Decay of two-dimensional superfluid turbulence over pinning surface

TL;DR

This work probes the decay of quasi-2D quantum turbulence in superfluid $^4$He confined to nanofluidic channels, revealing a universal fast transient $L(t) \propto t^{-2}$ followed by a slower, geometry-dependent regime. A two-mode pump–probe setup in nanofluidic Helmholtz resonators enables precise measurement of vortex line density from fourth-sound attenuation, while a depinning-based model recasts wall pinning as a velocity-dependent effective mutual friction with parameters $\hat{\alpha}$ and $\hat{\alpha}'$; numerical simulations incorporating pinning and image boundaries reproduce the observed decay features across geometries. The results show that pinning can dramatically modify dissipation and decay, producing non-self-similar dynamics and slow approaches to $L \propto t^{-1}$, with implications for other pinning-dominated 2D vortex systems and potentially informing understanding of phenomena like pulsar glitches. Overall, the work provides a quantitative framework linking surface roughness, pinning, and vortex dynamics in confined 2D quantum turbulence.

Abstract

We report on the free decay of quasi-two-dimensional turbulence in superfluid $^4$He confined within nanofluidic channels. Using a pump-probe technique, we observe a complex decay of the vortex density $L(t)$ that deviates from a simple power law. The decay exhibits a universal fast transient, scaling as $L\propto t^{-2}$, followed by a slower non-universal regime that depends on the geometry and flow conditions. We demonstrate that this behavior is governed by the interplay between vortex pinning on the disordered topography of the channel walls and the mobilizing effect of the weak probe flow. A numerical model that treats pinning as a velocity-dependent effective mutual friction successfully reproduces the essential features of our experimental observations.

Figures (6)

• Figure 1: (a, b) Acoustic simulation of the superfluid resonant modes inside the nanofluidic cavity used in pump-probe measurement for the three used resonators. The frequencies are approximately 2 kHz for probe and 30 kHz for pump modes novotny_inter_2025. (c) A scheme of the measurement sequence, the bottom sketch shows the detuning of the probe and pump frequencies on the corresponding resonances. The inset axes show a zoomed view of $t > 5$ s data (shaded area in the main plot). (d,e) Vortex depinning model in the numerical simulation (see text).
• Figure 2: Vortex line density decay from all three geometries for a range initial densities $L_0$ and similar probe velocity amplitudes $v_p\approx 12$ cm/s for all, except for the two gray curves in panel (b), for which $v_p\approx 4.2$ cm/s. The dashed straight lines indicate $50/t^2$ (for $t < 0.5$ s) and $100/t$ (for $t > 0.5$ s) and are identical for all three. The delay on the order of 10 ms is consistent with the inverse linewidth of the 4th sound resonance.
• Figure 3: Decay of vortex density in the numerical simulation. (a-c) The initial conditions used in the simulations, from left to right: random or vortex pairs (see text), $2\times 2$ and $2 \times 10$. (d) The decay of the vortex density without probe flow: without pinning (lower four curves, same legend as in (e)) and with pinning (depinning velocity $v_d$ in the legend in cm/s). (e) The effect of initial conditions on the decay with $v_\mathrm{pin} = 10$ cm/s and $v_\mathrm{p} = 11$ cm/s. The dotted line shows $1[\mathrm{mm}^{-2}\mathrm{s}^2]/t^2$. (f) Effective mutual friction parameters. The dashed lines are bare $\alpha$ and $\alpha^\prime$ at 1.3 K donnelly_1998. The gray dashed line corresponds to $v/v_\mathrm{d} = 12/11$.
• Figure 4: (a) - A photo of the Helmholtz resonator. (b) - Sketches of resonator channels with dimensions, which are then used to calculate the probe velocity $v_{probe}$. (c) - A sketch of the measurement circuit.
• Figure 5: Decay of vortex line density in the G device. As Fig. \ref{['fig:decay-data-all']}c, but for $T = 1.450$ K; $v_p = 0.16$ms$^{-1}$. The black dashed lines indicate $100/t$, $50/t^{2}$ as in Fig. \ref{['fig:decay-data-all']}.