Passive Dispersion from Steady Sources in a Turbulent Environment

211

CHAPTER 9 PASSIVE DISPERSION F R O M STEADY SOURCES IN A TURBULENT

The problem of interest

ENVIRONMENT

is the

passive tracer) in a turbulent

spreading of a contaminant

flowfield

(or

such as the atmospheric wind.

It i s a d v e c t e d w i t h t h e flow a n d s p r e a d s l a t e r a l l y a n d v e r t i c a l l y a s a result o f turbulence in the e n v i r o n m e n t . T h e m o l e c u l a r diffusion is generally

small and

can

be

neglected, but

the

effects

of initial

m o m e n t u m a n d b u o y a n c y can b e important a n d will b e discussed in Chapter

10. For the p r o b l e m c o n s i d e r e d the m e t h o d s d i s c u s s e d

in

Chapter 8 can be used to calculate the concentration as function of d i s t a n c e f r o m t h e s o u r c e . T o t h i s e n d , it i s n e c e s s a r y t o e s t a b l i s h t h e relevant values of the apparent the directions normal to the

turbulent diffusivities K

y

flow;

and K

z

the contribution from K

in

to the

x

c o n t a m i n a n t transport is a s s u m e d negligible in c o m p a r i s o n w i t h that from advection. T h e m e t h o d described is general a n d w o u l d apply to releases in any turbulent

flow;

application

ocean, river or the test

of primary

interest,

flow

of a w i n d tunnel. The

however,

is

the

spreading

of

pollutants in the atmosphere. A l t h o u g h the K-theory is widely used for

this purpose,

the

complexity of the

flow

in the

atmospheric

boundary layer m a k e s our

simplifications questionable. Even

seemingly safe a s s u m p t i o n

o f negligible streamwise diffusion m a y

the

h a v e t o b e r e c o n s i d e r e d for flows o v e r c o m p l e x t e r r a i n , b u t p r o b l e m s arise

also

classical

for

flat

approach

terrain by

in

nominally steady

Pasquill

and

Gifford

conditions.

(see Turner

The

1973)

a t t e m p t e d t o e s t a b l i s h d i s p e r s i o n c o e f f i c i e n t s v a l i d for b r o a d c l a s s e s of p r o b l e m s , thereby m a k i n g dispersion analysis practical b u t at the expense of details a n d accuracy. The publication of Turner's Workbook

(1973) made

dispersion

a n a l y s i s s i m p l e a n d p r a c t i c a l a l s o for u s e r s n o t t r a i n e d a s b o u n d a r y layer meteorologists. T h e m e t h o d s

described therein

are

n o w so

widely u s e d a n d h a v e p r o v e d to b e so useful to so m a n y , that recent critique that the b o o k is t w e n t y y e a r s o u t of date is n o t likely to r e m o v e it f r o m c i r c u l a t i o n . W e n o t e h e r e t h e d i f f e r e n t r e q u i r e m e n t s p e r t a i n i n g t o p r e d i c t i o n m o d e l s for a t m o s p h e r i c d i s p e r s i o n . F o r r a p i d screening of the c o n s e q u e n c e s of an accidental release, approximate but simple m e t h o d s c a n b e v e r y useful. M e t h o d s w i t h h i g h accuracy are

meaningful

only

when

all

the

input

can

be

specified

with

Chapter 9

212

precision, an

impossible requirement

for a c c i d e n t a l r e l e a s e s .

For

r e g u l a t o r y p u r p o s e s , o n t h e o t h e r h a n d , it w i l l b e p o s s i b l e t o c a l c u l a t e the dispersion of a given release quite accurately b a s e d on k n o w n emission rates and local w e a t h e r statistics. For this purpose w e see n e w methods coming into use. I n w h a t f o l l o w s w e w i l l d i s c u s s first t h e d e v e l o p m e n t s l e a d i n g u p to the K-theory w i t h s o m e applications. In the last section w e will r e t u r n t o t h e c r i t i q u e o f t h i s a p p r o a c h a n d t h e p r e s e n t o u t l o o k for improvements.

9.1

T h e Problem of Turbulent Diffusion

Let us consider the p r o b l e m sketched in Fig. 9 - 1 . A steady source Q emits a contaminant into a turbulent stream. For our purposes w e m a y consider the emitted material as small particles w h i c h follow w i d e l y different a n d

irregular

trajectories

as

indicated. They

are

Collector

x

Wind

Concentration

Trajectories

L

u Fig.

9-1:

X

u

Particle trajectories and track angles experiment in turbulent diffusion.

X

for

hypothetical

Passive Dispersionfrom Steady Sources (Turbulent)

213

collected at a station l o c a t e d a d i s t a n c e L d o w n w i n d . If the

number

and radial positions of the collected particles are recorded, w e obtain after

long

time

a

distribution

as

indicated.

The

hypothetical

e x p e r i m e n t is m e a n i n g f u l o n l y a s l o n g a s t h e w i n d a n d t h e t u r b u l e n t properties

do not

change

during

the

s a m p l i n g p e r i o d . T h i s is

problem in a w i n d tunnel, but in the atmosphere

the conditions

o f t e n n o t fulfilled. I n t h e w i n d t u n n e l o n e w o u l d e x p e c t a nearly

symmetric

about

the

center

axis

inasmuch

no are

distribution as

the

flow

properties are statistically the s a m e in all m e r i d i a n planes. In

the

a t m o s p h e r e this is n o t the case. T h e velocity is nearly constant in the lateral direction but large distances statistical

v a r i e s significantly in the vertical at

from the ground. T h e turbulence,

not-too-

as m e a s u r e d

averages, depends therefore on position, and w e can

expect a symmetric distribution

by not

downstream.

T h e discussion h a s t o u c h e d o n s o m e properties w h i c h are u s e d to c h a r a c t e r i z e t u r b u l e n t f l o w f i e l d s ; first, s t a t i o n a r i t y , i . e . t h e

statistical

properties should not change with time, and homogeneity,

neither

s h o u l d they d e p e n d o n position a n d finally, isotropy, w h i c h i m p l i e s invariance to c h a n g e s in direction. The

diurnal

obviously

in

(day and conflict

night) variations

with

the

in the

stationarity

atmosphere

are

requirement,

and

h o m o g e n e i t y is c l e a r l y i n f l u e n c e d b y t h e t o p o g r a p h y . B u t i n r e s t r i c t e d p e r i o d s o f t i m e ( d u r a t i o n h o u r s ) , a n d o v e r flat t e r r a i n t h e f l o w c a n b e considered stationary a n d h o m o g e n e o u s . Isotropy on the other hand, is rarely possible inside the g r o u n d b o u n d a r y change in w i n d velocity, but

layer w i t h its

it i s o f t e n a s s u m e d

for s c a l e s

rapid much

s m a l l e r t h a n t h e d i s t a n c e to t h e g r o u n d ( l o c a l i s o t r o p y ) . If w e k n o w the s o u r c e strength,

the w i n d velocity, the distance L

and the resulting particle distribution,

w e could determine the K ' s

from the o b s e r v e d s t a n d a r d deviations a

y

discussion

and a

z

as indicated in the

of the diffusion equation in the preceding chapter. ( W e

considered here only the symmetric case implying isotropy.) T h e m e t h o d , a s p r o p o s e d , w o u l d b e n e i t h e r p r a c t i c a l n o r e l e g a n t . It is natural

to ask

if one could replace downstream

location

measurements with

a

of

single

the

at

measurement

at t h e s o u r c e . T h e quantities to b e m e a s u r e d w o u l d b e

the fluctuations

a

the

distribution

point

in w i n d direction and the m e a n w i n d velocity. O n e

w o u l d feel i n t u i t i v e l y t h a t t h e r e o u g h t t o b e a r e l a t i o n b e t w e e n wind characteristics

at the source at a g i v e n time, a n d the

the

resulting

lateral t r a n s p o r t o f e m i t t e d m a t e r i a l d o w n w i n d at a later t i m e . T h i s is another character

w a y of saying that the turbulent eddies maintain

their

as they are a d v e c t e d d o w n w i n d . T h e idea is formalized in

Chapter 9

214

meteorology as Taylor's hypothesis w h i c h expresses the equivalence of a point m e a s u r e m e n t

of a turbulent quantity

i n t i m e At t o t h e

s p a t i a l v a r i a t i o n o f t h e s a m e q u a n t i t y o v e r t h e l e n g t h L = U At w h e r e U is the m e a n w i n d speed (Stull, 1988). T h e variations in local w i n d directions can b e m e a s u r e d b y m e a n s of the so-called bivanes w h i c h give the instantaneous deviations from the

mean

wind direction

in

the

lateral

and

vertical plane.

