Ergodicity and spectral cascades in point vortex flows on the sphere

David G. Dritschel, Marcello Lucia, and Andrew C. Poje
Phys. Rev. E 91, 063014 – Published 29 June 2015

Abstract

We present results for the equilibrium statistics and dynamic evolution of moderately large [n=O(102103)] numbers of interacting point vortices on the sphere under the constraint of zero mean angular momentum. For systems with equal numbers of positive and negative identical circulations, the density of rescaled energies, p(E), converges rapidly with n to a function with a single maximum with maximum entropy. Ensemble-averaged wave-number spectra of the nonsingular velocity field induced by the vortices exhibit the expected k1 behavior at small scales for all energies. Spectra at the largest scales vary continuously with the inverse temperature of the system. For positive temperatures, spectra peak at finite intermediate wave numbers; for negative temperatures, spectra decrease everywhere. Comparisons of time and ensemble averages, over a large range of energies, strongly support ergodicity in the dynamics even for highly atypical initial vortex configurations. Crucially, rapid relaxation of spectra toward the microcanonical average implies that the direction of any spectral cascade process depends only on the relative difference between the initial spectrum and the ensemble mean spectrum at that energy, not on the energy, or temperature, of the system.

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  • Received 9 July 2014
  • Revised 4 May 2015

DOI:https://doi.org/10.1103/PhysRevE.91.063014

©2015 American Physical Society

Authors & Affiliations

David G. Dritschel1, Marcello Lucia2, and Andrew C. Poje2

  • 1Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, United Kingdom
  • 2Graduate Faculty in Physics & Department of Mathematics, City University of New York–CSI, Staten Island, New York 10314, USA

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Issue

Vol. 91, Iss. 6 — June 2015

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