Condensate turbulence in two dimensions
Abstract
The nonlinear Schrödinger equation with repulsion (also called the GrossPitaevsky equation) is solved numerically with damping at small scales and pumping at intermediate scales and without any largescale damping. Inverse cascade creating a wave condensate is studied. At moderate pumping, it is shown that the evolution comprises three stages: (i) short period (few nonlinear times) of setting the distribution of fluctuations with the flux of waves towards large scales, (ii) long intermediate period of selfsaturated condensation with the rate of condensate growth being inversely proportional to the condensate amplitude, the number of waves growing as √t, the total energy linearly increasing with time and the level of overcondensate fluctuations going down as 1/√t, and (iii) final stage with a constant level of overcondensate fluctuations and with the condensate linearly growing with time. Most of the waves are in the condensate. The flatness initially increases and then goes down as the overcondensate fluctuations are suppressed. At the final stage, the second structure function <\ψ_{1}ψ_{2}\^{2}>~lnr_{12} while the fourth and sixth functions are close to their Gaussian values. Spontaneous symmetry breaking is observed: turbulence is much more anisotropic at large scales than at pumping scales. Another scenario may take place for a very strong pumping: the condensate contains 2530 % of the total number of waves, the harmonics with small wave numbers grow as well.
 Publication:

Physical Review E
 Pub Date:
 November 1996
 DOI:
 10.1103/PhysRevE.54.5095
 Bibcode:
 1996PhRvE..54.5095D
 Keywords:

 47.10.+g;
 47.27.Gs;
 Isotropic turbulence;
 homogeneous turbulence