Resonant and near-resonant internal wave triads for non-uniform stratifications. Part 2: Vertically bounded domain with mild-slope bathymetry

Weakly nonlinear internal wave-wave interaction is a key mechanism that cascades energy from large to small scales, leading to ocean turbulence and mixing. Oceans typically have a non-uniform density stratification profile; moreover, submarine topography leads to a spatially varying bathymetry . Under these conditions and assuming mild-slope bathymetry, we employ multiple-scale analysis to derive the wave amplitude equations for weakly nonlinear wave-wave interactions. The waves are assumed to have a slowly (rapidly) varying amplitude (phase) in space and time. For uniform stratifications, the horizontal wavenumber condition for waves (1, 2, 3), given by, is unaffected as is varied, where denote the mode number. Moreover, the nonlinear coupling coefficients (NLC) are proportional to, implying that triadic waves grow faster while travelling up a seamount. For non-uniform stratifications, triads that do not satisfy the condition may not satisfy the horizontal wavenumber condition as is varied, and unlike uniform stratification, the NLC may not decrease (increase) monotonically with increasing (decreasing). NLC, and hence wave growth rates for weakly nonlinear wave-wave interactions, can also vary rapidly with. The most unstable daughter wave combination of a triad with a mode-1 parent wave can also change for relatively small changes in. We also investigate higher-order self-interactions in the presence of a monochromatic, small-amplitude topography; here, the topography behaves as a zero-frequency wave. We derive the amplitude evolution equations and show that higher-order self-interactions might be a viable mechanism of energy cascade.