Testing theories of atmospheric gravity wave saturation and dissipation

Journal of Atmospheric

and Terresrrial

Physics,

Pergamon PII: SOO21-9169(96)00027-X

Vol. 58, No. 14, pp. 1575-1589, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved OtX-9169/96 $15.00+0.00

Testing theories of atmospheric gravity wave saturation and dissipation Chester Department

of Electrical

S. Gardner

and Computer Engineering, Urbana-Champaign

University of Illinois at Urbana-Champaign, IL, U.S.A.

(Received 30 November 1995 ; accepted 29 January 1996)

Abstract-Numerous

mechanisms have been proposed to account for the dissipation and saturation of gravity waver; in the atmosphere. We review the leading wave dissipation paradigms and identify the experimental data required to test definitively the fundamental physics upon which these theories are based. We also examine the separability of the joint vertical wave number (m) and temporal frequency (0) spectrum and the separability of the unambiguous two-dimensional horizontal wave number spectrum. Definitive tests of the Linear Instability (Dewan and Good, 1986) SaturatedCascade (Dewan, 1994), and Diffusive Filtering Theories (Gardner, 1994) and separability of the (m, o) spectrum are within the observational capabilities of modern remote sensing instruments. The Diffusive Damping (Weinstock, 1990) and Doppler Spreading Theories (Hines, 1991) are untestable in their present forms. Separability of the two-dimensional horizontal wave number spectrum is difficult to test using current technology, although analysis of airglow images may provide some insight. Copyright 0 1996 Elsevier Science Ltd

INTRODUCTION

Gravity waves exert a major influence on the largescale circulation and structure of the atmosphere and are responsible for much of the spatial and temporal variability above the tropopause. Gravity waves have been studied extensively during the past 35 yr since Hines (1960) first suggested they may play important roles in establishing the wind and thermal structure of the upper atmosphere. In spite of the enormous amount of theoretical and observational research which has been published since Hines’ seminal work, there is still considerable disagreement about which processes dominate wave saturation and dissipation. Numerous theories have been proposed yet few experiments have been conducted to test definitively, the various saturation and dissipation mechanisms. Consequently, gravity wave effects have at best only been crudely parameterized in existing global circulation models. Most theoretical and observational studies have concentrated on explaining and characterizing the apparent universal power law behavior of the vertical wave number (m) spectrum of horizontal winds F,(m). The canonical spectrum is characterized by a vertical wave number m. that partitions the spectrum into a low wave number regime which is believed to be dominated by the gravity wave source characteristics.

In this region the spectrum is usually assumed to be proportional to ms, where s N 1. At large wave numbers greater than m., saturation and dissipation processes are believed to control the spectrum. In this region the spectrum is proportional to N2/m3. Because m, is proportional to N/t&,,,, where N is the buoyancy frequency and uh, is the rms horizontal wind perturbation, m. decreases with increasing altitude as uh, increases. The existing theories invoke a variety of different physical mechanisms for dissipating wave energy including shear and convective instabilities, cascade processes, wave-induced Doppler effects, and wave-induced diffusion. However, all models predict the same form and behavior for F,(m). Each theory incorporates an m. which is proportional to N/I&,,,,, each theory, under certain conditions, predicts a spectrum which is approximately proportional to Nz/m3 in the region m. < m, and each theory can be tuned so that the predicted spectral magnitudes are consistent with observations. Consequently, it is not possible to test any of the theories by comparing their predicted m-spectra of horizontal winds with observed spectra. In this paper we review the leading wave dissipation paradigms and identify the experimental data required to test the fundamental physics upon which these theories are based, as well as to test some of their most important predictions. This work is based on papers 1575

1576

C. S. Gardner

presented by the author at the Canadian Network for Space Research International Conference on Gravity Waves in the Atmosphere (Chateau Lake Louise, Alberta, 22-26 March 1994) and the 1995 NSF CEDAR Annual Meeting (Boulder, CO, 2630 June 1995).

LINEAR INSTABILITY THEORY

The Linear Instability Theory (LIT) of the gravity wave saturation was originally proposed by Dewan and Good (1986) to explain the vertical wave number spectrum of horizontal wind fluctuations. It is based on earlier work of Hodges (1967) who showed that shear and convective instabilities limit the horizontal perturbation velocities of quasi-monochromatic waves to values approximately equal to the intrinsic horizontal phase speeds of the waves. The gravity wave dispersion relation can be used to express this amplitude as N/m, where N is the buoyancy frequency and m is the vertical wave number. Dewan and Good argued that as a gravity wave packet propagates upward in the atmosphere, the horizontal wind amplitude increases until the limit N/m is reached, at which point the wave breaks dissipating excess energy as turbulence in such a way that the wave amplitude is subsequently maintained at this so-called saturation limit. Because the saturation limit is small for large m, at low altitudes only the shortest scale waves are saturated. As the altitude increases, saturation progresses to larger scales (i.e., smaller m) as these waves reach their amplitude limits. Dewan and Good defined m. as the wave number of the largest scale saturated wave. For m < m., wave amplitudes are controlled by source characteristics while for m. < m, they are limited by saturation to values -N/m. By using several approaches including dimensional analysis, Dewan and Good showed that in the saturation regime (m. < m) the m-spectrum of horizontal wind fluctuations, F,(m), is proportional to N2/m3. In the source region, the spectrum is usually assumed to be proportional ms, where s - 1. Thus, the linear instability model for the vertical wave number spectrum is given by 2

m < m* F,(m)=

(1) m. I m I mb

where the buoyancy wave number mb marks the transition between waves and turbulence. For mb < m the

wind fluctuations are dominated by turbulence and the spectrum is proportional to m-513. Notice that we have used the standard engineering convention of writing the spectrum in units of (m’/s’)/(cyc/m) where m, m*, mb, and N have units of rad/m or rad/s. The wave variance is obtained by integrating F,(m) with respect to dm/(2n). The shape and magnitude of the model gravity wave spectrum in (1) is governed by three parameters CI, m., and s. The parameter s is determined by source characteristics while t( and m. can be related to the total variance of horizontal winds ((u’)‘) and the variance of the vertical shear of the horizontal winds ((&‘a~)‘) by integrating the model spectrum in (1) and solving for m. and CI. (s+3)crN2 m* =

1

tl=

~ Ri ^[ (Sf3)

1 I”

2(s+ 1)((U’)2)

N

- uI,,s

1 + In (mJm*)

where hi is a form of the Richardson N2

13=((adjaz)*)

1

(2) (3)

number. N2

=

I

mbm2F

-1 27r0

(m)dm

(4)

U

Dewan and Good argued that I?i - 1 while Smith et al. (1987) observed that when wave perturbations just reach the point of convective instability, /2 = ((U>

(5)

where 6, is the vertical gradient or lapse rate of potential temperature, which leads to Ri = 2/p,

(6)

where p is the spectral index of the gravity wave temporal frequency (w) spectrum. The canonical spectrum given by (1) is plotted in Fig. 1. For LIT, saturation is assumed to be responsible for the high m limit 2nuN2/m3 which is essentially constant with altitude, location, and time. The parameter m. is adjusted so that the integral over the complete wave spectrum is equal to the total horizontal wind variance which does vary throughout the atmosphere. For example, as the wave field propagates upward, the total variance increases and so m. must decrease (equation (2)). As we shall see later, several other theories, which rely on entirely different physical mechanisms for dissipating wave energy, predict this same form and behavior for F,(m). While much attention has been given to predicting and

Testing theories of atmospheric gravity wave saturation and dissipation

104

10-5

lo.4

Vertical Wave Number

IO” m/k

10”

