Testing the consistency of the resonant wave interaction approximation with simulated dynamics of idealised 2-D internal wave fields

Nonlinear interactions of internal ocean waves play an important role in ocean mixing and are of theoretical interest even in idealised settings. According to the resonant wave interaction approximation, which is commonly used to derive the kinetic equation for the energy spectrum, the dominant interactions are between wave triads whose wavevectors satisfy bold italic k equals bold italic p plus bold italic q k = p + q $\\boldsymbol{k}=\\boldsymbol{p}+\\boldsymbol{q}$ and whose frequencies satisfy omega Subscript bold italic k Baseline equals StartAbsoluteValue omega Subscript bold italic p Baseline plus or minus omega Subscript bold italic q Baseline EndAbsoluteValue ω k = | ω p ± ω q | $\\omega _{\\boldsymbol{k}} = |\\omega _{\\boldsymbol{p}}\\pm \\omega _{\\boldsymbol{q}}|$ . 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 idealised direct two-dimensional (2-D) non-rotating numerical simulations, representing near-observed energy levels of the oceanic internal wave field. We show that the slow-amplitude assumptions underlying the derivations are inconsistent with the simulated dynamics in this particular set of simulations. In addition, most of the triads satisfying the resonant conditions do not contribute significantly to nonlinear wave energy transfer in our idealised 2-D simulations, while some interactions that are dominant in nonlinear energy transfers do not satisfy the resonance conditions. Understanding the relevance of these results to the observed three-dimensional rotating ocean internal wave field requires further study.