Turbulence in quantum hydrodynamics (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

By numerically solving the Gross-Piteavskii equation with the energy injection and dissipation, we study quantum turbulence comprised of quantized vortices with discrete circulations. We consider two opposite cases of fully developed quantum turbulence with strong energy injections and transition from quantum turbulence with vortices to vortex-free state with weak energy injections. With strong energy injections, the obtained energy spectrum for fully developed quantum turbulence has two scaling regions separated by the scale of the mean inter-vortex distance. In scales larger than the mean inter-vortex distance, the energy spectrum shows the well-known Kolmogorov law with the exponent −5/3. In scales smaller than the mean inter-vortex distance, on the other hands, the exponent of the energy spectrum changes to −7/5 suggesting that quantum turbulence is dominated by the Kelvin-wave cascade process of a single vortex line. With weak energy injections, the vortex-line density monotonically decreases with decreasing the strength of the energy injection and vanishes at the certain value of the energy injection. Close to the transition point, the vortex-line density can be regarded as the order parameter of the transition, showing its power-law behavior with the critical exponent 0.81. This value of the critical exponent is consistent with that for the (3+1)-dimensional directed percolation, which suggests the same underlying physics for two transitions of the directed percolation and quantum turbulence.