[Manuscript received November 26, 1929]
The existence of the ozone, and many facts regarding its distribution and changes, are now well established,. McLennan's identification of the green auroral line , shown in the spectra both of polar and non-polar aurorae, as due to atomic oxygen, proves that oxygen is present in the atomic state at heights of 100 km. and above during polar auroral displays, and, in non-polar regions, at all times, in varying measure and at a height which is as yet unknown.
No theoretical discussion of the presence and changes of these gases in the atmosphere appears to have been attempted hitherto. The production of the ozone has been alternatively attributed to solar corpuscular radiation, possibly associated with aurorae and magnetic storms, or to ultra-violet radiation; recently the latter view has fallen into disfavour, because ozone is least abundant at the end of the summer, and most abundant at the end of the winter. One of the main results of this paper is the proof (on the basis of certain assumptions which are considered reasonable) that such an annual variation of ozone is quite compatible with the production of ozone solely by ultra-violet radiation. Hence the latter view cannot at present be rejected. But neither, on the other hand, can it be accepted definitely as yet; the present theory neglects certain factors, partly because their magnitude is quite uncertain: but they may possibly be dominant factors in the ozone changes and, if so, the theory would need substantial modification. Fortunately the theory makes certain predictions (cf. §§ 26 , 27) capable of being tested by observation and experiment, so that further light on the points in doubt may be hoped for.
The discussion involves the consideration of the amount of atomic oxygen at the level of maximum ozone density. In another paper I hope to deal with the question of the proportion of ozone and atomic oxygen at greater heights, showing that atomic oxygen probably becomes an important constituent beyond 100 km. This result is of obvious interest in connection with the green light of the polar and non-polar aurorae; it may also have an important bearing on the state of ionisation of the outer atmosphere.
A very small systematic component of this motion of the upper air might suffice to bring about the slow annual variation of ozone. There appears unfortunately to be no direct way of testing whether or not the annual variation actually is produced in this way. It may prove easier to settle the point by examining other possible hypotheses. In the following discussion the bodily transport of ozonised air is completely left out of the account, partly because at present it cannot be estimated quantitatively, and also in order to see whether a consistent explanation of the facts about atmospheric ozone can be arrived at without assuming such motion.
The relative importance of these various causes of ozone cannot yet be ascertained in any simple way. The intensity of the corpuscular radiation is quite unknown, while that of the ultra-violet radiation can only be more or less plausibly conjectured.
The greater ozone content in high latitudes than in low, and, in high latitudes, in spring than in autumn (corresponding to a large increase in ozone during the winter, when these latitudes receive but little ultra-violet radiation), has led several writers to conclude that corpuscular radiation, rather than ultra-violet, must be the chief source of atmospheric ozone: and that the principal role of ultra-violet radiation, which is most abundant during the summer half year, is to reduce the amount of ozone. But, as will be shown, it is possible to reconcile the above facts with the hypothesis that ultra-violet radiation is the sole or main source of ozone; and there is at least one reason for regarding this view as more probable than the other. For according to Fabry the radiation in the band 1300-1850 Å would be principally absorbed approximately at the same level, about 45 km., as that at which the ozone is observed to be abundant. The auroral corpuscular radiation, on the other hand, does not descend below about 90 km., and no particles are known which are so penetrating as to get down to 45 km. and be there absorbed. Ozone formed at 90 km. would descend to 45 km. only very slowly, whether by convective mixing or by steady fall under gravity--in the latter case the time of descent would be reckoned in years.
The height of the ozone layer is thus slightly favourable to the ultra-violet theory of its formation, and it appears worth while to find whether, and under what conditions, ultra-violet radiation alone is capable of accounting for the observed facts about ozone, leaving corpuscular radiation, and bodily transport of ozonised air, entirely out of account, without necessarily suggesting that they are unimportant. It may be possible to test whether the conditions imposed in the present theory are fulfilled; if not, it will definitely indicate that one, at least, of the neglected factors is of importance. The theory is found also to lead to certain predictions which can be tested by further observations on atmospheric ozone.
The ozone layer may extend upwards throughout the atmosphere from about 40 km., but it is known that in the lower atmosphere there is a relatively small proportion of ozone. The maximum density of O3 seems to occur at about 40 km. The decrease in the ozone density below this level requires explanation, for if the atmosphere were uniformly mixed the ozone concentration should be uniform, and the actual density should increase downwards in proportion to the total density of the atmosphere. Twisted meteor trails give clear indication of convection and mixing in the upper atmosphere, but the rate at which the ozone would be transferred downwards from the level at which it is newly produced cannot be calculated at present. It seems likely that mixing will be less effective just below the layer of maximum ozone density than elsewhere, because this is a region where the actual (and not merely the potential) temperature is increasing upwards, so that the air there is more than usually stable. But even were there no convective mixing, the ozone should descend steadily, owing to its excess weight as compared with nitrogen and oxygen; the rate of such descent can be estimated.
