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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 ≈ 1708 has been calculated up to ≈ 5.106 and its Nusselt number is ∼ 0.143  0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes over the horizontal wavenumber has been calculated up to = 109 and scales as ∼ 0.115  0.31 for 107 < ≤ 109, quite similar to 3D turbulent data that show ∼ 0.105  0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133  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.


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