Evolution of a Network of Vortex Loops in the Turbulent Superfluid Helium; Derivation of the Vinen Equation

The evolution a network of vortex loops due to the fusion and breakdown in the turbulent superfluid helium is studied. We perform investigation on the base of the "rate equation" for the distribution function $n(l)$ of number of loops in space of their length $l$. There are two mechanisms for change of quantity $n(l)$. Firstly, the function changes due to deterministic process of mutual friction, when the length grows or decreases depending on orientation. Secondly, the change of $n(l)$ occurs due to random events when the loop crosses itself breaking down into two daughter or two loops collide merging into one larger loop. Accordingly the "rate equation" includes the "collision" term collecting random processes of fusion and breakdown and the deterministic term. Assuming, further, that processes of random colliding are fastest we are in position to study more slow processes related to deterministic term. In this way we study the evolution of full length of vortex loops per unit volume-so called vortex line density ${\cal L}(t)$. It is shown this evolution to obey the famous Vinen equation. In conclusion we discuss properties of the Vinen equation from the point of view of the developed approach.Comment: Presentation at QFS2006, submitted to JLT

Reconnections of Vortex Loops in the Superfluid Turbulent HeII. Rates of the Breakdown and Fusion processes

Kinetics of merging and breaking down vortex loops is the important part of the whole vortex tangle dynamics. Another part is the motion of individual lines, which obeys the Biot-Savart law in presence of friction force and of applied external velocity fields if any. In the present work we evaluate the coefficients of the reconnection rates $A(l_{1},l_{2},l)$ and $B(l,l_{1},l_{2})$. Quantity $A$ is a number (per unit of time and per unit of volume) of events, when two loops with lengths $l_{1}$and $l_{2}$ collide and form the single loop of length $l=l_{1}+l_{2}$. Quantity $$ describes the rate of events, when the single loop of the length $l$ breaks down into two the daughter loops of lengths $l_{1}$ and $l_{2}$. These quantities ave evaluated as the averaged numbers of zeroes of vector $\mathbf{S}$ connecting two points on the loops of $\xi_{2}$ and $\xi_{1}$ at moment of time $t$. Statistics of the individual loops is taken from the Gaussian model of vortex tangle. PACS-number 67.40Comment: 9 pages, 5 figures, To be submitted to JLT

Probing of quantum turbulence with the emitting vortex loops

The statistics of vortex loops emitted from the domain with quantum turbulence is studied. The investigation is performed on the supposition that the vortex loops have the Brownian or random walking structure with the generalized Wiener distribution. The main goal is to relate the properties of the emitted vortex loops with the parameters of quantum turbulence. The motivation of this work is connected with recent studies, both numerical and experimental, on study of emitted vortex loops. This technique opens up new opportunities to probe superfluid turbulence. We demonstrated how the statistics of emitted loops is expressed in terms of the vortex tangle parameters and performed the comparison with numerical simulations

Torsional oscillation of the vortex tangle. Possible applications to oscillations of solid ⁴He

Torsional oscillation of the vessels with quantum fluids is one of oldest and most popular methods for the study of quantized vortices. The recent and very bright example is the discovery of the supersolidity of the solid helium. In the torsion oscillation experiments the drop in the period of oscillations with achievement of some small temperature has been observed. This effect was attributed to the appearance of the superfluid component. This phenomenon depends on many various factors and has various explanations. But, if to adopt (at least hypothetically, at this stage) that the phenomenon of “supersolidity” (dissipativeless flow) is realized, we must consider the relaxation of the vortex system (we can call it as vortex tangle, vortex fluid, chaotic set of vortices, etc.). We have to do it for the very simple reason, that the only way to involve the superfluid component into rotation is the presence of the polarized vortices (with nonzero mean polarization along the axis of rotation). In the present work we submit the approach describing the vortex tangle relaxation model for the torsional oscillation responses of quantum systems, having in mind to apply it for the study of solid ⁴He. It is shown that the rotation of the superfluid component occurs in the relaxation-like manner with the relaxation time dependent on the amplitude of oscillation (as well as on the temperature and pressure). The study of this problem shows that there is a quasi-linear solution explaining the (amplitude dependent) shift of period. There is also an imaginary shift of the frequency (also the amplitude-dependent), which describes an additional dissipation. The results of the theory are compared with the recent measurements

