Kelvin waves and the decay of quantum superfluid turbulence

We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid $^4$He at low and ultra-low temperatures, $0\leqslant T \leqslant 1.1\,$K. Using high resolution vortex filament simulations with full Biot-Savart vortex dynamics, we show that for ultra-low temperatures $T\lesssim 0.5 \,$K, when the mutual friction parameters $\alpha\simeq \alpha' < 10^{-5}$, the vortex reconnections excite Kelvin waves with wave lengths $\lambda$ of the order of the inter-vortex distance $\ell$. These excitations cascade down to the resolution scale $\Delta\xi$ which in our simulations is of the order $\Delta \xi\sim \ell/100$. At this scale the Kelvin waves are numerically damped by a line-smoothing procedure, that is supposed to mimic the dissipation of Kelvin waves by phonon and roton emission at the scale of the vortex core. We show that the Kelvin waves cascade is statistically important: the shortest available Kelvin waves at the end of the cascade determine the mean vortex line curvature $S$, giving $S \gtrsim 30 /\ell$ and play major role in the tangle decay at ultra-low temperatures below $0.6\,$K. The found dependence of $\ell S$ on the resolution scale $\Delta \xi$ agrees with the L'vov-Nazarenko energy spectrum of weakly-interacting Kelvin waves, $E\Sb{LN}\propto k^{-5/3}$ rather than the spectrum $E\Sb{LN}\propto k^{-1}$, suggested by Vinen for turbulence of Kelvin waves with large amplitudes. We also show that already at $T=0.8\,$K, when $\alpha$ and $\alpha'$ are still very low, $\alpha\simeq \alpha'<10^{-3}$, the Kelvin wave cascade is fully damped, vortex lines are very smooth, $S \simeq 2 /\ell$ and the tangle decay is predominantly caused by the mutual friction.