Unsteady Adjustment Problems

Abstract

The basic approximations are formulated with a view to filtering acoustic waves out of the solutions of equations for atmospheric motions, because such waves are of no importance concerning weather prediction. On the other hand, when considering the approximate, simplified set of equations (primitive equations, Boussinesq equations or quasi-geostrophic model equation), one is allowed to specify a set of initial conditions less in number than for the “exact” equations. This is due to the fact that the limiting process which leads to the approximate model, filters out some time derivatives. Due to this one encounters the problem of deciding what initial conditions one may prescribe and in what way these are related to the initial conditions associated with the exact, full, equations? The latter are not in general consistent with the estimates of basic orders of magnitude implied by the asymptotic model. A physical process of time evolution is necessary to bring the initial set to a consistent level as far as the orders of magnitude is concerned. Such a process is called one of ADJUSTMENT of the initial set of data to the asymptotic structure of the model under consideration. The process of adjustment, which occurs in many fields of Fluid Mechanics besides Meteorology, is short on the time scale of the asymptotic model considered, and at the end of it, in an asymptotic sence, we obtain values far the set of initial conditions suitable to the model.