Existence of energy maximizing vortices in a three-dimensional quasigeostrophic shear flow with bounded height

The existence of an energy maximizer relative to a class of rearrangements of a given function is proved. The maximizers are stationary and stable solutions of the quasigeostrophic equation, which governs the time evolution of large-scale three-dimensional geophysical flow in a vertically bounded domain. The background flow is unidirectional, with linear horizontal shear. The theorem proved implies the existence of a family of stationary and stable vortices that rotate in the same direction as the background shear. It extends an earlier theorem by Burton and Nycander, which is valid for a vertically unbounded domain.