10-l

(cyc/m)

Fig. 1. The canonical gravity wave vertical wave number spectrum of horizont,al wind fluctuations. WI. is inversely proportional to the rms horizontal wind fluctuations and partitions the spectrum into a small wave number region m -Cm. where the shape and magnitude are dominated by source characteristics and a high wave number region m. < m dominated by saturation and dissipation effects.All leading theories predict that the spectrum is proportional to N2/m3in the high m region. m. decreases as the wave field propagates upward because the total variance increases. These characteristics of F,(m)are generally consistent with observations throughout the atmosphere.

measuring the m-spectrum of horizontal winds, the temporal frequency (o), zonal wave number (k), and meridional wave number (1) spectra, as well as the joint spectra, are also of considerable interest. By assuming saturated waves obey the polarization and dispersion relations,, all forms of the wave spectra (1-, 2-, and 3-D) can be predicted using the LIT mspectrum model in (1) (Gardner et al., 1993a). No theory can ever be proved by comparing predictions with observations. However, experimental verification does increase confidence in a paradigm while contradictions can expose flaws. The most crucial experimental tests are those which address the fundamental physics upon which the theory is based. If the fundamental physical assumptions are shown to be valid, then other experimental tests can be used to refine the theory and its predictions. For LIT, the fundamental assumption is that for all waves with m. < m, the horizontal wind amplitudes are limited by shear and convective instabilities to values N N/m. Stated another way, all waves for which the vertical wavelength satisfies, the condition 1, < 1: are saturated and the horizontal wind amplitudes of these waves are proportional to N/m = 1,/T,, where TB is the buoyancy period. Their variances or kinetic energies per unit mass are proportional to Ai, i.e., u’(m)’ N N2/m2 N Aa.

(7)

The definitive test of the fundamental physics underpinning LIT, IISto measure the horizontal wind

1577

amplitudes and intrinsic vertical wavelengths of quasimonochromatic gravity waves and confirm the assumed wavelength dependence. It has been nearly 30 yr since Hodges (1967) first derived the shear and convective instability limits for gravity waves and almost 10 yr since Dewan and Good (1986) proposed the LIT model for the wave spectrum. Even though wave amplitudes and vertical wavelengths are easily derived from horizontal wind profiles, there have been no published direct measurements of the wavelength dependencies of the wind amplitudes or variances of saturated gravity waves. Numerical simulations of Kelvin-Helmholtz instabilities in monochromatic gravity waves and numerous case studies of breaking waves observed in the atmosphere have confirmed that indeed wave amplitudes are ultimately limited to the values originally established by Hodges. Extensive radar observations of horizontal winds have shown that the canonical spectrum plotted in Fig. 1 provides a reasonably accurate description of observed spectra throughout the troposphere, stratosphere, and mesosphere (e.g. Tsuda et al., 1989). LIT requires all waves with 1, < 1: to be saturated so the few case studies of breaking waves that have been reported do not constitute a definitive test of the theory. These waves may simply be unusual events, generated by especially energetic sources, that reached the instability limit before other dissipation mechanisms had a chance to dominate. Nonbreaking waves are also frequently observed in temperature, wind, and density profiles. How do the horizontal wind amplitudes of these waves vary with wavelength? Extensive Na and Rayleigh lidar measurements from a variety of locations have been used to infer wave amplitudes and vertical wavelengths of quasimonochromatic waves in the stratopause and mesopause regions (Gardner and Voelz, 1987; Beatty et al., 1992; Collins et al., 1994, 1995). These studies have all shown that the inferred wave variances in the saturation regime are proportional to Al, not 1:. Although these results clearly contradict LIT, they are not definitive because the horizontal wind amplitudes were inferred from atmospheric and Na density perturbations. Rayleigh and Na density lidars provide only indirect experimental evidence that the fundamental assumption of LIT is not valid. The definitive test would be to repeat these observations with Doppler radars and lidars which can measure the horizontal wind profiles directly. There is considerable value in doing so. If the previous lidar results are confirmed, then it is likely the linear instability paradigm would have to be abandoned. On the other hand, if the new tests support LIT, confidence in most of the competing theories would be weakened since few

1578

C. S. Gardner

predict amplitude limits proportional in the so-called saturation regime.

to AZfor waves

obtain for the total gravity wave variance <(u’)‘> -

SATURATED-CASCADETHEORY

Recently Dewan (1994) proposed a modification to LIT to explain an observed relationship between the vertical wavelengths and periods of some saturated gravity waves (e.g. Gardner and Voelz, 1987; Manson, 1990). The theory is an extension of earlier work which employed dimensional analysis to predict various forms of the 1-D spectra (Dewan, 1991). Dewan argued that waves at saturation amplitudes established by shear and convective instabilities simultaneously transfer energy to higher frequency waves via cascade processes. Because saturated wave amplitudes are comparable to the horizontal phase speed, the wave variance is given by u’(m)’ - N2/m2 - A:

(8)

Simultaneously these waves are assumed to dissipate energy in a cascade process by transferring excess energy to smaller temporal scales, i.e. to higher frequency waves. Thus the wave variance is also given by U’(W)’- E/W - T

(9)

where w is the intrinsic frequency, T is the intrinsic period, and E is the energy dissipation rate which has units of m’js’. By equating these two expressions the wavelength-period relations are obtained. m - N(w/E)“’

(10)

ii - TB(~T)‘j2

(11)

Saturated-Cascade Theory (SCT) appears to apply to a subset of waves, viz., those that satisfy AL- T’12. Spectra are derived by using the LIT model for the mspectrum, the wavelength-period relations given by (10) and (1 I), and the gravity wave dispersion relation. Because the vertical wave number of each wave depends on its frequency through (lo), SCT invokes a severe form of non-separability on the joint (m, w) spectrum but yields all forms of the 1-D gravity wave spectra. The predicted spectra are similar to those of Diffusive Filtering Theory (Gardner, 1994) which is also a non-separable model. The SCT model for the temporal frequency spectrum of horizontal wind perturbations is derived from (9) using dimensional arguments (Dewan, 1991) F,(w)-E/W~

flw
(12)

where f is the inertial frequency. By integrating the spectrum over all frequencies between f and N, we

E/f

(13)

The parameter m. for SCT is defined as the vertical wave number of the largest vertical scale, lowest frequency saturated-cascade wave. It can be related to the rms wind perturbations by setting w = f in (10) and using (13). m. - N( f

je)‘”- N/u&,,

(14)

Notice that this is the same form for m. predicted by LIT (equation (2)) using different physical arguments. While SCT predicts all of the 1-D spectra, those predictions are not unique so it is not possible to test the theory definitively by comparing spectra models with observations. Recall the fundamental assumption of SCT is that wave amplitudes are simultaneously limited by linear instabilities and cascade processes. Saturatedxascade waves have wind variances that are proportional to 1: (equation (8)) and to A;- T”2 (equation (9)) so that for these waves (11)). The crucial test is to measure 2, - T”‘(equation the horizontal wind amplitudes, vertical wavelengths, and intrinsic periods of quasi-monochromatic waves for which m, < m and confirm the relationships given in equations (SHll). Reid (1986) and Manson (1990) summarized the characteristics of numerous quasi-monochromatic gravity waves measured in the mesopause region using radars, lidars, and airglow imagers. Their data exhibit features similar to the wavelength-period relations predicted by SCT, viz., there is an absence of waves at the shortest vertical scales. The cutoff vertical wave number is approximately proportional to o’/‘. Dewan (1994) attributes the scatter in the observed waves reported by Reid and by Manson to statistical fluctuations in E. Muraoka et al. (1988), Tsuda et al. (1990) and Nakamura et al. (1993) used radar wind observations to determine the intrinsic parameters of numerous large scale waves in the upper mesosphere. Their data are clustered near the inertial frequency at m values near those predicted by Dewan’s wavelength-period relation. Gardner and Voelz (1987), Beatty et al. (1992), and Collins et al. (1994, 1995) used Na/Rayleigh lidars to infer amplitudes, vertical wavelengths, and observed periods of quasi-monochromatic gravity waves in the mesopause and stratopause regions. Their results show a clear wavelengthperiod relationship of the type predicted by SCT (equation (11)) in both the stratosphere and mesosphere. However, the variances of these waves are proportional to 1j and T3/*. Although the lidar results clearly contradict the fundamental SCT assumptions,

