Self-organization in He4 near the superfluid transition in heat flow and gravity

We investigate the nonlinear dynamics of He4 slightly below the superffluid transition by integrating model F equations in three dimensions. When heated from above under gravity, a vortex tangle and a sheetlike phase slip are generated near the bottom plate. Then a self-organized superfluid containing high-density vortices and phase slips grows upward into an ordinary superfluid. The thermal resistance due to these defects produces a constant temperature gradient equal to the gradient of the pressure-dependent transition temperature $T_{\lambda}(p)$. In this self-organized region, the temperature deviation $T-T_{\lambda}(p)$ consists of a negative constant independent of the height and time-dependent fluctuations. Its time-average is calculated in good agreement with the experimental value (W.A. Moeur {\it et al.}, Phys. Rev. Lett. 78, 2421 (1997)).