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

Recently the interest in thermal counterflow of superfluid $^{4}\mathrm{He}$, 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 $7\ifmmode\times\else\texttimes\fi{}7\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}$ 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 $(L\ensuremath{\lesssim}{10}^{5}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2})$ 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 $(Lg{10}^{5}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2})$ 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.