Does confined turbulent convection ever attain the 'asymptotic scaling' with 1/2 power?

We examine turbulent thermal convection at very high Rayleigh numbers using helium gas in a cylindrical container of diameter-to-height aspect ratios of 1 and 4, and confirm that the Nusselt number, Nu, follows an approximately 1/3 power of the Rayleigh number, Ra, up to some value of Ra that depends on various experimental conditions, such as the nearness to the critical point of helium as well as the aspect ratio. An enhancement of heat transport occurs for higher Ra, coinciding with a substantial increase in Prandtl number as well as in various measures of Boussinesq conditions, and marking the transition from the region in which Nu ≈ 0.064Ra^1/3 to another in which Nu ≈ 0.078Ra^1/3. By necessity, the Nusselt number in the transition region increases more steeply than 1/3. The transitional slope, which happens to be close to 1/2, does not occur at unique values of Ra for given Pr and so should not be mistaken for the 'ultimate regime' of Kraichnan. By comparing various experiments performed under different conditions (but in the apparatus with the same horizontal surfaces), we empirically find that substantially negative values of a non-dimensional parameter related to fluid conductivity and viscosity correlate well with the observed enhancement of heat transport.