More recent theoretical, numerical, and laboratory studies have expanded our understanding of gravity wave instability processes considerably. Several studies have begun to clarify the diversity of instability character with wave amplitude and propagation angle, including identifying links between the resonant interactions arising at small wave amplitudes and the local instabilities accompanying wave breaking and turbulence generation at large amplitudes [Klostermeyer, 1991; Lombard and Riley, 1996; Sonmor and Klaassen, 1997; Dunkerton, 1997a]. Other efforts have demonstrated the role of wave–wave interactions at small and large wave amplitudes [Dunkerton, 1987; Klostermeyer, 1991; Thorpe, 1994; Vanneste, 1995] and defined the dynamics of the transition to turbulence for gravity waves that are locally convectively or dynamically unstable [Winters and D'Asaro, 1994; Andreassen et al., 1994, 1998; Fritts et al., 1994, 1996a, 1998; Fritts and Werne, 2000]. Further studies have addressed the processes accounting for more general spectral energy transfers, effects of finite amplitude and wave–mean flow interactions on wave propagation, dispersion, and instability, and the competition among and finite-amplitude responses to initial instabilities of varying amplitudes. Our purpose in this section is to review the more recent work in these areas.
6.1. Wave-Wave Interactions
 Wave–wave interactions as a source of middle atmosphere gravity waves were discussed briefly in section 3. The purpose here is to review their expected influences on the shape and evolution of the gravity wave spectrum more broadly. A number of authors have addressed wave–wave interactions and spectral evolution from a statistical perspective; others have approached the problem deterministically. There remain, however, substantial uncertainties and disagreements over the roles these interactions play in spectral evolution and wave amplitude constraints.
 Early studies of wave–wave interactions in the atmosphere identified the dominant interactions previously recognized in oceanic applications [McComas and Bretherton, 1977; Yeh and Liu, 1981; Müller et al., 1986]. Three-wave (second order) resonant interactions require a match of wave numbers and frequencies among the participating waves satisfying
where subscripts denote the secondary wave quantities, and include (1) elastic scattering, (2) induced diffusion, and (3) parametric subharmonic instability (PSI). Elastic scattering refers to the backscattering of an upward propagating wave into a downward propagating wave (or vice versa) of comparable vertical wave number by a low-frequency motion (or mean flow) having twice the vertical wave number (Bragg scattering). Induced diffusion refers to the transfer of energy from one wave to another having a nearly identical wave number through interaction with a low-frequency (or mean) structure at a much smaller vertical wave number. Alternatively, induced diffusion may be regarded as the evolution of a wave packet in a large-scale shear flow. The third interaction, and arguably the most important in terms of spectral broadening or wave amplitude constraints, is PSI. In its most studied form, PSI represents an exchange of energy from a dominant motion of intermediate frequency to two motions having approximately half the parent frequency and large and nearly opposite wave number vectors, though other wave number combinations exhibit preferred growth in certain circumstances, such as the presence of a mean shear.
 More recent efforts have identified other possibilities for energy exchange and further modes of interaction likely to occur among atmospheric waves and other motions. Broutman and Young  and Broutman et al. , for example, noted that an energetic resonant-triad member can result in an irreversible transfer of energy to small wave numbers not anticipated by McComas and Bretherton . Dong and Yeh  relaxed the assumptions of Yeh and Liu  and considered wave–wave interactions not confined to a vertical plane or a resonance surface (nonresonant interactions were also discussed by Müller et al. ). Further efforts revealed a potential for nonresonant gravity wave–vortical mode interactions for which the threshold amplitude is reduced or removed in the presence of rotation [Yeh and Dong, 1989]. As wave amplitudes increase, additional interactions become energetically viable. Dunkerton  performed numerical studies to assess the impact of wave–wave interactions on wave momentum transport, with and without mean shear. He found a rapid energy transfer on the timescale of the forced waves following attainment of large amplitudes, followed by a nonlinear cascade toward increasing complexity as additional interactions became possible. When critical levels were present for the forced waves, Dunkerton  found weaker resonant wave–wave interactions, the occurrence of additional nonresonant interactions, and the excitation of a wave having higher intrinsic frequency than the forced waves corresponding to a different branch of the PSI.
 Klostermeyer  likewise found that larger primary wave amplitudes enable a cascade of interactions that quickly populate the spectrum, while waves that approach or exceed a convectively unstable amplitude support various modes of instability [Hines, 1971, 1988b; Lombard and Riley, 1996; Sonmor and Klaassen, 1997; Dunkerton, 1997a]. Vanneste  performed both analytic and numerical studies of wave–wave interactions localized in space and noted good agreement between the two methods. In particular, Vanneste  found that spatially localized interactions transferred significant energy on a timescale of ∼10 primary wave periods for a primary wave amplitude of . Vanneste  also noted that mean shear suppresses the dominant unsheared wave–wave interaction but enables a “difference instability” which occurs near the primary wave critical level and has a secondary wave frequency larger than that of the primary wave (like Dunkerton ), downward secondary wave propagation, and a growth rate that increases with shear strength. Such interactions are necessarily only resonant at specific locations but, nevertheless, enable rapid energy transfers under certain conditions [Grimshaw, 1988].
 An example of resonant excitation via PSI in the laboratory study by Thorpe  is shown in Figure 13. Here a primary wave with phase aligned along the mean tilt of the tank transfers energy to a lower-frequency wave that becomes the predominant wave at later times. The wave–wave interaction proceeds quickly because of the large forcing wave amplitude, and the secondary wave achieves local convective instability. A numerical wave–wave interaction study performed by Vanneste  is displayed in Figure 14 and exhibits the decay of a primary wave at higher frequency (Figure 14b) into two motions having lower frequencies (Figures 14c and 14d). The mean flow also experiences significant modifications in the presence of strong wave–wave interactions and can result in critical-level interactions and convective instability of the wave field [Sutherland, 2001]. Hence wave–wave and wave–mean flow interactions should be expected to play increasing roles in spectral evolution and energy and momentum fluxes as wave amplitudes increase with altitude.
