f’het. Space Sci.. Vol. 38, No. 9. pp. 1105-1119, 1990 ~32~33/~ %3.oo+o.w Printed in Great Britain. pergamon Press plc

THE OBSERVABLE EFFECTS OF GRAVITY WAVES ON AIRGLOW EMISSIONS

DAVID W. TARASICK’ snd COLIN 0. HINFSf

Centre for Research in Barth and Space Science, York University, North York, Ontario, Canada M3J IP3

(Received 26 March 1990)

Abstract-The influence of gravity waves on airglow emissions is explored with a view to elucidating the observable effects of such waves. The results obtained are applicable to ground-based photometric obser- vations, where the total line-of-sight emission, at an arbitrary zenith angle, is measured. Explicit relations are developed for the dependence of the magnitude and phase of the observable parameter q, the ratio of brightness fluctuations to temperature fluctuations. Both magnitude and phase depend on dynamical parameters (wavelength and period), the zenith angle at which observations are made, and a chemical parameter which is specific to the reaction chemistry. Methods of (i) distinguishing between evanescent and internal gravity wave modes, and (ii) determining the vertical wavelength and sense of vertical propagation for internal modes, are proposed. The former may be of considerable importance obser- vationaliy, since the effects of evanesxmt and internal modes on airglow are quite different. The latter is a measurement that is typically difficult to make directly.

It is shown that, for an isothermal atmosphere, the dependence of v on the characteristics of the perturbing wave is partially separable from the dependence on the emission chemistry, making it possible to draw conclusions about the effects of gravity waves on airglow brightness and temperature that are independent of any specific emission mechanism, as well as conclusions that are independent of specific wave characteristics.

1. INTRODUCI’ION

The existence of irregular variability in airglow bright- ness has been known for many years (Berthier, 1956), and the suggestion that this variability could in part be due to atmospheric waves was apparently first made at about the same time (Krassovsky, 1957; reported in Krassovsky, 1972). Hines (1964) recognized that these variations had length scales and phase speeds similar to those to be expected of internal acoustic-gravity waves, but it was not until Krassovsky (1972) reported that oscillations observed in the intensity and ro- tational tem~rature of the OH (4,l) band had periods and phase speeds appropriate to such gravity waves that variations in nightglow emissions began to be commonly attributed to such waves. Since that time, there have been many reports of successful detection of atmospheric gravity waves in the OH airglow (Moreels and Hers& I977 ; Clairemidi et al., 1985 ; Taylor et al., 1987; Sivjee et al., 1987), in the Na 5892 A emission (Taylor et al., 1987; Takahashi et al., 1985 ; Molina. 1983), in the atomic oxygen green line

* Present address : ARQX, Atmospheric Environment Service, 4905 Dufferin Street, North York, Ontario M3H 5T4.

TPresent address: Arecibo Observatory, Box 995, Arecibo, Puerto Rico 00613, U.S.A.

(Armstrong, 1982 ; Teitelbaum et al., 198 1 ; Taylor et al., 1987; Gavrilov and Shved, 1982) and red line (Misawa et al., 1984; Porter et al., 1974), and in the O,(biC) emission (Noxon, 1978 ; Viereck and Deehr, 1989).

Gravity waves have also been identified in lidar data (Gardner and Shelton, 1985) and in radar studies (Reid, 1986; Hunsucker, 1982). as well as in satellite mass spectrometer measurements (Dudis and Reber, 1976; Reber et al., 1975), and they are now generally considered a ubiquitous and important dynamical feature of the middle and upper atmosphere. Interest in these waves amongst aeronomers has increased greatly in the last decade or so, due in large part to an increasing recognition that their ability to transport energy and momentum causes them to play an impor- tant role in the heat budget and the general circulation of the upper atmosphere (Garcia and Solomon, 1985 ; Ebel, 1984; Richmond, 1978; Hines, 1972).

While a fairly comprehensive body of theory exists on the subject of the atmospheric gravity waves themselves, and their behaviour (e.g. Hines, 1974 ; Lighthill, 1978), comparatively little work has been done on the subject of the interaction of these waves with the chemical processes producing airglow emis- sion. Krassovsky (1972) presented a simple theory for the explanation of his OH observations in terms of gravity waves, but found that he was unable to

1106 D. W. TARASICK and C. 0. HINES

obtain agreement between theory and observations. Weinstock (1978) attempted a full theoretical treat- ment of the effect of gravity waves on O,(b’Z) emission, in which he found it necessary to invoke non-linear theory because of the large density gradient of the atomic oxygen concentration. Some theoretical work has also been done on the effects of gravity waves on F-region airglow {Porter et al., 1974), on OH intensity profiles (Hatfield et al., 1981), and on OH intensity and temperature (Walterscheid et al., 1987 ; Frederick, 1979). These studies, too, found minor-species density gradients and their potential non-linear effects to be a matter of concern.

Hines and Tarasick (1987) presented a more sophis- ticated analysis of the effect of gravity waves on airglow brightness and temperature as measured by a ground-based photometer. Beginning with the (Eulerian) form that is appropriate to line-of-sight integrated measurements, they employed a coordinate transformation within the integral to show that : (1) Krassovsky’s (incomplete) Lagrangian theoretical form requires correction, but is then reconciled with his observations; (2) Weinstock’s analysis is seriously in error ; and (3) the non-linea~ty that may arise from vertical movement of a minor species layer with a large density gradient, for local, in situ measurements, does not appear for vertically-integrated measure- ments. In keeping with this third conclusion, Schubert and Walterscheid (1988), through building on the work of Walterscheid et al. (1987), expressed no con- cern for the non-linearity once they, too, turned to vertically-integrated formulae.

