Two-Particle Separation in the Point Vortex Gas Model of Superfluid Turbulence

Abstract

We study numerically statistical properties of the random mean square separation, Δ(t)=〈Δ2〉1/2 of fluid particle pairs in the Onsager’s point vortex gas, i.e. in a random, inviscid, two-dimensional flow generated by the motion of quantized point vortices of random polarities. This model is perhaps the simplest, two-dimensional model of superfluid turbulence in helium II at low temperatures when the normal fluid is absent. The existence of the inverse energy cascade through scales in such a system was discovered in the pioneering work by Onsager (Nuovo Cimento Suppl. 6:279, 1949). We found that the mean diffusivity of particle pairs, d〈Δ2〉/dt obeys, to a good degree of accuracy, the Richardson’s diffusion ‘four-third’ law, d〈Δ2〉/dt∼Δ4/3. We also analyzed the details of time evolution of the rms particle separation. At small time the particle separation is ballistic, while at large time, when the motion of individual particles in the considered pair becomes uncorrelated, the mean square separation follows the diffusion law, 〈Δ2〉∼t. For intermediate time we found that the root mean square separation, Δ=〈Δ2〉1/2 is very sensitive to the initial separation, due to the memory of initial separation kept by the particle pair during the initial period of exponential ballistic separation, so that the rms separation never follows the Richardson’s t3/2 scaling law. We characterized the time evolution of the rms particle separation in the vicinity of the inflexion point of the curve Δ(t) by the power law, \(\Delta\sim\gamma t^{\alpha(\Delta_{0})}\) , and found the dependence of α and γ on the initial separation Δ0, thus generalizing the Richardson’s t3/2-law for the considered two-dimensional point vortex gas.