Based on a weakly nonlinear WKB-like theory for two-dimensional atmospheric gravity waves (GWs), wavetrain solutions are derived and examined with respect to stability. Systematic multiple-scale analysis, starting from the fully compressible Euler equations with a distinguished limit favorable for GWs near breaking level, reveals that pseudo-incompressible theory applies rather than Boussinesq theory if we allow for non-uniform stratification. A spectral expansion including a mean flow, combined with the additional WKB assumption of slowly varying phases and amplitudes, is used to find general asymptotic solutions. Particular leading order solutions of the form $U(t-\mathbf{C}\cdot\mathbf{x})$ (wavetrains) are derived. Such wavetrains only exist if the stratification is uniform or if the vertical component of $\mathbf{C}$ is zero. The wavetrains are characterized by nonlinear phase functions. To gain insight into stability properties, the solutions are perturbed by small wave-like deviations in a reference frame moving with $\mathbf{C}$. We obtain a Hamilton-Jacobi equation for the perturbation phase function which can be solved analytically. Perturbations with wave modes parallel to the underlying GW phase speed are always stable whereas perturbations with orthogonal modes grow exponentially causing instabilities with growth rates depending on the (non-uniform) stratification. These instabilities are triggered by the nonlinear advection terms. The results of the stability analysis are compared with direct numerical simulations.

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*email: mark.schlutow@fu-berlin.de
*Preference: Oral