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Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer
~rmosph~rrc Enaironmcwt Vol. 17. No. 12. pp. 2505-2512.
1983
000&4981/83 33.00 + 0.00 Pergamon Press Ltd.
Prmted m Great Entam.
MEAN CONCENTRATION AND FLUX PROFILES FOR CHEMICALLY REACTIVE SPECIES IN THE ATMOSPHERIC SURFACE LAYER DAVID R. FITZIARRALDand JJONALD H. LENSCHOW National Center for Atmospheric Research,t Boulder, CO 80307,U.S.A. (First
9 February
received
1983)
Abstract-We use Monin-Obukhov similarity theory to study the vertical structure of turbulent fluxesand concentrations of reactivespecies in the surface layer. A surface layer flux-gradient relation that includes firstand second-order chemical reactions is derived from the flux budgets. The theory is then applied specifically to the O,-NO-NO, photochemical reaction triad. In this case it is found that the non-reactive flux-gradient relation usually suffices, but that chemical reactions can seriously alter the surface layer turbulent flux and mean concentration profiles. It is demonstrated that six observations of fluxes or concentrations are necessary to determine the surface fluxes when reactions are significant. Significant deviations from the predicted nonreactive flux are found to be quite plausible for NO, levels characteristic of moderately polluted air.
1. INTRODUCTION Surface sources and sinks play a major role in the budgets of many atmospheric trace reactive species. Estimating surface sources and sinks, along with time changes and mean and turbulent transport over a specified area, allows us to estimate as a residual the chemical transformation rate of trace reactive species. One important element of surface sinks is dry deposition, which can have significant effects on vegetation and on the acidity of surface water. A common simplification in chemical modeling studies to describe dry deposition is the use of a deposition velocity (Sehmel, 1980). It is assumed that the rate of surface deposition is proportional to the mean concentration at a particular reference level. That is, the deposition velocity is independent of the turbulence structure of the atmospheric surface layer. Sehmel (1980), for example, recommends that mean concentrations for estimating deposition velocity be presented at a standardized height of 1 m. Although this concept may apply to relatively slowly reacting species, we show here that it does not apply to rapidly reacting species. Because observational techniques have developed to the point where accurate measurements of both mean profiles and turbulent fluxes of reactive species are now possible (e.g. Lenschow et al., 1982; Wesely et al., 1978; Delany and Davies, 1983), it is important to consider details of the surface layer structure so that the observations can be related to the physical characteristics of the underlying surface, independent of surface layer transport properties. In this paper we discuss the possibility that reactions occurring in the
t The National Center for Atmospheric Research is sponsored by the National Science Foundation.
surface layer in the vicinity of the observations may change flux estimates substantially even below 1 m. Results presented here also apply to model studies in which the lower boundary condition is specified using a deposition velocity at fixed height (e.g. H$v, 1982). We show that vertical turbulent transfer in the surface layer can maintain a system of reacting species in a state away from chemical equilibrium. Since the turbulent mixing time scale increases with height in the surface layer, this effect is more pronounced as one approaches the ground. If reaction time scales are comparable to or less than the turbulent mixing time scales, significant departures from chemical equilibrium can occur. In this situation we show that it is not sufficient to use only measurements of the species concentration and gradient for estimating the surface flux of a constituent, but the concentrations and gradients of the other reacting species in the system must be considered as well. The problem discussed in this paper is not, however, related only to cases in which profiles are used to infer fluxes. Even if the flux is directly measured (say by the eddy-correlation method), the results must still be extrapolated to some surface transfer height, the effective minimum height to which surface layer turbulence can transport the species (Wesely and Hicks, 1977), to determine the transport of a quantity into or out of the surface. Very near the surface (say within 1 cm) molecular diffusion becomes significant. The time scale for diffusion through this layer is of the order of 1 s. Therefore, to justify ignoring chemical reactions in the molecular sublayer we limit our consideration to reaction times of at least several seconds. In the Appendix we use the Monin-Obukhov similarity theory (Businger, 1973) and covariance budget equations, with the addition of first- and second-order chemical reactions, to develop a modified flux-profile
2505
relationship in the surface layer. Example calculations in Section 2 illustrate the differences between reactive and non-reactive profiles for specified surface fluxes. In Section 3 a procedure for determining surface fluxes from profile data is presented. This theory is used to examine the range of possible fluxes that can be inferred from the 0, and NO, mean concentration data of Delany and Davies (1983).
2. THEORETICAL
DISCUSSION
The similarity hypothesis applied to surface layer structure has been extensively verified by results from field experiments. Basically, this hypothesis, commonly known as the Monin-Obukhov (M-O) similarity hypothesis, states that mean and turbulence variables in the surface layer (the lowest part of the boundary layer where the fluxes of non-reactive species can be considered constant) are functions of height z above the ground and hydrodynamic stability. Stability is specified by a length scale,
concentration of S, through chemical reactions. tor a single-component first-order reaction in the neutral surface layer the height at which the reaction time , ’ is equal to the diffusion time (Lenschow, 19X?i. z -= u,k,‘l,jl.
is proportional to the height IO which the constituent flux can be considered constant. Chemical reactions can also affect the flux-gradient relation. In the appendix we develop a flux-gradient relation that includes the effects of chemical reactivity and discuss conditions under which the non-reactive flux-gradient relation is a good approximation for a reactive species. We now consider specifically the photochemical triad of 03, NO* and NO: NOz+hv+NO+O 0+0,-o,
nSi/c’t + IJf%!$/ax+ wasi/az
+ aG)az
= Qi,
(1)
where U and W are the mean horizontal and vertical velocity, respectively and Qi is the sum of the internal sources and sinks. Capital letters refer to mean quantities and small letters to turbulent fluctuations. The overbar denotes a temporal or spatial average over a time or length sufficient to give a stable estimate. If we limit our consideration to first- and second-order chemical reactions, the chemical source term can be written as: Qi =
(4)
NO + 0,3 --*NO1 + Oz. The second reaction occurs much more rapidly than the other two (Leighton, 1961), and thus (4) can be written as QI1=jC-kAB
where u* is the friction velocity, g/T is the buoyancy parameter, Kis the von Karman constant and wf3,is the virtual temperature flux at the surface. Surface layer variables can then be empirically determined as functions of the normalized ratio z/L. This approach applies to trace constituents that have no sources or sinks in the surface layer. For species that are not conserved, surface layer relations will, in general, depend upon the source/sink terms, as well as on z/L. We develop a formulation for the flux-gradient relation for non-conserved constituents and examine the conditions under which the M-O flux-gradient relation may be considered a valid surface layer approximation. The mean budget of a non-conserved constituent Si can be written (Lenschow, 1982) as
(31
= Qb = -Q,,
where A, B and C are the mean concentrations of 03, NO and NOz, respectively, and the second-moment term kab is assumed negligible. This is justified if the fluctuation levels for these species are much less than their means. Lenschow (1982) shows that this is true for Oa in well-mixed air over central U.S. These reactions occur so rapidly that, for the most part, other competing reactions are not significant for the time scales considered here. At equilibrium, called the photostationary state (Leighton, 1961) QS = 0. In the lower part of the surface layer the chemical system (4) will not, in general, be in equilibrium if surface fluxes are not zero because of vertical turbulent diffusion. For the remainder of this section we consider the ease when the height is well below the height where the chemical and turbulent time scales are comparable; that is, z + u*~,/K, where 7,, the inverse of the reactivity coefficient, is the effective reactivity time constant (Lenschow, 1982). Thus the standard M-O fluxgradient function 4s is assumed. We apply this analysis to the estimation of fluxes based on mean profiles measured near the ground. Fluxes of 03, NO and NO2 are related to their mean gradients by the relations: 4,: ‘;A+I,
= - wa
(6a)
4;
= - wb
(W
=
(W
‘zB,Ku,
~>~‘ZC,KU,
-NY,
(2) where the subscript z denotes differentiation Cjir& + C hnn[SmSn+smsnIv
where ja and kimnare the reactivity coefficients and the Sj are the concentrations of species that affect the
(5)
with respect to height. One feature of these relations is that certain sums and differences of fluxes are constant with height. Lenschow (1982) showed that the O3 oudget
Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer equations can be written, in the present notation, as: (u,rcz~;‘A,),
= jC -kAB
(74
(U,KZf$;lBz)z
=jC
(7b)
(u,K$;~C~)~
=
-kAB
-jC+kAB.
