  We consider a shear flow of a scale invariant homogeneous Gaussian random
velocity field which does not depend on coordinates in the direction of the
flow. We investigate a heat advection coming from a Gaussian random homogeneous
source. We discuss a relaxation at large time of a temperature distribution
determined by the forced advection-diffusion equation. We represent the
temperature correlation functions by means of the Feynman-Kac formula. Jensen
inequalities are applied for lower and upper bounds on the correlation
functions. We show that at finite time there is no velocity dependence of long
range temperature correlations (low momentum asymptotics) in the direction of
the flow but the equilibrium heat distribution has large distance correlations
(low momentum behaviour) with an index depending on the scaling index of the
random flow and of the index of the random forcing. If the velocity has
correlations growing with the distance (a turbulent flow) then the large
distance correlations depend in a crucial way on the scaling index of the
turbulent flow. In such a case the correlations increase in the direction of
the flow and decrease in the direction perpendicular to the flow making the
stream of heat more coherent.
