Generation of Vortices and Observation of Quantum Turbulence in an Oscillating Bose-Einstein Condensate

We report on the experimental observation of vortex formation and production of tangled vortex distribution in an atomic BEC of ⁸⁷Rb atoms submitted to an external oscillatory perturbation. The oscillatory perturbations start by exciting quadrupolar and scissors modes of the condensate. Then regular vortices are observed finally evolving to a vortex tangle configuration. The vortex tangle is a signature of the presence of a turbulent regime in the cloud. We also show that this turbulent cloud has suppression of the aspect ratio inversion typically observed in quantum degenerate bosonic gases during free expansion.     

Universal Critical Velocity for the Onset of Turbulence of Oscillatory Superfluid Flow

The critical velocity v _c for the onset of turbulent drag of small spheres oscillating in superfluid ⁴He is frequency dependent (ω/2π from 100 Hz to 700 Hz) and is described by $v_{c}=2.6\sqrt{\kappa \omega}$ , where κ is the circulation quantum. A qualitative analysis based on a recent theory of the onset of superfluid turbulence gives $v_{c}\approx \sqrt{8\kappa \omega/\beta}$ , where β～1 depends on the coefficients of mutual friction. This agrees well with the data and implies that v _c is a universal critical velocity that is independent of geometry, size, and surface properties of the oscillating body. This is confirmed by comparing our data on spheres with v _c obtained with other oscillating structures by other groups. Numerical simulations indicate somewhat larger critical velocity, above which a rapid increase in vortex length is observed.     

Gradual Eddy-Wave Crossover in Superfluid Turbulence

We revise the theory of superfluid turbulence near the absolute zero of temperature and suggest a differential approximation model for the energy fluxes in the k-space, ε _HD(k) and ε _KW(k), carried, respectively, by the collective hydrodynamic (HD) motions of quantized vortex lines and by their individual uncorrelated motions known as Kelvin waves (KW). The model predicts energy spectra of the HD and the KW components of the system, ?_HD(k) and ?_KW(k), which experience a smooth crossover between different regimes of motion over a finite range of scales. For an experimentally relevant range of Λ≡ln?(?/a) (? is the mean intervortex separation and a is the vortex core radius) between 10 and 15 the total energy flux ε=ε _HD(k)+ε _KW(k) and the total energy spectrum ?(k)=?_HD(k)+?_KW(k) are dominated by the HD motions for k<2/?. In this region ?(k) follows the HD spectrum with constant energy flux ε?ε _HD=const.: ?(k)∝ k ^?5/3 for smaller k and tends to equipartition of the HD energy ?(k)∝ k ² for larger k. This bottleneck accumulation of the energy spectrum is milder than the one predicted before in (L’vov et al. in Phys. Rev. B 76:024520, 2007) based on a model with sharp HD-KW transition. For Λ=15, it results in a prediction for the effective viscosity ν ^?′?0.004κ (κ is the circulation quantum) which is in a reasonable agreement with its experimental value in ⁴He low-temperature experiment ≈0.003κ (Walmsley et al. in Phys. Rev. Lett. 99:265302, 2007). For k>2/?, the energy spectrum is dominated by the KW component: almost flux-less KW component close to the thermodynamic equilibrium, ?≈?_KW≈const at smaller k and the KW cascade spectrum ?(k)→?_KW(k)∝ k ^?7/5 at larger k.     

Quantum Turbulence in 4He, Oscillating Grids, and Where Do We Go Next?

Experimental approaches to the study of quantum turbulence (QT) in superfluid ⁴He in the low temperature limit, where the normal fluid density is effectively zero, are considered. A succinct general introduction covers liquid ⁴He, superfluidity, critical velocities for the onset of dissipation, quantized vortex lines and QT. The QT can be created mechanically by the oscillation of wires or grids above characteristic critical velocities. The interesting dynamics of the oscillating grid are discussed. It exhibits an enhanced effective mass due to backflow, as expected from classical hydrodynamics. It is found that the critical velocity attributable to the onset of QT production rises with increasing temperature. Oscillating objects like grids or wires create QT that is not well-characterized in terms of length scale, and the QT is not spatially homogeneous. The QT can be detected by the trapping of negative ions on vortex cores. Although the corresponding capture cross-section has not yet been measured, it is evidently very small, so that the technique cannot be expected to be a very sensitive one. In the future it is hoped to create well-characterized, homogeneous QT by means of a drawn grid. Improved sensitivity in the detection of QT is being sought through calorimetric techniques that monitor the temperature rise of the liquid caused by the decay of the vortex lines.     

An Introduction to Quantum Turbulence

Turbulence in a superfluid differs from that in a classical fluid because the flow of a superfluid is strongly influenced by quantum effects. Such turbulence is therefore often described as quantum turbulence. We give a brief historical account of the study of quantum turbulence, explaining how our understanding of it has developed. Particular attention is then paid to developments during the past ten years, which have seen the study of types of quantum turbulence that have close classical analogues. Similarities and differences between the classical and quantum cases are discussed, and aspects that are either not understood or the subject of speculation are emphasized. The paper provides an introduction to other and more detailed presentations in the Symposium.      

