On the Existence of Solitary Wave Solutions to the Shallow-Water Equations on the f Plane

Abstract

Abstract In a recent paper Li argued that the inviscid shallow-water equations on the f plane exhibit a class of zonally propagating solitary wave solutions, referred to by the author as “geosolitary waves.” These solutions have propagation speeds that are slower than the shallow-water speed gH, and as constructed by Li, have a cusp in the height and zonal velocity field at the origin. The meridional velocity is found to be discontinuous at the origin, though its magnitude is much smaller than the magnitude of the zonal velocity. The authors reconsider the existence of geosolitary waves from the point of view of dynamical system theory and demonstrate that the cusp discontinuity at the origin is an unavoidable property of the solutions. Moreover, it is shown that, in contrast to suggestions made by Li, adding viscosity and diffusivity does not yield smooth, bounded solitary wave solutions. Last, the reason why geosolitary waves cannot be expected to exist in a continuously stratified fluid is discussed.