On Conditional Statistics in Scalar Turbulence: Theory vs. Experiment

We consider turbulent advection of a scalar field $T(\text{B}\mathrm{.}r)$, passive or active, and focus on the statistics of gradient fields conditioned on scalar differences $\mathrm{\Delta }T(R)$ across a scale $R$. In particular we focus on two conditional averages $⟨{\mathrm{\nabla }}^{2}T\mid \mathrm{\Delta }T(R)⟩$ and $⟨\mathrm{\mid }\mathrm{\nabla }T{\mathrm{\mid }}^2\mid \mathrm{\Delta }T(R)⟩$. We find exact relations between these averages, and with the help of the fusion rules we propose a general representation for these objects in terms of the probability density function $P(\mathrm{\Delta }T,R)$ of $\mathrm{\Delta }T(R)$. These results offer a new way to analyze experimental data that is presented in this paper. The main question that we ask is whether the conditional average $\langle\nabla^2 T\big| \Delta T®\rangle$islinearin$\Delta T$. We show that there exists a dimensionless parameter which governs the deviation from linearity. The data analysis indicates that this parameter is very small for passive scalar advection, and is generally a decreasing function of the Rayleigh number for the convection data.