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OF CHEMISTRY
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VILA-GUERAU for
Marine
FOR DE
and Atmospheric
(Received
ON THE THE
NO-03-NO2
ARELLANO Research,
FLUX-GRADIENT
and PETER Utrecht
in final form
University,
8 May,
SYSTEM G. DUYNKERKE Utrecht,
The Netherlands
1992)
Abstract. It is shown that K-theory has to be modified for chemical systems that react with time scales similar to the turbulence time scale. In such systems, the value of the exchange coefficient depends not only on the turbulence parameters, but also on the chemical reaction rates. As an example, the NO-01-NO2 chemical system is studied. Using second-moment equations, new flux-gradient relationships for the neutral atmospheric surface layer are obtained which depend on the time scale ratios of turbulence (7,) and chemical reactions (TV,,). i.e., reactive K-theory. Within the framework of this reactive K-theory, the flux of a chemical species is both a function of the concentration gradients of the three chemical species involved and of the ratio of the time scale of turbulence to the time scale of chemistry. In the special case of slow chemistry (7, e TV+,) inert K-theory is applicable. The reactive exchange coefficients are implemented in a surface-layer model that calculates the flux and concentration profiles of the three chemical species. The results of the calculations of the effective exchange coefficients are different for reactive K-theory and inert K-theory; the differences are largest for nitric oxide, but smaller for ozone and nitrogen dioxide.
1. Introduction In this paper we investigate the influence that turbulence and chemical reactions have on the parameterizations of turbulent fluxes in the neutral surface layer. The mixing length or K-theory is revised to include the effect produced by chemical species that react with time scales similar to the time scale of turbulence. Under these circumstances, chemistry and turbulence are coupled and consequently one obtains modified parameterizations for the turbulence fluxes. Turbulence transport can be described approximately by introducing exchange or K-coefficients which relate the turbulence fluxes to the concentration gradients. A review of K-theory can be found elsewhere (Nieuwstadt and van Dop, 1982; Panofsky and Dutton 1984). K-theory has been thoroughly tested for the atmospheric boundary layer and has been found to have severe limitations in convective conditions where the transport is dominated by large buoyant eddies. For this case, modifications of the theory were introduced by Wyngaard (1982). However, in the surface layer, where the size of the transporting eddies is limited, there is reasonable evidence that K-theory does work (Hunt and Weber, 1979). At present in most studies it is assumed that the exchange coefficient is not altered by chemical reactions. In other words, the same values are used for the exchange coefficient for inert and reactive species. Lamb (1973) questioned the validity of mixing-length theory when chemical transformations with time scales similar to the turbulence time scale are involved. He concluded that K-theory can Boundary-Layer Q 1992 Kluwer
Meteorology 61: 375-387, 1992. Academic Publishers. Printed in the Netherlands.
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only be used for slow chemical reactions. Corrsin (1974) presented the limitations of K-theory and identified some conditions under which K-theory is valid. He suggested the introduction of correction terms because of chemical reactions. Other studies on the revision of K-theory in connection with chemical reactions in the atmospheric surface layer were conducted by Brost et al. (1988) and Fitzjarrald and Lenschow (1983). Both developed modified versions of the K-theory for reactive chemical species. Brost et al. (1988) based their study on a new exchange coefficient that depends on the time scales of chemical reactions and turbulence. However, they found that no significant deviation from the inert exchange coefficients can be expected if the chemical species are both produced and destroyed. For the same chemical species as we treat in this study (the NO-OX-NO2 system), Fitzjarrald and Lenschow (1983) developed new relations between fluxes and gradients in a non-neutral surface layer. They concluded that at least for one of the chemical species (NO), the reactive exchange coefficient would deviate significantly from the inert exchange coefficient. Using the inert expressions for the exchange coefficients, Lenschow and Delany (1987) found an analytical solution for the flux and concentration profiles of NO and N02. However, these solutions are only found if one assumes that the ozone concentration is constant with height