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arXiv:2504.21416 · Physical Review Fluids

Kolmogorov cascade as the governing mechanism for intervortex spacing in quantum turbulence

Clément Bret, Pantxo Diribarne, Jérôme Duplat, B. Rousset

Published Jul 21, 2025• 2 citations
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Abstract

In this paper, we investigate inertially forced isothermal quantum turbulence (coflow of normal and superfluid components) at temperatures of 1.6 and 2 K. The experiments are carried out in a large optical cryostat, where quasi-isotropic, homogeneous turbulence is generated using a double oscillating grid. Turbulence intensity is tuned by varying the grid stroke and frequency. The flow is probed via 2D and 3D reconstructions of quasi-isodense microsphere trajectories, from which we extract the large-scale properties of the flow in the fully turbulent regime for different Reynolds numbers: the turbulent velocity fluctuations, the energy transfer rate, and the integral length scale. Additionally, we determine the mean vortex line density $\mathcal{L}$ via attenuation of a second sound standing wave across the entire measurement volume. Our results confirm with an improved accuracy that the intervortex spacing $\ell =1/\sqrt{\mathcal{L}}$ scales with the Reynolds number $\text{Re}_κ$ (based on the quantum of circulation $κ$) as $\ell\propto \text{Re}_κ^{-3/4}$, with a well-defined numerical prefactor and no observed temperature dependence. This scaling recalls that of the dissipative length scale in classical Kolmogorov (K41) turbulence and it lead previous authors to interpret $\ell$ as an effective dissipation length scale. However, in our temperature range, this interpretation is not consistent with the apparent temperature independence of the prefactor. Based on those arguments, we propose an alternative interpretation that suggests that the inter-vortex distance in coflow turbulence is the consequence of the quantum restricted depth of the superfluid component energy cascade.

