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

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The Nature of High Reynolds Number Turbulence

We examine the evolution of a two—dimensional vortex which initially consists of an axisymmetric monopole vortex with a perturbation of azimuthal wavenumber m=2 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. 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.

We consider 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 diagnostics, the appearance of an inflection 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 profiles based on the tanh function.

This talk is part of the Isaac Newton Institute Seminar Series series.

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