The Hines and Tarasick (1987) model treats the restricted case of vertical viewing. The primary pur- pose of the present work is to extend this model to off-zenith observations and to make an initial assess- ment of the consequences. This extension is required to make the model a truly practical one for com- parison with a wide range of data. Hines and Tarasick (1987) stressed the importance of the simultaneous measurement of such parameters as wave period, propagation speed, and horizontal and vertical wave- length, in addition to that of n, the ratio of brightness fluctuations to temperature fluctuations. Any prac- tical system to do this involves taking observations at several zenith angles. Such observations necessarily introduce two new parameters, zenith angle of view and wave azimuth relative to viewing azimuth ; the theory will be richer in consequence. One example of this new wealth is a simple method for dete~ining whether the waves being observed are evanescent or internal wave modes. Analyses of airglow data in which gravity waves have been detected have generally assumed that the waves being observed were internal

waves (e.g. Krassovsky, 1972 ; Battaner and Molina, 1980; Gavrilov and Shved, 1982; Molina, 1983; Myrabar et al., 1987; Sivjee et al., 1987 ; Viereck and Deehr, 1989). Gravity waves detected in photo- graphs of meteor trails and in radar and lidar studies have in fact been internal modes almost exclusively (Hines, 1960; Manson and Meek, 1976; Chanin and Hauchecorne, 1981). Lack of vertical resolution, how- ever, makes the situation for airglow observations somewhat different. Vertically evanescent waves are not subject to antiphase cancellation for ground- based vertical viewing in the way that internal waves are, since their phase does not vary with height; for this reason it is possible that a significant degree of observational bias exists in favour of their detection over that of internal waves. The effects of the two types of wave on airglow can be quite different, as the following analysis shows, and their misidentification in data analysis could lead to serious errors of interpretation.

It will be shown that, for an isothermal atmosphere, the dependence of q on the characteristics of the per- turbing wave may be partially separated from the dependence on the emission chemistry. The con- ceptual advantage that this affords will be evident : it will be possible to draw conclusions about the effects of gravity waves on airglow brightness and tem- perature that are independent of any specific emission mechanism, as well as some that are independent of specific wave characteristics. An important feature of this approach is the ease with which the formalism can be applied to different emission mechanisms, for example to examine the effects of different production hypotheses.

We believe that the results obtained will be of direct significance to the experimentalist. Several obser- vational tests are, in fact, immediately suggested by them. We compare our results with the limited set of observational data that is available, in part to dem- onstrate the kind of conclusions that may be drawn without the assumption of a specific emission mech- anism. To accurately draw more detailed conclusions from the data will require a more complex chemical mode1 than that employed by Hines and Tarasick (1987) : this analysis is already in hand and will be presented in two subsequent publications (Tarasick and Shepherd, 1990a,b).

2. BRIGHTNESS AND TEMPR~TURE

FLU~A~ONS

Basic formulation The apparent emission rate, as measured by an

observer on the ground (or from a nadir-oriented

The observable effects of gravity waves on airglow 1107

satellite), can be expressed (Cham~rlain, 1961) as

where E is the volume emission rate, dr is an element of path length along the line of sight, and S is a constant which will depend upon the units of measure- ment and the instrument employed, and will in any case be seen to cancel from the formulae that follow. For S = 10e6, B will be in Rayleighs. In the interest of continuity with previous work, we will refer to the apparent emission rate B by the more colloquia1 term “brightness”.

We take the measured temperature to be the bright- ness-weighted average of the actual temperature along the line of sight :

S s

m sTdr

T,= D B ’

(2)

The exact form of the expression for E will depend upon the chemicai reactions producing the emission. The individual reaction rates will depend upon the density of the background atmospheric species, N, and of the chemically reactive minor species, n, and upon the reaction rate constants, k, which last, in general, depend upon temperature, T. All these four may vary with time and position in space. Thus, we begin by considering wave-modified parameters of the form

N&Y, ~6 = No(z) + N’(x, P, Z, t)

T&Y, z, r) = 7-r&? + T’(& y, G0, (3)

where zero subscripts indicate unperturbed values, assumed to be variable in height only, and primes indicate wave-induced fluctuations. Corresponding to these there will be a wave-modified volume emission rate,

s(X,y,Z,t) = so(Z)+a’(X,.Y,z,0. (4)

Standard linearization procedure would demand at this point the assumption that all primed parameters be small in comparison with the corresponding unper- turbed values. Such an assumption would seriously restrict the validity of the theory in application, how- ever, as a number of previous authors have noted (Weinstock, 1978 ; Walterscheid et al., 1987 ; Frederick, 1979 ; Molina, 1983). This is because in the presence of strong height gradients of one or more of these

parameters (as may typically occur for no), this assumption may be violated severely even when the perturbing wave is relatively small. Our own analysis, given in Hines and Tarasick (1987) for vertical viewing and generalized below, avoids this unnecessary assumption and assumes only that the perturbing gravity waves may be treated by linear theory.

We assume that observations of airglow brightness are made by a photometer with a field of view sufficiently narrow that (1) is valid, and inclined at some zenith angle t? to the vertical. We further assume that B is not too large, so that we may take the atmo- sphere to be plane-parallel. We can choose coor- dinates such that the observing system is at the origin and the central axis of the viewing column lies in the x-z plane, with z directed vertically upward. Then equations (1) and (2) may be written

I m B= (tzo +E’) set B dz

0

and

03 s (so +d)(To + T’) xc 6 dz

Tm= a B

(6)

The unperturbed brightness will then be

s 02

B, = a0sec9dz D

(7)

and the unperturbed measured temperature will be

We are interested not in the absolute values of the brightness or the measured temperature, but in how their values vary when the atmosphere is perturbed by a gravity wave. For this purpose we define a fractional brightness perturbation as

B’ (B--B,)

Bo - Bo

and a fractional ~~urbation of measured tem~ratu~ as

Krassovsky (1972) introduced the ratio

00)

1108 D. W. TARASICK and C. 0. HINES

as a useful parameter for airglow study, one he found (somewhat inaccurately) to depend in a simple fashion on the chemistry of the emission process. Subsequent work (Hines and Tarasick, 1987) has shown q to be dependent on gravity wave parameters as well as the emission chemistry. It remains a useful parameter for the study both of gravity waves and of airglow, and we now wish to investigate its dependence on parameters related to off-zenith observations.

