Gravity wave dynamics and effects in the middle atmosphere

2.1.1. High-Frequency Waves:

[13] The dispersion relation (23) is derived by neglecting acoustic wave solutions, but the terms proportional to retain one of the larger compressibility effects associated with gravity wave motions. These arise when the vertical wave number m becomes small so that is a nonnegligible factor. With 7 km in the middle atmosphere, this term becomes significant for waves with vertical wavelengths of ∼30 km or longer, where the change in background density becomes significant over the vertical wavelength of the wave. This compressibility effect is important in the interpretation of airglow images of gravity wave perturbations near the mesopause [e.g., Swenson et al., 2000]. As the vertical wavelength grows very large and , the wave will undergo total internal reflection where the vertical group velocity can change sign (called a turning level). Above this level, the wave is evanscent. If turning levels occur both above and below, the wave is said to be ducted. Observational and modeling studies associated with these effects will be described further in 15.

[14] Neglecting rotation effects but retaining the scale-height compressibility term and setting in equation (23) yields an equation describing the maximum horizontal intrinsic phase speed prior to total internal reflection as a function of horizontal wavelength:

This gives a useful approximation to the fully compressible solution [Marks and Eckermann, 1995]. Figure 1 compares solutions for and the corresponding maximum intrinsic frequency versus horizontal wavelength for the fully compressible, incompressible, and approximate equation (29) solutions.

[15] Figure 1a shows that short horizontal wavelengths are much more easily reflected and will likely remain trapped at lower altitudes. Thus only waves with horizontal wavelengths >10 km are generally considered important to middle-atmosphere dynamics. Although the vertical wavelength theoretically goes to infinity at the turning level (and therefore the WKB approximation should be violated), the theory embodied in equation (29) appears to provide a good approximation even for waves observed near this theoretical limit [Swenson et al., 2000].

[16] For high-frequency waves for which the Coriolis force can be neglected and for which , the dispersion relation simplifies to

where α is the angle between lines of constant phase and the vertical. These are simple plane waves with zero velocity perturbations perpendicular to the plane of propagation. Group velocity simplifies to

where is the horizontal component of group velocity, the horizontal wind in the direction of propagation, and is the horizontal wave number. Note that phase propagation is downward () for upward group velocity. In this form it can be seen that group velocity is parallel to lines of constant phase and perpendicular to phase propagation.

[17] High intrinsic frequency waves are observed not only in mesopause airglow image data [Swenson and Espy, 1995; Taylor and Garcia, 1995; Gardner et al., 1996; Taylor et al., 1997; Isler et al., 1997; Hickey et al., 1998; Haque and Swenson, 1999] but also in sublimb viewing infrared and microwave satellite observations in the stratosphere [Wu and Waters, 1996a, 1996b; Alexander, 1998; Dewan et al., 1998; Picard et al., 1998; McLandress et al., 2000] and in middle-atmosphere radar data [Reid et al., 1988; Fritts et al., 1990a, 1992; Fritts and Wang, 1991; Hoppe and Fritts, 1995; Mitchell and Howells, 1998].

2.1.2. Medium-Frequency Waves:

[18] For gravity waves with midrange intrinsic frequencies, very simplified relationships emerge that lend valuable insight into gravity wave properties and the effects due to changes in the background wind and stability. Letting represent the horizontal phase speed and , the dispersion relation then simplifies to

and the vertical wave number can now be very simply related to the background wind and stability:

Note that for , and for . Equations (32) and (33) demonstrate that vertical wavelength and intrinsic frequency are both proportional to the intrinsic phase speed . The level where is a critical level for the wave where vertical wavelength shrinks to zero. This theoretical limit is never achieved in the real atmosphere because a host of instability and dissipation mechanisms become more likely as a wave approaches a critical level.

[19] The group velocity for medium-frequency gravity waves becomes

and has the same magnitude as the phase propagation but opposite sign in the vertical. The time-dependent ray equations [Lighthill, 1978] show that a wave packet requires an infinite time to reach a critical level (with ), implying that time-dependent and nonlinear effects will become important as a critical level is approached. Note, however, that wave transience and time dependence of the background flow can allow different interactions near the critical level. Departures from linear theory due to transience and nonlinearity are discussed in 35.

[20] The polarization relations under the medium-frequency approximation take simple forms:

Note that group velocity, vertical wavelength, and intrinsic frequency increase as a wave propagates “upshear,” defined as an altitude region where the intrinsic phase speed is increasing. These simplified forms demonstrate that vertical velocity will grow proportionately larger than the horizontal velocity perturbations in these conditions, and pressure perturbations become increasingly important. The opposite is true for waves propagating “downshear” such as those approaching a critical level (). In fact, it is the tendency for to increase as the wave approaches a critical level that drives the tendency for instability and turbulence generation. These approximations for waves with midrange frequencies are inappropriate for most detailed analyses, but the set of very simple relationships that results provides an extremely valuable intuitive view of gravity wave properties and their relationship to background winds and stability.

2.1.3. Low-Frequency Waves:

[21] Low-frequency gravity waves, or inertia-gravity waves, are those for which the rotation of the Earth has an important influence. These wave perturbations have an interesting three-dimensional helical structure. Velocity perturbations in the direction perpendicular to the direction of propagation are no longer zero but grow proportionately larger as intrinsic frequency decreases toward f. For a zonally propagating wave the meridional velocity amplitude is

An approximation to the dispersion relation appropriate for both low- and medium-frequency gravity waves is

Any wave nearing a critical level (where the background wind speed approaches the horizontal phase speed and ) will move toward lower intrinsic frequency where rotation effects become important. The true critical level is therefore where , somewhere below the critical level described for medium-frequency waves ().

[22] Because the ratio of vertical to horizontal group velocity,

becomes quite small for inertia-gravity waves relative to their higher-frequency counterparts, they can be found at large horizontal distances from their sources [Dunkerton, 1984; Eckermann, 1992]. They are very commonly seen in lower stratosphere observations.

[23] A number of observational techniques are sensitive to inertia-gravity waves. Balloon-borne measurements of horizontal wind profiles in the lower stratosphere enable inertia-gravity wave hodograph and rotary spectral analyses [Kitamura and Hirota, 1989; Barat and Cot, 1992; Tsuda et al., 1994a; Ogino et al., 1995, Hamilton and Vincent, 1995; de la Torre et al., 1996; Sato et al., 1994; Sato and Dunkerton, 1997; Shimizu and Tsuda, 1997; Vincent and Alexander, 2000]. Limb viewing satellite measurements can also be sensitive to inertia-gravity waves [Fetzer and Gille, 1994, 1996; Eckermann and Preusse, 1999; Tsuda et al., 2000], and they are prominent signals in stratospheric radar and lidar observations [Sato, 1994; Mitchell et al., 1994; Sato et al., 1997; Riggin et al., 1997] and middle-atmosphere rocket sounding data [Hirota and Niki, 1985; Hamilton, 1991; Tsuda et al., 1992; Eckermann et al., 1995; Fritts et al., 1997a].

3.1.1. Topographic Generation

[33] Mountain waves have been studied extensively over the past three decades using theoretical, numerical, and observational methods. As a result, idealized two-dimensional (2-D) responses to 2-D topography are reasonably well understood in terms of nonlinear wave dynamics and hydraulic theory [Long, 1955; Smith, 1985; Durran and Klemp, 1987; Nance and Durran, 1998; Lott, 1998; Farmer and Armi, 1999]. Responses to three-dimensional (3-D) topography have also been quantified increasingly in a number of measurement programs and numerical studies, yielding estimates of mountain wave scales, amplitudes, and momentum fluxes. Horizontal wavelengths for vertically propagating waves are typically tens to hundreds of kilometers, while amplitudes vary from small to breaking [Lilly and Kennedy, 1973; Nastrom and Fritts, 1992; Chan et al., 1993; Dörnbrack et al., 1999; Leutbecher and Volkert, 2000]. Horizontal and vertical cross sections of a wave structure anticipated over northern Scandinavia in a 3-D mesoscale simulation by Dörnbrack et al. [1999] are shown in Figure 2 and exhibit the large-scale thermal structure (areas of low temperature in Figure 2) believed to trigger polar stratospheric clouds (PSC) and ozone depletion. Such simulations, however, do not yet capture the responses often observed over smaller-scale topography [Nastrom and Fritts, 1992].

[34] Momentum fluxes likewise vary greatly but are typically in the range of ∼0.1 to a few m² s⁻² (per unit mass, or ∼0.01–0.5 Pa) in the upper troposphere and lower stratosphere based on in situ aircraft and radar measurements and large-scale cross-mountain pressure distributions [Lilly and Kennedy, 1973; Lilly and Lester, 1974; Smith, 1978; Kennedy and Shapiro, 1979; Shutts et al., 1988; Fritts et al., 1990b; Sato, 1994]. Momentum fluxes are also preferentially associated with the smaller horizontal scale waves that are vertically propagating because of their larger vertical velocities. Maximum (generally negative in Figure 2) momentum fluxes were inferred at horizontal wavelengths between ∼10 and 100 km by Nastrom and Fritts [1992] (see Figure 3).

