On quantum turbulence in superfluid 4He (Archives of Mechanics - Test, 2001)
The alternative approach to the homogeneous quantum turbulence is proposed in order to derive the evolution equation for vortex line-length density. Special attention is paid to reconnections of vortex lines. According to our previous paper, the summary line-length change ΔS of two vortex lines resulting from the reconnection (in the presence of counterflow Vns) ban be approximated by the expression: ΔS = -a t1/2 + b Vns2 t3/2, with a>0, b≥0. The dynamics of vortex lines in the tangle is considered as a sequence of reconnections followed by "free" evolutions. For the steady-state turbulence, the average line-length change between reconnections has to be zero. If, for a given value of the counterflow, the line density is smaller than the equilibrium one, the reconnections occur less frequently and becomes positive. As a result, the line density grows until the equilibrium is restored. On the other hand, when the line-density is too large, the reconnections are very frequent, so the lines shorten between reconnections and the line density becomes smaller. The time derivative of total line density is proportional to the reconnection frequency multiplied by the average line-length change due to a single reconnection. The evolution equation obtained in the proposed approach resembles the alternative Vinen equation.