in the whole atmospheric surface layer. Gao et al. (1991) found different values for the exchange coefficients of the NO--O,-NO, system in a second-order closure study. They calculated the exchange coefficients from the profiles of fluxes and gradients obtained from a one-dimensional model applied to the atmospheric surface layer. Experiments designed to determine the exchange coefficient for ozone in the atmospheric surface layer were carried out at Pawnee site (Colorado, USA) by Zeller et al. (1990). They found differences in the exchange coefficient of momentum, heat and ozone. Their ozone observations show that chemical reactions can be one of the factors which influences the exchange coefficient K and the subsequent calculation of ozone fluxes. Schumann (1989), using a large-eddy simulation model, studied the oxidation of nitric oxide by ozone in a convective boundary layer and found different values for the exchange coefficients of the two chemical species. In this study, we first show that the exchange coefficients need to be modified due to chemical reactions. Secondly, we calculate explicitly the expressions for the exchange coefficients for one of the key chemical reactions in atmospheric chemistry studies. Thirdly, we implement these new exchange coefficients in a model which calculates the fluxes and concentration profiles in the atmospheric surface layer. The chemical cycle considered is the NO-03-NO2 system, which can be summarised in the following way:
NO+O,
+kz NO2 + hv (A s 420 nm) , kl +
(1)
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where ki and k2 are second and first-order chemical reaction rates, respectively. The first chemical reaction is the oxidation of nitric oxide by ozone. The second reaction is the photodissociation of nitrogen dioxide by solar radiation (hv). The chemical system (1) plays a very important role in atmospheric chemistry processes because it is the only ozone source in the planetary boundary layer. In this paper, chemical system (1) will be used to investigate the effect of turbulence on chemical reactions, especially for the case where the chemical transformation time scales are of the same order as the turbulence time scale (Fitzjarrald and Lenschow, 1983: Schumann, 1989; Vila-Guerau de Arellano et al., 1992a). To analyse the flux-gradient relationships, we consider the flux equations for the three chemical species in question. The analysis of a (second-order) flux equation has proved very successful for studying the heat flux under various atmospheric conditions (Wyngaard, 1982). The second-order equations for the three chemical species are derived from the continuity equation, which can be formulated in the following way:
= vj(klCICz
- kzC3
i= 1,2,3.
In this equation Ci is the instantaneous concentration where the values i = 1, 2 and 3 represent nitric oxide, ozone and nitrogen dioxide respectively and u, is the component of the instantaneous wind field in the direction x,. The molecular diffusivity of the chemical species is yj and Si accounts for the sources and sinks. Kronecker’s delta (6,,,,) is equal to 1 if m = n and equal to 0 if m # y1 The vector (v,, v2, v3) is equal to (- 1, - 1,l). From Equation (2) one can derive the flux equations for the atmospheric surface layer. These flux equations yield new relationships for the chemically active species. These new expressions contain not only the turbulence characteristics of the atmospheric surface layer but also the chemical reaction rates of the different species.
2. Reactive K-Theory
Lamb (1973) and Corrsin (1974) already pointed to the limitations of inert Ktheory for chemical species that react with time scales similar to the turbulence time scale. Using a similar approach, we show how the exchange coefficients are modified and depend on both the turbulence and chemical parameters. In mixinglength theory, the flux between the two patches containing the chemical species with different concentrations C, and Cz is calculated in the following way. It is assumed that both patches are separated by a typical distance L, which is crossed by a transporting a-eddy with a velocity w, in time 7,. The chemical species
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decays non-linearly. If a decay is assumed to be linear, one obtains for the exchange coefficient a value which depends only on the turbulence characteristics. For simplicity, a two-dimensional turbulent field is considered. The generalization to a three-dimensional field will give similar results. The instantaneous concentration at level 2 (C,) is the average concentration at level 2 (c) plus a concentration fluctuation (c,) caused by the arrival of an a-eddy that originated at level 1. Thus, the concentration fluctuation can be written
c, = c, - G.
