Quantum turbulence

The time evolution of the ground state wave function of a zero-temperature Bose-Einstein condensate (BEC) gas is well described by the Hamiltonian Gross-Pitaevskii (GP) equation. Using a set of appropriately interleaved unitary collision-stream operators, a qubit lattice gas algorithm is devised, which on taking moments, recovers the Gross-Pitaevskii (GP) equation under diffusion ordering (time scales as length 2 ). Unexpectedly, there is a class of initial states whose Poincaré recurrence time is extremely short and which, as the grid resolution is increased, scales with diffusion ordering (and not as length 3 ). The spectral results of J. Yepez et al. [Phys. Rev. Lett. 103, 084501 (2009).] for quantum turbulence are revised and it is found that it is the compressible kinetic energy spectrum that exhibits three distinct spectral regions: a small-k classical-like Kolmogorov k -5/3 , a steep semiclassical cascade region, and a large-k quantum vortex spectrum k -3 . For most evolution times the incompressible kinetic energy spectrum exhibits a somewhat robust quantum vortex spectrum of k -3 for an extended range in k with a k -3.4 spectrum for intermediate k. For linear vortices of winding number 1 there is an intermittent loss of the quantum vortex cascade with its signature seen in the time evolution of the kinetic energy E kin (t), the loss of the quantum vortex k -3 spectrum in the incompressible kinetic energy spectrum as well as the minimalization of the vortex core isosurfaces that would totally inhibit any Kelvin wave vortex cascade. In the time intervals around these intermittencies the incompressible kinetic energy also exhibits a multicascade spectrum.

Quantum vortex structures and energy cascades are examined for two dimensional quantum turbulence (2D QT) at zero temperature. A special unitary evolution algorithm, the quantum lattice algorithm (QLA), is employed to simulate the Bose-Einstein condensate (BEC) governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a short Poincaré recurrence time. It is demonstrated that such short recurrence times are destroyed by stronger nonlinear interaction. The similar loss of Poincaré recurrence is also seen in the 3D GP equation . Various initial conditions are considered in an attempt to discern if 2D QT exhibits inverse cascades are is seen in 2D CT (classical turbulence). In our simulation parameter regimes, no dual cascades spectra were observed for 2D QT-unlike that seen in 2D CT.

The intrinsic coupling between local polarization vectors and superfluid vorticity represents afundamental aspect of quantum fluid dynamics governed by geometric Berry phase mechanisms. Thisreview examines the unified theoretical framework derived from spinor Gross-Pitaevskii equations,demonstrating how vortex-induced velocity fields induce polarization precession through geometricphase accumulation. The phenomenological evolution equation dP/dt = v_vor × P + Γ·P + F(P,ψ) emerges as the central result, incorporating conservative Berry phase effects and dissipativeprocesses. Applications span spinor Bose-Einstein condensates, superfluid ³He systems, and quantumturbulence in ⁴He. Numerical investigations using pseudo-spectral codes and experimental proposalsprovide pathways for validation of this geometric coupling mechanism with broader implications forquantum hydrodynamics and quantum information processing.

We propose a novel framework for the development of a true infinite persistence guidance system (IPGS) leveraging quantum vortex technology. Unlike conventional inertial navigation systems, which drift due to accumulated errors and entropic degradation, a vortex-anchored guidance architecture utilizes topologically protected quantum states as an unchanging reference. This paper outlines four candidate platforms-superconducting vortex lattices, superfluid helium toroidal vortices, Bose-Einstein condensate (BEC) rings, and synthetic amniotic gels-as viable technological pathways. Engineering models, control theory integration, persistence protocols, experimental design, simulations, and validation workflows are examined under the Unified Resonant Framework (URF), with applications ranging from aerospace navigation to medical resonance technologies.

An oscillating object can stretch quantized vortices attached to it in superfluids due to the relative superflow, steadily generating quantum turbulence, even in the zero-temperature limit. We report the emission and propagation of quantized vortices from quantum turbulence generated in superfluid 4 He at low temperatures. A vortex-free vibrating wire enables us to detect the first collision of vortex rings and therefore to measure the time-of-flight of a vortex emitted from a generator to a detector. The detection times from the start of turbulence generation exhibit an exponential distribution, suggesting that the detection is a Poisson process. Vortices are emitted continuously, but each vortex has a random flight velocity and direction. We estimated the nondetection time and mean detection period from the distribution for two flight distances. By estimating the flight velocity, we find that only vortices with velocities lower than the detector velocity can be detected, even if the sizes of the emitted vortices are smaller than the wire thickness or the vibration amplitude. The ratio of the detection rate as a function of vortex velocity suggests anisotropic emission of vortices from the quantum turbulence.

