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3.11.2 Discrete Numerical Configuration

Due to the pressure coordinate, the model can only be hydrostatic [12]. The domain is discretized with a uniform grid spacing in latitude and longitude on the sphere $ \Delta \phi=\Delta \lambda=4^{\circ}$, so that there are ninety grid cells in the zonal and forty in the meridional direction. The internal model coordinate variables $ x$ and $ y$ are initialized according to

$\displaystyle x=r\cos(\phi),~\Delta x$ $\displaystyle =$ $\displaystyle r\cos(\Delta \phi)$ (3.54)
$\displaystyle y=r\lambda,~\Delta y$ $\displaystyle =$ $\displaystyle r\Delta \lambda$ (3.55)

Arctic polar regions are not included in this experiment. Meridionally the model extends from $ 80^{\circ}{\rm S}$ to $ 80^{\circ}{\rm N}$. Vertically the model is configured with fifteen layers with the following thicknesses

$\displaystyle \Delta p_{1}$ $\displaystyle =$ $\displaystyle 7103300.720021$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{2}$ $\displaystyle =$ $\displaystyle 6570548.440790$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{3}$ $\displaystyle =$ $\displaystyle 6041670.010249$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{4}$ $\displaystyle =$ $\displaystyle 5516436.666057$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{5}$ $\displaystyle =$ $\displaystyle 4994602.034410$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{6}$ $\displaystyle =$ $\displaystyle 4475903.435290$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{7}$ $\displaystyle =$ $\displaystyle 3960063.245801$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{8}$ $\displaystyle =$ $\displaystyle 3446790.312651$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{9}$ $\displaystyle =$ $\displaystyle 2935781.405664$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{10}$ $\displaystyle =$ $\displaystyle 2426722.705046$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{11}$ $\displaystyle =$ $\displaystyle 1919291.315988$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{12}$ $\displaystyle =$ $\displaystyle 1413156.804970$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{13}$ $\displaystyle =$ $\displaystyle 1008846.750166$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{14}$ $\displaystyle =$ $\displaystyle 705919.025481$ Pa$\displaystyle ,$  
$\displaystyle \Delta p_{15}$ $\displaystyle =$ $\displaystyle 504089.693499$ Pa$\displaystyle ,$  

(here the numeric subscript indicates the model level index number, $ {\tt k}$; note, that the surface layer has the highest index number 15) to give a total depth, $ H$, of $ -5200{\rm m}$. In pressure, this is $ p_{b}^{0}=53023122.566084$ Pa. The implicit free surface form of the pressure equation described in Marshall et al. [39] with the nonlinear extension by Campin et al. [8] is employed. A Laplacian operator, $ \nabla^2$, provides viscous dissipation. Thermal and haline diffusion is also represented by a Laplacian operator.

Wind-stress forcing is added to the momentum equations in (3.56) for both the zonal flow, $ u$ and the meridional flow $ v$, according to equations (3.50) and (3.51). Thermodynamic forcing inputs are added to the equations in (3.56) for potential temperature, $ \theta $, and salinity, $ S$, according to equations (3.52) and (3.53). This produces a set of equations solved in this configuration as follows:


$\displaystyle \frac{Du}{Dt} - fv + \frac{1}{\rho}\frac{\partial \Phi^{'}}{\part... ...{h}u - (g\rho_0)^2\frac{\partial}{\partial p}A_{r}\frac{\partial u}{\partial p}$ $\displaystyle =$ \begin{displaymath}\begin{cases} {\cal F}_u & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath} (3.56)
$\displaystyle \frac{Dv}{Dt} + fu + \frac{1}{\rho}\frac{\partial \Phi^{'}}{\part... ...{h}v - (g\rho_0)^2\frac{\partial}{\partial p}A_{r}\frac{\partial v}{\partial p}$ $\displaystyle =$ \begin{displaymath}\begin{cases} {\cal F}_v & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath} (3.57)
$\displaystyle \frac{\partial p_{b}}{\partial t} + \nabla_{h}\cdot \vec{u}$ $\displaystyle =$ 0 (3.58)
$\displaystyle \frac{D\theta}{Dt} - \nabla_{h}\cdot K_{h}\nabla_{h}\theta - (g\rho_0)^2\frac{\partial}{\partial p}\Gamma(K_{r})\frac{\partial\theta}{\partial p}$ $\displaystyle =$ \begin{displaymath}\begin{cases} {\cal F}_\theta & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath} (3.59)
$\displaystyle \frac{D s}{Dt} - \nabla_{h}\cdot K_{h}\nabla_{h}s - (g\rho_0)^2\frac{\partial}{\partial p}\Gamma(K_{r})\frac{\partial S}{\partial p}$ $\displaystyle =$ \begin{displaymath}\begin{cases} {\cal F}_s & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath} (3.60)
$\displaystyle \Phi_{-H}'^{(0)} + \alpha_{0}p_{b}+ \int^{p}_{0}\alpha' dp$ $\displaystyle =$ $\displaystyle \Phi'$ (3.61)

where $ u=\frac{Dx}{Dt}=r \cos(\phi)\frac{D \lambda}{Dt}$ and $ v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$ are the zonal and meridional components of the flow vector, $ \vec{u}$, on the sphere. As described in MITgcm Numerical Solution Procedure 2, the time evolution of potential temperature, $ \theta $, equation is solved prognostically. The full geopotential height, $ \Phi$, is diagnosed by summing the geopotential height anomalies $ \Phi'$ due to bottom pressure $ p_{b}$ and density variations. The integration of the hydrostatic equation is started at the bottom of the domain. The condition of $ p=0$ at the sea surface requires a time-independent integration constant for the height anomaly due to density variations $ \Phi_{-H}'^{(0)}$, which is provided as an input field.

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Next: 3.11.3 Experiment Configuration Up: 3.11 P coordinate Global Previous: 3.11.1 Overview   Contents
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