Heat transport by turbulent rotating Rayleigh-Benard convection and its dependence on the aspect ratio

We report on the influence of rotation about a vertical axis on heat transport by turbulent Rayleigh-Benard convection in a cylindrical vessel with an aspect ratio Gamma equivalent to D/L=0.50 (D is the diameter and L the height of the sample) and compare the results with those for larger. The working fluid was water at T-m=40 degrees C where the Prandtl number Pr is 4.38. For rotation rates Omega less than or similar to 1 rad s(-1), corresponding to inverse Rossby numbers 1/Ro between zero and twenty, we measured the Nusselt number Nu for six Rayleigh numbers Ra in the range 2.2 x 10(9)less than or similar to Ra less than or similar to 7.2 less than or similar to 10(10). For small rotation rates and at constant Ra, the reduced Nusselt number Nu(red) equivalent to Nu(1/Ro)/Nu(0) initially increased slightly with increasing 1/Ro, but at 1/Ro=1/Ro(0)similar or equal to 0.5 it suddenly became constant or decreased slightly depending on Ra. At 1/Ro(c)approximate to 0.85 a second sharp transition occurred in Nu(red) to a state where Nu(red) increased with increasing 1/Ro. We know from direct numerical simulation that the transition at 1/Ro(c) corresponds to the onset of Ekman vortex formation reported before for Gamma=1 at 1/Ro(c)similar or equal to 0.4 and for Gamma=2 at 1/Ro(c)=0.18 (Weiss et al., Phys. Rev. Lett., vol. 105, 2010, 224501). The Gamma-dependence of 1/Ro(c) can be explained as a finite-size effect that can be described phenomenologically by a Ginzburg-Landau model; this model is discussed in detail in the present paper. We do not know the origin of the transition at 1/Ro(0). Above 1/Ro(c), Nu(red) increased with increasing Gamma up to similar to 1/Ro=3. We discuss the Gamma-dependence of Nu(red) in this range in terms of the average Ekman vortex density as predicted by the model. At even larger 1/Ro greater than or similar to 3 there is a decrease of Nu(red) that can be attributed to two possible effects. First, the Ekman pumping might become less efficient when the Ekman layer is significantly smaller than the thermal boundary layer, and second, for rather large 1/Ro, the Taylor-Proudman effect in combination with boundary conditions suppresses fluid flow in the vertical direction.