Near-resonant instability of geostrophic modes: beyond Greenspan's theorem.

We explore the near-resonant interaction of inertial waves with geostrophic modes in rotating fluids via numerical and theoretical analysis. When a single inertial wave is imposed, we find that some geostrophic modes are unstable provided that the wave amplitude, or Rossby number $Ro$, is sufficiently large. We show this instability to be caused by triadic interaction involving two inertial waves and a geostrophic mode such that the sum of their eigen frequencies is non-zero. We derive theoretical scalings for the growth rate of this near-resonant instability which is proportional to $Ro^2$ at low $Ro$ and transitions to a $Ro$ law at moderate $Ro$. These scalings are in excellent agreement with direct numerical simulations. This instability could explain recent experimental observations of geostrophic instability driven by waves.