Suppression of Turbulence in Rotational Flows

Technical Physics Letters - The capabilities of controlling turbulence in a spherical Couette flow have been experimentally investigated. It is shown that, with increasing modulation amplitude of...     

Model of Turbulent Oscillating Flows in Smooth Tubes

Turbulent oscillating flows in smooth tubes are considered. A group of flows in which the logarithmic boundary layer grows monotonically with time is distinguished; the conditions of quasistationarity of these flows are determined. A model of a quasistationary turbulent oscillating flow in a smooth tube is constructed; the model is in satisfactory agreement with experiment.     

Universal Crossover Approach to Equation of State for Fluids

It is well known that any classical equation of state fails to describe the properties of fluids in the critical region, where the behavior of fluids is strongly affected by density fluctuations. In the present work, a universal approach to incorporate the effects of density fluctuations in the global behavior of one-component fluids is proposed. As an illustration of our general approach, a crossover generalization of a four-parametric cubic equation of state, which can be useful for engineering applications, is demonstrated. The obtained crossover equation reproduces Ising-like singular scaling behavior in the critical region and reduces to the original cubic equation of state far away from the critical point. In addition, the crossover equation of state is applied to describe thermodynamic properties of methane, ethane, carbon dioxide, and water. It is shown that incorporation of critical fluctuations leads to a significant improvement in the ability of the cubic equation to represent thermodynamic properties and liquid–vapor equilibrium of one-component fluids.     

Quantum Turbulence: Aspects of Visualization and Homogeneous Turbulence

Quantum turbulence is now a mature field of study, which cannot be surveyed easily in a single presentation. The paper therefore focusses on reviews in two areas: the visualization of quantum turbulence, which has the potential to transform our knowledge of the subject; and homogeneous quantum turbulence, which, although much studied, still presents us with interesting and fundamental problems. The latest results based on the use of He₂ metastable excimer molecules as tracers are presented. Other topics addressed include the behaviour of the normal fluid in thermal counterflow, fluctuations in vortex-line density, and the mechanisms by which quantum turbulence on a small scale can evolve in various types of flow to a larger scale.     

Hydrodynamic Instability and Turbulence in Quantum Fluids

Superfluid turbulence consisting of quantized vortices is called quantum turbulence (QT). Quantum turbulence and quantized vortices were discovered in superfluid ⁴He about 50 years ago, but innovation has occurred recently in this field. One is in the field of superfluid helium. Statistical quantities such as energy spectra and probability distribution function of the velocity field have been accessible both experimentally and numerically. Visualization technique has developed and succeeded in the direct visualization of quantized vortices. The other innovation is in the field of atomic Bose-Einstein condensation. The modern optical technique has enabled us to control and visualize directly the condensate and quantized vortices. Various kinds of hydrodynamic instability have been revealed. Even QT is realized experimentally. This article describes such recent developments as well as the motivation of studying QT.     

Numerical Studies of Quantum Turbulence

We review numerical studies of quantum turbulence. Quantum turbulence is currently one of the most important problems in low temperature physics and is actively studied for superfluid helium and atomic Bose–Einstein condensates. A key aspect of quantum turbulence is the dynamics of condensates and quantized vortices. The dynamics of quantized vortices in superfluid helium are described by the vortex filament model, while the dynamics of condensates are described by the Gross–Pitaevskii model. Both of these models are nonlinear, and the quantum turbulent states of interest are far from equilibrium. Hence, numerical studies have been indispensable for studying quantum turbulence. In fact, numerical studies have contributed to revealing the various problems of quantum turbulence. This article reviews the recent developments in numerical studies of quantum turbulence. We start with the motivation and the basics of quantum turbulence and invite readers to the frontier of this research. Though there are many important topics in the quantum turbulence of superfluid helium, this article focuses on inhomogeneous quantum turbulence in a channel, which has been motivated by recent visualization experiments. Atomic Bose–Einstein condensates are a modern issue in quantum turbulence, and this article reviews a variety of topics in the quantum turbulence of condensates, e.g., two-dimensional quantum turbulence, weak wave turbulence, turbulence in a spinor condensate, some of which have not been addressed in superfluid helium and paves the novel way for quantum turbulence researches. Finally, we discuss open problems.     

Energy Spectra in Quantum Fluid Turbulence

The Gross–Pitaevskii (GP) equation describes the dynamics of quantum fluids such as superfluids and Bose–Einstein condensates. Numerical simulations of turbulence obeying the GP equation with forcing and dissipation are performed. The interaction energy spectrum obeys thescaling law E^int(k) α k^−3/2, which is consistent with the weak turbulence analysis. However, in contradiction to the assumptions in the weak turbulence analysis, it is found that the density fluctuation is not small and that the frequency spectrum does not have narrow peaks. Another possibility to explain the scaling law is discussed.     

