Atmospheric Environment Vol. 26A, No. 5, pp. 947 949, 1992. 0004 6981/92 $5.00+0.00 Printed in Great Britain. Pergamon Press pie

S H O R T C O M M U N I C A T I O N

VERTICAL DISPERSION OF GROUND-LEVEL RELEASES IN THE SURFACE BOUNDARY LAYER

AKULA VENKATRAM

ENSR Consulting and Engineering, Camarillo, CA 93012, U.S.A.

(First received 28 March 1991 and in final form 28 June 1991)

Abstract--By assuming that the eddy diffusivity of mass is proportional to that of heat, this paper derives simple expressions for the asymptotic behavior of cross-wind integrated ground-level concentrations under neutral, stable, and unstable conditions. We show that:

C~, ~ x , 1 under neutral conditions; x , 1/3 under stable conditions; x , 2 under unstable conditions;

where ~r. = CYu. [LIQ and x. =x/ILI, and (~Y is the cross-wind integrated concentration, u. is the surface friction velocity, L is the Monin-Obukhov length, and Q is the pollutant release rate.

We show that simple interpolations between the asymptotic limits provide excellent fits to the Prairie Grass (Barad, 1958, Paper No. 59, Geophysics Research Directorate, MA) diffusion data. Our analysis of surface dispersion in unstable conditions indicates that the concentration decrease with distance is not consistent with that predicted by free convection theory (Yaglom, 1972, Atmos. Ocean Phys. 8, 333-340). Under asymptotically unstable conditions, the concentration falls off as x-2 rather than as x-3/2 predicted by the theory.

Key word index: Surface layer dispersion, cross-wind integrated concentration, asymptotic limits, Prairie Grass data.

l. I N T R O D U C T I O N

It is generally accepted that dispersion in the surface layer can be described by the diffusion equation, van Ulden (1978) showed that using the eddy diffusivity of heat to model mass transfer yielded results for surface concentrations that com- pared well with observations from the Prairie Grass experi- ment (Barad, 1958). In his paper, van Ulden (1978) presents analytical expressions that can be used to estimate surface concentrations. This paper expands upon the work of van Ulden and others (Briggs and McDonald, 1978; Venkatram, 1982) by clarifying the behavior of surface concentrations under the limits of neutral, very stable, and very unstable conditions. This examination of the asymptotic behavior serves two purposes: it provides insight into dispersion, and it allows the formulation of simple expressions for concentra- tions that interpolate between the asymptotic limits. These expressions are in, many ways, simpler than those presented by other authors. While several of the results presented here are not new, the derivation of the equations presented here is original as far as the author knows.

2. THE EXPRESSIONS

The asymptotic expressions for ground-level concentra- tions are derived using the eddy diffusivity relationship (see Venkatram, 1988):

da~ - - = 2 K z , (1) dt

where az is the vertical spread of the plume, t is the travel

time, and K~ is the appropriate eddy diffusivity. Equation (1) can be written in terms of the distance, x, from the source, as follows:

da~ u - - = 2Kz. (2)

dx

In Equation (2), u and Ks are evaluated at a height propor- tional to a~. As long as this height is a constant fraction of try, the results presented in this paper do not depend on the value of the fraction; we will assume that u and K~ are evaluated at a height z = a~.

2.1. The neutral boundary layer

When the boundary layer is neutral, K z depends only on the surface friction velocity u. and the height z, while the wind speed u is a function of u., z, and the roughness length z 0. Evaluating the expressions for Kz and u at z = a~, we find

Kz~u . z and u~u . ln(aJZo). (3)

Substituting Equation (3) in Equation (2) results in

da~ In (a~/Zo)-~x ~ a,. (4)

Integrating Equation (4), we find that

a~[ln(a~/Zo)- 1] ~x. (5)

Using Equation (3), we can replace the logarithmic term in Equation (5) to obtain

947

948 Short Communication

Now, u, is usually a small fraction of u except at small distances from the source. Thus, Equation (6) can be approx- imated by

a~u~u,x . (7)

The cross-wind integrated ground-level concentration is given by

¢7,~~. (8) O'zU

Using Equation (8), we obtain the relationship

C ' ~ Q (9) U , X "