Figure 13. Dyed isopycnal surfaces in a rocked inclined tube showing the development and instability of a wave excited by the PSI. Note that the primary wave (with uniform structure along the tube) contributes largely at early times, whereas the subharmonic wave dominates the instability at later stages. After Thorpe  (reprinted with permission of Cambridge University Press).
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Figure 14. Numerical simulation of PSI in an unsheared atmosphere with a scale height of 7 km. Shown are the (first panel) total, (second panel) primary wave, and (third and fourth panels) two secondary wave vertical velocities at altitudes of 10.7 (solid lines) and 17.75 km (dashed lines). Note that both secondary waves grow to amplitudes exceeding that of the primary wave as the interaction proceeds. After Vanneste  (reprinted with permission of Springer-Verlag).
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 Theories of middle atmosphere spectral evolution employing statistical wave–wave interactions have also borrowed from the oceanic literature. The Lagrangian arguments by Allen and Joseph  were employed in the Doppler spread theory by Hines [1991, 1993] to account for the form of the saturated “tail” spectrum. Hines  extended these arguments and used the apparent success of this model to argue against the relevance of other existing saturation theories, most specifically “linear” saturation theory [Dewan and Good, 1986; Smith et al., 1987] and the “nonlinear diffusion” theory by Weinstock [1976, 1982, 1984, 1990]. The Hines theory attributes the large majority of spectral energy transfers to Doppler spreading by a “broad” spectrum of waves, with wave amplitudes and interactions increasing with altitude. The theory presumes that dissipation generally does not occur except at small scales having vertical wave numbers , where is the maximum vertical wave number for which spectral character is determined largely by Doppler spreading, and infers a spectral shape approaching the value most often cited by other saturation theories and various observations cited above. Advantages of the Hines theory include a recognition of the increasing statistical importance of nonlinear wave–wave interactions with increasing wave amplitudes and a natural transition from “linear” source spectra at lower altitudes to a nonlinear “saturated” spectrum at higher altitudes.
 The Hines Doppler spread theory has its detractors, however. Several authors have performed numerical studies that appear to seriously undermine the assumptions of Doppler spread theory [Zhong et al., 1995; Bruhwiler and Kaper, 1995; Broutman et al., 1997; Eckermann, 1997; Buckley et al., 1999; Walterscheid, 2000]. Essentially, these studies show that when time dependence and vertical motions of the underlying wave field are accounted for, the tendency for transfer of spectral energy to ever smaller vertical scales is substantially reduced (or reversed). Zhong et al.  employed ray-tracing techniques to examine tidal modulation of wave propagation and noted that time dependence caused critical levels to become transient. Bruhwiler and Kaper  formulated the problem from a Hamiltonian perspective and showed that the theory was in good agreement with ray-tracing results for a spectrum of waves with various background wave amplitudes. They found, in particular, a tendency for low-m waves to remain at low m and for high-m waves to move preferentially to lower m, representing a sharp departure from expectations based on Doppler spread arguments.
 Broutman et al.  performed both ray-tracing and full numerical simulations, demonstrating the equivalence of the two approaches and the various influences of spatial and temporal variability on short-wave propagation. These results are illustrated in Figure 15. In particular, they exhibit a scattering of short-wave (large m) energy to both smaller and larger wave numbers, consistent with previous assessments based only on ray tracing [Broutman, 1986; Broutman and Young, 1986; Bruhwiler and Kaper, 1995], and a spectral form close to that is consistent with the results of Eckermann . Eckermann  and Walterscheid  also considered single waves in backgrounds having various character. Eckermann  addressed specifically the various assumptions underlying the Doppler spread theory by Hines [1991, 1993, 1996], concluding that wave field transience significantly reduces the tendency to transfer wave energy to smaller vertical scales and eliminates a cutoff in unspread vertical wave numbers near , itself a key element of Doppler spread theory. Walterscheid  obtained similar results, emphasizing both the mitigating effects of time dependence on Doppler spreading and the role of vertical velocities in allowing wave packets to penetrate anticipated critical levels. Hines  raised strong objections to the analysis by Eckermann , arguing that the multiple-wave background offset the results of a single-wave background to a significant degree.
Figure 15. Perturbation density illustrating the encounter of a small-scale wave packet with an inertia-gravity wave of much larger scale. In this case, the small-scale wave has a horizontal phase speed 0.2 of the maximum of the inertia-gravity wave motion, no critical level trapping occurs, and both the vertical wavelength and horizontal phase speed of the small-scale wave increase dramatically. After Broutman et al. .
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 Buckley et al.  and Sonmor and Klaassen  extended the analyses by Eckermann, Broutman et al., and Walterscheid by considering the effects of a spectrum of background waves, isolated wave packets, and a mean shear. Both studies suggest a weaker tendency for refraction toward large m than with the steady assumptions underlying Doppler spread theory. Sonmor and Klaassen  also found that multiple waves induce caustics under more general conditions than supported for single-wave environments, while both studies found that mean shears have significant cumulative refraction effects over timescales of a few inertial periods.
 Doppler spread theory, as it is presently developed, precludes competitive instability processes (i.e., linear saturation theory) from contributing to the shape of the vertical wave number spectrum except, to quote from Hines , at wave numbers “a little less than ; but, arising as they almost certainly do in small-scale ‘white-cap’ regions, their effect is unlikely to extend very far”. However, an increasing number of numerical and observational studies are providing evidence that local wave instability, via convective or shear instability, at large vertical scales is more the rule than the exception (see section 6.2.1), despite the clear potential for wave–wave interactions to contribute importantly to energy transfers as wave amplitudes increase. Additional observations suggesting that the atmospheric wave spectrum is often not broad but is composed of a single or a few dominant motions further undermine the ability of Doppler spreading among multiple waves to provide large enough spectral transfers and to reproduce the observed amplitude limits.
