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11.4 Ozone photochemistry: the Chapman model
Figure 2.9 demonstrates that the upper stratosphere in the tropics and summer
hemisphere undergoes strong heating. This heating is due to absorption of Solar
ultra-violet radiation by oxygen and ozone. We have treated this subject very
schematically in sections 2.12 and 2.13. This section will expand upon this theme,
including the actual formation of ozone in the layer between 100 hPa and 1 hPa. With
this theory we are able to understand why there is an ozone layer and why its mixing
ratio is greatest at a particular height (approximately 35 km), although it should be
stressed directly that an explanation of the exact distribution of ozone requires
knowledge of the large scale circulation, which is in fact partly induced by the heating
due absorption of solar radiation by ozone. Therefore, we will be concerned in this
section with a small part of a non-linear problem involving the interaction of the
chemistry, radiation and fluid dynamics of the atmosphere.
A relatively detailed account of the theory behind the formation of the ozone
layer is presented in the following pages. Particular attention is given to explaining the
role of absorption of Solar ultraviolet radiation (important for the energy balance of
the upper atmosphere).
The basic mechanism that gives rise to the ozone layer was put forward by
Sidney Chapman in 1930. It involves the following (photo-) chemical reactions.
!
!
!
!
O2 + h"1 # O + O ;
O + O2 + M " O3 + M ;
O3 + h" 2 # O + O2 ;
O + O3 " 2O2 ;
(11.49a)
(11.49b)
(11.49c)
(11.49d)
where h is Planck’s constant and ν is the frequency of the incident radiation that is
absorbed in the particular photochemical reaction and M is any other air molecule
needed for the energy balance of the reaction. The wavelength bands involved in the
two photochemical reactions are (roughly)
"1 = 0.19 # 0.24 µm (dissociation of O2 )
(11.50a)
and
!
!
" 2 = 0.2 # 0.3 µm (dissociation of O3 ) .
The molecule number concentrations (per cubic meter) of O, O2 and O3 are denoted
by, respectively, n1, n2 and n3. The number concentration of O2-molecules is assumed
to be constant fraction of the total number of molecules, which can be computed from
the equation of state in the following form.
p = nkT .
!
(11.52)
The rates of change of the number concentrations of O and of O3, as a result of the
four reactions (11.49a-d) involved in what is now referred to as the Chapman cycle
are
dn1
= 2 j a n 2 " kb n1n 2 n + jc n 3 " k d n1n 3 ;
dt
!
(11.51)
In this equation n is the air-molecule number concentration and k is Boltzman’s
constant (1.38066 " 10-23J K-1 molecule-1). Assuming that 21 % of the air molecules are
oxygen molecules, we find
! p.
n 2 = 0.21
kT
!
(11.50b)
(11.53)
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dn 3
= kb n1n 2 n " jc n 3 " k d n1n 3 .
dt
(11.54)
In these equations kb and kd are constants determining the rate of, respectively,
reactions (11.49b) and (11.49d). The exact values of these so-called rate constants
are determined in laboratory experiments164. The parameters ja and jc are so-called first
order photolysis rate coefficients.
The magnitudes of ja and jc, respectively, depend on the probability that a photon
will be absorbed by, respectively, an oxygen molecule and an ozone molecule (given
in terms of an absorption cross section, defined in sections 2.10 and 2.12), the
probability that (if absorption occurs) the oxygen or ozone molecule will dissociate
(the so-called “quantum yield”), and on the radiation flux, I. The values of ja and jc are
thus calculated from
!
#2
j a = $ " O2 ( #,T )%O2 ( #,T ) I ( # ) d# ,
(11.55)
#1
#2
jc = $ " O3 ( #,T )%O3 ( #,T ) I ( # ) d# ,
!
(11.55)
#1
where " O2 and " O3 are the wavelength- and temperature-dependent absorption cross
sections (in m2) (see e.g. figure 2.36), "O2 and "O3 are the quantum yields for,
!
respectively, reaction (11.49a) and reaction (11.49c). The units of I are now photons
! per square
! meter per second per meter of wavelength.
