Self-resonances in uniformly stratified uniform shear flow.

Internal waves (IWs), propagating inside the stably stratified system, are a significant component of ocean mixing and energy redistribution among ocean layers. This talk presents resonance interactions among primary IWs and their second harmonics, i.e., self-resonances. A self-resonance occurs when the resonating primary (k_m, ω_m) wave and its second harmonic (k_r, ω_r) satisfy k_r ± 2k_m = 0 and ω_r ± 2 ω_m = 0, where k’s and ω's are the wavenumber and frequency, respectively. A prototypical scenario exhibiting self-resonances among IWs in a uniformly stratified uniform shear flow in a finite depth two-dimensional vertically bounded channel is considered. Such resonances are not possible in a uniformly stratified media without shear. The existence of self-resonances for different buoyancy frequencies is shown by solving the linear wave dispersion relation. After identifying the resonating modes, the amplitude equations are derived for analyzing the stability of such modes. The effect of shear rate and stratification on the self-resonances is addressed by varying the Richardson number, i.e., the squared ratio of the buoyancy frequency and shear rate. This study will help understand the role of stratification and shear in generating self-resonances among IWs.