The

standard deviations of these directional variations can be evaluated; for s m a l l a n g l e s a a n d 0, t h e y a r e r e l a t e d t o t h e s t a n d a r d d e v i a t i o n s of the local velocities v a n d w as follows:

For

very

short

distances

downstream,

the

following

linear

relationships appear obvious (Fig. 9-1): °y = a a

To

develop a

relationship

>

x

valid

a

z

=

6

(~)

x

9

for l o n g e r

Taylor (1921, see Panofsky and Dutton, "Diffusion

a

distances

or

1984) suggested his

Theorem" which assumes homogeneous

and

2

time, famous

stationary

turbulence. For the lateral dispersion, these conditions are better m e t t h a n for t h e v e r t i c a l . I f t h e s t a n d a r d d e v i a t i o n o f t h e l a t e r a l direction

a

a

i s k n o w n , (it c o u l d b e m e a s u r e d

wind

at a n y point in this

restrictive case), the s t a n d a r d d e v i a t i o n (or w i d t h ) of the d o w n w i n d lateral distribution is given approximately b y Pasquill (1977): o

y

= a XJ{T)

(9-3)

a

w h e r e T is the diffusion time, X the d o w n w i n d d i s t a n c e

reached

in

t i m e T a n d / (T) a f u n c t i o n t a b u l a t e d b e l o w : Table 9-1:

PasquilVsfunctionf(T).

X[km]

0

0.1

0.2

0.4

1.0

2.0

4.0

10

>10

f(T)

1

0.8

0.7

0.65

0.6

0.4

0.4

0.33

0.33 V l O / X

( A s t h e w i n d v e l o c i t y v a r i e s r a p i d l y u p w a r d s , X (T) w i l l d e p e n d o n h e i g h t . O n a s s u m i n g a c o n s t a n t w i n d s p e e d , w e c a n r e p l a c e X (T) w i t h x, the coordinate aligned w i t h the w i n d vector.)

Passive Dispersionfrom Steady Sources (Turbulent) Although turbulence

the

assumption

of

homogeneous

215

and

stationary

is e v e n m o r e q u e s t i o n a b l e in the v e r t i c a l direction,

the

r e l e v a n t r e l a t i o n s h i p s b e t w e e n t h e d i r e c t i o n a l v a r i a b l e OQ a t a p o i n t and

the

vertical diffusion

Panofsky and Dutton,

scale

o , z

have been

worked out

(see

1984, p. 2 4 1 ) . W h e n e v e r local

measurements

are available, s u c h relations are v e r y useful since the

measurements

take

into

account

all the

special circumstances

pertaining

to

a

specific site, s u c h as t o p o g r a p h y , r o u g h n e s s a n d stability (yet to b e discussed). In w h a t follows w e will consider primarily the alternate

approach,

t h a t o f u s i n g s e m i - e m p i r i c a l r e l a t i o n s h i p s , v a l i d for b r o a d c l a s s e s o f problems.

Considerable

efforts

have

been

expended

by

highly

c o m p e t e n t meteorologists to measure, process and classify the data which

are

now

in

widespread

use

for

atmospheric

dispersion

e s t i m a t e s . W e w i l l d i s c u s s t h e i r u s e after w e h a v e d e t e r m i n e d w h a t information is n e e d e d to predict the d o w n w i n d c o n c e n t r a t i o n o n the basis of the diffusion equation.

9.2

T h e Turbulent Diffusion Equation for a Steady Source

T h e equation of interest is obtained b y omitting the unsteady in Eq. (8-28). A further

term

simplification is possible b y neglecting the

turbulent diffusion in the direction of the w i n d in c o m p a r i s o n w i t h the w i n d - d r i v e n transport. A l l the material emitted is a s s u m e d

to

c o m e from a n u p s t r e a m s o u r c e (or several s o u r c e s ) a n d n o additional sources or sinks are present in the d o m a i n of interest. T h e general flow situation is depicted in Fig. 9-2. T h e equation to b e solved is

dx

dy

ay

dX dz

dz

(9-4)

T h e boundary conditions require that the concentration

vanishes

at large distances from the s o u r c e ( s ) . In addition w e m u s t satisfy the global continuity condition that the flux of emitted material,

through

a n y c r o s s - p l a n e x = c o n s t . , m u s t e q u a l t h e s t r e n g t h o f t h e s o u r c e Q. S o m e difficulties associated w i t h this equation h a v e a l r e a d y b e e n discussed. unknown,

The but

turbulent

diffusivities

K

y

and

likely to b e c o m p l i c a t e d functions

K

z

are

not

only

of position

and

environmental factors; weather, topography, ground roughness

etc.

T h e m e a n w i n d U is not constant,

but

varies significantly in

the

v e r t i c a l d i r e c t i o n b o t h i n m a g n i t u d e a n d d i r e c t i o n . B u t it i s p o s s i b l e

Chapter 9

216

Fig.

9-2;

Steady ground the wind.

source

in coordinate

system

aligned

with

to i n t r o d u c e a p p r o x i m a t i o n s w h i c h m a k e s the task o f s o l v i n g equation

much

easier without

sacrificing too m u c h

in

terms

the of

a c c u r a c y or validity. T h e first a p p r o x i m a t i o n i s t o a s s u m e t h a t t h e m e a n w i n d v e l o c i t y is constant. T h e m e a s u r e m e n t s

of w i n d characteristics are normally

a v a i l a b l e a t a h e i g h t o f 10 m . I t a p p e a r s

reasonable

to u s e

the

a v e r a g e d v a l u e a t t h i s h e i g h t for s o u r c e s n o t t o o far f r o m t h e g r o u n d . A t greater heights an improved value could b e obtained b y m e a n s of k n o w n c o r r e l a t i o n s for t h e w i n d p r o f i l e , s u c h a s t h e s i m p l e p o w e r l a w or the l o g a r i t h m i c velocity profile familiar from

boundary-layer

theory, (Panofsky a n d Dutton, 1984). T h e error in the velocity will b e partially compensated in the semi-empirical exchange K

y

coefficients,

a n d K , a s t h e s e a r e e v a l u a t e d u s i n g t h e s a m e a p p r o x i m a t i o n for z

the w i n d speed. B y assuming that the K ' s are independent of y and z, Eq. (9-4) b e c o m e s linear a n d m o r e o v e r is satisfied b y the G a u s s i a n

distribution

f u n c t i o n f a m i l i a r f r o m t h e l a m i n a r t h e o r y o f C h a p t e r 8. W e r e w r i t e the governing equation as follows:

(9-5)

Passive Dispersionfrom Steady Sources (Turbulent)

217

with the generic solution in the form of the product of two Gaussian distributions, in the y - a n d z-directions respectively, given b y

exp

2

°y°z

exp

\a

(9-6)