Let us estimate the values of
! ja and jc.!We therefore simplify matters strongly by
assuming that the quantum yields corresponding to the two photochemical reactions in
the Chapman cycle are equal to 1. The reaction (11.49c), for instance, is sensitive to
radiation in a wavelength band between 0.2 and 0.3 µm. The global average and
yearly average Solar flux at the top of the atmosphere within this wavelength band is
about 10 W m-2. Let us assume that absorption cross section corresponding to this
photochemical reaction is constant and equal to 5 " 10-23 m2 molecule-1. In that case
jc = 5 "10#23 "10 "
!
!
(11.56)
where " /hc is the average energy of one photon within the spectral interval (11.50b)
of interest, where c is the speed of light (h=6.63 " 10-34 J s and c=3 " 108 m s-1). In the
case of reaction 11.49c, the average wavelength is " =0.25 µm.
Reaction (11.49a) is sensitive to radiation in a wavelength band between 0.19 and
!
0.24 µm. The global average and yearly
flux within this wavelength
! average Solar !
band at the top of the atmosphere is about
! 2.15 W m-2. Assuming that the absorption
cross section corresponding to photochemical reaction (11.49a) is constant and equal
to 3 " 10-28 m2 molecule-1
!
!
$
% 6 "10#3 s-1
hc
!
j a = 3 "10#28 " 2.15 "
$
% 7 "10#10 s-1
hc
(11.57)
( " =0.215 µm). It is important to remember that the above estimates ja and jc apply to
the top of the atmosphere. Obviously, since the radiation is depleted quickly as one
goes downward into the atmosphere, the value of the photolysis coefficient will also
decrease strongly with decreasing height. This is illustrated in figure 12.1. In any case
ja<< jc at all levels. Therefore, we conclude that reaction (11.49a) is relatively very
slow compared to reaction (11.49c). We may therefore assume that reaction (11.49a)
is “slaved” to reaction (2.49c), i.e.
164
see http://jpldataeval.jpl.nasa.gov/
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Fig. 11.1. Photolysis rates as a function of height for reactions (2.49a) and (2.49c). The symbols k1 and
k3 are used here instead of ja and jc, respectively. (Source: Daniel Jacob, 1999: An Introduction to
Atmospheric Chemistry. Princeton University Press).
dn1
= 0.
dt
Therefore, from (11.53) we obtain,
!
n1 =
2 j a n 2 + jc n 3
.
kd n 3 + kb n 2 n
(11.58)
Substituting this into (11.54) gives
!
!
2n (2 j n + j n )
dn 3
= 2 ja n2 " 3 a 2 c 3 .
dt
n 3 + n 2 n ( kb /kd )
(11.59)
The first term on the r.h.s. of (11.59) represents the ozone-formation rate, while the
second term on the r.h.s. of (11.59) represents the ozone-destruction rate. If these two
processes are in equilibrium we have
dn 3
= 0.
dt
!
From this we obtain the following expression or the steady state value [n3]0 of the
ozone concentration:
[n3 ] 0 =
!
)
# 4 j k n &1/2 -+
ja n2 +
*"1+ %1+ c b ( . [molecules per m3]
2 jc +
ja kd ' +
$
,
/
The values of the reaction coefficients kb and kd can be found in the NASA-publication
given in a footnote earlier in this section. We thus find that
$ -2060 ' 3 -1
kb = 6 "10#46 m6s-1 and kd = 8 "10#18 exp&
)m s .
% T (
!
(11.60)
(11.61)
With (11.51) and (11.52) and the estimates of the values of the photolysis coefficients
in (11.56) and (11.57), we can obtain a rough estimate of the steady state ozone
concentration as a function of pressure and temperature. For instance, at a height of 20
km, where p=55 hPa, T=217 K (table 2.1), ja=10-13 s-1, jc=5×10-4 s-1 and kb and kd given
by (11.61) we obtain an ozone molecule concentration of 19.5 m-3 (10.6 ppm).
According the observations (figure 2.34) the global average ozone concentration at 55
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hPa (20 km) is approximately 3 m-3. The value found from (11.60), incidently, is quite
sensitive to the temperature.
The Chapman theory systematically over-estimates the ozone concentrations,
but nevertheless, qualitatively it does very well in explaning why there is a maximum
in the ozone concentration at some height in the stratosphere. Basically, this is due to
the fact that there are very few oxygen molecules at very high levels to form enough
oxygen atoms, while there is insufficient ultraviolet radiation at low levels to form
ozone from oxygen atoms. The ozone concentration in the stratosphere does not only
depend on the reactions (11.49a-d). There are other important reactions that promote
the destruction of ozone. Also, transport of ozone by the circulation is an important
factor, which explains the fact that most ozone is found over the poles (figure 11.2),
while most ozone is produced in the tropics.
