + 12 , k − 12 ), W U1
i+ 2 , j+ 12 ,k
is defined as U Wi+ 1 , j+ 1 ,k = 2
2
Wi,Tj,k Ni, j,k− 1
+
2
T Wi+1, j,k
Ni+1, j,k− 1
+
2
Wi,Tj+1,k Ni, j+1,k− 1 2
+
T Wi+1, j+1,k
Ni+1, j+1,k− 1
.
(5.2)
2
U
Since W T are continuous at the boundary of vertically adjacent Tcells, W i+ 1 , j+ 1 ,k and W U 1 1 seem to be i+ 2 , j+ 2 ,k 2 2 discontinuous at the boundary when N are vertically different, for example, Ni, j,k+ 1 < Ni, j,k− 1 , over the bottom 2
2
relief. However, this apparent discrepancy can be consistently interpreted by introducing diagonally upward or downward mass fluxes as shown below.
a. Onedimensional variation of bottom relief We first consider a case in which the bottom depth varies in one direction like a staircase (Figure 5.1(a)). Assume a barotropic current flows over the topography. The Upoints are just intermediate between Tpoints. This indicates the mass continuity for Tcells. The barotropic flow comes from the left and barotropy is conserved in shallow regions. Figure 5.1(b) depicts the mass continuity for Ucells, derived from those for adjacent Tcells. Except for fluxes just on the bottom slope, each flux is obtained as a mean value of neighboring fluxes for Tcells. Just on the bottom slope, we must introduce a flux that flows along the slope to ensure mass continuity. The lowermost Ucells at the slope have nonzero vertical flux at the bottom. The barotropy of the flow and the distribution of vertical velocity are thereby kept reasonable for Ucell fluxes. (a)
(b)
U, V
U, V
3
3
U, V
3
9
3
3
4.5
U, V
3
U, V
6.75
3.75 0.75
3 3
U, V
3
4.5
1.5 3
U, V
U, V
4.5
9
3 2.25
3.75 1.5
3
3
3
1.5
Figure 5.1. (a) Twodimensional mass fluxes for Tcells on a stairlike topography. (b) Twodimensional mass fluxes for Ucells on the same topography.
b. Twodimensional variation of bottom depth The diagonally upward or downward mass fluxes introduced in the previous simple case are generalized for flows over bottom topography that varies twodimensionally. For simplicity, we consider a twolayer case without losing generality. First, we consider three examples of bottom relief, then generalize the results. Example 1 Consider a case in which all cells are sea cells in the upper and lower layers except for cell d in the lower layer (cell dl ) (Figure 5.2). We use suffixes l and u to designate the lower and the upper layer. The central Tpoint and Tcell are represented by A. The vertical mass flux W T should be continuous at the interface between cells Al and Au , though the area of cell Al (3/4 measured in grid area units) differs from that of Au (1 unit). Let us – 50 –
5.1. Advection terms consider how this Tcell mass flux W T should be distributed to the mass flux W U of neighboring cells represented by a, b, c, and d. In the lower layer, WT is shared by three cells, al , bl , and cl , so the contribution of W T to each W U is W T /3, but in the upper layer, it is W T /4 because part of WT should also be shared by W U at cell du . Here W U at the bottom of cell du is no longer zero. Therefore, W T /4 of the W T /3 shared by each of the three lower sea grid cells al , bl , and cl is purely vertical, and the remaining W T /12 (= W T /3 − W T /4) flows to cell du through the interface. Gathering these diagonal fluxes from the lower three cells, the total amount entering cell du is certainly W T /4 (=W T /12 × 3). The advected momentum value should be the mean of those at the starting and ending cells of the flux, if the centered difference scheme is used, which is necessary to conserve the total kinetic energy.
UPPER
LOWER
du
cu 䃁
cl
dl 䃁
䃁
T
W Al
Au 䃁
䃁
䃁
au
bu
T T
al
bl
T
䃁 1/4W T
䃁
䃁 1/3W T
1/3W 䋭 1/4W = 1/12W
Figure 5.2. First example of landsea patterns, in which all four upper cells are sea cells, with three sea cells and one land cell in the lower layer.
Example 2
Next, consider an example in which only bl is a sea cell in the lower layer, and all four cells are sea cells in the upper layer (Figure 5.3). In the lower layer, W T is shared only by bl but in the upper layer, it is shared by all four cells. Therefore, W T /4 of W T at cell bl is carried vertically upward and the remaining 3W T /4 is distributed to the other three cells in the upper layer (au , cu , and du ), each receiving W T /4. Example 3 A third example holds that the upper layer also has land area. In this example, cells cl , dl , and du are land cells and the others are sea cells (Figure 5.4). In the lower layer, W T is shared by two cells (al , and bl ) while it is shared by three cells (au , bu , and cu ) in the upper layer. Therefore, from each of al and bl , W T /3 of W T /2 goes vertically upward and the remaining W T /6 (= W T /2 −W T /3) goes diagonally upward to cell cu with a total amount of W T /3 (= W T /6 × 2).
–
51
–
Chapter 5
Equations of motion (baroclinic component)
UPPER
LOWER
du
cu 䃁
cl
䃁
dl T
W
Au
Au 䃁
䃁
䃁
au
bu
T
T
䃁 1/4W T
al
bl
䃁 W T
T
1/3( W 䋭 1/4W ) = 1/4W
Figure 5.3. Second example of landsea patterns, in which all four upper cells are sea cells, with one sea cell and three land cells in the lower layer. c. Generalization The relationship between the landsea distribution and the vertically and diagonally upward fluxes stated above is generalized for an arbitrary landsea distribution. Assume cell dl is a land but cell du is a sea cell and consider the diagonally upward fluxes coming to cell du . We take Nl as the number of sea cells around point A in the lower layer and Nu as the number in the upper layer (1 ≤ Nl < Nu ≤ 4). Each cell in the lower layer carries W T /Nl , and W T /Nu of it goes vertically upward. The remaining W T /Nl −W T /Nu = W T (Nu − Nl )/(Nl Nu )
(5.3)
should be distributed as diagonally upward fluxes to sea cells in the upper layer at which the lower layer is land. The number of such upper sea cells is Nu − Nl including cell du . Thus, each diagonally upward flux coming to cell du is W T (Nu − Nl )/(Nl Nu ) × 1/(Nu − Nl ) = W T /(Nl Nu ). (5.4) The number of such fluxes coming to the cell du is Nl , so their total is W T /(Nl Nu ) × Nl = W T /Nu .
(5.5)
Based on these discussions we understand the difference between (5.1) and (5.2). We regard the name of each cell such as al also as a landsea index. If we assume that al = 1(0) when cell al is a sea (land) cell, then the diagonally upward mass flux and momentum flux coming from cell al to cell du are al W T /(Nl Nu ) and
al W T (ual + udu )/(2Nl Nu ),
(5.6)
where ual and udu are the velocity at cells al and du , respectively. Purely vertical mass flux and momentum flux from cell al to cell au are expressed as al W T /Nu
and
al W T (ual + uau )/(2Nu ), – 52 –
(5.7)
5.1. Advection terms
UPPER
LOWER
cu
dl
cl
du 䃁
T
Au 䃁
W 䃁
䃁
bu
䃁
al
bl
au
T
T
䃁 1/3W T
Al
䃁 1/2W T
1/2W 䋭 1/3W = 1/6W
T
Figure 5.4. Third example of landsea patterns, in which one of the upper cells is a land cell, with two land and two sea cells in the lower layer. respectively, where uau is the velocity at cell au . Similar formulations apply to cells bl and cl . Mass and momentum fluxes for W T at other Tpoints around cell du should be calculated similarly to complete vertically and diagonally upward momentum advections around cell du . When Nu = Nl , diagonally upward fluxes need not be considered and only vertical fluxes (5.7) apply.
5.1.2
Horizontal mass flux and its momentum advection
a. Horizontal mass fluxes We next consider the generalization of the Ucell horizontal mass fluxes for arbitrary coast lines. To do this, we start with the generalization of the Tcell mass continuity (3.3)(3.9). Assuming that ei+ 1 , j+ 1 is a landsea index 2 2 T T (unity for sea and zero for land) for Ucell (i + 12 , j + 12 ), the general formulae for Ui+ 1 , j and Vi, j+ 1 are given as 2
2
1 T ∗ Ui+ 1 , j = (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )ui+ 1 , j Δyi+ 1 , j Δz 2 2 2 2 2 2 2 2 and 1 Vi,Tj+ 1 = (ei− 1 , j+ 1 + ei+ 1 , j+ 1 )v∗i, j+ 1 Δxi, j+ 1 Δz 2 2 2 2 2 2 2 2
(5.8)
Here we neglect the vertical subscript (k + 12 ). Substituting these formulae into the Tcell mass continuity (3.3), the X (zonal) component of the mass continuity
–
53
–
Chapter 5
Equations of motion (baroclinic component)
for Ucell (i + 12 , j + 12 ) (3.10), XMCUi+ 1 , j+ 1 , multiplied by its own landsea signature ei+ 1 , j+ 1 , is expressed as 2
2
2
2
Δyi+ 1 , j+ 1 2 2 XMCUi+ 1 , j+ 1 = ei+ 1 , j+ 1 Δz 2 2 2 2 2 1 [(e 1 1 + ei− 1 , j+ 1 )u∗i− 1 , j − (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )u∗i+ 1 , j ] × 2 2 2 2 2 2 Ni, j i− 2 , j− 2 2 2 1 + [(e 1 1 + ei+ 1 , j+ 1 )u∗i+ 1 , j − (ei+ 3 , j− 1 + ei+ 3 , j+ 1 )u∗i+ 3 , j ] 2 2 2 2 2 2 Ni+1, j i+ 2 , j− 2 2 2 1 + [(e 1 1 + ei− 1 , j+ 3 )u∗i− 1 , j+1 − (ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )u∗i+ 1 , j+1 ] 2 2 2 2 2 2 Ni, j+1 i− 2 , j+ 2 2 2 1 + [(ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )u∗i+ 1 , j+1 − (ei+ 3 , j+ 1 + ei+ 3 , j+ 3 )u∗i+ 3 , j+1 ] 2 2 2 2 2 2 2 2 Ni+1, j+1 2 2 1 Δyi+ 1 , j+ 1 2 2 = ei+ 1 , j+ 1 (e 1 1 + ei− 1 , j+ 1 )u∗i− 1 , j Δz × 2 2 2 2 2 Ni, j i− 2 , j− 2 2 1 1 + − + (ei+ 1 , j− 1 + ei+ 1 , j+ 1 )u∗i+ 1 , j 2 2 2 2 Ni, j Ni+1, j 2 1 − (e 3 1 + ei+ 3 , j+ 1 )u∗i+ 3 , j 2 2 Ni+1, j i+ 2 , j− 2 2
1 + (e 1 1 + ei− 1 , j+ 3 )u∗i− 1 , j+1 2 2 Ni, j+1 i− 2 , j+ 2 2 1 1 + − + (ei+ 1 , j+ 1 + ei+ 1 , j+ 3 )u∗i+ 1 , j+1 2 2 2 2 Ni, j+1 Ni+1, j+1 2 1 − (e 3 1 + ei+ 3 , j+ 3 )u∗i+ 3 , j+1 . 2 2 Ni+1, j+1 i+ 2 , j+ 2 2
(5.9)
Here, recalling Ni, j = ei− 1 , j− 1 + ei+ 1 , j− 1 + ei− 1 , j+ 1 + ei+ 1 , j+ 1 , 2
2
2
2
2
2
2
(5.10)
2
we have,
−
1 1 + (ei+ 1 , j− 1 + ei+ 1 , j+ 1 ) 2 2 2 2 Ni, j Ni+1, j 1 1 = (e 1 1 + ei− 1 , j+ 1 ) − (e 3 1 + ei+ 3 , j+ 1 ) 2 2 2 2 Ni, j i− 2 , j− 2 Ni+1, j i+ 2 , j− 2 (5.11)
and −
1 Ni, j+1
+
1 Ni+1, j+1
(ei+ 1 , j+ 1 + ei+ 1 , j+ 3 ) 2
=
1 Ni, j+1
2
2
2
(ei− 1 , j+ 1 + ei− 1 , j+ 3 ) − 2
2
2
2
1 Ni+1, j+1
(ei+ 3 , j+ 1 + ei+ 3 , j+ 3 ). 2
2
2
2
(5.12)
– 54 –
5.1. Advection terms Thus, based on (3.13) and (3.14), 1 (e 1 1 + ei− 1 , j+ 1 )(u∗i− 1 , j + u∗i+ 1 , j ) 2 2 2 2 Ni, j i− 2 , j− 2 2 2 (ei+ 3 , j− 1 + ei+ 3 , j+ 1 )(u∗i+ 1 , j + u∗i+ 3 , j )
XMCUi+ 1 , j+ 1 = ei+ 1 , j+ 1 2
2
2
−
1 Ni+1, j
1
Δyi+ 1 , j+ 1 2
2
2
Δz
2
2
2
2
2
(e 1 1 + ei− 1 , j+ 3 )(u∗i− 1 , j+1 + u∗i+ 1 , j+1 ) 2 2 Ni, j+1 i− 2 , j+ 2 2 2 1 − (ei+ 3 , j+ 1 + ei+ 3 , j+ 3 )(u∗i+ 1 , j+1 + u∗i+ 3 , j+1 ) 2 2 2 2 Ni+1, j+1 2 2
1 1 = ei+ 1 , j+ 1 ei− 1 , j+ 1 Ui,Uj + UU 2 2 2 2 Ni, j Ni, j+1 i, j+1 1 1 1 U U U − ei+ 3 , j+ 1 Ui+1, + U +ei− 1 , j− 1 Ui, j j i+1, j+1 2 2 Ni+1, j 2 2 Ni, j Ni+1, j+1 1 1 1 U U U − ei+ 3 , j+ 3 Ui+1, + e U − e U 1 3 3 1 j+1 i, j+1 i+1, j . i− 2 , j+ 2 N i+ 2 , j− 2 N 2 2 Ni+1, j+1 i, j+1 i+1, j +
(5.13)
Adding the Y (meridional) component to the above formula, we obtain the horizontal part of the Ucell mass continuity as follows:
HMCUi+ 1 , j+ 1 = ei+ 1 , j+ 1 2 2 2 2 1 1
1 1 U U × ei− 1 , j+ 1 Ui,Uj + Ui,Uj+1 − ei+ 3 , j+ 1 Ui+1, + Ui+1, j j+1 2 2 Ni, j 2 2 Ni+1, j Ni, j+1 Ni+1, j+1 1 1 1 1 U U − ei+ 1 , j+ 3 + ei+ 1 , j− 1 Vi,Uj + Vi+1, Vi,Uj+1 + Vi+1, j j+1 2 2 Ni, j 2 2 Ni, j+1 Ni+1, j Ni+1, j+1 1 1 U U + ei− 1 , j− 1 (Ui,Uj + Vi,Uj ) − ei+ 3 , j+ 3 (Ui+1, j+1 + Vi+1, j+1 ) 2 2 Ni, j 2 2 Ni+1, j+1 1 1 U U (5.14) + ei− 1 , j+ 3 (Ui,Uj+1 −Vi,Uj+1 ) − ei+ 3 , j− 1 (Ui+1, −V ) j i+1, j . 2 2 Ni, j+1 2 2 Ni+1, j Assuming mass fluxes ME , MN , MNE , MSE as follows:
ME MN
i, j+ 21
i+ 21 , j
MNEi, j MSEi, j
1 1 Ui,Uj + Ui,Uj+1 , 2 2 2 2 Ni, j Ni, j+1 1 1 U , = ei+ 1 , j+ 1 ei+ 1 , j− 1 Vi,Uj + Vi+1, j 2 2 2 2 Ni, j Ni+1, j 1 = ei+ 1 , j+ 1 ei− 1 , j− 1 (Ui,Uj + Vi,Uj ), 2 2 2 2 Ni, j 1 = ei− 1 , j+ 1 ei+ 1 , j− 1 (Ui,Uj −Vi,Uj ), 2 2 2 2 Ni, j
= ei+ 1 , j+ 1 ei− 1 , j+ 1
(5.15)
then, HMCUi+ 1 , j+ 1 2
2
= ME
i, j+ 21
− ME
i+1, j+ 21
+ MN
i+ 21 , j
− MN
i+ 12 , j+1
+MNEi, j − MNEi+1, j+1 + MSEi, j+1 − MSEi+1, j .
(5.16)
Here, ME and MN are axisparallel mass fluxes, and MNE and MSE are horizontally diagonal ones (Figure 5.5).
–
55
–
Chapter 5
Equations of motion (baroclinic component)
MSE i, j+1
MN i+1/2, j+1
MNE i+1, j+1
ME i, j+1/2
X
ME i+1, j+1/2
MN i+1/2, j
MSE i+1, j
i, j MNE i, j
Figure 5.5. Distribution of generalized mass fluxes for Ucell (i + 1/2, j + 1/2) If we derive the formula for the standard case from (5.14) (all of N are 4), 1 1 U 1 (Ui, j +UiU, j+1 ) − (UiU+1, j +UiU+1, j+1 ) 2 2 2 1 U 1 U + (Vi, j +Vi, j+1 ) − (ViU+1, j +ViU+1, j+1 ) 2 2 1 1 U 1 U U (U +Vi, j ) − (Ui+1, j+1 +ViU+1, j+1 ) + 2 2 i, j 2 1 U 1 U + (Ui, j+1 −Vi, j+1 ) − (UiU+1, j −ViU+1, j ) . 2 2
HMCUi+ 1 , j+ 1 = 2
2
(5.17)
This expression means that the horizontal mass flux convergence is a mean of those of the axisparallel mass fluxes (3.12) and of the diagonal ones (3.17). However, their weighting factors α and β are both 1/2 in the present case, while (α , β ) = (2/3, 1/3) for the generalized Arakawa scheme, which conserves quasienstrophy such as (δ v/δ x)2 and (δ u/δ y)2 in a horizontally nondivergent flow.
b. Horizontal momentum advection For the standard case away from land, we have the freedom to choose weights (α : β ) to average the convergences of the axisparallel and the horizontally diagonal mass fluxes, as long as α + β = 1, as seen in (5.17). In MRI.COM α = 2/3 and β = 1/3 are chosen for the standard case so that the momentum advection terms lead to
– 56 –
5.1. Advection terms the generalized Arakawa scheme. In this case the zonal momentum advection is expressed as follows: 2 1 1 (u 1 1 + ui+ 1 , j+ 1 ) (Ui,Uj + Ui,Uj+1 ) 2 2 2 3 2 i− 2 , j+ 2 1 U 1 U − (ui+ 1 , j+ 1 + ui+ 3 , j+ 1 ) (Ui+1, j + Ui+1, j+1 ) 2 2 2 2 2 2 1 1 U + (ui+ 1 , j− 1 + ui+ 1 , j+ 1 ) (Vi,Uj + Vi+1, j) 2 2 2 2 2 2 1 1 U − (ui+ 1 , j+ 1 + ui+ 1 , j+ 3 ) (Vi,Uj+1 + Vi+1, ) j+1 2 2 2 2 2 2 1 1 1 + (u 1 1 + ui− 1 , j− 1 ) (Ui,Uj + Vi,Uj ) 2 2 2 3 2 i+ 2 , j+ 2 1 U 1 U − (ui+ 1 , j+ 1 + ui+ 3 , j+ 3 ) (Ui+1, j+1 + Vi+1, j+1 ) 2 2 2 2 2 2 1 1 + (ui+ 1 , j+ 1 + ui− 1 , j+ 3 ) (Ui,Uj+1 −Vi,Uj+1 ) 2 2 2 2 2 2 1 U 1 U − (ui+ 1 , j+ 1 + ui+ 3 , j− 1 ) (Ui+1, −Vi+1, ) j j 2 2 2 2 2 2
CADi+ 1 , j+ 1 (u) = 2
2
(5.18)
This scheme under Arakawa’s Bgrid arrangement conserves the quasienstrophies ((∂ u/∂ y)2 and (∂ v/∂ x)2 ) in a horizontally nondivergent flow. To merge the generalized Arakawa scheme for the standard case into the general form of the horizontal mass flux expressed in Figure 5.5 and related momentum flux, let us examine the axisparallel and horizontally diagonal mass flux associated with Ui,Uj . Look at Figure 5.2b, where letters a, b, c, and d designate the landsea index and names of Ucells. Cell d is assumed to be a sea cell (d = 1). We analyze two kinds of mass fluxes associated with Ui,Uj under different combinations of a, b, and c (eight cases), as indicated in the first column in Table 5.1. Column (A) corresponds to an index, which is unity for its own combination and zero for all other combinations. Column (B) lists the coefficient of Ui,Uj in the axisparallel mass flux of the Ucell mass continuity (5.14). Column (C) indicates the coefficient of Ui,Uj in the horizontally diagonal mass flux of (5.14). The generalized coefficient of Ui,Uj in the axisparallel mass flux (c1 ) is obtained by summing the product of A and B over the eight cases. Similarly, the generalized coefficient in the horizontally diagonal mass flux (c2 ) is obtained by the summing the product of A and C. That is, c1
=
8
1
∑ An Bn = 6 c(ab − a − b+3)
n=1
and c2
=
8
1
∑ An Cn = 6 b(3−a − c).
(5.19)
n=1
Then, the axisparallel and the horizontally diagonal flux of zonal momentum (u) related with Ui,Uj , multiplied by the landsea index d, are d (uc + ud )c1Ui,Uj 2 d (ub + ud )c2Ui,Uj 2
1 1 (uc + ud ) cd(ab − a − b+3)Ui,Uj , 2 6 and 1 1 = (ub + ud ) bd(3−a − c)Ui,Uj , 2 6
=
respectively.
–
57
–
(5.20)
Chapter 5
Equations of motion (baroclinic component)
Table 5.1. Definition of a landsea index, the index identifying each case (column A), the coefficient of Ui,Uj in the axisparallel mass flux (column B), and the coefficient of Ui,Uj in the horizontally diagonal mass flux (column C), for eight combinations of indices a, b, and c in Figure 5.2b. Cell d is assumed to be a sea cell, and the momentum advection by means of Ui,Uj into and from cell d is generalized. B Coefficient of Ui,Uj
C Coefficient of Ui,Uj
CASE n
Landsea index a b c
A
(axisiparallel) +
(horizontallydiagonal) ×
1
1 1 1
abc
1/3
1/6
2
1 1 0
ab(1−c)
0
1/3
3
1 0 1
ab(1−b)c
1/3
0
4
0 1 1
(1−a)bc
1/3
1/3
5
1 0 0
−
0
0
6
0 1 0
(1−a)b(1−c)
0
1/2
7
0 0 1
(1−a)(1−b)c
1/2
0
8
0 0 0
−
0
0
The resultant momentum fluxes are as follows: FE
i, j+ 21
FN
i+ 21 , j
1 (u 1 1 2 i− 2 , j+ 2 1 (u 1 1 2 i+ 2 , j− 2 1 (u 1 1 2 i+ 2 , j+ 2 1 (u 1 1 2 i− 2 , j+ 2
(u) = (u) =
FNEi, j (u) = FSEi, j (u) =
+ ui+ 1 , j+ 1 )ME 2
i, j+ 21
2
+ ui+ 1 , j+ 1 )MN 2
i+ 21 , j
2
, ,
+ ui− 1 , j− 1 )MNEi, j , 2
2
+ ui+ 1 , j− 1 )MSEi, j , 2
(5.21)
2
where ME
=
MN
=
i, j+ 12 i+ 21 , j
MNEi, j
=
MSEi, j
=
1 (CXNi, j Ui,Uj + CXSi, j+1 Ui,Uj+1 ), 6 1 U (CY Ei, j Vi,Uj + CYWi+1, j Vi+1, j ), 6 1 CNEi, j (Ui,Uj + Vi,Uj ), 6 1 CSEi, j (Ui,Uj −Vi,Uj ), 6
(5.22)
and CXNi, j
= ei+ 1 , j+ 1 ei− 1 , j+ 1 (ei+ 1 , j− 1 ei− 1 , j− 1 − ei+ 1 , j− 1 − ei− 1 , j− 1 + 3),
CXSi, j
= ei+ 1 , j− 1 ei− 1 , j− 1 (ei+ 1 , j+ 1 ei− 1 , j+ 1 − ei+ 1 , j+ 1 − ei− 1 , j+ 1 + 3),
CY Ei, j
= ei+ 1 , j+ 1 ei+ 1 , j− 1 (ei− 1 , j+ 1 ei− 1 , j− 1 − ei− 1 , j+ 1 − ei− 1 , j− 1 + 3),
CYWi, j
= ei− 1 , j+ 1 ei− 1 , j− 1 (ei+ 1 , j+ 1 ei+ 1 , j− 1 − ei+ 1 , j+ 1 − ei+ 1 , j− 1 + 3),
CNEi, j
= ei+ 1 , j+ 1 ei− 1 , j− 1 (3 − ei− 1 , j+ 1 − ei+ 1 , j− 1 ),
CSEi, j
= ei− 1 , j+ 1 ei+ 1 , j− 1 (3 − ei+ 1 , j+ 1 − ei− 1 , j− 1 ).
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2
2 2 2 2
2 2
2 2 2 2
2 2
– 58 –
2
2
2
2
2
2
2
2
2 2
2 2 2 2
2 2 2 2
2 2 2 2
2 2
(5.23)
5.2. Viscosity Finally, convergence of the horizontal momentum fluxes is written as CADi+ 1 , j+ 1 (u) = FE 2
2
i, j+ 21
(u) − FE
i+1, j+ 21
(u) + FN
i+ 21 , j
(u) − FN
i+ 21 , j+1
(u)
+FNEi, j (u) − FNEi+1, j+1 (u) + FSEi, j+1 (u) − FSEi+1, j (u).
5.2
(5.24)
Viscosity
For horizontal viscosity, the harmonic (default) or biharmonic (option VISBIHARM) scheme can be selected. Anisotropy of viscosity with respect to the flow direction can be applied (option VISANISO) when harmonic viscosity is chosen. The viscosity coefficient is a constant by default but can be a function of local velocity gradients and gridsize (option SMAGOR). For vertical viscosity, the harmonic scheme is used and the local coefficient is the larger of a background constant and that calculated through a turbulence closure scheme. A parameterization of bottom friction (Weatherly, 1972) is adopted at the lowest layer.
5.2.1
Horizontal viscosity
Horizontal tension DT and horizontal shear DS are defined as follows. ∂ ∂ u v − hμ , DT = hψ hμ ∂ μ hψ hψ ∂ ψ hμ ∂ ∂ v u DS = hψ . + hμ hμ ∂ μ hψ hψ ∂ ψ hμ The friction terms are
(5.25)
(5.26)
Vu =
1 ∂ 2 1 ∂ 2 h σT + 2 h σS , h2ψ hμ ∂ μ ψ h μ hψ ∂ ψ μ
(5.27)
Vv =
1 ∂ 2 1 ∂ 2 h σS − 2 h σT , h2ψ hμ ∂ μ ψ hμ hψ ∂ ψ μ
(5.28)
where νH is the horizontal viscosity coefficient, σT = νH DT , and σS = νH DS . In the geographical coordinate system, where (μ , ψ ) = (λ , φ ), hλ = a cos φ , and hφ = a, tension and shear are DT =
1 ∂u 1 ∂v v − − tan φ , a cos φ ∂ λ a ∂ φ a
(5.29)
DS =
1 ∂v 1 ∂u u + + tan φ . a cos φ ∂ λ a ∂ φ a
(5.30)
Vu =
1 ∂ ∂ 1 2 tan φ σT + σS − σS , a cos φ ∂ λ a ∂φ a
(5.31)
Vv =
1 2 tan φ ∂ 1 ∂ σS − σT + σT , a cos φ ∂ λ a ∂φ a
(5.32)
The friction terms in this case are
where the third term on the r.h.s. is called the metric term. When the VISBIHARM option is selected, the above operation is repeated twice using a viscosity coefficient νBH . The friction terms Vu and Vv given by (5.27) and (5.28) are signreversed and substituted as u and v in equations (5.25) and (5.26). A biharmonic scheme dissipates noise only on scales near the grid size. This scale selectivity – 59 –
Chapter 5
Equations of motion (baroclinic component)
allows the explicitly represented eddies to survive without unphysical damping in eddyresolving models, although we must note that a biharmonic operator produces overshootings and spurious oscillations of variables (Delhez and Deleersnijder, 2007). A biharmonic viscosity scheme is not suitable for coarse resolution models that cannot resolve mesoscale eddies.
5.2.2 Horizontal anisotropic viscosity The viscosity in an ocean general circulation model seeks to attenuate numerical noise rather than parameterizing the subgrid scale momentum transport. The momentum advection scheme should conserve the total kinetic energy in the general threedimensional flows and the total enstrophy in the twodimensional flows. Therefore, spatially and temporally centered discretization should be used, although this inevitably produces neargridsize noise accompanying numerical dispersion. In eddyresolving models, the current velocity and the numerical noise are greater than in eddyless models. A biharmonic viscosity scheme has been widely used to reduce numerical noise while maintaining the eddy structure. Smith and McWilliams (2003) proposed a method of making a harmonic viscosity scheme anisotropic in an arbitrary direction. Setting σT and σS in equations (5.27) and (5.28) to 1 2 − n2 σT ( ν + ν ) 0 n n −2n D μ ψ T 0 1 μ ψ 2 , (5.33) = + (ν0 − ν1 )nμ nψ σS ν1 n2μ − n2ψ 2nμ nψ DS 0 where nˆ = (nμ , nψ ) is a unit vector in an arbitrary direction and ν0 (ν1 ) is the viscosity coefficient parallel (perˆ When option VISANISO is selected, nˆ is set to the direction of local flow in MRI.COM. Given pendicular) to n. the harmonic viscosity only in the direction of flow (ν1 = 0), the numerical noise is erased while the swift currents and eddy structures are maintained. The following is a note on usage. The ratio ν1 /ν0 is read from namelist: nmlvisaniso (variable name: cc0; default value is 0.2). The ratio at the lateral boundary is also read from the namelist (cc1; default is 0.5). When the variable flgvisequator is set as a positive number in the namelist, the ratio ν1 /ν0 is tapered linearly from cc0 at the latitude flgvisequator (in degrees) to zero at the equator. The ratio is not tapered when a negative number is set, and the default value of flgvisequator is −1.
5.2.3 Smagorinsky parameterization for horizontal viscosity To give the necessary but minimum viscosity to reduce numerical noise, the viscosity coefficient is made proportional to the local deformation rate (Smagorinsky, 1963; Griffies and Hallberg, 2000). When this parameterization is used with the biharmonic scheme, the scale selectivity of the viscosity scheme becomes more effective. Defining deformation rate D: T −1 = D =
D2T + D2S ,
(5.34)
the viscosity coefficients are set as follows.
2
νH
=
CΔmin π
νBH
=
Δ2min νH , 8
D,
(5.35) (5.36)
where C (cscl) is a dimensionless scaling parameter set by considering numerical stability and Δmin is the smaller of the zonal and meridional gridspacings. –
60
–
5.2. Viscosity The parameter C should be selected to satisfy the following conditions. • Restriction of grid Reynolds number: Δmin νH > U , 2 • Restriction on the width of the lateral boundary layer:
(5.37)
νH > β Δ3min ,
(5.38)
• CFL condition:
Δ2min , (5.39) 2Δt where β = d f /dy is the meridional gradient of the Coriolis parameter. Scaling the deformation rate D by U/Δmin √ gives the condition for stability: C > π / 2 ≈ 2.2 from (5.37) (Griffies and Hallberg, 2000).
νH <
5.2.4
Discretization
Using the notations
δμ Ai, j
≡
δψ Ai, j
≡
Ai+ 1 , j − Ai− 1 , j 2
2
Δμ Ai, j+ 1 − Ai, j− 1
,
δi Ai, j
2 , Δψ ≡ Ai+ 1 , j − Ai− 1 , j ,
δ j Ai, j
≡ Ai, j+ 1 − Ai, j− 1 ,
2
2
2
2
2
and Ai, j Ai, j
μ
≡
ψ
≡
1 (A 1 + Ai+ 1 , j ), 2 2 i− 2 , j 1 (A 1 + Ai, j+ 1 ), 2 2 i, j− 2
(5.40)
deformation rates are discretized as follows: DT i, j DSi, j
= =
hψ i, j δμ hμ i, j hψ i, j δμ hμ i, j
u hψ v hψ
ψ
i, j
hμ i, j − δψ hψ i, j
ψ
+ i, j
hμ i, j δψ hψ i, j
v hμ u hμ
μ
, i, j
μ
.
(5.41)
i, j
Viscosity forces are discretized as follows: Fx i+ 1 , j+ 1 2
2
= ×
1
× ΔVi+ 1 , j+ 1 2 2 ⎤ ⎡ 1 1 ψ μ ⎦, ⎣ δi ΔyΔzh2ψ νH DT + 2 δ j ΔxΔzh2μ νH DS i+ 12 , j+ 12 i+ 12 , j+ 12 h2ψ i+ 1 , j+ 1 hμ i+ 1 , j+ 1 2
Fy i+ 1 , j+ 1 2
2
= ×
2
2
2
1
× (5.42) ΔVi+ 1 , j+ 1 2 2 ⎤ ⎡ 1 1 ψ μ ⎦. ⎣ δi ΔyΔzh2ψ νH DS − 2 δ j ΔxΔzh2μ νH DT i+ 12 , j+ 12 i+ 12 , j+ 12 h2ψ i+ 1 , j+ 1 hμ i+ 1 , j+ 1 2
2
2
–
61
–
2
Chapter 5
Equations of motion (baroclinic component)
When the grid point (i − 12 , j + 12 ) is defined as a (vertically partial) land (Figure 5.6a), the velocity gradients at the wall are calculated as follows.
∂u ∂x ∂v ∂x
=
i, j+ 12
=
i, j+ 12
ui+ 1 , j+ 1 2
Δxi−j
2
vi+ 1 , j+ 1 2
Δxi−j
2
, ,
(5.43)
where Δxi−j is the length between the points (i, j + 12 ) and (i + 12 , j + 12 ). The contribution of this wall to the force is: FxW i+ 1 , j+ 1 2
2
=
−
ui+ 1 , j+ 1 1 2 2 2 Δy Δ˜ z h ν , 1 1 1 1 ψ − ΔVi+ 1 , j+ 1 hψ 2i+ 1 , j+ 1 i, j+ 2 i, j+ 2 i− 2 , j Hi− 2 , j Δxi+ 1 , j+ 1 2
FyW i+ 1 , j+ 1 2
2
=
2
2
2
2
2
vi+ 1 , j+ 1 1 2 2 2 − Δy Δ˜ z h ν , 1 1 1 1 ψ − ΔVi+ 1 , j+ 1 hψ 2i+ 1 , j+ 1 i, j+ 2 i, j+ 2 i− 2 , j Hi− 2 , j Δxi+ 1 , j+ 1 2
2
2
2
2
(5.44)
2
where Δ˜zi, j+ 1 is the wall height. 2
5.2.5 Vertical viscosity Only the harmonic scheme is considered. The vertical momentum flux is assumed to be proportional to the vertical gradient of velocity. For the upper part of a Ucell at the (k + 12 )th vertical level, the momentum flux (positive upward) is calculated as follows: uk− 1 − uk+ 1 ∂u 2 2 = −νvk , − νv ∂z k Δzk where Δzk = (Δzk− 1 + Δzk+ 1 )/2, and Δzk+ 1 is the thickness of the Ucell (dzu). Similarly, the momentum flux in 2 2 2 the lower part of the Ucell is calculated as follows: uk+ 1 − uk+ 3 ∂u 2 2 = −νvk+1 , − νv ∂ z k+1 Δzk+1 where νvk+1 is set to zero if the (k + 32 )th level is the solid Earth. The bottom friction is calculated independently (see the next subsection). Also note that the variations of the grid thickness at the bottom and near the sea surface are not considered when evaluating fluxes for simplicity. To calculate viscosity, the divergence of the momentum flux is first calculated. The expression for the vertical viscosity term is ∂u ∂u νv ∂ z k − νv ∂ z k+1 νvk (uk− 12 − uk+ 12 ) νvk+1 (uk+ 12 − uk+ 32 ) ∂ ∂u νv = = − % % 1 ∂z ∂ z k+ 12 Δzk Δz Δzk+1 % Δz 1 Δz 1 k+ 2
k+ 2
(5.45)
k+ 2
% that is, where the variation of the grid thickness for the Ucell is now taken into account and is represented by Δz, & 1 (Figure 5.6(b)). Note that the first term on the r.h.s. of equation (5.45) is set to zero in % 1 = Δz 1 − Δz Δz k+ 2
k+ 2
k+ 2
calculating the viscosity term for the vertical level of
1 2
(k = 0).
– 62 –
5.2. Viscosity
a u i1/2, j+1/2
u i+1/2, j+1/2 㼺y i+1/2, j
㼺x i, j+1/2
u i+1/2, j1/2 㼺z i
u i1/2 u i1/2
~ 㼺
u i+1/2
zi
㼺z i+1 㼺z~ i+1
u i+3/2
~
u k+1/2
㼺z k+1/2
㼺z k+1/2
~ 㼺z k+3/2
㼺z k+1
u k+1/2
u k1/2
㼺z k+3/2
u k+1/2
㼺z k
u k1/2
㼺z k+1/2
b
Figure 5.6. (a) A schematic distribution of grids for horizontal viscosity. Upper: Plan view. Lower: Side view. The shadings denote solid earth. (b) A schematic distribution of grids for vertical viscosity. Side views. Left: The lower adjacent layer (k + 32 ) has a sea bed. Right: The Ucell (k + 12 ) has a sea bed.
–
63
–
Chapter 5
Equations of motion (baroclinic component)
5.2.6 Bottom friction When a Ucell in the (k + 12 )th layer contains solid earth (Figure 5.6(b) right), the stress from the lower boundary (τxb , τyb ) is calculated following Weatherly (1972). The specific expression is as follows.
τxb τyb
=
−ρ0Cbtm
u2
k+ 12
+ v2
k+ 12
cos θ0 sin θ0
− sin θ0 cos θ0
uk+ 1 2
vk+ 1
,
2
where Cbtm is a dimensionless constant. Viscous stress at the lower boundary has a magnitude proportional to the square of the flow speed at the Ucell and an angle (θ0 + π ) relative to the flow direction. In MRI.COM, Cbtm = 1.225 × 10−3 θ0 = ±π /18 [rad] (≡ 10[◦ ]), where θ0 is positive (negative) in the northern (southern) hemisphere. The variables are designated in the model as Cbtm = abtm, cos θ0 = bcs, and sin(±θ0 ) = isgn ∗ bsn, where isgn = 1 in the northern hemisphere and isgn = −1 in the southern hemisphere.
References Delhez, E. J. M., and E. Deleersnijder, 2007: Overshootings and spurious oscillations caused by biharmonic mixing., Ocean Modell., 17, 183–198. Griffies, S., and R. Hallberg, 2000: Biharmonic friction with a Smagorinskylike viscosity for use in largescale eddypermitting ocean models, Mon. Weather Rev., 128, 2935–2946. Ishizaki, H., and T. Motoi, 1999: Reevaluation of the TakanoOonishi scheme for momentum advection on bottom relief in ocean models, J. Atmos. Oceanic Tech., 16, 1994–2010. Smagorinsky, J., 1963: General circulation experiments with the primitive equations: I. The basic experiment, Mon. Weather Rev., 91, 99–164. Smith, R. D., and J. C. McWilliams, 2003: Anisotropic horizontal viscosity for ocean models., Ocean Modell., 5, 129–156. Weatherly, G. L., 1972: A study of the bottom boundary layer of the Florida current, J. Phys. Oceanogr., 2, 54–72.
–
64
–
Chapter 6
Temperature and salinity equations
This chapter formulates a basic part of the temperature and salinity prediction procedure, except for ocean surface processes (related to mixed layer and surface fluxes). First, the finite difference expression of the flux form is explained (Section 6.1). The formulation of each component of the flux (advection and diffusion) is then given (Sections 6.2 and 6.3). The advection scheme mainly focuses on the Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979) scheme. Other advection schemes, QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979; option QUICKEST), the Uniformly ThirdOrder Polynomial Interpolation Algorism (UTOPIA; Leonard et al., 1993; option UTOPIA), and the Second Order Moment (SOM; Prather, 1986; option SOMADVEC) are explained in Chapter 13. Finally, a convective adjustment scheme is explained (Section 6.4).
6.1
Flux form
The equations governing the time change of temperature and salinity (differential form) are presented in equations (2.14) to (2.29) of Chapter 2. The finite difference forms of (2.14) and (2.15) are given by calculating fluxes through each cell face, and setting their divergence and convergence to be the time change at the grid cell (Figure 6.1) as follows: ΔVi,n+1 = T n−1 1 ΔV n−1 1 Ti,n+1 j,k+ 1 j,k+ 1 i, j,k+ i, j,k+ 2
2
2
2
+ 2 Δt {FX i− 1 , j,k+ 1 − FX i+ 1 , j,k+ 1 + FY i, j− 1 ,k+ 1 − FY i, j+ 1 ,k+ 1 + FZ i, j,k+1 − FZ i, j,k }, (6.1) 2
2
2
2
2
2
2
2
where T is temperature or salinity, ΔV is the volume of the grid cell, and FX, FY, and FZ represent (flux) × (area of the grid boundary). The fluxes include contributions by advection and diffusion. Note that ΔV from k = 1 to k = ksgm varies with time (Chapter 4). The volume of the grid cell is written as follows: ΔVi, j,k = a tri− 1 , j− 1 Δzi− 1 , j− 1 ,k + a tli+ 1 , j− 1 Δzi+ 1 , j− 1 ,k 2
2
2
2
2
2
2
2
+ a bri− 1 , j+ 1 Δzi− 1 , j+ 1 ,k + a bli+ 1 , j+ 1 Δzi+ 1 , j+ 1 ,k , 2
2
2
2
2
2
2
2
(6.2)
where Δz is the thickness of the corresponding velocity cell (in which fluctuations of the surface height are considered). The horizontal area of the leftlower quarter of the Tcells at (i, j) (corresponding to the southwestern part in geographical coordinates) is represented by a tri− 1 , j− 1 . Similarly, a tli+ 1 , j− 1 is the rightlower (southeastern), 2 2 2 2 a bri− 1 , j+ 1 is the leftupper (northwestern), and a bli+ 1 , j+ 1 is the rightupper (northeastern) quarter of a Tcell 2 2 2 2 at (i, j). See also Section 3.5.
6.2
Advection
There are several choices for tracer advection schemes. For horizontal advection, the Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979), the Uniformly ThirdOrder Polynomial Interpo–
65
–
Chapter 6
Temperature and salinity equations
V Ti, j+1/2 UVi+1/2, j+1/2
UVi1/2, j+1/2 U Ti1/2, j
U Ti+1/2, j
Ti, j
UVi1/2, j1/2
UVi+1/2, j1/2 V Ti, j1/2 W Ti, k
UVi1/2, k+1/2
Ti, k+1/2
UVi+1/2, k+1/2
W Ti, k+1 Figure 6.1. Grid arrangement around TSBox (Upper: Views from the upper, Below: views from the horizontal). Fluxes represented by an arrow are calculated.
lation Algorithm (UTOPIA; Leonard et al., 1993), and the Second Order Moment (SOM; Prather, 1986) schemes are available. For vertical advection, QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979), is available in addition to QUICK and SOM. When SOM is selected, it should be used for both horizontal and vertical directions. Refer to Leonard (1979) for a comparison between QUICK and QUICKEST. Leonard et al. (1993) provides a detailed explanation of the extension from QUICKEST to UTOPIA. The procedure for calculating UTOPIA is based on Leonard et al. (1994). This section explains the QUICK scheme. Explanations for QUICKEST, UTOPIA, and SOM are given in Chapter 13. Fluxes due to advection are given as follows: FXAi+ 1 , j,k+ 1 2
T = Ui+ 1 , j,k+ 1 Ti+ 1 , j,k+ 1 ,
(6.3)
FYAi, j+ 1 ,k+ 1
= Vi,Tj+ 1 ,k+ 1 Ti, j+ 1 ,k+ 1 ,
(6.4)
= Wi,Tj,k+1 Ti, j,k+1 ,
(6.5)
2
2
2
FZAi, j,k+1
2
2
2
2
2
2
2
2
where horizontal volume transport U T and V T are defined as follows, using (3.4) to (3.9), T Ui+ 1 , j,k+ 1 = 2
2
Δyi+ 1 , j 2 ui+ 1 , j− 1 ,k+ 1 Δzi+ 1 , j− 1 ,k+ 1 + ui+ 1 , j+ 1 ,k+ 1 Δzi+ 1 , j+ 1 ,k+ 1 , 2 2 2 2 2 2 2 2 2 2 2 2 2 – 66 –
(6.6)
6.2. Advection Vi,Tj+ 1 ,k+ 1 = 2
2
Δxi, j+ 1 2 vi− 1 , j+ 1 ,k+ 1 Δzi− 1 , j+ 1 ,k+ 1 + vi+ 1 , j+ 1 ,k+ 1 Δzi+ 1 , j+ 1 ,k+ 1 . 2 2 2 2 2 2 2 2 2 2 2 2 2
(6.7)
Vertical volume transport W T is then obtained by diagnostically solving (3.3). Moreover, the vertical velocity w, which is necessary for using QUICKEST, is calculated as follows (w is not needed, except for QUICKEST): Wi,Tj,k+1
=
wi, j,k+1 × areati, j,k+ 1 ,
(6.8)
2
where = a tri− 1 , j− 1 × eui− 1 , j− 1 ,k+ 1 + a tli+ 1 , j− 1 × eui+ 1 , j− 1 ,k+ 1
areati, j,k+ 1 2
2
2
2
2
2
2
2
2
2
2
+ a bri− 1 , j+ 1 × eui− 1 , j+ 1 ,k+ 1 + a bli+ 1 , j+ 1 × eui+ 1 , j+ 1 ,k+ 1 . 2
2
2
2
2
2
2
2
2
2
(6.9)
An array eu is set to be unity if the corresponding Ucell is a sea cell and set to be zero otherwise. This formulation does not depend on the choice of the advection scheme. The difference arises from the way of determining grid boundary values of tracer Ti+ 1 , j,k+ 1 , Ti, j+ 1 ,k+ 1 , and Ti, j,k+1 . The centered finite difference 2 2 2 2 scheme uses the average between two neighboring points of temperature and salinity as the grid boundary value. In the QUICK scheme, the grid boundary value is interpolated by a quadratic function, using three points, with one of them added from the upstream side (Figure 6.2). T Ti+1
quadratic function that passes three points
Ti Ti1
Ti+1/2 Ui+1/2
i1
i+1
i
X
Tbox
Figure 6.2. Schematic for interpolation: Ti indicates the representative temperature currently calculated at Tpoint, and Ti+ 1 is the grid boundary value. In the QUICK scheme, Ti+ 1 is interpolated by a quadratic function that passes 2 2 Ti and the neighboring Tpoint values, Ti−1 and Ti+1 .
Originally, grid boundary values in the QUICK scheme are given as follows: Ti+ 1 , j,k+ 1
=
Ti, j+ 1 ,k+ 1
=
Ti, j,k+1
=
2
2
2
2
Δxi Ti+1, j,k+ 1 + Δxi+1 Ti, j,k+ 1 2
2
Δxi+1 + Δxi Δy j Ti, j+1,k+ 1 + Δy j+1 Ti, j,k+ 1
−
Δxi+1 Δxi ci+ 1 , j,k+ 1 , 2 2 4
Δy j+1 Δy j di, j+ 1 ,k+ 1 , 2 2 Δy j+1 + Δy j 4 Δzk+ 1 Ti, j+1,k+ 3 + Δzk+ 3 Ti, j,k+ 1 Δzk+ 1 Δzk+ 3 2 2 2 2 2 2 − ei, j,k+1 , Δzk+ 3 + Δzk+ 1 4 2
2
2
–
2
67
–
−
(6.10) (6.11) (6.12)
Chapter 6
Temperature and salinity equations
where c, d, and e are defined depending on the direction of the mass flux as follows:
2
Δxi δx δx Ti, j,k+ 1
=
ci+ 1 , j,k+ 1 2
2 (= c p ), x 2Δxi Δxi+1 δx δx Ti+1, j,k+ 1 2 (= cm ), x 2Δxi+1 Δy j δy δy Ti, j,k+ 1 2 (= d p ), y 2Δy j Δy j+1 δy δy Ti, j+1,k+ 1 2 (= dm ), y 2Δy j+1 Δzk+ 3 δz δz Ti, j,k+ 3 2 2 (= e p ), z 2Δzi, j,k+ 3 2 Δzk+ 1 δz δz Ti, j,k+ 1 2 2 (= em ), z 2Δzi, j,k+ 1
= =
di, j+ 1 ,k+ 1 2
2
= =
ei, j,k+1
=
T if Ui+ 1 , j,k+ 1 > 0, 2
2
T if Ui+ 1 , j,k+ 1 < 0, 2
2
if Vi,Tj+ 1 ,k+ 1 > 0, 2
2
(6.13)
if Vi,Tj+ 1 ,k+ 1 < 0, 2
2
if Wi,Tj,k+1 > 0, if Wi,Tj,k+1 < 0.
2
The finite difference operators are defined as follows (definitions in y and z directions are the same):
δx Ai ≡
Ai+ 1 − Ai− 1 2
, δx Ai+ 1 ≡
2
Δxi
2
Ai+1 − Ai , Δxi+ 1 2
Ai+ 1 + Ai− 1
Ai+1 + Ai x 2 , Ai+ 1 ≡ . (6.14) 2 2 2 Letting c p , d p , and e p represent their values for positive velocity at the grid boundary and cm , dm , and em represent their values for negative velocity at grid boundary and taking x
Ai ≡
we obtain
T FXAi+ 1 , j,k+ 1 =Ui+ 1 , j,k+ 1 2
2
FYAi, j+ 1 ,k+ 1 2
2
FZAi, j,k+1
2
2
ca
= cm + c p
(6.15)
cd
= cm − c p
(6.16)
da
= dm + d p
(6.17)
dd
= dm − d p
(6.18)
ea
= em + e p
(6.19)
ed
= em − e p ,
(6.20)
Δxi, j Ti+1, j,k+ 1 + Δxi+1, j Ti, j,k+ 1 2
2
Δxi+1, j + Δxi, j
2
Δxi+1, j Δxi, j − ca i+ 1 , j,k+ 1 2 2 8
Δxi+1, j Δxi, j T + Ui+ cd i+ 1 , j,k+ 1 , 1 , j,k+ 1  2 2 8 2 2 Δyi, j Ti, j+1,k+ 1 + Δyi, j+1 Ti, j,k+ 1 Δyi, j+1 Δyi, j T 2 2 =Vi, j+ 1 ,k+ 1 − da i, j+ 1 ,k+ 1 2 2 Δyi, j+1 + Δyi, j 8 2 2 Δyi, j+1 Δyi, j + Vi,Tj+ 1 ,k+ 1  dd i, j+ 1 ,k+ 1 , 2 2 8 2 2 Δzi, j,k+ 1 Ti, j,k+ 3 + Δzi, j,k+ 3 Ti, j,k+ 1 Δzi, j,k+ 3 Δzi, j,k+ 1 T 2 2 2 2 2 2 =Wi, j,k+1 − eai, j,k+1 Δzi, j,k+ 3 + Δzi, j,k+ 1 8 2
+ Wi,Tj,k+ 1  2
Δzi, j,k+ 3 Δzi, j,k+ 1 2
8
2
(6.21)
(6.22)
2
ed i, j,k+1 .
(6.23)
Equation (6.21) can be rewritten as T ˜ FXAi+ 1 , j,k+ 1 Ui+ 1 , j,k+ 1 Ti+ 1 , j,k+ 1 2 2 2 2 2 2
– 68 –
+ AQ
∂ 3 Ti+ 1 , j,k+ 1 2
∂ x3
2
,
(6.24)
6.3. Diffusion where T˜i+ 1 , j,k+ 1 is the value of T at the grid boundary interpolated by the cubic polynomial, and 2
2
T AQ = Ui+ 1 , j,k+ 1  2
Δxi+1 Δxi+ 1 Δxi 2
8
2
.
(6.25)
Although the time integration for advection is done by the leapfrog scheme, the second term on the r.h.s. of (6.24) has a biharmonic diffusion form, and thus the forward scheme is used to achieve calculation stability (Holland et al. 1998). A similar procedure is applied for the northsouth and vertical directions. The weighted upcurrent scheme is used for vertical direction if wi, j,k > 0 and the Tpoint at (i, j, k + 32 ) is below the bottom. The upstreamside weighting ratio is given by the user as the namelist parameter vupp.
6.3
Diffusion
Historically, a harmonic diffusion operator is applied in each direction of the model coordinates to express mixing of tracers. In the real ocean, transport and mixing would occur along neutral (isopycnal) surfaces. Thus, horizontal mixing along a constant depth surface is generally inappropriate since neutral surfaces are generally slanting relative to a constant depth surface. Neutral physics schemes are devised as substitutes for the harmonic scheme in the horizontal direction, while the harmonic scheme continues to be used for vertical diffusion. Three types of horizontal diffusion, harmonic horizontal diffusion (default), biharmonic horizontal diffusion (option TRCBIHARM), and isopycnal diffusion (option ISOPYCNAL), are available in MRI.COM. When isopycnal diffusion (Redi, 1982) is selected, the parameterization scheme for eddy induced advection by Gent and McWilliams (1990) (GM scheme) is used with it. This is realized by merely setting two mixing coefficients for the isopycnal diffusion tensor (Griffies, 1998). An anisotropic GM scheme (Smith and Gent, 2004, option GMANISOTROP), which gives greater diffusivity only in the direction of the current vector, is also available. The following is a guide to selecting a horizontal diffusion scheme. Biharmonic diffusion is appropriate for a high resolution model that can resolve eddies because it is more scaleselective than harmonic diffusion and does not unnecessarily suppress disturbances in resolved scales. However, biharmonic diffusion is not recommended in eddyless models because this would result in numerical instability. Harmonic horizontal diffusion is not recommended because this scheme would cause unrealistic crossisopycnal (diapycnal) mixing as mentioned above. Instead, both isopycnal diffusion and the GM scheme should be used. Isopycnal diffusion mixes tracers along neutral surfaces. The GM scheme represents eddyinduced transports in isopycnal layers, mimicking transport caused by baroclinic instability. Using an anisotropic GM scheme can maintain the mesoscale eddy structures and swift currents by restricting the direction of diffusion, and thus may be usable even for a high resolution model.
