Near-Resonant Triadic Instability of Internal Waves in Non-uniform Stratification

One proposed mechanism for the transfer of energy from large-scale internal tides to small dissipative scales is through triadic resonant instability, whereby a pair of sibling waves interact with a parent internal tide so that they grow in amplitude, extracting energy from the parent. In uniform stratification the sibling waves can grow in pure resonance such that their frequencies and both horizontal and vertical wavenumbers add or subtract to those of the parent. However, in realistic ocean stratification, a low vertical mode internal tide does not have sinusoidal vertical structure. So the sibling waves can only be in near resonance with the parent. In this work we develop a theory for the growth of sibling waves in near resonance with a vertical mode-1 parent wave, testing its predictions against fully nonlinear numerical simulations and laboratory experiments in which the background buoyancy frequency decreases linearly with depth. Theory, simulations and experiments show that near-resonance can occur if the parent wave has frequency close to the maximum buoyancy frequency, N0. However, the maximum growth rate is negligibly small if the parent wave has frequency less than approximately 0.5 N0.