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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). Hydrostatic pressure

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

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

and so

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

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

$\displaystyle \phi _{hyd}(r=R_{o})=loading$ (1.34) Surface pressure

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

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


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

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:

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

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

$\displaystyle \mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right)$ (1.36)

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. 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:

$\displaystyle \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\lef...
...^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla . \vec{\mathbf{F}}$ (1.37)

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; Williams, 1969; Potter, 1976]. 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:

$\displaystyle \vec{\mathbf{v}}.\widehat{n}=0$ (1.38)

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:

$\displaystyle \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}$ (1.39)


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

presenting inhomogeneous Neumann boundary conditions to the Elliptic problem (1.37). As shown, for example, by Williams (1969), 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 \begin{displaymath}
\end{array}=0\end{displaymath} and $ \widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following self-consistent but simpler homogenized Elliptic problem is obtained:

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

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:

$\displaystyle \widehat{n}.\nabla \phi _{nh}=0$ (1.40)

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, a,b).

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

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Next: 1.3.7 Forcing/dissipation Up: 1.3 Continuous equations in Previous: 1.3.5 Solution strategy   Contents
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