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(-alpha). Our numerical results show that the exponent a strongly depends on how the inverse particle cascade is attenuated at ks lower than the forcing wave-number. If the inverse cascade is arrested by a friction at low ks, 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. (C) 2011 Elsevier B.V. All rights reserved.