Energy spectrum of thermal counterflow turbulence in superfluid helium-4

Recent preliminary experiments [A. Marakov et al., Phys. Rev. B 91, 094503 (2015).] using triplet-state ${\mathrm{He}}_{2}$ excimer molecules as tracers of the motion of the normal fluid have shown that, in thermal counterflow turbulence in superfluid $^{4}\mathrm{He}$, small-scale turbulence in the superfluid component is accompanied, above a critical heat flux, by partially coupled large-scale turbulence in both fluids, with an energy spectrum proportional to ${k}^{\ensuremath{-}m}$, where $m$ is greater than the Kolmogorov value of 5/3. Here we report the results of a more detailed study of this spectrum over a range of temperatures and heat fluxes using the same experimental technique. We show that the exponent $m$ varies systematically with heat flux but is always greater than 5/3. We interpret this as arising from the steady counterflow, which causes large-scale eddies in the two fluids to be pulled in opposite directions, giving rise to dissipation by mutual friction at all wave numbers, mutual friction tending also to oppose the effect of the counterflow. Comparison of the experimental results with a simple theory suggests that this process may be more complicated than we might have hoped, but experiments covering a wider range of heat fluxes, which are technically very difficult, will probably be required before we can arrive at a convincing theory.