Dissipation in quantum turbulence in superfluid ${}^4\mathrm{He}$ above 1 K

There are two commonly discussed forms of quantum turbulence in superfluid ${}^4\mathrm{He}$ above 1 K: in one there is a random tangle of quantized vortex lines, existing in the presence of a nonturbulent normal fluid; in the second there is a coupled turbulent motion of the two fluids, often exhibiting quasiclassical characteristics on scales larger than the separation between the quantized vortex lines in the superfluid component. The decay of vortex line density, $L$, in the former case is often described by the equation $dL/dt=-{\chi }_2(\kappa /2\pi )L^2$, where $\kappa$ is the quantum of circulation and ${\chi }_2$ is a dimensionless parameter of order unity. The decay of total turbulent energy, $E$, in the second case is often characterized by an effective kinematic viscosity, ${\nu }^{\prime }$, such that $dE/dt=-{\nu }^{\prime }{\kappa }^2{L}^2$. We present values of ${\chi }_2$ derived from numerical simulations and from experiment, which we compare with those derived from a theory developed by Vinen and Niemela. We summarize what is presently known about the values of ${\nu }^{\prime }$ from experiment, and we present a brief introductory discussion of the relationship between ${\chi }_2$ and ${\nu }^{\prime }$, leaving a more detailed discussion to a later paper.

Figure 1

Observed decays of vortex line density in decaying TCT (1.65 K).

Figure 2

Observed decay in line density from a small heat flux.

Figure 3

Observed (open circles) and theoretical (filled circles) values of ${\chi }_2$, the theoretical values being derived from Eq. (12).

Figure 4

Value of $c_2$ derived from simulations of the approach of counterflow to a steady state, and the corresponding value of ${\chi }_2$ derived from Eq. (12).

Figure 5

$\left(1/L\right)$ plotted against time from simulations at $T=1.4\mathrm{K}$ and $v_n=9$ mm ${\mathrm{s}}^{-1}$.

Figure 6

(a) The variation of the parameter $c_2$ with time from simulations of the decaying line density at $T=1.4\mathrm{K}$ and $v_n=9$ mm ${\mathrm{s}}^{-1}$. (b) The variation of ${\chi }_2$ with time, obtained by substituting $c_2$ from Fig. 6 into Eq. (12).

Figure 7

Plots of ${\chi }_2$ against time derived as explained in the text.

Figure 8

Plots showing how $c_2$, averaged over time in the steady state, varies with position across the channel. Blue line: flow between parallel plates; red line: spatially uniform flow. Temperature = 1.4 K.

Figure 9

Diagram showing how $c_2$ varies with position across the channel and with time during a decay. Temperature = 1.4 K.

Figure 10

Plots of ${\chi }_2$ against time for flow between parallel plates.

Figure 11

Values of ${\nu }^{\prime }$ for decaying TCT derived from measurements of both the decaying turbulent energy and the decaying vortex line density. Values of ${\nu }^{\prime }$ derived from the decay of line density alone, based on Eq. (14), are included for comparison.

Figure 12

Values of ${\nu }^{\prime }$ for various types of decaying coupled turbulence. $▴$, Ref. [12]; $\circ$, Ref [11]; $♦$, Ref. [13] decay of superflow in channel D7; $•$, Ref. [13] decay of superflow in channel D10; $\blacksquare$, Ref. [13] decay of counterflow in channel D10; $\times$, Ref. [24] with no grid; $\ast$, Ref. [24] with grid.