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Algorithm for calculation of the chain length and rate of stratospheric ozone depletion in O_{x}, HO_{x}, NO_{x}, ClO_{x} and BrO_{x} cycles has been developed. The most important new element in the theory of stratospheric chain processes is the correct determination the propagation rate, taking into account all reactions involved rather than a single reaction, which has the lowest rate, as it was usually done before. The role of null chain processes in the cycles has been considered and shown that these processes play a decisive role in formation of families of the odd oxygen, nitrogen, chlorine and bromine in the daytime while at night they play no role. Using two-dimensional model Socrates, and algorithm developed correct rate of ozone depletion and chain length in the cycles above for model conditions of June 2020 at 50^{。}N have been calculated.

For the first time the photochemical theory of stratospheric ozone was developed by the outstanding English geophysicist Sydney Chapmen [

The chain length (in this case it’s a number of molecules of ozone, destroyed by one active particle in its life time in the stratosphere) is one of the most important characteristics of chain process. Special importance this index has acquired in the contemporary conditions, when it was convincingly demonstrated, that because of the chain processes the anthropogenic factors proved to be capable of competing with the natural processes. Therefore it is important to understand, what a future potential participants of the chain stratospheric processes are danger for the ozone layer and what is their effectiveness in comparison with the chlorine components, having played the dominant role in the anthropogenic depletion of the ozone layer in the end of the past century. A definition of the chain length makes it possible to successfully solve the task as well as task of a comparison of the effectiveness of these cycles, similarly, as this is done with respect to various halogen-containing chemicals, for which an index of Ozone Depletion Potential is used (see, for example, [_{x}, HO_{x}, NO_{x}, ClO_{x} and BrO_{x} cycles which are considered now to be the main cycles of destruction of stratospheric ozone. New key points in our analysis are a correct determination of the rate of chain propagation and finding algorithm for the chain length calculations in the cycles mentioned above.

In the simplest case the chain process of the ozone depletion in the stratosphere can be written as follows [

Here (R1) is a reaction of chain initiation, (R2), (R3) are the ones of chain propagation and (R4) is the one of chain termination. The length of chain, ν, which in this case is a number of odd oxygen particles, destroyed by one active particle X in time of its stratospheric life, can be calculated using Equation (1):

If reactions (R2) and (R3) are unique processes of a mutual exchange between X and XO, their rates become identical already after several cycles. However, as a rule, besides reactions (R2) and (R3) other processes of exchange between X and XO run in the stratosphere. As a result, it does not lead to alignment of the reaction rates even at considerable chain length. But in any case because the reactions of chain propagation are consecutive ones, the rate of destruction of odd oxygen (i.e. O_{3} + O) are defined by the rate of so-called limiting step of the process. Usually the limiting step refers to the reaction with the lowest rate. But in most real situations in the stratosphere it is difficult to determine a true limiting step (as a single reaction) for all heights. For example, in the NO_{x} cycle the chain propagation reactions are

and reaction (R6) is considered to be a limiting step in NO_{x} cycle because it has a minimal rate in the stratosphere [12-14]. But as it seen from

more height reaction (R5) is getting slower than reaction (R6).

In addition, around 40 km reaction rates are the same and formally a limiting step here is missing.

To resolve the problem we offer a simple rule for calculating the rate of limiting step (=the rate propagation)— in case of two reactions of propagation, (R2) and (R3), the rate of the limiting step, should be defined by Equation (2):

where, and the rate of ozone depletion, , should be defined by Equation (3):

where factor 2 appears as two particles of odd oxygen are destroyed in the chain process. Rate of limiting step by Equation (2) is shown in _{2} = W_{3} the rate of destruction of odd oxygen equals 0.5 × W_{2} or 0.5 × W_{3}. The last conclusion is a logical consequence of the fact that two successive steps require two times longer than one.

From here it is easy to conclude that under three reactions of chain propagation run with rates W_{2}, W_{3} and W_{4}, is expressed by Equation (4):

Similarly, it is possible to calculate the rate of a limiting step at any number of propagation reactions at any small differences in their rates that would represent a significant challenge for the limiting step in usual formulation. The concept of a limiting step of chain process introduced and its definition through inverse rates of chain propagation reactions is a basic new feature of stratospheric chain process, which was not considered till now at description of the chain stratospheric processes. We’ll show how to use the new concept at consideration of main cycles of stratospheric ozone depletion in the next section.

This section presents the results of calculation of the chain lengths, as well as the rates of limiting steps and chain limitation in O_{x}, HO_{x}, NO_{x}, ClO_{x} and BrO_{x} cycles. The chain length has been determined by Equation (1). All numerical data have been obtained using 2D model Socrates [

It’s assumed that odd oxygen [O_{x}] = [O_{3}] + [O(^{3}P)] + [O(^{1}D)] where O(^{3}P) is atom O in the ground state and O(^{1}D) is the one in exited state with energy 1.96 eV. According [

where M is molecules of air, , and are coefficients of photodissociation of O_{2} and O_{3}, respectively.