We report the critical behavior of steady quantum turbulence in superfluid 4 He. By using a thin vibrating wire with no bridge vortices in the superfluid, we find that lifetime of turbulent state for a given driving force reveals an exponential distribution. The mean lifetime estimated from the distribution decreases exponentially but greatly below a critical injection power. We estimate the vortex line density in the turbulent state and find that this critical behavior arises when the interdistance between vortex lines becomes as large as the turbulent region.

The drag force on a sphere and a cylinder moving in Hell is considered on the basis of the full system of the two-fluid dissipative hydro-dynamical equations. It is shown, that due to the thermomechanical effect in superfluid there exists a specific correction to the drag force on a body. The value of this thermomechanical correction AF has the order of the ratio IT/R or I~/R 2 (R is the size of the body, and 1T is the length of the formation of the convective heat flow). The value of AF grows at low temperatures, where 1T is proportional to the characteristic time of the phonon-roton scattering process, and in the region near the A-point, where 1T has the same temperature dependence as the order parameter correlation length l T ~ ~ ~ (T;~ -T) -2/3. The effects under discussion can be investigated experimentally by measurements of the mobility of small objects. A comparison is given with the data of the mobility of positive and negative ions in Hell near the A-point.

We show unequivocal evidence for formation of 𝐻𝑒 2 * excimers in liquid He II created by ionizing radiation produced through neutron capture. Laser beams induced fluorescence of the excimers. The fluorescence was recorded at a rate of 55.6 Hz by a camera. The location of fluorescence was determined with an uncertainty of 5 m. The technique provides an opportunity to record the flow of 𝐻𝑒 2 * excimers in a medium with very small viscosity and enables measurement of turbulence around macroscopic liter size objects or vortex matter in three dimensions.

Time dependent observations of point-to-point correlations of the velocity vector field (structure functions) are necessary to model and understand fluid flow around complex objects. Using thermal gradients, we observed fluid flow by recording fluorescence of \({He}_{2}^{*}\) excimers produced by neutron capture throughout a ~ cm3 volume. Because the photon emitted by an excited excimer is unlikely to be recorded by the camera, the techniques of particle tracking (PTV) and particle imaging (PIV) velocimetry cannot be applied to extract information from the fluorescence of individual excimers. Therefore, we applied an unsupervised machine learning algorithm to identify light from ensembles of excimers (clusters) and then tracked the centroids of the clusters using a particle displacement determination algorithm developed for PTV. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a n...

A pair of perturbed anti-parallel vortices is simulated using the three-dimensional Gross-Pitaevski equations. The simulations show decay of the vortex line length along with the depletion of local kinetic energy in a manner analogous to the decay of kinetic energy in the viscous, Navier-Stokes equations, even though the governing equations are Hamiltonian and energy conserving. The stages of development include vortex line stretching, conversion of kinetic energy into of the interaction component, the generation of vortex waves, multiple reconnections along the waves leading to the creation of vortex rings, kinetic energy transfer to high wavenumbers, evidence for energy transfer into phonons and finally the emission of the vortex rings. Some of this has been seen before, but not in a single calculation, which demonstrates the existence of a mechanism for an energy cascade that does not rely on statistical arguments. A four vortex example is given to demonstrate that these steps might be general. The wave generation and reconnection steps all depend upon interactions between distinct vortices, unlike vortex wave models where self-interactions along the vortices generate these effects. A variety of additional cases are noted whose analysis is in progress. So far, none have generated the dynamics that vortex wave models require to generate a cascade. Further cases are being pursued.

A grid of bars towed through a sample of He II produces both superfluid turbulence and classical hydrodynamic turbulence. The two velocity fields-in the normal fluid and in the superfluid-have been observed to have the same energy spectral density over a large range of scales. Here, we introduce a characteristic scale l q ϭ2(/ 3 ) Ϫ1/4 , where is the rate of turbulent energy dissipation per unit volume, and note that the energy spectrum in superfluid turbulence depends also on the quantum of circulation , for wave numbers k Ͼk q ϵ2/l q . We propose that the spectral density in this range is of the form (k)ϭC Ϫ1 k Ϫ3 , where C is the three-dimensional Kolmogorov constant in classical turbulence. This form is consistent with recent experiments in the temperature range 1.2 KϽTϽ2 K on the temporal decay of the vortex line density in the grid-generated He II turbulence.

