arXiv:physics/0007032v1 [physics.ao-ph] 11 Jul 2000 Chapman’s model for ozone concentration: earth’s slowing rotation effect in the atmospheric past J. C. Flores† and S. Montecinos‡ † Universidad de Tarapacá, Departamento de Fı́sica, Casilla 7-D, Arica, Chile. ‡ Universidad de la Frontera, Departamento de Fı́sica, Casilla 54-D, Temuco, Chile. Chapman’s model for ozone concentration is studied. In this nonlinear model, the photodissociation coefficients for O2 and O3 are time-depending due to earth-rotation. From the Kapitsa’s method, valid in the high frequency limit, we find the criterion for the existence of equilibrium solutions. These solutions are depending on the frequency, and require a rotation period T which satisfies T < T1 or T > T2 . Where the critical periods T1 and T2 , with T2 > T1 , are a function of the parameters of the system (reaction rates and photodissociation coefficients). Conjectures respect to the retardation of the earth’s rotation, due to friction, suggest that the criterion was not even verified in the atmospheric past. Key words: Atmospheric Physics. Chemical Physics. Nonlinear Dynamic Systems. Oscillations. 1 1.- Introduction The dynamics of the ozone layer in the atmosphere has different basic process like: chemical reactions, photochemical reactions and transport (diffusion, convection, etc.). In a general point of view, this dynamics is complex and requires some approximations to be studied. In this sense, we consider a photochemical model proposed by S. Chapman (Brasseur, 1986; Chapman, 1930; Wayne, 1991). This model considers a set of reactions between the oxygen components. Explicitly, R1) R2) R3) R4) O + O2 + M → O3 + M, O + O3 → 2O2 , O2 + hν → O + O, O3 + hν → O + O2 . In the reaction of the ozone production R1, M denotes any atmospheric element acting as catalyzer. The reaction R2 denotes the loss of oxygen O and ozone O3 producing molecular oxygen O2 . R3 and R4 correspond to photochemical destruction process related to the solar radiation (symbolized by hν). The time evolution equations for the constituents consider the above reactions for variable concentration assuming the concentration of O2 being stationary. Let X and Y be the concentration of O and O3 respectively. Then, the Chapman’s model (see for instance Brasseur, 1986; Montecinos, 1998; 1999) considers the time evolution equations for the concentrations given by dX = J1 + J2 Y − (k1 + k2 Y ) X, dt 2 (1) dY = k1 X − (J2 + k2 X) Y, (2) dt where, on the right hand, the positive terms are production rates, and the negative ones are loss rates. In the nonlinear system (1,2), the quantities J1 and J2 are related to the reactions R3 and R4 and correspond to the photodissociation of O2 and O3 respectively. The important fact is that they dependent on the sun’s radiation, and then, they are periodic in time with a period T = 24 hours. In this paper, and by simplicity, we assume Ji (t) = Jio (1 − cos ωt), i = 1, 2 (3) where ω = 2π/T , and Jio are positive constants. On the other hand, the positive constants k1 and k2 in (1,2), are temperature dependent. Also they are dependent on the O2 concentration, and related to the reaction velocity in R1 and R2 respectively (DeMore, 1994). In this paper we propose an analytical study of the nonlinear model (13). In a general point of view, the study of this system is a difficult task, nevertheless, some interesting results can be find in the high frequency (ω) limit. In fact, we use a method proposed originally by Kapitsa (Landau, 1982) for a mechanical particle in a field with rapid temporal oscillations in the parameters and nonlinear terms. We find explicitly the solutions of the systems (1-3) in the high frequency regime (19,20). They are the equilibrium solution for the time-averaged concentration. Calculations tell us that no solution exist in some frequency range. The study of the behavior of this model, for different frequencies, has a physical interest because earth period of rotation changes with the geological age. It is a known fact that the rotation of earth is gradually slowing down by friction. In fact, the estimated rotation velocity diminishes by 4.4 hours every billon of years (Shu, 1982). For explicit calculations, we assume the following values for the parameters: J1o ∼ 107 [1/s]; J2o ∼ 10−3 [1/s]; k1 ∼ 10[1/s]; 3 k2 ∼ 2.5×10−15 [1/s], (4) corresponding to the values for the ozone layer altitude at, more or less, 35 km. Finally we note that the case of small frequency (ω → 0) can be solved. In fact, here the parameters Ji (t) evolve slowly with time and then, they can be assumed as constant in the integration process of (1,2). So the solutions corresponding to the ‘fixed point’ ( dx ∼ dy ∼ 0, for definitions see (Seydel, dt dt 1988; Wio, 1997)) are Y = Jo − 1o 4J2 + v u u t J1o 4J2o !2 + k1 J1o , 2k2 J2o (5) Y J2 (t) . (6) k1 − k2 Y Namely, in this approximation, the variable Y is constant and X varies linearly with the dissociation coefficient. Moreover, the solutions (5,6) are consistent with the numerical solution in the stratosphere (Fabian, 1982; Montecinos, 1996). We note that, from (5), it is easy to show that the variable Y satisfies Y ≤ kk12 . In fact, it is a general bound for the solution of the systems (1,2) (Montecinos, 2000). X= 2.