2 Or

t

T h e v a r i a n c e s o f these distributions are related to the K ' s as follows: 2K

a

2K

=

— r

• a

=

x

(9-7)

T h i s c h a n g e f r o m K t o a* i s n o t j u s t a c h a n g e i n n o t a t i o n . It i s v e r y t

difficult t o o b t a i n a c c u r a t e d i s p e r s i o n d a t a i n t h e a t m o s p h e r e . It i s also v e r y costly to obtain m a n y point m e a s u r e m e n t s W h e n a set of v a l u e s of x measurements,

i

n

space

has

been

the corresponding values of K

t

simultaneously.

obtained

through

can be found only b y

numerical differentiation, a very inaccurate procedure. T h e a, t

on the

o t h e r h a n d , c a n b e f o u n d b y fitting a G a u s s i a n c u r v e t h r o u g h t h e d a t a . F r o m the viewpoint of numerical problem. The constant

accuracy,

this is a m u c h

C is d e t e r m i n e d from the global

easier

continuity

condition; the flux of x through a n y cross-plane x = constant,

must

equal the rate at w h i c h material is e m i t t e d at the source:

9 O n s u b s t i t u t i n g x (x,y,z)

•fir-"'

dy

dz

(9-8)

from ( 9 - 6 ) into the double integral, w e obtain

the solution o f interest:

x{x,y,z)

=

Q jtUO O y

exp

z

W

2

°;

(9-9)

N o t e t h a t i n t h e i n t e g r a t i o n o f t h e flux, E q . ( 9 - 8 ) , t h e r a n g e i n z i s from z = 0 to » . T h i s takes into account the perfect reflection from the g r o u n d plane, n o material is d e p o s i t e d or absorbed. A s written, the

solution

represents

a

ground

source

only,

w e will

consider

e l e v a t e d s o u r c e s a n d t h e i r r e f l e c t i o n s l a t e r i n t h i s c h a p t e r . B u t first

Chapter 9

218

w e will discuss the K ' s w h i c h determine the widths, a

y

a n d a , of the z

p l u m e in the lateral a n d vertical directions. T h e m o s t obvious choice w o u l d b e to a s s u m e constant values of K

y

and K , w h i c h according to Eq. (9-7) w o u l d give p l u m e

growth

z

proportional to V x \ i.e. a parabolic form. T h i s is not in a c c o r d w i t h the p l u m e forms o b s e r v e d . D i s p e r s i o n p l u m e s g r o w m o s t often m u c h faster

with x

particularly

t

in u n s t a b l e air.

The case K = c o n s t ,

c o r r e s p o n d s p h y s i c a l l y t o a s i t u a t i o n w h e r e all e d d i e s a r e o f t h e s a m e size, a n d hence the rate of g r o w t h constant as in laminar flow. In reality the atmospheric turbulence sizes, and

i s c h a r a c t e r i z e d b y e d d i e s o f all

as the p l u m e g r o w s , the larger eddies gain in relative

i m p o r t a n c e g i v i n g a h i g h e r t r a n s v e r s e r a t e o f g r o w t h . T h i s effect

can

b e reproduced in our theory b y m a k i n g the K ' s functions of x, the d o w n w i n d d i s t a n c e . A s i m p l e a n d a t t r a c t i v e c h o i c e w o u l d b e to m a k e K a power of x, but there is n o universal value of this power w h i c h w o u l d fit all e n v i r o n m e n t a l s i t u a t i o n s . T h e p a r t i c u l a r c h o i c e o f K (or equivalently o

y

and a ) , h i n g e s o n w h a t is d e n o t e d the z

atmospheric

"stability". T h e t e r m stability atmosphere. enhanced

refers to the turbulent exchange processes in the

These are

suppressed

w h e n the atmosphere

under

stable

is unstable.

conditions

and

The general ideas of

stability, as related to density gradients in the vertical direction, w e r e discussed

in Chapter 2. W i t h r e g a r d to p l u m e f o r m b o t h the w i n d

s p e e d a n d its g r a d i e n t ( t h e w i n d s h e a r ) a r e i m p o r t a n t . W i t h i n c r e a s i n g w i n d s p e e d the p l u m e b e c o m e s m o r e elongated or slender, a n d m o r e rapidly

diluted

turbulence.

irrespective

of

the

nature

of the

atmospheric

But the w i n d also influences the turbulence.

At wind

speeds in excess o f 6 m / s , the stability is determined b y the w i n d . T h e interaction of the w i n d w i t h vegetation, buildings a n d

topography,

also gives rise to turbulent e x c h a n g e processes w h i c h can b e

taken

into account only crudely in any theory. A nice physical discussion of the atmospheric processes w h i c h control the turbulent exchange, with r e f e r e n c e t o p l u m e b e h a v i o u r , i s g i v e n b y S t u l l ( 1 9 8 8 , C h a p t e r 1). A very useful compilation o f data w i t h brief discussions,

can also be

found in Turner's W o r k b o o k (1973). T h e W o r k b o o k m a k e s use of the c l a s s i f i c a t i o n s c h e m e for a t m o s p h e r i c s t a b i l i t y p r e p a r e d b y P a s q u i l l and

Gifford.

The

atmospheric

conditions

of interest

are

divided into

five

c a t e g o r i e s , A , B , C , D , E a n d F, w h e r e A i s t h e m o s t u n s t a b l e ( s t r o n g thermal

convection),

D represents

the

neutral

condition

(purely

m e c h a n i c a l turbulence), a n d F the stably stratified case w h e r e mechanical turbulence

the

is strongly d a m p e d . T h e v a r i o u s categories

Passive Dispersion from Steady Sources (Turbulent)

219

a n d their characteristics, are t a b u l a t e d in T a b l e 9-2 ( r e p r o d u c e d

from

Panofsky and Dutton, 1984). Table 9-2; Key to Pasquill categories. Day Surface wind speed (at 10 m) (m/s)

Night

Incoming Solar Radiation

< 2 2 - 3 3 -5 5 -6 > 6

Strong

Moderate

Slight

Thinly Overcast or * 4/8 Low Cloud

A A- B B C C

A- B B B -C C -D D

B C C D D

E D D D

Clear or ^ 3 / 8 Cloud

F E D D

Note: The neutral class, D, should be assumed for overcast conditions during day or night.

For

each

variation

category,

of a

prepared the

y

and

a

Pasquill z

as

and

function

diagrams shown

Gifford

determined

of downwind

the

"best"

distance

x

and

in Fig. 9-3. W i t h this information

it

b e c o m e s possible to calculate the d o w n w i n d concentration, due to a s t e a d y s o u r c e , for a w i d e r a n g e o f c o n d i t i o n s .

Table 9-3: Formulas for a (x) and a (x) y

Pasquill Type

z

(10? < x < 10 m). 4

a (m)

a

a (m)

y

z

Open-Country Conditions A

0.22 x (1 + 0.0001 x)'

B

0.16 x (1 + 0.0001 x) '

C

1

/

2

0.20 x 0.12 x

1

/

2

0.11 x (1 + 0.0001 A : ) '

1

/

2

0.08 A : (1 + 0.0002 x ) "

/

2

0.06 x (I + 0.0015 * ) "

1

/

D

0.08 A: (1 +0.0001 x)'

1

E

0.06 x (1 + 0.0001 x ) '

1

/

2

0.03 x (1 + 0.0003 x ) "

F

0.04 x (1 + 0.0001 x ) '

1

/

2

0.016 x (1 + 0.0003 x ) "

1

/

2

2

1

1

Urban Conditions A -B

0.32 J C ( 1 + 0.0004 A : ) "

1

/

2

/

2

0.24 A: (1 + 0.001 x)

1

/

2

C

0.22 x (1 + 0.0004 A : ) -

D

0.16 A : ( 1 + 0.0004 A : ) "

1

/

2

0.14 x {I + 0.0003 x ) "

E-F

0.11 x ( l +0.0004 A : ) "

1

/

2

0.08 x (I +0.00015 A : ) "

a

f r o m Briggs (1973).

