Energy transfer in internal wave triads for non-uniform stratifications. Part II: Bounded domain with varying topography

Weakly nonlinear triadic wave-wave interactions is a mechanism by which energy from large scale oceanic internal waves cascades to small scales, finally leading to ocean mixing. Due to variations in submarine topography, ocean depth ($h$) is also variable, which in turn can impact the formation of resonant triads. Using multiple scale analysis, amplitude evolution equations of the waves forming a triad are derived in the presence of weakly varying $h$, assuming the waves slowly vary with amplitude but rapidly vary in phase both in space and time. For triads interacting in a medium of varying $h$ and uniform stratification, the horizontal wavenumber condition for waves (1,2,3), given by ${k}_{(1,a)}+{k}_{(2,b)}+{k}_{(3,c)}=0$ is unaffected, where $(a,b,c)$ are integers denoting the modenumber. For nonuniform stratification, triads (and self-interactions) that do not satisfy the condition $a=b=c$ can violate the horizontal wavenumber condition as $h$ varies. In nonuniform stratification, the nonlinear coupling coefficients (NLC) do not decrease (increase) monotonically with increasing (decreasing) $h$. Also NLC may change by one order of magnitude with a slow change in $h$. Moreover, the most unstable triad was found to change with relatively small changes in $h$.