The spreading and diffusion of two-dimensional vortices subject to weak external\ud random strain fields is examined. The response to such a field of given angular\ud frequency depends on the profile of the vortex and can be calculated numerically.\ud An effective diffusivity can be determined as a function of radius and may be\ud used to evolve the profile over a long time scale, using a diffusion equation that\ud is both nonlinear and non-local. This equation, containing an additional smoothing\ud parameter, is simulated starting with a Gaussian vortex. Fine scale steps in the\ud vorticity profile develop at the periphery of the vortex and these form a vorticity\ud staircase. The effective diffusivity is high in the steps where the vorticity gradient is\ud low: between the steps are barriers characterized by low effective diffusivity and high\ud vorticity gradient. The steps then merge before the vorticity is finally swept out and\ud this leaves a vortex with a compact core and a sharp edge. There is also an increase\ud in the effective diffusion within an encircling surf zone.\ud \ud In order to understand the properties of the evolution of the Gaussian vortex, an\ud asymptotic model first proposed by Balmforth, Llewellyn Smith & Young (J. Fluid\ud Mech., vol. 426, 2001, p. 95) is employed. The model is based on a vorticity distribution\ud that consists of a compact vortex core surrounded by a skirt of relatively weak\ud vorticity. Again simulations show the formation of fine scale vorticity steps within\ud the skirt, followed by merger. The diffusion equation we develop has a tendency to\ud generate vorticity steps on arbitrarily fine scales; these are limited in our numerical\ud simulations by smoothing the effective diffusivity over small spatial scales
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