It is sufficient to take the coefficient of diffusion D for O3 in air to be the same as that of O2 in nitrogen, namely, 0.17 at normal temperature and pressure. At a height where the total air density is a times that at ground level, D = 0.17/a. A correction factor roughly equal to T/273 is also needed (T being the absolute temperature at the given level), but this can be ignored since only the order of magnitude of D is here required. The coefficient of mobility is  D/kT, or, taking T=300, it is 4x1012/a. The excess force of gravity on an O3 molecule in air is approximately 3x10-20 dyne, so that the downward velocity v of the ozone will be 10-7/a cm. sec.-1. At 40 km. a is about 1/300, and v=3x10-5 cm sec.-1, or about 3 cm. per day. Above or below the level, v is greater or less in inverse proportion to the density; the rate of transfer of O3 molecules across any horizontal surface is equal to the product of v into the number of O3 molecules per c.c., and it is therefore the same at all levels so long as the concentration of ozone is uniform, while if the concentration increases (or decreases) upwards, the rate of transfer alters proportionately. At 100 km. v is about 50 metres per day; these values of v show how long a time would be required for O3 formed at auroral levels to sink to 40 km. level owing solely to gravity.
If the ozone be uniform in concentration above 40 km., the rate of descent, at any level in this region, is about 3x108 O3 molecules per sq. cm. per sec.; if the same amount of ozone were spread uniformly above 30 km., the corresponding number would be about 107.
The lower limit of the ozone layer will naturally be somewhat indefinite, owing to diffusion or mixing. In the boundary layer, where the O3 concentration is decreasing downwards, the O3 will be steadily sinking owing to gravity, as in the upper layers, and there will be an additional downward flow due to the concentration gradient. At the top of the boundary layer, as has just been seen, the number of molecules per sq. cm. entering the layer per sec. is about 3x108; at lower levels the number due to gravity fall will progressively decrease, so that between any two levels more molecules enter from above than flow out from below. Thus O3 molecules must in this region be continuously transformed into something not ozone, by reactions between themselves or, more probably, with other atmospheric gases. The total rate of disappearance in the whole boundary layer is equal to the number entering from above, i.e. about 3x108. This is a very small number; the loss per day would be about 2x1013, or about 4x1015 per half year--a number quite inappreciable compared with the actual decrease in the ozone content at Lerwick from spring to autumn, namely, about 2x1018. The loss of ozone at the base of the layer in which it is formed therefore seems negligible in comparison even with the slow annual variation of ozone, and still more so in comparison with the large changes, over Europe, from day to day. The loss at the lower boundary might, however, be greater than has here been calculated, if ozone is carried downwards by convective mixing of the air as well as by steady diffusive fall subject to gravity.
The amount of solar radiation received in this band, at the outside of the atmosphere, is unknown; it cannot be measured because almost all of it is absorbed at a high level, nor can it be estimated, at present, from the theory of the sun's radiation. If the sun were a complete radiator, at the temperature 6000°K, the amount of energy received at the earth at the equator at noon would be about 3.7x104 ergs. cm.-2 sec.-1, or about 5x1015 quanta, sufficing, if completely absorbed by O3 molecules, and if each quantum absorbed decomposes one ozone molecule, to dissociate 5x1015 per cm.2 sec. This is far greater than the rate of destruction of O3 molecules at the base of the layer, as estimated in §4; moreover, if it continued for 1600 seconds, or about half an hour, unchecked by any compensating process, all the 8x1018 O3 molecules per cm.2 would disappear. Since this does not happen, either the photo-electric efficiency of the process must be extremely small (that is, only a very small proportion of the quanta absorbed are effective in dissociating ozone molecules), or some restoring process must go on, and the most obvious is the reformation of O3 molecules by union between O atoms and O2 molecules.
As has just been stated, it is not possible to conclude that the number of O3 molecules dissociated per cm.2 sec. actually is 5x1015, because it is uncertain whether the sun radiates as a black body in this region, and because the proportion of quanta effective for dissociation is unknown; but the region is sufficiently near the limit up to which the sun's radiation is measurable, for the extrapolation to be not very unsafe. It is therefore likely that the actual rate of dissociation of O3 is not much, if any, greater than 5x1015, and also perhaps not very much less.
The O atoms formed by dissociation of O2 may be different from one another, and from those formed from O3; they may be ionised or excited, and unequally ready to attach themselves to an O2 molecule to form ozone. A complete theory would have to take account of such differences, but at present the necessary knowledge is lacking. It is therefore sufficient to suppose that of the whole number of O atoms present at any time, certain fractions take part, each second, in the reactions
(a) O + O = O2In the average over any sufficiently long time, O atoms of each kind produced must disappear in numbers equal to those formed; and the same applies to the O3 molecules.