(3)
During its transport from level 1 to level 2, the chemical species decays nonlinearly at a chemical reaction rate equal to k:
%&-kc2 dt
Integration concentration
(4)
a
of Equation
(4) provides the following value for the instantaneous
c,
c, =
1 + T,kG Substituting
’
(5) in (3) yields
c, =
-_ [G - (1 + ~akCJCz1. 1 + T&C, 1
The concentration G can be expressed as a function of the concentration c using Taylor series (the series is expanded about the point x, z + L,, t - ~~a):
-
acl 7,-f
c,=c,-&+ where second- and higher-order rearranging the terms yields c, =
1
*..,
(7)
at
dZ
terms have been truncated.
Placing Cz in (6) and -
(,r, + L,T,kC,)
2
- T&C:
- (~a + ~2,kC’l) y
1
1 + r,kC,4
. (8)
The flux of the chemical species can be obtained by multiplying expression (8) by the characteristic velocity of the eddy (wa). Finally, averaging over a large number of eddies (denoted by a bar) yields the following expression for the concentration flux: WC
=
5
-
- (WT + wT2kCI)
$
-
- wdC:
-1 (9)
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where L, r and w are the average length, time and velocity respectively of a large number of eddies. Assuming steady-state and neglecting third-order correlation terms because they are smaller than second-order correlation terms (Wyngaard and CotC, 1971), one finds the flux of the chemical species between levels 1 and 2
WC =
1
-dC,
-
2
l+rkC,
wLF-
WrkC1
To analyse the terms WT and wL, one needs heuristic arguments. The magnitudes w and T are expected to be uncorrelated since T is always positive and w can randomly have either positive or negative values, i.e. wr is equal to zero. The term WL is expected to have a negative value because positive values of w are correlated with negative values of L and vice versa, due to the definition of coordinates. Therefore, the final expression for the flux is
wc=-KaCl
(11)
a.7 where
K is the reactive exchange coefficient K=-
defined by
wL lhkc’
(12)
It is clear from expression (12) that the value of K depends not only on the turbulence characteristics w, L and T, but also on the chemical characteristics represented by the time scale (kc,)-‘. If this time scale is larger than T (slow chemistry), the value of K depends only on the turbulence characteristics, i.e., K is equal to -wL. If this is not the case, one obtains new expressions for the exchange coefficient which take the chemical transformation into account.
3. Flux-Gradient
Relationships
One of the most important chemical systems in atmospheric chemistry studies is the NO-O,-NO, cycle. The time scale of this cycle is of the order of minutes and consequently one expects turbulence mixing to affect the chemical reactions. Therefore, it is an important chemical system by which to study the effect of turbulence on chemical reactions. The three chemical species, C,(NO), C,(O,) and C3(N02), react with a second-order chemical reaction and a first-order chemical reaction (chemical reaction 1). The flux equations for this triad in a neutral
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boundary layer can be written in the following way:
(4
(b)
(c)
(4
= vi i (1 - &,,,,)k,C, WC, - k2 ii+ m,n (4 (f 1
i= 1,2,3.
(13)
The concentration (CJ has been decomposed into a time-averaged term (c) and a fluctuating term (cJ; w is the vertical velocity, p is the pressure and p is the density. The bar indicates that Equation (13) has been averaged over time. The molecular diffusivity and the source terms are neglected. The terms of Equation (13) can be interpreted physically as the temporal change of the flux (a), the flux produced by the mean gradient concentration (b), the flux produced by the pressure gradient (c), the flux transport caused by vertical velocity (d), the flux produced/destroyed by second-order chemical reaction (e) and the flux produced/destroyed by first-order chemical reaction (f). The advection and the buoyancy terms are omitted since the surface layer is assumed to be horizontally homogeneous and neutral. The calculation of the expressions in a non-horizontal homogeneous surface layer (Tsarenko and Yaglom, 1991) will complicate the final solution, without adding relevant information. To obtain an expression which relates the flux to the concentration gradient, it is assumedthat: first, terms (a) and (d) are smaller than the other terms (Wyngaard and CotC, 1971), and second, term (c) can be parameterized using the well-known assumption which relates the flux production caused by the pressure gradient to the flux (Wyngaard, 1982). Therefore, term (c) can be expressed as:
-1- $P
=!!!5
P ( ‘dz >
Tr ’
(14)
where T is the turbulence time scale. The pressure gradient term is parameterized as a function only of the turbulence time scale (G-J. For a passive scalar in a neutral surface layer, terms (b) and (c) of Equation (13) balance one another. Therefore, substituting term (c) for term (b) into Equation (14) one obtains the turbulence time scale (7J Tt =
k(z + zo> Au*
(15)
where the von Karman (K) constant is equal to 0.4, U* is the friction velocity and z, is the height above the roughness length (zO). The ratio of the vertical velocity variance and the friction velocity squared (A = w*Iu**) is estimated experimentally; in this study a value 1.56 is used for this ratio (Panofsky and Dutton, 1984).
