On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet

Abstract

Motivated by the work of previous authors on vortex sheets and their applications, the intrinsic inviscid evolution equations of a closed vortex sheet in a plane, separating two piecewise constant density fluids, and their Hamiltonian form are investigated. The model has potential applications to problems involving the dynamics of interfaces of two immiscible fluids. A boundary Poisson bracket, which appears to be new and related to the KdV bracket, is obtained containing the curve-tangential derivative $\partial /\partial s$. Lagrangian invariants of the sheet motion by its self-induced velocity–the Cauchy principal value of the Biot–Savart integral–are also derived.