6.3.1
Vertical diffusion
Vertical diffusion assumes that vertical diffusion flux is proportional to the vertical gradient of temperature and salinity. By default, vertical diffusivity is given as a function of depth. Its profile is stored in the one dimensional array vdbg(1 : km) (e.g., Tsujino et al., 2000). A three dimensional distribution can be set by selecting option VMBG3D to incorporate locally enhanced mixing processes induced by interaction between the bottom topography and tidal currents (e.g, St. Laurent et al., 2002). With this choice, three dimensional distributions for vertical diffusivity and viscosity should be prepared in advance. In addition to these static profiles, the following processes give vertical diffusivity coefficients for every model time step. –
69
–
Chapter 6
Temperature and salinity equations
• Surface mixed layer models. • Vertical component of isopycnal diffusion, • Enhanced diffusivity where the stratification is unstable (option DIFAJS). • Enhanced diffusivity around rivermouths to avoid negative salinity (by setting flg enhance vm rivmouth of namelist nrivermouth to be .true.). This scheme is needed when positive definiteness is not guaranteed by a tracer advection algorithm. The vertical diffusion for “this” time step is taken as the largest of the above estimations. When one or more of the above mixing schemes is employed, a backward (implicit) scheme is used in the time integration (Section 12.5) because high diffusivity is expected. Otherwise, a forward scheme is used. The finite difference form is as follows: FZDi, j,k+1 = −κz areati, j,k+1 δz Ti, j,k+1 , where
δz Ti, j,k+1 ≡
(6.26)
Ti, j,k+ 1 − Ti, j,k+ 3 2
2
Δzk+1
(6.27)
Noted that, for simplicity, the change of the grid thickness at the bottom and fluctuations of the surface height are not considered in the grid distance Δzk+1 when calculating the gradient.
6.3.2 Harmonic horizontal diffusion Harmonic horizontal diffusion assumes that diffusion fluxes are proportional to the gradient of temperature and salinity. The finite differences of the fluxes are as follows: y
= −κH Δyi+ 1 , j Δzi+ 1 , j,k+ 1 δx Ti+ 1 , j,k+ 1 ,
FXDi+ 1 , j,k+ 1 2
2
2
2
2
2
x
= −κH Δxi, j+ 1 Δzi, j+ 1 ,k+ 1 δy Ti, j+ 1 ,k+ 1 ,
FYDi, j+ 1 ,k+ 1 2
2
2
2
2
2
2
2
(6.28) (6.29)
where
δx Ti+ 1 , j,k+ 1 2
2
Ti+1, j,k+ 1 − Ti, j,k+ 1
≡
2
Δxi+ 1 , j
2
,
(6.30)
.
(6.31)
2
δy Ti, j+ 1 ,k+ 1 2
2
Ti, j+1,k+ 1 − Ti, j,k+ 1
≡
2
2
Δyi, j+ 1 2
6.3.3 Biharmonic horizontal diffusion Biharmonic horizontal diffusion assumes that diffusion flux is proportional to the gradient of the Laplacian of temperature and salinity. The finite difference of the flux is as follows: y
= κb Δyi+ 1 , j Δzi+ 1 , j,k+ 1 δx ∇2 Ti+ 1 , j,k+ 1 ,
FXDi+ 1 , j,k+ 1 2
2
2
where ∇2 Ti, j,k+ 1 = 2
2
2
2
x
= κb Δxi, j+ 1 Δzi, j+ 1 ,k+ 1 δy ∇2 Ti, j+ 1 ,k+ 1 ,
FYDi, j+ 1 ,k+ 1 2
2
2
2
2
2
2
2
Δxi, j Δyi, j y x (δx Δzi, j,k+ 1 δx Ti, j,k+ 1 + δy Δzi, j,k+ 1 δy Ti, j,k+ 1 ). 2 2 2 2 ΔVi, j,k+ 1 2
– 70 –
(6.32) (6.33)
(6.34)
6.3. Diffusion
6.3.4
Isopycnal diffusion
In isopycnal diffusion, diffusion flux is expressed by high diffusivity along the isopycnal surface κI , low diffusivity across the isopycnal surface κD , and their product with the gradient of temperature and salinity in each direction. Using diffusion tensor K, each flux component is written as F m (T ) = −K mn ∂n T , and then ⎛ K=
1 + Sy2 + ε Sx2
κI ⎜ ⎝ (ε − 1)Sx Sy 1 + S2 (1 − ε )Sx
⎞ (1 − ε )Sx ⎟ (1 − ε )Sy ⎠ , ε + S2
(ε − 1)Sx Sy 1 + Sx2 + ε Sy2 (1 − ε )Sy
(6.35)
where ⎞ ∂ρ ∂ρ ⎜ ∂x ∂y ⎟ ⎟ S = (Sx , Sy , 0) = ⎜ ⎝− ∂ ρ , − ∂ ρ , 0⎠ , ∂z ∂z S =  S , κD ε = κI ⎛
(6.36) (6.37) (6.38)
(Redi, 1982). The finite difference is based on Cox (1987) except for the small isopycnal slope approximation. The finite difference form of three components of the gradient of temperature and salinity in calculating the eastwest component of flux FXDi+ 1 , j,k is as follows: 2
(δx T )i+ 1 , j,k+ 1 2
2
(δy T )i+ 1 , j,k+ 1 2
2
(δz T )i+ 1 , j,k+ 1
= δx Ti+ 1 , j,k+ 1 , 2
2
xy
= δy Ti+ 1 , j,k+ 1 , 2
2
xz
(6.39) (6.40)
2
= δz Ti+ 1 , j,k+ 1 .
(δx T )i, j+ 1 ,k+ 1
= δx Ti, j+ 1 ,k+ 1 ,
(6.42)
(δy T )i, j+ 1 ,k+ 1
= δy Ti, j+ 1 ,k+ 1 ,
(6.43)
2
2
2
(6.41)
Similarly, the northsouth component is given by 2 2
2 2
(δz T )i, j+ 1 ,k+ 1 2
2
xy
2
2
2
2
yz
= δz Ti, j+ 1 ,k+ 1 , 2
2
(6.44)
and the vertical component is given by (δx T )i, j,k+1
xz
= δx Ti, j,k+1 , yz
(6.45)
(δy T )i, j,k+1
= δy Ti, j,k+1 ,
(6.46)
(δz T )i, j,k+1
= δz Ti, j,k+1
(6.47)
(Figure 6.3). The density gradient in the calculation of each component of the diffusion tensor can be obtained by replacing T in the above equation with ρ . However, density is calculated at the reference level k + 12 for the eastwest and northsouth components, and at the reference level k + 1 for the vertical component. In addition, the upper bound on the isopycnal slope Smax is set because a nearly vertical isopycnal slope and the resultant low horizontal diffusivity could cause numerical instability. If S > Smax , ∂z ρ in all flux components is replaced so as to satisfy S = Smax . – 71 –
Chapter 6
Temperature and salinity equations
Ti, j+1
Ti+1, j+1
Ti, j
Ti+1, j
Ti, j1
Ti+1, j1
Figure 6.3. The way of calculating the gradient at the circle (i + 12 , j, k + 12 ) in isopycnal diffusion: the eastwest gradient is indicated by an arrow through the circle, and the northsouth gradient is given by averaging four arrows in the vertical direction.
Griffies et al. (1998) noted a problem in the finite difference expression of the isopycnal diffusion as implemented in the GFDLmodel by Cox (1987). The problem is that two properties, the downgradient orientation of the diffusive fluxes along the neutral directions and the zero isoneutral diffusive flux of locally referenced density, are not guaranteed because of the nature of the finite difference method and the nonlinearity of the equation of state. Griffies et al. (1998) proposed a remedy, but this remains to be implemented in MRI.COM.
6.3.5 Gent and McWilliams parameterization The Gent and McWilliams (1990) parameterization represents transports of temperature and salinity due to disturbances smaller than the grid size, assuming that a flux proportional to the gradient of the layer thickness exists along the isopycnal surface. The isopycnal diffusion stated above does not produce any flux when the isopycnal surface coincides with the isotherm and isohaline surface. This parameterization, however, produces fluxes in such a case, and acts to decrease the isopycnal slope. Flux convergence due to diffusion is expressed as follows: R(T ) = ∂m (J mn ∂n T )
(6.48)
Diffusion tensor J mn is expressed as the sum of the symmetric component K mn = (J mn + J nm )/2 and the antisymmetric component Amn = (J mn − J nm )/2. Isopycnal diffusion has the form of a symmetric diffusion tensor.
– 72 –
6.3. Diffusion m = −Amn ∂ T is as follows: Convergence of a skew flux caused by the antisymmetric component Fskew n
RA (T ) =
∂m (Amn ∂n T )
= ∂m (Amn )∂n T = ∂n (∂m Amn T ),
(6.49)
where Amn ∂m ∂n T = 0 and ∂m ∂n Amn = 0 are used. If we set a virtual velocity un∗ ≡ −∂m Amn , then the flux due to the antisymmetric component could be regarded as the advection due to this virtual velocity u∗ . In this case, the flux is Fadv = u∗ T and RA (T ) = −u∗ · ∇T since u∗ is divergence free by definition. The Gent and McWilliams parameterization is given by u∗ = −
∂h ∂ (κT ∇ρ h) , ∂ρ ∂ρ
(6.50)
where h is the depth of the neutral surface (ρ = const). This velocity is expressed in the depth coordinate as u∗ = (−∂z (κT Sx ), −∂z (κT Sy ), ∇h · (κT S))
(6.51)
(Gent et al., 1995). Generally, the corresponding antisymmetric tensor A can not be uniquely determined. Here, from u1∗ = −∂y A21 −
∂z A31 = −∂z (κT Sx ), we choose A21 = 0, A31 = κT Sx . Likewise, we adopt A12 = 0, A32 = κT Sy . A specific form of A for the Gent and McWilliams parameterization is given as follows: ⎞ ⎛ 0 0 −κT Sx ⎟ ⎜ (6.52) A=⎝ 0 0 −κT Sy ⎠ . κT Sx
κT Sy
0
The flux due to advection can be expressed using a vector streamfunction, Ψ = (−κT Sy , κT Sx , 0), which produces u∗ in (6.51): Fadv = u∗ T = T (∇ × Ψ). The skew diffusive expression for the flux using tensor A in (6.52) is Fskew = −A∇T = −(∇T ) × Ψ = Fadv − ∇ × (T Ψ). Thus, the convergence of the flux expressed in tensorial form matches that of the advective expression. In other words, the Gent and McWilliams parameterization is realized by only adding A to the tensor of the isopycnal diffusion K (Griffies, 1998).
6.3.6 Anisotropic GentMcWilliams scheme Using unit vector nˆ = (nx , ny ) in an arbitrary direction, the twodimensional anisotropic diffusion tensor is
defined as follows: K2 =
L M
M N
=
κA n2x + κB n2y κB nx ny
κB nx ny κB n2x + κA n2y
,
(6.53)
ˆ This is applied to the where κA is the diffusivity in the nˆ direction, and κB is that in the direction normal to n. antisymmetric tensor in the GentMcWilliams scheme, and the following expression is obtained (Smith and Gent, 2004), ⎛ ⎞ 0 0 −LSx − MSy ⎜ ⎟ (6.54) A = ⎝ 0 0 −MSx − NSy ⎠ . LSx + MSy
MSx + NSy – 73 –
0
Chapter 6
Temperature and salinity equations
In the choice of option GMANISTROP, nˆ is set in the direction of the local horizontal velocity. The value of κA is read from the namelist njobpi (variable name aitd). The ratio of κB /κA is read from the namelist nmlgmanisotrop (variable name cscl isotrop). The default value of cscl isotrop is set to 1/2.
6.4 Convective adjustment Convective adjustment is realized by replacing the density (temperature and salinity) that is statically unstable (the upper density exceeds the lower density) in a water column with the averaged density between neighboring levels (neutralization), considering that interior convection occurs in that place. Most of the realistic phenomena represented by the convective adjustment include the developing mixed layer due to surface cooling during winter. Convective adjustment also includes the case in which dense bottom water flows out the sill and flows down along the slope. Moreover, the convective adjustment includes the practical effect that it suppresses disturbances caused by the numerical calculation error and smoothes the distribution. In general, there are three numerical schemes for convective adjustment. In the simplest one, adjustment is done for a pair of two neighboring levels, and then for a pair of another two neighboring levels. By repeating this procedure, it attempts to neutralize the density in the unstable part. This procedure is simple at each step, but it has a defect that the finitetime repetition does not necessarily guarantee reaching the complete averaged value. In the second scheme, adjustment is done by assigning a high vertical diffusivity between the two levels that are statically unstable and by solving the vertical diffusion term using an implicit method. This method cannot remove the unstable condition completely in one procedure. However, it has good calculation efficiency for the case where the model has a high vertical diffusivity already due to the mixed layer or isopycnal diffusion schemes and thus needs an implicit method to solve it. In MRI.COM, this scheme is invoked by specifying the option DIFAJS. The vertical diffusivity between the unstable grid points is set to 10000 cm2 s−1 . In the third scheme, the unstable part is first neutralized. The stability at the top and bottom of the neutralized column is then reexamined, If the unstable condition remains, the part including the alreadyneutralized column is reneutralized. This procedure continues until the instability at the top and bottom of the neutralized column disappears. This method can remove the unstable part completely and thus is called ”Complete Convection,” but it requires a number of iterations, the vertical level size minus one, at maximum. The third method, which is the default scheme in MRI.COM, is explained below (Ishizaki, 1997).
6.4.1 Algorithm In order to minimize the judgment process (”IF” statement) and replace it by arithmetic calculation, this scheme defines two integer indices, αk and βk , at the layer boundaries, and six real variables TU k , TLk , SU k , SLk , VU k , and VLk , (k = 1, KM −1), in addition to the vertical rows of temperature, salinity, and density Tk , Sk , Rk , (k = 12 , KM − 12 ) (KM is the number of levels; see Figure 6.4). The level at the vertical boundary of a Tcell corresponds to the integer k. The index αk indicates an unstable part within a water column: αk = 1 if it is unstable at the level between k − 12 and k + 12 , and αk = 0 if it is neutral or stable. The index βk memorizes the mixed part: βk = 1 at the boundary where it is neutral as a result of mixing, and βk = 0 elsewhere. Variables TU k , SU k , and VU k and TLk , SLk , and VLk are temperature, salinity and volume accumulated by multiplying α above the level k and below the level k, respectively, and are expressed by the following recursive relation.
–
74
–
6.4. Convective adjustment
VU 1
= α1V 1 ,
VU 2
= α2 (V1+ 1 + α1V 1 ) = α2 (V1+ 1 + VU 1 ),
2
2
2
2
· · ·, = αk (Vk− 1 + VU k−1 ),
VU k
2
· · ·, = αKM−1 (VKM−1− 1 + VU KM−2 ),
VU KM−1
(6.55)
2
and VLKM−1
= αKM−1VKM− 1 ,
VLKM−2
= αKM−2 (VKM−1− 1 + αKM−1VKM− 1 ) = αKM−2 (VKM−1− 1 + VLKM−1 ),
2
2
2
2
· · ·, VLk
= αk (Vk+ 1 + VLk+1 ), 2
· · ·, VL1
= α1 (V1+ 1 + VL2 ),
(6.56)
2
where Vk+ 1 denotes a volume of the cell at the level k + 12 . In a similar way, other quantities are expressed as 2 follows: TU 1 = α1 T 1 V 1 ,
TU k = αk (Tk− 1 Vk− 1 + TU k−1 ),
SU 1 = α1 S 1 V 1 ,
SU k = αk (Sk− 1 Vk− 1 + SU k−1 ),
2
2
2
2
2
2
2
2
TLKM−1 = αKM−1 TKM− 1 VKM− 1 , T Lk = αk (Tk+ 1 Vk+ 1 + TLk+1 ), 2
2
2
2
SLKM−1 = αKM−1 SKM− 1 VKM− 1 , SLk = αk (Sk+ 1 Vk+ 1 + SLk+1 ), 2
2
2
2
(6.57)
where Tk+ 1 and Sk+ 1 are temperature and salinity at the level k + 12 . 2 2 According to this definition, if αk = 1 and elsewhere 0, we get VU k + VLk = Vk− 1 +Vk+ 1 , 2
2
TU k + TLk = Tk− 1 Vk− 1 + Tk+ 1 Vk+ 1 , 2
2
2
2
SU k + SLk = Sk− 1 Vk− 1 + Sk+ 1 Vk+ 1 , 2
2
2
2
indicating a volume and accumulated temperature and salinity in an unstable part and TU k + TLk , VU k + VLk SU k + SLk = , VU k + VLk
TM k− 1 ,k+ 1 = 2
2
SM k− 1 ,k+ 1 2
2
(6.58)
are volume averaged temperature and salinity, respectively. If the level k constitutes a series of the unstable part, the same equation holds for the averaged temperature and salinity. For example, let αk−1 = αk = 1 and αk−2 = αk+1 = 0,
–
75
–
Chapter 6
Temperature and salinity equations
Surface T1/2
k=1/2
α1
k=1
α2
k=2
αk1
k1
αk
k=1+1/2
T1+1/2
k1/2
Tk1/2
k+1/2
Tk+1/2
k
αk+1
k+1
αKM2 k=KM2
TKM11/2 k=KM11/2
αKM1 k=KM1
TKM1/2 k=KM1/2 k=KM
Bottom Figure 6.4. Reference vertical grid points in Section 6.4
VU k−1 + VLk−1
= VU k + VLk = Vk−1− 1 +Vk− 1 +Vk+ 1 , 2
TU k−1 + TLk−1
2
2
= TU k + TLk = Tk−1− 1 Vk−1− 1 + Tk− 1 Vk− 1 + Tk+ 1 Vk+ 1 , 2
SU k−1 + SLk−1
2
2
2
2
2
= SU k + SLk = Sk−1− 1 Vk−1− 1 + Sk− 1 Vk− 1 + Sk+ 1 Vk+ 1 , 2
2
2
2
2
2
(6.59)
and TU k−1 + TLk−1 TU k + TLk = , VU k−1 + VLk−1 VU k + VLk SU k−1 + SLk−1 SU k + SLk = = . VU k−1 + VLk−1 VU k + VLk
TM k−1− 1 ,k+ 1 = 2
2
SM k−1− 1 ,k+ 1 2
2
– 76 –
(6.60)
6.4. Convective adjustment These are averages of the three layer, k − 1 − 12 , k − 12 , and k + 12 .
6.4.2
Numerical procedure
[1] Density is calculated at the intermediate depth between adjacent levels using a pair of temperature and salinity and is judged to be statically stable or unstable. If an instability occurs, α (α 1 ) is replaced by 1, otherwise by 0. At this stage, β (β 0 ) is set to 0, where the superscript denotes the number of the iteration. After this preprocessing, the following procedure (represented by nth) is repeated until the instability is removed. [2] Based on equations (6.55) to (6.57), VU, TU, SU, VL, TL, and SL are calculated using α n for a water column that includes an unstable part. [3] The vertical mean TM and SM are calculated for the unstable part using equation (6.58) and substituted for the original values of T and S. This change modifies the density at the intermediate depth in [1]. [4] The value of α n is stored in β n . β n = 1 is set if α n = 1, or α n = 0 and β n−1 = 1, and otherwise β n = 0. This is presented by the following: (6.61) βkn = αkn + βkn−1 (1 − αkn ). [5] The static stability is judged only for βkn = 0. Let αkn+1 = 1 if statically unstable, and 0 otherwise. If there is no unstable part, the procedure for that water column is completed. [6] For a water column which still includes an unstable part, modification for αkn+1 is done using βkn by the following. After the procedure [2], any instability will be found only at the bottom of the part that is neutral as a result of prior mixing. In that case, the neutral part must be treated as an unstable part, that is, αkn+1 = 1. On the other hand, no more procedure is needed if the upper and lower end is stable, giving αkn+1 = 0. This is done by a recursive formula going down and up in the following. (n+1)
γ1 = α1 (n+1)
(n+1)
+ (1 − αk
(n+1)
= γk + (1 − γk )βk αk+1 ,
, γ k = αk
αKM−1 = γKM−1 , αk
(n+1)
(n)
)βk γk−1 (n)
(n+1)
(6.62)
where γ is a work variable, but may be treated as α itself in a FORTRAN program. Then, the procedure goes back to [2]. Table 6.1 shows an example of the case with six levels. Static instability is removed after the threetime iteration. The second column of α in the table is the result of the corrected αkn+1 using βkn based on equation (6.62), as described in [6]. Note that βk0 = 0, though there is no description in the table.
– 77 –
Chapter 6
n
1
VU
Table 6.1. Example of the convective adjustment procedure VL VU+VL TU+TL V1 1 + V2 1
T 1 V 1 + T1 1 V1 1 + T2 1 V2 1
1
V 1 + V 1 1 + V2 1
T 1 V 1 + T1 1 V1 1 + T2 1 V2 1
1
0
0
0
0 V5 1
0 V 4 1 + V5 1
0 + T5 1 V5 1
0 1
V1 1 + V2 1 + V3 1 2
V 1 + V1 1 + V 2 1 + V3 1
V 2 1 + V3 1
1
1
V1
2
1
1
V 1 + V1 1
3
0
0
0
0
4 5
0 1
0 1
0 V4 1
1
0
1
3
0 1
1 1
2 2
2
2
2
V2 1
2
2
2
V1
2
2
V 1 + V1 1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
V 1 + V1 1 + V 2 1 + V3 1
V 1 + V 1 1 + V2 1
V3 1
V 1 + V1 1 + V 2 1 + V3 1
2
2
β
V 1 + V 1 1 + V2 1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
T4 1 V4 1 2
2
2
2
3
∑ Tk+ 12 Vk+ 12
1
∑ Tk+ 12 Vk+ 12
1
∑ Tk+ 12 Vk+ 12
1
k=0 3 k=0 3 k=0
4
0
0
0
0
0
0
0
5
0
0
0
0
0
0
1
1
0
1
V1
V1 1 + V2 1 + V3 1 + V4 1 + V5 1
∑ Tk+ 12 Vk+ 12
1
∑ Tk+ 12 Vk+ 12
1
∑ Tk+ 12 Vk+ 12
1
∑ Tk+ 12 Vk+ 12
1
∑ Tk+ 12 Vk+ 12
1
2
3
α
k
2
2
Temperature and salinity equations
3 4 5
0 0 1 0
1 1 1 1
2
2
V 1 + V1 1 2
2
2
2
2
2
2
2
2
2
V 4 1 + V5 1
V 1 + V 1 1 + V2 1 + V 3 1 + V4 1
V5 1
2
2
2
2
2
2
2
2
2
2
2
V3 1 + V4 1 + V5 1
V 1 + V 1 1 + V2 1 + V 3 1 2
2
V 2 1 + V3 1 + V 4 1 + V5 1
V 1 + V 1 1 + V2 1 2
2
2
2
2
5
∑ Vk+ 12
5
k=0 5
k=0 5
k=0 5
k=0 5
k=0 5
k=0 5
k=0 5
k=0 5
k=0
k=0
∑ Vk+ 12 ∑ Vk+ 12 ∑ Vk+ 12 ∑ Vk+ 12
– 78 –
6.4. Convective adjustment
References Cox, M. D., 1987: Isopycnal diffusion in a zcoordinate ocean model, Ocean Modell., 74, 15. Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20, 150155. Gent, P. R., J. Willebrand, T. J. McDougall, and J. C. McWilliams, 1995: Parameterizing eddyinduced tracer transports in ocean circulation models, J. Phys. Oceanogr., 25, 463–474. Griffies, S. M., 1998: The GentMcWilliams skew flux, J. Phys. Oceanogr., 28, 831841. Griffies, S. M., A. Gnanadesikan, R. C. Pacanowski, V. D. Larichev, J. K. Dukowicz, and R. D. Smith., 1998: Isoneutral diffusion in a zcoordinate ocean model., J. Phys. Oceanogr., 28, 805830. Holland, W. R., J. C. Chow, and F. O. Bryan, 1998: Application of a thirdorder upwind scheme in the NCAR ocean model., J. Climate, 11, 1487–1493. Ishizaki, H., 1997: A massive treatment scheme of complete convection for ocean models, unpublished manuscript. Leonard, B. P., 1979: A stable and accurate convective modeling procedure based upon quadratic upstream interpolation, J. Comput. Methods Appl. Mech. Eng., 19, 5998. Leonard, B. P., M. K. MacVean, and A. P. Lock, 1993: PositivityPreserving Numerical Schemes for Multidimensional Advection, NASA Tech. Memo., 106055, ICOMP9305, 62pp. Leonard, B. P., M. K. MacVean, and A. P. Lock, 1994: The flux integral method for multidimensional convection and diffusion, NASA Tech. Memo., 106679, ICOMP9413, 27pp. Prather, M. J., 1986: Numerical advection by conservation of secondorder moments, J. Geophys. Res., 91, 66716681. Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation, J. Phys. Oceanogr., 12, 1154–1158. Smith, R. D., and P. R. Gent, 2004: Anisotropic GM parameterization for ocean models, J. Phys. Oceanogr., 34, 2541–2564. St. Laurent, L. C., H. L. Simmons, and S. R. Jayne, 2002: Estimating tidally driven mixing in the deep ocean, Geophys. Res. Lett., 29,, 2106,doi:10.1029/2002GL015633. Tsujino, H., H. Hasumi, and N. Suginohara, 2000: Deep pacific circulation controlled by vertical diffusivity at the lower thermocline depth, J. Phys. Oceanogr., 30, 2853–2865.
–
79
–
Part III
Additional Processes
–
81
–
Chapter 7
Mixed layer model
The surface boundary layer is made turbulent by wind and other factors injecting momentum, so vertical mixing could be induced even if the stratification is stable. However, this phenomenon is not expressed by the fundamental equation of the model.∗ Therefore, general circulation models express the effects of turbulent phenomena by using synoptic states (variables of velocity and temperature in the model). In MRI.COM, the vertical viscosity and diffusivity are set to large values in the uppermost layer or calculated at each time step by using mixed layer models. Three mixed layer models are supported: the turbulence closure model by Mellor and Yamada (1982) and Mellor and Blumberg (2004) introduced in Section 7.1, the turbulence closure model by Noh and Kim (1999) in Section 7.2, and Kprofile parameterization by Large et al. (1994) in Section 7.3. Table 7.1 lists the notation of the vertical viscosity and diffusivity as well as physical properties that are predicted or diagnosed in each mixed layer model. Table 7.1. Physical properties predicted and diagnosed in each mixed layer model and their notations used for explanation in each section Physical property (name of variable in MRI.COM)
7.1
Mellor and Yamada
Noh and Kim
KPP
coefficient of diffusion (avdsl)
KH
KH
Kx
coefficient of viscosity (avm) turbulent velocity (q: only MellorYamada) turbulent kinetic energy (eb: only Noh and Kim)
KM q 2 q /2
KM E
Kx 
vertical turbulent scale (alo: except for KPP)
l
l

Mellor and Yamada’s Turbulence Closure Model
7.1.1
Turbulence Closure Model
The physical properties in the basic equations of motion for a Boussinesq fluid are separated into averaged components and perturbed components, and then the equations are time averaged. The expressions for averaged velocity U, averaged pressure P, and averaged potential temperature Θ are
∂ Ui = 0, ∂ xi DU j ∂ ρ ∂ 1 ∂ Uk ∂ U j 1 ∂P + ε jkl fkUl = 2ν , (− uk u j ) − −gj + + Dt ∂ xk ρ0 ∂ x j ρ0 ∂ xk 2 ∂xj ∂ xk ∂ ∂ ∂Θ DΘ , (− uk θ ) + κ = Dt ∂ xk ∂ xk ∂ xk ∗ It
could be expressed by relaxing the hydrostatic approximation but this is for future work.
–
83
–
(7.1) (7.2) (7.3)
Chapter 7
Mixed layer model
where D( )/Dt ≡ Uk ∂ ( )/∂ xk + ∂ ( )/∂ t, g j is the gravity vector, fk is the Coriolis vector, εi jk is the alternating tensor, ν is viscosity, and κ is diffusivity. Averaged quantities (resolved by the general circulation model) are represented by capital letters, and turbulent components (unresolved by the general circulation model) are represented by lowercase letters. The statistical averages of the turbulent components are represented by . The equation for salinity is similar to that for temperature (7.3). If density is calculated from temperature and salinity, the whole expression would become complicated. Therefore, density is assumed to be a function only of temperature here (we set ρ = β θ , where β is the coefficient of thermal expansion). The evolution of large scale physical quantities could be obtained by calculating the statistically averaged turbulent component ( ) expressed above for each time step, but the calculation is very complicated and unknown higher order terms arise when evolution equations for statistically averaged turbulent components are derived (e.g., to calculate the covariance of turbulent velocities, the equation for a turbulent velocity component is multiplied by another component of turbulent velocity and then statistically averaged). Thus, the expression is not closed, although it should be closed at a certain level (usually called closure). The expression closed in the second order of the turbulent component is named second moment closure and is popular in modeling turbulence. Following Kantha and Clayson (2000), the second moment closure is expressed as
∂ ui u j 1 ∂ pui ∂ pu j D ui u j ∂
+
uk ui u j − ν + + fk (ε jkl ul ui + εikl ul u j ) + Dt ∂ xk ∂ xk ρ0 ∂ x j ∂ xi '∂u ∂u ( ' p ∂u
∂Uj ∂ u j ( ∂ Ui β j i i = − uk ui − (g j ui θ + gi u j θ ) + − 2ν , (7.4) − uk u j + ∂ xk ∂ xk ρ0 ρ0 ∂ x j ∂ xi ∂ xk ∂ xk ) ∂θ * ) ∂ u j * 1 ) ∂ p * D u j θ ∂
uk u j θ − κ u j −ν θ + + ε jkl fk ul θ θ + Dt ∂ xk ∂ xk ∂ xk ρ0 ∂ x j '∂u ∂θ (
∂Uj β ∂Θ j − g j θ 2 − (κ + ν ) , (7.5) − uk θ = − u j uk ∂ xk ∂ xk ρ0 ∂ xk ∂ xk ' ∂θ ∂θ ( ) ∂ θ 2 * D θ 2 ∂
∂Θ
uk θ 2 − κ . (7.6) = −2 uk θ − 2κ + Dt ∂ xk ∂ xk ∂ xk ∂ xk ∂ xk Mellor and Yamada (1982) reduced the higher order terms as follows. It should be noted that it is not unique. Based on Rotta’s (1951a,b) hypothesis of energy redistribution, the covariances of pressure and velocity gradients are assumed to be linear functions of Reynolds stress: ∂U ' p ∂u ∂ u j ( δi j ∂Uj q i i =−
ui u j − q2 +C1 q2 , + + ρ0 ∂ x j ∂ xi 3l1 3 ∂xj ∂ xi
(7.7)
where q2 ≡ u2i , l1 is the length scale, C1 is a nondimensional constant, and δi j is Kronecker’s delta, which is unity for i = j and zero for i = j. Using Kolmogolov’s hypothesis of local isotropy in small eddies the energy dissipation is modeled as follows, 2ν
'∂u ∂u ( 2 q3 j i =− δi j , ∂ xk ∂ xk 3 Λ1
(7.8)
where Λ1 is the length scale. The redistribution of temperature and the dissipation of heat are modeled as follows in the same form as above, ' p ∂θ ( q = − u j θ ρ0 ∂ x j 3l2
(7.9)
'∂u ∂θ ( j = 0, (κ + ν ) ∂ xk ∂ xk
(7.10)
–
84
–
7.1. Mellor and Yamada’s Turbulence Closure Model where l2 is the length scale. The dissipation of temperature variance is 2κ
' ∂θ ∂θ ( q = −2 θ 2 , ∂ xk ∂ xk Λ2
(7.11)
where Λ2 is the length scale. In order to avoid the higher order problems, the turbulent velocity diffusion term and the other higher order terms are modeled as follows: ∂ u u ∂ u u ∂ u u 3 i j j k i k lqSq , (7.12) + +
uk ui u j = 5 ∂ xk ∂xj ∂ xi ∂ u θ ∂ u θ j k , (7.13) +
uk u j θ = −lqSuθ ∂xj ∂ xk
uk θ 2 =
−lqSθ
∂ θ 2 , ∂ xk
(7.14)
where Sq , Suθ , and Sθ are nondimensional numbers and can be set as constants or functions of certain parameters. Other relations are pθ = 0 and pui = 0. The essence of the MellorYamada mixed layer model is that the above length scales are related linearly to each other: (7.15) (l1 , Λ1 , l2 , Λ2 ) = (A1 , B1 , A2 , B2 )l, where l is the vertical scale of turbulence (also called the master length scale), and A1 , B1 , A2 , B2 , and C1 are empirical constants and are determined from experiment data. Mellor and Yamada (1982) employ (A1 , B1 , A2 , B2 ,C1 ) = (0.92, 16.6, 0.74, 10.1, 0.08).
7.1.2 Level 2.5 Model The turbulence model that solves the evolution of the statistically averaged values of the secondorder turbulent components based on the simplification described in the previous subsection is called the level4 model. The level3 model solves the evolution of the turbulent kinetic energy (q2 /2) and the variance of potential temperature ( θ 2 ) (in some cases, the covariance of potential temperature and salinity ( θ s) and the variance of salinity ( s2 )). The other statistically averaged values are solved diagnostically through algebraic equations assuming them to be in the steady state. In the level2.5 model, the variance of the potential temperature is also assumed to be in a statistically steady state (see expression (7.33) that appears later). In the level2 model, the turbulent kinetic energy is also assumed to be in a statistically steady state. The level2.5 model is employed as the surface boundary layer model by MRI.COM and is further simplified by applying the following boundary layer approximations. • Neglect the Coriolis term in the equations of motion for the turbulent components. • Neglect the molecular viscosity and diffusivity. • Use the hydrostatic assumption in the vertical component of the equation of motion. • Consider only vertical differentiation (direction perpendicular to the boundary) in the spatial differentiation for the term involving turbulent velocity.
–
85
–
Chapter 7
Mixed layer model
The equations for the large scale physical quantity become
∂ DU 1 ∂P + uw = − + fV, Dt ∂z ρ0 ∂ x ∂ DV 1 ∂P + vw = − − fU, Dt ∂z ρ0 ∂ y ρ 1 ∂P 0 = − −g , ρ0 ∂ z ρ0 DΘ ∂ + ( wθ ) = 0. Dt ∂z
(7.16) (7.17) (7.18) (7.19)
The level2.5 system consists of the time evolution equation for turbulent kinetic energy and algebraic equations for other secondmoment turbulent quantities. The time evolution equation for the turbulent kinetic energy is
∂ q2 D q2 ∂
− lqSq = Ps + Pb − ε , Dt 2 ∂z ∂z 2
(7.20)
where
∂U ∂V − wv ∂z ∂z is the term for energy produced by the vertical shear of the averaged flow, Ps = − wu
Pb = −g wρ /ρ0
(7.21)
(7.22)
is the term for energy produced by buoyancy, and
ε = q3 /Λ1
(7.23)
is the energy dissipation term. The algebraic equations for the statistically averaged values, which are expressed by other secondmoment turbulent quantities, are given below. ∂U ∂V q2 l1
−4 wu + + 2 wv − 2Pb , 3 q ∂z ∂z
2 ∂ U ∂ V q l 1 2 wu
v2 = + − 4 wv − 2Pb , 3 q ∂z ∂z
2 ∂ U ∂ V l q 1 2 wu
w2 = + + 2 wv + 4Pb , 3 q ∂z ∂z ∂V ∂U 3l1
− uw ,
uv = − vw q ∂z ∂z 3l1
∂U
wu = −( w2 −C1 q2 ) − g uρ , q ∂z
∂V 3l1
vw = −( w2 −C1 q2 ) − g vρ , q ∂z
∂Θ ∂U 3l2 − uw ,
uθ = − wθ q ∂z ∂z ∂Θ ∂V 3l2
vθ = − vw , − wθ q ∂z ∂z ∂Θ 3l2
wθ = − w2 − g θ ρ , q ∂z ∂Θ Λ2
θ 2 = − wθ . q ∂z
u2 =
– 86 –
(7.24) (7.25) (7.26) (7.27) (7.28) (7.29) (7.30) (7.31) (7.32) (7.33)
7.1. Mellor and Yamada’s Turbulence Closure Model Some of the terms in these equations can be further simplified as follows:
∂U , ∂z ∂V , − vw = KM ∂z ∂Θ − θ w = KH , ∂z − uw = KM
(7.34) (7.35) (7.36)
KM = lqSM ,
(7.37)
KH = lqSH .
(7.38)
This simplification means that the vertical turbulent fluxes are proportional to the gradient of the large scale values. The ultimate purpose of solving the mixed layer model is to determine the coefficients of momentum and heat fluxes, KM and KH , using (7.37) and (7.38). Assuming that the potential density is linearly related to the potential temperature (and salinity), the simultaneous equations for SM and SH are derived as follows: SM [6A1 A2 GM ] + SH [1 − 3A2 B2 GH − 12A1 A2 GH ] = A2 , SM [1 + 6A21 GM − 9A1 A2 GH ] − SH [12A21 GH + 9A1 A2 GH ] = A1 (1 − 3C1 ),
(7.39)
where GM
≡
GH
≡
l 2 ∂ U 2 ∂ V 2 , + q2 ∂z ∂z l 2 g ∂ ρ˜ , q 2 ρ0 ∂ z
(7.40) (7.41)
and ∂ ρ˜ /∂ z is the vertical gradient of potential density. Using SM and SH , KM and KH are then obtained from (7.37) and (7.38) by determining q and l. The turbulent velocity q is obtained by solving the following expression that is modified from (7.20) using the above results:
∂ U 2 ∂ V 2 g ∂ q2 ∂ ∂ q2 ∂ ρ˜ − KE = KM + KH + (7.42) − ε, ∂t 2 ∂z ∂z 2 ∂z ∂z ρ0 ∂z where KE = lqSq
(7.43)
and advection terms are neglected. In MRI.COM, Sq is set proportional to SM (Sq ∝ SM ). We adopt the form Sq = Sqc SM /SMn , where Sqc = 0.2 and SMn = 0.3927. With this choice, Sq = 0.2 when the stratification is neutral (GH = 0). The sea surface boundary condition for the turbulent kinetic energy follows Mellor and Blumberg (2004): Kq
∂ q2 = 2αCB u3τ , ∂z
(7.44)
where αCB = 100 and uτ is the frictional velocity defined as uτ ≡ (τs /ρs )1/2 by using the surface stress (τs ) and the sea surface density (ρs ). This flux boundary condition is analytically converted to the condition for q at the sea surface: q2 = (15.8αCB )2/3 uτ2 .
–
87
–
(7.45)
Chapter 7
Mixed layer model
The vertical scale of the turbulence (master length scale) is estimated by many formulae such as a time evolution equation (which is usually empirical and is not completely based on physics) and a diagnosis. The formula used in MRI.COM is a diagnosis for cases without surface wind wave effects: l=γ
0 zb
z qdz /
0 zb
qdz ,
(7.46)
where γ = 0.2, and zb is the depth of the bottom. This is recognized as the averaged depth with the weight of the kinetic energy, which is sufficient for the ocean boundary layer according to Mellor and Yamada (1982). Roughness parameter zw due to surface wind waves is given by Mellor and Blumberg (2004) as follows: z w = βw
u2τ , βw = 2.0 × 105 . g
(7.47)
For a depth of z < zw , l = max(κ zw , κ z),
(7.48)
where κ is the von Karman constant (κ = 0.4). For depths exceeding zw , (7.46) is employed.
7.1.3
Implementation
This section briefly describes the solution procedure. The mixed layer model (subroutine name mysl25 in my25.F90) is called as the last procedure of each time step that proceeds from n to n+1. After the master length scale (l) for the present time step (n) is determined using (7.46), (7.47), and (7.48), the turbulent kinetic energy (q2 /2) is solved using (7.42), (7.15), (7.23), and (7.45), where the forward finite difference（n → n+1）is used in time. The implicit method is used for the vertical diffusion of the turbulent kinetic energy and energy dissipation term, since these terms could become significantly large (see Section 12.5). The vertical viscosity and diffusivity for the time step n+1 are estimated using q, l, and (7.37) to (7.39). The vertical scale of the turbulence (master length scale) based on (7.46) is calculated to prepare for the next time step. The turbulent kinetic energy and the master length scale are defined at the bottom of the tracer grid cell (i, j, k). The specific expression for the discretized form of the turbulent kinetic energy (E = q2 /2) equation is as follows: Ekn+1 − Ekn 1 KE k− 12 (Ek−1 − Ek = Δt Δzk Δzk− 1 n+1
n+1
)
−
n+1 KE k+ 1 (Ekn+1 − Ek+1 )
2
2
Δzk+ 1 2
(U n+11 −U n+11 )(U˜ k− 1 − U˜ k+ 1 ) (V n+11 −V n+11 )(V˜k− 1 k− 2 k+ 2 k− 2 k+ 2 2 2 2 + KM k + K Mk Δz2k Δz2k Bn+11 − Bn+11 k− 2 k+ 2 − KH k − 2Ekn+1 qnk /B1 lkn , Δzk
− V˜k+ 1 ) 2
(7.49)
where u˜ = (un+1 + un )/2 and B is buoyancy (= − gρρ0 ). The discrete expression for shear production (the second and third terms on the r.h.s.) and the buoyancy sink (the fourth term on the r.h.s.) follows Burchard (2002), which is consistent with the conservation law of the sum of mean and turbulent energy. To summarize, the numerical operations proceed in the following order: 1. Calculate the master length scale for the present time step using (7.46), (7.47), and (7.48). 2. Update the turbulent kinetic energy using (7.49). – 88 –
7.2. Turbulent mixed layer model by Noh and Kim (1999) 3. Solve the algebraic equation for SM and SH using (7.39). 4. Calculate the vertical viscosity and diffusivity for the next time step using (7.37), (7.38), and (7.43). 5. Calculate the master length scale using (7.46) for the next time step.
7.2
Turbulent mixed layer model by Noh and Kim (1999)
The mixed layer model proposed by Mellor and Yamada was originally developed for the atmospheric boundary layer, and its surface boundary is treated as a solid wall. When they applied this model to the ocean, they regarded that the turbulent kinetic energy is injected into the ocean by the wind stress at the solidwall sea surface. The model by Mellor and Yamada could therefore be considered to insufficiently represent the oceanic turbulent mixed layer. Noh and Kim (1999) presented a model that can resolve this insufficiency. The model is basically the same as that by Mellor and Yamada and is classified as a second moment closure model.
7.2.1
Fundamental equation
The equations for the zonal and meridional components of the velocity, U, V , buoyancy B = −gΔρ /ρo , and turbulent energy E in the large scale fields are DU Dt DV Dt DB Dt DE Dt
∂ 1 ∂P
uw − + fV, ∂z ρ0 ∂ x ∂ 1 ∂P = − vw − − fU, ∂z ρ0 ∂ y ∂ ∂R = − bw + , ∂z ∂z * ∂ ) ∂U ∂V p + uu + vv + ww − uw =− w − vw + bw − ε , ∂z ρ0 ∂z ∂z
=−
(7.50) (7.51) (7.52) (7.53)
where R is the downward short wave radiation and ∂ R/∂ z is its convergence. The turbulent flux is expressed by using the large scale fields (in capital letters) as follows, ∂ ∂U 1 ∂ P DU = KM − + fV, (7.54) Dt ∂z ∂z ρ0 ∂ x ∂ ∂V 1 ∂ P DV = KM − − fU, (7.55) Dt ∂z ∂z ρ0 ∂ y ∂ ∂B ∂R DB KH − = , (7.56) Dt ∂z ∂z ∂z ∂ ∂E ∂U ∂U ∂V ∂V ∂ B DE KE + KM − ε. (7.57) = + KM − KH Dt ∂z ∂z ∂z ∂z ∂z ∂z ∂z The central problem is how to determine the viscosity, diffusivity (KM , KH , KE ), and turbulent energy dissipation rate (ε ). By using the typical velocity scale (q = (2E)1/2 ) and the vertical length scale (l) of the turbulence, we assume the following KM
=
Sql,
(7.58)
KH
=
SB ql,
(7.59)
KE
=
SE ql,
(7.60)
ε
= Cq3 l −1 . – 89 –
(7.61)
Chapter 7
Mixed layer model
The constants (S, SB , SE ,C) are obtained from experiments and it is assumed that S = S0 = 0.39, Pr = Pr0 = S/SB = 0.8, σ = S/SE = 1.95, and C = C0 = 0.06 for neutral stratification. As for the influence of the stratification, we assume that the vertical scale of turbulence is limited by the vertical scale of buoyancy lb = q/N (N 2 = ∂ B/∂ z). That is, K ∼ qlb ∼ qlRit −1/2 ,
(7.62)
Rit = (Nl/q)2 .
(7.63)
where Rit is the turbulent Richardson number
This means that when the stratification is strong (N is large, Rit is large, and K is small), the turbulent energy is not transported downwards since the internal waves are induced by the injected turbulent energy, resulting in their horizontal propagation. It could also be considered that the local turbulent energy dissipation becomes large. The following equation is used for S so that it satisfies (7.62) when Rit is large: S/S0 = (1 + α Rit )−1/2 ,
(7.64)
where α is a tuning parameter. Noh and Kim (1999) recommend α ∼ 120.0, but α ∼ 5.0 is the default value of MRI.COM. The effect of stratification on the energy dissipation (C) is set as follows: C/C0 = (1 + α Rit )1/2 .
(7.65)
The effect of stratification on the Prandtl number (Pr) is set following Noh et al. (2005): Pr/Pr0 = (1 + β Rit )1/2 ,
(7.66)
where β is a tuning parameter and 0.5 is used following Noh et al. (2005). The vertical scale of turbulence is given by l=
κ (z + z0 ) , (1 + κ (z + z0 )/h)
(7.67)
where z0 is the sea surface roughness (z0 = 1[m]), z is the depth, and h is the mixed layer depth. The vertical scale becomes longer as the mixed layer becomes deeper. The boundary conditions are as follows:
∂U ∂z ∂B KH ∂z ∂E KE ∂z
KM
τ , ρ0
(7.68)
= Q0 ,
(7.69)
= mu3∗ ,
(7.70)
=
where m is a tuning parameter and m = 100 is recommended by Noh and Kim (1999). In the case of unstable stratification (N 2 < 0), KM = KH = 1.0[m2 s−1 ] and KE is estimated from the turbulent velocity scale and the vertical length scale in the model. This treatment is due to the difference between the time scales of the vertical convection and the development of turbulence.
–
90
–
7.3. K Profile Parameterization (KPP)
7.2.2
Implementation
Equation (7.57) is solved for the prognostic variable E in the subroutine nkoblm in nkoblm.F90 as the last procedure of each time step. The forward finite difference is used in the time evolution. The implicit method is used for the vertical diffusion of the turbulent kinetic energy and energy dissipation term, since these terms could become significantly large (see Section 12.5). The new E is used to determine the coefficients of viscosity and diffusivity for the next time step. The turbulent kinetic energy and the master length scale are defined at the center of the tracer grid cell (i, j, k − 12 ). The specific expression for the discretized form of the turbulent kinetic energy (E) equation is as follows: n E n+11 − Ek+ 1 k+ 2
Δt
2
=
n+1 n+1
KE k (Ek− 1 − Ek+ 1 )
1
2
Δzk+ 1
Δzk
2
2
−
KE k+1 (E n+11 − E n+13 ) k+ 2
Δzk+1
k+ 2
(un+11 − un+11 )(un+11 − unk+ 1 ) 1 (un+11 − un+13 )(unk+ 1 − un+13 ) 1 k− 2 k+ 2 k− 2 k+ 2 k+ 2 k+ 2 2 2 + KM k + KM k+1 2 Δzk Δzk+ 1 2 Δzk+1 Δzk+ 1 2
1 + KM k 2
2
(vn+11 − vn+11 )(vn+11 − vnk+ 1 ) k− 2
k+ 2
k− 2
Δzk Δzk+ 1
2
2
1 + KM k+1 2
(vn+11 − vn+13 )(vnk+ 1 − vn+13 ) k+ 2
k+ 2
2
Δzk+1 Δzk+ 1
k+ 2
(7.71)
2
Bn+11 − Bn+11 1 Bn+11 − Bn+13 1 k− 2 k+ 2 k+ 2 k+ 2 n − KH k+1 − 2CE n+11 qnk+ 1 /lk+ − KH k 1. k+ 2 2 Δzk 2 Δzk+1 2 2 The discrete expression for the shear production (the second through fifth terms on the r.h.s.) and the buoyancy sink (the sixth term on the r.h.s.) follows Burchard (2002), which is consistent with the conservation law of the sum of mean and turbulent energy. To summarize, the numerical operations proceed in the following order: 1. Update the master length scale using (7.67), 2. Update the turbulent kinetic energy using (7.71), 3. Calculate the vertical viscosity and diffusivity for the next time step.
7.3 7.3.1
K Profile Parameterization (KPP) Outline
The Kprofile parameterization (KPP) (Figure 7.1, Equation 7.79) is a method to determine the coefficients of vertical viscosity and diffusivity. First, the turbulent vertical velocity scale in the mixed layer (the boundary layer) is determined following the MoninObukhov similarity law near the boundary. This scale is then multiplied by the mixed layer thickness and the nondimensional profile function that are separately obtained to produce the vertical viscosity and diffusivity. The coefficient νx below the mixed layer is set to different values from those within the mixed layer and connected continuously. In this way, KPP differs from a series of turbulent closure model represented in the previous subsections. The KPP scheme is an application of the nonlocal Kprofile model (Treon and Mahrt, 1986) used in the atmospheric model to the ocean model by Large et al. (1994). The KPP subroutine in MRI.COM is based on that of the NCEP ocean model (NCOM). The paramterization for mixing due to salt fingering is not adopted and a vertically onedimensional background profile is given at vdbg.F90 by user (see Chapter 16). – 91 –
Chapter 7
Mixed layer model
The time tendency of an averaged value X due to the turbulent eddy is expressed by the vertical divergence of turbulent flux wx, ∂t X = −∂z wx, (7.72) where X represents the time averaged components of the velocity component U, V , temperature T , salinity S, buoyancy B, and so on, and x represents the turbulent components of the velocity component u, v, temperature t, salinity s, buoyancy b, and so on. Furthermore, w is the vertical velocity due to the turbulent eddy (positive upward). Hereinafter, the momentum component is expressed as m and the scalar property as s in some cases. In the KPP scheme, the turbulent flux within the mixed layer is expressed by the vertical gradient term of X and the nonlocal transport term.† That is (7.73)
wx = −Kx (∂z X − γx ). The vertical viscosity and diffusivity coefficients Kx and the nonlocal transport γx are calculated in the following order by the KPP subroutine of MRI.COM. • Calculation of the sea surface fluxes (momentum and buoyancy) wx0 • Calculation of the stability scale L • Calculation of the mixed layer thickness h • Calculation of the nondimensional universal function φx • Calculation of the turbulent vertical velocity scale wx • Calculation of the vertical viscosity and diffusivity coefficient within the mixed layer Kx • Connection of the coefficient in the mixed layer to the coefficient νx below the mixed layer base Kx∗ • Calculation of the vertical viscosity and diffusivity coefficient due to shear instability νxs • Adopt the larger value of Kx∗ and νxs for the vertical viscosity and diffusivity coefficient • Calculation of the nonlocal transport γx
7.3.2 MoninObukhov similarity law The MoninObukhov similarity law applies to the boundary layer near the sea surface. In this boundary layer, only the distance from the sea surface d(= −z) and the sea surface flux wx0 are important. From these parameters, the following three basic turbulent parameters are determined, • Frictional velocity: u∗2 = ( wu0 2 + wv0 2 )1/2 = τ0 /ρ0 ,
(7.74)
• Turbulent scale of scalar (temperature and salinity): s∗ = − ws0 /u∗ ,
(7.75)
† Nonlocal means the phenomenon in which the material is transported upwards (that is opposite to the gradient of the averaged fields) due to the gradient induced by the turbulent component ∂ x/∂ z even though the gradient of the averaged fields ∂ X/∂ z is locally positive. See Figure 7.1(a).
– 92 –
7.3. K Profile Parameterization (KPP) (a)
(b)
buoyancy(solid) and its ﬂux (dotted)
diﬀusion coeﬃcient
mixed layer(solid), interior(dotted)
mixed layer
0
dk he h
0
0
Figure 7.1. Schematic diagram of KPP. (a) Relative buoyancy (solid line) and buoyancy flux (dotted line) profiles. Upward buoyancy flux (nonlocal transport; Section 7.3.7) could occur in the boundary layer even if the stratification is neutral. Also indicated are the entrainment depth (he ) and mixed layer base (h). (b) The boundary layer diffusivity profile Kx (solid line; equation (7.79), and the interior diffusivity profile (dotted line). The diffusivity at depth (dk ) immediately above the mixed layer base (h) is set to the value marked by 2, and the boundary layer profile and the interior profile are connected by the dashed line (see Section 7.3.4).
• MoninObukhov stability scale: L = u∗3 /(κ B f ),
(7.76)
where τ0 is the sea surface wind stress, ρ0 is density, κ = 0.4 is the von Karman constant, and B f is the surface buoyancy flux (positive downward and thus negative means unstable). It should be noted that the direction of B f is opposite to that within the ocean. Note also that the fluxes need not be constant in the boundary layer (d < ε h [ε 1, in general ε ∼ 0.1]), but are affected by the sea surface flux wx0 and derived properties u∗ , s∗ , L. In this case, the nondimensional profiles of velocity and scalar are defined as functions of stability parameter
ζ = d/L: κd ∂z (U 2 +V 2 )1/2 , u∗ κd φs (ζ ) = ∗ ∂z S. s
φm (ζ ) =
(7.77) (7.78)
These functions are determined empirically based on observations.
7.3.3
Coefficients of vertical viscosity and diffusivity
The profile of vertical viscosity and diffusivity Kx in the mixed layer is defined as the product of the turbulent vertical velocity scale wx and the nondimensional vertical shape function G(σ ). Since the mixing becomes more effective due to the turbulent eddies as the mixed layer becomes thicker, the coefficient Kx is set linearly proportional to h: Kx (σ ) = hwx (σ )G(σ ), – 93 –
(7.79)
Chapter 7
Mixed layer model
where σ = d/h (fractional depth within the mixed layer) is the nondimensional vertical coordinate. The vertical shape G(σ ) is approximated by a third order polynomial (O’Brien, 1970), G(σ ) = a0 + a1 σ + a2 σ 2 + a3 σ 3 .