(74
Conserved combinations of fluxes result from adding and subtracting relations in (7) and substituting (6): (wn - wb), = 0
(84
(wa + w(‘)Z = 0 -(wb + WC), = 0.
(W
2501
the non-reactive case requires only measurement of two quantities. Mean 0s measurements, or even simultaneous Oz and NO, measurements at two levels, are not sufficient mathematically to determine the fluxes of these quantities. In the limit of low NO, levels, of course, the reactive solution for Oj must approach the non-reactive solution. In Section 3 we present numerical examples of the inaccuracy of the flux estimates when chemical reactions are neglected. Figure 1 illustrates model solutions for the flux and concentration of O3 for a particular value of mean O3
(84
Thus (wb + WC), which is proportional - to the flux of NO + NOz = NO,., as well as (G - wb) and (& +wc), are constant. This means that the deposition velocity, w,, for NO,,
is a well-defined quantity of a conserved species. Knowing the flux and concentration of NO, at one height, the profile of NO, is uniquely determined. This is not true for any one of the constituents unless concentrations of the other two are negligible. Since the quantities in parentheses in (8) are conserved, their profiles and fluxes are given by integrating the fluxgradient relations (Paulson, 1970). Profiles of two of the conserved quantities whose fluxes are given in (8) are written as,
I
I
60 OZONE
I 70
[
ppbv]
A-B=D(z)=D(z,)+[(wa-wb)/~u.Jj-(z,zl,L) (104 -B + C = E(Z) = E(z,) + [ (wb + WC,/KU,] f(z, zl, L), (lob)
wheref(z, zr, L) = s:, (4,/z)dz, and where zr is some reference level. We note that it is sufficient to solve explicitly for only one nonconserved quantity, and then use (10) to find the remaining two. Equations (6a) and (7a) can then be written as: A, = (wa),
=
ii&,/KU,Z
(116)
(kAB-jC).
(lib)
Using (lo), (11) becomes two coupled ordinary differential equations, one for the concentration, the other for the flux of 03. We note that it is not possible to attain photostationary equilibrium at more than one point in the lower surface layer since the photostationary state is equivalent to having zero turbulent flux divergence. If reactions are occurring, then six quantities must be determined simultaneously (in addition to the atmospheric stability, surface roughness and friction velocity) to find the profiles of concentration and fluxes for the three constituents in this system. By contrast,
I’
0
’
I
I
J
5
IO
I5
FLUX
RATIO
Fig. 1. Sample profiles for u, = 0.2 m s-r
and z0 = 0.01 m at neutral stability with the following conditions at ze: [0,] = 4Oppbv, flux of 0, = -0.23 ppbv m s-r; [NO] = 6 ppbv, flux of NO = -0.001 ppbv m s-r; [NO,] = 20 ppbv, flux of NO1 = - 0.20 ppb m s- ‘. The non-reactive solution is given by the solid line, the reactive solution with the standard M-O flux-gradient relation is given by the dashdot line and the reactive solution with the modified fluxgradient relation obtained from the flux budget is given by the dotted line. The top figure is 0, concentration and the bottom is the ratio of 0s flux to its constant non-reactive value.
L)Aw~
3508
R.
I.‘ITLJARRAL.Dand
concentration and flux at the surface transfer height. We see that, for this example, the reactive solutions are quite different from the non-reactive solution, with the constant flux layer extending only to about 4 m. We also note that it is not necessary, in this case, to use the corrected flux-gradient relation from the appendix. We have assumed here a small negative NO surface flux for this example; the solutions are similar for other small NO fluxes, including a positive surface flux.
Doh41 D H. Lwww\r. and zL.
a--J Measured variables for the test case are given in ‘Table 1. To illustrate how reactions change the flux and concentration profiles, we first arbitrarily assume that r(z2) is at its photostationary value (zero flux divergence at zLI: I‘(_;) = (1 -c/i.4 ,/I ‘.
3. DETERMINING CONSTITUENTS
FLUXES OF REACTIVE
USING THE PROFILE
METHOD
In this section we discuss the profile technique used to infer the fluxes of 03, NO and NO1. Results of the procedure are then used to interpret observations of mean NO, and 0, at two levels in the surface layer reported by Delany and Davies (1983). The profile method requires at least two levels of mean concentration measurements along with auxiliary meteorological data to find u* and L. Assuming constant flux with height, the flux ofa constituent can then be determined from empirical profile functions. This is the method used by Galbally (1971) and others to find the turbulent O3 flux in the surface layer. We have demonstrated, however, that source terms in the mean budget of a reactive substance require that the surface layer turbulent flux vary with height. Thus, when these terms are important we must solve for all fluxes and concentrations in the system simultaneously in such a way that observed concentrations are satisfied at the observation points. Near the surface, solutions of relations (1 la, b) can be efficiently obtained from the iterative ‘shooting’ method of solving two-point boundary value problems (see Roberts and Shipman. 1972, Ch. 2). Two integrations, one up and the other down, are done for each iteration. Equations (1 la, b) are integrated from the lowest level using the observed concentration and an assumed flux, usually the nonreactive flux. The flux at the lower level is corrected using the resulting deviation between observed and predicted concentrations at the upper level until a specified tolerance is achieved. This method converged in two to eight iterations, depending on tolerance levels. Delany and Davies (1983) present a few cases of simultaneous measurements of concentrations of O3 and NO, at two levels (0.125 and 1.75 m) above a grasscovered surface in moderately polluted air in England. We use one of their profile observations here to demonstrate the range of possible efTects that chemical reactions can have on the flux estimates. For simplicity we use constant values of the reaction coefficientsj and k. In this case we have only four of the six measurements needed to determine the fluxes completely. Thus the system has two degrees of freedom. We have chosen the missing variables to be the NO/NO, concentration ratio at the two observation levels z1
(13)
For the reaction coefficients assumed here, r(zJ x 0.127 and the reactivity time constant (Lenschow, 1982) is T, % 107 s. The remaining free variable is r(zl). Solutions for r(zI) ranging from zero to one and for the non-reactive case are shown in Fig. 2. The change in concentration profiles is relatively small as r(-l, ) is varied over its range. However, the flux profiles change drastically as r(zl) gets large. Note that all of the calculated O3 fluxes equal the non-reactive solution at the same height. If the NO/NO, concentration ratio does not exceed 0.1, the calculated fluxes may not differ from the nonreactive fluxes by more than 10~~20’~,,.In some cases, though, the reactive terms introduce such curvature in the flux profiles that the flux estimate at a height equal to the roughness length [assumed here to be equal to the surface transfer length as defined by Wesely and Hicks (1977)] may be greatly diff‘erent from the nonreactive estimate. The entire range of ambiguity in the test case is presented in Fig. 3. Contours of the ratio of the reactive to the non-reactive fluxes appear as straight lines on the r(zI ). r(z2) plane. II is even theoretically possible for the reactive flux to be of the opposite sign of the non-reactive flux in the limit where extremely large NO concentrations occur. We have demonstrated that reaction eft‘ects may seriously alter the value of a turbulent flux inferred using the profile method. We therefore recommend that when NO, levels are high (say > 20 ‘i, of [O,] ), at least six measurements be made to determine the flux, either as concentrations at two levels or as some
Table 1. Observations of Delany and Davies (1983) used here as an example 19.5 20.6 64.0 69.1 O.CW4 s-l) 0.004 0.18 u,(ms-‘) ‘X L (m) __ _______~ The heights of observation are z, = 0.125 m and zz = 1.75 m above the surface. Concentrations are given in ppbv.
