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Warning, /doc/overview/finding_pressure.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 UTCf67abf1ee3 Jeff*0001 .. _finding_the_pressure_field: 0002 0003 Finding the pressure field 0004 -------------------------- 0005 0006 Unlike the prognostic variables :math:`u`, :math:`v`, :math:`w`, 0007 :math:`\theta` and :math:`S`, the pressure field must be obtained 0008 diagnostically. We proceed, as before, by dividing the total 0009 (pressure/geo) potential in to three parts, a surface part, 0bad585a21 Navi*0010 :math:`\phi _{s}(x,y)`, a hydrostatic part :math:`\phi _{\rm hyd}(x,y,r)` 0011 and a non-hydrostatic part :math:`\phi _{\rm nh}(x,y,r)`, as in f67abf1ee3 Jeff*0012 :eq:`phi-split`, and writing the momentum equation as in :eq:`mom-h`. 0013 0014 Hydrostatic pressure 0015 ~~~~~~~~~~~~~~~~~~~~ 0016 0017 Hydrostatic pressure is obtained by integrating :eq:`hydrostatic` vertically from :math:`r=R_{o}` 0bad585a21 Navi*0018 where :math:`\phi _{\rm hyd}(r=R_{o})=0`, to yield: f67abf1ee3 Jeff*0019 0020 .. math:: 0021 0bad585a21 Navi*0022 \int_{r}^{R_{o}}\frac{\partial \phi _{\rm hyd}}{\partial r}dr=\left[ \phi _{\rm hyd} f67abf1ee3 Jeff*0023 \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr 0024 0025 and so 0026 0bad585a21 Navi*0027 .. math:: \phi _{\rm hyd}(x,y,r)=\int_{r}^{R_{o}}bdr f67abf1ee3 Jeff*0028 :label: hydro-phi 0029 0030 The model can be easily modified to accommodate a loading term (e.g** Warning **
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0031 atmospheric pressure pushing down on the ocean’s surface) by setting: 0032 0bad585a21 Navi*0033 .. math:: \phi _{\rm hyd}(r=R_{o})= \text{loading} f67abf1ee3 Jeff*0034 :label: loading 0035 0036 Surface pressure 0037 ~~~~~~~~~~~~~~~~ 0038 0039 The surface pressure equation can be obtained by integrating continuity, 0bad585a21 Navi*0040 :eq:`continuity`, vertically from :math:`r=R_{\rm fixed}` to :math:`r=R_{\rm moving}` f67abf1ee3 Jeff*0041 0042 .. math:: 0bad585a21 Navi*0043 \int_{R_{\rm fixed}}^{R_{\rm moving}}\left( \nabla _{h}\cdot \vec{\mathbf{v} f67abf1ee3 Jeff*0044 }_{h}+\partial _{r}\dot{r}\right) dr=0 0045 0046 Thus: 0047 0048 .. math:: 0bad585a21 Navi*0049 \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}} \cdot \nabla \eta 0050 +\int_{R_{\rm fixed}}^{R_{\rm moving}} \nabla _{h}\cdot \vec{\mathbf{v}} f67abf1ee3 Jeff*0051 _{h}dr=0 0052 0bad585a21 Navi*0053 where :math:`\eta =R_{\rm moving}-R_{o}` is the free-surface f67abf1ee3 Jeff*** Warning **
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0054 :math:`r`-anomaly in units of :math:`r`. The above can be rearranged to yield, using Leibnitz’s theorem: 0055 0056 .. math:: 0bad585a21 Navi*0057 \frac{\partial \eta }{\partial t}+ \nabla _{h}\cdot 0058 \int_{R_{\rm fixed}}^{R_{\rm moving}}\vec{\mathbf{v}}_{h}dr=\text{source} f67abf1ee3 Jeff*0059 :label: free-surface 0060 0061 where we have incorporated a source term. 0062 0063 Whether :math:`\phi` is pressure (ocean model, :math:`p/\rho _{c}`) or 0064 geopotential (atmospheric model), in :eq:`mom-h`, the horizontal gradient term can be written 0065 0066 .. math:: 0bad585a21 Navi*0067 \nabla _{h}\phi _{s}= \nabla _{h}\left( b_{s}\eta \right) f67abf1ee3 Jeff*0068 :label: phi-surf 0069 0070 where :math:`b_{s}` is the buoyancy at the surface. 0071 0bad585a21 Navi*0072 In the hydrostatic limit (:math:`\epsilon _{\rm nh}=0`), equations f67abf1ee3 Jeff*0073 :eq:`mom-h`, :eq:`free-surface` and :eq:`phi-surf` can be solved by 0bad585a21 Navi*0074 inverting a 2-D elliptic equation for :math:`\phi _{s}` as described in f67abf1ee3 Jeff*** Warning **
