Internal wave mode resonant triads in an arbitrarly stratified finite-depth ocean with background rotation

Internal tides generated by barotropic tides on bottom topography or the spatially compact near-inertial mixed layer currents excited by surface winds can be conveniently represented in the linear regime as a superposition of vertical modes at a given frequency in an arbitrarily stratified ocean of finite depth. Considering modes $(m,n)$ at a frequency $\omega$ in the primary wave field, we derive the weakly nonlinear solution, which contains a secondary wave at $2\omega$ that diverges when it forms a resonant triad with the primary waves. In nonuniform stratifications, resonant triads are shown to occur when the horizontal component of the classical RTI criterion $\vec{k}_1+\vec{k}_2+\vec{k}_3=0$ is satisfied along with a non-orthogonality criterion. In nonuniform stratifications with a pycnocline, infinitely more pairs of primary wave modes $(m,n)$ result in RTI when compared to a uniform stratification. Further, two nearby high modes at around the near-inertial frequency often form a resonant triad with a low mode at $2\omega$, reminiscent of the features of PSI near the critical latitude. The theoretical framework is then adapted to investigate RTI in two different scenarios: low-mode internal tide scattering over topography, and internal wave beams incident on a pycnocline.