1

0.20 A : 1

/

2

1

/

2

Chapter 9

220

Fig.

9-3;

Pasquill and (b) vertical

distance/mm

source

[m ]

distancefrom

source

[m ]

Gifford's

dispersion

diagrams,

(a)

lateral;

Passive Dispersionfrom Steady Sources (Turbulent)

221

I n t h e a g e o f t h e P C t h e d i a g r a m s a r e p r o b a b l y l e s s useful t h a n t h e a n a l y t i c fits s u g g e s t e d b y B r i g g s ( 1 9 7 3 ) . T h e s e r e v i s e d d a t a

also

i n c l u d e v a l u e s r e l e v a n t for r o u g h ( u r b a n ) t e r r a i n . B r i g g s ' fits a r e g i v e n in Table 9-3, (reproduced from Panofsky a n d Dutton, 1984).

9.3

Plumes from Elevated Sources and Associated Phenomena

Problems

with

ground

pollution

can

be

alleviated simply

by

r e l e a s i n g the effluent at s o m e h e i g h t f r o m the g r o u n d . A familiar situation is illustrated in Fig. 9-4. T h e s m o k e from a stack is carried d o w n w i n d in the form of a growing p l u m e . A t a certain point plume

outline

concentration

touches

the

ground,

is o b s e r v e d further

and

an

increased

d o w n w i n d as

level

the of

result of ground

reflection. (The visible outline of a s m o k e p l u m e corresponds r o u g h l y to a ten per cent concentration level, Gifford, 1959.) W e c a n calculate t h e effect o f r e f l e c t i o n ( a n d s a t i s f y t h e b o u n d a r y

condition in

the

g r o u n d plane) b y placing an i m a g e source at an equal (negative) height b e l o w the g r o u n d as indicated in the figure. E q . ( 9 - 5 ) is linear, so the

Image

Fig.

9-4:

Stack plume and its reflection effective stack height is the sum and the added height AH.

in the ground plane; of the geometric height

the H g

Chapter 9

222

solution of interest

is the s u m

of the two contributions,

from

the

s o u r c e a n d its i m a g e , r e s p e c t i v e l y , i . e . :

X{x,y,z;H)

9

=

exp

2jta a U y

2

z

exp

1

\o,

tz-H^

2

o

+ exp

(9-10)

2

7

T h e h e i g h t H is h e r e t h e effective h e i g h t o f r e l e a s e e q u a l t o t h e s u m o f the geometric height H

g

a n d t h e a d d e d h e i g h t AH d u e to initial i m p u l s e

a n d b u o y a n c y . V a r i o u s empirical relations are available to calculate AH.

Examples

are

found

in

Turner's

W o r k b o o k as

Williamson

(1973). W e will discuss the

momentum

later.

P l u m e s f r o m tall s t a c k s e x h i b i t m a n y i n t e r e s t i n g a n d flow

phenomena.

D o w n w a s h from the

well

as

effects of b u o y a n c y

in and

unexpected

stack itself and

downdraft

from buildings a n d structures nearby, can lead to critical levels of c o n c e n t r a t i o n s c l o s e r to t h e g r o u n d t h a n e x p e c t e d . A n i n i t i a l v e r t i c a l v e l o c i t y , t w e n t y t o t h i r t y p e r c e n t h i g h e r t h a n t h e w i n d s p e e d , suffices t o c u r e t h e p r o b l e m i n m o s t c a s e s . I n difficult s i t u a t i o n s w i n d t u n n e l t e s t i n g c o u l d b e c a l l e d for to o b t a i n s a t i s f a c t o r y s o l u t i o n s . S o m e interesting p h e n o m e n a associated w i t h the diurnal cycle, are i l l u s t r a t e d i n F i g . 9 - 5 . T h e looping

is associated w i t h the

daytime

o c c u r r e n c e o f w a r m p o c k e t s o f air ("thermals") rising t h r o u g h atmosphere.

From

continuity,

these

are

accompanied

by

the

colder

downdrafts, a n d the result is visible in the f o r m of the w a v y outline o f t h e p l u m e . T h e n e t r e s u l t i s a v e r y r a p i d m i x i n g o f t h e effluent w i t h a t m o s p h e r i c air. ( B u o y a n t p l u m e s c a n a l s o o s c i l l a t e i n h e i g h t d u e t o stratification,

the basic p h e n o m e n a

and

the "buoyancy frequency"

w e r e discussed in Chapter 2.) A s the daytime heating of the g r o u n d ceases, n e w large eddies (generated b y the thermals) are n o longer formed, and the decays

into

smaller

and

smaller

eddies.

A

turbulence

"residual

eventually formed in w h i c h the turbulence has near equal

layer"

is

intensity

i n all d i r e c t i o n s . A p l u m e r e l e a s e d i n t h i s r e s i d u a l l a y e r w i l l s p r e a d i n all d i r e c t i o n s at t h e s a m e r a t e ; w e s e e t h e p h e n o m e n o n o f Later in the night, the residual layer is further

coning.

stabilized b y its

contact with the (cold) ground. T h e rate of spreading in the horizontal direction will in the e n d greatly e x c e e d that in the vertical, a n d w e see the p h e n o m e n o n of

fanning.

Passive Dispersion from Steady Sources (Turbulent)

223

Coning

Looping Stable

Mixing Stable

Fanning

Mixing Stable

Stable

New Mixing

Lofting Fig.

9-5;

Layer

Fumigation Plume flow phenomena the atmosphere.

associated

with diurnal

changes

in

A s this stable layer g r o w s from the g r o u n d up, a p l u m e released in t h e r e s i d u a l l a y e r a b o v e t h e s t a b l e l a y e r , w i l l g r o w faster u p w a r d . T h e d o w n w a r d dispersion is b l o c k e d b y the stable layer. In this case w e see the p h e n o m e n o n of

lofting.

A n o t h e r p h e n o m e n o n w h i c h is also c a u s e d b y the interaction of t w o l a y e r s , i s t h a t o f f u m i g a t i o n . T h i s o c c u r s , t y p i c a l l y , s h o r t l y after sunrise. The new well-mixed

layer w h i c h forms, serves to

bring

Chapter 9

224

d o w n w a r d s to the g r o u n d level, the pollutants trapped in the l a y e r s a b o v e . T h i s "old" p o l l u t e d m a t e r i a l

could have

stable

undergone

c h e m i c a l c h a n g e s w h i c h w o u l d a d d to its n u i s a n c e v a l u e . The situation of a well-mixed layer b e l o w and a stable layer above, w h i c h effectively p u t s a lid o n the h e i g h t o f the p l u m e , is l o o s e l y r e f e r r e d to a s a n i n v e r s i o n . ( M o r e p r e c i s e l y , a n i n v e r s i o n

represents

the case w h e n the temperature increases w i t h height.) In the presence of s u c h a n u p p e r l i d o n p l u m e g r o w t h , t h e d i s p e r s i o n a n a l y s i s m u s t b e modified. A r o u g h p r o c e d u r e is o u t l i n e d in T u r n e r ' s W o r k b o o k . T h e s o l u t i o n g i v e n b y E q . ( 9 - 1 0 ) i s u s e d d o w n w i n d o n l y to a d i s t a n c e x ' w h e r e the estimated value of o

z

is about half the height o f the stable

layer as measured from the p l u m e centerline. T h e concentration the

level of the

stable

layer, is t h e n a b o u t

centerline value. A t a d o w n w i n d distance

ten per

of 2 x

cent

of

at the

t h e flow i n t h e

"channel" b e t w e e n t h e g r o u n d a n d t h e l i d , i s a s s u m e d to b e w e l l m i x e d in the vertical w i t h a uniform concentration. Further dilution occurs only as a result of lateral p l u m e g r o w t h a n d the Gaussian applies only to the

distribution

y-direction.