Testing theories of atmospheric gravity wave saturation and dissipation they are not definitive because the horizontal wind amplitudes were inferred from atmospheric and Na density perturbations and the measured wave periods were not the intrinsic periods. The definitive test would be to directly measure the horizontal wind amplitudes, intrinsic: vertical wavelengths, and intrinsic periods of quasi-monochromatic waves using Doppler radars and lidars and confirm the assumed wind variance dependencies (equations (8) and (9)) and the derived wavelength--period relations (equations (10) and (11)). The most difficult wave parameter to measure is the intrinsic period. Doppler effects of the mean wind can shift the observed frequencies to values significantly different from the intrinsic frequencies. Analysis techniques based upon hodographs of the horizontal wind field have been used successfully to measure the intrinsic periods of large vertical scale waves (e.g. Muraoka et al., 1988; Tsuda el al., 1990; Nakamura et al., 1993). Airglow imagers in combination with radar and lidar derived horizontal winds have also been used to determine the intrinsic parameters of small horizontal scale waves near the mesopause (e.g. Swenson et al., 1995; the imager based Taylor et al., 1995). Unfortunately, technique can only compensate for Doppler effects of the mean wind field. Groundbased observers would also see a frequency shift caused by the velocity of the source that generated the wave, e.g. a moving storm front. If the source is unknown it will not be possible to compensate the imager data for source motion effects.

DOPPLER SPREADING AND DIFFUSIVE DAMPING THEORIES

For the linear instability and saturated-cascade paradigms, individual wave packets are assumed to reach their saturation amplitudes more or less independently of all the other waves in the spectrum. Another group of theories has been developed by assuming that the m-spectrum is controlled by nonlinear effects arising from strong wave-wave interactions. Hines (199 1) suggested that the form of mspectrum is a consequence of Doppler spreading of the vertical wavelengths by the irregular winds of the wave system itself. The observed Doppler shifted frequency w,,~~is given by %bs

= w+h,ucosQ

= ,(l+

ycose)

(15)

where u is the horizontal wind, h = (@+1’)” is the magnitude of the h’orizontal wave number vector, k is the zonal wave number, f3 is the meridional wave number, and q is the angle between the horizontal

1579

wind and wave propagation direction. In deriving the right-hand side of (15), we have used the approximate gravity wave dispersion relation h N corn/N. Doppler spreading is greatest for those waves with small horizontal phase speeds (w/h = N/m), viz., waves with the largest values of m. The rms spread in the observed frequency can be calculated from (15) by replacing u by the wind perturbation u’. Therms frequency spread is comparable to or greater than the intrinsic frequency (i.e., w I co’,,) when m. - N/I&

I m.

(16)

Hines argued that high m waves satisfying (16) are often Doppler shifted into critical layer conditions by the horizontal wind fluctuations of the more energetic large-scale waves and then are dissipated as turbulence. Other waves having smaller values of m near m. are Doppler shifted into a large m “tail region” which exhibits characteristics of the canonical m-spectrum. Waves in the tail region are eventually Doppler shifted into critical layer interactions and dissipated as turbulence. Under the Doppler Spreading Theory (DST) paradigm, the high m edge of the source spectrum near m. is continuously eroded as waves are Doppler shifted first into the tail region and then into the turbulence region. Notice that m. given by (16) decreases as z& increases with increasing altitude which is the same behavior predicted by LIT and SCT. In all three theories, m. identifies the high m edge of the source spectrum. Saturation and cascade processes are assumed to establish this edge in the LIT and SCT paradigms, while Doppler effects caused by strong wave-wave interactions are assumed to be responsible in the DST paradigm. Weinstock (1990) suggested that the amplitudes of waves of all scales are limited by diffusion-like processes related to the chaotic motions imposed by all smaller scale waves. Diffusion contributed by nonlinear wave-wave interactions, can be described mathematically by retaining the appropriate nonlinear terms in the wave equation. By following this approach, Weinstock derived an expression for the nonlinear dispersion relation which he then used to determine the growth in wave amplitude with altitude. Weinstock also argued that only the small-scale motions contribute to the diffusion of the large-scale waves. The large scale motions are assumed to act mostly like the mean flow in their influence on the smaller scale motions. These assumptions lead to a differential equation which describes the evolution of the m-spectrum as the wave field propagates upward. By assuming a source spectrum in the lower atmosphere, the gravity wave spectrum in the upper atmo-

C. S. Gardner

1580

sphere can be computed by solving the differential equation with the source spectrum as the input. Zhu (1994) extended the diffusive damping paradigm by also considering the effects of radiative damping which may be important in the middle atmosphere. Medvedev and Klaassen (1995) recently examined the combined effects of Doppler spreading and diffusive damping on F,(m). Weinstock’s Diffusive Damping Theory (DDT) and Hine’s Doppler Spreading Theory both assume that wave amplitudes are controlled by nonlinear wavee wave interactions. According to DDT, the small-scale waves control the large-scale waves through diffusionlike processes, while for DST, the large-scale waves control the small-scale waves through Doppler effects. Unfortunately, these mathematically elegant but complex theories have only been developed in sufficient detail to model the vertical wave number spectrum of horizontal winds. Because both predict the standard N2/m3 behavior for the high m region of the spectrum, it will not be possible to test these theories experimentally until they have been used to model other forms of gravity wave spectra or to make other testable predictions.

DIFFUSIVE FKTERING

THEORY

Diffusion effects caused by wave-wave interactions can also be modeled as a filtering process which selectively removes waves from the source spectrum as the wave field propagates upwards (Gardner, 1994,1995). Dissipation is most severe when the vertical phase speed of a wave is comparable to or smaller than the effective vertical velocity of momentum diffusion. In the Diffusive Filtering Theory (DFT) paradigm, all components of the wave spectrum are assumed to grow with increasing altitude in response to decreasing density until they are damped by diffusion. In this way the joint (m, w) spectrum is partitioned into regions corresponding to damped or undamped waves. By assuming an appropriate (m, w) source spectrum in the lower atmosphere, the m- and w-spectra at higher altitudes are computed by letting the amplitudes of all wave components grow with altitude and then including only those components lying in the undamped region. In the DFT paradigm, the shapes and magnitudes of the spectra are a consequence of the combined effects of the gravity wave source characteristics in the lower atmosphere and diffusive damping. The partitioning of the (m, CO)spectrum is related to the effective vertical diffusivity of the atmosphere D;,. While molecular and turbulent eddy diffusion contribute to DZL, it is assumed that wave induced

Gravity Waves

I

HP

I

I

Id

d

Vertical Wave Numbern1/271

I

I

(cychn)