 Clearly, the jury is still out on these issues. There is clear evidence, as noted, of discrete, large-amplitude events or superpositions of a few waves which yield canonical spectral shapes in an Eulerian frame but which suggest a simpler Lagrangian viewpoint. Indeed, Hines  stated “that the occurrence of the large-wave number Eulerian tail has nothing whatever to do with any physical process. . If one insists on defining ‘waves’ according to their Eulerian linear description, then one is forced to admit to the existence of ‘nonlinear wave–wave interactions’. .but these are mere mathematical artifacts and have no physical import.” However, while transformation to a Lagrangian coordinate does remove the advective nonlinearity in the Navier-Stokes equations, it does not remove the coupling of wave scales that nonlinearity implies. For example, claims of Lagrangian linearity cannot explain the nonlinear coupling of scales observed in nonlinear simulations or laboratory studies revealing significant wave–wave interactions [McEwan, 1971; Klostermeyer, 1991; Thorpe, 1994; Vanneste, 1995]. Likewise, shears due to mean winds and large-scale waves imply critical levels, small vertical scales, and instability processes that are demonstrably not mathematical artifacts. Indeed, Broutman et al. (personal communication, 2002) have pointed out that the dispersion relation employed by Hines [2001, 2002a] does not include the influences of mean wind shear on vertical wave structure and thus does not describe the full effects of Doppler shifting. The important issue is not whether nonlinear wave–wave (and wave–mean flow) interactions are important, as they surely are, but whether they are the dominant mechanism in constraining wave amplitudes and shaping the spectrum for , as argued by Hines  and Chunchuzov , or whether they are one of several key processes acting in this fashion, which seems a more defensible perspective, given the various evidence available to date. The identified links between wave instabilities at small and large amplitudes [Sonmor and Klaassen, 1997], however, may render such distinctions obsolete.
 The Weinstock [1976, 1982, 1985] theory was discussed in previous reviews and has itself served as the basis for additional saturation theories by Zhu  and Medvedev and Klaassen  having various attributes. All of these theories view nonlinearity as spectral diffusion of one form or another. There is no clearly defined link, however, with the predictions of wave–wave interaction theory or direct numerical studies [e.g., Dunkerton, 1987; Klostermeyer, 1991; Vanneste, 1995], so testing of the physical basis for the theory has proved challenging. Recently, Hines [2002b] has raised serious objections to assumptions in the original work by Weinstock and its successors by association. However, while Weinstock's theory is clearly an approximation to a class of nonlinear effects, like Doppler spread theory, it calls attention to the importance of nonlinear interactions in shaping the wave spectrum with increasing altitude. Indeed, Hines's criticism has itself been recently challenged (Klaassen and Medvedev, personal communication, 2000). In summary, nonlinear wave–wave interactions must be regarded, at present, as a viable means of exchanging energy among gravity waves at various scales and frequencies throughout the middle atmosphere. Evidence suggests, however, that they cannot account, by themselves, for the general shape or amplitude of the gravity wave spectrum. Rather, they represent one of several nonlinear processes that act jointly to define gravity wave spectral character and evolution with altitude. Wave–wave interactions likewise do not contribute directly to energy dissipation, except through their links to specific instabilities at larger wave amplitudes [McEwan, 1971].
6.2. Instability and Turbulence Dynamics
 Theoretical, numerical, and observational studies have made substantial contributions to our understanding of instability and turbulence dynamics accompanying gravity waves in recent years. Perhaps the greatest advances were made possible by the continuing evolution of high-performance computers, which are now capable of direct numerical simulations (DNS) of stratified and sheared flows having resolutions of ∼10003 and above. Such simulations are capable of describing both the transition to turbulence in a geophysical flow and the vorticity dynamics driving the turbulence cascade. Observational capabilities have also progressed significantly over the last decade or so and have yielded high-resolution in situ and ground-based measurements of wave and turbulence structures. Other numerical and laboratory studies have contributed to our understanding of general turbulence dynamics in applications not specific to atmospheric gravity waves.
 Turbulence is generally believed to arise locally within a gravity wave (or a field of superposed waves) when the flow is either convectively or dynamically unstable and the timescale for instability growth is sufficiently shorter than that describing the evolution of the wave field. More recent analyses have shown, however, that a wide spectrum of instabilities is possible, with specific instability character depending on wave amplitude and intrinsic frequency, mean shear and stability profiles, and the form and amplitude of perturbations triggering flow instability. In particular, local instability can occur at wave amplitudes below that often considered necessary for convective instability in many instances [McEwan, 1971; Hines, 1988b; Lombard and Riley, 1996; Sonmor and Klaassen, 1997]. It is also believed, based on simple stability arguments, that dynamical instability should predominate for intrinsic frequencies and that convective instabilities should predominate for intrinsic frequencies [Dunkerton, 1984; Fritts and Rastogi, 1985] because of the very different relative amplitude thresholds in each case. These expectations are supported by the numerical simulations performed to date [Andreassen et al., 1994, 1998; Fritts et al., 1994, 1998, 2003; LeLong and Dunkerton, 1998a, 1998b].
 Because local instabilities at wave amplitudes below that normally identified as the “convective” instability limit,
generally exhibit slow growth, however [Lombard and Riley, 1996; C. Bizon, personal communication, 2001], we assume for our discussion that the nominal threshold amplitudes for dynamical and convective instability are those which yield and , respectively, with the Richardson number (Ri) defined to be
where and are the local mean zonal and meridional wind shears and N is the local buoyancy frequency. For most intrinsic frequencies the two threshold amplitudes are nearly identical [Dunkerton, 1984; Fritts and Rastogi, 1985], departing significantly only for .
