Quasiclassical and ultraquantum decay of superfluid turbulence

We address a question which, after a decade-long discussion, still remains open: what is the nature of the ultraquantum regime of decay of quantum turbulence? The model developed in this work reproduces both the ultraquantum and the quasiclassical decay regimes and explains their hydrodynamical natures. In the case where turbulence is generated by forcing at some intermediate length scale, e.g., by the beam of vortex rings in the experiment of Walmsley and Golov [Phys. Rev. Lett. 100, 245301 (2008)], we explain the mechanisms of generation of both ultraquantum and quasiclassical regimes. We also find that the anisotropy of the beam is important for generating the large-scale motion associated with the quasiclassical regime.

Figure 1

Vortex line density $L$ (${\mathrm{cm}}^{-2}$) vs time $t$ (s) corresponding to small (left) and large (right) times of vortex ring injection. Fitting the decay as $L\sim t^{-\alpha }$, we obtain $\alpha =1.07$ and 1.48, respectively. The dashed lines show the ultraquantum $L\sim t^{-1}$ (left) and quasiclassical $L\sim t^{-3/2}$ (right) behaviors.

Figure 2

Probability distribution functions of the vortex line curvature $C$ (${\mathrm{cm}}^{-1}$). Solid line, ultraquantum decay at $t=0.08\mathrm{s}$; dashed line, quasiclassical decay at $t=0.7\mathrm{s}$.

Figure 3

Energy spectrum (arbitrary units) vs wave number $k$ (${\mathrm{cm}}^{-1}$) corresponding to the quasiclassical decay (left) at $t=0.1\mathrm{s}$ (solid line), $t=1.1\mathrm{s}$ (dot-dashed line), and $t=3.4\mathrm{s}$(dashed line); and to the ultraquantum decay (right) at $t=0.07\mathrm{s}$ (solid line), $t=0.12\mathrm{s}$ (dashed line), and $t=0.6\mathrm{s}$ (dot-dashed line). In the former, note the formation and decay of the Kolmogorov spectrum (indicated by the straight dashed line).

Figure 4

Head-on reconnection of two vortex rings of similar size (bottom left) tends to produce two vortex loops of similar size (bottom right), whereas the reconnection of two rings traveling in the same direction (top left) produces one vortex loop which is much smaller and one which is much bigger (top right).

Figure 5

Probability distribution functions (PDFs) [$\mathcal{F}\left(R\right)$] of loop sizes $R$ ($\mathrm{cm}$). Dashed line: initial PDF. Gray line: resulting PDF for isotropic initial condition: no large loops are created by the vortex reconnections. Black line: resulting PDF for anisotropic initial condition: note the generation of large loops.