Exact Results for Conditional Means of a Passive Scalar in Certain Statistically Homogeneous Flows

For a passive scalar T(r, t) randomly advected by a statistically homogeneous flow, the probability density function (pdf) of its fluctuation can in general be expressed in terms of two conditional means: 〈?² T|T〉 and 〈|?T|²|T〉. We find that in some special cases, either one of the two conditional means can be obtained explicitly from the equation of motion. In the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping, 〈?² T|T〉=?(〈|?T|²〉/〈T ²〉)T. The linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear. On the other hand, when the scalar is supported by a homogeneous white-in-time external source, 〈|?T|²|T〉=〈|?T|²〉.