3.9.3 Discrete Numerical Configuration

The domain is discretised with a uniform grid spacing in latitude and longitude $\Delta \lambda=\Delta \varphi=1^{\circ}$ , so that there are sixty grid cells in the zonal and meridional directions. Vertically the model is configured with four layers with constant depth, $\Delta z$ , of

m. The internal, locally orthogonal, model coordinate variables

and

are initialized from the values of $\lambda$ , $\varphi$ , $\Delta \lambda$ and $\Delta \varphi$ in radians according to

$\displaystyle x=r\cos(\varphi)\lambda,~\Delta x$	$\displaystyle =$	$\displaystyle r\cos(\varphi)\Delta \lambda$	(3.22)
$\displaystyle y=r\varphi,~\Delta y$	$\displaystyle =$	$\displaystyle r\Delta \varphi$	(3.23)

The procedure for generating a set of internal grid variables from a spherical polar grid specification is discussed in section

$\fbox{ \begin{minipage}{5.5in} {\em S/R INI\_SPHERICAL\_POLAR\_GRID} ({\em model... ...ta x_v$, $\Delta y_u$: {\bf DXv}, {\bf DYu} ({\em GRID.h}) \par \end{minipage} }$

As described in 2.15, the time evolution of potential temperature, $\theta$ , (equation 3.17) is evaluated prognostically. The centered second-order scheme with Adams-Bashforth time stepping described in section 2.15.1 is used to step forward the temperature equation. Prognostic terms in the momentum equations are solved using flux form as described in section

. The pressure forces that drive the fluid motions, ( $\frac{\partial p^{'}}{\partial \lambda}$ and $\frac{\partial p^{'}}{\partial \varphi}$ ), are found by summing pressure due to surface elevation $\eta$ and the hydrostatic pressure. The hydrostatic part of the pressure is diagnosed explicitly by integrating density. The sea-surface height, $\eta$ , is diagnosed using an implicit scheme. The pressure field solution method is described in sections 2.3 and

3.9.3.1 Numerical Stability Criteria

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

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

(3.24)

of $\approx 100$ km. This is greater than the model resolution in mid-latitudes $\Delta x=r \cos(\varphi) \Delta \lambda \approx 80~{\rm km}$ at $\varphi=45^{\circ}$ , 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

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

(3.25)

evaluates to 0.012, which is well below the 0.3 upper limit for stability for this term under ABII time-stepping.

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

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

(3.26)

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.

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

(3.27)

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

$\displaystyle C_{a} = \frac{\vert \vec{u} \vert \delta t}{ \Delta x}$

(3.28)

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

The stability parameter for internal gravity waves propagating at $2~{\rm m}~{\rm s}^{-1}$

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

(3.29)

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