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Crosswind shear effects on atmospheric diffusion
Atmospheric Environment Pergamon Press 1972. Vol. 6, pp. 221-232. Printed in Great Britain.
C R O S S W I N D SHEAR EFFECTS ON A T M O S P H E R I C DIFFUSION* G. T. CSANADY Environmental Fluid Mechanics Laboratory, University of Waterloo, Waterloo, Ontario, Canada (First recieved 25 May, 1971 and in final form 4 October 1971)
Abstract---Some experimental data on atmospheric diffusion over a "medium" distance range (order 30 kin) have been analysed in an attempt to discover any effects on lateral spread attributable to the skewed wind profile of the planetary boundary layer. The observations analysed were obtained over "natural" terrain (containing woodlots and villages) in Arkansas. Wind shear effects on diffusion were considered "significant" when the cloud decisively outgrew the size predicted by the Hay-Pasquill technique from fixed point cross-wind velocity fluctuations Such "significant" effects were found on few occasions only, associated with quite extreme rates of cross-wind shear, accompanied by strong stability. INTRODUCTION IT MAY be expected that when an atmospheric cloud of tracer or pollutant is deep enough to occupy a substantial fraction of the E k m a n layer, its lateral spread will be influenced or possibly dominated by the variations of the mean velocity vector along the vertical ("directional shear"), instead of being determined entirely by atmospheric turbulence. When the tracer is released at ground level, the upper edge of the cloud moves vertically at a velocity of the order of the friction velocity u* = ~¢/(~'o/p) (which is also the order of the r.m.s, turbulent velocity). The E k m a n layer thickness being approximately 0.2 u * / f ( f = Coriolis parameter), it thus takes a period of order 0.2f 1/2 h at mid-latitudes for such a cloud to become large enough for shear effects to be appreciable. Typically, a cloud travels 10 km f r o m its point of release in a period of this order. Thus shear effects may begin to develop at such diffusion distances and could become dominant at distances of the order of 100 kin. Similar conclusions may be derived by more detailed argument based on an idealized E k m a n layer model (CSANADY,1969). In an analysis of the few available experimental data PASQUILL 0969) has concluded that effects at ground level, attributable to cross-wind shear, were not of practical importance at distances below about 12 kin, but they began to dominate diffusion beyond about 25 km. These conclusions were based mainly on the data of FUQUAY et al. (1964) and it would be desirable to subject them to further checks. Recently, the staff of the Travelers Research Centre have carried out a series of diffusion experiments, code named " W o o d l o t " over " n a t u r a l " rural terrain in Arkansas to downwind distances of the order of 30 k m (HaLST et al., 1969). According to the above estimates, shear effects on diffusion should be noticeable at such distances The purpose of the present paper is to analyse the Woodlot long-distance diffusion data and exhibit the effects of shear on diffusion, where such effects may be found. The experiments were not designed exclusively for the purpose of providing long-term diffusion data: there was as much or greater interest in effects on diffusion close to the * This work was carried out during a visit to Travelers Research Center, Hartford, Connecticut. 221
222
G.T. CSANADY
source due to finite source size and to finite size obstacles such as isolated woodlots. Thus the data are not ideal for our purpose here, but in view of the scarcity of information on thisttopic their analysis seemed worthwhile.
DETERMINATION OF LATERAL SPREAD In the long-distance diffusion experiments of the Woodlot series ground level dosages were determined at a grid of stations extending some 30 km downwind of the release line. At distances beyond about 10 km the tracer cloud was usually wide enough for the gridpoints to yield a cross-wind dosage profile with reasonable resolution, so that the cross-windgrowth of the cloud could be followed, provided that the wind was indeed of the right direction for the cloud to pass over the grid and provided that not too many sampling instruments failed. A substantial number of trials yielded useful information of this kind, although in some cases only a single cross-section was fully covered by the sampler grid as actually laid out. In the following the x-coordinate axis is taken to be in the surface wind direction, the y-axis horizontally across wind, and the centre of the source is at the origin. The release took place along a line of 800 m length, appropriately chosen upwind of the grid, from a moving truck carrying a dispenser, for a release period of about 25 min. The angle between the release line and the wind direction varied from 0 to 45 ° or so, so that the "effective" length (perpendicular to the wind, i.e. along the y-axis) varied from about 500 to 800 m. At large enough distances from the release line one would expect the cloud to behave as if it was released from a point source. In the distance-range of interest this was not entirely the case: in some instances a "solid core" of high dosage could be identified at the edges o f which the dosage dropped very rapidly to zero. Diffusion from a line-source of finite length may be treated by regarding each element of the line as a point source, and superimposing the many point source clouds, in each of which the lateral ( = along the y-axis) dosage distribution is Gaussian with standard deviation %. This point source standard deviation is the fundamental quantity with which simple theories of atmospheric diffusion are concerned (see e.g. PASQUILL, 1962). In extracting data on ey from the observed spread of a line source cloud some care is necessary, although the procedure is straightforward and is outlined in the Appendix. The distinction between point-source % and "standard deviation" of an entire cloud released from an extended source is not always made clear and may lead to confusion (RAYNOR et al., 1970). At large distances from the source, of course, the distinction becomes unimportant. The primary variable obtained in the analysis of the experimental data is the " I 0 per cent width" of a cross-wind profile at a given distance from the source, i.e. the distance between the two points at the edges where the concentration drops to 10 per cent of the maximum in that section. As there were often only few dosage measurements per cross-section to define a profile (some of these being, moreover, subject to a certain random scatter) it was only possible to obtain a 10 per cent width estimate from a more or less subjectively fitted curve. The accuracy of this procedure probably varies considerably, but in the case of the better defined profiles (usually at the larger distances) it is quite reasonable, perhaps of order I0 per cent. The actual magnitudes of % derived from the 10 per cent width by the method outlined in Appendix I should
C r o s s - w i n d S h e a r Effects o n A t m o s p h e r i c D i f f u s i o n
223
thus be, on the whole, reasonable estimates. However, in attempting to assess the growth of % with distance, the inaccuracy of the individual readings is a more serious handicap. The gradient, d%/dx could not, in fact, be determined for individual trials with any accuracy. The errors were particularly large at the smaller distances, where the cloud was sometimes only of the order of twice the release line width. Sample dosage profiles (across wind, at ground level), are shown in FIG. 1. Values of ~y determined in the above manner from the results of the trials are shown in TABLE 1, the starred values being judged to be rather less accurate than the others (say, typically -f- 30 per cent, although some may be even worse.)
o X =6km I 0 % width : 1 5 0 0 rn A - X = 16 km
I 0 % width: 4 6 0 0 m
E -%.
