Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp. 4711-4715, December 1975
Geophysics
Estimates of stratospheric pollution by an analytic model (eddy diffusion/NO/supersonic transport)
DONALD M. HUNTEN Kitt Peak National Observatory, Tucson, Arizona 85726
Communicated by Harold S. Johnston, September 22, 1975
ABSTRACT With suitable choices of the height profile of eddy diffusion coefficient, the vertical flow of an inert tracer is given by an analytic solution. Odd nitrogen, or NO., from aircraft exhausts can be regarded as such a tracer, and the amount in the stratosphere resulting from a source of a given strength can be immediately calculated. The resulting destruction of ozone is then estimated with the help of a formula obtained from earlier work.
effect on ozone as the source flies higher and higher above the tropopause. Quantitative estimates of this effect are given below. In an eddy-diffusion model of vertical transport, it is assumed that the flux of tracer is proportional to the gradient of mixing ratio, df/dz. For such a formulation to make sense, the mixing ratio must be a global,. or at least hemispheric, average; it is best if f is uniform worldwide. Natural substances, notably CH4 and N20, exist that obey this last restriction very well, for they originate in the ground and are well distributed by motions in the troposphere. As they move up through the stratosphere, these molecules are slowly destroyed by photochemical processes at a calculable rate. The resulting gradient of their mixing ratio has been observed, and they are therefore particularly valuable as tracers for the rate of the mixing process. This application is discussed below in some detail. An artificial source of odd nitrogen within the stratosphere is fairly concentrated geographically, and the applicability of a one-dimensional eddy treatment must be questioned. The situation is not nearly as bad as might be expected, as seen in the behavior of radioactive clouds from nuclear tests (5, 6, *). In a few months, material spreads all around a latitude circle, with somewhat slower north-south spreading. This horizontal spreading is able to give fair uniformity within a hemisphere before a large amount of vertical motion can take place. The reduction of ozone is still further smoothed, as discussed below. A more serious difficulty is in defining the nature of the tropopause and the stratospheric motions just above it. The tropopause is the point at which the lapse rate of several degrees Celsius per kilometer gives way to a height-independent temperature, characteristic of the stratosphere. The height is around 9 km in polar regions, and 17 km over the equator. To a first approximation, horizontal mixing takes place along surfaces parallel to the tropopause (7, 8). Thus, at least at the heights reached by aircraft, the important height is the distance above the tropopause, not above the ground. In the present context, the important aspect of the tropopause is not its position on a temperature profile, but the fact that it is the beginning of a region in which vertical transport is very sluggish. In the language of the next section, the eddy diffusion coefficient suddenly becomes smaller by a factor of 33. The lower stratosphere is therefore a barrier to vertical transport of pollutants. Eddy diffusion The vertical flux X (molecules cm-2 sec') of tracer is assumed to be proportional to the gradient of its mixing ratio. It is convenient to write the proportionality constant as Kna,
0 In 1970, Crutzen (1) published a suggestion that NO and NO2 might catalyze the destruction of ozone in the stratosphere, and a year later he, and also Johnston, pointed out that this effect would probably be enhanced by the operation of a fleet of supersonic transports, or SSTs (2, 3). The problem has been studied by the Climatic Impact Assessment Program of the Department of Transportation, and in parallel efforts elsewhere. Reduced to its simplest terms, the catalytic cycle is NO + 03 o NO2 + 02 NO2 + 0 - NO + 02. Since the 0 atom is produced by photolysis of another ozone molecule, the net result is to convert the two ozone molecules into three oxygen molecules. An important part of the problem is to describe the removal of the catalysts NO and NO2 from their sources in the stratosphere. This removal can be described by simplified models of the atmosphere that use the concept of vertical "eddy diffusion." These models are usually treated in a computer, and their basic simplicity is therefore hidden. This paper expounds a method that gives an analytic solution, and removes the need to take on faith the results of an elaborate computer program. A representation of the atmosphere is chosen that permits this simple analytic solution of the flow of an inert tracer. The source is at some height within the stratosphere, and the sink is in the troposphere. The distribution with height of the tracer can thus be found for any given source. The oxides of nitrogen are by no means inert: the problem is important because of their chemical activity. But this activity merely turns them into one another without destroying their basic character, which involves the presence of a single nitrogen atom in a molecule. The entity odd nitrogen approximates closely an inert tracer within the stratosphere, for computations of transport. The principal constituents are NO, NO2, and HNO3. The last of these, nitric acid, is rapidly removed in the troposphere by solution and rainout. The end product of the method discussed here is a mixing ratio f (molecules of odd nitrogen per molecule of air) as a function of height.-This quantity is the basic input to chemical models of ozone destruction, such as those of Crutzen (2), Johnston (3), and McElroy et al. (4). We do not treat the chemistry, but rather quote the published results. Of partic* H. S. Johnston, D. ular interest is the effect of the flight altitude of aircraft Kattenhorn, and G. Whitten, J. Geophys. Rev., sources. A given amount of odd nitrogen has more and more in press. 4711
4712
Geo'physics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Hunten
100 1000 RELATIVE VALUES
FIG. 1. The simple two-layer model discussed here. (a) The eddy-diffusion coefficient K; (b) the density of the minor constituent for equal sources at zq = 16 and 22 km; also the residence time from ref. 9; (c) the corresponding mixing ratios.
where K is the eddy-diffusion coefficient, and na is the number-density of air molecules: df = -Kna,. [1] K is a purely empirical quantity, to be determined (as shown in examples below) by substituting observed quantities in [1]. For an initial discussion, we shall take K8 - i0) cm2 sec1 in the troposphere and K8 = 3 X 103 cm2 sec-1 in the lower stratosphere, as illustrated in the left panel of Fig. 1. A source of molecules, Q cm2 sec'1, is assumed at height zq. We consider only the steady state; since there is no sink at high altitudes, the flux is zero above zq. As the material flows down from the source to a sink in the troposphere, its flux below zq is X =-Q. Eq. [1] may be solved for df/dz, and immediately integrated if suitable functional forms are chosen for X1, K, and na. Exponential functions of height are particularly convenient; 4 is taken constant in this paper. If Zt is the height of a reference level, we write = nat exp (-h) na = nfa, exp - (z - z,)/H [2] K - K, exp - (z - z,)/HK = Kt exp (ah) Kna = Ktnat exp - (z - z,)/Hp = Ktnat exp -(1 - a)h.
