Surface waves on steady perfect‐fluid flows with vorticity

This is a theory of two‐dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem: the volume of fluid per period, the circulation per period on the free stream line, and the rearrangement class (equivalently, the distribution function) of the vorticity field. A minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function. This decreasing function can be thought of as an infinite‐dimensional Lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem. (Note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady.) To illustrate the idea with a minimum of technical difficulties, the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved. The theory applies equally to irrotational flows and to flows with locally square‐integrable vorticity. © 2011 Wiley Periodicals, Inc.