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 above a threshold value of the Rossby number $kRo$ based on the wavenu...

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Bibliographic Details
Published in:Journal of fluid mechanics 2020-10, Vol.900, Article R2
Main Authors: Le Reun, T., Gallet, B., Favier, B., Le Bars, M.
Format: Article
Language:English
Subjects:
Computational fluid dynamics
Computer simulation
Energy
Fluid mechanics
Fluids
Growth rate
Inertial waves
Instability
JFM Rapids
Modes
Numerical analysis
Physics
Resonant frequencies
Resonant interactions
Rossby number
Rotating fluids
Scaling
Simulation
Stability
Theoretical analysis
Wave amplitude
Wavelengths
Waves
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Summary: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 eigenfrequencies 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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2020.454