Problems associated with inversions in valleys can be particularly severe a n d n o e a s y p r e d i c t i o n m e t h o d is available w h i c h c a n this

case.

More

detailed

discussions

on

air

pollution

and

solve the

properties a n d effects o f c o m m o n pollutants, will b e found in L y o n s and Scott (1990).

9.4

Line Sources, Area Sources and Approximations

T h e s o l u t i o n s d i s c u s s e d s o far r e p r e s e n t p o i n t s o u r c e s o n t h e g r o u n d or from e l e v a t e d stacks etc. T h e r e are m a n y situations o f practical interest, w h e r e the point-source m o d e l w o u l d b e u n a c c e p t a b l e for d i s p e r s i o n e s t i m a t e s . A l a r g e b u r n i n g d u m p w o u l d b e o n e e x a m p l e . A t l a r g e d i s t a n c e s it c o u l d b e r e p r e s e n t e d a s a p o i n t s o u r c e , b u t at d i s t a n c e s n o t m u c h g r e a t e r t h a n t h e w i d t h o f t h e d u m p , better m e t h o d s w o u l d b e called for. T h i s is a n e x a m p l e o f a n a r e a source with continuous (distributed) strength. A large area containing m a n y different sources, such as an u r b a n region, can also b e considered an area source although the emission really c o m e s from m a n y isolated (stationary or m o v i n g ) s o u r c e s . Line s o u r c e s c a n b e associated w i t h stretches of h i g h w a y or airport r u n w a y s , w h e r e the large n u m b e r of m o v i n g sources can b e converted to a constant line strength per unit length a n d time. The linearity of the

diffusion

equation

makes

it e a s y t o

find

Passive Dispersion from Steady Sources (Turbulent)

225

approximate a n s w e r s . B y replacing the c o n t i n u o u s area or line source with a large number

of point sources,

solution can be found.

In practice

a mathematically

accurate

only a few point sources

required to obtain an acceptable answer. ( T h e p r o b l e m of relevant dispersion coefficients remains,

the tabulated

are

finding

the

values have

b e e n v e r i f i e d o n l y for p o i n t s o u r c e s . ) For a line source of constant strength,

it i s s t r a i g h t f o r w a r d

to

obtain an analytic solution. This solution shows, moreover, w h e n the a p p r o x i m a t i o n o f "infinite l e n g t h " b e c o m e s a c c e p t a b l e . W e will consider a line g r o u n d source o f strength q (per unit length and time) a n d of total length L . Let us a s s u m e that the w i n d b l o w s n o r m a l to the line. A s before, let the c o o r d i n a t e x b e aligned w i t h the m e a n w i n d a n d t h e o r i g i n o f o u r c o o r d i n a t e s y s t e m (x,y,z)

fixed

at t h e

midpoint of the line, (Fig. 9-6). T h e solution of interest is obtained f r o m E q . ( 9 - 9 ) b y i n t e g r a t i n g t h e k n o w n s o l u t i o n dx

for t h e

point

s o u r c e o f s t r e n g t h q dy o v e r t h e l e n g t h o f t h e l i n e L .

a) Line source. Fig.

9-6:

Coordinate system point sources.

b) Approximate sources. for

line source

and

system

of

approximation

point by

Chapter 9

226

c h a n g i n g t o t h e n e w v a r i a b l e p = y /(a

On

y

solution as

y[2 ) w e c a n r e w r i t e

the

follows

"^)lf ^ e

w h e r e p \ - L/(a

y

"

2]dP

(9 12)

2>f2).

O n c h e c k i n g that the (integral) continuity c o n d i t i o n is satisfied, w e find

Uxdz Jo The

integral solution, E q . ( 9 - 1 2 ) , is familiar from our

discussions

i n C h a p t e r s 6 a n d 8. W e k n o w t h a t w h e n p\ = 2 o r g r e a t e r , t h e i n t e g r a l c a n b e a p p r o x i m a t e d b y i t s v a l u e for p\

= oo a s u p p e r l i m i t , i . e . ^/~JI/2.

Estimates b a s e d o n infinite length are therefore reasonable as long as L exceeds about six times o . y

The case w h e n the w i n d direction is not n o r m a l to the line, c a n b e a p p r o x i m a t e d b y r e p l a c i n g U b y its n o r m a l c o m p o n e n t U . n

This gives

reasonable answers as long as the angle b e t w e e n the line source

and

the w i n d vector is greater than 45 degrees. W h e n a h i g h w a y o r r u n w a y i s m o d e l l e d b y a l i n e s o u r c e , it w i l l b e necessary

to take into a c c o u n t

the m e c h a n i c a l m i x i n g due

to

the

m o v i n g v e h i c l e s . A s i m p l e f i x i s t o m a k e u s e o f a "virtual" l i n e s o u r c e l o c a t e d f u r t h e r u p w i n d t h a n t h e r e a l s o u r c e . T h e d i l u t i o n w i t h air a t t h e r e a l s o u r c e , d e t e r m i n e s h o w far u p w i n d t h e v i r t u a l s o u r c e s h o u l d be located. Area sources with source strength a function of both coordinates x a n d y , are m o r e difficult to solve analytically. W e c a n b u i l d s i m p l e m o d e l s where w e m a k e use not only of point sources, but also of line sources,

or a

combination

of both,

as

indicated

in Fig. 9-7.

An

attractive alternative, in cases w h e n the emitted material is m i x e d w i t h air a l r e a d y i n t h e s o u r c e r e g i o n , i s t o m a k e u s e o f o n e (or m o r e ) virtual

source(s) located upwind.

With

this

model

a

reasonable

e s t i m a t e far d o w n w i n d c a n b e o b t a i n e d w i t h a m i n i m u m o f effort. B u t all d i s p e r s i o n e s t i m a t e s o f t h e t y p e d i s c u s s e d h e r e , a r e b a s e d o n l o n g - t i m e a v e r a g e d o b s e r v a t i o n s . T h e r e is l i t t l e c h a n c e t h a t s h o r t t e r m or i n s t a n t a n e o u s o b s e r v a t i o n s w o u l d a g r e e w i t h these results.

Passive Dispersion from Steady Sources (Turbulent)

a)

Area source source.

approximated

b)

Area source strength.

modelled

by six line sources

227

and

one

point

U

Fig.

9.5

9-7a,b:

Approximate

by a virtual point

models for area

source

of

equivalent

sources.

T h e Outlook for I m p r o v e d Dispersion Models

V a r i o u s s h o r t c o m i n g s o f t h e p r e d i c t i o n m e t h o d s for a t m o s p h e r i c dispersion, b a s e d on the Pasquill-Gifford approach, have b e e n k n o w n for s o m e t i m e . T h e s t a b i l i t y c l a s s i f i c a t i o n a p p e a r s t o b e s t r o n g l y biased towards neutral stability, e v e n w h e n convective conditions actually exist. T h e empirical data are furthermore b a s e d o n passive

Chapter 9

228

releases

from ground

dispersion

level

sources,

from elevated sources,

and

such

their as

use

tall

in

predicting

stacks,

is

highly

questionable. In convective conditions the atmosphere has regions of updraft

(plumes, thermals) a n d from continuity also downdraft. T h e

d o w n w a r d m o v e m e n t extends over the larger area ( 6 0 % ) and a passive tracer

released from an

elevated source

is m o r e likely to

spread

d o w n w a r d s towards the ground. A g r o u n d release will on the

other

h a n d initially rise and return to g r o u n d level highly diluted from the m i x i n g aloft. T h i s d i f f e r e n c e b e t w e e n t h e t w o t y p e s o f r e l e a s e s i s n o t a p p a r e n t f r o m t h e P a s q u i l l - G i f f o r d a p p r o a c h , b u t it h a s

considerable

practical consequences. The

dispersion

different

models, in present

w a y s . A s i m p l e fix, w i t h

use,

some

could be i m p r o v e d in

success,

is to shift

the

classification s c h e m e t o w a r d s the u n s t a b l e side to eliminate the bias towards

neutral

Panofsky

and

conditions.

