/fl- 46 

[ KAS A-Cfi-1799 34) MID CLE A'IMOSEti ERE MODEL FNG 

Zinal Eie~ort [Jchns HcFkics Uriv,) 23 p 

CSCL 048 

N8 7-13023 

Unclas 

G3/46 44714 

Final Report on 

Mi ddl e Atmosphere Model i ng 

Darrell F. Strobe1 

Principal Investigator 

Dept. of Earth and Planetary Sciences 

The Johns Hopki ns University 

Baltimore, MD 212128 

December 1, 1984 to November 30, 1985 

NAG 5-489 
Brief Summary 

The pri nci pal research effort accomplished under NASA Grant NAG 5-489 

was the research reported in Attachment 1 entitled "Energy Balance Constraints 

on Gravity Wave Induced Eddy Diffusion in the Mesosphere and Lower 

Thermosphere." 

issue of Journal of Geophysical Research--Atmospheres. 

This paper is in press and should appear in the November 1985 

In this paper we make a strong case for turbulent diffusive transport 

of constituents and potential temperature at rates significantly less than used 

by most workers in the field. Recent analysis of microwave derived water vapor 

mixing ratio profiles in the upper mesosphere by the PI substantiate this 

conclusion of the paper. 

We also argued in this paper that larger diffusion coefficients are 

perrni tted for chemical ly active species than the constraints deri ved from 

thermodynamic considerations. 

fusion coefficients used by researchers to transport odd oxygen can be supplied 

Recent work by the PI shows that the large difby n-vity Y' u waves whcse amplitudes are near their saturation value. 

In addition some preliminary research was carried out in non-LTE 

radiative transfer. This research will be continued in subsequent NASA grants 

and used in the analysis of SME data. 
Energy Balance Constraints on Gravity Wave Induced Eddy Diffusion 

in 

The Mesosphere and Lower Thermosphere 

bY 

Darrell F, Strobel' 

John P. Apruzese2 

and 

Mark R. Schoeber13 

June 1985 

' Department of Earth and Planetary Sciences 

The Johns Hopki ns Uni versi ty 

Baltimore, MD 21218 

Plasma Physics Division 

Naval Research Laboratory 

Washi ngton, DC 20375 

Laboratory for Atmospheres 

Goddard Space Flight Center 

Greenbelt, MD 20771 
Abstract L 

Breaking gravity waves generate and maintain a background level of turbulence which is capable of producing substantial cooling and/or heating in the 

upper mesosphere and lower thermosphere. 

breaking gravity waves is critically dependent on the eddy Prandtl number (Pt) 

applicable to mesospheric turbulence. When Pt - 1, our calculations of the heat 

budget for the mesopause region imply that the globally averaged eddy or turbulent diffusion coefficient cannot exceed lo6 cm s . This upper limit on turbulent diffusion applies 1oo”’both potential temperature transport and chemically 

inert tracer transport when radiative damping is negligible. For chemically 

active species larger diffusion coefficients are permitted, because the effective eddy diffusion coefficient is increased by an additive term L/2y2, where L 

is the chemical loss rate and y is the vertical wavenumber. 

The net thermodynamic effect of 

2 -1 

For Pt - 4-6, the turbulent diffusion of momentum (DM) is sufficiently 

greater than the turbulent .diffusion of heat (DH) that the conversion of gravity 

wave energy to heat with high efficiency nearly balances the divergence of the 

downward eddy heat f ux in the wave breaking zone. Therefore the heat budget of 

the mesopause region would no longer provide a powerful and useful constraint on 

DH. If Pt exceeds 6 with high efficiency for energy conversion to heat, gravity 

waves would heat the mesosphere throughout the wave breaking region. 

