Quantum Turbulence Research Articles

Quantum turbulence of bellows-driven4He superflow: Steady state

Quantum turbulence in superfluid ${}^{4}$He is studied by the attenuation of second sound in flow channels of 7-mm and 10-mm side square cross sections, and 115-mm length. The ends of the channels are plugged by sintered silver superleaks to allow a pure superflow (i.e., a net flow of the superfluid component only). Flows are generated by mechanically operating a low temperature bellows assembly, as opposed to the helium fountain pump commonly used for previous superflow turbulence studies. The temperature range is $1.35\phantom{\rule{0.28em}{0ex}}\mathrm{K}\ensuremath{\le}T\ensuremath{\le}1.95\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, at the saturated vapor pressure. The observed turbulent steady state is characterized by the vortex line density ${L}^{1/2}=\ensuremath{\gamma}(T)(v\ensuremath{-}{v}_{\mathrm{c}})$, where $v$ is the mean superflow velocity and ${v}_{\mathrm{c}}$ is the critical velocity for the onset of turbulence. The character of the steady state agrees with the Vinen phenomenological model for thermal counterflow turbulence. The coefficient $\ensuremath{\gamma}(T)$ is in fair agreement with previous thermal pure superflow and counterflow experiments. The critical velocity ${v}_{\mathrm{c}}\ensuremath{\approx}0.2$ cm/s is roughly temperature independent.

Energy Spectra of Vortex Distributions in Two-Dimensional Quantum Turbulence

Theoretical analysis of a confined turbulent quantum fluid reveals features of the clustering of quantized vortices and permits a new way to compute the Kolmogorov constant, which captures the nature of energy flow across different length scales.

Numerical study of decay of vortex tangles in superfluid helium at zero temperature

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.

NONLINEAR SECOND SOUND WAVES IN SUPERFLUID HELIUM: INSTABILITIES, TURBULENCE AND ROGUE WAVES

Recent work on nonlinear second sound wave propagation and acoustic turbulence in superfluid 4 He is reviewed. Observations of direct and inverse turbulent energy cascades are described. The direct cascade arises due to the strong nonlinear dependence of the second sound wave velocity on its amplitude. The flux of energy injected at the driving frequency is transformed via successively higher harmonics until it is eventually attenuated by viscous dissipation at the short-wavelength edge of the spectrum. The onset of the inverse cascade occurs above a critical driving energy density, and it is accompanied by giant waves that constitute an acoustic analogue of the rogue waves that occasionally appear on the surface of the ocean. The theory of the phenomena is outlined and shown to be in good agreement with the experiments.

Time-of-Flight Measurements of Vortices Emitted from Quantum Turbulence in Superfluid 4He

An oscillating obstacle generates quantum turbulence in superfluids, when vortices remained attached to obstacle surfaces or vortex rings collided with it during oscillation. Turbulence provides a source of vortices; however, the characteristics of these vortices are not clear. In the present work, we report the flight of vortices emitted from quantum turbulence in superfluid 4He at low temperatures, using vibrating wires as a generator and a detector of vortices. A vortex-free vibrating wire can detect only the first colliding vortex ring, though it will be refreshed after low vibration and be able to detect a vortex ring again. By measuring a period from the start of turbulence generation to the vortex detection repeatedly, we find an exponential distribution of time-of-flights with a non-detection period t0 and a mean detection period t1, suggesting a Poisson process. Both periods t0 and t1 increase with increasing distance between a generator and a detector. A vortex flight velocity estimated from period t0 suggests that the sizes of the emitted vortex rings distribute to a range smaller than a generator thickness or a generator vibration amplitude. Vortices are emitted radially from a turbulence region, at least in the direction of oscillator vibration.

Thermally and mechanically driven quantum turbulence in helium II

In most experiments with superfluid helium, turbulence is generated thermally (by applying a heat flux, as in thermal counterflow) or mechanically (by stirring the liquid). By modeling the superfluid vortex lines as reconnecting space curves with fixed circulation, and the driving normal fluid as a uniform flow (for thermal counterflow) and a synthetic turbulent flow (for mechanically driven turbulence), we determine the difference between thermally and mechanically driven quantum turbulence. We find that in mechanically driven turbulence, the energy is concentrated at the large scales, the spectrum obeys Kolmogorov scaling, vortex lines have large curvature, and the presence of coherent vortex structures induces vortex reconnections at small angles. On the contrary, in thermally driven turbulence, the energy is concentrated at the mesoscales, the curvature is smaller, the vorticity field is featureless, and reconnections occur at larger angles. Our results suggest a method to experimentally detect the presence of superfluid vortex bundles.

