Heat transport by coherent Rayleigh-Bénard convection

Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number Ra ≈ 1708 has been calculated up to Ra ≈ 5.10⁶ and its Nusselt number is Nu ∼ 0.143 Ra^0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes Nu over the horizontal wavenumber has been calculated up to Ra = 10⁹ and scales as Nu ∼ 0.115 Ra^0.31 for 10⁷ < Ra ≤ 10⁹, quite similar to 3D turbulent data that show Nu ∼ 0.105 Ra^0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra^0.217. That solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength, and spacing of elemental plumes.