Fig. 2. Gravity wave damping limit imposed by vertical diffusion. Wave intrinsic frequencies extend from the inertial frequency (f) to the buoyancy frequency (N). The diagonal line represents the diffusive damping limit. Waves to the right of this line are severely attenuated by diffusion because the effective vertical velocity of momentum diffusion (m&J exceeds the vertical phase speed (o/m) of the waves. The damping limit moves to the left as the wave field propagates upward because D,, increases with increasing altitude. The plotted limit is typical of the mid-latitude mesopause region where the inertial period is -20 h, the buoyancy period is - 5 min, and D,, - 350 m’/s.

diffusion dominates, at least below the turbopause. A wave of intrinsic frequency o and vertical wave number m will be severely damped when the effective vertical diffusion velocity (mDzz) of particles experiencing the wave motion exceeds the vertical phase velocity of the wave (w/m). Thus only waves satisfying mD,, I w/m

(17)

are permitted to grow in amplitude with increasing altitude. Waves not satisfying (17) are completely damped. The boundary between the damped and undamped regions of the (m, w) spectrum is defined by the equality in (17) and is represented by the diagonal line on the log-log plot in Fig. 2. The lowfrequency, high wave number waves (i.e. slow vertical phase speed waves) to the right of the diagonal line are eliminated from the spectrum by diffusive damping. Notice that the resulting (m, w) spectrum cannot be separable for m. < m because the cutoff condition o = D,,m2 depends on m and w. The m- and w-spectra are obtained by integrating the (m, co) source spectrum over the undamped region illustrated in Fig. 2. The resulting m-spectrum is similar to the canonical form plotted in Fig. 1. rn. is the wave number corresponding to the intersection of the diagonal damping limit line and the w = f line in Fig. 2,

(18) where the right-hand

side of (18) follows because

Testing theories of atmospheric gravity wave saturation and dissipation

D =i(w’T) 1z

~

I-

-

fw)*>/N’

(19)

where i?i is defined by (4), w’ is the vertical wind perturbation, T’ the temperature perturbation and r the adiabatic lapse rate of dry air. For m < m. none of the waves are damped so the m-spectrum in this region is identical to the source spectrum. Form. < m the source spectrum is integrated over D,& I o I N and the m-spectrum decreases with increasing m as progressively more Iof the low frequency waves are damped. The spectrum falls to zero above m = md = (N/D,) ‘jz because in this region all the waves are damped by diffusion. If the temporal frequency spectrum FU(w) is proportional to l/w”, DFT predicts that F,,(m) is proportional to l/rn’P- ’ for m. < m. For p = 2, the spectra are proportional to l/w* and N2/m3. Although the spectrum predicted by DFT exhibits many of the features predicted by the various saturation theories, none of the wave components, including those in the region m. < m where the spectrum is proportional to N2/m3, are saturated. The m-spectrum rem,ains constant with increasing altitude for m. < m, belcause successively more low frequency waves are removed by diffusive damping as D,, increases with height. DFT predicts a dependence of wave variance on vertical wavelength and intrinsic period which is different from that assumed in LIT and SCT. For p = 2, DFT predicts (u’(m)2> - N2/Hm3 - 1,’

(20)

(~‘(0)~) - (N2/Hm?)(f/co)3’2 - T3/*

(21)

where the expectations denote averages over all temporal frequencies in (20) and over all vertical wave numbers in (21). Rayleigh and Na lidar measurements of quasi-monochromatic wave variances are consistent with these predicted relationships (Gardner and Voelz, 1987; Beatty et al., 1992; Collins et al., 1994, 1995), although the wind variances in all these studies were inferred from measurements of atmospheric and Na density variances. The fundamental assumption of DFT is the wave damping criterion given by (17). This assumption can be tested by measuring the intrinsic periods and vertical wave lengths of quasi-monochromatic gravity waves to confirm that all waves lie in the undamped region denoted in Fig. 2. Because D, is proportional to the total wave variance or thermal flux (w’T’ ) (equation (19)) and therefore varies with altitude, location, and time, the boundary between the damped and undamped region is variable. A rigorous test of the DFT must also include measurements of D,. The

1581

radar, lidar, and airglow data summarized by Manson (1990) and Collins et al. (1995) suggest that the DFT damping criterion is valid, at least in the upper mesosphere where the inferred effective vertical diffusivity is N 350 m*/s. However, these data do not constitute a definitive test because in most cases the reported wave periods were observed periods, not intrinsic periods. The validity of the DFT damping criterion can also be tested by confirming that the joint (m, co) spectrum is not separable. This issue is discussed in the following section.

SEPARABILITY

OF THE (m, o) SPECIWJM

In many analytical studies of atmospheric gravity waves it is assumed that the joint (m, w) spectrum of horizontal winds is separable, i.e. F&r, w) = (2n)2((u’)2)A(m)B(o)

(22)

F,(m) = 2rr(rQ2)A(m)

(23)

E;(w) = 2n((u’)*)B(w)

(24)

where

Separability is a direct mathematical consequence of the physical mechanisms that control energy dissipation in Linear Instability Theory. Dewan and Good (1986) assumed that the saturation amplitude of each wave packet is N/m, regardless of the frequency or horizontal wave number. This leads to the familiar N2/m3 form for the m-spectrum of horizontal winds. Because wave packets reach their saturation limits more or less independently of each other, the derivations employed by Dewan and Good for the mspectrum can also be applied selectively to all waves of any given frequency with the same N2/m3 result. Thus, LIT implies that the shape of the m-spectrum does not depend on wave frequency, at least for m. < m, although the magnitude may. In other words, the joint (m, o) spectrum of horizontal winds is separable. By making similar arguments it follows that the joint (m, h) spectrum, where h is the magnitude of the horizontal wave number vector, must also be separable. These attributes are a direct consequence of the assumption that saturation is independent of frequency and horizontal wave number, i.e. depends only on N/m. It is not surprising then that separability of the (m, w) spectrum is frequently invoked in theoretical treatments of gravity wave spectra. In Saturated-Cascade Theory, Dewan (1994) added an additional constraint on saturation which depends on wave frequency, viz., that saturated waves dissipate

1582

C. S. Gardner

energy in a cascade process by transferring excess energy to higher frequency waves. This constraint given by (9) enforces a severe form of non-separability on the joint (m, w) spectrum in the form of the wavelength-period relations (equations (10) and (11)). Only waves for which the vertical wave number and frequency are related by equation (10) contribute to the spectrum. In the (m, w) plane, these waves lie along a diagonal line similar to that plotted in Fig. 2. The position of the line is related to the energy dissipation rate a, and according to Dewan can vary significantly because of large statistical fluctuations in a. Diffusive Filtering Theory also yields a non-separable (m, w) spectrum because the damping criterion depends on both frequency and wave number (equation (17)). Weinstock’s (1990) Diffusive Damping Theory and Hines’ (1991) Doppler Spreading Theory have not yet been developed in sufficient detail to determine if the joint (m, w) spectra arising from these paradigms are separable. However, Gardner (1994) has shown that Weinstock’s spectral models are consistent with the assumption that wave-induced diffusion selectively eliminates waves from the spectrum in a way that can be modeled as a filtering process. Thus, it seems likely that the DDT paradigm will also yield non-separable (m, w) spectra. It is difficult to test directly for separability, because both the m and w spectra, as well as the joint spectra measured by stationary groundbased observers, are distorted by Doppler effects arising from the mean horizontal winds (e.g. Fritts and Van Zandt, 1987; Lefrere and Sidi, 1990; Gardner et al., 1993b). The observed Doppler shifted frequency webs given by (15) is a function of the intrinsic frequency, vertical wave number, and horizontal wind. It can be shown, that in the presence of a mean horizontal wind, the joint (m, w,J will not be separable even if the intrinsic spectrum is separable (Gardner et al., 1993a). The few direct observational tests of (m, w,,,J separability that have been reported are consistent with this theoretical prediction (e.g. Fritts and Chou, 1987; Senft and Gardner, 1991). The vertical wind spectrum F, is related to the horizontal wind spectrum through the gravity wave polarization relations (w - em/N) F&r,

w) = (olN2F,(m,

0)

