The role of non-normality in overreflectiontheoryNikolaos A. Bakas∗Harvard UniversityCambridge, MA 02138Brian F. FarrellHarvard UniversityCambridge, MA 02138March 24, 2010Submitted to the Journal of Atmospheric Sciences∗corresponding author : Nikolaos Bakas, National and Kaposistrian University of Athens, Build. IV, office34, Panepistimiopolis, Zografos, Athens, Greece, email: [email protected] role of non-normality in the overreflection of gravity waves is investigated. In thelimit of weak stratification, wave packets having a critical level inside a shear layer of finitedepth are reflected with amplified energy. This process, which exhibits the characteristics ofstimulated emission, occurs in three stages: first the incoming wave enters the shear layer andexcites non-propagating perturbations leaning towards and against the shear. Subsequently, theenergy of perturbations leaning against the shear grows in a manner similar to energy growthof perturbations in constant shear flows, indicating that the Orr mechanism that is slightlymodified by stratification underlies the observed growth. Finally, the amplified perturbationsexcite propagating waves originating from the vicinity of the shear layer boundary. The role ofnon-normality in this process is also investigated from the perspective of the associated non-orthogonality of the modes of the dynamical system. It is found that the incident wave packetprojects on non-orthogonal analytic modes having the structure of a downward propagatingwave in the far field below the shear layer and overreflection results from the interaction amongthese non-orthogonal modes.1. IntroductionShear instability of stably stratified fluids is a common feature of atmospheric and oceanicflows that has been studied for over a century. However, despite the fact that the conditionsunder which shear instability occurs are now well known, the physical mechanism underlyingthe instability is not comprehensively understood. Conceptual frameworks that have been ad-vanced for a physical, mechanistic basis for shear instability are the overreflection theory andthe counter-propagating Rossby wave theory.Overreflection theory (see Lindzen (1988) for a review) describes the instability of stablystratified flows in terms of continuous overreflection of gravity waves. Overreflection is a processinitially noted by Jones (1968) who studied the scattering properties of a shear region withRichardson number less than 1/4. Jones (1968) found that a gravity wave propagating towardsa critical level embedded in a shear region is overreflected (that is the reflection coefficient exceedsone) due to extraction of energy from the mean flow. Lindzen (1974) further suggested that ifthe overreflected waves were fully reflected back by a containing surface and if the fully reflectedand overreflected waves interfered constructively (by satisfying a proper quantization condition),then every time they traveled across the region between the overreflecting critical level and thecontaining surface, they would be amplified by the same factor and by being continuouslyoverreflected, they could lead to the formation of an exponentially growing eigenmode. Lindzenand Rosenthal (1976, 1981, 1983), Rosenthal and Lindzen (1983a,b) and Lindzen and Tung(1978) found the necessary conditions for overreflection and for quantization of the waves andwere able to relate all unstable modes with continuously overreflected waves in the cases ofbarotropic, baroclinic and stratified shear flow instabilities. Hence, shear flow instability appears3to be closely related to the overreflection process.In order to shed more light on the mechanism of overreflection, Lindzen and Barker (1985)studied the time evolution of the scattering of an incident wave train under a specific wavegeometry and found that the reflected and transmitted wave amplitudes at steady state areequal and that the characteristic time scale needed for the reflected and transmitted waveamplitudes to reach their steady state values is almost identical. This result is consistent withoverreflection being a stimulated emission instigated by the incident wave. Lindzen and Barker(1985) also found that this characteristic time scale depended only on the variation of velocityacross the critical level and was independent of the Richardson number in the vicinity of thislevel. This indicated that the stimulated emission is a kinematic effect. Based on this result,Lindzen (1988) hypothesized that the Orr mechanism (Orr 1907), in which a perturbationleaning against the shear intensifies transiently as it is sheared over, provides the necessaryenergy extraction mechanism from the mean flow in the process of overreflection. However,he did not support this hypothesis with numerical or analytical evidence. Transient growthof small perturbations in stratified flows was investigated by Farrell and Ioannou (1993) whowere able to identify the Orr mechanism, that was slightly modified by stratification, as theunderlying mechanism for rapid transient perturbation growth. Farrell and Ioannou (1993) alsotraced the observed energy growth to the non-normality of the dynamical operator governing theperturbation dynamics. However, since their analysis was performed for a flow lacking the wavegeometry for overreflection (Lindzen 1988), these results do not directly apply to perturbationgrowth during overreflection. In this work we undertake this task to study the scattering of awave packet by a finite stably stratified shear layer bounded by two regions of uniform velocityand clarify the role of non-normality and the Orr mechanism in the overreflection process.4A very different approach to understanding modal instability is through the framework ofcounter propagating Rossby waves that was pioneered by Bretherton (1966a) for Rossby waves.Bretherton (1966a) showed that unstable modes can be associated with a constructive interac-tion between two counter propagating Rossby edge waves that become phase locked in a mutualreenforcing configuration. This approach of interacting edge waves that are supported by regionsof vorticity or density gradient discontinuity was followed by studies of baroclinic (Hoskins et al.1985; Davies and Bishop 1994), Kelvin-Rossby wave (Sakai 1989), planar shear flow (Heifetzand Methven 2005) and Holmboe instabilities (Baines and Mitsudera 1994). In addition to ap-plication of the counter-propagating Rossby wave idea to models in which