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 homogeneo us flow, the probability density function (pdf) of its fluctuation can in g eneral be expressed in terms of two conditional means: [del(2)T\T] and [del T\(2)\T]. We find that in some special cases, either one of the two condit ional 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 dampi ng, [(VT)-T-2\T]=-([\del T\(2)]/ [T-2]) T. The linearity of the conditional mean in these cases follows directly from the fact that the advection equa tion of a passive scalar is linear. On the other hand, when the scalar is s upported by a homogeneous white-in-time external source, [\del T\(2)\T] = [ \del T del(2)].