Observations of Vortex Emissions from Superfluid 4He Turbulence at High Temperatures

An immersed object with high velocity oscillations causes quantum turbulence in superfluid ⁴He, even at very low temperatures. The continuously generated turbulence may emit vortex rings from a turbulent region. In the present work, we report vortex emissions from quantum turbulence in superfluid ⁴He at high temperatures, by using three vibrating wires as a turbulence generator and vortex detectors. Two detector wires were mounted beside a generator wire: one in parallel and the other in perpendicular to the oscillation direction of the generator. The detection times of vortex rings represent an exponential distribution with a delay time t ₀ and a mean detection period t ₁. The delay time includes the generation time of a fully developed turbulence and the time-of-flight of a vortex ring. At high temperatures, vortices are dissipated by relative motion between a normal fluid component and the vortices, resulting that only large vortex rings are reachable to the detectors. Using this method, we detected vortex rings with a diameter of 100 μm, comparable to a peak-to-peak vibration amplitude of 104 μm of the generator. The large vortices observed here are emitted anisotropically from the generator. The emissions parallel to the vibrating direction are much less than those perpendicular to the direction.     

Time-of-Flight Experiments of Vortex Rings Propagating from Turbulent Region of Superfluid 4He at High Temperature

We report the time-of-flight of quantized vortex rings generated by a vibrating wire in superfluid ⁴He which contains normal fluid component. A cover box of vibrating wires and slow cooling of superfluid reduce the number of vortices attached to wire surfaces, enabling us to study vortex rings propagating from a turbulent region. Using two vibrating wires as a generator and a detector of vortices, the time-of-flight of vortices propagating a distance of 0.88 mm was measured at 1.25 K. We find that the time-of-flights distribute from 0.06 s to 27.4 s, much larger than the lifetimes of circular vortex rings limited in the size of a generator amplitude. These results imply that large vortex rings with non-circular shape or vortex tangles are created by the generator, propagating slowly and colliding with the detector before complete disappearance.     

Vortex Rings in Superfluid 3He–B at Low Temperatures

Turbulence in classical fluids has far-reaching technological implications but is poorly understood. A better understanding might be gained from studying turbulence in quantum systems. In a pure superfluid (at low temperatures), there is no viscosity and vortex lines are quantised. Quantum turbulence consists of a tangle of quantised vortex lines which interact via their self-induced flow. We have recently developed techniques for detecting vortices in superfluid ³He–B in the low temperature limit. We find that the transition to turbulence from a moving grid occurs by the entanglement of emitted vortex rings. Here, we discuss the propagation of the ballistic vortex rings emitted at low grid velocities. We have measured the temperature at which the rings decay before reaching the detectors. Our results, at two different pressures, confirm that the vortex rings decay in accordance with mutual friction.     

Propagation of Quantized Vortices Driven by an Oscillating Sphere in Superfluid 4He

We performed numerical simulation of quantum turbulence at 0 K generated from remnant vortices attached to an oscillating sphere. The remnant vortices are extended by the sphere motion and form a tangle with emitting vortex loops. As time passes, the length of vortices in a computational volume becomes statistically steady. We investigate in the statistical steady state the distribution of the length of vortex loops and anisotropy of their propagation direction caused by the sphere oscillation. The propagation direction of the emitted vortex loops is anisotropic along the oscillation direction of the sphere. The obtained results are consistent with results obtained in the experimental study using vibrating wires in superfluid ⁴He.     

Transition to Quantum Turbulence and the Propagation of Vortex Loops at Finite Temperatures

We performed numerical simulation of the transition to quantum turbulence and the propagation of vortex loops at finite temperatures in order to understand the experiments using vibrating wires in superfluid ⁴He by Yano et al. We injected vortex rings to a finite volume in order to simulate emission of vortices from the wire. When the injected vortices are dilute, they should decay by mutual friction. When they are dense, however, vortex tangle are generated through vortex reconnections and emit large vortex loops. The large vortex loops can travel a long distance before disappearing, which is much different from the dilute case. The numerical results are consistent with the experimental results.     