We have studied a Bose-Einstein condensate of $^{87}Rb$ atoms under an oscillatory excitation. For a fixed frequency, we observe the formation of different structures in the cloud as a function of two parameters: time and amplitude of the excitation. Our results are summarized in a diagram of amplitude versus time in which we identify four zones exhibiting different behaviors. We concentrate on the regimes generated by the larger values of the parameters of the excitation, which we identify as the turbulent and granular regimes. We also present simulations based on the Gross-Pitaevskii equation which show the overall behavior of the diagram, supporting our observations.

A novel concept of quantum turbulence in finite size superfluids, such as trapped bosonic atoms, is discussed. We have used an atomic 87 Rb BEC to study the emergence of this phenomenon. In our experiment, the transition to the quantum turbulent regime is characterized by a tangled vortex lines formation, controlled by the amplitude and time duration of the excitation produced by an external oscillating field. A simple model is suggested to account for the experimental observations. The transition from the non-turbulent to the turbulent regime is a rather gradual crossover. But it takes place in a sharp enough way, allowing for the definition of an effective critical line separating the regimes. Quantum turbulence emerging in a finite-size superfluid may be a new idea helpful for revealing important features associated to turbulence, a more general and broad phenomenon.

In this paper, we determine the instability effects of a phase twist superposed on a quantum vortex defect governed by the Gross–Pitaevskii equation. For this, we consider the modified form of the equation in two cases: when a uniform phase twist is present everywhere in the condensate, and when the defect is subject to a localized phase twist confined to the defect healing region. In the first case, we show that a secondary, new defect is produced as manifestation of an Aharonov–Bohm type effect. In the second case, we prove that due to energy minimization, the defect changes its configurational energy by converting localized twist to writhe. This mechanism, typical of classical elastic systems, is shown to occur also in quantum defects, and it may find useful applications in science and technology.

The quantization of vortex lines in superfluids requires the introduction of their density L(r, t) in the description of quantum turbulence. The space homogeneous balance equation for L(t), proposed by Vinen on the basis of dimensional and physical considerations, allows a number of competing forms for the production term P. Attempts to choose the correct one on the basis of time-dependent homogeneous experiments ended inconclusively. To overcome this difficulty we announce here an approach that employs an inhomogeneous channel flow which is excellently suitable to distinguish the implications of the various possible forms of the desired equation. We demonstrate that the originally selected form which was extensively used in the literature is in strong contradiction with our data. We therefore present a new inhomogeneous equation for L(r, t) that is in agreement with our data and propose that it should be considered for further studies of superfluid turbulence.

Transport phenomena in superfluid helium can be described using the two-fluid Landau-Khalatnikov model and the Gorter-Mellink mutual friction. Here we discuss a mathematical formulation of the two-fluid model that uses macroscopic conservation balances of mass, momentum and energy of each species, and assumes local thermodynamic equilibrium. A particularity of this model is that it describes the state of He II as well as that of each of the two-fluid components in terms of pressure p and temperature T, which is convenient for stable numerical solution. The equations of the model form a system of partial differential equations (PDE) that can be written in matrix form for convenience. On this base, a three-dimensional numerical model using a complete and consistent, while still practical, system of PDEs was developed. In the form described, the PDE can be solved using threedimensional Lagrangian finite element in space supplemented by a Beam-Warming time-2 marching algorithm. Once validated, this solver will allow to simulate He II thermal counterflow applied to arbitrary geometry.

The widely discussed result of derivation for the Casimir temperature has been published by the authors of this paper. Recently we noticed that the unsolved problem of Navier-Stokes equation's existence and the smoothness can be thought by the idea of the globular protein (Einstein's colloidal particles (1905)) placed in liquid 3 He at the Casimir temperature. Although the globular protein is not dissolved in liquid 3 He and is as the cusp but it is exactly due to such nature of cusp in liquid 3 He, leads to an interesting phenomenon if it is differentiable in the Navier-Stokes equation in a peculiar case, one can understand the existence and the smoothness are established as to a promising candidate answer.

