Theory of energy spectra in superfluid ${}^4\mathrm{He}$ counterflow turbulence

In the thermally driven superfluid ${}^4\mathrm{He}$ turbulence, the counterflow velocity $U_{\text{ns}}$ partially decouples normal and superfluid turbulent velocities. Recently, we suggested [J. Low. Temp. Phys. 187, 497 (2017)] that this decoupling should tremendously increase the turbulent energy dissipation by mutual friction and significantly suppress the energy spectra. Comprehensive measurements of the apparent scaling exponent $n_{\text{exp}}$ of the second-order normal-fluid velocity structure function $S_2\left(r\right)\propto r^{n_{\text{exp}}}$ in the counterflow turbulence [J. Gao et al., Phys. Rev. B 96, 094511 (2017)] confirmed our scenario of gradual dependence of the turbulence statistics on flow parameters. We develop an analytical theory of the counterflow turbulence, accounting for a twofold mechanism of this phenomenon: (i) a scale-dependent competition between the turbulent velocity coupling by mutual friction and the $U_{\text{ns}}$-induced turbulent velocity decoupling and (ii) the turbulent energy dissipation by mutual friction enhanced by the velocity decoupling. The suggested theory predicts the energy spectra for a wide range of flow parameters. The mean exponents of the normal-fluid energy spectra ${〈m〉}_{\text{10}}$, found without fitting parameters, qualitatively agree with the observed $n_{\text{exp}}+1$ for $T\gtrsim 1.85\mathrm{K}$.

Figure 1

The compensated energy spectra ${\mathcal{E}}_j\left(q\right)=q^{5/3}{E}_j\left(q\right)/E_j\left(q_0\right)$ vs $q=k/k_0$ for ${\mathrm{Re}}_j=\infty$ and different combinations of $T$ and ${\tilde{U}}_{\text{ns}}$. The lines corresponding to different values of ${\tilde{\mathrm{\Omega }}}_{\text{ns}}$ are shown by different colors (from top to bottom): $\tilde{\mathrm{\Omega }}=0$ (the horizontal gray lines), ${\tilde{\mathrm{\Omega }}}_{\text{ns}}=1.0$ (black lines), ${\tilde{\mathrm{\Omega }}}_{\text{ns}}=5.0$ (blue lines), ${\tilde{\mathrm{\Omega }}}_{\text{ns}}=10.0$ (green lines), and ${\tilde{\mathrm{\Omega }}}_{\text{ns}}=20.0$ (the lowest red lines). The normal-fluid energy spectra are shown by solid lines, the superfluid spectra by dashed lines. Note that in the left column $q^{5/3}{\mathcal{E}}_j\left(q\right)$ varies from 0.05 to 1.0, while in the middle and the right columns from 0.01 to 1.0. In these panels, the level 0.05 is shown by the horizontal dotted lines. The labels “$\mathrm{\Omega }$“ in the figures mark one of the lines of the corresponding color (solid or dashed lines) for further clarity.

Figure 2

The outer-scale scaling exponents of the normal component $m_{\text{n}}\left(1\right)$ [Eq. (19)] vs ${\tilde{\mathrm{\Omega }}}_{\text{ns}}$ for ${\tilde{U}}_{\text{ns}}=1$ (left panel), ${\tilde{U}}_{\text{ns}}=4$ (middle panel), and ${\tilde{U}}_{\text{ns}}=8$ (right panel). The lines from top to bottom correspond to $T=1.65\mathrm{K}$ (black lines), $T=1.85\mathrm{K}$ (blue lines), $T=2.0\mathrm{K}$ (green lines), and $T=2.10\mathrm{K}$ (red lines).

Figure 3

The ${\tilde{\mathrm{\Omega }}}_{\text{ns}}$ dependence of the scaling exponents $m_j\left(1\right)$ and the mean exponent ${〈m_j〉}_{10}$. The normal-fluid component exponent $m_{\text{n}}\left(1\right)$ is marked as red solid line, and ${〈m_{\text{n}}〉}_{10}$ by red full circles. The superfluid exponent $m_{\text{s}}\left(1\right)$ is marked by blue dashed line, and the mean exponent ${〈m_{\text{s}}〉}_{10}$ by empty blue circles. The exponents were calculated for $T=2.0\mathrm{K}$ and ${\tilde{U}}_{\text{ns}}=4$.

Figure 4

The viscous corrections to the energy spectra at different temperatures: $T=1.65\mathrm{K}$ (left panel), $T=1.95\mathrm{K}$ (middle panel), and $T=1.21\mathrm{K}$ (right panel). The normal-fluid spectra are shown by solid lines, the superfluid spectra by dashed lines. The spectra are shown for ${\mathrm{Re}}_{\text{n}}=\infty$ (black lines), ${\mathrm{Re}}_{\text{n}}={10}^3$ (blue lines), and ${10}^2$ (red lines). All spectra were calculated with ${\tilde{U}}_{\text{ns}}=4$ and ${\tilde{\mathrm{\Omega }}}_{\text{ns}}=10$.

Figure 5

The (uncompensated) energy spectra of counterflow turbulence ${\mathcal{E}}_{\text{s}}\left(q\right)=E\left(q\right)/E\left(q_0\right)$ vs $q=k/k_0$ (dashed lines) and the sketch of the quantum peak (dotted-dashed lines) for $T=1.65\mathrm{K}$, $Q=300\mathrm{mW}$/${\mathrm{cm}}^2$ (upper black lines), $T=1.85\mathrm{K}$, $Q=497\mathrm{mW}$/${\mathrm{cm}}^2$ (blue lines), $T=2.00\mathrm{K},Q=586\mathrm{mW}$/${\mathrm{cm}}^2$ (green lines) and $T=2.10\mathrm{K},Q=350\mathrm{mW}$/${\mathrm{cm}}^2$ (lowest red lines). Note that here the $q$ range, where ${\mathcal{E}}_{\text{s}}\left(q\right)$ are valid, is limited by $q\lesssim 200$.

Figure 6

The compensated energy spectra for experimental conditions at $T=1.65\mathrm{K}$ (a), $T=1.85\mathrm{K}$ (b), $T=2.0\mathrm{K}$ (c), and $T=2.1\mathrm{K}$ (d). The spectra for large heat fluxes $Q$ are shown by red lines [upper lines in (a)–(c)], for moderate heat fluxes by green lines, and for small heat fluxes by lower blue lines. The explicit values of $Q$ are given in Table 2. The normal-fluid spectra are shown by solid lines, the superfluid spectra by dashed lines. Note that all spectra here are shown for $q<100$, within the applicability range of Eqs. (4) for the relevant flow parameters.

Figure 7

The second-order structure functions [19] $S_2\left(r\right)$ as a function of separation $r$ for $T=1.85\mathrm{K}$ and different heat fluxes. The black dashed lines indicate the scaling behavior $r^{2/3}$ and $r^{3/2}$ and serve to guide the eye only.