Local and nonlocal energy spectra of superfluid ${}^3\mathrm{He}$ turbulence

Below the phase transition temperature $T_{\text{c}}\simeq {10}^{-3}\mathrm{K}$ ${}^3\mathrm{He}$-B has a mixture of normal and superfluid components. Turbulence in this material is carried predominantly by the superfluid component. We explore the statistical properties of this quantum turbulence, stressing the differences from the better known classical counterpart. To this aim we study the time-honored Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid turbulence. We combine pseudospectral direct numerical simulations with analytic considerations based on an integral closure for the energy flux. We avoid the assumption of locality of the energy transfer which was used previously in both analytic and numerical studies of the superfluid ${}^3\mathrm{He}$-B turbulence. For $T<0.37T_{\text{c}}$, with relatively weak mutual friction, we confirm the previously found “subcritical” energy spectrum $E\left(k\right)$, given by a superposition of two power laws that can be approximated as $E\left(k\right)\propto k^{-x}$ with an apparent scaling exponent $\frac{5}{3}<x\left(k\right)<3$. For $T>0.37T_{\text{c}}$ and with strong mutual friction, we observed numerically and confirmed analytically the scale-invariant spectrum $E\left(k\right)\propto k^{-x}$ with a $(k$-independent) exponent $x>3$ that gradually increases with the temperature and reaches a value $x\sim 9$ for $T\approx 0.72T_{\text{c}}$. In the near-critical regimes we discover a strong enhancement of intermittency which exceeds by an order of magnitude the corresponding level in classical hydrodynamic turbulence.

Figure 1

The normalized energy spectra ${\mathcal{E}}_s\left(k\right)=E\left(k\right)/E_0$ compensated by $k^3$: subcritical [panel (a)] and supercritical [panel (b)] (solid lines) for different values of $\mathrm{\Omega }$. The critical spectrum (with $\mathrm{\Omega }=0.9)$ is shown in both panels. The dashed lines in panel (a) are the LNV prediction (A2a) for the subcritical spectra with one fitting parameter in Eq. (25) $\left(b=0.5\right)$ for all $\mathrm{\Omega }<0.9$. The subcritical spectra are not scale invariant and are not described by a single exponent. The horizontal dashed lines in both panels show the critical spectrum. Other dashed lines in panel (b) represent the scale-invariant spectra (16a) with an $\mathrm{\Omega }$-dependent exponent $x$.

Figure 2

The second and forth-order velocity $S_2\left(\tilde{r}\right)$ and $S_4\left(\tilde{r}\right)$ [panels (a),(c)] and vorticity $T_2\left(\tilde{r}\right)$ and $T_4\left(\tilde{r}\right)$ [panels (b),(d)] structure functions for different $\mathrm{\Omega }$. The straight dashed lines with the estimates of the apparent scaling exponents serve to guide the eye only.

Figure 3

The velocity $F_{\text{v}}\left(\tilde{r}\right)=S_4\left(\tilde{r}\right)/S_2^2\left(\tilde{r}\right)$ and vorticity $F_{\omega }\left(\tilde{r}\right)=T_4\left(\tilde{r}\right)/T_2^2\left(\tilde{r}\right)$ flatness vs $\tilde{r}$ for different $\mathrm{\Omega }$ [panels (a) and (b)] and vs $\mathrm{\Omega }$ for different $\tilde{r}$ [panels (c) and (d)]. The straight dashed lines with the estimates of the apparent scaling exponents serve to guide the eye only.

Figure 4

The differential [panels (a),(b),(c)] and the integral [panels (d),(e),(f)] energy balances in the subcritical regimes with $\mathrm{\Omega }=0$ [panels (a),(d)], $\mathrm{\Omega }=0.5$ [panels (b),(e)] and $\mathrm{\Omega }=0.7$ [panels (c),(f)]. The nonlinear energy transfer is shown by green lines, the viscous dissipation by red lines, and the dissipation by mutual friction by blue lines.

Figure 5

The differential [panels (a),(b),(c)] and the integral [panels (d),(e),(f)] energy balances in the critical and the supercritical regimes with $\mathrm{\Omega }=0.9$ [panels (a),(d)], $\mathrm{\Omega }=1.1$ [panels (b),(e)], and $\mathrm{\Omega }=5.0$ [panels (c),(f)]. The nonlinear energy transfer is shown by green lines, the viscous dissipation by red lines, and the dissipation by mutual friction by blue lines.

Figure 6

The energy (red lines) and enstropy (blue lines) time evolutions in the subcritical regime normalized by mean-in-time values [panels (a),(b),(c) with $\mathrm{\Omega }=0,0.5,0.7]$, the critical [panel (d), $\mathrm{\Omega }=0.9]$, and the supercritical regime [panels (e),(f) with $\mathrm{\Omega }=1.1,5.0]$.

Figure 7

Temperature dependence of the mutual friction parameter $\tilde{\alpha }\left(T\right)=\alpha /(1-{\alpha }^{\prime })$, taken from Ref. [28].