Pageoph, Vol. 118 (1980), Birkhfiuser Verlag, Basel

Systems Modelling of Stratospheric Ozone Transport and Photochemistry By I. L. KAROL1)

Abstract - A scheme of a system of physical and chemical processes controlling the production, transport and destruction of ozone and its gaseous catalysts, as well as other related gases in the low and high stratosphere is presented. An account is made of temperature variations of the stratospheric layer resulting from changes in ozone content; also included is the effect of temperature variations on photochemical reaction rates and ozone and other gases transport between atmospheric layers. Parameters describing major relations of the system are inferred from the analysis of ozone and trace gas data and from the results of model calculations of interdependence between variations in temperature and ozone content of the layer. An analysis of minor fluctuations of the linearized system shows that photochemical processes are responsible for its aperiodic stability and that gas transport between atmospheric layers destabilizes the system. Key words: Ozone; Photochemistry; System modelling; Transport.

1. Introduction The basic a p p r o a c h used in the studies o f the stratospheric ozone a n d m i n o r gases p h o t o c h e m i s t r y is m a t h e m a t i c a l m o d e l l i n g o f their t r a n s p o r t in the a t m o s p h e r e with the i n v o l v e m e n t in dozens a n d h u n d r e d s o f p h o t o c h e m i c a l reactions (see e.g. C I A P , 1975; Db~WER et al., 1977). Even the simplest p a r a m e t r i z a t i o n o f the a t m o s p h e r i c t r a n s p o r t by vertical m a c r o t u r b u l e n t diffusion ( o n e - d i m e n s i o n a l models) results in c o m p l i c a t e d systems o f dozens o f n o n l i n e a r equations. Coefficients o f these systems (reaction rates a n d diffusion coefficients) are n o t k n o w n sufficiently a n d d e p e n d on the t e m p e r a t u r e o f the considered stratospheric layer to be d e t e r m i n e d b y the ozone content. This d e p e n d e n c e u p o n t e m p e r a t u r e is significant, nevertheless it is n o t t a k e n into a c c o u n t in m o s t o f the models. The present p a p e r describes first a t t e m p t s on the use o f the system analysis a p p r o a c h to the stratospheric ozone p r o b l e m characterized by a c o m p l i c a t e d system o f feedbacks. Basic c o m p o n e n t s o f the system a n d their relations are d e t e r m i n e d ; the equations describing these relations are o b t a i n e d , as well as p a r a m e t e r s o f these relations for different a s s u m p t i o n s on i n v o l v e m e n t in the system o f various groups o f 1) Main Geophysical Observatory, Leningrad, USSR.

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minor gases in the atmosphere. Preliminary results of the quantitative and qualitative research into system behaviour at the major external disturbances are presented.

2. 'Stratosphere-ozone-minor gases' system Figure 1 gives a block diagram of the system. It demonstrates the relations between the ozone content (OC) and the contents of gases chemically active (AC) and chemically passive (PC) in respect to ozone together with their 'source gases' (SC) in the upper (U: 30-50 km and lower (L:10-30 km) stratosphere. These relations are determined by the temperature of these layers (UT and LT) which affect transport rate of gases between the layers (TR), and also by sets of chemical reactions responsible for the formation (F) and destruction (D) of ozone and gases of interest. The latter depend both on the temperature and concentration of these gases. Captions in Fig. 1 are common in the field of system analysis (see e.g. FORRESTER, 1968). Among a great variety of chemical reactions the following main sets are selected (CIAP, 1975): I. Ozone-forming reactions in the upper (UOFR) and lower stratosphere (LOFR): 1. O 2 + h v ~ 2 0 ( ~ <

242)

9

2. 0 + O 2 + M - + O 3 + M (M = N2, 02, Ar);

-41

r'-

UPPER SP~ ~ ~E

" I

5S~

-"x

LOWER

i

SPH~,E

Figure 1 B l o c k d i a g r a m of the s y s t e m of o z o n e t r a n s p o r t a n d p h o t o c h e m i s t r y in the u p p e r a n d l o w e r s t r a t o sphere. T h e s y m b o l s s t a n d for: O = o z o n e ; A = gases active in r e l a t i o n to o z o n e ; P = gases passiv e to o z o n e ; S = ' s o u r c e g a s e s ' ; T = t e m p e r a t u r e ; C = gas c o n t e n t ; T R = t r a n s p o r t r a t e ; SR = supply rate; RR = removal rate; FR = formation rate; DR = destruction rate; U = u p p e r (L = lower) s t r a t o s p h e r e ; (e.g. U O D R = ozone d e s t r u c t i o n r a t e in the u p p e r s t r a t o s p h e r e ) ; solid lines = gas flows; d a s h e d lines = i n f o r m a t i o n flows.

