UBC Theses and Dissertations
Parametric subharmonic instability and the β-effect Chan, Ian
Parametric subharmonic instability (PSI) is a nonlinear interaction between a resonant triad of waves, in which energy is transferred from low wavenumber, high frequency modes to high wavenumber, low frequency modes. In the ocean, PSI is thought to be one of the mechanisms responsible for transferring energy from M₂ internal tides (internal gravity waves with diurnal tidal frequency) to near-inertial waves (internal gravity waves with frequency equal to the local Coriolis frequency) near the latitude of 28.9 degrees. Due to their small vertical scale, near-inertial waves generate large vertical shear and are much more efficient than M₂ internal tides at generating shear instability needed for vertical mixing, which is required to maintain ocean stratification. The earlier estimate of the time-scale for the instability is an order of magnitude larger than the time-scale observed in a recent numerical simulation (MacKinnon and Winters) (MW). An analytical model was developed by (Young et al. 2008) (YTB), and their findings agreed with the MW estimation; however as YTB assumed a constant Coriolis force, the model cannot explain the intensificaiton of PSI near 28.9 degrees as observed in the model of MW; in addition, the near-ineartial waves can propagate a significant distance away from the latitude of 28.9 degrees. This thesis extends the YTB model by allowing for a linearly varying Coriolis parameter (β-effect) as well as eddy diffusion. A linear stability analysis shows that the near-inertial wave field is unstable to perturbations. We show that the β-effect results in a shortening in wave length as the near-inertial waves propagate south; horizontal eddy diffusion is therefore enhanced to the south, and limits the meridional extent of PSI. The horizontal diffusion also affects the growth rate of the instability. A surprising result is that as the horizontal diffusion vanishes, the system becomes stable; this can be demonstrated both analytically and numerically. Mathematically, the β-effect renders the spatial differential operator nonnormal, which is characterized with the aid of pseudo-spectra. Our results suggest the possibility of large amplitude transient growth in near-inertial waves in regimes that are asymptotically stable to perturbations.
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