2ooo
400
'E
§
E c
.-.9.
i~. I000
t~ 0
I ,7 l I
2
3
I
0
I
2
Cross wind distoncekm,
3
4
5
o
6
orbJtraryorgin
FIo. 1. Sample cross-wind dosage profiles. CONVECTION
VELOCITY
In order to decide whether the observed spread can be attributed to atmospheric turbulence alone (with no appreciable contribution from wind shear) it is necessary to have some information on wind speed and horizontal turbulence. The spread due to atmospheric turbulence may be calculated from the relationship, discussed in detail by PASQUILL(1962): ~, x
_
vr.~
(I)
U
where U is the (assumedly constant) convection velocity of a diffusing cloud and vr.s is a root mean square turbulent velocity in the y-direction, obtained from moving average velocities of drifting particles over "travel time" T, the mean-square being formed over the "sampling time" s. In the atmosphere, of course, wind speed is not constant but varies with height and the choice of a suitable convection velocity from a wind profile measured at one location would present serious problems. Therefore, estimates of the convection velocity were obtained from the dosage data collected at the few "sequentializer" stations deployed for the trials, which gave
224
G.T.
CSANADY
c),~E
c~
~J
~ ~ oc ~ co ~ ~
o o ~o
'=~ "=~ ~
o o
~-=~ '=~ "==~ ,"~ ¢'~
I
t~
~A Z "O
z
I Z
,-I
O N O
~
oo~
~
,=~ ¢'~
o o o ~kO
I I
Cross-wind Shear Effects on Atmospheric Diffusion ~4 O
b$
o o o
ooo$ I
~ o ~ o o. . . .~. .o o o $
~ oooo$oo$$ o o
I
I
E i .= ¢.J 0J
o O O O O O O O O O O O O O O O O O O O O O O O O O
dddddNNNNoNNddNNdNNNNdNNd~
°ooooooooooooooo z
o
oddddc~ddc~dddddo¢~
E
~oo~o~
OO
o~o
**~
**
~ O O
6 Z O
o L
.O
E
O
Z
225
226
G . T . C~ANADY
dosages in 10 half-hour periods following release. The time of the maximum reading provided a crude estimate (-b 15 rain) of the effective "arrival time" for the dosage cloud. At least for travel times of 2 h and more the velocity calculated from the distance, release time--sequentializer station, and the travel time should be a reasonable approximation to a convection velocity. It may be noted, however, that the estimates provided by different sequentializer scattered strongly and that there was also a fairly systematic increase of convection velocity with distance (which makes sense physically for a growing cloud). TABLE 1 lists what appeared to be best estimates of convection velocity determined on this basis. Where this estimate was not available or was obviously wrong, an estimate of the wind speed on the basis of Rawinsonde data is given (starred), using data of the 30 m level. In general, the accuracy of these latter estimates of convection velocity or more directly, of the travel times they imply must be classed as quite low (perhaps within a factor of 2). HORIZONTAL TURBULENCE DATA As originally defined Vr.s in equation (1) refers to the Lagrangian velocity of drifting particles. It is, however, normally approximated by a moving average of fixed point (Eulerian) velocities over a fraction of the travel time, T/fl, the constant being an uncertain quantity. A common assumption is to take/3 ---- 4, and it is known that the exact value of fl has relatively little influence on the results." The ratio v/U is to first order equal to the azimuth angle 0 of the wind, so that equation (1) may be approximately converted into x
= Or/B.s
(2)
where 0r/B.~ is now the (Eulerian) r.m.s, azimuth angle (measured from the " m e a n " wind direction), averaged over T/~ and then "sampled" (i.e.r.m.s. value of the moving average determined) over period s. The practical merit of this particular formulation is that r.m.s, azimuth angle determined in this manner changes only slowly with the averaging time T/E, a quantity the appropriate value of which is uncertain: we mentioned before the inaccuracy of determining the travel time T, to which comes the arbitrariness inherent in the choice of 8. Thus, while the estimated value of T/B may easily be in error by as much as a factor of 4, this may not cause an error in our estimate of 0r/B.~ greater than by a factor of 1.5 or 2. Azimuth angle readings obtained at the 90 m level at the centre of the experimental grid were utilized to estimate values of 0r/B.~ for comparison with observed %Ix ratios. The data available were 5-rain average wind directions. By averaging 2, 3, etc. successive readings 10, 15, etc. rain moving averages were obtained. The release period (equivalent to the theoretical "sampling'"period) was a little under 30 min so that there were only 6 moving averages with their beginnings contained within the release period or close to it. To obtain a r.m.s, value from 6 readings is not possible with any accuracy; thus the difference between maximum and minimum was determined and divided by 4 to provide a crude estimate. This is the value 0r/B.~ listed in TABLE 1. fl = 4 was used and some crude interpolation was also necessary to obtain 0r/B.s for a desired T/ft. For all releases, 6 readings were used in arriving at an estimated 0riB.~ the closest available to the recorded release period in each case, regardless of the exact length of the release period.
