We collect and describe the observed geometrical and dynamical properties of turbulence in quantum fluids, particularly superfluid helium and atomic condensates for which more information about turbulence is available. Considering the spectral features, the temporal decay, and the comparison with relevant turbulent classical flows, we identify three main limiting types of quantum turbulence: Kolmogorov quantum turbulence, Vinen quantum turbulence, and strong quantum turbulence. This classification will be useful to analyze and interpret new results in these and other quantum fluids.

## I. INTRODUCTION

Turbulence, ubiquitous in nature and technology, is still a major problem of classical physics. As new contexts of this problem are investigated, turbulence is better understood and new physics is discovered. A context of current interest is turbulence in quantum fluids, or *quantum turbulence* (QT), the study of which was pioneered by the late Vinen.^{1} Quantum fluids are fluids that, due to Bose–Einstein condensation of the constituent particles, exhibit quantum mechanical effects at the macroscopic level. Example systems are liquid helium (bosonic ^{4}He and fermionic ^{3}He), ultracold atomic gases, polariton condensates, magnons, electrons in metals, the interior of neutron stars, and models of dark matter.

The main property of quantum fluids^{2} is that their vorticity is confined to individual (*discrete*) vortex lines of fixed circulation $ \kappa = h / m$, where *h* is Planck's constant and *m* is the mass of the relevant boson. This quantization of the vorticity, conjectured by Onsager^{3} and experimentally demonstrated in liquid helium by Vinen,^{4} is a fundamental property, which arises from the existence of a governing macroscopic wavefunction. It also makes quantum turbulence visibly different from ordinary (classical) turbulence: whereas classical turbulence contains a *continuous* distribution of eddies of arbitrary sizes and strengths, quantum turbulence consists of a disordered tangle of *discrete* vortex lines of fixed circulation and thickness, as shown in Fig. 1.

A second property of quantum fluids is the two-fluid nature.^{2} The Bose–Einstein condensate, which exists when the temperature of the system is below a critical temperature, constitutes a fluid, which is free from viscosity. At non-zero temperature, this condensate coexists with an incoherent thermal part, which plays an important role in turbulence as it dissipates the kinetic energy of the vortex lines. In liquid helium at sufficiently high temperature, the mean free path of thermal excitations is short enough to form what is effectively a classical fluid called the normal fluid component. Liquid helium, therefore, becomes a mixture of an inviscid superfluid (associated with the condensate) and a viscous normal fluid, as described by Landau's two-fluid theory. The thermal excitations (phonons and rotons) are scattered by the velocity field of the vortex lines, creating a mutual friction force between normal fluid and superfluid components. When stirred, both fluid components may become turbulent, creating an unusual doubly turbulent state^{5,6} consisting of continuous normal fluid eddies and discrete superfluid vortex lines, which interact with each other.

Despite these differences between quantum fluids and ordinary fluids, experiments with turbulent liquid helium have revealed that, under certain conditions, there are remarkable similarities between quantum turbulence and classical turbulence. However, under other conditions, quantum turbulence is wholly unlike classical turbulence.

As the study of quantum turbulence is currently being extended from superfluid helium to atomic Bose–Einstein condensates and other quantum fluids, the existence (or not) of a classical limit for these out-of-equilibrium quantum systems is an important problem. At this stage, it is, therefore, useful to critically review the main observed properties of turbulence in quantum fluids, to consider the conditions under which these properties appear, and to compare the different regimes of quantum turbulence with each other and with classical turbulence. Putting together information which is scattered over the literature, our aim in this paper is to identify and characterize the types of quantum turbulence, which currently exist.

## II. PHENOMENOLOGY OF QUANTUM TURBULENCE

Experimental, theoretical, and numerical studies suggest that the phenomenology of quantum turbulence can be organized in the following main types: (i) *Kolmogorov quantum turbulence*, (ii) *Vinen quantum turbulence*, and (iii) *strong quantum turbulence*. These names and this classification are motivated by comparisons with classical turbulence. We stress that we limit our attention to the quantum fluids, which, until now, have received most of the attention from the point of view of turbulence: superfluid ^{4}He, superfluid ^{3}He-B, and atomic condensates.

The two main theoretical and numerical methods for quantum turbulence^{2} are the Gross–Pitaevskii equation (GPE) and the vortex filament method (VFM). To appreciate the assumptions, advantages, and shortcomings of these methods, we start by briefly describing them.