 With the high-resolution simulations now possible on current supercomputers, DNS studies of gravity wave breaking at sufficiently high intrinsic frequencies (i.e., more nearly comparable, rather than disparate, horizontal and vertical wavelengths) can capture both gravity wave scales and a broad range of turbulence scales simultaneously. On the other hand, the large disparity between inertia-gravity wave and turbulence scales (∼106 or more) prevents simulations that span this range of scales. Instead, relevant simulations have addressed the KH instability of inertia-gravity waves and the turbulence arising from KH instability separately.
6.2.1. Gravity Wave Breaking
 A number of numerical studies addressed gravity wave breaking and the accompanying wave amplitude limits in 2-D for lack of adequate computational resources to perform full 3-D studies. Because the instability processes are inherently 3-D, however, 2-D studies are either limited in their utility (e.g., in quantifying wave amplitude limits and momentum flux divergence) or complete misrepresentations of the relevant dynamics (e.g., in modeling wave breaking and turbulence dynamics) [Andreassen et al., 1994]. Thus, we will review here only those studies addressing the 3-D character of wave instability or the occurrence and scales of such events in the atmosphere.
 The first 3-D numerical studies of gravity wave instability dynamics in the atmosphere were performed by Andreassen et al. , Fritts et al. , and Isler et al. . A parallel study examining similar dynamics in an oceanic context was performed by Winters and D'Asaro . These studies addressed gravity wave breaking via convective instability, which appears to be the preferred instability for gravity waves at relatively high intrinsic frequencies [Dunkerton, 1984; Fritts and Rastogi, 1985]. These simulations succeeded in describing the character of the primary wave instability process; this comprises counterrotating streamwise (along the flow) convective rolls or vortices (with spanwise, or normal, wave number). These convective rolls occupy the full depth of the convectively unstable region within the gravity wave, derive their eddy energy (or vorticity) from baroclinic and shear sources within the 2-D flow, and trigger a turbulence cascade via mutual vortex interactions as they interact with adjacent shear layers (or vortex sheets). Indeed, the initial study of gravity wave breaking in 3-D provided a plausible explanation for apparent streamwise instability structures observed in NLC near the summer mesopause [Fritts et al., 1993b] (see Figure 16). Additional observational evidence of such instabilities has come from recent analyses of airglow imaging data and correlative wind measurements [Swenson and Mende, 1994; Hecht et al., 1997, 2000].
Figure 16. (top) Observed and (bottom) simulated gravity wave and instability structures in NLC. Photo was taken at Turku, Finland, on 21–22 July 1989, courtesy of Pekka Parvianien. Streamwise-aligned instability structures are believed to have accounted for the smaller-scale bright bands oriented approximately normal to the gravity wave phase fronts in the upper image. After Fritts et al. [1993b].
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 Subsequent numerical studies at higher resolution addressed wave breaking in streamwise and transverse shear flows [Fritts et al., 1996a], with superposed low- and high-frequency motions [Fritts et al., 1997b], and the vorticity dynamics driving the turbulence cascade [Andreassen et al., 1998; Fritts et al., 1998]. The latter studies addressed in detail the vorticity dynamics of the initial shear-aligned instabilities as well as their subsequent interactions with adjacent vorticity sheets (the mean and 2-D gravity wave shears having spanwise vorticity).
 The vorticity field arising because of convective instability of the gravity wave, the subsequent transition to turbulence, and the dynamics within the turbulent flow are illustrated in Figure 17 at various times throughout the simulation described by Andreassen et al.  and Fritts et al. [1998, 1999]. The simulation was performed for a gravity wave of horizontal wavelength ∼30 km and intrinsic frequency at the level of wave breaking of . A mean shear was imposed to confine the breaking to the region below the initial critical level for the wave. The entire transition from a 2-D overturning gravity wave to quasi-isotropic turbulence occupies approximately a buoyancy period in this simulation. Vortex structures are displayed using a quantity representing the rotational or “tube-like” character of the motion field analogous to the minimum pressures within the flow (see Jeong and Hussain  and Andreassen et al.  for further details). The volumes are viewed from below, with the streamwise direction and wave propagation to the right.
Figure 17. Vorticity structures in a breaking gravity wave viewed from below with the direction of wave propagation to the right. The images display vortex structures from left to right and top to bottom, spanning one buoyancy period following attainment of convective instability of the wave field. See text for details. After Fritts et al.  (reprinted with permission of Terra Scientific Publishing Company).
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 The transition from 2-D laminar flow to 3-D quasi-isotropic turbulence accompanying wave breaking involves several distinct phases: (1) the initial shear-aligned convective instability within the gravity wave (first two images in Figure 17), (2) a second phase in which divergent spanwise motions below adjacent streamwise vortices stretch and thin the adjacent spanwise vortex sheets, causing them to become locally dynamically unstable, (3) the formation of secondary, spanwise-localized KH billows on the intensified vortex sheets and their linkage to the overlying streamwise vortices to form series of intertwined vortex loops (second, third, and fourth images in Figure 17), and (4) the subsequent interactions of neighboring vortices, the excitation of twist waves on the various vortices, and the unraveling, fragmentation, and breakup of the vortices that comprises the cascade to smaller scales of motion (last four images in Figure 17). Indeed, the various stages in the transition to turbulence often occur nearly simultaneously in different (or the same) portions of the flow. Similar simulations having a spanwise mean shear component and more recent higher-resolution simulations having no mean shear confirm the general nature of the transition and turbulence dynamics described above [Fritts et al., 1996a, 2003].
 Various stages in the turbulence evolution noted above also parallel in important respects vortex dynamics observed in other turbulent flows. Intertwined vortex loops, or similar “hairpin” or “horseshoe” vortices, are seen to arise in sheared boundary layers [Gerz et al., 1994; Adrian et al., 2000]; twist waves, so easily excited in the wave breaking simulations, were also observed, but not recognized as such, in laboratory studies of vortex dynamics [Cadot et al., 1995]. Finally, the sequence of vortex dynamics seen in gravity wave breaking exhibits striking parallels to that observed in the transition to turbulence accompanying the KH instability to be discussed below [see also Arendt et al., 1997, 1998; Fritts et al., 1999]).