The Hs are scale heights, negative if the quantity increases with height, and 1/Hp = 1/H + 1/HK = (1 - a)/H; a is defined as -HI/HK, and h is a dimensionless height variable. The solution is then f = A
qHp Kna
n =flffla=Afla fn = An,, -
-A
4t'H
_
=
OHP K
Kna(1 - a)
An,
OH -
K(1- a)
low values of the troposphere. -This exponential curve is the same for a given source at any height, and its extension is shown by dashed lines. The curves for number density become asymptotic to a constant value n = QH/Ks The curve for mixing ratio rises indefinitely, because a fixed amount of pollutant is being mixed with a rarer and rarer atmosphere. "Residence times" can be worked out for one definition (9), and are shown in Fig. lb by an extra scale. It is useful to represent the results of a transport model by an "injection coefficient" a, following a suggestion by McElroy et al. (4). As Fig. lc illustrates, the mixing ratio above the source is independent of height; we shall call it fq. By definition, fq is related to the source strength Q by
fq
[5]
aQ.
=
A typical value for at at 18 km is 5 X 10-17 cm2 sec. A source strength Q = 108 molecules cm-2 sect then gives a mixing ratio 5 X 10-9, or 5 ppb. If several sources are present, perhaps including natural ones, their effects are simply added. The reduction of ozone by odd nitrogen occurs mainly by reactions above 25 km. Thus, for a source at any lower height, the injection coefficient permits an immediate calculation of the corresponding change of odd-nitrogen mixing ratio. A curve of a as a function of source height has the same form as the envelope in Fig. 1c. Fig. 2 shows two such curves for different assumptions about the height of the tropopause and the variation of eddy coefficient with height. The four points are taken from ref. 4. With that exception, the models have been chosen to be analytically soluble, and
[3]
[4]
where A is the integration constant and n is the numberdensity of tracer. A is determined by matching solutions at the boundary of each layer. For Fig. 1, K is constant in each layer and Hp = H. A is zero in the troposphere (solutions are also indicated for a positive value), and in the stratosphere, with K = K8 and Zt at the tropopause,
E
:
20
_
w I
~~~~~~2
C.)
0
QHa
n= KaKs [1 - 0.97 exp (-h)]. (The factor 0.97 = 1 - K/Kt is obtained by matching the solutions at the tropopause.) This expression holds below the source, where 4 =-Q. Above the source height zq, the mixing ratio is constant, and the number density is proportional to na. Below the source, there is an exponential rise from the
,
U,
lo 0
IO19
lo-is
-1017
lo-16
INJECTION COEFFICIENT acm sec
FIG. 2. Injection coefficients for the two models of Table 1. The squares were taken from the papers of McElroy et al. (4, 20).
Geophysics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Hunten
EDDY COEFFICIENT K, cm sec
4713
~~I
I
40H E
CH4
NO 2
30
it
xCrL 20_
E
C,w
CO0
10 2
le6 2 1 .5 VOLUME MIXING RATIO f, ppm
.2
a =
4
Z-
Hp
)
-
D] -
a)hq
-
D]
FIG. 4. Loss time constants, r, for use in Eq. [8]. The N20 results are from ref. 17, and the others were calculated with the OH and O(1D) concentrations of ref. 4.
(Z) = J(nl/r)dz.
[6]
Expression [6] is obtained by evaluating [3] at the source height Zq (or hq), and dividing by Q -4. C is equal to Hp/(Ktnat), and D is usually near 1. The algebraic expression for D is often fairly complicated, and it may better be determined by matching numerical values at the boundary with the next lower layer. The parameters C, D, Zt, and Hp are given in Table 1. The choice of Ks is discussed in the next section, as is the reasoning behind the adoption of the 3-layer model. The reference height Zt for each layer is at its base, which is also the tropopause in the 2-layer model. Use of natural tracers The eddy-diffusion coefficient K in Eq. [1] is a parameter to be determined from observations of appropriate natural tracers. Here we shall focus on methane, CH4, which is the best-observed of the suitable substances. The relevant data are shown as points in Fig. 3. Inspection of these results shows a small but distinct gradient of the mixing ratio starting at the tropopause. The measurements, obtained by Martell, Ehhalt, and collaborators (10, 11), are of exceptional quality. Samples, obtained by balloons or, at 50 km, by a rocket, were analyzed on the ground by gas chromatography. Methane can also be observed spectroscopically (12, 13); the results confirm the ones shown, but do not define the gradients well enough for the present analysis. From the slope in Fig. 3 is obtained Hf, the scale height of the mixing ratio. Then Eq. [1] may be written =
lo'
z
C[exp (1
K (Z)
Id0
duced, in the stratosphere. The photochemical lifetime T(Z) is computed as described below and is shown in Fig. 4. The flux is then obtained by integrating the destruction rate per unit volume from infinity down to height z:
for the injection coefficient is
C[exp (zq
109
DESTRUCTION TIME CONSTANT, SEC
FIG. 3. Methane mixing ratios from the balloon data and the 50-km rocket measurement of refs. 10 and 11. The curved line, drawn by eye, was used to obtain the profile of K. an expression
I7
lo8
fl(z/) n, (df ldz)
(z)Hf(z) n(z)
[8]
Substitution into Eq. [7] gives a somewhat scattered set of points, which are well represented by the straight line and upper scale of Fig. 3. Many other representations of the data could have been picked, but this one, an exponential function of height, is convenient because it allows the analytic solution of the eddy equation used here. The eddy coefficient shown in Fig. 3 is represented by K = 2200 exp (z - 14)/9.43 cm2sec-',
which is 3.5 X 104 cm2 sec1 at 40 km. In the form of [2], = -9.43 km. If H = 6.2 km, then Hp = 18.1 km and a =0.657. The same methane data have already been analyzed by Wofsy and McElroy (14); their technique was to postulate various profiles of K and pick the one that predicted the best methane profile. The resulting Ks differ only slightly below 20 km, and are greater by a factor 4 above 30 km; correspondingly, they predict too much methane at 50 km. Another analysis, with data from an additional balloon flight, has been mentioned by Ehhalt et al. (15); their values are somewhat larger yet, partly from use of daytime values for r(z) instead of a diurnal mean (16). In any case, the lifetime is still an important cause of uncertainty. The important loss processes for methane are attack by hydroxyl OH and metastable O(ID), both of whose concentrations must be calculat-
HK
Table 1. Model parameters for use in Eq.