The GPE (a Schroedinger equation with a cubic nonlinearity) is a mean-field model of a Bose–Einstein condensate at zero temperature. The GPE accounts for vortex lines and their dynamics (including Kelvin waves, vortex reconnections and the finite vortex core radius), and for the dynamics of sound waves. The GPE model is, hence, a compressible model. If a suitable external potential is included in the GPE, the observed behavior of trapped atomic condensate is accurately described (including oscillatory modes, density profiles, and expansion of the condensate), provided that the temperature is much less than the critical temperature. In the GPE context, a homogeneous condensate is usually meant to model superfluid helium: In this case, the dispersion relation includes phonons at small wavenumbers, but lacks the roton dip at large wavenumbers, which is characteristic of ^{4}He; therefore, the GPE is considered a qualitative model of liquid helium at very low temperatures. There are variants of the GPE, which include rotons.^{7}

The VFM is based on the remark that the vortex core radius, *a*_{0}, is many order of magnitude smaller than the average distance between vortex lines, $\u2113$, in typical experiments. This remark allows the classical description of a vortex line as a space-curve of infinitesimal thickness, which carries the circulation *κ* and moves according to Schwarz's equation.^{2} Schwarz's equation determines the velocity of the vortex line at each point along it in terms of the local and global geometry of the vortex configuration (via the Biot–Savart law^{2}) and the friction with a prescribed normal fluid velocity field. The VFM assumes that the fluid is incompressible. The dynamics of vortex lines (including Kelvin waves) is captured very well, but vortex reconnections must be performed algorithmically. The major advantage of the VFM is that finite-temperature effects are accounted by known friction coefficients; the main disadvantage is that sound generation is not included in the model. A modern variant^{8} of the VFM couples the Schwarz equation to a Navier–Stokes equation for the normal fluid modified by the presence of friction, yielding a fully self consistent description of the two-fluid hydrodynamics of ^{4}He at intermediate and high temperatures.

Before we describe the three types of quantum turbulence, in Sec. II A, we recall the basic properties of classical turbulence, which serves as a reference.

### A. Classical turbulence

The paradigm of classical turbulence is the statistically steady, homogeneous, and isotropic turbulence (HIT) of a viscous incompressible fluid.^{9} This state of disorder is achieved, by continual supply of kinetic energy and removal of heat to compensate for the dissipation arising from the fluid's viscosity, for example, in a wind tunnel.

Theoretically and numerically, classical turbulence is studied by solving the Navier–Stokes equation. HIT is the conceptual pillar, which supports the toolkit for engineering applications (the $ k \u2212 \u03f5$ model and large eddies simulations). The theory of HIT makes successful predictions even outside the range for which it was originally designed,^{11} e.g., inhomogeneous and unsteady flows. Although most flows are not homogeneous and isotropic at the large length scales, it is reasonable to expect that, if the turbulence is intense enough and is probed away from boundaries, the properties at scales which are smaller than the anisotropic large scales are independent of the large-scale flow structure. (The effect of large-scale anisotropy on the small-scale properties has been investigated in detailed experiments.^{10}) Quantum turbulence is often investigated in configurations with clear asymmetry, such as helium flows along channels and cigar-shaped atomic condensates, so it is natural to ask if HIT is a useful paradigm. The answer is that since the study of quantum turbulence is still at an early stage, the properties of HIT are the natural reference, but it is important to determine experimentally or numerically the degree of anisotropy of a particular turbulent flow, as we shall see.

Classical HIT has two characteristic length scales: the large length scale *D* at which kinetic energy is injected and the small length scales *η* (called the Kolmogorov length scale) at which kinetic energy is dissipated by viscous forces. Vortices (eddies), continually created at large length scales, are unstable and break up into smaller and smaller eddies to which they transfer their kinetic energy. It is convenient to introduce the wavenumber $ k = 2 \pi / r$ corresponding to eddies of size *r*. In the inertial range of wavenumbers $ k D = 2 \pi / D \u226a k \u226a k \eta = 2 \pi / \eta $, the energy transfer is self-similar, viscosity playing no role. Here, the turbulence displays the celebrated kinetic energy spectrum $ E \u0302 ( k ) \u223c k \u2212 5 / 3$ (called the Kolmogorov spectrum), which is interpreted as the signature of a dissipationless energy cascade from large eddies to small eddies. For $ k \u226b k \eta $, viscous forces dominate the dynamics and the energy spectrum decays exponentially with *k*. Whereas the kinetic energy is concentrated at the large length scale *D*, the vorticity is concentrated at the small length scale *η*; more precisely, the distribution of enstrophy $ \Omega \u0302 ( k )$ (vorticity squared) scales as $ \u223c k 1 / 3$, peaking at $ k \eta $ and then decaying for $ k > k \eta $.

The intensity of the turbulence is quantified by the non-dimensional Reynolds number $ Re = U D / \nu $, where *U* is the flow speed at the large length scale *D* and *ν* is the fluid's kinematic viscosity. Physically, the Reynolds number is a measure of the ratio of inertial and viscous forces in the Navier–Stokes equation. It can also be used to express the extent of the linear separation between the smallest length scale and the largest length scale according to $ \eta / D \u2248 Re \u2212 3 / 4$.

Another property of HIT that is relevant to quantum turbulence is that the distributions of the values of velocity components are Gaussian, as confirmed by experiments^{12} and numerical simulations.^{13} (Velocity increments, however, follow power-law statistics at small scales.)