(7.80)
Since the turbulent eddy does not cross the sea surface, Kx = 0 at σ = 0. Thus, a0 = 0 in equation (7.80). The MoninObukhov similarity law is applied in the boundary layer (σ < ε [= 0.1]). Assuming that the turbulent flux wx is linear (Lumley and Panofski, 1964; Tennekes, 1973), we have wx (σ )(a1 + a2 σ ) =
κ u∗ wx(d) φx (ζ ) wx0
(7.81)
from (7.73) with γx = 0, (7.78), and (7.79). Equation (7.81) holds if wx (σ ) =
κ u∗ . φx (ζ )
(7.82)
To avoid wx becoming too large, wx is set to be constant below the boundary layer (σ = ε (∼ 0.1)) under the unstable condition (ζ (= d/L) < 0),
κ u∗ φx (ε h/L) κ u∗ wx (σ ) = φx (σ h/L)
wx (σ ) =
ε < σ < 1 ζ < 0, otherwise.
(7.83)
The nondimensional profile function φx is determined as a function of the stability parameter ζ (= d/L) based on experiments. They are determined so that wx could be scaled by κ u∗ for a neutrally stable case (h/L ≤ 0), and so that wx could be larger (smaller) than κ u∗ for an unstable (stable) case. Large et al. (1994) employ the following expressions.
φm φm φm φs φs
= φs = 1 + 5ζ = (1 − 16ζ )−1/4 = (am − cm ζ )−1/3 =
(1 − 16ζ )−1/2
= (as − cs ζ )−1/3
0 ≤ ζ,
ζm ≤ ζ < 0, ζ < ζm , ζs ≤ ζ < 0, ζ < ζs .
(7.84)
Here, (ζs , cs , as , ζm , cm , am ) = (−1.0, 98.96, −28.86, −0.2, 8.38, 1.26). Assuming a linear profile for the turbulent flux wx (flux decreases linearly from the sea surface value), equation (7.81) is
wx(σ )/ wx0 = 1 − βr σ /ε = a1 + a2 σ .
(7.85)
From this, we have a1 = 1 and a2 = −βr /ε . Using the condition at the mixed layer base, G(1) = ∂σ G(1) = 0, and assuming ε = 0.1, we have a2 = −2, a3 = 1, and βr = 0.2.
7.3.4 Coefficients of vertical viscosity and diffusion at the base of the mixed layer In the KPP scheme, viscosity and diffusion coefficients in the mixed layer (Kx ) are connected to the background profile (νx ) at the base of the mixed layer. The coefficient at the boundary (level k) is determined by using the following equations:
δ = (h − dk− 1 )/(dk+ 1 − dk− 1 ) 2
2
2
dk− 1 < h < dk+ 1 , 2
2
(7.86)
Kx∗ = (1 − δ )2 Kx (dk− 1 ) + δ 2 νx (dk )
dk− 1 < h ≤ dk ,
(7.87)
Kx∗ = (1 − δ )2 Kx (dk− 1 ) + δ 2 Kx (dk )
dk < h ≤ dk+ 1 ,
(7.88)
2 2
2
Λx = (1 − δ )νx (dk− 1 ) + δ Kx∗ .
2
(7.89)
2
– 94 –
7.3. K Profile Parameterization (KPP) The vertical viscosity and diffusion coefficients at depth dk that corresponds to the boundary between the mixed layer base (h) and the interior region is presented by Λx (2 in Figure 7.1(b)). The coefficient Kx∗ is set to yield large coefficients there. When dk− 1 < h < dk (when the mixed layer is shallower than dk ), Kx (dk ) that appears in (7.88) 2 is not defined and is replaced by the background value νx (dk ).
7.3.5
Thickness of the mixed layer
The thickness h of the mixed layer is estimated from the vertical profiles of buoyancy B(d) and velocity V (d). Here, it is determined as the depth where the bulk Richardson number referred to the sea surface, Rib (d) =
(Br − B(d))d , Vr − V (d)2 +Vt2 (d)
(7.90)
equals the reference value Ric (0.3 in MRI.COM), where Br and Vr are the buoyancy and the velocity in the uppermost layer. The quantity Vt /d is called the turbulent velocity shear and is expressed as follows: Vt2 (d) =
Cv (−βT )1/2 (cs ε )−1/2 dNws , Ric κ 2
(7.91)
where Cv = 1.8 and βT = −0.2. The turbulent velocity shear is introduced to achieve entrainment at the mixed layer base under strong stratification.
7.3.6
Mixing due to shear instability
In the stratified interior, mixing could occur locally due to shear instabilities. The tendency for shear instability is measured by the local gradient Richardson number, Rig =
N2 (∂zU)2 + (∂zV )2
.
(7.92)
When Rig is below a critical value Ri0 , the vertical velocity shear overcomes the stabilizing effect of the buoyancy gradient. This process is parameterized using a vertical mixing coefficient νxs given by the following formula and is the same for viscosity and diffusivity.
νxs /ν 0 = 1 Rig < 0 νxs /ν 0
= [1 − (Rig /Ri0
)2 ] p1
0 < Rig < Ri0
νxs /ν 0 = 0 Ri0 < Rig
(7.93) (7.94) (7.95)
Here, ν 0 = 50 × 10−4 [m2 s−1 ], Ri0 = 0.7, and p1 = 3. The final vertical mixing coefficient for “this” time step is taken as the larger of this shear instability mixing coefficient and the one obtained by connecting boundary layer and background interior (Section 7.3.4).
7.3.7
Nonlocal Transport
If the stratification is locally stable or neutral, the turbulent buoyancy flux should be downward or zero according to the r.h.s. of (7.73 [γx = 0]). However, an upward turbulent buoyancy flux exists if the sea surface buoyancy flux is unstable (upward; Figure 7.1(a)). This is called a nonlocal transport (or counter gradient transport).
–
95
–
Chapter 7
Mixed layer model
In the mixed layer, the turbulence exhibits a nonlocal character, and the local buoyancy flux depends on the boundary layer parameters such as the sea surface flux wb and the thickness of the mixed layer h in addition to the local gradient. Furthermore, nonlocal transport has a nonzero value only for the scalar under unstable forcing (Deardroff, 1972). Here, using the parameterization of Mailhˆot and Benoit (1982), the nonlocal transport γs for the scalar under the unstable forcing is calculated by
γs = C∗
ws0 , w∗ h
(7.96)
where C∗ = 10. These are summarized as follows:
γx γm
=
0
=
0
γs
= Cs
γθ
ws0 ws (σ )h
wθ0 + wθR = Cs ws (σ )h
ζ ≥ 0, ζ < 0, ζ < 0,
(7.97)
ζ < 0,
where Cs = C∗ κ (cs κε )1/3 ,
(7.98)
and wθR expresses the absorption of short wave radiation, which will be detailed in Chapter 8.
References Burchard, H., 2002: Energyconserving discretization of turbulent shear and buoyancy production, Ocean Modell., 4, 347361. Deardroff, J. W., 1972: Theoretical expression for the countergradient vertical heat flux, J. Geophys. Res., 77, 59005904. Kantha, L. H., and C. A. Clayson, 2000: Small Scale Processes in geophysical Fluid Flows, Academic Press, 888pp. Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization, Rev. Geophys., 32, 363403. Lumley, J. A., and H. A. Panofsky, 1964: The structure of the atmospheric turbulence, 239pp., John Wiley, New York. Mailhˆot, J., and R. Benoit, 1982: A finiteelement model of the atmospheric boundary layer suitable for use with numerical weather prediction models, J. Atmos. Sci., 39, 22492266. Mellor, G. L., and A. Blumberg, 2004: Wave breaking and ocean surface layer thermal response, J. Phys. Oceanogr., 34, 693698. Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys. Space Phys., 20, 851875. Noh, Y., Y.J. Kang, T. Matsuura, and S. Iizuka, 2005: Effect of the Prandtl number in the parameterization of vertical mixing in an OGCM of the tropical Pacific, Geophys. Res. Lett., 32, L23609, doi:10.1029/2005GL024540.
–
96
–
7.3. K Profile Parameterization (KPP) Noh, Y., and H.J. Kim, 1999: Simulations of temperature and turbulence structure of the oceanic boundary layer with the improved nearsurface process, J. Geophys. Res., 104, 15,62115,634. O’Brien, J. J., 1970: A note on the vertical structure of the eddy exchange coefficient in the planetary boundary layer, J. Atmos. Sci., 27, 12131215. Rotta, J. C., 1951a: Statistische Theotie nichthomogener Turbulenz, Z. Phys., 129, 547572. Rotta, J. C., 1951b: Statistische Theotie nichthomogener Turbulenz, Z. Phys., 131, 5177. Tennekes, H., 1973: A model for the dynamics of the inversion above a convective boundary layer, J. Atmos. Sci., 30, 558567. Treon, I. B., and L. Mahrt, 1986: A simple model of the atmospheric boundary layer; sensitivity to surface evaporation, Boundary Layer Meteorol., 37, 129148.
–
97
–
Chapter 8
Sea surface fluxes
Sea surface momentum, heat, and fresh water fluxes are needed to drive an ocean model. This chapter explains how those fluxes are treated in MRI.COM. The treatment is designed for the standalone simulation driven by atmospheric reanalysis data or observed data and simulation using a coupled atmosphereocean model. By default, a standalone simulation is driven by momentum fluxes and temperature and salinity fluxes derived by restoring the model sea surface temperature and salinity to the observed ones. It would be natural to use sea surface momentum, heat and fresh water fluxes based on observation when a realistic simulation is intended. However, the simulated sea surface temperature and salinity do not always agree with the observed ones due to the following reasons. 1. Sea surface heat and fresh water fluxes derived from observations include large errors. 2. The ocean standalone simulations do not include feedback mechanisms from the ocean to the atmosphere. 3. Since ocean models have dynamical errors, they do not necessarily capture all of the phenomena. Thus, it is not common in a standalone simulation to drive the model by fluxes without a feedback process, such as flux data derived from atmospheric reanalysis data. Rather, it is common to calculate fluxes using a bulk formula with sea surface meteorological elements derived from atmospheric reanalysis data. For a simulation using a coupled atmosphereocean model, all surface fluxes are calculated by the atmospheric component and fluxes on the oceanic grid points are given by a flux coupler. Users are referred to Yukimoto et al. (2010) for the treatment of fluxes in a coupled model. Section 8.1 describes momentum flux. The surface forcing terms for temperature and salinity are presented in Section 8.2, and their components, (heat flux (Section 8.3), fresh water flux (Section 8.4), and the equivalent fluxes under invariable first layer volume condition (Section 8.5)) are detailed in the succeeding sections. Several options are available for the choice of the bulk formula to calculate momentum, latent and sensible heat, and evaporative fluxes. Each bulk formula is detailed in Section 8.6 with description about a general formulation for the bulk transfer coefficient. Finally, the work flow in MRI.COM is summarized in Section 8.7.
8.1
Momentum flux (surface stress)
Surface forcing to the momentum equation, or surface momentum flux into the ocean, is in the form of wind stress (or stress from sea ice below sea ice), and is expressed as a body force to the first level velocity in the model algorithm (Fx , Fy ) =
(τx , τy ) , ρ0 Δz 1 2
where Δz 1 is the thickness of the first layer and τx and τy are zonal and meridional stress. 2
–
99
–
(8.1)
Chapter 8
Sea surface fluxes
8.1.1 Input of wind stress data By default, wind stresses are given as external data and are read from a file. (This input file is always required, and thus a file filled with zeros is necessary when wind stress is not applied.) The input data should be in units of [dyn · cm−2 ]. The order of operations is as follows: • Wind stress data are read from a file in force.F90 in the following format: read(unit = iwind) τx , τy . • Timeinterpolated in mkflux.F90 to produce surface stress. • A part of the surface stress might be modified by the stress from sea ice. • Surface stress is applied to the first level horizontal velocity in clinic.F90 as follows:
∂ 1 (τx , τy ). (u1 , v1 ) = ... + ∂t ρ0 Δz 1
(8.2)
2
In the last of the above operations, (u1 , v1 ) is horizontal velocity at the first level and Δz 1 is the thickness of the 2
first layer.
8.1.2 Calculating wind stress using a bulk formula If the model option TAUBULK is specified, the wind stress is calculated based on a bulk formula. In this option, zonal and meridional wind velocities at 10 m in units of [cm · s−1 ] should be prepared as external data, instead of the default wind stress data. Each component of wind stress is calculated with a bulk transfer coefficient CD (see Section 8.6) as follows:
τ = ρaCD Ua − Us (Ua − Us ),
(8.3)
where ρa is the density of air, Ua is the wind vector at 10 m height, and Us is the first level velocity.
8.2 8.2.1
Sea surface forcing for temperature and salinity Temperature
The contribution of surface forcing (FzT ) to the first layer temperature is expressed as FzT ∂ T = ... + , 1 ∂ t k= 2 Δz 1
(8.4)
2
where Δz 1 is the thickness of the first layer of the Tcell. The term (FzT ) consists of the net sea surface heat flux 2
gained by the ocean (QNET ), heat transport due to fresh water flux, and restoring of SST (To ) to a specific value (T ∗ ): QNET 1 FzT = + (WAO +WIbot ) · To +WIsurf · Tfreeze − (To − T ∗ )Δz 1 . (8.5) 2 ρ0 C p γt The heat flux term is converted to heat transport using the reference density (ρ0 ) and specific heat (C p ) of sea water. The water transported through the airsea interface (WAO ) and ice bottom (WIbot ) is assumed to have the first level temperature (To ). The water transported from the ice surface (WIsurf > 0) is assumed to have the freezing point –
100
–
8.2. Sea surface forcing for temperature and salinity temperature (Tfreeze ). The fourth term of the r.h.s. is a restoring term to the observed SST. The parameter γt is a restoring time (in units of seconds) and should be specified in the namelist njobbdy as rtmsc in units of days. The restoring term is converted to heat transport by multiplying the thickness of the first layer of Tcell (Δz 1 ). 2 By default, surface temperature forcing consists only of the restoring term. By specifying the option HFLUX, sea surface heat fluxes are calculated using the observed atmospheric variables. The fresh water flux terms are included when the option WFLUX is specified. If the option SFLUXW is specified, fresh water flux is not added to the ocean and the expression for the heat transport term is modified as follows: FzT =
QNET 1 −WIsurf · (To − Tfreeze ) − (To − T ∗ )Δz 1 . 2 ρ0 C p γt
(8.6)
See Section 8.5 for the derivation. Net sea surface heat flux QNET is given by QNET = QSH + QLO + QLA + QSN + Qice ,
(8.7)
where QSH is the net shortwave radiation flux, QLO is the net longwave radiation flux, QLA is the latent heat flux, and QSN is the sensible heat flux. The last term on the r.h.s., Qice , is due to exchange of heat with sea ice and will be detailed in Chapter 9. Downward heat flux is defined as positive. Three of the first four components of the heat flux, QLO , QLA , and QSN , depend on sea surface temperature, and thus they are calculated using the model SST.
8.2.2
Salinity
The contribution of surface forcing (FzS ) to the first layer salinity is expressed as FzS ∂ S , 1 = ... + k= 2 ∂t Δz 1
(8.8)
2
where Δz 1 is the thickness of the first layer of the Tcell. 2 The explicit surface forcing (FzS ) consists of salt transport due to formation and melting of sea ice and restoring
of SSS (So ) to a specific value (S∗ ):
FzS = (WIbot +WIsurf ) · SI −
1 (So − S∗ )Δz 1 . 2 γs
(8.9)
The water exchanged with ice (WIbot + WIsurf ) is assumed to have low salinity (SI = 4.0 [psu]). The parameter γs is a restoring time (in units of seconds) and should be specified in the namelist njobbdy as rtmsc in units of days (the same as temperature). The restoring term is converted to salinity transport by multiplying the thickness of the first layer of the Tcell (Δz 1 ). The fresh water flux modifies the volume but not the salt content of the surface layer, 2
changing the salinity of the surface layer (see also Chapter 4). By default, surface salinity forcing consists only of the restoring term. When the option WFLUX is specified, the effect of the fresh water flux on salinity could be included. If the option SFLUXW is specified, the fresh water flux is not added to the ocean and its effect on salinity is now explicitly expressed as the salinity flux: FzS = −WAO · So − (WIbot +WIsurf ) · (So − SI ) − where So is the first level salinity. See Section 8.5 for the derivation. –
101
–
1 (So − S∗ )Δz 1 , 2 γs
(8.10)
Chapter 8
Sea surface fluxes
8.3 Heat flux 8.3.1 Shortwave radiation flux By default, the downward shortwave radiation (Qdown SH ) is read as external data. A part of the insolating shortwave down radiation (αo QSH ) is reflected at the sea surface, and the remainder penetrates into the ocean interior as the net shortwave radiation flux (QSH (0) = (1 − αo )Qdown SH ), where αo is the albedo at the ocean surface. More than 50% of the insolating radiation (nearinfrared band) is absorbed within a depth of 1 m below the sea surface, but the remainder (visible and ultraviolet bands) penetrates further into the ocean interior and affects the subsurface temperature structure. According to Paulson and Simpson (1977), the shortwave radiation flux penetrating into the ocean interior is given by QSH (z) = QSH (0)[R exp(z/ζ1 ) + (1 − R) exp(z/ζ2 )],
(8.11)
where we set R = 0.58, ζ1 = 35 [cm], and ζ2 = 2300 [cm], using the optical properties of Water Type I based on the classification by Jerlov (1976). Vertical convergence of the penetrating shortwave radiation energy ∂ QSH (z)/∂ z is converted to heat within each layer. Using option SOLARANGLE enables us to represent the shortwave radiation flux including the effect of the insolation angle. This scheme is based on Ishizaki and Yamanaka (2010). In this case, if the depth (z) on the r.h.s. of Equation (8.11) is replaced by the penetrating distance from the sea surface, the shortwave radiation is given by QSH (z) = QSH (0)[R exp(z/ζ1 sin θw ) + (1 − R) exp(z/ζ2 sin θw )],
(8.12)
where θw is the penetrating angle in the ocean interior. When option SOLARANGLE is specified in coupled models, Equation (8.12) is replaced by (8.13) QSH (z) = QSHb (z) + QSHd (z), where QSHb (z) and QSHd (z) are the shortwave radiation fluxes due to direct solar radiation and diffuse solar radiation. Those fluxes are expressed as follows: QSHb (z) = QSHb (0)[Rb exp(z/ζ1 sin θw ) + (1 − Rb ) exp(z/ζ2 sin θw )],
(8.14)
QSHd (z) = QSHd (0)[Rd exp(z/ζ1 ) + (1 − Rd ) exp(z/ζ2 )],
(8.15)
where Rb and Rd are the ratio of nearinfrared radiation to total radiation for direct solar radiation and diffuse solar radiation. QSHb (0) and QSHd (0) are the net direct and diffuse solar radiation fluxes at the sea surface. There are three options for the sea surface albedo. The first option (albedo choice = 1 in the namelist njobalb) is a constant value, which should be specified as alb in the namelist njobalb. The second option (albedo choice = 2) is based on Large and Yeager (2008) and is given by
αo = 0.069 − 0.011 cos(2φ ),
(8.16)
where φ is latitude. The third option (albedo choice = 3) is based on Baker and Li (1995) and is given by
αo = 0.06 + 0.0421x2 + 0.128x3 − 0.04x4 + (
3.12 0.074x + )x5 , 5.68 +U 1.0 + 3.0U
where x = 1 − sin θa (θa is a height angle of the Sun), and U is the surface wind speed [m · s−1 ].
–
102
–
(8.17)
8.3. Heat flux
8.3.2
Shortwave radiation flux based on chlorophyll concentration
Recent studies indicate that solar radiation absorption and local heating within the upper ocean are strongly influenced by the chlorophyll concentration. Using both the CHLMA94 and NPZD options enables us to use the shortwave penetration model with the chlorophyll concentration (Morel and Antoine, 1994). In this scheme, the shortwave radiation flux penetrating into the ocean interior is given by QSH (z) = QSH (0)[R exp(−z/0.267 sin θw ) + (1 − R){V1 exp(−z/ζ1 ) +V2 exp(−z/ζ2 )}],
(8.18)
The first exponential is for the infrared waveband (> 750nm), which is not influenced by biological materials. The second and third exponentials are for the ultraviolet and visible bands (< 750nm). V1 , V2 , ζ1 , and ζ2 are calculated from an empirical relationship as a function of chlorophyll concentration (chl [mg · m−3 ]) as follows: V1 = 0.321 + 0.008C + 0.132C2 + 0.038C3 − 0.017C4 − 0.007C5 ,
(8.19)
V2 = 0.679 − 0.008C − 0.132C2 − 0.038C3 + 0.017C4 + 0.007C5 ,
(8.20)
ζ1 = 1.540 − 0.197C + 0.166C − 0.252C − 0.055C + 0.042C ,
(8.21)
ζ2 = 7.925 − 6.644C + 3.662C2 − 1.815C3 − 0.218C4 + 0.502C5 ,
(8.22)
2
3
4
5
where C = log10 (chl). It is noted that V1 +V2 = 1. When the option SOLARANGLE is added to the options mentioned above, the shortwave radiation is slightly modified by QSH (z) = QSH (0)[R exp(−z/0.267 sin θw ) + (1 − R){V1 exp(−z/ζ1 sin θw ) +V2 exp(−z/ζ2 sin θw )}].
(8.23)
In coupled models, Equation (8.23) is replaced by QSH (z) = QSHb (z) + QSHd (z),
(8.24)
where QSHb (z) and QSHd (z) are the shortwave radiation fluxes due to direct solar radiation and that due to diffuse solar radiation. Those fluxes are expressed as follows: QSHb (z) = QSHb (0)[Rb exp(−z/0.267 sin θw ) + (1 − Rb ){V1 exp(−z/ζ1 sin θw ) +V2 exp(−z/ζ2 sin θw )}], (8.25) QSHd (z) = QSHd (0)[Rd exp(−z/0.267) + (1 − Rd ){V1 exp(−z/ζ1 ) +V2 exp(−z/ζ2 )}],
(8.26)
where Rb and Rd are the ratios of the nearinfrared part to the total radiation for direct and diffuse solar radiation, respectively.
8.3.3 Longwave radiation flux Longwave radiation is calculated by removing the upward radiation due to the observed SST ToOBS (external data) from the net longwave radiation QOBS LO (external data) and by adding the upward radiation due to the model SST. OBS QLO = QOBS + 273.16)4 − (To + 273.16)4 ). LO + em σ ((To
(8.27)
Here, em = 0.97 is the emissivity for sea water, and σ = 5.67 × 10−5 [erg·s−1 ·cm−2 ·K−4 ] is the StefanBoltzmann constant. Using option LWDOWN enables us to use the downward longwave radiation as external data. In this case, dummy data should be prepared for the SST data as external data. –
103
–
Chapter 8
Sea surface fluxes
8.3.4 Latent and sensible heat fluxes The bulk method is used to calculate latent and sensible heat fluxes. In the bulk method, latent heat flux QLA and sensible heat flux QSN are calculated using bulk transfer coefficients CE and CH (see Section 8.6). QLA = −ρa LCE U10 (qs − qa )
(8.28)
QSN = −ρaC paCH U10 (To − Ta ).
(8.29)
and Here, ρa is the atmospheric density, L is the latent heat for evaporation, qa is the specific humidity at the sea surface, qs is the saturated specific humidity of the sea surface temperature, Ta is the sea surface air temperature, and U10 is the scalar wind speed. The quantity Cpa is the specific heat for the atmosphere. MRI.COM can provide three calculation methods for bulk transfer coefficients: Kondo (1975), Large and Yeager (2004), and Kara et al. (2002). In the following, we explain the elements necessary for calculating these bulk transfer coefficients and calculation of the latent and sensible heat fluxes. (a) Kondo (1975)ᲢBULKKONDO2Უ The bulk formula based on Kondo (1975) uses sea surface pressure Ps [hPa], sea surface air temperature Ta [◦ C], sea surface specific humidity qa [g·g−1 ], and scalar wind speed U10 [cm·s−1 ] as input data. Elements necessary for calculating the latent and sensible heat fluxes are obtained as follows: 1. Saturated water vapor pressure es [hPa] for sea surface temperature (To [◦ C]) es = 0.98 × 6.1078 × 107.5To /(237.3+To ) ,
(8.30)
2. Saturated specific humidity for sea surface temperature qs [g·g−1 ] qs =
0.62197es , Ps − 0.378es
(8.31)
3. Latent heat of evaporation L [erg·g−1 ] L = 4.186 × 107 (594.9 − 0.5To ),
(8.32)
ρa = 1.205 × 10−3 .
(8.33)
4. Atmospheric density ρa [g·cm−3 ]
(b) Large and Yeager (2004)ᲢBULKNCARᲣ The bulk formula based on Large and Yeager (2004) uses the same input data and elements as those of Kondo (1975). (c) Kara et al. (2002)ᲢBULKKARAᲣ The bulk formula based on Kara et al. (2002) uses four elements: sea surface pressure Ps [hPa], sea surface air temperature Ta [◦ C], sea surface dew point temperature Td [◦ C], and scalar wind speed U10 [cm·s−1 ]. Elements necessary for calculating the latent and sensible heat fluxes are obtained as follows: 1. Water vapor pressure at the sea surface ea [hPa] ea = 6.1121 exp[(18.729 − Td /227.3)Td /(Td + 257.87)], –
104
–
(8.34)
8.4. Freshwater flux 2. Saturated water vapor pressure es [hPa] for sea surface temperature (To [◦ C]) es = 0.9815 × 6.1121 exp[(18.729 − To /227.3)To /(To + 257.87)],
(8.35)
3. Specific humidity at the sea surface qa [g·g−1 ] qa =
0.62197ea , Ps − 0.378ea
(8.36)
4. Latent heat of evaporation L[erg·g−1 ] L = 1010 (2.501 − 0.00237To ),
(8.37)
ρa = 103 Ps /[Rg (Ta + 273.16)(1.0 + 0.61qa )].
(8.38)
5. Air density ρa [g·cm−3 ] Here, Rg is the gas constant for dry air. The saturated specific humidity for sea surface temperature qs [g·g−1 ] is given by Equation (8.31), which is equivalent to Kondo (1975).
8.4 Freshwater flux 8.4.1 Introduction Freshwater flux through the sea surface is caused mainly by precipitation (P), evaporation (E), river discharge (R), and formationmelting of sea ice (I). The fresh water fluxes are included in the model when option WFLUX is specified. In this case, precipitation and river discharge data should be prepared as input data. A free surface model can deal with freshwater flux explicitly and calculate the salinity change using a volume change of the first layer. Even with the WFLUX option, the salinity restoring term could be added to the water flux by specifying the option SFLUXR to avoid unexpected salinity drift. The salinity restoring is converted to the equivalent fresh water flux in this case. It is possible to transform all freshwater fluxes to salinity flux by specifying option SFLUXW, in order to prevent the thickness of the first layer (nominally several meters) from becoming zero or unrealistically large due to excess precipitation or evaporation.
8.4.2 Calculating freshwater flux Freshwater flux through the sea surface F W [cm·s−1 ] is given by F W = P − E + R + I,
(8.39)
where precipitation P is given as external data, and evaporation E is calculated in the model, E = ρaCE U10 (qs − qa ) = −QLA /L.
(8.40)
River discharge R is given as external data when RUNOFF is selected. Freshwater flux I due to the formation and melting of the sea ice is added when the sea ice model is used (ICE) (see Chapter 9 for the sea ice model). We get F W = P − E when only WFLUX is selected. – 105
–
Chapter 8
Sea surface fluxes
Since F W is not related directly to the sea surface salinity, the model sea surface salinity might be far from the observed value. Hence, an adjustment is sometimes needed to restore the model sea surface salinity to the observed one (see the last term on the r.h.s. of the next equation). Generally, the model salinity is restored to the observed climatological sea surface salinity since no reliable data set of historical sea surface salinity is available at present. By converting the restoring term to the fresh water flux, the expression for F W becomes FW = P − E + R + I +
1 So − S∗ . γs S0
(8.41)
In this case, the salinity flux should be FzS = (WIbot +WIsurf ) · SI ,
(8.42)
where WIbot is the water transported from the ice bottom, WIsurf is the water transported through the ice surface, and SI is the salinity of the water exchanged with ice. The global mean of F W should not necessarily be zero, and thus the volume of the first layer averaged globally may continue to increase or decrease. To avoid this, the integral of the globally averaged freshwater flux is set to zero by selecting option WADJ, W = FW − FW , FADJ
(8.43)
where F W is the global mean of F W (= P − E(+R) + γs (So − S∗ )/S0 ).
8.5 Equivalent surface temperature and salinity fluxes for constant first layer volume Precipitation, evaporation, and river discharge (WAO = P − E + R) are assumed to have the first level temperature (To ) and zero salinity. The water transported through the ice bottom (WIbot ) is also assumed to have the first level temperature. The water transported from the ice surface (WIsurf > 0) is assumed to have the freezing point temperature (Tfreeze ). The water exchanged with ice is assumed to have low salinity (SI = 4.0 [psu]). Note that the freezing point temperature (Tfreeze ) is given by mSI , where m = −0.0543 [K / psu]. If fresh water flux is simply added to the first layer volume, surface temperature and salinity fluxes due to the fresh water flux are given by the following. T FWF
= (WAO +WIbot ) · To +WIsurf · Tfreeze ,
(8.44)
S FWF
= (WIbot +WIsurf ) · SI .
(8.45)
If the surface fresh water flux is not added to the first layer to avoid an unexpected sea level rise or fall during a longterm integration, equivalent temperature and salinity fluxes should be imposed. The conservation equations for the first layer heat and salinity are considered. Temperature and salinity are assumed to evolve from the old values (Told , Sold ) to the new values (Tnew , Snew ) due to fresh water flux. If a volume change is allowed, Vnew Tnew
= Vold Told + (WAO Told +WIbot Told +WIsurf Tfreeze ) · ΔA · Δt,
(8.46)
Vnew Snew
= Vold Sold + (WIbot SI +WIsurf SI ) · ΔA · Δt,
(8.47)
= Vold + (WAO +WIbot +WIsurf ) · ΔA · Δt,
(8.48)
Vnew where ΔA is the horizontal area.
– 106
–
8.6. Bulk transfer coefficient If a volume change is not allowed, Vold Tnew
T = Vold Told + FWF · ΔA · Δt,
(8.49)
Vold Snew
S Vold Sold + FWF · ΔA · Δt,
(8.50)
=
T and F S are temperature and salinity flux for the constant volume case. where FWF WF Removing Tnew , Snew , and Vnew from the above equations, we have,
Vold {Vold Told + (WAO Told +WIbot Told +WIsurf Tfreeze ) · ΔA · Δt} T · ΔA · Δt), = {Vold + (WAO +WIbot +WIsurf ) · ΔA · Δt}(Vold Told + FWF
(8.51)
Vold {Vold Sold + (WIbot SI +WIsurf SI ) · ΔA · Δt} S · ΔA · Δt). = {Vold + (WAO +WIbot +WIsurf ) · ΔA · Δt}(Vold Sold + FWF
(8.52)
The fluxes for the constant volume case are given by T FWF S FWF
Vold WIsurf · (Told − Tfreeze ), Vold +W · ΔA · Δt Vold = − {WAO · Sold + (WIbot +WIsurf ) · (Sold − SI )}, Vold +W · ΔA · Δt = −
(8.53) (8.54)
where W = WAO +WIbot +WIsurf . If Vold  W · ΔA · Δt, we have,
8.6
T FWF
= −WIsurf · (Told − Tfreeze ),
(8.55)
S FWF
= −WAO · Sold − (WIbot +WIsurf ) · (Sold − SI ).
(8.56)
Bulk transfer coefficient
This section briefly describes how to calculate sea surface fluxes using a bulk formula. For details, refer to Kantha and Clayson (2000) and Large and Yeager (2004).
8.6.1
Formulation of bulk formula
Transfer processes through atmosphere and ocean boundaries are governed by complicated turbulent processes. Traditionally, these turbulent fluxes are parameterized as a bulk transfer law. This method does not require specific interactions between the atmosphere and the ocean, and attributes all unknown processes to bulk transfer coefficients. Momentum (τ ), sensible heat (HS ), latent heat (HL ), and water vapor (E) fluxes are written in terms of turbulent components as follows:
τ
= −ρa wu = ρa u2∗ ,
(8.57)
HS
= −ρa c p wθ = ρa c p u∗ θ∗ ,
(8.58)
HL
= −ρa LE wq = ρa LE u∗ q∗ ,
(8.59)
= −ρa wq = ρa u∗ q∗ = HL /LE ,
(8.60)
E
where a bar over a variable denotes a covariance between the turbulent component of vertical velocity and the turbulent component of each physical property, ρa is air density, u∗ is friction velocity, θ∗ ≡ HS /(ρa c p u∗ ) is the –
107
–
Chapter 8
Sea surface fluxes
temperature scale in the boundary layer, q∗ ≡ HL /(ρa LE u∗ ) = E/(ρa u∗ ) is the scale of specific humidity in the boundary layer, c p is the specific heat of air, and LE is the latent heat of water vapor evaporation. Sea surface fluxes are also represented using bulk formulae as follows:
τ
= ρaCDh (Ua −Us )2 ,
(8.61)
HS
= −ρa c p Ua −Us CHh (Ts − Ta ),
(8.62)
HL
= −ρa LE Ua −Us CEh (qs − qa ),
(8.63)
= ρa Ua −Us CEh (qs − qa ),
(8.64)
E
where Ua is the wind velocity at height of zh , and Us is the velocity component in the Ua direction at the sea surface. The subscript “a” means a value of each physical property at z = zh , and the subscript “s” means the value just above the sea surface. Parameter CDh is called a drag coefficient. CHh and CEh are transfer coefficients for heat and water vapor and are called the Stanton coefficient and the Dalton coefficient, respectively. These can be estimated by observed atmospheric elements (wind velocity, air temperature, and humidity) at a height in the boundary layer, not by observed turbulent fluxes, using the following Equations (8.65), (8.66), and (8.67) and the similarity law of MoninObukhov mentioned below, CDh CHh CEh
u2∗ , (Ua −Us )2 HS /ρa c p u∗ θ∗ = − = , Ua −Us (Ts − Ta ) Ua −Us (Ts − Ta ) E/ρa u∗ q∗ = = . Ua −Us (qs − qa ) Ua −Us (qs − qa ) =
(8.65) (8.66) (8.67)
The similarity law of MoninObukhov assumes that physical properties in the atmosphereocean boundary layer (a layer with a thickness of several tens of meters located below the lower mixed layer of the atmosphere) have similar vertical profiles when they are scaled with the stability parameter and sea surface fluxes. Vertical profiles of wind velocity, air temperature, and humidity can be written as follows:
κz ∂U u∗ ∂ z κz ∂ T θ∗ ∂ z κz ∂ q q∗ ∂ z
z , L z , = φH L z , = φE L
= φM
(8.68) (8.69) (8.70)
where κ = 0.4 is the von Karman constant, and L is the MoninObukhov length scale L=
u3 Tv u3∗ =− ∗ . κ Qb κ g wθv
(8.71)
In (8.71), Qb = −g wθv /Tv is the buoyancy flux, Tv is the virtual temperature (Tv = T (1 + 0.608 q)), and wθv is the net heat flux including the water vapor flux: wθv = wθ (1 + 0.608 q) + 0.608 T wq.
(8.72)
In (8.68) to (8.70), ζ = z/L is the MoninObukhov similarity variable, and φM,H,E is a nondimensional function for wind velocity, air temperature, and specific humidity. The nondimensional function is assumed to be a mathematically simple function. – 108
–
8.6. Bulk transfer coefficient Integrating Equations (8.68), (8.69), and (8.70) vertically, we have the following. + , z u∗ ln − ΨM (ζ ) U(z) −Us = κ z0 + , θ∗ z ln T (z) − Ts = − ΨH (ζ ) κ z0T + , z q∗ ln − ΨE (ζ ) , q(z) − qs = κ z0E where ΨM,H,E (ζ ) =
ζ
0
dζ 1 − φM,H,E (ζ ) ζ
(8.73) (8.74) (8.75)
.
(8.76)
In (8.73) to (8.75), z0 , z0T , and z0E are roughness lengths for each physical property. When the stability of the boundary layer is already known, bulk transfer coefficients can be estimated using these roughness lengths,
κ2 CDh = + ,2 , zh ln − ΨM (ζh ) z0 κ2 ,+ ,. CHh,Eh = + zh zh ln − ΨM (ζh ) ln − ΨH,E (ζh ) z0 z0T,0E
(8.77)
(8.78)
If neutrally stable (ζ = 0), the bulk transfer coefficient is a function of the roughness length only and is expressed as follows:
κ2 CDNh = + , , zh 2 ln z0N CHNh,ENh
=
=
κ2 + ,+ , zh zh ln ln z0N z0TN,0EN CDNh + , 1 1/2 z0TN,0EN 1 − CDNh ln κ z0N
(8.79)
(8.80)
(8.81)
1/2
=
+ ln
κ CDNh ,. zh z0TN,0EN
(8.82)
Normally, these neutral bulk transfer coefficients are estimated at a height of 10 m. Nonneutral bulk transfer coefficients at an arbitrary height (zh ) are connected with the neutral and stable bulk transfer coefficients at a height of 10 m by eliminating the roughness length as follows: CDN10 CDh = + ,2 , 1 zh −1/2 1 + CDN10CDN10 ln − ΨM (ζh ) κ z10 CDh 1/2 CHN10,EN10 C DN10 , , CHh,Eh = + 1 zh −1/2 1 + CHN10,EN10CDN10 ln − ΨH,E (ζh ) κ z10
(8.83)
– 109
–
(8.84)
Chapter 8
Sea surface fluxes
where z10 means z = 10[m]. The neutral bulk transfer coefficients at a height of 10 m (CDN10 , CHN10 , and CEN10 ) are often estimated, according to the stability, as a function of velocity at 10 m. Various formulations can be used (see the following subsections). To be more realistic, the bulk transfer coefficients should be regarded as a function of wave age, but its general formulation has not been achieved yet. In principle, if we have a complete set of atmospheric elements in the boundary layer, we can get the bulk coefficients and fluxes because height can be adjusted using the similarity law. This requires an iteration of the calculation as follows (you can select an appropriate way for your data): • Convert wind speed, air temperature, and specific humidity into those at 10 m in the neutral and stable case and use a bulk coefficient at 10 m in the neutral case, • Convert a bulk coefficient at 10 m into one with a height and a stability at which atmospheric elements are observed. • Convert air temperature and specific humidity into those at a height where wind speeds are observed, and convert the bulk coefficient at 10 m and neutral case into one with a height and a stability at which wind speeds are observed.
8.6.2 Kondo (1975) BULKKONDO2 When atmospheric elements are observed at various heights, latent and sensible heat fluxes are estimated at a height where air temperature and humidity are observed. Momentum fluxes are estimated at the same height. In this case, wind speed and bulk transfer coefficients should be corrected to ones at the height where the air temperature and humidity are observed. In the program of bulk.F90, you may specify BULKITER in configure.in to do this iteration. The observed heights are specified as altu (wind), altt (temperature), and altq (humidity) in the namelist njbkondo. The following procedure is executed: 1. Estimate the neutral bulk transfer coefficient at 10 m assuming that the observed wind speed is at 10 m in the neutral case. When the 10 m wind speed is already known, the neutral bulk transfer coefficient is given as follows: CDN10 = 10−3 {ad + 10−2 bd U10d }, p
(8.85)
where ad , bd , and pd are nondimensional numbers depending on the wind speed (Table 8.1). 2. Recalculate the roughness length using this neutral bulk transfer coefficient at 10 m and correct the height for wind speed once again:
−1/2 z0 = exp ln z10 − κ CDN10 ,
(8.86)
u10 = u(z) ln(10/z0 )/ ln(z/z0 ).
(8.87)
3. Reestimate the neutral bulk transfer coefficient at 10 m. 4. Convert the bulk transfer coefficient to that in the neutral case at the observed height:
κ2 CDNh =
2 −1/2 κ CDN10 − ln(z10 /zh ) –
110
–
(8.88)
8.6. Bulk transfer coefficient 1/2
CHNh,ENh =
κ CDNh 1/2
κ CDN10 /CHN10,EN10 + ln(zh /z10 )
,
(8.89)
where z10 means z = 10[m]. 5. Estimate the bulk transfer coefficient in which stability is considered using the neutral bulk transfer coefficient obtained above. The neutral bulk transfer coefficient at 10 m is given as follows: CDN10
=
CEN10
=
CNN10
=
10−3 {ad + 10−2 bd U10d }, p
−3
(8.90)
−2
pe 10 {ae + 10 beU10 + ce (10−2U10 − 8)2 }, p 10−3 {ah + 10−2 bhU10h + ch (10−2U10 − 8)2 },
(8.91) (8.92)
where CDN10 , CEN10 , and CHN10 are bulk transfer coefficients for neutral atmospheric stability. and ae,h , be,h , ce,h , and pe,h are nondimensional numbers depending on wind speed (Table 8.1). Table 8.1. Nondimensional parameters used in calculating the bulk transfer coefficient based on Kondo (1975)
U10
ad
ae
ah
bd
be
bh
ce
ch
pd
pe
ph
0.32.2 2.25
0 0.771
0 0.969
0 0.927
1.08 0.0858
1.23 0.0521
1.185 0.0546
0 0
0 0
0.15 1
0.16 1
0.157 1
58 825
0.867 1.2
1.18 1.196
1.15 1.17
0.0667 0.025
0.01 0.008
0.01 0.0075
0 0.0004
0 0.00045
1 1
1 1
1 1
2550
0
1.68
1.652
0.073
0.016
0.017
0
0
1
1
1
(m · s−1 )
The bulk transfer coefficient depending on the stability is estimated in the following operations. First, the stability of the atmospheric boundary layer is defined as follows: s = s0
s0  , s0  + 0.01
(8.93)
s0 =
To − Ta . 2 10−4U10
(8.94)
where
The atmospheric boundary layer is unstable if s > 0 (To − Ta > 0) and stable if s < 0 (To − Ta < 0). Finally, the bulk transfer coefficient depending on the stability is given as follows: ⎧ ⎪ s < −3.3 ⎨ 0 (8.95) CD = CDNh {0.1 + 0.03s + 0.9 exp(4.8s)} −3.3 ≤ s < 0 ⎪ √ ⎩ 0 ≤ s, CDNh (1.0 + 0.47 s) ⎧ ⎪ s < −3.3 ⎨ 0 CE = (8.96) CENh {0.1 + 0.03s + 0.9 exp(4.8s)} −3.3 ≤ s < 0 ⎪ √ ⎩ 0 ≤ s, CENh (1.0 + 0.63 s) ⎧ ⎪ s < −3.3 ⎨ 0 CH = (8.97) CHNh {0.1 + 0.03s + 0.9 exp(4.8s)} −3.3 ≤ s < 0 ⎪ √ ⎩ 0 ≤ s. CHNh (1.0 + 0.63 s) –
111
–
Chapter 8
Sea surface fluxes
8.6.3 Large and Yeager (2004) BULKNCAR In the bulk formula based on Large and Yeager (2004), each flux is estimated at a height where wind speed is observed, by transforming air temperature and humidity from the height where they are observed (zθ and zq , respectively) to the height where wind speed is observed. To do this iteration in the program bulk.F90, specify the option BULKITER in configure.in. The observed heights are specified as altu (wind) altt (temperature), and altq (humidity) in the namelist njbncar. The bulk transfer coefficient at 10 m in the neutral case is given as follows: 2.70 U10N + 0.142 + , U10N 13.09 103CHN10,EN10 = 18.0 CDN10 , stable ζ > 0, 103CHN10,EN10 = 32.7 CDN10 , unstable ζ ≤ 0, 103CDN10 =
(8.98) (8.99) (8.100)
Each physical property is estimated at a height where wind speed is observed in the following operations. First, calculate the virtual temperature θv as follows:
θv = θ (zθ )(1 + 0.608 q(zq ))
(8.101)
Next, calculate the bulk transfer coefficient at 10 m in the neutral and stable cases assuming that the first guess for the 10 m wind speed in the neutral and stable cases is U10N = U(zu ) (Equations (8.98) to (8.100)). The first guesses of the scales for the friction velocity, air temperature, and specific humidity are estimated assuming that these bulk coefficients are at the observed height and stability, 1 τ u∗ = = CDN10 U(zu ), (8.102) ρa CHN10 HS θ∗ = = (θ (zθ ) − Ts ), (8.103) ρa c p u ∗ CDN10 E CEN10 = (q(zq ) − qsat (Ts )), (8.104) q∗ = ρa u ∗ CDN10 where qsat (Ts ) is the saturated specific humidity at sea surface temperature Ts . Next, perform the iteration using the three MoninObukhov similarity variables, ζu = zu /L, ζθ = zθ /L, and ζq = zq /L, and an integral of the nondimensional profile function for the vertical gradient of each physical property, ΨM (ζ ) for momentum, and ΨH (ζ ) for scalars. The MoninObukhov similarity variables are calculated as follows, , + κ gz θ∗ q∗ . (8.105) ζ= 2 + u∗ θv (q(zq ) + 0.608−1 ) The integral of the nondimensional profile function is expressed as ΨM (ζ ) = ΨH (ζ ) = −5ζ ,
(8.106)
if it is stable (ζ ≥ 0), and ΨM (ζ ) = 2 ln
1+X 2
+ ln 1 + X2 , ΨH (ζ ) = 2 ln 2 – 112
1 + X2 2
− 2 tan−1 (X) +
π , 2
(8.107) (8.108)
–
8.6. Bulk transfer coefficient if it is unstable (ζ < 0). In the above, X = (1 − 16ζ )1/4 .
(8.109)
Using these values, convert the wind speed to that at 10 m in the neutral and stable cases, and convert the temperature and specific humidity to those at a height where the wind speed is observed, ,−1 + CDN10 zu ln − ΨM (ζu ) , U10N = U(zu ) 1 + κ z10 + , θ∗ zθ ln + ΨH (ζu ) − ΨH (ζθ ) , θ (zu ) = θ (zθ ) − κ zu , + z q∗ q q(zu ) = q(zq ) − ln + ΨH (ζu ) − ΨH (ζq ) , κ zu
(8.110) (8.111) (8.112) (8.113)
where z10 means z = 10[m]. Estimate the bulk coefficient at 10 m in the neutral and stable cases using U10N , and then obtain the bulk coefficient at a height (zu ) where wind speed is observed, CDu
=
CHu
=
CEu
=
CDN10 + ,2 , 1 zu −1/2 1 + CDN10CDN10 ln − ΨM (ζu ) κ z10 CDu 1/2 CHN10 CDN10 , , + 1 zu −1/2 1 + CHN10CDN10 ln − ΨH (ζu ) κ z10 CDu 1/2 CEN10 CDN10 + , . 1 zu −1/2 1 + CEN10CDN10 ln − ΨH (ζu ) κ z10
(8.114)
(8.115)
(8.116)
Repeat the procedures to calculate the bulk coefficients using these bulk coefficients with temperature and specific humidity at z = zu , and recalculate the scales for virtual temperature, friction velocity, temperature, and specific humidity (Equations of (8.102), (8.103), and (8.104)).
8.6.4
Kara et al. (2002) BULKKARA
In the bulk formula based on Kara et al. (2002), bulk transfer coefficients CD (momentum), CE (latent heat), and CH (sensible heat) are calculated using a polynomial fitting as follows to reduce the calculation cost due to iteration: 2 ), C0D = 10−3 (0.692 + 0.071U10 − 0.00070U10 −3
C1D = 10
2 (0.083 − 0.0054U10 + 0.000093U10 ),
CD = C0D +C1D (To − Ta ), −3
C0H = 10
(8.117) (8.118) (8.119)
2 (0.8195 + 0.0506U10 − 0.0009U10 ),
2 ), C1H = 10−3 (−0.0154 + 0.5698/U10 − 0.6743/U10
(8.120) (8.121)
CE = C0H +C1H (To − Ta ),
(8.122)
CH = 0.95CE ,
(8.123) –
113
–
Chapter 8
Sea surface fluxes
where 2.5[m · s−1 ] ≤ U10 ≤ 32.5[m · s−1 ]. Flux correction factors may be specified by setting close budgets, corr factor n, and corr factor s in namelist njbkara.
8.6.5 Bulk coefficient over sea ice A similar treatment is necessary even over sea ice, but it is common to replace the neutral and stable bulk coefficients at 10 m with constants, as described in Large and Yeager (2004). Then, bulk coefficients and wind speed at each height are calculated based on the stability. Large and Yeager (2004) uses CDN10 = CHN10 = CEN10 = 1.63 × 10−3 ,
(8.124)
but the default in MRI.COM uses, following Mellor and Kantha (1989), CDN10 = 3.0 × 10−3
(8.125)
CHN10 = CEN10 = 1.5 × 10−3 .
(8.126)
The wind stress could be separately calculated over sea ice under the option TAUBULK. Users should specify flg strsi = .true. in ice main cat.F90.
8.7 Work flow in MRI.COM 8.7.1 Momentum flux • Use of bulk formula (TAUBULK) 1. Input of external data (velocity vectors at 10 m height) in force.F90. (a) Time interpolation of external data in mkflux.F90. (b) Calculation of wind stress (τx , τy ) (Equation (8.3)) in bulk.F90 (called from mkflux.F90). (c) Calaculation of icewater stress and total surface stress based on areaweighted average in ice dyn.F90 (called from the sea ice part (ICE), see Chapter 9). 2. Update velocity for the first layer (Equation (8.2)) in clinic.F90. • Use of external data (default) 1. Input of external data in force.F90. 2. Time interpolation of external data in mkflux.F90. 3. Calaculation of icewater stress and total surface stress based on areaweighted average in ice dyn.F90 (called from the sea ice part (ICE), see Chapter 9). 4. Update velocity for the first layer (Equation (8.2)) in clinic.F90. • Coupled model 1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm scup get a2o (cgcm scup.F90). – 114
–
8.7. Work flow in MRI.COM 2. Getting fluxes for this step from get flux a2o (get fluxes.F90). 3. Calaculation of icewater stress and total surface stress based on areaweighted average in ice dyn.F90 (called from the sea ice part (ICE), see Chapter 9). 4. Update velocity for the first layer (Equation (8.2)) in clinic.F90.
8.7.2
Temperature (heat) flux
• Heat flux type (HFLUX) 1. Input of external data in force.F90 OBS OBS (a) read(unit = ihflx) QOBS SH , QLO , To
(b) read(unit = ibulk) Ta , qa (Td , if TDEW), U10 , Ps 2. Calculation of FT in mkflux.F90 (a) Time interpolation of external data. (b) Calculation of QSH (Equation (8.11)). (c) Calculation of QLO （Equation (8.27)). (d) Calculation of QLA and QSN (Equation (8.28)(8.29)) in bulk.F90 (called from mkflux.F90). (e) Preparation for QSH and (QLO + QLA + QSN ). (f) Heat flux between sea ice and sea water (Qice ) in ice main cat.F90 (called from the sea ice part (ICE), see Chapter 9). T (Equation (8.44) or (8.55)). (g) Calculation of FWF
3. Update first layer temperature using F T (Equation (8.4)) in tracer.F90. – Update temperature at the depth deeper than the first layer using shortwave absorption (Equation (8.11)). • Restoring sea surface temperature type (default) 1. Calculation of F T (the last term on the r.h.s. of Equation (8.5)) in mkflux.F90. 2. Update the first layer temperature using F T (Equation (8.4)) in tracer.F90. • Coupled model 1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm scup get a2o (cgcm scup.F90). 2. Calculation of FT in mkflux.F90 (a) Getting fluxes for this step from get flux a2o (get fluxes.F90). (b) Preparation for QSH and (QLO + QLA + QSN ). (c) Heat flux between sea ice and sea water (Qice ) in ice main cat.F90 (called from the sea ice part (ICE), see Chapter 9). T (Equation (8.44) or (8.55)). (d) Calculation of FWF
3. Update first layer temperature using F T (Equation (8.4)) in tracer.F90. – Update temperature at the depth deeper than the first layer using shortwave absorption (Equation (8.11)). –
115
–
Chapter 8
Sea surface fluxes
8.7.3 Salinity and fresh water flux • Freshwater flux type WFLUX 1. Input of external data in force.F90: (a) read(ipcpr) P (b) read(irnof) R, when RUNOFF 2. Calculation of F W and F S in mkflux.F90: (a) Time interpolation of external data. (b) Calculation of E (Equation (8.40)) in bulk.F90 (called from mkflux.F90). (c) Freshwater and salinity fluxes between sea ice and sea water I and F S (Equation (8.42)) in ice main cat.F90 (called from the sea ice part (ICE), see Chapter 9). (d) Restoring term for sea surface salinity and conversion to F W (Equation (8.41)). W for the option WADJ (Equation (8.43)). (e) Calculation of FADJ
(f) Set F W = 0 and reevaluate F S for option SFLUXW (Equation (8.10)). 3. Update salinity of the first layer: W ) in surface integ.F90. (a) Update the first layer volume using F W (or FADJ
(b) Update the first layer salinity using F S (Equation (8.8)) in tracer.F90. • Restoring sea surface salinity (default) 1. Calculation of F S (the last term on the r.h.s. of Equation (8.9)) in mkflux.F90 2. Update the first layer salinity (Equation (8.8)) in tracer.F90 • Coupled model 1. Getting fluxes from the atmospheric model at the beginning of a coupling cycle in cgcm scup get a2o (cgcm scup.F90). 2. Calculation of F W and F S in mkflux.F90: (a) Getting fluxes for this step from get flux a2o (get fluxes.F90). (b) Freshwater and salinity fluxes between sea ice and sea water I and F S (Equation (8.42)) in ice main cat.F90 (called from the sea ice part (ICE), see Chapter 9). (c) Calculation of restoring term for sea surface salinity and conversion to F W (Equation (8.41)). W for the option WADJ (Equation (8.43)). (d) Calculation of FADJ
(e) Set F W = 0 and reevaluate F S for the option SFLUXW (Equation (8.10)). 3. Update salinity of the first layer: W ) in surface integ.F90. (a) Update the first layer volume using F W (or FADJ
(b) Update the first layer salinity using F S (Equation (8.8)) in tracer.F90.
