Sustained turbulence in the three-dimensional Gross–Pitaevskii model

We study the three-dimensional 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 − α . Our numerical results show that the exponent α 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 4-wave to a 3-wave weak turbulence acoustic regime with evidence of 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.