[35] The Global Atmospheric Sampling Program (GASP) and more recent aircraft observations also enabled assessments of the statistical importance of mountain waves over rough and smooth terrain and relative to other significant sources. These showed horizontal velocity and temperature variances to be ∼2–3 times higher over significant topography compared to plains and oceans, independent of other sources [Nastrom et al., 1987; Jasperson et al., 1990; Bacmeister et al., 1990a] and ∼5 times higher than regions having no obvious meteorological sources [Fritts and Nastrom, 1992]. Because of the importance of such waves for mean circulations a number of authors have attempted to parameterize their specific effects in large-scale models (see 42). The importance of mountain waves relative to other significant sources is discussed further in 23.

[36] Of relevance to the middle atmosphere, then, mountain waves have dominant scales (in terms of momentum fluxes) of ∼10–100 km, phase speeds near zero and vertical wavelengths dictated by the local static stability and mean wind in the plane of wave propagation [Bacmeister et al., 1990b]. Indeed, Bacmeister [1993] has argued that mountain waves likely account for a large fraction of zonally averaged wave-induced force (wave drag) in the mesosphere, although this result is based on predictions of Lindzen-type parameterizations (see 42) and remains to be demonstrated observationally.

3.1.2. Convective Generation

[37] Although it has been known for decades that convection can excite gravity waves, it has only been in more recent years that observations and models have begun to characterize the waves generated by this source and the mechanisms within convection responsible for their generation. In fact, there is at the time of this review still considerable controversy and ongoing research aimed at understanding this wave generation mechanism.

[38] One of the difficulties in characterizing this source is its inherent intermittency. Observations of high-frequency waves in the stratosphere have shown a close correspondence with deep convective clouds (see Figure 4) [Sato, 1992, 1993; Alexander and Pfister, 1995; Sato et al., 1995; Dewan et al., 1998; McLandress et al., 2000; Alexander et al., 2000]. Observational estimates of momentum flux are fewer but show significant variability. Local magnitudes range up to ∼0.03–0.2 Pa [Sato, 1993; Alexander and Pfister, 1995; Alexander et al., 2000], and longer-term averages peak at Pa [Sato and Dunkerton, 1997; Sato et al., 1997; Vincent and Alexander, 2000].

[39] Waves generated by convection are not characterized by a single prominent phase speed or frequency as is the case for topographic waves. Instead, convection can generate waves throughout the full range of phase speeds, wave frequencies, and vertical and horizontal scales. The low-frequency waves, in particular, may be observed in the middle atmosphere at large horizontal distances from the convective source, making correlation with clouds or other indicators of convection more difficult. In the tropics, however, far from topography and regions of baroclinic instability, the occurrence of inertia-gravity waves has been linked to convection as the source [Pfister et al., 1986; Tsuda et al., 1994a; Karoly et al., 1996; Shimizu and Tsuda, 1997; Wada et al., 1999; Vincent and Alexander, 2000].

[40] Model studies of waves generated by convection (Figure 5) have greatly enhanced our understanding of these phenomena. Convection involves a time-varying thermal forcing associated with latent heat release that can interact with overlying stable layers and shear in complex ways that are not fully understood. Three simplified mechanisms have been proposed to describe convective generation: (1) pure thermal forcing, (2) an “obstacle” or “transient mountain” effect, and (3) a “mechanical oscillator” effect.

[41] For pure thermal forcing in the absence of strong shear the heating depth and the vertical wavelength of the gravity waves generated are surprisingly approximately equal [Alexander et al., 1995; Piani et al., 2000]. This relationship (demonstrated by Salby and Garcia [1987] for larger-scale waves) arises because the heating projects most strongly onto a wave with vertical wavelength twice the depth of the heating, but the wave is then refracted to half that vertical wavelength as it propagates across the twofold increase in buoyancy frequency at the tropopause (see equation (33)). Bergman and Salby [1994] used global cloud imagery at tropical latitudes to estimate the planetary-scale and long-wavelength, low-frequency gravity wave responses with this model, and the far-field effects of the waves generated have been extended through the mesosphere by Garcia and Sassi [1999]. The linear mechanism used in these studies has been validated for tropical planetary-scale waves, but the amplitudes are generally larger than those in corresponding nonlinear models [Manzini and Hamilton, 1993]. The gravity wave momentum fluxes derived from these cloud data sets are still highly uncertain. Pandya and Alexander [1999] further describe how the local mesoscale environment surrounding the storm can modify the smaller-scale gravity wave response in the stratosphere from a latent heat source. Observational evidence for the thermal forcing mechanism of gravity wave generation is given by McLandress et al. [2000] (Figure 6), although its importance relative to other mechanisms could not be quantified. Pure thermal forcing would be expected to generate an isotropic wave field, with anisotropies occurring primarily via background wind filtering effects.

[42] The obstacle effect was descibed by Clark et al. [1986] for short horizontal wavelength waves in the troposphere generated by boundary layer convection [Kuettner et al., 1987], but the ideas developed in their study have also contributed to understanding the larger horizontal scale waves in the middle atmosphere generated by convection. The obstacle effect can be thought of as analogous to topographic wave generation where the convective heating modifies the shape of isentropes at the bottom of a stable layer which is moving relative to the convective obstacle. Clark et al. [1986] found that both the pure thermal forcing and the obstacle effect produced gravity waves in the overlying stable layer but that the obstacle effect generated much larger amplitude waves. Pfister et al. [1993a, 1993b] explicitly modeled a linear version of this mechanism and found evidence for it in stratospheric aircraft observations over convection. Pfister et al. [1993b] also described how the time dependence of the obstacle (or transient mountain) affects the phase speed spectrum of the waves that are generated. Observational evidence for the obstacle effect has appeared in radiosonde analyses of low-frequency waves [Vincent and Alexander, 2000; Alexander and Vincent, 2000]. This mechanism is associated with an anisotropic wave spectrum with a preference for generating waves propagating opposite to the mean wind relative to the obstacle.

[43] Fovell et al. [1992] described the “mechanical oscillator” mechanism as a periodic and localized momentum source term that generates waves with a frequency equal to the oscillation frequency. This mechanism is not distinct from the time-varying thermal source when that source oscillates with a regular period [Clark et al., 1986; Alexander et al., 1995; Pandya and Alexander, 1999]. Lane et al. [2001] describe a similar mechanism acting in their 3–D model that had little or no shear in the forcing region, with some added constraints. In their conceptual model the parcel oscillates about its level of neutral buoyancy at the local tropospheric buoyancy frequency which limits this mechanism to generating only very high frequency gravity waves in the upper troposphere at altitudes above any lower minima that occur in the N profile. For the mechanical oscillator mechanism, anisotropies in the wave spectrum can result from background wind effects in a manner similar to the obstacle effect [Fovell et al., 1992].

[44] In summary, the three simplified excitation mechanisms that have been proposed may all be important, depending strongly upon the local shear and the vertical profile and time dependence of the latent heating. In reality, the three mechanisms are not distinct but coupled. However, one mechanism or another may serve to largely explain a set of observations with certain wind shear conditions or may serve to explain the generation of a certain class of waves, such as long vertical wavelength wave generation associated with deep convective heating.

3.1.3. Shear Generation

[45] The excitation of gravity waves by unstable shears has been studied for many years but remains one of the least quantified sources of gravity wave activity. A major challenge in these efforts was to account for the emergence of gravity waves that are propagating away from the shear layer on a timescale competitive with the most rapidly growing Kelvin-Helmholtz (KH) instability. This proved difficult for linear modes of instability for which the growth rate depends on the horizontal wave number. The subharmonic interaction, or pairing, of KH modes was proposed by Davis and Peltier [1979] to provide such excitation, but Fritts [1984b] showed pairing not to be an efficient mechanism when the exterior flow is stably stratified. By invoking a nonlinear interaction between KH and propagating modes, however, Fritts [1984b] and Chimonas and Grant [1984] were able to account for rapid excitation of propagating gravity waves. This “envelope” radiation is essentially excitation of a gravity wave by the packet-scale motions accompanying a packet of coherent KH billows evolving together in an unstable shear layer of finite horizontal extent. Two aspects of this mechanism have been assessed more recently in greater detail. Bühler et al. [1999] and Bühler and McIntyre [1999] examined wave radiation from the collapse of a mixed region due to KH billows of finite extent and concluded that this source could not be neglected in the momentum budget at greater altitudes. The mechanism envisioned by Bühler et al. [1999] is represented schematically in Figure 7.Scinocca and Ford [2000] reexamined the nonlinear radiation from spatially localized KH billows and concluded, like Fritts [1984b], that envelope radiation is a viable mechanism for gravity wave generation and likely an important contributor to the mesospheric momentum budget.

