Dual cascade and dissipation mechanisms in helical quantum turbulence

While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulent flows, in quantum turbulence its role is not fully understood. We present numerical simulations of the free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at high spatial resolution. The evolution has remarkable similarities with classical flows, which go as far as displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatiotemporal analysis indicates that both quantities are dissipated at small scales through nonlinear excitation of Kelvin waves and the subsequent emission of phonons. At the onset of the decay, the resulting turbulent flow displays polarized large scale structures and unpolarized patches of quiescence reminiscent of those observed in simulations of classical turbulence at very large Reynolds numbers.

Figure 1

Evolution of the incompressible energy $E_k^i$ and of the regularized helicity $H_r$ in the ${1024}^3$ and ${2048}^3$ GPE runs, and in Navier-Stokes. Note the early “inviscid” phase in which quantities are approximately constant. The solid black line shows the total vortex length in the ${2048}^3$ GPE run. Inset: $H_r\left(t\right)$ (dashed blue line) and the nonregularized helicity $H\left(t\right)$ (solid red line) in the ${2048}^3$ GPE run.

Figure 2

Spectrum of the (a) incompressible kinetic energy, and (b) helicity in the ${2048}^3$ GPE run. At large scales both follow a scaling compatible with a classical dual cascade (thick dashed lines). At scales smaller than the intervortex scale ($k_{\ell }\approx 80$) a second range compatible with a Kelvin wave cascade is observed in $E_k^i$ (thick dash-dotted line). The helicity spectrum broadens in time indicating a direct transfer.

Figure 3

Compensated helicity spectra in the ${2048}^3$ GPE run. The spectrum is compatible with $\eta {\epsilon }^{-1/3}{k}^{-5/3}$ scaling. The time-averaged spectrum is also shown.

Figure 4

Spatiotemporal power spectra for the ${512}^3$ GPE run between $t=0$ and 2 for (a) the mass density $\rho$, and zooms between $k=0$ and 100 for (b) the incompressible and (c) compressible velocity. Same for late times ($t\in [8,10]$) are shown in (d), (e), and (f). The solid (green) curve is the dispersion relation of Kelvin waves, while the dotted (white) curve corresponds to sound waves.

Figure 5

Three-dimensional rendering of vortex lines at the onset of the decay in the ${2048}^4$ GPE run of (a) a slice of the full box, and successive zooms in (b) and (c) into the regions indicated by the (red) rectangles. (d) Sketch of the transfer of helicity from writhe to twist in a bundle of vortices, and for an individual vortex.

Figure 6

Correlation function of $\rho$ in the ${2048}^3$ GPE run. At $t=0$ it decays rapidly in units of the healing length $\xi$, but then quickly develops long-range correlations. Inset: Ratio of eigenvalues ${\lambda }_{\max }/{\lambda }_{\min }$ as a function of $2\pi /d$, with $d$ the size of the box used for the average (blue triangles: regions with structures; red triangles: regions of quiescence).