– 116
–
8.8. Remarks
8.8
Remarks
Although recent satellite observations enable us to obtain sea surface fluxes with high resolution in space and time, even higher accuracy is necessary for practical use. For example, a bias of several W·m−2 in heat flux greatly affects the thickness of sea ice, meaning that accuracy on the order of several W·m−2 is necessary to clarify climatic change (WGASF 2000). Efforts in enhancing observations and evaluating sea surface fluxes based on various methods have been expanded globally. In the future, a highresolution model and a new scheme for advection and diffusion may be developed to improve the simulation capability. It is noted, however, that increased observation frequency does not necessarily guarantee improved accuracy of the fluxes (for example, it is unlikely that the accuracy of a bulk coefficient would be improved). Hence, it should be kept in mind that sea surface fluxes presently involve large uncertainties.
8.9
Appendix
8.9.1
Unit of constants Symbol and Name
ρo ρa Cp C pa αo em σ Rg
8.9.2
sea water density air density sea water specific heat air specific heat sea surface albedo sea water emissivity StefanBoltzmann constant gas constant for dry air
Unit
Numeral
Model(CGS)
1.0 1.205 × 10−3
SI
×g·cm−3
×103 Kg·m−3
×g·cm−3
×103 Kg·m−3
3.99 1.00467
107 erg·g−1 ·K−1 107 erg·g−1 ·K−1
103 J·Kg−1 ·K−1 103 J·Kg−1 ·K−1
alb 0.97 5.67
10−5 erg·s−1 ·cm−2 ·K−4
10−8 J·s−1 ·m−2 ·K−4
2.871
106 erg·g−1 ·K−1
102 J·Kg−1 ·K−1
Unit of variables Unit
Symbol and Name Q∗∗
Model(CGS) ×g·s−3
Sea surface heat flux
SI ×10−3 W·m−2
Freshwater flux
cm·s−1
10−3 J·s−1 ·m−2 10−2 m·s−1
τ∗ Δz 1
Momentum flux (wind stress) the first layer thickness
dyn · cm−2 cm
10−1 N · m−2 10−2 m
T∗
temperature
◦C
◦C
Ps U10
sea surface pressure scalar wind speed
hPa (not in CGS) cm·s−1
hPa 10−2 m·s−1
ρ∗ q∗ e∗ L
density specific humidity water vapor pressure evaporation latent heat
g·cm−3 g·g−1 hPa (not in CGS)
103 Kg·m−3 Kg·Kg−1 hPa
erg·g−1
J·Kg−1
FW
2
–
117
–
Chapter 8
Sea surface fluxes
References Baker, H. W., and Z. Li, 1995: Improved simulation of clearsky shortwave radiative transfer in the CCCGCM., J. Climate, 8, 22132223. Ishizaki, H., and G. Yamanaka, 2010: Impact of explicit sun altitude in solar radiation on an ocean model simulation, Ocean Modell., xxx, in press. Jerlov, N. G., 1976: Marine Optics, 231pp., Elsevier. Kantha, L. H., and C. A. Clayson, 2000: Small Scale Processes in Geophysical Fluid Flows., 888pp., Academic Press. Kara, A. B., P. A. Rochford, and H. E. Hurlburt, 2002: Airsea flux estimates and the 19971998 ENSO event., BoundaryLayer Meteorol., 103, 439458. Kondo, J., 1975: Airsea bulk transfer coefficients in diabatic conditions., BoundaryLayer Meteorol., 9, 91112. Large, W. G., and S. G., Yeager, 2008: The global climatology of an interannually varying airsea flux data set., Climate Dynamics, DOI 10.1007/s0038200804413, 24pp. Large, W. G., and S. G., Yeager, 2004: Diurnal to decadal global forcing for ocean and seaice models: the data sets and flux climatologies., NCAR Technical Note: NCAR/TN460+STR. CGD Division of the National Center for Atmospheric Research. Mellor, G. L., and L. Kantha, 1989: An iceocean coupled model, J. Geophys. Res., 94, 1093710954. Morel, A., and D. Antoine, 1994: Heating rate within the upper ocean in relation to its biooptical state, J. Phys. Oceanogr., 24, 16521665. Paulson, C. A., and J. J. Simpson, 1977: Irradiance measurements in the upper ocean, J. Phys. Oceanogr., 7, 952956. WGASF, 2000: Intercomparison and validation of oceanatmosphere energy flux fields., Final report of the Joint WCRP/SCOR Working Group on AirSea Fluxes (WGASF), pp.306.. Yukimoto, S., and coauthors, 2010: Meteorological Research InstituteEarth System Model v1 (MRIESM1) Model Description , Technical Reports of the Meteorological Research Institute, No.64, in press.
– 118
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Chapter 9
Sea ice
This chapter describes the sea ice part. The sea ice part of MRI.COM treats formation, accretion, melting, and transfer of sea ice and snow. Heat, water, salt, and momentum fluxes are exchanged with the ocean. Sea ice is categorized by its thickness, but it has a single layer. Snow does not have heat capacity (socalled zerolayer). Thus, it might be regarded as an intermediate complexity ice model. This chapter is organized as follows. Section 9.1 outlines the model. The following sections describe details of the solution procedure. According to the order of solving the equations, we deal with thermodynamics in Section 9.2, remapping among thickness categories in Section 9.3, dynamics in Section 9.4, advection in Section 9.5, and ridging in Section 9.6. Discretization issues are described in Section 9.7, and finally some technical issues are presented in Section 9.8.
9.1
Outline
The sea ice part of an ocean model gives surface boundary conditions. Heat, fresh water, salt, and momentum are exchanged at their interfaces. The sea ice part solves fractional area, heat content, thickness, and their transport of ice categorized by its thickness and dynamics of the gridcell averaged ice pack. The ice model of MRI.COM is based on the iceocean coupled model of Mellor and Kantha (1989). For processes that are not explicitly discussed nor included there, such as categorizing by thickness, ridging, and rheology, we adopt those of the Los Alamos sea ice model (CICE; Hunke and Lipscomb, 2006). The fundamental property that defines the state of sea ice is the fractional area as a function of location (x, y) and thickness (hI ). The equation for this distribution function (g(x, y, hI )) is
∂g ∂ 1 ∂ (ghψ uI ) ∂ (ghμ vI ) + χ, ( f g) − + = ∂t ∂ hI h μ hψ ∂μ ∂ψ
(9.1)
where f is the thermodynamic growth rate of thickness, (uI , vI ) is the velocity vector of ice pack, and χ is the rate of change of distribution function caused by ridging. The growth rate of ice thickness is computed by solving thermodynamic processes. Using this rate ( f ), thickness categories are remapped according to the first term on the r.h.s. To compute the velocity of the ice pack (uI , vI ), we have to solve the momentum equation. On transporting the ice distribution (second and third terms on the r.h.s.), other conservative properties such as volume and energy are also transported. Using the transported ice distribution function, the ridging process (χ ; fourth term on the r.h.s.) is solved. The formulation and solving procedure of each process are presented in later sections. We discretize the thickness in several categories. If an ice pack is divided into n categories separated at Hn with H0 = 0[m], the fractional area of each category (an ) is defined as follows: an =
Hn Hn−1
gdh.
(9.2)
Other major variables, ice and snow thickness, surface temperature, bottom temperature and salinity, and internal energy of ice, are defined for each category. Velocity is defined for an ice pack, the total ice mass in a grid cell. In – 119
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Chapter 9
Sea ice
the vertical direction, both ice and snow have one layer with the heat capacity for sea ice but without heat capacity for snow. The heat capacity for sea ice is due to brine and is represented by the temperature at the center of the ice. It is assumed that sea ice has the same energy (temperature) throughout the whole layer. Figure 9.1 and Table 9.1 summarize symbols used in this chapter and their variable names in the source code of MRI.COM. Table 9.1. Physical quantities used in the sea ice part (cf. Figure 9.1) and their variable names in the source code. Meaning Variable name Variable name of average hI hs
ice thickness snow thickness
hicen hsnwn
hiceo hsnwo
A AhI
area fraction (compactness) average ice thickness (volume)
aicen hin
a0iceo hi
Ahs T3 T1
average snow thickness (volume) skin temperature of upper surface temperature of ice
hsn tsfcin t1icen
hsnw tsfci 
T0 T0L S0 S0L
skin temperature of lower surface skin temperature of sea surface at open leads skin salinity of lower surface
t0icen t0icen s0n
t0iceo (under the sea ice) t0icel (open leads) s0
skin salinity of sea surface at open leads heat flux on the ice side of the ice bottom
s0n fheatn
s0l 
QIO QAO FT I FT L FSI , FSL
heat flux on the air side at open leads heat flux on the ocean side of the ice bottom heat flux on the ocean side at open leads
fheat ftio ftao

salinity flux driving first layer of the ocean model

sfluxi = FSI + FSL

WAI
fresh water flux driving first layer of the ocean model fresh water flux due to snow fall
wfluxi = WIO +WAO +WRO +WFR snowfall
WAI
at the upper surface of ice fresh water flux due to sublimation
W
sublim

WIO
at the upper surface of ice fresh water flux due to freezing and melting at the bottom of ice
wio

WAO WRO
fresh water flux due to freezing at open leads fresh water flux due to melting
wao 
wrss, wrsi (snow, ice)
WFR uI
at the upper surface of ice fresh water flux due to formation of frazil ice zonal component of ice pack velocity

wrso uice
vI
meridional component of ice pack velocity

vice
– 120
–
9.2. Thermodynamic processes
short wave
QAI
WAI T3
snow
hS
QS T2
Ss = 0.0
S0L FSL
FTL
QI2
ice
WAO
QAO
T0L
Si = 4.0
T1
hI QIO
S0
T0 FTI
FSI
T(1), S(1)
WFR
WRO WIO
Figure 9.1. Meaning of symbols and their locations. The left side is related to heat flux, and the right side is related to fresh water flux. Sea ice is separated into the part that originated from sea water (thickness: hI and salinity: SI = 4.0 [psu]) and the part that originated from snow (thickness: hs ). The former is further divided into upper and lower halves. Sea ice thus has three vertical layers. The temperatures at the layer boundaries are T0 , T1 , T2 , and T3 from the bottom. Heat fluxes within each layer are QIO , QI2 , and QS from the bottom. The heat flux at the airice interface is QAI and that at the iceocean interface is FT I . Sea ice is in fact categorized by thickness, and each symbol should have a suffix (n) of the category number. At open leads or open water, symbols have the suffix L. For the definitions of fresh water fluxes, see Table 9.1.
9.2
Thermodynamic processes
In considering thermodynamics, thermal energy of sea ice should be defined. The basis of energy (i.e., zero energy) is defined here as that of sea water at 0◦ C. The thermal energy (enthalpy; E(T, r)) of sea ice that has temperature T (< 0)◦ C and brine (salt water) fraction r is the negative of the energy needed to raise the temperature to 0◦ C and melt all of it: E(T, r) = r(C po T ) + (1 − r)(−LF +C pi T ), (9.3) where C po and C pi are the specific heats of sea water and sea ice, and LF is the latent heat of melting/freezing. Thus defined, the energy of sea ice is negative definite. The brine represents the fact that the salt with salinity (SI ) in sea ice exists in a liquid state. If the ice temperature is T1 , the brine is assumed to have the same temperature, and its salinity is S = T1 /m, where m defines the freezing temperature as a function of salinity. Hence, the brine fraction of sea ice is r = SI /S = mSI /T1 . As in Mellor and Kantha (1989), the specific heat of snow is not considered. Although the surface fluxes are positive downward (positive toward the ocean) in the ocean model, the sea ice part is coded such that fluxes are positive upward. In this section, we assume that the fluxes are positive upward.
–
121
–
Chapter 9
Sea ice
9.2.1 Formation of new sea ice a. From a sea water below the freezing point temperature Sea ice is formed when sea surface temperature is below the freezing point. If the temperature of the first layer of the ocean model is below the freezing point as a function of salinity, the temperature is set to the freezing point (Tfreeze ), and the heat needed to raise the temperature is regarded as the release of latent heat and is used to form new ice. The thickness of the new ice (hI ) is computed by assuming that the total thermal energy of the first layer of the ocean model (whose layer thickness is Δz 1 ) is conserved before and after the sea ice formation: 2
ρo Δz1 [C po T ] = ρI hI {r(C po Tfreeze ) + (1 − r)(−LF +C pi Tfreeze )} + ρo (Δz − ρI hI /ρo )(Cpo Tfreeze ),
(9.4)
where r = mSI /Tfreeze . Using the above equation, we compute the thickness of the new ice: hI =
C po (Tfreeze − T ) ρo Δz . ρI (1 − r) LF + Tfreeze (C po −C pi )
(9.5)
With the ice thickness known, the fractional area is determined by the following procedure: = 10[cm] and the fractional area is computed • For a grid cell without sea ice, the first thickness is set to hnew I new = h and A = 1. . If A > 1 (i.e., h > 10[cm]), h as A = hI /hnew I I I I • For a grid cell where sea ice already exists, hI is added to each category and open water. Note that this operation practically eliminates supercooling in the ocean interior. Hence, the formation of frazil ice is not considered in this model. The medium that contacts the sea ice differs at the upper (air) and lower (sea water) surfaces. Even for an ocean grid cell with sea ice, not all the surface is covered by sea ice, i.e., there may be open water. We treat processes at each interface separately. Heat flux and thermal energy are computed by solving the balance equation at each interface and each layer in Figure 9.1.
b. Input of iceberg Input of iceberg (Ficeberg ) is given as a water mass flux per unit area (kg m−2 s−1 ). The thickness of the new ice (hI ) is give by (9.6) hI = Ficeberg Δt/ρI , where Δt is the unit time step. With the ice thickness known, the fractional area is determined by the same procedure used for ice created from supercooled water as explained above.
9.2.2
Airice interface
a. Heat flux at the upper surface of ice (QAI ) The surface heat flux (QAI ) is expressed as follows: QAI = QSI + QLI − (1 − αI )SW − LW + εI σ (T3 + 273.16)4 . We will now examine each component. –
122
–
(9.7)
9.2. Thermodynamic processes i) Short wave The downward short wave radiation is represented by SW. The albedo of sea ice is αI , which is 0.82 for cold (T3 < −1◦ C) snow, 0.73 for melting snow, and 0.64 for bare ice (while melting). One might use the more sophisticated albedo scheme of the Los Alamos sea ice model (CICE; Hunke and Lipscomb, 2006) by choosing model option CALALBSI. We briefly describe how the incoming short wave radiation is treated by CICE. The downward short wave radiation is treated at each interface as follows: • Among the net absorbed shortwave flux (= (1 − αI )SW), some fraction (i0 ) penetrates into ice and the rest is absorbed at the surface and used to warm the upper interface. See Table 9.2 for the specific value of i0 . • The part penetrating into the ice (= (1 − αI )i0 SW ) is attenuated according to Beer’s Law with the bulk extinction coefficient κi = 1.4m−1 . The attenuated part is used to warm the ice interior. • The rest is absorbed into the ocean. The albedos and penetration coefficients of CICE are listed on Table 9.2. The property fsnow is the snow fraction of the upper surface of the ice, which is expressed as follows: fsnow =
hs , hs + hsnowpatch
(9.8)
where hs is the snow thickness and hsnowpatch = 0.02m. If the ice thickness (hI ) is less than href = 0.5[m], the albedo of thin ice is computed as
αthinice = αo + β (αcoldice − αo ),
(9.9)
arctan(ar hI ) , ar = 4.0, arctan(ar href )
(9.10)
where
β=
and αo is the albedo of the ocean. If the surface temperature T3 becomes −1 < T3 < 0[◦ C], the albedo of melting ice and snow is computed as
αmeltice αmeltsnow
= αthinice − γi (T3 + 1.0),
(9.11)
= αcoldsnow − γs (T3 + 1.0),
(9.12)
where the condition αmeltice > αo is imposed. Using the snow fraction on the surface of the ice fsnow , the total albedo is computed as
αi = αmeltice (1 − fsnow ) + αmeltsnow fsnow .
(9.13)
The albedos for visible and near infrared wave lengths are computed separately. If the short wave flux is given as the sum of all four components (direct and diffuse for visible and near infrared wave lengths), a constant ratio (visible) : (near infrared) = 0.575 : 0.425 is assumed, and the total albedo is computed as the weighted average. ii) Long wave The downward long wave radiation from the atmosphere is represented by LW in (9.7). The black body radiation from the sea surface is εI σ (T3 + 273.16)4 , where εI is emissivity, and σ is the StefanBoltzmann constant. Hereinafter, we use LW I = LW − εI σ (T3 + 273.16)4 as the net longwave radiation. iii) Sensible heat flux –
123
–
Chapter 9
Sea ice
Table 9.2. Albedo and surface transparency of the albedo scheme of CICE.
albedo for cold snow αcoldsnow (T3 < −1◦ C) albedo for cold ice αcoldice (T3 < −1◦ C, hi > 0.5m) reduction rate of albedo for melting ice γi (−1◦ C < T3 < 0, hi < 0.5m) reduction rate of albedo for melting snow γs (−1◦ C < T3 < 0, hi < 0.5m) fraction of transparent short wave flux through the ice surface (i0 )
near infrared
visible
(> 700nm)
(< 700nm)
0.70 0.36
0.98 0.78
−0.075/◦ C −0.15/◦ C
−0.075/◦ C −0.10/◦ C
0.0
0.7 × (1.0 − fsnow )
The sensible heat flux (QSI in (9.7)) is computed using a bulk formula: QSI = ρaC paCHAI U10 (T3 − TA ),
(9.14)
where ρa is the density of air, C pa is the specific heat of air, CHAI is the bulk transfer coefficient at the airice interface (Section 8.6), U10 is the scalar wind speed at 10 [m], TA is the surface air temperature, and T3 is the ice surface temperature (Figure 9.1). iv) Latent heat flux The latent heat flux (QLI in (9.7)) is computed using a bulk formula: QLI = ρa LsCHAI U10 (qi − qA ),
(9.15)
where Ls is the latent heat of sublimation, qi is the saturation humidity at the ice surface temperature T3 , and qA is the specific humidity of air. Section 9.9.1 details a computing method for qi . Fresh water loss due to sublimation is computed as WAI = ρaCHAI U10 (qi − qA )/ρo .
(9.16)
b. Heat flux in the snow (QS ) If we neglect the heat capacity of snow, the heat flux is constant within the snow layer and is computed as QS =
ks (T2 − T3 ), hs
(9.17)
where hs is the thickness of the snow layer and ks is the thermal conductivity of snow. c. Heat flux in the ice interior (QI2 , QIO ) In the upper half of the ice layer, the heat flux is computed as follows: QI2 =
kI (T1 − T2 ), hI /2
(9.18)
where kI is the thermal conductivity of sea ice. If we neglect heat capacity of snow, QS = QI2 . Using this relation, the interface temperature is computed as T2 =
ks hs T3 + ks hs +
– 124
k1 hI /2 T1 . k1 hI /2
–
(9.19)
9.2. Thermodynamic processes In the lower half of the ice layer, the heat flux is computed as follows: QIO =
kI (T0 − T1 ). hI /2
(9.20)
d. Melting at the upper surface (WRO ) The melting rate at the upper surface is computed as follows: First, the surface temperature (T3 ) is computed by equating the fluxes at the upper surface (QAI = QS ). If T3 is lower than the freezing temperature (0.0◦ C for snow and mSI [◦ C] for sea ice), melting does not occur. If T3 is higher than the freezing temperature, melting occurs. In this case, T3 is set to the freezing temperature, and the interior heat flux just below the ice surface QS is recalculated. The imbalance between QAI and QS is used to melt snow or ice. A melting rate is computed as WRO = (QAI − QS )/(ρo L3 )
(9.21)
L3 ≡ [E(T3 , 1) − E(T1 , r1 )]
(9.22)
where L3 = LF for snow melt and for ice melt. The brine fraction for sea ice is r1 = mSI /T1 . The temperatures of the melted water are mSI [◦ C] for sea ice and 0 [◦ C] for snow.
e. Notes • All precipitation on sea ice is assumed to be snow. • Melted water is assumed to run off to the ocean. • If all preexisting ice is melted, the residual surface heat flux QAI is added to the iceocean flux QIO .
f. Procedure The solution procedures are basically the same with or without snow. To be exact, the interface fluxes should be computed iteratively by adjusting surface temperature T3 until a balance is achieved. We adopt the semiimplicit method described below. A situation without snow (hs = 0, T3 = T2 ) is considered. First, the surface temperature (T3 ) is computed by assuming that the fluxes on both sides are the same. Inserting (9.7) and (9.18) into QAI = QI2 with T3 → T3 + δ T3 , kI {T1 − (T3 + δ T3 )} hI /2
= QLI (T3 + δ T3 ) + QSI (T3 + δ T3 ) −(1 − αI )(1 − i0 )SW − LW + εI σ {(T3 + δ T3 ) + 273.16}4 .
(9.23)
By expanding the specific heat, latent heat, and black body radiation in a Taylor series, we have, kI ∂ QLI ∂ QSI δ T3 + δ T3 − (1 − αI )(1 − i0 )SW (T1 − (T3 + δ T3 )) = QLI + QSI + hI /2 ∂ T3 ∂ T3 −LWI + 4εI σ (T3 + 273.16)3 δ T3 .
(9.24)
Using this, we compute δ T3 and add it to T3 to obtain a new temperature:
δ T3 =
I (T1 − T3 ) −QSI − QLI + (1 − αI )(1 − i0 )SW + LWI + hIk/2
∂ QLI ∂ T3
I + ∂∂QTSI + 4εI σ (T3 + 273.16)3 + hIk/2 3
–
125
–
,
(9.25)
Chapter 9
Sea ice
and T3new = T3old + δ T3 ,
(9.26)
where
∂ QSI ∂ T3 ∂ QLI ∂ T3
= ρaC paCHAI U10 , = ρa LsCHAI U10
∂ qi . ∂ T3
(9.27) (9.28)
Note that the dependency of Ls on temperature (Section 9.9.1) is not considered in the partial differentiation with respect to temperature. The specific form for the partial derivative of specific humidity (∂ qi /∂ T3 ) is presented in Section 9.9.1. If the new surface temperature (T3new ) is below the freezing point, melting does not occur. If not, T3new is set to the freezing temperature (= mSI ), and the heat flux in the ice interior is reevaluated. The amount of melting (WRO ) is obtained using the imbalance: QAI QI2 L3 ΔhI
= −QSI − QLI + (1 − αI )(1 − i0 )SW + LWI , kI = (T1 − T3new ), hI /2 = [E(T3new , 1) − E(T1 , r1 )], (QAI − QI2 )Δt = = WRO Δt. ρi L 3
(9.29)
If there is snow, the above procedure is performed for the snow surface (T3new = 0[◦ C], L3 = LF ). If all the snow melts away, the residual heat (Eres ) is used to melt the ice: Eres
= (QI2 − QAI )Δt − hs ρs LF ,
L3
= [E(mSI , 1) − E(T1 , r1 )],
ΔhI
9.2.3
=
−Eres /ρi L3 = WRO Δt.
(9.30)
Heat balance in the ice interior
The thermal energy of the ice is affected by vertical heat fluxes and horizontal heat transport due to advection. The equation for the thermal energy (enthalpy) is written as follows:
∂ ∂ ρI hI E(T1 , r1 ) = QIO − QI2 + [SWsurface − SWbottom ], E(T1 , r1 ) + uI i ∂t ∂ xi
(9.31)
where SWsurface
= (1 − αI )i0 SW,
(9.32)
SWbottom
= (1 − αI )i0 SW × exp(−κi hI ).
(9.33)
The above equation can be solved explicitly without causing serious problems when the time step is not too long.
9.2.4 Iceocean interface Melting and freezing at the iceocean interface is computed using heat fluxes at the interface as depicted in Figure 9.1. The icecovered area and the open water are treated separately. The solution method slightly differs from that of Mellor and Kantha (1989). – 126
–
9.2. Thermodynamic processes In open water, the heat flux on the air side of the airocean interface (QAO ) is computed as in Chapter 8, 4 Qall AO = QSO (T ) + QLO (T ) − (1 − αo )SW − LW + εo σ (T + 273.16) .
(9.34)
The temperature at the first level of the ocean model (T ) is used. All the heat and fresh water fluxes are evaluated using the temperature and salinity at the first level of the ocean model. By doing so, the equation to compute melting and freezing rates becomes linear. Here, short wave radiation is assumed to pass through the skin layer without absorption and is excluded from the evaluation of the freezing rate in open water: QAO = QSO + QLO − LW + εo σ (T + 273.16)4 .
(9.35)
This operation causes the short wave radiation to be absorbed in the ocean interior. In reality, the heat stored in the skin layer in open water is used to melt ice laterally (edge melting). To include this effect, some fraction (Ψ) of the bottom melting (WIO ) may be used for the edge melting. The details are described in the last part of this section. In the icecovered area, the heat flux on the ice side of the iceocean interface (QIO ) is computed according to (9.20). Melting and freezing occur due to the imbalance between fluxes above and below the interface: FTI
=
QIO −WIO ρo Lo ,
(9.36)
FTL
= QAO −WAO ρo Lo ,
(9.37)
where Lo ≡ [E(T0 , 1) − E(T1 , r1 )](= LF ).
(9.38)
The heat flux that drives the first level of the ocean model is given by FT = (AQIO + (1 − A)QAO ) −WO ρo Lo ,
(9.39)
WO ≡ AWIO + (1 − A)WAO .
(9.40)
where
The flux balance for salt is written as follows: FSI
= WIO (SI − S),
(9.41)
FSL
= WAO (SI − S).
(9.42)
Unlike Mellor and Kantha (1989), the salinity at the first level of the ocean model (S) is used instead of the salinity at the skin layer (S0 ). By doing so, the equation to solve for S0 becomes linear. It could also be said that it is natural to use the first level salinity itself in evaluating the salt flux that drives the first level of the ocean model. Note that only fresh water fluxes that are relevant to freezing and melting at the iceocean interface are included in the above equations. The restoration to climatological salinity and fresh water fluxes caused by surface melting, precipitation, and evaporation are excluded in the above balance. The salinity flux caused by melting and freezing at the ocean surface that drives the first level of the ocean model is given by FS = (AWIO + (1 − A)WAO )(SI − S).
(9.43)
Note that the effect of melting at the upper surface of the ice could also be included in the driving salt flux for the ocean model: FS = −A{WROice (SI − S) −WROsnow S}. – 127
–
(9.44)
Chapter 9
Sea ice
However, fluxes on the oceanic side of the interface (FTI , FTL , FSI , and FSL ) could be obtained as the boundary conditions (z → 0) for the molecular boundary layer: FTI /(ρoC po ) = −CTz (T0I − T ),
(9.45)
FTL /(ρoC po ) = −CTz (T0L − T ),
(9.46)
and FSI
= −CSz (S0I − S),
(9.47)
FSL
= −CSz (S0L − S),
(9.48)
uτ , −1 (Prt k ln(−z/z0 ) + BT )
(9.49)
where CTz = −1/2
2 + τ 2 )1/4 ρ is the friction velocity, k = 0.4 is von Karman’s constant, z0 is the roughness parameter, uτ ≡ (τIO o IOy x (τIOx , τIOy ) is the stress vector at the oceanice interface, and z u 1/2 0 τ BT = b Pr2/3 , (9.50) ν
with Pr ≡ ν /αt = 12.9. The specific values for other parameters are given in Section 9.9.2. Parameters related to salinity are uτ CSz = , −1 (Prt k ln(−z/z0 ) + BS ) and z u 1/2 0 τ Sc2/3 , BS = b ν where Sc ≡ ν /αb = 2432. Roughness parameter z0 is computed as follows: lnz0 = Alnz0I + (1 − A)lnz0L , where z0I = 0.05
hI , hIlim = 3.0 [m], hIlim
(9.51)
(9.52)
(9.53)
(9.54)
and
ρo u2τ . (9.55) ρa g The roughness parameter below ice (9.54) is also used for open water in MRI.COM. The above equations are solved simultaneously to obtain melting and freezing temperatures at the upper surface of the ocean under the following constraint: z0L = 0.016
WO = 0 T0 = mS0
A=0 A > 0,
(9.56)
where m defines the freezing line as a function of salinity. Solving procedure We first solve for S0I and S0L using (9.36), (9.41), (9.45), (9.47), (9.37), (9.42), (9.46), and (9.48): S0I
=
S0L
=
CSz S + (ρoC poCTz T − QIO )(SI − S)/ρo Lo , CSz + ρ0C poCTz m(SI − S)/ρo Lo CSz S + (ρoC poCTz T − QAO )(SI − S)/ρo Lo . CSz + ρoC poCTz m(SI − S)/ρo Lo –
128
–
(9.57) (9.58)
9.2. Thermodynamic processes Using S0I and S0L , T0I and T0L are computed from (9.56), FTI and FTL are computed from (9.45) and (9.46), FSI and FSL are computed from (9.47) and (9.48), and finally WIO and WAO are computed from (9.36) and (9.37). Since WAO is positive (only freezing is allowed in open water), WAO = 0 when WAO < 0. In this case, S0L = S from (9.42) and (9.48). Furthermore, FTL = QAO in (9.37) and T0L = T assuming that there is no ice effect in open water if freezing does not occur. Finally, the surface boundary conditions for heat and salt fluxes are computed from (9.39) and (9.43). z → 0:
∂T = FT , ∂z ∂S −κV = FS , ∂z
−κV
(9.59) (9.60)
where κV is the vertical diffusivity for the surface level of the ocean model. If −WIO Δt > hI , all ice would melt away. In this case, the amount of heat needed to melt all the ice is consumed, and the residual is returned to the ocean. That is, the heat flux necessary to melt all the ice is removed from the ocean. Specifically, WIO is computed by hI = −
ρo WIO Δt ρI
(9.61)
and FTI is obtained as FTI = QIO −WIO ρo Lo .
(9.62)
When −WIO Δt < hI , reduction of the fractional area is allowed by edge melting. The procedure is as follows: The fraction used for edge melting is defined as Ψ, and −(1 − Ψ)WIO Δt is used to reduce the thickness holding the fractional area. The new thickness is hI and ΨWIO Δt is used to reduce compactness holding the thickness:
A = A(1 + ΨWIO Δt/hI ).
(9.63)
It does not seem that there is a widely accepted parameterization scheme for edge melting. According to Steele (1992), bottom and top melting are dominant processes, and thus Ψ ∼ 0.1 is usually used in MRI.COM.
9.2.5
Archimedes’ Principle
From Archimedes’ principle, the part of snow that is below freeboard absorbs sea water to become sea ice. The following equality will be achieved at equilibrium:
ρI hI + ρs hs = ρo hI .
(9.64)
At the end of the time step, the above equality is checked, and if
ρI hI + ρs hs > hI , ρ0 then the new ice thickness is set as:
(9.65)
= hnew I
ρI hI + ρs hs . ρo
(9.66)
δ hs = −
− hI )ρI (hnew I , ρs
(9.67)
Since the change in snow thickness is
the new snow thickness is obtained as follows, = hs + δ hs . hnew s – 129
–
(9.68)
Chapter 9
Sea ice
Since the salinity of sea ice is SI = 4.0 [psu], salt is removed from the first layer of the ocean model: Snew = Sold − SI
9.3
δ hs ρs . Δz1 ρo
(9.69)
Remapping in thickness space
After the thermodynamic processes are solved, the resultant ice thickness in some thickness categories might not be within the specified bound. Following the method adopted by CICE, we assume that there is a thickness distribution function in each category and use it to redistribute the new thickness distribution into original categories. This procedure corresponds to the first term on the r.h.s. of (9.1):
∂g ∂ ( f g). = ∂t ∂ hI
(9.70)
In practice, a thickness category is regarded as a Lagrange particle, and the category boundaries are displaced as a result of thermodynamics. A linear thickness distribution function is assumed within each displaced category, and ice is remapped into the original categories using these functions. First, boundaries of thickness categories are displaced. If the ice thickness in category n (hn ) changes from hm n to (m is the time step index), the growth rate ( fn ) at thickness hn is represented as fn = (hm+1 − hm hm+1 n n n )/Δt. Using this, the growth rate at the upper category boundary Hn is obtained by linear interpolation: Fn = fn +
fn+1 − fn (Hn − hn ). hn+1 − hn
(9.71)
If the fractional area is zero in either category n or n + 1, Fn is set to the growth rate at the nonzero category. When the fractional area is zero on both categories, Fn = 0. The new category boundary after thermodynamics is obtained as Hn∗ = Hn + Fn Δt.
(9.72)
∗ , H ∗ ] is determined. For simplicNext, the thickness distribution function g within the displaced category [Hn−1 n ∗ and HR = Hn∗ . Function g should satisfy the following equality for fractional area and ity, we write HL = Hn−1 volume:
HR
HL HR
HL
gdh
hgdh
= an ,
(9.73)
=
(9.74)
vn .
We adopt a linear function of thickness for g. The thickness space is transformed to η = h−HL , and the thickness distribution function is written as g = g1 η + g0 . These are substituted into (9.73) and (9.74) to yield
ηR2 + g0 ηR 2 η3 η2 g 1 R + g0 R 3 2 g1
= an ,
(9.75)
= an ηn ,
(9.76)
where ηR = HR − HL and ηn = hn − HL . These are algebraically solved for g0 and g1 as g0
=
g1
=
6an 2ηR η − n , 3 ηR2 ηR 12an . η − n 2 ηR3 – 130
–
(9.77) (9.78)
9.3. Remapping in thickness space The values of the thickness distribution function at category boundaries are given as follows 6an 2ηR − ηn , 2 3 ηR ηR 6an g(ηR ) = 2 ηn − . 3 ηR
g(0) =
(9.79) (9.80)
Equation (9.79) gives g(0) < 0 when the thickness is in the right third of the thickness range or ηn > 2ηR /3. Equation (9.80) gives g(ηR ) < 0 when the thickness is in the left third of the thickness range or ηn < ηR /3. Since a negative g is physically impossible, we redefine the range of the thickness distribution function. Specifically, when the thickness is within the left third of the thickness range, a new right boundary is set at HC = 3hn − HL and g is set to zero for [HC , HR ]. In this case, ηR = HC − HL in (9.77) and (9.78). When the thickness is within the right third of the thickness range, a new left boundary is set at HC = 3hn − 2HR and g is set to zero for [HL , HC ]. In this case, ηR = HR − HC and ηn = hn − HC in (9.77) and (9.78). Finally, we remap ice into the original categories using the above thickness distribution function. If Hn∗ > Hn , ice is transferred from category n to n + 1. The transferred area Δan and volume Δvn are Δan = and Δvn =
H∗ n
gdh
(9.81)
hgdh.
(9.82)
Hn
H∗ n Hn
If Hn∗ < Hn , ice is transferred from category n + 1 to n. The transferred area Δan and volume Δvn are Δan = and Δvn =
Hn
gdh
(9.83)
hgdh.
(9.84)
Hn∗
Hn Hn∗
Snow and thermal energy are also transferred in proportion to the transferred volume. For example, Δvsn = vsn (Δvin /vin ) for snow and Δein = ein (Δvin /vin ) for thermal energy. If ice is created in open water, the left boundary of category 1 (H0 ) is moved to F0 Δt, where F0 is the growth rate in open water. After area and volume are remapped in higher categories, ice area, volume, and energy are added to category 1. If ice is not created in open water, H0 remains zero, but the growth rate at the left boundary of category 1 is set to F0 = f1 . If F0 < 0, the fractional area of category 1 thinner than Δh0 = −F0 Δt is added to open water area. In this operation, volume and energy are invariant. The area to be added to open water is Δa0 =
Δh0
gdh.
(9.85)
0
The right boundary of the thickest category N (HN ) is a function of its mean thickness hN . When hN is given, HN is computed as HN = 3hN − 2HN−1 . It is guaranteed that g(h) > 0 for HN−1 < h < HN and g(h) = 0 for HN < h.
–
131
–
Chapter 9
Sea ice
9.4 Dynamics 9.4.1 Momentum equation for ice pack The momentum equation for an ice pack with mass ρI AhI is
∂ (AhI uI ) − ρI AhI f vI ∂t ∂ ρI (AhI vI ) + ρI AhI f uI ∂t ρI
1 ∂h + Fμ (σ ) + A(τAIx + τIOx ), hμ ∂ μ 1 ∂h = −ρI AhI g + Fψ (σ ) + A(τAIy + τIOy ), hψ ∂ ψ = −ρI AhI g
(9.86) (9.87)
where (uI , vI ) is the velocity vector, h is the sea surface height, (Fμ , Fψ ) is the ice’s internal stress (which is a function of internal stress tensor (σ )), and τAI and τIO are stresses exerted by the atmosphere and ocean.
9.4.2
Stresses at top and bottom
The stress at the top is wind stress:
τAI = ca ρa Ua − uI [(Ua − uI ) cos θa + k × (Ua − uI ) sin θa ],
(9.88)
where Ua is the surface wind vector, ca is the bulk transfer coefficient between air and ice, ρa is the density of air, and θa is the angle between the wind vector and the ice drift vector, which is set to zero. Stress at the bottom is ocean stress:
τIO = cw ρo Uw − uI [(Uw − uI ) cos θo + k × (Uw − uI ) sin θo ],
(9.89)
where Uw is the velocity of the first level of the ocean model, cw is the bulk transfer coefficient between the ice and ocean, ρo is the density of sea water, and θo is the angle between the ice drift vector and the surface velocity of the ocean, which is set to 25◦ in the northern and 25◦ in the southern hemisphere.
9.4.3
Internal stress
In a highly concentrated icepack, the effect of the internal stress is as large as the Coriolis effect and the surface stresses. The expression of the internal stress is derived by regarding the ice as continuous media. The elasticplasticviscous (EVP) model by Hunke and Ducowicz (1997, 2002) is adopted for the constitutive law (the relation between stress and strain rate). The EVP model is a computationally efficient modification of the viscousplastic (VP) model (Hibler, 1979). In the VP model, the internal stress could be very large when the concentration is high and strain rate is near zero, which makes the explicit integration infeasible. An alternative, the implicit method, is usually adopted, but it is not suitable for parallel computing. The EVP model treats the ice as an elastic media and a large local force is released by elastic waves, which would be damped within the time scale of the wind forcing. The constitutive law of the EVP model is
η −ζ P 1 1 ∂ σi j + σi j + σkk δi j + δi j = ε˙i j , i, j = 1, 2, E ∂t 2η 4ηζ 4ζ
(9.90)
where ζ and η are viscous parameters, P represents ice strength, and E is an elastic parameter (mimics Young’s modulus). In the VP model, tendency terms are zero.
– 132
–
9.4. Dynamics The r.h.s. (ε˙i j ) is the strain rate tensor, expressed in Cartesian coordinate as:
ε˙i j =
1 ∂ uI i ∂ uI j . + 2 ∂xj ∂ xi
(9.91)
The divergence, tension, and shear of the strain rate are defined as follows: DD = ε˙11 + ε˙22 , DT = ε˙11 − ε˙22 , DS = 2ε˙12 .
(9.92)
The equation for the stress tensor for σ1 = σ11 + σ22 and σ2 = σ11 − σ22 is given by 1 ∂ σ 1 σ1 P + + E ∂t 2ζ 2ζ 1 ∂ σ 2 σ2 + E ∂t 2η 1 ∂ σ12 σ12 + E ∂t 2η
= DD ,
(9.93)
= DT ,
(9.94)
=
1 DS . 2
(9.95)
In generalized orthogonal coordinates, divergence, tension, and shear of the strain rate are expressed by 1 ∂ (hψ u) ∂ (hμ v) DD = , (9.96) + hμ hψ ∂μ ∂ψ hψ ∂ u hμ ∂ v − , (9.97) DT = hμ ∂ μ hψ hψ ∂ ψ hμ hμ ∂ u hψ ∂ v DS = + . (9.98) hψ ∂ ψ hμ hμ ∂ μ hψ The internal stress is obtained as the divergence of the internal stress tensor, Fμ
=
3 1 2 1 ∂ σ1 1 ∂ (h2ψ σ2 ) 2 ∂ 2 hμ σ12 , + + 2 2 2 hμ ∂ μ hμ hψ ∂ μ h μ hψ ∂ ψ
(9.99)
Fψ
=
3 1 2 1 ∂ σ1 1 ∂ (h2μ σ2 ) 2 ∂ 2 hψ σ12 . − 2 + 2 2 hψ ∂ ψ hμ hψ ∂ ψ h μ hψ ∂ μ
(9.100) (9.101)
The viscous parameters are obtained from the concentration and velocity as follows: P , 2Δ P η = , 2e2 Δ 1/2
1 . Δ = D2D + 2 (D2T + D2S ) e
ζ
=
(9.102) (9.103) (9.104)
The pressure of the ice is a function of ice concentration and thickness: P = P∗ AhI e exp[−c∗ (1 − A)],
(9.105)
where P∗ is the scaling factor for pressure, c∗ is a parameter that defines the dependency on concentration, and e is the axis ratio of the elliptic yield curve (e = 2). The elastic parameter E is given by E=
2Eo ρI AhI min(Δx2 , Δy2 ), Δte2
(9.106)
where Eo is a tuning factor that satisfies 0 < Eo < 1, Δte is the time step for ice dynamics, and Δx2 and Δy2 are the zonal and meridional grid widths. –
133
–
Chapter 9
Sea ice
9.4.4 Boundary conditions Surface stresses on the ice are exerted for the fractional area of the ice within a grid cell. The ice concentration is multiplied by the wind and ocean stresses. For the stress on the ocean, the iceocean stress is exerted for the icecovered area, and the wind stress is exerted for the nonice area:
νV
∂ U ∂ V (1 − A) A (τAOx , τAOy ). = − (τIOx , τIOy ) + , ∂ z ∂ z k=0 ρo ρo
(9.107)
Note that (τIOx , τIOy ) is reversed in sign because it is defined by (9.89) as the stress on the ice.
9.4.5
Solution procedure
Given the surface wind vector and the surface velocity of the ocean needed to compute surface stresses, the momentum equations ((9.86) and (9.87)) and the equations for stress tensors ((9.93), (9.94), and (9.95)) are solved. First, the stress tensor is computed using the equations for stress tensors, the momentum equation is then solved using the stress tensor. Basically, the implicit method is used for prognostic variables for each equation. For example, stress tensor σ1 is solved for σ1m+1 as follows: 1 σ1m+1 − σ1m σ1m+1 P + m = Dm + D. m E Δt 2ζ 2ζ
(9.108)
Note that strain rate tensors and viscous parameters are updated every time step using a new velocity. The momentum equations are solved using σ m+1 above:
ρI AhI
um+1 − um I I Δt
− ρI AhI g = ρI AhI f vm+1 I
1 ∂h + Fμ (σ m+1 ) + AτAIx hμ ∂ μ
(9.109)
m+1 ) cos θo + (Vw − vm+1 ) sin θo ], +Acw ρo Uw − um I [(Uw − uI I
ρI AhI
vm+1 − vm I I Δt
=
−ρI AhI f um+1 − ρI AhI g I
1 ∂h + Fψ (σ m+1 ) + AτAIy hψ ∂ ψ
(9.110)
m+1 ) cos θo − (Uw − um+1 ) sin θo ]. +Acw ρo Uw − um I [(Vw − vI I
Note that the surface velocity of the ocean (Uw ,Vw ) is constant during the integration. The starting time level of the ocean model is used, n − 1 for the leapfrog time step, and n for the Matsuno scheme. For the leapfrog time step of the ocean model, m+1 m+1 m+1 τIO = cw ρo Un−1 [(Un−1 ) cos θo + k × (Un−1 ) sin θo ]. w − uI w − uI w − uI
(9.111)
The time step of the ice dynamics is limited by the phase speed of the elastic wave. To damp the elastic waves during the subcycle, the ice dynamics is subcycled several tens of steps during one ocean model time step.
9.5 Advection Fractional area, snow volume, ice volume, ice energy, and ice surface temperature (optional; set flg advec tskin to be .true. in mod seaice cat.F90) of each category are advected. A multidimensional positive definite advection transport algorithm (MPDATA; Smolarkiewicz, 1984) is used.
– 134
–
9.6. Ridging The advection equation for a property (α ) is given by
∂α + A (α ) = 0, ∂t
(9.112)
where A is an advection operator defined as 1 A (α ) = hμ hψ
∂ (hψ uα ) ∂ (hμ vα ) . + ∂μ ∂ψ
(9.113)
The specific representation for α is A for fractional area, Ahs for snow volume, AhI for ice volume, and AhI E for ice energy. In MPDATA, (9.112) is first solved to obtain a temporary value using the upstream scheme with a midpoint velocity between time levels n and n + 1. Using this temporary value, an antidiffusive velocity is computed as , + ∗ ∗ 1 1 n+ 1 1 n+ 12 2 ∂ α n+ 12 n+ 12 ∂ α 2 u Δx − Δt(u ) v − Δtu u˜i+ 1 , j = 2 α∗ 2 ∂x 2 ∂y 1 1 1 ∂ u n+ 2 ∂ v n+ 2 1 , (9.114) − Δtun+ 2 + 2 ∂x ∂y , + ∂ α∗ 1 ∗ 1 1 1 n+ 1 n+ 12 n+ 12 ∂ α 2 Δy − Δt(vn+ 2 )2 v v v˜i, j+ 1 = Δtu − 2 α∗ 2 ∂y 2 ∂x n+ 12 n+ 12 1 ∂u ∂v 1 . (9.115) − Δtvn+ 2 + 2 ∂x ∂y This velocity is used to compute a new value using the upstream scheme starting from the above temporary value. Since MPDATA is positive definite, the new area and thickness should be positive. If the sum of the fractional area exceeds one, the ridging scheme will adjust the fractional area. Since energy is negative definite, the sign is reversed just before advection and returned to a negative value after the advection.
9.6
Ridging
As a result of advection, the sum of the fractional area might exceed one, especially where the velocity field is convergent. In such a case, it is assumed that ridging occurs among ice to yield a sum equal to or less than one. Even if the sum is less than one, ridging or rafting might occur where the concentration is high. The ridging scheme of MRI.COM follows that of CICE, which is briefly summarized in this section. First, we determine a fractional area that undergoes ridging: aP (h) = b(h)g(h). The weighting function b(h) is chosen such that the ridging occurs for thin ice: G(h) 2 1− ∗ if G(h) < G∗ ∗ G G = 0 otherwise,
b(h) =
(9.116) (9.117)
where G(h) is the area of ice thinner than h and G∗ is an empirical constant with G∗ = 0.15. The participation function for category n (aPn ) is obtained by integrating for a range [Hn−1 , Hn ] as aPn =
2 Gn−1 − Gn . (G − G ) 1 − n n−1 G∗ 2G∗
(9.118)
The property aPn is the fractional contribution from the category n among the total area of ice subject to ridging. The property Gn is the area summed from category 0 to n. This equation is used for the category that satisfies –
135
–
Chapter 9
Sea ice
Gn < G∗ . If Gn−1 < G∗ < Gn , then G∗ is replaced by Gn . If Gn−1 > G∗ , then aPn = 0. If a0 > G∗ , then ridging does not occur. Ridging occurs such that the total area is reduced while conserving ice volume and energy. It is assumed that √ ice of thickness hn is homogeneously distributed between Hmin = 2hn and Hmax = 2 H ∗ hn after ridging, where H ∗ = 25 [m]. The thickness ratio before and after the ridging is kn = (Hmin + Hmax )/(2hn ). Therefore, when an area of category n is reduced by ridging at a rate rn , the area of thicker categories is increased by rn /kn . Among the new ridges, the fractional area that is distributed in category m is: fmarea =
HR − HL , Hmax − Hmin
(9.119)
where HL = max(Hm−1 , Hmin ) and HR = min(Hm , Hmax ). The fractional volume that is distributed in category m is: (HR )2 − (HL )2 . (9.120) fmvol = (Hmax )2 − (Hmin )2 The snow volume and ice energy are distributed by the same ratio as the ice volume. The net area lost by ridging and open water closing is assumed to be a function of the strain rates. The net rate of area loss of the ice pack (Rnet ) is given by Rnet =
Cs (Δ − DD ) − min(DD , 0), 2
(9.121)
where Cs is the fraction of shear dissipation energy that contributes to ridge building (0.5 is used in MRI.COM),
1/2 DD is the divergence, and Δ = D2D + e12 (D2T + D2S ) . These strain rates are computed by the dynamics scheme. The total rate of area loss due to ridging, Rtot = ∑Nn=0 rn , is related to the net rate as follows:
N 1 Rnet = aP0 + ∑ aPn 1 − Rtot . kn n=1
(9.122)
Since Rnet is computed from (9.121), Rtot is computed from (9.122). Thus, the area subjected to ridging from category n is computed as arn = rn Δt (rn = aPn Rtot ). The area after ridging is arn /kn and the volume after ridging is arn hn . Using these, the ice subjected to ridging is first removed from category n, and the ridged ice is then redistributed into each category. In practice, we require that arn ≤ an . If A > 1 after ridging, Rnet is adjusted to yield A = 1, and the ridging procedure is repeated.
9.7 Discretization 9.7.1 Advection (MPDATA) In MPDATA, tracer (α in Figure 9.2) is updated following a threestep procedure. 1. A temporary value is computed using an upstream scheme. The tracer fluxes at the side boundaries of the unit grid cell where the tracer is defined are:
Δy n n n αi,n j (uni+ 1 , j + uni+ 1 , j ) + αi+1, , j (ui+ 1 , j − ui+ 1 , j ) 2 2 2 2 2 2
Δx Fy (αi,n j , αi,n j+1 , vni, j+ 1 ) = αi,n j (vni, j+ 1 + vni, j+ 1 ) + αi,n j+1 (vni, j+ 1 − vni, j+ 1 ) , 2 2 2 2 2 2
n n Fx (αi,n j , αi+1, j , ui+ 1 , j ) =
– 136
–
(9.123)
9.7. Discretization where
1 ui+ 1 , j+ 1 + ui+ 1 , j− 1 , (9.124) 2 2 2 2 2 2 1 v 1 1 + vi− 1 , j+ 1 . vi, j+ 1 = (9.125) 2 2 2 2 i+ 2 , j+ 2 The zonal flux is defined at the closed circle and the meridional flux is defined at the closed square in Figure 9.2. Using this, the temporary value (α ∗ ) is computed using an upstream scheme: ui+ 1 , j =
(αi,∗ j − αi,n j )ΔSi, j Δt
n n = Fx (αi−1, j , αi, j , u
n+ 12
i− 12 , j
+ Fy (αi,n j−1 , αi,n j , v
n ) − Fx (αi,n j , αi+1, j, u
n+ 12
i, j− 12
n+ 12
i+ 12 , j
) − Fy (αi,n j , αi,n j+1 , v
)
n+ 12
i, j+ 12
).
(9.126)
2. Compute the antidiffusive transport velocity. Using the temporary value computed in the first step, the antidiffusive transport velocity is computed as follows: ⎡ ⎤ ∗ ∗ 1 1 1 1 ∂ α ∂ α 1⎣ 1 1 n+ n+ n+ ⎦ u 12 Δx − Δt(u 12 )2 u˜i+ 1 , j = − Δtu 12 vn+ 2 i+ 2 , j i+ 2 , j i+ 2 , j 2 2 α ∗(2) ∂ x i+ 12 , j α ∗(6) ∂ y i+ 21 , j i+ 12 , j
v˜i, j+ 1 2
∂u
i+ 12 , j
∂ v n+ 1
1 n+ 1 2 , (9.127) − Δtu 12 + i+ 2 , j ∂ x 2 ∂ y i+ 12 , j ⎡ ⎤ ∗ ∗ 1 ∂α ∂ α 1⎣ 1 1 n+ 12 n+ 12 2 n+ 12 ⎦ v 1 Δy − Δt(v 1 ) = − Δtv 1 un+ 2 i, j+ 2 i, j+ 2 i, j+ 2 2 α ∗(2) ∂ y i, j+ 12 α ∗(6) ∂ x i, j+ 12 1 1 i, j+ 2
1 n+ 1 − Δtv 2 1 2 i, j+ 2
∂u ∂v + ∂x ∂y
i, j+ 2
n+ 1
2
i, j+ 12
,
(9.128)
where ∗(2) i+ 12 , j
=
∗(2) i, j+ 12
=
∗(6) i+ 12 , j
=
∗(6) i, j+ 12
=
α α α α
1 ∗ ∗ (α + αi+1, j ), 2 i, j 1 ∗ (α + αi,∗ j+1 ), 2 i, j 1 ∗ ∗ ∗ ∗ ∗ ∗ + αi+1, (α j−1 + 2αi, j + 2αi+1, j + αi, j+1 + αi+1, j+1 ), 6 i, j−1 1 ∗ ∗ ∗ ∗ ∗ ∗ + αi−1, (α j+1 + 2αi, j + 2αi, j+1 + αi+1, j + αi+1, j+1 ), 6 i−1, j ∂ α∗
∂x
i+ 12 , j
∂y
i+ 12 , j
∂ α∗
1
vn+ 2
1
un+ 2
∂ α∗ ∂ y i+ 12 , j ∂ α∗ ∂ x i, j+ 12
=
=
= =
∗ ∗ αi+1, j − αi, j
Δx ∗ αi, j+1 − αi,∗ j Δy
137
–
(9.130) (9.131) (9.132)
,
(9.133)
,
(9.134)
∂ α∗ 1 n+ 12 ∂ α ∗ n+ 12 v 1 + v 1 i, j− 2 4 i, j+ 2 ∂ y i, j+ 12 ∂ y i, j− 12 ∂ α∗ ∂ α∗ 1 1 n+ n+ 2 +v 2 1 + v , 1 i, j+ 2 i, j− 2 ∂ y i+1, j+ 12 ∂ y i+1, j− 12
∗ 1 1 ∂ α∗ 1 n+ 2 ∂ α n+ u 1 + u 12 1 i− 2 , j 4 i+ 2 , j ∂ x i+ 2 , j ∂ x i− 12 , j ∂ α∗ ∂ α∗ n+ 1 n+ 1 , +u 12 + u 12 1 1 i+ 2 , j+1 i− 2 , j+1 ∂ x i+ 2 , j+1 ∂ x i− 2 , j+1 –
(9.129)
(9.135)
(9.136)
Chapter 9
Sea ice
∂u ∂v + ∂x ∂y
n =
uni+ 1 , j Δyi+ 1 , j − uni− 1 , j Δyi− 1 , j + vni, j+ 1 Δxi, j+ 1 − vni, j− 1 Δxi, j− 1 2
2
2
2
ΔSi, j
i, j
∂ v n+ 12 + ∂ x ∂ y i+ 12 , j ∂ u ∂ v n+ 1 2 + ∂ x ∂ y i, j+ 12
∂u
= =
2
2
2
2
,
+ , 1 ∂ u ∂ v n+ 21 ∂ u ∂ v n+ 12 , + + + 2 ∂ x ∂ y i, j ∂ x ∂ y i+1, j + , 1 ∂ u ∂ v n+ 21 ∂ u ∂ v n+ 12 . + + + 2 ∂ x ∂ y i, j ∂ x ∂ y i, j+1
(9.137)
(9.138) (9.139)
3. Update tracer using the antidiffusive velocity starting from the temporary value. ∗ (αi,n+1 j − αi, j )ΔSi, j
Δt
∗ ∗ ∗ ∗ = Fx (αi−1, j , αi, j , u˜i− 1 , j ) − Fx (αi, j , αi+1, j , u˜i+ 1 , j ) 2
2
+ Fy (αi,∗ j−1 , αi,∗ j , v˜i, j− 1 ) − Fy (αi,∗ j , αi,∗ j+1 , v˜i, j+ 1 ). 2 2
Ui1/2, j+3/2
α i+1, j+1
α i, j+1
Ui+3/2, j+1/2
Ui+1/2, j+1/2 α i, j
Ui1/2, j1/2
Ui+3/2, j+3/2
Ui+1/2, j+3/2
Ui1/2, j+1/2
(9.140)
α i+1, j Ui+3/2, j1/2
Ui+1/2, j1/2
Figure 9.2. Position for tracer (α ) and velocity (U). Area and thickness are defined at tracer points. Zonal fluxes are computed at closed circles and meridional fluxes are computed at closed squares. The budget is computed for a unit grid cell for α .
