Kraichnan flow in a square: an example of integrable chaos

Abstract

The Kraichnan flow provides an example of a random dynamical system accessible to an exact analysis. We study the evolution of the infinitesimal separation between two Lagrangian trajectories of the flow. Its long-time asymptotics is reflected in the multiplicative large deviation regime of the statistics of stretching exponents. In the flow in a two-dimensional periodic square where the anisotropy persists at small scales, the calculation of the large deviation rate function of the stretching exponents reduces to the study of the ground state energy of an integrable periodic Schroedinger operator of the Lame type. The underlying integrability permits to explicitly exhibit the non-Gaussianity of the multiplicative large deviations, to analyze the time-scales at which the large deviation regime sets in and to identify the general scenario for the appearance of multiplicative large deviations and the restrictions on its applicability.