This result has been derived using other methods by van Ulden (1978) and Briggs (1982). In anticipation of the other expressions derived in the paper, we rewrite Equation (9) using the following definitions:

~y, = CYu, ILl, x , = x/ILI, (10) Q

where L is the Monin-Obukhov length. Then, the behavior of ground-level concentrations under neutral conditions is described by

C ~ x . 1 . (11)

2.2. The unstable boundary layer

Under asymptotically unstable conditions, we can assume that the speed is independent of height, and that (u, /u) approaches a constant ( ~ 0.4). The eddy diffusivity of heat as a function of height is given by (see Businger, 1973a):

Ka( z )~ u , z ( - z / L ) 1/2. (12)

Following van Ulden (1978), we assume that Kn describes the transfer of mass. Then Equation (2) for asymptotically unstable conditions becomes

d~ U ~-X ~ U, az( -- Cfz/L) 1/2, (13)

o r

do- z ~x ~ a~/2lL]- 1/2, (14)

because (u , /u) approaches a constant value. Integrating Equation (14) we find that

a~~xZlL1-1. (15a)

Notice that a~ grows as x 2 rather x 3/2 as predicted by Yaglom (1972). However, Equation (15a) is more consistent with the analysis of Briggs (1988), who showed that the vertical spread of the plume grew faster than the 'free convec- tion' prediction. He explained his empirical result with the 'sweepout' of particles by convective updrafts. Here, we find that it is not necessary to invoke this mechanism explicitly. We only have to assume that the mechanisms of heat and mass transfer are similar to obtain the x 2 growth of the plume.

Briggs (1982) points out that Equation (15a) implies that, at a given x, a~ decreases with an increase in u,. This counterintuitive result can be seen by writing the equation as

X2 ~O t r ~ 3 ' (15b)

U ,

by using the identity for L. This sort of result is a feature of the surface unstable boundary layer (see Tennekes, 1973). To maintain the flow of the arguments presented here, we will discuss this in more detail in the last section of the paper.

Using Equation (15a) for a, and taking (u, /u) as a con- stant, we can write the expression for the cross-wind integ- rated concentration as

CY~ Q ~ ~ QILI (16) O'zl/ U , X 2 "

Equation (16) can be expressed in terms of non-dimensional variables as

C ~ ~ x , 2. (17)

2.3. The stable boundary layer

Under highly stable conditions, u and KH can be expressed as (Businger, 1973a):

u ~ u , z / L and K n ~ u , L . (18)

Substituting these expressions in Equation (2) for the rate of growth of tr~ we obtain

da~ u.(~JL)~-~-x ~u,L, (19)

o r d az3 L2"

dx

Integrating Equation (20), we obtain 3 2 q z ~ x L ,

o r

tYz~ Xl/3 L2/3.

(20)

(21)

(22)

Note that az grows as x 1/3 under very stable conditions. Using this relationship for try, we can write u(az) as

u( az) ~ u, tr z/ L ~ u, ( x / L ) 1/3. (23)

Then, UtTz~U,X2/3 L 1/3. (24)

The expression for the cross-wind integrated ground-level concentration becomes

UtTz IA,X2/3 L1/3 , (25)

and in terms of non-dimensional variables, Equation (25) can be written as

C~ ~ x ; 2/3. (26)

This completes our analysis of the asymptotic behavior of ground-level concentrations. The next section describes the evaluation of the expressions derived here.

3. EVALUATION OF EXPRESSIONS AGAINST OBSERVATIONS

The expressions derived in the previous sections refer to asymptotically neutral, stable, and unstable conditions. We expect the behavior of plumes in conditions between these limits to be described by expressions that are also functions of the surface length, z 0. Because the z 0 dependence is weak (Briggs, 1982), we will examine whether this intermediate behavior can be described by a simple interpolation of the limiting expressions.