- The method of Kapitsa As said before, we shall study the system (1-3) in the high frequency regime. High frequency means here small period of oscillations T respect to the relaxation-time TR for the slow variables. Assume the separation of the concentrations X and Y in a slow temporal variation (x and y) and other fast (ε and η, respectively). Namely, X = x + ε; Y = y + η, (7) where the fast variables are periodic with temporal average zero, namely, 4 hεiT = hηiT = 0. (8) The equation (1) can be re-written like dx dε + = J1o + J2o y − (k1 + k2 y) x − (J1o + J2o y) cos ωt+ dt dt (J1o − J2o cos ωt − k2 x) η − (k1 + k2 y) ε − k2 εη, (9) and equation (2) becomes dy dη + = k1 x − (J2o + k2 x) y + J2o y cos ωt− dt dt o (J2 + k2 x − J2o cos ωt) η + (k1 − k2 y) ε − k2 εη. (10) On the other hand, the fast variables are only related to rapid oscillation (Landau, 1982). In this way, from the above expression (9,10), they are assumed to be a solution to the differential equations: dε dη = − (J1o + J2o y) cos ωt; = J2o y cos ωt. (11) dt dt At this point a remark becomes necessary. The expression (7), complemented with the above equations (11), defines a change of variables without approximations. Nevertheless, the differential equations (11) are suggested by the direct oscillatory term in (9) and (10). Kapitsa’s method consider the equations for dε and dη as approximated. dt dt In one period, the slow variables are essentially constants and the fast have zero average (8), then, the time average of the equation (9) becomes. dx = J1o + J2o y − (k1 + k2 y) x − J2o hη cos ωtiT − k2 hεηiT , dt and, for equation (10), we obtain dy = k1 x − (J2o + k2 x) y + J2o hη cos ωtiT − k2 hεηiT . dt 5 (12) (13) Nevertheless, since equations (11) can be solved exactly, ε(t) = − 1 o (J1 + J2o y) sin ωt; ω η(t) = J2o y sin ωt, ω (14) the evolution equations for the slow variables become J o k2 dx = J1o + J2o y − (k1 + k2 y) x + 2 2 y (J1o + J2o y) , dt 2ω (15) and J o k2 dy = k1 x − (J2o + k2 x) y + 2 2 y (J1o + J2o y) . (16) dt 2ω This set of equations are the basis for our analytical results. They are restricted to the high frequency approximation. This approximation becomes given by the ‘expansion’ in 1/ω 2 related to the last term in (15,16). Remark that it is an autonomous nonlinear systems and then, without the explicit temporal dependence. This transformation, from a set of equations with time-periodic parameters, to other autonomous, is related to the Kapitsa original ideas (Landau, 1982). In a general frame of work, the system (15,16) is complex. Moreover, the approximation of high frequency is valid when the relaxation time TR , of the equations (15,16), is bigger than 2π/ω. This comparison is a difficult task, nevertheless, the case J2o = 0 can be solved exactly to estimate the validity of the approximation. It corresponds formally to eliminate the dissociation of O3 . The asymptotic solution of the non-autonomous system (1-3) is (Montecinos, 2000). X= J1o Jo cos(ωt − φ), −q 1 2k1 4k12 + ω 2 Y = k1 , k2 (17) where the phase φ is given by the relation: tan φ = ω/2k1. On the other hand, combining the equations (15,16) with (7), in the high frequency approximation we found that the equilibrium solution given by the Kapitsa’s method is: Jo k1 Jo Y = . (18) X = 1 − 1 sin ωt; 2k1 ω k2 6 It is direct to show that the exact solution (17) reduces to (18) in the high frequency limit. Moreover, the relaxation time TR can be calculated here analytically. It is given by TR = 1/2k1 . So, we expect that the high frequency approximation (15,16) is valid when 4πk1 ≪ ω. 3.