Finally, in classical turbulence, the rate of the kinetic energy dissipation, *ϵ*, tends to a non-zero constant in the limit $ \nu \u2192 0$ (i.e., the limit $ Re \u2192 \u221e$). This result is in sharp contrast to what happens in laminar flows, where the dissipation goes to zero with the viscosity. Physically, in a turbulent flow, the reduction of the viscosity is compensated by the creation of motions at smaller length scales containing much vorticity but little energy. This property (called the dissipation anomaly^{14}) is clearly relevant to the task of comparing classical and quantum turbulence because superfluids have zero viscosity.

We note that although the scenario that we have described is robust, it is only a first approximation: Higher order statistics depart from the Kolmogorov scaling, representing intermittency corrections,^{9} which are beyond the scope of this paper.

### B. Kolmogorov quantum turbulence

There is general consensus^{15} that, under certain conditions, quantum turbulence takes a form, which is similar to classical turbulence, and therefore, we examine this type first.

In any turbulent quantum fluid, there is necessarily a third characteristic length scale (besides the large length scale of the energy injection and the small length scale of the energy dissipation): the average distance between vortex lines, $\u2113$. In the experiments, this parameter is usually estimated as $ \u2113 \u2248 L \u2212 1 / 2$, where *L* is the vortex line density (vortex length per unit volume). The vortex line density can be experimentally measured in liquid helium by a number of techniques, such as second sound attenuation, ion trapping, and Andreev reflection.^{16} The vortex line density is often considered a measure of the intensity of quantum turbulence, although, for the sake of comparison between experiments, a better measure^{17} would be the dimensionless parameter $ D / \u2113$.

Direct evidence of the classical $ k \u2212 5 / 3$ Kolmogorov scaling in quantum turbulence was provided by experiments in which ^{4}He was stirred by rotating propellers^{18,19} and towed grids^{20} or was driven along wind tunnels.^{21} The local velocity fluctuations were measured by miniature Pitot tubes^{18} or cantilever anemometers.^{19} This type of quantum turbulence has been called “quasi-classical” or *Kolmogorov quantum turbulence*. The experimental evidence was obtained over a wide temperature range, from the critical temperature down to temperatures where the normal fluid fraction is only a few percent.

At intermediate to high temperatures (relative to the critical temperature), the role of the mutual friction is crucial, allowing energy exchange between normal fluid and superfluid. The mutual friction depends on the relative velocity of the two-fluid components, the density of vortex lines, and dimensionless temperature-dependent friction coefficients *α* and $ \alpha \u2032$. Therefore, at large length scales, the turbulent normal fluid and superfluid tend to move together, locked by the mutual friction, a situation referred to in the literature as “co-flow” to distinguish it from “counterflow” (see Sec. III). The situation in ^{3}He-B is different because the normal component is so viscous that its flow is laminar in all experiments; unlike in ^{4}He, the normal fluid of ^{3}He plays no dynamical role and simply provides a friction to the motion of the vortex lines. Nevertheless, even in ^{3}He-B, a Kolmogorov turbulence is predicted^{22,23} when the cascading dynamics is significantly faster than the dissipative action arising from the mutual friction. In both ^{4}He and ^{3}He-B, the proportion of normal fluid becomes less and $ \alpha , \alpha \u2032 \u2192 0$ as the temperature is lowered, leaving what is effectively a pure superfluid.

Kolmogorov quantum turbulence at very low temperature was created in both ^{4}He and ^{3}He-B by injecting charged vortex rings^{24,25} and by oscillating grids and forks.^{26,27} In a turbulent superfluid, the rms vorticity is usually identified^{28} as $ \u2248 \kappa L$. The prediction^{29} that Kolmogorov quantum turbulence decays as $ L \u223c t \u2212 3 / 2$ for large *t* was experimentally verified^{20,24} in ^{4}He using second sound or ion techniques and in ^{3}He-B using Andreev scattering.^{26,30} The corresponding decay of the total kinetic energy scales as $ t \u2212 2$ at large *t*.

The numerical simulations, which contributed to the evidence of the Kolmogorov $ k \u2212 5 / 3$ scaling (in both statistically steady and decaying regimes), modeled the superfluid using the two models described in Sec. II: the GPE^{31–33} and the VFM.^{36–41} VFM simulations also reproduced the observed $ L \u223c t \u2212 3 / 2$ temporal decay of the turbulence.

