The long view of triadic resonance instability in finite-width internal gravity wave beams

This article presents our exploration into how a finite-width internal gravity wave beam is modified by triadic resonance instability. We present both experimental and weakly non-linear modelling to examine this instability mechanism, in which a primary wave beam generates two secondary wave beams of lower frequencies and shorter length scales. Through a versatile experimental set-up, we examine how this instability evolves over hundreds of buoyancy periods. Unlike predictions from previous 0D weakly non-linear theory, we find that the wave does not monotonically approach a saturated equilibrium of triadic interactions; rather, the amplitudes and structures of the constituent beams continue to modulate without ever reaching a steady equilibrium. To understand this behaviour, we develop a weakly non-linear approach to account for the spatio-temporal evolution of the amplitudes and structures of the beams over slow time-scales and long distances, and explore the consequences using a numerical scheme to solve the resulting equations. Through this approach, we establish that the evolution of the instability is remarkably sensitive to the spatio-temporal triadic configuration for the system and how part of the observed modulations can be attributed to a competition between the linear growth rate of the secondary wave beams and the finite residence time of the triadic perturbations within the underlying primary beam.