 The scales at which gravity wave breaking occurs are becoming better known based on both direct and indirect measurements. Such observations also address the current debate over the mechanisms constraining wave amplitudes and imposing saturation of the gravity wave spectrum (see sections 6.1 and 6.3). Direct measurements of density or temperature (via lidar and balloon- or rocket-borne instrumentation) reveal frequent occurrences of near-adiabatic or superadiabatic lapse rates, with occurrence frequency and vertical scale increasing with altitude. Recent examples include balloon and Rayleigh lidar measurements of large-amplitude wave motions at lower altitudes [Hauchecorne et al., 1987; Shutts et al., 1988; Wilson et al., 1991a; Whiteway and Carswell, 1995; Hoppe et al., 1999] and sodium resonance lidar measurements at greater altitudes [Hecht et al., 1997; Williams et al., 2002]. Rocket measurements employ various techniques, including falling spheres [Fritts et al., 1988b], chaff [Wu and Widdel, 1991], ionization gauges [Lübken, 1997], and Rayleigh lidars [Hoppe et al., 1999]. These observations reveal unstable or nearly unstable layers typically up to a few kilometers in depth, suggesting convectively unstable gravity waves having vertical wavelengths of ∼3–10 km or more, with the larger scales more prevalent near the mesopause. The recent sodium lidar measurements by Williams et al.  represent a particularly striking example, with multiple superadiabatic layers a few kilometers in depth within a near-adiabatic layer extending more than 10 km and persisting for more than 4 hours (see Figure 18). As noted above, these observations suggest, at least in this instance, wave saturation via local convective instability, though wave superposition or wave–wave interactions, specifically the PSI displayed in Figure 13, may play a role in the vertical fine structure within the overturning wave field. While this event is exceptional, the frequency of observation of such features suggests that they are far from the infrequent or pathological cases implied by Hines . Indirect inferences of convective overturning scales are provided by NLC and airglow measurements of horizontal instability scales [Fritts et al., 1993b; Swenson and Mende, 1994; Hecht et al., 1997, 2000], where the observed roll spacing is suggested to be comparable to the unstable layer depth in numerical simulations. Additional inferences of gravity wave amplitudes and instability depths are obtained from radar wind measurements via the gravity wave dispersion relation [Muraoka et al., 1988].
Figure 18. Contours of (a) temperature, (b) sodium density, (c) temperature perturbations as a percent of the mean value, and (d) profile of mean temperature for the 8-hour wave observation using the Fort Collins sodium lidar on day 307 of 1997. Also shown are the amplitude of the 8-hour wave in the temperature field (e) and the phase of the maximum temperature (f). The 1σ uncertainties are shown with horizontal bars and the dashed and dash-dotted lines in Figure 18d are the diurnal means for a 24-hour campaign in November 1998 and nightly mean for November averaged from 1990 to 1997. After Williams et al. .
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6.2.2. Kelvin-Helmholtz Instability
 KH instability is among the most common sources of turbulence in the atmosphere. In many instances it owes its existence, in part at least, to wind shears due to inertia-gravity waves because such motions contribute preferentially to wind shears throughout the atmosphere. Earlier evidence of KH instability due to inertia-gravity waves was reviewed by Fritts and Rastogi . More recent stability analyses have identified the preferred modes of instability of inertia-gravity waves with and without mean shears [Fritts and Yuan, 1989c; Yuan and Fritts, 1989; Dunkerton, 1997a]. Numerical simulations of unstable inertia-gravity waves revealed the character of the KH instability and its preferred orientation within the wave field [LeLong and Dunkerton, 1998a, 1998b]. The numerical studies showed a preferred direction consistent with the stability analysis by Fritts and Yuan [1989c] but with an increasing degree of isotropy as . These simulations were also used to assess the theoretical prediction of the wave amplitude required for onset of shear instability by Dunkerton  and Fritts and Rastogi . Results are displayed in Figure 19 and reveal that time dependence of the wave field increases the wave amplitude required to initiate instability, as anticipated by Lombard and Riley  and Sutherland  at higher intrinsic frequencies.
Figure 19. Threshold amplitude a for shear (KH) instability of inertia-gravity waves as a function of R = f in the absence of mean shear based on theory for a time-independent flow (solid line) and DNS simulations (thick solid line with data points). After LeLong and Dunkerton [1998a] (reprinted with permission of the American Meteorological Society).
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 Other evidence of the importance of KH instability and its relation to inertia-gravity waves comes from observations of wind and temperature profiles in the lower and middle stratosphere. Radar measurements at several sites have revealed persistent layers of enhanced radar reflectivity with spacings of a few kilometers, often exhibiting slow vertical motions and inferred unstable wave amplitudes [Sato and Woodman, 1982; Yamamoto et al., 1987]. Likewise, high-resolution balloon measurements have revealed multiple layers yielding signatures of local turbulent mixing, with near-adiabatic layers sandwiched between sharp temperature inversions [Cot and Barat, 1986; Coulman et al., 1995]. Examples of the latter, with profiles of the corresponding Richardson number, are displayed in the top and middle panels of Figure 20. Note, in particular, that the observations suggest a minimum Richardson number of ∼1/4 following instability and mixing, well below what was often assumed to accompany restratification in the past. It will be seen below that these measurements are in good agreement with recent high-resolution DNS studies of KH instability.
Figure 20. Profiles of , θ, and Ri obtained by (top and middle) high-resolution balloon measurements and (bottom) DNS showing the aftermath of KH instability and mixing events. Note that the magnitude of and the gradient of θ maximize at the edges of the mixed region and that Ri ≈ 1/4 throughout the mixed region. Modified after Coulman et al.  and Fritts et al. .