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Modellingof Stratospheric Ozone Transport and Photochemistry

697

II. Ozone-destroying reactions in these layers UODR, LODR): 3. 03 + h ~ - + O 2 + O*(1 < 310) "a O2 + O (1 < 1140) 5. Oa + NO--+ NO 2 + 02 7. Oa + O H - + H O 2 + 02 9. Oa + HO2-+ OH + 202 11. O3 + C1--+C10 + 02

4. 03 + O --~ 209. 6. O + NO2--+ NO + 02 8. O + O H - + H + 02 10. O + HO2--+ OH + 02 12. O + C10-+ C1 + 02;

III. Reactions which form ozonewise gases m the lower (LAFR) and upper (UAFR) stratosphere from passive gases: 13. HNO3 + h ~ NO2 + OH (h < 320) 14. H202 + hv-+ 2OH (~ < 580)

15. CIONO2 + hv ~ C10 + NO2 (h < 430);

IV. Reactions destroying ozonewise gases in the lower (LADR) and upper (UADR) stratosphere: 16. NO2 + OH + M - + HNO3 + M 18. HO2 + OH--~ H20 + 02 20. CH4 + Cl-+ HC1 + CH3

17. CIO + NO2 + M - + CIONO2 + M 19. CH4 + O H - + H20 + CHa 21. C1 + HO2--+ HC1 + 02;

V. Reactions which form ozonewise gases by destroying 'source' gases transported into the stratosphere (USDR and LSDR): 22. 24. 25. 27. 28.

N + O2-+ NO + O H20 + O * - + 2 O H CH4 + O*-+ CH3 + OH CHxFyCI~ + h~--+ C1 + CHxFyCI~_ 1 N20 + h~-+ N2 + 02;

23. N20 + O*---~ 2NO H N2 + 02 26. H2 + O*-+ H + OH

VI. Reactions between different ozonewise gases: 29. 31. 33. 35.

CO + O H - + CO2 + H NO + HO2-+ NO2 + OH NO2 + h~ - + N O + O (h < 400) O* + M--+ O + M

30. 32. 34. 36.

HO2 + HO2-+ H202 + 02 H202 + OH--+H20 + HO2 C10 + N O - + C1 + NO2 HC1 + O H - + C1 + H20.

Explanations to the above list are: A is the wavelength of solar radiation in nanometers (rim = 10 -9 m); 1-4 indicate the reactions of the known Chapman's cycle; 5-12 are the reactions of the catalytic transformation of O + Oa -+ 202 by means of NO ~ NO2, H ~- HO ~- HO2 and C1 ~ C10 mutual transition; 22 is the reaction of NO formation from N being neutral or ionized by cosmic particles. The compounds HNO3, H202, HC1, C1ONO2 are passive in relation to ozone, they are partially transported to the troposphere out of the stratosphere, but they also decompose again into ozonewise compounds (active to ozone) by dissociations 13-15 and reaction 36, while reactions 22-28 form such compounds from 'source' gases transported to the stratosphere from the troposphere. The constants for most of these

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reactions are not k n o w n with sufficient accuracy, in particular this is true for their dependence u p o n temperature (DUWWER et al., 1977). The basic equations of the system are:

dUOC/dt = U O F R - U O D R + S O T R

(1)

dLOC/dt = L O F R - L O D R + T O T R - S O T R dUAC/dt --- U A F R - U A D R + S A T R

(2)

dLAC/dt = L A F R - L A D R - S A T R + T A T R dUPC/dt = U P F R - U P D R + S P T R

(3)

dLPC/dt = L P F R - L P D R - S P T R + T P T R dUSC/dt = - U S D R

+ SSTR; dLSC/dt = - L S D R

- SSTR + TSTR,

(4)

SXTP and T X T R (X = O, A, P, S) are the transport rates between the U and L layers and through the tropopause respectively. I f the two stratospheric layers of interest are merged into one layer and the transport between these layers is neglected, the system is transformed into four equations for O C = U O C + L O C ; AC=UAC+ L A C ; P C = U P C + LPC; SC = U S C + LSC.