We derive a type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature, that describes step-by-step (local) energy cascade over scales caused by 4-wave interactions. Resulting new energy spectrum ELN(k) ∝ k -5/3 must replace in future theory (e.g. in finding the quantum turbulence decay rate) the previously used spectrum EKS(k) ∝ k -7/5 , which was recently shown to be inconsistent due to nonlocality of the 6-wave energy cascade.

We report on direct measurements of the energy dissipated in the spinup of the superfluid component of 3 He-B. A vortex-free sample is prepared in a cylindrical container, where the normal component rotates at constant angular velocity. At a temperature of 0.20 T c , seed vortices are injected into the system using the shear-flow instability at the interface between 3 He-B and 3 He-A. These vortices interact and create a turbulent burst, which sets a propagating vortex front into motion. In the following process, the free energy stored in the initial vortexfree state is dissipated leading to the emission of thermal excitations, which we observe with a bolometric measurement. We find that the turbulent front contains less than the equilibrium number of vortices and that the superfluid behind the front is partially decoupled from the reference frame of the container. The final equilibrium state is approached in the form of a slow laminar spin-up as demonstrated by the slowly decaying tail of the thermal signal.

Below the phase transition temperature Tc ≃ 10 -3 K 3 He-B has a mixture of normal and superfluid components. Turbulence in this material is carried predominantly by the superfluid component. We explore the statistical properties of this quantum turbulence, stressing the differences from the better known classical counterpart. To this aim we study the time-honored Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid turbulence. We combine pseudo-spectral direct numerical simulations with analytic considerations based on an integral closure for the energy flux. We avoid the assumption of locality of the energy transfer which was used previously in both analytic and numerical studies of the superfluid 3 He-B turbulence. For T < 0.37 Tc, with relatively weak mutual friction, we confirm the previously found "subcritical" energy spectrum E(k), given by a superposition of two power laws that can be approximated as E(k) ∝ k -x with an apparent scaling exponent 5 3 < x(k) < 3. For T > 0.37 Tc and with strong mutual friction, we observed numerically and confirmed analytically the scale-invariant spectrum E(k) ∝ k -x with a (k-independent) exponent x > 3 that gradually increases with the temperature and reaches a value x ∼ 9 for T ≈ 0.72 Tc. In the near-critical regimes we discover a strong enhancement of intermittency which exceeds by an order of magnitude the corresponding level in classical hydrodynamic turbulence.

We report measurements of the dissipation in the Superfluid helium high REynold number von Kármán flow experiment for different forcing conditions. Statistically steady flows are reached; they display a hysteretic behavior similar to what has been observed in a 1:4 scale water experiment. Our macroscopical measurements indicate no noticeable difference between classical and superfluid flows, thereby providing evidence of the same dissipation scaling laws in the two phases. A detailed study of the evolution of the hysteresis cycle with the Reynolds number supports the idea that the stability of the steady states of classical turbulence in this closed flow is partly governed by the dissipative scales. It also supports the idea that the normal and the superfluid components at these temperatures (1.6 K) are locked down to the dissipative length scale.

We present the Fully cOUpled loCAl model of sUperfLuid Turbulence (FOUCAULT) that describes the dynamics of finite temperature superfluids. The superfluid component is described by the vortex filament method while the normal fluid is governed by a modified Navier–Stokes equation. The superfluid vortex lines and normal fluid components are fully coupled in a self-consistent manner by the friction force, which induces local disturbances in the normal fluid in the vicinity of vortex lines. The main focus of this work is the numerical scheme for distributing the friction force to the mesh points where the normal fluid is defined (stemming from recent advances in the study of the interaction between a classical viscous fluid and small active particles) and for evaluating the velocity of the normal fluid on the Lagrangian discretisation points along the vortex lines. In particular, we show that if this numerical scheme is not careful enough, spurious results may occur. The new scheme whic...

We stir vortices into a trapped quasi two-dimensional atomic Bose-Einstein condensate by moving three laser stirrers. We apply stirring protocols introduced by Boyland et al. (2000), that efficiently build in topological chaos in classical fluids and are classified as Pseudo-Anosov stirring protocols. These are compared to their inefficient mixing counterparts, finite-order stirring protocols. We investigate if inefficient stirring protocols result in a more clustered distribution of vortices. The efficiency with which vortices are 'mixed' or distributed in a condensate is important for investigating dynamics of continuously forced quantum turbulence and the existence of the inverse cascade in turbulent two-dimensional superfluids.