9.7.2 Momentum equation Specific forms of discretization for properties related to internal stress are given here. The strain rate tensor (ε˙ ) and stress tensor (σ ) are defined at tracer points (Figure 9.3). Components (divergence, tension, and shear) of the strain rate tensor are expressed in a discretized form as follows: (DD )i, j
=
Δyi+ 1 , j Δyi− 1 , j 1 2 2 (uI i+ 1 , j+ 1 + uI i+ 1 , j− 1 ) − (uI i− 1 , j+ 1 + uI i− 1 , j− 1 ) 2 2 2 2 2 2 2 2 Δxi, j Δyi, j 2 2 Δxi, j+ 1 Δxi, j− 1 2 2 + (vI i+ 1 , j+ 1 + vI i− 1 , j+ 1 ) − (vI i+ 1 , j− 1 + vI i− 1 , j− 1 ) , 2 2 2 2 2 2 2 2 2 2
– 138
–
9.7. Discretization (DT )i, j
uI i− 1 , j+ 1 Δyi, j− 1 uI i+ 1 , j− 1 uI i− 1 , j− 1 1 Δyi, j+ 12 uI i+ 12 , j+ 12 2 2 2 2 2 2 2 + − − 2 Δxi, j+ 1 Δyi+ 1 , j+ 1 Δyi− 1 , j+ 1 Δxi, j− 1 Δyi+ 1 , j− 1 Δyi− 1 , j− 1
=
2
−
(DS )i, j
2
2
1 Δxi+ 1 , j vI i+ 1 , j+ 1 2
2
2
2
2
2
2 Δyi+ 1 , j Δxi+ 1 , j+ 1
2
−
2
vI i+ 1 , j− 1 2
2
2
2
Δxi+ 1 , j− 1
2
2
2
Δxi, j− 1 vI i− 1 , j+ 1
+
2
Δyi, j− 1 2
2
2
2
2
Δxi− 1 , j+ 1
2
−
2
vI i− 1 , j− 1 2
2
2
2
Δxi− 1 , j− 1
,
vI i− 1 , j+ 1 Δyi, j− 1 vI i+ 1 , j− 1 vI i− 1 , j− 1 1 Δyi, j+ 12 vI i+ 12 , j+ 12 2 2 2 2 2 2 2 + − − 2 Δxi, j+ 1 Δyi+ 1 , j+ 1 Δyi− 1 , j+ 1 Δxi, j− 1 Δyi+ 1 , j− 1 Δyi− 1 , j− 1
=
2
+
2
2
1 Δxi+ 1 , j uI i+ 1 , j+ 1 2
2
2
2
2
2
2 Δyi+ 1 , j Δxi+ 1 , j+ 1
2
−
2
uI i+ 1 , j− 1 2
2
2
2
Δxi+ 1 , j− 1
2
+
2
2
Δxi, j− 1 uI i− 1 , j+ 1 2
Δyi, j− 1 2
2
2
2
2
Δxi− 1 , j+ 1
2
−
2
uI i− 1 , j− 1 2
2
2
2
Δxi− 1 , j− 1
.
The internal stress is defined at velocity points and computed from stress tensor as follows: (Fμ )i+ 1 , j+ 1 2
2
=
1 2 1 (σ1 )i+1, j+1 + (σ1 )i+1, j − (σ1 )i, j+1 − (σ1 )i, j 2 2 Δxi+ 1 , j+ 1 2
(9.141)
2
2 2 1 Δyi+1, j+ 12 [(σ2 )i+1, j+1 + (σ2 )i+1, j ] − Δyi, j+ 21 [(σ2 )i, j+1 + (σ2 )i, j ] + 2 Δy2 1 1 Δxi+ 1 , j+ 1 i+ 2 , j+ 2
2
2
Δx2i+ 1 , j+1 [(σ12 )i+1, j+1 + (σ12 )i, j+1 ] − Δx2i+ 1 , j [(σ12 )i+1, j + (σ12 )i, j ] 2 2 , + Δx2i+ 1 , j+ 1 Δyi+ 1 , j+ 1 2 2 2 2 (Fψ )i+ 1 , j+ 1 2
2
=
1 2 1 (σ1 )i+1, j+1 + (σ1 )i, j+1 − (σ1 )i+1, j − (σ1 )i, j 2 2 Δyi+ 1 , j+ 1 2
−
2
2 2 1 Δxi+ 1 , j+1 [(σ2 )i+1, j+1 + (σ2 )i, j+1 ] − Δxi+ 1 , j [(σ2 )i+1, j + (σ2 )i, j ] 2
2
Δx2i+ 1 , j+ 1 Δyi+ 1 , j+ 1
2
2
2
2
2
Δy2i+1, j+ 1 [(σ12 )i+1, j+1 + (σ12 )i+1, j ] − Δy2i, j+ 1 [(σ12 )i, j+1 + (σ12 )i, j ] 2 2 . + Δy2i+ 1 , j+ 1 Δxi+ 1 , j+ 1 2 2 2 2
Ui1/2, j+3/2
Ui+3/2, j+3/2
Ui+1/2, j+3/2
σi, j+1,εi, j+1 Ui1/2, j+1/2
σi+1, j+1,εi+1, j+1 Ui+3/2, j+1/2
Ui+1/2, j+1/2 σi+1, j,εi+1, j
σi, j,εi, j Ui1/2, j1/2
Ui+3/2, j1/2
Ui+1/2, j1/2
Figure 9.3. Position of variables used by dynamics scheme.
–
139
–
(9.142)
Chapter 9
Sea ice
9.8 Technical issues 9.8.1 Source codes The sea ice part consists of the following programs. ice bulk.F90: computes fluxes at the airice interface ice cat albedo.F90: computes surface albedo and fraction of short wave penetrating into ice ice cat com.F90: defines variables and sets parameters ice cat bulk.F90: computes surface fluxes using bulk formula ice date.F90: computes date and time ice dyn.F90: computes ice dynamics ice flux.F90: computes airice interface processes ice grid.F90: sets grid cells (substitution from ocean model) ice hist.F90: computes and outputs history ice main cat.F90: calls subroutines (main part of the ice model) ice mpdata.F90: computes advection term using MPDATA ice remapv.F90: remaps ice into thickness categories after thermodynamics ice restart.F90: reads and writes restart ice ridge.F90: computes ridging ice time.F90: manages calendar mod seaice cat.F90: computes thermodynamics and various adjustment processes The ice model could only be used as a part of the ocean model. Among the subroutines included in the above programs, only si initialize and simain in ice main cat.F90 are called from the ocean model. si initialize initializes the ice model by reading parameters, creating grid cells, and reading restart, which is called from the subroutine ogcm ini of ogcm.F90. simain is the main program and calls subroutines of the ice model, which is called from the subroutine mkflux of mkflux.F90. To use the ice part, the ocean model options ICE, SIDYN, ICECAT, should be selected. If SIDYN is not selected, dynamics is not solved and ice drifts with a third of the surface ocean velocity. By selecting CALALBSI, the albedo scheme of CICE is used. Otherwise, the constant albedo is used. Other things to be noted are: • The ice model uses the forward scheme in time integration. The ice model is not called in the backward part of the Matsuno (Euler backward) scheme. • The output of history and restart for the category integrated/averaged state is automatically done at the same time as the ocean model for separate files (Chapter 16). – 140
–
9.8. Technical issues
9.8.2
Coupling with an atmospheric model
In a coupled mode, the boundary between the atmospheric component and the oceanice component is at the airice(snow) boundary. The fluxes above the airice boundary are computed by the atmospheric component and passed to the oceanice component via the coupler scup (Yoshimura and Yukimoto, 2008). To use all the heat fluxes given by the atmospheric component, fluxes are globally adjusted at each subcycle time step of the oceanic component to account for changes in ice area during the subcycle. See Yukimoto et al. (2010) for details. In the atmospheric component, temperature in the atmospheric boundary layer and at the ice surface (T3 (tsfcin)) are computed along with heat flux in snow layer (QS = QI2 (fheatu)) using ice temperature (T1 (t1icen)), snow thickness (hs (hsnwn)) and ice thickness (hI (hicen)) given by the oceanic component. All the informations needed by the oceanic component are received via cgcm scup get a2o in ogcm.F90 at the beginning of the coupling cycle. The information needed by the ice part is extracted via get fluxi a2o in ice flux.F90. The main part of the ice is solved using surface fluxes and ice surface temperature (optional) from the atmospheric component. The properties needed by the atmospheric component are sent via cgcm scup put o2a in ogcm.F90 at the end of the coupling cycle.
9.8.3
Job parameters (namelist)
The runtime job parameters (namelist) are listed on Table 9.3. Table 9.3. namelist for the ice model variable name
group
description
usage
file ice restart in
infli
restart file of average for input
used only in OGCM
file ice restart out temp
outflir
restart file of average for output
used only in OGCM
file ice hist temp
outflih
history file of average for output
used only in OGCM
irstrt
njobpsi
set initial state
nstepi
njobpsi
time steps to be proceed
1:read restart, 0: start without ice same as OGCM if not speci
the interval of time step by which
fied 0 for no output
int bgtice
njobpsi
water budget is written ibyri ibmni ibdyi
njobpsi njobpsi njobpsi
start year of this series of integra
same as OGCM if not speci
tion
fied
start month of this series of integra
same as OGCM if not speci
tion
fied
start day of this series of integration
same as OGCM if not specified
adtdi
njobidyn
time step interval for dynamics (in
about a tenth of the baro
minutes)
clinic time step of the ocean model
hbound(0:ncat)
njobpscat
category boundary
lsicat volchk
njobpscat
flag for checking mass conservation
.false. by default
num hint ic
nhsticint
the number of time step intervals of
not exceed 3
history output of categorized ice
–
141
–
Chapter 9
Sea ice
variable name
group
description
usage
maxnum hist ic
nhsticfile nhsticfile
maximum histories allowed to be written to one file the interval of time step for history
given for each num hint ic (same for below) the number of time steps or
nwrt hist ic
nhsticfile
output western end for history output
1 for monthly output
imin hist ic imax hist ic jmin hist ic jmax hist ic
nhsticfile nhsticfile nhsticfile
eastern end for history output southern end for history output northern end for history output
file ice hist ic temp
nhsticfile
core part of the file name for history
file icecat restart in
inflic
restart file name of categorized ice for input
num rst ic
outflic
the number of files allowed to be output
maxnum rst ic
outflic
nwrt rst ic
outflic
the maximum snap shots allowed to be written to one file the interval of time step for snap
outflic
shot output core part of the file name for snap
file icecat restart out temp
the number of time steps or 1 for monthly output
shot alb ice visible t0 alb ice nearIR t0
njobalbsi njobalbsi
visible ice albedo for thicker ice near infrared ice albedo for thicker
0.78 0.36
alb snw visible t0 alb snw nearIR t0
njobalbsi njobalbsi
ice visible, cold snow albedo near infrared, cold snow albedo
0.98 0.70
alb ice visible dec ratio alb ice nearIR dec ratio
njobalbsi njobalbsi
visible ice albedo declination rate near infrared ice albedo declination
0.075 [◦ C−1 ] 0.075 [◦ C−1 ]
njobalbsi njobalbsi
rate visible snow albedo declination rate near infrared ice albedo declination
0.10 [◦ C−1 ] 0.15 [◦ C−1 ]
hi ref
njobalbsi
rate the maximum ice thickness to
0.50 [m]
atan ref
njobalbsi
which connection function is used the base value of the tangent hyper
4.0
alb snw visible dec ratio alb snw nearIR dec ratio
bolic connection function tsfci t0
njobalbsi
the temperature at which ice albedo
0.0 [◦ C]
is equated to that of ocean tsfci t1
njobalbsi
the temperature at which ice albedo is started to decline to that of the ocean
– 142
–
1.0 [◦ C]
9.8. Technical issues variable name
group
fsnow patch
njobalbsi
description
usage
area fraction of snow on melting bare ice
0.02
–
143
–
Chapter 9
Sea ice
9.9 Appendix 9.9.1 Saturation water vapor pressure and latent heat The latent heat of sublimation and the saturation specific humidity is computed according to Appendix 4 of Gill (1982) as follows. The saturation vapor pressure ew (in units [hPa]) of pure water vapor over a plane water surface is given by log10 ew (t) = (0.7859 + 0.03477t)/(1 + 0.00412t).
(9.143)
In air, the partial pressure ew of water vapor at saturation is not exactly ew but is given by
ew = fw ew .
(9.144)
fw = 1 + 10−6 p(4.5 + 0.0006t 2 ),
(9.145)
The value of fw is given by
where p is the pressure (in units [hPa]). The saturation vapor pressure ei of pure water vapor over ice is given by,
log10 ei (t) = log10 ew (t) + 0.00422t.
(9.146)
The current version does not include this correction. Thus, ei = ew . The saturation specific humidity q is by definition:
ei /ps = q/(ε + (1 − ε )q),
(9.147)
where ε is the molecular weight ratio between water vapor and air:
ε = mw /ma = 18.016/28.966 = 0.62197,
(9.148)
and ps is sea level pressure (units in [hPa]). Using this, the saturation specific humidity is given as
q = ε ei /(ps − (1 − ε )ei ).
(9.149)
Ls = 2.839 × 106 − 3.6(T3 + 35)2 Jkg−1 .
(9.150)
The latent heat of sublimation is given by
Surface temperature of ice is computed using semiimplicit method, where an expression for the partial derivative of the saturation specific humidity by temperature is needed as in (9.28). First, using equation (9.149),
∂ ei ∂ qi ε ps = , 2 ∂t {ps − (1 − ε )ei } ∂ t
(9.151)
where ∂ ei /∂ t is expressed by setting fw = 1 as
∂ ei = ln 10 · 10g(t) · g (t), ∂t
where g(t) = (0.7859 + 0.03477t)/(1 + 0.00412t) and g (t) = ∂ g (t)/∂ t. –
144
–
(9.152)
9.9. Appendix
9.9.2
Physical constant, parameters
Since the ice part is coded in SI units, constants and parameters are written in SI units.
Thermodynamics parameter
value
variable name in MRI.COM
Thermal ice conductivity Thermal snow conductivity Specific heat of sea water
= 2.04 J m−1 s−1 K−1
kI ks = 0.31 J m−1 s−1 K−1 Cpo = 3990 J kg−1 K−1
CKI CKS CP0
Specific heat of air Specific heat of ice
Cpa = 1004.67 J kg−1 K−1 C pI = 2093 J kg−1 K−1
CPAIR CPI
Specific heat of snow Stefan Boltzmann constant Albedo of open ocean surface
Cps = 0.0 J kg−1 K−1 σ = 5.67 × 10−8 W m−2 K−4 αo = 0.1
Albedo of ice Albedo of snow
αI = 0.6 αs = 0.75 εo = 0.97 εI = 1.0 εs = 1.0 CHAI = 1.5 × 10−3 LF = 3.347 × 105 J kg−1 equation (9.150) m = −0.0543 K/ppt n = −0.000759 K m−1 zoI = 0.05hI /3 SI = 4.0 psu k = 0.4 κH = 1.0 × 103 m2 s−1 North : 3.0, South : 3.0 ν = 1.8 × 10−6 m2 s−1 αt = 1.39 × 10−7 m2 s−1 αb = 6.8 × 10−10 m2 s−1 Prt = 0.85 b = 3.14
Emissivity of ocean surface Emissivity of ice surface Emissivity of snow surface bulk transfer coefficient Latent heat of fusion Latent heat of sublimation constants for fusion phase equation: T f = mS + nz Ice roughness parameter Salinity of sea ice von Karman’s constant Thickness/compactness diffusion of ice Scaling factor for κH Seawater kinematic viscosity Seawater heat diffusivity Seawater salinity diffusivity Turbulent Prandtl number b in eqs (9.50),(9.52)
–
145
–
— STBL ALBW ALBI ALBS EEW EEI EES CHAI ALF RLTH XMXM XNXN Z0 SI XK AKH FKHDN, FKHDS ANU AT AS PRT AB
Chapter 9
Sea ice
Dynamics parameter
value
Reference water density Reference air density Reference ice density Reference snow density (ratio between
ρo = 1036 kg m−3 ρa = 1.205 kg m−3 ρI = 900 kg m−3 ρs = 330 kg m−3
variable name in MRI.COM ROAIR RICE
snow and water) efolding constant for ice pressure
c∗ = 20.0
CSTAR
pressure scaling factor drag coefficient (airice)
P∗
drag coefficient (iceocean) yield curve axis ratio
cw = 5.5 × 10−3 e = 2.0 Eo = 0.25 θo = ±25◦ (positive/negative in the northern/southern hemisphere) θa
scaling factor for Young’s modulus water turning angle air turning angle
= 2.75 × 104 N m−2
ca =
1.5 × 10−3
RO
RDSW
PRSREF CDRGAI CDRGIW ELIPS EYOUNG WIANGL –
References Gill, A. E., 1982: AtmosphereOcean Dynamics, Academic Press, 662pp. Hibler, W. D., III, 1979: A dynamic thermodynamic sea ice model, J. Phys. Oceanogr., 9, 815–846. Hunke, E. C, and J. K. Dukowicz, 2002: The ElasticViscousPlastic Sea Ice Dynamics Model in General Orthogonal Curvilinear Coordinates on a Sphere – Incorporation of Metric Terms, Mon. Weather Rev., 130, 1848–1865. Hunke, E. C, and J. K. Dukowicz, 1997: An ElasticViscousPlastic Model for Sea Ice Dynamics, J. Phys. Oceanogr., 94, 1849–1867. Hunke, E. C, and W. H. Lipscomb, 2006: CICE: the Los Alamos Sea Ice Model Documentation and Software User’s Manual, available at http://climate.lanl.gov/source/projects/climate/Models/CICE/index.shtml, 59pp. Mellor, G. L., and L. Kantha, 1989: An IceOcean Coupled Model, J. Geophys. Res., 94, 10,937–10,954. Smolarkiewicz, P. K., 1984: A fully multidimensional positive definite advection transport algorism with small implicit diffusion, J. Comput. Phys., 54, 325–362. Steele, M., 1992: Sea ice melting and floe geometry in a simple iceocean model, J. Geohys. Res., 97, 17,729– 17,738. Yoshimura, H., and S. Yukimoto, 2008: Development of simple couplar (Scup) for earth sysyem modeling, Pap. Meteor. Geophys., 59, 1929, doi:10.2467/mripapers.59.19. Yukimoto, S., and coauthors, 2010: Meteorological Research InstituteEarth System Model v1 (MRIESM1), Technical Reports of the Meteorological Research Institute, No.64, in press.
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Chapter 10
Bottom Boundary Layer (BBL)
MRI.COM has the option of adopting a simple bottom boundary layer model (the option BBL). This chapter describe the formulation and usage of this model.
10.1
General description
In general, zcoordinate models cannot properly reproduce either dense overflows from the NordicScotlandGreenland ridges to the Atlantic oceans or the dense downslope flows from the continental slope around Antarctica. The former becomes the core water of the North Atlantic Deep Water, and the latter becomes that of the Antarctic Bottom water. As a result, the abyssal waters in the world ocean tend to exhibit a warming bias. To mitigate these deficiencies, MRI.COM incorporates a simple bottom boundary layer (BBL) model, used in Nakano and Suginohara (2002). This BBL model lies at the bottom of the normal ocean grid along the bottom topography. The following components of the overflow/downslope flows are (partly) incorporated into this BBL model: • The advection along the bottom topography. • The pressure gradient terms when the dense water lies on slope. • The eddy activity to create the flow crossing f /h contours.
10.2
Grid arrangement
In zcoordinate models, the flow along the bottom topography is expressed as a sequence of horizontal and vertical movements along a staircaselike topography. When the number of vertical grid points representing the bottom topography is roughly the same as that of the horizontal grid points, the downslope flow can be reasonably represented even in the zcoordinate model (Winton et al., 1998). In general, however, the number of the vertical grids used in ocean general circulation models is not large enough. Even if the number is large enough, the concentrations of tracers might be significantly diffused during their movement owing to the horizontal diffusion with the surrounding waters. The incorporation of BBL remedies these problems. The BBL grid cells in MRI.COM are arranged as in Figure 10.1. The grid cells in MRI.COM are composed of Ugrid cells, but it is intuitively easier to understand the grid arrangement of BBL as if they are located at Tgrid cells. This is because the depth where the pressure gradient is evaluated is defined as the average depth between the neighboring BBL cells. (In Figure 10.1a, the pressure gradient used to calculate the velocity between the BBLs with different depths is evaluated at the averaged depth between these BBLs.) Assume that the number of the vertical grids, km, is 50 before the BBL option is applied. When the BBL option is applied, the number of vertical grids, km, becomes 51 (50 + 1). The master BBL cells are set at the lowest level of each array regardless of the actual depths of the BBL cells. This arrangement is suitable for calculating the horizontal advection between the BBL cells. In addition, to easily express the interaction between the BBL cells and the inner ocean cells, we place –
147
–
Chapter 10
Bottom Boundary Layer (BBL)
=⇒
(a)
(b)
Figure 10.1. (a) Arrangement of BBL(Tbox). (b) Schematic arrangement that is identical to (a).
a dummy BBL cell in each vertical column just below the bottom grid of the inner cell and copy the temperature, salinity, and velocities from the master BBL cell.
10.3 Pressure gradient terms (a) <<
ǹ$
ǹ+
<* Z
7 <* Z
ǹ$
Figure 10.2. Evaluation of horizontal pressure gradient.
Consider the case in which water in the inner grids is uniform, and there is a dense layer along the bottom slope. From the physical point of view, the horizontal pressure gradient should move the dense water downward along the slope in the nonrotational frame. However, in the original MRI.COM code without the BBL option, the pressure gradient is zero in the lowermost inner cells along the bottom slope. In the BBL cells, the horizontal pressure gradient is calculated as follows when the slope is smooth: ∂ p ∂ p ∂ p ∂H , = + ∂ x z=const ∂ x z=H(x) ∂ z ∂ x ∂ p ∂H = . − g ρ z=H(x) ∂x ∂x
(10.1) (10.2)
z=H(x)
In the above case, the density of the water in the inner cells is uniform (ρ = ρI ), and there is a dense layer (ρ = ρB )
– 148
–
10.4. Eddy effects along the bottom slope. The horizontal pressure gradient becomes ∂ [pz=z0 + gH(x)ρI ] ∂ p ∂H ∂H − g ρB = − g ρB , ∂ x z=H(x) ∂x ∂x ∂x ∂ p ∂H = − g(ρB − ρI ) , ∂x ∂x
(10.3) (10.4)
z=z0
where pz=z0 is the the pressure at z = z0 in Figure 10.2. The second term on the r.h.s. of Eq. (10.4) represents the effect of the slope. The larger the difference in the density or the steeper the slope of the topography, the larger the horizontal pressure gradient becomes. When there is no difference in the density between the inner cells and BBL cells, the pressure gradient is zero regardless of the slope of the bottom topography.
10.4
Eddy effects
In the rotational frame, the dense water along the continental slope flows along f /h without the eddy effects. In this case, introducing the BBL model does not lead to a better representation of the overflow/downslopeflow in the Nordic Seas or on the continental shelf around Antarctica. In the real world, eddy effects create the cross f /h flow, resulting in the overflow/downslopeflows. Jian and Garwood (1996) investigated the threedimensional features of downslope flows using an eddyresolving model and demonstrated that the dense water descends roughly 45◦ left to the geostrophic contour (in the northern hemisphere) with vigorous eddy activity. The observation of significant eddy activity south of the Denmark Strait is consistent with this result. In order to incorporate this effect into the noneddy resolving models, we apply Rayleigh drag whose coefficient is nearly equal to the Coriolis parameter, α f . In this case, the geostrophic balance is modified and written as follows:
∂p − α u, ∂x ∂p fu = − − α v, ∂y
−fv
=
−
(10.5) (10.6)
After some algebra we obtain
α ∂p − f v − u = − f v = − , f ∂x α ∂p f u + v = f u = − , f ∂y
(10.7) (10.8)
where v = v − (α / f )u and u = u + (α / f )v correspond to the geostrophic velocity for the pressure gradient. If we put α = f , the direction of the flow is 45◦ to the right of the pressure gradient. In general, the horizontal pressure field is parallel to the topographic contour near the deep water formation area. Thus, incorporating the Rayleigh drag coefficient, α = f , leads to the dense water descending 45◦ to the left of the geostrophic contour. Because this Rayleigh drag is thought to be caused by the eddy activity and is observed where the dense water descends to the deep layer, the coefficient of the Rayleigh drag should be parameterized as a function of the local velocity and topography. However it is very difficult to appropriately determine this function in coarseresolution models. Accordingly, in the default setting of the BBL option in MRI.COM, the depth range where α = f is arbitrarily set above 2500 m in the northern Atlantic and above 4000 m around Antarctica to represent the observed dense overflow/downslopeflow. Below those depths, α is set to zero.
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149
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Chapter 10
Bottom Boundary Layer (BBL)
10.5 Usage In this section, we show how to use the BBL model in the MRI.COM. • Add the BBL option in the file configure.in. • Set kbbl=1 in the file configure.in. (kbbl is the number of BBL layers. Presently, only one layer (kbbl = 1) is available). • Set the lowest layer of dz.F90 as the depth of BBL. Fifty to one hundred meters is recommended for the depth of BBL, which roughly corresponds to the observed thickness of the BBL layer near the Denmark Strait. • Add ho4bbl, exnnbbl to file topo after ho4, exnn. These two variables have the same format as ho4, exnn. Variable ho4bbl corresponds to the thickness of the BBL, and variable exnnbbl corresponds to the number of BBL layers; only one layer is presently available. Thus exnnbbl should be equal to or less than one. • Set the area where the Rayleigh drag coefficient is applied. The default setting is 2500m in the North Atlantic and 4000 m around the Southern Ocean. In the default setting, the temperature and salinity in the BBL cells are set to those in the cells k=km. When the option INILEV is used, the temperature and salinity in the BBL cells are set to those in the lowermost inner cells. Example of configure.in with the BBL option DEFAULT_OPTIONS="OMIP FREESURFACE UTOPIA ULTIMATE ZQUICKEST ZULTIMATE CYCLIC ISOPYCNAL MELYAM HFLUX WFLUX RUNOFF CLMFRC HIST HISTFLUX ICE SIDYN INILEV BBL" IMUT=184 JMUT=171 KM=48 KSGM=1 KBBL=1 SLAT0=84.D0 SLON0=0.D0 DXTDGC=2.D0 DYTDGC=1.D0 ITMSC=0 ITMSCB=0 ISRSTB=NSDAY NPARTA=4 NPLAT=75.D0 NPLON=320.D0
The above example includes BBL in DEFAULT OPTIONS and KBBL = 1.
A sample Fortran program of the topography data follows. A Fortran program to create topography data with BBL
integer(4) :: ho4(imut,jmut),exnn(imut,jmut) integer(4) :: ho4bbl(imut,jmut),exnnbbl(imut,jmut) write(inidt) ho4, exnn write(inidt) ho4bbl,exnnbbl
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10.6
Usage notes
10.6.1
Limit of the area where BBL model should be applied
Ideally, the BBL model should be applied to the globe, and its behavior should change due to temperature, salinity, velocity, and topography. In the real configuration, the BBL model is only effective near the abyssal water formation areas. When used in other areas, such as the near the equator, the BBL model does not improve the tracer and velocity fields. Furthermore, because the BBL model connects the model cells along the topography, it inevitably induces a water mass exchange between the cold abyssal water and the warm shallow water. Thus, unphysical diapycnal diffusion could occur with the BBL model. This effect is not severe in high latitudes where the difference in temperature between the shallow continental shelves and the deep layer is expected to be small, but it is extremely problematic for the cells in low latitudes. This problem is similar to the problem for typical σ layer models. To prevent this, we choose to apply the BBL model in areas where the BBL model is important.
10.6.2 Limits of the BBL Linear interpolation of the temperature and salinity along an extremely steep slope may cause problems. For the default setting of MRI.COM, the BBL model is not applied in such places and isolated grids.
10.6.3 Notes for the program code In each vertical column, the BBL exists both at the bottom cell (k=km) and at the lowest ocean cell (k=exn(i, j)). In general, the lowest ocean cell of the Tpoint and Upoint for the same horizontal indices might differ. Thus, the treatment of atexl and aexl is always very confusing. At the bottom cell (k=km), we set atexl=1 and aexl=1 while we set atexl=1 and aexl=0 at the lowest ocean cells. In the program energy.F90, the treatment of atexl=1 in the BBL model needs special care to avoid double counts.
Reference Jiang, L., and R. W. Garwood Jr., 1996: Threedimensional simulations of overflows on continental slopes, J. Phys. Oceanogr., 26, 12141233. Nakano, H., and N. Suginohara, 2002: Effects of bottom boundary layer parameterization on reproducing deep and bottom waters in a world ocean model, J. Phys. Oceanogr., 32, 1209–1227. Winton, M., R. Hallberg, and A. Gnanadesikan, 1998: Simulation of densitydriven frictional flow in zcoordinate ocean models, J. Phys. Oceanogr., 28, 2163–2174.
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Chapter 11
Biogeochemical model
There are several options for Biogeochemical models in MRI.COM. These biogeochemical models have been developed for both oceanonly and coupled oceanatmospherevegetation carbon cycle studies. They feature an explicit representation of a marine ecosystem, which is assumed to be limited by light, temperature, and nutrients availability. This chapter describes the details of the biogeochemical models.
11.1
Inorganic carbon cycle and biological model
Biogeochemical models are composed of inorganic carboncycle and ecosystem component models. In the inorganic carboncycle component, the partial pressure of CO2 at the sea surface (pCO2 ) is diagnosed from the values of dissolved inorganic carbon (DIC) and Alkalinity (Alk) at the sea surface, which should be calculated in the ecosystem component. The difference in pCO2 between the atmosphere and ocean determines uptake or release of CO2 from the ocean to the atmosphere and is essential for simulating the CO2 concentration in the atmosphere. Inorganic carbonate chemistry and partial pressure physics are well understood and can be reproduced with fair accuracy. The ecosystem component deals with various biological activities, and gives sources and sinks of the nutrients, DIC, Alk, and dissolved oxygen through these activities. Our knowledge of these activities is far from complete, and they are difficult to estimate even in stateoftheart models. There are many biological models and methods for calculating the ecosystem components. One of the simplest biological models has only one nutrient component (such as PO4 ) as a prognostic variable and calculates neither phytoplankton nor zooplankton explicitly. In these cases, the export of biologically generated soft tissue (organic matter) and hard tissue (carbonate) to the deep ocean, collectively known as the biological pump, is parameterized in terms of temperature, salinity, short wave radiation, and nutrients. A NutrientPhytoplanktonZooplanktonDetritus (NPZD) model is more complex than the above model, but still a simple biological model. The NPZD model has four prognostic variables (nutrient, phytoplankton, zooplankton, and detritus). Though parameterized in a simple form, basic biological activities, such as photosynthesis, excretion, grazing, and mortality are explicitly calculated. More complex models classify phytoplankton and zooplankton into several groups, and deal with many complex interactions between them. In general, it is expected that the more complex the biological model becomes the more realistic pattern the model can simulate. However, because of our incomplete knowledge about the biological activities, the complex models do not always yield better results, even though they require more computer resources. To simulate the carbon cycle in the ocean, some biological process should be calculated in the ecosystem component to obtain DIC at the sea surface. However, the carbon cycle component is not always necessary when our interests are to simulate the ecosystem itself. The Ocean CarbonCycle Model Intercomparison Project (OCMIP) protocols and studies of Yamanaka and Tajika (1996) and Obata and Kitamura (2003) focus on the former carbon cycle in the ocean, and their ecosystem components in these studies are quite simple. Biogeochemical models adopted in MRI.COM are classified in this category. The latter studies usually use complex biological models such
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Biogeochemical model
as NEMURO (Kishi et al., 2001). Of course, this type of model could be adopted as an ecosystem component of the biogeochemical model in the former studies in hopes of better simulation of carbon cycle. The carbon cycle component follows the OCMIP protocols (Orr et al., 1999) whose authority is recognized in the community. MRI.COM has several options for the ecosystem component. At present, MRI.COM can incorporate the Obata and Kitamura model (Obata and Kitamura, 2003) or an NPZD model based on Oschlies (2001), and they cannot be used without the carbon cycle component. The biogeochemical model of MRI.COM is largely based on Schmittner (2008) when an NPZD model is adopted as an ecosystem component. Units in MRI.COM are cgs, but in these biogeochemical subroutines, we use MKS units for the sake of future development. We use mol/m3 for the units of nutrients. The unit of mmol/m3 is used in some models such as Oschlies (2001). When the coefficients of their model are applied, they should be converted to the corresponding units.
11.2 Governing equations Here we describe the biogeochemical models of MRI.COM. When an NPZD model is incorporated as the ecosystem component, the governing equations are as follows. When Obata and Kitamura model is used instead of the NPZD model, the first four biogeochemical compartments (DIC, Alk, PO4 , and O2 ) are used.
∂ [DIC] ∂t ∂ [Alk] ∂t ∂ [PO4 ] ∂t ∂ [O2 ] ∂t ∂ [NO3 ] ∂t ∂ [PhyPl] ∂t ∂ [ZooPl] ∂t ∂ [Detri] ∂t
= A ([DIC]) + D([DIC]) + Sb([DIC]) + Jv([DIC]) + Jg([DIC]),
(11.1)
= A ([Alk]) + D([Alk]) + Sb([Alk]) + Jv([Alk]),
(11.2)
=
A ([PO4 ]) + D([PO4 ]) + Sb([PO4 ]),
(11.3)
= A ([O2 ]) + D([O2 ]) + Sb([O2 ]) + Jg([O2 ]),
(11.4)
= A ([NO3 ]) + D([NO3 ]) + Sb([NO3 ]),
(11.5)
=
A ([PhyPl]) + D([PhyPl]) + Sb([PhyPl]),
(11.6)
=
A ([ZooPl]) + D([ZooPl]) + Sb([ZooPl]),
(11.7)
= A ([Detri]) + D([Detri]) + Sb([Detri]),
(11.8)
where A () is advection, D() is diffusion, and Sb() is source minus sink due to the biogeochemical activities. The square brackets mean dissolved concentration in mol/m3 of the substance within them. The terms represented by Jg() and Jv() are the airsea gas fluxes at the sea surface, and they appear only in the uppermost layer. The term Jg() is calculated based on the OCMIP protocol by using the airsea gas transfer velocity and concentration in the seawater. The term Jv() appears only when the salinity flux is given virtually instead of the increase or decrease of the volume at the surface layers due to evaporation and precipitation.
11.3
Carbon cycle component
To estimate Jg and Jv, we follow the protocol of OCMIP, which is described in detail in Najjar and Orr (1998), and Orr (1999). The program to calculate them is based on the subroutine downloaded from the OCMIP website. We have modified this subroutine so that it can be used in the vector oriented calculation of the MRI.COM code. –
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11.3. Carbon cycle component
11.3.1
Airsea gas exchange fluxes at the sea surface (Jg )
The airsea gas transfer must be calculated for [DIC] and [O2 ]. The terms Jg([DIC]) and Jg([O2 ]) appear only in the uppermost layer. When these fluxes are expressed as Fg([DIC]) and Fg([O2 ]), Jg([DIC]) and Jg([O2 ]) are given as follows: Jg([DIC]) = Jg([O2 ]) =
Fg([DIC]) , Δz1 Fg([O2 ]) , Δz1
(11.9) (11.10)
where 2 Fg([DIC]) = KCO w ∗ ([CO2 ]sat − [CO2 ]surf ), 2 Fg([O2 ]) = KO w ∗ ([O2 ]sat − [O2 ]surf ),
[CO2 ]sat [O2 ]sat
(11.11) (11.12)
= α ∗ pCO2 atm ∗ P/Po ,
(11.13)
=
(11.14)
C
[O2 ]sato ∗ P/Po .
Here a standard gas transfer formulation is adopted. Next, we elaborate on the above equations.
Piston velocity O2 2 Parameters KCO w and Kw are the airsea gas exchange transfer (Piston) velocity and are diagnosed as follows. 2 KCO w 2 KO w
2 = (1 − A)[Xconv aU10 ](ScCO2 /660)−1/2 ,
=
2 (1 − A)[Xconv aU10 ](ScO2 /660)−1/2 ,
(11.15) (11.16)
where • A is the fraction of the sea surface covered with ice, • U10 is 10m scalar wind speed, • a is the coefficient of 0.337, consistent with a piston velocity in cm/hr, and specified in the OCMIP protocol, • Xconv = 1/(3.6 × 105 ), is a constant factor to convert the piston velocity from [cm/hr] to [m/s], • ScCO2 and ScO2 are the Schmidt numbers for CO2 and O2 . They are computed using the formulation of Wannikhof (1992) for CO2 and that of Keeling et al. (1998) for O2 .
Computing CO2 and O2 concentrations at the surface The concentration of CO2 is computed as follows. • [CO2 ]surf is diagnosed every step from [DIC],[Alk], temperature, and salinity at the surface. • α C is the solubility of CO2 , which is diagnosed from sea surface temperature and salinity. • [O2 ]sato is the saturation oxygen concentration before the variations in total pressure are taken into account and is diagnosed from the sea surface temperature and salinity. –
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• pCO2 atm is specified. (In the OCMIP climatology, it is set to 280ppm.) • P is the sea surface pressure in units of [hPa] and Po = 1013.25 [hPa] is the standard surface atmospheric pressure.
Diagnosis of [CO2 ] at the surface Diagnosis of [CO2 ] at the surface is the most complex of the above calculations and has the heaviest computational burden. To be precise, diagnosis of [CO2 ] actually means diagnosing [CO2 ] + [H2 CO3 ], which are difficult to distinguish analytically. These two species are usually combined and the sum is expressed as the concentration of a hypothetical species, [CO∗2 ] or [H2 CO∗3 ]. Here, the former notation is used. The relationship between this [CO∗2 ] and DIC is as follows: 2− [DIC] = [CO2 ] + [H2CO3 ] + [HCO− 3 ] + [CO3 ]
= [CO∗2 ]
2− + [HCO− 3 ] + [CO3 ].
(11.17) (11.18)
In the OCMIP protocol, the following equations are solved to obtain [CO∗2 ]. The equilibrium expressions for dissociation are: K1 =
[H + ][HCO− 3] ∗ [CO2 ] KB =
K1P =
[H + ][H2 PO− 4] [H3 PO4 ]
[H + ][CO2− 3 ] , − [HCO3 ]
(11.19)
[H + ][B(OH)− 4] , [B(OH)3 ]
K2P = KSi =
K2 =
[H + ][HPO2− 4 ] − [H2 PO4 ]
(11.20) K3P =
[H + ][PO3− 4 ] , 2− [HPO4 ]
[H + ][SiO(OH)− 3] , [Si(OH)4 ]
(11.21) (11.22)
KW = [H + ][OH − ],
(11.23)
[H + ]F [SO2− 4 ] , − [HSO4 ]
(11.24)
[H + ]F [F − ] , [HF]
(11.25)
KS = and
KF =
where [H+ ] is the hydrogen ion concentration in seawater and [H+ ]F is the free hydrogen ion concentration. There is another scale for the hydrogen ion concentration, the total hydrogen ion concentration [H+ ]T . The subscript T means “total” and F means “free.” These three hydrogen ion concentrations are related as follows: ST FT , + [H + ] = [H + ]F 1 + KS KF ST and [H + ]T = [H + ]F 1 + . KS
(11.26) (11.27)
There are three pH scales corresponding to these three hydrogen ion concentrations. The equilibrium constants Kx are given as a function of temperature, salinity, and pH. Note that the equilibrium constants are given in terms of concentrations, and that all constants are referenced to the seawater pH scale, except for KS , which is referenced to the free pH scale. – 156
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11.3. Carbon cycle component The total dissolved inorganic carbon, boron, phosphate, silicate, sulfate, and fluoride are expressed as follows: 2− [DIC] = [CO∗2 ] + [HCO− 3 ] + [CO3 ],
(11.28)
[BT ] = [B(OH)3 ] + [B(OH)− 4 ],
(11.29)
2− 3− [PT ] = [H3 PO4 ] + [H2 PO− 4 ] + [HPO4 ] + [PO4 ],
(11.30)
[SiT ] = [Si(OH)4 ] + [Si(OH)− 3 ],
(11.31)
2− [ST ] = [HSO− 4 ] + [SO4 ],
(11.32)
[FT ] = [HF] + [F − ].
(11.33)
and
Alkalinity used in this calculation is defined as follows: 2− − − 2− 3− − [Alk] = [HCO− 3 ] + 2[CO3 ] + [B(OH)4 ] + [OH ] + [HPO4 ] + 2[PO4 ] + [SiO(OH)3 ]
(11.34)
−[H + ]F − [HSO− 4 ] − [HF] − [H3 PO4 ]. These expressions exclude the effect of NH3 , HS− , and S2− . If we assume that [DIC], [Alk], [PT ], and [SiT ] are known, this system contains 18 equations with 18 unknowns, so they can be solved using the NewtonRaphson method. The concentration [SiT ] is not predicted in the biogeochemical model adopted in MRI.COM but rather is specified as a typical value of 7.68375×10−3 mol/m3 . (The sensitivity to [SiT ] is much less than that to other variables).
11.3.2
Dilution and concentration effects of evaporation and precipitation on DIC and Alk
The dilution and concentration effects of evaporation and precipitation significantly impact the concentrations of some chemical species in seawater. This is particularly true for DIC and Alk, which have large background concentrations compared with their spatial variability. MRI.COM uses a free surface, so the impact of evaporation and precipitation is straightforward to model unless the option SFLUXW or SFLUXR is used. In these options, salinity flux is diagnosed and applied instead of the freshwater flux. In this case, the dilution and concentration effect of evaporation (E) and precipitation (P) should be taken into account. Here, they are parameterized as virtual DIC and Alk fluxes, similar to the virtual salt flux used in physical ocean GCMs. In MRI.COM, the tendency of salinity due to the virtual salt flux is given by sflux(i, j) = −(P − E) ∗ S(i, j, 1)/Δz,
(11.35)
where S(i, j, 1) and Δz are the salinity and thickness of the uppermost layer. Note that the variable sflux(i, j) is not the salinity flux but the time change rate of salinity due to the flux even though the spelling brings up the image of the flux. In MRI.COM, DIC and Alk are modified by the virtual salt flux as follows: Jv(DIC(i, j, 1)) = sflux(i, j)/S(i, j, 1) ∗ DIC(i, j, 1),
(11.36)
Jv(Alk(i, j, 1)) = sflux(i, j)/S(i, j, 1) ∗ Alk(i, j, 1).
(11.37)
Strictly speaking, airsea fluxes of fresh water impact other species. However, these modifications are not usually applied because their spatial variabilities are significantly greater than those of DIC and Alk, –
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In the OCMIP protocol, the global averaged salinity Sg is used instead of S(i, j, 1) in equations(11.36,11.37). In addition, globally integrated Jv(DIC) and Jv(Alk) are set to 0. In MRI.COM, these modifications are not the default considering the use in regional ocean models.
11.4 Obata and Kitamura model This section was contributed by A. Obata. The Obata and Kitamura model used in MRI.COM simply represents the source and sink terms of [DIC], [Alk], [PO4 ], and [O2 ] due to the biogeochemical activities: new production driven by insolation and phosphate concentration in the surface ocean, its export to depth, and remineralization in the deep ocean. According to the MichaelisMenten kinetics (Dugdale, 1967), phosphorus in the new production exported to depth (ExprodP) is parameterized as rL[PO4]2 /([PO4] + k), where r is a proportional factor (r = 0.9[yr−1 ]), L is the insolation normalized by the annual mean insolation on the equator, and k is the halfsaturation constant (k = 0.4[molkg−1 ]). The values of r and k are adjusted to reproduce the optimum atmospheric CO2 concentration and ocean biogeochemical distribution for the preindustrial state of the model. The relationship between the changes in the chemical composition of seawater and the composition of particulate organic matter (POM) is assumed to follow the Redfield ratio P : N : C : O2 = 1 : 16 : 106 : −138 (Redfield et al., 1963). The rain ratio of calcite to particulate organic carbon (POC) is 0.09, which is in the range proposed by Yamanaka and Tajika (1996). The surface thickness where the export production occurs is fixed at 60 m. The vertical distribution of POM and calcite vertical flux below a depth of 100 m is proportional to (z/100m)−0.9 and exp(−z/3500m) (z is the depth in meters), respectively, following the work of Yamanaka and Tajika (1996). The remineralization of POM (RemiP for phosphorus) and the dissolution of calcite (SolnCa) at depth are parameterized by these fluxes. Oxygen saturation is prescribed at the sea surface. The solubility of oxygen is computed from the formula of Weiss (1970). Source and sink terms of Sb() representing the above processes are as follows.
Sb([DIC]) = 106 ∗ RemiP + SolnCa − 106 ∗ ExprodP
(11.38)
Sb([Alk]) = 2 ∗ SolnCa + 16 ∗ ExprodP − 16 ∗ RemiP
(11.39)
Sb([PO4 ]) = RemiP − ExprodP
(11.40)
Sb([O2 ]) = − 138 ∗ Sb([PO4 ])
(11.41)
11.5 NPZD model The NPZD model used in MRI.COM is constructed on the assumptions that the biological composition ratio is nearly constant (Redfield ratio) and that the concentration of the biology can be estimated by nitrogen or phosphorus. The prognostic variables of nitrogen (NO3 ), phytoplankton (PhyPl), zooplankton (ZooPl), and detritus (Detri) are normalized in terms of nitrogen 1 mol/m3 . For example, [PhyPl] represents the concentration of phytoplankton estimated in terms of nitrogen in one cubic meter (N mol /m3 ). The increase and decrease of carbon can be diagnosed by multiplying by Rcn (ratio of C:N). Source and sink terms Sb() calculated in the NPZD model are as follows. Those for DIC and Alk, Sb(DIC) and
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11.5. NPZD model Sb(Alk), used for calculating the carbon cycle, are described later in this section. Sb([PhyPl]) = Priprod − MortP1 − MortP2 − GrP2Z
(11.42)
Sb([ZooPl]) = assim ∗ GrP2Z − Excrtn − MortZ
(11.43)
Sb([Detri]) = [(1 − assim) ∗ GrP2Z + MortP2 + MortZ] − RemiD − wdetri
∂ Detri ∂z
(11.44)
Sb([NO3 ]) = MortP1 + Excrtn + RmeiD − PriProd
(11.45)
Sb([PO4 ]) = R pn ∗ Sb([NO3 ])
(11.46)
Sb([O2 ]) = − Ron ∗ Rnp ∗ Sb([PO4 ])
(11.47)
There is no input from the atmosphere such as nitrogen fixation in the above equations, so the sum of these five equations becomes zero at each grid point except for the term for detritus sinking (−wdetri ∂ Detri ∂ z ). The term for detritus sinking expresses the biological pump, whose role is to remove nutrients from the upper layers and transport them into the deep ocean where the plankton cannot use the nutrients. When vertically integrated, the sum of each grid is 0 even though this sinking term is included. The nutrients are transported horizontally through physical processes such as advection and diffusion. In general, the nitrate limit is more severe than the phosphate limit so it is not always necessary to calculate phosphate. However, in the simpler model of Obata and Kitamura (2003), phosphate is used as a prognostic variable, so to be consistent, phosphate is calculated in the ecosystem component of MRI.COM. Next, we elaborate on the above equations.
11.5.1 Description of each term • Priprod = J(I, N, P) ∗ [PhyPl] Primary production expresses photosynthesis (described in detail in the next subsection). • MortP1 = φP ∗ [PhyPl] The conversion of mortality phytoplankton directly into nutrients. This term was introduced by Oschlies (2001) to increase the primary production of subtropical gyre, where the nutrient limit is severe. • MortP2 = φPP ∗ [PhyPl]2 The conversion from phytoplankton to detritus (normal mortality of phytoplankton). • GrP2Z = G(P) ∗ [ZooPl]2 The grazing of zooplankton. There are a number of parameterizations of grazing. In this formulation, this is given as G(P) = g ∗ ε ∗ [PhyPl]2 /(g + ε ∗ [PhyPl]2 ). Among the grazing, the ratio assim is used for the growth of zooplankton, and the remainder (1 − assim) is converted to detritus. • Excrtn = d ∗ [ZooPl] Excretion of zooplankton. The excretion is dissolute and directly returned to nutrients (NO3 ). • MortZ = φZ ∗ [ZooPl] The conversion from the zooplankton to detritus (mortality of zooplankton). • RemiD = φD ∗ [Detri] Remineralization of detritus. This is converted to nutrients through the activity of bacteria.
–
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Chapter 11
Biogeochemical model 2TKOCT[2TQFWEVKQP
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Figure 11.1. Schematic of NPZD model
11.5.2 Primary Production The growth rate of phytoplankton is limited by the irradiance (I ) and nutrients. This limitation is expressed in several ways. Here we adopt an expression with a minimum function: J(I, N, P) = min (JI , JN , JP ) ,
(11.48)
where JI denotes the purely lightlimited growth rate, and JN and JP are nutrientlimited growth rates that are functions of nitrate or phosphate. The lightlimited growth is calculated as follows: JI =
Jmax α I . 2 [Jmax + (α I)2 ]1/2
(11.49)
Here, Jmax is the lightsaturated growth, which depends on temperature based on Eppley (1972) as Jmax = a · bcT ,
(11.50)
where a = 0.6day−1 , b = 1.066, and c = 1(◦ C)−1 . Note that the default values in MRI.COM are based on Schmittner (2008) and differ from these values (see Table 11.1). Equation (11.49) is called Smithtype growth. The coefficient α in the equation is “the initial slope of photosynthesis versus irradiance (PI) curve,” that is,
∂ JI . I→0 ∂ I
α = lim
(11.51)
Thus, it represents how sensitive JI is to the irradiance when the light is weak. Irradiance (I ) depends on the angle of incidence and the refraction and absorption in the seawater. z˜ I = Iz=0 PAR exp −kw z˜ − ke Pdz ,
(11.52)
0
where Iz=0 denotes the downward shortwave radiation at the sea surface, PAR is the photosynthetically active radiation ratio (0.43) and z˜ = z/ cos θ = z/ 1 − sin2 θ /1.332 is the effective vertical coordinate (positive downward) with 1.33 as the refraction index according to Snell’s law relating the zenith angle of incidence in air (θ ) to the angle of incidence in water. The angle of incidence θ is a function of the latitude φ and declination δ . – 160
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11.5. NPZD model For the nutrientlimited growth rate (JN and JP ), we adopt the Optimal Uptake (OU) equation instead of the classic MichaelisMenten (MM) equation. For the classic MM equation, the nitratelimited growth rate is expressed as
Jmax N , (11.53) KN + N where KN is a halfsaturation constant for NO3 uptake. In contrast, the Optimal Uptake (OU) equation for a nitrate is expressed as follows: V0 N JN = JOU = , (11.54) N + 2 VA00 N + VA00 JN = JMM =
where A0 and V0 are the potential maximum values of affinity and uptake rate, respectively (see Smith et al. (2009) for details). Optimal Uptake (OU) kinetics assumes a physiological tradeoff between the efficiency of nutrient encounter at the cell surface and the maximum rate at which a nutrient can be assimilated (Smith et al., 2009). The key idea is that phytoplankton alters the number of its surface uptake sites (or ion channels), which determines the encounter timescale, versus internal enzymes, which assimilate the nutrients once encountered. We set parameters VO and AO so that the rates of uptake, JMM and JOU , are equal at N = KN . In addition, we fix the ratio V0 /A0 = αOU , where αOU is determined from fitting the data. This requires 1 2 αOU Jmax . (11.55) V0 = 0.5 1 + KN Finally, we obtain
V0 N . (11.56) N + 2 αOU N + αOU = 0.19, which is determined from the fitting of log KN vs log N in the wide range of N by Smith et al. √
JOU =
We use αOU (2009).
11.5.3 Variation of DIC and Alk due to biological activity Production of DIC and Alk is controlled by changes in inorganic nutrients and calcium carbonate (CaCO3 ), in molar numbers according to (11.57) Sb([DIC]) = Sb([PO4 ])Rcp − Sb([CaCO3 ]), Sb([Alk]) = −Sb([NO3 ]) − 2 · Sb([CaCO3 ]).
(11.58)
Thus, only these source and sink terms of DIC and Alk are estimated. Since [PO4 ] and [NO3 ] are prognostic variables, their source and sink are explicitly calculated by the biological model. In contrast, the downward movement of CaCO3 is much faster than the modeled downward velocity of water mass, so [CaCO3 ] is not a prognostic variable, and its source (Pr) and sink (Di) are diagnosed by the following equation, Sb([CaCO3 ]) = Pr([CaCO3 ]) − Di([CaCO3 ]).