3. A case of Chapman's photochemical cycle Let us start f r o m the simplest case, when blocks with all gases active and passive in respect to ozone are neglected. As the mixing ratio of ozone in the U layer and the density of ozone (Oa) in the L layer slightly vary with height, one m a y adopt that UOC

= [(O) + (O3)]I(M) - ( O 3 ) / ( M ) ;

L O C = [(O) + (O3)]lm ~-

(03)lm,

where (A) is the n u m b e r of molecules of any gas A in cm 3, and the normalizing factor m = (Mh) = 1018 is equal to standard air density (M) at h = 24 k m level. W h e n only reactions 1--4 are taken into account, the following equations for ozone formation and destruction rates are easily obtained: U O F R = 0.42. J1 ;

L O F R = 2. m - 1. J1(O2)

U O D R = 9.SSa. k ( m O C ) 2" ( M ) - I ;

(5)

L O D R = 9.5. m J a ' k(LOC) 2- ( M ) - 2,

(6)

where

J~(z) = ~ I(A)~(~) exp { - see q~. [%(~) V2(z) + -a(h) Va(z)l} d~

(7)

are the rates of photodissociations o f 02 and 03 (i = 1 and 3) at the levels Z with the energy absorption cross-sections ~2(A) and ~a(A) respectively; I(A) is the solar radiation intensity at the upper b o u n d a r y of the atmosphere;

V~(z) =

(05) dz;

V~(z) =

(0~) dz;

~o is the zenith angle of the Sun; ki = a~ exp (bdT) is the constant of i reaction,

k = k4/k~ = (a4/a2) exp (b2 - b~)/T;

a~/a2 = 1.7 102a;

b2 - b~ = 2800~

(8)

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Modelling of Stratospheric Ozone Transport and Photochemistry

699

For a single-layer model which does not incorporate transport, the equation for OC = (Oa) from equations (1), (5) and (6) may be presented as d(Oa) = 2J1(O2) [1 (03)2]" at L - ~ ] '

(Oa)~ JI(M) (02) 2 = J 3 . k '

(9)

where (03), is the ozone density at the photochemical equilibrium between the rates of ozone formation and destruction OFR = O D R in the Chapman's scheme. If J~ and k (and consequently (O,)) are assumed to be constant, the general solution of the equation is: [(Oa) - (03),]/[(03) + (O3),] = C exp [ - 2 t ~ / J 1 J 3 . k / ( M ) ] with arbitrary constant C. It follows from the solution that at any initial condition, for t--~ oo: (O3)-+ (03), exponentially, i.e. (03) is aperiodically stable. In the general case when J, and k depend on (Oa), the solution of equation (9) can be expressed as an integral. Its qualitative analysis reveals that its behaviour does not significantly change as t -+ ~ .

4. Linearized model o f Chapman's cycle and its analysis

In the general case equation (9) and equations (1)-(5), (6) of the system are integrodifferential and nonlinear, therefore general methods of their qualitative analysis are unknown. It is possible and reasonable, however, to study small variations of this system after its linearization. Introducing small deviations of all variables considered by addition of letter D to the symbols, we have the following equations after some transformations and simplifications: L O F R D = - 0.42[F1UL. UOCD + F1L. LOCD]-/z U O F R D = - 0.42. F l U . UOCD; FiU U O D R D

=

=

EU=

sec ~0.J~3

f?

M) dz;

FiL = sec q~-Jia(h - z). m; G U L = sec q~J~3

[EU - GU]. UOCD; + [EL - G L ] . L O C D

(M) dz

L O D R D = - G U L . UOCD

(10)

2 - U O D R [ ( U O C ) - I + (b2 - b , ) . UTD/2. UOCD. U T ~]

E L = 2. L O D R [ ( L O C ) - I + (b2 - b4). LTD/2- LOCD. LT z] GU=

9.5- k. FaU. (UOC)2/(M);

G L = 9.5. m- k. FaL. (LOC)Z/(M) 2

G U L = 9.5. m. k. FsUL-(LOC)2/(M) 2,

/z = (M)/m.