Cross-wind Shear Effects o n Atmospheric Diffusion DETECTION
OF SHEAR
227
EFFECTS
Given the crudeness of the method used in arriving at an estimate of 0T/B,s it is surprising that the values of this quantity are generally as close to ~/x as they are. F o r example, the eleven readings of ~y/x in TABLE 1 for 16-17 km distance, for which 0r/B, s values are available for comparison, average out at 0.0587 as against an average 0r/n,~ of 0.0514 for the same entries. Thus the procedure adopted above as estimating a "relevant" value of 0r/a, ~ was generally successful, even if the individual estimates are not always accurate and apparently subject to considerable random scatter. One would also be tempted to conclude that shear effects are evidently minor, if we are able to account for observed spread by an analysis of turbulence data alone. A closer examination of the data reveals, however, that the grossest discrepencies between %Ix and OriB.~in TABLE 1 may be certainly attributed to shear effects and that it is also probable that cloud growth was effectively increased in some other tests less obviously out of line in TABLE 1. T o decide which of the trials were in the shear-dominated category we start by focussing attention on those in which %Ix was substantially (several times) greater than 0r/a.~.* The most obvious such case in TABLE 1, for example is trial WT12-Y-2, where the ratio of the two quantities in question is 23; there are, however, many others in which the ratio is high, and in favor of %Ix .Such a trial will be regarded as "shear influenced" if the following two further criteria are satisfied: (1) Rawinsonde data taken during the experiment show substantial directional shear of wind in the lowest few hundred meters of the atmosphere. (2) The cross-wind dosage profile exhibits evident skewness (this is a marked property of shear-augmented diffusion), which often also appears as a two-peaked distributionJ" Examples of such distributions with substantial skewness are shown in FIG. 2.
. - 400 E
=-
'YI
A
"G
21-Y-I A
g g 200 o a
0
25
5,0
0
25
5.0
Cross- wind distance in km, arbitrary origin
FIG. 2. Cross-wlnd dosage profiles showing substantial skewness. * Because there were a number of substantial deviations in the opposite direction, Orta,, >
~y/x,
i.e. overestimatesof spread, further criteria were sought to support the hypothesis that a given underestimatewas at least partly caused by wind shear. t Due, presumably, to the irregularity of individual diffusing clouds; one part of the cloud may have travelled at a different level from the rest, moving in a slightly different direction.
G. T. CSANADY
228 Testl2, 20 °0 hours
Test 16, 20°°hours
"~ 1.5 0
+1.5
NN~561 3O 45
366
76
274 198 213
Height, m I
I I
Test 21, 17°0 hours
I 2
] 3
m s-~
I 4
I 5
FIG. 3. Hodographs of velocity vector on occasions of shear-im~luenced diffusion (from Rawinsonde. Test 21 17"0hours
Test 12, 20 °`' hours IA°C Test 16, 20 °o hoursll.0°C Test 21, IT°°hours 12.0°C
40C
300 E
J= c~ T 200 "~
. Test 12 ~ 2 0 °°hours Test 16 1
~
0°°hours
100
I -
0
2
I 4
I
T
6
8
Teml~emture differenceegeinsfgroundlevel, =C
F[o. 4. Temperature profiles corresponding to Hodographs of FIG. 3.
Cross-wind Shear Effects on Atmospheric Diffusion
229
The results from 5 runs, responsible for 8 entries in TABLE 1 satisfy these criteria; an idea of the magnitude of wind-shear for these is given in the Table's last column. Hodographs of the velocity vector derived from Rawinsonde observations closest in time to these trials are shown in FIG. 3. It may be seen that the most pronounced shear occurred on occasion of trials 12-Y-2 and 12-G-2, which, as pointed out before, are characterized by a ~r/x ratio furthest from their 0r/~. s estimates. The average value of %,Ix for the eight shear-influenced entries in TADLE 1 is 0.10, while the average 0r/a, s estimate for these is 0.023. One derives the estimate that cross-wind spread was increased typically 4-5 fold in these "shear-influenced" trials, as compared to the expected effects of atmospheric turbulence alone. As to the frequency of shear-influenced diffusion we note here that only 5 out of the 26 trials that could be analysed in this manner exhibited dominant shear effects. In those cases when the shear effect was evident, the change of wind direction with height, as recorded by the Rawinsonde, was marked indeed and was not of the classical Ekman layer type (in Test 12 the sense of the change was opposite to that in an equilibrium E k m a n layer) but was associated with a moderate to strong inversion. The temperature profiles taken on the same Rawinsonde flight as the velocity vectors in FIG. 3 are shown in FIG. 4, which lists times of the tests and ground level temperatures. A referee drew attention to the discrepency between the average 0r/D,s for the shearinfluenced trials (0.023) and the average value of the same quantity for the remaining trials (about 0.05). A low average turbulence level does not seem inconsistent with the rapid variation of wind with height. TURBULENCE-CAUSED DIFFUSION When the "shear-influenced" readings in TABLE 1 are deleted one may attempt to remove the remaining r a n d o m scatter (presumably caused by the considerable errors possible in our estimates) by averaging over more or less coherent groups of readings. All the entries fall naturally into one of three distance ranges, " s h o r t " = < 11 km, " m e d i u m " = 13-18 kin, and "long" > 23 km. Averaged values of oy/x and the quantity OT/a.sfor these groups are shown in TABLE 2. Clearly, the averaged values of %/x and 0r/0,s agree quite well, both as regards absolute magnitude and variation with distance. One may conclude that in these trials the diffusion may be attributed to atmospheric turbulence alone (with no shear contribution) and that the Hay-Pasquill prediction method based on the azimuth angle variance supplies good estimates of cloud spread out to distances of order 30 km. This is further p r o o f (if such is indeed necessary) for the proposition that any effects the increased roughness of " n a t u r a l " terrain may have on diffusion are exerted through influencing atmospheric turbulence. TABLE 2. TURBULENCE-CAUSED DIFFUSION, GROUPED READINGS
Distance-range (km) Average distance (km) No. of readings in group Average o~[x Average Or/a,s Typical oy (m)
4.5-11 9 6 0.059 0.063 530
13-18 16 15 0.054 0.051 860
23-30 24 8 0.038 0.036 910
230
G.T. CSANADY
