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

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

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

and write (1.1) in the form:

(1.26)\[\frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi _{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}\]
(1.27)\[\frac{\partial \phi _{hyd}}{\partial r}=-b\]
(1.28)\[\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{ \partial r}=G_{\dot{r}}\]

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

The \(\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right)\) in (1.26) and (1.28) represent advective, metric and Coriolis terms in the momentum equations. In spherical coordinates they take the form [1] - see Marshall et al. (1997a) [MHPA97] for a full discussion:

(1.29)\[ \begin{align}\begin{aligned}G_{u} = & -\vec{\mathbf{v}}.\nabla u && \qquad \text{advection}\\& -\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} && \qquad \text{metric}\\& -\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} && \qquad \text{Coriolis}\\& +\mathcal{F}_{u} && \qquad \text{forcing/dissipation}\end{aligned}\end{align} \]
(1.30)\[ \begin{align}\begin{aligned}G_{v} = & -\vec{\mathbf{v}}.\nabla v && \qquad \text{advection}\\& -\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\} && \qquad \text{metric}\\& -\left\{ 2\Omega u\sin \varphi\right\} && \qquad \text{Coriolis}\\& +\mathcal{F}_{v} && \qquad \text{forcing/dissipation}\end{aligned}\end{align} \]
(1.31)\[ \begin{align}\begin{aligned}G_{\dot{r}} = & -\underline{\underline{\vec{\mathbf{v}}.\nabla \dot{r}}} && \qquad \text{advection}\\& -\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} && \qquad \text{metric}\\& +\underline{2\Omega u\cos \varphi} && \qquad \text{Coriolis}\\& +\underline{\underline{\mathcal{F}_{\dot{r}}}} && \qquad \text{forcing/dissipation}\end{aligned}\end{align} \]

In the above ‘\({r}\)’ is the distance from the center of the earth and ‘\(\varphi\) ’ is latitude (see Figure 1.20).

Grad and div operators in spherical coordinates are defined in Coordinate systems.

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. 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.

1.3.4.2. Hydrostatic and quasi-hydrostatic forms

These are discussed at length in Marshall et al. (1997a) [MHPA97].

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:

\[\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos \varphi\]

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) [MHPA97]. As in HPE only a 2-d elliptic problem need be solved.

1.3.4.3. Non-hydrostatic and quasi-nonhydrostatic forms

MITgcm 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 (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) [WB95]; Marshall et al. (1997a) [MHPA97].

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:

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

where \(p_{hy}\) is the hydrostatic pressure.

1.3.4.4. 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 Hydrostatic Primitive Equations for the Atmosphere in Pressure Coordinates - see eqs. (1.59) to (1.63).

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. (1.98) to (1.103).

[1]In the hydrostatic primitive equations (HPE) all underlined terms in (1.29), (1.30) and (1.31) are omitted; the singly-underlined terms are included in the quasi-hydrostatic model (QH). The fully non-hydrostatic model (NH) includes all terms.