Experimental observations of sub-threshold triadic resonances in internal wave beams
This work examines sub-threshold triadic responses to a vortex-ring-induced perturbation of an internal gravity-wave field, using sparsity-promoting dynamic mode decomposition (spDMD) to diagnose wave structures from experimental observations. Compared with a standard approach to dynamic mode decomposition (DMD) that selects modes by truncation of the singular value spectrum, spDMD, an alternative approach that instead seeks to minimise reconstruction error, is shown here to robustly identify previously indistinguishable oscillatory signals of high frequency in cases of both triadic resonance instability (TRI) and sub-threshold sustained triadic response (STR). These signals are consistent with wave--wave interaction products composed of a primary internal wave with frequency $ω_0$, and secondary TRI waves with frequencies $ω_1$ and $ω_2$ respectively, leading to signals with frequencies $ω_3 = ω_0 + ω_1$ and $ω_4 = ω_0 + ω_2$. We show that these signals satisfy conditions for resonance, but fail to satisfy the dispersion relation for internal waves, indicating that they are confined, non-propagating oscillatory signals rather than freely propagating internal waves. A hierarchical analysis of wave--wave interactions provides evidence that feedback between both branches of the TRI secondary waves gives rise to these signals. Whilst we expect our observations to generalise to many classes of wave--wave interaction, we cannot exclude the possibility that the particular observations we make in the laboratory may arise due to reflections of the primary wave from a free surface. However, beneath strongly density-stratified regions of the ocean we would expect to find these interaction pathways faced with the same geometric configuration, so we are confident our findings will transfer directly to the natural environment.