Abstract
The evolution of a quantised vortex tangle in superfluid 4He 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 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 (in the contrary to the both Vinen equation alone and alternative Vinen equation) is applicable to an analysis of transients in which the counter-flow changes its direction as well as to the processes with counter-flow changing periodically with various frequencies.
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