Strange behavior of a passive scalar in a linear velocity field

Damping (or growth) rates of a typical realization, mean-field and high-order correlation functions of a passive scalar (e.g., a number density of particles) advected by a linear velocity fields are estimated. It is shown that all statistical moments higher than the first moment and a typical realization of a passive scalar without an external pumping decay for both laminar and random incompressible linear velocity fields. Strong compressibility of a laminar linear velocity field can result in a growth of a typical realization and the high-order moments of a passive scalar. It is demonstrated that for a laminar compressible linear velocity field the flux of particles from the infinity does not vanish and the total number of particles is not conserved. For a random compressible linear velocity field a typical realization decays whereas the high-order moments of a passive scalar can grow. Comparison of the obtained results with those for dynamics of a passive scalar advected by a homogeneous isotropic and compressible turbulent flow with a given longitudinal two-point correlation function $F=1\mathrm{}-r^2$ is performed (where r is the distance between two points measured in the units of the maximum scale of turbulent motions).