Optimal Growth in Inertia–Gravity Wave Packets: Energetics, Long-Term Development, and Three-Dimensional Structure

Abstract

Abstract Using a hierarchy of three models of increasing realism and complexity, and expanding on a previous study, optimal perturbations of inertia–gravity wave (IGW) packets are studied with respect to several aspects. It is shown that normal modes are comparatively less able to extract energy from the IGW over finite time due to their time-invariant structure, while singular vectors (SVs) can adjust their dynamical fields flexibly so as to optimize the statically enhanced roll and Orr mechanisms by which they grow. On longer time scales, where the time dependence of the IGW packet precludes a normal-mode analysis, optimal growth is found to further amplify suitable perturbations. The propagation characteristics of these exhibit critical layer interactions for horizontal propagation directions transverse with respect to the IGW, preventing significant vertical propagation, while parallel and obliquely propagating perturbations of sufficiently long horizontal scales are found to radiate gravity waves into altitudes not directly affected by the IGW. The SVs with shorter wavelengths, as found for short optimization times, stay confined via a linear wave duct near the altitude of least static stability where they are excited. At optimization times of the order of the IGW period the leading SVs, with an energy growth by about three orders of magnitude, propagate obliquely, possibly in correspondence to previous results by others from simulations of nonlinear IGW breakdown. The three-dimensional structure of SVs shows an amplitude modulation strictly confining the perturbations also to the horizontal location of least static stability, suggesting a picture of turbulence onset in IGW packets where local patches of growing perturbations initially dominate.