NO&,) NO,@,) %@I) O,(G) k (ppbv-’ i (s-l)
Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer
2509
Fig. 3. Contours of the ratio of the re&ve 0s flux to the ~on-r~~ti~ ffux ( :d) plotted on ther~z*~~z*)plane for the exampie discussed in the text.
FLUX RATIO Fig. 2. Profilesthat fit ~n~tmtio~s of0s observed by Delany and Davies (1983) (Tabie 1)at two levels for values of the ratio r(zl) ranging from zero to on& The nonreactive case is given by the dashed line. Parameter r(zs) is fIxed at its pbotostationary value of 0.127. The top figure is 0s concentration profiles and the bottom is the ratio of Q3 &ix to its constant non-reactive value.
of con~ntrati~ns with dire&y measured fluxes (e.g. by the ~dy-~orreiatian method). It is important to recognize that in obse~at~~n~ and model studies one wishes to estimate the surface flux ofa substance into or out of the system. This flux properly should be evaluated at a height equal to the transfer length, though obse~ati~~s at this level are not normally available since it is typically of the order of the height of the roughness elements. Over grassland, for examPIe, it may be of the order of 1 cm. At this height m~urements of ~adients or direct 3ux measurements are not feasible by the techniques discussed here. The results in Fig. 2 iliustrate that even if Buxes are measured directly, along with coneentrations, the corm&ion procedure outlined here may be required to estimate the flux at the surface. c~~bi~atj~n
The results presented here question the validity of using deposition velocities, defined by the ratio offlux to #n~~t~tion, at some surface layer reference levej when the chemical reaction time is fess than or comparable to the turbulent diffusion time. A more reasonable approach to estimate surface deposition rates under there conditions would be to define a deposition velocity at a height equal to the surface transfer height*with concentrations and Ruxesat this level determined by the technique prescribed here. The ideas discussed here are not limited to chemicai reactions. They also have applications to other nonconserved constituents in the atmospheric surface layer. Over an evaporating surface upward-moving air will, on average, be more humid than dawnwardmoving air. Therefore particles which wouid be the same size at the same h~dity will be larger in the upward-moving air than in the downward-moving air. Thus the layer in which the drier downward-moving air is mixed with the moisture upward-rnovi~~air near the surface can be insides, on average, as a net particle sizeconverter. If the particle size distribution is not uniform, the number of particles in some size ranges will grow at the expeuse of others, and vice versa. Thus the Aux profife for particles within a specified size range is not n~sarily constant with height, and the flux-gradient relation may not necessarily follow the ~onin~bnkhov s~rnjlari~yprediction. ~~r~ermore, the net particle mass flux, measured by the eddycorrelation technique, will be too large. Wesely et al. (1982) observed a net upward flux of pa&les over the ocean, possibly due to this effect. The theory presented here relies heavily on Moni~~bukhov similarity formulations and there-
t’ore is nol applicable In
particular,
this
throughout formulation
achieve a photostationary
the boundary cannot
equilibrium
layer. Such a state. character&d scale much shorter
may
be
a
than a turbulent
useful
layer.
be made to
throughout
a
b! a chemical time mixmg time scale.
upper-boundar?
condition.
Development
of a theory to describe the fluxes and
concentrations
01‘ reactive species m the boundary
layer is a necessary
future
complement
to the work
presented here. A~~fl~ledyrrnun~.s J. C‘. Wyngaard made several useful suggesttons concerning the flux budget and flux-gradient rclattonshtps m the surface layer: Brian Ftedler also provtded helpful comments. David R. Filljarrdld did this work as an Advanced Study Program Fellow at the Nattonal Center for Atmosphcrtc Research. The manuscript was expertly typed by Jo Ann Fankhauser with extcnstvc rcvtsinns typed hy Daloris Ildming
Wesely M. I... (‘oak D R. Harl R L. and Wtlltams R M 11982) Au sea exchange of CO, dnd evidence for enhanced upward fiuxe\. .I !~twpk I’\. Rc\ 87. XX?7 X83? Wyngaard J C‘ t I’)821 Roundarj-layer modchng 111 .1rmosphrrlc 7urhulw t‘ trod Arr P~~llurwtt ~~&//rm~ (Fdned by Nlewstadt I I M xnd Van Dop H I D Rcidel. Dordrccht Wyngaard J C‘. and C‘ote 0. R. 119?1) The hudger\ 01 turbulent kinetic energq dnd temperature varlancc 111the atmospheric surface layer Bountlar~-f.cl~c,r t1~1 9. .LI I 460. Wqngaard J. C‘.. C‘otC:0. R. and lzumt Y. (1971) L~GII free convectton. stmtlaritv and the budgets of shear \tress and heat flux. J. afmcx. &I. 28, I171 1182. Wyngaard J. C.. Pennell W. T. Lenschow D. H. and LcMonc M. A. (19781 The temprarure humidity covariance budget in the convective boundary layer J. ufmos. SC,;. 35. 47 5X. Zeman 0. (I981 J Progress in the modeling of planetary boundary layers. In .I Kl I I Kurd.Clvc-h 13, 25.3 2”.
APPENDIX
REFEREKC‘ES Brutsaert W. H. (1982) Esaporution mro [he Atmosphere. D. Reidel. Dordrecht. Businger J. A. (1973) Turbulent transfer in the atmospheric surface layer. In Workshop on Mirromerrorology (Edited by Haugen D. A.). American Meteorological Society, Boston, MA. Businger J. A. (1982) Equations and concepts. In Armosphertc Turbulence and Air PoUurion Modelling (Edited by Niewstadt F. T. M. and Van Dop H.). D. Reidel, Dordrecht. Businger J. A.. Wyngaard J. C., lzumi Y. and Bradley E. F. (1971) Flux profile relationships in the atmospheric surface layer. J. armos. Sci., 28, 181-189. Delany A. C. and Davies T. D. (1983) Dry deposition of NO, to grass in rural East Anglia. Atmospheric Encironmenf 17. 1391 1394. Galbally I. E. (1971) Ozone profiles and ozone fluxes in the atmospheric surface layer. Q. JI R met. Sot. 97, 18-29. Hlv 0. (1982) One-dimensional vertical model for ozone and other gases in the atmospheric boundary layer. Atmospheric Environmenr 17, 535550. Leighton P. A. (1961) Phorochemisrrr o/ Air Pollurion. Academic Press, New York. Lenschow D. H. (1982) Reactive trace species in the boundary layer from a micrometeorological perspective. J. mef. Sot. Jpn 60, 472480. Lenschow D. H.. Pearson R. and Stankov B. B. (1982) Measurements of ozone vertical flux to ocean and forest. J. geophys. Rex 87, 8833-8837. Nieuwstadt F. T. M. (1983) Observations on the turbulent structure of the stable boundary layer. Q. JI R. mer. Sot. Submitted. Panofsky H. A.,Tennekes H., Lenschow D. H. and Wyngaard J. C. (1977) The characteristtcs of turbulent velocity components in the surface layer under convective condittons. Boundary-Layer Mrr. 11, 355 361. Paulson C. A. (1970) The mathematical representation ol wind speed and temperature profiles in the unstable atmospheric surface layer. J. oppl. Mer. 9. 857 861. Roberts S. M. and Shipman J. S. (1972) Two-poinf Boundary Value Problems: Shooring Methods. Elsevier, New York. Sehmel G. A. (1980) Particle and gas dry deposition: a review. Armospheric Enc;ironmml 14, 983-1011. Wesely M. L. and Hicks 9. 9. (1977) Some factors that affect the deposit rates of sulfur dioxide and similar gases on vegetation. J. Air Polluf. Control Ass. 27, 11 l&l 116. Wesely M. L., Eastman J. A., Cook D. R. and Hicks B. B. (1978) Daytime variations of ozone eddy fluxes to maize. Boundary-Layer Mtv. 20. 459-47 I.