Since reaction (R11) converts O(^{1}D) to O(^{3}P), it follows that the reaction (R12) determines the loss of all three component of odd oxygen, i.e. (R12) may be seen as a reaction of chain termination, W_{d}(−O_{x}). On the other hand, reaction (R12) should be considered as a chain propagation reaction in O_{x} cycle because just this one does destroy odd oxygen. It follows that chain length in O_{x} cycle, is equal to 1 (see Equation (1)) and the rate of chain propagation, W_{p}(−O_{x}), is:

It can be shown that stratospheric lifetime of O_{x}, τ(O_{x}), is much more than τ(O_{3}), τ(O(^{3}P)) and τ(O(^{1}D)) if the latter are defined through its individual ways of destruction. The situation is realized thank to so called O_{x} null cycle which doesn’t destruct O_{3} or O(^{3}P). The reactions of O_{x} null cycle are (R13) and (R9):

So using Equation (2) one can find that the rate of the limiting step in O_{x} null cycle

where and

. The rate of chain termination in null cycle is the same as in the main cycle, i.e. So we obtain that

Height profile of is shown in ^{6}, and at the altitude of 55 km it’s about 50. In our case the chain length means a number of mutual conversion of O_{x} components occurring during stratospheric lifetime of O_{x}, τ(O_{x}). So, a condition of means that mutual conversion of O_{3}, O(^{3}P) and O(^{1}D) occur so fast in comparison with τ(O_{x}) that they are getting chemically indistinguishable and one can consider lifetime of these components to be the same as τ(O_{x}). But it should be the case only for daytime conditions. At night, W_{9} and are getting to zero, O_{x} family disappears and lifetime of O_{3}, O(^{3}P) and O(^{1}D) are getting dependent on

its individual loss processes.

Finally, let’s underline that to understand a daytime chemistry of ozone it is possible only through a method of families. It is the only way to define correctly atmospheric lifetime of ozone as well as to show that a unique source of ozone in the stratosphere is photodissociation of О_{2}, and a unique sink is a reaction O_{3} with O(^{3}P).

Family of odd hydrogen. Destruction of odd oxygen in HO_{x} cycle occurs through the following chain mechanisms.

Cycle I:

Cycle II:

Cycle III:

Cycle IV:

Rate of the limiting steps in Cycles I-IV is determined by Equations (8)-(11):

where W_{#} is the rate of reaction (R#). Height profiles of and their sum are shown in

Chain termination processes in HO_{x} cycle and their rates are shown in _{x} cycle is due to the reaction (R20):

So the rate of chain termination in hydrogen cycle, , can be written as Equation (13):

Chain length in HO_{x} cycle one can find using Equation (14)

In conclusion it should be noted that under description of ozone depletion in HO_{x} cycle usually reactions with individual participation O_{3} or O are taking into account (see [11,14]). In this connection it should be underlined that as it follows from Chapman’s mechanism, ozone can be destructed only through depletion of the odd oxygen which requires at least two reaction of chain propagation. But in this case the rate of ozone depletion is expressed using limiting steps as it was done above. Using individual reaction with O_{3} or O means, in fact, an ignoring chain mechanism of ozone depletion in the stratosphere.

NO_{x} family includes NO and NO_{2}. Like HO_{x}, cycle NO_{x} includes three chain mechanisms Cycle I:

Cycle II:

Cycle III:

Height profiles of the rate of limiting steps in Cycles I-III are shown in

It’s seen that the most important cycles in that case are Cycles I and II. So the rate of ozone depletion in total NO_{x} cycle, can be expressed by Equation (15):

Chain limitation in NO_{x} cycle occur through chemical conversion of NO in N_{2}:

and physical process of the turbulent transport characterized by time τ_{d} expressed by Equation (16):

Here H is scale height, k_{zz} is a vertical coefficient of turbulent diffusion. So the rate of chain limitation in NO_{x} cycle, W_{d}(NO_{x}), can be expressed as

Height profiles of W_{d}(NO_{x}) are shown in

Knowledge of the rates of chain propagation and of chain limitation allows one to express a chain length in NO_{x} cycle,:

It should add that in the low and middle stratosphere NO_{x} cycle runs mainly not destroying odd oxygen. It’s explained by a null cycle, a chain length of which here is much higher than the one of main cycle determined by Equation (18). Reactions of the NO_{x} null cycle are the following:

Limiting step of null cycle includes reactions (R5) and (R23) and its rate can be easily calculated by the same rules as above. Chain limitation is the same as in the main cycle (see Equation (17)). Let’s remind, that the null cycle acts only in day conditions that provides a possibility of existence of NO_{x} family. At night the rate of the reaction (R23) is getting to zero and NO_{x} family disintegrates.

_{x} cycle.