* 
1 

1. Introduction 

Lindzen's (1981) parameterization of turbulence and stress generated by 

breaking gravity and tidal waves initiated considerable research on constituent 

transport and the momentum and heat budget of the mesosphere (e.g. Holton 1982, 

1983; Dunkerton, 1982; Schoeberl et al., 1983; Apruzese et al., 1984; Fritts and 

Dunkerton, 1985; Garcia and Solomon, 1985). The identification of a specific 

mechanism which produces turbulent mixing allows a quantitative description of 

associated effects even though it may be highly parameterized. 

the research referenced above it would appear that the intensity of the turbulence as represented by the eddy or turbulent diffusion coefficient, D(cm s 

must be greater than lo6 cm s 

tuent transport, whereas our previous study of the heat budget (Apruzese et al., 

1984) concluded that D < lo6 cm s 

On the basis of 

2 -1 ), 

2 -1 from studies of the momentum budget and consti2 -1 2 -1 

Taken at face value we might deduce that the effective eddy Prandtl 

with a preferred value of 6 x lo5 cm s . 

number nust be greater than one. 

diffusion coefficient, DM, divided by the thermal diffusion coefficient, DH,) 

In a turbulent atmosphere tracer transport and potential temperature transport 

should be described by the same coefficient, DH, in the limit of a chemically 

inert species and adiabatic motion, if K theory adequately describes turbulent 

transport processes (see Kraichnan, 1976, for comments on this simplification). 

The studies of Allen et al. (1981) and Garcia and Solomon (1985) clearly suggest 

2 -1 that DH exceed lo6 cm s for chemically active species in contrast to the upper 

limit on DH obtained by Apruzese et al. (1984). 

(1985) their net DH had only a weak dependence on the chemical loss rate and thus 

should be approximately equal to the heat diffusion coefficient. They assumed 

Pt = 1 and calculated DM with Holton's (1982) modification of Lindzen's (1981) 

parameterization of breaking gravity waves. 

> lo6 cm s 

(The eddy Prandtl number, Pt, is the momentum 

* 

In the case of Garcia and Solomon 

As indi cated above they obtai sed % 

2 -1 in the mesopause region, with largest values during solstice. 
2 

I. 

The principal objective of this report is to re-examine the constraints 

on turbulence imposed by the heat budget and determine whether there is sufficient theoretical evidence to support the hypothesis that the eddy Prandtl 

number is greater than one in the mesosphere. 

is calculated with turbulent diffusion coefficients commonly used in chemical 

The mesopause thermal structure 

models and deduced from mean zonal wind deceleration. As seen below, extreme 

mesopause temperatures of less than 100 K are produced by the large net cooling. 

2. Model 

The model adopted for this study includes the non-LTE version of the 

- *-d - 

Apruzese et al. (1982) two-stream IR algorithm, the physical processes of molecular 

and turbulent heat conduction, odd oxygen transport, gravity wave dissipation, 

and NO IR cooling as described in Apruzese et al. (1984). The approximate IR 

algorithm incorporates exact cooling to space escape probabilities from Fels and 

Schwarzkopf’s (1981) extensive tables on CO2 transmittances to generate 

accurate cooling to space rates. 

is treated approximately; consequently only thermal disturbances with vertical 

wavelengths greater than 8 km can be treated with sufficient accuracy in the 

model. 

Radiative exchange between atmospheric layers 

* 

None of the results presented below are sufficiently sensitive to our 

approximate IR cooling and heating rates to warrant a substantially more 

accurate treatment of IR radiative transfer at this time. 

Vertically propagating gravity waves carry‘ energy which is preferentially deposited as heat in the wave breaking zone as a result of the turbulence 

created. Schoeberl et al. (1983) gave a simple analytic expression for this 

heating in terms of the efficiency E with which gravity wave energy is converted 

to heat 

N~D~ 

H = -?c (1 + P;’)E K s-l 

P 9 
3 

where N is the buoyancy frequency, and cp is the specific heat at constant 

pressure. 

ward turbulent heat flux whose divergence (convergence) results in cooling 

(heating). 

field is 

Also turbulence generated by breaking gravity waves leads to a downThe usual expression for this process acting on the mean temperature 

where Ca > 0 for heating, p is mass density, T is temperature, g is gravita- .I 

-tional acceleration, and z is altitude. 

