  We report measurements of the temperature variance $\sigma^2(z,r)$ and
frequency power spectrum $P(f,z,r)$ ($z$ is the distance from the sample bottom
and $r$ the radial coordinate) in turbulent Rayleigh-B\'enard 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 $0.1$ $\sigma^2(z,r)$ 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
$f\tau_0$ less than or similar to $0.2$, had the form $P(f\tau_0) = P_0
(f\tau_0)^{-1}$ with $P_0 =0.208 \pm 0.008$ a universal constant. Here $\tau_0
= \sqrt{2R}$ where $R$ is the radius of curvature of the temperature
autocorrelation function $C(\tau)$ at $\tau = 0$. For $z/L \simeq 0.5$ the
measurements yielded $P(f\tau_0) \sim (f\tau_0)^{-\alpha}$ with $\alpha$ 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.