Turbulence in superfluid helium is unusual and presents a challenge to fluid dynamicists because it consists of two coupled, inter penetrating turbulent fluids: the first is inviscid with quantised vorticity, the second is viscous with continuous vorticity. Despite this double nature, the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those of ordinary turbulence. We present experimental, numerical and theoretical results which explain these similarities, and illustrate the limits of our present understanding of superfluid turbulence at smaller scales.

Turbulence in superfluid helium is unusual and presents a challenge to fluid dynamicists because it consists of two coupled, interpenetrating turbulent fluids: the first is inviscid with quantized vorticity, and the second is viscous with continuous vorticity. Despite this double nature, the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those of ordinary turbulence. We present experimental, numerical, and theoretical results that explain these similarities, and illustrate the limits of our present understanding of superfluid turbulence at smaller scales.

We show that a moving obstacle, in the form of an elongated paddle, can create vortices that are dispersed, or induce clusters of like-signed vortices in 2D Bose-Einstein condensates. We propose new statistical measures of clustering based on Ripley's K-function which are suitable to the small size and small number of vortices in atomic condensates, which lack the huge number of length scales excited in larger classical and quantum turbulent fluid systems. The evolution and decay of clustering is analyzed using these measures. Experimentally it should prove possible to create such an obstacle by a laser beam and a moving optical mask. The theoretical techniques we present are accessible to experimentalists and extend the current methods available to induce 2D quantum turbulence in Bose-Einstein condensates.

We have analysed the axisymmetric and non-axisymmetric modes of a continuum of vortices in a rotating superfluid. We have investigated how changing the temperature affects the growth rate of the disturbances. We find that, in the long axial wavelength limit the condition q = α/(1 -α ′ ) = 1, where α and α ′ are temperature-dependent mutual friction parameters, is the crossover between damped and propagating Kelvin waves. Thus at temperatures for which q > 1, perturbations on the vortices are unlikely to cause vortex reconnections and turbulence. These results are in agreement with the recent discovery of Finne et al 1 of an intrinsic condition for the onset of quantum turbulence in 3 He-B.

To model isotropic homogeneous quantum turbulence in superfluid helium, we have performed Direct Numerical Simulations (DNS) of two fluids (the normal fluid and the superfluid) coupled by mutual friction. We have found evidence of strong locking of superfluid and normal fluid along the turbulent cascade, from the large scale structures where only one fluid is forced down to the vorticity structures at small scales. We have determined the residual slip velocity between the two fluids, and, for each fluid, the relative balance of inertial, viscous and friction forces along the scales. Our calculations show that the classical relation between energy injection and dissipation scale is not valid in quantum turbulence, but we have been able to derive a temperature-dependent superfluid analogous relation. Finally, we discuss our DNS results in terms of the current understanding of quantum turbulence, including the value of the effective kinematic viscosity.

We report measurements of the dissipation in the Superfluid helium high REynold number von Kármán flow experiment for different forcing conditions. Statistically steady flows are reached; they display a hysteretic behavior similar to what has been observed in a 1:4 scale water experiment. Our macroscopical measurements indicate no noticeable difference between classical and superfluid flows, thereby providing evidence of the same dissipation scaling laws in the two phases. A detailed study of the evolution of the hysteresis cycle with the Reynolds number supports the idea that the stability of the steady states of classical turbulence in this closed flow is partly governed by the dissipative scales. It also supports the idea that the normal and the superfluid components at these temperatures (1.6 K) are locked down to the dissipative length scale.

We develop an analytic theory of strong anisotropy of the energy spectra in the thermally-driven turbulent counterflow of superfluid 4 He. The key ingredients of the theory are the three-dimensional differential closure for the vector of the energy flux and the anisotropy of the mutual friction force. We suggest an approximate analytic solution of the resulting energy-rate equation, which is fully supported by the numerical solution. The two-dimensional energy spectrum is strongly confined in the direction of the counterflow velocity. In agreement with the experiment, the energy spectra in the direction orthogonal to the counterflow exhibit two scaling ranges: a near-classical non-universal cascade dominated range and a universal critical regime at large wave-numbers. The theory predicts the dependence of various details of the spectra and the transition to the universal critical regime on the flow parameters. This article is part of the theme issue 'Scaling the turbulence edifice'.

