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Groundlevel concentration fluctuations from a buoyant and a non-buoyant source within a laboratory convectively mixed layer
Ar,,,os,,he,,c
Enc,rownen,
Vol IS.
No
7. pp
1297-1309.
1984
ooo4-6981184 $3 00 + 0.00 Q 1984 Pcrgamon Press Ltd
Pnntcd I” Great Bntam
GROUNDLEVEL CONCENTRATION FLUCTUATIONS FROM A BUOYANT AND A NON-BUOYANT SOURCE WITHIN A LABORATORY CONVECTIVELY MIXED LAYER J. W. DEARDORFF and G. E. WILLIS Department of Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, U.S.A. (Ftrsr receitied 18 July 1983 and infinolfirm
13 February 1984)
Abstract-With the use of mixed-layer scaling, the near-surface mean concentration and the concentration fluctuations at moderate distances downwind of a stack emitting a highly buoyant efRuent are obtained and compared quantitatively with those from a non-buoyant source that is otherwise Identical. The environment ISa laboratory mixed layer in a state of free convection with mean wind simulated by towing the stack along a horizontal line. Although the buoyant effluent has a much smaller maximum mean groundlevel concentration occurring at a greater downwind &stance than the non-buoyant eflluent, the decay of mean concentration from that pomt downwind is found to be exceedingly slow in the buoyant case. The cumulative frequency distributions of sampled concentrations greater than zero for these same two cases are found to be log-normal in only a bimodal sense, with the weaker concentrations having a greater logarithmic range in amplitude than the stronger concentrations. The ratio of the root-mean-square (r.m.s.) concentration fluctuation to the mean concentration, in the central portion of the mean plume, is found to decay with downwind distance. Homogenization associated with both downstream and vertical mixing IS believed responsible for the rapid decay observed. Key word index: Concentration fluctuations, ground level concentrations from buoyant and non-buoyant sources, mixed-layer dispersion, relative concentration intensity.
1. INTRODUCTION The effluent concentrations observed at ground level downwind of tall stacks under conditions of thermal convection are of primary concern to researchers of diffusion. Highest concentrations are believed to occur under these conditions (Bierly and Hewson, 1962; Irwin and Cope, 1979; Venkatram, 1980) for a given wind speed and source strength. The efficiency of the convective motions in carrying portions of elevated stack plumes downwards is responsible for their early appearance at the ground. The thermal-convection tank is proving to be a valuable laboratory apparatus for studying dispersion within a convectively mixed boundary layer (Lamb, 1982) under conditions when the lower region of shear induced turbulence (z < -L) occupies only a small fraction of the total boundary-layer depth (h). In that case h + - L, where L is the Monin-Obukhov length, and thermal convection dominates the turbulence production at heights exceeding -L (Wyngaard er al., 1971). The condition h % - L occurs frequently in the daytime over land under sunny or partially sunny conditions, and it is also common for the stack height itself to exceed -L. For these reasons, a laboratory convection tank is a useful tool even though no mean flow exists within its convective boundary layer (L = 0).
A re-analysis of Prairie Grass data using mixed-layer
scaling (Nieuwstadt, 1980) has provided supportive evidence for the laboratory results using mixed-layer scaling in the case of the mean dispersion from a nearsurface non-buoyant source. Also, recent measurements at the Boulder Atmospheric Observatory for a non-buoyant source located at the height z = h/2 have largely verified the laboratory results for this source height (Moninger et al., 1983). Of special interest in both cases is the lack of mean-plume Gaussianity in the vertical direction. The concentration fluctuations are also of great interest, since the mean concentration distribution (from data averaged over a time period with relatively uniform surface conditions and relatively constant wind velocity) gives little information about the nearpeak values of concentration and associated toxicity which may occur over short time periods and in small sample volumes. The present study therefore emphasizes the concentration fluctuations (near ground level) measured at various scaled distances downwind of the stack. Although the non-buoyant stack plume is a most important reference case for study, primary interest resides in the dispersion of buoyant plumes (e.g. Weil and Jepsen, 1977; Manins, 1979; Venkatram, 1980; Kerman, 1981) since the majority of elevated point sources are smoke stacks emitting warm gases. Therefore, in this study we compare near-surface concentration data (the mean and fluctuation statis-
1297
I W. DEARDOWI
1298
and G.
E.
WILLIS
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tics) from a highly buoyant source with those from an
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_
.:.