[7]
The methane number density n(z) follows directly from the measurements shown in Fig. 3; to get Hf(z) we must estimate the slopes, which are obviously rather uncertain. The curve shown is typical of those that can be drawn through the points; they all give similar results for K, within a factor 2 or so. The quantity that is not directly observed is the flux X, which requires a photochemical calculation. This calculation is simple for CH4, a gas that is only destroyed, not pro-
Zt,
Model
All 1. (2-layer) 2. (3-layer)
Layer t s b s
km 11
i5 10 14
H,Hp,
[61
km
C
D
6.2 18.1 6.2 18.1
0.082 20 0.29
0 0.992 0.7 0.979
16.7
In Column 2, t is the troposphere (K = 105 cm2 sec-), b is the
boundary layer (4 km deep, K = 3 x 104), and s is the stratosphere (K from Fig. 3). Units of C are 10- 17 cm sec.
4714
Geophysics:
Hunten
ed. [Transport is not significant for these short-lived constituents; the O(lD) is the product of ozone photolysis and the source of OH radicals by reaction with H20.] Fig. 4 shows the result, and also profiles for N20 (17) and CO. The lifetime of CO is too short for it to be a really useful tracer. N20 will be valuable once we have adequate observations, for it is destroyed primarily by photolysis, whose rate can be confidently calculated (17). The choice of a discontinuity of IK at the tropopause is not demanded by the methane data, as is obvious from Fig. 3, but it is permitted. Far more influential are observations of CO and O3, as pointed out by Seiler and Warneck (18). Other evidence about Ks for all regions of the atmosphere is discussed by Hunten (19). It is an obvious conclusion that the potential value of tracers for defining transport parameters in the stratosphere has barely begun to be realized. The techniques exist to make the measurements; they simply have not been used on an adequate scale. Although the situation is not so bad for methane, more data would still be valuable. The natural tracers have been observed almost exclusively at a latitude near 300 N, where the height of the tropopause is 15 km. It has been argued above that the relevant height scale for other latitudes should be based on the tropopause. In addition, a boundary layer can be inserted in the region 10-14 km, with an intermediate value of K. This 3-layer model (2 in Table 1) has been adopted with considerable success in a very recent study* of the transport of carbon-14 from nuclear tests, and will be used here as standard.
Odd nitrogen, natural and artificial Before one can have confidence in predicted effects of artificial sources, he should be able to show that he can adequately describe the natural situation. Such a description is a bit beyond the simple method discussed here, but it can be used to illuminate an existing computation, that of McElroy et al. (4, 20). N20, discussed above as a tracer, is also the principal source of odd nitrogen in the stratosphere. Its lifetime, shown in Fig. 4, is almost entirely determined by photolysis into N2 + O. This process gives even nitrogen, but a slower one, attack by O('D), gives two molecules of NO. According to ref. 20, the source strength Q is about 7 X 107 molecules cm-2 sec1 spread mainly over the 20- to 40-km range. For the same profile of K, the injection coefficient at 30 km is a = 17 X 10-17 cm2 sec; the product aQ gives a mixing ratio f = 12 X 10-9 or 12 ppb. This result is insensitive to the assumed values of K: with faster transport the source Q is larger, but a is smaller because the loss of odd nitrogen is greater. Another possible source of odd nitrogen is destruction of NH3, though the actual end product is not known and its mixing ratio near the surface is only 6 ppb. McConnell (21) finds that the source, if such it is, has an effective height around 20 kmi, and a magnitude very close to that for N20. But the injection coefficient is smaller by a factor of 7; thus, at most, the source from NH3 makes only a minor contribution, as McConnell also points out. Further study would be worthwhile, but is unlikely to reverse this conclusion. Some observational data exist for the various forms of odd nitrogen. The amounts computed for the N20 source are in fair agreement (4, 20), though there are still some discrepancies. Without the natural odd nitrogen, the total amount of ozone in the stratosphere would be greater by a factor of 2; with it, the calculated ozone profile agrees very well with observation (4). With less ozone, the competing destruction
Proc. Nat. Acad. Sci. USA 72 (1975) Table 2. Predicted effects of various assumed fleets of aircraft Aircraft
Subsonic
Number Q', kT/y Height, km Effective height (, 10-'7 cm2 sec
1000 600
0.35 0.34 0.47
03 loss, %
60 60 16 18 4.45 0.43 0.60
13 15 1.27 1.22 1.7
12 14
fq, ppb
Present Advanced SST SST
200 800 20 22 9.6 12.3 17
The "advanced SST" is arbitrarily taken to produce four times the odd nitrogen of a Concorde. The 2-km difference between the actual and effective heights allows for the translation from highlatitude flight paths to a model for 300 latitude. The last three lines give the injection coefficient for model 2, the odd-nitrogen mixing ratio, and the expected ozone reduction for one hemisphere.
process becomes less efficient, and odd nitrogen accounts for about %4 of the total destruction. Though the discrepancies still need to be studied, the general agreement supports the general validity of the description. The response of the ozone concentration [0s] to an increment of odd nitrogen is complicated by feedback effects, which are analyzed in ref. 4 and found to be small but not negligible. The final result of an elaborate model, summarized in Table 5 of that paper, can be represented by -
100
A 03
[03]
=
6
=
1.405/
-
0.0105f
,
[9]
where f, in parts per billion is equal to fq in Table 2. Thus,
an increment of 1 ppb would reduce the ozone abundance 1.4%. For a brief review of the chemical aspects, see
by 6 =
ref. 22. Artificial sources must be treated carefully because they are not spread worldwide like the natural tracers. Most of the remarks here will be confined to air traffic in the northern hemisphere at about 450 N. The material is assumed to spread uniformly over the whole hemisphere. In truth, there is some leakage across the equator, but there will also be a tendency for a maximum amount near the source. These two effects come close to compensating each other, and the estimates should be valid for middle latitudes. There may indeed be a considerable "corridor effect" at the heights where the aircraft fly. But the odd nitrogen must move upwards to 25-30 km to affect the ozone, and the ozone in turn spreads outwards and downwards from this same region. These two smoothing processes will almost certainly eliminate any substantial corridor effect in the ozone reduction. The area of a hemisphere is 2.55 X 1018 cm2. Odd-nitrogen emissions are usually expressed in i09 g/year, or kilotons/year, in the form NO2. One kT/y (which is the rate of production by a Concorde or TU-144 flying 7 hr/day) therefore corresponds to a source strength of 1.62 X 105 molecules cm-2 sec1. With the injection coefficient a in units of 10-17 cm-2 sec1 and the source Q' in units of kT/y, we obtain for the mixing ratio
fq
=
0.0016aQ' parts per billion.
