Slow modes in passive advection

The anomalous scaling in the Kraichnan model of advection of the passive scalar by a random velocity field with nonsmooth spatial behavior is traced to the presence of slow resonance-type collective modes of the stochastic evolution of fluid trajectories. We show that the slow modes are organized into infinite multiplets of descendants of the primary conserved modes. Their presence is linked to the nondeterministic behavior of the Lagrangian trajectories at high Reynolds numbers caused by the sensitive dependence on initial conditions within the viscous range where the velocity fields are more regular. Revisiting the Kraichnan model with smooth velocities, we describe the explicit solution for the stationary state of the scalar. The properties of the probability distribution function of the smeared scalar in this state are related to a quantum mechanical problem involving the Calogero-Sutherland Hamiltonian with a potential.