It is important to recall from Sec. II that the VFM is incompressible and the GPE is compressible. In the GPE model, vortex acceleration induces density waves,^{34} so a turbulent tangle of vortex lines moving under the influence of each other necessarily involves also sound waves, as early reported.^{31,33} Vortex reconnections and annihilation also induce sound waves.^{35} At relatively large temperatures, the most important mechanism to dissipate the kinetic energy of the vortex lines is the friction with the normal fluid (an effect which is observed in the experiments and is described well by the VFM), whereas in the limit of zero temperature it is sound generation (an effect which is described well by the GPE). The second effect has also been observed in the laboratory: At mK temperatures in ^{4}He, the normal fluid is utterly negligible, but a turbulent tangle of vortex lines still decays rather quickly with time.^{27}

At non-zero temperatures, the coupled Kolmogorov dynamics of normal fluid and superfluid was demonstrated also using other models, such as the coarse-grained Hall-Vinen-Bekarevic-Khalatnikov (HVBK) equations,^{42} a modified Leith model,^{43} and modified shell models.^{44,45} In particular, the HVBK model agreed with the striking experimental verification^{46} in ^{4}He of the so-called 4/5 law of classical turbulence, a statement about the third moments of velocity increments, which can be derived exactly from the Navier–Stokes equation for HIT in the inertial range. Remarkably, the Kolmogorov picture of classical turbulence is also able to capture in quantum turbulence the behavior of the scaling exponents of the low-order velocity circulation moments;^{7} higher order moments are described by a bifractal model, as in classical turbulence.^{47}

A snapshot of Kolmogorov quantum turbulence computed in a periodic domain is quite similar to vortex tangle displayed in Fig. 1, consisting of a disordered tangle of vortex lines whose average radius of curvature is of the order of magnitude of the average inter-vortex distance $\u2113$. In the statistical steady-state, the vortex lines continually collide and reconnect at the rate of $ \u2248 \kappa L 5 / 2$ reconnections per unit time per unit volume.^{48}

It is important to remark that the appearance of the classical Kolmogorov scalings in a quantum fluid is limited to the “classical length scales”^{49} corresponding to wavenumbers $ k D \u226a k \u226a k \u2113$ (where $ k \u2113 = 2 \pi / \u2113$). Only in this range, it is possible for vortex lines to locally polarize (even if only partially), effectively creating classical eddies that can undergo the process of vortex stretching (vortex stretching on individual vortex lines is prevented by the quantization of the circulation). Usually, this polarization is poorly visible in images such as Fig. 1, which do not display the orientation of the vortex lines, but it becomes apparent by computing a suitably defined coarse-grained vorticity field.^{39,50} The coarse-graining procedure reveals that bundles of vortex lines can spontaneously come together, parallel to each other in the mist of the random background of the other vortex lines, creating regions of relatively large velocity and energy. The remaining vortex lines, although containing most of the vortex length,^{51} contribute less to the energy because their velocity fields tend to cancel out. A similar effect^{52} takes place in classical HIT, where tubular regions of large enstrophy and energy are responsible for the $ k \u2212 5 / 3$ spectrum, and the rest of the flow is incoherent.

On the contrary, in the “quantum length scale”^{49} range of wavenumbers $ k \u226b k \u2113$, the dynamics is non-classical, because it strongly depends on the quantization of the circulation, which has no classical analogue. In this range, the energy spectrum scales as $ k \u2212 1$, which is the spectrum of an isolated straight vortex (at length scales shorter than $\u2113$, the dominant velocity field arises from the nearest vortex, which seems effectively straight at this scale).

The scenario that we have described (different dynamics of the turbulence at the classical and at the quantum length scales) is confirmed by studies of the statistics of velocity components. Measurements^{53} in ^{4}He performed using tracer particles smaller than $\u2113$ (as well as numerical simulations which focused on the velocity at given points in space^{54,55}) revealed velocity distributions, which scale as $ v \u2212 3$ at large *v*; this power-law behavior disagrees with the Gaussian statistics of classical HIT. More careful numerical^{56} and experimental^{57} studies verified that, if the measurement region (in time or space) extends to distances larger than $\u2113$, classical Gaussian statistics are recovered.

The disordered vortex tangle such as that shown in Fig. 1 suggests that quantum turbulence has a rich topology. A recent study^{58} has shown that this is indeed the case. In the statistical steady-state, the continual vortex reconnections knot and unknot vortex lines, sustaining a spectrum of vortex knots (closed vortex loops with non-trivial topology, the simplest of which is the trefoil). Surprisingly, the vortex tangle always contains some knots of very high order as well as a non-zero degree of linkage between vortex lines. The precise relation between the topology and the geometry/dynamics of the turbulence is still a mystery, but this result hints at energy implications (vortex reconnections, which create and destroy knots, represent kinetic energy loss in the form of sound radiation). It has even been suggested^{59} that the decay of superfluid turbulence may follow particular topological pathways.

