  Nonlinear interaction and breaking of internal ocean waves are responsible
for much of the interior ocean mixing, affecting ocean carbon storage and the
global overturning circulation. These interactions are also believed to dictate
the observed Garrett-Munk wave energy spectrum, which is still unexplained
after 50 years of studies. According to the resonant wave interaction
approximation, used to derive the kinetic equation for the energy spectrum, the
dominant interactions are between wave triads whose wavevectors satisfy
$\mathbf{k}=\mathbf{p}+\mathbf{q}$, and whose frequencies satisfy
$\omega_{\mathbf{k}}=|\omega_{\mathbf{p}}\pm\omega_{\mathbf{q}}|$. In order to
test the validity of the resonant wave interaction approximation, we examine
several analytical derivations of the theory. The assumptions underlying each
derivation are tested using idealized direct 2d numerical simulations,
representing near-observed energy levels of the oceanic internal wave field. We
show that the assumptions underlying the derivations are not consistent with
the simulated dynamics. In addition, most of the triads satisfying the resonant
conditions do not contribute significantly to nonlinear wave energy transfer in
our simulations, while some interactions that are dominant in nonlinear energy
transfers do not satisfy the resonance conditions. We also point to possible
self-consistency issues with some derivations found in the literature.
