Nonlinear interaction of gravity and acoustic waves

Here we have investigated the possibility of an inertio-acoustic wave-mode to be unstable with regards to gravity mode perturbations through non-linear triad interactions in the context of a shallow non-hydrostatic model. We have considered highly truncated Galerkin expansions of the perturbations around a resting, hydrostatic and isothermal background state in terms of the eigensolutions of the linear problem. For a single interacting wave triplet, we have shown that an acoustic mode cannot amplify a pair of inertio-gravity perturbations due to the high mismatch among the eigenfrequencies of the three interacting wave-modes, which requires an unrealistically high amplitude of the acoustic mode in order for pump wave instability to occur. In contrast, it has been demonstrated by analysing the dynamics of two triads coupled by a single mode that a non-hydrostatic gravity wave-mode participating in a nearly resonant interaction with two acoustic modes can be unstable to small amplitude perturbations associated with a pair of two hydrostatically balanced inertio-gravity wave-modes. This linear instability yields significant inter-triad energy exchanges if the nonlinearity associated with the second triplet containing the two hydrostatically balanced inertio-gravity modes is restored. Therefore, this inter-triad energy exchanges lead the acoustic modes to yield significant energy modulations in hydrostatic inertio-gravity wave modes. Consequently, our theory suggests that acoustic waves might play an important role in the transient phase of the three-dimensional adjustment process of the atmosphere to both hydrostatic and geostrophic balances.