[10 1
Most long-distance air traffic is at high latitudes, where the height of the tropopause is 10-12 km (23). It is assumed
Geophysics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Hunten
that the 12 km figure should be converted to 14 km for use with model 2 of Table 1; thus, 2 km is added to all heights. Predictions for several assumed fleets are given in Table 2. The number of "jumbo" jets is 2.5 times the 1974 fleet, and has been taken to fly somewhat higher. The prediction is extremely sensitive to the exact height, as illustrated by the two assumptions in the table. The effects shown are beginning to be appreciable; but one can be fairly sure that today's smaller fleet has not yet made a serious dent in the ozone. A relatively modest fleet of 60 Concordes and TU144s could be significant, and a larger fleet of advanced aircraft very serious indeed. Needless to say, the effects from different groups of aircraft are additive. The predictions for stratospheric flight are considerably greater than those of McElroy et al. (4), who deliberately adopted a highly conservative approach. Their tropopause was high, 16 km, and no adjustment was made to actual flight altitudes. Emissions were averaged over the globe, rather than half of it. For flights primarily over the North Atlantic, the present assumptions should be realistic. The impressive accuracy (4) in the prediction of mean stratospheric ozone suggests that the chemical scheme is correct to better than a factor of 2. The transport calculation is more uncertain, particularly for sources very near the tropopause where a factor-of-10 uncertainty is probable. The other predictions are thought to be good to a factor of 3. Summary The effect of odd nitrogen on stratospheric ozone can be estimated with good accuracy by breaking down the problem into its parts. The sources and sinks of odd nitrogen are first treated as discussed here. Second, the partitioning among the major forms NO, NO2, and HNO3 is calculated. Finally, the destruction of O3 by NO2 is obtained. The second and third steps can be represented by Eq. [9], obtained from the detailed results of McElroy et al. (4). The simplified method is more in the domain of a hand calculator than a computer. Although one can be confident regarding the general behavior of the eddy-diffusion coefficient up to 40 km or so, it would be valuable to have many more observations of natural tracers, especially CH4, N20, and CO. Another important quantity is the OH concentration, needed for interpreting the CH4 and CO distributions. The region just above the tropopause is especially tricky, and especially important because of the possible impact of subsonic aviation. At present this impact is negligible, but as traffic grows and flies higher, a noticeable depletion of ozone is a real possibility. To protect the stratosphere, subsonic jet engines, as well as their
4715
supersonic cousins, may have to be modified (24) to produce less odd nitrogen. Kitt Peak National Observatory is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. I thank A. J. Broderick and D. H. Ehhalt for valuable communications. 1. 2. 3. 4. 5. 6.
7.
8. 9. 10. 11. 12. 13. 14.
15.
16. 17. 18. 19. 20. 21. 22.
23. 24.
Crutzen, P. J. (1970) Q. J. R. Meteorol. Soc. 96,320-325. Crutzen, P. J. (1971) J. Geophys. Res. 76,7311-7327. Johnston, H. S. (1971) Science 173,517-522. McElroy, M. B., Wofsy, S. C., Penner, J. E. & McConnell, J. C. (1974) J. Atmos. Sci. 31, 287-303. Gudiksen, P. H., Fairhall, A. W. & Reed, R. J. (1968) J. Ceophys. Res. 73,4461-4473. Johnston, H., Whitten, G. & Birks, J. (1973) J. Geophys. Res. 78,6107-6135. Davidson, B., Friend, J. P. & Seitz, H. (1966) Tellus 18, 301315. Reed, R. J. & German, K. E. (1965) Mon. Weather Rev. 93, 313-321. Hunten, D. M. (1975) Geophys. Res. Lett. 2,26-28. Ehhalt, D. H. & Heidt, L. E. (1973) J. Geophys. Res. 78, 5265-5271. Ehhalt, D. H., Heidt, L. E. & Martell, E. A. (1972) J. Geophys. Res. 77,2193-2196. Kyle, T. G., Murcray, D. G., Murcray, F. H. & Williams, W. J. (1969) J. Geophys. Res. 74,3421-3425. Cumming, C. & Lowe, R. P. (1973) J. Geophys. Res. 78, 5259-5264. Wofsy, S. C. & McElroy, M. B. (1973) J. Geophys. Res. 78, 2619-2624. Ehhalt, D. H., Heidt, L. E., Lueb, R. H. & Roper, N. (1974) in Proceedings of the Third Conference on the Climatic Impact Assessment Program (U.S. Department of Transportation), pp. 153-160. Crutzen, P. J. (1974) Geophys. Res. Lett. 1, 205-208. Bates, D. R. & Hays, P. B. (1967) Planet. Space Sci. 15, 189197. Seiler, W. & Warneck, P. (1972) J. Geophys. Res. 77, 32043214. Hunten, D. M. (1975) in Atmospheres of the Earth and Planets, ed. McCormac, B. M. (Reidel, Dordrecht), pp. 59-72. McConnell, J. C. & McElroy, M. B. (1973) J. Atmos. Sci. 30, 1465-1480. McConnell, J. C. (1973) J. Geophys. Res. 78, 7812-7821. Schiff, H. I. & McConnell, J. C. (1973) Rev. Geophys.. Space Phys. 11, 925-934. Jocelyn, B. E., Leach, J. F. & Wardman, P. (1973) Water, Air, Soil Pollut. 2, 141-153. Ferri, A. & Agnone, A. (1974) Am. Inst. Aeronaut. Astronaut. paper no. 74-160.
