Energy and vorticity spectra in turbulent superfluid ${}^4\mathrm{He}$ from $T=0$ to $T_{\lambda }$

We discuss the energy and vorticity spectra of turbulent superfluid ${}^4\mathrm{He}$ in the entire temperature range from $T=0$ up to the phase transition “$\lambda$ point,” $T_{\lambda }\simeq 2.17\mathrm{K}$. Contrary to classical developed turbulence in which there are only two typical scales, i.e., the energy injection $L$ and the dissipation scales $\eta$, here, the quantization of vorticity introduces two additional scales, the vortex core radius $a_0$ and the mean vortex spacing $\ell$. We present these spectra for the super- and the normal-fluid components in the entire range of scales from $L$ to $a_0$ including the crossover scale $\ell$ where the hydrodynamic eddy cascade is replaced by the cascade of Kelvin waves on individual vortices. At this scale, a bottleneck accumulation of the energy was found earlier at $T=0$. We show that even very small mutual friction dramatically suppresses the bottleneck effect due to the dissipation of the Kelvin waves. Using our results for the spectra we estimate the Vinen “effective viscosity” ${\nu }^{\prime }$ in the entire temperature range and show agreement with numerous experimental observations for ${\nu }^{\prime }\left(T\right)$.

Figure 1

Log-log plots of the energy spectra in superfluid ${}^4\mathrm{He}$ compensated by the inertial range scaling at different temperatures (shown as labels). (a) Plots of the (compensated by $k^{5/3}$ and normalized by their values at $k\ell =0.1)$ full superfluid energy spectra (solid lines) ${\mathcal{E}}^{\text{s}}\left(\kappa \ell \right)$ and their hydrodynamic (large scale) parts ${\mathcal{E}}_{\mathrm{HD}}^{\text{s}}\left(\kappa \ell \right)$ (dashed lines) for the one-fluid model (Sec. 4b). (b) Plots of the (compensated by the anomalous scaling $k^{1.72}$ and normalized by their inertial range value) shell energies of the normal-fluid component $|u_m^{\text{n}}{|}^2=k_m{\mathcal{E}}^{\text{n}}\left(k_m\ell \right)$ (solid lines) and of the superfluid component $|u_m^{\text{s}}{|}^2=k_m{\mathcal{E}}^{\text{s}}\left(k_m\ell \right)$ (dashed lines) for the two-fluid shell model (Sec. 3b).

Figure 2

Linear-log plots of the vorticity spectral densities, $\left(k\ell \right){\Omega }^2\left(k\ell \right)$, normalized by their $k\ell =0.1$ values, at different temperatures. The total mean square vorticity $\langle {\omega }^2\rangle$ is proportional to the area under the plot. (a) Superfluid vorticity spectra $\left(k\ell \right){\Omega }_{\text{s}}^2\left(k\ell \right)$ at low temperatures in the one-fluid model, Sec. 4b. (b) Normal and superfluid vorticity spectra $\left(k\ell \right){\Omega }_{\text{n}}^2\left(k\ell \right)$ (solid lines) and $\left(k\ell \right){\Omega }_{\text{s}}^2\left(k\ell \right)$ (dashed lines) at high temperatures in the two-fluid model, Sec. 3b.

Figure 3

Cross-correlation coefficients $K_1\left(k\ell \right)$, Eq. (20a) (solid lines) and $K_2\left(k\ell \right)$, Eq. (20b) (dashed lines) for different temperatures. Color code is the same as in Figs. 1 and 2: $T=1.07$ (blue), $1.3$ (cyan), $1.8$ (green), and $2.16\mathrm{K}$ (red).

Figure 4

Temperature dependence of the ratios: ${\varepsilon }_{\text{s}}/{\varepsilon }_{\text{n}}$ (horizontal black line); $R_{\text{s}}$, Eq. (28a) (blue line with triangles); $R_{\text{n}}$, Eq. (28a) (green line with diamonds); and $R_{\text{s+n}}$, Eq. (28b) (red line with circles).

Figure 5

Comparison of the experimental, numerical, and analytical results for the temperature dependence of the effective kinematic viscosities: blue triangles—Manchester spin-down experiments [19]; green empty circles—Manchester ion-jet experiments [20]; sea-green diamonds with error-bars—Prague counterflow experiments [15]; cyan crosses with error-bars—Prague decay in grid co-flow experiments [16]; magenta empty squares—Oregon towed grid experiments [26]; and pink right triangles—Oregon towered grid experiments [27]. Solid green line—experimental results [66] for the normal-fluid kinematic viscosity ${\nu }_{\text{n}}=\mu /{\rho }_{\text{n}}$ (normalized by the normal-fluid density); and dashed green line—He-II kinematic viscosity $\nu \equiv \mu /\rho$ (normalized by the total density)—see also Table 1. Thin black dash line—effective viscosity for the random vortex tangle ${\nu }_{\text{rnd}}^{\prime }$, estimated by Eq. (31); and thick dot-dashed black line—the Vinen-Niemela estimate [10] of the effective superfluid viscosity, ${\nu }_{\text{s}}^{\prime }$, given by Eq. (32). Blue solid line—${\nu }^{\prime }\left(T\right)$ at $T<1.1\mathrm{K}$ from numerical solution of Eq. (51b) for the one-fluid differential model of gradual eddy-wave crossover; red solid line—${\nu }^{\prime }\left(T\right)\text{at}T>0.9\mathrm{K}$ from numerical simulations in Sec. 3b of gradually damped two-fluid HVBK equations (1) in the Sabra shell-model approximation (33).