We develop here a surface layer flux-gradient relation for a reactive trace species that uses a second-moment closure hypothesis and includes chemical effects. The magnitude of these chemical terms and conditions under which a modified flux-gradient relation can be a good approximation are then discussed. The budget of the vertical flux of a chemically reactive trace spectfies that includes first- and second-order chemical reactions in a horizontally homogeneous boundary layer can bc written as
t
~JlnH.\.
+
2
n
)_.k,,,S,ws,.
tA.1)
m.n
where p is kinematic pressure, T and 6 are mean and fluctuating temperature, respectively and g is gravitational acceleration. Summations are taken over the number of reactive species in the system. Coefficients ji. and k,,, represent effects of first- and second-order chemical reactions, respectively. Equation (A.l) is analogous to the temperature flux budget equation, but also includes terms due to chemical reactions. A derivation of the temperature flux budget equation is given by Businger (1982); Wyngaard (1982) discusses approximations applicable to a near-neutral surface layer. Surface layer measurements reported by Wyngaard and Cott (1971) indicate that the triple correlation and time rateof-change terms are usually small in the temperature and momentum flux equations, and we assume here that they are small for reactive species as well. A simple model that is frequently used for the pressure correlation term in (A.11 for non-reactive species (Wyngaard. 1982) is
(A.2) where rv is a characteristic time scale for the destruction of the flux by the pressure term. We find it convenient to use a length scale rather than a time scale. We assume that this length scale I is directly proportional to the height z. a reasonable surface layer hypothesis since z is the only scaling length in the neutral surface layer. Thus [ = U-’ =
u,r,
--
X
.
(A.3)
where k’ is the von Karman constant, taken here to bc 0.4, and
Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer u* is the friction velocity. Thus (A.l) can be rewritten as ($2)
= -x’(z)+;s+&,
(A.4)
Substituting (A.?+ (A.lO), (A.12), (A.13) and (A.14) into (A.3) and solving for the mean gradient,
as,- - (1-~ibi(z/L)4h)wsi + Rwi+ (gKbiZ/(Tu*))R,i at
where
2511
-
q/(1
u:f(l
- biz/L)waz
-
b+/L)
(A.15) 1 1 1 -G---_=l,i LIZ A;
a
1
and ais_ (A.5) Thus the mean gradient of a reactive species is related to its 1 - (az/&) ’ arz flux but is also affected by fluxes of other constituents and by temperature fluctuations via a second term in (A.15). This is a with complete formulation for the flux-profile relationship for a chemically reactive species with the closure conditions (A.3) (‘4.6) and (A.9). The first term of (A.15) varies inversely with height and thus dominates for small z. The relative importance of the and temperature species covariance term with respect to the species flux term in the numerator of the second term in (A. 15) Rwi = c (1 -Sin)jinws,+2 c (1 -Gi,)ki,,S,,,ws,, (A.7) can be found by taking their ratio and assuming that Rgi n m.n _ ooeoi and R,,+ _ b,oi, with similar proportionality factors, with 6,” = 1 and 6,. = 0 if m # n. The length scale Ali in (A.4) where a,, rr, and bi are standard deviations of temperature, includes the effects of chemical reactivity, represented in (A.5) vertical velocity and species concentrations, respectively. by the chemical length scale &. In a simple first-order decay Surface layer observations (Wyngaard et al., 1971) show that reaction we note that the effect of A: is to increase iii over its 3 C7,U*/w8 non-reactive value. (A.16) The buoyancy term in (A.4) may also be affected by ~ %I% = 1 + 5(2/ - L)2’3 chemical reactions. The equation for this covariance is Thus the ratio of the two R terms in (A.15) is grcb,z Roi
_--_ Tu,
_+ ~.A.s.B+ 2 1 kimnSmsn@,
(A.9) We define a length scale bz
1
1
bz
1;
1 b = where bi E ~ b,z’ 1 - bzll;
’
(A.lO)
and a chemical reactivity factor R,, = x(1-6,,)ji,s,B+2
n Thus (A.8) reduces to
2 (1-6i,)ki,,S,,,~.
(A.11)
m,”
(A.12) k &i The M-O flux-gradient relation for potential temperature is defined as &=
-%$.
(A.13)
Panofsky et al. (1977) show that the following expression relating u, to u+ is a good fit to unstable surface layer data: = f(z/L) = 1.6 + 2.9 (z/ - L)2’3.
z < I&l/Ui,
(A.17)
(A.14)
(A.18)
where
We note that the length scale &’ is proportional to a ratio of fluxes; therefore it can be positive, negative or zero. For a single-component first-order reaction, 1;’ is proportional to the reciprocal of the reaction coefficient. Assuming that a is of order one (subsequently to be justified), u., 1: 0.2 m s- 1and j = 0.004 s-’ (typical of NO2 photochemical decay) and that fluxes of various constituents are comparable, we obtain J.i = 125 m. Therefore, for this case we see that the second term in (A.15), which is solely a chemical reactivity term, has little effect on the flux-profile relation in the lowest 10 m. It is often assumed for a non-reactive species, with some experimental justification from humidity measurements (Brutsaert, 1982),that c#$‘~= c#J*, where +yR isa species fluxprofile function. The non-reactive form of (A.15) is
as,_-
u, s,o -asi -aao --= -weaZ-w"idz+R,i.
u;/u:
Ki
l+s(z/-L)2’s
We now limit ourselves to the case of a near-neutral boundary layer, with [z/L1 4 1. Then from (A.17), gKbi(zIT)u,Roi < Rwi, and the condition under which the second term on the right side in (A.15) is small in comparison (A.@ to the first term is given by
where 0 is the mean potential temperature, K and Di are the molecular conductivity and diffusion coefficients, respectively, and the subscript j in the molecular term on the right side denotes a summation over all three spatial coordinates. In contrast to the equation for a flux (A.l), the covariance equation involving two scalars does not have a pressure correlation term; thus the only way covariance is lost is by molecular destruction. We again assume steady state, neglect the third-moment term (Wyngaard et al., 1978) and make a closure assumption for the molecular dissipation term in (A.8) which is analogous to the pressure transport closure assumption, (A.2). Thus
1
L
m. n
n
lzi
3
-biz
aZ -
(1 -ub(z/L)c#~,)nq (1 - b(z/L)Uf
(A.19)
KU*Z
Since 4,,(O) ? 0.74 (Businger et al., 1971) and f(0) = 1.6, we find that a = 0.84 when [z/L 1is small. By substituting a wide range of values of b into (A.15), we find that when b = 0.25, e%pRIL a$, over a broad range of values of z/L, and that c#$‘~ departs very little from c#J,, over a broad range of values of b with u = 0.84. These values for a and b can be substituted into (A.5) and (A.lO), respectively. The resulting values of ui and bi can be then substituted into (A.15). For /z/L I @1 we obtain
s__p a2
0.74 wsi KUtZ
[~-O.‘+-~)]~ (A.20)
‘511
DAVID R. FITUARRALU and DCINALD H. LENSCH~W
is, ?Z
- 0.74 WS( - (1 -z/J.,). KU*i
(A.21)
where 1.19E.;I;’ i, F---_ 1.; - L:
(A.22)
is a single chemical reactivity length scale. Thus we have obtained an expression for the flux-gradient relationship for
reactive species near neutral stability. We note that n, is a function of z. For z/i, 4 1, however, the fluxes and mean concentrations are nearly constant with height, so that L, can be considered to be constant. As noted previously, I;’ can be of either sign or zero. If ii:’ 16 ij, then li = E.:‘.This can happen in the surface layer if the surface flux of a species is much less than that of a second species which is a chemical source or sink for the first. For the 03-NO-NO2 triad this may be the situation for NO. Thus the flux-gradient relationship for NO could show larger departures from M-O similarity than 03.