As in Hines and Tarasick (1987), we wish to avoid the assumptions n’ << n,, and E’ << sO. The difficulty with these assumptions lies in the fact that many species of aeronomical interest exhibit profiles of con- centration versus height that vary sharply at certain heights ; e.g. the underside of the atomic oxygen layer. At such heights, the local concentration at a fixed point in space may therefore vary sharply-violating locally these assumptions-if the layer is translated upward or downward by a gravity wave. But the desired quantities, B and T,,,, both result from height integrations through the entire emitting layer, and so are not affected by such local changes, which merely remove to some new height what they take away from another. Observable effects, in these line-of-sight inte- grated measurements as distinct from in situ measure- ments, will result only from movement of parts of the layer relative to each other, and not from bulk movement of the layer relative to the observer. Ver- tical translation of the entire emitting layer simply changes the domain of integration ; by employing a coordinate transformation to the integrand that reflects this fact, we can avoid intr~u~ng to the math- ematical integral non-linea~ties that clearly do not affect the physical integral.

Mapping The coordinate transformation that we employ maps

the deformed layer back to its undeformed position before the column integration is performed. The gas currently in a small neighborhood about a position (x, y, z) within the viewing column will have moved there through some small distance (Lg. h) from its unperturbed position at (<,n, 0 = (x-Jy-g, z-h). We would like to relate N(x, y, z, t) to N( 5, v, 5, to). We do so by integrating the equation of mass conservation,

;gL -v.v,

where D/Dt indicates the derivative following the fluid motion, a/at-t-v-V, and where v = (u, U, w) is the per- turbation velocity :

N(x,Y, z,t)=No(l)exp(-~~~~V.vdr’), (13)

with the integrand being evaluated at points (x’, y’, z’) corresponding to the position of the gas in our small neighborhood at time t’. Since we define (5,~ [) to be the unpertur~d position of the gas, i.e. the position occupied by it at time I,, where to is some time prior to the onset of the wave, we have written N&c, q, 4, to) as simply N,(c).

Expanding the exponential as a Taylor series and retaining only the zero and first order terms, we find

N(x,y,z,t) = N,(t) (I- lI;;;V.vdt’). (14)

Any difference between V * v(x, y, z, t) and V ’ v( 5, q, 1, t) will be of second order at most [as may be seen by similarly expanding V - v(x, y, z, t)]. To first order then, the integral may be taken to be over t only, with position in space held fixed, at (x, y, z), or at ([,q, [), or at any point within a perturbation distance of x, y,z. For convenience we choose the point (x- h tan @,O, r-h), which lies along the line of sight at height <. We thus write

where the integrand is to be evaluated at height [ along the line of sight.

We assume that at the time scale of the waves con- sidered, the various gas species move together; i.e. diffusion is ignored. This assumption will not, in most cases, restrict the validity of our results, since molec- ular and eddy diffusion time constants below 120 km or so are generally long with respect to gravity wave periods. [The characteristic time for eddy diffusion- the time required to diffuse through one scale height- is about 100 h at 100 km (Winick, 1983).] We further assume, for the present analysis, that changes in the concentration of the reactive species owing to chemi- cal reactions may be neglected as well ; this assump- tion may not always be justified, since photochemical time constants for airglow reactions vary widely, and may be comparable to gravity wave periods. The case where this assumption is not valid forms part of the subject of a paper currently in preparation (Tarasick and Shepherd, 199Ob).

Under the foregoing assumptions, the mass con- servation equation for the minor species will he

1 Dn --= n Dt

-v*v. (16)

The observable effects of gravity waves on airglow

This equation has the exact solution

n(x.p,z,i)=n,(Oexp(- [~~~~V*vdt’), (17)

like that for the major species; it permits the first order expansion

k(x, y, z, t) = k ( W, Y, z, t))

= k(To(i))+ T'(i)WldTl~=r~

correct to first order. The temperature dependence is usually approximated either by

1109

n,(z)+n’(x,y,z,t)=n,(i) (1- lf;V*vdl.) (18)

(where the integrand is to be evaluated at height < along the line of sight) for waves of small amplitude, quite independently of any assumption about the magnitude of the vertical gradient of n,. This is to say that although

f

s V-v dt

10

is necessarily small with respect to 1, under the assumption that the perturbing wave is small, it is not required that n’(x, y, z, t) be small with respect to

n,(z). The equation of adiabatic change of state may be

written as

DT Dt = (y-l)T&, (19)

where p is the atmospheric density, and y the ratio of specific heats, c,/c,. This may be combined with the mass conservation equation, written for p,

P -a!= -v.v

Dt

and integrated to yield

T(x,y,z,t) = T,(i)exp( -b--l][~~~~V.vdr.).

(21)

As before, this may be expanded to :

T&)+T’(x,y,z,t)= To(C) I-(y-1) V vdt ( 63

(22)

where the integrand is to be evaluated at height { along the line of sight. No assumption has been made regarding the magnitude of the vertical gradient of T,,, nor is it required that T/(x, y, z, t) be small with respect to T,,(z).

Photochemical rate constants are in general func- tions of temperature. We may therefore write

k = k* e-u/T (24)

in which case kO = k* e- or by

‘ITo and [dk/dT],_ TO = ak,,lT &

k = k,(T/T,)-4 (25)

in which case k, = k,(T,/TJ4 and [dk/dT],=,O = -qk,/T,,. The two forms are equivalent, in the

linear approximation, for appropriate choices of the constants k*, T,, q, k* and a. Adopting the latter form for purposes of illustration, we find

k,(z)+k’(x,y,z,t) =k,G’) l+q(y-1) V vdt ( L%

(26)

with the integrand being evaluated at height c along the line of sight.

