Nonlocality in homogeneous superfluid turbulence

Simulating superfluid turbulence using the localized induction approximation allows neighboring parallel vortices to proliferate. In many circumstances a turbulent tangle becomes unsustainable, degenerating into a series of parallel, noninteracting vortex lines. Calculating with the fully nonlocal Biot-Savart law prevents this difficulty but also increases computation time. Here we use a truncated Biot-Savart integral to investigate the effects of nonlocality on homogeneous turbulence. We find that including the nonlocal interaction up to roughly the spacing between nearest-neighbor vortex segments prevents the parallel alignment from developing, yielding an accurate model of homogeneous superfluid turbulence with less computation time.

Figure 1

Projections of vortex tangles from the present work, with applied velocity perpendicular to [(a)–(c)] or within [(d)–(f)] the viewing plane. All computations use $v_{ns}=11$ cm/s, in a periodic cube with side length $50\mu \mathrm{m}$. The first row shows full Biot-Savart law calculation at time 0.142 s. The second and third rows show the LIA calculation at times 0.9 s and 0.142 s, respectively. In panels (b) and (e) the tangle has begun to exhibit anisotropy; in panels (c) and (f) the system has reached an open-orbit state, with the vortices almost entirely straight and parallel.

Figure 2

Results from LIA and full Biot-Savart calculations for (a) $L\left(t\right)$ and (b) $I_{\parallel }\left(t\right)$, with $v_{ns}=11$ cm/s.

Figure 3

Evolution of the vortex line length density, $L\left(t\right)$, within each octant of the periodic cube. The curve marked by black asterisks gives the line length per total system volume, $D^3$. Each other curve gives $L\left(t\right)$ for one of the eight ${\mathbb{R}}^3$ octants. The applied velocity is $v_n=12$ cm/s.

Figure 4

Average equilibrated line length density, $\langle L\rangle$, within each octant of the periodic cube, at multiple velocities.

Figure 5

Average equilibrated line length density, $\langle L\rangle$, for the total system volume, at multiple velocities. The linear fit to our data yields a slope of $c_L=0.106$.

Figure 6

Left axis (red circles): The quantity $\langle \overline{\Gamma }\rangle$ taken from the mutual friction force density, ${\overline{F}}_{sn}$, along ${\widehat{v}}_{ns}$. The linear fit of this data yields a slope of $c_F=0.088$. Right axis (blue diamonds): The average vortex velocity, ${\overline{v}}_L$. The linear fit of this data yields a slope of $c_v=0.070$.

Figure 7

A plot of $ln\left(\dot{N}\right)$ versus $ln(\langle L\rangle )$, finding the exponent of $\dot{N}=dN/dt\propto {\langle L\rangle }^m,m=2.47$.

Figure 8

Effects of nonlocal distance, $d_{NL}$, on the equilibrated line length density, $\langle L\rangle$. Each curve is a series of trials at the $d_{NL}$, listed in the legend. The side length for the sample region is $D=50\mu \mathrm{m}$, and the largest $d_{NL}$ allows interactions throughout the volume. Averages at each $v_{ns}$ used the initial equilibrated range, similar to Aarts [10].

Figure 9

rms fractional deviation of $\langle L\rangle$ from the full Biot-Savart calculation for several nonlocal distances. Box side is $50\mu \mathrm{m}$. The trials contributing to each point are from Fig. 8, and use the initial equilibrated time domain for calculating the average of $L\left(t\right)$.

Figure 10

rms fractional deviation of the standard deviations $\sigma \left(L\right(t\left)\right)$ (red circles) and $\sigma \left(I_{\parallel }\left(t\right)\right)$ (blue diamonds) from the full Biot-Savart calculation plotted versus nonlocal distance. Inset: rms fractional deviation of the averages $\langle L\rangle$ (red circles) and $\langle I_{\parallel }\rangle$ (blue diamonds) from the full Biot-Savart calculation plotted versus nonlocal distance. Averages use the entire equilibrated time domain. Comparing $\langle L\rangle$ in the inset here to the data in Fig. 9 shows the effect of changing the time domain used for the averaging.

Figure 11

Fractional deviation of (a) $\langle L\rangle$ and (b) $\sigma \left(L\right(t\left)\right)$ from the full Biot-Savart calculation plotted versus driving velocity. Each curve is a different nonlocal distance, $d_{NL}$, indicated by the legend. Averages use the entire equilibrated time domain.

Figure 12

$\beta {\langle L\rangle }^{1/2}$ data for full Biot-Savart calculations at several system sizes $D$. Averages use the entire equilibrated time domain.

Figure 13

Square root of line length density as function of interaction distance $d_{NL}$, for three box sizes $D$. In each case $\alpha =0.1$ and $v_{ns}=9$ cm/s.

Figure 14

Excess line length as a function of interaction distance, for several combinations of friction parameter $\alpha$ and applied velocity $v_{ns}$. For each curve ${\langle L\rangle }^{1/2}$ for the largest $d_{NL}$ is subtracted. Main graph: Expanded view near critical interaction distance. Inset: Entire range. The legend identifies each curve by $\alpha$ and $v_{ns}$, the latter measured in cm/s.

Figure 15

For comparison with Fig. 10: Panel (a) has system size $D=25\mu \mathrm{m}$; panel (b) has $D=100\mu \mathrm{m}$. rms fractional deviation from the full Biot-Savart calculation of the standard deviations $\sigma \left(L\right(t\left)\right)$ (red circles) and $\sigma \left(I_{\parallel }\left(t\right)\right)$ (blue diamonds), plotted versus nonlocal distance. Insets: rms fractional deviation from the full Biot-Savart calculation of $\langle L\rangle$ (red circles) and $\langle I_{\parallel }\rangle$ (blue diamonds), plotted versus nonlocal distance. Time averages use the entire time domain after initial equilibration.