1.3.6. Finding the pressure field

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 (1.25), and writing the momentum equation as in (1.26). Hydrostatic pressure

Hydrostatic pressure is obtained by integrating (1.27) 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

(1.33)\[\phi _{\rm hyd}(x,y,r)=\int_{r}^{R_{o}}bdr\]

The model can be easily modified to accommodate a loading term (e.g atmospheric pressure pushing down on the ocean’s surface) by setting:

(1.34)\[\phi _{\rm hyd}(r=R_{o})= \text{loading}\] Surface pressure

The surface pressure equation can be obtained by integrating continuity, (1.3), vertically from \(r=R_{\rm fixed}\) to \(r=R_{\rm moving}\)

\[\int_{R_{\rm fixed}}^{R_{\rm moving}}\left( \nabla _{h}\cdot \vec{\mathbf{v} }_{h}+\partial _{r}\dot{r}\right) dr=0\]


\[\frac{\partial \eta }{\partial t}+\vec{\mathbf{v}} \cdot \nabla \eta +\int_{R_{\rm fixed}}^{R_{\rm moving}} \nabla _{h}\cdot \vec{\mathbf{v}} _{h}dr=0\]

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:

(1.35)\[\frac{\partial \eta }{\partial t}+ \nabla _{h}\cdot \int_{R_{\rm fixed}}^{R_{\rm moving}}\vec{\mathbf{v}}_{h}dr=\text{source}\]

where we have incorporated a source term.

Whether \(\phi\) is pressure (ocean model, \(p/\rho _{c}\)) or geopotential (atmospheric model), in (1.26), the horizontal gradient term can be written

(1.36)\[\nabla _{h}\phi _{s}= \nabla _{h}\left( b_{s}\eta \right)\]

where \(b_{s}\) is the buoyancy at the surface.

In the hydrostatic limit (\(\epsilon _{\rm nh}=0\)), equations (1.26), (1.35) and (1.36) 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. Non-hydrostatic pressure

Taking the horizontal divergence of (1.26) and adding \(\frac{\partial }{\partial r}\) of (1.28), invoking the continuity equation (1.3), we deduce that:

(1.37)\[\nabla_{3}^{2}\phi _{\rm nh}= \nabla \cdot \vec{\mathbf{G}}_{\vec{v}}-\left( \nabla_{h}^{2}\phi _{s}+ \nabla^2 \phi _{\rm hyd}\right) = \nabla \cdot \vec{\mathbf{F}}\]

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) [HW65]; Williams (1969) [Wil69]; Potter (1973) [Pot73]. 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:

(1.38)\[\vec{\mathbf{v}} \cdot \hat{\boldsymbol{n}} =0\]

where \(\widehat{n}\) is a vector of unit length normal to the boundary. The kinematic condition (1.38) 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. (1.38) implies, making use of (1.26), that:

(1.39)\[\hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}= \hat{\boldsymbol{n}} \cdot \vec{\mathbf{F}}\]


\[\vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \nabla _{h}\phi_{s}+ \nabla \phi _{\rm hyd}\right)\]

presenting inhomogeneous Neumann boundary conditions to the Elliptic problem (1.37). As shown, for example, by Williams (1969) [Wil69], 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 (1.37) 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 (1.39) the modified boundary condition becomes:

(1.40)\[\hat{\boldsymbol{n}} \cdot \nabla \phi _{\rm nh}=0\]

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) [MHPA97] [MAH+97].

The solution \(\phi _{\rm nh}\ \)to (1.37) and (1.39) does not vanish at \(r=R_{\rm moving}\), and so refines the pressure there.