Linearizedform If we assume steady-state first-order perturbations

with time variation eio’, appropriate to a sinusoidal wave, then the integrals in equations (15), (18), (22) and (26) will yield

rc--iW-‘V*v(t),

where K is a constant of first or higher order in wave amplitude, one that derives from integration through the transient interval between onset and steady-state conditions. Being constant in time, it cannot affect our results for first-order fluctuations. Ignoring such dc offsets as irrelevant, we therefore find :

N,(z)+N’(x,y,z, t) = NO(l+iw-‘V*v)(i

n,(z)+n’(x,y,z, t) = n,(l +io-‘V*~)]~

k,(z)+k’(x,y,z, t) = k,(l -qiw-‘)V*~)]~

TO(z)+T’(x,y,z,t) = TO(l+iw-‘(y-l)V~v)(l (27)

to first order, where the Ii notation indicates that the right-hand side is to be evaluated at height c along the line of sight. Each of the perturbation terms contains the factor io- ‘V - v. Since, in a linear theory, the per- turbation terms can only appear in linear com-

1110 D. W. TARASICK and C. 0. HINES

binations with each other, it will be possible to write

(4) as

~&)+~‘(x,Y,z,t) = sO(l+i@‘xV*v)li, (28)

where x is some linear combination of the coefficients of io- ‘V - v in (27). A simple example may be helpful at this point: consider a process that can be rep- resented by a simple rate equation of the form

E = kn”Nfi. (29)

This might be the following, originally proposed for the production of the Oz atmospheric bands,

O+O+M + O,(b’X)+M, (30)

with quenching and other loss pathways neglected, andu=2,andjI= l.Then,

E~+E’ = (k,+k’)(n,+n’)“(N,+N’)B (31)

so that (27) gives

erJ(z)+z’(X,Y,z,0

= k,&N{[l +io-‘(cr+B-q[y- l])V*v], (32)

to first order. Thus in this example

&g(z) = k&Ngli (33)

and

x = a+D-4(Y--)I,. (34)

Using the currently accepted value of q for reaction (30), namely q = 2.0 (Campbell and Gray, 1973), we obtain x = 2.2.

With (28) for the volume emission rate, we now have, from (5) :

B= 5

m so[l+i~-‘XV.v]rsec6dz. (35)

0

All that remains to complete our coordinate trans- formation is to find its Jacobian. We set

dz = (dz/db) 15 dc

= (l+dhldc)l, di

= (1 -io-’ dw/d<)l, dc. (36)

Then upon linearization we obtain for (35)

s

co B= E~[~+~w-‘xV*~-~W-’ dw/dz]secf?dz,

0

(37)

where we have renamed the c as z in order to continue employing z as our working symbol for height. No change in the limits of integration is implied, since n, = 0 at both. The d/dz operation must be applied

along the inclined line of sight, in keeping with our choice of this line as the path of integration, and so we will employ

d dxa a dz=drz;+iiz=tan8L+$

when we come to evaluate dw/dz. Making the same series of substitutions in equation

(6), we obtain

s

cc sOTo[l+io~‘(x+(~-l))V~v

0

T, = - io- ’ dw/dz] set 0 dz

Bo+B’ (38)

with evaluation as in (37). We note that nowhere in these two expressions, or anywhere in their develop- ment, does the concentration gradient for the reac- tive species appear. This is a significant difference from the work of authors who have used a thin-layer, rather than a vertically-integrated, model (Walterscheid et al., 1987; Frederick, 1979).

Equation (38) may be linearized and simplified to produce, for (10)

where

s

co io-’

TZ s,,TO(y- 1)V.v dz

0 -=

KY

s

m (40)

goTo dz 0

is the relative temperature fluctuation in a constant- To atmosphere, and

s

m . -I

T: lw eO(TO-T:){XV.v-dw/dz} dz

o p= m m

s &oTo dz

0

(41)

is an additional component, which appears along with (40) in a varying- To atmosphere.

The variation of To between 75 and 110 km is in fact not great. Since taking To to be independent of z greatly simplifies the following analysis, we do so ; we subsequently discuss briefly the consequences to the analysis of allowing To to vary with height.

3. OBSERVATIONAL EFFECI-S

Gravity wave relations For the case where To is approximately independent

of height, we can use the gravity wave relations for

The observable effects of gravity waves on airglow

an isothermal atmosphere (Hines, 1960) to relate dw/dz and V-v. For a single wave of the form ez’2H+i(w’-k‘x~ky~~k), the dispersion relation is

1111

@ E g 5 = (x-p--iv)(y-1)-l. i

(50) 0 m

For complex emission chemistries, x is not inde- pendent of z (Tarasick and Shepherd, 1990a); however, it remains approximately so. By taking x and To to be independent of z, we are afforded the very simple form of equation (50). As will shortly be apparent, the loss of predictive accuracy that this entails is outweighed by a considerable gain in the ease with which the analysis can be compared with observations.

k,’ = k&$/w2 - 1) +w2/C2 - 1/4H2, (42)

where

0,’ = (y - l)gZ/C2, (43)

and where k,, k, and k, are the horizontal and vertical wavenumbers, with ki = k,’ + k,f, g is the acceleration due to gravity, C2 = yp,,/pO is the square of the speed of sound, and H = C’/yg is the scale height of the atmosphere. In this formulation w is the intrinsic fre- quency of the wave, which is not necessarily the observed frequency, if there is a background wind. The components of v are given by

u v w

x=r=z

= A ,z/2H+I(or-k~~-kkyy-k~z) 3 w

where A is a constant of proportionality. The scale factors X, Y and 2 are given by

X = ok,C’(k, -i(l -y/2)g/C2) (45)

Y = wk,.C2(k, -i(l -y/2)g/C2) (46)

Z = o(w2 - C2k,2). (47)

Using these expressions we obtain :

dw

dz=

[-ik, tan 0-ikz+1/2H][C2k,f-w2]V v

k,2g+ik,w2-02/2H

= (p+iv)V*v (48)

say, where ~1 and v are real. The quantity p+iv is not, of course, an observable parameter : physically it represents the ratio of the vertical expansion rate (vertical velocity divergence) of a fluid element, as measured along the line of sight, to its overall expan- sion rate (total velocity divergence).

