Triadic resonances in the internal wave modes with background shear

We consider a monochromatic primary wave-field of discrete internal wave modes in a finite-depth, two dimensional uniformly stratified shear flow. Using a weakly-nonlinear theory, we study the resonances consisting of the superharmonic (2ω) part of the secondary wave field. In the absence of shear, it is known that a wave triad should satisfy a frequency condition, a horizontal wavenumber condition and a mode number condition for resonance. In the presence of weak-shear, using an asymptotic theory, we show that the mode number condition is not necessary. This results in the activation of several new resonances leading to the possibility of self-interaction and resonances close to ω = 0, even with arbitrarily weak shear. A similar asymptotic theory can be extended to other inhomogeneities (non-uniform stratification, non-cartesian geometry, etc.) as well. For an exponential background shear flow, we track the location of these resonances in the parameter space of Richardson number (Ri) and ω for a range of modal interactions and present their behaviour.