Initiation of ray tracing models: evolution of small-amplitude gravity wave packets in non-uniform background

Abstract

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.