This gives us for the fractional brightness per- turbation

-I

B’ lW s,,[X-p--iv]V’vdz

-= &

s

ou (49)

Ed dz 0

The factor in square brackets is independent of z, since C and H in (48) and therefore p+iv, are constant in a constant-T, atmosphere. Further, the To of the numerator and denominator in (40) is now constant and so may be cancelled from under the respective integral signs. Then (49) may be divided by (40), to yield for (11)

Fourier spectral analyses of real observations may yield the real and imaginary parts of @ directly, in which case (50) may be applied as it stands. However, since magnitude, 1 q 1, and phase, $J (the relative phase of brightness and temperature oscillations) are the quantities generally reported in the observational literature, we will express our results in terms of these parameters :

and

Iv1 = ((~-~)2+vz)“2(y-1)~’ (51)

( )i

rc x-p<o,v<o

4=tan-’ 2 + 0 X-P>0 X-P --n x-p<o,v>o.

(52)

The angle r#~ is positive where the oscillations in bright- ness lead those in temperature, and is defined by con- vention as lying between - 180” and 180”. Equations (51) and (52) take x to be real ; where chemical time constants are comparable to gravity wave periods, x may be complex, and these equations must be altered accordingly (Tarasick and Shepherd, 1990b).

Internal gravity waves Internal gravity waves, by definition, have k, real.

The range of values taken by p and v in such waves is not immediately apparent from (48). It is convenient to use the dispersion relation, (42), to rewrite (48) :

I -y/2+(1 -d/k;C2)w2C’/g

-k,k,w2C2 tan O/kzg’

1 -d/k,2g2 (k, real)

(53)

v= -yk,H-yk~~~~,~jb;yw2/2k:C’l ck, realj.

(54)

These two expressions, (53) and (54), are identical to

1112 D. W. TAMSCK and C. 0. HINES

0 6.3x10-

.3x lo-’

-6.3x10-’

FIG. 1. VARIATION OF p WITH HORIZONTAL AND VERTICAL WAVENUMBER ; tJ = 45”, k,, = 0. The value of C* used here is that appropriate to the mesopause and lower thermosphere, based on H = 6 km and g = 9.5 m s- ‘. Note that a wavenumber of 6.3 x 10m4 m- ’ corresponds to a wavelength of

10 km. The extrema occur at the edges of the domain.

(33) and (34) of Hines and Tarasick (1987) except for larger zenith angles the plane approximation for the the addition of the terms in k, tan 0. atmosphere is poor.

It may be shown that the denominator in (53) and (54) is z 1 for internal gravity waves. Further analysis shows that, for 0 = 0, ,U has a minimum of 0.3, at w2 = O,andamaximumof0.7,at~~ = coj,02/k,f = 0. The final term in ~1 depends upon the direction of propagation of the wave. Since the sense of horizontal propagation (the sign of k,) can be observed directly, this opens the possibility that, in principle, the sign of kZ, and therefore the sense of vertical energy propa- gation, may be determined. It may be shown, however, that

The variation of v is less complicated than that of p. It may be shown that the factor in square brackets in (54) has a minimum value of

I - 2(y - 1)/v N 0.43

and a maximum value of 1. For typical gravity waves, where k,2 >> kl, or where the zenith angle 0 is small, this will therefore give

max (k,k,co2C2 tan B/k,fg*) N w2c2 9 tan 0

2g2

(Y-l) = 2 tan 0 = 0.2 tan 0. (55)

(We take y = 1.4, here and elsewhere.) The potential addition to the range of p from this term, for 0 = 45”, is therefore only f0.2. The actual extension is less, since the extrema for this term do not occur at the same o as those for the remainder of (53) ; a numerical computation shows that for waves propagating in the azimuth of view (k, = 0), in the range I,, L; = l-1000 km :

v GZ -yk,H ‘v *8.8H/d, (kp real), (57)

where the positive sign corresponds to upward energy propagation (downward phase progression), and I, is the vertical wavelength 2n/Jk,(. This offers a practical method of measuring both the vertical wavelength and the direction of vertical energy propagation (the sign of k,). From (50) :

v = -(y--l))11 sin+, (58)

where $J is the phase angle between brightness and temperature oscillations, defined in (52). For positive values of x - p, if temperature oscillations lead bright- ness oscillations then the wave energy is propagating upwards.

0.22 N /L N 0.78. (56)

Figure 1 illustrates this variation. For smaller zenith angles or k, > 0 the range is, of course, less. For much

This result will be of interest to the observer, since measurements of these two parameters, vertical wave- length and direction of energy propagation, are typi- cally difficult to make. Since horizontal wavelength and propagation direction may be observed directly,

The observable effects of gravity waves on airglow 1113

and vertical wavelength and propagation direction may now be inferred, one may also infer from gravity wave theory the intrinsic frequency of the wave (i.e. that obtained in a coordinate system moving with the local background wind) : the wave may be completely characterized by a single scanning photometer. If the background wind is known (from Doppler inter- ferometer measurements, say), direct observation of the apparent frequency of the wave permits a second evaluation of the intrinsic frequency. Such a set of measurements will be overdetermined, and our theor- etical results may then be tested.

returning instead to equation (48), we see immediately that

where the denominator is now real, and

-kX[C2ki-w’]tan 13

’ = k~g+ik,w*--d/2H (k: < 0). 161)

The range of p and v is much larger in this case. A Lamb wave, for which

We can be more precise about the variation of the phase angle (52) with wave parameters and zenith angle. From (50) (53) and (54) the phase angle is (with 1 -d/k&~* N 1):

C’ki-co’=0 and w=O (62)

(Lamb, 1945), yields g = v = 0, while a wave such that

Q, = tan-’

i

yk,H+yk,H tan @[l -yw2,Qk~C2]

(x-l+y,‘2-(t--02/k~CZ)w2Cz/g2 . f k,kzw2Cz tan B/k,fg*) 1

(59)

k: = m4/g1 and kz = i(o’/g- l/ZH) (63)

The denominator of this expression is roughly constant, and typically equal to about 1 or 2, by the previous analysis. Except for waves for which lkxl 3 IkJ, the phase angle will have only a small dependence on 8. Moreover, by (.56), unless ;t for the emission process being observed is less than ~0.8, the angle will be less than f90”.

