Variational methods and nonlinear quasigeostrophic waves

Abstract

In this paper, we discuss zonally periodic steady quasigeostrophic waves in a β-plane channel, by using variational methods. A class of steady quasigeostrophic waves are determined by the potential vorticity field profile, g(⋅), which is a function of the stream function. We show that zonally periodic steady quasigeostrophic waves exist when the bottom topography and the potential vorticity field are bounded. We also show that these waves are unique if, in addition, the potential vorticity field profile is increasing and passes through the origin. Finally, we demonstrate that the zonal periodic wave in the case with g(ψ)=arctan(ψ) is nonlinearly stable in the sense of Liapunov, under a boundedness condition for the potential vorticity field, or equivalently, under suitable conditions on the bottom topography, β parameter, and zonal period T.