(26)

so that separability can also be tested by measuring F,(m).If the (m, o) spectrum of horizontal winds is separable (equation (22)), then

Fdm) = F,(m)

s

& N(4fV2B@)AI = ((w’)2>F&4,

I

(27)

and so the m-spectrum of vertical winds has the same shape as the horizontal wind spectrum. LIT predicts that F,,,(m) is proportional to mm3 form. < m. On the other hand, the SCT and DFT non-separable models predict Fw(m) -m and -m 5-2p, respectively. Because these predictions are quite different, measurements of F,.(m) can provide a definitive test of separability and of the validity of the wave dissipation paradigms. Most research conducted on gravity waves during the past 30 yr has focused on the wave induced perturbations in horizontal winds, temperatures, and densities. Because of considerable observational difficulties, little experimental work has been published on vertical winds, especially observations of the vertical wave number spectrum. Vertical velocities are typically very small compared to horizontal velocities. Even in the upper mesosphere where wave amplitudes are known to be large, the vertical wind fluctuations are not expected to exceed a few m/s. These small velocities are difficult to measure with the accuracy and resolution sufficient for gravity wave studies. Most attempts to measure vertical velocities have utilized radars with beamwidths of several degrees or more. Because of the aspect sensitivity of the scatterers within these broad beams, it is difficult to insure that the measurements are not contaminated by the strong horizontal winds. Because horizontal velocities can approach and even exceed 100 m/s in the mesosphere, pointing accuracies of a few tenths of a degree or a few mrad are required to insure that horizontal contamination does not exceed a few tenths of m/s. Another complication is the distortion caused by Doppler effects of the mean horizontal wind field. When observations are made using stationary groundbased instruments with limited frequency responses, Doppler effects can reduce disproportionately the magnitudes of m-spectra measured at high values of m. This distortion is most significant for the vertical wind spectrum. Because the high m waves are the ones most susceptible to Doppler shifting (see equation (15)), instruments with limited high and low frequency responses will not be sensitive to the high m energy that is Doppler shifted out of the frequency range of the instrument. The measured m-spectrum magnitudes will be smaller than the intrinsic magnitudes with the distortion most pronounced at the highest values of m where Doppler shifting is most severe. Consequently, the observed m-spectra will be steeper (i.e. indices will be more negative) than the intrinsic spectra. The effect is illustrated in Fig. 3 using DFT models of the wave spectra. Figures 3a and b are contour plots of F,,,(m, cuobs) predicted by DFT when the mean horizontal winds are respectively, zero and

Testing theories of atmospheric gt-avity wave saturation and dissipation

lO.‘~

10-2.

g

m-3-‘/

P

(a) u = 0

IO-‘-

Ids

/ , lo-4-~g / /=’ 10-S

(b) u = 30 m/s

-N

/,,+

3” l(y5.

10..6>

1583

111 *

-f

r’

10-3 10-4 m/2x (eyc/nr)

10-2 m/2x (cyc/m)

Vertical Winds

/

W-

IO4

m/(*x) (cyc/m)

II

Fig. 3. (a) Contour plots of the joint (m, mobs)spectrum of vertical wind fluctuations predicted by Diffusive Filtering Theory for an isotropic wave field when the mean horizontal winds are zero and (b) 30 m/s. (c) Observed m-spectrum of vertical winds for several values of instrument frequency response for the case where the mean horizontal wind velocity is 30 m/s. wIoW is the instrument low frequency cutoff and mhighis the high frequency cutoff. The computed spectra include the contributions of all waves with observed frequencies between wloWand ahigh. The spectra in all three figures are representative of the mid-latitude mesopause region.

30 m/s. Notice that for z? = 30 m/s, considerable energy has been Doppler shifted to frequencies below f and above N. Notice also that considerable energy has been Doppler shifted into the damped region of the intrinsic spectrum and that the distortion is most significant at the highest values of m. Only the wave energy at those frequencies within the response of the instrument will contribute to the measured vertical wave number spectrum. In Fig. 3c are plots of the predicted observed spectra which were obtained by integrating the modlel F,(m, oobs) over wloW< w,,,~ < are respectively the low and whight where wow and Whi.& high frequency cutoff responses of the instrument or processed data. Significant distortion occurs even for data averaging times (N l/mhigh) as short as 10 min. Therefore, accurate high temporal resolution vertical wind data are required for definitive tests of (m, w) separability.

Among the few reported measurements of vertical wind profiles and their m-spectra are the papers of Kuo et al. (1985), Larsen et al. (1986, 1987), and Cornish (1988). These authors used ST radars to measure vertical wind spectra in the troposphere and stratosphere. Kuo et al. (1985) reported spectral indices near - 1.7 for data in the troposphere between 2.85 and 7.5 km altitude and near -0.34 in the lower stratosphere between 8.4 and 13 km. Larsen et al. (1986, 1987) reported values near - 1.3 for data between 3.6 and 18 km altitude. Sidi et al. (1988) analyzed a single balloon profile of temperature and vertical velocity in the lower stratosphere. The mspectra indices were - 2.1 for temperature and - 1.8 for vertical velocity which is consistent with a separable (m, w) spectrum. However, these indices are considerably smaller than the canonical - 3 predicted by all the leading theories for temperature and horizontal

1584

C. S. Gardner

wind m-spectra. Cot and Barat (1990) presented similar data for a single balloon sounding of the troposphere and stratosphere. These data also exhibited similar shapes for the m-spectra of temperature and vertical winds. Spectral indices for both parameters were near -5/3. Fritts and Hoppe (1995) recently published m-spectra of vertical winds computed from EISCAT observations of PMSE near 85 km altitude. They reported spectral indices varying between - 1 and - 3 for two different data sets spanning a height range of slightly more than 6 km. Because of strong scattering from the PMSE, they were able to derive vertical wind profiles at high temporal resolution, although the lengths of the data sets were short (- 1 h and 3 h). Gardner et al. (1995b) used a Na wind/temperature lidar to measure vertical wind profiles between 84 and 100 km above Haleakala, Maui. They analyzed a single 6 h data set and reported a spectral index of - 1.4 when the vertical wind profiles were averaged for 20 min before computing spectra. They also showed that the spectrum steepened when the averaging times were increased and attributed the effect to Doppler distortion caused by the mean horizontal winds. The quasi-monochromatic wave data sets summarized by Manson (1990) Gardner (1995) and Collins et al. (1995) all exhibit an absence of waves at the shortest vertical scales where the cutoff is proportional to WI/~. These data sets also provide evidence that the (m, w) spectrum is not separable, although the majority of the reported frequencies were observed not intrinsic. Although the published radar and lidar observations of F&r) suggest that the (m, w) spectrum is not separable, the observed spectra are not consistent with the predictions of any of the non-separable theories. Accurate, high temporal resolution observations are required to conduct definitive tests of the dissipation theories and (m, w) separability. Unfortunately much of the previously reported data suffer from one or more experimental limitations. To be definitive, future studies must address these limitations. Potential contamination from horizontal winds must be minimized and well characterized. The pointing accuracies of both radars and lidars must be maintained to within a few tenths of a degree of zenith. For radars, it is also necessary to demonstrate that horizontal wind contamination resulting from the aspect sensitivity of scatterers within the main beam and strong off-zenith scatterers in the beam side-lobes are both negligible. The temporal resolution should be comparable to or shorter than the buoyancy period to minimize Doppler distortion and the vertical resolution should extend over at least one decade, preferably from 1;’to il/lO (i.e., m. < m < 10 m.). And

of course, the rms error of the vertical wind profiles should be much smaller than W&