Grid Turbulence in Superfluid 3He-B at Low Temperatures

Quantum turbulence consists of a tangle of quantised vortex lines which interact via their self induced flow. At very low temperatures there is no normal fluid component and no associated viscosity. These are very simple conditions in which to study turbulence which might eventually lead to a better understanding of turbulence in general. There are a number of interesting questions, such as how closely does quantum turbulence resemble classical turbulence and how does it decay in the absence of the viscous dissipation. We have recently developed techniques for detecting quantum turbulence in superfluid ³He-B in the low temperature limit. Using a vibrating grid, we find an unexpected sharp transition to turbulence via the entanglement of emitted vortex rings. Measurements also suggest that the quantum turbulence produced by the grid decays in a manner similar to that expected for classical turbulence, but the decay rate appears to be governed by the circulation quantum rather than viscosity.      

Instability of Quantized Vortices Attached to Oscillating Boundaries in Superfluid 4He

The motion of quantized vortices is studied using a vibrating wire in superfluid ⁴He. A vortex filtering method provides a superfluid practically free of remanent vortices in which the vibration of a wire cannot generate turbulence. Vortex lines are produced by cooling through the superfluid transition and remain forming bridges between a wire and a surrounding wall. Bridged remanent vortices increase the resonance frequency of a vibrating wire: the rate of an increase due to the remanent vortices is constant in a laminar flow regime and steeply increases in a turbulent flow regime with increasing wire velocity. These results suggest that oscillation of the bridged vortices provides a linear contribution to the wire vibration in the laminar flow regime, until instability occurs in the oscillation of the vortices, causing turbulence.      

Decay of Counterflow Quantum Turbulence in Superfluid 4He

We have simulated the decay of thermal counterflow quantum turbulence from a statistically steady state at T=1.9 K, with the assumption that the normal fluid is at rest during the decay. The results are consistent with the predictions of the Vinen equation (in essence the vortex line density decays as t ^?1). For the statistically steady state, we determine the parameter c ₂, which connects the curvature of the vortex lines and the mean separation of vortices. A formula connecting the parameter χ ₂ of the Vinen equation with c ₂ is shown to agree with the results of the simulations. Disagreement with experiment is discussed briefly.     

A Simple Phenomenological Model for the Effective Kinematic Viscosity of Helium Superfluids

A number of experiments where quantum turbulence in helium superfluids has been generated by various means (such as towed/oscillating grids, thermal counterflow, pure superflow, spin-down, ion/vortex rings emission) displays a temporal decay of the observed vortex line density, of the power law form L=Γ t ^?3/2 at late times. The prefactor, Γ, in analogy with classical homogeneous isotropic turbulence, allows deducing the temperature dependent effective kinematic viscosity, ν_eff, for turbulent helium superfluids. It appears to be a robust quantity, independent of methods of generating quantum turbulence and detecting the decaying vortex line density. We present a simple phenomenological model to estimate ν_eff based on comparison of dissipation terms in equations of motion for classical viscous flow and vortex flow of a superfluid in a stationary normal fluid. This model leads to ν_eff≈κ q, where q=α/(1?α′); α and α′ being dimensionless mutual friction parameters. Within the temperature range where mutual friction dissipation mechanism is dominant this simple model prediction agrees well with the experimental data and with the recent theoretical estimate of Roche, Barenghi and Leveque (Europhys. Lett. 87:54006, 2009).     

Dynamic Remanent Vortices in Superfluid 3He-B

We investigate the decay of vortices in a rotating cylindrical sample of ³He-B, after rotation has been stopped. With decreasing temperature vortex annihilation slows down as the damping in vortex motion, the mutual friction dissipation α(T), decreases almost exponentially. Remanent vortices then survive for increasingly long periods, while they move towards annihilation in zero applied flow. After a waiting period Δt at zero flow, rotation is reapplied and the remnants evolve to rectilinear vortices. By counting these lines, we measure at temperatures above the transition to turbulence ∼0.6 T _c the number of remnants as a function of α(T) and Δt. At temperatures below the transition to turbulence T≲0.55 T _c, remnants expanding in applied flow become unstable and generate in a turbulent burst the equilibrium number of vortices. Here we measure the onset temperature T _on of turbulence as a function of Δt, applied flow velocity v=v _n−v _s, and length of sample L.     

