The spiral wind-up of vorticity in an inviscid planar vortex

The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as 'axisymmetrization', is studied asymptotically and numerically. The vortex is perturbed at t = 0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t(-lambda) with lambda = 1 + (n(2) + 8)(1/2), where n is the azimuthal wavenumber of the perturbation and n greater than or equal to 1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).