In regard to the growth of % with distance in these turbulence-dominated trials we have earlier pointed out the unreliability of d % / d x estimates derived from individual trials. We may expect to obtain somewhat better results from the average values in TABLE 2 ; % calculated f r o m average distance x and average % I x ratio is shown in the last row of the table. While the numerical values cannot be accepted as entirely accurate, they do suggest that beyond about 16 k m the rate of growth of % slows down and may even approach the theoretical x ~/2 law. I f values of some additional trials in TABLE I (for which 0r/B, s estimates were not available) are included in the averaging, one obtains the modified "typical" values % = 815 m at 16 k m and 930 m at 24 km, a ratio of 1.15, as against 1.23 for a square-root relationship. It may be noted here that FUQUAY et al. (1964) also claim that at distances of this order of magnitude the square-root relationship is approached, although this does not appear to be obvious f r o m the experimental data they quote. Typical values of % I x reported by FUQUAY et al. at distances comparable to those discussed here (12-25 km) are, generally, very similar to the values of TABLE 2, although as PASQUILL(1969) has shown, they begin to show demonstrable shear effects beyond 12 km. As far as one can judge from the data in TABLE 2, a similar influence is not evident in the majority of the Arkansas trials. I f we accept that an asymptotic state ( x ~/2 law for %) has been reached at the 16 km distance from the release line, we can calculate an effective eddy diffusivity from the relationship U O'y2
K,
-
2x
(3)
Using the apparently "typical" convection velocity U = 3 m s -1, % / x = 0.051, % = 815 m we find Ky : 6.2 × l0 s cm z s - L Such an asymptotic value of the diffusivity also implies a certain value of the Lagrangian time-scale of eddies because by Taylor's well known theorem Ky = v 2 tL
(4)
where v is the mean square turbulent velocity across wind. Estimating the typical (vz) I/2 at 30 cm s - 1 we find a Lagrangian time scale of 11.5 min, which is about half the average release ("sampling") period. The order of magnitude equality of the sampling period and the Lagrangian time scale is on intuitive grounds a reasonable result. F o r longer releases one would certainly expect larger % and _Ky. We may add here for comparison that the value of K, derived f r o m the data of trial 12-Y-2 (greatest shear effect) by the above equation (3) is close to l0 T cm 2 s -~, or about 15 times more than our "typical" Ky caused by atmospheric diffusion at distances of 15-30 km, for a release period of order 30 min. REFERENCES CSANADYG. T. (1969) Diffusion in an Ekman layer, or. atmos. Sei. 26, 414-426. FUQUAYJ. J., Sr~mSONC. L. and HINDSW. T. (1964) Prediction of environmental exposures from sources near the ground based on Ha~ord experimental data. or. appl. Met. 3, 761-770. HmST G. R. et al. (1969) Final report on "Woodlot" experiments. Travelers Research Center, Hartford, Conn.
Cross-wind Shear Effects on Atmospheric Diffusion
231
PASQUILLF. (1962) Atmospheric Diffusion, 267 pp. Van Nostrand, New York. PASQtnLL F. (1969) The influence of the turning of wind with height on crosswind diffusion. Phil. Trans. R. Soe. A 265, 173-181o RA~,tOR G. S., OGDEN E. C. and HAYES J. V. (1970) Dispersion and deposition of ragweed pollen from experimental sources. 3". appL Met. 9, 885-895.
APPENDIX Diffusion from a finite line source The line source may be considered to be a superposition of point sources, each of strength q dy', if q is the strength of the line-source per unit length. As diffusion along x (the wind direction) may be neglected in such problems, the length of the source perpendicular to the wind (projected to the normal) is the relevant quantity. Let this "effective" length be b; than a Gaussian diffusion model yields for ground-level release: bl2
X
q
~rayozU
f
exp --
20, z
--b[2
J
dy'.
Here the origin of the coordinates is at the center of the release line, X is concentration, U is wind speed and ay and tr= are standard deviations of dispersion respectively in the crosswind and vertical directions (both functions of x). Carrying out the integrations yields: q
U [eft (b[2 -- y~
. [b/2 + y~]
On the axis of the plume (y = 0) we have b xo = (V2,0 ,,, At large distances x the argument of the error function becomes small and X, may be approximated by: qb -+ oo). Xa ~r try tr= U (x This is equivalent to a point-source formula with qb being the total source strength. The width w of a diffusing plume is often defined as the distance between the two points where the concentration drops to 10 per cent of the axial value. Here we find:
x... =0, X,
=
1
[
w
2 erf(b/Z(v'2)try) err ( ~ )
(
+ w
+ erf \ 2 ( a / 2 ) % / j .
In observations we measure w from a fitted curve to the cross-wind profile, while b is known. The last relationship then represents an implicit equation for the determination of try. At large enough distances from the source we have b/try ~ 0; there the concentration ratio tends to the point-source formula: 0.1
Xe
W2
~ ----exp { - - ( ~ ) }
from which we find W
2 (V'2) oy
1.52
or
4.3 W
232
G . T . CSANADV
2.75
ASYMPTOTE: w
b --+0.906
2.50
2.25 -r I.(3 1:3 :3 0 _J (J
2.0
1.75
'Sl{= 0
q 0.25
I
r
r
O. 5
0.75
1.0
LINE SOURCE LENGTH
, 1.25
r
l
1.5
1.75
2.0
b 2~O"y~/:~
FXG. AI.
This is, of course, a well-known rule. Closer to the source, however, where a "solid core" still exists, the more complete implicit equation has to be used to arrive at %. This implicit relationship may be regarded (by the equation quoted above) to be a function of the form: w 2 ( v/2) %
f ( / b \ ~ , "
The relationship in this form is shown in Fro. A1. At large values b/2(v/2)a~ the function tends to 2 (v'2) % -- 2 (~/2) % + 0.906
~ oo .
When, for a given section, w is observed and b is known the straight line W
2 (~/2) oy b
w b
known constant
2 (,/2) ,,, intersects the curve shown in FIG. A1 at a point which yields the value of % as the solution for that section. It should be noted here that the entire theoretical argument applies to the ensemble-average con¢entration field of an (instantaneous) line source. In the observations the width w is that of a single realization, which differs from the ensemble average cloud width in a random manner. To use this observed width is clearly far from ideal, but this is usually all one can do in dealing with field data. The repetition of the experiment many times under identical ambient conditions is rarely possible.