Dutton,

(Weil

1984,

and

B r o w e r as

Chapter

10.)

discussed

More

radical

by new

approaches h a v e b e e n suggested, h o w e v e r , to m a k e better use of the improved understanding of atmospheric boundary-layer physics; Weil (1985)

lists

the

"impingement" model.

probability-density

model

as

Neither procedure

alternatives can

take

method to

into

the

as

well

improved

account

the

as

an

Gaussian streamwise

diffusion; for t h i s p u r p o s e C s a n a d y ' s p u f f m o d e l , d i s c u s s e d i n C h a p t e r 8, h a s b e e n p r o p o s e d . F o r t h e a v e r a g e u s e r o f m e t h o d s o r c o m p u t e r c o d e s for d i s p e r s i o n estimates,

based

on

Pasquill-Gifford's ideas,

i m p r o v e m e n t of the present

procedures

appear

the

updating

most

and

attractive.

A

m o r e r e f i n e d a p p r o a c h r e q u i r e s m o r e t h a n a n e w s e t o f c u r v e s for t h e dispersion

coefficients,

however.

New parameters

have

to

be

introduced w h i c h reflect i m p o r t a n t physical p h e n o m e n a . T o evaluate these

parameters

above

the

additional

limited

data

information will be needed

(windspeed

and

qualitative

o b s e r v a t i o n s ) r e q u i r e d for a P a s q u i l l - G i f f o r d d i s p e r s i o n That

part

of the

atmosphere,

friction a n d h e a t transfer boundary

over

weather

estimate.

w h i c h is strongly influenced

from the ground, is called the

layer ( P B L ) or the a t m o s p h e r i c

boundary

and

by

planetary

l a y e r ( A B L ) . It

extends from the g r o u n d to a h e i g h t o f the order of o n e kilometer. Over

this height

the

wind vector changes

from

its

velocity

and

direction near the g r o u n d to its geostrophic m a g n i t u d e a n d direction. (The E k m a n layer w a s discussed briefly in Chapter 7.) T h e top o f the boundary

layer is c h a r a c t e r i z e d

also b y the vanishing of vertical

transport of heat and m o m e n t u m , b u t a precise value of the height h can not b e given. Close to the g r o u n d there are strong gradients

in

Passive Dispersionfrom Steady Sources (Turbulent)

velocity and

potential

temperature,

but

the

229

shear stress is

nearly

constant as reflected in the logarithmic w i n d velocity profile familiar from turbulent wall boundary layers. T h e turbulence

in this part of

the layer is g e n e r a t e d b y wind-shear. The

convective (day-time) boundary

layer,

(sometimes

CBL), includes a second source of turbulent production,

denoted a

strong

v e r t i c a l m i x i n g d r i v e n b y t h e g r o u n d h e a t flux w h i c h i n t u r n d e p e n d s on insolation, ground

temperature

and

moisture,

re-radiation

etc.

A b o v e a certain height L, generally referred to as the M o n i n - O b u k h o v length,

the

convective turbulence

dominates

over w i n d shear. A t

m i d d a y the vertical extent o f the c o n v e c t i v e r e g i o n is typically ten times the height o f the region of w i n d shear close to the g r o u n d . T h e understanding of CBL physics has improved more rapidly than

the

u n d e r s t a n d i n g of its counterpart, the stable (night-time) b o u n d a r y layer

(SBL). Accurate prediction

dispersion

of the

rates) is still b e y o n d reach,

CBL character but

(including

w h e n properly scaled,

s e e m i n g l y different b e h a v i o r can b e described b y the

same

set of

(scaled) variables. In technical applications of turbulent boundary-layer theory, i d e a o f s c a l i n g is o f f u n d a m e n t a l

the

importance. Distances from the wall

are scaled b y the thickness of the b o u n d a r y layer, a n d the velocity b y the

free-stream

velocity. A more refined investigation of the

wall

r e g i o n r e v e a l s t h e seeding v e l o c i t y u . = Vr/p

a n d t h e s c a l i n g l e n g t h (or

viscous length) z . = v/u.. These quantities

a r e r e l e v a n t p r i m a r i l y for

smooth

walls.

In

the

presence

of wall

roughness

the

derived

( l o g a r i t h m i c ) v e l o c i t y p r o f i l e is s h i f t e d u p w a r d b y a r o u g h n e s s

height

z . T h i s scaling c a n b e carried over at the least to the lower p a r t of Q

t h e C B L w h e r e t h e flow i s d o m i n a t e d b y w i n d s h e a r . A b o v e the height corresponding to the M o n i n - O b u k h o v value L , the turbulence r a t i o z/L

is d o m i n a t e d b y b u o y a n t convection. T h e

can be seen as a stability parameter,

related to the gradient

Richardson number

dimensionless

a n d it c a n i n f a c t b e

defined in Chapter

2.

A c c o r d i n g t o P a n o f s k y a n d D u t t o n ( p . 1 4 1 ) w e h a v e a p p r o x i m a t e l y Ri = z/L

i n u n s t a b l e a i r . T h e u p p e r l i m i t o f t h e s c a l e d v a r i a b l e z/L

obviously

h/L

which

becomes

an

important

parameter

c o m p a r i n g d i f f e r e n t flow s i t u a t i o n s . A t h e i g h t s o f t h e o r d e r o f h, Coriolis

parameter*/

has

some importance,

and

to relate

f r i c t i o n a l effects, w e c a n f o r m t h e d i m e n s i o n l e s s p a r a m e t e r

is

when it

the to

u./h/.

The Coriolis parameter is proportional to the earth's speed of rotation at the latitude of interest; J- 1 Q sin 0 where Q is one revolution per day. It follows t h a t / is of the order of 10" radians per second. 4

Chapter 9

230

This discussion ( h / L , u . / h / , z /h) and

parameters,

brought

forth three

dimensionless

just from the dynamics of the

Q

of heat

has

mass

transfer

(humidity),

flow.

groups,

Considerations

would produce

additional

m a k i n g t h e h o p e f o r s i m p l e i m p r o v e m e n t s r e m o t e . It is

very fortunate

t h a t , for t h e p u r p o s e o f m o d e l i n g a t m o s p h e r i c

t h e p a r a m e t e r h/L

flows,

i s b y far t h e m o s t i m p o r t a n t ( H u n t e t al., 1 9 9 1 ) . B u t

d i f f i c u l t i e s r e m a i n a s ft a n d L a r e n o t a v a i l a b l e f r o m r o u t i n e w e a t h e r o b s e r v a t i o n s , sufficient for t h e P a s q u i l l - G i f f o r d

approach,

and

will

A d e t a i l e d p r o c e d u r e for c a l c u l a t i n g a t m o s p h e r i c p a r a m e t e r s ,

not

have to b e determined. determined directly from routine measurements,

is p r o p o s e d b y v a n

Ulden a n d Holtslag (1985). A m o n g the parameters

o f interest is the

h e a t flux t o t h e s u r f a c e , H . T h e h e a t c o n t e n t o f a u n i t v o l u m e o f air is p c

p

T whereas H represents

the heat transferred

per unit area

time. O n f o r m i n g the ratio o f the p r o d u c t g H z to p c

p

and

T, w e get a

q u a n t i t y w i t h d i m e n s i o n s v e l o c i t y c u b e d . T h i s v e l o c i t y i s , i n fact, t h e convective

velocity

scale

w+

of prime

importance

when

scaling

velocities in the convective layer.

