  We present a renormalization-group calculation based on model F for the
thermal conductivity lambda_T(T,Q) in the presence of a homogeneous heat
current Q and gravity. For temperatures below T_lambda we obtain a large but
finite thermal conductivity corresponding to superfluid 4He with dissipation.
Furthermore, we consider the self-organized critical state where the effects of
gravity and heat current cancel each other so that the distance from
criticality delta T = T(z) - T_lambda(z) is constant in space and a function of
Q. We compare our theoretical results with recent experiments.