C R O S S W I N D SHEAR EFFECTS ON A T M O S P H E R I C DIFFUSION* G. T. CSANADY Environmental Fluid Mechanics Laboratory, University of Waterloo, Waterloo, Ontario, Canada (First recieved 25 May, 1971 and in final form 4 October 1971)
Abstract---Some experimental data on atmospheric diffusion over a "medium" distance range (order 30 kin) have been analysed in an attempt to discover any effects on lateral spread attributable to the skewed wind profile of the planetary boundary layer. The observations analysed were obtained over "natural" terrain (containing woodlots and villages) in Arkansas. Wind shear effects on diffusion were considered "significant" when the cloud decisively outgrew the size predicted by the Hay-Pasquill technique from fixed point cross-wind velocity fluctuations Such "significant" effects were found on few occasions only, associated with quite extreme rates of cross-wind shear, accompanied by strong stability. INTRODUCTION IT MAY be expected that when an atmospheric cloud of tracer or pollutant is deep enough to occupy a substantial fraction of the E k m a n layer, its lateral spread will be influenced or possibly dominated by the variations of the mean velocity vector along the vertical ("directional shear"), instead of being determined entirely by atmospheric turbulence. When the tracer is released at ground level, the upper edge of the cloud moves vertically at a velocity of the order of the friction velocity u* = ~¢/(~'o/p) (which is also the order of the r.m.s, turbulent velocity). The E k m a n layer thickness being approximately 0.2 u * / f ( f = Coriolis parameter), it thus takes a period of order 0.2f 1/2 h at mid-latitudes for such a cloud to become large enough for shear effects to be appreciable. Typically, a cloud travels 10 km f r o m its point of release in a period of this order. Thus shear effects may begin to develop at such diffusion distances and could become dominant at distances of the order of 100 kin. Similar conclusions may be derived by more detailed argument based on an idealized E k m a n layer model (CSANADY,1969). In an analysis of the few available experimental data PASQUILL 0969) has concluded that effects at ground level, attributable to cross-wind shear, were not of practical importance at distances below about 12 kin, but they began to dominate diffusion beyond about 25 km. These conclusions were based mainly on the data of FUQUAY et al. (1964) and it would be desirable to subject them to further checks. Recently, the staff of the Travelers Research Centre have carried out a series of diffusion experiments, code named " W o o d l o t " over " n a t u r a l " rural terrain in Arkansas to downwind distances of the order of 30 k m (HaLST et al., 1969). According to the above estimates, shear effects on diffusion should be noticeable at such distances The purpose of the present paper is to analyse the Woodlot long-distance diffusion data and exhibit the effects of shear on diffusion, where such effects may be found. The experiments were not designed exclusively for the purpose of providing long-term diffusion data: there was as much or greater interest in effects on diffusion close to the * This work was carried out during a visit to Travelers Research Center, Hartford, Connecticut. 221
222
G.T. CSANADY
source due to finite source size and to finite size obstacles such as isolated woodlots. Thus the data are not ideal for our purpose here, but in view of the scarcity of information on thisttopic their analysis seemed worthwhile.
DETERMINATION OF LATERAL SPREAD In the long-distance diffusion experiments of the Woodlot series ground level dosages were determined at a grid of stations extending some 30 km downwind of the release line. At distances beyond about 10 km the tracer cloud was usually wide enough for the gridpoints to yield a cross-wind dosage profile with reasonable resolution, so that the cross-windgrowth of the cloud could be followed, provided that the wind was indeed of the right direction for the cloud to pass over the grid and provided that not too many sampling instruments failed. A substantial number of trials yielded useful information of this kind, although in some cases only a single cross-section was fully covered by the sampler grid as actually laid out. In the following the x-coordinate axis is taken to be in the surface wind direction, the y-axis horizontally across wind, and the centre of the source is at the origin. The release took place along a line of 800 m length, appropriately chosen upwind of the grid, from a moving truck carrying a dispenser, for a release period of about 25 min. The angle between the release line and the wind direction varied from 0 to 45 ° or so, so that the "effective" length (perpendicular to the wind, i.e. along the y-axis) varied from about 500 to 800 m. At large enough distances from the release line one would expect the cloud to behave as if it was released from a point source. In the distance-range of interest this was not entirely the case: in some instances a "solid core" of high dosage could be identified at the edges o f which the dosage dropped very rapidly to zero. Diffusion from a line-source of finite length may be treated by regarding each element of the line as a point source, and superimposing the many point source clouds, in each of which the lateral ( = along the y-axis) dosage distribution is Gaussian with standard deviation %. This point source standard deviation is the fundamental quantity with which simple theories of atmospheric diffusion are concerned (see e.g. PASQUILL, 1962). In extracting data on ey from the observed spread of a line source cloud some care is necessary, although the procedure is straightforward and is outlined in the Appendix. The distinction between point-source % and "standard deviation" of an entire cloud released from an extended source is not always made clear and may lead to confusion (RAYNOR et al., 1970). At large distances from the source, of course, the distinction becomes unimportant. The primary variable obtained in the analysis of the experimental data is the " I 0 per cent width" of a cross-wind profile at a given distance from the source, i.e. the distance between the two points at the edges where the concentration drops to 10 per cent of the maximum in that section. As there were often only few dosage measurements per cross-section to define a profile (some of these being, moreover, subject to a certain random scatter) it was only possible to obtain a 10 per cent width estimate from a more or less subjectively fitted curve. The accuracy of this procedure probably varies considerably, but in the case of the better defined profiles (usually at the larger distances) it is quite reasonable, perhaps of order I0 per cent. The actual magnitudes of % derived from the 10 per cent width by the method outlined in Appendix I should
C r o s s - w i n d S h e a r Effects o n A t m o s p h e r i c D i f f u s i o n
223
thus be, on the whole, reasonable estimates. However, in attempting to assess the growth of % with distance, the inaccuracy of the individual readings is a more serious handicap. The gradient, d%/dx could not, in fact, be determined for individual trials with any accuracy. The errors were particularly large at the smaller distances, where the cloud was sometimes only of the order of twice the release line width. Sample dosage profiles (across wind, at ground level), are shown in FIG. 1. Values of ~y determined in the above manner from the results of the trials are shown in TABLE 1, the starred values being judged to be rather less accurate than the others (say, typically -f- 30 per cent, although some may be even worse.)
o X =6km I 0 % width : 1 5 0 0 rn A - X = 16 km
I 0 % width: 4 6 0 0 m
E -%.