Submitted for the MAR16 Meeting of The American Physical Society The interplay between universal scaling laws and vortex clustering in two-dimensional quantum turbulence AUDUN SKAUGEN, LUIZA ANGHELUTA, Univ of Oslo-The relationship between vortex dynamics and the turbulent energy spectrum is an active research topic in quantum turbulence of superfluids and Bose-Einstein condensates. The energy spectra in quantum turbulence exhibit a Kolmogorov-5/3 scaling law, analogous to classical turbulence. Recent developments show that in two-dimensional quantum flows, this energy spectrum corresponds to an inverse energy cascade, which is realized by clustering of likesigned quantized vortices. We investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal-5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and velocity, which also show a similar scaling behavior. The-5/3 scaling law appearing in the power spectrum of the vortex number is consistent with a scenario of isolated vortices passively advected by a turbulent superfluid velocity, which is again generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial correlations, and exhibits a power-law tail with a-5/3 exponent that we can predict analytically from the point vortex model.

We investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal −5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and vortex velocity, which is also characterized by a similar scaling behavior. The −5/3 scaling law appearing in the power spectrum of vortex number fluctuations is consistent with the scenario of passive advection of isolated vortices by a turbulent superfluid velocity generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial configurations, and exhibits a power-law tail distribution with a −5/3 exponent.

A theory of the nuclear magnetic susceptibility of dilute solutions of 3He in 4He at low temperatures is presented. It is shown that ~-1 is a linear function of the corresponding inverse free fermion magnetic susceptibility. It is also shown that the experimental Curie constant differs from the theoretical one by an amount which is of the order of the 3He concentration. It follows that the existing experimental results are not adequate for the calculation of the Fermi liquid magnetic susceptibility enhancement factor.

Energy cascade is ubiquitous in systems far from equilibrium. Facilitated by particle interactions and external forces, it can lead to highly complex phenomena like fully developed turbulence, characterized by power law velocity correlation functions. Yet despite decades of research, how these power laws emerge from first principle remains unclear. Recently, experiments show that when a Bose condensate is subjected to periodic shaking, its momentum distribution exhibits a power law behavior. The flexibility of cold atom experiments has provided new opportunities to explore the emergence of these power laws, and to disentangle different sources of energy cascade. Here, we point out that recent experiments in cold atoms imply that classical turbulence is part of a larger family of scale invariant phenomena that include ideal gases. Moreover, the property of the entire family is contained in the structure of its Floquet states. For ideal gases, we show analytically that its momentum di...

We perform a study on the evolution of helical quantum turbulence at different temperatures by solving numerically the Gross-Pitaevskii and the Stochastic Ginzburg-Landau equations, using up to 4096 3 grid points with a pseudospectral method. We show that for temperatures close to the critical the fluid described by these equations can act as a classical viscous flow, with the decay of the incompressible kinetic energy and the helicity becoming exponential. The transition from this behavior to the one observed at zero temperature is smooth as a function of temperature. Moreover, the presence of strong thermal effects can inhibit the development of a proper turbulent cascade. We provide anzats for the effective viscosity and friction as a function of the temperature.

We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.

Quantum turbulence deals with the phenomenon of turbulence in quantum fluids, such as superfluid helium and trapped Bose-Einstein condensates (BECs). Although much progress has been made in understanding quantum turbulence, several fundamental questions remain to be answered. In this work, we investigated the entropy of a trapped BEC in several regimes, including equilibrium, small excitations, the onset of turbulence, and a turbulent state. We considered the time evolution when the system is perturbed and let to evolve after the external excitation is turned off. We derived an expression for the entropy consistent with the accessible experimental data, that is, using the assumption that the momentum distribution is well-known. We related the excitation amplitude to different stages of the perturbed system, and we found distinct features of the entropy in each of them. In particular, we observed a sudden increase in the entropy following the establishment of a particle cascade. We a...