In this case /J,3(z) -

- ~J' - Ja I(A)rx{(A),3(A).exp { - sec q~. [c%(A)V2(z) + %(A) Va(z)]} dA, sec q~. ~ v3

(11) where h is the boundary level height between upper and lower stratosphere and

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ratios U T D / U O C D ; L T D / L O C D of deviations of temperature and ozone content in these layers are assumed to be the known constants. Dependence of the ozone transport rate O T R between lower and upper stratosphere upon the ozone content and the temperature gradients (i.e. upon differences U O C - LOC, U T - LT) is also linearized as: S O T R D = OTL. L O C D . m - O T U . U O C D . (M)

(12)

where the above dependence of U T D and L T D upon U O C D and L O C D is used. Coefficients O T U and O T L are evaluated in the following way: vertical ozone fluxes II through the level of 28 km during all the seasons and in different latitudes were determined using generalized data on seasonal distribution of ozone content (CLAP, 1975) and values of seasonal vertical velocities and coefficients of vertical turbulent diffusion (KAROL,1972). II values are satisfactorily correlated (correlation coefficient is 0.78 for 36 cases) with the magnitude of the vertical gradient of seasonal temperature F of the layer 10-30 mb in the same latitudes during the same seasons. The ozone flux from the upper stratosphere to the lower stratosphere tends to increase with the decrease of the temperature gradient (in high latitudes mostly in winter) and changes its sign at the increase of P up to 2~ (in the tropical zone and in summer). The regression equation of this correlation is: II = p(F - 1.6~

P -

O. 1 g/km 2 sec ~

(13)

The ozone transported across the boundary z = h of the upper and lower stratosphere then disseminates over the depths of the layers U H = 20 km and L H = 15 kin, thus affecting its content in the layers and their temperature U T D and LTD. It is reasonable to assume that the deviation of the gradient F is the difference of deviations U T D and L T D at some average levels U Z and LZ in these layers when U Z - LZ = 15 kin. The relationship between deviations of temperature and ozone content in the layer was inferred from the results of computations using the models of radiationconvection equilibrium of MANABE and WETHBRALD (1967) and RECK (1976). In the upper stratosphere for the standard U T D = I~ U O C D . (M) equals approximately 5% of the mean density U O C . ( M ) = 0.14 mg m -8 at 1.3 m m of 03 (STP) in the layer 28--48km and yields U T D / U O C D . ( M ) = 140~ in the lower stratosphere for L T D = I~ L O C D . m equals approximately 20% o f the mean density L O C . m = 0.28.mg.m -3 at 2.0ram of 03 in the layer of 10-28 km and yields L T D / L O C D . m = 18~ m - 3 (all these figures refer to mean annual conditions in temperate latitudes). Now we can write O T U = - p U T D / U O C D . ( M ) + K = - 4 . 7 10 -8 sec -1 U H ( U Z - LZ) (14) O T L = - p L T D / L O C D - m + K = - 8 . 0 10 -9 sec -x, L H ( U Z - LZ)

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Modelling of Stratospheric Ozone Transport and Photochemistry

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where K is the p a r a m e t e r describing the d e p e n d e n c e o f gas t r a n s p o r t rate u p o n its c o n c e n t r a t i o n gradient. Physical m e a n i n g o f K is close to the coefficient o f vertical t u r b u l e n t diffusion a n d its value at the level considered is a b o u t 1 m2/sec. This is m u c h less t h a n values o f the terms to which K is a d d e d a n d thus it m a y be neglected here. Deviations o f ozone (and o t h e r gases) r e m o v a l rates f r o m the T X T R D m a y be d e t e r m i n e d in a similar way. T h e y are a s s u m e d to L O C D (and L X C D ) with the coefficient L R R . L R R -1 is a residence time o f tracer in the lower stratosphere relative to its t r o p o s p h e r e a n d its value is a b o u t 1 year ( C I A P , 1975).

lower s t r a t o s p h e r e to be p r o p o r t i o n a l well-known m e a n t r a n s p o r t into the

N o w the system o f two equations (1) for U O C D a n d L O C D will be: dUOCD/dt = all.UOCD

+ a12.LOCD (15)

d L O C D / d t = a 2 1 " U O C D + a22' L O C D ,

where alx = G U - 0.42-FLU - E U - O T U ; ax2 = O T L / ~ ;

a2~ = G U L - 0 . 4 2 ( M ) . F 1 U L + O T U

a22 = G L - 0 . 4 2 . ( M ) . F ~ L / m - E L - O T L - L R R .