-_
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otherwise identical non-buoyant source. Our emphasis here is on near-surface concentrations. For the effects of plume buoyancy upon plume rise and upon the mean plume envelope aloft as measured within the laboratory convectively mixed layer, the reader is directed to a recent paper by Willis and Deardorff (1983). The degree of plume buoyancy chosen for study here IS relatively large, so that most observed buoyant plumes from tall stacks will undergo behavior intermediate to the non-buoyant and buoyant cases studies. It would obviously be desirable to study the quantitative behavior of plumes of intermediate buoyancy also, but this must await future multi-year investigations.
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STACK \
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2. EXPERIMENTAL METHODS The laboratory model is descnbcd in detail elsewhere (Wittis and Deardoti, 1974) Briefly, it consists of a water tank, 1.24m on a side, with a convectively mixed layer of depth, It, of 0.232 m mean height, and an overlying stable layer with stratitI&on of 0.05 C mm-’ in the experiments to be reported. This m&Mayer turbulence is driven by a Iarge surface heating rate of about 5 kW m- ‘. The model includes no mean flow and thus no mean shear-generated turbulence. Favorable comparisons of atmospheric mixed-layer turbufence statistics with those from the model appear in Willis and Deardozff (1974), Caughey and Palmer ( 1979),Wyngaard and LeMone (f980), Lenschow et ai. (1980). Nichoils and LeMone (1980) and Caughey (1982). The effects of a uniform mean wind m bending over a vertically emitted plume are simulated in the water tank by towing a mode1 stack of height 2, = 0.031 m at a horizontal speed of U = 0.2 w,, where w, is the stack-exit velocity of 0.1 m s - I. Thedimensionfess release height is zJh = 0. I34. A dimensionless ‘stack buoyancy’ parameter, F, . is defined as F, = w,rf (S/PO)(PO-P&W:
uh),
(1)
where r, is the stack radius of I.0 mm, (p. - pS) is the initial effiuent density deficit relative to the ambient fluid, equal to O&41p0 for the buoyant plumes to be treated and zero for the non-buoyant plumes; and w+ is the free-convection velocity scale (Deardorff, 1970)whose value avera8ed 9 mm s- ’ in the present experiments. Values of F, used in this study are 0 for the non-buoyant releases and 0.11 for the buoyant releases. The latter value is typical for a large industrial plant stack, such as that at Sudbury, Ontario, ona summer day with winds near 3-4 m s-l. Willis and Deardorff (1983) found F, to be the most appropriate dimensionle~ stack-buoyancy parameter, under convective conditions, in governing plume behavior for X exceeding a few tenths. In (I), the numerator relative to U determines the potential for stack e!Iluent to rise due to its buoyancy, while the denominator, w: h, is a measure of the ability of convective ~un~r~layer turbulence to break up and dilute the plume over the height h. A planform top view of the tank and expertmental arrangement appears In Fig. 1. A high intensity argon-ion (blue) laser beam extended the w&h of the tank in the J direction at a height of 0.020 m. Due to index of refraction variations near the heated lower boundary the beam could not be positioned iower without encountering excessive vertical displacements. This height of 0.086 h, where concentration measurements were made, is here referred to as ‘ground-level”. After the lower-surface heating had been imtiated and the convective motions had become fuily established, the plume emission and stack translation were begun. When the stack
Fig. 1. Top planform depiction showing the towed stock, diffusing stack phime, and the tiscr beam intersecting the plume @a&cd line) where fluorescence occurs. Tests by Willis and Deardorff (1983) showed that there was no significant effect upon the phune dispersion or ambient turbulence associated with the moving towing tube (to left of stack) which rested on the bottom surface of the convection tank. had been towed some distance in Y beyond the locatmn of the laser beam the turbulent convective motions will have carried portions of the plume down to a height where they can intercept the blue beam {dashed line of Fig. 1). The stack plume (water-alcohol mixture) contained a small amount of RhodamineS dye which fluoresces when energized by the laser beam. A photo-multiplier detector, equip@ with an optical fitter whose window passed light in the fluorescent wavelengths, viewed a narrow section of the laser beam along y. This detector was located a small height above the water level in the tank, and was periodically traversed along the axis of the beam (see Fig. 2). The intensity of the emitted light is linearly proportional to the efftuent concentration, c. The minimum concentration which couki be measured, determined by the system noise teve1, was about 59; of the vertically well mixed value at y = 0 near the end of the experiment. Details of the laser-dye-fluorescence technique have been reported by Liu et ni. (1977). Traverses across the dispersing plume by the remote detector were made every 7-10s giving a series of concenTroverstng track
P,hotomultlpller
II
tf
II
Stab@
2 ..