Given the variety of physical and radiative processes outlined above 

and discussed in depth in Apruzese et al. (1984) the globally averaged 

equi 14 brium temperatures in the mesosphere and lower thermosphere were calculated as a function of the 'eddy diffusion coefficient, DM, and eddy Prandtl 

number, Pt. 

tion between the turbulent diffusion and temperatures could be represented by 

the product of their global averages. 

In computing these temperatures it was presumed that the correlaIn other words if <> repcesents a global 

average then we assumed 

a D - aT,.= - a <D> - a <T> <- 

az az az az 

We have tested this assumption under solstitial conditions using an analytic - 

model of the mesospheric wind and temperature structure. Using Lindzen's formula D a (U-C)~ we found that using global averages to represent the heat flux 

is accurate to better than 1%. 

, 
4 .i 

The global heat budget is also affected by the convergence of the down- 

_- 

ward heat flux associated with the anticorrelation of the zonal mean vertical 

- 

velocity, w, (a result of gravity wave breaking) and the zonally averaged ternperature, i.e. the term -- ' a <pGi> in the thermodynamic energy equation. This 

term is substantially smaller than the net heating term due to gravity waves, Hg 

+ Cg. 

Garcia and Solomon (1985) with a mean meridional circulation driven by gravity 

P az 

For example at the mesopause under solstitial conditions the model of 

wave breaking predicts < D > = 1.5 x 106cm s-l and < Gf > = 16 K cm s-l. The w 

- 

and fields have an approximate height dependence of p In. The convergence of 

thistrrrat flux is spread over at least 2 or 3 scale heights and has a magnitude 

I 

of at most 10% of Hg + Cg. This is not surprising as w and f have large arnplitudes only at polar latitudes; the global average is accordingly small. 

3. Results 

In our initial set of calculations, four representative turbulent diffusion coefficients were constructed and are shown in the top panel of Fig. 1. 

Model A is our "best fit" profile from Apruzese et al. (1984) (their model 4), 

although no real effort was expended to generate a truly excellent fit. 

has an asymptotic value for D of 6x10' em's-' at high altitudes and is characModel A 

terized by very low values throughout the mesosphere. 

Model B is an average of the summer and winter diffusion coefficients 

estimated by Llndzen (1981) for breaking gravity waves. Model C includes, in 

addition to gravity wave generated turbulent diffusion, Lindzen's estimate of the 

contribution from the breaking diurnal tide. 

regarded as 1 lustrati ve rather than rt cjorous globally averaged values of the 

turbulent diffusion coefficient. Finally Model D is the envelope of the highly 

structured vertical eddy diffusion coefficient used by Allen et al. (1981) in a 

These two profiles should be 
5 

1U model of mesospheric chemistry. Most 1D models of mesospheric chemistry 

require eddy diffusion coefficients in excess of lo6 cm s and possibly as 

high as lo7 cm s , Model D is certainly not 

2 -1 

2 -1 2 -1 . With a peak value of 2x106 cm s 

an extreme case. 

In the middle panel of Fig. 1 the four globally averaged temperature 

profiles calculated with their respective turbulent diffusion coefficients are 

compared with the CIRA 1972 temperature profile, which is representative of the 

observed globally averaged temperature. 

that Pt = 1, i.e. DM = DH E D, and E = 1. 

for conversion of gravity wave energy to heatis _based on numerical simulations 

of Fritts and Dunkerton (1984). Note that Model A underestimates the temperature 

by up to 20°K between 65 and 75 km, the region where IR cooling is dominated by 

&hot bands and isotqpjc bands (Dickinson, 1973; Wehrbein and Leovy, 1982). 

comparison of our cooling rates with the very recent computations of Dickinson 

(1985) indicates that our rates are too large in the 63-82 km region, 

shown below a 40% reduction in the CO, cooling rate eliminates the discrepancy 

In these calculations it was assumed 

The assumption of a high efficiency 

A 

As will be 

between observed and calculated temperatures. 

calculated temperature profile above 95 km from CIRA 1972 is due to the uncertainty in the C02(u2) deactivation rate by 0 collisions as discussed at length in 

Apruzese et al. (1984). The thermosphere is not of fundamental importance in 

The departure ,of the Model A 

this study and is not discussed further. 