Evanescent waves

is sufficient to make p and v infinite. A more detailed examination of the variation of p is contained in Figs 2 and 3. Figure 2, for k, negative imaginary (energy density increasing downwards), includes both the zero of equation (62) and the infinity of equation (63), and is irregular in consequence. Figure 3, for k: positive imaginary (energy density increasing upwards), is remarkably smooth, and devoid of infinities since the k2 in (63) does not become positive imaginary until w > CB, (the acoustic wave cutoff frequency). In fact, p z 1 for w < w,. In both figures the planar area on the left represents the domain of internal gravity wave modes, which are not shown.

In the case of vertically evanescent waves (i.e. kz The phase of evanescent waves is independent of ima~nary) equations (53) and (54) are not valid; height, and in the vertical direction B’ and T’ will be

FIG. 2. VARIATION OF p WITH FREQUENCY AND HORIZONTAL WAVELENGTH FOREVANESCENT WAVES, WITH ENERGYDENSInANDHORIZONTALENERGYFLUXlNCREASlNGDOWNWARDS.

The large planar area on the left represents the domain of propagating internal gravity wave modes, which are not shown, The zero line described by equation (62) lies just to the right of the shallow ridge that borders this area. The sharp cleft results from the singularity described by equation (63). Values near the

cleft have been truncated to facilitate viewing of the remaining portion of the figure.

1114 D. W. TARASCK and C. 0. Him

FIG. 3. VARIATION OF p WITH FREQUENCY AND HORIZONTAL WAVELENGTH FOR F~~ANIBCENT WAVES, WITH ENERGY DENSITY AND HORIZONTAL ENERGY FLUX INCREASING UPWARDS.

The large planar area on the left represents the domain of propagating internal gravity wave modes, which are not shown. Except for w 1 o,, I( w 1. The values of w8 and w, used here and in the previous figure are

those appropriate to the mesopause and lower thermosphere, based on H = 6 km and 9 = 9.5 m s- ‘.

in phase or in antiphase. At non-zero zenith angles,

however, equation (61) for v reveals that there will be a phase difference between B’ and T’ that increases with zenith angle and with wavenumber in the azi- muth of the viewing cohrmn. Examination of (52) shows that this phase difference is in general quite large. The e-&z factor severely limits the amplitude, and hence the observability, of an evanescent wave away from the interface or reflection height that sup- ports it. It is probable, then, that typical observed evanescent waves will have I& x 0. On this simplifying assumption (61) reduces to

(W as does (54) after application of the dispersion relation. The quotient within brackets can be shown to lie in the range 0.6-l .4 for o < w,. Taking this x 1 we obtain :

v s - k,H tan 8 N +6.3H tan @/LX (k; z 0),

(65)

where the sign indicates direction of horizontal propa- gation relative to viewing azimuth, and 1, is the hori- zontal wavelength 27r/(k,l. Under the same conditions (52) approximates to

4 = tan- 1 ($!g!). (66)

The k, tan @ factor in the numerator implies that the observed phase angle will depend on the zenith angle 6, to an extent that will depend on the wave azimuth of propagation, and (inversely) on the horizontal wave- length. This is in marked contrast to the behaviour of the phase angle for internal waves, equation (59), where the phase angle is relatively insensitive to any of these parameters, except of course in the limiting case of waves of very small k,. Figures 4 and 5 show this phase behaviour for internal and evanescent modes for different horizontal wavelengths. It will be seen that these two classes of wave should, in general, be easily distinguishable in this manner.

The preceding analysis makes it apparent that the distinction between the two should be considered essential to any comparison of theory with experi- ment, since the effects of the two classes of wave on airglow can be quite different. Although gravity waves detected by other, ve~ically-resoIv~ methods have typically been internal modes, airglow observations may preferentially show evanescent waves, since these are not subject to line-of-sight phase cancellation in the way that internal waves are. Alternatively, they may show near-evanescent (i.e. large vertical wave-

The observable effects of gravity waves on airglow I115

-90 I ! 1, , r, I, I, , , I, I,, 1 -75 -60 -45 -30 -15 0 15 30 45 60 ‘75

Zenith angle 0 (degrees)

FIG. 4. VARIATIONOFTHERELATIVEPHASEANGLE$ BETWEEN

BRIGHTNESS AND TEMPERATURE FLUCTUATIONS WITH ZENITH

ANGLE@.

x. = 2. Dashed curve : for an evanescent wave of horizontal wavelength 50 km, t = 5.2 min. Solid curve : for an internat wave of horizontal wavelength 50 km, z = 25.3 min. Nega- tive values of C#J indicate that the sense of horizontal phase

progression is toward the observer.

90-‘L-L”““““‘“I’1

70- i

,' 50- ,' -

,'

3 30- ,_I'

8 __,'

b io- ,__- _,-_

____

$lO- __--

__.- __-_

___-

s-30- _,_. ,_1'

-50 - ,,*,'

-7o-

-90,. / 1,. I. /, * I (. / * I r IS. *, c -75 -60 -45 -30 -15 0 15 30 45 sb 75

Zenith angle 8 (degrees)

FIGS. VARIATIONOFTHRRELATIVEPHASEANGLE$ BETWEEN

BRIGHTNESS AND TEMPERATURE FLUCTUATIONS WITH ZENITH

ANGLE fl.

x = 2. Dashed curve: for an evanescent wave of horizontal wavelength 200 km, T = 13.2 min. Solid curve : for an internal wave of horizontal wavelength 200 km, z = 99.7 min. Nega tive values of 4 indicate that the sense of horizontal phase

progression is toward the observer.

length, or k, very small) internal waves, since the phase of these waves would not vary significantly over the emission layer thickness. Most of the studies in which 4 observations are reported find C#I % 0” (Krassovsky, 1972 ; Shagaev, 1974; Takeuchi and Misawa, 1981; Myrabo et al., 1983, 1987; Sivjee et al., 1987; Hecht et at., 1987; Viereck and Deehr, 1989). This is indica- tive of evanescent or near-evanescent waves. More- over, the suggestion by Hers6 et al. (1980) that their photogrammetric observations may be interpreted by assuming that the emissive layer is “ruffled like the

wavy surface of the sea” is intriguingly suggestive of evanescent waves. Future experiments should be designed to distinguish the two wave classes.

