  We consider the flow of a Newtonian fluid in a three-dimensional domain,
rotating about a vertical axis and driven by a vertically invariant horizontal
body-force. This system admits vertically invariant solutions that satisfy the
2D Navier-Stokes equation. At high Reynolds number and without global rotation,
such solutions are usually unstable to three-dimensional perturbations. By
contrast, for strong enough global rotation, we prove rigorously that the 2D
(and possibly turbulent) solutions are stable to vertically dependent
perturbations: the flow becomes 2D in the long-time limit.
  These results shed some light on several fundamental questions of rotating
turbulence: for arbitrary Reynolds number and small enough Rossby number, the
system is attracted towards purely 2D flow solutions, which display no energy
dissipation anomaly and no cyclone-anticyclone asymmetry. Finally, these
results challenge the applicability of wave turbulence theory to describe
stationary rotating turbulence in bounded domains.
