Self-resonance and stability in uniformly stably stratified shear flow

Stratified flows are of great importance from a geophysical point of view because of their ability to support generating and propagating internal waves (IWs). For instance, oceanic IWs act as a primary source for transferring heat, energy, and momentum throughout the ocean. Understanding interactions among IWs are necessary to know the energy transfer mechanism among IWs or IWs and surrounding flow. We aim to study self-resonances among IWs in an ideal situation consisting of uniform shear flow with uniform stratification in a vertically confined two-dimensional channel. In the absence of shear, self-resonance is not possible among IWs in a uniformly stratified medium. In contrast, shear instigates self-resonance among IWs, and thus, plays a crucial role in energy transfer. First, we show exact self-resonances and obtain associated parameters by solving the dispersion relation. Next, we study the stability analysis of self-resonating IWs using the amplitude evolution equations, which is the first step towards understanding the energy transfer mechanism through IWs in the ocean. From the analytical solution of amplitude equations, we analyze the role of stratification on the stability of self-resonating IWs and hence, the energy transfer among IWs and their higher-order harmonics.