## 1. Introduction

[2] Internal wave generation by tide-topography interaction exhibits a rich range of behaviour. In this paper a new weakly-nonlinear process, near-resonant triads, is reported in this context for the first time. This mechanism may be important for transferring energy in the oceanic internal wave field.

[3] Exact phase-locked resonant triads occur between three waves whose wave vectors _{j} = (*k*_{j}, *m*_{j}) and frequencies *ω*_{j} satisfy the resonant conditions _{1} + _{2} + _{3} = 0 and *ω*_{1} + *ω*_{2} + *ω*_{3} = 0 where the wave frequencies satisfy the dispersion relation appropriate for the type of wave in question. The nonlinearly-coupled evolution equations for the envelopes of the three waves, which vary slowly in time and space, have soliton solutions [*Craik*, 1985; *Degasperis et al.*, 2006]. Resonant internal wave triads are of great interest to oceanographers because they are believed to play a fundamental role in nonlinear energy transfer across the oceanic internal wave spectrum [*Hibiya et al.*, 2002; *Furuichi et al.*, 2006; *MacKinnon and Winters*, 2005; *Gerkema et al.*, 2006].

[4] Near-resonant triads occur when there is a slight frequency or wave number mismatch, i.e.,

or

where the wave number and frequency mismatches are small. Evolution equations for the amplitudes of waves in a near-resonant triad were first derived in the context of light waves in nonlinear dielectrics [*Armstrong et al.*, 1962] and were discussed in a general setting by *Bretherton* [1964]. They have since been described in other contexts including shoaling surface gravity waves in shallow water [*Freilich and Guza*, 1984], waves in plasma [*Drysdale and Robinson*, 2002] and nonlinear optics [*Robinson and Drysdale*, 1996]. The theory of near-resonant interactions is summarized by *Craik* [1985]. Near-resonant interactions among discrete internal wave modes have received little attention in the study of internal waves, one exception being a study of the stochasticities of two-triad interactions [*Kim and West*, 1996]. Near-resonant internal wave interactions among plane waves are also briefly mentioned by *Koudella and Staquet* [2006].

[5] In this Letter the occurrence of phase-locked, near-resonant internal wave triads are reported for the first time. The strength of the interaction is surprisingly strong which demonstrates that under certain conditions near-resonant interactions may be an important factor in determining the internal wave spectrum in the ocean. The waves are generated in numerical simulations of tide-topography interaction at a shelf break. The form of the forcing generates waves of all vertical modes at predominantly tidal frequency. The mode-one and -three waves are part of a near-resonant triad which includes a mode-two wave with twice the frequency. Because the mode-one and -three waves have a localized source from which they propagate, the amplitudes of the three waves in the near-resonant triads vary predominantly in space rather than in time.