Local and nonlocal dynamics in superfluid turbulence

In turbulent superfluid He II, the quantized vortex lines interact via the classical Biot-Savart law to form a complicated vortex tangle. We show that vortex tangles with the same vortex line density will have different energy spectra, depending on the normal fluid which feeds energy into the superfluid component, and identify the spectral signature of two forms of superfluid turbulence: Kolmogorov tangles and Vinen tangles. By decomposing the superfluid velocity field into local and nonlocal contributions, we find that in Vinen tangles the motion of vortex lines depends mainly on the local curvature, whereas in Kolmogorov tangles the long-range vortex interaction is dominant and leads to the formation of clustering of lines, in analogy to the “worms” of ordinary turbulence.

Figure 1

Magnitude of the driving normal fluid velocity field, $|{\mathbf{v}}_n^{\mathrm{ext}}|$, plotted on the $xy$ plane at $z=0$ corresponding to model 2 (synthetic normal flow turbulence, left) and model 3 (frozen Navier-Stokes turbulence, right). The velocity scales (cm/s) are shown at right of each panel. Note the more localized, more intense regions of velocity which are present in model 3.

Figure 2

Evolution of the vortex line density $L\left({\text{cm}}^{-2}\right)$ vs time $t$ (s) for model 1 (red [gray] line, uniform normal flow), model 2 (black line, synthetic normal fluid turbulence), and model 3 (dashed blue [gray] line, frozen Navier-Stokes turbulence). The inset displays the oscillations of $L$ vs $t$ in more detail. Parameters: temperature $T=1.9\mathrm{K}$, $V_n=1\mathrm{cm}/\mathrm{s}$ (for model 1), $\mathrm{Re}=79.44$ (for model 2), and $\mathrm{Re}=3025$ (for model 3).

Figure 3

Snapshot of the vortex tangle for model 1 (uniform normal fluid, left), model 2 (synthetic turbulence, middle), and model 3 (frozen Navier-Stokes turbulence, right) at time $t=20\mathrm{s}$ (parameters as in Fig. 2).

Figure 4

Average curvature $C$ (${\mathrm{cm}}^{-1}$) vs time $t$ ($\mathrm{s}$) for model 1 (uniform normal flow, red [gray] line), model 2 (synthetic turbulence, black line), and model 3 (frozen Navier-Stokes turbulence, dashed blue [gray] line). Parameters as in Fig. 2.

Figure 5

Left: Probability density function of the curvature, $\mathrm{PDF}\left(C\right)$, vs curvature, $C$ (${\mathrm{cm}}^{-1}$), corresponding to model 1 (uniform normal flow, red [gray] line), model 2 (synthetic turbulence, black line), and model 3 (frozen Navier-Stokes turbulence, dashed blue [gray] line). Right: the same data plotted on a log-log scale, where the matching slopes on the plot illustrate that we have the same Kelvin waves in all three models (because they are all at the same temperature). Parameters as in Fig. 2.

Figure 6

Energy spectrum $E\left(k\right)$ (arbitrary units) vs wave number $k$ (${\mathrm{cm}}^{-1}$) (time averaged over the saturated regime) corresponding to vortex tangles generated by uniform normal fluid (model 1, top), synthetic normal fluid turbulence (model 2, middle), and frozen Navier-Stokes turbulence (model 3, bottom). The dashed lines indicate the $k^{-1}$ (top) and the $k^{-5/3}$ dependence (middle and bottom), respectively. Parameters as in Fig. 2. The compensated spectra $k{E}_s\left(k\right)$ and $k^{5/3}{E}_s\left(k\right)$ in the insets show the regions of $k$ space where the approximate scalings $k^{-1}$ and $k^{-5/3}$ apply.

Figure 7

Energy spectrum $E\left(k\right)$ (arbitrary units) vs wave number $k$ (${\mathrm{cm}}^{-1}$) (time averaged over the saturated regime) as in Fig. 6, but the simulations are performed using the reconnection algorithm of Kondaurova et al. [35]. Note that there is no significant difference from spectra obtained using our standard algorithm; see Fig. 6. Vortex tangles generated by uniform normal fluid (model 1, top), synthetic normal fluid turbulence (model 2, middle), and frozen Navier-Stokes turbulence (model 3, bottom). The dashed lines indicate the $k^{-1}$ (top) and the $k^{-5/3}$ dependence (middle and bottom), respectively. Parameters: temperature $T=1.9\mathrm{K}$, $V_n=0.75\mathrm{cm}/\mathrm{s}$ (for model 1), $\mathrm{Re}=81.59$ (for model 2), and $\mathrm{Re}=3025$ (for model 3).

Figure 8

Superfluid energy density ${\epsilon }_s={\left|{\mathbf{v}}_s\right|}^2/2$ smoothed over the average intervorex spacing $\ell$, plotted on the $xy$ plane and averaged over $z$. Left: model 1 (uniform normal fluid); middle: model 2 (synthetic turbulence); right: model 3 (frozen Navier-Stokes turbulence). Parameters as in Fig. 2.

Figure 9

Ratio of nonlocal to total self-induced velocity as a function of time $t$ $\left(\mathrm{s}\right)$ for tangles generated by uniform normal fluid (model 1, red [gray] line, bottom), synthetic normal fluid turbulence (model 2, black line, middle), and frozen Navier-Stokes turbulence (model 3, blue [gray] line, top). Parameters as in Fig. 2.

Figure 10

Evolution of the vortex line density $L\left({\text{cm}}^{-2}\right)$ vs time $t$ (s) for model 1 [red [gray] line, uniform normal flow, respectively at $V_n=1(\mathrm{cm}/\mathrm{s})$ (top), $V_n=0.75(\mathrm{cm}/\mathrm{s})$ (middle) and $V_n=0.55(\mathrm{cm}/\mathrm{s})$ (bottom)], model 2 [black line, synthetic normal fluid turbulence, respectively at $\text{Re}=79.44$ (top), $\text{Re}=81.59$ (middle), and $\text{Re}=83.86$ (bottom)], and model 3 (dashed blue [gray] line, frozen Navier-Stokes turbulence, at $\text{Re}=3025$).

Figure 11

Ratio $v^{\mathrm{non}}/v^{\mathrm{self}}$ as a function of vortex line density $L\left({\mathrm{cm}}^{-2}\right)$ corresponding to model 1 (uniform normal flow, red [gray] circles), model 2 (synthetic normal flow turbulence, black crosses), and model 3 (frozen Navier-Stokes equation, blue [gray] stars).