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 While observations help define the scales at which KH instability and mixing occur and the consequences of mixing for the temperature and wind fields, they cannot identify the dynamics of the turbulence transition or the evolution of the shear layer as KH breakdown and mixing occur. The only means of understanding these aspects of inertia-gravity wave instability is via DNS studies spanning KH growth, breakdown, and restratification.
 There have been many numerical studies of KH instability in the past. Only recently, however, have computational capabilities permitted 3-D studies of high enough resolution to confirm the predictions and laboratory observations of secondary instability [Klaassen and Peltier, 1985; Thorpe, 1987; Palmer et al., 1994, 1996; Caulfield and Peltier, 1994; Fritts et al., 1996b; Smyth, 1999]. Early 3-D KH simulations were not adequate to describe the transition to turbulence, the structure and anisotropy within the inertial-range of turbulence, and the implications for dissipation and mixing. However, recent simulations are now adequate to investigate these turbulence effects [Werne and Fritts, 1999, 2001; Smyth and Moum, 2001]. The vorticity dynamics accompanying this evolution and its relation to the vorticity dynamics of wave breaking were described by Fritts and Werne .
 The KH simulations performed by Werne and Fritts [1999, 2001] have illustrated the dynamics of the transition to turbulence, the expansion of turbulence throughout the KH billow and the shear layer, and the turbulence statistics, dissipation, anisotropy, and decay. Most striking, perhaps, is the great degree of similarity of the turbulence dynamics to those observed in previous simulations of gravity wave breaking. Common features include (1) the initial streamwise, or shear-aligned, convective instability within the outer regions of the KH billow, (2) the stretching and wrapping of spanwise vortex sheets around the streamwise vortices, (3) the dynamical instability and rollup of the localized and intensified vortex sheets, and (4) the subsequent vortex interactions and twist wave excitation driving the cascade to smaller scales of motion [Fritts and Werne, 2000]. Major differences in the turbulence transitions accompanying wave breaking and shear instability are the timescale and the character of mixing. As noted above, the transition due to high-frequency wave breaking requires approximately one buoyancy period for the wave parameters simulated. The transition is somewhat longer due to KH instability, however, because turbulence requires time to expand from the initial site of instability throughout the KH billow and the full depth of the evolving shear layer. Implications for mixing are probably very different as well since KH instability mixes the shear layer vigorously prior to restratification, whereas turbulence due to wave breaking is quickly advected out of the unstable phase of the wave motion, allowing vigorous turbulence to act within the stably stratified portions of the wave field.
 The vorticity dynamics and impact on the thermal field of KH turbulence are illustrated in Figure 21 at four times throughout the simulation described by Werne and Fritts . In this simulation, a buoyancy period corresponds to 28 time units. Note here the gradual penetration of turbulence throughout the KH billow, the rapid obliteration of thermal gradients by turbulence shortly after it occurs, the homogenization of the expanded shear layer, the very sharp thermal gradients that evolve as a result of efficient mixing within the shear layer, and the continuing small-scale dynamics and mixing within the edge regions of the mixed layer at later times. Profiles obtained in the KH DNS at are displayed in the fourth panel of Figure 20 and exhibit good agreement with those observed by Coulman et al.  (first and second panels), despite the markedly different Reynolds numbers of the two flows. The implications for inertia-gravity waves include (1) instability, turbulence, and mixing on timescales short compared to wave periods, hence confined to the unstable phase of the wave, (2) reduced shears and wave amplitudes due to mixing, and (3) alternating layers of low and high static stability that may influence subsequent instability and mixing processes.
Figure 21. The 2-D streamwise-vertical cross sections of (left) 3-D vorticity magnitude and (right) temperature fields at four times throughout the turbulence transition and restratification of a KH billow. See text for details. After Werne and Fritts  (reprinted with permission of Elsevier Science).
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6.2.3. Synthesis and Other Instability and Large-Amplitude Processes
 Studies cited above have expanded greatly our understanding of gravity wave instability processes over the last decade. Importantly, these studies have also suggested links between specific modes of instability of small- and large-amplitude gravity waves. Klostermeyer  found the PSI to be related to a 2-D parametric instability at finite amplitude and suggested, based on his analysis, that resonance may underlie all gravity wave instability processes. Klostermeyer  and Lombard and Riley  also identified transverse modes of instability at finite amplitudes below those required for local convective or dynamical instability.
 The links between instabilities at small and large wave amplitudes were further clarified and generalized by Sonmor and Klaassen  to include all instability types and all wave amplitudes and intrinsic frequencies (or phase propagation angles) assuming no rotation. Specifically, they found a link between the PSI and the “slantwise static instability” (SSI) of Hines [1971, 1988b] at small wave amplitudes and with convective instability and high wave number parametric instability at amplitudes (see equation (58)). Sonmor and Klaassen  further linked the “branch-C” instability of Yeh and Liu  to the resonant elastic scattering and induced diffusion interactions originally identified by McComas and Bretherton  but noted quite different growth rates than inferred in the earlier study. The “shear-aligned instability” (SA, having spanwise wave number, as noted in the wave breaking simulations discussed in section 6.2.1) is itself linked to higher-order resonant instabilities at smaller wave amplitudes and exhibits an amplitude threshold near the convective limit for waves at high intrinsic frequencies [Winters and Riley, 1992; Sonmor and Klaassen, 1997]. A 2-D dual-mode instability represents a generalization of the KH instability within an inertia-gravity wave and competes favorably with SA or convective instability at large amplitudes [Sonmor and Klaassen, 1997] but is itself stabilized by weak environmental shear [Thorpe, 1994]. Dunkerton [1997a] examined the relative roles of these instabilities in the presence of rotation, employing a steady, plane-parallel approximation to the wave structure and obtained similar results concerning the dominant instabilities at large wave amplitudes.