DOI
10.1103/4vhk-vgn1

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In this paper, we investigate inertially forced isothermal quantum turbulence (coflow of normal and superfluid components) at temperatures of 1.6 and 2 K. The experiments are carried out in a large optical cryostat, where quasi-isotropic, homogeneous turbulence is generated using a double oscillating grid. Turbulence intensity is tuned by varying the grid stroke and frequency. The flow is probed via 2D and 3D reconstructions of quasi-isodense microsphere trajectories, from which we extract the large-scale properties of the flow in the fully turbulent regime for different Reynolds numbers: the turbulent velocity fluctuations, the energy transfer rate, and the integral length scale. Additionally, we determine the mean vortex line density L\mathcal{L}L via attenuation of a second sound standing wave across the entire measurement volume. Our results confirm with an improved accuracy that the intervortex spacing ℓ=1/L\ell =1/\sqrt{\mathcal{L}}ℓ=1/L​ scales with the Reynolds number Reκ\text{Re}_κReκ​ (based on the quantum of circulation κκκ) as ℓ∝Reκ−3/4\ell\propto \text{Re}_κ^{-3/4}ℓ∝Reκ−3/4​, with a well-defined numerical prefactor and no observed temperature dependence. This scaling recalls that of the dissipative length scale in classical Kolmogorov (K41) turbulence and it lead previous authors to interpret ℓ\ellℓ as an effective dissipation length scale. However, in our temperature range, this interpretation is not consistent with the apparent temperature independence of the prefactor. Based on those arguments, we propose an alternative interpretation that suggests that the inter-vortex distance in coflow turbulence is the consequence of the quantum restricted depth of the superfluid component energy cascade.
The typical spacing between superfluid vortices in an isothermal turbulent tangle is proportional to the integral scale H rescaled by the quantum Reynolds number Re_K=H.V/K to the power of 3/4, where K is the quantum of circulation around of single vortex [Salort et al.,EPL 2011]. This empirical relation can be seen as the quantum-turbulence version of the corresponding well-know equation giving Kolmogorov dissipative scale in classical turbulence. In 2014, we studied the temperature dependence of the numerical factor (d/H).Re_K^{3/4} in 4He by joint numerical and experimental analysis of steady state turbulence over a wide temperature interval (1.2 - 2.16 K) [Babuin et al., EPL 2014]. Agreement between the two analyses was found good except at the very ends of this temperature interval. We will discuss this issue by presenting additional experimental data obtained by post-processing of superfluid experiments published between 1975 and 1998.
Turbulence, the irregular motion of fluids, is a challenging problem in physics. Yet some properties of turbulence appear to be universal, independent of the underlying host fluid supporting the motion. Recent studies have found that turbulence in superfluid helium, a quantum fluid, exhibits two of the most fundamental laws of classical fluid turbulence: the Kolmogorov −5/3 law, and the dissipation anomaly. These laws appear despite the fluid being highly constrained by quantum mechanical effects, and completely lacking kinematic viscosity. Such findings suggest further insight into the universal features of turbulent phenomena can be gained by studying analogies between classical and quantum turbulence.\nAtomic Bose-Einstein condensates (BECs) offer a new platform for the study of quantum turbulence; the geometric control available in BEC experiments offers the possibility of studying quantum turbulence in effectively two-dimensional fluids. As two-dimensional turbulence exhibits dramatically different features from its 3D counterpart, BEC systems allow for further study of the analogies between classical and quantum turbulence. In this thesis we numerically and theoretically study 2D quantum turbulence in BECs within the framework of the Gross-Pitaevskii model. We focus on analogies with classical 2D turbulence, with the aim of identifying common or universal features.\nFirst we investigate coherent vortex structures in negative temperature equilibria via an experimentally accessible flow-field measure. Coherent vortices are shown to produce a clear signal in this measure that is independent of the confinement geometry, and we demonstrate that it can be observed in dynamical simulations.\nSecond, studying a quantum analogue of the two-dimensional cylinder wake, we investigate the phenomenon of Strouhal oscillations. We find that the Strouhal number obeys a universal relation, similar to the classical form, upon introducing a modified superfluid Reynolds number that accounts for the critical velocity for vortex nucleation. Like the classical Reynolds number, the superfluid Reynolds number is found to govern the transition from laminar to turbulent behaviour in the quantum fluid.\nFinally, simulating decaying 2D quantum turbulence for very large vortex numbers, we show that quantum fluids are capable of supporting the direct enstrophy cascade, a fundamental feature of two-dimensional turbulent flows. The quantum fluid manifests key features of the classical cascade, including Batchelor’s −3 law of the inertial range, scaling of the inertial range against the superfluid Reynolds number, and the value of the Kraichnan-Batchelor constant.\nThe findings from this work thus provide some new insight into the universality of fundamental turbulence concepts, and their applicability to quantum fluids.
