Triadic resonances in internal wave modes with background shear

Abstract

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

Superharmonic and triadic resonances in a horizontally oscillated stably stratified square cavity

“…2014; Wunsch 2015; Sutherland 2016; Aksu 2017; Liang, Zareei & Alam 2017; Shmakova, Ermanyuk & Flór 2017; Wunsch 2017; Husseini et al. 2020; Le Dizès 2020; Varma, Chalamalla & Mathur 2020; Boury, Peacock & Odier 2021; Dobra, Lawrie & Dalziel 2021, 2022; Patibandla, Mathur & Roy 2021), as well as horizontally forced interfacial waves (Marcotte, Gallaire & Bongarzone 2023). Our focus here is on the dynamic roles of the superharmonics in a square container with initially linearly stratified fluid that is subjected to periodic horizontal oscillations.…”

“…Here, we consider the responses as the amplitude of the horizontal forcing is increased and the contributions of the temporal harmonics to the structure of the forced response flow become more important. There has been much recent interest in higher temporal harmonics, particularly with possible resonant interactions between them, in various studies involving internal waves (Ermanyuk & Gavrilov 2008;Jiang & Marcus 2009;Ermanyuk, Flór & Voisin 2011;Rodenborn et al 2011;Diamessis et al 2014;Wunsch 2015;Sutherland 2016;Aksu 2017;Liang, Zareei & Alam 2017;Shmakova, Ermanyuk & Flór 2017;Wunsch 2017;Husseini et al 2020;Le Dizès 2020;Varma, Chalamalla & Mathur 2020;Boury, Peacock & Odier 2021;Dobra, Lawrie & Dalziel 2021, 2022Patibandla, Mathur & Roy 2021), as well as horizontally forced interfacial waves (Marcotte, Gallaire & Bongarzone 2023). Our focus here is on the dynamic roles of the superharmonics in a square container with initially linearly stratified fluid that is subjected to periodic horizontal oscillations.…”