1983
000&4981/83 33.00 + 0.00 Pergamon Press Ltd.
Prmted m Great Entam.
MEAN CONCENTRATION AND FLUX PROFILES FOR CHEMICALLY REACTIVE SPECIES IN THE ATMOSPHERIC SURFACE LAYER DAVID R. FITZIARRALDand JJONALD H. LENSCHOW National Center for Atmospheric Research,t Boulder, CO 80307,U.S.A. (First
9 February
received
1983)
Abstract-We use Monin-Obukhov similarity theory to study the vertical structure of turbulent fluxesand concentrations of reactivespecies in the surface layer. A surface layer flux-gradient relation that includes firstand second-order chemical reactions is derived from the flux budgets. The theory is then applied specifically to the O,-NO-NO, photochemical reaction triad. In this case it is found that the non-reactive flux-gradient relation usually suffices, but that chemical reactions can seriously alter the surface layer turbulent flux and mean concentration profiles. It is demonstrated that six observations of fluxes or concentrations are necessary to determine the surface fluxes when reactions are significant. Significant deviations from the predicted nonreactive flux are found to be quite plausible for NO, levels characteristic of moderately polluted air.
1. INTRODUCTION Surface sources and sinks play a major role in the budgets of many atmospheric trace reactive species. Estimating surface sources and sinks, along with time changes and mean and turbulent transport over a specified area, allows us to estimate as a residual the chemical transformation rate of trace reactive species. One important element of surface sinks is dry deposition, which can have significant effects on vegetation and on the acidity of surface water. A common simplification in chemical modeling studies to describe dry deposition is the use of a deposition velocity (Sehmel, 1980). It is assumed that the rate of surface deposition is proportional to the mean concentration at a particular reference level. That is, the deposition velocity is independent of the turbulence structure of the atmospheric surface layer. Sehmel (1980), for example, recommends that mean concentrations for estimating deposition velocity be presented at a standardized height of 1 m. Although this concept may apply to relatively slowly reacting species, we show here that it does not apply to rapidly reacting species. Because observational techniques have developed to the point where accurate measurements of both mean profiles and turbulent fluxes of reactive species are now possible (e.g. Lenschow et al., 1982; Wesely et al., 1978; Delany and Davies, 1983), it is important to consider details of the surface layer structure so that the observations can be related to the physical characteristics of the underlying surface, independent of surface layer transport properties. In this paper we discuss the possibility that reactions occurring in the
t The National Center for Atmospheric Research is sponsored by the National Science Foundation.
surface layer in the vicinity of the observations may change flux estimates substantially even below 1 m. Results presented here also apply to model studies in which the lower boundary condition is specified using a deposition velocity at fixed height (e.g. H$v, 1982). We show that vertical turbulent transfer in the surface layer can maintain a system of reacting species in a state away from chemical equilibrium. Since the turbulent mixing time scale increases with height in the surface layer, this effect is more pronounced as one approaches the ground. If reaction time scales are comparable to or less than the turbulent mixing time scales, significant departures from chemical equilibrium can occur. In this situation we show that it is not sufficient to use only measurements of the species concentration and gradient for estimating the surface flux of a constituent, but the concentrations and gradients of the other reacting species in the system must be considered as well. The problem discussed in this paper is not, however, related only to cases in which profiles are used to infer fluxes. Even if the flux is directly measured (say by the eddy-correlation method), the results must still be extrapolated to some surface transfer height, the effective minimum height to which surface layer turbulence can transport the species (Wesely and Hicks, 1977), to determine the transport of a quantity into or out of the surface. Very near the surface (say within 1 cm) molecular diffusion becomes significant. The time scale for diffusion through this layer is of the order of 1 s. Therefore, to justify ignoring chemical reactions in the molecular sublayer we limit our consideration to reaction times of at least several seconds. In the Appendix we use the Monin-Obukhov similarity theory (Businger, 1973) and covariance budget equations, with the addition of first- and second-order chemical reactions, to develop a modified flux-profile
2505
relationship in the surface layer. Example calculations in Section 2 illustrate the differences between reactive and non-reactive profiles for specified surface fluxes. In Section 3 a procedure for determining surface fluxes from profile data is presented. This theory is used to examine the range of possible fluxes that can be inferred from the 0, and NO, mean concentration data of Delany and Davies (1983).
2. THEORETICAL
DISCUSSION
The similarity hypothesis applied to surface layer structure has been extensively verified by results from field experiments. Basically, this hypothesis, commonly known as the Monin-Obukhov (M-O) similarity hypothesis, states that mean and turbulence variables in the surface layer (the lowest part of the boundary layer where the fluxes of non-reactive species can be considered constant) are functions of height z above the ground and hydrodynamic stability. Stability is specified by a length scale,
concentration of S, through chemical reactions. tor a single-component first-order reaction in the neutral surface layer the height at which the reaction time , ’ is equal to the diffusion time (Lenschow, 19X?i. z -= u,k,‘l,jl.
is proportional to the height IO which the constituent flux can be considered constant. Chemical reactions can also affect the flux-gradient relation. In the appendix we develop a flux-gradient relation that includes the effects of chemical reactivity and discuss conditions under which the non-reactive flux-gradient relation is a good approximation for a reactive species. We now consider specifically the photochemical triad of 03, NO* and NO: NOz+hv+NO+O 0+0,-o,
nSi/c’t + IJf%!$/ax+ wasi/az
+ aG)az
= Qi,
(1)
where U and W are the mean horizontal and vertical velocity, respectively and Qi is the sum of the internal sources and sinks. Capital letters refer to mean quantities and small letters to turbulent fluctuations. The overbar denotes a temporal or spatial average over a time or length sufficient to give a stable estimate. If we limit our consideration to first- and second-order chemical reactions, the chemical source term can be written as: Qi =
(4)
NO + 0,3 --*NO1 + Oz. The second reaction occurs much more rapidly than the other two (Leighton, 1961), and thus (4) can be written as QI1=jC-kAB
where u* is the friction velocity, g/T is the buoyancy parameter, Kis the von Karman constant and wf3,is the virtual temperature flux at the surface. Surface layer variables can then be empirically determined as functions of the normalized ratio z/L. This approach applies to trace constituents that have no sources or sinks in the surface layer. For species that are not conserved, surface layer relations will, in general, depend upon the source/sink terms, as well as on z/L. We develop a formulation for the flux-gradient relation for non-conserved constituents and examine the conditions under which the M-O flux-gradient relation may be considered a valid surface layer approximation. The mean budget of a non-conserved constituent Si can be written (Lenschow, 1982) as
(31
= Qb = -Q,,
where A, B and C are the mean concentrations of 03, NO and NOz, respectively, and the second-moment term kab is assumed negligible. This is justified if the fluctuation levels for these species are much less than their means. Lenschow (1982) shows that this is true for Oa in well-mixed air over central U.S. These reactions occur so rapidly that, for the most part, other competing reactions are not significant for the time scales considered here. At equilibrium, called the photostationary state (Leighton, 1961) QS = 0. In the lower part of the surface layer the chemical system (4) will not, in general, be in equilibrium if surface fluxes are not zero because of vertical turbulent diffusion. For the remainder of this section we consider the ease when the height is well below the height where the chemical and turbulent time scales are comparable; that is, z + u*~,/K, where 7,, the inverse of the reactivity coefficient, is the effective reactivity time constant (Lenschow, 1982). Thus the standard M-O fluxgradient function 4s is assumed. We apply this analysis to the estimation of fluxes based on mean profiles measured near the ground. Fluxes of 03, NO and NO2 are related to their mean gradients by the relations: 4,: ‘;A+I,
= - wa
(6a)
4;
= - wb
(W
=
(W
‘zB,Ku,
~>~‘ZC,KU,
-NY,
(2) where the subscript z denotes differentiation Cjir& + C hnn[SmSn+smsnIv
where ja and kimnare the reactivity coefficients and the Sj are the concentrations of species that affect the
(5)
with respect to height. One feature of these relations is that certain sums and differences of fluxes are constant with height. Lenschow (1982) showed that the O3 oudget
Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer equations can be written, in the present notation, as: (u,rcz~;‘A,),
= jC -kAB
(74
(U,KZf$;lBz)z
=jC
(7b)
(u,K$;~C~)~
=
-kAB
-jC+kAB.