(b) O + O2 = O3 ,
(c) O + O3 = 2 O2 .
(d) O3 = O2 + O ,which occurs (§ 5) through absorption of radiation in the Hartley band, and possibly also spontaneously, or by reason merely of collisions with other molecules in the course of their "thermal" motions; secondly by the reaction (c), and thirdly by the reaction
(e) 2 O3 = 3 O2 .The latter is generally supposed to be the mode of purely thermal decomposition of ozone, but this view has recently been contested by Riesenfeld and his collaborators, who assert that it also decomposes monomolecularly, presumably according to the formula (d).
According to their experiments, made between 85° and 95°C., the decomposition occurs according to the equation
dc/dt = -k1c - k2c2 ,where c denotes the ozone concentration in molecules per litre, and the coefficients of monomolecular and bimolecular reaction, k1 and k2, depend on temperature: and have the values
k1 = 2.5x10-3 min.-1; k2 = 4.1 litre mol.-1 min.-1 ,at 95°C., increasing with the temperature by respectively 2x10-3 and 2.5 per 10°, over the range 80° to 100°C. The bimolecular constant k2 is independent of the partial pressure of admixed O2, or argon, but is increased by the presence of N2 and, still more of CO2, the values at 95°C. in the presence of a considerable excess of N2 and CO2 being respectively about 5 and 7. The monomolecular constant, on the other hand, they consider to be independent of the presence of any of these gases.
At the very low pressures existing in the layer of atmospheric ozone, where c is of the order 2x10-8 or less, the bimolecular decomposition would be quite negligible compared with the monomolecular, if at the temperature T of the layer the ratio of the magnitudes of k1 and k2 is similar to that given by Riesenfeld for 95°C. The value of T is not known accurately, but is unlikely to be less than about 30°C., and may be 100°C. or even more. Riesenfeld's experiments give but little indication of the probable value of k1 at 30°C., but perhaps suggests 10-4 as the order of magnitude. This would correspond to a reduction of ozone in the ratio 10 to 1 in about 120 days, which would be insensible in the course of a single day and night, but would affect the annual ozone variation.
If, however, T is of the order 100°C., the observed constancy of the ozone during the day and night seems incompatible with Riesenfeld's value of k1, which would imply a reduction in the ratio 8 to 1 in the course of 12 hours. Should observation confirm that T is of this order, I should regard it as a disproof of Riesenfeld's value of k1, or rather as an indication that his k1does not correspond to a real homogeneous gas reaction. He himself states that part of k1 is due to a wall reaction in his vessel, though he considers the major part to depend on a real homogeneous gas reaction. The difficulties of laboratory experiments on the thermal decomposition of ozone, and the discrepancies (which he discusses) between his results and those of other workers to whom he refers, make it unsafe to rely on the experimental values of k1 and k2 at present. It is, however, desirable to bear in mind the possibility of a slow thermal decomposition, whether bimolecular or monomolecular, of upper-atmospheric ozone, in connection with the annual and 11-year variations, though it is unlikely that it affects the daily variation.
Even if the ozone layer is at 400°K instead of 300°K, the same conclusion holds good; in this case e-E/RT is about 10-12.5 or 3x10-13, and the number of effective collisions between O3 molecules is about 3x10-23 n32 per cm3 sec. The time required for all the ozone to disappear by this means, at this constant rate, would be more than thirty years.
Riesenfeld's value of k2, about 6 litre/mol. min. at 95°C., is equivalent to 3x10-22 n32 effective collisions per cm3 sec., or about ten times as many as here estimated for the higher temperature of 400°K or 123°C.; the discrepancy therefore amounts to more than a factor of 10, and if Riesenfeld's value is correct, the bimolecular thermal decomposition would require consideration in connection with the 11-year ozone variation if T = 400°, but probably not if T = 300°; in either case it would be without appreciable influence on the diurnal and annual variations.
In the reaction (d), or in the dissociation O2, i.e.,
(f) O2 = O + O ,producing absorption of radiation, the energy absorbed is partly used in separating the two parts of the original molecule, while some may appear as excess kinetic energy of the products of dissociation. Such excess kinetic energy rapidly becomes distributed among the other gas molecules, and goes to raise the temperature. The products of dissociation (O, or O and O2) may be excited or ionised, and may radiate energy (of longer wave-length than the original) in returning to their normal state, or in recombining. This energy may be absorbed by the gas and further raise the temperature. If all the dissociated particles recombine, the gas is left in its original form except that its temperature has been raised by the absorption of radiant energy and its conversion mainly into molecular kinetic energy. It is thus that the relatively high temperature of the ozone layer is explained.