Quantum turbulence in superfluids appears as a stochastic tangle of quantized vortex lines. Interest to this system extends beyond the field of superfluid helium to include a large variety of topics both fundamental and engineering problem. In the article we present a discussion of the hot topic, which is undoubtedly mainstream in this field, and which deals with the quasi-classical properties of quantum turbulence. The idea that classical turbulence can be modeled by a set of slim vortex tubes (or vortex sheets) has been discussed for quite a long time. In classical fluids, the concept of thin vortex tubes is a rather fruitful mathematical model. Quantum fluids, where the vortex filaments are real objects, give an excellent opportunity for the study of the question, whether the dynamics of a set of vortex lines is able to reproduce (at least partially) the properties of real hydrodynamic turbulence. The main goal of this article is to discuss the current state of this activity. We cover such important topics as theoretical justification of this model, and experimental and numerical evidence for quasi-classical turbulence.

The term ''quantum turbulence'' (QT) unifies the wide class of phenomena where the chaotic set of one dimensional quantized vortex filaments (vortex tangles) appear in quantum fluids and greatly influence various physical features. Quantum turbulence displays itself differently depending on the physical situation, and ranges from quasiclassical turbulence in flowing fluids to a near equilibrium set of loops in phase transition. The statistical configurations of the vortex tangles are certainly different in, say, the cases of counterflowing helium and a rotating bulk, but in all the physical situations very similar theoretical and numerical problems arise. Furthermore, quite similar situations appear in other fields of physics, where a chaotic set of one dimensional topological defects, such as cosmic strings, or linear defects in solids, or lines of darkness in nonlinear light fields, appear in the system. There is an interpenetration of ideas and methods between these scientific topics which are far apart in other respects. The main purpose of this review is to bring together some of the most commonly discussed results on quantum turbulence, focusing on analytic and numerical studies. We set out a series of results on the general theory of quantum turbulence which aim to describe the properties of the chaotic vortex configuration, starting from vortex dynamics. In addition we insert a series of particular questions which are important both for the whole theory and for the various applications. We complete the article with a discussion of the hot topic, which is undoubtedly mainstream in this field, and which deals with the quasi-classical properties of quantum turbulence. We discuss this problem from the point of view of the theoretical results stated in the previous sections. We also included section, which is devoted to the experimental and numerical suggestions based on the discussed theoretical models.

The energy spectra of the 3D velocity field, induced by various vortex filaments configurations are reviewed. The especial attention is paid to configurations generating the Kolmogorov type energy spectrum E(k) ∝ k-5/3. The motivation of this work is related to the problem of modeling classical turbulence with a set of chaotic vortex filaments. The quantity <v(k)v(-k)> can be exactly calculated, provided that we know the probability distribution functional ({ (,)}) t ξ s  of vortex loops configurations. The knowledge of ({ (,)}) t ξ s  is identical to the full solution of the problem of quantum turbulence and, in general,  is unknown. One of the simplifications is to investigate various truthful vortex configurations which can be elements of real vortex tangles. These configurations are: the uniform and nonuniform vortex arrays, the straight lines with excited Kelvin waves on it and the reconnecting vortex filaments. We demonstrate that the spectra E(k), generated by the these configurations, are close to the Kolmogorov dependence ∝ k-5/3 , and discuss the reason for this as well as the reason for deviation. PACS: 67.25.dk Vortices and turbulence; 47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids; 03.75.Kk Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow.

The energy spectrum of the 3D velocity field, induced by collapsing vortex filaments is studied. One of the aims of this work is to clarify the appearance of the Kolmogorov type energy spectrum E(k) ∝ k −5/3 , observed in many numerical works on discrete vortex tubes (quantized vortex filaments in quantum fluids). Usually, explaining classical turbulent properties of quantum turbulence, the model of vortex bundles, is used. This model is necessary to mimic the vortex stretching, which is responsible for the energy transfer in classical turbulence. In our consideration we do not appeal to the possible "bundle arrangement" but explore alternative idea that the turbulent spectra appear from singular solution, which describe the collapsing line at moments of reconnection. One more aim is related to an important and intensively discussed topic-a role of hydrodynamic collapse in the formation of turbulent spectra. We demonstrated that the specific vortex filament configuration generated the spectrum E(k) close to the Kolmogorov dependence and discussed the reason for this as well as the reason for deviation. We also discuss the obtained results from point of view of the both classical and quantum turbulence.