(74
Conserved combinations of fluxes result from adding and subtracting relations in (7) and substituting (6): (wn - wb), = 0
(84
(wa + w(‘)Z = 0 -(wb + WC), = 0.
(W
2501
the non-reactive case requires only measurement of two quantities. Mean 0s measurements, or even simultaneous Oz and NO, measurements at two levels, are not sufficient mathematically to determine the fluxes of these quantities. In the limit of low NO, levels, of course, the reactive solution for Oj must approach the non-reactive solution. In Section 3 we present numerical examples of the inaccuracy of the flux estimates when chemical reactions are neglected. Figure 1 illustrates model solutions for the flux and concentration of O3 for a particular value of mean O3
(84
Thus (wb + WC), which is proportional - to the flux of NO + NOz = NO,., as well as (G - wb) and (& +wc), are constant. This means that the deposition velocity, w,, for NO,,
is a well-defined quantity of a conserved species. Knowing the flux and concentration of NO, at one height, the profile of NO, is uniquely determined. This is not true for any one of the constituents unless concentrations of the other two are negligible. Since the quantities in parentheses in (8) are conserved, their profiles and fluxes are given by integrating the fluxgradient relations (Paulson, 1970). Profiles of two of the conserved quantities whose fluxes are given in (8) are written as,
I
I
60 OZONE
I 70
[
ppbv]
A-B=D(z)=D(z,)+[(wa-wb)/~u.Jj-(z,zl,L) (104 -B + C = E(Z) = E(z,) + [ (wb + WC,/KU,] f(z, zl, L), (lob)
wheref(z, zr, L) = s:, (4,/z)dz, and where zr is some reference level. We note that it is sufficient to solve explicitly for only one nonconserved quantity, and then use (10) to find the remaining two. Equations (6a) and (7a) can then be written as: A, = (wa),
=
ii&,/KU,Z
(116)
(kAB-jC).
(lib)
Using (lo), (11) becomes two coupled ordinary differential equations, one for the concentration, the other for the flux of 03. We note that it is not possible to attain photostationary equilibrium at more than one point in the lower surface layer since the photostationary state is equivalent to having zero turbulent flux divergence. If reactions are occurring, then six quantities must be determined simultaneously (in addition to the atmospheric stability, surface roughness and friction velocity) to find the profiles of concentration and fluxes for the three constituents in this system. By contrast,
I’
0
’
I
I
J
5
IO
I5
FLUX
RATIO
Fig. 1. Sample profiles for u, = 0.2 m s-r
and z0 = 0.01 m at neutral stability with the following conditions at ze: [0,] = 4Oppbv, flux of 0, = -0.23 ppbv m s-r; [NO] = 6 ppbv, flux of NO = -0.001 ppbv m s-r; [NO,] = 20 ppbv, flux of NO1 = - 0.20 ppb m s- ‘. The non-reactive solution is given by the solid line, the reactive solution with the standard M-O flux-gradient relation is given by the dashdot line and the reactive solution with the modified fluxgradient relation obtained from the flux budget is given by the dotted line. The top figure is 0, concentration and the bottom is the ratio of 0s flux to its constant non-reactive value.
L)Aw~
3508
R.
I.‘ITLJARRAL.Dand
concentration and flux at the surface transfer height. We see that, for this example, the reactive solutions are quite different from the non-reactive solution, with the constant flux layer extending only to about 4 m. We also note that it is not necessary, in this case, to use the corrected flux-gradient relation from the appendix. We have assumed here a small negative NO surface flux for this example; the solutions are similar for other small NO fluxes, including a positive surface flux.
Doh41 D H. Lwww\r. and zL.
a--J Measured variables for the test case are given in ‘Table 1. To illustrate how reactions change the flux and concentration profiles, we first arbitrarily assume that r(z2) is at its photostationary value (zero flux divergence at zLI: I‘(_;) = (1 -c/i.4 ,/I ‘.
3. DETERMINING CONSTITUENTS
FLUXES OF REACTIVE
USING THE PROFILE
METHOD
In this section we discuss the profile technique used to infer the fluxes of 03, NO and NO1. Results of the procedure are then used to interpret observations of mean NO, and 0, at two levels in the surface layer reported by Delany and Davies (1983). The profile method requires at least two levels of mean concentration measurements along with auxiliary meteorological data to find u* and L. Assuming constant flux with height, the flux ofa constituent can then be determined from empirical profile functions. This is the method used by Galbally (1971) and others to find the turbulent O3 flux in the surface layer. We have demonstrated, however, that source terms in the mean budget of a reactive substance require that the surface layer turbulent flux vary with height. Thus, when these terms are important we must solve for all fluxes and concentrations in the system simultaneously in such a way that observed concentrations are satisfied at the observation points. Near the surface, solutions of relations (1 la, b) can be efficiently obtained from the iterative ‘shooting’ method of solving two-point boundary value problems (see Roberts and Shipman. 1972, Ch. 2). Two integrations, one up and the other down, are done for each iteration. Equations (1 la, b) are integrated from the lowest level using the observed concentration and an assumed flux, usually the nonreactive flux. The flux at the lower level is corrected using the resulting deviation between observed and predicted concentrations at the upper level until a specified tolerance is achieved. This method converged in two to eight iterations, depending on tolerance levels. Delany and Davies (1983) present a few cases of simultaneous measurements of concentrations of O3 and NO, at two levels (0.125 and 1.75 m) above a grasscovered surface in moderately polluted air in England. We use one of their profile observations here to demonstrate the range of possible efTects that chemical reactions can have on the flux estimates. For simplicity we use constant values of the reaction coefficientsj and k. In this case we have only four of the six measurements needed to determine the fluxes completely. Thus the system has two degrees of freedom. We have chosen the missing variables to be the NO/NO, concentration ratio at the two observation levels z1
(13)
For the reaction coefficients assumed here, r(zJ x 0.127 and the reactivity time constant (Lenschow, 1982) is T, % 107 s. The remaining free variable is r(zl). Solutions for r(zI) ranging from zero to one and for the non-reactive case are shown in Fig. 2. The change in concentration profiles is relatively small as r(-l, ) is varied over its range. However, the flux profiles change drastically as r(zl) gets large. Note that all of the calculated O3 fluxes equal the non-reactive solution at the same height. If the NO/NO, concentration ratio does not exceed 0.1, the calculated fluxes may not differ from the nonreactive fluxes by more than 10~~20’~,,.In some cases, though, the reactive terms introduce such curvature in the flux profiles that the flux estimate at a height equal to the roughness length [assumed here to be equal to the surface transfer length as defined by Wesely and Hicks (1977)] may be greatly diff‘erent from the nonreactive estimate. The entire range of ambiguity in the test case is presented in Fig. 3. Contours of the ratio of the reactive to the non-reactive fluxes appear as straight lines on the r(zI ). r(z2) plane. II is even theoretically possible for the reactive flux to be of the opposite sign of the non-reactive flux in the limit where extremely large NO concentrations occur. We have demonstrated that reaction eft‘ects may seriously alter the value of a turbulent flux inferred using the profile method. We therefore recommend that when NO, levels are high (say > 20 ‘i, of [O,] ), at least six measurements be made to determine the flux, either as concentrations at two levels or as some
Table 1. Observations of Delany and Davies (1983) used here as an example 19.5 20.6 64.0 69.1 O.CW4 s-l) 0.004 0.18 u,(ms-‘) ‘X L (m) __ _______~ The heights of observation are z, = 0.125 m and zz = 1.75 m above the surface. Concentrations are given in ppbv.
