Sustained turbulence in the three-dimensional Gross-Pitaevskii model

TL;DR

This paper investigates turbulence in the 3D Gross-Pitaevskii model, revealing how different dissipation mechanisms affect energy spectra and observing a transition to weak turbulence with a condensate formation.

Contribution

It provides new insights into how inverse cascade attenuation influences turbulence spectra and characterizes the transition to weak turbulence in the Gross-Pitaevskii system.

Findings

Inverse cascade attenuation determines spectral exponent

Weak wave turbulence prediction matches observed $k^{-1}$ spectrum

Condensate formation leads to a transition to weak turbulence with $k^{-3/2}$ spectrum

Abstract

We study the 3D forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form $k^{-\alpha}$. Our numerical results show that the exponent $\alpha$ strongly depends on how the inverse particle cascade is attenuated at $k$'s lower than the forcing wave number. If the inverse cascade is arrested by a friction at low $k$'s, we observe an exponent which is in good agreement with the weak wave turbulence prediction $k^{-1}$. For a hypo-viscosity, a $k^{-2}$ spectrum is observed which we explain using a critical balance argument. In simulations without any low-$k$ dissipation, a condensate at $k=0$ is growing and the system goes through a strongly-turbulent transition from a four-wave to a three-wave weak turbulence acoustic regime with $k^{-3/2}$ Zakharov-Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov $k^{-5/3}$, but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.