High Rayleigh number convection in a one-dimensional model

A model for one-dimensional convection is proposed by adding a buoyancy term to the Burgers' equation and including an equation for the temperature perturbation. A linear stability analysis shows onset of instability at a critical Rayleigh number. Computation in the unstable region shows steady convection with only one convection cell. Computations up to 10(5) times the critical Rayleigh number do not show transition to an oscillatory state or to turbulence. Using a large Rayleigh number approximation, closed form solutions for the spectrum and the scaling for the heat transport due to nonlinear convection are obtained up to two orders. These are shown to be in good agreement with numerical results at high Rayleigh number.