Kraichnan Flow in a Square: An Example of Integrable Chaos

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 large deviation regime of the statistics of stretching exponents. Whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is Gaussian, this does not have to be the case in the presence of an anisotropy. We analyze in detail the flow in a two-dimensional periodic square where the anisotropy generally persists at small scales. The calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic Schrödinger operator of the Lamé type. The underlying integrability permits to explicitly exhibit the non-Gaussianity of the multiplicative large deviations and to analyze the time-scales at which the large deviation regime sets in. In particular, we indicate how the divergence of some of those time scales when the two Lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow. The analysis of the two-dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability.