The relaxation of a smooth two-dimensional vortex to axisymmetry, also
known as 'axisymmetrization', is studied asymptotically and numerical
ly. The vortex is perturbed at t = 0 and differential rotation leads t
o 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-grain
ed 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 pertu
rbation and n greater than or equal to 1. The far-field stream functio
n of the perturbation decays with the same exponent. To obtain these r
esults 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).