(9-13)

T h e s c a l i n g v e l o c i t y u«, i n t h e w i n d - s h e a r d o m i n a t e d s u r f a c e l a y e r is already k n o w n . A t the transition height b e t w e e n the t w o layers, the t w o s c a l i n g v e l o c i t i e s s h o u l d b e r o u g h l y e q u a l , i . e . w h e n z = L , w„ = u.. F r o m (9-13) w e obtain in this w a y the M o n i n - O b u k h o v length. A m o r e rigorous derivation based on wall-layer similarity, reveals that von Karman constant hence

k s h o u l d b e i n c l u d e d (k is d i m e n s i o n l e s s

of no consequence

analysis).

The

similarity

quantity w h e n the heat

in an

argument

analysis

flux

also

based

defines

on L as

the and

dimensional a

negative

i s p o s i t i v e or u p w a r d . ( F o r a d e t a i l e d

derivation, see Panofsky and Dutton, p 131-2.)

L = -

kgH

(9-14)

W e s e e t h a t i n s t a b l e a i r ( H < 0 ) , L > 0, i n n e u t r a l c o n d i t i o n s H = 0, L - » oo a n d i n u n s t a b l e air L < 0. Practical use o f the scaling p a r a m e t e r s requires that their v a l u e s are k n o w n , b u t the e v a l u a t i o n is n o t trivial, v a n U l d e n a n d H o l t s l a g (1985) suggest to derive L from M o n i n - O b u k h o v similarity theory. W e

Passive Dispersionfrom Steady Sources (Turbulent)

231

n o t e h e r e t h a t t h e s i m p l e r e l a t i o n for u . g i v e n a b o v e , w i l l h a v e t o b e c o r r e c t e d for h e a t t r a n s f e r a n d r o u g h n e s s e f f e c t s a s i n d i c a t e d b y v a n U l d e n a n d Holtslag. P a n o f s k y a n d D u t t o n s u g g e s t to obtain L v i a the gradient Ricardson

number.

D u e to the a s y m p t o t i c nature of the atmospheric b o u n d a r y layer, the specification o f height h is not simple. T h e layer height

changes

moreover with time. For the unstable layer, v a n Ulden a n d Holtslag s u g g e s t a r a t e e q u a t i o n for h. empirical relation relation is h It

n

=c

= c yju*L/J,

s

to

include

information

in

the

and direct numerical

dispersion parameters are

an

preferred dispersion

simulations of

s h o w excellent agreement

normalized w.r.t. h

nondimensional t i m e x/U

stable layer the

A detailed discussion is given in Weil ( 1 9 8 5 ) . H e notes

atmospheric values

For the

this

that laboratory measurements the

For the neutral layer they prefer

u*/f.

n

w h e r e / again is the Coriolis parameter.

s

remains

parameters.

h

d i s t a n c e X = w+x/(U

and

given as functions

of the

h), w h i c h i s t h e r a t i o o f t r a v e l

[x i s d o w n w i n d d i s t a n c e , U i s w i n d s p e e d ) t o t h e

t i m e s c a l e h/w*.

when

turbulent

F o r t h e s t r o n g l y c o n v e c t i v e c a s e w h e r e h / - L > 10, t h e

f o l l o w i n g r e l a t i o n i s s u g g e s t e d for t h e l a t e r a l d i s p e r s i o n

, °

h

6

coefficient:

(9-15)

X

+0.7X

w h e r e t h e v a l u e o f t h e coefficient 0 . 7 i s s o m e w h a t u n c e r t a i n . For a

z

the effects o f source height m a y b e important.

For

source

h e i g h t s a b o v e z = 0 . 1 h, a v a i l a b l e r e s u l t s i n d i c a t e t h e s i m p l e r e l a t i o n

^ = 0.6X h

(9-16)

b u t b e y o n d X = 0.7 the v a l u e c a n b e a s s u m e d to b e c o n s t a n t d u e to p l u m e trapping in the convective layer. For g r o u n d level sources Weil cites a s o m e w h a t complicated interpolation formula. W e n o t e h e r e t h a t a c o r r e c t i o n for a n e l e v a t e d s o u r c e a p p e a r s m o s t urgent as the conditions in N o r t h w e s t E u r o p e are close to neutral 70% o f t h e t i m e w i t h b o u n d a r y - l a y e r h e i g h t h, r a n g i n g f r o m 5 0 0 - 1 0 0 0 (Hunt

et al.,

1991). Dispersion

ground-level sources, typically

is

only one

calculations

are valid at best tenth

o f h.

based

on

in the surface

This

means

that

data

m

from

layer which dispersion

calculations b a s e d o n the Pasquill-Gifford dispersion coefficients, are m e a n i n g f u l o n l y for s t a c k h e i g h t s l e s s t h a n 5 0 m . T h e a c t u a l h e i g h t s

Chapter 9

232

of large industrial stacks are m u c h higher, and herein lies an obvious n e e d for b e t t e r m e t h o d s . Research

and

development of simple

validated

methods

currently u n d e r w a y w i t h the h o p e of arriving at international as

pollutants

important

in

the

atmosphere

know no

national

are

norms,

borders.

An

event in this regard, w a s the W o r k s h o p organized b y

E P A / A M S in 1984 (reviewed b y Weil, 1985).

Special Nomenclature /

Coriolis

J(T)

f u n c t i o n o f d i f f u s i o n t i m e , T a b l e 9-1

parameter

h

height of convective boundary layer

H

effective s o u r c e h e i g h t , H = H

g

+ AH

h e a t flux t o s u r f a c e k

von Karman constant

K

t u r b u l e n t diffusivity, d i r e c t i o n n

L

distance d o w n w i n d from source

n

length of line source M o n i n - O b u k h o v length, Eq. (9-14) p

argument, exponential function

q

strength of line source

Q

strength of point source

Ri

(gradient) Richardson

number

T

diffusion time, Eq. (9-3)

u*

wall velocity scale,

U

m e a n w i n d velocity

w*

convective velocity scale, Eq. (9-13)

X

n o n d i m e n s i o n a l d o w n w i n d d i s t a n c e , X = w„x/(U

X (T)

d i s t a n c e r e a c h e d i n t i m e T, T a b l e 9-1

y„

wall length scale,

z

Q

roughness height

a

n

d i s p e r s i o n coefficient, direction n

o ,OQ

standard deviation, direction of w i n d vector

X

concentration

a

=V

h)

= v/u.

REFERENCES B r i g g s , G . A . ( 1 9 7 3 ) D i f f u s i o n e s t i m a t e s for s m a l l e m i s s i o n s .

ATDL

Passive DispersionfromSteady Sources (Turbulent) Contribution

No.

79.

Atmospheric

233

Turbulence

and

Diffusion

Laboratory, Oak Ridge, Tennessee. C s a n a d y , G . T . ( 1 9 7 3 ) Turbulent Publishing, Dordrecht, Gifford,

Diffusion

F.A. (1959) Smoke plumes

i n d i c e s . Intern.

in the Environment.

D. Reidel

Holland.

J. Air Pollution

and

q u a n t i t a t i v e air

pollution

2, p p 4 2 - 5 2 .

