Reduced Systems for the Shallow Water Equations

Abstract

We consider smooth solutions of the shallow water equations restricted only by the assumption that the magnitude of the deviation of the geopotential from the mean is small relative to that mean. For such solutions we find only two possible reduced systems. The first system is equivalent to the nondivergent barotropic system. A typical flow satisfying this system is the external mode with geopotential deviations on the order of 1% of the mean, and velocity on the order of 10 m s⁻¹. The second reduced system is similar to the balance equations. An example of a flow satisfying the second system is the largest internal mode with geopotential deviations on the order of 10% of the mean, and velocity on the order of 10 m s⁻¹. The error incurred by using a reduced system can be decreased by expanding the smooth solutions of the corresponding full system in an asymptotic series of solutions of the reduced system. We discuss an equivalent method to accomplish this reduction in error and derive improved approximating systems for both cases. To support the analysis we compare a numerical solution of the initialized shallow water equations with the corresponding numerical solutions of the reduced and improved approximating systems.