In the lower panel of Fig. 1 the divergence of the turbulent heat flux 

is illustrated. For Model A below the mesopause the cooling rate does not 

exceed 5 K d'l . 

the respective mesopause temperatures are 65 and 46 K. 

pause the value of D in Model C is significantly larger than Model B's value. 

With Models B and C which have a peak value of D - 3x106 crn2-l, 

Note that at the meso- 
. 6 

The reason for such cold mesopause temperatures is the extreme divergence of the 

turbulent heat fluxes. In Model B it peaks at - 30 K d’’, whereas in Model C it 

approaches 350’K d-’. With solar heating rates of less 5 K d-l, the equilibrium 

temperature mst become very cold to minimize IR cooling and maximize convergence of downward heat flux from the thermosphere carried by molecular conduction. Model D also yields a cold mesopause (-100 K) but up to 65 km is in 

close agreement with Model A, which suggests that below 65 km the atmospheric 

heat budget is insensitive to values of D that do not exceed the Model D profile. Above 80 km the large values of D in Model D generate a divergence of the 

turbulent heat flux in excess of - 63K& 

To analytically illustrate the strong sensitivity of temperature to 

--**. 

turbulent diffusion, we write the net heating due to gravity waves as 

when the small 

for a diatomic 

-4 -2 3.5 x 10 s , 

assumed in Fig. 1 

a ’T 1 aDM 

az DM az term proportional to 7 is neglected, H:= ---) and Cp is 

gas. For typical mesospheric values N2 =+(.i.i+$) = 

-I 

6 2 -1 H = 6 kin, DM in units of 10 cm s , with Pt = 1 and E = 1 as 

6 2 -1 DM(10 cm s ) K d-’ 

Thus for constant DM, Hg + Cg = -7.5 K d’ when DM = lo6 cGs-1 or -22 K d-1 for 

DM = 3x106 cm2s’l, which may be compared against a maximum solar heating rate in 

this region of 5 K d’’ Thus when Hg + Cg < -5 K d’l (DM > 7x105 crn2s’l >, the mesopause rust cool down to decrease IR cooling and increase convergence of the 

dcxilr.ward thermal conduction heat flux. Note that when nm - 0.7, i.e. DM a 

exp(z/l.4H), then Hg + Cg - 0. 

H 

’ ‘U 

If DM increases more rapidly with altitude than 
7 

this, gravity waves will produce a net heating of the region. Similarly if DM 

.+-.- 

increases more slowly with height than exp (z/1.4H) net gravity wave cooling 

occurs . 

Returning to Fig. 1, it is apparent why Model A with an asymptotic 

value of DM = 6x105 cm2s-1 is close to the upper limit on h, beyond which 

disastrous consequences on the mesopause temperature result from gravity wave 

cooling. 

generates net gravity wave cooling over this region. Only when 91 > lo6 C$S-’, 

does this cooling rate become excessive and the mesospheric temperature plunges. 

In Models B and C, h > 106 cm2s-1 from 50 to 75 km and 91 does not increase 

with height more rapidly than exp(z/ZH). 

this region drives the mesospheric temperatures below 100°K above 75 km. 