(11.59)
Following Schmittner et al. (2008), the source term (Pr([CaCO3 ])) of calcium carbonate is determined by the production of detritus as follows: Pr([CaCO3 ]) = [(1 − assim) ∗ [GrP2Z] + [MortP2] + [MortZ]] RCaCO3 /POC RC:N ,
(11.60)
where assim, GrP2Z, MortP2, and MortZ are as described above. The sink term (Di([CaCO3 ])) of calcium carbonate is parameterized as Di([CaCO3 ]) =
Pr([CaCO3 ])dz · –
161
–
d exp (−z/DCaCO3 ) , dz
(11.61)
Chapter 11
Biogeochemical model
which expresses an instantaneous sinking with an eholding depth of DCaCO3 =3500m. In this equation, z is positive downward. This depth of 3500m was estimated by Yamanaka and Tajika (1996) to reproduce the observed nutrient profile. This value is standard and also is used in the simple biological model in the protocol of OCMIP. The vertical integral of the source minus sink should be zero. Thus, when the sea bottom appears before the sum becomes zero, the remaining calcium carbonate is assumed to be dissolved in the lowermost layer. By using the ratio RCaCO3 /POC = 0.035 used by Schmittner et al. (2008), the resultant global mean Rain ratio should be roughly consistent with the recently estimated range (0.07 to 0.11) based on various observations.
11.6 Usage There are three options, CARBON, NPZD, and CBNHSTRUN. Options NPZD and CBNHSTRUN require CARBON. • When option NPZD is not used in configure.in, the simple biological model of Obata and Kitamura (2003) is applied as the ecosystem component. – numtrc p=4 should be specified in configure.in. Using a different number causes the program to stop. – When used in the NEC super computer, the compile option of FFLAGS = −pvctlexpand = 6 should be used to activate the inline expansion. If this option is not used, the model works but quite slowly due to the low efficiency in the vectorization. The configurations in restart and history file are set in namelist inflpt, outfpt, outfph: namelist /inflpt/ fn_ptrc_in
(1:numtrc_p+np_inout)
namelist /outfpt/ fn_ptrc_out (1:numtrc_p+np_inout) namelist /outfph/ fn_ptrc_hist (1:numtrc_p+np_hist) np_inout = 2 np_hist = 5 (when NPZD is used, this number is 6) fn_ptrc_in (1) : restart file for dic (input) (2) : restart file for alk (3) : restart file for po4
(input) (input)
(4) : restart file for o2
(input)
(5) : restart file for pco2o
(input)
(6) : restart file for pco2a
(input)
fn_ptrc_out (1) : restart file for dic (2) : restart file for alk
(output) (output)
(3) : restart file for po4
(output)
(4) : restart file for o2
(output)
(5) : restart file for pco2o
(output)
(6) : restart file for pco2a fn_ptrc_hist(1) : history file for dic
(output)
(2) : history file for alk (3) : history file for po4 (4) : history file for o2 (5) : history file for pco2o (6) : history file for pco2a (7) : history file for oaco2 – 162
–
11.6. Usage (8) : history file for gswd
np inout and np hist is automatically set in ogfile.F90. • When CBNHSTRUN is used, the partial pressure of atmospheric CO2 , pCO2a, should be applied as additional atmospheric forcing, and the namelist inflpco2a that indicates file name should be specified. The input intervals and the file size should be the same as other atmospheric forcing files for heat flux (see Chapter 16). namelist /inflpco2a/ file_pco2a_ref file_pcof2a_ref: Partial pressure of atmospheric CO2 (ppt) • When CBNHSTRUN is not used in the oceanonly model, the ocean interacts with the onebox model that contains the uniform partial pressure of atmospheric CO2 . In this case, namelist njobco2io should be specified as follows. namelist /njobco2io/ file_atmco2_in, file_atmco2_out file_atmco2_in : Partial pressure of atmospheric CO2 (ppt) in the onebox model at the start of the integration file_atmco2_out: Partial pressure of atmospheric CO2 (ppt) in the onebox model at the end of the integration
• When NPZD is used, an NPZD model is used as the ecosystem component. – numtrc p=8 should be described in configure.in. When another value is specified, the model stops. – The compile option related to the inline expansion should appear as FFLAGS = −pvctlexpand = 10. – When the model option CHLMA94 is used, the chlorophyll concentration is considered to calculate the shortwave penetration following Morel and Antoine (1994). See section 8.3.2. – The settings of restart and history files are specified in namelist inflpt, outfpt, outfph as follows: namelist /inflpt/ fn_ptrc_in
(1:numtrc_p+np_inout)
namelist /outfpt/ fn_ptrc_out
(1:numtrc_p+np_inout)
namelist /outfph/ fn_ptrc_hist (1:numtrc_p+np_hist) np_inout = 2 np_hist = 6 fn_ptrc_in (1) : restart file for dic
(input)
(2) : restart file for alk
(input)
(3) : restart file for po4 (4) : restart file for o2 (5) : restart file for no3
(input) (input) (input)
(6) : restart file for PhyPl
(input)
(7) : restart file for ZooPl
(input)
(8) : restart file for Detri (9) : restart file for pco2o
(input) (input)
(10) : restart file for pco2a
(input)
fn_ptrc_out (1) : restart file for dic –
163
–
(output)
Chapter 11
Biogeochemical model (2) : restart file for alk (3) : restart file for po4 (4) : restart file for o2
(output) (output) (output)
(5) : restart file for no3 (6) : restart file for PhyPl
(output) (output)
(7) : restart file for ZooPl
(output)
(8) : restart file for Detri
(output)
(9) : restart file for pco2o
(output)
(10) : restart file for pco2a fn_ptrc_hist(1) : history file for dic (2) : history file for alk
(output)
(3) : history file for po4 (4) : history file for o2 (5) : history file for no3 (6) : history file for PhyPl (7) : history file for ZooPl (8) : history file for Detri (9) : history file for pco2o (10) : history file for pco2a (11) : history file for oaco2 (12) : history file for eprdc (13) : history file for gswd (14) : history file for pprdc The parameters of NPZD are set in namelist nbioNPZD. The default values are based on Schmittner et al. (2008) and listed on Table 11.1. If the parameters of Oschlies (2001) are used, the high nutrientlow chlorophyl (HNLC) region in the North Pacific is not appropriately expressed. This may be because the parameters of Oschlies (2001) are calibrated for the North Atlantic biological model. The commonly used unit of time in biological models is [day]. Thus, in the namelist, the time unit of the biological parameter is specified by using the unit [day]. In the model, the time unit is converted to seconds, [s]. namelist /nbioNPZD/ & & alphabio,abio,bbio,cbio,dkcbio,dkwbio,rk1bioNO3,rk1bioPO4,& & gbio,epsbio,phiphy,phiphyq,a_npz,phizoo,d_npz,remina, & & w_detr,fac_wdetr, & & c_mrtn, Rcn, Ron,Rnp, dp_euph, dp_mrtn, dp_eprdc, Rcaco3poc, Dcaco3
11.7 Program structure ogcm__ini  + rdinit – 164
–
11.7. Program structure  + ptrc_init  + obgc_init  + (obgcinit0)  + cbn_readdt  + co2calc ogcm__run  + part_1   
+ cbn_flx
  
  
 
 
  
  + tracer
   + part_2 
 + o2flux  + co2flux  + co2calc
 + bio_calc

 + cbn_rewrit  + writdt  + cbn_writdt
Reference Dugdale, R. C., 1967: Nutrient limitation in the sea: dynamics, identification, and significance, Limnol. Oceanogr., 12, 685–695. Eppley, R., 1972: Temperature and phytoplankton growth in the sea, Fish. Bull., 70, 1063–1085. Keeling, R. F., B. B. Stephens, R. G. Najjar, S. C. Doney, D. Archer and M. Heimann, 1998: Seasonal variations
–
165
–
Chapter 11
Biogeochemical model
in the atmospheric O2 /N2 ratio in relation to the airsea exchange of O2 , Global Biogeochem. Cycles, 12, 141164. Kishi, M. J., H. Motono, M. Kashiwai and A. Tsuda, 2001: An ecologicalphysical coupled model with ontogenetic vertical migration of zooplankton in the northwestern Pacific, J. Oceanogr., 57, 499507. Morel, A., and D. Antoine, 1994: Heating rate within the upper ocean in relation to its biooptical state, J. Phys. Oceanogr., 24, 16521665. Najjar, R., J. C. Orr, 1998: Design of OCMIP2 simulations of chlorofluorocarbons, the solubility pump and common biogeochemistry, Internal OCMIP Report, LSCE/CEA Saclay, 25 pp, GifsurYvette, France. Obata, A., and Y. Kitamura, 2003: Interannual variability of the seaair exchange of CO2 from 1961 to 1998 simulated with a global ocean circulationbiogeochemistry model, J. Geophys. Res., 108, 3377,doi:10.1029/2001 JC001088,2003. Orr, J., R. Najjar, C. Sabine, and F. Joos, 1999: AbioticHOWTO, Internal OCMIP Report, LSCE/CEA Saclay, 25 pp, GifsurYvette, France. Oschlies, A., 2001: Modelderived estimates of new production: New results point toward lower values, DeepSea Res. II, 48, 2173–2197. Redfield, A. C., B. H. Ketchum, and F. A. Richards, 1963: The influence of organisms on the composition of sea water, in The sea, vol. 2, edited by M. N. Hill, pp. 2677, WileyIntersci., New York.. Schmittner, A., A. Oschlies, H. Matthews, and E. Galbraith, 2008: Future changes in climate, ocean circulation, ecosystems, and biogeochemical cycling simulated for a businessasusual CO2 emission scenario until year 4000 AD, Global Biogeochem. Cycles, 22, GB1013,doi:10.1029/2007GB002953. Schmittner, A., A. Oschlies, H. Matthews, and E. Galbraith, 2009: Correction to “Future changes in climate, ocean circulation, ecosystems, and biogeochemical cycling simulated for a businessasusual CO2 emission scenario until year 4000 AD”, ,, Global Biogeochem. Cycles, 23. GB3005,doi:10.1029/2009GB003577 Smith, L., Y. Yamanaka, M. Pahlow, and A. Oschlies, 2009: Optimal uptake kinetics: physiological acclimation explains the pattern of nitrate uptake by phytoplankton in the ocean, Mar., Ecol. Prog. Ser., , in press. Wanninkhof, R., 1992: Relationship between wind speed and gas exchange over the ocean, J. Geophys. Res., 97, 73737382. Weiss, R. F., 1970: The solubility of nitrogen, oxygen and argon in water and seawater, DeepSea Res., 17, 721–735. Yamanaka, Y., and E. Tajika, 1996: The role of the vertical fluxes of particulate organic matter and calcite in the oceanic carbon cycle: Studies using an ocean biogeochemical general circulation model, Global Biogeochem. Cycles, 10, 361382.
– 166
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11.7. Program structure
Table 11.1. Parameters used for the NPZD ecosystem component (NPZD). description unit
namelist name alphabio abio bbio
: : :
cbio
:
PARbio dkcbio dkwbio rk1bioNO3
: : : :
Initial slope of PI curve Maximum growth rate parameter Maximum growth rate = a * b ** (c * T)
m−2 )−1
[(W [day−1 ]
# 0.1d0 # 0.2d0 # 1.066d0 # 1.d0
Photosynthetically active radiation Light attenuation due to phytoplankton Light attenuation in the water Halfsaturation constant for NO3 uptake
[m−1
(mol [m−1 ] [mol m−3 ]
rk1bioPO4 : alpha_ou :
Halfsaturation constant for PO4 uptake Fitting constant for Optical Uptake kinetics
[mol m−3 ]
gbio
Maximum grazing rate
[day−1 ]
:
default value
day−1 ]
m−3 )−1 ]
m−3 )−2
# 0.43d0 # 0.03d3 # 0.04d0 # 0.7d3 # 0.0d0 # 0.19d0
day−1 ]
# 1.575d0 # 1.6d6 # 0.014d0
epsbio phiphy
: :
Prey capture rate Specific mortality/recycling rate
[(mol [s−1 ]
phiphyq a_npz phizoo
: : :
Quadratic mortality rate Assimilation efficiency Quadratic mortality of zooplankton
[(mol m−3 )−1 day−1 ]
# 0.05d3 # 0.925d0
d_npz remina
: :
Excretion Remineralization rate
[(mol m−3 )−1 day−1 ] [day−1 ] [day−1 ]
# 0.34d3 # 0.01d0 # 0.048d0
[m day−1 ]
# 2.0d0 # 3.d0
w_detr : fac_wdetr : : : : c_mrtn :
Sinking velocity Arbitrary parameter for numerical stability. When the concentration of detritus in the n+1 st level is higher than fac wdetri times that in the n th level, w detri is set to 0 between n and n+1 level. Dimensionless scaling factor for Martin et al (1987)
# 0.858d0
:
Phi(z) = Phi(zo) * (z/dp mrtn)**(c mrtn)
Rcn Ron Rnp
: : :
Molar elemental ratio (C/N) Molar elemental ratio (O2 /N) Molar elemental ratio (N/P)
dp_euph dp_mrtn
: :
Maximum depth of euphotic zone Characteristic depth of martin curve
[m] [m]
# 150.d0 # 400.d0
dp_eprdc
:
[m]
# 126.d0
Rcaco3poc : Dcaco3 :
The depth where the bioexport is diagnosed This value should be less than dp mrtn. CaCO3 over nonphotosynthetical POC production ratio CaCO3 remineralization efolding depth
[m]
# 3500.d0
shwv_intv :
Interval for calculating the irradiance and lightlimited
[min]
# 10.d0
: :
# 7.d0 # 10.d0 # 16.d0
growth rate. This must be a divisor of the time step for tracer.
–
167
–
# 0.05d0
Part IV
Miscellaneous
–
169
–
Chapter 12
Basics of the finite difference method
This chapter describes the basics of finite difference methods for solving differential equations. The general principles of the finite differencing methods are introduced using the diffusion equation as an example in Section 12.1. Sections 12.2 and 12.3 describe applying finite difference methods of time and space derivatives in differential equations. Considerations in finitedifference methods for advectiondiffusion equations are discussed in Section 12.4. An implicit method for solving the diffusion equation is described in Section 12.5. Durran (1999) treats the basics of finite difference methods for solving differential equations of advection and diffusion in geophysical fluid dynamics.
12.1
Diffusion equation
As an example, consider an initialboundary value problem expressed by a onedimensional diffusion equation (heat conductive equation), ∂T ∂ 2T (12.1) =κ 2. ∂t ∂x Given T (x, 0) = f (x) as the initial distribution and T (0,t) = T (L,t) = 0 as the boundary condition, the analytical solution is ∞ 2 (12.2) T (x,t) = ∑ fm e−κ km t sin(km x), m=0
where
2 L mπ f (x) sin(km x)dx, km = . (12.3) L 0 L Next, consider the finite difference method to get the solution numerically. In the finite difference method, grids fm =
are set with a finite increment in space and time, and each term in the equation is evaluated at each grid using T jn = T (x j ,tn ). For example, n n T jn+1 − T jn T j+1 − 2T jn + T j−1 , (12.4) =κ Δt Δx2 where Δt = tn+1 − tn and Δx = x j+1 − x j . Distribution at the new time level T n+1 can then be calculated if T n is known. This finite difference equation is identical to the original differential equation (12.1) in the limit Δt → 0, Δx → 0 (consistency). If the initial distribution is assumed to be f (x) = T0 sin k1 x, the solution of the finite difference equation (12.4) for t = tn is (12.5) T jn = λ n T0 sin k1 x j , where
2κ Δt (1 − cos k1 Δx). (12.6) Δx2 In order to suppress oscillation and divergence of the solution (stability), 0 < λ < 1 is necessary and Δx and Δt
λ = 1−
must be set to satisfy this condition. This solution is identical to the analytical solution in the limit of Δt → 0, Δx → 0 (convergence). – 171
–
Chapter 12
Basics of the finite difference method
To summarize, the finite difference method that satisfies consistency, stability, and convergence is the necessary condition for an accurate solution.
12.2 Finite difference expressions for time derivatives The following four finite difference expressions are employed for the time derivatives in MRI.COM:
forward : backward : Matsuno : leapfrog :
T n+1 − T n = F(T n ) Δt T n+1 − T n = F(T n+1 ) Δt T ∗ n+1 − T n T n+1 − T n = F(T n ), = F(T ∗ n+1 ) Δt Δt T n+1 − T n−1 = F(T n ). 2Δt
(12.7) (12.8) (12.9) (12.10)
The scheme used in the previous section is the forward scheme. The forward, backward, and Matsuno schemes use the values at two time levels and are accurate to O(Δt), while the leapfrog scheme uses three time levels and is accurate to O(Δt 2 ). Basically, the leapfrog scheme is employed in MRI.COM because of its higher order accuracy. However, the leapfrog scheme cannot be applied to the diffusion equation. A solution by the finite difference method using the leapfrog scheme is given by T jn = (Ta λan + Tb λbn ) sin k1 x j , where
α2 + 4
, λb = −
−α −
4κ Δt (1 − cos k1 Δx)). (12.12) (α ≡ 2 2 Δx2 Because λb < −1 for arbitrary values of α , the divergent mode with oscillation is always included (computational mode). In order to avoid this computational mode, the forward scheme is employed for diffusion and viscosity
λa = −
−α +
(12.11)
α2 + 4
terms in MRI.COM. When the diffusion and viscosity coefficients are very large as in the surface mixed layer, the time step has to be unusually small for the stability of the forward scheme according to (12.6). In such a case, the backward scheme is used for vertical diffusion and viscosity (implicit method; see Section 12.5). Though the time integration at each point can proceed without referring to the result of other points by the forward, leapfrog, and Matsuno schemes, it must be done by solving combined linear equations in the backward scheme (see Section 12.5). The Matsuno scheme is useful for suppressing the computational mode in the leapfrog scheme. By defaults, the Matsuno scheme is used once per twelve steps of the leapfrog scheme in MRI.COM. This interval can be changed at run time using a namelist parameter (matsuno int of the namelist group njobp). It should be noted that the Matsuno scheme needs twice as many numerical operations as the forward and leapfrog schemes.
12.3 Finite difference expression for space derivatives Let us consider a onedimensional advection equation,
∂T ∂T = −u , ∂t ∂x – 172
–
(12.13)
12.3. Finite difference expression for space derivatives where u is a constant velocity. The solution is T (x,t) = T (x − ut, 0).
(12.14)
Using the leapfrog scheme for time differencing, the finite difference equation can be written as follows: T jn+1 − T jn−1 2Δt
= −u
n n T j+ 1 − T j− 1 2
2
Δx
,
(12.15)
n n where T j− 1 and T j+ 1 are the values at the left and right (in the same way for updown and frontrear grid cells) 2 2 faces of the grid cell for x j , i.e., values at x j− 1 and x j+ 1 . The point x j− 1 is defined as the central point between x j 2
2
2
and x j−1 . Because the transport of T at the boundary that enters a grid cell is identical to that leaving the adjacent grid cell, the total T in the whole system is conserved in this finite difference equation. n There are several methods to decide T j− 1 using a value at a single or multiple grid points. The following are two simple and popular formulations,
2
upstream finite difference :
n n n n T j− 1 = T j−1 (u > 0), T j− 1 = T j (u < 0), 2
central finite difference :
n T j− 1 =
(12.16)
2
n T j−1 + T jn
2
2
.
(12.17)
The former is accurate to O(Δx), and the latter is accurate to O(Δx2 ). In central finite differencing, the expression for (12.15) is T jn+1 − T jn−1 2Δt
= −u
n n − T j−1 T j+1 . 2Δx
(12.18)
Assuming the solution to be T (x,t) = τ (t)e−ikx ,
τ n+1 = τ n−1 + 2iατ n , where α ≡
uΔt sin kΔx. Δx
It is stable (neutral) if α  ≤ 1. To be stable for any wave number, uΔt Δx ≤ 1
(12.19)
(12.20)
must be satisfied (CFL condition). However, if τ n = τ 0 e−inΔθ , Δθ = − sin−1 [μ sin kΔx](where μ ≡
uΔt ). Δx
(12.21)
Expanding the r.h.s. by a Taylor expansion we obtain Δθ
1 −μ sin kΔx − (μ sin kΔx)3 6 μ (kΔx)3 μ 3 (kΔx)3 −μ kΔx + − 6 6 (kΔx)2 = −μ kΔx 1 − (1 − μ 2 ) . 6
(12.22)
This means that the phase of the solution from this finite difference scheme is delayed relative to that of analytical solution, depending on its wavenumber (numerical dispersion). Therefore, a distribution with maxima and minima that do not exist in the initial distribution arises. However, this method is popularly used since the kinetic energy is conserved by employing the central difference in the advection term in the equation of motion. Moreover, –
173
–
Chapter 12
Basics of the finite difference method
the “Arakawa method,” which can nearly conserve the enstrophy (squared vorticity) for horizontally nondivergent flows, is adopted in MRI.COM by using the central difference. This topic is treated in Chapter 5. Using the upstream finite difference, the finite difference equation (12.15) is T jn+1 − T jn−1
n T jn − T j−1 . Δx
(12.23)
∂ T uΔx ∂ 2 T + O(Δx2 ). + ∂x 2 ∂ x2
(12.24)
2Δt
= −u
Expanding the r.h.s. by a Taylor expansion we obtain: −u
The second term has the diffusion (heat conductive) form (which disappears in the central finite differencing). Actually, the initial distribution diffuses when the advection equation is solved by the upstream finite difference (numerical diffusion). The third order schemes (QUICK, QUICKEST, and UTOPIA) can be used in MRI.COM to suppress the numerical dispersion and diffusion somewhat in the advection calculation for tracers, but not completely. The grid boundary value is set in QUICK as n T j− 1 = 2
n n n n −T j−2 3T j−1 + 6T j−1 + 3T jn + 6T jn − T j+1 n = (u > 0), T j− (u < 0). 1 8 8 2
(12.25)
The QUICKEST method uses the time averaged value at the grid boundary as the tracer value to be transported, considering the change of the value there by advection during one time step. UTOPIA is a multidimensional extension of QUICKEST. The details of these schemes are described in Chapter 13.
12.4 Finite differencing of advectiondiffusion equation According to the above restriction, when advectiondiffusion equations (2.14) and (2.15) are expressed in finite difference form using the leapfrog scheme, it is necessary to use present (previous) time level for advection (diffusion) term. The following finite difference equation is then employed: T n+1 − T n−1 2Δt Sn+1 − Sn−1 2Δt
= −A (T n ) + D(T n−1 )
(12.26)
= −A (Sn ) + D(Sn−1 ).
(12.27)
When the vertical diffusion term is very large, D(T n+1 ) is used instead of D(T n−1 ). This formula is an implicit scheme and is described in the next section.
12.5 Implicit method for vertical diffusion equation Turbulent mixing is parameterized by using high vertical diffusivity and viscosity determined by boundary layer models, which was treated in Chapter 7. The time step must be set very small to keep the calculation stable when viscosity and diffusivity are very high, since the time tendency becomes very large due to the rapid mixing. To avoid this problem, the implicit method uses the advanced (mixed) state for evaluating viscosity and diffusivity, unlike the normal explicit method where previous or present values are used. Expressing the present time step as n and the time step before and after as n ± 1, the finitedifference method is applied to the temperature equation using the leapfrog scheme. The diffusion term is written separately using the – 174
–
12.5. Implicit method for vertical diffusion equation (n − 1) step for the horizontal direction and the (n + 1) step for the vertical direction, (T n+1 − T n−1 )/2Δt = −A (T n ) + DH (T n−1 ) + DV (T n+1 ).
(12.28)
Putting all the terms involving T n+1 on the l.h.s., T n+1 − 2ΔtDV (T n+1 ) = T n−1 + 2Δt(−A (T n ) + DH (T n−1 ))
(12.29)
is obtained, which is an algebraic equation for T n+1 . The finite difference form is rewritten specifically as Tkn+1 − 2Δt
1 n+1 n+1 κk− 1 (Tk−1 − Tkn+1 )/Δzk− 1 − κk+ 1 (Tkn+1 − Tk+1 )/Δzk+ 1 2 2 2 2 Δzk
(12.30)
= Tkn−1 + 2Δt(−A (Tkn ) + DH (Tkn−1 )). By putting a=
2Δt κk− 1 2
Δzk Δzk− 1
, b = 1 + a + c, c =
2
2Δt κk+ 1 2
Δzk Δzk+ 1
,
(12.31)
2
we get, n+1 n+1 + bTkn+1 − cTk+1 = Tkn−1 + 2Δt(−A (Tkn ) + DH (Tkn−1 )). −aTk−1
(12.32)
Setting Fk ≡ −A (Tkn ) + DH (Tkn−1 ), this is expressed in the matrix form as ⎛
b ⎜ ⎜ −a ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
−c b −a
⎞⎛ −c b .. .
−c .. . −a
..
.
b −a
T1n+1
⎞
⎛
T1n−1 + 2ΔtF1
⎞
⎟ ⎜ n+1 ⎟ ⎜ ⎟ ⎟ ⎜ T2 ⎟ ⎜ T2n−1 + 2ΔtF2 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ T n+1 ⎟ ⎜ T n−1 + 2ΔtF ⎟ 3 ⎟⎜ 3 ⎟ ⎜ ⎟ 3 ⎟⎜ . ⎟ ⎜ ⎟ .. ⎟⎜ . ⎟ = ⎜ ⎟. . ⎟⎜ . ⎟ ⎜ ⎟ ⎟ ⎜ n+1 ⎟ ⎜ n−1 ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ −c ⎟ ⎜ TKM−2 ⎟ ⎜ TKM−2 + 2ΔtFKM−2 ⎟ ⎟ ⎜ n+1 ⎟ ⎜ n−1 ⎟ b −c ⎠ ⎝ TKM−1 ⎠ ⎝ TKM−1 + 2ΔtFKM−1 ⎠ n−1 n+1 −a b TKM TKM + 2ΔtFKM
(12.33)
The l.h.s. has the form of the tridiagonal matrix.
12.5.1
A solution of tridiagonal matrix
In general, simultaneous linear equations for n variables with tridiagonal matrix coefficients ⎛
B1
⎜ ⎜ A2 ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎞⎛
C1 B2 .. .
C2 .. . An−1
..
.
Bn−1 An
X1
⎞
⎛
D1
⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ X2 ⎟ ⎜ D2 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎟⎜ . ⎟ = ⎜ . ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ Cn−1 ⎠ ⎝ Xn−1 ⎠ ⎝ Dn−1 ⎠ Bn Xn Dn
are solved using the Thomas method, which is modified from LU decomposition,
–
⎞
175
–
(12.34)
Chapter 12
Basics of the finite difference method
P1
= C1 /B1
(12.35)
Q1
(12.36)
Xn
= D1 /B1 Ck = Bk − Ak Pk−1 Dk − Ak Qk−1 = Bk − Ak Pk−1 = Qn
Xk
= Qk − Pk Xk+1
Pk Qk
(2 ≤ k ≤ n − 1) (2 ≤ k ≤ n)
(12.37) (12.38) (12.39)
(1 ≤ k ≤ n − 1).
(12.40)
References Durran, D. R., 1999: Numerical methods for wave equations in geophysical fluid dynamics, SpringerVerlag, 465pp.
– 176
–
Chapter 13
Tracer advection schemes
The default tracer advection scheme of MRI.COM is the Quadratic Upstream Interpolation for Convective Kinematics (QUICK; Leonard, 1979) as described in Chapter 6. Other options with higher accuracy are described in this chapter. Section 13.1 describes the QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979) for vertical advection (option QUICKEST). Section 13.2 describes the Uniformly ThirdOrder Polynomial Interpolation Algorithm (UTOPIA; Leonard et al., 1993) for horizontal advection (option UTOPIA). The UTOPIA scheme is a twodimensional generalization of the QUICKEST scheme. For these schemes, a flux limiter that prevents unrealistic extrema should be used (Leonard et al., 1994). A flux limiter for the QUICKEST scheme is used when option ZULTIMATE is specified. A flux limiter for the UTOPIA scheme is used when option ULTIMATE is specified. The above schemes seek to improve the accuracy by refining the finitedifference expression at the cell faces. There is another approach that seeks to improve the accuracy by considering the distribution within the cell. The second order moment (SOM; Prather, 1986) scheme takes this approach and is available in MRI.COM through the option SOMADVEC (Section 13.3).
13.1
QUICKEST for vertical advection
This section describes the specific expression and the accuracy of the QUICK with Estimated Streaming Terms (QUICKEST; Leonard, 1979) for vertical advection. Consider a onedimensional equation of advection for incompressible fluid
∂T ∂ + (wT ) = 0, ∂t ∂z
(13.1)
where w is a constant. Although the velocities are not constants in the real three dimensional ocean, we assume a constant velocity for simplicity. Following the notation of vertical grid points and their indices (Section 3.2), tracers are defined at the center (k + 12 ) of the vertical cells and vertical velocities are defined at the top (k) and bottom (k + 1) faces of the vertical cells. The following relation holds for the vertical grid spacings: Δzk =
Δzk+ 1 + Δzk− 1 2
2
2
.
(13.2)
In QUICKEST, the distribution of tracer T is defined using the second order interpolations, and the mean value during a time step at the cell face (boundary of two adjacent tracer cells) is calculated. The coefficients for the
–
177
–
Chapter 13
Tracer advection schemes
second order interpolation are calculated first. A Taylor expansion of T about point zk gives Tk− 3 2
Tk− 1 2
Tk+ 1 2
Tk+ 3 2
= c0 + c1 = c0 + c1 = c0 − c1
Δzk− 3 2
+ Δzk− 1
Δzk− 1
Δz2k− 1
2
+ c2
2
+ c2
2
2
2
4
Δzk− 3 2
2
2 + Δzk− 1 2
+ O(Δz3 ),
+ O(Δz3 ),
(13.4)
Δz2k+ 1
Δzk+ 1
2 + c2 + O(Δz3 ), 2 4 2 Δzk+ 3 Δzk+ 3 2 2 = c0 − c1 + O(Δz3 ). + Δzk+ 1 + c2 + Δzk+ 1 2 2 2 2 2
(13.3)
(13.5) (13.6)
Coefficients c0 , c1 , and c2 can be solved using three of the four equations (13.3), (13.4), (13.5), and (13.6). The three upstreamside equations are chosen. When w > 0 (w < 0), equations (13.4), (13.5), and (13.6) ((13.3), (13.4), and (13.5)) are used. The solution is as follows. c0
=
c1
=
c2
=
Tk− 1 Δzk+ 1 + Tk+ 1 Δzk− 1 2
2
2
2
2Δzk
−
Δzk+ 1 Δzk− 1 2
2
4
Δzk− 1 − Δzk+ 1 2 2 − c2 , Δzk 2 T −T T 1 −T 3 k− 21 k+ 21 k+ 2 k+ 2 1 − Δz Δzk +Δzk+1 Δz k+1 T −Tk T 1 −T 1 3 1 k− k− k− k+ 2 1 2 2 2 − Δz +Δz Δz Δz
c2 ,
Tk− 1 − Tk+ 1 2
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
2
k−1
k
k−1
k
(13.7) (13.8)
(w > 0), (13.9) (w < 0).
Next, equation (13.1) is integrated over one time step and one grid cell. t n+1
dt
tn
zk zk+1
dz
The r.h.s. of (13.10) can be written as −
∂T =− ∂t
t n+1 tn
t n+1
dt
tn
zk zk+1
dz
∂ (wT ). ∂z
dt(wu Tu − wl Tl ),
(13.10)
(13.11)
where subscript u (l) denotes z = zk (z = zk+1 ). Assuming w does not depend on time, t n+1 tn
dtTl =
0 −wl Δt
[cn0 + cn1 ξ + cn2 ξ 2 + O(Δz3 )]
dξ . wl
(13.12)
Thus expression (13.11) becomes −Δt(wu Tun − wl Tln ) + O(Δz3 wΔt),
(13.13)
where Tln
=
1 wl Δt
= cn0 −
0
(cn0 + cn1 ξ + cn2 ξ 2 )d ξ −wl Δt cn1 cn wl Δt + 2 w2l Δt 2 . 2
3
(13.14)
Using up to the second order terms of a Taylor expansion, the l.h.s. of (13.10) can be written as follows: ⎡ ⎤ t n+1 zk Δz2k+ 1 ∂T n+1 n n+1 n 3 2 dt dz (Tzz 1 − Tzz k+ 1 ) + O(Δz )⎦ , (13.15) = Δzk ⎣Tk+ 1 − Tk+ 1 + k+ 2 ∂t 24 2 2 2 tn zk+1 – 178
–
13.2. UTOPIA for horizontal advection where Tzz n+1 k+ 12
− Tzz nk+ 1 2
∂ Tzz n = Δt + O(Δt 2 ) ∂ t k+ 1 2 ,n + ∂2 ∂ + O(Δt 2 ) = −Δt 2 (uT ) 1 ∂z ∂z k+ 2
∂ = −Δt (wTzz )nk+ 1 + O(Δt 2 ) ∂z 2 n wu Tzz u − wl Tzz nl + O(wΔtΔx) + O(Δt 2 ). = −Δt Δzk+ 1
(13.16)
2
The expression for the r.h.s. of (13.15) becomes ⎡ ⎤ Δz2k+ 1 w T n − u T n u zz u l zz l n 2 Δt Δzk ⎣T n+11 − Tk+ + O(Δz3 )⎦ + O(wΔtΔx3 ) + O(Δz3 Δt 2 ). 1 − k+ 2 24 Δzk+ 1 2
(13.17)
2
Based on (13.13) and (13.17), the discretized forecasting equation is expressed as follows: ⎡ ⎤ Δz2k+ 1 Δt 2 ⎣wu Tun − wl T n − Tkn+1 = Tkn − (wu Tzz nu − wl Tzz nl )⎦ + O(α Δz3 ) + O(Δz2 Δt 2 ), l Δzk+ 1 24
(13.18)
2
where
α
≡
Tzz nl
=
wΔt < 1, Δz 2c2 + O(Δz).
(13.19) (13.20)
The accuracy of equation (13.18) is max(O(Δz3 ), O(Δz2 Δt 2 )).
13.2
UTOPIA for horizontal advection
The Uniformly Third Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al., 1993) is an advection scheme that can be regarded as a multidimensional version of QUICKEST. In MRI.COM, horizontally twodimensional advection is calculated using UTOPIA. Vertical advection is calculated separately using QUICKEST. Since grid intervals could be variable in both zonal and meridional directions in MRI.COM, UTOPIA is formulated based on a variable grid interval. It is assumed that the tracer cell is subdivided by the boarderlines of the velocity cells into four boxes with (almost) identical area. Consider an equation of advection:
∂T 1 ∂ 1 ∂ (hψ uT ) + (hμ vT ) = 0. + ∂t hμ hψ ∂ μ hμ hψ ∂ ψ
(13.21)
Integrated over a tracer cell and for one time step, ψL +ΔψL /2
=
dψ
μL +ΔμL /2
d μ (χ n+1 − χ n )
ψL −ΔψL /2 μL −ΔμL /2 n n −Δt(ur Tr Δyr − unl Tln Δyl + vnu Tun Δxu − vnd Tdn Δxd ),
(13.22)
where χ ≡ hμ hψ T and Trn etc. on the r.h.s. are the face values described later. On the l.h.s. of (13.22), the secondorder interpolation of χ is used to integrate the terms. The Taylor expansion of χ about L is given as follows (see Figure 13.1 for the label of the point):
χ = χL + a10 (μ − μL ) + a20 (μ − μL )2 + a01 (ψ − ψL ) + a02 (ψ − ψL )2 + a11 (μ − μL )(ψ − ψL ). – 179
–
(13.23)
Chapter 13
Tracer advection schemes
Then values at points E, W, N, and S are
NW
N
W
L
E
SW
S
SE
WW
u
r
SS l
d
Figure 13.1. Labels of tracer grid points (upper case characters) and faces (lower case characters).
χE
=
χL + a10 Δμr + a20 Δμr2 ,
(13.24)
χW
=
χL − a10 Δμl + a20 Δμl2 ,
(13.25)
χN
=
χL + a01 Δψu + a02 Δψu2 ,
(13.26)
=
χL − a01 Δψd + a02 Δψd2 ,
(13.27)
χS where
ΔψL + ΔψN , 2 ΔψL + ΔψS , Δψd ≡ 2 Δ μL + Δ μE Δ μr ≡ , 2 ΔμL + ΔμW Δ μl ≡ . 2 Using these known values, the following parameters are obtained, χ E − χL χL − χW Δ μl + Δμr Δ μr Δ μl , a10 = Δμr + Δμl χE − χL χL − χW − Δ μr Δ μl a20 = , Δ μr + Δ μl χ N − χL χL − χ S Δψd + Δψu Δψ u Δψd a01 = , Δψu + Δψd χN − χL χ L − χ S − Δψu Δψd a02 = . Δψu + Δψd Δ ψu
≡
Substituting (13.23) into the l.h.s. of (13.22) yields 2 2 Δ Δ μ ψ L n+1 − an20 ) + − an02 ) . ΔμL ΔψL χLn+1 − χLn + L (an+1 (a 12 20 12 02 – 180
–
(13.28) (13.29) (13.30) (13.31)
(13.32)
(13.33)
(13.34)
(13.35)
(13.36)
13.2. UTOPIA for horizontal advection Using equation (13.21), the following approximation is allowed: , + hψ r unr Tμnμ r − hψ l unl Tμnμ l hμ u vnu Tμnμ u − hμ d vnd Tμnμ d n+1 n , + a20 − a20 = −Δt Δ μL ΔψL , + n n n n n n hμ u vnu Tψψ hψ r unr Tψψ r − hψ l ul Tψψ l u − hμ d vd Tψψ d n , + − a = −Δt an+1 02 02 Δ μL ΔψL
(13.37) (13.38)
where Tμnμ r is the value of the second order derivative at the right face r, whose expression is similar to that of c20 described later. Therefore, under a suitable approximation, TLn+1 = TLn −
Δt n ˜n (u T Δyr − unl T˜ln Δyl + vnu T˜un Δxu − vnd T˜dn Δxd ), ΔSL r r
(13.39)
where T˜ln T˜dn
ΔμL2 n ΔψL2 n Tμ μ l − T , 24 24 ψψ l 2 2 Δμ ΔψL n = Tdn − L Tμnμ d − . T 24 24 ψψ d = Tln −
(13.40) (13.41)
Next, the expressions for Tln and Tdn are required. The term Tln is the average over the hatched area of Figure 13.2, and the values of T n are given as the second order interpolation about l of Figure 13.1. Similar operations will be used to obtain the expression for Tdn .
W
L
vlΔt
ulΔt
Figure 13.2. Area used to average tracer values for the face l First, Taylor expansions of T n about l and d are written as follows:
T n l =c00 + c10 (μ − μl ) + c20 (μ − μl )2 + c01 (ψ − ψL ) + c02 (ψ − ψL )2 + c11 (μ − μl )(ψ − ψL ),
(13.42)
T d =d00 + d10 (μ − μl ) + d20 (μ − μl ) + d01 (ψ − ψL ) + d02 (ψ − ψL ) + d11 (μ − μl )(ψ − ψL ).
(13.43)
n
2
2
–
181
–
Chapter 13
Tracer advection schemes
The T n values at eight points around l are, Δ μL Δμ 2 + c20 L , 2 4 ΔμW ΔμW2 c00 − c10 + c20 , 2 4 Δ μE Δ μE 2 + c20 ΔμL + c00 + c10 ΔμL + , 2 2 ΔμWW ΔμWW 2 + c20 ΔμW + c00 − c10 ΔμW + , 2 2 Δ μL TLn + c01 Δψu + c02 Δψu2 + c11 Δψu , 2 Δ μL TLn − c01 Δψd + c02 Δψd2 − c11 Δ ψd , 2 ΔμW TWn + c01 Δψu + c02 Δψu2 − c11 Δψu , 2 ΔμW TWn − c01 Δψd + c02 Δψd2 + c11 Δψ d . 2
TLn
= c00 + c10
(13.44)
TWn
=
(13.45)
TEn
=
n TWW
=
TNn
=
TSn
=
n TNW
=
n TSW
=
(13.46) (13.47) (13.48) (13.49) (13.50) (13.51)
To obtain all six coefficients, six of these equations (points) are used. The equations are chosen according to the following flow direction. unl > 0,
vnl > 0
⇒ L, W, WW, S, NW, SW
(13.52)
unl < 0,
vnl > 0
⇒ L, W, E, N, S, SW
(13.53)
unl
vnl
⇒ L, W, WW, N, NW, SW
(13.54)
⇒ L, W, E, N, S, NW
(13.55)
> 0,
unl < 0,
<0
vnl < 0
From equations (13.44) and (13.45), c00
=
c10
=
ΔμL ΔμW ΔμW TLn + ΔμL TWn − c20 , 2Δμl 4 ΔμL − ΔμW TLn − TWn − c20 . Δμl 2
(13.56) (13.57)
When unl > 0, from (13.44) and (13.47),
c20
=
n TLn − TWn TWn − TWW − Δ μl Δμll , Δμl + Δμll ΔμW + ΔμWW where Δμll ≡ . 2
(13.58)
Using equations (13.50) and (13.51), n n − TWn TWn − TSW TNW − Δψu Δψd c02 = . Δψu + Δψd
(13.59)
When unl < 0, from (13.45) and (13.46), TEn − TLn TLn − TWn − Δ μr Δ μl c20 = . Δμr + Δμl –
182
–
(13.60)
13.2. UTOPIA for horizontal advection Using equations (13.48) and (13.49),
TNn − TLn TLn − TSn − Δψu Δψd c02 = . Δψu + Δψd
(13.61)
When vnl > 0, from (13.49) and (13.51), c01
=
c11
=
n ) ΔμW (TLn − TSn ) + ΔμL (TWn − TSW + c02 Δψd , 2Δμl Δψd n − TWn − TSn + TLn TSW . Δμl Δψd
(13.62) (13.63)
When vnl < 0, from (13.48) and (13.50), c01
=
c11
=
n ΔμW (TNn − TLn ) + ΔμL (TNW − TWn ) − c02 Δψu , 2Δμl Δψu n + TWn TNn − TLn − TNW . Δ μl Δ ψ u
(13.64) (13.65)
Next, using equation (13.43), the T n values at eight points around d are TLn
=
TSn
=
TNn
=
n TSS
=
TEn
=
TWn
=
n TSE
=
n TSW
=
ΔψL Δψ 2 + d02 L , 2 4 ΔψS Δψ 2 d00 − d01 + d02 S , 2 4 ΔψN ΔψN 2 + d02 ΔψL + d00 + d01 ΔψL + , 2 2 ΔψSS ΔψSS 2 + d02 ΔψS + d00 − d01 ΔψS + , 2 2 Δψ L TLn + d10 Δμr + d20 Δμr2 + d11 Δ μr , 2 ΔψL TLn − d10 Δμl + d20 Δμl2 − d11 Δμl , 2 Δψ S TSn + d10 Δμr + d20 Δμr2 − d11 Δ μr , 2 ΔψS TSn − d10 Δμl + d20 Δμl2 + d11 Δ μl . 2 d00 + d01
(13.66) (13.67) (13.68) (13.69) (13.70) (13.71) (13.72) (13.73)
From equations (13.66) and (13.67), d00
=
d01
=
ΔψL ΔψS ΔψS TLn + ΔψL TSn , − d02 2Δψd 4 ΔψL − ΔψS TLn − TSn . − d02 Δψd 2
(13.74) (13.75)
When vnd > 0, from (13.66) and (13.69),
d02
From (13.72) and (13.73),
=
n TLn − TSn TSn − TSS − Δψd Δψdd , Δψd + Δψdd ΔψS + ΔψSS where Δψdd ≡ . 2
n n − TSn TSn − TSW TSE − Δμr Δ μl . d20 = Δ μr + Δ μl
–
183
–
(13.76)
(13.77)
Chapter 13
Tracer advection schemes
When vnd < 0, from (13.67) and (13.68), TNn − TL TLn − TSn − Δψu Δψd d02 = . Δψu + Δψd From (13.70) and (13.71),
(13.78)
TEn − TLn TLn − TWn − Δ μr Δ μl . d20 = Δμr + Δμl
(13.79)
When und > 0, from (13.71) and (13.73), d10
=
d11
=
n ΔψS (TLn − TWn ) + ΔψL (TSn − TSW ) + d20 Δμl , 2Δψd Δμl n TLn − TSn − TWn + TSW . Δψd Δμl
(13.80) (13.81)
When und < 0, from (13.70) and (13.72), d10
=
d11
=
n − TSn ) ΔψS (TEn − TLn ) + ΔψL (TSE − d20 Δμr , 2Δψd Δμr n − TLn + TSn TEn − TSE . Δψd Δμr
(13.82) (13.83)
The value of Tln is the average of T n over the hatched area of Figure 13.2. Defining
ξln = we have Tln
= +
a ξln ΔtΔψL
+
ψL −ΔψL /2
unl , hμ l
ψL +ΔψL /2 μl
ψL −ΔψL /2
μl −ξln Δt
ηln =
vnl , hψ l
(13.84)
T nd μ dψ ⎤
n
μl + ξln (ψ −ψL +ΔψL /2) η l
ψL −ΔψL /2−ηln Δt μl −ξln Δt
{T n (ψ ) − T n (ψ + ΔψL )}d μ d ψ ⎦
, + 1 n 1 1∗ n 2 2 = c00 − ηl Δt c01 + Δψ + (ηl Δt) c02 2 12 L 3 1 1 1 − ξln Δt c10 + (ξln Δt)2 c20 + † ξln ηln Δt 2 c11 . 2 3 3
(13.85)
This is the result for unl > 0 and vnl > 0. The result is the same independent of the sign of unl and vnl . Similarly, Tdn
=
1 1 d00 − ηdn Δt d01 + (ηdn Δt)2 d02 2 , + 3 1 n 1 1 1 − ξd Δt d10 + ΔμL2 + ‡ (ξdn Δt)2 d20 + § ξdn ηdn Δt 2 d11 , 2 12 3 3
where
ξdn = ∗1 2 †1 4 ‡1 2 §1 4
und , hμ d
ηdn =
in CCSR model in CCSR model in CCSR model in CCSR model
– 184
–
vnd . hψ d
(13.86)
(13.87)
13.3. Second Order Moment (SOM) scheme Therefore, T˜ln
1 n 2 ΔμL2 1 n c20 = c00 − ξl Δt c10 + (ξl Δt) − 2 3 12
T˜dn
1 1 1 − ηln Δt c01 + (ηln Δt)2 c02 + ξln ηln Δt 2 c11 , 2 3 3 1 n 2 ΔψL2 1 n d02 = d00 − ηd Δt d01 + (ηd Δt) − 2 3 12 1 1 1 − ξdn Δt d10 + (ξdn Δt)2 d20 + ξdn ηdn Δt 2 d11 . 2 3 3
(13.88)
(13.89)
Finally, we describe how to derive the boundary conditions. Since the face values of the tracers are calculated through the second order interpolation, the value of a tracer at a point over land is sometimes necessary. For that case, the value should be appropriately decided by using the tracer values at the neighboring points in the sea. Since ocean models generally assume that there is no flux of tracers across landsea boundary, the provisional value over land should be given so as not to create a normal gradient at the boundary. When the face value of a tracer at boundary l is calculated, W and L are not land, but either N or S may be land, and either NW or SW may be land. When N or S is land, the landsea boundary runs at the center of L in the zonal direction. It is reasonable to assume that the value of land grid N or S must not cause any meridional tracer gradient at L set by second order interpolation using the values at grids N, L, and S. Thus, we set (TNn − TLn )Δψd2 = (TSn − TLn )Δψu2 .
(13.90)
When NW or SW is a land grid, the following should be assumed. n n − TWn )Δψu2 (TNW − TWn )Δψd2 = (TSW
(13.91)
n (TWW − TWn )Δμl2 = (TLn − TWn )Δμll2 .
(13.92)
(TEn − TLn )Δμl2 = (TWn − TLn )Δμr2 .
(13.93)
When WW is a land grid,
When E is a land grid,
Similar boundary conditions are specified for face d.
13.3 Second Order Moment (SOM) scheme 13.3.1 Outline The Second Order Moment (SOM) advection scheme by Prather (1986) seeks to improve the accuracy by treating the tracer distribution within a grid cell, unlike the scheme that aims to calculate the tracer flux at the boundary of grid cells with high accuracy. It is assumed that the distribution of tracer f in a grid cell (0 ≤ x ≤ X, 0 ≤ y ≤ Y, 0 ≤ z ≤ Z; volume V = XY Z) can be represented using second order functions as follows: f (x, y, z) = a0 + ax x + axx x2 + ay y + ayy y2 + az z + azz z2 + axy xy + ayz yz + azx zx.
(13.94)
Prather (1986) expressed the above as a sum of orthogonal functions Ki (x, y, z); f (x, y, z) = m0 K0 + mx Kx + mxx Kxx + my Ky + myy Kyy + mz Kz + mzz Kzz + mxy Kxy + myz Kyz + mzx Kzx , – 185
–
(13.95)
Chapter 13
Tracer advection schemes
where the orthogonal functions are given as follows: = 1,
K0
Kx (x) = x − X/2, Kxx (x) = x2 − Xx + X 2 /6, Ky (y) = y −Y /2, Kyy (y) = y2 −Y y +Y 2 /6, Kz (z) = z − Z/2,
(13.96)
Kzz (z) = z − Zz + Z /6, 2
2
Kxy (x, y) = (x − X/2)(y −Y /2), Kyz (y, z) = (y −Y /2)(z − Z/2), Kzx (z, x) = (z − Z/2)(x − X/2), and
Ki K j dV = 0 (i = j).
The constants for normalization are decided using
Kx2 dV = V X 2 /12,
2 Kxy dV
Ky2 dV
= VY /12, 2
Kz2 dV = V Z 2 /12,
= V X Y /144, 2 2
2 Kyz dV
(13.97)
2 Kxx dV = V X 4 /180, 2 Kyy dV = VY 4 /180,
Kzz2 dV = V Z 4 /180,
= VY Z /144, 2 2
2 Kzx dV = V Z 2 X 2 /144.
The moments are set by the following expressions:
S0
=
Sx
= (6/X)
Sxx Sy Syy
f (x, y, z)K0 dV = m0V,
= (30/X 2 ) = (6/Y )
f (x, y, z)Kx (x)dV = mxV X/2,
= (30/Y 2 )
f (x, y, z)Kxx (x)dV = mxxV X 2 /6,
f (x, y, z)Ky (y)dV = myVY /2,
Sz
= (6/Z)
Szz
= (30/Z 2 )
Sxy
= (36/XY )
Syz
= (36/Y Z)
Szx
= (36/ZX)
f (x, y, z)Kyy (y)dV = myyVY 2 /6,
f (x, y, z)Kz (z)dV = mzV Z/2,
(13.98)
f (x, y, z)Kzz (x)dV = mzzV Z 2 /6, f (x, y, z)Kxy (x, y)dV = mxyV XY /4, f (x, y, z)Kyz (y, z)dV = myzVY Z/4, f (x, y, z)Kzx (z, x)dV = mzxV ZX/4.
All these moments are transported with the upstream advection scheme. The procedure is carried out in one direction at a time. The second and third procedures use the results of the last procedure. For simplicity, we – 186
–
13.3. Second Order Moment (SOM) scheme describe the change of each moment caused by an advection procedure in one direction (x) in a two dimensional plane (xy) in the following. You may replace (y,Y ) with (z, Z). When velocity c in the x direction is positive, the right part of the grid cell, X − ct ≤ x ≤ X, 0 ≤ y ≤ Y, 0 ≤ z ≤ Z,
(13.99)
is removed from the cell and added to the adjacent cell on the right during time interval t. This part is expressed using superscript R. The remaining part, 0 ≤ x ≤ X − ct, 0 ≤ y ≤ Y, 0 ≤ z ≤ Z,
(13.100)
is expressed by superscript L. New orthogonal functions Ki R (Ki L ) are calculated in the part R (L) with the volume V R = ctY Z (V L = (X − ct)Y Z). The orthogonal functions are given as follows: K0 L = K0 R = 1, Kx L = x − (X − ct)/2, Kx R = x − (2X − ct)/2, Kxx L = x2 − (X − ct)x + (X − ct)2 /6, Kxx R = x2 − (2X − ct)x + (X − ct)X + (ct)2 /6, Ky L = Ky R = y −Y /2,
(13.101)
Kyy L = Kyy R = y2 −Y y +Y 2 /6, Kxy L = [x − (X − ct)/2](y −Y /2), Kxy R = [x − (2X − ct)/2](y −Y /2). The basic quantities for calculating the moments are m0 R
=
R m0 + K¯ xR mx + K¯ xx mxx ,
mx R
=
mx + 2K¯ xR mxx ,
mxx R
=
mxx ,
my R
=
my + 2K¯ xR mxy ,
myy R
=
myy ,
mxy R
=
mxy ,
(13.102)
where K¯ is the average of the new orthogonal function: K¯ xL = −ct/2, K¯ xR = (X − ct)/2, L R = ct(2ct − X)/6, K¯ xx = (X − ct)(X − 2ct)/6. K¯ xx
The moments in the right part to be removed are expressed as follows: S0 R
= α [S0 + (1 − α )Sx + (1 − α )(1 − 2α )Sxx ],
Sx R
= α 2 [Sx + 3(1 − α )Sxx ],
Sxx R Sy R Syy
R
Sxy R
= α 3 Sxx ,
(13.103)
= α [Sy + (1 − α )Sxy ], = α Syy , = α 2 Sxy , – 187
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Chapter 13
Tracer advection schemes
where α = α R = ct/X = V R /V . The moments in the remaining part are expressed as follows: S0 L Sx
L
Sxx L Sy
L
Syy L Sxy
L
= (1 − α )[S0 − α Sx − α (1 − 2α )Sxx ], = (1 − α )2 (Sx − 3α Sxx ), = (1 − α )3 Sxx ,
(13.104)
= (1 − α )(Sy − α Sxy ), = (1 − α )Syy , = (1 − α )2 Sxy .
As the final step of the procedure, the orthogonal functions and moments transported from the adjacent cell and those in the remaining part of the original cell are combined to create new united moments in the cell. The calculation is terribly complex, and only the results are presented: S0 = S0 R + S0 L , Sx = α Sx R + (1 − α )Sx L + 3[(1 − α )S0 R − α S0 L ], Sxx = α 2 Sxx R + (1 − α )2 Sxx L + 5{α (1 − α )(Sx R − Sx L ) + (1 − 2α )[(1 − α )S0 R − α S0 L ]}, Sy = Sy R + Sy L , Syy = Syy R + Syy L , Sxy = α Sxy R + (1 − α )Sxy L + 3[(1 − α )Sy R − α Sy L ], where
α = α R = V R /(V R +V L ).
(13.105)
The original moments are conserved through these operations.