Geophysics
Estimates of stratospheric pollution by an analytic model (eddy diffusion/NO/supersonic transport)
DONALD M. HUNTEN Kitt Peak National Observatory, Tucson, Arizona 85726
Communicated by Harold S. Johnston, September 22, 1975
ABSTRACT With suitable choices of the height profile of eddy diffusion coefficient, the vertical flow of an inert tracer is given by an analytic solution. Odd nitrogen, or NO., from aircraft exhausts can be regarded as such a tracer, and the amount in the stratosphere resulting from a source of a given strength can be immediately calculated. The resulting destruction of ozone is then estimated with the help of a formula obtained from earlier work.
effect on ozone as the source flies higher and higher above the tropopause. Quantitative estimates of this effect are given below. In an eddy-diffusion model of vertical transport, it is assumed that the flux of tracer is proportional to the gradient of mixing ratio, df/dz. For such a formulation to make sense, the mixing ratio must be a global,. or at least hemispheric, average; it is best if f is uniform worldwide. Natural substances, notably CH4 and N20, exist that obey this last restriction very well, for they originate in the ground and are well distributed by motions in the troposphere. As they move up through the stratosphere, these molecules are slowly destroyed by photochemical processes at a calculable rate. The resulting gradient of their mixing ratio has been observed, and they are therefore particularly valuable as tracers for the rate of the mixing process. This application is discussed below in some detail. An artificial source of odd nitrogen within the stratosphere is fairly concentrated geographically, and the applicability of a one-dimensional eddy treatment must be questioned. The situation is not nearly as bad as might be expected, as seen in the behavior of radioactive clouds from nuclear tests (5, 6, *). In a few months, material spreads all around a latitude circle, with somewhat slower north-south spreading. This horizontal spreading is able to give fair uniformity within a hemisphere before a large amount of vertical motion can take place. The reduction of ozone is still further smoothed, as discussed below. A more serious difficulty is in defining the nature of the tropopause and the stratospheric motions just above it. The tropopause is the point at which the lapse rate of several degrees Celsius per kilometer gives way to a height-independent temperature, characteristic of the stratosphere. The height is around 9 km in polar regions, and 17 km over the equator. To a first approximation, horizontal mixing takes place along surfaces parallel to the tropopause (7, 8). Thus, at least at the heights reached by aircraft, the important height is the distance above the tropopause, not above the ground. In the present context, the important aspect of the tropopause is not its position on a temperature profile, but the fact that it is the beginning of a region in which vertical transport is very sluggish. In the language of the next section, the eddy diffusion coefficient suddenly becomes smaller by a factor of 33. The lower stratosphere is therefore a barrier to vertical transport of pollutants. Eddy diffusion The vertical flux X (molecules cm-2 sec') of tracer is assumed to be proportional to the gradient of its mixing ratio. It is convenient to write the proportionality constant as Kna,
0 In 1970, Crutzen (1) published a suggestion that NO and NO2 might catalyze the destruction of ozone in the stratosphere, and a year later he, and also Johnston, pointed out that this effect would probably be enhanced by the operation of a fleet of supersonic transports, or SSTs (2, 3). The problem has been studied by the Climatic Impact Assessment Program of the Department of Transportation, and in parallel efforts elsewhere. Reduced to its simplest terms, the catalytic cycle is NO + 03 o NO2 + 02 NO2 + 0 - NO + 02. Since the 0 atom is produced by photolysis of another ozone molecule, the net result is to convert the two ozone molecules into three oxygen molecules. An important part of the problem is to describe the removal of the catalysts NO and NO2 from their sources in the stratosphere. This removal can be described by simplified models of the atmosphere that use the concept of vertical "eddy diffusion." These models are usually treated in a computer, and their basic simplicity is therefore hidden. This paper expounds a method that gives an analytic solution, and removes the need to take on faith the results of an elaborate computer program. A representation of the atmosphere is chosen that permits this simple analytic solution of the flow of an inert tracer. The source is at some height within the stratosphere, and the sink is in the troposphere. The distribution with height of the tracer can thus be found for any given source. The oxides of nitrogen are by no means inert: the problem is important because of their chemical activity. But this activity merely turns them into one another without destroying their basic character, which involves the presence of a single nitrogen atom in a molecule. The entity odd nitrogen approximates closely an inert tracer within the stratosphere, for computations of transport. The principal constituents are NO, NO2, and HNO3. The last of these, nitric acid, is rapidly removed in the troposphere by solution and rainout. The end product of the method discussed here is a mixing ratio f (molecules of odd nitrogen per molecule of air) as a function of height.-This quantity is the basic input to chemical models of ozone destruction, such as those of Crutzen (2), Johnston (3), and McElroy et al. (4). We do not treat the chemistry, but rather quote the published results. Of partic* H. S. Johnston, D. ular interest is the effect of the flight altitude of aircraft Kattenhorn, and G. Whitten, J. Geophys. Rev., sources. A given amount of odd nitrogen has more and more in press. 4711
4712
Geo'physics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Hunten
100 1000 RELATIVE VALUES
FIG. 1. The simple two-layer model discussed here. (a) The eddy-diffusion coefficient K; (b) the density of the minor constituent for equal sources at zq = 16 and 22 km; also the residence time from ref. 9; (c) the corresponding mixing ratios.
where K is the eddy-diffusion coefficient, and na is the number-density of air molecules: df = -Kna,. [1] K is a purely empirical quantity, to be determined (as shown in examples below) by substituting observed quantities in [1]. For an initial discussion, we shall take K8 - i0) cm2 sec1 in the troposphere and K8 = 3 X 103 cm2 sec-1 in the lower stratosphere, as illustrated in the left panel of Fig. 1. A source of molecules, Q cm2 sec'1, is assumed at height zq. We consider only the steady state; since there is no sink at high altitudes, the flux is zero above zq. As the material flows down from the source to a sink in the troposphere, its flux below zq is X =-Q. Eq. [1] may be solved for df/dz, and immediately integrated if suitable functional forms are chosen for X1, K, and na. Exponential functions of height are particularly convenient; 4 is taken constant in this paper. If Zt is the height of a reference level, we write = nat exp (-h) na = nfa, exp - (z - z,)/H [2] K - K, exp - (z - z,)/HK = Kt exp (ah) Kna = Ktnat exp - (z - z,)/Hp = Ktnat exp -(1 - a)h.