Decay of the vortex tangle at zero temperature and quasiclassical turbulence

We review and analyze a series of works, both experimental and numerical and theoretical, dealing with the decay of quantum turbulence at zero temperature. Free decay of the vortex tangle is a key argument in favor of the idea that a chaotic set of quantum vortices can mimic classical turbulence, or at least reproduce many of the basic features. The corresponding topic is referred as the quasiclassical turbulence. Appreciating significance of the challenging problem of classical turbulence it can be expressed that the idea to study it in terms of quantized line is indeed very important and may be regarded as a breakthrough. For this reason, the whole theory, together with the supporting experimental results and numerical simulations should be carefully scrutinized. One of the main arguments, supporting the idea of quasiclassical turbulence is the fact that vortex tangle decays at zero temperature, when the mutual friction is absent. Since all other possible mechanisms of dissipation of the vortex energy, discussed in literature, are related to the small scales, it is natural to suggest that the Kolmogorov cas-cade takes place with the flow of the energy in space of scales, just like as in the classical turbulence. In the present work we discuss an alternative mechanism of decay of the vortex tangle, which is not associated with dissipation at small scales. This mechanism is a diffusive-like spreading of the vortex tangle due to evaporation of small vortex loops. We discuss a number of experiments and numerical simulations, considering them from the point of view of alternative mechanism

Method of trial distribution function for quantum turbulence

Studying quantum turbulence the necessity of calculation the various characteristics of the vortex tangle (VT) appears. Some of "crude" quantities can be expressed directly via the total length of vortex lines (per unit of volume) or the vortex line density L(t) and the structure parameters of the VT. Other more “subtle” quantities require knowledge of the vortex line configurations {s(ξ,t)}. Usually, the corresponding calculations are carried out with the use of more or less truthful speculations concerning arrangement of the VT. In this paper we review other way to solution of this problem. It is based on the trial distribution functional (TDF) in space of vortex loop configurations. The TDF is constructed on the basis of well established properties of the vortex tangle. It is designed to calculate various averages taken over stochastic vortex loop configurations. In this paper we also review several applications of the use this model to calculate some important characteristics of the vortex tangle. In particular we discussed the average superfluid mass current J induced by vortices and its dynamics. We also describe the diffusion-like processes in the nonuniform vortex tangle and propagation of turbulent fronts

Energy spectrum of the quantum vortices configurations

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 can be exactly calculated, provided that we know the probability distribution functional P({s(ξ,t)}) of vortex loops configurations. The knowledge of P({s(ξ,t)}) is identical to the full solution of the problem of quantum turbulence and, in general, P 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

Numerical study of the diffusive-like decay of the vortex tangle without mutual friction

The numerical simulation of the diffusive-like decay of the vortex tangle without mutual friction was performed. The simulation was made with the use of the localized induction approximation. The early developed by authors algorithm, which is based on consideration of crossing lines, was used for vortex reconnection processes. We have determined the influence of different factors on decay of an inhomogeneous vortex tangle: a diffusion (large vortex loops break up to smaller ones which go away from initial volume), change of length owing to reconnection processes, the eliminations of small vortices below the space resolution, the insertion and removing of points to supply numerical algorithm stability. The obtained numerical results demonstrate that the vortex tangle, initially localized in the small region, is smearing into ambient space. The time evolution of vortex line density inside the initial domain satisfactory agrees with the ones, obtained from the solution of diffusion equation

Simulation of stochastic vortex tangle

We present the results of simulation of the chaotic dynamics of quantized vortices in the bulk of superfluid He II. Evolution of vortex lines is calculated on the base of the Biot-Savart law. The dissipative effects appeared from the interaction with the normal component, or/and from relaxation of the order parameter are taken into account. Chaotic dynamics appears in the system via a random forcing, e.i. we use the Langevin approach to the problem. In the present paper we require the correlator of the random force to satisfy the fluctuation-dissipation relation, which implies that thermodynamic equilibrium should be reached. In the paper we describe the numerical methods for integration of stochastic differential equation (including a new algorithm for reconnection processes), and we present the results of calculation of some characteristics of a vortex tangle such as the total length, distribution of loops in the space of their length, and the energy spectrum

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