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Applying the assumptions mentioned the results in the following way
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above to the flux equations, one can write
WCi= -Kil
(16)
(Kii) is a second-rank tensor which can be ex-
where the exchange coefficient pressed in the form of a matrix
1 + rl + y3 K-
= "
KU*(Z
+ zo>
-rl
1+
1+r,+r2+r3
r3
r2 +
r3
-r2
t
r2
1+
rl
(17)
f-3 rl
+
r2 1.
The dimensionless time-scale ratios between the time scales of turbulence chemical reactions are rI, r2 and r3 and are defined as:
p11_= TChl
K(Z + zo)k,NO Au, '
and the
,rChl= (k,NO)-' (19)
K(Z + Zo)kz
,,=L= TCh3
Au,
'
TCh3
=
(b-’
.
(20)
The intensity of the coupling between the three chemical species is given by the off-diagonal terms of the matrix K,, (17). It is important to notice the difference between Equations (16) and (17) and the flux-gradient relationships if no chemical transformations were taken into account (Businger 1982; Panofsky and Dutton, 1984). In the latter case, the chemical species can be treated as inert; then the solution can be written as a matrix with only diagonal terms, i.e., Kll = K22 = K33
=
KW+(Z
+
zo).
The inert flux-gradient relationships can be recovered from Equations (16) and (17) if rI, r2 and r3 < 1. This is possible when the time scales of the chemical transformations are larger than the turbulence time scale, i.e. slow chemistry. Similar solutions are also found for inert species. From the NO-03-NO2 system, two inert species can be defined: NO, = NO + NO2 and 0, = O3 + N02. Adding row 1 and row 3 in the matrix (17), one obtains the flux of NO,. Similarly, the flux of 0, is equal to row 2 plus row 3. In both cases, one obtains for the exchange COeffiCient a Value equal to KU*(Z + Zo), i.e., the value of K depends only upon the turbulence. The advantage of these new flux-gradient relations compared to the expressions derived by Fitzjarrald and Lenschow (1983) is their simplicity and their dependence on dimensionless parameters. Reactive K-theory can thus be studied as a function of the ratio of the time scales of turbulence and chemical transformations, i.e. rI, r2 and r3
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4. Limiting
PETER
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Cases
Three different limits of Equations (16) and (17) are studied. These cases depend upon the value of the dimensionless time scales and are called slow chemistry, moderate chemistry and fast chemistry (Donaldson and Hilst, 1972; Carmichael and Peters, 1981). To simplify the notation, only the Kij matrix is given. 4.1.
SLOW
CHEMISTRY
The time scales of the chemical reactions are larger than the turbulence time scale, i.e. rl, r2 and r3 + 1. Therefore the solution can be written as: (21)
This is the solution for the flux-gradient relationships when chemical reactions are not influenced by turbulence. Chemical transformations are much slower than the turbulence time scale and consequently are not affected by this process. For the NO-03-NO2 system in the surface layer, this solution only holds when chemical species are near ground level. 4.2.
