Vortex-Density Fluctuations, Energy Spectra, and Vortical Regions in Superfluid Turbulence

Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.

Figure 1

The time evolution of the total vortex line density (black dashed line) $L$ (${\mathrm{cm}}^{-2}$) and the polarized component $L_{\parallel }$ (${\mathrm{cm}}^{-2}$) (solid red line).

Figure 2

Power spectral density (PSD) (arb. units) of fluctuations of the total vortex line density $L$ (black upper solid line) and of the polarized vortex line density $L_{\parallel }$ (red lower solid line) versus frequency $f$ (${\mathrm{s}}^{-1}$). The dashed (magenta) line shows the $f^{-5/3}$ scaling.

Figure 3

Left: Snapshot of the vortex tangle at $t=7\mathrm{s}$. Vortex lines are locally colored according to the local magnitude of the smoothed vorticity field $\mathit{\omega }$. Light grey regions correspond to high vorticity, dark regions to low vorticity. Right: The same snapshot but split into the locally polarized [top, $\omega ({\mathbf{s}}_i)>1.4{\omega }_{\mathrm{rms}}$] and random components [bottom, $\omega ({\mathbf{s}}_i)<1.4{\omega }_{\mathrm{rms}}$], respectively. The color scale is normalized by the maximum value ${\omega }_{\max }=38.4{\mathrm{s}}^{-1}$, ${\omega }_{\mathrm{rms}}=15.7{\mathrm{s}}^{-1}$.

Figure 4

Energy spectra $E(k)$ (arb. units) versus wave number $k$ (${\mathrm{cm}}^{-1}$) corresponding to Fig. 3. Top: The upper solid line (a) is the energy spectrum corresponding to the flow induced by all vortex lines. Lower curves correspond to the flow induced by vortices with smoothed vorticity below the following thresholds: (b) $\omega <1.7{\omega }_{\mathrm{rms}}$, (c) $1.4{\omega }_{\mathrm{rms}}$, (d) $1.2{\omega }_{\mathrm{rms}}$, and (e) ${\omega }_{\mathrm{rms}}$. Bottom: Energy spectra corresponding to vortex lines with smoothed vorticity, respectively, below (a) and above (b) the threshold $1.4{\omega }_{\mathrm{rms}}$. The dashed lines display the $k^{-1}$ and the $k^{-5/3}$ scalings.