A derivation is given of the heat flux induced by large amplitude gravity waves; the case where nonlinear or turbulent damping exceeds molecular damping. This case is typical of the middle atmosphere. The derivation is based on a simple algorithm for converting the linear dispersion relation for gravity waves into a nonlinear one, and generalizes a recent derivation by Walterscheid. The derived heat deposition flux is shown to cause local cooling where the waves are close to saturation, but could cause heating where the waves are far from saturation. It is then proven that the heat flux is related to the diffusivity and to the wave momentum deposition flux by simple expressions. Hence, knowledge of one transport coefficient is practically equivalent to knowledge of the other two. The derivation is limited to waves that have some randomicity.