Rotating quantum turbulence in superfluid4He in theT=0limit

Observations of quantum turbulence in pure superfluid ${}^{4}$He in a rotating container are reported. Large-scale forcing through rotational oscillations of the cubic container and detecting of turbulence via monitoring ion transport along the axis of rotation were used. Near the axial walls, with increasing forcing the vortex tangle grows without an observable threshold. This tangle gradually develops into bulk turbulence at a characteristic amplitude of forcing that depends on the forcing frequency and rotation rate. At higher amplitudes, the total vortex line length increases rapidly. Resonances of inertial waves are observed in both laminar and turbulent bulk states. On such resonances, the turbulence appears at smaller amplitudes of forcing.

Unitary qubit lattice simulations of complex vortex structures

A quantum vortex is a topological singularity with quantized circulation, unlike a classical vortex with its continuous circulation strength. Quantum turbulence, envisaged as strong tangle of quantum vortices, of a Bose–Einstein condensate is examined by developing a unitary qubit lattice algorithm for the solution of the Gross–Pitaevskii equation. Earlier, it was shown that a certain class of initial conditions had a very short Poincare recurrence time for this Hamiltonian system. Here it is shown quantitatively that increasing the internal energy of the initial state leads to a systematic degradation of this class of solutions. Coupled Bose–Einstein condensate systems are investigated for a Hopf link class of initial conditions in which a vortex ring core is threaded by a linear vortex core that then closes toroidal around the vortex ring. These states are known as skyrmions and play a role in particle physics, astrophysics and condensed matter physics.

Unusual Behavior of a MEMS Resonator in Superfluid 4He

Novel mechanical resonators based on micro-electro-mechanical systems (MEMS) technology were developed for the study of superfluid 4He. The MEMS device is composed of two parallel plates, the movable plate suspended by four serpentine springs above the substrate, forming a shear mechanical oscillator. A specific device with a 1.25 μm gap was tested in the superfluid phase of 4He. At temperatures below 400 mK the device exhibits nonlinear and hysteretic behavior when the excitation exceeds a threshold. The anomalies are reminiscent of quantum turbulence and vorticity effects observed in other mechanical oscillators such as tuning forks or vibrating grids.

Temperature suppression of Kelvin-wave turbulence in superfluids

Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length scale separating a quasi-inertial range of Kelvin-wave turbulence from a far-dissipation range. The letter culminates with analytical predictions for the energy spectrum of the Kelvin-wave turbulence in both of these regimes.

Creation and characterization of vortex clusters in atomic Bose-Einstein condensates

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.

Critical dynamics of a two-dimensional superfluid near a nonthermal fixed point

Critical dynamics of an ultracold Bose gas far from equilibrium is studied in two spatial dimensions. Superfluid turbulence is created by quenching the equilibrium state close to zero temperature. Instead of immediately re-thermalizing, the system approaches a meta-stable transient state, characterized as a non-thermal fixed point. A focus is set on the vortex density and vortex-antivortex correlations which characterize the evolution towards the non-thermal fixed point and the departure to final (quasi-)condensation. Two distinct power-law regimes in the vortex-density decay are found and discussed in terms of a vortex binding-unbinding transition and a kinetic description of vortex scattering. A possible relation to decaying turbulence in classical fluids is pointed out. By comparing the results to equilibrium studies of a two-dimensional Bose gas, an intuitive understanding of the location of the non-thermal fixed point in a reduced phase space is developed.

Quantized vortex reconnection: Fixed points and initial conditions

Quantized vortices are phase singularities in complex fields. In superfluids, they appear as mobile interacting defects that may cross and reconnect by exchanging tails. Reconnection is a topology-changing event that allows vortex tangles to decay; it is a defining signature of quantum turbulence. We report a family of fixed points (i.e., stationary solutions), including planar forms, that capture reconnection in the Gross-Pitaevskii model in contrast to previous suggestions of pyramidal structures. These are obtained using a well known, systematic method for generating low-energy relaxed initial conditions for Gross-Pitaevskii simulations.

Turbulent Superfluid Profiles and Vortex Density Waves in a Counterflow Channel

In this paper we study the two-dimensional profiles of the superfluid component velocity and the quantized vortex-points density in a counterflow channel where the influence of the walls cannot be neglected. The numerical results obtained show the presence of vortex density waves in the channel, as shown in a recent paper by means of the one-fluid model.

Testing the performance of a cryogenic visualization system on thermal counterflow by using hydrogen and deuterium solid tracers

An experimental apparatus has been designed to analyze by visualization cryogenic flows of liquid (4)He and consequently address unresolved problems of quantum turbulence. The newly implemented flow visualization setup is described and its specific features discussed. Thermal counterflow experiments have been performed and the motion of solid hydrogen and deuterium tracers studied by using the particle tracking velocimetry technique in order to probe the system capabilities. It is shown that the obtained results are consistent with the two-fluid model describing the behavior of superfluid (4)He. A number of technical and fundamental issues, such as particles' aggregation, role of rotating particles in counterflow and evidence of non-Gaussian distribution of tracers' velocities, are also discussed. The apparatus appears to be well-suited to the task of analyzing cryogenic flows and potentially capable of obtaining new results stimulating further understanding of the underlying physics.