I Stock’
2 I L Loser beam
Fig. 2. End view (in y-z plane) of convection tank containing mixed layer and stable water hyer above, and schematically showing buoyant e&ent within the m&d layer. The photomuItiplier detector and its traversing track at top are mounted dire&y above the laser be&n, while thr stack moves along -x (directed into the paper).
Groundlevel concentration fluctuations from a buoyant and a non-buoyant source tration profdes in yand t (time). The plume-fluctuation pattern is assumed frozen at the traverse midpoint time, since the detector translation speed was 20 times greater than the r.m.s. turbulence velocity. Through the use of mixed-layer scaling (e.g. see Willis and Deardorff, 1976) the measured variables were converted into the dimensionless distance downwind from the stack X = (w*/V) (x/h) and dimensionless concentration C = ch= U/S
(2)
where S is the source emission rate. The concentration measurements were averaged over an ensemble of seven experiments for the buoyant case and six experiments for the non-buoyant case, to yield the mean values. From these measurements the lateral spread of the plume at ground level, the cross-wind integrated concentrations, and concentration fluctuation statistics were also obtained. In five additional sets of experiments, both with buoyant and non-buoyant effluent, y-integrated concentrations at the same near-surface height were obtained through continuous measurement of the net in-tank absorption of a He-Ne (red) laser beam directed along y at this height when the stack eflluent contained some added blue due. The dimensionless y-integrated concentration is defined by m CY = s -m
C dylh,
(3)
1299
which can be seen to have unit value if the effluent IS uniformly distributed in the vertica! up to the height (h) utilized in the definition of C, with zero value above. The respective calibration constants (to relate sensor voltages to dimensionless concentrations or to their y-integrated value) made use of this condition, upon thoroughly mixing all the water ofdepth H in the tank at the end of each experiment and substituting H in place of h in (2) or (31.With this method of calibration, the source emission rate, 5, need not be known since only the dimensionless concentrations (involving c normalized by S) were obtained. However, for large values of X in the non-buoyant case (6 < X < 8) it was found that TY from the calibration lacked l&15”,, of reaching unity. Therefore, the calibration constants were increased by lo”, in both the non-buoyant and buoyant cases. There is consequently an absolute uncertainty of about this amount in the values of results to be presented, presumably associated with uncertainty in the effective value of h. The tank side walls begin to influence the diffusion after a particle travel time corresponding to X L 4; hence results will not be presented here for values of X appreciably greater than this. 3.
VISUAL APPEARANCE OF THE PLUMES
Side-view photographs of the buoyant and nonbuoyant stack plumes are shown in Figs. 3(a) and (b), respectively. In looking for effects of stack plume
a
b
no
Fig. 3. Side-view photographs of a buoyant plume (a) and a non-buoyant plume (b). The dark line appearing in the background near height z/h = 0.1 is the intersection of the tank rear wall and bottom plate.
J W DEARDORFFand G. E. WILLIS
1300
buoyancy it is most instructive to view the photographs as a unit. For both cases the stack variables, r,, w,, h, and U were the same, and on a statistical basis both plumes encountered similar environmental turbulence fields. The only quantity that varied for the two cases is (pO- p,), so that F, = 0.11 in (a) and 0 in (b). Immediately after stack exit the plumes assume the familiar bent-over appearance associated with real stack plumes in the atmosphere. Plume looping, characteristic of plumes released into an unstable atmospheric boundary layer, is evident. The plume rise due to the added buoyancy is clearly evident in (a). A quantitative estimate of its average magnitude, as well as that of plumes of lesser buoyancy, is reported in Willis and Deardorff (1983). Perhaps the most striking comparative feature of the photographs is the effect of added plume buoyancy in inhibiting large-scale downward motions of the plume. Clearly, large portions of the non-buoyant plume are carried to ground level at distances near X = 0.5 before encountering plume lift-off due to the surface sweep-out effect of vigorous large-scale updrafts. Initial plume descent to ground level (in the mean) followed by sweep-out is absent for the buoyant plume. With F, = 0.11 the plume buoyancy and associated vertical momentum are still sufficiently great at distances out to X = 1 to counteract largely the downdraft motions of the convection. Later, of course, the stack plume becomes more dilute and can then be carried to ground level by the turbulent motions. For the buoyant case, large portions of the plume apparently become trapped in the upper reaches of the boundary layer (near the inversion level), and then serve for some time as a continuous source of material to be carried downward, as in the process of fumigation. This effect is quite distinct from the elevated concentration maximum occurring in a convectively 1.0
I
mixed layer with a non-buoyant ehluent (see Willis and Deardorff, 1976, 1978). In that case the eievated maximum resulting from the sweep-out effect is often too weak to be very noticeable from individual photographs, and occurs at z/h = 0.75. For the highly buoyant release, the stronger elevated maximum is estimated to lie near z/h = 1.0. when the quantity (gadtJ/&)“’ (we/h) = 8, where the numerator 1s the Brunt-Vaisala frequency within the inversion, 2 IS the coefficient of thermal expansion. and /Z/M+ lb the convective time scale within the mixed layer. A typical value for this ratio is 10 for atmospheric convective conditions. In our laboratory model this ratio can be varied within the range l-20. The smaller this ratio, the greater is the expected penetration of the buoyant effluent above the average top of the mixed layer, all else being equal. The existence of a distinct, identifiable stack plume obviously breaks down for unstable conditions at relatively short downstreamdistances when the turbulent motions break up the plume so effectively that the concept of a plume entity becomes nebulous except in the statistical sense.
