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.

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

Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints

We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, for the listed different settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule. This means that neither stepsize nor stopping rule require to know the Lipschitz constant of the objective or constraint. We also construct Mirror Descent for problems with objective function, which is not Lipschitz continuous, e.g. is a quadratic function. Besides that, we address the problem of recovering the solution of the dual problem

General model selection estimation of a periodic regression with a Gaussian noise

This paper considers the problem of estimating a periodic function in a continuous time regression model with an additive stationary gaussian noise having unknown correlation function. A general model selection procedure on the basis of arbitrary projective estimates, which does not need the knowledge of the noise correlation function, is proposed. A non-asymptotic upper bound for quadratic risk (oracle inequality) has been derived under mild conditions on the noise. For the Ornstein-Uhlenbeck noise the risk upper bound is shown to be uniform in the nuisance parameter. In the case of gaussian white noise the constructed procedure has some advantages as compared with the procedure based on the least squares estimates (LSE). The asymptotic minimaxity of the estimates has been proved. The proposed model selection scheme is extended also to the estimation problem based on the discrete data applicably to the situation when high frequency sampling can not be provided

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

Scattering of first and second sound waves by quantum vorticity in superfluid Helium

We study the scattering of first and second sound waves by quantum vorticity in superfluid Helium using two-fluid hydrodynamics. The vorticity of the superfluid component and the sound interact because of the nonlinear character of these equations. Explicit expressions for the scattered pressure and temperature are worked out in a first Born approximation, and care is exercised in delimiting the range of validity of the assumptions needed for this approximation to hold. An incident second sound wave will partly convert into first sound, and an incident first sound wave will partly convert into second sound. General considerations show that most incident first sound converts into second sound, but not the other way around. These considerations are validated using a vortex dipole as an explicitely worked out example.Comment: 24 pages, Latex, to appear in Journal of Low Temperature Physic

Parametric generation of second sound in superfluid helium: linear stability and nonlinear dynamics

We report the experimental studies of a parametric excitation of a second sound (SS) by a first sound (FS) in a superfluid helium in a resonance cavity. The results on several topics in this system are presented: (i) The linear properties of the instability, namely, the threshold, its temperature and geometrical dependencies, and the spectra of SS just above the onset were measured. They were found to be in a good quantitative agreement with the theory. (ii) It was shown that the mechanism of SS amplitude saturation is due to the nonlinear attenuation of SS via three wave interactions between the SS waves. Strong low frequency amplitude fluctuations of SS above the threshold were observed. The spectra of these fluctuations had a universal shape with exponentially decaying tails. Furthermore, the spectral width grew continuously with the FS amplitude. The role of three and four wave interactions are discussed with respect to the nonlinear SS behavior. The first evidence of Gaussian statistics of the wave amplitudes for the parametrically generated wave ensemble was obtained. (iii) The experiments on simultaneous pumping of the FS and independent SS waves revealed new effects. Below the instability threshold, the SS phase conjugation as a result of three-wave interactions between the FS and SS waves was observed. Above the threshold two new effects were found: a giant amplification of the SS wave intensity and strong resonance oscillations of the SS wave amplitude as a function of the FS amplitude. Qualitative explanations of these effects are suggested.Comment: 73 pages, 23 figures. to appear in Phys. Rev. B, July 1 st (2001