Classical and quantum regimes of the superfluid turbulence

We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter $q$ which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with $q^{-1}$ playing the same role as the Reynolds number ${\rm Re}=UR/\nu$ in classical hydrodynamics. It marks the transition between the «laminar» and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when ${\rm Re}\gg 1$ in classical hydrodynamics, and $q\ll 1$ in the superfluid hydrodynamics. Another parameter of the superfluid turbulence is the superfluid Reynolds number ${\rm Re}_s=UR/\kappa$, which contains the circulation quantum $\kappa$ characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. The phase diagram of the dynamical vortex states is suggested.