Unlike the prognostic variables u, v, w, \theta and 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, \phi _{s}(x,y), a hydrostatic part \phi _{\rm hyd}(x,y,r) and a non-hydrostatic part \phi _{\rm nh}(x,y,r), as in :eq:`phi-split`, and writing the momentum equation as in :eq:`mom-h`.
Hydrostatic pressure is obtained by integrating :eq:`hydrostatic` vertically from r=R_{o} where \phi _{\rm hyd}(r=R_{o})=0, to yield:
\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
\phi _{\rm hyd}(x,y,r)=\int_{r}^{R_{o}}bdr
The model can be easily modified to accommodate a loading term (e.g atmospheric pressure pushing down on the ocean’s surface) by setting:
\phi _{\rm hyd}(r=R_{o})= \text{loading}
The surface pressure equation can be obtained by integrating continuity, :eq:`continuity`, vertically from r=R_{\rm fixed} to r=R_{\rm moving}
\int_{R_{\rm fixed}}^{R_{\rm moving}}\left( \nabla _{h}\cdot \vec{\mathbf{v}
}_{h}+\partial _{r}\dot{r}\right) dr=0
Thus:
\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 \eta =R_{\rm moving}-R_{o} is the free-surface r-anomaly in units of r. The above can be rearranged to yield, using Leibnitz’s theorem:
\frac{\partial \eta }{\partial t}+ \nabla _{h}\cdot
\int_{R_{\rm fixed}}^{R_{\rm moving}}\vec{\mathbf{v}}_{h}dr=\text{source}
where we have incorporated a source term.
Whether \phi is pressure (ocean model, p/\rho _{c}) or geopotential (atmospheric model), in :eq:`mom-h`, the horizontal gradient term can be written
\nabla _{h}\phi _{s}= \nabla _{h}\left( b_{s}\eta \right)
where b_{s} is the buoyancy at the surface.
In the hydrostatic limit (\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 \phi _{s} as described in Chapter 2. Both ‘free surface’ and ‘rigid lid’ approaches are available.
Taking the horizontal divergence of :eq:`mom-h` and adding \frac{\partial }{\partial r} of :eq:`mom-w`, invoking the continuity equation :eq:`continuity`, we deduce that:
\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}}
For a given rhs this 3-D elliptic equation must be inverted for \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.
We apply the condition of no normal flow through all solid boundaries - the coasts (in the ocean) and the bottom:
\vec{\mathbf{v}} \cdot \hat{\boldsymbol{n}} =0
where \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 r=R_{\rm moving}. No-slip \left( v_{T}=0\right) \ or slip \left( \partial v_{T}/\partial n=0\right) \ conditions are employed on the tangential component of velocity, 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:
\hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}= \hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}}
where
\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 \delta-function sheet of ‘source-charge’, replace the inhomogeneous boundary condition on pressure by a homogeneous one. The source term rhs in :eq:`3d-invert` is the divergence of the vector \vec{\mathbf{F}}. By simultaneously setting \hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}}=0 and \hat{\boldsymbol{n}} \cdot \nabla \phi_{\rm nh}=0\ on the boundary the following self-consistent but simpler homogenized Elliptic problem is obtained:
\nabla ^{2}\phi _{\rm nh}= \nabla \cdot \widetilde{\vec{\mathbf{F}}}\qquad
where \widetilde{\vec{\mathbf{F}}} is a modified \vec{\mathbf{F}} such that \widetilde{\vec{\mathbf{F}}} \cdot \hat{\boldsymbol{n}} =0. As is implied by :eq:`inhom-neumann-nh` the modified boundary condition becomes:
\hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}=0
If the flow is ‘close’ to hydrostatic balance then the 3-d inversion converges rapidly because \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 \phi _{\rm nh}\ to :eq:`3d-invert` and :eq:`inhom-neumann-nh` does not vanish at r=R_{\rm moving}, and so refines the pressure there.