Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics

The Reynolds stress, or equivalently the average of the momentum flux, is key to understanding the statistical properties of turbulent flows. Both typical and rare fluctuations of the time averaged momentum flux are needed to fully characterize the slow flow evolution. The fluctuations are described by a large deviation rate function that may be calculated either from numerical simulation, or from theory. We show that, for parameter regimes in which a quasilinear approximation is accurate, the rate function can be found by solving a matrix Riccati equation. Using this tool we compute for the first time the large deviation rate function for the Reynolds stress of a turbulent flow. We study a barotropic flow on a rotating sphere, and show that the fluctuations are highly non-Gaussian. This work opens up new perspectives for the study of rare transitions between attractors in turbulent flows.