FIGURE 11.2. Zonal mean distribution of ozone molecule number density (1018 m-3) at the equinox (22
September), based on measurements taken in the 1960’s (From D.J. Jacob, 1999: Introduction to
Atmospheric Chemistry. Princeton University Press).
An analysis of the stability of the chemical equilibrium (11.60) is of great
interest, because it provides us with a criterion for the existence of the equilibrium as
well as an associated adjustment timescale, similar to the radiative equilibrium
timescale associated with adjustment to radiative equilibrium (section 2.4). Suppose
we perturb the state of chemical equilibrium (11.60) slightly. Mathematically this can
expressed as follows:
n3 = [n 3 ] 0 + n3 ' ,
!
where n3’ is the perturbation ozone molecule number concentration. For convenience,
equation (11.59) is written as
dn3
= an32 + bn3 + c ,
dt
where, because
!
!
(11.62)
k
n3 << n2 n b ,
kd
(11.63)
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a=
"2 jc kd
"4 ja kd
;b =
;c = 2 ja .
n2 nkb
nkb
(11.64)
Substituting (11.62) into (11.63) we obtain
!
dn3 '
2
= a(n3 ') + 2a[n3 ]0 n3 '+bn3 ' .
dt
(11.65)
If n3 '<< [n3 ]0 (i.e. the perturbation is small) we are left with the following simple
equation governing the time-evolution of the perturbation.
!
!
dn3 '
= 2a[n3 ]0 + b n3 ' .
dt
(
)
(11.66)
The solution of this equation is of the form
!
n3 '" exp(#t ) ,
(11.67)
with
!
!
(
(11.68)
implying that the equilibrium is a stable equilibrium. The time scale associated
with (re)adjustment to photochemical equilibrium is
"=
!
)
" = 2a[n3 ]0 + b < 0
#1
n 2 nkb
.
$
2a[ n 3 ] 0 + b 4k d j a n 2 + jc [ n 3 ]
0
(
)
(11.69)
We see that, since n decreases with height while jaand jc increase with height, τ is
smaller (adjustment is faster) at upper levels than at lower levels.
The estimates of ja and jc, given in, respectively (11.57) and (11.56) are most
appropriate for the upper atmosphere, say at 70 km, because the insolation intensity,
assumed to obtain these estimates, is representative for the top of the atmosphere.
Using the standard atmosphere values of p and T (table 2.1) for z=70 km, (11.69)
gives
" ( 70 km) # 1 hour .
!
The coefficients ja and jc decrease very quickly with decreasing height, because Solar
ultraviolet radiation with wavelenths smaller than 300 nm is depleted nearly
completely above the tropopause (figure 2.29). Therefore, the adjustment time of the
ozone layer below 70 km height is significantly greater than 1 day. In fact in the lower
stratosphere the adjustment time is in order of several months to one year! For
instance, at a height of 20 km, where p=55 hPa, T=217 K, ja=10-13 s-1, jc=5×10-4 s-1 and
kb and kd given by (11.61) we get
" (20 km) # 210 days .
!
Because transport by the circulation can alter the ozone distribution
significantly on a time scale of several weeks, we expect that the ozonedistribution in the lower stratosphere is not in photochemical equilibrium. This
explains a remarkable feature seen in figure 11.2, namely that the ozone molecule
number concentration is higher over the spring pole than over the equator, despite the
fact that, due to lack of insolation, ozone has hardly been produced over the spring
pole during the previous months. During the northern hemisphere winter the heating
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resulting from absorption of Solar radiation over the equator and summer hemisphere
has forced a meridional circulation bringing ozone-rich air form the tropics towards
the winter pole. Over the dark winter pole the ozone molecule number concentration
hardly changes because both destruction and production of ozone require Solar
radiation (eq. 11.59).
A more precise impression of the ozone concentration as a function of height,
according to the Chapman theory, is obtained when equation (11.59) is coupled to the
radiation model that is described in section (2.7). The radiation model provides the
temperature and radiation-fluxes that are needed to calculate the photolysis
coefficients. This is the subject the following section.