Temperature power spectra of turbulent Rayleigh-B\'enard convection with a Prandtl number $Pr = 12.3$ $^*$

We report on measurements of power spectra of temperature fluctuations in turbulent Rayleigh-B\'enard convection in a cylindrical sample with aspect ratio $\Gamma = D/L = 0.50$ (D is the diameter and L the height) as a function of the distance $z$ from the bottom or top plate. The working fluid was a fluorocarbon at a mean temperature $T_{m} = 25.00^{o}$C with a Prandtl number $Pr = 12.3$, and the Rayleigh number was $Ra \simeq 4 \times10^{11}$. Consistent with many previous investigations, there was a low-frequency range, spanning about a factor of twenty, where the spectra could be described by a power law $P(f) = P_0 f^{-\alpha}$. Contrary to the finding by He et al.\footnote{X. He, D.P.M. vanGils, E. Bodenschatz, and G.Ahlers, Phys. Rev. Lett. {\bf 112}, 174501 (2014).} for $Pr \simeq 0.8$ of a universal spectrum with $\alpha = 1.0$ in the near-wall range $z/L \leq 0.1$ and $\alpha \simeq 1.5$ for $z/L = 0.5$, we found that $\alpha$ varied with $z$ from about 0.6 near the plate ($z/L \simeq 0.01$) to about 1.1 at the cell center ($z/L = 0.5$). Along the sample center line and for $z/L \leq 0.1$ $\alpha$ could be described well by $\alpha =\alpha_0 \ln(z/L) + \alpha_1$ with $\alpha_0 \simeq 0.2$ and $\alpha_1 \simeq 1.5$.