Finally, as in classical turbulence, intermittency corrections are responsible for deviations from the self-similar Kolmogorov statistics.^{7,60,61}

### C. Vinen turbulence

If quantum turbulence were always of the Kolmogorov type described in Sec. II B, we would conclude that, since the turbulence contains enough quanta of circulation, the classical limit is indeed recovered in agreement with Bohr's correspondence principle. However, a different type of quantum turbulence, first envisaged by Volovik,^{62} was experimentally identified in ^{4}He by Walmsley and Golov^{25} and in ^{3}He-B by Bradley *et al.*^{26} This type of turbulence, which decays as $ L \u223c t \u2212 1$ at large times (corresponding to the total kinetic energy that decays as $ t \u2212 1$) is called “ultra-quantum” or *Vinen quantum turbulence*. Numerically, the $ L \u223c t \u2212 1$ decay was seen in simulations^{38} of Walmsley and Golov's experiment and in simulations of turbulence driven by a uniform normal flow^{48} (both using the VFM); it was also seen in simulations of the thermal quench of a Bose gas^{63} using the GPE. In addition to reproducing the $ L \u223c t \u2212 1$ decay, these simulations revealed that the energy spectrum of Vinen quantum turbulence is different from that of Kolmogorov quantum turbulence in two important respects: it lacks the concentration of energy at the large length scales near *k _{D}* typical of classical turbulence, peaking instead near $ k \u2248 k \u2113$, and it scales as $ k \u2212 1$ for large

*k*. The energy spectrum of Vinen quantum turbulence is thus reminiscent of the energy spectrum of a random gas of vortex rings of radius

*R*, which peaks at $ k \u2248 1 / R$ and scales as $ k \u2212 1$ for $ k > 1 / R$ (if the rings' radii are not the same but span a distribution of values, then the peak near $ 1 / R$ broadens). The absence of the $ k \u2212 5 / 3$ scaling suggests that in Vinen quantum turbulence, the energy transfer from small wavenumbers $ \u223c k D$ to wavenumbers $ \u223c k \u2113$ is weak or essentially absent.

The lack of polarization of the vortex lines in Vinen quantum turbulence becomes apparent by computing the coarse-grained vorticity, which is very small (i.e., vortex lines tend to be randomly oriented with respect to each other). For this reason, the local mesoscale helicity, $ h ( \zeta )$ (where *ζ* is the arc length), which measures the non-local vortex interaction,^{40} is smaller than in Kolmogorov quantum turbulence. For the sake of illustration, the vortex lines of Fig. 1 are color-coded according to $ | h ( \zeta ) |$: a low value (blue color) at a point along a vortex line means that, at that point, the line moves with velocity predominantly due to the local curvature; a high value (yellow color) means that the line's velocity is predominantly due to other vortex lines. The weak vortex interaction in Vinen quantum turbulence is reflected in the rapid decay with distance of the velocity correlation function, as found by Stagg *et al.*^{63}

A model that accounts for the generation and decay of both Kolmogorov and Vinen turbulence based on the fluxes of energy at the classical and the quantum length scales was presented by Zmeev *et al.*^{64} The key ingredient to create Vinen turbulence is either the lack of forcing at the large length scales (in the steady case) or an initial condition that lacks sufficient energy at the large length scales (in the unsteady case). The paradigm is the initial condition, which is used to model the thermal quench of a Bose gas and the formation of a condensate:^{63,65} The phase is spatially random, and the occupation number is uniform in *k*-space: Without any anisotropy, in three-dimensional flows, there is no mechanism to create large flow structures.^{66,67}

### D. Quantum turbulence at small length scales

In quantum turbulence, the physics of the small length scales deserves special attention. In general, the friction with the normal fluid damps the energy of the vortex lines. Small-scale vortex structures such as small vortex rings or loops shrink and vanish, passing their energy to the normal fluid, which turns it into phonons (heat) via viscous forces. Other vortex structures that are affected by the friction are Kelvin waves (helical oscillations of the centerline of an unperturbed vortex). Kelvin waves are created by vortex interactions and reconnections,^{68} as seen experimentally,^{69} or simply thermally.^{70} When two vortex lines collide,^{71} immediately after the reconnection both lines acquire the shape of a cusp. As the two cusps relax, Kelvin wavepackets are radiated away along the vortex lines, and the friction reduces their amplitude as they propagate. The frequency of a Kelvin wave of wavenumber *k* is $ \omega \u2248 \beta k 2$ (i.e., shorter Kelvin waves move faster), where $ \beta = \kappa / ( 4 \pi ) \u2009 ln \u2009 ( k a 0 )$ and *a*_{0} is the vortex core radius. The amplitude of the Kelvin wave decays exponentially with time as $ exp \u2009 ( \u2212 \alpha \beta k 2 t )$, where *α* is a temperature-dependent friction parameter. As a consequence, the geometrical appearance of the turbulent vortex tangle changes with temperature: At high temperatures, the vortex lines are very smooth, but as the temperature is reduced and the friction coefficient *α* decreases, the vortex lines display cusps, kinks and high frequency Kelvin waves.^{72}

Since the superfluid has zero viscosity, the classical definition of Reynolds number does not apply. However, using the same argument for which the Reynolds number measures the ratio of inertial to viscous forces in the Navier–Stokes equation, a superfluid Reynolds number $ Re s$ can be defined by the ratio of inertial and friction forces. One finds^{73} $ Re s = ( 1 \u2212 \alpha \u2032 ) / \alpha $, where *α* and $ \alpha \u2032$ are, respectively, the dissipative and non-dissipative friction coefficients. Note that $ Re s$ depends only on temperature, not on *U* or *D*. Since *α* and $ \alpha \u2032$ tend to zero for $ T \u2192 0$, the limit $ Re s \u2192 \u221e$ corresponds to the temperature $ T \u2192 0$.

