Variational methods and nonlinear quasigeostrophic waves

In this paper, we discuss zonally periodic steady quasigeostrophic waves in a beta-plane channel, by using variational methods. A class of steady quas igeostrophic waves are determined by the potential vorticity field profile, g(.), which is a function of the stream function. We show that zonally per iodic steady quasigeostrophic waves exist when the bottom topography and th e 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 peri odic wave in the case with g(psi)= arctan(psi) is nonlinearly stable in the sense of Liapunov, under a boundedness condition for the potential vortici ty field, or equivalently, under suitable conditions on the bottom topograp hy, beta parameter, and zonal period T. (C) 1999 American Institute of Phys ics. [S1070-6631(99)00804-1].