and so
The model can be easily modified to accommodate a loading term (e.g atmospheric pressure pushing down on the ocean's surface) by setting:
1.3.6.2 Surface pressureThe surface pressure equation can be obtained by integrating continuity, (1.3), vertically from to
Thus:
where is the freesurface anomaly in units of . The above can be rearranged to yield, using Leibnitz's theorem:
where we have incorporated a source term. Whether is pressure (ocean model, ) or geopotential (atmospheric model), in (1.26), the horizontal gradient term can be written where is the buoyancy at the surface. In the hydrostatic limit ( ), equations (1.26), (1.35) and (1.36) can be solved by inverting a 2d elliptic equation for as described in Chapter 2. Both `free surface' and `rigid lid' approaches are available.
1.3.6.3 Nonhydrostatic pressureTaking the horizontal divergence of (1.26) and adding of (1.28), invoking the continuity equation (1.3), we deduce that:
For a given rhs this 3d elliptic equation must be inverted for 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 3d problem does not need to be solved.
1.3.6.3.1 Boundary ConditionsWe apply the condition of no normal flow through all solid boundaries  the coasts (in the ocean) and the bottom:
where is a vector of unit length normal to the boundary. The kinematic condition (1.38) is also applied to the vertical velocity at . Noslip or slip conditions are employed on the tangential component of velocity, , 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:
where
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 function sheet of `sourcecharge', replace the inhomogeneous boundary condition on pressure by a homogeneous one. The source term in (1.37) is the divergence of the vector By simultaneously setting and on the boundary the following selfconsistent but simpler homogenized Elliptic problem is obtained:
where is a modified such that . As is implied by (1.39) the modified boundary condition becomes:
If the flow is `close' to hydrostatic balance then the 3d inversion converges rapidly because is then only a small correction to the hydrostatic pressure field (see the discussion in Marshall et al, a,b). The solution to (1.37) and (1.39) does not vanish at , and so refines the pressure there.
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