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 $Ri\sim O(10)$ .
In this paper, we study the linear stability of a two-dimensional shear flow of an air layer overriding a water layer of finite depth. The air layer is considered to be of an infinite extent with an exponential velocity profile. Three different background conditions are considered in the finite-depth water layer: a quiescent background, a linear velocity profile and a quadratic velocity profile. It is known that the cases of the quiescent water layer and the linear velocity profile allow for analytical treatment. We further provide an analytical solution for the case of the quadratic velocity field: we specifically consider a flow-reversal profile, although the result could be generalized to other quadratic profiles as well. The role of water layer depth on the growth rate of the Miles and rippling instabilities is studied in each of the three cases. Using asymptotic analysis, with the air–water density ratio being a small parameter, we obtain an analytical expression for the growth rate of the Miles mode and discuss the condition for the existence of a long-wave cutoff for these profiles. We provide analytical expressions for the stability boundary in the parameter space of inverse squared Froude number and wavenumber. In scenarios where a long-wave cutoff does not exist, we have carried out a long-wave asymptotic study to obtain the growth rate behaviour in that regime.
<p>In this work we study resonant triad interactions among discrete internal wave modes in a finite-depth, two dimensional uniformly stratified shear flow. The primary wave-field is considered to be a linear superposition of various internal wave modes. The weakly-nonlinear solution of the primary wave-field consists of a superharmonic (2&#969;) part and a mean-flow part (&#969;=0). &#160;For a given modal interaction, we study the location in the frequency (&#969;) -Richardson number (Ri) parameter space where the amplitude of the superharmonic part attains a maximum i.e, where two primary internal wave modes of modenumbers 'm' and 'n' resonantly excite a secondary wave mode of modenumber 'q'. Using asymptotic theory we show that, unlike the case of no-shear, the presence of weak-shear, doesn't require the vertical wavenumber condition to be satisfied for resonance. This entails an activation of several new resonances in the presence of arbitrarily weak shear, where only the frequency and the horizontal wavenumber conditions are satisfied. This also leads to the possibility of self-interaction and resonances close to &#969; = 0. A similar asymptotic theory can be extended to other inhomogeneities (eg: non-uniform stratification) as well. For an exponential background shear flow, we track the location of these resonances in the (&#969;, Ri) parameter space and present their behaviour.</p>
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