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
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.