Theoretical analysis of quantum turbulence using the Onsager ideal turbulence theory

We investigate three-dimensional quantum turbulence as described by the Gross-Pitaevskii model using the analytical method exploited in the Onsager "ideal turbulence" theory. We derive the scale independence of the scale-to-scale kinetic energy flux and establish a double-cascade scenario: At scales much larger than the mean intervortex ℓ_{i}, the Richardson cascade becomes dominant, whereas at scales much smaller than ℓ_{i}, another type of cascade is induced by quantum stress. We then evaluate the corresponding velocity power spectrum using a phenomenological argument. The relation between this cascade, which we call quantum stress cascade, and the Kelvin-wave cascade is also discussed.