Velocity Statistics Distinguish Quantum Turbulence from Classical Turbulence

By analyzing trajectories of solid hydrogen tracers, we find that the distributions of velocity in decaying quantum turbulence in superfluid ${}^4\mathrm{He}$ are strongly non-Gaussian with $1/v^3$ power-law tails. These features differ from the near-Gaussian statistics of homogenous and isotropic turbulence of classical fluids. We examine the dynamics of many events of reconnection between quantized vortices and show by simple scaling arguments that they produce the observed power-law tails.

Figure 1

Depiction of one-dimensional topological vortices that reconnect by merging at the moment $t_0$ and then separate [25]; the minimum separation distance is $\delta (t)$, with $\delta (t_0)=0$.

Figure 2

Time-varying pulsed counterflow velocity distributions at $T=1.90\mathrm{K}$ showing (a) $v_x$ and (b) $v_z$ for a portion of a heat pulse of $0.17\mathrm{W}/{\mathrm{cm}}^2$ with the heater turned off at $t=t_{\mathrm{off}}$ [30]. White denotes amplitudes with zero probability.

Figure 3

Local velocity and energy statistics derived from the data in Fig. 2 for all particle trajectories with $t>t_{\mathrm{off}}$ (computed from over $1.1\times {10}^6$ values of velocity). All distributions are scaled to give unit variance using ${\sigma }_{v_x}=0.066$, ${\sigma }_{v_z}=0.074\mathrm{cm}/\mathrm{s}$, or ${\sigma }_E=0.017(\mathrm{cm}/\mathrm{s}{)}^2$. (a) Probability distribution function of $v_x$ (black circles) and $v_z$ (red squares). The solid lines are fits to ${Pr}_v(v_i)=a|v_i-{\overline{v}}_i{|}^{-3}$, where $i$ is either $x$ (black) or $z$ (red) and ${\overline{v}}_i$ is the mean value of $v_i$. For comparison, the dashed (blue) line shows the distribution for classical turbulence in water [32] computed from over ${10}^7$ velocity values. The distribution is scaled using ${\sigma }_v=0.25\mathrm{cm}/\mathrm{s}$ and with a peak value matched to the $v_x$ data. The velocity statistics in water are close-to-Gaussian over five decades in probability. (b) Probability distribution function of $E=(v_x^2+v_z^2)/2$ with a fit of the form $\underset{E}{Pr}(E)=a{E}^{-2}$ shown (in red).

Figure 4

Statistics of the reconnection fits. (a) Four typical forward events. Symbols denote the measured separation $\delta (t)$ of pairs of particles on reconnecting vortices with an example error bar $\sigma =4\mu \mathrm{m}$, while solid lines show fits to (3). (b) Normalized distributions of the amplitude $A$ for 20 300 forward events (black circles) and 19 600 reversed events (red squares). (c) Normalized distributions of the correction amplitude $c$ for the same 20 300 forward events (black circles) and 19 600 reversed events (red squares).