4.~m
MEANCONCENTRATIONANDITSLATERAL SPREAD
Smoothed lateral plume spreads, obtained from m OJ
--m
-m
using measured values of c, are shown in Fig. 4 vs dimensionless downwind distance. The r.m.s. standard deviations of the mean (error bars) indicate that the difference between the buoyant and non-buoyant cases is statistically negligible, at least for smaii X, suggesting that the cross-wind spread of the plume is essen-
I
I
_F*=O 0.8
_
__
F,
= 0.11
0.6
Fig. 4. The dimensionless lateral plume spread [square-root of Equation (4) J vs dimensionless downwind distance for the buoyent and non-buoyant stack phrmes. Error bars represent f one standard deviation of the mean from the seven experiments in each ease..
Groundlevel concentration fluctuations from a buoyant and a non-buoyant source tially independent of F,. This conclusion would appear to be counter to the usual assumption of crosswind plume spread being proportional to plume rise which is dependent on F,. However, initial plumespread enhancement due to buoyancy is presumably present at the elevated plume close to the source but appears to be negligible compared to the total spread after the plume reaches ground level much later (X = 0.7). With a tow speed of Cl = 2.2 w* the stack moves a distance of about 1.5 h over the increment AX = 0.7. This distance is roughly equivalent to the characteristic dominant horizontal convection scale (Kaimal et al., 1976; Willis and Deardorff, 1979). Thus, any portion of the plume appearing at ground level within the distance X = 0.7 can be expected on the average to have been under the influence of and rapidly dispersed by a single large-scale (and horizontally diverging) downdraft. Its effects on the plume spread could dominate and mask any spread due to internally generated turbulence. The r.m.s. plume spreads are systematically about 10% smaller, at all values of X, than those we previously measured in the non-buoyant plume studies using the oil-droplet-in-water technique (Deardorff and Willis, 1975). Possibly, the present measurement method slightly underestimates plume spreads by missing extremities of the plume where the dye may be very dilute and consequent light emission weak and undetectable. The measured groundlevel, laterally integrated, dimensionless concentration, F, is shown in Fig. 5. The pronounced effect of plume buoyancy (F, = 0.11) in 4
I
0 r\ /
3-
I
/c : I :
‘\
delaying and hindering the appearance of groundlevel concentrations (glc) is evident. The added plume buoyancy is effective in reducing the observed maximum glc by an order of magnitude. Only for X exceeding 3 or 4 does the laterally integrated glc in the buoyant case reach levels comparable to those attained in the non-buoyant case. Mean centerline (p = 0) concentrations were obtained from the smoothed curves of Figs. 4 and 5 using the Gaussian assumption; i.e. by dividmg cy by (2n)“’ a,/h. They are shown in Fig. 6. The effect of plume buoyancy in reducing the maximum centerline glc is even more pronounced than in Fig. 5 for cY due to the earlier appearance of glc in the non-buoyant case when aYis smaller. Specifically, for F, = 0.11 the peak mean concentration is reduced by a factor of about 12 and the appearance of the maximum is delayed by AX = 0.4 at least, as compared with the non-buoyant case. In fact, this faint maximum in Fig. 6 at X = 1.0 may not be statistically significant, and could reflect small sampling errors or small departures of the assumed lateral distribution from the Gaussian. Also, quantitative comparisons such as this can depend sensitively upon the stack height and initial momentum jet of the stack effluent in the non-buoyant case. Nevertheless, it seems safe to conclude that the short-range benefits of effluent buoyancy of this magnitude in reducing the glc are mostly lost for X > 3. For typical values of mixedlayer scaling variables, this dimensionless distance corresponds to a distance downwind from the stack of from 5 to 20 km. I
I ,--F*=
0
x-F*=
0.11
co\, \ \ 0
1301
\G lJ \ 0 '\ \
Fig. 5. Mean values of ?? near ground level (r/h = 0.086; r,/h = 0.14) vs dimensionless downwind distance. X, for the buoyant case (x’s are data points through which dotted curve is faired) and the non-buoyant case (circles are the data points through which dashed curve is faired). p is defined in Equations (2) and (3).