2ooo
400
'E
§
E c
.-.9.
i~. I000
t~ 0
I ,7 l I
2
3
I
0
I
2
Cross wind distoncekm,
3
4
5
o
6
orbJtraryorgin
FIo. 1. Sample cross-wind dosage profiles. CONVECTION
VELOCITY
In order to decide whether the observed spread can be attributed to atmospheric turbulence alone (with no appreciable contribution from wind shear) it is necessary to have some information on wind speed and horizontal turbulence. The spread due to atmospheric turbulence may be calculated from the relationship, discussed in detail by PASQUILL(1962): ~, x
_
vr.~
(I)
U
where U is the (assumedly constant) convection velocity of a diffusing cloud and vr.s is a root mean square turbulent velocity in the y-direction, obtained from moving average velocities of drifting particles over "travel time" T, the mean-square being formed over the "sampling time" s. In the atmosphere, of course, wind speed is not constant but varies with height and the choice of a suitable convection velocity from a wind profile measured at one location would present serious problems. Therefore, estimates of the convection velocity were obtained from the dosage data collected at the few "sequentializer" stations deployed for the trials, which gave
224
G.T.
CSANADY
c),~E
c~
~J
~ ~ oc ~ co ~ ~
o o ~o
'=~ "=~ ~
o o
~-=~ '=~ "==~ ,"~ ¢'~
I
t~
~A Z "O
z
I Z
,-I
O N O
~
oo~
~
,=~ ¢'~
o o o ~kO
I I
Cross-wind Shear Effects on Atmospheric Diffusion ~4 O
b$
o o o
ooo$ I
~ o ~ o o. . . .~. .o o o $
~ oooo$oo$$ o o
I
I
E i .= ¢.J 0J
o O O O O O O O O O O O O O O O O O O O O O O O O O
dddddNNNNoNNddNNdNNNNdNNd~
°ooooooooooooooo z
o
oddddc~ddc~dddddo¢~
E
~oo~o~
OO
o~o
**~
**
~ O O
6 Z O
o L
.O
E
O
Z
225
226
G . T . C~ANADY
dosages in 10 half-hour periods following release. The time of the maximum reading provided a crude estimate (-b 15 rain) of the effective "arrival time" for the dosage cloud. At least for travel times of 2 h and more the velocity calculated from the distance, release time--sequentializer station, and the travel time should be a reasonable approximation to a convection velocity. It may be noted, however, that the estimates provided by different sequentializer scattered strongly and that there was also a fairly systematic increase of convection velocity with distance (which makes sense physically for a growing cloud). TABLE 1 lists what appeared to be best estimates of convection velocity determined on this basis. Where this estimate was not available or was obviously wrong, an estimate of the wind speed on the basis of Rawinsonde data is given (starred), using data of the 30 m level. In general, the accuracy of these latter estimates of convection velocity or more directly, of the travel times they imply must be classed as quite low (perhaps within a factor of 2). HORIZONTAL TURBULENCE DATA As originally defined Vr.s in equation (1) refers to the Lagrangian velocity of drifting particles. It is, however, normally approximated by a moving average of fixed point (Eulerian) velocities over a fraction of the travel time, T/fl, the constant being an uncertain quantity. A common assumption is to take/3 ---- 4, and it is known that the exact value of fl has relatively little influence on the results." The ratio v/U is to first order equal to the azimuth angle 0 of the wind, so that equation (1) may be approximately converted into x
= Or/B.s
(2)
where 0r/B.~ is now the (Eulerian) r.m.s, azimuth angle (measured from the " m e a n " wind direction), averaged over T/~ and then "sampled" (i.e.r.m.s. value of the moving average determined) over period s. The practical merit of this particular formulation is that r.m.s, azimuth angle determined in this manner changes only slowly with the averaging time T/E, a quantity the appropriate value of which is uncertain: we mentioned before the inaccuracy of determining the travel time T, to which comes the arbitrariness inherent in the choice of 8. Thus, while the estimated value of T/B may easily be in error by as much as a factor of 4, this may not cause an error in our estimate of 0r/B.~ greater than by a factor of 1.5 or 2. Azimuth angle readings obtained at the 90 m level at the centre of the experimental grid were utilized to estimate values of 0r/B.~ for comparison with observed %Ix ratios. The data available were 5-rain average wind directions. By averaging 2, 3, etc. successive readings 10, 15, etc. rain moving averages were obtained. The release period (equivalent to the theoretical "sampling'"period) was a little under 30 min so that there were only 6 moving averages with their beginnings contained within the release period or close to it. To obtain a r.m.s, value from 6 readings is not possible with any accuracy; thus the difference between maximum and minimum was determined and divided by 4 to provide a crude estimate. This is the value 0r/B.~ listed in TABLE 1. fl = 4 was used and some crude interpolation was also necessary to obtain 0r/B.s for a desired T/ft. For all releases, 6 readings were used in arriving at an estimated 0riB.~ the closest available to the recorded release period in each case, regardless of the exact length of the release period.
