Localized approximation method for vortical flows

An approximation method of Moore for Kelvin-Helmholtz instability is formulated as a general method for two-dimensional, incompressible, inviscid flows generated by a vortex sheet. In this method the nonlocal equations describing evolution of the sheet are approximated by a system of (local) differential equations. These equations are useful for predicting singularity formation on the sheet and for analyzing the initial value problem before singularity formation. The general method is applied to a number of problems: Kelvin-Helmholtz instability for periodic vortex sheets, motion of an interface in Hele-Shaw flow, Rayleigh-Taylor instability for stratified flow, and Krasny's desingularized vortex sheet equation. A new physically desingularized vortex sheet equation is proposed, which agrees with the finite thickness vortex layer equations in the localized approximation.