[46] Assuming envelope radiation to be the predominant shear excitation mechanism, we can infer characteristic horizontal scales of tens to ∼100 KH wavelengths or a few to tens of kilometers, with horizontal phase speeds comparable to the mean wind at the unstable shear layer. There is, nevertheless, some observational evidence suggesting that certain linear modes of shear instability can be excited [Mastrantonio et al., 1976; Lalas and Einaudi, 1976]. Under such circumstances the gravity wave scales are determined by the character of the shear flow and tend to be considerably larger than KH wavelengths.

3.1.4. Geostrophic Adjustment

[47] Most evidence for gravity wave excitation accompanying the restoration of balanced flow, commonly termed geostrophic adjustment, even though the balance need not be geostrophic, comes from theoretical or numerical treatments of adjustment processes. In such a process, an initial or evolving unbalanced flow relaxes to a new balanced state via both a redistribution of mean momentum, energy, and potential vorticity and a radiation of excess energy away as inertia-gravity waves. Such unbalanced initial flows can arise, for example, where cross-stream pressure gradients are varying along the fluid motion (requiring an acceleration or deceleration of the motion), accompanying frontal evolution and baroclinic instability, or in response to local body forces due to gravity wave dissipation and momentum flux divergence. The first of these was modeled in 2-D and 3-D by Fritts and Luo [1992] and Luo and Fritts [1993] as adjustments to unbalanced initial conditions and was found to lead to both new balanced 2-D or 3-D flows and a spectrum of inertia-gravity waves that owe their character to the geometry of the source perturbation. Observations of inertia-gravity wave activity by Hirota and Niki [1985] and Thomas et al. [1992] exhibiting downward phase progression above the jet stream and upward phase progression below the jet stream appear to provide some support for this conceptual model. Further support comes from the more recent study by Guest et al. [2000] who found both a high correlation of inertia-gravity wave observations with a synoptic pattern having a jet stream nearby and a tendency for reverse ray tracing to suggest the jet stream as the wave source. Our understanding of such adjustment processes is nevertheless poor at this time and will benefit from further research efforts.

[48] Frontogenesis and baroclinic instability were found by O'Sullivan and Dunkerton [1995], Griffiths and Reeder [1996], and Reeder and Griffiths [1996] to yield similar results, with both adjusted mean and waves scales largely defined by the character of the unbalanced flow. An example of the inertia-gravity waves emitted accompanying adjustment of an evolving baroclinic flow is shown in Figure 8 [O'Sullivan and Dunkerton, 1995]. As discussed above, gravity waves are seen to arise preferentially in regions of strong flow deceleration.

[49] Gravity waves arising from local body forces due to dissipating gravity waves were considered theoretically by Zhu and Holton [1987] and more recently by Vadas and Fritts [2001]. In all cases, the response is a combination of an adjusted mean (with redistributed momentum or potential vorticity), an altered mean thermal structure, and radiated inertia-gravity waves ensuring energy conservation in the absence of dissipation. The study by Vadas and Fritts [2001] considered a temporally evolving body force and showed (1) that the adjusted mean state depends on the momentum impulse and forcing geometry but not on the forcing duration and (2) that the radiated inertia-gravity waves are constrained by both the geometry and the temporal structure of the body forcing. Thus, while both long- and short-duration forcing (having the same geometry and momentum impulse) induce the same balanced mean flow, their gravity wave responses are quite different, with long-duration forcing strongly suppressing gravity waves having periods shorter than the forcing duration. The adjustment processes discussed here are also closely analogous to the radiation of gravity waves by a collapsing mixed layer due to shear instability examined by Bühler et al. [1999] and discussed above, though the source here is due to an addition, rather than a redistribution, of momentum.

[50] Scales of inertia-gravity waves arising from adjustment or body forcing processes are imposed by the source spatial scales and further constrained by the source timescales. Typical vertical scales are a few kilometers or more, with horizontal scales along the direction of propagation of ∼10–100 times larger. Wave periods likely vary from an hour or so out to the inertial period. In cases where the source is elongated in one direction horizontally, the predominant direction of propagation of the excited inertia-gravity waves is normal to the source axis.

[51] A comparison of gravity wave velocity and temperature variances associated with various sources computed using GASP data near the tropopause is shown in Figure 9 [Fritts and Nastrom, 1992]. These variances, for 64- and 256-km flight segments, suggest that topography, jet stream shears, and convective and frontal activity all contribute substantially to gravity wave excitation where these sources are prevalent.

3.1.5. Wave–Wave Interactions

[52] Nonlinear wave–wave interactions are arguably one of the more important mechanisms in energy exchanges, amplitude constraints, and spectral evolution for gravity waves in the middle atmosphere. Their roles and importance are not universally acknowledged, however, and there remains much to be done in quantifying their dynamics and effects.

[53] As a source of gravity waves the atmospheric community has taken guidance from oceanographers and attributed energy transfers to resonant and nonresonant interactions of various forms. There has been a long debate over the efficiency of such interactions and the validity of the various approaches in the oceanographic and atmospheric communities. Yet laboratory experiments and direct numerical simulations demonstrate that specific interactions, e.g., the parametric subharmonic instability (or PSI, see 35), can be efficient and robust under a variety of circumstances [McEwan, 1971; McEwan and Robinson, 1975; Mied, 1976; Dunkerton, 1987; Thorpe, 1994; Vanneste, 1995; Staquet and Sommeria, 1996]. Indeed, Klostermeyer [1984, 1990] has argued for evidence of PSI in radar measurements of the middle atmosphere. Thus wave–wave interactions are likely an important source of waves at a range of scales not directly linked to source scales at lower altitudes. Because of their roles in spectral energy transfers and amplitude constraints, a more complete discussion of wave-wave interactions is deferred until 35.

3.2.1. Upward and Downward Propagation

[55] It is useful for many purposes to distinguish what fractions of the gravity wave spectrum are propagating upward and downward. Two approaches have usually been used to do this. One employs 2-D frequency–vertical wave number spectra or direct measurements of phase progression for individual waves to make the distinction, while a second employs a rotary spectral analysis of individual profiles. Each method is hampered by certain assumptions but yields insightful, if not precise, results.

[56] The 2-D spectral approach employed by Fukao et al. [1985], Fritts and Cho [1987], and Wilson et al. [1991a] assumes that observed and intrinsic frequencies are the same, thus confusing vertical directionality for motions having intrinsic phase speeds comparable to and opposed to mean winds and erring in the assignment of wave frequency. These are not generally serious errors when mean winds are not extreme because the total wave variance at small intrinsic phase speeds (hence small vertical scales and amplitudes) is not large. The forms of the observed frequency spectra of horizontal velocity and temperature (slopes typically near −5/3) do not change dramatically for mean winds smaller than the intrinsic phase speeds of the most energetic waves [Fritts and VanZandt, 1987], however, while there is relatively less variance for for which Doppler-shifting effects are larger. These and other observations suggest that the fraction of upward propagating gravity wave energy is much greater than 1/2 and that the upward propagating fraction is likely an underestimate because of uncertainties over wave frequencies [Vincent, 1984b; Fukao et al., 1985; Fritts and Chou, 1987; Wilson et al., 1991a; Lintelman and Gardner, 1994; Gavrilov et al., 1996; de la Torre et al., 1999].

[57] The second method, rotary spectral analysis, was first employed by Vincent [1984b] and is related to Stokes analysis used to assess the degree of polarization of the motion field [Vincent and Fritts, 1987; Eckermann and Vincent, 1989; Eckermann and Hocking, 1989; Barat and Cot, 1992; Nakamura et al., 1993a; de la Torre et al., 1999; Zink and Vincent, 2001a]. Rotary spectral analysis decomposes all oscillatory motions in the vertical into a superposition of upward and downward propagating motions assumed to be circularly polarized. This assumption causes a single upward propagating inertia-gravity wave with amplitude A and intrinsic frequency (having elliptical polarization) to be misidentified as a sum of an upward propagating wave of amplitude and a downward propagating wave of amplitude . As , this yields and because the dynamics are then consistent with the assumptions. However, for the inferred (and errant) downward propagating fraction of the total variance of approaches 1/2 as (or as ) and is entirely errant [Eckermann and Vincent, 1989]. Thus the rotary spectral analysis likewise leads to a conservative estimate of the upward propagating fraction of gravity wave energy. Effects of wave superposition, while not always significant, may also influence the results of a Stokes analysis and contribute to a lack of precision in partitioning of wave energy between upward and downward propagating components [Eckermann and Hocking, 1989]. Studies that use the technique of ellipse fitting to hodographs [Tsuda et al., 1994a; Hamilton and Vincent, 1995; Shimizu and Tsuda, 1997] yield unambiguous estimates of the upward and downward propagating fractions, at least when a single dominant inertia-gravity wave is present in the profiles.

