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bf89a37abc Phob* .. _hydrostatic_quasihydrostatic_forms:
                 
f67abf1ee3 Jeff* Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and Non-hydrostatic forms
                 ------------------------------------------------------------------------------
                 
                 Let us separate :math:`\phi` in to surface, hydrostatic and
                 non-hydrostatic terms:
                 
                 .. math::
0bad585a21 Navi*    \phi (x,y,r)=\phi _{s}(x,y)+\phi _{\rm hyd}(x,y,r)+\phi _{\rm nh}(x,y,r)
f67abf1ee3 Jeff*    :label: phi-split
                 
                 and write :eq:`horiz-mtm` in the form:
                 
                 .. math::
0bad585a21 Navi*    \frac{\partial \vec{\mathbf{v}}_{h}}{\partial t}+ \nabla _{h}\phi
                    _{s}+ \nabla _{h}\phi _{\rm hyd}+\epsilon _{\rm nh} \nabla _{h}\phi
                    _{\rm nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  
f67abf1ee3 Jeff*    :label: mom-h
                 
                 .. math:: 
0bad585a21 Navi*    \frac{\partial \phi _{\rm hyd}}{\partial r}=-b
f67abf1ee3 Jeff*    :label: hydrostatic
                 
                 .. math::
0bad585a21 Navi*    \epsilon _{\rm nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{\rm nh}}{
f67abf1ee3 Jeff*    \partial r}=G_{\dot{r}}
                    :label: mom-w
                 
0bad585a21 Navi* Here :math:`\epsilon _{\rm nh}` is a non-hydrostatic parameter.
f67abf1ee3 Jeff* 
                 The :math:`\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right)` in
                 :eq:`mom-h` and :eq:`mom-w` represent advective, metric and Coriolis
                 terms in the momentum equations. In spherical coordinates they take the
                 form  [#]_ - see Marshall et al. (1997a) :cite:`marshall:97a` for a full discussion:
                 
                 .. math::
                    :label: gu-spherical
                 
0bad585a21 Navi*    G_{u} = & -\vec{\mathbf{v}} \cdot  \nabla  u && \qquad \text{advection} 
f67abf1ee3 Jeff* 
                    & -\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}
                 
                 .. math::
                    :label: gv-spherical
                 
0bad585a21 Navi*    G_{v} = & -\vec{\mathbf{v}} \cdot  \nabla  v && \qquad \text{advection}
f67abf1ee3 Jeff*  
                    & -\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\} && \qquad \text{metric}    
                 
2645b5126e Henr*    & -\left\{ 2\Omega u\sin \varphi\right\} && \qquad \text{Coriolis}  
f67abf1ee3 Jeff* 
                    & +\mathcal{F}_{v} && \qquad \text{forcing/dissipation}
                 
                 .. math::
                    :label: gw-spherical
                 
0bad585a21 Navi*    G_{\dot{r}} = & -\underline{\underline{\vec{\mathbf{v}} \cdot  \nabla  \dot{r}}} && \qquad \text{advection}
f67abf1ee3 Jeff*  
                    & -\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}
                 
                 
                 In the above ‘:math:`{r}`’ is the distance from the center of the earth
                 and ‘:math:`\varphi` ’ is latitude (see :numref:`sphere_coor`).
                 
                 Grad and div operators in spherical coordinates are defined in :ref:`operators`.
                 
                 
                 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 :math:`r` in, for example, :eq:`gu-spherical`, we do not
                 replace :math:`r` by :math:`a`, the radius of the earth.
                 
94151a9b18 Jeff* .. _hydro_and_quasihydro:
f67abf1ee3 Jeff* 
                 Hydrostatic and quasi-hydrostatic forms
                 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                 
                 These are discussed at length in Marshall et al. (1997a) :cite:`marshall:97a`.
                 
                 In the ‘hydrostatic primitive equations’ (**HPE**) all the underlined
                 terms in Eqs. :eq:`gu-spherical`
                 :math:`\rightarrow` :eq:`gw-spherical` are neglected and ‘:math:`{r}`’
                 is replaced by ‘:math:`a`’, the mean radius of the earth. Once the
0bad585a21 Navi* pressure is found at one level - e.g. by inverting a 2-D Elliptic
                 equation for :math:`\phi _{s}` at :math:`r=R_{\rm moving}` - the pressure
f67abf1ee3 Jeff* can be computed at all other levels by integration of the hydrostatic
                 relation, eq :eq:`hydrostatic`.
                 
                 In the ‘quasi-hydrostatic’ equations (**QH**) strict balance between
                 gravity and vertical pressure gradients is not imposed. The
                 :math:`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. :eq:`gu-spherical` :math:`\rightarrow` :eq:`gw-spherical` 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 :eq:`mom-w` becomes:
                 
0bad585a21 Navi* .. math:: \frac{\partial \phi _{\rm nh}}{\partial r}=2\Omega u\cos \varphi
f67abf1ee3 Jeff* 
                 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
0bad585a21 Navi* et.al. (1997a) :cite:`marshall:97a`. As in **HPE** only a 2-D elliptic problem need be solved.
f67abf1ee3 Jeff* 
                 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. :eq:`gu-spherical` :math:`\rightarrow` :eq:`gw-spherical` 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:`continuity` 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) :cite:`white:95`; Marshall et al. (1997a) :cite:`marshall:97a`.
                 
                 Quasi-nonhydrostatic Atmosphere
                 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                 
                 In the non-hydrostatic version of our atmospheric model we approximate
                 :math:`\dot{r}` in the vertical momentum eqs. :eq:`mom-w` and :eq:`gv-spherical` (but only here) by:
                 
                 .. math:: \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt}
                    :label: quasi-nh-w
                 
0bad585a21 Navi* where :math:`p_{\rm hy}` is the hydrostatic pressure.
f67abf1ee3 Jeff* 
                 Summary of equation sets supported by model
                 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                 
                 Atmosphere
                 ^^^^^^^^^^
                 
                 Hydrostatic, and quasi-hydrostatic and quasi non-hydrostatic forms of
                 the compressible non-Boussinesq equations in :math:`p-`\ coordinates are
                 supported.
                 
                 Hydrostatic and quasi-hydrostatic
                 '''''''''''''''''''''''''''''''''
                 
                 
                 
                 The hydrostatic set is written out in :math:`p-`\ coordinates in
                 :ref:`atmos_appendix` - see eqs. :eq:`atmos-prime` to :eq:`atmos-prime5`.
                 
                 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 :math:`z-`\ coordinates are supported.
                 
                 Non-hydrostatic
                 '''''''''''''''
                 
                 Non-hydrostatic forms of the incompressible Boussinesq equations in
                 :math:`z-` coordinates are supported - see eqs. :eq:`eq-ocean-mom` to :eq:`eq-ocean-salt`.
                 
                 
                 
                 .. [#] In the hydrostatic primitive equations (**HPE**) all underlined terms in :eq:`gu-spherical`, :eq:`gv-spherical` and :eq:`gw-spherical` are omitted; the singly-underlined terms are included in the quasi-hydrostatic model (**QH**). The fully non-hydrostatic model (**NH**) includes all terms.
                 

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