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Logarithmic Spatial Variations and Universal $f^{-1}$ Power Spectra of Temperature Fluctuations in Turbulent Rayleigh-Bénard Convection

We report measurements of the temperature variance $σ^2(z,r)$ and frequency power spectrum $P(f,z,r)$ ($z$ is the distance from the sample bottom and $r$ the radial coordinate) in turbulent Rayleigh-Bénard convection (RBC) for Rayleigh numbers $\textrm{Ra} = 1.6\times10^{13}$ and $1.1\times10^{15}$ and for a Prandtl number $\textrm{Pr} \simeq 0.8$ for a sample with a height $L = 224$ cm and aspect ratio $D/L = 0.50$ ($D$ is the diameter). For $z/L$ less than or similar to $0.1$ $σ^2(z,r)$ was consistent with a logarithmic dependence on $z$, and there was a universal (independent of $\textrm{Ra}$, $r$, and $z$) normalized spectrum which, for $0.02$ less than or similar to $fτ_0$ less than or similar to $0.2$, had the form $P(fτ_0) = P_0 (fτ_0)^{-1}$ with $P_0 =0.208 \pm 0.008$ a universal constant. Here $τ_0 = \sqrt{2R}$ where $R$ is the radius of curvature of the temperature autocorrelation function $C(τ)$ at $τ= 0$. For $z/L \simeq 0.5$ the measurements yielded $P(fτ_0) \sim (fτ_0)^{-α}$ with $α$ in the range from 3/2 to 5/3. All the results are similar to those for velocity fluctuations in shear flows at sufficiently large Reynolds numbers, suggesting the possibility of an analogy between the flows that is yet to be determined in detail.