Gaussian model of vortex tangle in He II

Abstract

A description of the chaotic vortex tangle in superfluid turbulent He II is developed. Unlike current phenomenological theory dealing with only the macroscopic variable, the vortex line density L v (t) and describing thereby only the macroscopic hydrodynamic phenomena, our approach allows us to describe effects due to the arrangement of the vortex tangle and the interaction of lines. To develop this approach we introduce a trial distribution function in the space of vortex loop configurations which absorbs all properties of superfluid turbulence known both from experiment and from numerical simulations. This trial distribution function is built in terms of the path integral. A number of allowed configurations is obtained evaluating the path integral with constraints connected with the established properties of the vortex tangle. Using the trial distribution function we also build the characteristic ͑generating͒ functional which allows us to evaluate any average over the vortex loop configuration. On the basis of the developed approach we briefly discuss some simple statistical characteristics of the vortex tangle. A more extended example of the developed approach studying superfluid mass current induced by vortex tangle is reported in a subsequent paper.

Energy Spectrum of the 3D Velocity Field, Induced by Vortex Tangle

“…(4.60),(4.61)) that the velocity v(r) scales as r 1 . Thus, the uniform vortex array creates the course-grained motion, which is rotation, as it should be.…”

“…This theory is based on the Gaussian model (see 1 ). The probability P Gauss ({s(ξ ,t)}) of finding a particular configuration {s(ξ ,t)} is expressed by the probability distribution functional (for details, see the paper by author 1 )…”

“…For this reason, the problem of the spectrum of the 3D velocity field induced by a chaotic vortex filament becomes one of the central questions. The formal relation, allowing of calculating E(k) = ρ s v k v −k via the vortex line configuration {s(ξ )}, is given by formula (see 1 )…”

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

“…We would like to recall that he supposed decay of the vortex tangle due to the cascade like breakdown of vortex loops with further disappearance of them on very small scales. Relation (9) shows that there is possible the inverse cascade, which corresponds to the cascade-like fusion of vortex loops. Unfortunately our approach has too approximate character to do any strong quantitative conclusion.…”

“…One way elaborated in context of lambda-transition ( 8 ) is to imagine vortex loops as a set of rings of different sizes and to take their radius as the only degrees of freedom. In paper of author (see 9 ) there is offered another way -to think of vortex loops as randomly walking chains.The physical ground for this supposition is the following one. During evolution the loops undergo huge amount of collisions with other loops (or with itself) with consequent reconnection.…”