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1.3.4 Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and Non-hydrostatic forms

Let us separate $ \phi $ in to surface, hydrostatic and non-hydrostatic terms:

$\displaystyle \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r)$ (1.25)

and write eq( 1.1) in the form:

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

Here $ \epsilon _{nh}$ is a non-hydrostatic parameter.

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 `$ {r}$' is the distance from the center of the earth and `$ \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.
} 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 $ r$ in, for example, (1.29), we do not replace $ r$ by $ a$, the radius of the earth. Hydrostatic and quasi-hydrostatic forms

These are discussed at length in Marshall et al (1997a).

In the `hydrostatic primitive equations' (HPE) all the underlined terms in Eqs. (1.29 $ \rightarrow $ 1.31) are neglected and `$ {r}$' is replaced by `$ a$', 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:

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

making a small correction to the hydrostatic pressure.

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. 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. 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. Quasi-nonhydrostatic Atmosphere

In the non-hydrostatic version of our atmospheric model we approximate $ \dot{
r}$ in the vertical momentum eqs(1.28) and (1.30) (but only here) by:

$\displaystyle \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}$ (1.32)

where $ p_{hy}$ is the hydrostatic pressure. Summary of equation sets supported by model Atmosphere

Hydrostatic, and quasi-hydrostatic and quasi non-hydrostatic forms of the compressible non-Boussinesq equations in $ p-$coordinates are supported. Hydrostatic and quasi-hydrostatic

The hydrostatic set is written out in $ p-$coordinates in appendix Atmosphere - see eq(1.59). Quasi-nonhydrostatic

A quasi-nonhydrostatic form is also supported. Ocean Hydrostatic and quasi-hydrostatic

Hydrostatic, and quasi-hydrostatic forms of the incompressible Boussinesq equations in $ z-$coordinates are supported. Non-hydrostatic

Non-hydrostatic forms of the incompressible Boussinesq equations in $ z-$ coordinates are supported - see eqs(1.99) to (1.104).

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