The Hs are scale heights, negative if the quantity increases with height, and 1/Hp = 1/H + 1/HK = (1 - a)/H; a is defined as -HI/HK, and h is a dimensionless height variable. The solution is then f = A
qHp Kna
n =flffla=Afla fn = An,, -
-A
4t'H
_
=
OHP K
Kna(1 - a)
An,
OH -
K(1- a)
low values of the troposphere. -This exponential curve is the same for a given source at any height, and its extension is shown by dashed lines. The curves for number density become asymptotic to a constant value n = QH/Ks The curve for mixing ratio rises indefinitely, because a fixed amount of pollutant is being mixed with a rarer and rarer atmosphere. "Residence times" can be worked out for one definition (9), and are shown in Fig. lb by an extra scale. It is useful to represent the results of a transport model by an "injection coefficient" a, following a suggestion by McElroy et al. (4). As Fig. lc illustrates, the mixing ratio above the source is independent of height; we shall call it fq. By definition, fq is related to the source strength Q by
fq
[5]
aQ.
=
A typical value for at at 18 km is 5 X 10-17 cm2 sec. A source strength Q = 108 molecules cm-2 sect then gives a mixing ratio 5 X 10-9, or 5 ppb. If several sources are present, perhaps including natural ones, their effects are simply added. The reduction of ozone by odd nitrogen occurs mainly by reactions above 25 km. Thus, for a source at any lower height, the injection coefficient permits an immediate calculation of the corresponding change of odd-nitrogen mixing ratio. A curve of a as a function of source height has the same form as the envelope in Fig. 1c. Fig. 2 shows two such curves for different assumptions about the height of the tropopause and the variation of eddy coefficient with height. The four points are taken from ref. 4. With that exception, the models have been chosen to be analytically soluble, and
[3]
[4]
where A is the integration constant and n is the numberdensity of tracer. A is determined by matching solutions at the boundary of each layer. For Fig. 1, K is constant in each layer and Hp = H. A is zero in the troposphere (solutions are also indicated for a positive value), and in the stratosphere, with K = K8 and Zt at the tropopause,
E
:
20
_
w I
~~~~~~2
C.)
0
QHa
n= KaKs [1 - 0.97 exp (-h)]. (The factor 0.97 = 1 - K/Kt is obtained by matching the solutions at the tropopause.) This expression holds below the source, where 4 =-Q. Above the source height zq, the mixing ratio is constant, and the number density is proportional to na. Below the source, there is an exponential rise from the
,
U,
lo 0
IO19
lo-is
-1017
lo-16
INJECTION COEFFICIENT acm sec
FIG. 2. Injection coefficients for the two models of Table 1. The squares were taken from the papers of McElroy et al. (4, 20).
Geophysics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Hunten
EDDY COEFFICIENT K, cm sec
4713
~~I
I
40H E
CH4
NO 2
30
it
xCrL 20_
E
C,w
CO0
10 2
le6 2 1 .5 VOLUME MIXING RATIO f, ppm
.2
a =
4
Z-
Hp
)
-
D] -
a)hq
-
D]
FIG. 4. Loss time constants, r, for use in Eq. [8]. The N20 results are from ref. 17, and the others were calculated with the OH and O(1D) concentrations of ref. 4.
(Z) = J(nl/r)dz.
[6]
Expression [6] is obtained by evaluating [3] at the source height Zq (or hq), and dividing by Q -4. C is equal to Hp/(Ktnat), and D is usually near 1. The algebraic expression for D is often fairly complicated, and it may better be determined by matching numerical values at the boundary with the next lower layer. The parameters C, D, Zt, and Hp are given in Table 1. The choice of Ks is discussed in the next section, as is the reasoning behind the adoption of the 3-layer model. The reference height Zt for each layer is at its base, which is also the tropopause in the 2-layer model. Use of natural tracers The eddy-diffusion coefficient K in Eq. [1] is a parameter to be determined from observations of appropriate natural tracers. Here we shall focus on methane, CH4, which is the best-observed of the suitable substances. The relevant data are shown as points in Fig. 3. Inspection of these results shows a small but distinct gradient of the mixing ratio starting at the tropopause. The measurements, obtained by Martell, Ehhalt, and collaborators (10, 11), are of exceptional quality. Samples, obtained by balloons or, at 50 km, by a rocket, were analyzed on the ground by gas chromatography. Methane can also be observed spectroscopically (12, 13); the results confirm the ones shown, but do not define the gradients well enough for the present analysis. From the slope in Fig. 3 is obtained Hf, the scale height of the mixing ratio. Then Eq. [1] may be written =
lo'
z
C[exp (1
K (Z)
Id0
duced, in the stratosphere. The photochemical lifetime T(Z) is computed as described below and is shown in Fig. 4. The flux is then obtained by integrating the destruction rate per unit volume from infinity down to height z:
for the injection coefficient is
C[exp (zq
109
DESTRUCTION TIME CONSTANT, SEC
FIG. 3. Methane mixing ratios from the balloon data and the 50-km rocket measurement of refs. 10 and 11. The curved line, drawn by eye, was used to obtain the profile of K. an expression
I7
lo8
fl(z/) n, (df ldz)
(z)Hf(z) n(z)
[8]
Substitution into Eq. [7] gives a somewhat scattered set of points, which are well represented by the straight line and upper scale of Fig. 3. Many other representations of the data could have been picked, but this one, an exponential function of height, is convenient because it allows the analytic solution of the eddy equation used here. The eddy coefficient shown in Fig. 3 is represented by K = 2200 exp (z - 14)/9.43 cm2sec-',
which is 3.5 X 104 cm2 sec1 at 40 km. In the form of [2], = -9.43 km. If H = 6.2 km, then Hp = 18.1 km and a =0.657. The same methane data have already been analyzed by Wofsy and McElroy (14); their technique was to postulate various profiles of K and pick the one that predicted the best methane profile. The resulting Ks differ only slightly below 20 km, and are greater by a factor 4 above 30 km; correspondingly, they predict too much methane at 50 km. Another analysis, with data from an additional balloon flight, has been mentioned by Ehhalt et al. (15); their values are somewhat larger yet, partly from use of daytime values for r(z) instead of a diurnal mean (16). In any case, the lifetime is still an important cause of uncertainty. The important loss processes for methane are attack by hydroxyl OH and metastable O(ID), both of whose concentrations must be calculat-
HK
Table 1. Model parameters for use in Eq.
