Dynamics of the density of quantized vortex lines in counterflow turbulence: Experimental investigation

Recently the interest in thermal counterflow of superfluid He-4, the most extensively studied form of quantum turbulence, has been renewed. Particularly, an intense theoretical debate has arisen about what form, if any, of the so-called Vinen equation accurately captures the dynamics of vortex line density, L. We address this problem experimentally, in a 21 cm long channel of square 7 x 7 mm(2) cross section. Based on large statistics of second-sound data measured in nonequilibrium square-wave modulated thermally induced counterflow we investigate the phase portrait of the general form of the governing dynamical equation and conclude that for sparse tangles (L less than or similar to 10(5) cm(-2)) all proposed forms of this equation based on the concept of a homogeneous random tangle of quantized vortices provide equally adequate descriptions of the growth of L, while for dense tangles (L > 10(5) cm(-2)) none of them is satisfactory or able to account for the significant slow-down in tangle growth rate as the steady state is approached. We claim, however, that agreement with theory is recovered if the geometrical parameter c(2) introduced in numerical studies by K. W. Schwarz [Phys. Rev. B 38, 2398 (1988)] is allowed to vary with vortex line density which also greatly improves the prediction of the observed early decay rate.