Ergodicity and spectral cascades in point vortex flows on the sphere
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We present results for the equilibrium statistics and dynamic evolution of moderately large [n=O(102-103)] 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 k-1 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.
Dritschel , D G , Lucia , M & Poje , A C 2015 , ' Ergodicity and spectral cascades in point vortex flows on the sphere ' Physical Review. E, Statistical, nonlinear, and soft matter physics , vol 91 , no. 6 , 063014 . DOI: 10.1103/PhysRevE.91.063014
Physical Review. E, Statistical, nonlinear, and soft matter physics
© 2015 American Physical Society. Reproduced in accordance with the Publisher's author reuse policy. Originally published by Physical Review E: http://dx.doi.org/10.1103/PhysRevE.91.063014
A.C.P. was supported under DOD (MURI) Grant No. N000141110087 ONR. The computations were supported by the CUNY HPCC under NSF Grants No. CNS-0855217 and No. CNS-0958379.
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