Inverse Energy Cascade in Forced Two-Dimensional Quantum Turbulence

We demonstrate an inverse energy cascade in a minimal model of forced 2D quantum vortex turbulence. We simulate the Gross-Pitaevskii equation for a moving superfluid subject to forcing by a stationary grid of obstacle potentials, and damping by a stationary thermal cloud. The forcing injects large amounts of vortex energy into the system at the scale of a few healing lengths. A regime of forcing and damping is identified where vortex energy is efficiently transported to large length scales via an inverse energy cascade associated with the growth of clusters of same-circulation vortices, a Kolmogorov scaling law in the kinetic energy spectrum over a substantial inertial range, and spectral condensation of kinetic energy at the scale of the system size. Our results provide clear evidence that the inverse energy cascade phenomenon, previously observed in a diverse range of classical systems, can also occur in quantum fluids.

Figure 1

Vortex configurations in a rightward-flowing superfluid stirred by a stationary grid of obstacle potentials, located at the left. Dots indicate the locations of positive (counterclockwise circulation) and negative (clockwise circulation) vortices at time $t=700\xi /c$. The vortices have been sorted into dipoles, free vortices, and clusters (see legend) using the cluster-finding algorithm (see text). Vortices in dipoles at the smallest length scales are not resolvable on the scale shown, and appear as isolated light-gray (green) dots. Streamlines (gray lines) give a visualization of the vortex-only velocity field by constructing the velocity field of an identical configuration of point vortices. The two right-hand panels only show half of the computational domain.

Figure 2

Spatiotemporal statistics of vortex clustering averaged over eight trajectories. For four values of $\gamma$ we show the mean charge ${\kappa }_c$ and radius $r_c$ of clusters identified by the cluster-finding algorithm and the 2nd and 4th order nearest-neighbor correlation functions $C_2$ and $C_4$ defined in Ref. 22. The left column shows clustering as a function of downstream distance from the grid $d$ at time $t=700\xi /c$, obtained by counting only clusters with center in (${\kappa }_c$ and $r_c$), or vortices lying in ($C_2$ and $C_4$), a box of dimensions ($64\xi$, $512\xi$) centered at ($d-L/2$, 0). The right column shows clustering as a function of time $t$. For ${\kappa }_c$ and $r_c$, we use a box centered at ($vt-32\xi$, 0) and hence comoving with the flow. Insets show the results for a delayed box centered at ($vt-96\xi$, 0). For $C_2$ and $C_4$, we show the average value for the entire system. Dashed horizontal lines at ${\kappa }_c=2.9$ and $C_2=C_4=0.5$ indicate the expected value of these quantities for a random arrangement of vortices.

Figure 3

Incompressible kinetic energy spectra at $t=550\xi /c$ for damping parameters (a) $\gamma =0.0009$, (b) $\gamma =0.003$, (c) $\gamma =0.009$ (circles), averaged over eight trajectories. Lines show theoretical predictions for $k^{-3}$ [Eq. (3)] and Kolmogorov $k^{-5/3}$ [Eq. (4)] spectra (see legend) in ideal 2D QT, with no fitted parameters. The Kolmogorov spectrum is shown using both the clustered vortex number $N_c$ (constant $C^{*}$), and the total vortex number $N$ (constant $C$) for comparison. We also show wave numbers associated with the core size (${\xi }^{-1}$), the forcing scale ($k_f$), and the mean diameter of the largest cluster ($k_c$). In an ideal IEC, $k_c$ approximately captures the lowest wave number of the Kolmogorov spectral region [21].

Figure 4

Signatures of spectral condensation. (a) Incompressible energy accumulated at the system scale ($E_0^i$, see text) and associated with the condensate. (b) Energy flux ${ϵ}_0^i=-d{E}_0^i/dt$ from the condensate to the rest of the system. (c) $E_0^i$ as a fraction of the total incompressible kinetic energy, $E_{\mathrm{tot}}^i={\int }_{\Delta k}^{\infty }{E}^i(k)dk$.