A Quasiparticle Detector for Imaging Quantum Turbulence in Superfluid ^3He-B

We describe the development of a two-dimensional quasiparticle detector for use in visualising quantum turbulence in superfluid $^3$ He-B at ultra-low temperatures. The detector consists of a $5 \times 5$ matrix of pixels, each a 1 mm diameter hole in a copper block containing a miniature quartz tuning fork. The damping on each fork provides a measure of the local quasiparticle flux. The detector is illuminated by a beam of ballistic quasiparticles generated from a nearby black-body radiator. A comparison of the damping on the different forks provides a measure of the cross-sectional profile of the beam. Further, we generate a tangle of vortices (quantum turbulence) in the path of the beam using a vibrating wire resonator. The vortices cast a shadow onto the face of the detector due to the Andreev reflection of quasiparticles in the beam. This allows us to image the vortices and to investigate their dynamics. Here we give details of the design and construction of the detector and show some preliminary results for one row of pixels which demonstrates its successful application to measuring quasiparticle beams and quantum turbulence.     

Energy Spectrum of the Superfluid Velocity Made by Quantized Vortices in Two-Dimensional Quantum Turbulence

We discuss the configurations of vortices in two-dimensional quantum turbulence, studying energy spectrum of superfluid velocity and correlation functions with the distance between two vortices. We apply the above method to quantum turbulence described by Gross-Pitaevskii equation in Bose-Einstein condensates. We make two-dimensional quantum turbulence from many dark solitons through the dynamical instability. A dark soliton is unstable and decays into vortices in two- and three-dimensional systems. In our work, we propose a method of discriminating between the uncorrelated turbulence and the correlated turbulence. We decompose the energy spectrum into two terms, namely the self-energy spectrum E _self (k) made by individual vortices and the interactive energy spectrum E _int (k) made by interference of two vortices. The uncorrelated turbulence is defined as turbulence with E _int (k)?E _self (k), while the correlated turbulence is turbulence where E _int (k) is not much smaller than E _self (k). Our simulations show that in the decay of dark solitons, the vortices created consist of correlated pairs of opposite circulation vortices, leading to the correlated turbulence.     

Vortex Line Density Fluctuations of Quantum Turbulence

We investigate vortex line density fluctuations of quantum turbulence generated by an oscillating grid in superfluid ³He-B. The scenario of quantum turbulence experimentally suggested by the Lancaster group is confirmed in the numerical simulation. The spectrum of the vortex line density fluctuations with respect to frequency obeyed a ?5/3 power law, which is consistent with the experiment of the Lancaster group. Based on the argument of time scales experienced by vortex rings with different sizes and on the power spectrum, the connection between the self-similar structure of the vortex tangle and the power spectrum is discussed.     

Decay of Turbulence Generated by Spin-Down to Rest in Superfluid 4He

We report on the extension of the experiments (P.M. Walmsley et al., Phys. Rev. Lett. 99:265302, 2007) on the decay of quasiclassical turbulence generated by an impulsive spin-down from angular velocity Ω to rest of superfluid ⁴He in a cubic container at temperatures 0.15 K–1.6 K. The density of quantized vortex lines L is measured by scattering negative ions. Following the spin-down, the maximal density of vortices is observed after time t～10Ω^?1. By observing the propagation of ions along the axis of the initial rotation, the transient dynamics of the turbulence spreading from the perimeter of the container into its central region is investigated. Nearly homogeneous turbulence develops after time t～100Ω^?1 and decays as L ∝ t ^?3/2. The effective kinematic viscosity in T=0 limit is ν=0.003κ, where κ=10^?3 cm²?s^?1 is the circulation quantum.     