The lower the temperature, the more freely the Kelvin waves propagate, and the further the distribution of curvatures extends to large values. Recent numerical simulations^{41} which carefully resolved numerically all Kelvin waves excited in the turbulence showed that, as $ Re s$ increases, the rate of kinetic energy dissipation arising from mutual friction, *ϵ*, at first decreases, then it flattens and becomes constant, as for the classical dissipation anomaly. As in classical turbulence, the generation of small-scale vortex structures as the turbulence becomes more intense prevents the dissipation from vanishing, unlike what happens in laminar flows. This is another remarkable similarity between classical and quantum turbulence. Being a property of the small length scales, this property applies to both Kolmogorov quantum turbulence and Vinen quantum turbulence.^{41}

If the temperature is further reduced, a different route to dissipate kinetic energy becomes possible, as discovered by Svistunov.^{74} The nonlinear interaction of finite-amplitude Kelvin waves creates shorter and shorter waves, which rotate more and more rapidly. This Kelvin wave energy cascade can transfer energy to length scales much smaller than $\u2113$, until they are sufficiently small that radiation of sound (phonons) takes place. Vinen^{75} estimated that in ^{4}He, the crossover from friction dissipation to sound dissipation occurs at approximately $ T \u2248 0.5 \u2009 K$ for $ L \u2248 10 10 \u2009 m \u2212 2$ (the precise value depending on whether it is dipole or quadrupole radiation).

At low temperatures, quantum turbulence may, thus, contain two energy cascades: a Kolmogorov cascade of bundled vortices (analog to classical eddies) in the range $ k D \u226a k \u226a k \u2113$, and a Kelvin wave cascade on individual vortex lines for $ k \u226b k \u2113$. Large GPE simulations of quantum turbulence show that the two cascade are separated by a bottleneck region^{76} and verified^{77} the predicted^{78} scaling behavior of the Kelvin cascade. Recently, experimental evidence in liquid helium of the Kelvin cascade has been announced.^{79}

### E. Strong turbulence

In three-dimensional atomic condensates, direct nondestructive visualization of vortex lines has been achieved only for one or two vortex lines at the time.^{80,81} The comparison between quantum turbulence in atomic Bose–Einstein condensates and in superfluid helium is, therefore, hindered by the lack of experimental techniques to visualize the turbulence, measure velocity fluctuations, and determine the vortex line density. In two-dimensional condensates, instead, vortices can be visualized and counted, but the physics of two-dimensional turbulence is very different and beyond the scope of this paper.

Another significant difference between atomic condensates and superfluid helium is that, currently, the ratio between the largest and the smallest length scales (the system's size *D* and the vortex core radius *a*_{0}, which is of the order of the healing length) is typically $ D / a 0 \u2248 10 2$. This value must be compared to $ D / a 0 \u2248 10 10$ in the largest turbulent ^{4}He facility (the SHREK facility^{82}) and $ D / a 0 \u2248 10 5$ in a small $ 5 \u2009 mm$ sample of ^{3}He. As a consequence, the range of *k*-space available to determine scaling laws in turbulent atomic condensates is quite limited. A final difference between atomic condensates and superfluid helium, again from the point of view of turbulence, is that most atomic condensates are experimentally confined by harmonic trapping potentials, which create a non-uniform density profile. The recent development of box trap potentials^{83} now allows condensates with uniform density, opening the way to better comparison with classical HIT and quantum turbulence in superfluid helium.

Although various techniques to create turbulent vortex lines in condensates have been proposed and realized (phase imprinting,^{54,84} rotation,^{85} and laser stirring^{86}), until now the most successful strategy has been shaking^{87} or oscillating the trap.^{88} Here, we concentrate on results obtained by shaking the trap back and forth.^{87}

The most direct signature of turbulent dynamics in atomic condensates is provided by two-dimensional density absorption images after expansion. These images have revealed large density fluctuations and fragmentation, and the lack of the expected inversion of the aspect ratio of an initially cigar-shaped condensate.^{87} However, it is the momentum distribution *n*(*k*) (determined, again, by expanding the condensate), which has given the most important physical information about the turbulence,^{88–90} revealing inter-scale energy transfer^{91,92} from small to large *k*.

Provided the temperature is sufficiently small compared to the critical temperature, the dynamics of atomic condensates can be accurately simulated by the GPE. The computed^{93} momentum distribution scales as $ n ( k ) \u223c k \u2212 2.55$ for large *k*, in good agreement with the experiments [ $ n ( k ) \u223c k \u2212 2.60$] in the range $ k D < k < k \xi $ (where *k _{D}* refers to the Thomas–Fermi radius and $ k \xi $ to the healing length).