We submit the results of the numerical experiment on the decay of the quantum turbulence in the absence of the normal component of the superfluid helium. Computations were fulfilled inside a fixed domain with the use of the vortex filament method. The purpose of this study was to ascertain the role of the various factors arising in the numerical procedure, such as change in length in the reconnection processes, the procedures regulating the amount of points on the lines, eliminations of very small loops below the space resolution as well as the evaporation of the loops from the volume. We would like to stress that the widely accepted mechanism-a cascadelike transfer of the energy by nonlinear Kelvin waves (and radiation of sound)-was not considered. One of the reasons is that the space resolution along the lines did not allow to detect generation of high harmonics, moreover, particularly to get harmonics, which really radiate sound. In addition, the use of the method assumes that the fluid is incompressible. Numerical simulations have been performed for the cubic domain with transparent walls embedded in an unbounded space, and for a cube with solid smooth walls. Calculations showed that in the case of unlimited space the decay of quantum turbulence caused by the evaporation of vortex loops, which is implemented in a diffusion-like manner. The rate of the attenuation of the vortex line density agrees with the one, predicted by the theory of diffusion of nonuniform vortex tangles. In the case of a cube with solid walls, the main decay is also due to the diffusion of the vortex loops to boundaries. The vortex loops, whose ends glide on a smooth wall, execute the sophisticated motion (especially when they jump from the one face to the other) with many subsequent reconnections. As a result, there appear smaller and smaller loops with a size of few spatial resolutions, which were removed from the calculation. Indirect comparison with the experiments shows that the time of decay agrees with the measured data.

The method of correlation functions and the method of quantum vortex configurations are used to calculate the energy spectrum of a three-dimensional velocity field that is induced by collapsing (immediately before reconnection) vortex filaments. The formulation of this problem is motivated by the idea of modeling classical turbulence by a set of chaotic quantized vortex filaments. Among the various arguments that support the idea of quasi-classical behavior for quantum turbulence, the most persuasive is probably the resulting Kolmogorov energy spectrum resembling EðkÞ / k À5=3 that was obtained in a number of numerical studies. Another goal is associated with an important and intensely studied theme that relates to the role of hydrodynamic collapse in the formation of turbulence spectra. Calculations have demonstrated that vortex filaments create a velocity field at the moment of contact, which has a singularity. This configuration of vortex filaments generates the spectrum E(k), which bears the resemblance to the Kolmogorov law. A possible cause for this observation is discussed, as well as the likely reasons behind any deviations. The obtained results are discussed from the perspective of both classical and quantum turbulence.

The hydrodynamic problems of superfluid fluid containing chaotic tangles of quantized vortex filaments are discussed. The construction of such hydrodynamics crucially depends on the statistics of vortex tangles, and two important cases are presented. The first corresponds to a tangle that consists of entirely chaotic vortex filaments. This case is implemented in counterflowing helium, and is referred to as Vinen turbulence. In the construction of macroscopic dynamics, the system of equations is closed by the Vinen equation for the density of vortex filaments. The second, referred to as the Hall-Vinen-Bekarevich-Khalatnikov case, corresponds to a situation where the system contains bundles of polarized vortex filaments. In this instance, the system is closed by the Feynman equation that relates density of vortex filaments with the vorticity of superfluid velocity. Problems related to the application of both approaches are discussed.

The statement of problem is motivated by the idea of modeling the classical turbulence with a set of chaotic quantized vortex filaments in superfluids. Among various arguments supporting the idea of quasi-classic behavior of quantum turbulence, the strongest, probably, is the k dependence of the spectra of energy, E(k) ∝ k −5/3 obtained in numerical simulations and experiments. At the same time the mechanism of classical vs quantum turbulence (QT) is not clarified and the source of the k −5/3 dependence is unclear. In this work we concentrated on the nonuniform vortex vortex bundles. This choice is related to actively discussed question concerning a role of collapses in the vortex dynamics in formation of turbulent spectra. We demonstrate that the nonuniform vortex vortex bundles, which appear in result of nonlinear vortex dynamics generates the energy spectrum, which close to the Kolmogorov dependence ∝ k −5/3 .

The problem of the statistics of a set of chaotic vortex lines in a counterflowing superfluid helium is studied. We introduced a Langevin-type force into the equation of motion of the vortex line in presence of relative velocity vns. This random force is supposed to be Gaussian satisfying the fluctuation-dissipation theorem. The corresponding Fokker-Planck equation for probability functional in the vortex loop configuration space is shown to have a solution in the form of Gibbs distribution with the substitution E{s} →E({s} − P(vn − vs), where E{s} is the energy of the vortex configuration {s}, and P is the Lamb impulse. Some physical consequences of this fact are discussed.