Quantum Turbulence: Achievements and Challenges

The paper is, for the most part, a review, written in its original form for an International Workshop on Vortices, Superfluid Dynamics, and Quantum Turbulence, held in Lammi, Finland, in 2010. Achievements are reviewed, and a strong emphasis is placed on problems that are either unsolved or still the subject of active discussion. These problems often call for more powerful experimental techniques, including local probes and actual visualization of the turbulent motion. Special emphasis is placed on recent progress in the development of visualization.     

Decay of Quantized Vortices in Quantum Turbulence

The decay of quantized vortex line length in quantum turbulence at zero temperature is studied numerically by solving the Gross–Pitaevskii equation with a small-scale dissipation. The obtained decay of the vortex line length L is consistent with the L ∝ t ^−3/2 behavior which supports the Kolmogorov energy spectrum of quantum turbulence. The mechanism of the decay is discussed.     

Quantum Turbulence in Trapped Atomic Bose-Einstein Condensates

Generally there are two kinds of cooperative phenomena comprised of quantized vortices. One is a vortex lattice under rotation, and the other is a vortex tangle (quantum turbulence) made by some flow. Both have been studied in the field of superfluid helium through the long research history. On the other hand, the research of atomic Bose-Einstein condensates (BECs) has been limited to the former case, namely a vortex lattice. In this work, we address for the first time quantum turbulence in atomic BECs theoretically and numerically. We propose how to make quantum turbulence in a trapped BEC by combining rotation around two axes, and confirm the Kolmogorov spectra by the Gross-Pitaevskii model.      

Thermoviscous Effects in Steady and Oscillating Flow of an Isotropic Superfluid: Theory

A calculation is presented of thermoviscous effects in both steady and oscillating flow of an isotropic superfluid through small apertures and channels. These calculations, which are based on the two-fluid model, are motivated by the work of Robinson and Atkins which included only the thermal effects of flow through a superleak. This paper extends these calculations to include the effects of normal fluid flow, compressibility, and thermal expansion. These effects are found to be both dissipative and reactive(nondissipative). The motivation for the extension is to provide a clear understanding of the reactive and dissipative forces at work in superfluid flow experiments. In the paper which immediately follows this one, predictions based on the results of this paper are compared with a wide array of experimental data. This work takes on importance due to the recent discovery of gyroscopic effects, and the possible development of sensitive gyroscopes in experimental cells whose geometry is similar to the one considered in this paper.     

Thermoviscous Effects in Steady and Oscillating Flow of Superfluid 4He: Experiments

The correct interpretation of superfluid flow experiments relies on the knowledge of thermal and viscous effects that can cause deviations from ideal behavior. The previous paper presented a theoretical study of dissipative and reactive(nondissipative) thermoviscous effects in both steady and oscillating flow of an isotropic superfluid through small apertures and channels. Here, a detailed comparison is made between the theory and a wide array of experimental data. First, the calculated resistance to steady superflow is compared with measurements taken in a constant pressure-head flow cell. Second, the resonant frequency and Q of three different helmholtz oscillators are compared with predictions based on the calculated frequency response. The resonant frequency and Q are extracted numerically from the frequency response, and analytical results are given in experimentally important limits. Finally, the measured and calculated frequency response are compared at a temperature where the Helmholtz oscillator differs significantly from a simple harmonic oscillator. This difference is used to explain how the thermal properties of the oscillator affect its response. The quantitative agreement between the theory and experiment provide an excellent check of the previously derived equations. Also, the limiting expressions shown in this paper provide simple analytical expressions for calculating the effects of the various physical phenomena in a particular experimental situation.     

Inertial Waves and Steady Flows in a Liquid Filled Librating Cylinder

The fluid flow in a non-uniformly rotating (librating) cylinder about a horizontal axis is experimentally studied. In the absence of librations the fluid performs a solid-body rotation together with the cavity. Librations lead to the appearance of steady zonal flow in the whole cylinder and the intensive steady toroidal flows near the cavity corners. If the frequency of librations is twice lower than the mean rotation rate the inertial waves are excited. The oscillating motion associated with the propagation of inertial wave in the fluid bulk leads to the appearance of an additional steady flow in the Stokes boundary layers on the cavity side wall. In this case the heavy particles of the visualizer are assembled on the side wall into ring structures. The patterns are determined by the structure of steady flow, which in turn depends on the number of reflections of inertial wave beams from the cavity side wall. For some frequencies, inertial waves experience spatial resonance, resulting in inertial modes, which are eigenmodes of the cavity geometry. The resonance of the inertial modes modifies the steady flow structure close to the boundary layer that is manifested in the direct rebuilding of patterns. It is shown that the intensity of zonal flow, as well as the intensity of steady flows excited by inertial waves, is proportional to the square of the amplitude of librations.