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O+Oa

Figure 3.1 Schematic of the Chapman odd-oxygen system.

3.2.1 Partitioning of odd oxygen

An important question still remains: what fraction of Ot is in the form of O, and how much is in the form of O? In other words, how is O, partitioned among its constituents? Considering just the reactions that interconvert O, (reactions (3.2) and (3.4)), and assuming photochemical steady state, then production of O, equals its loss:

where the left-hand side is the loss rate of O, and the right-hand side is the production rate of 03 (alternatively, production of O equals its loss; the left-hand side is the production rate of O atoms and the right-hand side is the loss rate of O atoms). Rearranging Equation (3.11), we get:

Figure 3.2 shows that this ratio ranges from 10 7 in the lower stratosphere to Iff 2 in the upper stratosphere. Thus, nearly all O, in the stratosphere is in the form of 0:,. This justifies our assumption (used in Equation (3.10)) that [OJ = [O,]. Note that, at night, photolysis ceases (/0 = 0) and all of the O atoms are converted to O, via reaction (3.2). Thus, all of tir is in the form of O, at night.

O/O, Ratio

O/O, Ratio Figure 3.2 Calculated profiles of the O, and O number density (solid lines, bottom axis), and the ratio lOJ/jO,] (dashed line, top axis). Calculation is for45°N equinoctial conditions based on daytime-average constituent abundances and photolysis rates.

Concentration (molecules/cm )

Figure 3.2 Calculated profiles of the O, and O number density (solid lines, bottom axis), and the ratio lOJ/jO,] (dashed line, top axis). Calculation is for45°N equinoctial conditions based on daytime-average constituent abundances and photolysis rates.

In the previous calculation, we assumed photochemical steady-state conditions for the members of O,. How good an assumption is this? To study this question, let us revisit the continuity equations for O, and O atoms (Equation (3.8)). Assuming that the reactions that interconvert O, (reactions (3.2) and (3.4)) are much faster than the reactions that produce and destroy O,, we can rewrite Equation (3.8), neglecting the Os production and loss terms:

These differential equations are appropriate for time periods short compared to the lifetime of O,. Over longer time periods, the terms representing production and loss of O, (as well as transport) can have a significant cumulative effect.

The differential equations in Equation (3.13) can be solved analytically to obtain:

where we have assumed that [O] = 0 at / = 0 and C is a constant of integration that is determined by the concentration of Or. Deviations from the steady state are represented by the exponential terms in Equation (3,14). The system approaches its steady-state value with an «»-folding time of l/(./0l + ¿VhqJC),]), which is the reciprocal of the sum of the loss frequencies of O, and O. Because the lifetime of O is much shorter than the lifetime of O,, £0+o, /03, and 1/(■/(>,+ &o+o,[C)21) = I/^o+oJOjJ). Thus, the perturbation term is damped on a time-scale equal to the lifetime of O atoms, which is much less than a second in the stratosphere. Conditions in the atmosphere do not change significantly on such short time-scales, so it is reasonable to assume that (). and O are always in the photochemical steady state.

Taking the limit of Equation (3.14) as time goes to infinity, the perturbation terms in Equation (3.14) are zero. In this case, the |0]/f0r] ratio becomes J(),/(£0+0J02l), which is the same result as was obtained when considering the solution to Equation

Why have we spent time and effort in this chapter defining Ov? The reason is that O, offers conceptual advantages versus thinking about the species O, and O as separate entities. The lifetime of O, is minutes or less in the stratosphere. Thus, one would expect that on time-scales much longer than this, the abundance of O, can change significantly. In this chapter, however, we have shown that (1) the lifetime of O, is much longer than the lifetime of 0-„ and (2) virtually all of Or is in the form of O,. As a result, the abundance of O, can in reality only change on time-scales comparable to or longer than the lifetime of Or. This is true because photolysis of O , creates an O atom that almost all of the time reacts with 0: to reform O,. Therefore, photolysis of O, does not represent a net sink of O,.

It is, in fact, quite common in both formal and colloquial situations for people to use "ozone" where they really mean "odd oxygen". For example, the literature is filled with statements such as: "the lifetime of ozone in the lower stratosphere is months." Strictly speaking, this is incorrect. Photolysis destroys O, rapidly, with a typical lifetime 1/L of minutes or less throughout the stratosphere. However, the lifetime of Ot is months, and virtually all of Ot is in the form of ozone. Thus, if you follow an air parcel in the lower stratosphere, you will see that its O, abundance is constant on time-scales shorter than the lifetime of O,. This does not mean that O , is not being destroyed

in the parcel. On the contrary, all of the O, in the parcel is destroyed many times every day that the parcel is in sunlight. However, Os is reformed in the parcel at almost exactly the same rate. Only on time-scales of the lifetime of O, are the effects of the imbalance between production and loss of O. large enough for its concentration to change noticeably. Thus, the statement that the lifetime of O, in the lower stratosphere is months, while technically incorrect, is true in a practical sense.

