Stability of atmospheric gravity waves

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Modern weather forecasting and climate prediction depend heavily on numerical simulations, which represent the physical quantities describing the atmosphere's state on a grid mesh partitioning the globe. The numerical models solve the fluid mechanical evolution equations, which are derived from first principles, for every state quantity on each grid point. The mesh size is limited by the computational power, such that phenomena with smaller scales than the mesh size remain unresolved by the models. Internal gravity wave breaking is one of those phenomena. Atmospheric gravity waves are most often excited in the troposphere, travel upwards, become unstable, as their amplitude grows due to the thinning background air, and break eventually. In particular, the altitudinal amplification is theoretically not well understood. Gravity wave breaking plays an important role for the precision of the forecast such that it cannot be neglected. A reliable workaround are parametrizations which estimate the impact of the unresolved effects by means of the resolved quantities. By definition, the quality of the parametrizations depends on the considered scales. As the computational power increases, the numerical models refine their resolution, hence the scales shorten, and more accurate parametrizations become necessary. In this project, a theory for gravity wave breaking applicable for next-generation parametrizations is developed combining methods from numerical, asymptotical, and functional analysis. First, asymptotic traveling wave solutions are derived from the scaled governing equations, which take the realistic altitudinal amplification into account for the first time. These solutions are numerically validated against the fully nonlinear Euler equations, which grasp the first principles, and the impact of dissipation is investigated. Traveling waves are a particular solution class that allows to examine stability analytically. From functional analysis, the method of spectral stability analysis is applied to derive criteria for the prediction of unstable waves. These criteria will serve as thresholds in the parametrizations for gravity wave breaking.