- Existence of equilibrium solutions In this section we are concerned with the fixed point solution (Seydel, 1988; Wio, 1997) of the autonomous set (15,16). This system have an equi= dy = 0, and given by the solution of librium point (xo , yo), defined by dx dt dt the equations: k2 J2o ! ! k1 J o k1 J o 2 − 22 yo2 + k2 J1o 1 − 22 yo − k1 J1o = 0, ω ω (19) and xo =  (J1o + J2o yo ) 1 + k2 J2o y 2ω 2 o (k1 + k2 yo )  . (20) Equations (19) and (20) define the homogeneous equilibrium solution of the autonomous system (15) and (16) and then, with (7) and (14), we have the solution of the systems (1,2) in the high frequency regime. The existence of real solutions, for the second degree equation (19), requires the inequality 4 k2 J1o 1 − 1 + ω 4 k2 J1o 2k1 J2o ! 8 k2 J1o 1 + 1 + ω 2 k1 k2 J1o J2o 8k1 J2o ! ≥ 0, (21) which corresponds to an inequality of second degree for 1/ω 2. Namely, there is no equilibrium solution of (15,16) if and only if, s s 1 2 1 2 1 2 2 k2 J1o k2 J1o + − ≤ ≤ + + . (22) 1 − 1 − k2 J1o k1 J2o k2 J1o k1 J2o ω2 k2 J1o k1 J2o k2 J1o k1 J2o 7 If we assume the condition k2 J1o ≪ 1, k1 J2o (23) valid for the parameters (4) of section 1, the inequality (22) can be re-written for the period T . In fact, there is no equilibrium solution of the system (15,16) when T1 ≤ T ≤ T2 (no − solution), (24) where s 2 ; T1 = 2π k1 J2o T2 = 4π s 1 . k2 J1o (25) 4.- Earth’s slowing rotation and the existence of solution It is interesting that the inequality (24) gives a region were no solution exist. Here we must take care because no oscillating solution like (7) exist. In fact, (24) splits the ω−space parameter in three regions: (i) The region defined by T < T1 where a real positive solution (yo > 0) of equation (15) exist, with a negative one (yo < 0) . (ii) The region defined by (24), where no solution of (15) exist. (iii) The region defined by T > T2 where solutions are negative (yo < 0). From (5,6), we known that in the slow frequency limit (region (iii)) a real positive solution exist. Then, Kapitsa’s method does not work well in this region, nevertheless, at least it says that an oscillating solution exist. At this point we can formulate the following question: since the earthrotation has diminished by friction, how has the change in rotation affected 8 the existence of the ozone layer ?. This question seems appropriate because the Kapitsa’s method tells us that the frequency of rotation and the ozone concentration are related. Using the parameter values (4), of section 1, we can estimate the critical period (24) : T1 ∼ 0.02 hours and T2 ∼ 22 hours. Is interesting that the actual period T = 24 hours, is in the region of permitted solution (T > T2 ). Moreover this is suggestive: from the retardation of earth rotation velocity data (4.4 hours/billon of years, (Shu, 1982)), a simple ∼ 0.46 billons years no solution existed calculation tells us that before 24−22 4.4 because we were in the region (24). This is a surprising estimation if we consider that actually the ozone layer is believed to have been in existence 0.7 billon years (Graedel, 1993). 5.- Conclusions We have considered the Chapman’s model for ozone production (1,2). In this nonlinear model, the parameters related to photodissociation are periodic in time (3). We were interested at the analytical study of this model by using the high frequency approximation, due to Kapitsa. Namely, we have considered the autonomous system (15,16), depending on the frequency, for the averaged variable concentrations. The existence of equilibrium solutions (fixed points (19,20)) is depending on the frequency. In fact, there are two critical period T1 and T2 so that for T1 < T < T2 there is no equilibrium solution (24). The values for the parameters (4), in section 1, give the condition of noexistence (24): 0.02 hours ≤ T ≤ 22 hours, and then compatible with the actual earth’s period of rotation, and existence of the ozone layer. Moreover, considering the earth’s slowing rotation motion due to friction (4.4 hours every billon of years, (Shu, 1982)), we estimate that the ozone existence condition is verified after 24−22 ∼ 0.46 billon years (section 4). This is a good 4.4 estimation if we consider the simplicity of the autonomous model given by equations (15,16). The age of the ozone layer is 0.7 billons of years (Graedel, 1993)). 9 Before to ending a remark, equations (15) and (16) are very adequate to the study of diffusion process, which was neglected in the original equations (1) and (2). In fact, because they are not time depending, when we add d2 d2 spatial diffusion terms D dx 2 X and D dx2 Y , they become similar to reactiondiffusion-equations. Acknowledgments: This work was possible thanks to Project UTAMayor 4725 (Universidad de Tarapacá). Useful discussion with professor H. Wio, D. Walgraef (visits supported by the FDI-UTA and PELICAN Projects) and M. Pedreros, are acknowledged. 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