Flux limiter Some limiters are necessary to guarantee that the tracer is positive (negative) definite. Prather (1986) proposed to set limits for the moments related to the direction of advection. For instance, when the moments are advected in the x direction, S0
≥ 0,
Sx
= min[+1.5S0 , max(−1.5S0 , Sx )],
Sxx
Sxy
(13.106)
(13.107)
= min[2S0 − Sx /3, max(Sx  − S0 , Sxx )],
(13.108)
= min[+S0 , max(−S0 , Sxy )].
(13.109)
It should be noted that the application of this limiter does not completely guarantee the tracer will be positive (negative) definite.
13.3.2 Calculating SOM advection in MRI.COM It should be noted that the coordinate system is not Cartesian in ocean models. Since the coordinate system covers a spherical surface, the x direction in a grid cell is not identical to that in the adjacent cell, for instance. –
188
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13.3. Second Order Moment (SOM) scheme Thus, the exact conservation of moments cannot be realized. In addition, a tracercell including solid earth (sea floor or lateral boundary) is not a cuboid, so the orthogonal functions cannot be defined precisely for such a cell. Nevertheless, the procedures described in the previous subsection can be carried out using the volume of seawater in the noncuboid grid cell, and the zeroth moment S0 (total amount of the tracer) is conserved. As indicated in the expressions in the previous subsections, the volumeintegrated moments (Si ) and the fraction of volume to be removed (α ) are used in the SOM advection scheme. There are 10 moments for each tracer. The fraction α is calculated using volume transports (USTARL, VSTARL, and WLWL), which are calculated in the subroutine cont). Following Prather (1986), the procedures in three directions are executed in order, not simultaneously. The procedure in the meridional direction (advec y) is called first, the zonal direction (advec x) next, and lastly the vertical direction (advec z). The procedure in the SOM scheme does not calculate the flux of the tracer across the boundary of grid cells, unlike other advection schemes. The change in the tracer value caused by SOM advection is estimated in the subroutine tracer and added to the variable trcal directly.
NAMELIST When the SOM advection scheme is used, the following namelist (njobsom) is required (see Table 13.1) on execution.
name
Table 13.1. Variables defined in namelist njobsom explanation string or value
file som in file som out limiter(numtrc)
base name of the restart file to be input base name of the restart file to be output flag to set limits of the moments
{file som in} i n * {file som out} i n * .true. / .false.
lrstinsom lrstoutsom
flag to read the restart file for the initial state flag to write the restart file for the final state
.true. / .false. .true. / .false.
flag to monitor the conservation of the moments .true. / .false. lsommonitor * i={x, xx, y, yy, z, zz, xy, yz, zx}, n={01, 02, ..., numtrc}; tracer number
References Leonard, B. P., 1979: A stable and accurate convective modeling procedure based upon quadratic upstream interpolation, J. Comput. Methods Appl. Mech. Eng., 19, 5998. Leonard, B. P., M. K. MacVean, and A. P. Lock, 1993: PositivityPreserving Numerical Schemes for Multidimensional Advection, NASA Tech. Memo., 106055, ICOMP9305, 62pp. Leonard, B. P., M. K. MacVean, and A. P. Lock, 1994: The flux integral method for multidimensional convection and diffusion, NASA Tech. Memo., 106679, ICOMP9413, 27pp. Prather, M. J., 1986: Numerical advection by conservation of secondorder moments, J. Geophys. Res., 91, 66716681.
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Chapter 14
Generalized orthogonal curvilinear coordinate grids
This chapter introduces general orthogonal curvilinear coordinates and presents related calculus.
14.1
Outline
An ocean model does not have any problem concerning the South Pole because it does not calculate around the South Pole. However, serious problems arise around the North Pole where the meridian concentrates to one point in the ocean. First, it is necessary to calculate the temporal evolution of the physical quantity in a special way only there, because the relations between Ucells that surround the North Pole and the northernmost Tcell are topologically peculiar. Next, even if a cell doesn’t touch the North Pole, its zonal lattice interval is extremely small near the North Pole. Therefore, a short time step for integration is required owing to the limitation of the CFL condition. This limitation is reflected directly in the increased calculation time required. Moreover, when the zonal grid intervals in low latitudes and the Arctic region are extremely different, the arguments about accuracies of numerical schemes and the parameters for diffusion and viscosity operators generally cannot be applied uniformly to a model domain. The following can be considered to avoid such problems concerning the North Pole. 1) Creating a huge island including the North Pole. The finitedifference calculation in the island is abandoned, and the lateral boundary values are restored to the climatology. 2) Shifting the singular points of the model to a continent or a huge island by changing the model’s horizontal grid system. The MRI.COM scheme adopts the latter approach, which is outlined in this section. Because the MRI.COM code is written based on the generalized orthogonal coordinate system, the geographic latitude (φ ) and longitude (λ ) are not of a great concern for the calculus in the model. However, it is necessary to know the land and sea distribution, sea depth, scale factor, and the Coriolis parameter given as a function of λ and
φ at every grid point of the model prior to the calculation. We describe the method of generating the orthogonal coordinate grid system of the model in Section 14.2. Using a conformal transformation in the general sense, the functions that describe the relation between model coordinates (μ , ψ ) and geographic coordinates (λ , φ ): λ (μ , ψ ), φ (μ , ψ ), μ (λ , φ ), and ψ (λ , φ ) are obtained. Because an atmospheric boundary condition is given in many cases at grid points in the geographic coordinates, it is necessary to prepare tables for converting the surface atmospheric temperature, the wind stress, and so on. To convert a vector quantity, we must remember that the direction of the μ contour differs from that of the λ contour (meridian). The difference is described in Section 14.3. We can use the functions : (λ , φ ) ⇐⇒ (μ , ψ ) to convert a scalar quantity as well as sea depth and the Coriolis parameter. The total flux that the ocean receives from the atmosphere should be equal to the total flux that the atmosphere gives to the ocean. The method for conserving the total flux is explored in Section 14.4. The vector operation in a generalized orthogonal coordinate system is concisely described in Section 14.5.
–
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Generalized orthogonal curvilinear coordinate grids
14.2 Generation of orthogonal coordinate system using conformal mapping We designate the plane that touches the sphere at the North Pole as SN . A polar stereographic projection is a conformal transformation in the general sense, so that an orthogonal coordinate system on the sphere is mapped onto an orthogonal coordinate system on SN and the orthogonality is preserved on the reverse transformation (Figure 14.1). Moreover, if SN is assumed to be a complex plane, various conformal transformations can be defined on it. Therefore, applying (i) the polar stereographic projection, (ii) a conformal transformation on the complex plane SN , and (iii) the reverse polar stereographic projection to a geographic coordinate grid point (λ , φ ) on the sphere, an orthogonal coordinate grid point on the sphere can be obtained (Bentzen et al., 1999).
Figure 14.1. Schematic illustration of a Polar stereographic projection (a conformal transformation in the general sense between the sphere and SN ).
The functions μ (λ , φ ) and ψ (λ , φ ) are obtained by the following procedure: 1. From a point (λ , φ ) on the sphere to a point z on SN (polar stereographic projection). Defining colatitude φ = π /2 − φ , φ eiλ , (14.1) z = tan 2 where the origin of SN corresponds to φ = 0 (φ = π /2), and the positive part of the real axis corresponds to λ = 0.
2. Conformal mapping MC on SN :
ζ = MC (z).
–
192
–
(14.2)
14.2. Generation of orthogonal coordinate system using conformal mapping 3. From a point ζ on SN to a point (μ , ψ ) on the sphere (reverse polar stereographic projection).
μ ψ
ψ
= arg(ζ ),
(14.3)
= 2 arctan ζ ,
(14.4)
= π /2 − ψ .
(14.5)
Functions λ (μ , ψ ) and φ (μ , ψ ) are obtained by reversing the above procedure. Defining ψ = π /2 − ψ , ψ ei μ , ζ = tan 2
(14.6)
z = MC−1 (ζ ),
(14.7)
= arg(z),
(14.8)
and
λ φ
φ
= 2 arctan z,
(14.9)
= π /2 − φ .
(14.10)
Thus, when a model coordinate grid point, (μ0 + Δμ × i, ψ0 + Δψ × j) is given, we know the geographic position of the point, Coriolis parameter, etc., at once. Bentzen et al. (1999) used the linear fraction conversion as a conformal transformation on SN . That is,
ζ = MC (z) =
(z − a)(b − c) , (c − a)(b − z)
where the three complex numbers a, b, and c expressed by φ φ a = tan a eiλa , b = tan b eiλb , 2 2
(14.11)
c = tan
φc 2
eiλc ,
(14.12)
correspond to the three geographic coordinate grid points (λa , φa ), (λb , φb ), and (λc , φc ), which are mapped to the model coordinate grid points, (μ , ψ ) = (0, π /2), (0, −π /2), (0, 0), respectively. Therefore, the singular point (μ , ψ ) = (0, π /2) in the model calculation can be put on Greenland, by setting (λa , φa ) at 75◦ N and 40◦ W. If the option TRIPOLAR or JOT is specified instead of SPHERICAL, then two singular points: (μ , ψ ) = (0, π /2) and (0, −π /2) can be put on arbitrary land locations by suitably setting (λa , φa ) and (λb , φb ), which are model parameters (NPLON, NPLAT, SPLON, and SPLAT in degree) that should be specified in configure.in. When option TRIPOLAR is specified, the parameters are set to φa = φb = 64◦ N, λa = 80◦ E, and λb = 100◦ W. The transformed grids are used for the region north of 64◦ N, and the geographic coordinates are used for the region south of 64◦ N. This tripolar coordinate system can express the Arctic Sea with a higher resolution than the Southern Ocean. The adoption of geographical coordinates south of 64◦ N enables us to do the assimilation and analysis with relative ease (Figure 14.2). When option JOT is specified, the Joukowski conversion is used as a conformal transformation on SN . That is, ψ 2 (14.13) z = MC−1 (ζ ) = ζ + 0 eiμ0 . ζ This Joukowski conversion maps the area outside the circle with a radius of ψ0 centered at the origin of the ζ plane to the whole domain of the zplane, and rotates it by μ0 . The left panel of Figure 14.2 presents an example where the coordinate system is created by setting ψ0 to 20◦ and μ0 to 80◦ . Because there is no discontinuity of grid – 193
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Generalized orthogonal curvilinear coordinate grids
Figure 14.2. Model coordinate grid arrangement in the Arctic sea. Left: Grid system made through the Joukowski conversion (JOT). Right: Combination of the coordinates made through the linear fraction conversion and conventional geographic coordinate (TRIPOLAR). spacing in this coordinate system, the singular points can be put at various positions. For instance, the singular point on the North American side can be put on the Labrador peninsula or in Greenland. Functions λ (μ , ψ ) and φ (μ , ψ ) are defined as subroutine mp2lp, and functions μ (λ , φ ) and ψ (λ , φ ) are defined as subroutine lp2mp. Module programs trnsfrm.{spherical, moebius, tripolar, jot}.F90 contain these internal subroutines. These functions, especially mp2lp, are frequently used when the topography and the surface boundary condition are made before starting the main integration of model.
14.3 Rotation of vector A vector expressed in geographic coordinates (λ , φ ) should be rotated when observed from model coordinates (μ , ψ ). First, we set z = f (ζ ),
z = x + iy,
ζ = u + iv,
(14.14)
∂x ∂y ∂y ∂x +i = −i . (14.15) ∂u ∂u ∂v ∂v At a certain point z0 = f (ζ0 ), the angle θ at which a curve v = v0 meets a straight line y = y0 is given by ∂y ∂y ∂x tan θ = = . ∂ x v0 ∂ u ∂ u v0 f (ζ )
=
Then (see Figure 14.3)
θ = arg( f (ζ0 )).
(14.16)
Assuming λ = arg(z) and μ = arg(ζ ), at point ζ0 the straight line (v = v0 ) meets the straight line (μ = μ0 ) at angle −μ0 and at point z0 the meridian (λ = λ0 ) meets the curve (v = v0 ) at angle λ0 − θ . The meridian (λ = λ0 )
– 194
–
14.4. Mapping a quantity from geographic coordinates to transformed coordinates
Figure 14.3. A meridian in geographical coordinate (λ , φ ) (left) and a meridian in model coordinate (μ , ψ ) (right). in λ φ coordinates meets the line (μ = μ0 ) in μ ψ coordinates at angle α given as follows::
α
= − μ 0 + λ0 − θ = λ0 − μ0 − arg( f (ζ0 )).
(14.17)
Subroutine rot mp2lp defined in trnsfrm. ∗ .F90 returns (cos α , sin α ) at a specified grid point of the model. A wind stress vector (τx , τy ) in geographical coordinates should appear in the model ocean described in the μ ψ coordinate system as (τx cos α − τy sin α , τx sin α + τy cos α ).
14.4 Mapping a quantity from geographic coordinates to transformed coordinates We consider a method to receive a quantity GI,J given at the geographic coordinate grids (I, J) as the quantity HM,N at the model coordinate grids (M, N) (Figure 14.4). The quantities are wind stress components after the vector rotation, precipitation per unit area, sea surface atmospheric temperature, and so on. In addition, the average depth at a model grid point can also be calculated by the following method because bottom topography (depths of sea floor) is usually given in geographic coordinates. Grids (I, J) and (M, N) are suitably subdivided into finer grids (i, j) and (m, n). We call these filter grids. A quantity Gi, j is assumed to be homogeneously distributed in the geographic filter grids (i, j) covered by grid (I, J), Gi, j = GI,J . Assume the quantity at a model filter grid Hm,n is equal to that at the nearest geographic filter grid, Hm,n = Gi(m,n), j(m,n) .
– 195
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Generalized orthogonal curvilinear coordinate grids
Figure 14.4. Grids (I, J) and (M, N) subdivided into finer grids (i, j) and (m, n).
The quantity at model grid (M, N) is obtained as the areaweighted average: HM,N =
1 AH M,N
, ∑ AH m,n Hm,n
(14.18)
m,n
where AH M,N is the area of model grid and AH m,n is the area of model filter grid. When the grid intervals of geographic filter grid (i, j) and model filter grid (m, n) are extremely small, the total quantity (flux) received on the model grids (M, N) is equal to the total quantity (flux) given by the geographic grids (I, J). The relation between the quantity in the geographic grids and that in the model grids is defined by weight w, HM,N = ∑ w(M, N, I, J)GI,J .
(14.19)
I,J
How is the quantity converted in an actual calculation in the model? 1) When the strict conservation of quantity (flux) is necessary: Fresh water is not permitted to be generated or vanish at the surface boundary in a run using an atmosphereocean coupled model, for instance. In this case, w(M, N, I, J) is prepared beforehand, and the flux is passed from the atmosphere through equation (14.19) to the ocean. The resolution of the filter grid need not be extremely fine, provided that every geographic filter grid is linked to morethanzero model filter grids and
∑ AG I,J = ∑ A G i, j = ∑ A H m,n = ∑ AH M,N . I,J
i, j
m,n
M,N
2) When conservation need not be guaranteed: When the ocean model is driven by the surface boundary condition based on atmospheric reanalysis data, the amount of fresh water entering the sea as precipitation and river discharge is not equal to that drawn from the ocean through evaporation and sublimation. Therefore, the global sea surface height rises or descends during years of integration. It is not very important to pursue complete conservation of freshwater under this condition. – 196
–
14.5. Vector operation and differentiation in a general orthogonal coordinate system In such a case, the flux at a model grid point can be prepared beforehand using equation (14.19), to avoid the timeconsuming flux conversions in the model calculation.
14.5
Vector operation and differentiation in a general orthogonal coordinate system
To formulate the model equations, we have to know the vector operation and differentiation in general orthogonal coordinates. Some basic formulae used in formulating primitive equations are presented here. The line element vector δ x at a certain point (μ , ψ , r) in an arbitrary general orthogonal coordinate system is expressed as
δ x = hμ δ μ eμ + hψ δ ψ eψ + hr δ rer ,
(14.20)
where basis vectors eμ , eψ , and er are mutually orthogonal unit vectors, and hμ , hψ , and hr are scale factors. Defining eψ ∂ eμ ∂ er ∂ + + ∇= , (14.21) hμ ∂ μ hψ ∂ ψ hr ∂ r the gradient of scalar A(μ , ψ , r) is ∇A =
eμ ∂ A eψ ∂ A er ∂ A + + , hμ ∂ μ hψ ∂ ψ hr ∂ r
(14.22)
and the divergence of vector A = Aμ eμ + Aψ eψ + Ar er is , + ∂ (hψ hr Aμ ) ∂ (hr hμ Aψ ) ∂ (hμ hψ Ar ) 1 + + ∇·A = . hμ hψ hr ∂μ ∂ψ ∂r The r component of curlA is
(14.23)
+ , ∂ (hψ Aψ ) ∂ (hμ Aμ ) 1 . − hμ hψ ∂μ ∂ψ
(14.24)
The calculation of velocity advection includes (a · ∇)A, where a is an arbitrary vector (a = aμ eμ + aψ eψ + ar er ). The μ component of (a · ∇)A is a · ∇Aμ +
Aψ hμ hψ
aμ
∂ hμ ∂ hψ − aψ ∂ψ ∂μ
+
Ar hr hμ
aμ
∂ hμ ∂ hr − ar ∂r ∂μ
.
(14.25)
The second and third terms are socalled ‘metric’ terms in the equation of motion in the spherical coordinates. These expressions in spherical coordinates (λ , φ , r) are shown next. Defining longitude λ , latitude φ , and radius of the earth r, scale factors are hλ = r cos φ , hφ = r, and hr = 1. Velocity vector v is v = ueλ + veφ + wer ,
(14.26)
where eλ , eφ , and er are the eastward, northward, and upward unit vectors, respectively, and (u, v, w) = (r cos φ λ˙ , rφ˙ , r˙). The gradient of scalar function A(λ , φ , r) is, ∇A = where ∇=
∂A eλ ∂ A eφ ∂ A + + er , r cos φ ∂ λ r ∂φ ∂r
(14.27)
eφ ∂ ∂ ∂ eλ + + er . r cos φ ∂ λ r ∂φ ∂r
(14.28)
–
197
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Chapter 14
Generalized orthogonal curvilinear coordinate grids
For vector A = Aλ eλ + Aφ eφ + Ar er , divergence is ∇·A =
+ , ∂ Aλ ∂ (cos φ Aφ ) ∂ (r2 Ar ) 1 + 2 + . r cos φ ∂ λ ∂φ r ∂r
(14.29)
and the r component of curlA is + , ∂ Aφ ∂ (cos φ Aλ ) 1 . − [curlA]r = r cos φ ∂ λ ∂φ
(14.30)
The λ component of (a · ∇)A is [(a · ∇)A]λ = a · ∇Aλ −
Aφ aλ tan φ Ar aλ + . r r
(14.31)
The Coriolis force in a generalized orthogonal coordinates (μ , ψ , r) system is given as 2Ω × v = (2Ωψ w − 2Ωr v)eμ + (2Ωr u − 2Ωμ w)eψ + (2Ωμ v − 2Ωψ u)er ,
(14.32)
where Ω = Ωμ eμ + Ωψ eψ + Ωr er is the rotation vector of the Earth, and v = ueμ + veψ + wer is the velocity vector. We designate f μ = 2Ωμ , fψ = 2Ωψ , and f = fr = 2Ωr in Chapter 2. The rotation vector of the Earth is (Ωλ , Ωφ , Ωr ) = (0, Ω cos φ , Ω sin φ ) in the geographic coordinate (λ , φ , r) system.
References Bentzen, M., G. Evensen, H. Drange, and A. D. Jenkins, 1999: Coordinate transformation on a sphere using conformal mapping, Mon. Weather Rev., 127, 27332740.
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198
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Chapter 15
Nesting
In MRI.COM, a highresolution regional model could be embedded in a lowresolution model using a nesting method. Only oneway nesting from a low to a highresolution model is implemented. This chapter briefly describes the nesting method employed by MRI.COM, its program structure, and how to construct a set of lowand highresolution models in nesting.
15.1
Feature
In a set of nested grid models, a highresolution model is embedded in a lowresolution model, and values at the side boundary of the highresolution model are given by the lowresolution model. The side boundary data could be exchanged both offline and online. In offline mode, the data needed to calculate side boundary values are output to files by first running the lowresolution model, and the highresolution model is executed reading these data and calculating the side boundary values. In online mode, the precommunicator of Scup (simple coupler) by Yoshimura and Yukimoto (2008) is used to exchange data, and the lowresolution and the highresolution models are run at the same time. The main region of the highresolution model (hereinafter called core region, the hatched region of Figure 15.1) is constructed by connecting tracer points (Tpoints) of the lowresolution model to form a rectangular region. Two rows or columns of velocity grid cells are added outside the core region at each side boundary (hereinafter called margin; the unhatched region just outside the core region in Figure 15.1), and the side boundary condition is imposed by the lowresolution model. Tracers at the boundary of the core region and one grid outside of it and velocities at two grid points outside the core region are replaced every time step using values from the lowresolution model and linear interpolation. These are referred to as the input interface (Figure 15.1). A problem arises for velocity points of the highresolution model between an ocean point and a land point of the lowresolution model, since only data at a single point is available for interpolation. In this case, the highresolution model velocity perpendicular to the coast line is assumed to be zero at the coast and linearly interpolated from the nearest ocean grid point of the lowresolution model. Furthermore, the highresolution model velocity tangent to the coast line is set to the same value at the nearest ocean grid point of the lowresolution model. By doing so, the interpolated velocity field satisfies the continuity equation. The boundary values supplied from the lowresolution model should be enough to compute values at the margin of the highresolution model using linear interpolation. In practice, three columns or rows are output; they are enough to compute the margin as well as the region one grid equivalent of the lowresolution model inside the boundary of the core region. For the northern boundary, T1T3 and U1U3 in Figure 15.1 are sent to the highresolution model. In this way, prognostic variables are directly replaced by those from the lowresolution model, which is usually called the clamped method (Cailleau et al., 2008). This method does not guarantee conservation of tracers in contrast to the method where fluxes are given at the boundary, but the integration is quite stable.
–
199
–
Chapter 15
Nesting T3 U3 Input interface for U
T2
Input interface for T
U2
T1
U1
:Temperature & Salinity points : Velocity points
Figure 15.1. Relation between the lowresolution (large symbols) and the highresolution model (small symbols) model. The hatched region is the main region of the highresolution model. The tracer points at the boundary of this region and at one grid point outside this boundary and the velocity points at two grid points outside this boundary are replaced by values from the lowresolution model. At the northern boundary, three rows (T1  T3 and U1  U3) are sent to the highresolution model from the lowresolution model.
15.2 Lowresolution model There is no particular problem in constructing the lowresolution model. Since the grid size ratio between the high and lowresolution models must be odd, the grid size of the lowresolution model should be determined according to that of the highresolution model. A grid size ratio of 1:3 or 1:5 is recommended. In offline mode, the following files are output from the lowresolution model to be read by the highresolution model. • Three rows or columns of baroclinic mode data for each side boundary. • Three rows or columns of barotropic mode data for each side boundary. • Landsea index of the lowresolution model. • Latitude and longitude of the lowresolution model. The filenames are set in parinit.F90 using namelist outflpar. In online mode, this namelist is not required. The daily output of side boundary data will work in running the highresolution model. In online mode, side boundary data are sent every time step.
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15.3. Highresolution model
15.3
Highresolution model
15.3.1
Required data
To run the highresolution model, prepare two data files that contain • The contribution ratio between low and highresolution models around the boundary, • Two dimensional distribution of horizontal diffusivity and viscosity. In addition, prepare the following data from the lowresolution model. • Three rows or columns of baroclinic mode data for each side boundary. • Three rows or columns of barotropic mode data for each side boundary. • Landsea index of the lowresolution model. • Latitude and longitude of the lowresolution model. Issues to be carefully considered in creating variable horizontal grid size information and bottom topography are detailed in the next subsection. Note the following when preparing barotropic boundary data. Since time filtering is used to feedback the result of the barotropic equations to baroclinic modes, the barotropic equations are integrated past the baroclinic time. Thus, the highresolution model needs future barotropic data in addition to the original output from the lowresolution model. To fulfill this need, prepare barotropic data in one file so that the second data in time sequence can be used as data after the last one for repeating year cycle run. For a historical run, append the second data of the following year of the lowresolution model to the present year data.
15.3.2
Creating data
We briefly describe how to create data that will be read from the highresolution model.
a. Horizontal grid size The same grid size as in the lowresolution model should be used in the vertical direction. The highresolution model is nested in the horizontal directions. The distance between the velocity points of the lowresolution model is divided equally in the highresolution model. There should be two marginal velocity cells outside the core region at the western and the southern boundary even if the western or the southern boundary is filled by land. However, if the eastern or the northern boundary is filled by land and does not receive boundary data, one marginal velocity cell will be enough.
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How to divide lowresolution cells
‘‘dxt’’ of the lowresolution model (Upoint distance: variable name is dxtdeg) should be divided equally in the highresolution model. There should be two marginal Upoints in the highresolution model (one cell will do if the eastern or northern boundary is filled by land) ***** lowresolution model ***** ifst is the western end grid point number of the lowresolution model to be used for interpolation. boundary  (i=ifst) (i=ifst) (i=ifst+1) (i=ifst+1) < dxt > U < dxt > U  ++ T < dxu > T < dxu > T (i=ifst) (i=ifst) (i=ifst+1) (i=ifst+1)  ***** highresolution model ***** (nesting ratio is 1:3)  (1) (1) (2) (2) (3) (3) <dxt>U<dxt>U<dxt>U<dxt>U +++T<dxu>T<dxu>T<dxu>T (1) (1) (2) (2) (3) (3)
How to create a grid size information file (for variable grid size only)
! the number of grid points of the highresolution model integer(4), parameter :: imut = 535, jmut = 431 integer(4), parameter :: ivgrid = 57 ! arbitrary real(8) :: dxt(imut), dyt(jmut) ! dxtdeg, dytdeg in the model write(unit=ivgrid) (dxt(i),i=1,imut), (dyt(j),j=1,jmut)
b. Topography of the highresolution model To avoid serious discontinuities in mass transport, the highresolution model should have the same topography as the lowresolution model around the side boundary of the highresolution model. It is recommended that the topography of the highresolution model should have the same topography as the lowresolution model in the two inner lowresolution velocity cells and the one outer lowresolution velocity cell from the boundary of the core region.
c. Weighting ratio between the low and highresolution model For smoothness, values of the highresolution model around the side boundary could be given as a weighted average of the low and highresolution models. However, we recommend that the weight for the lowresolution model to be unity at the input interface and zero elsewhere.
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15.4. Usage
weighting ratio between the low and highresolution model
! the number of grid points of highresolution model integer(4), parameter :: imut = 535, jmut = 431 integer(4), parameter :: ibuffer = 57 ! arbitrary real(8) :: wbuft(imut,jmut), wbufu(imut,jmut) write(unit=ibuffer) wbuft, wbufu
d. 2D distribution of diffusion and viscosity coefficient Diffusion and viscosity coefficients of a highresolution model should be the same as those of a lowresolution model around the side boundary. The coefficients are made small in the interior. 2D distribution of diffusion and viscosity coefficient
! the number of grid points of the highresolution model integer(4), parameter :: imut = 535, jmut = 431 integer(4), parameter :: ihdtsuv = 57 ! arbitrary real(8) :: hdts(imut,jmut), hduv(imut,jmut) write(unit=ihdtsuv) hdts, hduv
15.4
Usage
15.4.1
Compilation
The model option for the lowresolution model is PARENT, and that for the highresolution model is SUB. For online mode, the option NESTONLINE should be specified along with the name of the model such as NAME MODEL = modelname. For a calculation using parallel processors (option PARALLEL), only onedimensional partitioning (in the meridional direction) of model region is supported. The number of zonally partitioned regions (NPARTX) must be one. These model options and the model name should be specified in configure.in. The following describes parameters that a user should define before compilation.
a. Low resolution model The following model parameters should be defined in parent/parapar.F90 prior to compilation.
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ifsto ifedo jfsto jfedo
= = = =
35 157 61 159
: : : :
the the the the
western end point to be used to interpolate data (Tpoint) eastern end point to be used to interpolate data (Tpoint) southern end point to be used to interpolate data (Tpoint) northern end point to be used to interpolate data (Tpoint)
ipareo = ifedo  ifsto + 1 jpareo = jfedo  jfsto + 1 ... the total grid number to be sent to the highresolution model ioutrts = 5, ioutrtn = 0, ioutrtw = 5, ioutrte = 3 ... nesting ratio at each side boundary south(s), north(n), west(w), east(e)
b. High resolution model Model parameters that should be defined in sub2/parasub.F90 are listed on Table 15.1. Table 15.1. Parameters to be defined in sub2/parasub.F90 variable name ipmut, jpmut ifst, ifed, jfst, jfed
description the number of grid points of the lowresolution model define the boundary of the lowresolution model used for interpolation in the highresolution model (considered in terms of Tpoints ... should be the same as
ipare, jpare
iinrts, iinrtn, iinrtw, iinrte intbcl, intbtr
parapar.F90) the total grid number of the lowresolution model to be used for interpolation in the highresolution model (... should be the same as parapar.F90) nesting ratio at each side boundary the time interval in seconds of the boundary data (used only for offline mode)
e.g. iinrts=5, iintrn=3, iinrtw=5, iinrte=3 5 for nesting ratio 5:1 3 for nesting ratio 3:1
e.g., intbcl = 21600, intbtr = 21600
15.4.2 Running the models In offline mode, specify the names of data files and how boundary data are handled in namelist. Users should see parent/parinit.F90 for the lowresolution model and sub2/subinit.F90 for the highresolution model. The filename for the weighting ratio and the twodimensional distribution of horizontal diffusion and viscosity for the highresolution model (namelist inflsub) should be specified even in online mode.
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15.4. Usage a. Low resolution model The following namelists are read from parent/parinit.F90 on execution. Table 15.2. Namelist for the nested lowresolution model variable name
group
description
usage
nwrtpc
njobpar
The interval of time steps by which baroclinic mode is written to file
off line only
nwrtpt
njobpar
The interval of time steps by which barotropic mode is writ
off line only
ircfsto
njobpar
ten to file The record number in file at the beginning of the integration (baroclinic mode)
off line only
irtfsto
njobpar
The record number in file at the beginning of the integration (barotropic mode)
off line only
ibfirst file btrs out file btrn out
njobpar outflpar outflpar
Output (1) or nooutput(0) the initial state as boundary data the filename of the barotropic mode (southern end) the filename of the barotropic mode (northern end)
off line only off line only off line only
file btrw out file btre out file bcls out file bcln out
outflpar outflpar outflpar
the filename of the barotropic mode (western end) the filename of the barotropic mode (eastern end) the filename of the baroclinic mode (southern end)
off line only off line only off line only
outflpar outflpar
the filename of the baroclinic mode (northern end) the filename of the baroclinic mode (western end)
off line only off line only
outflpar outflpar outflpar
the filename of the baroclinic mode (eastern end) landsea index of the lowresolution model latitudelongitude of the low resolution model
off line only off line only off line only
file bclw out file bcle out file lidx out file pgrd out
b. High resolution model The following namelists are read from sub2/subinit.F90 on execution. Table 15.3. namelist for the nested highresolution model variable name
group
ircfst
njobsub
description
usage
The record number in file at the beginning of the integration
off line only
(baroclinic mode) irtfst
njobsub
The record number in file at the beginning of the integration (barotropic mode)
off line only
ircend
njobsub
The number of records written in the boundary file (baro
off line only
irtend
njobsub
clinic mode). The number of records written in the boundary file
off line only
irtcycle
njobsub
(barotropic mode). Return to the second record (1) or repeat the last data (0) if
off line only
inflp2s inflp2s
the last barotropic data is reached. the filename of the barotropic mode (southern end) the filename of the barotropic mode (northern end)
off line only off line only
file btrs file btrn
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variable name
group
description
usage
file btrw file btre file bcls
inflp2s inflp2s
the filename of the barotropic mode (western end) the filename of the barotropic mode (eastern end)
off line only off line only
inflp2s inflp2s inflp2s
the filename of the baroclinic mode (southern end) the filename of the baroclinic mode (northern end) the filename of the baroclinic mode (western end)
off line only off line only off line only
file lidx
inflp2s inflp2s
the filename of the baroclinic mode (eastern end) landsea index of the lowresolution model
off line only off line only
file pgrd file bwgt
inflp2s inflsub
latitudelongitude of the lowresolution model weighting ratio between the low and highresolution model around the boundary
off line only
file bhdf
inflsub
2D distribution of diffusion and viscosity of the highresolution model
file bcln file bclw file bcle
c. Online mode In online mode, the precommunicator of Scup (simple coupler) by Yoshimura and Yukimoto (2008) is used to exchange data and the low and highresolution models are run at the same time. User should tell the coupler how data are exchanged between the low and highresolution models via a namelist file NAMELIST SCUP. An example of NAMELIST SCUP is listed below. In this example, the model name of the lowresolution model is GLOBAL and that of the highresolution model is WNP01.
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15.4. Usage &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre &nam_scup_pre
an example of NAMELIST SCUP
model_put=’GLOBAL’, model_get=’WNP01’, type=’REAL8’ / var_put=’ALONTC’, var_get=’ALONTC’, dst_get=’ALL’ / var_put=’ALATTC’, var_get=’ALATTC’, dst_get=’ALL’ / var_put=’ALONUC’, var_get=’ALONUC’, dst_get=’ALL’ / var_put=’ALATUC’, var_get=’ALATUC’, dst_get=’ALL’ / var_put=’AEXLP’, var_get=’AEXLP’, dst_get=’ALL’ / var_put=’ATXLP’, var_get=’ATXLP’, dst_get=’ALL’ / var_put=’ULS’, var_get=’ULS’, dst_get=’FIRST’ / var_put=’VLS’, var_get=’VLS’, dst_get=’FIRST’ / var_put=’TLS’, var_get=’TLS’, dst_get=’FIRST’ / var_put=’SLS’, var_get=’SLS’, dst_get=’FIRST’ / var_put=’HTS’, var_get=’HTS’, dst_get=’FIRST’ / var_put=’UMS’, var_get=’UMS’, dst_get=’FIRST’ / var_put=’VMS’, var_get=’VMS’, dst_get=’FIRST’ / var_put=’ULN’, var_get=’ULN’, dst_get=’LAST’ / var_put=’VLN’, var_get=’VLN’, dst_get=’LAST’ / var_put=’TLN’, var_get=’TLN’, dst_get=’LAST’ / var_put=’SLN’, var_get=’SLN’, dst_get=’LAST’ / var_put=’HTN’, var_get=’HTN’, dst_get=’LAST’ / var_put=’UMN’, var_get=’UMN’, dst_get=’LAST’ / var_put=’VMN’, var_get=’VMN’, dst_get=’LAST’ / var_put=’ULW’, var_get=’ULW’, dst_get=’ALL’ / var_put=’VLW’, var_get=’VLW’, dst_get=’ALL’ / var_put=’TLW’, var_get=’TLW’, dst_get=’ALL’ / var_put=’SLW’, var_get=’SLW’, dst_get=’ALL’ / var_put=’HTW’, var_get=’HTW’, dst_get=’ALL’ / var_put=’UMW’, var_get=’UMW’, dst_get=’ALL’ / var_put=’VMW’, var_get=’VMW’, dst_get=’ALL’ / var_put=’ULE’, var_get=’ULE’, dst_get=’ALL’ / var_put=’VLE’, var_get=’VLE’, dst_get=’ALL’ / var_put=’TLE’, var_get=’TLE’, dst_get=’ALL’ / var_put=’SLE’, var_get=’SLE’, dst_get=’ALL’ / var_put=’HTE’, var_get=’HTE’, dst_get=’ALL’ / var_put=’UME’, var_get=’UME’, dst_get=’ALL’ / var_put=’VME’, var_get=’VME’, dst_get=’ALL’ / var_put=’ICECATS’, var_get=’ICECATS’, dst_get=’FIRST’ / var_put=’ICECATN’, var_get=’ICECATN’, dst_get=’LAST’ / var_put=’ICECATW’, var_get=’ICECATW’, dst_get=’ALL’ / var_put=’ICECATE’, var_get=’ICECATE’, dst_get=’ALL’ / var_put=’ICEDYNS’, var_get=’ICEDYNS’, dst_get=’FIRST’ / var_put=’ICEDYNN’, var_get=’ICEDYNN’, dst_get=’LAST’ / var_put=’ICEDYNW’, var_get=’ICEDYNW’, dst_get=’ALL’ / var_put=’ICEDYNE’, var_get=’ICEDYNE’, dst_get=’ALL’ / model_put=’WNP01’, model_get=’GLOBAL’, type=’REAL8’ / var_put=’ATEXSUB’, var_get=’ATEXSUB’, dst_get=’ALL’ / var_put=’TSUB’, var_get=’TSUB’, dst_get=’ALL’ / var_put=’SSUB’, var_get=’SSUB’, dst_get=’ALL’ /
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15.5 Program structure The program structure of a lowresolution model is as follows. ogcm__ini  + parinit (parinit.F90)  + offline mode  + if (ibfirst == 1)  + out[snwe]cli  + out[snwc]tro
(initialize lowresolution model)
(output boundary data of the initial state)
ogcm__run  + part_1 (online mode)    + surfce    +surfce_integ    +DO LOOP_N      + outmrgnt (send barotropic boundary data    to highresolution model)   + out[snwe]tro    + outmrgnt (send timefiltered barotropic boundary   data to highresolution model)  + out[snwe]tro  + part_2 (online mode)    + outmrgn (send baroclinic boundary data    to highresolution model)   + out[snwe]cli    + outmrgnvd (send viscosity/diffusivity   to highresolution model)  + out[snwe]vd  + outmrgn (offline mode at the end of each time step)    + if (mod(nnmats,nwrtpc)== 0)    + out[snwe]cli  + outmrgnvd (offline mode at the end of each time step)    + if (mod(nnmats,nwrtpc)== 0)    + out[snwe]vd  + outmrgn (offline mode at the end of each time step)  + if (mod(nnmats,nwrtpt)== 0)
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The program structure of a highresolution model is as follows. ogcm__ini  + subinit (subinit.F90)  + mknlvect  + mknlvecv  + offline mode  + rd[snwe]cli    + inplsubuf    + inplsubvf    + inplsubtf  + rd[snwc]tro  + inplsubus  + inplsubvs  + inplsubts
(initialize highresolution model) (make list vectors for interpolation of boundary data)
(read first two baroclinic boundary data) (interpolation of boundary data to highresolution model grid points)
(read first two baroclinic boundary data)
ogcm__run  + part_1    + surfce    +surfce_integ    + DO LOOP_N      + inpfacet (read/receive barotropic boundary data)      + rd[snwe]tro        + inplsubus        + inplsubvs        + inplsubts      + flmrgn[snwe]t (replace barotropic prognistic variables   to those from the lowresolution model)  + inpfacet (read/receive timefiltered barotropic boundary data)    + rd[snwe]tro      + inplsubus      + inplsubvs      + inplsubts
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15.5. Program structure    + flmrgn[snwe]t  + part_2  + inpface (read/receive baroclinic boundary data   after ‘‘trcimp’’ and before ‘‘stable’’)  + rd[snwe]cli      + inplsubuf      + inplsubvf      + inplsubtf    + flmrgn[snwe]c (replace baroclinic prognistic variables  to those from the lowresolution model)  + inpfacevd (read/receive viscosity/diffusivity data  after mixed layer model) + rd[snwe]vd    + inplsubtf  + flmrgn[snwe]vd (replace baroclinic prognistic variables to those from the lowresolution model)
References Cailleau, S., V. Fedorenko, B. Barnier, E. Blayo, and L. Debreu, 2008: Comparison of different numerical methods used to handle the open boundary of a regional ocean circulation model of the Bay of Biscay, Ocean Modell., 25, 116, doi:10.1016/j.ocemod.2008.05.009. Yoshimura, H., and S. Yukimoto, 2008: Development of simple coupler (Scup) for earth sysyem modeling., Pap. Meteor. Geophys., 59, 1929, doi:10.2467/mripapers.59.19.
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This chapter explains the procedure to run MRI.COM using a long term integration of a global model as a standard case. The description in this chapter is based on MRI.COM version 3.0 (MRICOM3 020091130) and may not correspond to the latest version of MRI.COM, since the run procedure (user interface) is continuously updated. It is recommended that users refer to README.First in src directory when they start setting up the model. The minimal information to prepare, run, and postprocess is presented in this chapter in the following order: • Model setup: User defined parameter files and compilation (Section 16.1 and Tables 16.1 and 16.2). • Input data: Topography, surface forcing, and climatology to be read at run time (Section 16.2 and Table 16.3). • Execution: An example shell script and namelist (Section 16.3). • Post process: A description of output files (Section 16.4 and Table 16.4). The input and main output files are listed in Tables 16.1 through 16.4. Note that cgs units are employed to express physical values in the model.
16.1
Model setup
This section describes the procedure necessary to set up the model and compile its programs. First, prepare configure.in that contains the information about the model options and grid sizes. In addition, edit several Fortran 90 files to specify parameters corresponding to the selected model options. Table 16.1. Files used for compilation and their related program files (see Section 16.1) parameter to be specified file name included from model size, model option, compile op
configure.in
param.F90 via configure
tion vertical resolution
dz.F90
param.F90.in
vertical diffusion coefficient
vdbg.F90
rdjobp.F90
grid of surface forcing data
intpolpar.F90
force.F90
vertical grid of climatological data (for model option TSINTPOL)
depclim.F90
tsclim.F90
3D
tsintpol.F90
tsclim.direct.F90
(for model option FLXINTPOL)
grid
of
climatological
data
(for model options TSINTPOL and TSCLDIRECT)
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16.1.1 Files needed for compilation a. configure.in The model options and configuration parameters should be specified in configure.in as follows. An example configure.in for Global tripolar 1◦ × 0.5◦ grid model DEFAULT_OPTIONS="ICE ICECAT SIDYN CALALBSI SFLUXW SFLUXR ISOPYCNAL SMAGOR VIS9P DIFAJS NOHKIM VVDIMP UTOPIA ULTIMATE ZQUICKEST ZULTIMATE BBL HIST HISTFLUX HFLUX TAUBULK WFLUX RUNOFF Y365D CLMFRC LWDOWN BULKNCAR BULKITER INILEV CYCLIC VARIABLE TRIPOLAR PARALLEL FREESURFACE" # IMUT=364 JMUT=368 KM=51 KSGM=5 KBBL=1 SLAT0=78.D0 SLON0=0.D0 NPARTX=8 NPARTY=4 DXTDGC=1.0D0 DYTDGC=0.5D0 NPLAT=64.D0 NPLON=80.D0 SPLAT=64.D0 SPLON=260.D0
The parameters required for each major option are listed in Table 16.2 (see also README.Options). Table 16.2. model parameters to be set in configure.in option name
variable name
description
default
IMUT, JMUT, KM
zonal/meridional/vertical grid number
SLON0, SLAT0 DXTDGC, DYTDGC
longitude/latitude of the first tracer grid point excluding the boundary land or ghost cells zonal/meridional grid sizes in degree for the case of uniform in
TRIPOLAR, JOT
NPLAT, NPLON
crement; unless VARIABLE geographical latitude and longitude of the displaced North Pole
FREESURFACE
SPLAT, SPLON KSGM NSFMRGN
geographical latitude and longitude of the displaced South Pole the number layers in the sea surface sigmalayer; see Chapter 4 the number of sideboundary ghost cells to reduce the communication cost in parallel computation (see Ishizaki and Ishikawa,
BBL PARALLEL
TSINTPOL
KBBL NPARTX, NPARTY
IMT, JMT, KK SLATC, SLONC DLATC, DLONC
2006) the number layers of bottom boundary layer model; must be 1 the number of zonally/meridionally partitioned region for a calculation using parallel processors: the number of parallel processes should be NPARTX × NPARTY 3D size of climatological data latitude/longitude of the first grid point data of climatology grid increment for latitude/longitude of climatology
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16.1. Model setup option name
variable name
description
FLXINTPOL (RUNOFF)
IMF, JMF IMROF, JMROF
2D size of sea surface data 2D size of river discharge data
INTPWIND passive tracer
NUMTRC P
horizontal interpolation type for the wind data; 1:linear, 2:third order spline only when any passive tracer is calculated (NUMTRC P ≥ 1)
b. dz.F90 Describe the discretization in the vertical direction in dz.F90. The widths of the discrete cells that fill the vertical column should be written from top to bottom. Usually the depth of the bottom level is determined first, and the column is separated into discrete cells. In general, the levels are finer from the sea surface to about 1000 m and become coarser towards the bottom. Tracer and velocity levels are placed at the middle level of the cell (Chapter 3). The number of vertical cells is described as KM in configure.in. The file dz.F90 is included by param.F90. Example for dz.F90 real(8), parameter :: dz(km) = (/& & 3.0d2, 4.0d2, 6.0d2, 8.0d2, 1.0d3, & ............ & 2.5d4, 2.5d4, 3.75d4, 7.50d4 /)
c. depclim.F90 For option TSINTPOL, define the information about the depth of climatological temperature and salinity data in depclim.F90. It is used to interpolate the data to model grid points. Enter the number of grid points (IMT, JMT, and , KK) and horizontal gridpoint information (DLATC, DLONC, SLATC, and SLONC) in configure.in (Table 16.2). A sample can be found in directory src/intpolpar.
d. tsintpol.F90 When option TSCLDIRECT is selected in addition to TSINTPOL, define the information about the 3D grid of climatological temperature and salinity data in tsintpol.F90. With this choice, the data should be prepared in direct access files. As in the default case, enter the number of grid points (IMT, JMT, and KK) and horizontal gridpoint information (DLATC, DLONC, SLATC, and SLONC) in configure.in (Table 16.2). But the horizontal grid of data may be defined in tsintpol.F90, overriding the above definition. A sample can be found in directory src/intpolpar.
e. intpolpar.F90 For option FLXINTPOL, define the information about the latitude and longitude of the surface data intpolpar.F90. It is used to interpolate the data to the model grids points for sea surface flux calculation. Enter the number of zonal and meridional grid points (IMF and JMF) in configure.in (Table 16.2). Samples for the datasets of NCEPNCAR and JRA25 reanalyses and CORE can be found in the directory src/intpolpar.
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f. vdbg.F90 The file vdbg.F90 is necessary if the vertical diffusion equation is solved using the implicit method (option VVDIMP). While the vertical diffusion coefficient is replaced by the large one from the mixed layer model and the convection scheme (option DIFAJS), the lower limit (vdbg(1 : km)) should be set in vdbg.F90. The sample of vdbg.F90 and the sample program to prepare it (vdmricom.F90) are in the directory src/vd. Here, the vertical diffusion coefficient used in the GFDLMOM is shown. Example for vdbg.F90
do k = 1, km vdbg(k) = .8d0 + 1.05d0 * datan(4.5d0*(dep(k+1)*1.d52.5d0)) / pi end do
16.1.2 Compilation of the model The standard compiling script is prepared as compile.sh in the src directory. The part depending on the system (OS, Fortran compiler, and compiler option) in compile.sh and configure have to be edited by users for their computer (hardware) environment. To compile the programs, execute compile.sh by issuing a shell prompt from the frontend machine. The function of compile.sh is to create param.F90 and Makefile from configure.in, param.F90.in, and Makefile.in by running configure and to execute the command make to create the executable file ogcm. The environment variables for compilation are set in configure using the options prescribed in configure.in. If configure.in is newer than param.F90, the parameter values defined in configure.in replace those in param.F90.in to create param.F90. The Makefile is created from Makefile.in. Finally, make is carried out, and the executable file ogcm is obtained. The program files that should be compiled are automatically selected according to the descriptions of the relationships in Makefile, but users should be careful since it might not be perfect. The compilation should be carried out after executing ./compile.sh clean when any compile option in configure.in is rewritten.
16.2 Preparation of input data files for execution The topographic data (file topo), temperature and salinity climatological data (file tscl), and wind stress data (file wind) are always necessary. According to user’s the specification of model options and job parameters, the following additional data files should be prepared: • zonal and meridional grid spacing (units in degrees, file vgrid; for VARIABLE) • information about the area and the length for each grid (file scale; except for SPHERICAL) • information about gridwise nudging “on” or “off” (file frc; for namelist parameter iforcev = 1) • sea surface radiation data (file hflx; for HFLUX) • sea surface meteorological data (file bulk; for HFLUX + selected bulk formula, see Chapter 8) • precipitation data (file prcp; for WFLUX) • river discharge data (file rnof; for RUNOFF) – 216
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16.2. Preparation of input data files for execution
Table 16.3. Main input data files and their related program files (see Section 16.2) file name specified in (namelist) read from
subject
rdjobp.F90 etc.
runtime job parameters (as namelist parameters)
standard input (or redirection from file)
variable horizontal grid spacing (for the model option VARIABLE)
file vgrid (inflg)
stmdlp.F90
topography
file topo (infltopo)
rdbndt.F90
area and length scale for each grid
file scale (inflscl)
rdbndt.F90
climatological data of temperature and salinity
file tscl (infltscl)
tsclim.F90
on/off of nudging for tracer points
file frc (njobbdy)
rstcoef.F90
wind stress
file wind (inflw)
force.F90
surface radiation data
file hflx (inflh)
force.F90
surface meteorological data
file bulk (inflh)
force.F90
precipitation data
file prcp (inflp)
force.F90
river discharge data
file rnof (inflo)
force.F90
seaice fractional area data
file icec (inflic)
force.F90
initial values for restart
file restart in (infla)
(for ice averaged over all categories)
file ice restart in (infli)
mod seaice.F90
(for ice for each thickness category)
file icecat restart in (inflic)
ice restart.F90
rdinit.F90
• seaice fractional area data (file icec; for ICECLIM)
16.2.1
Topographic and grid spacing data
a. Topography The topographic data consist of the single precision integer array HO4(IMUT, JMUT) that contains the sea floor depths of the velocity grid points (in cm) and the single precision integer array EXNN(IMUT, JMUT) that contains its corresponding vertical level. They should be written unformatted and sequentially as follows. Format of topographic data (file topo)
integer(4) :: ho4(imut,jmut),exnn(imut,jmut) write(unit=inidt) ho4, exnn
For an experiment with a realistic topography, the model topography is usually prepared by averaging the depth data of ETOPO2 over each grid cell. ETOPO2 is 2minutegrid topography data. An example of the topography for global 1◦ × 0.5◦ model created by this method is shown in Figure 16.1. On creating a model topography, especially for a lowresolution model, the user should be careful that the important gateways for the ocean circulation be kept open and that the land blocking the ocean circulation be kept closed. The topography data is read from the file file topo (namelist infltopo) in rdbndt.F90.
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Figure 16.1. Example of ocean model topography (global 1◦ × 0.5◦ grid model). b. Grid spacing The grid spacing should be prepared when variable grid spacing is used for either the zonal or meridional direction (the model option VARIABLE). The units are in degrees. It is read from the file file vgrid (namelist inflg) in stmdlp.F90. Format of grid spacing data (file vgrid; VARIABLE) real(8) :: dxtdeg(imut), dytdeg(jmut) ! grid increment for Tpoint write(ivgrid) dxtdeg, dytdeg
c. Grid cell area and distance When the model grid points are defined based on the general orthogonal coordinates, the quarter cell area and distance should be prepared. The units are in cgs. It is read from the file file scale (namelist inflscl) in rdbndt.F90. When spherical coordinates are used (option SPHERICAL), e.g., the grids are defined on geographical latitude and longitude, the grid information is analytically calculated in stmdlp.F90, and the file file scale is not necessary.
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Format of grid cell area and distance (file scale; if not SPHERICAL)
real(8) :: a_bl(imut,jmut), a_br(imut,jmut), a_tl(imut,jmut), a_tr(imut,jmut) real(8) :: dx_bl(imut,jmut), dx_br(imut,jmut), dx_tl(imut,jmut), dx_tr(imut,jmut) real(8) :: dy_bl(imut,jmut), dy_br(imut,jmut), dy_tl(imut,jmut), dy_tr(imut,jmut) write(unit=N_T) a_bl ! Ubox area of bottomleft 1/4 grid write(unit=N_T) a_br ! Ubox area of bottomright 1/4 grid write(unit=N_T) a_tl ! Ubox area of topleft 1/4 grid write(unit=N_T) a_tr ! Ubox area of topright 1/4 grid write(unit=N_T) dx_bl ! Ubox length of bottomleft 1/4 grid write(unit=N_T) dx_br ! Ubox length of bottomright 1/4 grid write(unit=N_T) dx_tl ! Ubox length of topleft 1/4 grid write(unit=N_T) dx_tr ! Ubox length of topright 1/4 grid write(unit=N_T) dy_bl ! Ubox length of bottomleft 1/4 grid write(unit=N_T) dy_br ! Ubox length of bottomright 1/4 grid write(unit=N_T) dy_tl ! Ubox length of topleft 1/4 grid write(unit=N_T) dy_tr ! Ubox length of topright 1/4 grid
16.2.2
Climatological data
a. Default By default, define the climatological temperature and salinity data at model grid points (i.e., IMT = IMUT, JMT = JMUT, KK = KM). However, when the option TSINTPOL is selected, data with uniform grid spacing are read and interpolated in the model. In this case, the grid point information (DLATC, DLONC, SLATC, and SLONC) should be specified in configure.in (Table 16.2), and the data grid points (alonc, alatc) are calculated in tsclim.F90. The climatological data are read from the file file tscl (namelist infltscl) in tsclim.F90. Format of climatological data (file tscl) for default case integer(4), parameter :: imn = 12 real(4) :: ttlev(imt,jmt,kk,imn),tslev(imt,jmt,kk,imn)
do m = 1, imn write(unit=itscl) (((ttlev(i,j,k),i=1,imt),j=1,jmt),k=1,kk), & & (((tslev(i,j,k),i=1,imt),j=1,jmt),k=1,kk) enddo
b. Option TSCLDIRECT When option TSCLDIRECT is selected, data should be prepared in direct access files. Monthly data should be prepared in separate files as follows. At run time, another namelist njobpts is required to handle data. The variable itsrepeat specifies whether the parepared data are repeatedly used (=1) or interannual variations are assumed (=0). When itsrepeat = 0, the month 00 (13) means the December (January) of the previous (next) year. The variable itsmonfst specifies the first data for “this” run, the start time is between mid(itsmonfst) and mid(itsmonfst − 1).
