I'm interested in a deeper understanding of Rayleigh-Benard convection that occurs when a shallow dish of water is heated from below. I more or less follow the derivation of convection cells using the Boussinesq approximation. But the resulting velocity distribution is so symmetrical that I can't see where the work necessary to generate and maintain the cyclic motion comes from. It would seem that each fluid element must act as a tiny heat engine, although not necessarily following a Carnot cycle. But with a highly symmetric velocity and temperature distribution, and the assumption that density is a function of temperature, it looks like the course of an individual fluid element on a V-p diagram would enclose zero area. That is, no net work is done by the fluid element. Anyone know of a treatment of this problem that takes thermodynamics into account?