Conditional temperature statistics in anisotropic turbulent thermal convection for Rayleigh numbers up to $10^{15}$

We present systematic measurements of conditional diffusion $r(x) = \langle \ddot{X} \vert X=x\rangle$ and dissipation $q(x) = \langle (\dot{X})^2 \vert X=x \rangle$ of the normalized temperature fluctuations $X=(T-\bar{T})/\sigma$ in turbulent Rayleigh-B\'enard convection (RBC) at several radial positions where the flow is anisotropic. The data cover the Rayleigh-number range $10^{13} \leq Ra \leq 10^{15}$ for a Prandtl number Pr $\simeq 0.80$. The sample was a right-circular cylinder with aspect ratio $\Gamma \equiv D/L = 0.50$ ($D= 1.12$ m is the diameter and $L = 2.24$ m is the height). We suggest analytic forms for the two conditional means and derived a general formula for the temperature probability-density function. Using $q(x)$ and $r(x)$, we calculated the normalized temperature dissipation $Q$.