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An accurate and efficient method to compute steady vortices without symmetry

Tuesday 24th July 2012 - 12:10 to 12:30
INI Seminar Room 1
Session Chair: 
Edriss Titi
When considering steady solutions of the Euler equations, it is often of interest to find isovortical flows, that is, solutions that can be obtained from rearrangements of a given vorticity distribution. Since inviscid transitions between such flows are, in principle, possible, these solutions may act as attractors in the unsteady dynamics (e.g. Dritschel 1986, 1995; Flierl & Morrison 2011). The computation of such steady vortex flows still presents some challenges. Existing Newton iteration methods become inefficient as the vortices develop fine-scale features; in addition, these methods do not, in general, find solutions from isovortical rearrangements. On the other hand, available relaxation approaches are more affordable, but their convergence is not guaranteed. In this work, we consider flows that may be approximated by a collection of uniform vortices, and overcome the limitations outlined above by using a discretization, based on an inverse-velocity mapping, which radically increases the efficiency of Newton iteration methods. In addition, we introduce a procedure to enforce the isovortical constraint in the solution method. We illustrate our methodology by exploring the solution structure of a wide range of unbounded flows. We uncover several families of lower-symmetry vortices. While asymmetric point vortex flows have been found by Aref & Vainchtein (1998), it appears that this is the first time that nonsingular, asymmetric steady vortices have been computed. In addition, we discover that, as the limiting vortex state for each flow is approached, each family of solutions traces a clockwise spiral in a bifurcation plot consisting of a velocity-impulse diagram. By the recently introduced ‘‘IVI diagram’’ stability approach (Luzzatto-Fegiz & Williamson 2010, 2011), each turn of this spiral is associated with a loss of stability. Such spiral structure is suggested to be a universal feature of steady, uniform-vorticity Euler flows.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons