  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.