Simulation of Counterflow Turbulence by Vortex Filaments

By using the vortex filament model with the full Biot-Savart law, we have succeeded for the first time in generating the statistically steady state of counterflow turbulence in superfluid ⁴He under periodic boundary conditions. This state exhibits the characteristic relation $L=\gamma^{2} v_{\mathit{ns}}^{2}$ between the line-length density L and the counterflow relative velocity v _ns and there is quantitative agreement between the coefficient γ and some measured values. Since we obtained the realistic state of quantum turbulence, we will use the numerical data to study the statistical property. We focus on the statistics of vortex reconnections from about 350 events in our steady counterflow turbulence simulation, and characterize the dynamics by the minimum separation distance between two reconnecting vortices, which was used in the visualization experiments using the solid hydrogen tracer particles by Paoletti et al. From our analysis we may conclude that the quantized circulation is still the dominant controlling feature and obtains the statistics of the correction factor.     

Impurity Effects in a Vortex Core in a Chiral p-Wave Superconductor Within the t-Matrix Approximation

We study the effects of non-magnetic impurity scattering on the Andreev bound states (ABS) in an isolated vortex in two-dimensional chiral p-wave superconductors numerically. We incorporate the impurity scattering effects into the quasiclassical Eilenberger formulation through the self-consistent t-matrix approximation. Within this scheme, we calculate the local density of states (LDOS) around two types of vortices: “parallel” (“anti-parallel”) vortex where the phase winding of the pair-potential coming from the vorticity and that coming from the chirality have the same (opposite) sign. When the scattering phase-shift δ ₀ of each impurity is small, we find that the impurities affect differently the spectra of quasiparticles localized around the two types of vortices in a way similar to that in the Born limit (δ ₀→0). For a larger δ ₀(?π/2), ABS in the vortex is strongly suppressed by the impurities for both types of vortices. We find that there are some correlations between the suppression of ABS near vortex cores and the low energy density of states due to the impurity bands in the bulk.     

The Dynamics of Vortex Generation in Superfluid 3He-B

A profound change occurs in the stability of quantized vortices in externally applied flow of superfluid ³He-B at temperatures ?0.6?T _c, owing to the rapidly decreasing damping in vortex motion with decreasing temperature. At low damping an evolving vortex may become unstable and generate a new independent vortex loop. This single-vortex instability is the generic precursor of turbulence. We investigate the instability with non-invasive NMR measurements on a rotating cylindrical sample in the intermediate temperature regime (0.3–0.6)?T _c. From comparisons with numerical calculations we interpret that the instability occurs at the container wall, when the vortex end moves along the wall in applied flow.     

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.      

Dissipation of Gross-Pitaevskii Turbulence Coupled with Thermal Excitations

Microscopic dissipation mechanism of quantized vortices in quantum fluid is studied numerically by solving the Gross-Pitaevskii equation coupled with the Bogoliubov-de-Gennes equation for thermal excitations. At low temperatures, dissipation works at smaller scales than the vortex core size, which supports the self-similar cascade process of quantized vortices at large scales and the Kolmogorov energy spectrum of quantum turbulence. On the other hand, this dissipation spreads to larger scales at high temperatures, and directly affects the vortex dynamics. This effect of dissipation at high temperatures is qualitatively similar to the mutual friction in superfluid ⁴He.     

On Developed Superfluid Turbulence

Superfluid turbulence is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the relative value of the friction force acting on a vortex with respect to the non-dissipative forces. The inverse parameter q^?1 plays the same role as the Reynolds number Re = U R/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence, described by a Kolmogorov cascade, occurs when Re ? 1 in classical hydrodynamics. In superfluids, the developed turbulence occurs at q ? 1. Another parameter of superfluid turbulence is the superfluid Reynolds number Re_s = U R/κ, which contains the circulation quantum κ characterizing quantized vorticity in superfluids. The two parameters q and Re_s control the crossover or transition between two classes of superfluid turbulence: (i) the classical regime, where the Kolmogorov cascade, probably modifed by the non-canonical dissipation due to mutual friction, is effective, vortices are locally polarized, and the quantization of vorticity is not important; and (ii) the Vinen quantum turbulence where the properties are determined by the quantization of vorticity. The phase diagram of these dynamical vortex states is suggested. PACS numbers: 43.37.+q, 47.32.Cc, 67.40.Vs, 67.57.Fg.