  An interesting question in turbulent convection is how the heat transport
depends on the strength of thermal forcing in the limit of very large thermal
forcing. Kraichnan predicted [Phys. Fluids {\bf 5}, 1374 (1962)] that the heat
transport measured by the Nusselt number (Nu) would depend on the strength of
thermal forcing measured by the Rayleigh number (Ra) as Nu $\sim$ Ra$^{1/2}$
with possible logarithmic corrections at very high Ra. This scaling behavior is
taken as a signature of the so-called ultimate state of turbulent convection.
The ultimate state was interpreted in the Grossmann-Lohse (GL) theory [J. Fluid
Mech. {\bf 407}, 27 (2000)] as a bulk-dominated state in which both the kinetic
and thermal dissipation are dominated by contributions from the bulk of the
flow with the boundary layers either broken down or playing no role in the heat
transport. In this paper, we study the dependence of Nu and the Reynolds number
(Re) measuring the root-mean-squared velocity fluctuations on Ra and the
Prandtl number (Pr) using a shell model for homogeneous turbulent convection
where 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 the
ultimate-state scaling behavior for fluids with moderate Pr, but the presence
of 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 of
turbulent convection at very high Ra, then the ultimate state cannot be a
bulk-dominated state.