[7]
The methane number density n(z) follows directly from the measurements shown in Fig. 3; to get Hf(z) we must estimate the slopes, which are obviously rather uncertain. The curve shown is typical of those that can be drawn through the points; they all give similar results for K, within a factor 2 or so. The quantity that is not directly observed is the flux X, which requires a photochemical calculation. This calculation is simple for CH4, a gas that is only destroyed, not pro-
Zt,
Model
All 1. (2-layer) 2. (3-layer)
Layer t s b s
km 11
i5 10 14
H,Hp,
[61
km
C
D
6.2 18.1 6.2 18.1
0.082 20 0.29
0 0.992 0.7 0.979
16.7
In Column 2, t is the troposphere (K = 105 cm2 sec-), b is the
boundary layer (4 km deep, K = 3 x 104), and s is the stratosphere (K from Fig. 3). Units of C are 10- 17 cm sec.
4714
Geophysics:
Hunten
ed. [Transport is not significant for these short-lived constituents; the O(lD) is the product of ozone photolysis and the source of OH radicals by reaction with H20.] Fig. 4 shows the result, and also profiles for N20 (17) and CO. The lifetime of CO is too short for it to be a really useful tracer. N20 will be valuable once we have adequate observations, for it is destroyed primarily by photolysis, whose rate can be confidently calculated (17). The choice of a discontinuity of IK at the tropopause is not demanded by the methane data, as is obvious from Fig. 3, but it is permitted. Far more influential are observations of CO and O3, as pointed out by Seiler and Warneck (18). Other evidence about Ks for all regions of the atmosphere is discussed by Hunten (19). It is an obvious conclusion that the potential value of tracers for defining transport parameters in the stratosphere has barely begun to be realized. The techniques exist to make the measurements; they simply have not been used on an adequate scale. Although the situation is not so bad for methane, more data would still be valuable. The natural tracers have been observed almost exclusively at a latitude near 300 N, where the height of the tropopause is 15 km. It has been argued above that the relevant height scale for other latitudes should be based on the tropopause. In addition, a boundary layer can be inserted in the region 10-14 km, with an intermediate value of K. This 3-layer model (2 in Table 1) has been adopted with considerable success in a very recent study* of the transport of carbon-14 from nuclear tests, and will be used here as standard.
Odd nitrogen, natural and artificial Before one can have confidence in predicted effects of artificial sources, he should be able to show that he can adequately describe the natural situation. Such a description is a bit beyond the simple method discussed here, but it can be used to illuminate an existing computation, that of McElroy et al. (4, 20). N20, discussed above as a tracer, is also the principal source of odd nitrogen in the stratosphere. Its lifetime, shown in Fig. 4, is almost entirely determined by photolysis into N2 + O. This process gives even nitrogen, but a slower one, attack by O('D), gives two molecules of NO. According to ref. 20, the source strength Q is about 7 X 107 molecules cm-2 sec1 spread mainly over the 20- to 40-km range. For the same profile of K, the injection coefficient at 30 km is a = 17 X 10-17 cm2 sec; the product aQ gives a mixing ratio f = 12 X 10-9 or 12 ppb. This result is insensitive to the assumed values of K: with faster transport the source Q is larger, but a is smaller because the loss of odd nitrogen is greater. Another possible source of odd nitrogen is destruction of NH3, though the actual end product is not known and its mixing ratio near the surface is only 6 ppb. McConnell (21) finds that the source, if such it is, has an effective height around 20 kmi, and a magnitude very close to that for N20. But the injection coefficient is smaller by a factor of 7; thus, at most, the source from NH3 makes only a minor contribution, as McConnell also points out. Further study would be worthwhile, but is unlikely to reverse this conclusion. Some observational data exist for the various forms of odd nitrogen. The amounts computed for the N20 source are in fair agreement (4, 20), though there are still some discrepancies. Without the natural odd nitrogen, the total amount of ozone in the stratosphere would be greater by a factor of 2; with it, the calculated ozone profile agrees very well with observation (4). With less ozone, the competing destruction
Proc. Nat. Acad. Sci. USA 72 (1975) Table 2. Predicted effects of various assumed fleets of aircraft Aircraft
Subsonic
Number Q', kT/y Height, km Effective height (, 10-'7 cm2 sec
1000 600
0.35 0.34 0.47
03 loss, %
60 60 16 18 4.45 0.43 0.60
13 15 1.27 1.22 1.7
12 14
fq, ppb
Present Advanced SST SST
200 800 20 22 9.6 12.3 17
The "advanced SST" is arbitrarily taken to produce four times the odd nitrogen of a Concorde. The 2-km difference between the actual and effective heights allows for the translation from highlatitude flight paths to a model for 300 latitude. The last three lines give the injection coefficient for model 2, the odd-nitrogen mixing ratio, and the expected ozone reduction for one hemisphere.
process becomes less efficient, and odd nitrogen accounts for about %4 of the total destruction. Though the discrepancies still need to be studied, the general agreement supports the general validity of the description. The response of the ozone concentration [0s] to an increment of odd nitrogen is complicated by feedback effects, which are analyzed in ref. 4 and found to be small but not negligible. The final result of an elaborate model, summarized in Table 5 of that paper, can be represented by -
100
A 03
[03]
=
6
=
1.405/
-
0.0105f
,
[9]
where f, in parts per billion is equal to fq in Table 2. Thus,
an increment of 1 ppb would reduce the ozone abundance 1.4%. For a brief review of the chemical aspects, see
by 6 =
ref. 22. Artificial sources must be treated carefully because they are not spread worldwide like the natural tracers. Most of the remarks here will be confined to air traffic in the northern hemisphere at about 450 N. The material is assumed to spread uniformly over the whole hemisphere. In truth, there is some leakage across the equator, but there will also be a tendency for a maximum amount near the source. These two effects come close to compensating each other, and the estimates should be valid for middle latitudes. There may indeed be a considerable "corridor effect" at the heights where the aircraft fly. But the odd nitrogen must move upwards to 25-30 km to affect the ozone, and the ozone in turn spreads outwards and downwards from this same region. These two smoothing processes will almost certainly eliminate any substantial corridor effect in the ozone reduction. The area of a hemisphere is 2.55 X 1018 cm2. Odd-nitrogen emissions are usually expressed in i09 g/year, or kilotons/year, in the form NO2. One kT/y (which is the rate of production by a Concorde or TU-144 flying 7 hr/day) therefore corresponds to a source strength of 1.62 X 105 molecules cm-2 sec1. With the injection coefficient a in units of 10-17 cm-2 sec1 and the source Q' in units of kT/y, we obtain for the mixing ratio
fq
=
0.0016aQ' parts per billion.
