Quantum and Quasiclassical Types of Superfluid Turbulence

By injecting negative ions in superfluid ${}^4\mathrm{He}$ in the zero-temperature limit ($T\le 0.5\mathrm{K}$), we have generated tangles of quantized vortex line with negligible large-scale flow. For this quantum regime of superfluid turbulence, the vortex line length $L$ was found to decay at late time $t$ as $L\propto t^{-1}$, the prefactor being independent of the initial value of $L$. The corresponding effective kinematic viscosity is 0.1 $\kappa$, where $\kappa$ is the circulation quantum. At $T>0.7\mathrm{K}$, a jet of ions generates quasiclassical tangles identical to those produced by mechanical means.

Figure 1

Cartoon of the vortex configurations (side view) at different stages. The shaded areas indicate the trajectories of ions used to probe the tangle along two orthogonal directions. To avoid contamination of the developing tangle with new ions and vortices, only one probe pulse of ions was fired for each realization of the tangle. (a) At $t=0$, a pulse of CVRs is injected from the left injector. While most CVRs make it to the collector as a sharp pulse, some got entangled near the injector. (b) At $t\sim 5\mathrm{s}$, the tangle spreads into the middle of the cell. (c) At $t\sim 20\mathrm{s}$, the tangle has occupied all volume. (d) For up to 1000 s, the homogeneous tangle is decaying further.

Figure 2

Transients of the current of CVRs to the top collector after different waiting times between the pulses (shown in seconds); the duration of injection is 0.3 s, $V=90\mathrm{V}$, $T=0.15\mathrm{K}$. The arrival time (leading edge indicated by an arrow), 0.7 s, is the same for all pulses. The inset shows the dependence of the arrival time on the driving voltage.

Figure 3

Free decay of a tangle at $T=0.15\mathrm{K}$. The injection direction and duration, and driving field are indicated. Probing with pulses of CVRs of duration 0.1–0.3 s were done in the same field as the initial injection, and also in the same direction except in one case (*). The line $L\propto t^{-1}$ corresponds to Eq. (2) with ${\nu }_V=0.1\kappa$.

Figure 4

Free decay of a tangle produced by a jet of ions (●) from the bottom injector into a $10\mathrm{V}/\mathrm{cm}$ field for 150 s, as well as by an impulsive spin-down to rest [8] from $1.5\mathrm{rad}/\mathrm{s}$ and $0.5\mathrm{rad}/\mathrm{s}$, at $T=1.60\mathrm{K}$. All tangles were probed by pulses of free ions in the horizontal direction (spin-down data: probe field $20\mathrm{V}/\mathrm{cm}$, pulse length 0.5 s; ion jet data: probe field $10\mathrm{V}/\mathrm{cm}$, pulse length 1.0 s ). The line $L\propto t^{-3/2}$ corresponds to Eq. (3) with ${\nu }_K=0.2\kappa$.

Figure 5

The effective kinematic viscosity for different types of superfluid turbulence. Vinen (quantum) type, generated by: CVRs (this work, $L(t)$ measured along the vertical (▵) and horizontal (▿) directions), buildup of counterflow [3], simulations [16], vibrating grid in ${}^3\mathrm{He}$-B vs temperatures at which ${}^4\mathrm{He}$ has comparable values of ${10}^{-4}<\alpha <2\times {10}^{-3}$ [20]. Kolmogorov (quasiclassical) type: ion jet (this work), spin-down [8] sampled by CVRs (▵) and free ions (▿), towed grid [11].