Dynamics of vortex tangle without mutual friction in superfluid 4^4He

A recent experiment has shown that a tangle of quantized vortices in superfluid $^4$He decayed even at mK temperatures where the normal fluid was negligible and no mutual friction worked. Motivated by this experiment, this work studies numerically the dynamics of the vortex tangle without the mutual friction, thus showing that a self-similar cascade process, whereby large vortex loops break up to smaller ones, proceeds in the vortex tangle and is closely related with its free decay. This cascade process which may be covered with the mutual friction at higher temperatures is just the one at zero temperature Feynman proposed long ago. The full Biot-Savart calculation is made for dilute vortices, while the localized induction approximation is used for a dense tangle. The former finds the elementary scenario: the reconnection of the vortices excites vortex waves along them and makes them kinked, which could be suppressed if the mutual friction worked. The kinked parts reconnect with the vortex they belong to, dividing into small loops. The latter simulation under the localized induction approximation shows that such cascade process actually proceeds self-similarly in a dense tangle and continues to make small vortices. Considering that the vortices of the interatomic size no longer keep the picture of vortex, the cascade process leads to the decay of the vortex line density. The presence of the cascade process is supported also by investigating the classification of the reconnection type and the size distribution of vortices. The decay of the vortex line density is consistent with the solution of the Vinen's equation which was originally derived on the basis of the idea of homogeneous turbulence with the cascade process. The obtained result is compared with the recent Vinen's theory.Comment: 16 pages, 16 figures, submitted to PR

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 (VLD) 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 \chi_2 of the Vinen equation with c_2 is shown to agree with the results of the simulations. Disagreement with experiment is discussed.Comment: 7 pages, 7 figure

Instability of vortex array and transitions to turbulent states in rotating helium II

We consider superfluid helium inside a container which rotates at constant angular velocity and investigate numerically the stability of the array of quantized vortices in the presence of an imposed axial counterflow. This problem was studied experimentally by Swanson {\it et al.}, who reported evidence of instabilities at increasing axial flow but were not able to explain their nature. We find that Kelvin waves on individual vortices become unstable and grow in amplitude, until the amplitude of the waves becomes large enough that vortex reconnections take place and the vortex array is destabilized. The eventual nonlinear saturation of the instability consists of a turbulent tangle of quantized vortices which is strongly polarized. The computed results compare well with the experiments. Finally we suggest a theoretical explanation for the second instability which was observed at higher values of the axial flow

From Coherent Modes to Turbulence and Granulation of Trapped Gases

The process of exciting the gas of trapped bosons from an equilibrium initial state to strongly nonequilibrium states is described as a procedure of symmetry restoration caused by external perturbations. Initially, the trapped gas is cooled down to such low temperatures, when practically all atoms are in Bose-Einstein condensed state, which implies the broken global gauge symmetry. Excitations are realized either by imposing external alternating fields, modulating the trapping potential and shaking the cloud of trapped atoms, or it can be done by varying atomic interactions by means of Feshbach resonance techniques. Gradually increasing the amount of energy pumped into the system, which is realized either by strengthening the modulation amplitude or by increasing the excitation time, produces a series of nonequilibrium states, with the growing fraction of atoms for which the gauge symmetry is restored. In this way, the initial equilibrium system, with the broken gauge symmetry and all atoms condensed, can be excited to the state, where all atoms are in the normal state, with completely restored gauge symmetry. In this process, the system, starting from the regular superfluid state, passes through the states of vortex superfluid, turbulent superfluid, heterophase granular fluid, to the state of normal chaotic fluid in turbulent regime. Both theoretical and experimental studies are presented.Comment: Latex file, 25 pages, 4 figure

Elementary Vortex Processes in Thermal Superfluid Turbulence

By solving pertinent mathematical models with numerical and computational methods, we analyze the formation of superfluid vorticity structures in a turbulent normal fluid with an inertial range exhibiting Kolmogorov scaling. We demonstrate that mutual friction forcing causes quantum vortex instabilities whose signature is spiral vortical configurations. The spirals expand until they accidentally meet metastable, intense normal fluid vorticity tubes of similar curvature and vorticity orientation that trap them by driving them towards low mutual friction sites where superfluid bundles are formed. The bundle formation sites are located within the tube cores, but, due to tube curvature and many-tube interaction effects, are displaced by variable distances from the tube centerlines as they follow the contours of the latter. We analyze possible implications of these processes in fully developed thermal superfluid turbulence dynamics

A method for driving an oscillator at a quasi-uniform velocity

We describe a simple way to drive an actuator, comprising a superconducting coil moving in a static magnetic field, at a quasi-uniform velocity. The main objective is to avoid oscillations in this system with low damping, as they undermine the uniformity of the velocity. The method consists in calculating the force that should be exerted on the coil from the equation of motion and programming a waveform generator to produce the corresponding current through the coil. The method was tested on a device towing a grid through a closely fitted channel filled with superfluid 4He at temperatures below 100 mK. The motion of the grid over the distance of 4.3 cm at 10 cm/s resulted in oscillations of less than 50 micron in amplitude (or less than 1 mm/s in terms of velocity). The method can be applied to any oscillator

Energy spectra of developed turbulence in helium superfluids

We suggest a "minimal model" for the 3D turbulent energy spectra in superfluids, based on their two-fluid description. We start from the Navier-Stokes equation for the normal fluid and from the coarse-grained hydrodynamic equation for the superfluid component (obtained from the Euler equation for the superfluid velocity after averaging over the vortex lines) and introduce a mutual friction coupling term, proportional to the counterflow velocity, the average superfluid vorticity and to the temperature dependent parameter q = alpha/(1+alpha'), where alpha and alpha' denote the dimensionless parameters characterizing the mutual friction between quantized vortices and the normal component of the liquid. We then derive the energy balance equations, taking into account the cross-velocity correlations. We obtain all asymptotical solutions for normal and superfluid energy spectra for limiting cases of small/big normal to superfluid density ratio and coupling. We discuss the applicability of our model to superfluid He II and to He-3-B

Grid turbulence in superfluid He-3-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 He-3-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