Turbulent convection at high Rayleigh numbers

Russ Donnelly had a vision of building a ten-meter tall Rayleigh-B\'enard convection cell for use at helium temperatures at one of the high-energy physics facilities with very large helium liquefaction capacity. It would have reached Rayleigh numbers in the $10^{20}$ range and had the promise of yielding detailed information about the so-called ultimate state of turbulent convection which is highly relevant to many geophysical and astrophysical problems as well as to oceanography and climate physics. Although this was not to happen for reasons beyond his control, a laboratory-sized precursor of this venture yielded data for $Ra$ up to $10^{17}$. $^{3}$ The results were interpreted to yield no definitive indication of a transition to the ultimate state. This talk will review some of these data and compare them with more recent measurements using SF$_6$ at ambient temperatures and high pressure. $^4$ This comparison suggests that the Donnelly group actually entered a transition range to the ultimate state near $Ra_1^* \simeq 6\times 10^{12}$, but re-entered the classical state at larger $Ra$ because with increasing $Ra$ the Prandtl number (which affects $Ra^*_1$) also increased in those experiments. In view of the above, one can estimate that, for the same parameter values, the originally envisioned ten-meter cell could have yielded a range of a couple of decades of $Ra$ in the ultimate state. \newline \vskip 0.05in \noindent 3.) J. J. Niemela, L. Skrbek, K. R. Sreenivasan, and R. J. Donnelly, Nature {\bf 404}, 837 (2000). \newline \noindent 4.) G. Ahlers, X. He, D. Funfschilling, and E. Bodenschatz, New J. Phys. {\bf 14}, 103012 (2012).