Energy spectrum of two-dimensional acoustic turbulence

TL;DR

This paper derives an exact analytical solution for the energy spectrum of 2D acoustic turbulence with a nearly linear dispersion law, applicable to systems like 2D Bose-Einstein condensates, and confirms it through numerical simulations.

Contribution

It provides the first exact analytical solution for the 2D acoustic turbulence spectrum, addressing a long-standing problem due to the singularity in 2D.

Findings

Derived the unique constant-flux power-law spectrum for 2D acoustic turbulence

Validated the spectrum through direct numerical simulations of the Gross-Pitaevskii equation

Established the spectrum's applicability to 2D Bose-Einstein condensates

Abstract

We report an exact unique constant-flux power-law analytical solution of the wave kinetic equation for the turbulent energy spectrum, $E(k)=C_1 \sqrt{\varepsilon\, a c_{\rm s} }/k$, of acoustic waves in 2D with almost linear dispersion law, $\omega_k = c_{\rm s} k[1+(ak)^2]$, $ ak \ll 1$. Here $\varepsilon$ is the energy flux over scales, and $C_1$ is the universal constant which was found analytically. Our theory describes, for example, acoustic turbulence in 2D Bose-Einstein condensates (BECs). The corresponding 3D counterpart of turbulent acoustic spectrum was found over half a century ago, however, due to the singularity in 2D, no solution has been obtained until now. We show the spectrum $E(k)$ is realizable in direct numerical simulations of forced-dissipated Gross-Pitaevskii equation in the presence of strong condensate.