MODERATE
CHEMISTRY
Since the time scales of the chemical transformations are similar to the turbulence time scale, it is necessary to use the full Ki, matrix to calculate the fluxes of the chemical species. However, normally in the surface layer, the O3 concentration is much higher than the NO concentration (03/N0 = 50); thus r2, r3 = O(1) % rl. The approximation of matrix (17) in this case becomes: 1+ -K&(Z
1+
+ Zo>
r3
-r 2
r2 + r3 i
r2
0
1+
r2 + r3 0
r3
r3
1+
.
(22)
r2
This case is usually encountered at the top of the surface layer (z = 100 m). Consequently, at this height the inert K-theory is modified by chemical reactions. 4.3.
FAST
CHEMISTRY
For the NO-03-NO2 system, fast chemistry is only possible at heights above the top of the surface layer (z s 100 m). However, it is interesting to study this case because it can give an indication of how chemical reactions are affected by turbulence. For fast chemistry, the time scales of the chemical transformations are smaller than the turbulence time scale. If one assumes again that the ozone concentration is much higher than the nitric oxide concentration, then rz, r3 % rl 9 1. Consequently, the solution for Equations (2) can be written as:
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FNO Fo,
FNoZ
the model top of the prescribed of the flux
conditions
8.0
NO (ppb) 03 (ppb) NO,
383
I
Boundary conditions and parameters used in runs. All the values are prescribed at the surface layer, except the flux of NO which is at the roughness length (zO). Positive values indicate upward direction Boundary
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(mb) (ppb m s-‘1 (ppb m ~~‘1 (ppb m SC’)
10.0 0.08 -0.125 -0.05
Constants
u* (m SC’) itI Cm) k, (ppm-’ min-‘) kz (min-‘)
0.25
0.1 25.0 0.5
K-theory is modified by the chemical transformations, what can happen in the mixed layer for the NO-03-NO2 5. Application
which again indicates system.
of the New Relationships
These new flux-gradient relationships can be studied by implementing Equations (16) and (17) in the surface-layer model developed by Vila-Guerau de Arellano et al. (1992b). Briefly, the model solves numerically the time-averaged Equations (2) for the three chemical species in the surface layer. Moreover, it is assumed that the concentrations are stationary and horizontally homogeneous. In that surface-layer model, the turbulent concentration flux is parameterized using inert K-theory. Now, the turbulent concentration flux is parameterized using reactive Ktheory. The model solves three second-order differential equations and therefore requires six boundary conditions. The concentrations of NO, O3 and NO1 and the fluxes of O3 and NOz are prescribed at the top (z,) of the surface layer and the flux of NO at the bottom (zh) of the surface layer. The concentrations at the top of the surface layer must satisfy the photostationary state relation equilibrium
(24) Table I summarises the values of the concentrations cal and chemical parameters used in the model.
and the other meteorologi-
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1
0.9 Kci [K U*(Zt
DUYNKERKE
z,)]
1.1
-I
Fig. 1. The ratio of the effective reactive exchange coefficient (KC,) to the inert (KU,(Z + 20)) exchange coefficient for NO (dotted-dashed line), O3 (dashed line) and NO2 (dotted line). The continuous line corresponds to the exchange coefficient for an inert chemical species.
Figure 1 shows the ratio between the effective reactive exchange coefficient (Kci) and the inert exchange coefficients KU*@. + lo). The values for the effective reactive exchange coefficient are calculated by dividing the flux by the gradient, both having been calculated with the surface-layer model. The maximum difference between the reactive and inert exchange coefficient is 30% for nitric oxide, 5% for nitrogen dioxide and 3% for ozone. These results agree with those of Fitzjarrald and Lenschow (1983) and Gao et al. (1991) who also suggested that NO showed the largest departure from inert K-theory. The asymptotic behaviour towards the inert values at the top of the surface layer is due to the assumption of photostationary equilibrium (Equation (24)). At that height, the flux divergence for the three chemical species is zero, i.e. chemical equilibrium holds, and therefore the exchange coefficient is determined only by turbulence. Future research will attempt to relax the boundary condition expressed in Equation (24). The concentration profiles calculated with the reactive and inert exchange coefficients of NO, O3 and NO2 are shown in Figure 2. The concentration values are made non-dimensional in terms of their respective concentrations at the roughness length. The largest deviations correspond to the NOz concentration profile. The differences in the concentration profiles suggest that if the flux of NO is calculated using K-theory, the O3 and NO2 concentration gradient should be taken into account in this calculation. Figure 2 also shows that the inert species NO, is a logarithmic function and consequently the vertical flux of NO, is constant with height. Similar behaviour is found for the inert species 0, (not plotted).
