  The ocean and the atmosphere, and hence the climate, are governed at large
scale by interactions between pressure gradient, Coriolis and buoyancy forces.
This leads to a quasi-geostrophic balance in which, in a two-dimensional-like
fashion, the energy injected by solar radiation, winds or tides goes to large
scales in what is known as an inverse cascade. Yet, except for Ekman friction,
energy dissipation and turbulent mixing occur at small scale implying the
formation of such scales associated with breaking of geostrophic dynamics
through wave-eddy interactions \cite{ledwell_00, vanneste_13} or frontogenesis
\cite{hoskins_72, mcwilliams_10}, in opposition to the inverse cascade. Can it
be both at the same time? We exemplify here this dual behavior of energy with
the help of three-dimensional direct numerical simulations of rotating
stratified Boussinesq turbulence. We show that efficient small-scale mixing and
large-scale coherence develop simultaneously in such geophysical and
astrophysical flows, both with constant flux as required by theoretical
arguments, thereby clearly resolving the aforementioned contradiction.
