  Most of the turbulent flows appearing in nature (e.g. geophysical and
astrophysical flows) are subjected to strong rotation and stratification. These
effects break the symmetries of classical, homogenous isotropic turbulence. In
doing so, they introduce a natural decomposition of phase space in terms of
wave modes and potential vorticity modes. The appearance of a new time scale
associated to the propagation of waves, in addition to the eddy turnover time,
increases the complexity of the energy transfers between the various scales;
nonlinearly interacting waves may dominate at some scales while balanced motion
may prevail at others. In the end, it is difficult to predict \emph{a priori}
if the energy cascades downscale as in homogeneous isotropic turbulence,
upscale as expected from balanced dynamics, or follows yet another
phenomenology.
  In this paper, we suggest a theoretical approach based on equilibrium
statistical mechanics for the ideal system, inspired from the restricted
partition function formalism introduced in metastability studies. In this
framework, we show analytically that in the presence of rotation, when the
dynamics is restricted to the slow modes, the equilibrium energy spectrum
features an infrared divergence characteristic of an inverse cascade regime,
whereas this is not the case for purely stratified flows.
