On Conditional Statistics in Scalar Turbulence: Theory vs. Experiment

We consider turbulent advection of a scalar field $T(\B.r)$, passive or active, and focus on the statistics of gradient fields conditioned on scalar differences $\Delta T(R)$ across a scale $R$. In particular we focus on two conditional averages $\langle\nabla^2 T\big|\Delta T(R)\rangle$ and $\langle|\nabla T|^2\big|\Delta T(R) \rangle$. 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 $\Delta T(R)$. 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 $\langle\nabla^2 T\big| \Delta T(R)\rangle$ 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.