  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.
