Non-modal growths of symmetric perturbations produced by paired normal modes

A complete set of normal modes is obtained for symmetric perturbations in a layer of vertically sheared basic flow. The modes exist in pairs and can be classified into three types: paired growing and decaying modes, paired slowly propagating modes, and paired fast propagating modes. For a growing (or decaying mode), the cross-band vertical circulation is tilted between the absolute-momentum surface (M-surface) and isentropic surface (B-surface) of the basic state, so the growth (or decay) of the mode is caused by the positive (negative) feedback of the inertial and buoyancy perturbation forces to the cross-band motions. For a slowly propagating mode, the cross-band circulation is tilted more slantwise than the M-surface and B-surface, so the mode propagation is driven by the inertial restoring force in the forward direction but retarded by the buoyancy restoring force in the backward direction. For a fast propagating mode, the cross-band circulation is tilted in the opposite direction with respect to the M-surface and B-surface, so the mode propagation is driven by the inertial and buoyancy restoring forces both in the forward direction. The cross-band velocity component modes are found to be orthogonal between different pairs, but the full-component modes are non-orthogonal (measured by the inner-product associated with the perturbation energy). Within each pair, the two modes have the same pattern in each component field but opposite polarization relationships between the cross-band velocity and the remaining component fields. Thus, large non-modal energy growths can be produced by paired normal modes. Four types of non-modal growths are identified based on their physical mechanisms. The detailed results will be presented at the conference.