Abstract
The shallow water equations are a symmetric hyperbolic system with two time scales. In meteorological terms, slow and fast scale motions are referred to as Rossby and inertial/gravity waves, respectively. We prove the existence of smooth solutions (solutions with a number of space and time derivatives on the order of the slow time scale) of the open boundary problem for the shallow water equations by the bounded derivative method. The proof requires that a number of initial time derivatives be of the order of the slow time scale and that the boundary data be smooth. If the boundary data are smooth and only have small errors, then we show that the solution of the open boundary problem is smooth and that only small errors are produced in the interior. If the boundary data are smooth but have large errors, then we show that the solution of the open boundary problem is still smooth. Unfortunately the boundary error propagates into the interior at the speed associated with the fast time scale and destroys the solution in a short time. Thus it is necessary to keep the boundary error small if the solution is to be computed correctly. We show that this restriction can be relaxed so that only the large-scale boundary data need be correct. We demonstrate the importance of these conclusions in several numerical experiments.
Highlights
In a previous paper (Browning et al, 1980) we proved the existence of smooth solutions of the equations of shallow water flow in a channel by the bounded derivative method (Kreiss, 1980)
In this paper we have proven the existence of smooth solutions of the shallow water equations in a region R with open boundary conditions
The importance of the form of the homogeneous boundary conditions and the smoothness of the inhomogeneous terms of the boundary conditions in maintaining the smoothness of the solution is evident in that proof
Summary
In a previous paper (Browning et al, 1980) we proved the existence of smooth solutions of the equations of shallow water flow in a channel by the bounded derivative method (Kreiss, 1980). We determine boundary conditions for the reduced system) To show this we integrate the full system which in the limit approach (3.24) and third equation of (3.1) over R obtaining (3.29) or (3.30) and which lead to an energy estimate of the right form for the corresponding solutions of (3.21). If we choose the initial data for Z correctly, Z will be close (to within an error term of order E) to a smooth solution for any time interval of order unity This tells us that (3.22) is the reduced system since as E + 0, Z d + 0 and Z' + 0. (4.9a) (4.9b) and choose the unique solution whose mean is the same as that of the observed data for $
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