Model D has Dpl increasing as exp (z/2H) from 50 to 80 km and hence 

The large gravity wave cooling rate in 

Examination of Eq. (3) reveals that gravity wave cooling is dominated 

t by the last term-; . Although over limited regions HD can be less than 

1.4H, eventually 

other terms with net gravity wave cooling. 

must level off (HD >> H) and the last term overwhelms the 

Only in the limit of large Pt can 

y reduced over the entire 

E = 1, then Hg + Cg - 0. 

gible gravity wave cooling. 

- 

With 

the contribution of this last term be substantial 

mesosphere. For constant ~(HD = =) when Pt = 6, 

finite HD, a lower Prandtl number will yield negl 

On the other hand, for large Pt(Zl0) gravity wave 

heating. 

To test the effect of the Prandtl number 

additional calculations were performed. In these 

rates between 63 and 82 km were reduced by 40% to 

breaking will only produce net 

dependence numeri cal ly , some 

calculations the C02 cooling 

bring them into closer 

agreement with Dickinson’s (1985) more accurate computations and to demonstrate 

that our conclusions do not depend critically on the IR cooling rates. In addition the C02(v2) deactivation rate by 0 coiiisions was adjusted to 
. 8 

2xlO-”exp(-8UO/T) cm3s-l (slightly larger than the value adopted by Gordiets et 

a]. (1982)) to bring the calculated temperatures in the thermosphere into 

agreement with CIHA 1972. For the reader interested in an extended discussion 

of the issues raised by this adjustment, Section 3.6 of Apruzese et al. (1984) 

should be consu 1 ted. 

In Fig. 2 Model 1 contains the above modifications with E = 1, Pt = 4 

and a 91 profile that is - 3.3 times the Model A diffusion coefficient in 

Fig. 1. 

exp [(z-65 km/H] cm s 

high altitudes given in figure caption.) 

cm2s‘1 for Model 1, slightly less than for Model A. The calculated mesospheric 

temperatures in the 65 to 75 km region are now in good agreement with CIRA 1972. 

Also the lower thermospheric temperatures above 95 km are also in good agreement 

with CIRA 1972. 

where our calculated temperatures exceed CIRA 1972 by up to 15” K, and would 

suggest that enhanced gravity wave cooling can be accommodated by lowering Pt. 

If Pt = 1, then the Model 2 temperature profile is obtained with a !%?sopause 

temperature of 140 K. 

sitivity to Pt that we would expect from our discussion of Eq.(3). 

(The precise expression for DM is DM = DoD,/(Do+D,), where Do = 3 x lo5 

2 -1 , H = scale height, and D,,, is the asymptotic value at 

The asymptotic value of DH is 5x105 

\-- 

The principal region of disagreement is between 80 and 95 km 

. Comparison of Models 1 and 2 i)lustrates the extreme senIt is confined to the region above 80 km where gravity wave cooling exceeds solar heating. 

In Model 3, Pt = 4 but DM is twice its value in Model 1. 

ature is only lowered by 10 K with this factor of 2 increase in DM. 

km the temperature change is insignif cant. 

when Pt = 4 is due to the large cance lation of gravity wave heating and cooling 

terms. For example at 90 km, Hg + Cg =, -2.7 K d-’ for Model 1 and -3.4 K d-’ 

for Model 3, with Cg = -6.3 and -10.6 K d-’, respectively. When compared with 

Above 100 km, the temperBelow 100 

The weak dependence on 9.1 (or DH) 

, 
9 . the solar heating rate of - 11 K d-’, which is rapidly increasing with height at 

90 km, the 0.7 K d-’ change in Hg + Cg is insignificant. 

4. Discussion and Summary 

The present study clearly demonstrates the importance of the Prandtl 

number for mesospheric turbulence. For large Pt(’6) and E = 1, gravity wave 

heating approximately balances gravity wave cooling and the globally averaged 

atmospheric temperature profile is controlled by radiative processes. 

6, gravity wave breaking produces only net heating. 