SEPARABILITY HORIZONTAL

OF THE TWO-DIMENSIONAL WAVE NUMBER

SPECTRUM

The horizontal wave number spectrum describes the distribution of wave energy as a function of propagation direction and horizontal scale size. It is a twodimensional function of the zonal (k) and meridional (1) wave numbers. The spectrum can also be expressed as a function of the polar coordinates (h, 4), where h = (k* + 12)‘/2is the magnitude of the horizontal wave number vector and 4 = tan-‘(k/l) is the azimuth angle of propagation (4 = 0 refers to due north). F&k, 1) = F*(h, 4)

(28)

Note in (28) we are referring to the unambiguous horizontal wave number spectrum where k and I can be positive or negative and 0 I 4 I 271. Positive values of k and I denote, respectively, eastward and northward propagation, while negative values denote westward and southward propagation. Because gravity waves accelerate the mean wind in the direction of the intrinsic horizontal phase velocity, the horizontal wave number spectrum can provide important insights into the interactions of dissipating waves with the mean flow (e.g. Walterscheid, 1995). In spite of its importance, F,(k, I) has not received much attention either from theoreticians or from experimentalists, perhaps because it is so difficult to measure. Only Linear Instability Theory and Diffusive Filtering Theory have been developed in sufficient detail to model the 2-D horizontal wave number spectrum (Gardner et al., 1993a; Gardner, 1994). The derivation involves using the standard Jacobian transformation and the gravity wave polarization relation to relate the (m, o, 4) spectrum to the (h, 4) spectrum. If we assume that waves propagating in different directions are statistically independent because they originate from different sources, then the unambiguous 2-D spectrum is given by F,(k, 0 = F,(h, 4) = 27rG,(h, 4)/h

(29)

where G,(h, $) is the 1-D horizontal wave number spectrum of only those waves propagating along azimuth 4. Gardner et al. (1993a) showed how to express G,(h, 4) in terms of F&z, w, 4) by assuming that saturated gravity waves obey the polarization relations.

Testing theories of atmospheric gravity wave saturation and dissipation 2-D Horizontal

:= s F,(m, Nh/m, 4)

(30)

1585

Wave Number Spearurn

h.(T)= f m, /N

z _~~rrr$$ted -Sum -

G,(h, 4) = ;,

m N F,(m, Nh/m, q5)dm n s om

--

Turbulence

(31)

Several theoretical treatments of gravity waves have assumed that the (h, 4) spectrum is separable F,(h, 4 = (2~)Z<(~‘)2)C(h)D(~)

(32)

1 2XF.(h, 4) d$ = 2n(z12)C(h) 2l.cs 0

(33)

where

Horizontal

& j-,=hFu(h, 4) dh = 27G’)2)~(~)

hC(h) dh = s

2n D(4)dc#J = 1. s0

(34)

(35)

When the spectrum is separable, only the magnitude of F,(h, 4) depends on 4. The shape is independent of azimuth angle. From (29H31) we see that F,(h, 4) will be separable if Ir;(m, o, 4) is separable in 4, that is if the magnitude but not the shape of F,(m, w, 4) depends on azimuth angle. For LIT, F,(m, w, 4) is separable in o but not in CJ~ because the breakpoint in the spectrum at m. is inversely proportional to u’,~, which depends on azimuth (see equation (2)). Recall that shear and convective instabilities are responsible for establishing the saturation limit 2n&/m3 for FJm) at high values of m. Because these imtability processes do not depend upon which direction the wave is propagating, the F,(m, 4) spectrum in the saturation regime must also be equal to 2nuN2/m3, regardless of the azimuth angle. However, the total wave variance does vary with propagation direction. Therefore m. must depend upon 4 so that the integral of F,,(m, 4) over all m equals (u’(4)), the horizontal wind variance contributed only by th’ose waves propagating in the 4 direction. The m-spectrum of the waves propagating at azimuth 4, F,(m, q5), is given by (lt(3), and (6) with the total variance ((u’)‘) replaced by (u’(4)‘). m.(4) and (u’(4)‘) can vary significantly with azimuth angle, especially if the wave field is dominated by just a few energetic sourc:es. Notice that in those directions where little wave energy is propagating, m.(4) will be much larger than m. given by (2) so that not all the waves with m. < m will be saturated. In fact the complete spectrum F,,(m), obtained by including the contributions from waves propagating at all azimuth angles (i.e. by integrating F,(m, 4) over q%),should be

Wave Number

h/2a

(cyc/m)

Fig. 4. 2-D horizontal wave number spectrum of horizontal winds predicted by Linear Instability Theory. The spectrum is not separable in h and 4 because the breakpoints m.(d) _ WG,,,,(~) and A*($) = .Md)/W m f/a,,(4) vary with azimuth angle. This spectrum is representative of the mid-latitude mesopause region.

considerably different from the standard model given by (1). The total spectrum will be consistent with (1) only if all the waves are propagating in the same direction or the wave field is isotropic. If not, the spectrum will be proportional to N2/m3 only when m.(4) < m for all 4 and proportional to m”only when m < m.(4) for all 4. For wave numbers between these extreme values of m.(4), the spectrum transitions gradually from the ms to rnm3. This feature of LIT has important theoretical implications which have not been explored by its proponents. Non-separability of the LIT models for the (m, 4) and (h, 4) spectra is a direct consequence of the assumption that wave packets saturate independently. The LIT (h, 4) spectrum is plotted in Fig. 4. Because the breakpoints m*(4) - N/U&~) and h.(4) = fm@)/N - &,A$) vary with azimuth angle, the shape as well as the magnitude of F,(h, 4) depends on azimuth. For DFT the joint (m, w, 4) spectrum was derived by (Gardner, 1994)

e

f’dm, w, 4) = (‘W2
for m < (w/D,)“‘~ I w I N

(36)

where D, is given by (19). Because the effective waveinduced diffusivity arises from the nonlinear interaction of all the waves in the spectrum, regardless of their propagation directions. D,, and m. = (f/Dzz)“’

C. S. Gardner

1586

presented k-spectra of density perturbations between 60 and 90 km inferred from shuttle reentry data over the mid-Pacific. Kwon et al. (1990) Hostetler and Gardner (1994), Qian et al. (1995) and Gardner et al. (1995a) published k-spectra of density perturbations in the upper stratosphere (2540 km) and upper mesosphere (S&l05 km) inferred from airborne Rayleigh and Na lidar data. These measurements all revealed a power law dependence with spectral indices varying between about - 1.5 to -2.2. Hecht et al. (1994) recently published the first h-spectra measurements inferred from nightglow images. None of these studies addressed the issue of (h, qb) separability. Airglow imager data may provide useful insights. However for several reasons, spectra derived from intensity and temperature images do not accurately represent the wave field characteristics. Airglow layers respond most strongly to waves with vertical wavelengths comparable to or greater than the layer thicknesses (- 10 km). The effects of small vertical scale waves are significantly attenuated. The oblique viewing geometry for off-zenith waves can result in substantial enhancements of the wave amplitudes when the induced perturbations are in-phase along the line-of-sight. Initial attempts to model the horizontal wave number spectrum of OH intensities have not been entirely successful (Qian et al., 1995; Gardner et al., 1995a). It will not be possible to test for (h, 4) separability experimentally using airglow imager data until accurate analytic models are developed which

are functions of the total wave variance (equations (18) and (19)) and do not depend on 4. If the parameters s and p, which depend on source characteristics, are also independent of azimuth, then the DFT spectrum given by (36) is separable in 4 and the 2-D horizontal wave number spectrum can be expressed in the form given by (32) with

r 2(p-l&+1)

1 (2p+3s+l)

C(h) =

2(p-

1

h '-l

h I h.

hl C-J h.