NO&,) NO,@,) %@I) O,(G) k (ppbv-’ i (s-l)
Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer
2509
Fig. 3. Contours of the ratio of the re&ve 0s flux to the ~on-r~~ti~ ffux ( :d) plotted on ther~z*~~z*)plane for the exampie discussed in the text.
FLUX RATIO Fig. 2. Profilesthat fit ~n~tmtio~s of0s observed by Delany and Davies (1983) (Tabie 1)at two levels for values of the ratio r(zl) ranging from zero to on& The nonreactive case is given by the dashed line. Parameter r(zs) is fIxed at its pbotostationary value of 0.127. The top figure is 0s concentration profiles and the bottom is the ratio of Q3 &ix to its constant non-reactive value.
of con~ntrati~ns with dire&y measured fluxes (e.g. by the ~dy-~orreiatian method). It is important to recognize that in obse~at~~n~ and model studies one wishes to estimate the surface flux ofa substance into or out of the system. This flux properly should be evaluated at a height equal to the transfer length, though obse~ati~~s at this level are not normally available since it is typically of the order of the height of the roughness elements. Over grassland, for examPIe, it may be of the order of 1 cm. At this height m~urements of ~adients or direct 3ux measurements are not feasible by the techniques discussed here. The results in Fig. 2 iliustrate that even if Buxes are measured directly, along with coneentrations, the corm&ion procedure outlined here may be required to estimate the flux at the surface. c~~bi~atj~n
The results presented here question the validity of using deposition velocities, defined by the ratio offlux to #n~~t~tion, at some surface layer reference levej when the chemical reaction time is fess than or comparable to the turbulent diffusion time. A more reasonable approach to estimate surface deposition rates under there conditions would be to define a deposition velocity at a height equal to the surface transfer height*with concentrations and Ruxesat this level determined by the technique prescribed here. The ideas discussed here are not limited to chemicai reactions. They also have applications to other nonconserved constituents in the atmospheric surface layer. Over an evaporating surface upward-moving air will, on average, be more humid than dawnwardmoving air. Therefore particles which wouid be the same size at the same h~dity will be larger in the upward-moving air than in the downward-moving air. Thus the layer in which the drier downward-moving air is mixed with the moisture upward-rnovi~~air near the surface can be insides, on average, as a net particle sizeconverter. If the particle size distribution is not uniform, the number of particles in some size ranges will grow at the expeuse of others, and vice versa. Thus the Aux profife for particles within a specified size range is not n~sarily constant with height, and the flux-gradient relation may not necessarily follow the ~onin~bnkhov s~rnjlari~yprediction. ~~r~ermore, the net particle mass flux, measured by the eddycorrelation technique, will be too large. Wesely et al. (1982) observed a net upward flux of pa&les over the ocean, possibly due to this effect. The theory presented here relies heavily on Moni~~bukhov similarity formulations and there-
t’ore is nol applicable In
particular,
this
throughout formulation
achieve a photostationary
the boundary cannot
equilibrium
layer. Such a state. character&d scale much shorter
may
be
a
than a turbulent
useful
layer.
be made to
throughout
a
b! a chemical time mixmg time scale.
upper-boundar?
condition.
Development
of a theory to describe the fluxes and
concentrations
01‘ reactive species m the boundary
layer is a necessary
future
complement
to the work
presented here. A~~fl~ledyrrnun~.s J. C‘. Wyngaard made several useful suggesttons concerning the flux budget and flux-gradient rclattonshtps m the surface layer: Brian Ftedler also provtded helpful comments. David R. Filljarrdld did this work as an Advanced Study Program Fellow at the Nattonal Center for Atmosphcrtc Research. The manuscript was expertly typed by Jo Ann Fankhauser with extcnstvc rcvtsinns typed hy Daloris Ildming
Wesely M. I... (‘oak D R. Harl R L. and Wtlltams R M 11982) Au sea exchange of CO, dnd evidence for enhanced upward fiuxe\. .I !~twpk I’\. Rc\ 87. XX?7 X83? Wyngaard J C‘ t I’)821 Roundarj-layer modchng 111 .1rmosphrrlc 7urhulw t‘ trod Arr P~~llurwtt ~~&//rm~ (Fdned by Nlewstadt I I M xnd Van Dop H I D Rcidel. Dordrccht Wyngaard J C‘. and C‘ote 0. R. 119?1) The hudger\ 01 turbulent kinetic energq dnd temperature varlancc 111the atmospheric surface layer Bountlar~-f.cl~c,r t1~1 9. .LI I 460. Wqngaard J. C‘.. C‘otC:0. R. and lzumt Y. (1971) L~GII free convectton. stmtlaritv and the budgets of shear \tress and heat flux. J. afmcx. &I. 28, I171 1182. Wyngaard J. C.. Pennell W. T. Lenschow D. H. and LcMonc M. A. (19781 The temprarure humidity covariance budget in the convective boundary layer J. ufmos. SC,;. 35. 47 5X. Zeman 0. (I981 J Progress in the modeling of planetary boundary layers. In .I Kl I I Kurd.Clvc-h 13, 25.3 2”.
APPENDIX
REFEREKC‘ES Brutsaert W. H. (1982) Esaporution mro [he Atmosphere. D. Reidel. Dordrecht. Businger J. A. (1973) Turbulent transfer in the atmospheric surface layer. In Workshop on Mirromerrorology (Edited by Haugen D. A.). American Meteorological Society, Boston, MA. Businger J. A. (1982) Equations and concepts. In Armosphertc Turbulence and Air PoUurion Modelling (Edited by Niewstadt F. T. M. and Van Dop H.). D. Reidel, Dordrecht. Businger J. A.. Wyngaard J. C., lzumi Y. and Bradley E. F. (1971) Flux profile relationships in the atmospheric surface layer. J. armos. Sci., 28, 181-189. Delany A. C. and Davies T. D. (1983) Dry deposition of NO, to grass in rural East Anglia. Atmospheric Encironmenf 17. 1391 1394. Galbally I. E. (1971) Ozone profiles and ozone fluxes in the atmospheric surface layer. Q. JI R met. Sot. 97, 18-29. Hlv 0. (1982) One-dimensional vertical model for ozone and other gases in the atmospheric boundary layer. Atmospheric Environmenr 17, 535550. Leighton P. A. (1961) Phorochemisrrr o/ Air Pollurion. Academic Press, New York. Lenschow D. H. (1982) Reactive trace species in the boundary layer from a micrometeorological perspective. J. mef. Sot. Jpn 60, 472480. Lenschow D. H.. Pearson R. and Stankov B. B. (1982) Measurements of ozone vertical flux to ocean and forest. J. geophys. Rex 87, 8833-8837. Nieuwstadt F. T. M. (1983) Observations on the turbulent structure of the stable boundary layer. Q. JI R. mer. Sot. Submitted. Panofsky H. A.,Tennekes H., Lenschow D. H. and Wyngaard J. C. (1977) The characteristtcs of turbulent velocity components in the surface layer under convective condittons. Boundary-Layer Mrr. 11, 355 361. Paulson C. A. (1970) The mathematical representation ol wind speed and temperature profiles in the unstable atmospheric surface layer. J. oppl. Mer. 9. 857 861. Roberts S. M. and Shipman J. S. (1972) Two-poinf Boundary Value Problems: Shooring Methods. Elsevier, New York. Sehmel G. A. (1980) Particle and gas dry deposition: a review. Armospheric Enc;ironmml 14, 983-1011. Wesely M. L. and Hicks 9. 9. (1977) Some factors that affect the deposit rates of sulfur dioxide and similar gases on vegetation. J. Air Polluf. Control Ass. 27, 11 l&l 116. Wesely M. L., Eastman J. A., Cook D. R. and Hicks B. B. (1978) Daytime variations of ozone eddy fluxes to maize. Boundary-Layer Mtv. 20. 459-47 I.
