Stability of atmospheric gravity waves

现代天气预报和气候预报在很大程度上依赖于数值模拟，数值模拟代表了在划分全球的网格网格上描述大气状态的物理量。对于每个网格点上的每个状态量，数值模型求解由第一原理导出的流体力学演化方程。网格大小受计算能力的限制，因此比网格大小小的现象不会被模型解决。重力内波破裂就是其中一种现象。大气重力波最常在对流层激发，向上传播，变得不稳定，因为背景空气变薄，其幅度增加，最终破裂。特别是，高度放大在理论上还没有得到很好的理解。重力波破碎对预报精度起着不可忽视的重要作用。一种可靠的解决办法是通过已解析量估计未解决影响的影响的参数化。根据定义，参数化的质量取决于所考虑的比例。随着计算能力的增加，数值模式的分辨率不断提高，因此尺度变小，需要更精确的参数化法。在这个项目中，结合数值分析、渐近分析和泛函分析的方法，发展了一种适用于下一代参数化的重力波破裂理论。首先，从尺度控制方程出发，首次考虑了实际的高度放大效应，得到了渐近行波解。用掌握第一原理的完全非线性欧拉方程对这些解进行了数值验证，并研究了耗散的影响。行波是一种特殊的解类，它允许以解析的方式检查稳定性。从泛函分析出发，应用谱稳定性分析的方法，推导出预报不稳定波的判据。这些标准将作为重力波破裂参数的阈值。

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.