I)@+ 1) 1 h. (2p+4)‘3 h. I h I md _= (N/D,,)“* h.2 0 h

(2p+3s+l)

(38)

where h. = fine/N = f 3’2/(NDf!2) is independent of azimuth. Because of the considerable observational difficulties, there have been few reported measurements of horizontal wave number spectra, especially in the middle atmosphere. Nastrom and Gage (1985) and Gage and Nastrom (1986) computed 1-D spectra (kspectra) from in situ aircraft observations of horizontal winds in the lower stratosphere. Bacmeister et al. (1995) analyzed extensive ER-2 aircraft observations of the k-spectra of winds, potential temperature, and 0, and N20 mixing ratios. Fritts et al. (1989)

Table 1. Key assumptions and predictions of several gravity wave saturation and dissipation theories Linear Instability Dewan and Good (1986)

N ljm”- 1N ,p-’ -l/m’ -12, p=2

,N2/m2 - 1:

u’(m)* u’(w)’ Wavelengthperiod relations

_

l/o’2P-

N

1/0”~ m

m*

- Nl4.m.

F,(m), m. < m

- N’/m’

I)/2 N -

<

T3’2,

-l/d

F,,.(m), m. <

m

-l/m’

- N2/m2 - 1:

p2P1’/2 p

=

(f ID:,) “2 - W4ms _m?‘P-2’N2/mZP- 1

-

T

Doppler

Spreading Hines (1991)

?

?

?

?

z

N(o/E)‘~’

?

?

N(f 18)‘j2 - N/u&,, - N2/m3

? - N2/m3

- NlhW - N2/m3

p = 2

-I/d -l/02, p= _ms-2P -m,

-E/O

Diffusive Damping Weinstock (1990)

2 m

(o/DA”~

-N2/m’,

F,(w)

Saturated-Cascade Dewan (1994)

Diffusive Filtering Gardner (1994)

-&/CL?

- l/oP

- l/op

-m

?

?

? ?

? ?

2

p=2

(m, o) separable

Yes

No

No

(h, 4) separable

No

Yes

?

Testing theories of atmospheric gravity wave saturation and dissipation relate the intensity and temperature fluctuations to the relative atmospheric density or temperature perturbations. Even then the tests may not be definitive because airglow imensities and rotational temperatures are only sensitive to the large vertical scale waves.

SUMMARY AND CONCLUSIONS

The key assumptions and predictions of the gravity wave theories discussed in the previous sections are summarized in Table 1. Blank entries imply that the theory makes no assumption or prediction, while question marks imply that the predictions are presently unknown. Notice that all theories predict that F,(m) - N2/m3 in the region m. < m so that none can be tested by comparing predicted and measured mspectra of horizontal winds. The Diffusive Damping and Doppler Spreading Theories are untestable in their present forms because they have only been developed in sufficient detail to predict F”(m) and those predictions are not unique. The key assumption for Linear Instability Theory is the vertical wavelength dependence of the horizontal wind variance while the key assumption for .Diffusive Filtering Theory is the so-called wavelength-period relation or diffusive damping criterion. For Saturated-Cascade theory, the key assumptions are the wavelength and period dependencies of the wind variance as well as the wavelengthperiod relations. The fundamental physics underpinning all three theories can be tested definitively by measuring the horizontal wind amplitudes, intrinsic vertical wavelengths, and intrinsic periods of quasimonochromatic waves. These measurements are well within the capabilities of existing remote sensing

1587

instruments and should be a major observational focus during the next several years. It is crucial that the dominant wave dissipation mechanism (or mechanisms) be identified so that accurate parameterizations of gravity wave effects can be developed and then incorporated in global circulation models. The separability of ,F,(m, w) can be tested by measuring F,,,(m), the m-spectrum of the vertical wind perturbations. If (m, w) separability holds, then the shapes of the horizontal and vertical wind spectra will be identical. Only the magnitudes will differ. Although the vertical winds are difficult to measure with the accuracy and resolution needed to make a definitive test of (m, w) separability, carefully designed experiments using existing atmospheric radars and lidars should be capable of resolving this issue. Existing data on quasi-monochromatic waves (e.g. Manson, 1990) and the m-spectra of vertical winds (e.g. Larsen et al., 1986, 1987; Gardner et al., 1995a) suggest that the (m,w) spectra are not separable. However, the observed vertical wind spectra are not consistent with the predictions of any of the non-separable theories. (h, 4) separability of the wave spectra is the most difficult feature to test using existing technology. While it may be possible to gain some insight by studying horizontal wave number spectra of airglow intensity and rotational temperature images, these data are distorted by the viewing geometry and are not sensitive to the smaller vertical scales (< 10 km). Furthermore, interpretation of the airglow data will continue to be hampered until adequate analytical models are developed which relate the airglow observations to the concomitant atmospheric density and temperature fluctuations. Acknowledgement-This

work was supported in part by the National Science Foundation grant ATM 94-03036.

REFERENCES Bacmeister J. T., Eckermann S. D., Newman P. A., Lait L., Chan K. R., Lowenstein M., Proffitt M. H. and Gary B. L.

1995

Beatty T. J., Hostetler

I”,. A. and Gardner

1992

Collins R. L., Nomura

A. and Gardner

Collins R. L., Tao X. and Gardner

Cornish

C. R.

C. S.

C. S.

C. S.

1994

1995

1988

Stratospheric horizontal wave number spectra ofwinds, potential temperature, and atmospheric tracers observed by high altitude aircraft. J. geophys. Res., in press. Lidar observations of gravity waves and their spectra near the mesopause and stratopause at Arecibo. J. atmos. Sci. 49,477497. Gravity waves in the Upper Mesosphere over Antarctica: Lidar observations at the South Pole and Syowa. .I. geophys. Res., 99(D3), 5475-5485. Lidar observations of gravity wave activity in the upper mesosphere over Urbana, IL: Implications for gravity wave propagation in the middle atmosphere. J. atmos. terr. Phys., in press. Observations of vertical velocities in the tropical upper troposphere and lower stratosphere using the Arecibo 430 MHz radar. J. geophys. Rex 93,94199431.

1588

C. S. Gardner

Cot C. and Barat J.

1990

Dewan E. M.

1991

Dewan E. M. and Good R. E.

1986

Fritts D. C. and Chou H. G.

1987

Fritts D. C. and Hoppe

1995

U. P.

Fritts D. C. and Van Zandt

Fritts D. C., Blanchard Gage K. S. and Nastrom

T. E.

1987

1989

R. C. and Coy L. G. D.

1986

Gardner

C. S.

1994

Gardner

C.S.

1995

Gardner

C. S. and Voelz D. G.

1987

Gardner

C. S., Hostetler

C. A. and Franke

Gardner

C. S., Hostetler

C. A. and Lintleman

Gardner

C. S., Qian J., Coble M. R. and Papen G. C.

1995a

Gardner

C.S., Tao X. and Papen G. C.

1995b

Hecht J. H., Walterschied

S. J.

S.

