3.8.2 Discrete Numerical Configuration

The domain is discretised with a uniform grid spacing in the horizontal set to $\Delta x=\Delta y=20$ km, so that there are sixty grid cells in the $x$ and $y$ directions. Vertically the model is configured with a single layer with depth, $\Delta z$, of $5000$ m.

3.8.2.1 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],

$$M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}}$$ (3.6)

of $\approx 100$km. This is greater than the model resolution $\Delta x$, 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]

$$S_{l} = 4 \frac{A_{h} \delta t}{{\Delta x}^2}$$ (3.7)

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

The numerical stability for inertial oscillations [1]

$$S_{i} = f^{2} {\delta t}^2$$ (3.8)

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

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

$$S_{a} = \frac{\vert \vec{u} \vert \delta t}{ \Delta x}$$ (3.9)

evaluates to 0.12. This is approaching the stability limit of 0.5 and limits $\delta t$ to $1200s$.