Enstrophy Cascade in Decaying Two-Dimensional Quantum Turbulence

We report evidence for an enstrophy cascade in large-scale point-vortex simulations of decaying two-dimensional quantum turbulence. Devising a method to generate quantum vortex configurations with kinetic energy narrowly localized near a single length scale, the dynamics are found to be well characterized by a superfluid Reynolds number ${\mathrm{Re}}_{\mathrm{s}}$ that depends only on the number of vortices and the initial kinetic energy scale. Under free evolution the vortices exhibit features of a classical enstrophy cascade, including a $k^{-3}$ power-law kinetic energy spectrum, and constant enstrophy flux associated with inertial transport to small scales. Clear signatures of the cascade emerge for $N\gtrsim 500$ vortices. Simulating up to very large Reynolds numbers ($N=32768$ vortices), additional features of the classical theory are observed: the Kraichnan-Batchelor constant is found to converge to $C^{\prime }\approx 1.6$, and the width of the $k^{-3}$ range scales as ${\mathrm{Re}}_{\mathrm{s}}^{1/2}$.

Figure 1

Kinetic energy spectra and fluxes (in units of ${\mathrm{\Gamma }}^2/4\pi L^2$) and vorticity distributions for $N=16$, 384 at (a) $t\approx 0.25\tau$, (b) $t\approx 1.85\tau$, and (c) $t\approx 4\tau$ and (d) $t\approx 20\tau$. In (a), at high $k$, $E_{\mathrm{int}}(k)$ is comparatively small and oscillates about zero, causing the broken data line on the log-log scale. Inset shows the full spectrum $E_{\text{self}}(k)+E_{\mathrm{int}}(k)$ at $t=0$. Spectra and fluxes are averaged over 4 runs and 9 time samples with time-spacing $\delta t\approx \tau /20$. Note that only the interaction term $E_{\mathrm{int}}(k)$ is shown, as the self-energy term only adds a configuration independent $N/k$ scaling. In (d) dash-dot lines are guides to the eye proportional to $k^{-3.2}$ and $k^{-2.2}$.

Figure 2

Compensated kinetic energy spectra for a range of $N$, averaged over ensemble and a time window $\sim 0.5\tau$. For clarity the spectra are vertically shifted by increasing powers of 4. The horizontal dashed lines shows the value $C^{\prime }=1.6$ (also vertically shifted for comparison at different $N$). Dash-dot lines indicate the intervortex distance wave number $k_{\ell }$ at different $N$ (see text). In the legend the bracketed superscripts indicate the number of independent realizations used in the ensemble average.