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integer(4), parameter :: imn = 12 character(128) :: file_tscl_month real(4) :: ttlev(imt,jmt,kk,imn), tslev(imt,jmt,kk,imn)
! separate file for each month do m = 1, imn write(file_tscl_month,’(1a,i2.2)’) trim(file_tscl), m open (unit=itscl,file=file_tscl_month,access=direct,recl=4*imt*jmt*kk) write(unit=itscl,rec=1) (((ttlev(i,j,k),i=1,imt),j=1,jmt),k=1,kk) write(unit=itscl,rec=2) (((tslev(i,j,k),i=1,imt),j=1,jmt),k=1,kk) close(itscl) end do
16.2.3 Nudging (body forcing) data The time integration could be conducted restoring temperature and salinity of the specified grid points to the climatological data read from the file file tscl (often referred to as nudging or body forcing). This is done by setting the namelist parameter iforcev (namelist njobp) to one. In this case, the data that contain information about the gridwise “on (=1)” or “off (=0)” of nudging should be prepared in a default case. In a case of TSINTPOL, data that contain the grid number of points where nudging is done should be prepared. It is read from the file file frc (namelist njobbdy) in rstcoef.F90. The restoring time for nudging should be specified as rtmscb (namelist njobbdy) in units of days. Format for nudging grid points (file frc) for TSINTPOL case #ifdef OGCM_TSINTPOL
! when using interpolated climatological data, ! for example a large size model
! numbf : the number of grid points for nudging ! numbf is a namelist parameter in the model (rstcoef.F90) integer(4), save :: numbf = 1000 namelist /nblarge/ numbf read(unit=5,nml=nblarge,iostat=ios) ! iposbf : the grid number of grids where nudging is done integer(4):: iposbf(numbf) write(ifrcdt) (iposbf(n),n=1,numbf)
! serial form
! in the subroutine rstcoef, iposbf(numbf) is treated as follows and ! the nudging grid points at (i,j) are defined. do n = 1, numbf j = iposbf(n)/imut + 1 i = iposbf(n)  (j1)*imut chfb(i,j,1:km) = chfbc ! coefficient for nudging ! (chfbc = 1.0d0 / rtmscb / 86400) enddo #endif /* OGCM_TSINTPOL */
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Format for nudging grid points (file frc) for default case
#ifndef OGCM_TSINTPOL #ifdef OGCM_BF2D
! for a small size model ! for two dimensional distribution
! chf2d : 2D on(=1) off(=0) information for nudging ! imut, jmut are the numbers of grid points in the model domain real(4) :: chf2d(imut,jmut) write(ifrcdt) chf2d
#else /* OGCM_BF2D */
! ! ! ! !
! single precision
serial form distributed to each node and stored in chfb(imx,jmx,km) multiplied by chfbc (chfbc = 1.0d0 / rtmscb / 86400)
! for three dimensional distribution
! chf3d : 3D on(=1) off(=0) information for nudging ! imut, jmut, km are the numbers of grids in the model domain real(8)
:: chf3d(imut,jmut,km)
write(ifrcdt) chf3d
! ! ! ! ! !
serial form read using the subroutine restart_read distributed to each node and stored in chfb(imx,jmx,km) multiplied by chfbc (chfbc = 1.0d0 / rtmscb / 86400)
#endif /* OGCM_BF2D */ #endif /* OGCM_TSINTPOL */
16.2.4
Atmospheric forcing data
By default, the surface forcing data are read at a uniform time interval (isrstb; in seconds), which should be specified in namelist ifrcd (force.F90) along with a parameter ifna that specifies the record number of surface forcing data corresponding to the nearest future from the start time of ”this” run. A leap year is set according to the calendar subroutine. When climatological data is repeatedly used, specify the option CLMFRC. When monthly data are used, specify the option MONFRC. The following data files should be prepared according to the chosen model options. Each file is opened only once at the beginning of the runtime and thus should contain all data needed for that run.
a. Wind stress data (mandatory): file wind (namelist inflw) zonal component: wsx4(imf,jmf) [dyn · cm−2 ] meridional component: wsy4(imf,jmf) [dyn · cm−2 ]
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real(4) :: wsx4(imf,jmf),wsy4(imf,jmf)
do irec = 1, nrec ! ‘‘nrec’’ is the number of data ! data formation write(unit=iwind,rec=irec) wsx4, wsy4 ! direct form enddo
b. Radiation data (HFLUX): file hflx (namelist inflh) short wave: qsh4(imf,jmf) [erg · s−1 · cm−2 = 10−3 W · m−2 ] long wave: qlo4(imf,jmf) [erg · s−1 · cm−2 = 10−3 W · m−2 ] sea surface temperature: sst4(imf,jmf) [◦ C]; in the case of LWDOWN (only downward radiation) dummy data are used
real(4) :: qsh4(imf,jmf), qlo4(imf,jmf) real(4) :: sst4(imf,jmf)
do irec = 1, nrec ! ‘‘nrec’’ is the number of data ! data formation write(unit=ihflx,rec=irec) qsh4, qlo4, sst4 ! direct form enddo
c. Data needed to calculate the latent and sensible heat fluxes using bulk formula (HFLUX): file bulk (namelist inflh) air temperature: sat4(imf,jmf) [◦ C] specific humidity: qar4(imf,jmf) wind speed: wdv4(imf,jmf) [cm · s−1 ] sea surface pressure: slp4(imf,jmf) [hPa]
real(4) :: sat4(imf,jmf), qar4(imf,jmf) real(4) :: wdv4(imf,jmf), slp4(imf,jmf)
do irec = 1, nrec ! ‘‘nrec’’ is the number of data ! data formation write(unit=ibulk,rec=irec) sat4,qar4,wdv4,slp4 enddo
d. Precipitation data (WFLUX): file prcp (namelist inflp) precipitation: pcp4(imf,jmf) [cm · s−1 ] Note: fresh water flux data should be scaled by pure water density (1.036 g cm−3 )
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real(4) :: pcp4(imf,jmf)
do irec = 1, nrec ! ‘‘nrec’’ is the number of data ! data formation write(unit=ipcpr,rec=irec) pcp4 enddo
e. River discharge data (RUNOFF): file rnof (namelist inflo) river discharge: rof4(imrof,jmrof) [cm · s−1 ] Note: fresh water flux data should be scaled by pure water density (1.036 g cm−3 )
real(4) :: rof4(imrof,jmrof) ! imrof, jmrof could be different than imf, jmf
do irec = 1, nrec ! ‘‘nrec’’ is the number of data ! data formation write(unit=irnof,rec=irec) rof4 enddo
f. Seaice fractional area data (ICECLIM): file icec (namelist inflic) seaice fractional area: aic4(imf,jmf)
real(4) :: aic4(imf,jmf)
do irec = 1, nrec ! ‘‘nrec’’ is the number of data ! data formation write(unit=iicec,rec=irec) aic4 enddo
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16.3 Execution To run a model, a shell script that handles input/output files, executes the compiled binary ogcm, and postprocesses is usually prepared. The namelist parameters that control the job have to be given to the standard input or redirection from a file. This section presents an example shell script for the global tripolar grid model. The following three files are used in this example. • run.sh :: A shell script that submits the job to a batch request controlling system NQS. • runogcm.sh :: The shell script that includes the command ogcm, the binary that executes the model. • NAMELIST :: The namelist file that is sent to ogcm via the standard input. For example, use the following command to submit a job to the batch request controlling system. % qsub run.sh
Example of shell script that should be submitted (run.sh)
#!/bin/sh #PBS l cpunum_job=8 #PBS l memsz_job=10gb #PBS l cputim_job=120000 #PBS A K0001 #PBS N coreI # RUNDIR=${HOME}/MRICOM/coreI/run # The directory of this shell script cd ${RUNDIR} echo echo START date echo pwd /usr/bin/mpirun np 8 ./runogcm.sh # this example is for the 8MPI job echo echo END date echo 
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An example shell script runogcm.sh that is called from run.sh
#!/bin/sh # # RUNDIR ... the directory of this shell script # set to reconfirm the path of the environmental variables # for the sub processes RUNDIR=$HOME/MRICOM/coreI/run cd $RUNDIR if [ ! s logs ]; then mkdir ./logs # output directory for execution logs (standard output) fi if [ ! s logs/ftrace ]; then mkdir ./logs/ftrace # output directory for job diagnoses by the system fi F_FTRACE=FMT1; export F_FTRACE F_SETBUF06=0; export F_SETBUF06 F_RECLUNIT=BYTE; export F_RECLUNIT FTRACEDIR=logs/ftrace; export FTRACEDIR ./ogcm
> logs/out.txt 2> logs/stderr.txt < NAMELIST
=======================
Example for NAMELIST ========================== (see README.Namelist for details)
# basic parameters for model time integration &njobp nfirst=1, nstep=48 neng=1, nneng=1, nwrit=48 nwrt2=1, mampai=1, mmpai2=1, nkeisu=48 adtuv=30, adtts=30, adtsf=30, hdts=1.0d7, hduv=0.0d0, hduv2=0.0d0, vduv=0.1d0, alpha=1.0d0, gamma=51*1.0d0, hupp=0.5d0, vupp=0.7d0, ifrcsf=1, iforcv=0, ispin=0, matsuno_int=12, nocbgt=0, flg_timemonitor=.false. /
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# (ibyr,ibmn,ibdy) model integration starting date (year, month, day) &njobpt ibyr=1001, ibmn=1, ibdy=1, / # file name for variable grid spacing (option VARIABLE) &inflg file_vgrid=’../data/vgrid.d’ / # Smagorinsky parameterization for horizontal viscosity (option SMAGOR) # (cscl) scaling factor of Smagorinsky viscosity coefficient &njobsmg cscl = 3.5d0 / # parameters for isopycnal diffusion, diapycnal diffusion, layer thickness diffusion &njobpi ai=1.0d7 ad=0.1d0 aitd=5.0d6 / # the number of restart and history files which is output during the job &numfl num_restart=1, num_hist=12 / # the names of the mandatory input data files # snapshot data to start integration &infla file_restart_in=’result/restart.1000’ / # topography &infltopo file_topo=’../data/topo.d’ / #scale factor &inflscl file_scale=’../data/scale_factor.d’ / # temperature and salinity climatology &infltscl file_tscl=’../data/tsclim.d’ / # # # # # #
nudging if iforcv (/njobp/) = 0, dummy file name for file_frc should be specified (rtmsc) restoring time for surface forcing [day] (rtmscb) restoring time for body forcing [day] (kmb) the vertical level below which body forcing (nudging) is applied for all area (kmb_cnst) the vertical level below which body forcing (nudging) is applied
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16.3. Execution # for the area specified in (file_frc) &njobbdy file_frc=’../data/bforce.d’, rtmsc=8.0d0, rtmscb=30.0d0, kmb=52, kmb_cnst=2, / # the name of the final state (restart) file &outfr file_restart_out_temp=’result/restart.1001’, / # The time step (in minutes) for the free surface equation and the way of timefiltering &njobpf adttr=1, ntflt=1, / # surface forcing data # (isrstb) the time interval of the surface forcing data # (ifna) the first record number for the sea surface data used for ‘‘this’’ run # the start time is between (ifna) and (ifna  1) &ifrcd isrstb=21600 ifna=1, / # wind stress &inflw file_wind=’../data/file_wind_core.grd’, / # the names of the input data files for the sea surface flux &inflh file_hflx=’../data/file_hflux_core.grd’, file_bulk=’../data/file_bulk_core.grd’, / # the name of the input data file for precipitation data &inflp file_prcp=’../data/file_prcp_core.grd’, / # the name of the input data file for river discharge data &inflo file_rnof=’../data/file_rnof_core.grd’, / # These parameters set up the diffusion around the river mouth &nrivermouth CFLlim=0.2d0, nspreadnum=0, flg_enhance_vm_rivmouth=.false., avdrmax=1.0d4, dep_rivmix=30.0d2, afc1=10.0d0,
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afc2=7.0d0, sal_lim=5.d0, hdsal=1.d8, / # the name of the output data file for mean states &outfh file_hist_temp=’result/hist.1001’, / # the name of the output data file for flux mean states &outff file_hflux_temp=’result/hflux.1001’, / # These parameters set the sea surface albedo &njobalb alb=0.066d0, albedo_choice=1 / # basic properties for sea ice &njobpsi irstrt=1, akh=1.0d2, int_bgtice=0, nstepi=1488, ibyri=1001, ibmni=1, ibdyi=1, / # the time step interval for ice dynamics &njobidyn adtdi=1.0d0 / # These parameters set the seaice albedo &njobalbsi alb_ice_visible_t0=0.8d0, alb_ice_nearIR_t0=0.52d0, alb_snw_visible_t0=0.98d0, alb_snw_nearIR_t0=0.70d0, alb_ice_visible_dec_ratio=0.075d0, alb_ice_nearIR_dec_ratio=0.075d0, alb_snw_visible_dec_ratio=0.10d0, alb_snw_nearIR_dec_ratio=0.15d0, hi_ref=0.5d0, atan_ref=3.0d0, tsfci_t0=1.0d0, tsfci_t1=0.0d0, fsnow_patch=0.02d0, / # The name of the sea ice restart file (input) &infli file_ice_restart_in=’result/ice_restart.1000’, /
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16.3. Execution # The name of the sea ice restart file (output) &outfir file_ice_restart_out_temp=’result/ice_restart.1001’, / # The name of the sea ice history file &outfih file_ice_hist_temp=’result/ice_hist.1001’, / # These parameters set the seaice thickness category &njobpscat hbound=0.0d0,0.6d0,1.4d0,2.4d0,3.6d0,30.0d0, lsicat_volchk=.false. / # the number of intervals by which mean states of thicknesscategorized ice are calculated &nhsticint num_hint_ic=1 / # These parameters set output of the mean states for thicknesscategorized seaice &nhsticfile maxnum_hist_ic=1, nwrt_hist_ic=1, imin_hist_ic=1, imax_hist_ic=364, jmin_hist_ic=1, jmax_hist_ic=368, file_ice_hist_ic_temp=’result/sicathsta.1001’ / # The name of the restart file (input) of the thicknesscategorized sea ice &inflic file_icecat_restart_in=’result/sicatrsta.1000’ / # These parameters set the output of final state for the thicknesscategorized sea ice &outflic num_rst_ic=1, maxnum_rst_ic=1, nwrt_rst_ic=1, file_icecat_restart_out_temp=’result/sicatrsta.1001’ / =====================================================================================
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Table 16.4. Main output data files and their related program files (see Section 16.4) file name specified in namelist as output from
restart data
file restart out temp
writdt.F90
(for ice averaged over all categories)
file ice restart out temp
writdt.F90
(for ice for each thickness category)
file icecat restart out temp
mean values
file hist temp
writdt.F90
(for sea surface fluxes)
file hflux temp
writdt.F90
(for ice averaged over all categories)
file ice hist temp
writdt.F90
(for ice for each thickness category)
file ice hist ic temp
ice restart.F90
ice hist.F90
16.4 Structure of output files This section summarizes the format of the final state (restart) data to resume the model integration and that of the mean state (history) data used for monitoring and analyses (Table 16.4). In addition to these, we are developing a simple module that outputs a mean state or a snap shot by only specifying a namelist at run time. For example, to have monthly mean temperature, add a following namelist entry, An example namelist for monthly mean temperature output &nmlhs_t fname=’result/hs_tt’ wrtint=1 [undef_mask=9.99e33] [ymdhm=2] /
where fname specifies the basename of file, wrtint specifies the output interval in terms of integration time step (1 for monthly output), and ymdhm specifies the depth of calendar date used in the file name. In the above example, the temperature averaged for a month of mm of a year yyyy is output to file result/hs tt.yyyymm. See README.Namelist for items available.
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16.4.1
Snapshot (restart)
Restart for the ocean model
integer(4) real(8) :: real(8) :: real(8) :: real(8) :: real(8) :: real(8) :: real(8) :: integer(4)
:: last, month, iday, ihour, imin ahour(km), aday(km), ayear(km), pd(km), pm(km+1), dmn(km) ddmna, over u(imut,jmut,km), v(imut,jmut,km) t(imut,jmut,km), s(imut,jmut,km) avd(imut,jmut,km), avm(imut,jmut,km) avq(imut,jmut,km), eb(imut,jmut,km) dbuf(imut,jmut,3) :: nu ! device number
! The informations below are written to a single file in the following order ! by serial form for the default case. ! ! ! ! ! ! ! ! ! ! ! ! !
last
: the total number of the integrated time steps : at the time of writing the restart data month, iday, ihour, imin : month, day, hour and minute for restart data over : the parameter of relaxation for barotropic stream function : (dummy data for FREESURFACE) ahour, aday, ayear : total integrated hour, day and year from the start : of the time integration (ahour is master) pd, pm : averaged pressure at each vertical level : used for the calculation of the equation of state : pd: Tpoints : pm: middle point between adjacent Tpoints ddmna : averaged density of the whole ocean dmn : averaged density at each vertical level
write(nu) last, month, iday, ihour, imin, over, & & ahour, aday, ayear, pd, pm, ddmna, dmn ! In the case of SPLITREST the followings are written in the separate files ! after the above informations about time. write(nu) write(nu) write(nu) write(nu) write(nu)
u v t s dbuf
write(nu) avd write(nu) avm write(nu) avq write(nu) eb
! ! ! ! ! ! ! ! ! ! ! !
zonal velocity meridional velocity temperature salinity 1: sea surface height 2: vertically integrated zonal transport 3: vertically integrated meridional transport the vertical diffusion coeff. in the case of VVDIMP the vertical viscosity coeff. in the case of VVDIMP the vertical diffusion coeff. of turbulent kinetic energy in the case of NOHKIM in the case of NOHKIM turbulent kinetic energy
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For sea ice, the gridaveraged state and the thicknesscategorized states are written to separate files. Restart for gridaveraged seaice
integer(4) :: last, month, iday, ihour, imin ! from the ocean main part real(8) :: a0iceo(imut,jmut) ! seaice concentration real(8) :: hiceo(imut,jmut) ! seaice thickness real(8) :: hi (imut,jmut) ! averaged thickness = a0iceo * hiceo real(8) :: hsnwo(imut,jmut) ! snow depth real(8) :: hsnw (imut,jmut) ! averaged snow depth = a0iceo * hsnwo real(8) :: t0iceo(imut,jmut) ! skin temperature beneath the sea ice real(8) :: t0icel(imut,jmut) ! skin temperature in the open leads real(8) :: tsfci (imut,jmut) ! seaice surface temperature real(8) :: uice(imut,jmut), vice(imut,jmut) ! drift vector ! stress tensor real(8) :: sigma1(imut,jmut), sigma2(imut,jmut), sigma3(imut,jmut) real(8) :: dbuf(imut,jmut,10) integer(4) :: nui ! device number write (nui) last, month, iday, ihour, imin dbuf(1:imut,1:jmut,1) = a0iceo(1:imut,1:jmut) dbuf(1:imut,1:jmut,2) = hi(1:imut,1:jmut) dbuf(1:imut,1:jmut,3) = hiceo(1:imut,1:jmut) dbuf(1:imut,1:jmut,4) = hsnw(1:imut,1:jmut) dbuf(1:imut,1:jmut,5) = hsnwo(1:imut,1:jmut) dbuf(1:imut,1:jmut,6) = t0iceo(1:imut,1:jmut) dbuf(1:imut,1:jmut,7) = t0icel(1:imut,1:jmut) dbuf(1:imut,1:jmut,8) = tsfci(1:imut,1:jmut) dbuf(1:imut,1:jmut,9) = uice(1:imut,1:jmut) dbuf(1:imut,1:jmut,10) = vice(1:imut,1:jmut) write(nui) dbuf(1:imut,1:jmut,1:10) dbuf(1:imut,1:jmut,1) = sigma1(1:imut,1:jmut) dbuf(1:imut,1:jmut,2) = sigma2(1:imut,1:jmut) dbuf(1:imut,1:jmut,3) = sigma3(1:imut,1:jmut) write(nui) dbuf(1:imut,1:jmut,1:3)
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16.4. Structure of output files
Restart data for thicknesscategorized seaice
integer(4) :: nstep_job, month, iday, ihour, imin real(8) :: aihour ! total integrated time integer(4), parameter :: ncat = 5 ! number of thickness categories ! ice concentration real(8) :: aicen (1:imut,1:jmut,0:ncat), a0iceo(imut,jmut) ! ice thickness real(8) :: hicen (1:imut,1:jmut,0:ncat), hiceo (imut,jmut) ! averaged seaice thickness real(8) :: hin (1:imut,1:jmut,0:ncat), hi (imut,jmut) ! snow depth real(8) :: hsnwn (1:imut,1:jmut,0:ncat), hsnwo (imut,jmut) ! averaged snow thickness real(8) :: hsn (1:imut,1:jmut,0:ncat), hsnw (imut,jmut) ! ice surface temperature real(8) :: tsfcin(1:imut,1:jmut,0:ncat), tsfci (imut,jmut) ! ice temperature real(8) :: t1icen(1:imut,1:jmut,0:ncat) ! sea surface skin temperature real(8) :: t0icen(1:imut,1:jmut,0:ncat) ! sea surface skin salinity real(8) :: s0n (1:imut,1:jmut,0:ncat) ! skin temperature beneath the sea ice real(8) :: t0iceo(imut,jmut) ! skin temperature in the open leads real(8) :: t0icel(imut,jmut) ! stress tensor real(8) :: sigma1(imut,jmut), sigma2(imut,jmut), sigma3(imut,jmut) integer(4) :: nu_icecat_rst ! device number write (nu_icecat_rst) nstep_job, month, iday, ihour, imin, aihour do m = 0, ncat write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) end do write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst) write(nu_icecat_rst)
aicen(1:imut,1:jmut,m) hin (1:imut,1:jmut,m) hsn (1:imut,1:jmut,m) hicen(1:imut,1:jmut,m) hsnwn(1:imut,1:jmut,m) tsfcin(1:imut,1:jmut,m) t1icen(1:imut,1:jmut,m) t0icen(1:imut,1:jmut,m) s0n(1:imut,1:jmut,m)
a0iceo(1:imut,1:jmut) hi(1:imut,1:jmut) hiceo(1:imut,1:jmut) hsnw(1:imut,1:jmut) hsnwo(1:imut,1:jmut) t0iceo(1:imut,1:jmut) t0icel(1:imut,1:jmut) tsfci(1:imut,1:jmut) uice(1:imut,1:jmut) vice(1:imut,1:jmut) sigma1(1:imut,1:jmut) sigma2(1:imut,1:jmut) sigma3(1:imut,1:jmut)
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16.4.2 Averaged value (history)
Ocean model history data (HIST)
integer(4) :: nkai, month, iday, ihour, imin, mdays real(4) :: um(imut,jmut,km), vm(imut,jmut,km) real(4) :: tm(imut,jmut,km), sm(imut,jmut,km) real(4) :: hm(imut,jmut) integer(4) :: nu_hst ! device number ! The informations below are written to a single file in the following order ! by serial form for the default case. ! nkai ! ! month, ! ! mdays
: the total number of the integrated time steps : at the time of writing the data iday, ihour, imin : month, day, hour and minute : at the time of writing the data : the data are averaged over mdays
write(nu_hst) nkai, month, iday, ihour, imin, mdays ! In the case of SPLITHIST the data below are written in a different file ! after the above information about time write(nu_hst) write(nu_hst) write(nu_hst) write(nu_hst) write(nu_hst)
um vm tm sm hm
! ! ! ! !
zonal velocity meridional velocity temperature salinity sea surface height
sea surface flux averaged value history file (HISTFLUX)
integer(4) :: nkai, month, iday, ihour, imin, mdays real(4) real(4) real(4) real(4) real(4) real(4)
:: :: :: :: :: ::
sqlw (imut,jmut) ! longwave radiation (downward minus upward) sqsn (imut,jmut) ! sensible heat flux sqla (imut,jmut) ! latent heat flux shflux(imut,jmut) ! total heat flux swflux(imut,jmut) ! total fresh water flux sstrx(imut,jmut), sstry(imut,jmut) ! sea surface stress
integer(4) :: nuf_hst ! device number write(nuf_hst) write(nuf_hst) write(nuf_hst) write(nuf_hst) write(nuf_hst) write(nuf_hst) write(nuf_hst) write(nuf_hst)
nkai, month, iday, ihour, imin, mdays sqlw (1:imut,1:jmut) sqsn (1:imut,1:jmut) sqla (1:imut,1:jmut) shflux(1:imut,1:jmut) swflux(1:imut,1:jmut) sstrx (1:imut,1:jmut) sstry (1:imut,1:jmut)
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16.4. Structure of output files For sea ice, the gridaveraged state and the thicknesscategorized states are written to separate files. Gridaveraged seaice history file (ICE)
integer(4) :: nkai, month, iday, ihour, imin, mdays real(4) real(4) real(4) real(4) real(4)
:: :: :: :: ::
sar(imut,jmut) ! seaice concentration shi(imut,jmut) ! averaged seaice thickness ssn(imut,jmut) ! averaged snow depth sti(imut,jmut) ! seaice surface temperature uice(imut,jmut), vice(imut,jmut) ! drift vector
real(4) :: sbuf(imut,jmut,6) integer(4) :: nui_hst ! device number write (nui_hst) nkai, month, iday, ihour, imin, mdays sbuf(1:imut,1:jmut,1) = shi(1:imut,1:jmut) sbuf(1:imut,1:jmut,2) = ssn(1:imut,1:jmut) sbuf(1:imut,1:jmut,3) = sar(1:imut,1:jmut) sbuf(1:imut,1:jmut,4) = sti(1:imut,1:jmut) sbuf(1:imut,1:jmut,5) = sui(1:imut,1:jmut) sbuf(1:imut,1:jmut,6) = svi(1:imut,1:jmut) write(nui_hst) sbuf(1:imut,1:jmut,1:6)
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Thicknesscategorized seaice history file (ICECAT)
integer(4) :: nstep_job, month, iday, ihour, imin, mdays integer(4), parameter :: ncat = 5 ! number of thickness category real(4) :: shi0(1:imut,1:jmut) ! sum of averaged seaice thickness ! for all categories real(4) :: shs0(1:imut,1:jmut) ! sum of snow depth for all category real(4) :: sai0(1:imut,1:jmut) ! sum of concentration for all category real(4) :: sui0(1:imut,1:jmut), svi0(1:imut,1:jmut) ! seaice drift velocity real(4) real(4) real(4) real(4) real(4) real(4) real(4)
:: :: :: :: :: :: ::
sain(1:imut,1:jmut,0:ncat) st0n(1:imut,1:jmut,0:ncat) ss0n(1:imut,1:jmut,0:ncat) shin(1:imut,1:jmut,1:ncat) shsn(1:imut,1:jmut,1:ncat) stsn(1:imut,1:jmut,1:ncat) st1n(1:imut,1:jmut,1:ncat)
! ! ! ! ! ! !
seaice concentration sea surface skin temperature sea surface skin salinity averaged seaice thickness averaged snow depth seaice surface temperature seaice temperature
integer(4) :: nu_icecat_hst ! device number write(nu_icecat_hst) nstep_job, month, iday, ihour, imin, mdays write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst)
= = = = =
shi0(1:imut,1:jmut) shs0(1:imut,1:jmut) sai0(1:imut,1:jmut) sui0(1:imut,1:jmut) svi0(1:imut,1:jmut)
write(nu_icecat_hst) = sain(1:imut,1:jmut,0) write(nu_icecat_hst) = st0n(1:imut,1:jmut,0) write(nu_icecat_hst) = ss0n(1:imut,1:jmut,0) do m = 1, ncat write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) write(nu_icecat_hst) end do
= = = = = = =
shin(1:imut,1:jmut,m) shsn(1:imut,1:jmut,m) sain(1:imut,1:jmut,m) stsn(1:imut,1:jmut,m) st1n(1:imut,1:jmut,m) st0n(1:imut,1:jmut,m) ss0n(1:imut,1:jmut,m)
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16.5. Appendix
16.5
Appendix
16.5.1
Model options
The model options are as follows. Only major options are listed here. Description about those related to biogeochemical models can be found in chapter 11. The description of all options for the latest version can be found in src/README.Options. In the source program an expression like OGCM PARALLEL is used but here OGCM is omitted for option expressions. Table 16.5. Description of Model Options Model option
Description
BBL BF2D
uses the bottom boundary layer model
BIHARMONIC
BULKKARA BULKKONDO2 BULKNCAR
reads 2D distribution of ”on”/”off” for nudging uses biharmonic operator for both horizontal viscosity and diffusion (*) If ISOPYCNAL is also selected, the biharmonic form is used only for viscosity and not for diffusion. Kara (2000) is used for the surface flux bulk formula. Kondo(1975) is used for the surface flux bulk formula. Large and Yeager (2004) is used for the surface flux bulk formula. This option corresponds to the COREs.
BULKITER
(*) BULKKARA, BULKKONDO2, BULKNCAR is available only for HFLUX case. Bulk transfer coefficient is calculated using iterative method
CALALBSI
if the observed wind speed is not at 10m. (*) use with BULKKONDO2 and BULKNCAR Seaice albedo is calculated using seaice conditions according to LosAlamos model instead
CALPP CARBON
of using a constant value considers the time variation of pressure for the equation of state
CBNHSTRUN
biogeochemical process is included (*) NUMTRC P=4 for ObataKitamura model; NUMTRC P=8 for NPZD model atmospheric pCO2 is given from file
CHFDIST
(*) use with CARBON employs horizontal distribution of surface restoring time for SST and SSS
CHLMA94 CLMFRC CYCLIC DIAGTRANSP
shortwave penetration scheme with chlorophyll concentration by Morel and Antoine (1994) (*) use with NPZD and SOLARANGLE uses climatological wind stress and surface heat flux (no leap year, repeating one year cycle) uses zonally cyclic condition outputs time averaged vertically integrated transport UM, VM
DIFAJS
(*) available only for FREESURFACE sets large vertical diffusion coefficient (1.0 m2 · s−1 ) between unstable points instead of convective adjustment
FLXINTPOL
interpolates the surface forcing data to model grid points
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Model option
description
FREESURFACE FSEB FSMOM
free surface (if undefined, rigidlid) free surface equations are advanced using the Eularbackward timeintegration
FSVISC
uses MOM free surface scheme calculates viscosity explicitly in the barotropic momentum equation (*) FSEB, FSMOM, FSVISC are available only in the case of FREESURFACE
GMANISOTROP
Anisotropic horizontal variation of thickness diffusion is used (*) use with ISOPYCNAL
GMVAR HFLUX HIST
Horizontal thickness diffusion is allowed to vary in horizontal calculates sea surface heat flux using bulk formula outputs averaged state of temperature, salinity and velocity
HISTFLUX HISTVAR
outputs averaged state of surface fluxes outputs variance of temperature, salinity and velocity
ICE ICECAT
(*) available only for HIST sea ice is included sea ice is categorized according to its thickness
ICECLIM INILEV
reading climatological seaice fractional area from file sets Levitus climatological threedimensional data for the initial value
ISOPYCNAL
uses isopycnal diffusion and GentMcWilliams’ parameterization for eddy induced tracer transport velocity (thickness diffusion) uses KPP for mixed layer model
KPP LWDOWN MELYAM
Long wave radiation data include only downward component instead of default net radiation uses Mellor and Yamada Level 2.5 for mixed layer model
MON30D MONFRC NESTONLINE
sets one month as 30 days and one year as 360 days uses monthly forcing for surface data low resolution (PARENT) and high resolution (SUB) models exchange data online
NOHKIM
(*) use with SCUP uses Noh’s mixed layer model
NOMATSUNO NPZD
uses forward finite difference instead of Matsuno scheme NPZD process is included (*) use with CARBON
PARALLEL
parallel calculation using MPI. The number of zonally and meridionally partitioned regions should be specified as NPARTX and NPARTY, respectively.
PARENT PRAJS
executed as low resolution model of the nesting calculation adjusts the freshwater flux to suppress the increase/decrease of total ocean volume (*) available only for WFLUX
RUNOFF
uses river runoff data (*) available only for WFLUX
SALCNSVRS
Salinity restoring flux is corrected so that globally integrated salinity flux become zero at each time step
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16.5. Appendix Model option SFLUXR SFLUXW
description (*) use with ”SFLUXR” SSS is restored to the climatological sea surface salinity as the salinity flux
SIDYN
calculates salinity flux converting from the surface freshwater flux (*) available only for WFLUX seaice dynamics model (EVP)
SLIMIT
(*) available only for ICE Tapers thickness diffusion near the sea surface
SMAGHD
uses the Smagorinsky viscosity coefficient multiplied by a constant ratio as the horizontal diffusion coefficient (*) available only for SMAGOR
SMAGOR SOLARANGLE
uses the Smagorinsky parameterization for horizontal viscosity solar insolation angle is considered in calculating short wave penetration
SOMADVEC SPHERICAL SPLITHIST
uses second order moment advection by Prather (1986) calculates scale factor semianalytically for the spherical coordinates output averaged temperature, salinity, velocity and sea surface height to separate files
SPLITREST
(*) available only for HIST input(output) initial(final) state data from(to) separate files for each properties
SUB TAUBULK TDEW
executed as a high resolution model of the nesting calculation calculates the wind stress using bulk formulae by reading wind speed over the ocean reads dewpoint temperature and converts to specific humidity
TRCBIHARM
(*) available only for HFLUX uses biharmonic operator for horizontal diffusion
TSCLDIRECT
TSINTPOL
(*) Should not be used with ISOPYCNAL Temperature and salinity climatology is read from direct access files naming convention is (file tscl)mm (mm = month) see README.Namelist interpolates climatological temperature and salinity data to model grids
ULTIMATE
applies ultimate limiter for the calculation of the horizontal advection of temperature and salinity (*) available only for for UTOPIA
UTOPIA VARHID
uses UTOPIA scheme for horizontal advection of temperature and salinity sets the horizontal (isopycnal) diffusion coefficient and viscosity as functions of grid size
VARIABLE VISANISO
variable horizontal grid spacing Anisotropic viscosity coefficients are used (*) use with VIS9P
VIS9P VISBIHARM VMBG3D
calculates the viscosity using adjacent 9 grid points uses biharmonic operator for both horizontal viscosity
VVDIMP
reads 3D vertical viscosity and diffusion coefficients from a file calculates the vertical diffusion/viscosity by implicit method (*) it is automatically loaded if any mixed layer model is used or option ISOPYCNAL is selected
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Model option
description
WADJ
adjusts sea surface freshwater flux every time step to keep its global sum to be zero (*) available only for WFLUX
WFLUX
uses the sea surface freshwater flux to force the model no leap year uses QUICKEST for the vertical advection of temperature and salinity
Y365 ZQUICKEST ZULTIMATE
applies ultimate limiter for the vertical advection of temperature and salinity (*) available only for ZQUICKEST
CGCM MOVE SCUP
used as an ocean module for a coupled model used as ocean module for data assimilation (MOVE) system use simple coupler (SCUP) library
SCUPCGCM
used as an ocean module for a coupled model using scup for communication
References Ishizaki, H., and I. Ishikawa, 2006: High parallelization efficiency in barotropicmode computation of ocean models based on multigrid boundary ghost area, Ocean Modell., 13, 238254.
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Acknowledgments The development of MRI.COM was first proposed by Yoshiteru Kitamura. During the course of its development, significant contributions were made by Seiji Yukimoto and Atsushi Obata. The authors owe special thanks to Hiroyasu Hasumi of the University of Tokyo for his valuable suggestions. Application of the EVP dynamics to the sea ice model, of σ coordinates to the free surface formulation, and of highaccuracy advection schemes to the tracer equations would not have been successful without his support. The experiences of coupling with other component models, such as atmospheric models and data assimilation schemes, greatly helped improve this model. The authors are grateful to the developers of those models. Among them, various comments made by Yosuke Fujii during his development of an adjoint code of this model were indispensable for refining the model codes. Thanks are extended to Shunji Konaga, Tatsushi Tokioka, Masahiro Endoh, Yoshihiro Kimura, and Noriya Yoshioka for their numerous contributions to the original models that MRI.COM is based on. Most figures of this manual were drawn by Yukiko Suda. The people listed above and all other members and former members of the Oceanographic Research Department of MRI contributed to the development and refinement of this model. Since the first version of this manual was published in Japanese, some authors had been requested or encouraged to publish an English version on many occasions to introduce this model at international scientific conferences and workshops. This fact strongly motivated us to write a new version in English. Continuous encouragement and numerous suggestions by Stephen Griffies at Geophysical Fluid Dynamics Lab., NOAA, USA and Gurvan Madec at LODYC, Institute Pierre Simon Laplace, France, during panel meetings of the working group for ocean model development (WGOMD) of Climate Variability and Predictability (CLIVAR) are gratefully acknowledged.
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気象研究所技術報告一覧表 第１号
バックグラウンド大気汚染の測定法の開発（地球規模大気汚染特別研究班，1978） Development of Monitoring Techniques for Global Background Air Pollution. (MRI Special Research Group on Global Atmospheric Pollution, 1978) 第２号 主要活火山の地殻変動並びに地熱状態の調査研究（地震火山研究部，1979） Investigation of Ground Movement and Geothermal State of Main Active Volcanoes in Japan. (Seismology and Volcanology Research Division, 1979) 第３号 筑波研究学園都市に新設された気象観測用鉄塔施設（花房龍男・藤谷徳之助・伴野 登・魚津 博，1979） On the Meteorological Tower and Its Observational System at Tsukuba Science City. (T. Hanafusa, T. Fujitani, N. Banno, and H. Uozu, 1979) 第４号 海底地震常時観測システムの開発（地震火山研究部，1980） Permanent Ocean－Bottom Seismograph Observation System. (Seismology and Volcanology Research Division, 1980) 第５号 本州南方海域水温図－400m（又は 500m）深と 1,000m 深－（1934－1943 年及び 1954－1980 年）（海洋研究部， 1981） Horizontal Distribution of Temperature in 400m (or 500m) and 1,000m Depth in Sea South of Honshu, Japan and Western －North Pacific Ocean from 1934 to 1943 and from 1954 to 1980. (Oceanographical Research Division, 1981) 第６号 成層圏オゾンの破壊につながる大気成分及び紫外日射の観測（高層物理研究部，1982） Observations of the Atmospheric Constituents Related to the Stratospheric ozon Depletion and the Ultraviolet Radiation. (Upper Atmosphere Physics Research Division, 1982) 第７号 83 型強震計の開発（地震火山研究部，1983） Strong－Motion Seismograph Model 83 for the Japan Meteorological Agency Network. (Seismology and Volcanology Research Division, 1983) 第８号 大気中における雪片の融解現象に関する研究（物理気象研究部，1984） The Study of Melting of Snowflakes in the Atmosphere. (Physical Meteorology Research Division, 1984) 第９号 御前崎南方沖における海底水圧観測（地震火山研究部・海洋研究部，1984） Bottom Pressure Observation South off Omaezaki, Central Honsyu. (Seismology and Volcanology Research Division and Oceanographical Research Division, 1984) 第 10 号 日本付近の低気圧の統計（予報研究部，1984） Statistics on Cyclones around Japan. (Forecast Research Division, 1984) 第 11 号 局地風と大気汚染質の輸送に関する研究（応用気象研究部，1984） Observations and Numerical Experiments on Local Circulation and Medium－Range Transport of Air Pollutions. (Applied Meteorology Research Division, 1984) 第 12 号 火山活動監視手法に関する研究（地震火山研究部，1984） Investigation on the Techniques for Volcanic Activity Surveillance. (Seismology and Volcanology Research Division, 1984) 第 13 号 気象研究所大気大循環モデル－Ⅰ（MRI・GCM－Ⅰ）（予報研究部，1984） A Description of the MRI Atmospheric General Circulation Model (The MRI・GCM－Ⅰ). (Forecast Research Division, 1984) 第 14 号 台風の構造の変化と移動に関する研究－台風 7916 の一生－（台風研究部，1985） A Study on the Changes of the Three  Dimensional Structure and the Movement Speed of the Typhoon through its Life Time. (Typhoon Research Division, 1985) 第 15 号 波浪推算モデル MRI と MRI－Ⅱの相互比較研究－計算結果図集－（海洋気象研究部，1985） An Intercomparison Study between the Wave Models MRI and MRI － Ⅱ － A Compilation of Results － (Oceanographical Research Division, 1985) 第 16 号 地震予知に関する実験的及び理論的研究（地震火山研究部，1985） Study on Earthquake Prediction by Geophysical Method. (Seismology and Volcanology Research Division, 1985) 第 17 号 北半球地上月平均気温偏差図（予報研究部，1986） Maps of Monthly Mean Surface Temperature Anomalies over the Northern Hemisphere for 1891－1981. (Forecast Research Division, 1986) 第 18 号 中層大気の研究（高層物理研究部・気象衛星研究部・予報研究部・地磁気観測所，1986） Studies of the Middle Atmosphere. (Upper Atmosphere Physics Research Division, Meteorological Satellite Research Division, Forecast Research Division, MRI and the Magnetic Observatory, 1986) 第 19 号 ドップラーレーダによる気象・海象の研究（気象衛星研究部・台風研究部・予報研究部・応用気象研究部・海 洋研究部，1986） Studies on Meteorological and Sea Surface Phenomena by Doppler Radar. (Meteorological Satellite Research Division, Typhoon Research Division, Forecast Research Division, Applied Meteorology Research Division, and Oceanographical Research Division, 1986) 第 20 号 気象研究所対流圏大気大循環モデル（MRI・GCM－Ⅰ）による 12 年間分の積分（予報研究部，1986） Mean Statistics of the Tropospheric MRI・GCM－Ⅰbased on 12－year Integration. (Forecast Research Division, 1986)
第 21 号 第 22 号 第 23 号 第 24 号 第 25 号 第 26 号 第 27 号 第 28 号 第 29 号
第 30 号
第 31 号
第 32 号 第 33 号 第 34 号
第 35 号 第 36 号 第 37 号 第 38 号 第 39 号 第 40 号
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宇宙線中間子強度 1983－1986（高層物理研究部，1987） Multi－Directional Cosmic Ray Meson Intensity 1983－1986. (Upper Atmosphere Physics Research Division, 1987) 静止気象衛星「ひまわり」画像の噴火噴煙データに基づく噴火活動の解析に関する研究（地震火山研究部， 1987） Study on Analysis of Volcanic Eruptions based on Eruption Cloud Image Data obtained by the Geostationary Meteorological satellite (GMS). (Seismology and Volcanology Research Division, 1987) オホーツク海海洋気候図（篠原吉雄・四竃信行，1988） Marine Climatological Atlas of the sea of Okhotsk. (Y. Shinohara and N. Shikama, 1988) 海洋大循環モデルを用いた風の応力異常に対する太平洋の応答実験（海洋研究部，1989） Response Experiment of Pacific Ocean to Anomalous Wind Stress with Ocean General Circulation Model. (Oceanographical Research Division, 1989) 太平洋における海洋諸要素の季節平均分布（海洋研究部，1989） Seasonal Mean Distribution of Sea Properties in the Pacific. (Oceanographical Research Division, 1989) 地震前兆現象のデータベース（地震火山研究部，1990） Database of Earthquake Precursors. (Seismology and Volcanology Research Division, 1990) 沖縄地方における梅雨期の降水システムの特性（台風研究部，1991） Characteristics of Precipitation Systems During the Baiu Season in the Okinawa Area. (Typhoon Research Division, 1991) 気象研究所・予報研究部で開発された非静水圧モデル（猪川元興・斉藤和雄，1991） Description of a Nonhydrostatic Model Developed at the Forecast Research Department of the MRI. (M. Ikawa and K. Saito, 1991) 雲の放射過程に関する総合的研究（気候研究部・物理気象研究部・応用気象研究部・気象衛星・観測システム 研究部・台風研究部，1992） A Synthetic Study on Cloud－Radiation Processes. (Climate Research Department, Physical Meteorology Research Department, Applied Meteorology Research Department, Meteorological Satellite and Observation System Research Department, and Typhoon Research Department, 1992) 大気と海洋・地表とのエネルギー交換過程に関する研究（三上正男・遠藤昌宏・新野 宏・山崎孝治，1992） Studies of Energy Exchange Processes between the Ocean－Ground Surface and Atmosphere. (M. Mikami, M. Endoh, H. Niino, and K. Yamazaki, 1992) 降水日の出現頻度からみた日本の季節推移－30 年間の日降水量資料に基づく統計－（秋山孝子，1993） Seasonal Transition in Japan, as Revealed by Appearance Frequency of PrecipitatingDays. －Statistics of Daily Precipitation Data During 30 Years－(T. Akiyama, 1993) 直下型地震予知に関する観測的研究（地震火山研究部，1994） Observational Study on the Prediction of Disastrous Intraplate Earthquakes. (Seismology and Volcanology Research Department, 1994) 各種気象観測機器による比較観測（気象衛星・観測システム研究部，1994） Intercomparisons of Meteorological Observation Instruments. (Meteorological Satellite and Observation System Research Department, 1994) 硫黄酸化物の長距離輸送モデルと東アジア地域への適用（応用気象研究部，1995） The Long－Range Transport Model of Sulfur Oxides and Its Application to the East Asian Region. (Applied Meteorology Research Department, 1995) ウインドプロファイラーによる気象の観測法の研究（気象衛星・観測システム研究部，1995） Studies on Wind Profiler Techniques for the Measurements of Winds. (Meteorological Satellite and Observation System Research Department, 1995) 降水・落下塵中の人工放射性核種の分析法及びその地球化学的研究（地球化学研究部，1996） Geochemical Studies and Analytical Methods of Anthropogenic Radionuclides in Fallout Samples. (Geochemical Research Department, 1996) 大気と海洋の地球化学的研究（1995 年及び 1996 年）（地球化学研究部，1998） Geochemical Study of the Atmosphere and Ocean in 1995 and 1996. (Geochemical Research Department, 1998) 鉛直２次元非線形問題（金久博忠，1999） Vertically 2dmensional Nonlinear Problem (H. Kanehisa, 1999) 客観的予報技術の研究（予報研究部，2000） Study on the Objective Forecasting Techniques（Forecast Research Department, 2000） 南関東地域における応力場と地震活動予測に関する研究（地震火山研究部，2000） Study on Stress Field and Forecast of Seismic Activity in the Kanto Region（Seismology and Volcanology Research Department, 2000） 電量滴定法による海水中の全炭酸濃度の高精度分析および大気中の二酸化炭素と海水中の全炭酸の放射性炭素 同位体比の測定（石井雅男・吉川久幸・松枝秀和，2000） Coulometric Precise Analysis of Total Inorganic Carbon in Seawater and Measurements of Radiocarbon for the Carbon Dioxide in the Atmosphere and for the Total Inorganic Carbon in Seawater（I.Masao, H.Y.Inoue and H.Matsueda, 2000） 気象研究所／数値予報課統一非静力学モデル（斉藤和雄・加藤輝之・永戸久喜・室井ちあし，2001） Documentation of the Meteorological Research Institute / Numerical Prediction Division Unified Nonhydrostatic Model (Kazuo Saito, Teruyuki Kato, Hisaki Eito and Chiashi Muroi，2001)
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大気および海水中のクロロフルオロカーボン類の精密測定と気象研究所クロロフルオロカーボン類標準ガスの 確立（時枝隆之・井上(吉川)久幸，2004） Precise measurements of atmospheric and oceanic chlorofluorocarbons and MRI chlorofluorocarbons calibration scale (Takayuki Tokieda and Hisayuki Y. Inoue，2004) PostScript コードを生成する描画ツール"PLOTPS"マニュアル（加藤輝之，2004） Documentation of "PLOTPS": Outputting Tools for PostScript Code (Teruyuki Kato，2004) 気象庁及び気象研究所における二酸化炭素の長期観測に使用された標準ガスのスケールとその安定性の再評価 に関する調査・研究（松枝秀和・須田一人・西岡佐喜子・平野礼朗・澤 庸介・坪井一寛・堤 之智・神谷ひ とみ・根本和宏・長井秀樹・吉田雅司・岩野園城・山本 治・森下秀昭・鎌田匡俊・和田 晃，2004） Reevaluation for scale and stability of CO2 standard gases used as longterm observations at the Japan Meteorological Agency and the Meteorological Research Institute (Hidekazu Matsueda, Kazuto Suda, Sakiko Nishioka, Toshirou Hirano, Yousuke, Sawa, Kazuhiro Tuboi, Tsutumi, Hitomi Kamiya, Kazuhiro Nemoto, Hideki Nagai, Masashi Yoshida, Sonoki Iwano, Osamu Yamamoto, Hideaki Morishita, Kamata, Akira Wada，2004) 地震発生過程の詳細なモデリングによる東海地震発生の推定精度向上に関する研究（地震火山研究部, 2005） A Study to Improve Accuracy of Forecasting the Tokai Earthquake by Modeling the Generation Processes (Seismology and Volcanology Research Department, 2005) 気象研究所共用海洋モデル（MRI.COM）解説（海洋研究部, 2005） Meteorological Research Institute Community Ocean Model (MRI.COM) Manual (Oceanographical Research Department, 2005) 日本海降雪雲の降水機構と人工調節の可能性に関する研究（物理気象研究部・予報研究部, 2005） Study of Precipitation Mechanisms in Snow Clouds over the Sea of Japan and Feasibility of Their Modification by Seeding (Physical Meteorology Research Department, Forecast Research Department, 2005) 2004 年日本上陸台風の概要と環境場（台風研究部, 2006） Summary of Landfalling Typhoons in Japan, 2004 (Typhoon Research Department, 2006) 栄養塩測定用海水組成標準の 2003 年国際共同実験報告（青山道夫, 2006） 2003 Intercomparison Exercise for Reference Material for Nutrients in Seawater in a Seawater Matrix (Michio Aoyama, 2006) 大気および海水中の超微量六フッ化硫黄(SF6)の測定手法の高度化と SF6 標準ガスの長期安定性の評価（時枝隆 之、石井雅男、斉藤 秀、緑川 貴, 2007） Highly developed precise analysis of atmospheric and oceanic sulfur hexafluoride (SF6) and evaluation of SF6 standard gas stability (Takayuki Tokieda, Masao Ishii, Shu Saito and Takashi Midorikawa, 2007) 地球温暖化による東北地方の気候変化に関する研究（仙台管区気象台, 環境・応用気象研究部, 2008） Study of Climate Change over Tohoku District due to Global Warming (Sendai District Meteorological Observatory, Atmospheric Environment and Applied Meteorology Research Department, 2008) 火山活動評価手法の開発研究（地震火山研究部, 2008） Studies on Evaluation Method of Volcanic Activity (Seismology and Volcanology Research Department, 2008) 日本における活性炭冷却捕集およびガスクロ分離による気体計数システムによる 85Kr の測定システムの構築お よび 1995 年から 2006 年の測定結果（青山道夫, 藤井憲治, 廣瀬勝己, 五十嵐康人, 磯貝啓介, 新田 済, Hartmut Sartorius, Clemens Schlosser, Wolfgang Weiss, 2008） Establishment of a cold charcoal trapgas chromatographygas counting system for 85Kr measurements in Japan and results from 1995 to 2006 (Michio Aoyama, Kenji Fujii, Katsumi Hirose, Yasuhito Igarashi, Keisuke Isogai, Wataru Nitta, Hartmut Sartorius, Clemens Schlosser, Wolfgang Weiss, 2008) 長期係留による 4 種類の流速計観測結果の比較（中野俊也, 石崎 廣, 四竈信行, 2008） Comparison of Data from Four Current Meters Obtained by LongTerm DeepSea Moorings (Toshiya Nakano, Hiroshi Ishizaki and Nobuyuki Shikama, 2008) CMIP3 マルチモデルアンサンブル平均を利用した将来の海面水温・海氷分布の推定（水田 亮, 足立恭将, 行本 誠史, 楠 昌司, 2008） Estimation of the Future Distribution of Sea Surface Temperature and Sea Ice Using the CMIP3 Multimodel Ensemble Mean (Ryo Mizuta, Yukimasa Adachi, Seiji Yukimoto and Shoji Kusunoki, 2008) 閉流路中のフローセルを用いた分光光度法自動分析装置による海水の高精度 pHT 測定（斉藤 秀, 石井雅男, 緑 川 貴, 井上（吉川）久幸, 2008） Precise Spectrophotometric Measurement of Seawater pHT with an Automated Apparatus using a Flow Cell in a Closed Circuit（Shu Saito, Masao Ishii, Takashi Midorikawa and Hisayuki Y. Inoue, 2008） 栄養塩測定用海水組成標準の 2006 年国際共同実験報告（青山道夫,J. BarwellClarke, S. Becker, M. Blum, Braga E.S., S. C. Coverly, E. Czobik, I. Dahllöf, M. Dai, G. O Donnell, C. Engelke, GwoChing Gong, GiHoon Hong, D. J. Hydes, MingMing Jin, 葛西広海, R. Kerouel, 清本容子, M. Knockaert, N. Kress, K. A. Krogslund, 熊谷正光, S. Leterme, Yarong Li, 増田真次, 宮尾 孝, T. Moutin, 村田昌彦, 永井直樹, G. Nausch, A. Nybakk, M. K. Ngirchechol, 小川浩史, J. van Ooijen, 太田秀和, J. Pan, C. Payne, O. PierreDuplessix, M. PujoPay, T. Raabe, 齊藤一浩, 佐藤憲一郎, C. Schmidt, M. Schuett, T. M. Shammon, J. Sun, T. Tanhua, L. White, E.M.S. Woodward, P. Worsfold, P. Yeats, 芳村 毅, A. Youénou, JiaZhong Zhang, 2008） 2006 Interlaboratory Comparison Study for Reference Material for Nutrients in Seawater（M. Aoyama, J. BarwellClarke, S. Becker, M. Blum, Braga E. S., S. C. Coverly, E. Czobik, I. Dahllöf, M. H. Dai, G. O. Donnell, C. Engelke, G. C. Gong, GiHoon Hong, D. J. Hydes, M. M. Jin, H. Kasai, R. Kerouel, Y. Kiyomono, M. Knockaert, N. Kress, K. A. Krogslund, M.
Kumagai, S. Leterme, Yarong Li, S. Masuda, T. Miyao, T. Moutin, A. Murata, N. Nagai, G. Nausch, M. K. Ngirchechol, A. Nybakk, H. Ogawa, J. van Ooijen, H. Ota, J. M. Pan, C. Payne, O. PierreDuplessix, M. PujoPay, T. Raabe, K. Saito, K. Sato, C. Schmidt, M. Schuett, T. M. Shammon, J. Sun, T. Tanhua, L. White, E.M.S. Woodward, P. Worsfold, P. Yeats, T. Yoshimura, A. Youénou, J. Z. Zhang, 2008）