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Weakly nonlinear internal wave-wave interaction is a key mechanism thatcascades energy from large to small scales, leading to ocean turbulence andmixing. Oceans typically have a non-uniform density stratification profile;moreover, submarine topography leads to a spatially varying ocean depth ($h$).Under these conditions and assuming mild-slope bathymetry, we employmultiple-scale analysis to derive the wave amplitude equations for triadic- andself-interactions. The waves are assumed to have a slowly (rapidly) varyingamplitude (phase) in space and time. For uniform stratifications, thehorizontal wavenumber ($k$) condition for waves ($1$,$2$,$3$), given by${k}_{(1,a)}+{k}_{(2,b)}+{k}_{(3,c)}=0$, is unaffected as $h$ is varied, where$(a,b,c)$ denote the modenumber. Moreover, the nonlinear coupling coefficients(NLC) are proportional to $1/h^2$, implying that triadic waves grow fasterwhile travelling up a seamount. For non-uniform stratifications, triads that donot satisfy the condition $a=b=c$ may not satisfy the horizontal wavenumbercondition as $h$ is varied, and unlike uniform stratification, the NLC may notdecrease (increase) monotonically with increasing (decreasing) $h$. NLC, andhence wave growth rates for both triads and self-interactions, can also varyrapidly with $h$. The most unstable daughter wave combination of a triad with amode-1 parent wave can also change for relatively small changes in $h$. We alsoinvestigate higher-order self-interactions in the presence of a monochromatic,small amplitude bathymetry; here the bathymetry behaves as a zero frequencywave. We derive the amplitude evolution equations and show that higher-orderself-interactions might be a viable mechanism of energy cascade.

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