3.14.3 Set-up description

The model is configured in hydrostatic form, using non-boussinesq

coordinate. The vertical resolution is uniform, $\Delta p^* = 50.10^2 Pa$ , with 20 levels, from

to 0 at the top. The domain is discretised using C32 cube-sphere grid [Adcroft et al., 2004a] that cover the whole sphere with a relatively uniform grid-spacing. The resolution at the equator or along the Greenwitch meridian is similar to the $128 \times 64$ equaly spaced longitude-latitude grid, but requires $25\%$ less grid points. Grid spacing and grid-point location are not computed by the model but read from files.

The vector-invariant form of the momentum equation (see section 2.15) is used so that no explicit metrics are necessary.

Applying the vector-invariant discretization to the atmospheric equations 1.59, and adding the forcing term (3.61, 3.62) on the right-hand-side, leads to the set of equations that are solved in this configuration:

$\displaystyle \frac{\partial \vec{\mathbf{v}}_h}{\partial t} +(f + \zeta)\hat{\mathbf{k}} \times \vec{\mathbf{v}}_h +\mathbf{\nabla }_{p} ($ KE $\displaystyle ) + \omega \frac{\partial \vec{\mathbf{v}}_h }{\partial p} +\mathbf{\nabla }_p \Phi ^{\prime }$	$\displaystyle =$	$\displaystyle -k_\mathbf{v}\vec{\mathbf{v}}_h$	(3.65)
$\displaystyle \frac{\partial \Phi ^{\prime }}{\partial p} +\frac{\partial \Pi }{\partial p}\theta ^{\prime }$	$\displaystyle =$	0	(3.66)
$\displaystyle \mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_h+\frac{\partial \omega }{ \partial p}$	$\displaystyle =$	0	(3.67)
$\displaystyle \frac{\partial \theta }{\partial t} + \mathbf{\nabla }_{p}\cdot (\theta \vec{\mathbf{v}}_h) + \frac{\partial (\theta \omega)}{\partial p}$	$\displaystyle =$	$\displaystyle -k_{\theta}[\theta-\theta_{eq}]$	(3.68)

where $\vec{\mathbf{v}}_h$ and $\omega =\frac{Dp }{Dt}$ are the horizontal velocity vector and the vertical velocity in pressure coordinate, $\zeta$ is the relative vorticity and

the Coriolis parameter, $\hat{\mathbf{k}}$ is the vertical unity vector, KE is the kinetic energy, $\Phi$ is the geopotential and $\Pi$ the Exner function ( $\Pi = C_p (p/p_c)^\kappa ~{\rm with}~ p_c = 10^5 Pa$ ). Variables marked with ' corresponds to anomaly from the resting, uniformly stratified state.

As described in MITgcm Numerical Solution Procedure 2, the continuity equation is integrated vertically, to give a prognostic equation for the surface pressure

The implicit free surface form of the pressure equation described in Marshall et al. [1997b] is employed to solve for

; Integrating vertically the hydrostatic balance gives the geopotential $\Phi'$ and allow to step forward the momentum equation 3.65. The potential temperature, $\theta$ , is stepped forward using the new velocity field (staggered time-step, section 2.8).

3.14.3.1 Numerical Stability Criteria

evaluates to $4.\times10^{-3}$ at the poles, for $f=2\Omega\sin(\pi / 2) =1.45\times10^{-4}~{\rm s}^{-1}$ , which is well below the $S_{i} < 1$ upper limit for stability.

The advective CFL Adcroft [1995] for a extreme maximum horizontal flow speed of $\vert \vec{u} \vert = 90. {\rm m/s}$ and the smallest horizontal grid spacing $\Delta x = 1.1\times10^5 {\rm m}$ :

The stability parameter for internal gravity waves propagating with a maximum speed of $c_{g}=100~{\rm m/s}$ Adcroft [1995]

evaluates to $4 \times 10^{-1}$ . This is close to the linear stability limit of 0.5.