Stability, transient perturbation, separatrix and search for chaotic dynamics in non-linear double triadic resonances : the case of four internal gravity waves

Triadic instability is a generic mechanism by which a primary wave of finite amplitude is destabilized by two secondary waves forming a resonant triad. For gravity wave in the ocean (Phillips, 1967) the resonant triads form continuous families represented in two-dimension as lines in the wave vector space of the secondary waves. We show here that the crossing of two of these lines results in a double triadic resonance where the instability is reduced for initially unstable triads and increased for stable triad. Using detuning (McEwan. & Plumb, 1977), we show that this double triadic interaction expends from a singular point to a finite region scaling with the Froude number F based on the primary wave amplitude. For long wave perturbations double triad interaction should be taken into account in an extanded region close to zero scaling as F^1/2 since several resonant branches originate from wave number zero with the same slope. The nonlinear evolution is also modified by the exact or detuned double triadic interaction and, using the invariants discussed in Bustamante & Kartashova (2009) for this nearly resonant four waves interaction, we explore the possibility of complete transfer of energy from the initial wave to resonant waves and search for chaotic dynamics.