On the role of optimal perturbations in the instability of monochromatic gravity waves

Abstract

Motivated by the useful new insights from optimal-perturbation theory into the onset of turbulence in other fields singular vectors ͑SVs͒ in stable and unstable gravity waves have been determined within the framework of the Boussinesq equations on an f plane. The difference between the dynamics of normal modes ͑NMs͒ and SV is characterized by a time invariance in the comparative role of the various possible exchange processes between NM and basic wave, while SV can have a highly time-dependent structure, allowing a more efficient energy exchange over a finite time. Both inertia-gravity waves ͑IGWs͒ and high-frequency gravity waves ͑HGWs͒ have been considered. At Reynolds numbers typical for the middle to upper mesosphere IGW admit rapid nonmodal growth even when no unstable NMs exist. SV energy growth within one Brunt-Vaisala period can cover two orders of magnitude, suggesting the possibility of turbulence onset under conditions where this would not be predicted by a NM analysis. HGWs show a dependence of short-term optimal growth on the direction of propagation of the perturbation with respect to the wave which is, at weak to moderate wave amplitudes, quite different from that of NM but reproduced in ensemble integrations from random initial perturbations. Their SVs are sharply peaked pulses with negligible group velocity which are repeatedly excited as the rapidly propagating wave passes over them. The transition of these to the leading NM, which is not moving with respect to the wave and which is typically broader in structure, is very slow, so that in many cases the turbulence onset via local perturbations of a gravity wave might be more appropriately described using optimal-perturbation theory. This might contribute to a better understanding of the often observed occurrence of thin turbulent layers in the middle atmosphere.

Spatial and temporal variability in MLT turbulence inferred from in situ and ground-based observations during the WADIS-1 sounding rocket campaign

“…It is, for example, known from radar observations of PMSEs, whose existence indicates the presence of turbulence (whether active or fossil), that MLT turbulence is a mesoscale phenomenon (e.g., . On the other hand, it is also known that turbulence is highly intermittent in both space and time (e.g., Fritts et al, 2009aFritts et al, , b, 2015Achatz, 2005). However, the degree of this intermittency is not yet quantified by measurements.…”

Gravity-wave breaking: Linear and primary nonlinear dynamics

“…As will be shown below this can be relevant for the breaking of IGWs. For a more thorough discussion of NMs and SVs, and the difference between the two one could also consult Farrell and Ioannou (1993b,a), Achatz (2005), and Achatz and Schmitz (2006a,b).…”

“…4 shows the factor by which either the leading NM or the leading SV grows within an optimization time s = 5 min (Achatz, 2005). The IGW has a wavelength K = 2p/K = 6 km, an inclination angle H = 89.5°, and amplitude a = 0.87, the ambient Brunt-Vaisala frequency is N = 2 · 10 À2 s À1 (so that s = 5 min roughly corresponds to one Brunt-Vaisala period), the f-plane is centered at latitude 70°N, so that the smallest Richardson number in the wave is Ri = 0.28.…”

“…It is important for the observational community to recognize this so as to avoid misinterpretations of their data. On this background the purpose of this review is twofold: (1) It summarizes the most important aspects of our knowledge under which conditions a GW will start breaking, and (2) it gives a summary of recent respective work of the author which hopefully might also help getting a clearer conceptual picture of GW breaking itself (Achatz, 2005;Achatz and Schmitz, 2006a,b;Achatz, 2007a,b). The central paradigm of wave-turbulence interaction addressed in this work is the breaking of a single wave, leaving aside the dynamics of a whole spectrum of GWs.…”

Gravity wave propagation through a large semidiurnal tide and instabilities in the mesosphere and lower thermosphere during the winter 2003 MaCWAVE rocket campaign

“…This proves to be a significant (and inaccurate) simplification, in general, as there are a multitude of instabilities that can arise, and these depend on the specific characteristics of the flow, the amplitudes and intrinsic properties of the involved waves, the initial conditions at the time of instability occurrence, and the influences of viscosity and thermal diffusivity (Achatz, 2005;Fritts et al, 2006;Lombard and Riley, 1996;Sonmor and Klaassen, 1997; also see Fritts and Alexander, 2003, for a review of these dynamics). Despite this complexity, the simpler view of these instabilities often provides a convenient and qualitative, if imperfect, context in which to assess instability structures, energetic sources, and occurrence.…”