Steady state and transient solutions of the nonlinear forced shallow water equations in one space dimension

Abstract

The one-dimensional shallow water equations permit steady state solutions for a given forcing that is independent of time. When the forcing is sufficiently small one obtains three periodic solution of which two have numerically large velocities and the third a somewhat lower velocity. Examples of solutions for simple forcing patterns are given. The nonlinear one-dimensional equations are simplified in the usual way by neglecting the advection term in the continuity equation and by replacing the geopotential by a constant when undifferentiated. It is shown that these equations have only one steady state solution which is similar to the low velocity solution in the more general system. The approximations act thus as a filter of large velocity solutions. Solutions of the simplified equations are obtained by formulating the equations in wave number space. Examples indicate that a maximum wave number of 30 is sufficient to obtain solutions of sufficient accuracy.