(16)

B e h a v i o u r o f this system solution is d e t e r m i n e d by the m a t r i x eigenvalues o f its coefficient, i.e. b y r o o t s o f the characteristic e q u a t i o n (h

-

a l l ) ' ( Z - a22) = a ~ ' a 2 1

(17)

with D = (all - a22) 2 + 4alzam

being the discriminant. O t h e r terms c o m p r i s i n g expressions for a~j are c o m p u t e d f r o m equations (10) a n d (11) a p p l y i n g s t a n d a r d m e a n a n n u a l p a r a m e t e r s o f the stratosphere in t e m p e r a t e latitudes at sec q~ = 2; they are given in T a b l e 1 for several levels which m a y be considered as m e a n ones for the layers u n d e r study. Table 1 Values of parameters for U layer (in 10 -7 see -1) and for L layer (in 10 -9 sec-O

Level (km)

GU

FU

EU

a12

ali

al~t)

36 32

1.63 0.29

56.7 26.0

8.91 2.32

-0.62 -0.31

-63.5 -27.5

-64.0 -28.0

Level (km)

GUL

GL

FUL

FL

EL

a22 + LRR

a22 + LRRt)

24 20

0.421 0.067

0.200 0.042

40.0 0.895

19.0 0.533

11.4 3.43

-22.2 4.08

-30.2 - 3.92

FU = 0.42-FLU; FUL = 0.42.F1UL-(M)/m; FL = 0.42.F1L.(M)/m. ~) For SOTRD = 0.

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As seen from the Table, coefficients al~ and a~2 in the upper stratosphere do not significantly depend on the selected mean level of the layer, while in the lower stratosphere this dependence is significant and the contribution of ozone transport (OTL) even changes the sign of a22 + L R R at the level of 20 km. Values of a2~ and eigenvalues of the system h~,2 for the levels considered and for L R R = 3.10 -8 sec -~ will be (in 10 -7 sec-1): a~x and OTU for Z = 32 km

al~ and OTU for Z = 36 km

Z (km)

a2i

~1

A2

a2i

A1

A2

20 24

-0.129 -0.516

-27.5 -27.5

-0.258 -0.516

-0.0687 -0.454

-63.5 -63.5

-0.259 -0.518

One may note that hi ~ all and h2 -~ a22 for the values considered here. The analysis of several cases with different combinations of the initial U O C D and L O C D shows, that in the u p p e r stratosphere the basic regime of the change of ozone content deviation U O C D is a rapid aperiodic establishment of photochemical equilibrium in ' m e a n time' of Ai-~ = 2-4 days, while in the lower stratosphere the equilibrium of LOC is attained slower (As 1 _ 220-450 days), and slightly unstable regimes may take place (e.g. for 'switching off' L R R - the removal from the lower stratosphere), which are characterized by positive small values of A2 at certain levels. The analysis of the above values of a~j shows that the ozone transport from the upper to the lower stratosphere (terms O T U and OTL in a2~ and a22) is the main 'destabilizing' factor. The change of the photodissociation rate of 02 which is the source of atomic oxygen, at the deviations of ozone content in the stratosphere, as well as the change in the rate of reactions 2 and 4 at the temperature deviation of LT (terms FU, FL and EL) are the main 'stabilizers'. The remaining factors affecting the photochemical source and sink of ozone in the stratosphere are of secondary importance.

5. Linearized model of'ozone-nitrogen oxides' scheme and its analysis Linearization in this scheme incorporating the reactions 1-6, 13, 16, 22, 23, 28, 33 and 35 and substantiated in CIAP (1975), produces the system of eight equations of the form (1)-(4) for the minor deviations x~ = U O C D ;

x2 = U A C D ;

x~ = L O C D ;

x6 = L A C D ;

x3 = U P C D ; x7 = L P C D ;

x4 = U S C D x8 = LSCD

(18)

from the gas contents: U A C = [(NO) + (NO2)]/(M);

LAC = [(NO) + (NO2)]/m;

U P C = (HNO3)/(M) LPC = (HNO3)/m;

(19) USC = (N20)/(M);

LSC = (N20)/m.

