Critical Velocities for Roton and Super-Flow Quantum Turbulence in Liquid $^4$He

Two different types of transitions of the superfluid $^4$He to quantum turbulence regimes are studied for $1{\rm D}$ geometry in the case when the influence of the normal fluid on superfluid flow is suppressed. It is shown that the roton mechanism of transition to quantum turbulence leads to a critical velocity satisfying the relation $v_c\propto d^{-1/4}$. In the super-flow mechanism, the transition to quantum turbulence arises when the "quantum Reynolds number" is about $10^3$ and the critical velocity depends on channel size $d$ as $v_c\propto d^{-1}$ in agreement with the equations of motion for a superfluid component of the liquid $^4$He being disturbed by small fluctuations of the normal fluid.