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Global Bifurcation for Corotating and Counter-Rotating Vortex Pairs.

Claudia García1,2, Susanna V Haziot3

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This study extends local vortex pair solutions to a global scale, revealing that fluid velocity diminishes or vortex patches intersect as solutions evolve. This advances understanding of fluid dynamics and vortex behavior.

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Area of Science:

  • Fluid Dynamics
  • Mathematical Physics
  • Nonlinear Analysis

Background:

  • Local existence of corotating and counter-rotating vortex pairs was previously established using desingularization of point vortices.
  • Existing local analysis relies on linear equations at trivial solutions, limiting understanding of complex vortex dynamics.

Purpose of the Study:

  • To construct a global continuation of previously identified local vortex pair solutions.
  • To analyze solutions that are significant perturbations beyond trivial cases.
  • To investigate the behavior of vortex pairs in a global context.

Main Methods:

  • Adaptation of the analytic global bifurcation theorem by Buffoni and Toland.
  • Inclusion of singularity handling at the bifurcation point within the global analysis.
  • Topological analysis of nonlinear properties of the fluid system.

Main Results:

  • Global continuation of local vortex pair solutions was successfully constructed for both corotating and counter-rotating pairs.
  • Along the global solution curve, a critical transition occurs: either the angular fluid velocity becomes zero.
  • Alternatively, the two vortex patches undergo self-intersection as the solution evolves globally.

Conclusions:

  • The study provides a comprehensive global analysis of vortex pair solutions, extending prior local findings.
  • The identified conditions (vanishing angular velocity or self-intersection) represent fundamental behaviors of these vortex structures.
  • This work deepens the understanding of nonlinear phenomena in fluid dynamics through advanced mathematical techniques.