Abstract Quantum turbulence — the chaotic, non-equilibrium state of a superfluid threaded by a dense tangle of quantized vortex lines — provides a uniquely clean testbed for the universal statistical mechanics of turbulent energy transfer. We develop a theoretical account of the energy cascade across all relevant length scales: from macroscopic, quasi-classical Richardson–Kolmogorov eddies, through the dimensionally frustrated L’vov–Nazarenko–Rudenko (LNR) bottleneck at the mean inter-vortex spacing, down to the weakly nonlinear Kelvin-wave cascade on individual filaments, and finally to the zero-temperature dissipation cutoff. Starting from the Gross–Pitaevskii equation and its Madelung hydrodynamic decomposition, we separate the compressible and incompressible components of the flow to isolate the vortical and acoustic energy channels. Vortex reconnection is identified as the microscopic mechanism coupling these regimes, and the competing predictions of the Vortex Filament Model and the Gross–Pitaevskii equation for reconnection dynamics are compared. We review the theoretical controversy between the Kozik–Svistunov six-wave cascade and the L’vov–Nazarenko four-wave cascade, together with the associated tilt-symmetry debate, and present the finite-temperature two-fluid (HVBK) description, showing how mutual friction locks the normal and superfluid components at large scales while damping Kelvin waves at small scales. PACS numbers: 67.25.dk, 47.37.+q, 03.75.Kk, 47.32.C−
Identified as a unique regime of fluid flow some 500 years ago, turbulence has fascinated and confounded for centuries. Indeed, a fundamental understanding of the turbulence phenomenon remains one of the most important unsolved problems in classical physics. Interestingly, superfluid atomic Bose-Einstein condensates (BECs) can also be agitated into highly excited states of chaotic vortex motion, suggestive of turbulence. However, in contrast to classical fluids, the vortices in these quantum systems exhibit topological stability, the restriction of quantized circulation, and a well defined length scale characterized by the so-called healing length of the superfluid. Such simplifying aspects motivate the study of turbulence in these systems, which may shed new light on the longstanding classical problem. \n\nWhilst a considerable number of studies have been conducted on turbulence in three-dimensional BEC systems, and such studies have uncovered similarities between the classical and quantum counterparts, including the presence of the Kolmogorov k−5/3k^{-5/3}k−5/3 law in the kinetic energy spectrum and the associated direct energy cascade, comparatively few studies have been conducted on two-dimensional quantum turbulence. An important question for such systems is whether the inverse energy cascade, a defining feature of classical two-dimensional turbulent fluid flows, may occur in these systems. In addition to a Kolmogorov kinetic energy spectrum, this process is signified by the clustering of same-circulation vortices, which results in the emergence of macroscopic vortex structures from the turbulent flow. \n\nAlthough intermittent few-vortex clustering has been observed in the break-down of superfluid flow around an obstacle, current attempts to observe a sustained inverse cascade have been unsuccessful, largely due to complications which arise due to the compressibility of the dilute Bose gas - a feature not considered in the classical theory.\n\nWe present a study of two-dimensional quantum turbulence in atomic BECs within the framework of damped Gross-Pitaevskii theory. With the motivation of determining the existence of the inverse cascade phenomenon in such systems, we focusing on experimentally realistic forcing procedures, and study the details of kinetic energy spectra, the statistics of vortex clustering, and the transport of kinetic energy through scale space in the resulting turbulent flows.\nWe investigate a range of vortex injection regimes produced through the circular stirring of a harmonically confined system with a repulsive Gaussian obstacle potential, identifying distinct regimes of turbulence. In the incompressible regime relevant to the inverse cascade, we identify numerous problems with the system and forcing procedure. Motivated by these findings, we investigate turbulence in a minimal model of the spindown of a toroidal persistent current. In this system, we identify a regime of forcing and damping in which the inverse cascade is corroborated by three independent measures: I) merging of vortex clusters; II) a Kolmogorov scaling in the incompressible kinetic energy spectrum; and III) Spectral condensation, associated with the emergence of large scale rotating structures from the turbulent flow. We provide clear evidence that the inverse cascade phenomenon, previously observed in a range of classical fluid systems, can also occur in quantum turbulence.
We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with q −1 playing the same role as the Reynolds number Re= UR/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re≫1 in classical hydrodynamics, and q ≪1 in superfluid hydrodynamics. Another parameter of superfluid turbulence is the superfluid Reynolds number Re s = UR/κ , which contains the circulation quantum κ characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. A phase diagram of the dynamical vortex states is suggested.
We study freely decaying quantum turbulence by performing high-resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from 10243{1024}^{3}10243 to 40963{4096}^{3}40963 grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale $\ensuremath{\ell}$, and a second inertial range for scales smaller than $\ensuremath{\ell}$. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite-temperature effects in decaying quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions under the GPE. We use finite-temperature GPE simulations to extract mean-free path by measuring the spectral broadening in the Bogoliubov dispersion relation that we obtain from the spatiotemporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, we perform low-Reynolds-number simulations of the Navier-Stokes equations, in order to compare the decay of high-temperature quantum flows with their classical counterparts, and to further calibrate the estimations of the effective viscosity (based on the mean-free-path computations).