Cross-wind Shear Effects o n Atmospheric Diffusion DETECTION
OF SHEAR
227
EFFECTS
Given the crudeness of the method used in arriving at an estimate of 0T/B,s it is surprising that the values of this quantity are generally as close to ~/x as they are. F o r example, the eleven readings of ~y/x in TABLE 1 for 16-17 km distance, for which 0r/B, s values are available for comparison, average out at 0.0587 as against an average 0r/n,~ of 0.0514 for the same entries. Thus the procedure adopted above as estimating a "relevant" value of 0r/a, ~ was generally successful, even if the individual estimates are not always accurate and apparently subject to considerable random scatter. One would also be tempted to conclude that shear effects are evidently minor, if we are able to account for observed spread by an analysis of turbulence data alone. A closer examination of the data reveals, however, that the grossest discrepencies between %Ix and OriB.~in TABLE 1 may be certainly attributed to shear effects and that it is also probable that cloud growth was effectively increased in some other tests less obviously out of line in TABLE 1. T o decide which of the trials were in the shear-dominated category we start by focussing attention on those in which %Ix was substantially (several times) greater than 0r/a.~.* The most obvious such case in TABLE 1, for example is trial WT12-Y-2, where the ratio of the two quantities in question is 23; there are, however, many others in which the ratio is high, and in favor of %Ix .Such a trial will be regarded as "shear influenced" if the following two further criteria are satisfied: (1) Rawinsonde data taken during the experiment show substantial directional shear of wind in the lowest few hundred meters of the atmosphere. (2) The cross-wind dosage profile exhibits evident skewness (this is a marked property of shear-augmented diffusion), which often also appears as a two-peaked distributionJ" Examples of such distributions with substantial skewness are shown in FIG. 2.
. - 400 E
=-
'YI
A
"G
21-Y-I A
g g 200 o a
0
25
5,0
0
25
5.0
Cross- wind distance in km, arbitrary origin
FIG. 2. Cross-wlnd dosage profiles showing substantial skewness. * Because there were a number of substantial deviations in the opposite direction, Orta,, >
~y/x,
i.e. overestimatesof spread, further criteria were sought to support the hypothesis that a given underestimatewas at least partly caused by wind shear. t Due, presumably, to the irregularity of individual diffusing clouds; one part of the cloud may have travelled at a different level from the rest, moving in a slightly different direction.
G. T. CSANADY
228 Testl2, 20 °0 hours
Test 16, 20°°hours
"~ 1.5 0
+1.5
NN~561 3O 45
366
76
274 198 213
Height, m I
I I
Test 21, 17°0 hours
I 2
] 3
m s-~
I 4
I 5
FIG. 3. Hodographs of velocity vector on occasions of shear-im~luenced diffusion (from Rawinsonde. Test 21 17"0hours
Test 12, 20 °`' hours IA°C Test 16, 20 °o hoursll.0°C Test 21, IT°°hours 12.0°C
40C
300 E
J= c~ T 200 "~
. Test 12 ~ 2 0 °°hours Test 16 1
~
0°°hours
100
I -
0
2
I 4
I
T
6
8
Teml~emture differenceegeinsfgroundlevel, =C
F[o. 4. Temperature profiles corresponding to Hodographs of FIG. 3.
Cross-wind Shear Effects on Atmospheric Diffusion
229
The results from 5 runs, responsible for 8 entries in TABLE 1 satisfy these criteria; an idea of the magnitude of wind-shear for these is given in the Table's last column. Hodographs of the velocity vector derived from Rawinsonde observations closest in time to these trials are shown in FIG. 3. It may be seen that the most pronounced shear occurred on occasion of trials 12-Y-2 and 12-G-2, which, as pointed out before, are characterized by a ~r/x ratio furthest from their 0r/~. s estimates. The average value of %,Ix for the eight shear-influenced entries in TADLE 1 is 0.10, while the average 0r/a, s estimate for these is 0.023. One derives the estimate that cross-wind spread was increased typically 4-5 fold in these "shear-influenced" trials, as compared to the expected effects of atmospheric turbulence alone. As to the frequency of shear-influenced diffusion we note here that only 5 out of the 26 trials that could be analysed in this manner exhibited dominant shear effects. In those cases when the shear effect was evident, the change of wind direction with height, as recorded by the Rawinsonde, was marked indeed and was not of the classical Ekman layer type (in Test 12 the sense of the change was opposite to that in an equilibrium E k m a n layer) but was associated with a moderate to strong inversion. The temperature profiles taken on the same Rawinsonde flight as the velocity vectors in FIG. 3 are shown in FIG. 4, which lists times of the tests and ground level temperatures. A referee drew attention to the discrepency between the average 0r/D,s for the shearinfluenced trials (0.023) and the average value of the same quantity for the remaining trials (about 0.05). A low average turbulence level does not seem inconsistent with the rapid variation of wind with height. TURBULENCE-CAUSED DIFFUSION When the "shear-influenced" readings in TABLE 1 are deleted one may attempt to remove the remaining r a n d o m scatter (presumably caused by the considerable errors possible in our estimates) by averaging over more or less coherent groups of readings. All the entries fall naturally into one of three distance ranges, " s h o r t " = < 11 km, " m e d i u m " = 13-18 kin, and "long" > 23 km. Averaged values of oy/x and the quantity OT/a.sfor these groups are shown in TABLE 2. Clearly, the averaged values of %/x and 0r/0,s agree quite well, both as regards absolute magnitude and variation with distance. One may conclude that in these trials the diffusion may be attributed to atmospheric turbulence alone (with no shear contribution) and that the Hay-Pasquill prediction method based on the azimuth angle variance supplies good estimates of cloud spread out to distances of order 30 km. This is further p r o o f (if such is indeed necessary) for the proposition that any effects the increased roughness of " n a t u r a l " terrain may have on diffusion are exerted through influencing atmospheric turbulence. TABLE 2. TURBULENCE-CAUSED DIFFUSION, GROUPED READINGS
Distance-range (km) Average distance (km) No. of readings in group Average o~[x Average Or/a,s Typical oy (m)
4.5-11 9 6 0.059 0.063 530
13-18 16 15 0.054 0.051 860
23-30 24 8 0.038 0.036 910
230
G.T. CSANADY