For Pt > 

In the other limit Pt d 1, 

gravity wave ’ breaking is general ly characterized by very strong cooling but with 

small heating at the bottom of the breaking zone. In this limit, the cooling is 

sufficiently strong to impose a powerful constraint on the maximum permissible 

level of turbulence in the mesosphere without plunging the mesopause temperature 

down to - 100 K. With allowance for uncertainty in our radiative transfer model 

this upper limit is D = lo6 cm s . Our preferred asymptotic value is (5-6)x105 

cm s Below 70 km, the D profile from Model D is an absolute upper 

limit. 

2 -1 

2 -1 when Pt = 1. 

From a close examination of Figs. 5 and 6 in Apruzese et al. (1984), 

one concludes that deducing a local upper limit to D independently of the total 

D profile is risky below the mesopause. 

(1984) almost satisfies the upper limit constraint DH < lo6 cm2s-l (DH = 

1.2~10~ cm2s’l for high altitudes) and is less than the diffusion profile of 

For example, Model 3 in Apruzese et al. 

Model D below 70 km. But the calculated mesopause temperature is - 130 K. , 

As discussed above, more rapid transport of constituents is needed than 

allowed by the heat transport. 

zation of linear wave chemical transport by eddy diffusion given in Strobe1 

Schoeberl et al. (1983) extended the parameteri- 

’ 

(1981) to include the effect of breaking gravity waves. Their Eqs. (27) and 
L 10 

(28) IiIay be combined to yield an effective vertical eddy diffusion coeffcient of 

where y is the vertical wave number and L is the chemical loss rate is units of 

s-1. 

by the breaking gravity wave and the second term represents eddy transport by 

the gravity wave for chemically active (L#O) species. Chao and Schoeberl (1984) 

have demonstrated that the last term, which arises from direct diffusive 

transport by the breaking gravity wave, should be negligible rather than 1/2 DH 

because the perturbation vertical velocity and mixing ratio are-almost in 

quadrature. 

The first term represents diffusion by the background turbulence created 

The eddy transport of heat and constituents by a gravity wave in 

the breaking region cannot be represented by a linear diffusie?oce<s 

according to their analysis, Schoeberl et al. (1983) showed for chemically 

active species in the mesosphere and lower thermosphere that the second term in 

Eq.(5) can equal or exceed the first term when OH obeys our upper limit 

constraint. The authors are not ware of any ID chemical model that has tested 

whether FZz with DH < lo6 cm s 

in the mesopause region. 

bulence, then the L/2y2 contribution to Kzz with DH < lo6 cm s 

2 -1 can adequately account for ionstituent transport 

If it can be proven that Pt = 1 for mesospheric tur2 -1 would be essential. 

k is the horizontal wave number, u is the mean zonal wind, and c is the zonal 

phase speed. 

Note that expression (5) is valid only when k(u-c) >> y2D~, and L, where 

The determination of the eddy Prandtl number in the mesosphere is an 

exceedingly difficult experimenta measurement. To date only inferences are 

available that are not definitive or compelling, Although Justus (1967) determined Pt to be -3 from photograph c tracking of rocket released chemical clouds 
11 . 

and analysis of turbulent wind data in the 90 - 110 km region, there was considerable scatter in his data points. Recently Fritts and Dunkerton (1985) have 

theoretically examined constituent and heat fluxes driven by localized gravity 

wave breaking where the breaking zones are small in vertical extent in comparison to the vertical wavelength. 

bulence leads to an eddy Prandtl number given by 

In their analysis this localization of tur2an + a2 Pf’ = 1 -n+r 

where a is a measure of supersaturation of the breaking gravity wave (a = 1 for 

saturation), n is a measure of the localization of turbulence, i.e. 

D = nl-tos$ 

these quantities. 

2, Pt - 7. 

6. 