 A further class of instability, “oblique instabilities” (having nonzero streamwise and spanwise wave numbers), was found by Lombard and Riley  and Sonmor and Klaassen [1996, 1997] to represent the most rapidly growing instability for an important physical range of wave amplitudes at high intrinsic frequencies. Sonmor and Klaassen [1996, 1997] argue that oblique instabilities connect smoothly to SA instabilities at higher frequencies and larger wave amplitudes but are the preferred form when shear is the primary source of instability energy, whereas SA instabilities are preferred when buoyancy is the dominant source. The parameter ranges in which these various instabilities predominate are displayed in Figure 22. Here the dash-dotted line denotes an overturning amplitude, and high intrinsic frequencies have large phase elevation angles. We note, however, that recent DNS studies [Fritts et al., 2003] reveal SA instabilities to predominate over 2-D dual-mode and oblique instabilities at finite amplitude for a wide range of the wave parameters displayed in Figure 22. This may be due either to the Sonmor and Klaassen analysis being inviscid (while the DNS studies are viscous) or to the selection of a finite-amplitude response in the DNS from among a variety of initial instabilities at infinitesimal amplitude.
Figure 22. Fastest growing linear instabilities as a function of wave amplitude and phase elevation angle. The dash-dotted line corresponds to the amplitude of overturning (a = 1 in our terminology), and the dashed line represents the transition from oblique to shear-aligned instability and from shear to buoyancy as the dominant source of eddy energy. After Sonmor and Klaassen  (reprinted with permission of the American Meteorological Society).
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 Other types of instabilities are enabled as a result of wave packet localization or by the induced mean flows accompanying waves of finite amplitude. Sutherland  demonstrated numerically the modulational instability (and stability) of a localized wave packet anticipated by Whitham [1965, 1974] but noted that this instability does not necessarily imply wave breaking. Sutherland also argued that self-acceleration of a vertically localized wave packet can lead to local convective instability at sufficiently large amplitudes and high intrinsic frequencies, providing a means by which wave localization can contribute to instability. The mechanism by which instability occurs is the differential tilting of surfaces of potential temperature by the wave-induced mean flow. Similar effects accompany localized wave packets incident on a turning level and may lead to wave instability via either self-acceleration (as above) or as a result of enhanced wave amplitude due to incident and reflected wave superposition [Sutherland, 2000]. Interesting additional consequences of wave packet localization and finite amplitude include tendencies (1) for wave packets that are vertically localized and of large amplitude to yield permanent momentum flux divergence near a turning level and (2) for wave packets that are both horizontally and vertically localized to penetrate substantial distances beyond a turning level based on linear theory [Sutherland, 1999, 2000]. Indeed, both of these mechanisms may have important but as yet unquantified implications for gravity wave momentum transport and redistribution in the middle atmosphere. The tendency for spatially localized, large-amplitude packets to penetrate turning levels may be particularly relevant for middle atmosphere momentum transport, where such packets may be more the rule than the exception. A demonstration of the tendency for penetration of a turning level as wave amplitude increases is shown in Figure 23. In these cases, total and partial reflection occur for small and intermediate wave amplitudes, while wave packet propagation appears to be increasingly unaffected by mean shear (and increasing intrinsic phase speed and frequency) as wave amplitude increases. Wave packet transmission and reflection for the largest wave amplitude are displayed at several stages throughout the event in Figure 24. Note, in particular, the phase distortions at the leading edge of the wave packet above the turning level which exhibit the self-acceleration effects discussed by Fritts and Dunkerton  and Sutherland [1999, 2000].
Figure 23. Time series of wave momentum flux for wave amplitudes of various vertical velocities incident on a turning level. The turning level anticipated by linear theory is shown by the horizontal line in each panel. After Sutherland  (reprinted with permission of the Royal Meteorological Society).
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 A final topic relevant to our understanding of instability dynamics in gravity waves is the optimal perturbation theory pioneered by Farrell and Ioannou [1996a, and references therein]. This theory has provided considerable insights into the occurrence of or competition among various instability modes in Couette, Poiseuille, and other time-independent shear flows known to be asymptotically stable below a threshold (or for any) Reynolds number. However, the theory also can be applied to time-dependent flows and to more general stratified problems including secondary KH and gravity wave instabilities for which the equations of motion are non-self-adjoint [Farrell and Ioannou, 1996b]. In such flows, particular superpositions of linear modes, termed “optimal perturbations”, may exhibit rapid transient growth exceeding the growth rates of all eigenfunctions of the linear system. If this growth achieves a sufficient amplitude to have nonlinear consequences, either among the perturbations or for the mean flow, then it will be the optimal perturbation which dictates the finite-amplitude behavior of the nonlinear system. In applications to the KH instability problem, the optimal perturbation methodology has shown that the structures of both the initial 2-D instability and the secondary 3-D instabilities depend sensitively on the form and amplitude of the initial perturbations (J. Werne, personal communication, 2001). Optimal perturbation theory also reveals why the details of the turbulence transition and cascade discussed above depend so sensitively on the initial noise spectrum [Werne and Fritts, 1999; Fritts and Werne, 2000]. In the atmosphere, then, we anticipate that instability structures accompanying wave breaking or shear instability may often be dictated more by other finite amplitude perturbations than by the exact form of the most rapidly growing linear eigenmode of the unstable flow. While initial perturbations dictate the details of the transition to turbulence, they appear not to alter the statistical effects of turbulence on the larger-scale flow [Fritts and Werne, 2000].
6.3. Saturation Theories
 The dynamics underlying gravity wave saturation in the atmosphere, i.e., instability and wave–wave interaction processes, appear to impose a near-universal spectral form far from the dominant sources at lower altitudes (but subject to all of the caveats about departures from universality noted above). Our purpose in this section is to review the various theories for the shape and evolution of the wave spectrum and to identify how they are linked to the instability dynamics discussed above.
