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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. [ABSTRACT FROM AUTHOR]

Shear flows with a free surface possess diverse branches of modal instabilities. By approximating the mean flow with a piecewise linear profile, an understanding and classification of the instabilities can be achieved by studying the interaction of the edge waves that arise at the density discontinuity at the surface and the vorticity waves that are supported at the mean vorticity gradient discontinuities in the interior. The various branches of instability are identified and their physical origin is clarified. The edge waves giving rise to the modal instabilities can also lead to a modest transient growth that extends into the regions of neutrality of the flow. However, when the continuous spectrum is excited substantial transient growth can arise and the optimal perturbations attain greater energy when compared with the energy of the fastest modal growing perturbation. These optimal perturbations utilize the continuous spectrum to excite at large amplitude the neutral or amplifying modes of the system. [ABSTRACT FROM AUTHOR]

Layered flows that are commonly observed in stratified turbulence are susceptible to the Taylor–Caulfield Instability. While the modal stability properties of layered shear flows have been examined, the non-modal growth of perturbations has not been investigated. In this work, the tools of Generalized Stability Theory are utilized to study linear transient growth within a finite time interval of two-dimensional perturbations in an inviscid, three-layer constant shear flow under the Boussinesq approximation. It is found that, for low optimization times, small-scale perturbations utilize the Orr mechanism and achieve growth equal to that in the case of an unstratified flow. For larger optimization times, transient growth is much larger compared to growth for an unstratified flow as the Kelvin–Orr waves comprising the continuous spectrum of the dynamical operator and the gravity edge-waves comprising the discrete spectrum interact synergistically. Maximum growth is obtained for perturbations with scales within the region of instability, but significant growth is maintained for modally stable perturbations as well. For perturbations with scales within the unstable region, the unstable normal modes are excited at high amplitude by their bi-orthogonals. For perturbations with modally stable scales, the Orr mechanism is utilized to excite at high amplitude neutral propagating waves resembling the neutral Taylor–Caulfield modes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]