3.2.2. Vertical Propagation Versus Ducting

[58] Another issue that has arisen recently because of its implications both for estimates of gravity wave fluxes of heat and momentum and for the parameterization of gravity wave effects in large-scale models is the occurrence of gravity wave ducting. Waves that propagate vertically (with ) but having reflection (or turning) levels above and below are trapped or ducted. From the dispersion relation, equation (24) or equation (33), we see that variations in the thermal structure, as measured by the buoyancy frequency N, influence waves having any direction of propagation in the same manner, whereas the wind profile affects waves having different propagation directions very differently. In general, wind maxima in a particular direction favor ducting at that level (and in that direction) because of the relative maxima they imply in , while maxima of favor ducting at this level for all propagation directions [Chimonas and Hines, 1986; Fritts and Yuan, 1989a]. Observational evidence of wind and thermal () ducts was obtained by Isler et al. [1997] and Walterscheid et al. [1999], while Taylor et al. [1995] and Mitchell and Howells [1998] inferred ducting from limited vertical wave extent. In all these cases, gravity waves having horizontal wavelengths of ∼20 km or less were judged to have a high probability of being ducted by the wind and/or thermal structures of the atmosphere. The probability of ducting remains finite but decreases as horizontal wavelengths increase [Swenson et al., 2000].

[59] Gravity wave structure within a duct typically exhibits very little vertical phase variation across the duct and is evanescent on either side, though more complex wave structures are also possible. Thus typical motions are largely in phase and vertical, have intrinsic frequencies near N, may have only a gradual decay of wave amplitude away from the duct, and have little or no associated momentum flux. Yet because of their structure such motions may make large contributions to vertical velocity, temperature, and airglow intensity variances.

[60] Despite our appreciation of the implications of gravity wave ducting, there have been several efforts to associate airglow intensity and/or temperature variances with gravity wave momentum fluxes (to which ducted waves contribute nothing, according to linear theory). These efforts by Swenson et al. [1999] and Gardner et al. [1999a] have been critiqued by Fritts [2000] and are flawed because of their neglect of ducting. Swenson et al. [2000] likewise recognized the lack of sensitivity to ducting behavior in earlier formulations and discussed the implications of the more complete dispersion relation, reaching similar conclusions to those of Isler et al. [1997] and Walterscheid et al. [1999] regarding susceptible horizontal wavelengths. The implications of these studies are that turning levels and ducting are important processes that impact gravity wave structure and inferred effects where wind and thermal variations are significant, particularly for shorter horizontal wavelengths.

3.2.3. Gravity Waves Versus Two-Dimensional Turbulence

[61] While quasi-2-D turbulence (hereafter 2DT, also called geostrophic or stratified turbulence under related assumptions) clearly makes contributions to atmospheric dynamics and velocity variances at some altitudes, the role it plays in the middle atmosphere appears to be small, at least for scales well below synoptic and planetary scales (∼500 km) [Nastrom and Gage, 1985]. Gage [1979] and Lilly [1983] have advocated 2DT as an explanation for the −5/3 power law in lower stratospheric horizontal wave number spectra, and Lilly [1983] has suggested that deep convection may be a viable source for 2DT. However, subsequent assessments using radar and balloon data by Smith et al. [1985], Vincent and Eckermann [1990], and Eckermann and Vincent [1993] suggested that only a small fraction of horizontal velocity variance could not be attributed to gravity waves. Increasing energy and momentum fluxes (per unit mass) and energy dissipation rates with altitude appear, at first sight, to imply a reduced role at greater altitudes. Without a more complete understanding of middle-atmosphere turbulence dynamics and energetics, however, it would be a mistake to assume that no strong middle-atmosphere sources of 2DT are present. This is particularly true given the potential role of gravity wave–vortical mode interactions as wave amplitudes increase [Dong and Yeh, 1988; Yeh and Dong, 1989], the departures of ratios of kinetic to potential energy from those predicted by gravity wave theories [Tsuda et al., 1991; Nastrom et al., 1997; de la Torre et al., 1999], and the increasing scales and energies of turbulence with altitude (see 19). We must be careful in attributing kinetic to potential energy ratios as evidence against a purely gravity wave interpretation, however, because such departures from theory can also arise from excess gravity wave energy near the inertial frequency [Thompson, 1978; Sato et al., 1999].

5.1.1. Remote Sensing Measurements From Space

[85] Observations from satellite platforms hold great promise for providing the kind of global coverage at frequent time intervals that is needed to understand gravity wave variability. However, gravity waves can have quite small vertical and horizontal scales and short periods. They can therefore be challenging to observe remotely from space. New techniques have emerged, yet each has been capable of observing only some fraction of the gravity wave spectrum that may be present in the atmosphere, while missing other portions of the spectrum.

[86] Infrared limb-viewing profiles like the Limb Infrared Monitor of the Stratosphere (LIMS) instrument [Gille and Russell, 1984] retrieve temperature profiles from 15 to 60 km at high vertical resolution (∼1.5 km). However, line-of-sight (LOS) integration effects and the spacing between profiles limit these data to observation of only longer horizontal wavelengths. Fetzer and Gille [1994] retrieved global fields of gravity wave temperature perturbations by subtracting Kalman-filtered temperature fields from unfiltered temperature profiles. The Kalman-filtered data included planetary waves with zonal wave number . The difference field, assumed to be gravity wave perturbations, included information on waves with vertical wavelengths of ∼6–50 km and horizontal wavelengths longer than ∼200 km. (Note that aliasing from shorter unresolved horizontal wavelengths was evident, but these represented a small fraction of the total gravity wave variance observed [Fetzer and Gille, 1994].) Fetzer and Gille [1994] noted that the temperature variances derived from these data are likely dominated by low-frequency inertia-gravity waves. Maps of the LIMS gravity wave temperature variance (Figure 11a) as a function of latitude and height showed a peak near the equator in the lower stratosphere below 20 km and a peak in the winter stratospheric jet above 20 km. Seasonal variation above 40 km at the equator related to the semiannual oscillation (SAO) was also evident in the gravity wave variance.

[87] Maps of gravity wave temperature variance have been similarly derived from Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere (CRISTA) data [Preusse et al., 1999]. These data give temperature profiles at finer horizontal resolution and slightly better vertical resolution than LIMS. CRISTA's altitude range varies with observational mode but can extend from ∼20 to 80 km. The CRISTA data were used to infer that topography was the source for waves observed in the stratosphere from space [Eckermann and Preusse, 1999; Preusse et al., 2002]. Global maps of the gravity wave temperature variance show similar features as those from LIMS (Figure 11b).

[88] Tsuda et al. [2000] derived global maps of gravity wave potential energy derived from GPS occultation data temperature profiles [Rocken et al., 1997]. These observations include gravity waves with vertical wavelengths of ∼2–10 km in the 15- to 40-km altitude range. The LOS integration path length is also ∼200 km, so the observed wave characteristics are similar to LIMS and CRISTA. The gravity wave energy shows a similar equatorial maximum below ∼25 km and midlatitude winter maxima at higher altitudes. At lower latitudes (), the gravity wave energy was particularly enhanced over regions of active convection. At midlatitudes, gravity wave energy was observed to be larger over continents than over oceans.

[89] Observations from the Microwave Limb Sounder (MLS) have been used to derive global maps of gravity wave temperature variance in the stratosphere and mesosphere [Wu and Waters, 1996a, 1996b]. These data include only gravity waves with long vertical wavelengths 12 km because of the deeper weighting functions associated with microwave profilers compared to infrared profilers. One observation mode (the scanning mode) reportedly included gravity waves with horizontal wavelengths of ∼60–150 km [Wu and Waters, 1996a], while another (the tracking mode) included gravity waves with horizontal wavelengths up to 1000 km. McLandress et al. [2000] noted that the integrated variance observed by the two modes is similar, suggesting that aliasing effects may be extending the range of wavelengths contributing to the variance in data from the scanning mode. Wu and Waters [1997] note that most of the observed waves propagate opposite to the direction of the wind. This, together with the long vertical wavelength restrictions, suggests these waves have much higher intrinsic frequencies than those observed via the other space-based observations described above. The global maps show peaks in variance at middle to high latitudes in winter, with a minor peak at subtropical latitudes in summer. Variances at the equator were indistinguishable from noise [McLandress et al., 2000]. Longitudinal variations suggest deep convection as a source of the waves observed in the summertime and topography as the source for waves at southern midlatitudes in winter [McLandress et al., 2000] (see also Figure 6).