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0075 Chapter 2. Both ‘free surface’ and ‘rigid lid’ approaches are available. 0076 0077 Non-hydrostatic pressure 0078 ~~~~~~~~~~~~~~~~~~~~~~~~ 0079 0080 Taking the horizontal divergence of :eq:`mom-h` and adding 0081 :math:`\frac{\partial }{\partial r}` of :eq:`mom-w`, invoking the 0082 continuity equation :eq:`continuity`, we deduce that: 0083 0084 .. math:: 0bad585a21 Navi*0085 \nabla_{3}^{2}\phi _{\rm nh}= \nabla \cdot \vec{\mathbf{G}}_{\vec{v}}-\left( 0086 \nabla_{h}^{2}\phi _{s}+ \nabla^2 \phi _{\rm hyd}\right) = 0087 \nabla \cdot \vec{\mathbf{F}} f67abf1ee3 Jeff*0088 :label: 3d-invert 0089 0bad585a21 Navi*0090 For a given rhs this 3-D elliptic equation must be inverted for 0091 :math:`\phi _{\rm nh}` subject to appropriate choice of boundary conditions. f67abf1ee3 Jeff*0092 This method is usually called *The Pressure Method* [Harlow and Welch 0093 (1965) :cite:`harlow:65`; Williams (1969) :cite:`williams:69`; Potter (1973) :cite:`potter:73`. In the hydrostatic primitive 0bad585a21 Navi*0094 equations case (**HPE**), the 3-D problem does not need to be solved. f67abf1ee3 Jeff*0095 0096 Boundary Conditions 0097 ^^^^^^^^^^^^^^^^^^^ 0098 0099 We apply the condition of no normal flow through all solid boundaries - 0100 the coasts (in the ocean) and the bottom: 0101 0bad585a21 Navi*0102 .. math:: \vec{\mathbf{v}} \cdot \hat{\boldsymbol{n}} =0 f67abf1ee3 Jeff*0103 :label: nonormalflow 0104 0105 where :math:`\widehat{n}` is a vector of unit length normal to the 0106 boundary. The kinematic condition :eq:`nonormalflow` is also applied to 0bad585a21 Navi*0107 the vertical velocity at :math:`r=R_{\rm moving}`. No-slip f67abf1ee3 Jeff*0108 :math:`\left( v_{T}=0\right) \ `\ or slip :math:`\left( \partial v_{T}/\partial n=0\right) \ `\ conditions are employed 0109 on the tangential component of velocity, :math:`v_{T}`, at all solid 0110 boundaries, depending on the form chosen for the dissipative terms in 0111 the momentum equations - see below. 0112 0113 Eq. :eq:`nonormalflow` implies, making use of :eq:`mom-h`, that: 0114 0115 .. math:: 0bad585a21 Navi*0116 \hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}= \hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}} f67abf1ee3 Jeff*0117 :label: inhom-neumann-nh 0118 0119 where 0120 0121 .. math:: 0bad585a21 Navi*0122 \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \nabla _{h}\phi_{s}+ \nabla \phi _{\rm hyd}\right) f67abf1ee3 Jeff*0123 0124 presenting inhomogeneous Neumann boundary conditions to the Elliptic 0125 problem :eq:`3d-invert`. As shown, for example, by Williams (1969) :cite:`williams:69`, one 0126 can exploit classical 3D potential theory and, by introducing an** Warning **
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0127 appropriately chosen :math:`\delta`-function sheet of ‘source-charge’, 0128 replace the inhomogeneous boundary condition on pressure by a 0129 homogeneous one. The source term :math:`rhs` in :eq:`3d-invert` is the 0bad585a21 Navi*0130 divergence of the vector :math:`\vec{\mathbf{F}}`. By simultaneously setting :math:`\hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}}=0` 0131 and :math:`\hat{\boldsymbol{n}} \cdot \nabla \phi_{\rm nh}=0\ `\ on the boundary the f67abf1ee3 Jeff*0132 following self-consistent but simpler homogenized Elliptic problem is obtained: 0133 0bad585a21 Navi*0134 .. math:: \nabla ^{2}\phi _{\rm nh}= \nabla \cdot \widetilde{\vec{\mathbf{F}}}\qquad f67abf1ee3 Jeff*0135 0136 where :math:`\widetilde{\vec{\mathbf{F}}}` is a modified :math:`\vec{\mathbf{F}}` 0bad585a21 Navi*0137 such that :math:`\widetilde{\vec{\mathbf{F}}} \cdot \hat{\boldsymbol{n}} =0`. As is implied by f67abf1ee3 Jeff*0138 :eq:`inhom-neumann-nh` the modified boundary condition becomes: 0139 0bad585a21 Navi*0140 .. math:: \hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}=0 f67abf1ee3 Jeff*0141 :label: hom-neumann-nh 0142** Warning **
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0143 If the flow is ‘close’ to hydrostatic balance then the 3-d inversion 0bad585a21 Navi*0144 converges rapidly because :math:`\phi _{\rm nh}\ `\ is then only a small f67abf1ee3 Jeff*0145 correction to the hydrostatic pressure field (see the discussion in 0146 Marshall et al. (1997a,b) :cite:`marshall:97a` :cite:`marshall:97b`. 0147 0bad585a21 Navi*0148 The solution :math:`\phi _{\rm nh}\ `\ to :eq:`3d-invert` and 0149 :eq:`inhom-neumann-nh` does not vanish at :math:`r=R_{\rm moving}`, and so f67abf1ee3 Jeff*0150 refines the pressure there. 0151
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