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Warning, /doc/overview/hydrostatic.rst is written in an unsupported language. File is not indexed.
view on githubraw file Latest commit 0bad585a on 2022-02-16 18:55:09 UTCbf89a37abc Phob*0001 .. _hydrostatic_quasihydrostatic_forms: 0002 f67abf1ee3 Jeff*0003 Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and Non-hydrostatic forms 0004 ------------------------------------------------------------------------------ 0005 0006 Let us separate :math:`\phi` in to surface, hydrostatic and 0007 non-hydrostatic terms: 0008 0009 .. math:: 0bad585a21 Navi*0010 \phi (x,y,r)=\phi _{s}(x,y)+\phi _{\rm hyd}(x,y,r)+\phi _{\rm nh}(x,y,r) f67abf1ee3 Jeff*0011 :label: phi-split 0012 0013 and write :eq:`horiz-mtm` in the form: 0014 0015 .. math:: 0bad585a21 Navi*0016 \frac{\partial \vec{\mathbf{v}}_{h}}{\partial t}+ \nabla _{h}\phi 0017 _{s}+ \nabla _{h}\phi _{\rm hyd}+\epsilon _{\rm nh} \nabla _{h}\phi 0018 _{\rm nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} f67abf1ee3 Jeff*0019 :label: mom-h 0020 0021 .. math:: 0bad585a21 Navi*0022 \frac{\partial \phi _{\rm hyd}}{\partial r}=-b f67abf1ee3 Jeff*0023 :label: hydrostatic 0024 0025 .. math:: 0bad585a21 Navi*0026 \epsilon _{\rm nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{\rm nh}}{ f67abf1ee3 Jeff*0027 \partial r}=G_{\dot{r}} 0028 :label: mom-w 0029 0bad585a21 Navi*0030 Here :math:`\epsilon _{\rm nh}` is a non-hydrostatic parameter. f67abf1ee3 Jeff*0031 0032 The :math:`\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right)` in 0033 :eq:`mom-h` and :eq:`mom-w` represent advective, metric and Coriolis 0034 terms in the momentum equations. In spherical coordinates they take the 0035 form [#]_ - see Marshall et al. (1997a) :cite:`marshall:97a` for a full discussion: 0036 0037 .. math:: 0038 :label: gu-spherical 0039 0bad585a21 Navi*0040 G_{u} = & -\vec{\mathbf{v}} \cdot \nabla u && \qquad \text{advection} f67abf1ee3 Jeff*0041 0042 & -\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan \varphi}{{r}}\right\} && \qquad \text{metric} 0043 0044 & -\left\{ -2\Omega v\sin \varphi+\underline{2\Omega \dot{r}\cos \varphi}\right\} && \qquad \text{Coriolis} 0045 0046 & +\mathcal{F}_{u} && \qquad \text{forcing/dissipation} 0047 0048 .. math:: 0049 :label: gv-spherical 0050 0bad585a21 Navi*0051 G_{v} = & -\vec{\mathbf{v}} \cdot \nabla v && \qquad \text{advection} f67abf1ee3 Jeff*0052 0053 & -\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan \varphi}{{r}}\right\} && \qquad \text{metric} 0054 2645b5126e Henr*0055 & -\left\{ 2\Omega u\sin \varphi\right\} && \qquad \text{Coriolis} f67abf1ee3 Jeff*0056 0057 & +\mathcal{F}_{v} && \qquad \text{forcing/dissipation} 0058 0059 .. math:: 0060 :label: gw-spherical 0061 0bad585a21 Navi*0062 G_{\dot{r}} = & -\underline{\underline{\vec{\mathbf{v}} \cdot \nabla \dot{r}}} && \qquad \text{advection} f67abf1ee3 Jeff*0063 0064 & -\left\{ \underline{\frac{u^{_{^{2}}}+v^{2}}{{r}}}\right\} && \qquad \text{metric} 0065 0066 & +\underline{2\Omega u\cos \varphi} && \qquad \text{Coriolis} 0067 0068 & +\underline{\underline{\mathcal{F}_{\dot{r}}}} && \qquad \text{forcing/dissipation} 0069 0070** Warning **
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0071 In the above ‘:math:`{r}`’ is the distance from the center of the earth** Warning **
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0072 and ‘:math:`\varphi` ’ is latitude (see :numref:`sphere_coor`). 0073 0074 Grad and div operators in spherical coordinates are defined in :ref:`operators`. 0075 0076 0077 Shallow atmosphere approximation 0078 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0079** Warning **
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0080 Most models are based on the ‘hydrostatic primitive equations’ (**HPE**’s) 0081 in which the vertical momentum equation is reduced to a statement of** Warning **
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0082 hydrostatic balance and the ‘traditional approximation’ is made in which 0083 the Coriolis force is treated approximately and the shallow atmosphere** Warning **
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0084 approximation is made. MITgcm need not make the ‘traditional** Warning **
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0085 approximation’. To be able to support consistent non-hydrostatic forms 0086 the shallow atmosphere approximation can be relaxed - when dividing 0087 through by :math:`r` in, for example, :eq:`gu-spherical`, we do not 0088 replace :math:`r` by :math:`a`, the radius of the earth. 0089 94151a9b18 Jeff*0090 .. _hydro_and_quasihydro: f67abf1ee3 Jeff*0091 0092 Hydrostatic and quasi-hydrostatic forms 0093 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0094 0095 These are discussed at length in Marshall et al. (1997a) :cite:`marshall:97a`. 0096** Warning **