The analysis of Fritts and Dunkerton (1985) strongly suggests that Pt 

large supersaturation (a > 2) results in the msosphere from breaking gravity 

waves. 

by Theon et al. (1967) and Philbrick et al. (1983) indicate supersaturation by 

up to a - 1.3, but not 2. 

and + = kx+mz-ct is the wave phase with the usual definitions for 

3.5 and n = 

(2)n 

They recommend a = 1.3; thus for n = 1.5, Pt = 

With a = 1, n = 1.5 and 2, the respective values of Pt are 3.3 and 

With even modest localization of turbulence (n = l), Pt 4 2 for a 4 1-1.5. 

2 unless 

t 

Then the eddy Prandtl number can be reduced to Pt = 1. Observations 

The Fritts and Dunkerton (1985) model is, however, an analytic treatment of convection effects for a single breaking gravity wave discussed by Chao 

and Schoeberl (1984) and may not accurately represent the breakdown of a gravity 

wave packet. For example, the turbulent region, initially set up by the folding 

of the potential temperature surface, will expand in time as stable air is 

entrained into the turbulent zone. .Thus the heat flux will vary in time and a 
12 

turbulent zone which best represents the time mean heat flux may be quite different froin the zone where breaking initially begins. The actual question of 

heat transport by breaking gravity waves may not be settled until detailed 

numerical sirmlations are performed. 

Finally, it is important to recall that the arguments put forth by Chao 

and Schoeberl (1984) and Fritts and Dunkerton (1985) for heat transport induced 

by a single breaking wave do not include heat transport produced by stable waves 

encountering a turbulent zone and/or being radiatively damped. The effect of 

these processes gives rise to an eddy Prandtl number of only 1. This discussion 

raises the possibility that Pt may indeed be variable with an initial value that 

Is large at the onset of wave breaking and as the turbulence develops dissipation of stable waves drives the eddy Prandtl number down to an asymptotic value 

of 1. i. - 

- 

-.&F.. 

Although this paper has been concerned p r i n c i p a l l y w i t h the globally 

averaged eddy diffusion coefficient, the constraints on it discussed above are 

also applicable locally. For example, the maximum values of occur above the 

cores of the zonal jets near the stratopause. If Pt were - 1,- then a secondary 

circulation would have to be set up in the atmosphere to locally balance the 

strong cooling when > 10 cm s . 6 2 -1 

Acknowledgments: This work was initiated while DFS was at the Naval Research 

Laboratory. DFS was supported by the Office of Naval Research 

and NASA's Upper Atmospheric Research Office. 

ported by NASA Contract NP.G 5-489. 

At JHU, DFS was supHe was also supported in part by 

the Naval Research Laboratory at Science Applications International 

Corporation. 
13 

REFERENCES 

Allen, M., Y. L. Yung, and 3. M. Waters, Vertical transport and photochemistry 

in the terrestrial mesosphere and lower thermosphere (50-120 km), Geophys. 

-- Res. 86:3617-3981. 

Apruzese, J. P., M. R. Schoeberl, and D. F. Strobel, Parameterization of IR 

cooling in a middle atmosphere dynmics model. 1. Effects on the zonally 

averaged circulation, - J. Geophys. -- Res., 87:8951-8966, 1982. 

Apruzese, J. P., D. F. Strobel, and M. R. Schoeberl, Parameterization of IR 

cooling in a middle atmosphere dynamics model. 2. Non-LTE radiative 

transfer and the globally averaged temperature of the mesosphere and lower 

thermosphere, J . Geophys . -- Res . , 89: 4917-4926, 1984 - 

Chao, W. C., and M. R. Schoeberl, A note on the linear approximation-of gravity 

L.. * 

wave saturation in the mesosphere, --- J . Atmos. Sci . , - 41: 1893-1898. 

Dickinson, R. E., Method of parameterization for infrared cooling between 30 and 

70 km, J. Geophys. Res., 7&:4451-4457, 1973. 

Dickinson, R. E., Infrared radiative cooling in the mesosphere and lower 

- - 

thermosphere, J. Atmos. Terr. Phys., in press, 1985. - --- 

Dunkerton, T. J., Wave transience in a compressible atmosphere, part 111, the 

saturation of internal gravity waves in the mesosphere, --- 3. Atmos. Sci., 

- 39~1042-1051, 1982. 