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1.8
2.2
Fig. 2. The dimensionless concentration profiles of NO, O3 and NO2 (made non-dimensional in terms of their respective concentrations at the roughness length) calculated using the reactive (dotted line) and inert (continuous line) exchange coefficients. The NO, concentration profile is represented by the dashed line.
Figure 3 shows the way in which the variation in the turbulence time scales influences the effective exchange coefficient for nitric oxide (KNO). Using the same boundary conditions as in Table I, the friction velocity is modified and consequently so are rl, r2 and r3. Decreasing u* increases the turbulence time scale. For the lowest value, u* = 0.2 m s-‘, moderate chemistry occurs at a lowest level in the atmospheric surface layer and consequently the influence of chemistry is more important. Maximum differences between inert and reactive K-values are also significant. Thus, for u* = 0.2 m ssl, the maximum difference is 35% and for u* = 0.6ms-‘, it is 15%. Similar results can be found by changing the chemistry time scales. This can be done either by varying the concentration of nitric oxide, ozone or nitrogen dioxide or by modifying the value of the chemical reaction rate k2, which depends on solar radiation. To complete this study, experimental evidence is required to validate the reactive K-theory. Experiments will have to be performed in cases where chemical reactions have time scales similar to the time scale of the turbulence mixing. 6. Conclusions
The parameterization of the concentration flux using K-theory has been studied for chemical systems which react with time scales similar to the turbulence time scale. The values of the exchange coefficients are found to depend upon the turbulence and chemical properties. As an example of the influence of turbulence
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0.8
0.9
1
K&K u* (z t zo)]-’ Fig. 3.
The ratio
of the effective reactive (KU*(Z + z,,)) for NO
exchange coefficient (KNo) as a function of the friction
to the inert exchange velocity u*.
coefficient
on chemical transformations, we have derived new flux-gradient relationships for the NO-03-NO2 system, i.e. reactive K-theory, from the second-moment equations in which chemistry is treated explicitly. These relations are established in a neutral and horizontally homogeneous surface layer. The new relationships show that the fluxes do not depend solely on their own gradient but they are coupled to the gradients of the other chemical species. Therefore, if turbulence is parameterized in terms of an exchange coefficient, the chemical reactions can severely affect the calculation of the fluxes. The new K-theory contains not only the usual parameters but also new parameters which describe the relation between turbulence and chemistry as a function of three dimensionless time-scale ratios. An analysis of these gradient-flux relationships shows that inert K-theory can be used for the first few metres of the surface layer. However, at higher altitudes where the time scales of turbulence and chemical reactions are of the same order of magnitude, the new relations are more appropriate. The reactive exchange coefficients are implemented in a surface-layer model which numerically solves the continuity equations for nitric oxide, ozone and nitrogen dioxide. The difference between the exchange coefficient calculated with the two K-theories is larger for nitric oxide than for nitrogen dioxide and ozone. Acknowledgements
Jordi Vila-Guerau de Arellano is supported by CIRIT (Catalonian Research Council, Spain) grant BE90-2114. The authors are very grateful for the comments of an anonymous referee who improved the contents of the paper considerably.
