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Thresholds for the formation of satellites\ud in two-dimensional vortices

By M R Turner and A D Gilbert


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

Publisher: Cambridge University Press
Year: 2008
DOI identifier: 10.1017/S0022112008003558
OAI identifier:

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