Kinetic equation for point vortices in a shear flow

Abstract

The kinetic equation for a set of Stewart point vortices moving on the background of a shear flow with uniform vorticity is derived. Besides the total number of vortices with certain amplitude values, this equation provides conservation of the total vorticity of vortex gas on each shear flow stream line. Such conservation laws prevent, in the general case, relaxation of the system to the Maxwell equilibrium distribution of vortices. In fact, the system is shown to tend to an equilibrium state, in which vortex gas vorticity on each stream line of shear flow is formed by vortices with equal amplitude signs, only the most intense vortices being concentrated in more exited regions, while less intensive vortices are concentrated at the periphery of perturbed domains. Applicability of the kinetic equation obtained for the description of Helmholtz vortices sets is discussed.