  We report on laboratory experiments of wave-driven rotating turbulence. A set
of wavemakers produces inertial-wave beams that interact nonlinearly in the
central region of a water tank mounted on a rotating platform. The forcing thus
injects energy into inertial waves only. For moderate forcing amplitude, part
of the energy of the forced inertial waves is transferred to subharmonic waves,
through a standard triadic resonance instability. This first step is broadly in
line with the theory of weak turbulence. Surprisingly however, stronger forcing
does not lead to an inertial-wave turbulence regime. Instead, most of the
kinetic energy condenses into a vertically invariant geostrophic flow, even
though the latter is unforced. We show that resonant quartets of inertial waves
can trigger an instability -- the ``quartetic instability'' -- that leads to
such spontaneous emergence of geostrophy. In the present experiment, this
instability sets in as a secondary instability of the classical triadic
instability.
