Temporal decay of vortex line density in rotating thermal counterflow of He II

TL;DR

This work investigates how rotation modifies thermal counterflow–driven quantum turbulence in superfluid $^4$He (He II) by measuring the vortex-line density $L(t,\Omega)$ via second-sound attenuation in two geometries: $\mathbf{\Omega} \perp \mathbf{v}_{ns}$ and $\mathbf{\Omega} \parallel \mathbf{v}_{ns}$. Using a square-channel setup and thermally driven counterflow, the authors confirm in steady state that $L$ obeys Feynman’s rule $L = \frac{2\Omega}{\kappa}$ in both geometries, with similar photos of $\sqrt{L}$ scaling with $v_{ns}$. In temporal decay, a robust late-time power law $L(t,\Omega) \propto (t-t_{vo})^{-\mu}$ with $\mu \approx \tfrac{3}{2}$ emerges, but horizontal rotation introduces an Ekman-layer screening via an effective Ekman time $T_{Ek}^{eff}=H (\nu_{eff}\Omega)^{-1/2}$ that shortens the self-similar decay window and rounds the decay at higher $\Omega$; axial rotation (with a longer $T_{Ek}^{eff}$) preserves the self-similar decay longer. The results elucidate the competition between rotation-induced two-dimensionalization and boundary-layer dissipation in rotating quantum turbulence and offer parallels to classical rotating turbulence, informing models of $\nu_{eff}$ and the nature of self-similar decay in He II.

Abstract

Horizontally ($\mathbfΩ \perp \mathbf{v}_{\rm{ns}}$) and axially ($\mathbfΩ \parallel \mathbf{v}_{\rm{ns}}$) rotating counterflow of superfluid $^4$He (He~II) generated thermally in a square channel is studied using the second sound attenuation technique, detecting statistically steady state and temporal decay of the density of quantized vortex lines $L(t,Ω)$. The array of rectilinear quantized vortices created by rotation at angular velocity $Ω$ strongly affects the transient regimes of quantum turbulence characterized by counterflow velocity $\mathbf{v}_{\rm{ns}}$, differently in both geometries. Two effects are observed, acting against each other and affecting the late temporal decay $L(t,Ω)$. The first is gradual decrease of the decay exponent $μ$ of the power law $L(t,Ω) \propto t^{-μ}$, associated with the fact that under rotation thermal counterflow acquires two-dimensional features, clearly observed and recently reported by us (Phys. Fluids \textbf{36}, 105121 (2024)) in the $\mathbfΩ \parallel \mathbf{v}_{\rm{ns}}$ geometry. It exists in the $\mathbfΩ \perp \mathbf{v}_{\rm{ns}}$ geometry as well, however, it is screened here by the influence of the effective Ekman layer built within the effective Ekman time of order seconds. For faster rotation rates $L(t,Ω)$ gradually ceases to display a clear power law. Instead, rounded and ever steeper decays occur, gradually shifted toward shorter and shorter times, significantly shortening the time range for a possible self-similar decay of vortex line density. This effect is not observed in $\mathbfΩ \parallel \mathbf{v}_{\rm{ns}}$ geometry, as here the much longer effective Ekman time of order minutes cannot affect the observed $L(t,Ω)$ decay appreciably.

Figures (3)

• Figure 1: Schematic diagram of the $7\times 7$ mm$^2$ square cross-section counterflow channel, as used in experiments on rotating channel counterflow in a) parallel ($\mathbf{\Omega} \parallel \mathbf{v}_{\rm{ns}}$) and b) perpendicular ($\mathbf{\Omega} \perp \mathbf{v}_{\rm{ns}}$) geometries. The resistive heater placed in its dead end generates the counterflow of normal and superfluid components. Vortex line density is measured using second sound attenuation.
• Figure 2: Square root of vortex line density in steady state counterflow turbulence, $\sqrt{L(v_{\rm{ns}},\Omega)}$, rotating in the $\mathbf{\Omega} \perp \mathbf{v}_{\rm{ns}}$ geometry at temperature 1.65 K, plotted as a function of counterflow velocity for several rotation speeds of the cryostat as indicated.
• Figure 3: Temporal decay of the vortex line density $L(t, \Omega)$ for rotating thermal counterflow in $\mathbf{\Omega} \perp \mathbf{v}_{\rm{ns}}$ geometry. The red dashed lines represent identical power law decays with the classical decay exponent $\mu=1.5$Stalp1999Morize2006. The vertical dashed lines indicate the effective Ekman times $T_{\rm{Ek}}^{\rm{eff}}=H (\nu_{{\rm{eff}}} \Omega)^{-1/2}$, where $\nu_{\rm{eff}} =\kappa/2$ is used as the value for the effective kinematic viscosity of turbulent He II. This gives effective Ekman times 30.6 s, 21.6 s and 17.6 s for rotation rates 60, 120 and 180 degrees per second.