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References Brost, R. A., Delany, A. C., and Huebert, B. J.: 1988, ‘Numerical Modeling of Concentration and Fluxes of HN03, NH3 and NH4N03 Near the Surface’, J. Geophys. Res. 93, 7137-7152. Businger, J. A.: 1982, ‘Equations and Concepts’. In Niuwstadt F. T. M. and van Dop, H. (eds.), Atmospheric Turbulence and Air Pollution Modelling, Reidel, Dordrecht, the Netherlands. Carmichael, G. R. and Peters, L. K.: 1981, ‘Application of the Mixing-Reaction in Series to NO,-03 plume chemistry’, Atmos. Environ. 15, 1069-1074. Corrsin, S.: 1974, ‘Limitations of Gradient Transport Models in Random Walks and in Turbulence’. Adv. Geophysics lSA, 25-60. Donaldson, C. dup. and Hilst, G. R.: 1972, ‘Effect of Inhomogeneous Mixing on Atmospheric Photochemical Reactions’, Environ. Sci. Technol. 6, 812-816. Fitzjarrald, D. R. and Lenschow, D. H.: 1983, ‘Mean Concentration and Flux Profiles for Chemically Reactive Species in the Atmospheric Surface Layer’, Atmos. Environ. 17, 2505-2512. Gao, W., Wesley, M. L., and Lee, I. Y.: 1991 ‘A Numerical Study of the Effect of Air Chemistry on Fluxes of NO, NO* and O3 Near the Surface’, J. Geophys. Res. 96, 18 761-18 769. Hunt, J. C. R. and Weber, A. H.: 1979, ‘A Lagrangian Statistical Analysis of Diffusion from a Ground-Level Source in a Turbulent Boundary Layer’, Quart. J. R. Meteorol. Sot. 105, 423-443. Lamb, R. G.: 1973, ‘Note on the Application of K-Theory to Diffusion Problems Involving Nonlinear Chemical Reactions’, Atmos. Environ. 7, 257-263. Lenschow, D. H. and Delany, A. C.:1987, ‘An Analytical Formulation for NO and NO2 Flux Profiles in the Atmospheric Surface Layer’, J. Atmos. Chem. 5, 301-309. Nieuwstadt, F. T. M. and van Dop, H.: 1982, Atmospheric Turbulence and Air Pollution Modelling, Reidel, Dordrecht, the Netherlands. Panofsky, H. A. and Dutton, J. A.: 1984, Atmospheric Turbulence, John Wiley & Sons, New York, USA. Schumann, U.: 1989, ‘Large-Eddy Simulation of Turbulent Diffusion with Chemical Reactions in the Convective Boundary Layer’, Atmos. Environ. 23, 1713-1727. Tsarenko, V. M. and Yaglom, A. M.: 1991, ‘Semipirical Theories of Turbulent Diffusion in a Neutrally Stratified Surface Layer’, Phys. Fluids A3, 2199-2206. Vi&Guerau de Arellano, J., Duynkerke, P. G., Jonker, P. J., and Builtjes, P. J. H .: 1992a, ‘An Observational Study on the Effects of Time and Space Averaging in Photochemical Models’, Atmos. Environ. (in press). Vila-Guerau de Arellano, J., Duynkerke, P. G., and Builtjes, P. J. H.: 1992b, ‘The Divergence of the Turbulent Diffusion Flux Due to Chemical Reactions in the Surface Layer’, submitted to J. Geophys. Res. Wyngaard, J. C.: 1982, ‘Boundary-Layer Modeling’, In Nieuwstadt, F. T. M. and van Dop, H. (eds.), Atmospheric Turbulence and Air Pollution Modelling, Reidel, Dordrecht, the Netherlands. Wyngaard, J. C. and Cot&, 0. R.: 1971, ‘The Budgets of Turbulent Kinetic Energy and Temperature Variance in the Atmospheric Surface Layer’, Boundary-Layer Meteorol. 9, 441-460. Zeller, K., Fox, D., and Massman, W.: 1990, ‘Simultaneous Measurements of the Eddy Diffusivities and Gradients of Ozone, Sensible Heat and Momentum’, Ninth Symposium on Turbulence Diffusion, Roskilde, Denmark, pp. 110-114.
