.. _finding_the_pressure_field: Finding the pressure field -------------------------- Unlike the prognostic variables :math:`u`, :math:`v`, :math:`w`, :math:`\theta` and :math:`S`, the pressure field must be obtained diagnostically. We proceed, as before, by dividing the total (pressure/geo) potential in to three parts, a surface part, :math:`\phi _{s}(x,y)`, a hydrostatic part :math:`\phi _{\rm hyd}(x,y,r)` and a non-hydrostatic part :math:`\phi _{\rm nh}(x,y,r)`, as in :eq:`phi-split`, and writing the momentum equation as in :eq:`mom-h`. Hydrostatic pressure ~~~~~~~~~~~~~~~~~~~~ Hydrostatic pressure is obtained by integrating :eq:`hydrostatic` vertically from :math:`r=R_{o}` where :math:`\phi _{\rm hyd}(r=R_{o})=0`, to yield: .. math:: \int_{r}^{R_{o}}\frac{\partial \phi _{\rm hyd}}{\partial r}dr=\left[ \phi _{\rm hyd} \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr and so .. math:: \phi _{\rm hyd}(x,y,r)=\int_{r}^{R_{o}}bdr :label: hydro-phi The model can be easily modified to accommodate a loading term (e.g atmospheric pressure pushing down on the ocean’s surface) by setting: .. math:: \phi _{\rm hyd}(r=R_{o})= \text{loading} :label: loading Surface pressure ~~~~~~~~~~~~~~~~ The surface pressure equation can be obtained by integrating continuity, :eq:`continuity`, vertically from :math:`r=R_{\rm fixed}` to :math:`r=R_{\rm moving}` .. math:: \int_{R_{\rm fixed}}^{R_{\rm moving}}\left( \nabla _{h}\cdot \vec{\mathbf{v} }_{h}+\partial _{r}\dot{r}\right) dr=0 Thus: .. math:: \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}} \cdot \nabla \eta +\int_{R_{\rm fixed}}^{R_{\rm moving}} \nabla _{h}\cdot \vec{\mathbf{v}} _{h}dr=0 where :math:`\eta =R_{\rm moving}-R_{o}` is the free-surface :math:`r`-anomaly in units of :math:`r`. The above can be rearranged to yield, using Leibnitz’s theorem: .. math:: \frac{\partial \eta }{\partial t}+ \nabla _{h}\cdot \int_{R_{\rm fixed}}^{R_{\rm moving}}\vec{\mathbf{v}}_{h}dr=\text{source} :label: free-surface where we have incorporated a source term. Whether :math:`\phi` is pressure (ocean model, :math:`p/\rho _{c}`) or geopotential (atmospheric model), in :eq:`mom-h`, the horizontal gradient term can be written .. math:: \nabla _{h}\phi _{s}= \nabla _{h}\left( b_{s}\eta \right) :label: phi-surf where :math:`b_{s}` is the buoyancy at the surface. In the hydrostatic limit (:math:`\epsilon _{\rm nh}=0`), equations :eq:`mom-h`, :eq:`free-surface` and :eq:`phi-surf` can be solved by inverting a 2-D elliptic equation for :math:`\phi _{s}` as described in Chapter 2. Both ‘free surface’ and ‘rigid lid’ approaches are available. Non-hydrostatic pressure ~~~~~~~~~~~~~~~~~~~~~~~~ Taking the horizontal divergence of :eq:`mom-h` and adding :math:`\frac{\partial }{\partial r}` of :eq:`mom-w`, invoking the continuity equation :eq:`continuity`, we deduce that: .. math:: \nabla_{3}^{2}\phi _{\rm nh}= \nabla \cdot \vec{\mathbf{G}}_{\vec{v}}-\left( \nabla_{h}^{2}\phi _{s}+ \nabla^2 \phi _{\rm hyd}\right) = \nabla \cdot \vec{\mathbf{F}} :label: 3d-invert For a given rhs this 3-D elliptic equation must be inverted for :math:`\phi _{\rm nh}` subject to appropriate choice of boundary conditions. This method is usually called *The Pressure Method* [Harlow and Welch (1965) :cite:`harlow:65`; Williams (1969) :cite:`williams:69`; Potter (1973) :cite:`potter:73`. In the hydrostatic primitive equations case (**HPE**), the 3-D problem does not need to be solved. Boundary Conditions ^^^^^^^^^^^^^^^^^^^ We apply the condition of no normal flow through all solid boundaries - the coasts (in the ocean) and the bottom: .. math:: \vec{\mathbf{v}} \cdot \hat{\boldsymbol{n}} =0 :label: nonormalflow where :math:`\widehat{n}` is a vector of unit length normal to the boundary. The kinematic condition :eq:`nonormalflow` is also applied to the vertical velocity at :math:`r=R_{\rm moving}`. No-slip :math:`\left( v_{T}=0\right) \ `\ or slip :math:`\left( \partial v_{T}/\partial n=0\right) \ `\ conditions are employed on the tangential component of velocity, :math:`v_{T}`, at all solid boundaries, depending on the form chosen for the dissipative terms in the momentum equations - see below. Eq. :eq:`nonormalflow` implies, making use of :eq:`mom-h`, that: .. math:: \hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}= \hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}} :label: inhom-neumann-nh where .. math:: \vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \nabla _{h}\phi_{s}+ \nabla \phi _{\rm hyd}\right) presenting inhomogeneous Neumann boundary conditions to the Elliptic problem :eq:`3d-invert`. As shown, for example, by Williams (1969) :cite:`williams:69`, one can exploit classical 3D potential theory and, by introducing an appropriately chosen :math:`\delta`-function sheet of ‘source-charge’, replace the inhomogeneous boundary condition on pressure by a homogeneous one. The source term :math:`rhs` in :eq:`3d-invert` is the divergence of the vector :math:`\vec{\mathbf{F}}`. By simultaneously setting :math:`\hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}}=0`  and :math:`\hat{\boldsymbol{n}} \cdot \nabla \phi_{\rm nh}=0\ `\ on the boundary the following self-consistent but simpler homogenized Elliptic problem is obtained: .. math:: \nabla ^{2}\phi _{\rm nh}= \nabla \cdot \widetilde{\vec{\mathbf{F}}}\qquad where :math:`\widetilde{\vec{\mathbf{F}}}` is a modified :math:`\vec{\mathbf{F}}` such that :math:`\widetilde{\vec{\mathbf{F}}} \cdot \hat{\boldsymbol{n}} =0`. As is implied by :eq:`inhom-neumann-nh` the modified boundary condition becomes: .. math:: \hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}=0 :label: hom-neumann-nh If the flow is ‘close’ to hydrostatic balance then the 3-d inversion converges rapidly because :math:`\phi _{\rm nh}\ `\ is then only a small correction to the hydrostatic pressure field (see the discussion in Marshall et al. (1997a,b) :cite:`marshall:97a` :cite:`marshall:97b`. The solution :math:`\phi _{\rm nh}\ `\ to :eq:`3d-invert` and :eq:`inhom-neumann-nh` does not vanish at :math:`r=R_{\rm moving}`, and so refines the pressure there.