Steeper momentum distributions [ $ n ( k ) \u223c k \u2212 3.5$, and, more recently,^{94}^{,} $ n ( k ) \u223c k \u2212 3.2$] were also observed in experiments in which the turbulence was excited by oscillating a box trap.^{88} The large density fluctuations, which are revealed by the numerical simulations^{93} of the experiments have never been observed in superfluid helium (where the velocity of vortex lines is much smaller than the speed of sound); the traditional reference problem—classical HIT—is also incompressible. Fortunately, another classical reference problem is available: *weak turbulence* of waves. Zakharov's statistical theory^{95} of interacting small-amplitude waves predicts $ n ( k ) \u223c k \u2212 3$, which is not far from values seen in condensates, particularly if one accounts for the finite-size of the system.^{94} The difference between the scaling exponents observed in harmonic traps and in box traps, and between experiments and wave turbulence theory, may be due to the different density profile (which is uniform in box traps), or the presence of vortex lines (which is not included in Zakharov's theory). In particular, the numerical simulations of shaken condensates^{93} show many small vortex rings and loops of size comparable to the healing length, which coexist with the large density fluctuations, as shown in Fig. 2. These vortex lines are not spread uniformly (they seem to concentrate at the back of the moving condensate) and are oriented randomly, making the turbulence non-homogeneous but isotropic. Issues worth exploring are the applicability of classical weak turbulence theory to turbulent systems (like these condensates) having very large density waves and the effect of the quantum pressure.

By comparing with turbulent superfluid helium, the absence of long vortex lines in turbulent condensates is striking (perhaps the reason is the way vortices nucleate from unstable solitons created by the shaking near the boundary). Perhaps other techniques to excite turbulence in atomic condensates may create vortex tangles more similar to turbulent helium. No scaling law is clearly visible in the spectrum of the incompressible kinetic energy of a shaken condensate at large scales—the radius of curvature of the small vortex loops is too small for the usual $ k \u2212 1$ range of isolated straight vortices to stand out; the only recognizable scaling is the $ k \u2212 3$ behavior near $ k \xi $ caused by the vortex core funnel.^{96} Nevertheless, the temporal decay of the vortex length scales as $ t \u2212 1$ for large *t*, which is the signature of random vorticity typical of Vinen turbulence. Here, the decay is clearly due to sound radiation.

Considering together the properties which have been observed in turbulent condensates and comparing them to known turbulence types, we conclude that quantum turbulence in atomic condensates fits neither the Kolmogorov type (for it lacks the characteristic polarized vortex lines) nor the Vinen type (the random vorticity typical of Vinen turbulence is dominated by strong density waves); it is also unlike the classical HIT scenario. Its closest classical analog is clearly Zakharov's *weak-wave turbulence*. However, it differs from it due to the much stronger nonlinearity of the waves, the additional presence of vortices whose effect on the waves' dynamics is still unexplored, and the mechanism for energy loss (sound radiation). It seems that quantum turbulence in atomic condensates is clearly a type of turbulence with its own characteristic properties. The name *strong quantum turbulence* highlights the role played by the large density fluctuations, making reference to the expression used in the classical literature for turbulent weakly interacting waves.

## III. DISCUSSION

The properties of the three types of quantum turbulence which we have reviewed are summarized in Table I; typical energy spectra are shown in Fig. 3. It must be stressed that this classification represents only a convenient way of organizing experimental and numerical observations at this stage of progress. The aim is to help identify and compare the underlying physics processes. As more quantum fluids and more quantum turbulent flows will be discovered and investigated, this classification may change, or more intermediate types will become known. A different way of organizing the properties of quantum turbulence would be to distinguish between incompressible quantum turbulence (comprising Kolmogorov type and Vinen type) in which density variations play no role, and compressible quantum turbulence, comprising the weak turbulent case (without vortices) and strong turbulence case (with vortices). The drawback of this classification is that density variations are present also in the Kolmogorov and Vinen types (large density changes are essential for vortex reconnections, and the generation of small density waves is the main sink of the kinetic energy of the vortices at very low temperatures).

. | Kolmogorov QT . | Vinen QT . | strong QT . |
---|---|---|---|

Energy peaks at | Small k | Intermediate k | Small k |

$ E \u0302 ( k ) \u223c$ | $ k \u2212 5 / 3$ | $ k \u2212 1$ | $ k \u2212 3$ |

scaling range | $ k D \u226a k \u226a k \u2113$ | $ k \u2248 k \u2113$ | $ k \u2009 \u2272 \u2009 k \xi $ |

$ L ( t ) \u223c$ | $ t \u2212 3 / 2$ | $ t \u2212 1$ | $ t \u2212 1$ |

Vortex configuration | Partially polarized | Random | Random |

. | Kolmogorov QT . | Vinen QT . | strong QT . |
---|---|---|---|

Energy peaks at | Small k | Intermediate k | Small k |

$ E \u0302 ( k ) \u223c$ | $ k \u2212 5 / 3$ | $ k \u2212 1$ | $ k \u2212 3$ |

scaling range | $ k D \u226a k \u226a k \u2113$ | $ k \u2248 k \u2113$ | $ k \u2009 \u2272 \u2009 k \xi $ |

