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

Note on the paper of Fu and Wong on strictly pseudoconvex domains with K\"ahler--Einstein Bergman metrics

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two

Numerical simulation of stochastic vortex tangles

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-disspation 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.Comment: 8 pages, 5 figure

Dynamics of coreless vortices and rotation-induced dissipation peak in superfluid films on rotating porous substrates

We analyze dynamics of 3D coreless vortices in superfluid films covering porous substrates. The 3D vortex dynamics is derived from the 2D dynamics of the film. The motion of a 3D vortex is a sequence of jumps between neighboring substrate cells, which can be described, nevertheless, in terms of quasi-continuous motion with average vortex velocity. The vortex velocity is derived from the dissociation rate of vortex-antivortex pairs in a 2D film, which was developed in the past on the basis of the Kosterlitz-Thouless theory. The theory explains the rotation-induced dissipation peak in torsion-oscillator experiments on $^4$He films on rotating porous substrates and can be used in the analysis of other phenomena related to vortex motion in films on porous substrates.Comment: 8 pages, 3 figures submitted to Phys. Rev.

Vortex dynamics in rotating counterflow and plane Couette and Poiseuille turbulence in superfluid Helium

An equation previously proposed to describe the evolution of vortex line density in rotating counterflow turbulent tangles in superfluid helium is generalized to incorporate nonvanishing barycentric velocity and velocity gradients. Our generalization is compared with an analogous approach proposed by Lipniacki, and with experimental results by Swanson et al. in rotating counterflow, and it is used to evaluate the vortex density in plane Couette and Poiseuille flows of superfluid helium.Comment: 18 pages, 2 figure

Equilibrium rotation of a vortex bundle terminating on a lateral wall

The paper investigates possibility of equilibrium solid-body rotation of a vortex bundle diverging at some height from a cylinder axis and terminating on a lateral wall of a container. Such a bundle arises when vorticity expands up from a container bottom eventually filling the whole container. The analysis starts from a single vortex, then goes to a vortex sheet, and finally addresses a multi-layered crystal vortex bundle. The equilibrium solid-body rotation of the vortex bundle requires that the thermodynamic potentials in the vortex-filled and in the vortex-free parts of the container are equal providing the absence of a force on the vortex front separating the two parts. The paper considers also a weakly non-equilibrium state when the bundle and the container rotate with different angular velocities and the vortex front propagates with the velocity determined by friction between vortices and the container or the normal liquid moving together with the container.Comment: 16 pages, 5 figure

A Kelvin-wave cascade on a vortex in superfluid 4^4He at a very low temperature

A study by computer simulation is reported of the behaviour of a quantized vortex line at a very low temperature when there is continuous excitation of low-frequency Kelvin waves. There is no dissipation except by phonon radiation at a very high frequency. It is shown that non-linear coupling leads to a net flow of energy to higher wavenumbers and to the development of a simple spectrum of Kelvin waves that is insensitive to the strength and frequency of the exciting drive. The results are likely to be relevant to the decay of turbulence in superfluid $^4$He at very low temperatures

Three dimensionality of pulsed second-sound waves in He II

Three dimensionality of 3D pulsed second sound wave in He II emitted from a finite size heater is experimentally investigated and theoretically studied based on two-fluid model in this study. The detailed propagation of 3D pulsed second sound wave is presented and reasonable agreement between the experimental and theoretical results is obtained. Heater size has a big influence on the profile of 3D second sound wave. The counterflow between the superfluid and normal fluid components becomes inverse in the rarefaction of 3D second sound wave. The amplitude of rarefaction decreases due to the interaction between second sound wave and quantized vortices, which explains the experimental results about second sound wave near [Phys. Rev. Lett. 73, 2480 (1994)]. The accumulation of dense quantized vortices in the vicinity of heater surface leads to the formation of a thermal boundary layer, and further increase of heating duration results in the occurrence of boiling phenomena. PACS numbers: 67.40.Pm 43.25.+y 67.40.BzComment: 30 pages, 9 figures. Physical Review B, Accepte

Kinetics of a Network of Vortex Loops in He II and a Theory of Superfluid Turbulence

A theory is developed to describe the superfluid turbulence on the base of kinetics of the merging and splitting vortex loops. Because of very frequent reconnections the vortex loops (as a whole) do not live long enough to perform any essential evolution due to the deterministic motion. On the contrary, they rapidly merge and split, and these random recombination processes prevail over other slower dynamic processes. To develop quantitative description we take the vortex loops to have a Brownian structure with the only degree of freedom, which is the length $l$ of the loop. We perform investigation on the base of the Boltzmann type kinetic equation for the distribution function $n(l)$ of number of loops with length $l$. By use of the special ansatz in the collision integral we have found the exact power-like solution to kinetic equation in the stationary case. This solution is not (thermodynamically) equilibrium, but on the contrary, it describes the state with two mutual fluxes of the length (or energy) in space of sizes of the vortex loops. The term flux means just redistribution of length (or energy) among the loops of different sizes due to reconnections. Analyzing this solution we drew several results on the structure and dynamics of the vortex tangle in the turbulent superfluid helium. In particular, we evaluated the mean radius of the curvature and the full rate of the reconnection events. We also studied the evolution of the full length of vortex loops per unit volume-the so-called vortex line density. It is shown this evolution to obey the famous Vinen equation. The properties of the Vinen equation from the point of view of the developed approach had been discussed.Comment: 34 pages, 9 Postscript figures, [aps,preprint,12pt]{revtex4