In a previous paper (Venkatram, 1982), we showed that the formulation for ground-level concentrations under stable conditions could be described by interpolating between the asymptotic descriptions of dispersion under neutral and stable conditions. We showed that the interpolation de- scribed the data from the Prairie Grass experiment (Barad, 1958). For the sake of completeness, we present the resulting expressions:

C~,=x~ 1 for x,~<l.4, (27a)

Short Communication 949

and

C~=0.89x, 2/3 for x ,> l . 4 . (27b)

The values of u, and L used in these expressions were taken from van Ulden (1978) who derived these variables from the Prairie Grass data. In this study, we analyzed the unstable data from the Prairie experiment using the neutral {Equation (11)} and the unstable {Equation(17~} asymp- totes. Figure 1 shows the data plotted in terms of C', and x,. We see that for x, ~< 10, (7' follows the neutral x , 1 behavior, while beyond x, = 10, we notice the approach to x , 2 unsta- ble asymptote. An adequate interpolation, between these two limits, that fits the data is

1 C ' , = x , ( l + o t x 2 ) l / 2 , ~t=6.0xl0 -3. (28)

Briggs (1982) derives an expression that is probably more accurate. However, we see that Equation (28) provides a more than adequate description of the Prairie Grass data, and at the same time retains the simplicity of the asympt~ic limits.

At large x, . C~, is approximately given by the expression

c~,=13/xL I29)

4, DISCUSSION

10

.01

l t-- ]

.001 .1 1 10 1 0 0 1 0 0 0

X*

Fig. 1. Plot of cross-wind integrated concentration against distance for the unstable runs of the Prairie Grass Experi- ment (_Barad, 1958). Data obtained from van Ulden (1978). C ", = C" u_, I LI /Q and x, = x / I L I. The curve corresponds to

Cr, = l / x , (1 "~0[X2) 1/2 where ct=6 × 10 -3.

The relationship between the growth of vertical spread and the eddy diffusivity of mass {Equation (2)} provides a convenient tool for the examination of dispersion in the limits of neutral, stable, and unstable conditions. Assuming that the eddy diffusivity of mass is similar in form to that of heat, we can show that dispersion at these limits of stability can be described in terms of the downwind distance, x, the surface friction velocity, u,, and the Monin-Obukhov length, L. These expressions are simple enough for practical applications.

The most significant result of our analysis is that the 'free convection' limit for dispersion might not be valid in the surface boundary layer. Under very unstable conditions, the concentration decreases as x- 2 rather than x- 3/2 as predic- ted by the free convection analysis.

Nieuwstadt (1980) shows that the x- 3/2 decrease holds for a range of distances he believes are governed by free convec- tion. However, a closer examination of his data plot indicates that at small x, there is a region of x- 1 behavior, while at large distances, the concentrations are described by x -2. Thus, the x -3/2 behavior might represent a transition be- tween the neutral and unstable asymptotes. Furthermore, his use of the wind speed at 8 m to represent the 'constant' wind speed in the free convection layer is likely to introduce uncertainty in his analysis.

The result on the concentration decrease in unstable conditions is consistent with the 'anomalous' behavior of the variance of the vertical velocity fluctuations and the temper- ature and velocity gradients in unstable conditions. These variables aro described by functions of ( z /L) that do not approach their ekpected asymptotic limits under unstable conditions. Tennekes (1973) and Businger (1973b) provide tentative explanations for these results. Tennekes (1973) shows that the functional forms of the temperature and velocity gradients suggest that the time-scale governing the return to isotropy of eddies generated by buoyancy is pro- portional to z and inversely proportional to u,.

Because K n is proportional to this time-scale, a decrease in u, will increase K H, and thus tr z as predicted by Equa- tion (15). We do recognize that this is little more than a rationalization of the observed behavior of a z. However, the lack of a good explanation is no reason for rejecting the facts on a s, even though it does not fit the 'convenient' free convection theory.

REFERENCES

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Briggs G. A. and McDonald K. R. (1978) Prairie Grass revisited. Optimum indicators of vertical spread. Proc. 9th Int. Technical Meeting on Air Pollution Modeling and its Application, No. 103, NATO/CCMS, 209-220.

Briggs G. A. (1982) Similarity forms for ground-source sur- face-layer diffusion. Boundary-Layer Met. 23, 489-502.

Briggs G. A. (1988) Analysis of diffusion field experiments. In Lectures on Air Pollution Modeling (edited by Venkatram A. and Wyngaard J. C.), pp. 63-118. American Meteoro- logical Society, Boston, MA,

Businger J. A. (1973a) Turbulent transfer in the atmospheric surface layer. In Workshop on Micrometeorology (edited by Haugen D. A.), pp. 67-100. American Meteorological Society, Boston, MA.

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