# 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) [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)$\vec{\mathbf{v}}.\widehat{n}=0$

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) [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 $$\widehat{n}.\vec{\mathbf{F}}=0$$ 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) [MHPA97] [MAH+97].

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