Slow modes in passive advection

TL;DR

This paper investigates the origin of anomalous scaling in passive scalar advection within the Kraichnan model, linking it to slow resonance modes and their organization, and explores the stationary state with smooth velocities using quantum analogies.

Contribution

It reveals the organization of slow modes into multiplets and connects the scalar's stationary distribution to a Calogero-Sutherland quantum problem.

Findings

Identification of slow resonance modes causing anomalous scaling

Explicit solution for the scalar's stationary state with smooth velocities

Connection between scalar distribution and Calogero-Sutherland Hamiltonian

Abstract

The anomalous scaling in the Kraichnan model of advection of the passive scalar by a random velocity field with non-smooth spatial behavior is traced down 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 non-deterministic 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.