 The earliest wave amplitude limits attributed to instability dynamics were proposed by Hodges  and involved the convective instability of a gravity wave due to its exponential growth with height. Subsequent studies by many authors considered the implications of such dissipation for momentum transports, turbulence, and mixing (see Fritts [1984a] for a review). Motivated by VanZandt's  suggestion of a universal gravity wave spectrum and their own observations of near universality in stratospheric wind spectra [Dewan et al., 1984], Dewan and Good  proposed the first theory of a saturated spectrum employing saturation (via local convective or shear instability) separately at each vertical wave number. Their dimensional analysis yielded a saturated range for gravity waves varying as with C dependent on spectral bandwidth (and a bandwidth ). The study by Smith et al.  extended this theory to account for attainment of saturated wave amplitudes by the wave spectrum as a whole, yielding a saturation spectrum amplitude of for velocities and a natural explanation for the evolution of the dominant vertical wave number, m*, toward smaller m with increasing altitude. Further refinements of this theory yielded a corresponding saturated amplitude for gravity wave temperature spectra of and implications for enhanced saturation and dissipation where N increases with altitude [Fritts et al., 1988a; VanZandt and Fritts, 1989]. Attributes of this “linear saturation theory” include a simple conceptual mechanism for amplitude limits for superposed waves of various scales and frequencies, general agreement of the predicted spectral form and amplitude and variations of wave energy density with altitude with observations throughout the atmosphere [Allen and Vincent, 1995], and a natural explanation for the increase in the dominant vertical wavelength with increasing altitude [Tsuda et al., 1989]. Failings of the theory include an inability of the spectral description to account for specific wave sources having character quite different from the canonical spectral form, an inability to account for amplitudes in the stratosphere below the saturation limit because of wind shear effects on vertical wavelengths [Eckermann, 1995a; Alexander, 1996], relative amplitudes of velocity and temperature spectra that depart from predictions [Tsuda et al., 1991; Nastrom et al., 1997; de la Torre et al., 1999], and the neglect of wave–wave interactions that are known to cause spectral energy transfers.
 An alternative explanation of the evolution of gravity wave spectral characteristics with increasing altitude was provided by Weinstock [1982, 1985, 1990] based on “nonlinear damping.” Weinstock argued that this mechanism could both constrain wave amplitudes and account for spectral broadening with increasing altitude. Accounting for nonlinear influences within the wave spectrum, even without accounting explicitly for resonant interactions, was an appealing approach and spawned a number of extensions to the theory by Zhu , Gardner , and Medvedev and Klaassen . As with linear saturation theory, however, the various extensions to Weinstock's theory have their detractors (see section 6.1), and clarification of the mechanisms controlling spectral amplitude and shape remains a high priority. Appealing aspects of the nonlinear damping theories include a recognition of the central role of nonlinearity in shaping the wave spectrum and approximate agreement with observed spectral shape and evolution with altitude. Liabilities include an inability to account explicitly for local wave field instability and turbulence and concerns by Hines [2002b] over the legitimacy of the underlying assumptions.
 A further approach to gravity wave saturation and spectral evolution was taken by Hines [1991, 1993, 1996], who modeled wave-wave interactions by “Doppler spreading” of any one member of the spectrum by all of the remaining members. The Hines theory relies on the Lagrangian description of wave-wave interactions by Allen and Joseph  and shares the attributes and liabilities of this formalism discussed at length in section 6.1. This approach accounts qualitatively for increasing interactions and effects with increasing altitude and yields predictions of a tail spectrum that is also in approximate agreement with observations, subject to correction of the earlier estimates of spectral slope [Hines, 2001; Chunchuzov, 2002]. The theory has also led to the development of a gravity wave parameterization that has experienced a number of successes (see section 7) but has been criticized (see section 6.1) for simplifications in the underlying assumptions that are inconsistent with the claimed cascade toward smaller vertical scales. Perhaps its major contribution has been a focusing of attention and renewed research activity on the role of wave-wave interactions in gravity wave spectral evolution in the atmosphere.
 A final approach to describing gravity wave spectral shape and evolution in the atmosphere is the “saturated-cascade similitude” theory by Dewan [1991, 1994, 1997]. The theory, as it has developed, embodies both the linear saturation ideas of Dewan and Good  and Smith et al.  as amplitude constraints and the notion that wave-wave interactions drive a cascade of wave energy from larger to smaller scales analogous to that observed in inertial-range turbulence. As in turbulence theory, spectral amplitudes are controlled by the energy dissipation rate ϵ, which represents both energy input at large scales via vertical wave propagation and energy dissipation at sufficiently small scales. The theory is appealing in its simplicity and its dependence on scaling arguments, and it makes a number of predictions of 1-D spectra that are either in reasonable agreement with observations or pose observational tests of the theory. The suggestion that saturation and cascade dynamics are jointly at work in shaping the gravity wave spectrum is more realistic than theories relying largely on one or the other. Dewan's theory is also more in line with recent results by Sonmor and Klaassen  indicating that local wave field instability and spectral energy transfers via wave-wave interactions are manifestations of related dynamics at different wave amplitudes. As such, further advances merging or generalizing saturation and cascade processes and effects may help to unify the divergent views of wave saturation and spectral evolution prevailing at present. There are, however, a number of shortcomings or uncertainties in the present formulation, hence motivation for further improvements to the theory. It is unclear but likely not the case that wave energy is transferred from large to small scales without dissipation, while the direction and/or efficiency of the cascade are uncertain, based on the discussion in section 6.1. Finally, the current theory does not predict 2-D spectra, although these would be valuable in assessing the interaction and cascade dynamics, and the prediction of the vertical wave number spectrum of vertical velocities has a slope of +1 which differs significantly from the various observations to date of order −1 to −3 [Kuo et al., 1985; Larsen et al., 1986; Wu and Widdel, 1990; Fritts and Hoppe, 1995].