Journal of Fluid Mechanics



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


ANDREW P. BASSOM a1 and ANDREW D. GILBERT a1
a1 School of Mathematical Sciences, University of Exeter, North Park Road, Exeter, Devon EX4 4QE, UK; email: drew@maths.exeter.ac.uk; adg@maths.exeter.ac.uk

Abstract

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−λ with λ=1+(n2+8)1/2, where n is the azimuthal wavenumber of the perturbation and n[gt-or-equal, slanted]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).

(Received October 8 1997)
(Revised March 30 1998)



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