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Dynamics of elliptical vortices with continuous profiles

Ling Xu*

Robert Krasny†

  • Department of Mathematics and Statistics, North Carolina Agricultural and Technical State University, Greensboro, North Carolina 27411, USA

  • Department of Mathematics, University of Michigan, Ann Arbor 48109, Michigan, USA

  • *lxu@ncat.edu
  • †krasny@umich.edu
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Phys. Rev. Fluids 8, 024702 – Published 21 February, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.024702

Abstract

This work examines the dynamics of elliptical vortices in two-dimensional ideal fluid using an adaptively refined and remeshed vortex method. Four examples are considered comprising two compact vortices denoted by MMZ (smooth) and POLY (nonsmooth), and two noncompact vortices denoted by Gaussian and smooth Kirchhoff. The vortices all have the same maximum vorticity and 2:1 initial aspect ratio, but unlike the top-hat Kirchhoff vortex, they have continuous profiles with different degrees of regularity. In each case the phase portrait of the vortex in a corotating frame has two hyperbolic points, and the separatrix divides space into four regions, a center containing the vortex core, two crescent-shaped lobes next to the core, and the exterior. As the vortices start to rotate, two spiral filaments emerge and form a halo of low-amplitude vorticity around the core; this is attributed to vorticity advection along the unstable manifolds of the hyperbolic points. In the case of the Gaussian vortex the core rapidly axisymmetrizes, but later it starts to oscillate and two small lobes enclosing weak vortical fluid form within the halo; this is attributed to a resonance stemming from the core oscillation. In the case of the MMZ, POLY, and smooth Kirchhoff vortices, the core remains elliptical for longer time, and the filaments entrain weak vortical fluid into two large lobes which together with the core form a nonaxisymmetric tripole; afterwards, however, the lobes repeatedly detrain some of their fluid into the halo; the repeated detrainment is attributed to a heteroclinic tangle near the hyperbolic points. While prior work suggested that elliptical vortices could evolve to become either an axisymmetric monopole or a nonaxisymmetric tripole, the current results suggest they may oscillate between these states.

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Outline

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Phys. Rev. Fluids 8, 024702– Published 21 February, 2023
  • Received 7 July 2022
  • Accepted 2 February 2023
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