13. It is an observed fact that n3 is nearly constant throughout the day and night, and even its annual range is not large in comparison with its mean value; moreover n2 can vary by only a very small fraction of its total amount. ...
14. ... Thus the number of O atoms is always small compared with the number of O3 molecules (in the main ozone layer).
The graph of n3 throughout the year consists of a main wave with maximum in spring and minimum in autumn, on which, according to the present theory, there is superposed a small daily fluctuation, not yet definitely determined. In considering the seasonal variation, we wish to ignore this small fluctuation, and consider the form of the main wave; this is conveniently done by finding the slope of the nearly identical curve drawn smoothly through the points on the actual curve corresponding to a particular hour on each day. This hour is here taken to be midnight ...
The observed annual variation of n3 is roughly of the form
(36) n31 = (n31)0 + (n31)1cos τ,
where (n31)1 is positive; corresponding to maximum n31 at the vernal equinox. ...
26. The predicted annual variation of n3 ... is zero at the equator ... and is reversed in southern latitudes, in accordance with observation.
In high latitudes ... the preceding analysis becomes inadequate ... . One of the most definite tests of the theory will be the observation (by the ultra-violet absorption of moonlight) of the ozone in the arctic circle during the period of continual darkness; according to the theory the ozone should be constant during this time, and the large increase from autumn to spring must occur during the prior and subsequent periods of continual darkness. If this is not verified, it will prove that some important factor concerned in the equilibrium and variations of ozone has been left out of this discussion.
It is not known how the height and density of the ozone layer vary with latitude. If the height is determined by the level of maximum absorption of the sun's ultra-violet radiation, it should be least at the equator; the increase in height at latitude φ should be -H loge cos φ, where H is the height of the "homogeneous" atmosphere. At latitude 60°, taking H as 8 km., this is about 5.5 km. The density of the air at the level of maximum absorption of radiation varies as cos φ.
The present theory can therefore be further tested by determining the variation of density of the ozone layer with latitude, by making careful observation of its height at different latitudes. ...
n31 -- § 17: midnight value of n3.
(n31)0 -- § 24: annual mean value of n31.
δn31 -- § 17: change of n31 from one midnight to the next.
N2, N3 -- § 11: the number per cc. of O2 and O3 molecules dissociated per sec. at or near the level of maximum ozone density.
k11, k12, k13, k33 -- § 10: coefficients of recombination.
K11 = k11n; K12 = k12n -- § 10.
In so far as dissociation (O3 = O + O2) and re-formation (O + O2 = O3) balance one another, they have no ultimate effect on the amount of ozone; but new O atoms are formed by dissociation of O2, and this tends to increase the amount of ozone. This rate of increase is supposed held in check by reactions which cause the reversion of some of the O (formed from O2 and O3) and O3 to O2, by the reactions 2 O = O2, O + O3 = 2 O2. These reactions occur mainly by day; most of the O atoms then present have been formed from O3. It is shown that the varying rates of these reactions can explain the observed annual variation of ozone, provided that the coefficients of reaction have suitable values.
,For references to the now extensive literature on atmospheric ozone, cf. (1) C. Fabry, Proc. Phys. Soc. 39, 1926, p. 1, and Second Report on Solar and Terrestrial Relationships (International Research Council), 1929, p. 49; and (2) G. M. B. Dobson, D. N. Harrison, and J. Lawrence, Proc. R. Soc., A., 122, 1929, p. 456.
J. C. McLennan, Bakerian Lecture, Proc. R. Soc., A., 120, 1928, p. 327, and references there cited.
Since communicated to the Philosophical Magazine, London, together with a further paper containing a discussion of another theory which has been proposed to account for the annual variation of ozone.
McLennan, J. C., Ruedy, R., and Krotkov, V. (Ottawa, Trans. R. Soc. Canada, 22, 1928, p. 300) state the contrary, apparently by an oversight in regard to the unit in which the ozone content is usually expressed.
Where k is Boltzmann's constant 1.37x10-16.
Y. Rocard, Paris, Comptes Rendus Acad. Sci. 188, 1929, p. 1336, estimates the velocity of descent of ozone in a nitrogen atmosphere, at 50 km., as 20 metres per day, and concludes that this obviously has no influence on the distribution of the ozone. While concurring in the conclusion, I believe the above method of estimating v to be the correct one.
See the discussion given by C. N. Hinshelwood in "Kinetics of Gas Reactions."
E. H. Riesenfeld and W. Bohnholtzer, Zs. physik. Chemie, 130, 1927, p. 241; E. H. Riesenfeld and H. J. Schumacher, ibid. 138, 1928, p. 268.
[Most of the rest of the paper is a mathematical development of the mechanism proposed in sections 6 and 7 . I will omit most of this derivation, but include some of the predictions that follow from it. --CJG]
[Italics in the original. --CJG]