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On upper bounds for infinite Prandtl number convection with or without rotation

Bounds for the bulk heat transport in Rayleigh–Benard convection for an infinite Prandtl number fluid are derived from the primitive equations. The enhancement of heat transport beyond the minimal conduction value (the Nusselt number Nu) is bounded in terms of the nondimensional temperature difference across the layer (the Rayleigh number Ra) according to Nu⩽cRa2/5, where c<1 is an absolute constant. This rigorous upper limit is uniform in the rotation rate when a Coriolis force, corresponding to the rotating convection problem, is included.