Counterflow-induced decoupling in superfluid turbulence

In mechanically driven superfluid turbulence, the mean velocities of the normal- and superfluid components are known to coincide: ${\mathbit{U}}_{\text{n}}={\mathbit{U}}_{\text{s}}$. Numerous laboratory, numerical, and analytical studies showed that under these conditions, the mutual friction between the normal- and superfluid velocity components also couples their fluctuations: ${\mathbit{u}}_{\text{n}}^{\ensuremath{'}}(\mathbit{r},t)\ensuremath{\approx}{\mathbit{u}}_{\text{s}}^{\ensuremath{'}}(\mathbit{r},t)$, almost at all scales. We show that this is not the case in thermally driven superfluid turbulence; here the counterflow velocity ${\mathbit{U}}_{\text{ns}}\ensuremath{\equiv}{\mathbit{U}}_{\text{n}}\ensuremath{-}{\mathbit{U}}_{\text{s}}\ensuremath{\ne}0$. We suggest a simple analytic model for the cross-correlation function $\ensuremath{\langle}{\mathbit{u}}_{\text{n}}^{\ensuremath{'}}(\mathbit{r},t)\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbit{u}}_{\text{s}}^{\ensuremath{'}}({\mathbit{r}}^{\ensuremath{'}},t)\ensuremath{\rangle}$ and its dependence on ${U}_{\text{ns}}$. We demonstrate that ${\mathbit{u}}_{\text{n}}^{\ensuremath{'}}(\mathbit{r},t)$ and ${\mathbit{u}}_{\text{s}}^{\ensuremath{'}}(\mathbit{r},t)$ are decoupled almost in the entire range of separations $|\mathbit{r}\ensuremath{-}{\mathbit{r}}^{\ensuremath{'}}|$ between the energy-containing scale and intervortex distance.