5.1.2. Radiosonde Profiles

[90] Observations of temperature and horizontal wind profiles from radiosondes have provided information on short vertical wavelength gravity waves in the lower stratosphere below ∼25 km. Several studies have inferred latitudinal, seasonal, and interannual variations in gravity wave activity in these data. The frequency of radiosonde launches rarely exceeds twice per day, so gravity wave perturbations are generally obtained by subtracting a low-order polynomial fit to the profiles and assuming the perturbations as a function of height are caused by gravity waves. In studies near the equator, however, where the inertial period is very long, radiosonde data have also been analyzed in the time domain for information on 1- to 3-day period oscillations as inertia-gravity waves [Sato et al., 1994; Sato and Dunkerton, 1997].

[91] Radiosonde profiles of horizontal wind plotted as hodographs generally show rotation indicative of upward propagating inertia-gravity waves [Ogino et al., 1995; Vincent and Alexander, 2000]. In fact, there is abundant evidence from analyses of radiosonde horizontal wind and temperature profiles that the perturbations in these vertical profiles are dominated by very low intrinsic frequency inertia-gravity waves, specifically those with frequencies between ∼1 and 3f [Vincent et al., 1997; Vincent and Alexander, 2000; Zink and Vincent, 2001a].

[92] Seasonal variations at northern midlatitudes and high latitudes show a maximum in gravity wave activity in winter and minimum in summer [Yoshiki and Sato, 2000; Kitamura and Hirota, 1989]. At very high northern latitudes, significant gravity wave variances were observed only in the absence of mountain wave critical levels [Bacmeister et al., 1990b; Worthington and Thomas, 1996; Whiteway and Duck, 1996; Whiteway et al., 1997]. At midlatitudes in the Southern Hemisphere, observations also show peak wave activity in winter [Allen and Vincent, 1995; Zink and Vincent, 2001a]. At Antarctic latitudes a maximum in spring has instead been observed [Allen and Vincent, 1995; Yoshiki and Sato, 2000].

[93] At the tropical latitudes of northern Australia, Indonesia, and the Indian Ocean a seasonal cycle in wave activity has been observed with a maximum in December to February [Allen and Vincent, 1995; Vincent and Alexander, 2000]. This is a period of intense deep convection in that part of the world. Waves observed in radiosonde data from that area also often tend to have long horizontal wavelengths, ∼1000 km [Tsuda et al., 1994a; Shimizu and Tsuda, 1997; Vincent and Alexander, 2000]. Propagation directions of gravity waves observed in radiosonde data at low latitudes are influenced by the winds of the quasibiennial oscillation (QBO). An excess of eastward propagating waves has been observed most prominently during periods with strong westward winds in the lower stratosphere and eastward shear aloft [Murayama et al., 1994; Tsuda et al., 1994a; Shimizu and Tsuda, 1997; Sato and Dunkerton, 1997; Vincent and Alexander, 2000]. Cadet and Teitelbaum [1979] observed the analogous situation in the opposite phase of the QBO: westward propagating gravity waves in westward QBO wind shear.

[94] Latitudinal variations are also apparent in gravity waves observed in radiosonde profiles. The data show a trend toward increasing wave energy equatorward of ∼30° [Kitamura and Hirota, 1989; Allen and Vincent, 1995; Ogino et al., 1995]. Note that a similar equatorial maximum was observed in the GPS and IR limb sounding satellite observations in the lower stratosphere described above.

5.1.3. Rocket Soundings

[95] Rocket soundings of temperature and horizontal wind have been used to infer seasonal and latitudinal variations in gravity wave activity in the stratosphere below ∼60 km at ∼1-km resolution. Rocket profiles from launches in the 1970s and 1980s at sites in the Northern Hemisphere and in the tropics have provided information on the seasonal and latitudinal variations in gravity wave activity in the stratosphere [Hirota, 1984; Hirota and Niki, 1985; Hamilton, 1991; Eckermann et al., 1995].

[96] The analysis method has varied between different studies, but the most common procedure is to subtract a background profile from the data and to assume that smaller-scale perturbations are gravity waves. In general, the rocket sounding analyses have shown a seasonal cycle in wave activity at high latitudes (∼40°–80°N) with a maximum in winter, similar to results of radiosonde analyses. Hodograph or rotary spectral analyses of rocket profiles have also shown a dominance of upward propagation, again similar to the radiosondes. At low latitudes, different analyses gave different seasonal cycles in wave activity, suggesting a sensitivity of this result to the vertical scales of waves considered. Mean kinetic and potential energies in the 20- to 40-km altitude range both showed increases at latitudes <30° [Eckermann et al., 1995]. A preference for eastward propagation at sites within 10° of the equator has been noted as a possible indication that the wave activity at these sites is dominated by equatorial Kelvin waves [Hamilton, 1991; Eckermann, 1995b; Holton et al., 2001].

5.1.4. Lidar

[97] Lidar observations of middle atmosphere gravity waves have been restricted to a few specific sites. Among these, only a few have operated for long enough periods to provide information on seasonal variations in gravity wave activity [Wilson et al., 1991b; Marsh et al., 1991; Mitchell et al., 1991; Whiteway and Carswell, 1995]. These are all Rayleigh lidar studies of gravity wave temperature or density perturbations at midlatitude sites. The data include gravity waves with vertical wavelengths as short as 1 km at altitudes in the upper stratosphere and lower mesosphere. The data have been analyzed both for vertical variations as well as in the time domain, though these observations are limited to nighttime only. The dominant pattern that has appeared in monthly mean averages of these data is a seasonal cycle with maximum in winter and minimum in summer: the same annual cycle that is observed in radiosonde and rocket sounding data at midlatitudes.

5.1.5. Radar

[98] Radar observations of gravity wave wind perturbations can be made in the lower stratosphere and in the mesosphere and lower thermosphere. Many of these observations have been made at a limited number of sites and on a short-term campaign basis. We focus here on studies that have compiled observations over long enough time periods to study seasonal variations. These long-term studies have, to date, all been at midlatitude sites.

[99] Analyses of gravity wave observations in the mesosphere in the ∼65- to 90-km region are generally reported for different ground-relative frequency bands for waves with periods ranging from 5 min to 24 hours. Gravity wave kinetic energy as well as variances in each of the three component wind directions have shown a semiannual variation at these altitudes with maxima in summer and winter and minima in spring and fall [Meek et al., 1985; Vincent and Fritts, 1987; Tsuda et al., 1990a; Manson and Meek, 1993; Nakamura et al., 1996]. These analyses included observations at both Northern and Southern Hemisphere sites, and the results show similar gravity wave activity in the two hemispheres. A seasonal asymmetry in the occurrence of polar mesosphere summer echos (PMSE) and the ice particles that cause them has been observed. Specifically, there is an absence of PMSE in the Southern Hemisphere that may be a result of warmer temperatures in the south due to a weaker mean meridional circulation and corresponding weaker gravity wave forcing [Balsley et al., 1995; Woodman et al., 1999]. Some mesopause temperature measurements support this idea [Huaman and Balsley, 1999] while others suggest similar thermal structures at high latitudes in each hemisphere [Lübken et al., 1999]. A recent chemical-dynamical model study [Siskind et al., 2003] supports the idea of hemispheric asymmetries in gravity wave forcing and an asymmetric narrow warm layer at the mesopause that may be difficult to resolve with some measurement techniques.

[100] Observations of horizontal and vertical wind covariance (momentum flux per unit density) in the mesosphere have shown an annual cycle, positive (i.e., eastward) in summer and negative (i.e., westward) in winter, so the flux tends to have the opposite sign as the winds at these altitudes [Tsuda et al., 1990b; Manson and Meek, 1993; Nakamura et al., 1993c, 1996]. The anticorrelation between flux direction and the background wind direction is a condition observed even more generally in shorter-term data records [Vincent and Reid, 1983; Reid and Vincent, 1987; Fritts and Vincent, 1987; Reid et al., 1988; Fritts and Yuan, 1989b; Meyer et al., 1989; Wang and Fritts, 1990; Nakamura et al., 1993c]. The peak magnitudes of monthly mean fluxes (per unit mass) at mesospheric heights have been reported at ∼1–4 m² s⁻². Reports of meridional fluxes have been variable, so no repeatable seasonal cycle is known, but reported magnitudes are often comparable to or slightly larger than the zonal fluxes.

