Properties of Helium-4 In and Near the Self-Organized Critical State

If a downward heat flux is imposed on a sample of $^{4}$He near $T_{\lambda}$, the sample can self-organize so that its temperature tracks the variation of $T_{\lambda}$ induced by the hydrostatic pressure head. This ``Self-Organized-Critical'' (SOC) state is the only means by which a uniform reduced temperature very close to $T_{\lambda}$ can be achieved on Earth in $^{4}$He. We recently reported preliminary analysis of extensive new measurements of the SOC state showing three new results: strong nonlinearity in the upward-going wave under high drive levels, the qualitative form of the breakdown of the SOC state with increasing downward heat flux greater than $\sim $12$\mu $W/cm$^{2}$, and, most intriguingly, we have corroborated and extended the as-yet unexplained result of Lee \textit{et al.} that the thermal resistivity of helium-II near $T_{\lambda }$ is larger under downward heat flux than it is under upward heat flux of equal magnitude (upward resistivity measured previously by Baddar \textit{et al.}). We find that the ``downward'' resistivity exceeds the ``upward'' by factors ranging from 18X at 20$\mu $W/cm$^{2}$ to 12X at 80$\mu $W/cm$^{2}$. Here we report results of further analysis which help to quantify and flesh out this intriguing picture.