Lyapunov exponent for inertial particles in the 2D Kraichnan model as a problem of Anderson localization with complex valued potential

Abstract

We exploit the analogy between dynamics of inertial particle pair separation in a random-in-time flow and the Anderson model of a quantum particle on the line in a spatially random real-valued potential. Thereby we get an exact formula for the Lyapunov exponent of pair separation in a special case, and we are able to generalize the class of solvable models slightly, for potentials that are real up to a global complex multiplier. A further important result for inertial particle behavior, supported by analytical computations in some cases and by numerics more generally, is that of the decay of the Lyapunov exponent with large Stokes number (quotient of particle relaxation and flow turn-over time-scales) as Stokes number to the power -2/3.