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 \(\phi _{hyd}(x,y,r)\) and a non-hydrostatic part \(\phi _{nh}(x,y,r)\), as in (1.25), and writing the momentum equation as in (1.26).

1.3.6.1. Hydrostatic pressure

Hydrostatic pressure is obtained by integrating (1.27) vertically from \(r=R_{o}\) where \(\phi _{hyd}(r=R_{o})=0\), to yield:

\[\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi _{hyd} \right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr\]

and so

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

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)\[\phi _{hyd}(r=R_{o})=loading\]

1.3.6.2. Surface pressure

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

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

Thus:

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

where \(\eta =R_{moving}-R_{o}\) is the free-surface \(r\)-anomaly in units of \(r\). The above can be rearranged to yield, using Leibnitz’s theorem:

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

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

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

In the hydrostatic limit (\(\epsilon _{nh}=0\)), 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.

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

For a given rhs this 3-d elliptic equation must be inverted for \(\phi _{nh}\) subject to appropriate choice of boundary conditions. This method is usually called The Pressure Method [Harlow and Welch (1965) [harlow:65]; Williams (1969) [williams:69]; Potter (1973) [potter:73]. In the hydrostatic primitive equations case (HPE), the 3-d problem does not need to be solved.

1.3.6.3.1. Boundary Conditions

We apply the condition of no normal flow through all solid boundaries - the coasts (in the ocean) and the bottom:

(1.38)\[\vec{\mathbf{v}}.\widehat{n}=0\]

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 \(r=R_{moving}\). 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)\[\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}\]

where

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

presenting inhomogeneous Neumann boundary conditions to the Elliptic problem (1.37). As shown, for example, by Williams (1969) [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 (1.37) is the divergence of the vector \(\vec{\mathbf{F}}.\) By simultaneously setting \(\widehat{n}.\vec{\mathbf{F}}=0\) and \(\widehat{n}.\nabla \phi _{nh}=0\ \)on the boundary the following self-consistent but simpler homogenized Elliptic problem is obtained:

\[\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad\]

where \(\widetilde{\vec{\mathbf{F}}}\) is a modified \(\vec{\mathbf{F}}\) such that \(\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0\). As is implied by (1.39) the modified boundary condition becomes:

(1.40)\[\widehat{n}.\nabla \phi _{nh}=0\]

If the flow is ‘close’ to hydrostatic balance then the 3-d inversion converges rapidly because \(\phi _{nh}\ \)is then only a small correction to the hydrostatic pressure field (see the discussion in Marshall et al. (1997a,b) [marshall:97a] [marshall:97b].

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