In regard to the growth of % with distance in these turbulence-dominated trials we have earlier pointed out the unreliability of d % / d x estimates derived from individual trials. We may expect to obtain somewhat better results from the average values in TABLE 2 ; % calculated f r o m average distance x and average % I x ratio is shown in the last row of the table. While the numerical values cannot be accepted as entirely accurate, they do suggest that beyond about 16 k m the rate of growth of % slows down and may even approach the theoretical x ~/2 law. I f values of some additional trials in TABLE I (for which 0r/B, s estimates were not available) are included in the averaging, one obtains the modified "typical" values % = 815 m at 16 k m and 930 m at 24 km, a ratio of 1.15, as against 1.23 for a square-root relationship. It may be noted here that FUQUAY et al. (1964) also claim that at distances of this order of magnitude the square-root relationship is approached, although this does not appear to be obvious f r o m the experimental data they quote. Typical values of % I x reported by FUQUAY et al. at distances comparable to those discussed here (12-25 km) are, generally, very similar to the values of TABLE 2, although as PASQUILL(1969) has shown, they begin to show demonstrable shear effects beyond 12 km. As far as one can judge from the data in TABLE 2, a similar influence is not evident in the majority of the Arkansas trials. I f we accept that an asymptotic state ( x ~/2 law for %) has been reached at the 16 km distance from the release line, we can calculate an effective eddy diffusivity from the relationship U O'y2
K,
-
2x
(3)
Using the apparently "typical" convection velocity U = 3 m s -1, % / x = 0.051, % = 815 m we find Ky : 6.2 × l0 s cm z s - L Such an asymptotic value of the diffusivity also implies a certain value of the Lagrangian time-scale of eddies because by Taylor's well known theorem Ky = v 2 tL
(4)
where v is the mean square turbulent velocity across wind. Estimating the typical (vz) I/2 at 30 cm s - 1 we find a Lagrangian time scale of 11.5 min, which is about half the average release ("sampling") period. The order of magnitude equality of the sampling period and the Lagrangian time scale is on intuitive grounds a reasonable result. F o r longer releases one would certainly expect larger % and _Ky. We may add here for comparison that the value of K, derived f r o m the data of trial 12-Y-2 (greatest shear effect) by the above equation (3) is close to l0 T cm 2 s -~, or about 15 times more than our "typical" Ky caused by atmospheric diffusion at distances of 15-30 km, for a release period of order 30 min. REFERENCES CSANADYG. T. (1969) Diffusion in an Ekman layer, or. atmos. Sei. 26, 414-426. FUQUAYJ. J., Sr~mSONC. L. and HINDSW. T. (1964) Prediction of environmental exposures from sources near the ground based on Ha~ord experimental data. or. appl. Met. 3, 761-770. HmST G. R. et al. (1969) Final report on "Woodlot" experiments. Travelers Research Center, Hartford, Conn.
Cross-wind Shear Effects on Atmospheric Diffusion
231
PASQUILLF. (1962) Atmospheric Diffusion, 267 pp. Van Nostrand, New York. PASQtnLL F. (1969) The influence of the turning of wind with height on crosswind diffusion. Phil. Trans. R. Soe. A 265, 173-181o RA~,tOR G. S., OGDEN E. C. and HAYES J. V. (1970) Dispersion and deposition of ragweed pollen from experimental sources. 3". appL Met. 9, 885-895.
APPENDIX Diffusion from a finite line source The line source may be considered to be a superposition of point sources, each of strength q dy', if q is the strength of the line-source per unit length. As diffusion along x (the wind direction) may be neglected in such problems, the length of the source perpendicular to the wind (projected to the normal) is the relevant quantity. Let this "effective" length be b; than a Gaussian diffusion model yields for ground-level release: bl2
X
q
~rayozU
f
exp --
20, z
--b[2
J
dy'.
Here the origin of the coordinates is at the center of the release line, X is concentration, U is wind speed and ay and tr= are standard deviations of dispersion respectively in the crosswind and vertical directions (both functions of x). Carrying out the integrations yields: q
U [eft (b[2 -- y~
. [b/2 + y~]
On the axis of the plume (y = 0) we have b xo = (V2,0 ,,, At large distances x the argument of the error function becomes small and X, may be approximated by: qb -+ oo). Xa ~r try tr= U (x This is equivalent to a point-source formula with qb being the total source strength. The width w of a diffusing plume is often defined as the distance between the two points where the concentration drops to 10 per cent of the axial value. Here we find:
x... =0, X,
=
1
[
w
2 erf(b/Z(v'2)try) err ( ~ )
(
+ w
+ erf \ 2 ( a / 2 ) % / j .
In observations we measure w from a fitted curve to the cross-wind profile, while b is known. The last relationship then represents an implicit equation for the determination of try. At large enough distances from the source we have b/try ~ 0; there the concentration ratio tends to the point-source formula: 0.1
Xe
W2
~ ----exp { - - ( ~ ) }
from which we find W
2 (V'2) oy
1.52
or
4.3 W
232
G . T . CSANADV
2.75
ASYMPTOTE: w
b --+0.906
2.50
2.25 -r I.(3 1:3 :3 0 _J (J
2.0
1.75
'Sl{= 0
q 0.25
I
r
r
O. 5
0.75
1.0
LINE SOURCE LENGTH
, 1.25
r
l
1.5
1.75
2.0
b 2~O"y~/:~
FXG. AI.
This is, of course, a well-known rule. Closer to the source, however, where a "solid core" still exists, the more complete implicit equation has to be used to arrive at %. This implicit relationship may be regarded (by the equation quoted above) to be a function of the form: w 2 ( v/2) %
f ( / b \ ~ , "
The relationship in this form is shown in Fro. A1. At large values b/2(v/2)a~ the function tends to 2 (v'2) % -- 2 (~/2) % + 0.906
~ oo .
When, for a given section, w is observed and b is known the straight line W
2 (~/2) oy b
w b
known constant
2 (,/2) ,,, intersects the curve shown in FIG. A1 at a point which yields the value of % as the solution for that section. It should be noted here that the entire theoretical argument applies to the ensemble-average con¢entration field of an (instantaneous) line source. In the observations the width w is that of a single realization, which differs from the ensemble average cloud width in a random manner. To use this observed width is clearly far from ideal, but this is usually all one can do in dealing with field data. The repetition of the experiment many times under identical ambient conditions is rarely possible.