Conditional statistics in scalar turbulence: Theory versus experiment

We consider turbulent advection of a scalar field $T\left(\mathrm{r}\right)$, passive or active, and focus on the statistics of gradient fields conditioned on scalar differences $\Delta T\left(R\right)\mathrm{}$ across a scale $R$. In particular we focus on two conditional averages $〈{\nabla }^{2}T\left|\Delta T\right(R)\mathrm{}〉$ and $〈{\left|\nabla T\right|}^2\left|\Delta T\right(R)\mathrm{}〉$. 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 $P(\Delta T, R)$ of $\Delta T\left(R\right)\mathrm{}$. These results offer a way to analyze experimental data that is presented in this paper. The main question that we ask is whether the conditional average $〈{\nabla }^{2}T\left|\Delta T\right(R)\mathrm{}〉$ is linear in $\Delta T$. 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.