[10 1
Most long-distance air traffic is at high latitudes, where the height of the tropopause is 10-12 km (23). It is assumed
Geophysics:
Proc. Nat. Acad. Sci. USA 72 (1975)
Hunten
that the 12 km figure should be converted to 14 km for use with model 2 of Table 1; thus, 2 km is added to all heights. Predictions for several assumed fleets are given in Table 2. The number of "jumbo" jets is 2.5 times the 1974 fleet, and has been taken to fly somewhat higher. The prediction is extremely sensitive to the exact height, as illustrated by the two assumptions in the table. The effects shown are beginning to be appreciable; but one can be fairly sure that today's smaller fleet has not yet made a serious dent in the ozone. A relatively modest fleet of 60 Concordes and TU144s could be significant, and a larger fleet of advanced aircraft very serious indeed. Needless to say, the effects from different groups of aircraft are additive. The predictions for stratospheric flight are considerably greater than those of McElroy et al. (4), who deliberately adopted a highly conservative approach. Their tropopause was high, 16 km, and no adjustment was made to actual flight altitudes. Emissions were averaged over the globe, rather than half of it. For flights primarily over the North Atlantic, the present assumptions should be realistic. The impressive accuracy (4) in the prediction of mean stratospheric ozone suggests that the chemical scheme is correct to better than a factor of 2. The transport calculation is more uncertain, particularly for sources very near the tropopause where a factor-of-10 uncertainty is probable. The other predictions are thought to be good to a factor of 3. Summary The effect of odd nitrogen on stratospheric ozone can be estimated with good accuracy by breaking down the problem into its parts. The sources and sinks of odd nitrogen are first treated as discussed here. Second, the partitioning among the major forms NO, NO2, and HNO3 is calculated. Finally, the destruction of O3 by NO2 is obtained. The second and third steps can be represented by Eq. [9], obtained from the detailed results of McElroy et al. (4). The simplified method is more in the domain of a hand calculator than a computer. Although one can be confident regarding the general behavior of the eddy-diffusion coefficient up to 40 km or so, it would be valuable to have many more observations of natural tracers, especially CH4, N20, and CO. Another important quantity is the OH concentration, needed for interpreting the CH4 and CO distributions. The region just above the tropopause is especially tricky, and especially important because of the possible impact of subsonic aviation. At present this impact is negligible, but as traffic grows and flies higher, a noticeable depletion of ozone is a real possibility. To protect the stratosphere, subsonic jet engines, as well as their
4715
supersonic cousins, may have to be modified (24) to produce less odd nitrogen. Kitt Peak National Observatory is operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. I thank A. J. Broderick and D. H. Ehhalt for valuable communications. 1. 2. 3. 4. 5. 6.
7.
8. 9. 10. 11. 12. 13. 14.
15.
16. 17. 18. 19. 20. 21. 22.
23. 24.
Crutzen, P. J. (1970) Q. J. R. Meteorol. Soc. 96,320-325. Crutzen, P. J. (1971) J. Geophys. Res. 76,7311-7327. Johnston, H. S. (1971) Science 173,517-522. McElroy, M. B., Wofsy, S. C., Penner, J. E. & McConnell, J. C. (1974) J. Atmos. Sci. 31, 287-303. Gudiksen, P. H., Fairhall, A. W. & Reed, R. J. (1968) J. Ceophys. Res. 73,4461-4473. Johnston, H., Whitten, G. & Birks, J. (1973) J. Geophys. Res. 78,6107-6135. Davidson, B., Friend, J. P. & Seitz, H. (1966) Tellus 18, 301315. Reed, R. J. & German, K. E. (1965) Mon. Weather Rev. 93, 313-321. Hunten, D. M. (1975) Geophys. Res. Lett. 2,26-28. Ehhalt, D. H. & Heidt, L. E. (1973) J. Geophys. Res. 78, 5265-5271. Ehhalt, D. H., Heidt, L. E. & Martell, E. A. (1972) J. Geophys. Res. 77,2193-2196. Kyle, T. G., Murcray, D. G., Murcray, F. H. & Williams, W. J. (1969) J. Geophys. Res. 74,3421-3425. Cumming, C. & Lowe, R. P. (1973) J. Geophys. Res. 78, 5259-5264. Wofsy, S. C. & McElroy, M. B. (1973) J. Geophys. Res. 78, 2619-2624. Ehhalt, D. H., Heidt, L. E., Lueb, R. H. & Roper, N. (1974) in Proceedings of the Third Conference on the Climatic Impact Assessment Program (U.S. Department of Transportation), pp. 153-160. Crutzen, P. J. (1974) Geophys. Res. Lett. 1, 205-208. Bates, D. R. & Hays, P. B. (1967) Planet. Space Sci. 15, 189197. Seiler, W. & Warneck, P. (1972) J. Geophys. Res. 77, 32043214. Hunten, D. M. (1975) in Atmospheres of the Earth and Planets, ed. McCormac, B. M. (Reidel, Dordrecht), pp. 59-72. McConnell, J. C. & McElroy, M. B. (1973) J. Atmos. Sci. 30, 1465-1480. McConnell, J. C. (1973) J. Geophys. Res. 78, 7812-7821. Schiff, H. I. & McConnell, J. C. (1973) Rev. Geophys.. Space Phys. 11, 925-934. Jocelyn, B. E., Leach, J. F. & Wardman, P. (1973) Water, Air, Soil Pollut. 2, 141-153. Ferri, A. & Agnone, A. (1974) Am. Inst. Aeronaut. Astronaut. paper no. 74-160.
