The horizontal momentum and continuity equations for the ocean
(1.99 and 1.101), or for the atmosphere
(1.45 and 1.47), can be summarized by:
where we are adopting the oceanic notation for brevity. All terms in the momentum equations, except for surface pressure gradient, are encapsulated in the vector. The continuity equation, when integrated over the fluid depth, , and with the rigidlid/no normal flow boundary conditions applied, becomes: Here, is the depth integral of , similarly for . The rigidlid approximation sets at the lid so that it does not move but allows a pressure to be exerted on the fluid by the lid. The horizontal momentum equations and vertically integrated continuity equation are be discretized in time and space as follows: As written here, terms on the LHS all involve time level and are referred to as implicit; the implicit backward time stepping scheme is being used. All other terms in the RHS are explicit in time. The thermodynamic quantities are integrated forward in time in parallel with the flow and will be discussed later. For the purposes of describing the pressure method it suffices to say that the hydrostatic pressure gradient is explicit and so can be included in the vector .
Substituting the two momentum equations into the depth integrated
continuity equation eliminates
and
yielding an
elliptic equation for
. Equations
2.5, 2.6 and
2.7 can then be rearranged as follows:
Equations 2.8 to 2.12, solved sequentially, represent the pressure method algorithm used in the model. The essence of the pressure method lies in the fact that any explicit prediction for the flow would lead to a divergence flow field so a pressure field must be found that keeps the flow nondivergent over each step of the integration. The particular location in time of the pressure field is somewhat ambiguous; in Fig. 2.1 we depicted as colocated with the future flow field (time level ) but it could equally have been drawn as staggered in time with the flow. The correspondence to the code is as follows:
In general, the horizontal momentum timestepping can contain some terms
that are treated implicitly in time,
such as the vertical viscosity when using the backward timestepping scheme
(implicitViscosity
=.TRUE.).
The method used to solve those implicit terms is provided in
section 2.6, and modifies
equations 2.5 and 2.6 to
give:
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