Stability of atmospheric gravity waves

现代天气预报和气候预测在很大程度上依赖于数值模拟，数值模拟是在划分地球仪的网格上描述大气状态的物理量。数值模型求解流体力学演化方程，这是来自第一性原理，为每个状态量在每个网格点。网格尺寸受到计算能力的限制，使得具有比网格尺寸更小尺度的现象仍然无法被模型解决。重力内波破碎就是其中之一。大气重力波最常在对流层中激发，向上传播，变得不稳定，因为它们的振幅由于背景空气变薄而增加，并最终破裂。特别是，高度放大在理论上还没有得到很好的理解。重力波破碎对预报的精度起着不可忽视的作用。一个可靠的解决方法是参数化，它通过已解决的量来估计未解决的效应的影响。根据定义，参数化的质量取决于所考虑的尺度。随着计算能力的提高，数值模型的分辨率提高，因此尺度缩短，更精确的参数化变得必要。在这个项目中，适用于下一代参数化的重力波破碎理论结合数值，渐近和功能分析的方法。首先，渐近行波解推导出的比例控制方程，其中考虑了现实的高度放大的第一次。这些解决方案进行了数值验证，对完全非线性欧拉方程，掌握第一性原理，和耗散的影响进行了研究。行波是一个特殊的解决方案类，允许检查稳定性分析。从泛函分析出发，应用谱稳定性分析方法，导出了不稳定波的预报准则。这些标准将作为重力波破碎参数化的阈值。

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.