We develop here a surface layer flux-gradient relation for a reactive trace species that uses a second-moment closure hypothesis and includes chemical effects. The magnitude of these chemical terms and conditions under which a modified flux-gradient relation can be a good approximation are then discussed. The budget of the vertical flux of a chemically reactive trace spectfies that includes first- and second-order chemical reactions in a horizontally homogeneous boundary layer can bc written as
t
~JlnH.\.
+
2
n
)_.k,,,S,ws,.
tA.1)
m.n
where p is kinematic pressure, T and 6 are mean and fluctuating temperature, respectively and g is gravitational acceleration. Summations are taken over the number of reactive species in the system. Coefficients ji. and k,,, represent effects of first- and second-order chemical reactions, respectively. Equation (A.l) is analogous to the temperature flux budget equation, but also includes terms due to chemical reactions. A derivation of the temperature flux budget equation is given by Businger (1982); Wyngaard (1982) discusses approximations applicable to a near-neutral surface layer. Surface layer measurements reported by Wyngaard and Cott (1971) indicate that the triple correlation and time rateof-change terms are usually small in the temperature and momentum flux equations, and we assume here that they are small for reactive species as well. A simple model that is frequently used for the pressure correlation term in (A.11 for non-reactive species (Wyngaard. 1982) is
(A.2) where rv is a characteristic time scale for the destruction of the flux by the pressure term. We find it convenient to use a length scale rather than a time scale. We assume that this length scale I is directly proportional to the height z. a reasonable surface layer hypothesis since z is the only scaling length in the neutral surface layer. Thus [ = U-’ =
u,r,
--
X
.
(A.3)
where k’ is the von Karman constant, taken here to bc 0.4, and
Mean concentration and flux profiles for chemically reactive species in the atmospheric surface layer u* is the friction velocity. Thus (A.l) can be rewritten as ($2)
= -x’(z)+;s+&,
(A.4)
Substituting (A.?+ (A.lO), (A.12), (A.13) and (A.14) into (A.3) and solving for the mean gradient,
as,- - (1-~ibi(z/L)4h)wsi + Rwi+ (gKbiZ/(Tu*))R,i at
where
2511
-
q/(1
u:f(l
- biz/L)waz
-
b+/L)
(A.15) 1 1 1 -G---_=l,i LIZ A;
a
1
and ais_ (A.5) Thus the mean gradient of a reactive species is related to its 1 - (az/&) ’ arz flux but is also affected by fluxes of other constituents and by temperature fluctuations via a second term in (A.15). This is a with complete formulation for the flux-profile relationship for a chemically reactive species with the closure conditions (A.3) (‘4.6) and (A.9). The first term of (A.15) varies inversely with height and thus dominates for small z. The relative importance of the and temperature species covariance term with respect to the species flux term in the numerator of the second term in (A. 15) Rwi = c (1 -Sin)jinws,+2 c (1 -Gi,)ki,,S,,,ws,, (A.7) can be found by taking their ratio and assuming that Rgi n m.n _ ooeoi and R,,+ _ b,oi, with similar proportionality factors, with 6,” = 1 and 6,. = 0 if m # n. The length scale Ali in (A.4) where a,, rr, and bi are standard deviations of temperature, includes the effects of chemical reactivity, represented in (A.5) vertical velocity and species concentrations, respectively. by the chemical length scale &. In a simple first-order decay Surface layer observations (Wyngaard et al., 1971) show that reaction we note that the effect of A: is to increase iii over its 3 C7,U*/w8 non-reactive value. (A.16) The buoyancy term in (A.4) may also be affected by ~ %I% = 1 + 5(2/ - L)2’3 chemical reactions. The equation for this covariance is Thus the ratio of the two R terms in (A.15) is grcb,z Roi
_--_ Tu,
_+ ~.A.s.B+ 2 1 kimnSmsn@,
(A.9) We define a length scale bz
1
1
bz
1;
1 b = where bi E ~ b,z’ 1 - bzll;
’
(A.lO)
and a chemical reactivity factor R,, = x(1-6,,)ji,s,B+2
n Thus (A.8) reduces to
2 (1-6i,)ki,,S,,,~.
(A.11)
m,”
(A.12) k &i The M-O flux-gradient relation for potential temperature is defined as &=
-%$.
(A.13)
Panofsky et al. (1977) show that the following expression relating u, to u+ is a good fit to unstable surface layer data: = f(z/L) = 1.6 + 2.9 (z/ - L)2’3.
z < I&l/Ui,
(A.17)
(A.14)
(A.18)
where
We note that the length scale &’ is proportional to a ratio of fluxes; therefore it can be positive, negative or zero. For a single-component first-order reaction, 1;’ is proportional to the reciprocal of the reaction coefficient. Assuming that a is of order one (subsequently to be justified), u., 1: 0.2 m s- 1and j = 0.004 s-’ (typical of NO2 photochemical decay) and that fluxes of various constituents are comparable, we obtain J.i = 125 m. Therefore, for this case we see that the second term in (A.15), which is solely a chemical reactivity term, has little effect on the flux-profile relation in the lowest 10 m. It is often assumed for a non-reactive species, with some experimental justification from humidity measurements (Brutsaert, 1982),that c#$‘~= c#J*, where +yR isa species fluxprofile function. The non-reactive form of (A.15) is
as,_-
u, s,o -asi -aao --= -weaZ-w"idz+R,i.
u;/u:
Ki
l+s(z/-L)2’s
We now limit ourselves to the case of a near-neutral boundary layer, with [z/L1 4 1. Then from (A.17), gKbi(zIT)u,Roi < Rwi, and the condition under which the second term on the right side in (A.15) is small in comparison (A.@ to the first term is given by
where 0 is the mean potential temperature, K and Di are the molecular conductivity and diffusion coefficients, respectively, and the subscript j in the molecular term on the right side denotes a summation over all three spatial coordinates. In contrast to the equation for a flux (A.l), the covariance equation involving two scalars does not have a pressure correlation term; thus the only way covariance is lost is by molecular destruction. We again assume steady state, neglect the third-moment term (Wyngaard et al., 1978) and make a closure assumption for the molecular dissipation term in (A.8) which is analogous to the pressure transport closure assumption, (A.2). Thus
1
L
m. n
n
lzi
3
-biz
aZ -
(1 -ub(z/L)c#~,)nq (1 - b(z/L)Uf
(A.19)
KU*Z
Since 4,,(O) ? 0.74 (Businger et al., 1971) and f(0) = 1.6, we find that a = 0.84 when [z/L 1is small. By substituting a wide range of values of b into (A.15), we find that when b = 0.25, e%pRIL a$, over a broad range of values of z/L, and that c#$‘~ departs very little from c#J,, over a broad range of values of b with u = 0.84. These values for a and b can be substituted into (A.5) and (A.lO), respectively. The resulting values of ui and bi can be then substituted into (A.15). For /z/L I @1 we obtain
s__p a2
0.74 wsi KUtZ
[~-O.‘+-~)]~ (A.20)
‘511
DAVID R. FITUARRALU and DCINALD H. LENSCH~W
is, ?Z
- 0.74 WS( - (1 -z/J.,). KU*i
(A.21)
where 1.19E.;I;’ i, F---_ 1.; - L:
(A.22)
is a single chemical reactivity length scale. Thus we have obtained an expression for the flux-gradient relationship for
reactive species near neutral stability. We note that n, is a function of z. For z/i, 4 1, however, the fluxes and mean concentrations are nearly constant with height, so that L, can be considered to be constant. As noted previously, I;’ can be of either sign or zero. If ii:’ 16 ij, then li = E.:‘.This can happen in the surface layer if the surface flux of a species is much less than that of a second species which is a chemical source or sink for the first. For the 03-NO-NO2 triad this may be the situation for NO. Thus the flux-gradient relationship for NO could show larger departures from M-O similarity than 03.