Diffusion and the formation of vorticity staircases in randomly strained two-dimensional vortices

The spreading and diffusion of two-dimensional vortices subject to weak externalrandom strain fields is examined. The response to such a field of given angularfrequency depends on the profile of the vortex and can be calculated numerically.An effective diffusivity can be determined as a function of radius and may beused to evolve the profile over a long time scale, using a diffusion equation thatis both nonlinear and non-local. This equation, containing an additional smoothingparameter, is simulated starting with a Gaussian vortex. Fine scale steps in thevorticity profile develop at the periphery of the vortex and these form a vorticitystaircase. The effective diffusivity is high in the steps where the vorticity gradient islow: between the steps are barriers characterized by low effective diffusivity and highvorticity gradient. The steps then merge before the vorticity is finally swept out andthis leaves a vortex with a compact core and a sharp edge. There is also an increasein the effective diffusion within an encircling surf zone.In order to understand the properties of the evolution of the Gaussian vortex, anasymptotic model first proposed by Balmforth, Llewellyn Smith & Young (J. FluidMech., vol. 426, 2001, p. 95) is employed. The model is based on a vorticity distributionthat consists of a compact vortex core surrounded by a skirt of relatively weakvorticity. Again simulations show the formation of fine scale vorticity steps withinthe skirt, followed by merger. The diffusion equation we develop has a tendency togenerate vorticity steps on arbitrarily fine scales; these are limited in our numericalsimulations by smoothing the effective diffusivity over small spatial scales.