Asymptotic localized solutions of the shallow water equations over a nonuniform bottom

Abstract

The present paper is an extended English version of the authors’ survey “Wave and vortex localized asymptotic solutions of linearized shallow water equations” [25] containing the results obtained by a group of employees of Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences and their colleagues from Germany, Italy and Mexico. It deals with the propagation of wave and eddies described by linearized shallow water equations with variable depth and excited by localized sources. We give fairly explicit asymptotic formulas for the solutions of the homogeneous and inhomogeneous problems with due account of the focal points and caustics for the solutions of the shallow water equations on the sphere, the solutions of equations taking into account weakly dispersive effects, the solution of the linear problem of run-up of long waves on a shallow curvilinear beach, and so on. Asymptotic formulas are obtained with the use of methods that were developed in the last decade and which are based on a modification of the Maslov canonical operator adapted for the construction of solutions localized in the vicinity of points and curves. Although the basic constructions are complicated, the final asymptotic formulas prove to be fairly simple and effcient and only require minimum information necessary for the description of qualitative and quantitative characteristics of waves and eddies. We discuss the applicability of the asymptotics obtained in the present paper in problems of propagation of tsunami waves and mesoscale eddies in atmosphere.