The idea that chaotic set of quantum vortices can mimic classical turbulence, or at least reproduce many main features, is currently actively being developed. Appreciating significance of the challenging problem of the classical turbulence it can be expressed that the idea to study it in terms of quantized line is indeed very important and may be regarded as a breakthrough. For this reason, this theory should be carefully scrutinized. One of the basic arguments supporting this point of view is the fact that vortex tangle decays at zero temperature, when the apparent mechanism of dissipation (mutual friction) is absent. Since the all possible mechanisms of dissipation of the vortex energy, discussed in the literature, are related to the small scales, it is natural to suggest that the Kolmogorov cascade takes the place with the flow of the energy, just as in the classical turbulence. In a series of recent experiment attenuation of vortex line density was observed and authors attribute this decay to the properties of the Kolmogorov turbulence. In the present work we discuss alternative possibility of decay of the vortex tangle, which is not related to dissipation at small scales. This mechanism is just the diffusive like spreading of the vortex tangle. We discuss a number of the key experiments, considering them both from the point of view of alternative explanation and of the theory of Kolmogorov turbulence in quantum fluids.
The idea that chaotic set of quantum vortices can mimic classical turbulence, or at least reproduce many main features, is currently actively being developed. Appreciating significance of the challenging problem of the classical turbulence it can be expressed that the idea to study it in terms of quantized line is indeed very important and may be regarded as a breakthrough. For this reason, this theory should be carefully scrutinized. One of the basic arguments supporting this point of view is the fact that vortex tangle decays at zero temperature, when the apparent mechanism of dissipation (mutual friction) is absent. Since the all possible mechanisms of dissipation of the vortex energy, discussed in the literature, are related to the small scales, it is natural to suggest that the Kolmogorov cascade takes the place with the flow of the energy, just as in the classical turbulence. In a series of recent experiment attenuation of vortex line density was observed and authors attribute this decay to the properties of the Kolmogorov turbulence. In the present work we discuss alternative possibility of decay of the vortex tangle, which is not related to dissipation at small scales. This mechanism is just the diffusive like spreading of the vortex tangle. We discuss a number of the key experiments, considering them both from the point of view of alternative explanation and of the theory of Kolmogorov turbulence in 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−
We investigate quantum vortex ring dynamics at scales smaller than the inter-vortex spacing in quantum turbulence. Through geometrical arguments and high-resolution numerical simulations, we examine the validity of simple estimates for the mean free path and the structure of vortex rings post-reconnection. We find that a large proportion of vortex rings remain coherent objects where approximately 75% of their energy is preserved. This leads us to consider the effectiveness of energy transport in turbulent tangles. Moreover, we show that in low density tangles, appropriate for the ultra-quantum regime, ring emission cannot be ruled out as an important mechanism for energy dissipation. However at higher vortex line densities, typically associated with the quasi-classical regime, loop emission is expected to make a negligible contribution to energy dissipation, even allowing for the fact that our work shows rings can survive multiple reconnection events. Hence the Kelvin wave cascade seems the most plausible mechanism leading to energy dissipation.
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−5/3 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.
There is a growing interest in the relation between classical turbulence and quantum turbulence. Classical turbulence arises from complicated dynamics of eddies in a classical fluid. In contrast, quantum turbulence consists of a tangle of stable topological defects called quantized vortices, and thus quantum turbulence provides a simpler prototype of turbulence than classical turbulence. In this paper, we investigate the dynamics and statistics of quantized vortices in quantum turbulence by numerically solving a modified Gross-Pitaevskii equation. First, to make decaying turbulence, we introduce a dissipation term that works only at scales below the healing length. Second, to obtain steady turbulence through the balance between injection and decay, we add energy injection at large scales. The energy spectrum is quantitatively consistent with the Kolmogorov law in both decaying and steady turbulence. Consequently, this is the first study that confirms the inertial range of quantum turbulence.
Quantum turbulence is characterised by the collective motion of mutually interplaying thin and discrete vortex filaments of fixed circulation which move in two mutually interacting fluid components. Despite this very peculiar nature determined by quantum-mechanical effects, turbulence in quantum fluids may exhibit very similar features to classical turbulence in terms of the vortex dynamics, energy spectrum and decay and intermittency. The recent work by Blaha et al. (2025 J. Fluid. Mech. 1015, A57) reveals an additional classical behaviour of quantum turbulence, by showing that the trajectories of starting vortices shed by accelerating airfoils in a quantum fluid are almost indistinguishable from their counterpart in classical viscous flows. These results strongly support the suggestive idea that turbulent flows, both classical and quantum, may be described by the collective dynamics of interacting, thin and discrete filaments of fixed circulation.
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.