The foregoing conclusions should not be greatly altered by taking T, to vary with height, so long as

T&tTi remains the dominant term in T,JTz. For waves of vertical wavelength less than 10 km or SO,

however, T$‘TE may be the dominant term (Hines and Tarasick, 1987). In this case we obtain from (41) and (49):

m . -I

rl B’ 3 _ ‘OJ

i’

s c,[X----iv]V*vdz

o “s&-- B, T: -

s

cn eg dz

0

s

m

e. To dz 0

X

s

m

iw-’ cO(TO- T$)[x--p-iv1V.v dz 0

s

m

s,[x--g-iv}V*vdz 0

=

s

m . (f-3) .sO(TO/T~--l)[~-~-iv]V~vdz

0

This is essentially identical to equation (41) of Hines and Tarasick (1987), since the quantity in square brackets, (y - l)n”, is again approximately independent of height. Their conclusions following that equation apply therefore to the case of off- zenith observations as well, and the reader is referred to the extensive discussion there.

When neither T;jTi nor T”,/Tz may be neglected, or where x is allowed to vary significantly with height

s ic

D a&--p--iv]V*vdz

= / rm eoTo/T,o(p 1)V.v dz

0

s co

+ e~(To/T~--l)[~-~-iv]V~vdz 0 >

(68)

This expression does not admit of analytical sim- plification in the manner of that for q”. Its numerical evaluation is quite simple, but requires the use of a model atmosphere and simultaneous assumptions about chemical and wave parameters, so that this

1116 D. W. TARASICK and C. 0. HINIB

exact form obscures the relative contributions of chemistry and dynamics that the approximate form (50) so neatly separates.

4. COMPARISON WITH OBSERVATIONS

We have avoided consideration of the parameter x in (50), since it is a quantity that depends in a complex manner on the specifics of the production mechanism for a particular airglow emission. (It is, of course, precisely this dependence that permits the study of airglow chemistry via gravity waves.) Some know- ledge of its range of possible values is, however, neces- sary in order to evaluate the range of possible values of 1~1 and $. Equation (28) defines x as the ratio of fractional fluctuations in the volume emission rate to fractional fluctuations in number density. A simple production process rate will vary as the second or third power of number density, for a two- or three- body reaction, less the variation with temperature, since this is usually negative. One might therefore expect typical values of x to be x2. Additional pro- duction paths augment this, and quenching reactions lower it; where chemical time constants are com- parable to gravity wave periods, x will be complex. Notwithstanding, detailed analysis Iinds x~,(~~) N 1.8, and 1.4 6 xoH < 2.8 (Iarasick and Shepherd, 1990a,b).

Magnitude of q Taking as a nominal value x = 2, equation (51)

indicates that minimum 1~1 for internal gravity waves, from (56), for v = 0 is ~3. Lower values might be produced from processes with x less than this nominal value, or by a superposition of several waves, in the manner proposed by Hines and Tarasick (1987) as an explanation of Noxon’s (1978) observations of anom- alously low b~(. Maximum 111, for v = 0, would be ~4.5. Larger values could be produced where v > 0; this would imply, however, a phase angle 4 different from 0, by (52). These values are in approximate agreement with published results, which have found (~1 generally in the range 2-8, and typically 3-5, with C$ typically z 0 (Krassovsky, 1972 ; Sivjee et al., 1987 ; Hecht et al., 1987; Viereck and Deehr, 1989).

For evanescent waves, equations (62) and (63) have demonstrated that any value of lrtl is in principle poss- ible. Examination of Fig. 2 shows that this is true for a restricted range of wave periods and wavelengths ; outside this range p x 1. This implies that for zenith viewing, in which case v = 0, )q( should most com- monly be ~2.5 for these waves, although it would become greater for non-zero zenith angles.

Phase of q The majority of observations show 4 x 0”. Where

the observations are in the zenith (e.g. Myrabra et al., 1983, 1987; Sivjee et al., 1987; Hecht et al., 1987; Viereck and Deehr, 1989), this is consistent either with evanescent waves or, with allowance made for experimental error, with internal waves of vertical wavelength 3 100 km. Where the observations are made at oblique viewing angles (e.g. Krassovsky, 1972; Shagaev, 1974; Takeuchi and Misawa, 1981), this is consistent with internal waves of large vertical wavelength or, by (66), (Figs 4 and 5) with evanescent waves of sufficiently long horizontal wavelength.

Larger phase angles would be produced by internal waves of shorter vertical wavelength, and as suggested previously, this allows the vertical wavelength to be calculated, by (57) and (58). The observation by Hecht et al. (1987), of C$ N 40”, for example, indicates a vertical wavelength of about 40 km.

The seminal observations of Krassovsky (1972) deserve particular attention, since Krassovsky reports a value for the wave propagation speed, of 250 m s- ‘. He also shows a portion of his data (Fig. 1) from which one may deduce that B’ and T’ were approxi- mately in phase, with an apparent wave period of z 6000 s. If one takes his quoted wave speed to be the horizontal phase trace speed, 1,/z, one finds 1, N 1500 km. At this large speed Doppler shifting should be small, and the intrinsic period close to the apparent one. Inserted in the dispersion equation, for H = 6 km andg = 9.5 m s- ‘, these figures yield iZ N 370 km. Krassovsky also argues that the actual wave speed may have been as high as 270 m SC’, for reasons that are not entirely clear; inserted in the dispersion equation, this figure yields k;Z < 0 : the wave in this case would have been evanescent. Neither an internal wave of such long vertical wavelength nor an evan- escent wave would show a phase angle Q, appreciably different from 0, for a horizontal wavelength of 1500 km, so that the two cases cannot be distinguished on this basis. From (53), the former case yields p = 0.30, while from (60) the latter gives p = 0.066. In both cases, v z 0. For our nominal value of x = 2, inserted in (51), this gives (~1 = 4.25 for the internal wave, and lr]J = 5.0 for the evanescent wave. Krassovsky found 1~1 between 4 and 5.