J. W DEARDORPFand G. E WILLIS
1302
\
_F,= _
F,=
0
1
0.11
6
Fig. 6. Mean values of dimensionless centerline (J’ = 0) groundlevel concentration vs X for the buoyant (dashed curve) and non-buoyant (solid curve) cases.
Of special interest in Fig. 6 is the prolonged downwind region 0.9 < X < 4 or greater over which C(X, 0, 0) varies only slightly. A similarly broad maximum in c(x, 40) at analogous distances from a buoyant source from the Kincaid power plant in Illinois was observed by Bowne (1981). In that case, F, is estimated to have averaged 0.1 (Bowne, personal communication). We attribute this extremely slow decay to the fumigation discussed in Section 3 as well as to residual plume buoyancy within the mixed layer. It is interesting that it can occur in such a manner as to produce a nearly constant glc over such a relatively great centerline distance. The 8atness of the non-buoyant glc curve of Fig. 6 for 2 < X c 4 is also interesting. For the smaller source height of 0.08 h the glc along y = 0 was found to increase slightly with X in this region by Willis and Deardorff (1976)due to the convective-eddy sweep-out effect which operates at smaller X. This effect diminishes with increasing source height.
5. CONCENTRATION
DISTRIBUTIONS
An example of the visual appearance of the fluorescing dye within the laser beam of the laboratory tank is shown in Fig. 7 (bottom), along with the remotely sensed crosswind proliIecentered at thesame time. The correspondence between the two is seen to be close but imperfect largely because of the finite time that was
required by the photomultiplier detector for each traverse. The averaging volume seen by this detector is estimated to have been defined by Ay = O.O2h,and Ax = AZ = 0.007h (for h = 0.23 m). The latter two dimensions were supplied by the effective diameter ofthe laser beam itself. Thus, the spatial resolution was rather high and extreme irregularities in individual lateral concentration profiles were typically observed, as in Fig. 7. The expected log-normal distribution
The concentration distribution could be expected to be log-normal, to first approximation, by thefollowing argument (Csanady, 1973, p. 225). After exiting the stack where the concentration is cO,a parcel of efIluent diffuses into increasingly numerous sub-parcels each undergoing repeated numbers (tat) of mixing events. After each mixing event each sub-parcel’s concentration is assumed to be diminished by a random factor p,(O -z p, < 1). At some later time, a particular subparcel which had undergone n diluting impulses would have a concentration of c = c,p,p,p, pm,so that In(c/c0)=lnp,+Inp2+lnp,+ ... + In p.. Although the distribution Of pi may not be normal, by the Central Limit Theorem the distribution of the sums of a large number of independent pi values, or their logarithms, is normal. Hence In c/c0 from a large collection of sub-parcels would be normally distributed, giving a log-normal distribution of c/co. It may be noted that this plausibility argument refers
Groundlevel concentration fluctuations from a buoyant and a non-buoyant source
-1.5
-1.0
0
1.0
1303
1.5
y/h Fig. 7. Upper: Profile of C m an expenment with non-buoyant effluent, at X = 1.33. Lower: photograph fluorescing dye wtthm the laser beam at the same time.