R. L. and Ross M. N.

1993

1994

1994

Hines C. 0.

1960

Hines C. 0.

1991

Hodges

1967

R. R.

Hostetler

Kuo F. S.

C. A. and Gardner

et al.

C. S.

1994

1985

A universal wave spectrum for atmospheric temperature and velocity fluctuations in the stratosphere? Geophys. Res. Lett. 17, 157771580. Similitude modeling of internal gravity wave spectra. Geophys. Res. Lett. 18, 1473-1476. Saturation and the ‘universal’ spectrum for vertical profiles of horizontal scalar winds in the atmosphere. /. geophys. Res. 91,2742-2748. An investigation of the vertical wave number and frequency spectra of gravity wave motions in the lower stratosphere. J. atmos. Sci. 44,3610-3624. High-resolution measurements of vertical velocity with the European incoherent scatter VHF radar, 2, SpecJ. geophys. tral observations and model comparisons. Res. 100,16827716838. The effects of Doppler shifting on the frequency spectra of atmospheric gravity waves. J. geophys. Res. 92, 9723-9732. Gravity wave structure between 60 and 90 km inferred from shuttle reentry data. J. atmos. Sci. 46,423434. Theoretical interpretation of atmospheric wave number spectra of wind and temperature observed by commercial aircraft during GASP. J. atmos. Sci. 43,729740. Diffusive filtering theory of gravity wave spectra in the J. geophys. Res. !J9,20601-20622. atmosphere. Scale-independent diffusive filtering theory of gravity The Upper Atmowave spectra in the atmosphere. sphere and Lower Thermosphere: A Review of Experiment and Theory. Geophysical Monograph 87, 155175. American Geophysical Union, Washington D.C. Lidar studies of the nighttime sodium layer over Urbana, Illinois, 2. Gravity waves. J. geophys. Res. 92,4673-4694. Gravity wave models for the horizontal wave number spectra of atmospheric velocity and density fluctuations. J. geophys. Res. 98, 1035-1049. Influence of the mean wind field on the separability of atmospheric perturbation spectra. J. geophys. Res. 98,8859-8872. High resolution horizontal wave number spectra of mesospheric wave perturbations observed during the 21 October triangular flight of ALOHA-93. Geophys. Res. Lett. 22, 2869-2872. Simultaneous lidar observations of vertical wind, temperature, and density profiles in the upper mesosphere: Evidence for nonseparability of atmospheric perturbation spectra. Geophys. Res. Lett. 22, 28772880. First measurements of the two-dimensional horizontal wave number spectrum from CCD images of the nightglow. J. geophys. Res. 99, 11449911460. Internal atmospheric gravity waves at ionospheric heights. Can. J. Phys. 38, 144-1481. The saturation of gravity waves in the middle atmosphere, II. Development of Doppler-spread theory. J. atmos. Sci. 48, 136Cl379. Generation of turbulence in the upper atmosphere by internal gravity waves. J. geophys. Res. 72, 34553458. Observations of horizontal and vertical wave number spectra of gravity wave motions in the stratosphere and mesosphere over the mid-Pacific. J. geophys. Res. 99, 1283-1302. Altitude dependence of vertical velocity spectra observed by VHF radar. Radio Sci. 20, 1349-1354.

Testing theories Kwon K. H., Senft D. C. and Gardner

of atmospheric

C. S.

1990

1986

Larsen M. F. et al.

Larsen

M. F., Rottger

gravity

J. and Holden

1987

D. N.

Lefrere J. and Sidi C.

1990

Manson

1990

A. H.

Medvedev

A. S. and Klaassen

Muraoka Y., Sugiyama T., Kawahira Tsuda T., Fukao S. and Kato S. Nakamura T., Tsuda T., Yamamoto Kato S. Nastrom

1995

G. P.

1988

K., Sato T.,

M., Fukao

S. and

1985

G. D. and Gage K. S.

Qian J., Gu Y. Y., Papen G. C. and Gardner

1995

C. S.

1986

Reid I. M. Senft D.C. and Gardner

1991

C.S.

Sidi C., Lefrere J., Dalaudier

1988

F. and Barat J.

Smith S. A., Fritts D. C. and Van Zandt

1987

T. E.

1995

Swenson G. R., Taylor M. J., Espy P. J., Gardner C. S. and Tao X.

Taylor M. J., Gu Y. Y., Tao X., Gardner Bishop M. B.

Walterscheid

Zhu X.

R. L.

J.

1995

C. S. and

Tsuda T., Inoue T., Fritts D. C., Vanandt T. E., Kato S., Sato T. and Fukao S. Tsuda T., Kato S., Yokoi T., Inoue T., Yamamoto Van Zandt T. E., Fukao S. and Sato T.

Weinstock

1993

1989 M.,

1990

1995

1990

1994

wave saturation

and dissipation

1589

Airborne sodium lidar observations of horizontal wave number spectra of mesopause density and wind perturbations. J. geophys. Res. 95, 13723-13736. Power spectra of vertical velocities in the troposphere and lower stratosphere observed at Arecibo, Puerto Rico. J. atmos. Sci. 43,223&2240. Direct measurements of vertical-velocity power spectra with the Sousy-VHF-Radar wind profiler system. J. atmos. Sci. 44,3442-3448. Distortion of gravity wave spectra measured from ground-based stations. Geophys. Res. Lett. 17, 15811584. Gravity wave horizontal and vertical wave lengths; an update of measurements in the mesopause region (N 8&100 km). J. atmos. Sci. 47,2765-2773. Vertical evolution of gravity wave spectra and the parameterization of associated wave drag. J. geophys. Res. 100,25841&25853. Cause of a monochromatic inertia gravity wave breaking observed by MU radar. Geophys. Res. Lett. 15, 134991352. Characteristics of gravity waves in the mesosphere observed with the middle and upper atmosphere radar. J. geophys. Res. %,8899-8910. A climatology of atmospheric wave number spectra observed by commercial aircraft. J. atmos. Sci. 42, 95&960. Horizontal wave number spectra of density and temperature perturbations in the mesosphere measured during the 4 August flight of ANLC-93. Geophys. Res. Letr. 22,2865-2868. Gravity wave motions in the upper middle atmosphere (60-l 10 km). J. atmos. terr. Phys. 48, 1057-1072. Seasonal variability of gravity wave activity and spectra in the mesopause region at Urbana. J. geophys. Res. 96,17229-l 7264. An improved atmospheric buoyancy wave spectrum model. J. geophys. Rex 93, 774790. Evidence for a saturated spectrum of atmospheric gravity waves. J. atmos. Sci. 44, 14041410. ALOHA-93 measurements of intrinsic AGW characteristics using the airborne airglow imager and groundbased Na wind/temperature lidar. Geophys. Res. Lett. 22,2841-2844. An investigation of intrinsic gravity wave signatures using coordinated lidar and nightglow imaging measurements. Geophys. Res. Lelt. 22,2853-2856. Radar observations of a saturated gravity wave spectrum. J. atmos. Sci. 46,244&2447. Gravity waves in the mesosphere observed with middle and upper atmosphere radar. Radio Sci. 26, 10051018. Gravity wave mean state interactions in the upper mesosphere and lower thermosphere. The Upper Atmosphere and Lower Thermosphere: A Review of Experiment and Theory. Geophysical Monograph 87, 133-144. American Geophysical Union, Washington D.C. Saturated and unsaturated spectra of gravity waves and scale-dependent diffusion. J. atmos. Sci. 47, 2211& 2225. A new theory of the saturated gravity wave spectrum for the middle atmosphere. J. ufmos. Sci. 51, 361.5 3626.