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The role of nonnormality in the overreflection of gravity waves is investigated. In the limit of weak stratification, wave packets having a critical level inside a shear layer of finite depth are reflected with amplified energy. This process, which exhibits the characteristics of stimulated emission, occurs in three stages: first, the incoming wave enters the shear layer and excites nonpropagating perturbations leaning with and against the shear. Subsequently, the energy of perturbations leaning against the shear grows in a manner similar to energy growth of perturbations in constant shear flows, indicating that the Orr mechanism that is slightly modified by stratification underlies the observed growth. Finally, the amplified perturbations excite propagating waves originating from the vicinity of the shear layer boundary. The role of nonnormality in this process is also investigated from the perspective of the associated nonorthogonality of the modes of the dynamical system. It is found that the incident wave packet projects on nonorthogonal analytic modes having the structure of a downward propagating wave in the far field below the shear layer and overreflection expressed by the interaction among these nonorthogonal modes. DOI: 10.1175/2010JAS3455.1

In this paper, the emission of internal gravity waves from a local westerly shear layer is studied. Thermal and/or vorticity forcing of the shear layer with a wide range of frequencies and scales can lead to strong emission of gravity waves in the region exterior to the shear layer. The shear flow not only passively filters and refracts the emitted wave spectrum, but also actively participates in the gravity wave emission in conjunction with the distributed forcing. This interaction leads to enhanced radiated momentum fluxes but more importantly to enhanced gravity wave energy fluxes. This enhanced emission power can be traced to the nonnormal growth of the perturbations in the shear region, that is, to the transfer of the kinetic energy of the mean shear flow to the emitted gravity waves. The emitted wave energy flux increases with shear and can become as large as 30 times greater than the corresponding flux emitted in the absence of a localized shear region. Waves that have horizontal wavelengths larger than the depth of the shear layer radiate easterly momentum away, whereas the shorter waves are trapped in the shear region and deposit their momentum at their critical levels. The observed spectrum, as well as the physical mechanisms influencing the spectrum such as wave interference and Doppler shifting effects, is discussed. While for large Richardson numbers there is equipartition of momentum among a wide range of frequencies, most of the energy is found to be carried by waves having vertical wavelengths in a narrow band around the value of twice the depth of the region. It is shown that the waves that are emitted from the shear region have vertical wavelengths of the size of the shear region.

Three-dimensional perturbations producing optimal energy growth in stratified, unbounded constant shear flow are determined. The optimal perturbations are intrinsically three-dimensional in structure. Streamwise rolls emerge as the optimally growing perturbations at long times, but their energy growth factor is limited by stratification to E = O(1/Ri), where Ri is the Richardson number. The perturbations that attain the greatest energy growth in the flow are combinations of Orr solutions and roll solutions that maximize their energy growth in typically O(10) advective time units. These optimal perturbations are localized in the high-shear regions of the boundary layer and are associated with strong updrafts and downdrafts that evolve into streamwise velocity streaky structures in the form of hairpin vortices in agreement with observations.