In addition, there is a computational advantage to using O, for numerically modeling the stratosphere. To predict the abundance of constituent X at a given point in time and space, models of the stratosphere integrate the differential equation lX1' = jf^F W^fr-LlXl-V-mxn)*^ (3.15)

To numerically solve Equation (3.15), the integral is broken into short time segments of length At and converted into a summation—a process often referred to as "discretization" (the literature on numerically solving differential equations is extensive, e.g. see Press et al. , Chapter 15). The difference in lifetimes of the stratospheric constituents means that there are several very different time-scales over which the dependent variables (O,, O, and other constituents) are changing (this is known as a "stiff set of differential equations). In order to maintain stability of the integration, one is required to follow the variation in the solution on the shortest time-scale, in this case the lifetime of O. Thus, a model of the lower stratosphere that considered O, and O as separate and independent species would have to have a time step At of less than X() (less than a millisecond in the lower stratosphere). With such a small time step, it is impractical for a model to integrate Equation (3.15) for seasonal or yearly time periods, and therefore it would be impossible to simulate many of the phenomena that are of interest. By modeling O, instead of both O, and O, however, one can take advantage of the much longer lifetime of O, , and use a bigger time step in the model.

3.3 Catalytic Loss Cycles

Calculations of O: abundance based on just the Chapman reactions yield estimates of O, that are considerably larger than measurements. The reason for this is that we have neglected an important O, loss pathway: radical catalysis [67|.

A catalyst is defined as a substance, often used in small amounts relative to the reactants, that increases the rate of a reaction without being consumed in the process. With this definition in mind, consider the chemical reaction sequence first suggested by Stolarski and Cicerone :

In this reaction set, CI acts as a catalyst by facilitating the conversion of O, to O, without being consumed. For this catalytic cycle to be important, it has to destroy O, at a rate comparable to or faster than the rate at which O and O, directly react. Otherwise the catalytic cycle will have little impact on the overall rate of O, destruction. Thus, we have to evaluate how fast the catalytic cycle (3.16) destroys O,.

3.3.1 Rate-limiting step

What determines the rate that a catalytic cycle destroys Ot? To figure this out, let us look more closely at the cycling between CI and CIO (Figure 3.3). CIO is formed only through reaction of CI with Ot. CI is reformed when CIO reacts with either O or with NO.

When the cycle is completed by CIO + O, then two O, are destroyed, as shown in reaction (3.16). When CIO reacts with NO, however, the following cycle results:

Note that this reaction scheme does not destroy O,; instead it merely converts O, to O.

In other words, every reaction between CIO and O leads to the loss of two O,. But not every reaction between CI and 03 leads to O, loss; if the resulting CIO reacts with NO, then a null cycle (reaction (3.17)) results. Therefore, the rate of O, loss in reaction (3.16) is set by the rate of the reaction between CIO and O, and not by the rate of reaction between CI and O,. In general, the rate at which a catalytic cycle destroys O, is determined by the slowest reaction in the cycle, which is known as the rate-limiting step. We can therefore write the rate of Ox loss through reaction (3.16) as 2/t(|0_0[CIO|| ()|. The factor of 2 accounts for the fact that every trip through the catalytic cycle destroys two Ov members (one O, and one O).

Figure 3.4 shows the ratio of the rate of O, loss through the CIO-CI catalytic cycle (2/tatHO[C10]|01) to the rate of O, loss through the Oyt-O reaction (2A.-()r,.o[0:i]) as a function of CIO abundance. If this ratio is much less than 1, then loss through the CIO-CI catalytic cycle is insignificant compared with loss through the direct 0,+0 reaction. If the ratio is much greater than 1. then the catalytic loss dominates the

CI + O, -> CIO + O, CIO + NO CI + N02 N02 + hv -> NO + O

No loss of odd oxygen' Figure 3.3 Schematic of the Cl-CIO system. Arrows denote pathways for conversion between CI and CIO. Each arrow is labeled by the reactant that accomplishes the conversion (e.g. the arrow going from CI to CIO labeled with O, means that CI is converted to CIO through reaction with (),).

Loss of two odd oxygens

Figure 3.3 Schematic of the Cl-CIO system. Arrows denote pathways for conversion between CI and CIO. Each arrow is labeled by the reactant that accomplishes the conversion (e.g. the arrow going from CI to CIO labeled with O, means that CI is converted to CIO through reaction with (),).

direct reaction. Figure 3.4 shows that these two loss pathways are equal for a CIO mixing ratio of -20 parts per trillion by volume. Typical stratospheric abundances of CIO are tens to hundreds of parts per trillion by volume [46,69—71 J, indicating that destruction of O, through this catalytic cycle is important. The importance of this conclusion cannot be overstated: the CIO radical, even in abundances as small as a Figure 3.4 Ratio of 2kvilM)[C\0]\0] to 2*(,rO|0,||0] versus the abundance of CIO. Computed for lower stratospheric conditions: temperature of 210 K, |O.J = 2 ppmv, and rate constants from DcMore ct al. [5|.

CIO (pptv)

Figure 3.4 Ratio of 2kvilM)[C\0]\0] to 2*(,rO|0,||0] versus the abundance of CIO. Computed for lower stratospheric conditions: temperature of 210 K, |O.J = 2 ppmv, and rate constants from DcMore ct al. [5|.

few tens of parts per trillion by volume, is capable of destroying significant amounts of Ot, despite O, abundances of several parts per million by volume.

3.3.2 Other catalytic cycles

The Cl-CIO cycle is just one of several catalytic cycles that destroy odd oxygen in the stratosphere. There is an NO-NO, cycle [72| that is quite similar to the CI CIO cycle:

The rate-limiting step of this cycle is the reaction between NO, and O. There is also an analogous OH-HO, cycle:

The rate-limiting step of this cycle is the reaction between HO, and O.

Note that these three cycles (reactions (3.16)—(3.19)) are rate limited by reactions involving O atoms. Other cycles, rate limited by reactions involving other species, tend to be (relatively) more important in regions where O atoms are rare (such as the lower stratosphere). The cycle

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