Statistics of Point Vortex Turbulence in Non-neutral Flows and in Flows with Translational and Rotational Symmetries

Abstract

A theory [Esler and Ashbee, J. Fluid Mech., 779, 275, 2015] describing the statistics of N freely-evolving point vortices in a bounded twodimensional domain is extended. First, the case of a non-neutral vortex gas is addressed, and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non-central chi-squared random variables, the details of which depend only on the shape of the domain. Equations for the equilibrium energy spectrum and other statistical quantities follow, the validity of which are verified against direct numerical simulations of the equations of motion. Second, domains with additional conserved quantities associated with a symmetry (e.g. circle, periodic channel) are investigated, and it is shown that the treatment of the non-neutral case can be modified to account for the additional constraint.

Chiral edge modes in Helmholtz-Onsager vortex systems

“…where the first sum contains N (N − 1) terms, the second sum contains N terms, and the strength of each vortex λ has been set to 1 as before. Previous work [36] has identified the limiting distribution of Ĥ in bounded domains. For the identical vortex systems considered here, this limiting distribution is Gaussian.…”

“…The second term in ( 4) is a sum of only N independent random variables and thus converges to 0. A central limit-type theorem [37] for the sequence of N (N − 1) random variables G(Z a , Z b ) shows that √ N ( H − μ) is asymptotically normal, where μ and the variance σ 2 are given by [36,37,39]…”

“…In other cases, the system is known to be nonergodic [56,57]. Numerical studies indicate evidence for ergodicity at intermediate energies for mixed vortex systems (with λ a = ±1) in bounded [26,36] and periodic [36] domains. On the sphere, the ergodic hypothesis appears to hold for a range of energies [58].…”

Chiral Edge Modes in Helmholtz-Onsager Vortex Systems

“…where the first sum contains N (N − 1) terms, the second sum contains N terms and the strength of each vortex λ has been set to 1 as before. Previous work [30] has identified the limiting distribution of Ĥ in bounded domains. For the identical vortex systems considered here, this limiting distribution is Gaussian.…”

“…The second term in ( 4) is a sum of only N independent random variables, and thus converges to 0. A central limit type theorem [31] for the sequence of N (N − 1) random variables G(Z a , Z b ) shows that √ N ( H − µ) is asymptotically normal, where µ and the variance σ 2 are given by [30,31,33]…”