Fels, S. B., and M. D. Schwarzkopf, An efficient, accurate algorithm for 

calculating CO, 15 band cooling rates, 3. Geophys. Res-, 86:1205-1232, 1981. 

Fritts, D. C. and T. J. Dunkerton, A quasi-linear study of gravity wave saturation and self-acceleration, --- 3. Atmos. Sci., in press, 1984. 

Fritts, D. C., and T. J. Dunkerton, Fluxes of heat and constituents due to 

convectively unstable gravity waves, --- J. Atmos. Sci., in press, 1985. 
14 

Garcia, R. R., and S. Solomon, The effect of breaking gravity waves on the 

dynamics and chemical composition of the mesosphere and lower thermosphere, 

- J. Geophys. - Res.,g:3850-3868,' 1985, 

Holton, 3. R., The role of gravity wave-induced drag and diffusion in the 

momentum budget of the mesosphere. --- J. Atmos. Sci ., - 39:791-799, 1982. 

Holton, 3. R., The influence of gravity wave breaking on the general circulation 

of the middle atmosphere, --- 3. Atmos. Sci., - 40:2497-2507, 1983. 

Justus. C. G., The eddy diffusivities, energy balance parameters, and heating 

rates of upper atmospheri c turbulence, - 3. Geophys. - Res., 72: 1035-1039, 1967. 

Kraichnan, R. H., Eddy viscosity in two and three dimen,sions, --- J. Atmos. Sci. -. -L 

- 33: 1521-1536, 1976. 

Lindzen, R, S., Turbulence and stress awing to gravity wave and tidal breakdown, 

- 3. Geophys. -- Res., 86:9707-9714, 1981. 

Philbrick, C. R., K. U. Grossmann, R. Hennig, G. Lange, D. Krankowsky, D. 

Offerma,nn, F. J. Schmidlin, and U, von Zahn, Vertical density and temperature 

structure over Northern Europe, Adv. Space -- Res., 2:121-124, 1983. 

Schoeberl, M. R., D. F. Strobel, and 

gravity wave breaking and stress 

- 88:5249-5259, 1983, 

Strobel, D. F., Parameterization of 

J. P. Apruzese, A numerical model of 

in the mesosphere, - 3. Geophys. - Res-, 

inear wave chemical transport in planetary 

atmospheres by eddy diffusion, J. Geophys. Res., 86:9806-9810, 1981. 

Theon, J. S., W. Nordberg, L. B. Katchen, and 3. J. Horvath, Some observations , 

on the thermal behavior of the mesosphere, --- 3. Atmos. Sci ., 24:428-438, 1967. 

Wehrhein, W. M., and C. B. Leovy, An accurate radiative heating and cooling 

algorithm for use in a dynamical model of the middle atmosphere. 

- Sei ., 2:1532-1544, 1982. 

-- 3. Atmos. 

. 
15 

Fi gure Captions 

Fig. 1 (Upper panel) Eddy or turbulent diffusion coefficients described in 

the text as a function of altitude and used to calculate temperature 

profiles (middle panel ) which are compared with CIRA 1972 temperature 

profile. (Lower panel) the cooling or heating rate associated with 

the divergence or convergence of the downward eddy heat flux. All 

- calculations with Pt = 1, 

Fig. 2 Calculated temperature profiles with modified CO, cooling rates as a 

function of altitude compared with CIRA 1972 temperature profile which \-. 

-1c. - 

illustrate sensitivity to Pt. The respective values of Pt and asymp- .. 

-- 

totic values of DM (D,) are: 

[l, 2 x lo6], and (Model 3) c4, 4 x lo6 cm s 

(Model 1) [4, 2 x lo6], (Model 2) 

1. 2 -1 
.. 

. - 

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