[101] Reports of seasonal changes in the anisotropy of wave propagation directions have varied among different studies [Manson and Meek, 1988; Nakamura et al., 1993d]. Maekawa et al. [1987] and Fan et al. [1991] inferred critical-level wind filtering as the mechanism responsible for anisotropies observed in the summer mesosphere wind field using the SOUSY and Arecibo VHF radars, respectively. Inferred propagation directions also depend greatly on the characteristics of gravity waves being observed. Vincent and Fritts [1987] and Nakamura et al. [1993b] observed a preference for meridional propagation of the dominant wave motions under winter conditions. Nakamura et al. [1993b] noted a tendency for propagation of the dominant motions toward either the NE or SW under summer conditions. Taylor et al. [1998] and Nakamura et al. [1998] observed propagation preferentially toward the NE for waves of shorter horizontal scales under summer conditions. Finally, Nakamura et al. [1993a] observed similar character in propagation directions at Adelaide and the MU radar at Kyoto in summer but with more isotropic propagation at Adelaide and more zonal propagation at Kyoto in winter.

[102] Observations in the stratosphere of the seasonal cycle of gravity wave activity measured by radar have been reported only for the MU radar near Kyoto in Japan for altitudes below 24 km [Tsuda et al., 1994b; Murayama et al., 1994; Sato, 1994]. These have shown an annual cycle in gravity wave variances in all three wind components with minima in summer and maxima in winter to early spring. Sato [1994] and Murayama et al. [1994] also reported zonal and meridional momentum fluxes. The zonal component showed a clear annual cycle with large negative values ∼ − 0.1 m² s⁻² in winter and near zero in summer. The meridional component also tended to be negative but with smaller magnitudes and no clear annual cycle. The observed wave characteristics at this site have been generally consistent with topographic gravity wave generation. Specific campaigns have, however, also observed waves associated with convection (see 8).

5.1.6. Aircraft

[103] Measurements of wind and temperature fluctuations by aircraft flying in the stratosphere have yielded information on the horizontal wavelength spectrum of gravity waves [Nastrom et al., 1987; Bacmeister et al., 1996] (see 19). Some aircraft studies have related gravity wave characteristics in the stratosphere to the wave sources below, including mountains [Gary, 1989; Bacmeister et al., 1990a, 1990b; Nastrom and Fritts, 1992; Leutbecher and Volkert, 2000] and convection and frontal systems [Fritts and Nastrom, 1992; Pfister et al., 1993a, 1993b; Alexander and Pfister, 1995; Alexander et al., 2000]. Several of these studies have provided in situ determination of gravity wave momentum fluxes. However, aircraft observations have been too limited in their duration to provide much information about the gravity wave climatology directly. Instead, they have contributed to our understanding of gravity wave variability by providing important quantitative constraints for theoretical and model studies of gravity wave sources (see 8 and Figure 9).

5.3.1. Sources

[109] There is convincing evidence for mountain waves in the stratosphere with fair quantitative agreement between high-resolution model studies and observations. Parameterized mountain waves, however, have a large effect in global models, and improved constraints on their properties and intermittency in occurrence are still needed. There is convincing evidence for convectively generated gravity waves in the stratosphere with qualitatively good agreement between observations and models. Observations that include both detailed information about the storm characteristics and about the waves that are generated are hard to find but are needed to make our understanding of convectively generated gravity waves more quantitative. There is evidence for low-frequency waves emanating from frontal zones in the jet stream and high-resolution model studies describing inertia-gravity wave generation under these conditions. Observational evidence detailed enough for a quantitative understanding of this wave source is lacking. A detailed understanding of the generation mechanisms and the resulting wave characteristics is needed for accurate description of gravity wave effects in global models.

5.3.2. Variability

[110] Observations of short vertical wavelength gravity waves in the lower stratosphere show increases in variance at latitudes <30°. This has been interpreted as evidence for very low intrinsic frequency waves at low latitudes that are excluded from higher latitudes by the variation in f and that have intermittent sources in time, such as would be expected from convection. Very close to the equator, at latitudes <10°, these measurements could also include contributions from short vertical wavelength Kelvin waves. Also in the tropics, there is evidence for geographical variations in long vertical wavelength gravity waves with high intrinsic phase speeds that are closely tied to deep convection. These observations support model studies that show how deep heat sources generate long vertical wavelength waves.

[111] The evidence for global variations in gravity wave sources from climatologies of gravity wave activity is currently weak because many of the most prominent variations observed can be explained by background atmosphere effects without any seasonal or geographical variations in sources. Yet we have clear evidence for the importance of topography, convection, and frontal generation of gravity waves, and these sources must clearly vary seasonally and geographically. The existing gravity wave climatologies that describe monthly mean gravity wave kinetic and potential energy have largely served their purpose. They have allowed us to test the linear theory of gravity waves and their interaction with the background atmosphere.

[112] Observational gravity wave studies in the future must begin to define the deviations in gravity wave activity from these climatological patterns. Observations and modeling tools must be used together to separate these variations due to the background atmosphere and observational biases from variations due to sources. Quantitative measures of the intermittency in wave activity are needed along with any monthly mean quantity since it is important to define whether a monthly mean value represents intermittent large bursts of wave activity or more constant, smaller-amplitude wave activity.

[113] Satellite observations hold the most promise for providing the kind of geographical and temporal coverage needed to understand gravity wave variability. However, inherent observational limitations mean that continued in situ measurements from aircraft and high-resolution ground-based profiling instruments will still be needed to understand the full spectrum of gravity waves present in the atmosphere.

6.2.1. Gravity Wave Breaking

[133] 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.

[134] The first 3-D numerical studies of gravity wave instability dynamics in the atmosphere were performed by Andreassen et al. [1994], Fritts et al. [1994], and Isler et al. [1994]. A parallel study examining similar dynamics in an oceanic context was performed by Winters and D'Asaro [1994]. 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].

[135] 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).

[136] 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. [1998] 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 [1995] and Andreassen et al. [1998] for further details). The volumes are viewed from below, with the streamwise direction and wave propagation to the right.

[137] 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].

[138] 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]).

[139] 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 35 and 40). 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. [2002] 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 [1996]. 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].

6.2.2. Kelvin-Helmholtz Instability

[140] 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 [1985]. 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 [1984] and Fritts and Rastogi [1985]. 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 [1996] and Sutherland [2001] at higher intrinsic frequencies.

[141] 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.

[142] 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.

[143] 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 [2000].

[144] 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.

[145] 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 [1999]. 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. [1995] (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.

6.2.3. Synthesis and Other Instability and Large-Amplitude Processes

[146] 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 [1991] 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 [1991] and Lombard and Riley [1996] also identified transverse modes of instability at finite amplitudes below those required for local convective or dynamical instability.

[147] The links between instabilities at small and large wave amplitudes were further clarified and generalized by Sonmor and Klaassen [1997] 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 [1997] further linked the “branch-C” instability of Yeh and Liu [1981] to the resonant elastic scattering and induced diffusion interactions originally identified by McComas and Bretherton [1977] 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 36) 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.

[148] A further class of instability, “oblique instabilities” (having nonzero streamwise and spanwise wave numbers), was found by Lombard and Riley [1996] 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.

[149] 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 [2001] 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 [1984] and Sutherland [1999, 2000].

[150] 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].

7.2.1. Commonalities and Differences

[166] All parameterizations assume a set of gravity wave properties, specified either as a continuous wave spectrum or as a collection of discrete waves at some source altitude. The spectral properties of the sources may vary considerably among the different parameterizations, and except for topographic waves (input at the ground), the source altitude for nonstationary waves will additionally vary even among different applications of the same parameterization. Differences in the specification of gravity wave sources among parameterizations may be responsible for some large part of the differences in their effects seen in models. However, no controlled comparisons have yet been made that address this issue.

[167] All parameterizations are one-dimensional in that they assume the waves considered propagate only vertically and that only vertical variations in the background atmosphere influence wave propagation. All parameterizations use linear theory in the absence of dissipation and use the midrange frequency approximation to the linear dispersion relation (32) for at least some portion of the calculation. Total internal reflection is included via equation (30) in one [Alexander and Dunkerton, 1999], but this is generally only important for gravity waves with horizontal wavelengths shorter than ∼50 km.

[168] Aside from gravity wave source specification, the main differences among parameterizations reside in their treatment of nonlinearity and mechanisms of wave dissipation. These differences result in different vertical variations in gravity wave effects.

7.2.2. Treatment of Nonlinearity and Wave Dissipation

[169] Most global model studies that have included gravity wave effects parameterized nonlinear effects using a scheme related to that proposed by Lindzen [1981] and formalized by Holton [1982]. These “Lindzen-type” parameterizations specify some discrete set of wave properties at a source level, use the midrange frequency approximation to calculate wave propagation with height, and then use the linear convective instability threshold to determine where wave dissipation will occur. Wave dissipation continues above the unstable level via a saturation condition (see Fritts [1984a] for a review). This basic formula is the basis of the widely used mountain wave parameterizations by Palmer et al. [1986] and McFarlane [1987], who added a Froude number condition for finding the altitude regions where wave saturation would occur. There are also nonstationary wave analogs to these mountain wave parameterizations [e.g., Kiehl et al., 1996; Norton and Thuburn, 1999].