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Let us assume t h a t the relations similar to (12) a n d (14) are valid: SATRD = -ATU. SSTRD

=

-

x~. (M) + A T L . x~. m + T L . x6 - T U . x~;

SPTRD = 0

S T [ / . x~. (M) + S T L . x~. m + T L . x8 - T U . x~;

(20)

T L - L H = T U . U H = K / ( U Z - LZ) a n d O~ is in p h o t o c h e m i c a l equilibrium with O a n d O*, as well as N O with NO~. In this scheme we o b t a i n O D R = 9.5.J3. [k4. (03) 2 + k6. (03).

(NO2)]/k2.(M) 2

A F R = J * . k * a . (03). (N20)/k35. (M) + J~3' ( H N O a ) + Q C O S

(21)

S D R = J*3 9k'23 9(O3). (N20)/k35. (M) + J26' ( N 2 0 ) A D R = P F R = k16. ( M ) . ( O H ) . (NO2);

P D R = J~3' (HNO3) + k29' (OH),

where J, a n d k~ are p h o t o d i s s o c i a t i o n rates a n d constants ( J * a n d k*3 refer to the u p p e r b r a n c h o f reactions 3 a n d 23) o f the a b o v e list o f reactions. The following values are a d o p t e d :

k23/k35 =

5/3;

k2o = 0.9.10-13;

k6 = 0.91.10-1~;

ks = 0.9.10 - ~ e -1~~176(cma/mol 9sec), k ~ values v a r y i n g with T a n d height are t a k e n f r o m C I A P (1975, Ch. 4). Q C O S is the N O source due to the extraterrestrial r a d i a t i o n i m p a c t on the atmosphere. The s t a n d a r d m e a n a n n u a l values (N~.O), (NO~,), (HNO3) a n d ( O H ) for the b o t h layers considered are inferred f r o m their vertical profiles for t e m p e r a t e latitudes presented in C I A P (1975). These quantities allow one to calculate the following n u m e r i c a l values o f the ' p h o t o c h e m i c a l p a r t ' o f m a t r i x terms a,y o f linearized system o f equations for levels o f 20 a n d 32 k m as representative for the L a n d U stratospheric layers respectively (i = 1. . . . , 4 - U layer; i = 5-8 - L layer, (n) = 10 -~ s e e - l ) : all+ am + a2~ = aaa + a41 + a~a =

O T U = - 2 . 8 8 (6); a12 = - 7 . 6 3 (5); a~5 = OTL/t~ A T U = - 7.97 (9); a22 + T U = - 1.02 (6); a2a = 7.29 (6); 1.24(9); a25 = ATL//x; a26 = TL//z; a3a + P T U = 7.97(9); T U = - 7 . 5 4 (6); a35 = PTL/t~; aa7 = T L / ~ ; S T U = 1.02(12); a44 + T U = - 5 . 3 6 ( 8 ) ; a ~ = STL//~; TL//~; a51 - OTU.t~ = - 3 . 8 5 (10); a55 + O T L + L R R = - 1 . 3 8 (8);

a56 a6~ a67 aTa aa~ a88

- 2 . 4 4 (5); a6~ - A T U ' t L = - 6 . 9 7 (11); a6e = TU.t~; (22) A T L = 5.10 (11); a66 + T L + L R R = - 1 . 2 3 (6) = - a 7 6 ; 2.70(7); a6a = 4.65(11); aT~ - P T U . / z = 7.30(11); TU't~; a75 + P T L = - 3 . 7 7 ( 1 1 ) ; a77 + T L + L R R = - 3 . 0 6 ( 7 ) ; STU.t~ = - 2 . 3 8 (12); aa4 = TU.t~; a85 + S T L = - 3 . 3 0 ( 1 2 ) ; T L + L R R = - 6 . 1 5 (10);

= + = = +

all o t h e r a~ = O.

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The direct evaluation of ATU . . . . . STL in relations (20), as has been done for ozone, is impossible due to poor knowledge of the vertical flows of gases of interest in the stratosphere. In such circumstances one can assume that the deviations of these transports are proportional to concentrations of gases at the boundary h of U and L layers: A T R D :OTRD = [(NO) + (NO2)]: (O3) 1z=h; STRD :OTRD = - (N20): (Os)[z ~h.

