Evolution of the line-length density and anisotropy of quantum tangle in4He

Abstract

The variety of dynamic phenomena exhibited by superfluid ${}^{4}\mathrm{He}$ involves the appearance and motion of quantized vortices. The evolution of a quantized vortex tangle depends on its line-length density, but it is also related to various geometrical measures of the vortex lines forming the tangle. In this paper the microscopic dynamics of the vortex tangle is studied analytically to derive an evolution equation for an average binormal to the vortex lines, which is an important measure controlling the growth of the vortex tangle. This equation supplements the Vinen equation for the line-length density. The resulting system of two equations is here examined both analytically and numerically. It is shown that the system is applicable to an analysis of transients in which the counterflow changes its direction as well as to the processes with counterflow changing periodically with various frequencies. The latter class of processes is particularly important since in this case both the Vinen equation alone and the alternative Vinen equation give unphysical solutions.