Nonthermal fixed points, vortex statistics, and superfluid turbulence in an ultracold Bose gas

Nonthermal fixed points of the dynamics of a dilute degenerate Bose gas far from thermal equilibrium are analyzed in two and three spatial dimensions. Universal power-law distributions, previously found within a nonperturbative quantum-field theoretical approach and recently shown to be related to vortical dynamics and superfluid turbulence [Phys. Rev. B 84, 020506(R) (2011)], are studied in detail. The results imply an interpretation of the scaling behavior in terms of independent vortex excitations of the superfluid and show that the statistics of topological excitations can be described in the framework of wave turbulence. The particular scaling exponents observed in the single-particle momentum distributions are found to be consistent with irreversibility as well as conservation laws obeyed by the wave interactions. Moreover, long-wavelength acoustic excitations of the vortex-bearing condensate, driven by vortex annihilations, are found to follow a nonthermal power law. Considering vortex correlations in a statistical model, the long-time departure from the nonthermal fixed point is related to vortex-antivortex pairing. The studied nonthermal fixed points are accessible in cold-gas experiments. The results shed light on fundamental aspects of superfluid turbulence and have strong potential implications for related phenomena, e.g., in early-universe inflation or quark-gluon plasma dynamics.

Symmetry of Kelvin-wave dynamics and the Kelvin-wave cascade in theT=0superfluid turbulence

The article considers implications of tilt symmetry (symmetry with respect to tilting of the coordinate axis with respect to which vortex motion is studied) in the non-linear dynamics of Kelvin waves. The conclusion is that although the spectrum of Kelvin wave is not tilt-invariant, this does not compromise tilt invariance of the Kelvin-wave cascade vividly argued now in the theory of superfluid turbulence. The article investigates the effect of strong kelvon interaction on the power-law exponent for the Kelvin-wave cascade and suggests a simple picture of the crossover from the classical Kolmogorov cascade to the quantum Kelvin-wave cascade, which does not encounter with mismatch of the energy distributions at the crossover and does not require a broad intermediate interval for realization of the crossover.

Energy spectra of superfluid turbulence in3He

In superfluid ${}^{3}$He-B, turbulence is carried predominantly by the superfluid component. To explore the statistical properties of this quantum turbulence and its differences from the classical counterpart, we adopt the time-honored approach of shell models. Using this approach, we provide numerical simulations of a Sabra shell model that allows us to uncover the nature of the energy spectrum in the relevant hydrodynamic regimes. These results are in qualitative agreement with analytical expressions for the superfluid turbulent energy spectra that were found using a differential approximation for the energy flux.

Instability of superfluid flow in the neutron star core

Pinning of superfluid vortices to magnetic flux tubes in the outer core of a neutron star supports a velocity difference of $\sim 10^5$ \cms\ between the neutron superfluid and the proton-electron fluid as the star spins down. Under the Magnus force that arises on the vortex array, vortices undergo {\em vortex creep} through thermal activation or quantum tunneling. We examine the hydrodynamic stability of this situation. Vortex creep introduces two low-frequency modes, one of which is unstable above a critical wavenumber for any non-zero flow velocity of the neutron superfluid with respect to the charged fluid. For typical pinning parameters of the outer core, the superfluid flow is unstable over wavelengths $\lambda\lap 10$ m and over timescales of $\sim (\lambda/{1 m})^{1/2}$ yr down to $\sim 1$ d. The vortex lattice could degenerate into a tangle, and the superfluid flow would become turbulent. We suggest that superfluid turbulence could be responsible for the red timing noise seen in many neutron stars, and find a predicted spectrum that is generally consistent with observations.

Generalized local induction equation, elliptic asymptotics, and simulating superfluid turbulence

We prove the generalized induction equation and the generalized local induction equation (GLIE), which replaces the commonly used local induction approximation (LIA) to simulate the dynamics of vortex lines and thus superfluid turbulence. We show that the LIA is, without in fact any approximation at all, a general feature of the velocity field induced by any length of a curved vortex filament. Specifically, the LIA states that the velocity field induced by a curved vortex filament is asymmetric in the binormal direction. Up to a potential term, the induced incompressible field is given by the Biot-Savart integral, where we recall that there is a direct analogy between hydrodynamics and magnetostatics. Series approximations to the Biot-Savart integrand indicate a logarithmic divergence of the local field in the binormal direction. While this is qualitatively correct, LIA lacks metrics quantifying its small parameters. Regardless, LIA is used in vortex filament methods simulating the self-induced motion of quantized vortices. With numerics in mind, we represent the binormal field in terms of incomplete elliptic integrals, which is valid for \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{3}$\end{document}R3. From this and known expansions we derive the GLIE, asymptotic for local field points. Like the LIA, generalized induction shows a persistent binormal deviation in the local field but unlike the LIA, the GLIE provides bounds on the truncated remainder. As an application, we adapt formulae from vortex filament methods to the GLIE for future use in these methods. Other examples we consider include vortex rings, relevant for both superfluid 4He and Bose-Einstein condensates.