Ergodicity and spectral cascades in point vortex flows on the sphere

TL;DR

This paper investigates the equilibrium and dynamic behavior of point vortex flows on a sphere, demonstrating ergodicity, spectral cascades, and the influence of temperature on large-scale flow structures.

Contribution

It provides new insights into the ergodic nature and spectral properties of point vortex systems on a sphere, including the transition of large-scale spectra with temperature.

Findings

Spectra exhibit a $k^{-1}$ behavior at small scales.

Large-scale spectra transition from positive to negative slope with temperature.

System dynamics are ergodic and relax to microcanonical ensemble measures.

Abstract

We present results for the equilibrium statistics and dynamic evolution of moderately large ($n = {\mathcal{O}}(10^2 - 10^3)$) numbers of interacting point vortices on the unit sphere under the constraint of zero mean angular momentum. We consider a binary gas consisting of equal numbers of vortices with positive and negative circulations. When the circulations are chosen to be proportional to $1/\sqrt{n}$, the energy probability distribution function, $p(E)$, converges rapidly with $n$ to a function that has a single maximum, corresponding to a maximum in entropy. Ensemble-averaged wavenumber spectra of the nonsingular velocity field induced by the vortices exhibit the expected $k^{-1}$ behavior at small scales for all energies. The spectra at the largest scales vary continuously with the inverse temperature $\beta$ of the system and show a transition from positively sloped to negatively sloped as $\beta$ becomes negative. The dynamics are ergodic and, regardless of the initial configuration of the vortices, statistical measures simply relax towards microcanonical ensemble measures at all observed energies. As such, the direction of any cascade process measured by the evolution of the kinetic energy spectrum depends only on the relative differences between the initial spectrum and the ensemble mean spectrum at that energy; not on the energy, or temperature, of the system.