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Numerical and analytical study of an asymptotic equation for deformation of vortex lattices

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Topological Dynamics in the Physical and Biological Sciences

It is known that when two-dimensional flows are subject to a suitable background rotation, formation of vortex lattices are observed. We can make use of critical points of the vorticity field and their connectivity (so-called, surface networks) to study reconnection of vorticity contours in 2D turbulence. In this talk we begin by noting how this method applies to the study of formation of vortex lattices.

We then study a coarse-grained, asymptotic equation which describes deformation vortex lattices derived by Smirnov and Chukbar, Sov. Phys. JETP vol 93, 126-135(2001). It reads $phi_t=phi_{xx} phi_{yy}-phi_{xy}^2,$ where $phi$ denotes displacement of vortex locations. This equation is particularly valid for geostrophic Bessel vortices with a screened interaction.

Numerical results are reported which indicate an ill-posed nature of the time evolution. Self-similar blow-up solutions were already given by those authors, which have an infinite total energy. We ask whether finite-time blow-up can take place developing from smooth initial data with a finite energy. More general self-similar blow-up solutions are sought, but all are found to have infinite total energy. Finally, remarks are made in connection with the Tkachenko-type lattice.

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

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