  Disentangling the evolution of a coherent mean-flow and turbulent
fluctuations, interacting through the non-linearity of the Navier-Stokes
equations, is a central issue in fluid mechanics. It affects a wide range of
flows, such as planetary atmospheres, plasmas or wall-bounded flows, and
hampers turbulence models. We consider the special case of a two-dimensional
flow in a periodic box, for which the mean-flow, a pair of box-size vortices
called \emph{condensate}, emerges from turbulence through an inverse cascade
process. As was recently shown, a perturbative closure describes correctly the
condensate when turbulence is excited at small scales. In this context, we
obtain explicit results for the statistics of turbulence, encoded in the
Reynolds stress tensor. We demonstrate that the two components of the Reynolds
stress, the momentum flux and the turbulent energy, are determined by different
mechanisms. It was suggested previously that the momentum flux is fixed by a
balance between forcing and mean-flow advection: using unprecedently long
numerical simulations, we provide the first direct evidence supporting this
prediction. By contrast, combining analytical computations with numerical
simulations, we show that the turbulent energy is determined only by mean-flow
advection, and obtain for the first time a formula describing its profile in
the vortex.
