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2.4 Pressure method with implicit linear freesurface
The rigidlid approximation filters out external gravity waves
subsequently modifying the dispersion relation of barotropic Rossby
waves. The discrete form of the elliptic equation has some zero
eigenvalues which makes it a potentially tricky or inefficient
problem to solve.
The rigidlid approximation can be easily replaced by a linearization
of the freesurface equation which can be written:

(2.15) 
which differs from the depth integrated continuity equation with
rigidlid (2.4) by the timedependent term
and freshwater source term.
Equation 2.7 in the rigidlid
pressure method is then replaced by the time discretization of
2.15 which is:

(2.16) 
where the use of flow at time level
makes the method implicit
and backward in time. This is the preferred scheme since it still
filters the fast unresolved wave motions by damping them. A centered
scheme, such as CrankNicholson (see section 2.10.1),
would alias the energy of the fast modes onto slower modes of motion.
As for the rigidlid pressure method, equations
2.5, 2.6 and
2.16 can be rearranged as follows:



(2.17) 



(2.18) 



(2.19) 



(2.20) 



(2.21) 



(2.22) 
Equations 2.17
to 2.22, solved sequentially, represent
the pressure method algorithm with a backward implicit, linearized
free surface. The method is still formerly a pressure method because
in the limit of large
the rigidlid method is
recovered. However, the implicit treatment of the freesurface allows
the flow to be divergent and for the surface pressure/elevation to
respond on a finite timescale (as opposed to instantly). To recover
the rigidlid formulation, we introduced a switchlike parameter,
(freesurfFac),
which selects between the freesurface and rigidlid;
allows the freesurface to evolve;
imposes the rigidlid. The evolution in time and location of variables
is exactly as it was for the rigidlid model so that
Fig. 2.1 is still
applicable. Similarly, the calling sequence, given in
Fig. 2.2, is as for the
pressuremethod.
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