Secondary Instabilities in Breaking Inertia–Gravity Waves

The three-dimensionalization of turbulence in the breaking of nearly vertically propagating inertia gravity waves is investigated numerically using singular vector analysis applied to the Boussinesq equations linearized about three two-dimensional time-dependent basic states obtained from nonlinear simulations of breaking waves: a statically unstable wave perturbed by its leading transverse normal mode, the same wave perturbed by its leading parallel normal mode, and a statically stable wave perturbed by a leading transverse singular vector. The secondary instabilities grow through interaction with the buoyancy gradient and velocity shear in the basic state. Which growth mechanism predominates depends on the time-dependent structure of the basic state and the wavelength of the secondary perturbation. The singular vectors are compared to integrations of the linear model using random initial conditions, and the leading few singular vectors are found to be representative of the structures that emerge in the randomly initialized integrations. A main result is that the length scales of the leading secondary instabilities are an order of magnitude smaller than the wavelength of the initial wave, suggesting that the essential dynamics of the breaking might be captured by tractable nonlinear three-dimensional simulations in a relatively small triply periodic domain.