3.10.2 Discrete Numerical Configuration

The model is configured in hydrostatic form. The domain is discretised 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

and

are initialized according to

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 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, ${\tt k}$ ) to give a total depth,

, of $-5450{\rm m}$ . 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 forcing is added to the momentum equations in (3.36) for both the zonal flow,

and the meridional flow

, according to equations (3.30) and (3.31). Thermodynamic forcing inputs are added to the equations in (3.36) for potential temperature, $\theta$ , and salinity,

, according to equations (3.32) and (3.33). This produces a set of equations solved in this configuration as follows:

$\displaystyle \frac{Du}{Dt} - fv + \frac{1}{\rho}\frac{\partial p^{'}}{\partial... ..._{h}\nabla_{h}u - \frac{\partial}{\partial z}A_{z}\frac{\partial u}{\partial z}$	$\displaystyle =$	$\begin{displaymath}\begin{cases} {\cal F}_u & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath}$	(3.36)
$\displaystyle \frac{Dv}{Dt} + fu + \frac{1}{\rho}\frac{\partial p^{'}}{\partial... ..._{h}\nabla_{h}v - \frac{\partial}{\partial z}A_{z}\frac{\partial v}{\partial z}$	$\displaystyle =$	$\begin{displaymath}\begin{cases} {\cal F}_v & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath}$	(3.37)
$\displaystyle \frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u}$	$\displaystyle =$	0	(3.38)
$\displaystyle \frac{D\theta}{Dt} - \nabla_{h}\cdot K_{h}\nabla_{h}\theta - \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial\theta}{\partial z}$	$\displaystyle =$	$\begin{displaymath}\begin{cases} {\cal F}_\theta & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath}$	(3.39)
$\displaystyle \frac{D s}{Dt} - \nabla_{h}\cdot K_{h}\nabla_{h}s - \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial s}{\partial z}$	$\displaystyle =$	$\begin{displaymath}\begin{cases} {\cal F}_s & \text{(surface)} \\ 0 & \text{(interior)} \end{cases}\end{displaymath}$	(3.40)
$\displaystyle g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz$	$\displaystyle =$	$\displaystyle p^{'}$	(3.41)

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 total pressure,

, is diagnosed by summing pressure due to surface elevation $\eta$ and the hydrostatic pressure.

3.10.2.1 Numerical Stability Criteria

The Laplacian dissipation coefficient, $A_{h}$ , is set to $5 \times 10^5 m s^{-1}$ . This value is chosen to yield a Munk layer width [1],

of $\approx 600$ km. This is greater than the model resolution in low-latitudes, $\Delta x \approx 400{\rm km}$ , ensuring that the frictional boundary layer is adequately resolved.

The model is stepped forward with a time step $\delta t_{\theta}=30~{\rm hours}$ for thermodynamic variables and $\delta t_{v}=40~{\rm minutes}$ for momentum terms. With this time step, the stability parameter to the horizontal Laplacian friction [1]

evaluates to 0.16 at a latitude of $\phi=80^{\circ}$ , which is below the 0.3 upper limit for stability. The zonal grid spacing $\Delta x$ is smallest at $\phi=80^{\circ}$ where $\Delta x=r\cos(\phi)\Delta \phi\approx 77{\rm km}$ .

The vertical dissipation coefficient, $A_{z}$ , is set to $1\times10^{-3} {\rm m}^2{\rm s}^{-1}$ . The associated stability limit

evaluates to

for the smallest model level spacing ( $\Delta z_{1}=50{\rm m}$ ) which is again well below the upper stability limit.

The values of the horizontal ( $K_{h}\$ ) and vertical ( $K_{z}$ ) diffusion coefficients for both temperature and salinity are set to $1 \times 10^{3}~{\rm m}^{2}{\rm s}^{-1}$ and $3 \times 10^{-5}~{\rm m}^{2}{\rm s}^{-1}$ respectively. The stability limit related to $K_{h}\$ will be at $\phi=80^{\circ}$ where $\Delta x \approx 77 {\rm km}$ . Here the stability parameter

evaluates to

for $f=2\omega\sin(80^{\circ})=1.43\times10^{-4}~{\rm s}^{-1}$ , which is close to the $S_{i} < 1$ upper limit for stability.

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

evaluates to $6 \times 10^{-2}$ . This is well below the stability limit of 0.5.

The stability parameter for internal gravity waves propagating with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$ [1]

evaluates to $3 \times 10^{-1}$ . This is close to the linear stability limit of 0.5.