Cascades in nonlocal turbulence

TL;DR

This paper analytically derives flux laws for 2D turbulence in the Gross-Pitaevsky model, confirming non-local wave interactions as the mechanism for constant flux in inverse and direct cascades, with spectra near thermal equilibrium.

Contribution

It provides the first analytical flux laws for nonlocal turbulence in the Gross-Pitaevsky model, validated by numerical simulations.

Findings

Flux laws analogous to Kolmogorov's 4/5-law are derived.

Constant flux in cascades is due to non-local wave interactions.

Spectra are close to logarithmically distorted thermal equilibrium.

Abstract

We consider developed turbulence in the 2D Gross-Pitaevsky model, which describes wide classes of phenomena from atomic and optical physics to condensed matter, fluids and plasma. The well-known difficulty of the problem is that the hypothetical local spectra of both inverse and direct cascades in the weak-turbulence approximation carry fluxes which are either zero or have the wrong sign; such spectra cannot be realized. We analytically derive the exact flux constancy laws (analogs of Kolmogorov's 4/5-law for incompressible fluid turbulence), expressed via the fourth-order moment and valid for any nonlinearity. We confirm the flux laws in direct numerical simulations. We show that a constant flux is realized by non-local wave interaction in both the direct and inverse cascades. Wave spectra (second-order moments) are close to slightly (logarithmically) distorted thermal equilibrium in both cascades.