Practical statistical models of environmental pollution

Mathl. Comput. Modellzng Vol. 21, No. 9, pp. 11-14, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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Practical Statistical Models of Environmental Pollution P.

C. CHATWIN, D.

M.

LEWIS

AND

N.

MOLE

School of Mathematics and Statistics, University of Sheffield P.O. Box 597, Sheffield S3 7RH, England

Abstract-The paper summarises recent theoretical work on turbulent diffusion that is baaed on a simple extension of exact results for the ideal case when there is no molecular diffusion. The results are consistent with data and appear to provide a new and promising approach for practical models that can be used in real time. Keywords-Turbulence, moments for concentration.

Atmospheric

dispersion, PDF modelling, Intermittency

factor,

Central

1. INTRODUCTION The way in which pollutants disperse in the natural environment is dominated turbulent diffusion. If I’(x, t) is the concentration of a pollutant not undergoing

by the process of chemical change,

then dr dt + u.vr

= KVV,

(I)

where IE is the molecular diffusivity and U = U(x,t) is the velocity field, determined by the Navier-Stokes equations and mass conservation. For pollutants undergoing chemical change, the processes of molecular diflusion and advection represented in (1) are still important, but other terms need to be added; this paper does not consider such pollutants. In almost all fluid flows (atmosphere, ocean, river, estuary), U is a random function of x and t, i.e., it is turbulent. It follows from (1) that I is also a random function, with statistical properties determined by IE, and by those of U. In view of the practical importance, and the scientific fascination, there has been much research on these statistical properties over the last 100 years or so. Despite this, the problem remains an unsolved one; the reasons for this are discussed

in [1,2].

2. SOME

PROPERTIES

The probability density function (pdf) defined by

OF THE CONCENTRATION of concentration

246 x, t) = -${prob[I(x,

will be denoted

t) 5 41,

PDF

by p(B; x, t), and is

(2)

We thank W. Zimmerman for his help. All our work on turbulent diffusion has benefited immensely from a long-standing collaboration with P. Sullivan of the University of Western Ontario. The financial support of the Ministry of Defence (UK), Health and Safety Executive (UK) and the Commission of the European Communities in various projects is gratefully acknowledged.

Typeset by AA&-W 11

12

P. C. CHATWIN

et al.

for 0 > 0. The equation governing p can be obtained from (1) and is discussed in [2]. There has been much emphasis on the (ensemble) mean concentration ~(x, t) defined by

Jrn ep(e;

Ax,t> =

t)

X,

de;

0

unfortunately, it has too often been forgotten that l? is a random variable and that of its statistical properties. In fact, o/p is never small, where ~(x, t) is the standard

~1 is but one deviation of

l?(x, t) and 9(x, An underlying “weak”

theme

process,

fluid element.

t) =

of this paper

it is the only agency For clarity,

consider

e2p(e;x, t) de is that,

although

which

changes

the common

- p2(~, t).

molecular

(4)

diffusion

is in some respects

the value of l? associated

practical

situation

a

with any given

when the concentration

of

pollutant at its sotirce (whether steady, or time-varying) is uniform (i.e., independent of position within the source) with value 81 and when, also, the pollutant is discharged into an ambient fluid where the concentration of pollutant is initially zero. In the hypothetical situation when there is no molecular diffusion, denoted below by a subscript zero, the concentration in any fluid element is either

81 or zero. Hence,

p(ei

x,

t) =

po(e; x, t) = (1 - T(~, t)) s(e) + 71(x,t) 6(e -

w,

(5)

where 7~is the intermittency factor in this hypothetical case and is, therefore, determined entirely by the statistical properties of the velocity field U and by the source geometry. It follows from (3), (4), and (5) that

14x, t) = P~(x,t) = dl, Three

(9

important

results

g2(x, t) =

&x,

t) = ~(1 - T) ep

=

po (el -

po).

(6)

are: ~(0; x, t) can be expressed

See [3]. The pdf in the real situation

exactly

in terms

of 7r. In

fact, p(e; x, t) = (1 - r(x, t))p,(e;

x, t) + ~(x, t)Ps(e;

x, t),

(7)

where p, and p, are conditional pdfs, conditional respectively on the point x and the time t being occupied by ambient or source fluid. the relationship between D’ and p is a simple generalisa(ii) See [4]. In many circumstances, tion of that in (6), viz. us 1 p$

(

o/JO) -

P> ’

(8)

where p(O) is a local scale for /J (e.g., the centre-line mean concentration in the plume from a smoke stack), and Q and ,D are positive parameters. For the data available (all from self-similar laboratory flows) when (8) was proposed, QI and p were constant [4] but it has since been shown that (8) h as wider applicability with a and ,0 varying with x (and perhaps t) [a]. extended to higher moments in [4], (iii) See [4,5]. The arguments leading to (8) were tentatively and it was subsequently shown in [5] that these extensions gave relationships between the non-dimensional shape parameters skewness S, kurtosis, K, and their higher-order analogues, that were (a) simple, and (b) independent of cx and p. The relationship between S and K is [5] K=S2+1.

(9)

Environmental

Pollution

13

Other algebraic relationships such as those between S and II, or K and a/p etc., are also simple, but involve CYand/or 0. The theory, therefore, predicts that a graph of K versus S should collapse onto a single curve whereas, for data where a and ,L3can be expected to vary-the normal situation, a graph of K versus o/p (for example) will not collapse. Examination of many datasets has confirmed these predictions [5], although the curve onto which the (K, S) datapoints collapse lies somewhat above that given by (9). (For datapoints from dosages, rather than concentrations, the collapse is even more remarkable [5], and closer to (9), but that is another story!)

