Diffusion of inhomogeneous vortex tangle and decay of superfluid turbulence

Abstract

The theory of inhomogeneous superfluid turbulence is developed on the basis of kinetics of merging and splitting vortex loops. Vortex loops composing the vortex tangle can move as a whole with some drift velocity depending on their structure and length. The flux of length, energy, momentum, etc., executed by the moving vortex loops takes place. The situation here is exactly the same as in usual classical kinetic theory, with the difference being that the "carriers" of various physical quantities are not the point particles but extended objects ͑vortex loops͒, which possess an infinite number of degrees of freedom, with highly involved dynamics. We suggest to complete our investigation, based on the supposition that vortex loops have a Brownian structure, with the only degree of freedom being, lengths of loops l. This concept allows us to study the dynamics of the vortex tangle on the basis of the kinetic equation for the distribution function n͑l , t͒-the density of a loop in the space of their lengths. Imposing the coordinate dependence on the distribution function n͑l , r , t͒ and modifying the "kinetic" equation with regard to an inhomogeneous situation, we are able to investigate various problems on the transport processes in superfluid turbulence. In this paper, we evaluate the flux of the vortex line density L͑x , t͒ due to the gradient of this quantity. The corresponding evolution of quantity L͑x , t͒ obeys the diffusion type equation, as it can be expected from dimensional analysis. The diffusion coefficient is arrived at from calculation of the ͑size-dependent͒ free path and drift velocity of the vortex loops, and takes the value 2.2, which exceeds approximately 20-fold the value obtained in early numerical simulation. We discuss the probable reason for this large discrepancy. We use the diffusion equation to describe the decay of the vortex tangle at a very low temperature. Comparison with recent experiments on decay of the superfluid turbulence is presented.

Spreading of superfluid vorticity clouds in normal-fluid turbulence

“…In our case, however, the normal-fluid inertia is large and the accompanying strong smoothing mutual friction effects do not allow the highly corrugated quantized vortex configurations that, via reconnections, lead to very small vortex loops. Notably, Nemirovskii has recently developed a theory of superfluid vortex cloud expansion in terms of a diffusion equation for the vortex-line density (Nemirovskii 2010). This theory is useful in the context of the pure superfluid Barenghi & Samuels (2002) or thermal superfluid with zero normal-fluid inertia (Tsubota et al 2003) cases discussed above, but it is not applicable in our case since cloud expansion here is driven by spiral structure forming instabilities that occur at the interface between the vortex cloud and highly energetic free normal-fluid turbulence.…”

Method of trial distribution function for quantum turbulence

“…2). To develop the theory of the transport processes fulfilled by vortex loops (in the spirit of classical kinetic theory) we need to calculate the drift velocity l V and the free path ( ) l λ for the loop of size l. Referring to the paper [20] we write down here the following result. The drift velocity l V for the loop of size l is…”

“…However, if we were to adopt the data grounded on the results of the theory described in [20,21] it is possible to conclude that 2.2 v D ≈ κ . The diffusion-like evolution of the vortex tangle on the base Eq.…”

“…However, since the authors studied the dilute tangle, they observed the "ballistic" regime, rather than pure diffusion [19]. The quantitative theory of the diffusion-like spreading of the VT was developed by the author; details can be found in paper [20]. The study is based on the TDF method stated in Sec.…”

“…Performing all prescribed computations (see for details [20]) we write the flux J of vortex line is proportional to ∇L , = v D ∇ J L and, correspondingly, the spatial-temporal evolution of quantity L obeys the diffusion-type equation .…”