Energy cascade with small-scale thermalization, counterflow metastability, and anomalous velocity of vortex rings in Fourier-truncated Gross-Pitaevskii equation.

The statistical equilibria of the (conservative) dynamics of the Gross-Pitaevskii equation (GPE) with a finite range of spatial Fourier modes are characterized using a new algorithm, based on a stochastically forced Ginzburg-Landau equation (SGLE), that directly generates grand-canonical distributions. The SGLE-generated distributions are validated against finite-temperature GPE-thermalized states and exact (low-temperature) results obtained by steepest descent on the (grand-canonical) partition function. A standard finite-temperature second-order λ transition is exhibited. A mechanism of GPE thermalization through a direct cascade of energy is found using initial conditions with mass and energy distributed at large scales. A long transient with partial thermalization at small scales is observed before the system reaches equilibrium. Vortices are shown to disappear as a prelude to final thermalization and their annihilation is related to the contraction of vortex rings due to mutual friction. Increasing the amount of dispersion at the truncation wave number is shown to slow thermalization and vortex annihilation. A bottleneck that produces spontaneous effective self-truncation with partial thermalization is characterized in the limit of large dispersive effects. Metastable counterflow states, with nonzero values of momentum, are generated using the SGLE algorithm. Spontaneous nucleation of the vortex ring is observed and the corresponding Arrhenius law is characterized. Dynamical counterflow effects on vortex evolution are investigated using two exact solutions of the GPE: traveling vortex rings and a motionless crystal-like lattice of vortex lines. Longitudinal effects are produced and measured on the crystal lattice. A dilatation of vortex rings is obtained for counterflows larger than their translational velocity. The vortex ring translational velocity has a dependence on temperature that is an order of magnitude above that of the crystal lattice, an effect that is related to the presence of finite-amplitude Kelvin waves. This anomalous vortex ring velocity is quantitatively reproduced by assuming equipartition of energy of the Kelvin waves. Orders of magnitude are given for the predicted effects in weakly interacting Bose-Einstein condensates and superfluid ^{4}He.