(23)

These relations are valid for similar vertical profiles of gas concentrations in the vicinity of this boundary. Analysis of profiles presented in CIAP (1975, Ch. 3) reveals that this is approximately true for (Oa) and (NO) + (NO2) but not for N20, which is transported from L to U layer in the direction opposite to the 08 transport; this is reflected by the sign of minus in the second equation (23). As the maximum ot" (HNOa)/(M) is situated approximately at the level h, its transport may be negligible and therefore P T R D = 0. For the numerical values of OTU, K and OTL in (14) we obtain for the levels U Z = 32 km; LZ = 20 km; K = lm2/sec;/~ = 0.256; (n) = 10 -n sec -1 OTU = - 4.67 (8);

ATU = -3.11 (11);

STU = 1.55 (9);

OTL = - 8.00 (9);

ATL = - 6.40 (12);

STL = 2.64 (10);

T U = - 3.33 (9);

TL

=

-4.45

(9).

A comparison of these numbers and L R R = 3.0.10-8 sec-1 with 'photochemical input' in a~j, presented in (22), leads to the following conclusions. The (NO) + (NO2) = AC transport and removal (LRR) are small in comparison to its photochemistry throughout the stratosphere, this is also true for (HNOs) = PC even for P T R D # 0 but proportional to O T R D like ATRD. Ozone transport and removal are essential in L layer but relatively unimportant in U layer. The (N20) = SC transport gives the basic input in a~j throughout the stratosphere. The ozone destruction by NO and NO2 is rather significant in L layer but comparatively small in U layer. The stability of the linearized system is investigated on the basis of sign determination of the real parts of matrix I]a~Jlleigenvalues h~using the known Routh's scheme for the roots of the characteristic polynomial of the 8th order of the system (GANTMACHER, 1967). In the general case at least one A~with Re A~ > 0 is present, as the last term D = la, I (the system determinant) of this polynomial is negative unlike its other positive terms. Therefore the system is unstable in this case and it maintains instability (D remains < 0) a t ' switching off' the OTRD, A T R D but retaining STRD (N20 transport). At STRD = 0 D becomes positive and all Re A, < 0, and the system is stable for small perturbations. 'Switching off' of the lower stratospheric removal (LRR = 0) maintains the system stability in the absence of other transports. Therefore, as in the Chapman's scheme, the photochemical transformations

Vol. 118, 1980)

Modelling of Stratospheric Ozone Transport and Photochemistry

705

stabilize the system and the involvement of the nitrogen oxide catalysts increases the stability. The transport is again the destabilizing element, especially the N20 transport (STRD) which is the worst modelled, and the situation is to be improved in the future.

6. Conclusions

The results of this study of the simplest linearized case appear rather promising for the application of system analysis methods in the problem of the stratospheric ozone dynamics. This approach provides a selection of the main factors, a discovery of'critical' parameters determining the dynamics of the event and requiring a special study. It would be helpful in the analysis of the ever complicating picture of transport and photochemical interrelations of many gases in the stratosphere, affecting its meteorology. Further investigations of the problem should include the general nonlinear case in the Chapman's scheme of reactions and in the above set of reactions. It is essential to make more careful study of the transport of ozone and other gases in the stratosphere, as well as of relations between the transport characteristics and the temperature field. Consideration of other and more than two gas reservoirs (e.g. upper and lower stratosphere in the tropical, polar and temperate latitudes) may be coupled with the improved transport modelling between them.

REFERENCES

CIAP (1975), Monograph I. The natural stratosphere of 1974. DOT-TST-75-51, Wash., D.C. DUEWER, W. H. et al. (1977), NOx catalytic ozone destruction: sensitivity to rate eoeffieients, J. Geophys. Res. 82, No. 6. FORRESTER,J. W. (1968), Principles of Systems. Wright-Allen Press, Camb. Mass. GANTMACHER,F. R. (1967), Teoria matritz, Moscow, 'Nauka'. KAROL, I. L. (1972), Radioaktivnye isotopy i globalnyi perenos v atmosfere. Leningrad, Gidrometeoizdat. (English translation: Radio-isotopes and global transport in the atmosphere. IPST, Jerusalem, 1974). MANABE,S. and WETHERALD,R. T. (1967), Thermal equilibrium of atmosphere with a given distribution of relative humidity, J. Atm. Sci. 24, No. 3. RECK, R. A. (1976), Stratospheric ozone effects on temperature, Science 192, No. 4239. (Received 12th December 1978)