To
illustrate how it is all done, we will rewrite governing equations
and the finite difference equivalents in a simplified form here
appropriate to a rectilinear coordinate system (as opposed to the
curvilinear system in equations (90) to (92)),
omitting for
simplicity the density terms, atmospheric pressure terms,
astronomical forcing terms and the horizontal diffusion terms. We
will also put
,
,
, and
).
are now the simplified set of governing equations. Their finite difference equivalents are
where
simplified relationships for the
other 
, 
and 
terms in equations
(109)
to (111). Note 
and
are the grid spacings in the x and y directions and
is the
time step. 
and 
are of course under modeler's
control,
subject only to physical and computer-resource considerations.
Once 
, 
are chosen, because the differencing
scheme is
explicit, 
must be chosen so that it is
This is the CFL condition for numerical stability of the scheme, which can only be avoided by employing an implicit scheme.
For the Persian Gulf
model, 
km. Maximum H is about 100m,
so that 
must be
less than 128 sec. We have put DTE=120 sec in the file pegtdu10. Try
running it with a larger value, say 240 sec, and watch the model blow
up rather quickly (of course DTE can be made lower but it only
wastes computer time!).
In the data assimilation runs, at time
step dictated by DTT in pegtdu10, the model generated 
is replaced (at gridpoints where observations are available) by a
weighted combination of model 
and observed 
.
In other words
is replaced by
, where a is given by
FRACTN in pegtdu10. By changing a and DTT we can control how
strongly the model is forced towards observations. Strictly
speaking, a should be a function of both the model and
observational data error statistics. That is a fancy way of saying
that we should look at how accurate the data are and how reliable
the model is before deciding what a should be. There has been a
tremendous amount of work done in the past 5 decades on how to do
data assimilation in dynamical models. There are extremely
sophisticated and highly resource-intensive methods (such as Kalman
filters, adjoint methods) that do this properly. However, we
prefer here to fix the a value
a priori. The results will not be too
different than when a more sophisticated approach is used.
