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Subsections


3.15.3 Discrete Numerical Configuration

The model is configured in hydrostatic form. The domain is discretised with a uniform grid spacing in latitude and longitude of $ \Delta x=\Delta y=4^{\circ}$, so that there are ninety grid cells in the $ x$ and forty in the $ y$ direction (Arctic polar regions are not included in this experiment). Vertically the model is configured with twenty layers with the following thicknesses $ \Delta z_{1} = 50\,{\rm m},\, \Delta z_{2} = 50\,{\rm m},\, \Delta z_{3} = 55\,{\rm m},\, \Delta z_{4} = 60\,{\rm m},\, \Delta z_{5} = 65\,{\rm m},\, $ $ \Delta z_{6}~=~70\,{\rm m},\, \Delta z_{7}~=~80\,{\rm m},\, \Delta z_{8}~=95\,{\rm m},\, \Delta z_{9}=120\,{\rm m},\, \Delta z_{10}=155\,{\rm m},\, $ $ \Delta z_{11}=200\,{\rm m},\, \Delta z_{12}=260\,{\rm m},\, \Delta z_{13}=320\,{\rm m},\, \Delta z_{14}=400\,{\rm m},\, \Delta z_{15}=480\,{\rm m},\, $ $ \Delta z_{16}=570\,{\rm m},\, \Delta z_{17}=655\,{\rm m},\, \Delta z_{18}=725\,{\rm m},\, \Delta z_{19}=775\,{\rm m},\, \Delta z_{20}=815\,{\rm m} $ (here the numeric subscript indicates the model level index number, $ {\tt k}$). The implicit free surface form of the pressure equation described in Marshall et. al [39] is employed. A Laplacian operator, $ \nabla^2$, provides viscous dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.

Wind-stress momentum inputs are added to the momentum equations for both the zonal flow, $ u$ and the meridional flow $ v$, according to equations (3.98) and (3.99). Thermodynamic forcing inputs are added to the equations for potential temperature, $ \theta $, and salinity, $ S$, according to equations (3.100) and (3.101). This produces a set of equations solved in this configuration as follows:


$\displaystyle \frac{Du}{Dt} - fv + \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} - A_{h}\nabla_{h}^2u - A_{z}\frac{\partial^{2}u}{\partial z^{2}}$ $\displaystyle =$ $\displaystyle {\cal F}_{u}$ (3.100)
$\displaystyle \frac{Dv}{Dt} + fu + \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} - A_{h}\nabla_{h}^2v - A_{z}\frac{\partial^{2}v}{\partial z^{2}}$ $\displaystyle =$ $\displaystyle {\cal F}_{v}$ (3.101)
$\displaystyle \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}$ $\displaystyle =$ 0 (3.102)
$\displaystyle \frac{D\theta}{Dt} - K_{h}\nabla_{h}^2\theta - \Gamma(K_{z})\frac{\partial^{2}\theta}{\partial z^{2}}$ $\displaystyle =$ $\displaystyle {\cal F}_{\theta}$ (3.103)
$\displaystyle \frac{D s}{Dt} - K_{h}\nabla_{h}^2 s - \Gamma(K_{z})\frac{\partial^{2} s}{\partial z^{2}}$ $\displaystyle =$ $\displaystyle {\cal F}_{s}$ (3.104)
$\displaystyle g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz$ $\displaystyle =$ $\displaystyle p^{'}$ (3.105)

where $ u$ and $ v$ are the $ x$ and $ y$ components of the flow vector $ \vec{u}$. The suffices $ {s},{i}$ indicate surface and interior model levels respectively. As described in MITgcm Numerical Solution Procedure 2, the time evolution of potential temperature, $ \theta $, equation is solved prognostically. The total pressure, $ p$, is diagnosed by summing pressure due to surface elevation $ \eta $ and the hydrostatic pressure.


3.15.3.1 Numerical Stability Criteria

The Laplacian dissipation coefficient, $ A_{h}$, is set to $ 400 m s^{-1}$. This value is chosen to yield a Munk layer width [1],


$\displaystyle M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}$     (3.106)

of $ \approx 100$km. This is greater than the model resolution in mid-latitudes $ \Delta x$, ensuring that the frictional boundary layer is well resolved.

The model is stepped forward with a time step $ \delta t=1200$secs. With this time step the stability parameter to the horizontal Laplacian friction [1]


$\displaystyle S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}$     (3.107)

evaluates to 0.012, which is well below the 0.3 upper limit for stability.

The vertical dissipation coefficient, $ A_{z}$, is set to $ 1\times10^{-2} {\rm m}^2{\rm s}^{-1}$. The associated stability limit


$\displaystyle S_{l} = 4 \frac{A_{z} \delta t}{{\Delta z}^2}$     (3.108)

evaluates to $ 4.8 \times 10^{-5}$ which is again well below the upper limit. The values of $ A_{h}$ and $ A_{z}$ are also used for the horizontal ($ K_{h}\ $) and vertical ($ K_{z}$) diffusion coefficients for temperature respectively.

The numerical stability for inertial oscillations [1]


$\displaystyle S_{i} = f^{2} {\delta t}^2$     (3.109)

evaluates to $ 0.0144$, which is well below the $ 0.5$ upper limit for stability.

The advective CFL [1] for a extreme maximum horizontal flow speed of $ \vert \vec{u} \vert = 2 ms^{-1}$


$\displaystyle S_{a} = \frac{\vert \vec{u} \vert \delta t}{ \Delta x}$     (3.110)

evaluates to $ 5 \times 10^{-2}$. This is well below the stability limit of 0.5.

The stability parameter for internal gravity waves [1]


$\displaystyle S_{c} = \frac{c_{g} \delta t}{ \Delta x}$     (3.111)

evaluates to $ 5 \times 10^{-2}$. This is well below the linear stability limit of 0.25.

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Next: 3.15.4 Code Configuration Up: 3.15 Centennial Time Scale Previous: 3.15.2 Overview   Contents
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