[170] Alexander and Dunkerton [1999] used Lindzen's convective instability criterion but assumed total wave breakdown at that level rather than employing the wave saturation assumption. This simplification reduced computation time, allowing a treatment of a detailed spectrum of waves rather than the typical small set of 5–10 waves employed in Lindzen-type schemes. They also made explicit the separation of the parameters describing local wave momentum flux magnitude (which determines breaking levels) and net wave momentum flux (which is generally smaller if there is intermittency in the wave occurrence). The parameterization by Norton and Thuburn [1999] instead locked these two parameters together so that smaller fluxes gave higher breaking levels and vice versa. This is appropriate for mountain waves where amplitude and net flux are naturally linked because the intermittency in the source is modeled directly via topography and surface wind variations but is inappropriate for nonstationary wave sources with globally averaged input properties.

[171] Warner and McIntyre [2001] used the midrange frequency approximation to the linear dispersion relation and a saturation condition to limit wave amplitudes to a “quasi-saturation” spectrum. They developed a spectral analogue to the Lindzen [1981] saturation criterion and simplified the computation with a three-part spectrum approximation. Short horizontal wavelength waves in their source spectrum [Warner and McIntyre, 1996] carry significant momentum flux, so their neglect of wave reflection is a potentially serious problem. Fritts and VanZandt [1993] and Fritts and Lu [1993] used saturation theory and empirical constraints on the gravity wave spectrum to describe the spectrally integrated energy density and momentum flux as functions of height.

[172] Hines [1991, 1997a, 1997b] used linear theory and the midrange frequency approximation to describe wave propagation with height but treated dissipation as resulting from wave-wave interactions which cause Doppler-shifting and refraction of portions of the wave spectrum to vertical wavelengths shorter than a prescribed cutoff wave number where dissipation is assumed to occur. Medvedev and Klaassen [1995, 2000] instead treated dissipation via the nonlinear diffusion ideas of Weinstock [1982, 1990]. Both of these mechanisms have, however, been questioned in the recent literature (see 35).

[173] Note that although many of the spectral gravity wave parameterization schemes [Fritts and Lu, 1993; Medvedev and Klaasen, 1995, 2000; Hines, 1997a, 1997b; Warner and McIntyre, 1996, 2001] have been constrained to reproduce the “universal” gravity wave energy spectrum observed at high m (19), these parameterizations can produce very different results (see 53). This leads to the natural conclusion that the shape and magnitude of the spectrum at high m alone do not sufficiently constrain parameterizations of gravity wave effects in the middle atmosphere.

[174] For a parameterization to be effective at both forcing the stratospheric QBO winds and driving the mesopause wind reversals, dissipation of nonstationary waves must occur for increasingly large intrinsic phase speeds and increasingly long vertical wavelengths at increasing altitudes. In other words, waves may dissipate fairly close to their critical levels in shear zones in the lower stratosphere but must dissipate far from their critical levels in the upper mesosphere. Intrinsic phase speed and vertical wavelength are related via equation (33). The result is that for nonstationary waves the force will tend to accelerate the mean wind (the force will have the same sign as the wind) in the lower stratosphere but decelerate the mean wind (the force will have opposite sign as the wind) in the upper mesosphere. (Note that this trend should not hold for the large amplitude mountain waves with phase speeds that have been observed. Such waves can break and cause decelerations at all levels.) The potential for gravity waves to accelerate the mean wind is one reason to avoid the term “gravity wave drag” since this is an example of “gravity wave push.”

[175] The vertical variation in wave dissipation described above is a natural consequence of linear instability theory for small amplitude waves propagating through exponentially decaying atmospheric density and the observed zonal mean middle atmosphere winds [Alexander and Rosenlof, 1996] (see also 19). It can thus occur using any of the parameterizations above based on convective instability for some range of source spectrum properties. That includes the Lindzen-type, Warner and McIntyre [1996, 2001], and Alexander and Dunkerton [1999] parameterizations. It can also arise naturally in the Hines [1997a, 1997b] parameterization for wave dissipation because the amplitudes of the waves that drive the dissipation in this mechanism grow with altitude. They can thus shift waves with larger and larger intrinsic phase speeds (and vertical scales) to the prescribed dissipation scale as a function of height. The Hines dissipation scale is further allowed to vary with height adding another means of tuning this effect.

[176] Vertical gradients of gravity wave energy and momentum fluxes give rise to important terms in the energy and momentum conservation equations in global models (see 47). All of the parameterizations will drive the deceleration of the winds in the mesosphere, but only a few have been shown to be effective at driving QBO circulations in global models. In addition to mean flow forcing effects of parameterized gravity waves, vertical mixing effects in the thermal energy and constituent transport equations are also important in the mesosphere of global models. Global model studies of both these effects are described in 47.

8.1.1. Zonal Mean Momentum Balance Studies

[180] The first method estimates the gravity wave contribution to the wave driving through examination of the zonal mean momentum balance. The difference between the total momentum forcing and the resolved EP flux divergence contribution is presumed to be due to gravity wave driving. The total force is calculated using global satellite observations of temperature and the important radiatively active gases (O₃, H₂O, etc.) input to a radiative transfer model to estimate the zonal mean radiative heating rate. This total force is what drives the zonal mean residual circulation (or transport circulation) in the middle atmosphere. Satellite temperature observations are then also used to compute global winds with a balance model, and the temperature and wind fields together are then used to define the planetary-scale wave fluxes and EP flux divergence. Examples of this method are described by Hamilton [1983], Smith and Lyjak [1985], Hitchman et al. [1989], Shine [1989], Marks [1989], Rosenlof [1995; 1996], and Fetzer and Gille [1996]. These studies have improved in accuracy over the years, partly because of improvements in the temperature and constituent measurements and partly through improvements in the models used to calculate the wind fields.

[181] Calculations of EP flux divergence give inherently noisy fields, so it is not surprising that most studies looking for the residual gravity wave term have focused on high altitudes near the stratopause and above where the gravity wave forcing becomes large [Hamilton, 1983; Smith and Lyjak, 1985; Shine, 1989; Marks, 1989; Fetzer and Gille, 1996]. At lower altitudes in the winter stratosphere the calculations of the residual gravity wave forcing have been highly variable, which could result both from errors in the calculations and true variability.

[182] Rosenlof [1996] and Alexander and Rosenlof [1996] focused on the summer stratosphere where resolved EP flux divergence is small and gravity wave driving is believed to dominate. A simple model of gravity wave propagation and dissipation was found to generate gravity wave driving in the summer stratosphere that was consistent with the observational constraints. The model used a constant spectrum of nonstationary gravity waves input in the troposphere. Ray et al. [1998] applied the same methods in the equatorial stratophere and inferred that gravity wave driving is important to the eastward phase of the stratopause SAO. The gravity wave driving in the summer stratosphere is westward and below ∼1 hPa has an estimated magnitude of order ∼1 m s⁻¹ d⁻¹. Similar westward forcing magnitudes have also been estimated near the tropopause in both summer and winter. The gravity wave driving at the onset of the stratopause SAO eastward phases (at equinoxes) has been estimated at ∼1–3 m s⁻¹ d⁻¹.

[183] Calculations of this type in the mesosphere extending up to ∼80 km altitude show the annual cycle in extratropical gravity wave driving of the summer-to-winter pole-to-pole zonal mean circulation. The wave driving is westward in winter and eastward in summer [Shine, 1989; Marks 1989; Fetzer and Gille, 1996]. Two of these studies suggested that the mesopheric gravity wave driving in the Southern Hemisphere winter is significantly larger than that in the Northern Hemisphere winter. These estimates of the mesosphere force peak at magnitudes ∼30–60 m s⁻¹ d⁻¹ at solstice seasons. This hemispheric asymmetry seems opposite to that suggested by PMSE and temperature observations described in 23. The differences could suggest a high degree of variability or could be due to errors in the estimates.

8.1.2. Radar Observations of Momentum Flux Convergence

[184] Radar observations of gravity wave momentum fluxes at a number of sites have been used to estimate the body force due to gravity waves in the lower stratosphere and in the mesosphere and lower thermosphere. Mean momentum fluxes in the lower stratosphere over the MU radar (at 35°N latitude) were ∼0.1–0.3 m² s⁻², with peak values of ∼1 m² s⁻² and implied mean flow forcing of ∼1 m s⁻¹ d⁻¹ [Fritts et al., 1990b]. Momentum fluxes in the mesosphere and lower thermosphere have been measured more widely. At extratropical sites, mean values of ∼1–10 m² s⁻² have been measured with mean flow forcing in the range ∼10–70 m s⁻¹ d⁻¹ that generally oppose the mean winds [Reid and Vincent, 1987; Fritts and Yuan, 1989b; Tsuda et al., 1990b; Nakamura et al., 1993c]. At the tropical Jicamarca radar (12°S) [Hitchman et al., 1992] fluxes of 1–4 × 10⁻⁴ Pa (∼2–8 m² s⁻²) were observed with mean flow forcing ∼10–60 m s⁻¹ d⁻¹. Measurements at 35° northern and southern latitudes have been compared [Nakamura et al., 1996], and like the residual force calculations, these suggest larger gravity wave driving in the Southern Hemisphere winter than in the Northern Hemisphere winter. Momentum fluxes observed in the mesosphere and lower thermosphere are highly variable, with peak values of ∼30–60 m² s⁻² greatly exceeding the mean values. These imply correspondingly larger and possibly localized mean flow forcing and have been correlated with larger-scale motions [Fritts and Vincent, 1987; Fritts et al., 1992; Murphy and Vincent, 1998].

