0 Non-hydrostatic forms

$\displaystyle \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)$

(1.25)

$\displaystyle \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }... ...yd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}$

(1.26)

$\displaystyle \frac{\partial \phi _{hyd}}{\partial r}=-b$

(1.27)

$\displaystyle \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{ \partial r}=G_{\dot{r}}$

(1.28)

The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right)$ in eq(1.26) and (1.28) represent advective, metric and Coriolis terms in the momentum equations. In spherical coordinates they take the form ^1.1 - see Marshall et al 1997a for a full discussion:

$\displaystyle \left. \begin{tabular}{l} $G_{u}=-\vec{\mathbf{v}}.\nabla u$\ \\ ... ...xtit{Coriolis} \\ \textit{\ Forcing/Dissipation} \end{tabular} \ \right. \qquad$

(1.29)

$\displaystyle \left. \begin{tabular}{l} $G_{v}=-\vec{\mathbf{v}}.\nabla v$\ \\ ... ...xtit{Coriolis} \\ \textit{\ Forcing/Dissipation} \end{tabular} \ \right. \qquad$

(1.30)

$\displaystyle \left. \begin{tabular}{l} $G_{\dot{r}}=-\underline{\underline{\ve... ... \\ \textit{Coriolis} \\ \textit{\ Forcing/Dissipation} \end{tabular} \ \right.$

(1.31)

In the above `

' is the distance from the center of the earth and ` $\varphi$ ' is latitude.

Grad and div operators in spherical coordinates are defined in appendix OPERATORS.

**Figure 1.16:** Spherical polar coordinates: longitude $\lambda$ , latitude $\varphi$ and r the distance from the center.
$\resizebox{5in}{!}{ \includegraphics*[0.5in,0.5in][7.5in,10.5in]{part1/sphere.ps} }$

1.3.4.1 Shallow atmosphere approximation

Most models are based on the `hydrostatic primitive equations' (HPE's) in which the vertical momentum equation is reduced to a statement of hydrostatic balance and the `traditional approximation' is made in which the Coriolis force is treated approximately and the shallow atmosphere approximation is made. The MITgcm need not make the `traditional approximation'. To be able to support consistent non-hydrostatic forms the shallow atmosphere approximation can be relaxed - when dividing through by

in, for example, (1.29), we do not replace

, the radius of the earth.

1.3.4.2 Hydrostatic and quasi-hydrostatic forms

In the `hydrostatic primitive equations' (HPE) all the underlined terms in Eqs. (1.29 $\rightarrow$ 1.31) are neglected and `

' is replaced by `

', the mean radius of the earth. Once the pressure is found at one level - e.g. by inverting a 2-d Elliptic equation for $\phi _{s}$ at $r=R_{moving}$ - the pressure can be computed at all other levels by integration of the hydrostatic relation, eq( 1.27).

In the `quasi-hydrostatic' equations (QH) strict balance between gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos \varphi$ Coriolis term are not neglected and are balanced by a non-hydrostatic contribution to the pressure field: only the terms underlined twice in Eqs. ( 1.29 $\rightarrow$ 1.31) are set to zero and, simultaneously, the shallow atmosphere approximation is relaxed. In QH all the metric terms are retained and the full variation of the radial position of a particle monitored. The QH vertical momentum equation (1.28) becomes:

$\displaystyle \frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi$

QH has good energetic credentials - they are the same as for HPE. Importantly, however, it has the same angular momentum principle as the full non-hydrostatic model (NH) - see Marshall et.al., 1997a. As in HPE only a 2-d elliptic problem need be solved.

1.3.4.3 Non-hydrostatic and quasi-nonhydrostatic forms

The MIT model presently supports a full non-hydrostatic ocean isomorph, but only a quasi-non-hydrostatic atmospheric isomorph.

1.3.4.3.1 Non-hydrostatic Ocean

In the non-hydrostatic ocean model all terms in equations Eqs.(1.29 $\rightarrow$ 1.31) are retained. A three dimensional elliptic equation must be solved subject to Neumann boundary conditions (see below). It is important to note that use of the full NH does not admit any new `fast' waves in to the system - the incompressible condition eq(1.3) has already filtered out acoustic modes. It does, however, ensure that the gravity waves are treated accurately with an exact dispersion relation. The NH set has a complete angular momentum principle and consistent energetics - see White and Bromley, 1995; Marshall et.al. 1997a.

1.3.4.3.2 Quasi-nonhydrostatic Atmosphere