Hunt, J.C.R., Holroyd, R.J., Carruthers, D.J., Robins, A . G . , Apsley, D.D., Smith, modelling Modeling

F.B., and

air

and

Thomson,

pollution

for

its Applications,

D.J.

regulatory

(1991) Developments uses.

I n Air

in

Pollution

viii ( E d s H . v a n D o p a n d D . G . S t e i n ) ,

p p 17-60. P l e n u m Press, N e w York, L o n d o n . Lyons,

T.J.

Meteorology.

and

Scott,

W . D . ( 1 9 9 0 ) Principles

of

Air

Pollution

Belhaven Press (Pinter Publishers), London.

P a n o f s k y , H . A . a n d D u t t o n , J . A . ( 1 9 8 4 ) Atmospheric

Turbulence.

John

Wiley and Sons, N e w York. P a s q u i l l , F . ( 1 9 7 7 ) s e e : H a n n a , S . R . , B r i g g s , G . A . Deardorff, J., B.A.,

Gifford,

F.A. and

Pasquill,

Egan,

F. ( 1 9 7 7 ) A M S W o r k s h o p

s t a b i l i t y c l a s s i f i c a t i o n s c h e m e s . Bull.

Am.

Meteorol.

Society

58,

on pp

1305-1309. S t u l l , R . B . ( 1 9 8 8 ) An

Introduction

to Boundary

Kluwer A c a d e m i c Publishers, Dordrecht, T u r n e r , D . B . ( 1 9 7 3 ) Workbook

of Atmospheric

Layer

Meteorology.

Holland. Dispersion

Estimates.

Environmental Protection A g e n c y R e p . N o A P - 2 6 , 6th van Ulden, A . P . and Holtslag, A . A . M . (1985) Estimation

of boundary

l a y e r p a r a m e t e r s for d i f f u s i o n a p p l i c a t i o n s . J. Clim. Appl. pp

US

Printing.

Met.

24,

1196-1207.

W e i l , J . C . ( 1 9 8 5 ) U p d a t i n g a p p l i e d diffusion m o d e l s . J. Clim. Appl. 24, p p

Met.

1111-1130.

W i l l i a m s o n , S.J. ( 1 9 7 3 ) Fundamentals Publishing Co., Reading, Mass.

of Air Pollution.

Addison Wesley

Chapter 9

234

PROBLEMS Problem

1. A p o i n t s o u r c e o f h e i g h t

100 m emits

nonbuoyant

material at the r a t e Q = 9 0 g / s . T h e w i n d velocity is 5 m / s a n d

the

w e a t h e r is m o d e r a t e l y s u n n y . F i n d t h e r e g i o n w i t h t h e h i g h e s t s u r f a c e concentration a n d the corresponding distance from the source. P r o b l e m 2. F i n d t h e h e i g h t r e q u i r e d t o r e d u c e t h e h i g h e s t s u r f a c e concentration to 2 . 5 - 1 0

4

g/m

for a w i n d s p e e d 3 m / s a n d a ( p o i n t )

3

source strength of 100 g / s . T h e w e a t h e r corresponds to B-stability. P r o b l e m 3 . A s s u m e t h a t t h e effective s o u r c e h e i g h t i s g i v e n b y t h e r e l a t i o n H = a + bin

w h e r e u is the w i n d s p e e d a n d a a n d b

are

constants. Find the w i n d speed w h i c h gives the highest concentration at surface level a n d the c o r r e s p o n d i n g c o n c e n t r a t i o n . E x p r e s s a n s w e r i n t e r m s o f a, b a n d

the

a . z

P r o b l e m 4. A p o i n t s o u r c e e m i t s p o l l u t a n t s a t t h e r a t e Q = 8 0 g / s at a n

effective

height of 80 m . T h e stability conditions vary with

height; b e t w e e n the surface and height

1 km, B-stability prevails.

A b o v e 1 k m , w e h a v e a stable layer. T h e w i n d s p e e d is 2 m / s . F i n d the concentration at surface level at distances 2 k m a n d 8 k m from

the

source. P r o b l e m 5. A p o w e r s t a t i o n b u r n s s u l f u r o u s

coal at the rate of

15,000 k g / h r . T h e coal contains 2 % Sulfur w h i c h in the c o m b u s t i o n p r o c e s s oxidizes to SO2. O n a given d a y w e h a v e southerly w i n d s at 2 m / s a n d w a r m s u n n y weather. F i n d the concentration of SO2 in a school yard

l o c a t e d 0.5

km

N - N E form

the

power

station.

The

effective s t a c k h e i g h t i s 10 m . Problem produces

a

6. A b u r n i n g w a s t e d u m p o f a b o u t foul

smelling gas

at

the

rate

1200 m

of 30

g/s.

2

fire a r e a , Find

the

c o n c e n t r a t i o n at the distance 3 0 0 m a n d 3 k m d o w n w i n d at n i g h t in overcast conditions and with winds of 7 m / s . Compare calculations b a s e d o n a ) four line sources, b ) a single line source through the center of the d u m p a n d c) a single (virtual) point source u p w i n d . T h e d u m p can b e considered a square w i t h o n e edge perpendicular to the w i n d direction. P r o b l e m 7. A m a j o r h i g h w a y r u n s i n d i r e c t i o n N o r t h - S o u t h . O n a day with overcast conditions and westerly wind of 4 m / s , about 8000

PASSIVE DISPERSIONFROM STEADY SOURCES (TURBULENT)235 automobiles

pass

through

the

1

straight-line segment o f interest w i t h an average velocity of 120 HC

km/hr. The emission as

function

typical

characteristics,

of velocity,

a u t o m o b i l e , is

for

given

a in

60

the diagram. T h e percentage scale can be reinterpreted in terms of

NO

absolute values on knowing that the

averaged

emission

40

hydrocarbon

( H C ) at

60

km/hr

X

is

2-lO- g/s. 2

Estimate

the

concentration at a distance m

d o w n w i n d from

20

hydrocarbon 300

the highway.

H o w m u c h can the concentration

60

be reduced w h e n w e legislate a

80

100

130

n e w speed limit o f 100 k m / h r ? Consider

the

same

problem

the

fc

NO -emission x

using

the

information given in the diagram. Problem elevated

8.

S o m e t i m e s it i s o f i n t e r e s t

concentration

to define a region of

above a specified level % , but

below

Q

the

m a x i m u m c o n c e n t r a t i o n XM- T h e c u r v e o f c o n s t a n t c o n c e n t r a t i o n XO = XO (x,y,H)

i s d e n o t e d a n i s o p l e t h a n d it e n c l o s e s t h e p o i n t o f m a x i m u m

c o n c e n t r a t i o n XM- C o n s i d e r t h e f o l l o w i n g p r o b l e m : W e h a v e a p o i n t source o f strength Q = 8 0 g / s . T h e w i n d s p e e d is 5 m / s , H = 2 0 m a n d the

stability

of Class

C.

Find

the

region on

the

ground

with

c o n c e n t r a t i o n e x c e e d i n g XO = 10~ g / m . 4

3

( H i n t : C h e c k first t h e m a x i m u m v a l u e a l o n g t h e c e n t e r l i n e y = 0, t o s e e i f XM > XO- W h e n n o t , s u c h a r e g i o n d o e s n o t e x i s t . W h e n XM > XO* y o u c a n f i n d t w o v a l u e s o f x , i . e . X\ a n d x x (XM)-

J

n

s

o

t h a t x\ <

t h e r a n g e x\ < x < X 2 , f i n d t h e v a l u e s o f y for

w h i c h X(X>Y>0) = XO)Problem

9.

Plot

and

compare

P r o b l e m 8 at h e i g h t s z = 0 a n d z = H .

the

isopleths

X ~ XO (x,y;H)

of