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0097 In the ‘hydrostatic primitive equations’ (**HPE**) all the underlined 0098 terms in Eqs. :eq:`gu-spherical`** Warning **
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0099 :math:`\rightarrow` :eq:`gw-spherical` are neglected and ‘:math:`{r}`’** Warning **
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0100 is replaced by ‘:math:`a`’, the mean radius of the earth. Once the 0bad585a21 Navi*0101 pressure is found at one level - e.g. by inverting a 2-D Elliptic 0102 equation for :math:`\phi _{s}` at :math:`r=R_{\rm moving}` - the pressure f67abf1ee3 Jeff*0103 can be computed at all other levels by integration of the hydrostatic 0104 relation, eq :eq:`hydrostatic`. 0105** Warning **
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0106 In the ‘quasi-hydrostatic’ equations (**QH**) strict balance between 0107 gravity and vertical pressure gradients is not imposed. The 0108 :math:`2\Omega u\cos\varphi` Coriolis term are not neglected and are balanced by a 0109 non-hydrostatic contribution to the pressure field: only the terms 0110 underlined twice in Eqs. :eq:`gu-spherical` :math:`\rightarrow` :eq:`gw-spherical` are set to 0111 zero and, simultaneously, the shallow atmosphere approximation is 0112 relaxed. In **QH** *all* the metric terms are retained and the full 0113 variation of the radial position of a particle monitored. The **QH** 0114 vertical momentum equation :eq:`mom-w` becomes: 0115 0bad585a21 Navi*0116 .. math:: \frac{\partial \phi _{\rm nh}}{\partial r}=2\Omega u\cos \varphi f67abf1ee3 Jeff*0117 0118 making a small correction to the hydrostatic pressure. 0119 0120 **QH** has good energetic credentials - they are the same as for 0121 **HPE**. Importantly, however, it has the same angular momentum 0122 principle as the full non-hydrostatic model (**NH**) - see Marshall 0bad585a21 Navi*0123 et.al. (1997a) :cite:`marshall:97a`. As in **HPE** only a 2-D elliptic problem need be solved. f67abf1ee3 Jeff*0124 0125 Non-hydrostatic and quasi-nonhydrostatic forms 0126 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0127 0128 MITgcm presently supports a full non-hydrostatic ocean isomorph, but 0129 only a quasi-non-hydrostatic atmospheric isomorph. 0130 0131 Non-hydrostatic Ocean 0132 ^^^^^^^^^^^^^^^^^^^^^ 0133 0134 In the non-hydrostatic ocean model all terms in equations 0135 Eqs. :eq:`gu-spherical` :math:`\rightarrow` :eq:`gw-spherical` are 0136 retained. A three dimensional elliptic equation must be solved subject 0137 to Neumann boundary conditions (see below). It is important to note that** Warning **
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0138 use of the full **NH** does not admit any new ‘fast’ waves in to the 0139 system - the incompressible condition :eq:`continuity` has already 0140 filtered out acoustic modes. It does, however, ensure that the gravity 0141 waves are treated accurately with an exact dispersion relation. The 0142 **NH** set has a complete angular momentum principle and consistent 0143 energetics - see White and Bromley (1995) :cite:`white:95`; Marshall et al. (1997a) :cite:`marshall:97a`. 0144 0145 Quasi-nonhydrostatic Atmosphere 0146 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 0147 0148 In the non-hydrostatic version of our atmospheric model we approximate 0149 :math:`\dot{r}` in the vertical momentum eqs. :eq:`mom-w` and :eq:`gv-spherical` (but only here) by: 0150 0151 .. math:: \dot{r}=\frac{Dp}{Dt}=\frac{1}{g}\frac{D\phi }{Dt} 0152 :label: quasi-nh-w 0153 0bad585a21 Navi*0154 where :math:`p_{\rm hy}` is the hydrostatic pressure. f67abf1ee3 Jeff*0155 0156 Summary of equation sets supported by model 0157 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0158 0159 Atmosphere 0160 ^^^^^^^^^^ 0161 0162 Hydrostatic, and quasi-hydrostatic and quasi non-hydrostatic forms of 0163 the compressible non-Boussinesq equations in :math:`p-`\ coordinates are 0164 supported. 0165 0166 Hydrostatic and quasi-hydrostatic 0167 ''''''''''''''''''''''''''''''''' 0168 0169 0170 0171 The hydrostatic set is written out in :math:`p-`\ coordinates in 0172 :ref:`atmos_appendix` - see eqs. :eq:`atmos-prime` to :eq:`atmos-prime5`. 0173 0174 Quasi-nonhydrostatic 0175 '''''''''''''''''''' 0176 0177 A quasi-nonhydrostatic form is also supported. 0178 0179 Ocean 0180 ^^^^^ 0181 0182 Hydrostatic and quasi-hydrostatic 0183 ''''''''''''''''''''''''''''''''' 0184 0185 Hydrostatic, and quasi-hydrostatic forms of the incompressible 0186 Boussinesq equations in :math:`z-`\ coordinates are supported. 0187 0188 Non-hydrostatic 0189 ''''''''''''''' 0190 0191 Non-hydrostatic forms of the incompressible Boussinesq equations in 0192 :math:`z-` coordinates are supported - see eqs. :eq:`eq-ocean-mom` to :eq:`eq-ocean-salt`. 0193 0194 0195 0196 .. [#] 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. 0197
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