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

Published online by Cambridge University Press:  25 September 1998

ANDREW P. BASSOM
Affiliation:
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
ANDREW D. GILBERT
Affiliation:
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[ges ]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).

Type
Research Article
Copyright
© 1998 Cambridge University Press

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