8.1.3. Model Studies of Zonal Mean Wave Driving

[185] A third method for estimating the gravity wave mean flow driving uses models of the middle atmosphere with parameterized radiative transfer and wave driving. The model results are then constrained by observations so that inferences about the gravity wave driving can be made. Holton [1982, 1983], Garcia and Solomon [1985], Holton and Schoeberl [1988], Hitchman et al. [1989], and Huang and Smith [1991, 1995] examined gravity wave driving at extratropical latitudes in the mesosphere. The gravity wave mean flow force in these models peaked at midlatitudes at solstice seasons with values ranging from ∼30 to 120 m s⁻¹ d⁻¹, with more recent work tending to report smaller values.

[186] Other global and mechanistic model studies have focused specifically on the magnitude of the zonal mean transport circulation required to account for the summer mesopause thermal structure [Garcia, 1989; McIntyre, 1989; Fritts and Luo, 1995, Zhu et al., 1997]. These have suggested vertical velocities near the mesopause of ∼5 cm s⁻¹ and meridional motions of ∼20–30 m s⁻¹; values that are comparable to radar and satellite measurements of mean meridional winds near the polar summer mesopause. The observed meridional winds differ somewhat in their latitudinal distribution with peak magnitudes observed at higher and lower latitudes [Nastrom et al., 1982; Lieberman et al., 1998] than the model studies' inferred meridional transport circulation winds. Variability in wave driving can also account for abrupt temperature transitions that have been observed at the summer mesopause [Luo et al., 1995; Lübken, 1999].

[187] Dunkerton [1997b] modeled the QBO and SAO and inferred that gravity waves likely contribute at least half of the wave momentum flux needed to drive the QBO. His estimates of the cross-tropopause momentum flux carried by gravity waves with phase speeds in the range of QBO wind speeds were ∼2–3×10⁻³ Pa.

8.4.1. Mesospheric Circulation

[196] Every gravity wave parameterization in use has been designed to yield the summer eastward and winter westward momentum forcing in the mesosphere that drives the pole-to-pole residual circulation (47). This role for gravity waves in the mesosphere was recognized decades ago [Lindzen, 1973, 1981; Holton, 1982] and has more recently been validated by full GCM studies [Roble and Ridley, 1994; Norton and Thuburn, 1996; Hamilton, 1997; Manzini and McFarlane, 1998].

[197] Two recent studies considered the effects of asymmetries in the eastward and westward gravity wave momentum fluxes input via parameterization but came to opposite conclusions. Manzini and McFarlane [1998] launched an isotropic spectrum of waves at two different source levels (at the ground and near the tropopause) in the MA/ECHAM4 model using the Hines [1997a, 1997b] parameterization. Launching the waves at the ground led to smaller eastward and larger westward momentum flux crossing the tropopause at midlatitudes because of filtering of eastward waves by the tropospheric jets (Figure 27). The larger westward fluxes in turn produced more realistic middle atmosphere zonal mean winds. Medvedev et al. [1998] applied their gravity wave parameterization [Medvedev and Klaassen, 1995] in the CMAM model. They compared results with an isotropic tropopause launch spectrum, with phase speeds in the range 0<c<60 m s⁻¹ in four azimuths, to results with only the eastward propagating waves with larger fluxes. The results with eastward waves alone showed weaker and more realistic zonal mean winds in both summer and winter (Figure 28). This is an apparently paradoxical result since the westward forcing needed to decrease the winter jet strength would be expected to decrease in the anisotropic case given the physics embodied in equation (42).

[198] Norton and Thuburn [1996, 1997] applied a Lindzen-type parameterization of gravity wave forcing in the UGAMP GCM. They found the model with gravity wave forcing developed a realistic 2-day wave while no 2-day wave developed in the model with only Rayleigh friction. They concluded that gravity wave forcing in their model was essential to maintaining the unstable zonal mean state that led to the 2-day wave generation.

[199] Several studies have employed mechanistic versions of GCMs to study the effects of various gravity wave parameterization schemes in the mesosphere. A comparison of the effects of the Hines [1997a, 1997b] and Fritts and Lu [1993] parameterizations on the diurnal tidal structure suggested that the Hines scheme tended to amplify tidal amplitudes, while the Fritts and Lu scheme caused excessive dissipation of the tides [McLandress, 1998]. Other mechanistic and simplified GCMs have been used to test gravity wave parameterization effects in the mesosphere [Hamilton, 1997].

8.4.2. High-Latitude Stratosphere

[200] Through the concept of “downward control” of the mean meridional circulation by wave-induced mean zonal forces [Haynes et al., 1991], Garcia and Boville [1994] stress that the winter mesospheric gravity wave forcing can have effects down to ∼30 km in the high-latitude stratosphere that can help to alleviate the “cold pole” problem that plagues GCM studies of the winter stratosphere. Hamilton [1997], however, also suggests that gravity wave forcing at stratospheric altitudes, although smaller in magnitude, is also important to the winter circulation.

[201] The role of gravity wave forcing in sudden stratosphere warming (SSW) events has been examined in several global model studies. Lawrence [1997] compared effects of Hines and Fritts and Lu parameterization on the winter stratosphere. Both parameterizations generated reasonable zonal mean mesosphere winds, but the Fritts and Lu parameterization was found to inhibit the occurrence of SSW events while Hines did not. Pawson [1997] found that topographic gravity wave drag in the lower stratosphere influenced the onset of SSW events but found little or no sensitivity to mesospheric drag parameterized as Rayleigh friction. Rind et al. [1988] found that gravity wave forcing experiments, designed to describe waves generated by different sources, affected both the character and existence of SSW events in the GISS GCM.

8.4.3. Equatorial Oscillations

[202] At the equator, where the Coriolis force vanishes, the atmospheric response to a zonal force is simply zonal acceleration. Dissipation of zonally propagating waves at tropical latitudes provides the forcing that drives or helps to drive the three dominant equatorial oscillations: (1) the QBO in the lower stratosphere, (2) the stratopause SAO, and (3) the mesopause SAO. Both gravity waves and planetary-scale waves likely participate in driving all three of these oscillations to differing degrees. What remains to be quantified is the spectrum of wave momentum flux in the tropics. Global model studies are therefore free to include adjustable parameterized gravity waves without adequate constraints either on the gravity wave fluxes or on the resolved planetary-scale wave fluxes. Mayr et al. [1997b] demonstrated the potential of the gravity-wave-driving mechanism for all three of these equatorial oscillations in a simplified global model with parameterized gravity wave fluxes.

[203] The QBO in the lower stratosphere zonal winds is a wave-driven phenomenon believed to be predominantly driven by dissipation of planetary-scale waves [Baldwin et al., 2001]. However, zonal forcing accompanying the dissipation of gravity waves is likely to contribute significantly, with recent estimates suggesting they may transport roughly half of the flux [Dunkerton, 1997; Piani et al., 2000]. Until very recently, obtaining a QBO in GCM studies had proved difficult. Recent GCM studies with increased horizontal and vertical resolution and decreased numerical diffusion have since developed QBO-like oscillations [Takahashi, 1996; 1999; Horinouchi and Yoden, 1998; Hamilton et al., 1999]. These successes are believed to be due to the increase in tropical wave fluxes allowed by the finer resolution and reduced numerical dissipation [Boville and Randel, 1992; Nissen et al., 2000]. Scaife et al. [2000] instead modeled a QBO in a GCM by including parameterized small-scale gravity wave fluxes.

[204] The SAO at the stratopause is driven in part by the meridional advection of summer hemisphere westward (easterly) winds across the equator and in part due to zonal wave driving particularly important during the eastward (westerly) wind phases of the SAO [Sassi et al., 1993; Hamilton et al., 1995; Muller et al., 1997; Garcia et al., 1997]. Rind et al. [1988], Jackson and Gray [1994], and Medvedev and Klaasen [2001] modeled realistic SAOs by including parameterized gravity wave fluxes, the latter with a model deep enough to include an SAO at the mesopause.