5. CONCLUSIONS

Hines and Tarasick (1987) stressed the need for proper measurement of such wave parameters as period and propagation speed, if observations of gravity waves in airglow are to provide insight into either the behaviour of atmospheric gravity waves or

The observable effects of gravity waves on airglow 1117

that of the airglow. The extension of the theory to oblique viewing angles has introduced two new par- ameters, zenith angle of view and wave azimuth of propagation relative to azimuth of view. The latter makes necessary the measurement of at least three of the four basic wave parameters : Ax,, k,, 2: and intrin- sic, or rest-frame period, if the wave is to be adequately described. Identifi~tion of the intrinsic period may in general require knowledge of the background wind, since gravity-wave propagation speeds in the middle and upper atmosphere are frequently of the same order as wind speeds. Doppler shifts may then be very large.

Such measurements must of course be made for each observation of /Q] and #. In the absence of such detailed info~ation about the gravity wave or waves producing the airglow fluctuations, the theory can account for almost any observed q, but it will yield correspondingly little new insight. Not least among the reasons that require proper identification of the wave characteristics is the necessity of distinguishing between evanescent and internal waves. Although published work to date has in general assumed that the observed fluctuations in brightness and temperature were due to internal gravity waves, it would appear that in some, if not many, cases they were in fact due to evanescent waves. Our analysis has shown that the effects of the two types of wave on airglow can be quite different, but it has also provided a simple obser- vational means of resolving the ambiguity in some cases. The same analysis has provided a means of deriving the approximate vertical wavelength and the sense of vertical energy propagation from # measure- ments. The latter may provide an indication of the relative frequency of occurrence of gravity waves due to aurora1 and tropospheric sources, respectively. The former will be of considerable use experimentally, since published reports have occasionally of necessity assumed values for the vertical wavelength of observed waves. Such values, usually of order 10 km (Myraba et al., 1987), are more than an order of magnitude smaller than the typical values we find from the analysis of published airglow data.

We therefore recommend that future observations of gravity waves in airglow should, as a minimum, consist of measurements of brightness and tempera- ture simultaneously at least at three (non-collinear) points in the sky. This is the minimum necessary to allow horizontal wave speed and propagation direo tion, as well as apparent wave period, to be determined from the data. A larger number of sampling points- ultimately, an imaging system-is of course prefer- able ; in particular observations in the zenith would be useful for approximate determination of the vertical

wavelength and sense of vertical propagation,by equa- tion (57) [although if zenith observations are not avail- able these parameters may be determined from (54)]. Observations at large (>45”) zenith angles would allow the unambiguous separation of evanescent from internal waves to be made for the widest possible range of waveleng~s (Figs 4 and 5).

If the horizontal wavelength and the wave period are known, then the dispersion equation (42) may be used to derive the vertical wavelength (or wave- number) and this may be compared with that derived from the C#J measurements by (57) or (54). The same calculation will also indicate whether the wave is evan- escent (kz < 0) or internal ; this can also be compared with C$ information via Figs 4 and 5, or equations (59) and (66). Such cross-checking of parameters may be quite important, since the experimental error in gravity wave observations is frequently large.

Once the wave parameters have been determined as accurately as possible, the parameters p and v may be calculated, either from (53) and (54) if the wave is internal, or from (60) and (61) if evanescent. The values found for p and v may be used in either of (51) and (52), along with the measured values for 1111 and (p, to solve for x, and the values found for x may in turn be compared with the values predicted by photochemical theory.

Evaluation of x for current models of O,(b’X) and OH chemistry has already been performed (Tarasick and Shepherd, 1990a,b), and may be necessary for other emissions in the future. In addition to such detailed chemical analysis, several extensions to the dynamical analysis suggest themselves. Where it is possible to observe rapid fluctuations of airglow inten- sity, it may be of interest to apply the analysis of Sections 2 and 3 to airglow fluctuations induced by waves in the acoustic regime. This would require some modification to the discussion following equations (53) and (54), and the approximate phase expressions (59) and (66), since at several points we have assumed o < We. The effects of non-steady-state chemistry are likely to be important for these much shorter wave periods as well. The inclusion of wave dissipation, which will also alter ,u and v, and the use of the full expression (68) to calculate the effect on q of a vari- able-temperature atmosphere, for different values of x, p and v would be of considerable value, if only to demonstrate that these effects are unimportant. Wave dissipation has been added to the Walterscheid et

aZ. (1987) non-integrated model by Hickey (1988a,b), who finds it to be an appreciable effect. A fourth extension, or adaption of the theory, interesting in its own right, may be rendered more so by the effects of non-steady-state atomic oxygen chemistry : investi-

1118 D. W. Tmsrc~ and C. 0. HI-

gation of the effect upon airglow of atmospheric tides. Measurement of the background wind requires

DoppIer-shift m~s~ment capab~ity such as that provided by an optical interferometer {Shepherd ed al., 1984 ; Rees et al., 1984). Such observations offer interesting possibihties for further analysis, since the relationship between the fluctuations in Doppier winds and t~rn~ratu~ caused by gravity waves will be quite different from that between brightness and rotational temperatures. Recent advances in Iidar technology make possible the detailed measurement of a~osphe~c density fluctuations during the passage of a gravity wave in the middle atmosphere. In com- bination with Doppler-shift and ordinary photo- metric observations, such measurements would pro- vide a powerful too1 for studying both gravity waves and airglow, and pose many new and interesting prob- lems for theoretical analysis.

Aekno~ie~~emen~~We wish to thank Dr Gordon Shepherd of C.R.E.S.S., York University, who provided the initial inspiration for this project and considerable support through its completion. We have benefited as well from the thought- ful criticism and advice of Dr Rudy Wiens of C.R.E.S.S., and from discussions with Dr Brian Solheim, also of C.R.E.S.S., Drs Peter Taylor and Jack McConnell, of York University, Dr G. G. Sivjee, of Embry Riddle University, and W.R.C. Underhill, of McMaster University.

We are indebted to Dr W. F. J. Evans of Environment Canada for supposing one of us (D.W.T.) as a post-doctoral fellow while this research report was written. This research was supported by the Natural Sciences and Engineering Research Council of Canada. The Arecibo Observatory is operated by Corner1 University under a cooperative agree- ment with the National Science Foundation.

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