Evolution of a superfluid vortex filament tangle driven by the Gross-Pitaevskii equation

The development and decay of a turbulent vortex tangle driven by the Gross-Pitaevskii equation is studied. Using a recently developed accurate and robust tracking algorithm, all quantized vortices are extracted from the fields. The Vinen's decay law for the total vortex length with a coefficient that is in quantitative agreement with the values measured in helium II is observed. The topology of the tangle is then investigated showing that linked rings may appear during the evolution. The tracking also allows for determining the statistics of small-scale quantities of vortex lines, exhibiting large fluctuations of curvature and torsion. Finally, the temporal evolution of the Kelvin wave spectrum is obtained providing evidence of the development of a weak-wave turbulence cascade.

Figure 1

Left: Isosurfaces of density field at different times. Low-density regions that correspond to vortex lines are plotted in red, while density fluctuations (sound) are rendered in light blue. Right: Corresponding tracked vortices. Different colors correspond to different vortices. Snapshots taken at $t=0$ [(a) and (b)], $t=12$ [(c) and (d)], and $t=105$ [(e) and (f)]. Resolution ${256}^3$.

Figure 2

(a) Temporal evolution of the vortex line density: tracked data are plotted using blue circles, and volume estimation in a solid red line. (b) Long time decay of $\mathrm{\Delta }\mathcal{L}$ (see text), together with Vinen's prediction $\mathrm{\Delta }{\mathcal{L}}_{\mathrm{Vinen}}={\left[{\chi }_2\frac{\mathrm{\Gamma }}{2\pi }(t-t_0)\right]}^{-1}$, setting $\chi =0.65$ (solid black line). Resolution ${256}^3$.

Figure 3

(a) Temporal evolution of the (normalized) total number of rings and crossing number. At $t=0,N_{\mathrm{rings}}\left(0\right)=128$ and $\overline{C}\left(0\right)=758$. (b) Temporal evolution of the total linking $\text{Lk}$, writhe $\text{Wr}$ and center-line helicity $H_c/{\mathrm{\Gamma }}^2$. (c) Visualization of two linked rings at $t=21$. (d) Visualization of a ring with high $\text{Wr}$ at $t=24.5$. Resolution ${256}^3$.

Figure 4

(a) PDFs of curvature $\kappa$ normalized by their respective mean values $\langle \kappa \rangle$ at different times [same legend as (b)]. The inset displays the temporal evolution of the mean and rms values of $\kappa$. (b) PDFs of torsion $\tau$ at different times. The inset emphasizes their ${\tau }^{-3}$ power-law tail. Resolution ${256}^3$.

Figure 5

(a) Temporal evolution of KW spectra (averaged over the 50 longest rings). Resolution ${256}^3$. (b) KW spectrum at $t\sim 5$ (averaged over the 50 longest rings) for run at resolution ${512}^3$. The dashed line displays the $k^{-11/3}$ scaling. The inset displays the respective $k^{11/3}$ (solid blue) and $k^{17/5}$ (dashed red) compensated spectra.