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An interesting question in turbulent convection is how the heat transportdepends on the strength of thermal forcing in the limit of very large thermalforcing. Kraichnan predicted [Phys. Fluids {\bf 5}, 1374 (1962)] that the heattransport measured by the Nusselt number (Nu) would depend on the strength ofthermal forcing measured by the Rayleigh number (Ra) as Nu $\sim$ Ra$^{1/2}$with possible logarithmic corrections at very high Ra. This scaling behavior istaken as a signature of the so-called ultimate state of turbulent convection.The ultimate state was interpreted in the Grossmann-Lohse (GL) theory [J. FluidMech. {\bf 407}, 27 (2000)] as a bulk-dominated state in which both the kineticand thermal dissipation are dominated by contributions from the bulk of theflow with the boundary layers either broken down or playing no role in the heattransport. In this paper, we study the dependence of Nu and the Reynolds number(Re) measuring the root-mean-squared velocity fluctuations on Ra and thePrandtl number (Pr) using a shell model for homogeneous turbulent convectionwhere buoyancy is acting directly on most of the scales. We find that Nu$\sim$Ra$^{1/2}$Pr$^{1/2}$ and Re$\sim$ Ra$^{1/2}$Pr$^{-1/2}$, which resemble theultimate-state scaling behavior for fluids with moderate Pr, but the presenceof a drag acting on the large scales is crucial in giving rise to such scaling.This suggests that if buoyancy acts on most of the scales in the bulk ofturbulent convection at very high Ra, then the ultimate state cannot be abulk-dominated state.

Ultimate-state scaling in a shell model for homogeneous turbulent convection