Conditional statistics in scalar turbulence: Theory versus experiment

We consider turbulent advection of a scalar field T(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 [del(2)T\del T(R)] and [\del T\(2)\Delta T(R)]. We find exact relations between these averages, and with the help of the fusion rules we propose a general representation far these objects in terms of the probability density function P(Delta T,R) of Delta T(R). 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 [del(2)T\Delta T(R)] 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.