$ L ( t ) \u223c$ | $ t \u2212 3 / 2$ | $ t \u2212 1$ | $ t \u2212 1$ |

Vortex configuration | Partially polarized | Random | Random |

The flexibility and versatility of the atomic condensates allow for the laboratory creation of more complex quantum fluids, which may in turn support novel types of quantum turbulence. One example are dipolar atomic condensates, in which the particles possess sizeable magnetic dipole moments.^{97} The quantum fluid then gains a magnetic character and a response to imposed magnetic fields analogous to a classical ferrofluid, leading to regimes of polarized and stratified quantum turbulence and a new modality to externally drive and control the fluid.^{98} A second example is the multi-component condensates,^{99} in which distinct condensates coexist and interact such that the system embodies the quantum analog of a multifluid. The fluids can either be miscible or immiscible depending on the atomic parameters,^{100} and the individual components can be independently addressed, e.g., to controllably induce co-flow or counterflow. For both the dipolar and multi-component systems, the additional interactions and degrees of freedom may provide additional channels for energy transfer and dissipation, as well as supporting distinct instabilities, which may lead to the onset of turbulence. Although quantum turbulence has yet to be experimentally probed in either system, advances have enabled vortex states to be experimentally created in both^{101,102} such that extending to turbulent regimes is within reach.

The recognition of different types of quantum turbulence may be particularly useful for problems, which cannot be directly accessed in the laboratory, and numerical modeling is constrained by scarce observations. Two examples of current astrophysical interest are worth mentioning. The first is neutron stars (pulsars). There is some evidence that the observed rotational glitches of these stars are related to quantum turbulence in the interior.^{103} Unfortunately the only experimental information about rotating quantum turbulence is from few studies of rotating thermal counterflow^{104,105} in helium, and thermal counterflow turbulence is an old problem still under investigation (as we shall see at the end of this section). Numerically, rotating quantum turbulence is still at early stage of investigation and appears to be very different from classical rotating turbulence.^{106} A better understanding of the spin-down and spin-up dynamics of quantum turbulence would help in setting up more efficient observational protocols for pulsar glitches.

The second example is dark matter. If it consists of light bosons, a possible model would be a self-gravitating condensate described by the Schroedinger–Poisson equation. Numerical simulations of dark matter galactic haloes have revealed tangles of vortex lines with a Vinen-like kinetic energy spectrum,^{107,108} alongside huge density fluctuations, which are even larger than in the strong turbulence of atomic condensates. Such model may provide information about micro-lensing, which would help to confirm or discard this approach to the dark matter problem.

Finally, we remark that the very first quantum turbulent flow, which was studied in Vinen's pioneering work—thermal counterflow in superfluid helium—is surprisingly hard to classify. This flow is created by an electric heater, which deposits a given heat flux into a sample of helium, resulting in the opposite motion (counterflow) of the normal fluid component (away from the heater) and the superfluid component (towards it). If this counterflow velocity exceeds a small critical value, a turbulent tangle of vortex lines is generated, which limits the otherwise ideal heat conducting property of liquid helium. Thermal counterflow has, therefore, important applications of cryogenic engineering. Although there is experimental and numerical evidence that at small heat flux the turbulence is Vinen-like (decaying as $ L \u223c t \u2212 1$), at large heat flux the decay is Kolmogorov-like^{109} (decaying as $ L \u223c t \u2212 3 / 2$). Because of the opposite motion of normal fluid and superfluid, this quantum turbulent flow, which has no classical analogy, does not fit our classification; indeed, it is still the subject of investigations.^{110}

In conclusion, the legacy of Vinen's early experiments on the quantization of circulation^{4} and turbulent heat conduction in liquid helium^{1} has been profound. Taken together, these two experiments have opened new doors to out-of-equilibrium quantum systems and the meaning of the classical limit, which are currently expanding in directions, which he could not have anticipated.

## ACKNOWLEDGMENTS

C.F.B. is indebted to W. F. Vinen for support and discussions over many years. The financial support of UKRI (UK Research and Innovation) under grant “Quantum simulators for fundamental physics” (ST/t006900/1) is acknowledged.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Carlo F. Barenghi:** Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (lead); Writing – review & editing (equal). **Holly AJ Middleton-Spencer:** Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Writing – review & editing (equal). **Luca Galantucci:** Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (lead); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). **Nick G. Parker:** Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created in this study.

## REFERENCES

*A Primer on Quantum Fluids*

*Turbulence: The Legacy of A. N. Kolmogorov*

*Superfluid Turbulence, Progress in Low Temperature*

*Kolmogorov Spectra of Turbulence I: Wave Turbulence*