3. IMPLICATIONS

FOR PRACTICAL

MODELS

We believe that it is inevitable that, one day, all practical models of environmental pollution will be inherently statistical. applied.

Several such models, e.g. [6], have already been proposed and

However, given the heavy computing time that tends to be associated with all work

involving turbulence, simple models of acceptable accuracy have a great advantage. It is therefore suggested that the ideas and results summarized above merit further investigation. The following brief comments are tentative (since they are based on work in progress) and complement those in [5]. It is shown in [5] that (9) holds exactly only when both p, and p, in (7) are delta functions; otherwise K > S2 + 1. The implication of collapses like those shown in [5] to curves near and somewhat above K = S2 + 1 is therefore that both pa and ps are close (in some sense) to delta functions. This is consistent with the theory in [4] that led to (8) and, later in [5], to (9). Let pu,, ~2, S,, and K, be the mean, variance, skewness and kurtosis of pa in (7) with an analogous notation for the corresponding parameters of p,. It follows from (7) after some algebra that /L = pa + nA, where A = pL,- pa, and that c2 = n(l-

7r)A2 + ~0; + (1 - n)a$

(lOa)

Sa3 = ~(1 - 7r)(l - 2n)A3 + 37r(l - r)A (0; - 0;) + {7&a;

+ (1 - +%a:}

1

(lob)

Kc4 = ~(1 - 7r) (1 - 37r+ 3~~) A4 + 6n(l - 7r)A2 { (1 - +; + 4~(1-

r)A (&CT:- &a;) + {nKsa,4 + (1 - r)K,a:}

+ ,cri} (1Oc)

Very near the source, before there has been time for molecular diffusion to act, pa z 02 z g: = 0, PL, z 81 for all x, and the pdf (7) reduces to (5). x(1 - $A2 M n(l - X) @, and SZ

(1 - 27r)

7#(1

-7+/z

Thus, from (lOa) to (IOc), u2 =

K ~ (1 - 37r+ 37r2) 7r(l -n)

.

(11)

From (11)) it is easily shown that, as expected, K = S2 + 1. If, at all distances downstream and all times, A2 stayed much greater than 0% and a:, (ll), and hence (9), would remain good approximations everywhere. But this cannot be true. As a result of molecular diffusion, g,” and 02 become nonzero, and both A = CL,- pa and 7r must approach zero as 1x1and t become large enough. There must, it appears, inevitably be a switch-over to a situation where the dominant term in the expression for g2 in (lOa) is ~7:; likewise, the dominant terms in (lob) and (10~) are eventually S,ai (unless S, = 0) and K,az. The overall statistics are dominated by the molecular diffusion of pollutant into initially ambient fluid. Then, provided ~~IT(T~/cT~ is sufficiently small, it can be shown that (K - S2) z (K, - Si) (1 + n).

(12)

To explain the data collapses, it therefore appears to be necessary (and perhaps sufficient) to find a family of pdfs p, with the following properties: - pa) to satisfy (8)-note that ,!L~, is a function of x and (in general) t; (i) ~2 ZZ/32&q&? (ii) (K, - S,“) is approximately independent of pa-to ensure collapse, using (12);

P. C. CHATWIN et al.

14

(iii) as the parameters of p, vary, a large spread of values of S, and K, must be generatedsince such a spread has been observed in all datasets that have been examined. Requirement (iii) rules out “simple” distributions (uniform, normal, exponential, . . . > for which S, and K, are universal constants. Nevertheless, there are many other candidate distributions (beta, gamma, . . . ) which are now being examined. Provided the search is successful, (7) gives p GZp, except very near the source. In practice, all the statistical properties of I’ can then be determined in terms of pa-and

this is relatively easy to predict. 4.

DATA ANALYSIS

No theoretical model of turbulent diffusion can be accepted until it has been successfully compared with data. The work on statistical models has highlighted some important problems that need attention in data analysis. These include the effects of instrument smoothing (important because of fine-scaIe intermittent structure}, and noise removal and base-line drift (important because low concentrations near ,LL~ are eventually predominant).

Recent work is described in [7,8].

REFERENCES 1. P.C. Chatwin and P.J. Sullivan, Quantitative models for environmental prediction: A review, In faxing and Transport in the Environment, (Edited by K.J. Beven, P.C. Chatwin and J. Millbank), pp. 353-369, Wiley, (1994). 2. N. Mole, P.C. Chatwin and P.J. Sullivan, Modelling concentration fluctuations in air pollution, In Modellzng Change m Environmental Systems, (Edited by B. Beck, A.J. Jakeman and M. McAleer), pp. 317-340, Wiley. (1993). 3. P.C. Chatwin and P.J. Sullivan, The intermittency factor of scalars in turbulence, Physics of Fluids Al, 761-763 (1989). 4. P.C. Chatwin and P.J. Sullivan, A simple and unifying physical interpretation of scalar fluctuation measurements from many turbulent shear flows, Journal of Flzlid Mechanics 212, 533-556 (1990). 5. N. Mole and E.D. Clarke, Relationships between higher moments of concentration and of dose in turbulent dispersion, Boundary-Layer Meteorology (to appear) (1995). 6. A.J. Jakeman, R.W. Simpson and J.A. Taylor, Modeling distributions of air poliutant concentrations: III The hybrid deterministic-statistical approach, Atrnos~~e~c ~n~zronrne~~ 22, 163-174 (1988). 7. I%. Mole, Some intersections between turbulent dispersion and statistics, ~~~~ronrne~~~ 1, 179-194 (1990). 8. D.M. Lewis and P.C. Chatwin, The treatment of atmospheric dispersion data in the presence of noise and baseline drift, Boundary-Layer Meteorology (to appear) (1995).