Nonlinear Waves of Vorticity

Abstract

This is a review of solutions of the vorticity equation for two‐dimensional flow of an inviscid incompressible fluid that represent nonlinear waves. Geophysical applications are emphasized. Some of the solutions are valid in the beta‐plane of Rossby. Some are related to weakly nonlinear perturbations of basic parallel flows and axisymmetric flows, to initial‐value problems of hydrodynamic instability and to variational principles of minimal enstrophy or maximal entropy. Some have been found by exploiting well‐known ideas of the theory of solitons. In addition to listing known solutions and presenting a synthesis of their relationship to other fluid dynamic results, we report a few new ideas and new solutions for strongly nonlinear waves.