May 1, 2009: Shane Keating, CIMS
Vorticity homogenization and the Prandtl-Batchelor theorem
In the inaugural volume of the Journal of Fluid Mechanics, G.K. Batchelor investigated 2D steady flow with closed streamlines and demonstrated that such flows expel gradients of relative vorticity from regions of weak viscosity. Equivalently, the vorticity can be thought to be homogenized within nested streamlines. This simple result has proven remarkably powerful and has been successfully applied to a variety of scalar fields, including: expulsion of magnetic field from thermal convection cells; homogenization of potential vorticity in oceanic gyres and Jupiter's Great Red Spot; and turbulent advection of toroidal angular momentum density in magnetic fusion. In this informal blackboard seminar, I will discuss this important result, its connection with `diascalar diffusivity', and its extension to (potential) vorticity homogenization in the presence of a magnetic field.