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Thresholds for the formation of satellites in two-dimensional vortices

By M.R. Turner and A.D. Gilbert

Abstract

This paper examines the evolution of a two-dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m = 2\ud added to it. If the perturbation is weak then the vortex returns to an axisymmetric state and the non-zero Fourier harmonics generated by the perturbation decay to zero.\ud However, if a finite perturbation threshold is exceeded, then a persistent nonlinear vortex structure is formed. This structure consists of a coherent vortex core with two satellites rotating around it.\ud \ud The paper considers the formation of these satellites by taking an asymptotic limit in which a compact vortex is surrounded by a weak skirt of vorticity. The resulting equations match the behaviour of a normal mode riding on the vortex with the evolution of fine-scale vorticity in a critical layer inside the skirt. Three estimates of inviscid thresholds for the formation of satellites are computed and compared: two estimates use qualitative\ud diagnostics, the appearance of an infection point or neutral mode in the mean profile. The other is determined quantitatively by solving the normal mode/critical-layer equations numerically. These calculations are supported by simulations of the full Navier-Stokes equations using a family of proles based on the tanh function

Topics: G100 Mathematics, H000 Engineering
Publisher: Cambridge University Press
Year: 2008
DOI identifier: 10.1017/S0022112008003558
OAI identifier: oai:eprints.brighton.ac.uk:6462

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