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Next: 2.6 Synchronous time-stepping: variables
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Vertical diffusion and viscosity can be treated implicitly in time
using the backward method which is an intrinsic scheme.
Recently, the option to treat the vertical advection
implicitly has been added, but not yet tested; therefore,
the description hereafter is limited to diffusion and viscosity.
For tracers,
the time discretized equation is:
|
(2.25) |
where
is the remaining explicit terms extrapolated
using the Adams-Bashforth method as described above. Equation
2.25 can be split split into:
where
is the inverse of the operator
|
(2.28) |
Equation 2.26 looks exactly as 2.23
while 2.27 involves an operator or matrix
inversion. By re-arranging 2.25 in this way we
have cast the method as an explicit prediction step and an implicit
step allowing the latter to be inserted into the over all algorithm
with minimal interference.
Fig. 2.3 shows the calling sequence for
stepping forward a tracer variable such as temperature.
In order to fit within the pressure method, the implicit viscosity
must not alter the barotropic flow. In other words, it can only
redistribute momentum in the vertical. The upshot of this is that
although vertical viscosity may be backward implicit and
unconditionally stable, no-slip boundary conditions may not be made
implicit and are thus cast as a an explicit drag term.
Next: 2.6 Synchronous time-stepping: variables
Up: 2. Discretization and Algorithm
Previous: 2.4 Explicit time-stepping: Adams-Bashforth
Contents
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