This study introduces a new computational scheme for the linear evolution of internal gravity wave packets passing over strongly non-uniform stratifications and background flows as found, e.g., near the tropopause. Focusing on linear dispersion, which is dominant at small wave amplitudes, the scheme describes general wave superpositions arising from wave reflections near strong variations of the background stratification. Thus, it complements WKB theory, which is restricted to nearly monochromatic waves but covers weakly nonlinear effects in turn. One envisaged application of the method is the formulation of bottom-of-the-stratosphere starting conditions for ray tracing parameterizations that follow nonlinear gravity wave packets into the upper atmosphere. A key feature in this context is the method’s separation of wave packets into up- and downward-propagating components. The paper first summarizes a multilayer method for the numerical solution of the Taylor–Goldstein equation. Borrowing ideas from Eliassen and Palm (Geophys Publ 22:1–23, 1961), the scheme is based on partitioning the atmosphere into several uniformly stratified layers. This allows for analytical plane wave solutions in each layer, which are matched carefully to obtain continuously differentiable global eigenmode solutions. This scheme enables rapid evaluations of reflection and transmission coefficients for internal waves impinging on the tropopause from below as functions of frequency and horizontal wavenumber. The study then deals with a spectral method for propagating wave packets passing over non-uniform backgrounds. Such non-stationary solutions are approximated by superposition of Taylor–Goldstein eigenmodes. Particular attention is paid to an algorithm that translates wave packet initial data in the form of modulated sinusoidal signals into amplitude distributions for the system’s eigenmodes. With this initialization in place, the state of the perturbations at any given subsequent time is obtained by a single superposition of suitably phase-shifted eigenmodes, i.e., without any time-stepping iterations. Comparisons of solutions for wave packet evolution with those obtained from a nonlinear atmospheric flow solver reveal that apparently nonlinear effects can be the result of subtle linear wave packet dispersion.

中文翻译：

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射线追踪模型的启动：非均匀背景下小振幅重力波包的演化

这项研究为内部重力波包的线性演化引入了一种新的计算方案，该波包通过强烈的非均匀分层和背景流，例如在对流层顶附近发现的。该方案侧重于在小波幅下占主导地位的线性色散，描述了由背景分层的强烈变化附近的波反射引起的一般波叠加。因此，它补充了 WKB 理论，后者仅限于近乎单色的波，但又涵盖了弱非线性效应。该方法的一个设想应用是为跟随非线性重力波包进入高层大气的光线追踪参数化制定平流层底部起始条件。在这方面的一个关键特征是该方法将波包分成向上和向下传播的分量。论文首先总结了泰勒-戈德斯坦方程数值解的多层方法。借鉴 Eliassen 和 Palm (Geophys Publ 22:1-23, 1961) 的想法，该方案基于将大气划分为几个均匀分层的层。这允许在每一层中分析平面波解，这些解被仔细匹配以获得连续可微的全局本征模式解。该方案能够快速评估从下方撞击对流层顶的内波的反射和透射系数，作为频率和水平波数的函数。然后，该研究涉及一种用于传播通过非均匀背景的波包的光谱方法。这种非平稳解通过泰勒-戈德斯坦本征模的叠加来近似。特别注意将调制正弦信号形式的波包初始数据转换为系统本征模式的幅度分布的算法。有了这个初始化，在任何给定的后续时间的扰动状态是通过适当相移的本征模式的单一叠加获得的，即没有任何时间步进迭代。波包演化的解与从非线性大气流求解器获得的解的比较表明，明显的非线性效应可能是微妙的线性波包分散的结果。特别注意将调制正弦信号形式的波包初始数据转换为系统本征模式的幅度分布的算法。有了这个初始化，在任何给定的后续时间的扰动状态是通过适当相移的本征模式的单一叠加获得的，即没有任何时间步进迭代。波包演化的解与从非线性大气流求解器获得的解的比较表明，明显的非线性效应可能是微妙的线性波包分散的结果。特别注意将调制正弦信号形式的波包初始数据转换为系统本征模式的幅度分布的算法。有了这个初始化，在任何给定的后续时间的扰动状态是通过适当相移的本征模式的单一叠加获得的，即没有任何时间步进迭代。波包演化的解与从非线性大气流求解器获得的解的比较表明，明显的非线性效应可能是微妙的线性波包分散的结果。在任何给定的后续时间的扰动状态是通过适当相移的本征模式的单个叠加获得的，即没有任何时间步进迭代。波包演化的解与从非线性大气流求解器获得的解的比较表明，明显的非线性效应可能是微妙的线性波包分散的结果。在任何给定的后续时间的扰动状态是通过适当相移的本征模式的单个叠加获得的，即没有任何时间步进迭代。波包演化的解与从非线性大气流求解器获得的解的比较表明，明显的非线性效应可能是微妙的线性波包分散的结果。