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

Abstract

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 above a threshold value of the Rossby number $kRo$ based on the wavenumber and wave amplitude. 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. The growth rate scaled by the global rotation rate is proportional to $(kRo{)}^2$ at low $kRo$ and transitions to a $kRo$ scaling for larger $kRo$. These scalings are in excellent agreement with direct numerical simulations. This instability could explain recent experimental observations of geostrophic instability driven by waves.

Accepted for publication in Journal of Fluid Mechanics Rapids. 13 pages, 4 figures

References in corpus (4)

• Intertial wave turbulence driven by elliptical instability
• Rotating Taylor-Green Flow
• Subcritical turbulent condensate in rapidly rotating Rayleigh-Bénard convection
• Shortcut to geostrophy in wave-driven rotating turbulence: the quartetic instability

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