Triadic resonances in internal wave modes with background shear

In this paper, we use asymptotic theory and numerical methods to study resonant triad interactions among discrete internal wave modes at a fixed frequency ( $\omega$ ) in two-dimensional, uniformly stratified shear flow. Motivated by linear generation mechanisms the ocean, assume primary field as superposition of various horizontally propagating vertical . The weakly nonlinear solution associated with is shown comprise superharmonic (frequency $2\omega$ zero fields, focus being on former. When two interacting $m$ $n$ are triadic resonance mode $q$ , it results divergence corresponding secondary amplitude. For given modal interaction $(m, n)$ amplitude plotted plane Richardson number $Ri$ locus locations shows how influenced background shear. limit weak $Ri\to \infty$ ), using an theory, show that horizontal wavenumber condition $k_m + k_n = k_q$ sufficient for resonance, contrast requirement additional $q |m-n|$ case no As result, resonances arbitrarily significantly larger than new occur presence include possibilities due self-interaction seemingly trivial $\omega \approx 0$ both which not possible limit. Our conclusions relevant other inhomogeneities such non-uniformity stratification well, thus providing understanding several recent studies have highlighted non-uniform stratifications. Extending our finite (finite ocean-like exponential flow profile, cograde–cograde interactions, significant curves start will extend below cutoff $\sim O(1)$ contrast, retrograde–retrograde all way from $Ri 0.5$ mixed appear $Ri\sim O(10)$ frequencies smaller Consequently, total $\text {cograde} \text {retrograde} {mixed}$ triads same order small $Ri\approx although attains maximum