Manikandan Mathur

We present an experimental study of resonant generation of superharmonic internal waves as a result of interaction between horizontally propagating vertical internal wave modes m and n at frequency ω0 in a uniformly stratified finite-depth fluid. Thorpe [J. Fluid Mech. 24, 737 (1966)JFLSA70022-112010.1017/S002211206600096X] has shown theoretically that modes m and n at frequency ω0 and mode p=|m-n| at frequency 2ω0 are in triadic resonance at specific values of ω0. We demonstrate the occurrence of this triadic resonance by forcing a primary wave field of modes m and n at various ω0 using a novel internal wave generator, and observing the spontaneous growth (or lack thereof) of the superharmonic mode p=|m-n| at frequency 2ω0. A superharmonic wave field with a predominantly mode-p=|m-n| structure is observed over a finite range of frequency (Δω0≃0.03N) around the resonant value, where N is the uniform buoyancy frequency. The spatial growth of the superharmonic wave field is then quantitatively measured, to subsequently compare with the predictions from amplitude evolution equations at resonance at various forcing amplitudes, thereby validating this model. It is furthermore shown that a large-scale spatial evolution of the wave field is more suited to describe our experiments than the slow temporal evolution approach. The paper concludes with a brief discussion of viscous effects

Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.

Internal waves in the ocean are well-recognized to play an important role in the global energy budget. Triadic resonance is one mechanism via which these internal waves transfer their energy to other spatial and temporal scales before dissipation, at locations blue both near and away from their generation sites. In this paper, we perform a combined theoretical and numerical study of triadic resonance in internal wave modes in a finite-depth ocean with an arbitrary stratification profile. Considering internal waves generated at spatially localized regions, the spatial evolution of the modal amplitudes within a resonant triad are derived based on the method of multiple scales. Two representative cases are considered: (i) modes 1 and 2 at a specific frequency ω0 in triadic resonance with the mode-1 superharmonic wave (frequency 2ω0) in a uniform stratification, and (ii) a self-interacting mode-3 at a specific frequency ω0 in triadic resonance with the mode-2 at frequency 2ω0 in an ocean-like nonuniform stratification. In case (ii), any initial energy in mode-3 at frequency ω0 gets permanently transferred to mode-2 at frequency 2ω0. Numerical simulations are performed to show the spontaneous excitation of superharmonic internal waves resulting from modal interactions in the aforementioned cases, and quantitatively validate the initial spatial evolution of the wave field predicted by the amplitude evolution equations. Furthermore, numerical simulations at off-resonant frequencies are used to identify the range of primary wave frequencies (around the resonant frequency) over which spontaneous superharmonic wave excitation occurs. Quantitative estimates of energy transfer rates within the resonant triads considered show that superharmonic wave excitation resulting from modal interactions should be an important consideration alongside other triadic resonances like parametric subharmonic instability (PSI). We conclude by giving estimates of the relative importance of superharmonic wave excitation in the ocean, and provide motivation for future studies.

We present a theoretical study of nonlinear effects that result from modal interactions in internal waves in a non-uniformly stratified finite-depth fluid with background rotation. A linear wave field containing modes m and n (of horizontal wavenumbers km and kn) at a fixed frequency ω results in two different terms in the steady-state weakly nonlinear solution: (i) a superharmonic wave of frequency 2ω, horizontal wavenumber km and kn and a vertical structure hmn(z) and (ii) a time-independent term (Eulerian mean flow) with horizontal wavenumber km and kn. For some (m, n), hmn(z) is infinitely large along specific curves on the .!=N0; f =!/plane, where N0 and f are the deep ocean stratification and the Coriolis frequency, respectively; these curves are referred to as divergence curves in the rest of this paper. In uniform stratifications, a unique divergence curve occurs on the (ω/N0, f/ω) plane for those (m, n ≠ m) that satisfy (m/3) < n < (3m). In the presence of a pycnocline (whose strength is quantified by the maximum stratification Nmax), divergence curves occur for several more modal interactions than those for a uniform stratification; furthermore, a given .m; n/interaction can result in multiple divergence curves on the (ω/N0, f/ω) plane for a fixed Nmax=N0. Nearby high-mode interactions in a uniform stratification and any modal interaction in a non-uniform stratification with a sufficiently strong pycnocline are shown to result in near-horizontal divergence curves around f/ω ≈ 1, thus implying that strong nonlinear effects often occur as a result of interaction within triads containing two different modes at the near-inertial frequency. Notably, self-interaction of certain modes in a non-uniform stratification results in one or more divergence curves on the (ω/N0,f/ω) plane, thus suggesting that even arbitrarily small-amplitude individual modes cannot remain linear in a non-uniform stratification. We show that internal wave resonant triads containing modes m and n at frequency ! occur along the divergence curves, and their existence is guaranteed upon the satisfaction of two different criteria: (i) the horizontal component of the standard triadic resonance criterion k1 + k2 + k3 = 0 and (ii) a non-orthogonality criterion. For uniform stratifications, criterion (ii) reduces to the vertical component of the standard triadic resonance criterion. For non-uniform stratifications, criterion (ii) seems to be always satisfied whenever criterion (i) is satisfied, thus significantly increasing the number of modal interactions that result in strong nonlinear effects irrespective of the wave amplitudes. We then adapt our theoretical framework to identify resonant triads and hence provide insights into the generation of higher harmonics in two different oceanic scenarios: (i) low-mode internal tide propagating over small-or large-scale topography and (ii) an internal wave beam incident on a pycnocline in the upper ocean, for which our results are in qualitative agreement with the numerical study of Diamessis et al. (Dynam. Atmos. Oceans., vol. 66, 2014, pp. 110-137).