3.10.3 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 2.11.4.

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

As described in 2.16, 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 2.16.1 is used to step forward the temperature equation. Prognostic terms in the momentum equations are solved using flux form as described in section 2.14. 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 2.4 and 1.3.6.

3.10.3.1 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.

Last update 2018-01-23