only to sub-parcels inal undilute
deriving ultimately from the origefIiuent, and therefore having a concen-
tration greater than zero or background level. Note also that since the above argument involves following particular parcels as they meander and dilute, there is no explicit requirement that y or z be fixed, although a fixed value of travel time, or of X, is implied. This simple theory, besides having nothing to say about the degree of plume meandering or pollutant intermittency within the plume, has at least two other drawbacks: (a) the assumption that pt < 1 requires that a weakly polluted parcel never mix with more polluted neighboring fluid and (b) allowing p, to vary randomly means that when p1 is small, for example, a mixing event creates a much larger sub-parcel that has the much smaller resultant concentration. When pr is close to unity, on the other hand, little additional volume is
of
Implied to be involved in the mixing event which leaves the concentration close to its previous value. The final sample drawn by the theory does not take this bias into account. Nevertheless, Csanady (1973) presents evidence in support of the log-normal distribution in flows not involving thermal convection. Effect offinite sampling volume One of several questions this argument does not address involves the averaging volume, or averaging period, of the individual con~ntration m~surement. If non-zero values of concentration on a relatively fine scale are distributed log-normally, what will be the distribution of concentrations measured with coarser resolution for which adjacent zero values may have been included in the average? To explore this question, we set up a two-dimensional numerical grid of 900
1304
_i W. DEARDORF~ and G. E. WILLIS
values of finite concentration prescribed randomly In space, with values satisfying a log-normal distribution. Then, non-overlapping compact sets of four adjacent concentrations were averaged together over the whole grid, and the distribution of the resulting values (225 of them) obtained. The result is shown in Fig. 8 by the dashed curves. The effect of the averaging when no zeroes were present is to reduce extreme values, decreasing the standard deviation of log C, but leaving the distribution still log-normal. If, however, a substantial fraction of zero values are present on the fine scale, namely 540 out of 900 in this case (with finite values again possessing a log-normal distribution), after averaging over sets of four the resulting smoothed values have the distribution shown by the solid curve labelled “after”. (There are then 200 finite values out of the 225 total.) The most dilute 20 3, of the finite population then no longer shares the log-normal distribution which the rest retain. In results to be presented, the fraction of the sampled population having zero concentration (in the region -by < y < oY)is estimated to have been about 40 y0 for X = 1.4, decreasing with X and being greater at small X due to lateral meandering of plume material. Thus, the fraction of zero values was not excessive, and the finite averaging volume used in this study is not cause for expecting log-normality to have been violated if present on a finer scale.
The sampling region Data samples were collected from individual remote-sensor traverses within the region - aY < y < a,,, where br is the usual mean-plume lateral spread obtained from the ensemble of six or seven experiments using the traversing remote detector. Measurements outside this region were excluded from this analysis in order to minimize the uncertainty
between a zero value and a finite value iymg below the threshold of measurement. The observations were not restricted to discrete values of y, such that a detailed _Vdependence could be examined, because insufficient sampling would have resulted. Using the convection tank, one cannot simply accumulate concentration samples over a period of time at a particular downstream distance at J I_-0, for example, during any one experiment. Instead, the elapsed time after passage of the stack by a particular point converts to x/U using Taylor’s translation hypothesis, and hundreds of separate experiments would then be required to collect the necessary data for given values of x and y. Hence, off-centerline values, out to y = +a,, will be included in our analyses for various X. although in examining the concentration variance this region will be subdivided into two equal portions: inner and outer. Observed lateral profiles of instantaneous concentration, C(y), from the seven experiments with F, = 0 for X = 1.4 f 0.1 are shown in Fig. 9(a); similar profiles from the six experiments with F, = 0.11 for X = 2.07 f 0.1 are shown in Fig. 9(b). As might be expected, the peak values often lie outside the analyzed range of - gY < y < oY;in Fig. 9(a) they do so four out of seven times, and in Fig. 9(b) two or three out of six times. Thus, the fluctuation statistics appear to be rather uniform within the analyzed region and can therefore be expected to be fairly representative of what occurs at y = 0 itself. It may also be noted that the concentration centrotd usually lies considerably off to one side of the y = 0 axis due to plume meandering. Thus, it is a definite advantage to know in advance the location of the true _V= 0 axis (along the stack towing line in the experiments) independently of what might be estimated from the limited supply of data samples at any given X. Observed cumulative distributions
T
C
05
02 I
1 5
IO 2030 CUMULATIVE
50
70 80
90 95
PERCENTAGE
Fig. 8. Cumulative frequency distribution (abscissa) of a numekally generated random distribution of log C values (see ordinate, arbitrary scale) before small-scale spatial averaging (steeper dashed line) and after averaging (less steep dashed line). Solid curves show same comparison when initial field contains 607, zero values. Distributions pertain only to finite C values.
Measured cumulative distributions for the case of the non-buoyant effluent for - u,, c y < uYare shown in Fig. 10 on a log-normal probability diagram. The number of samples entering each distribution at a particular mean value of X (= m increased from 470 for x = 0.65 to 3525 for x = 3.7. The rapid fall-off of the larger concentrations with Increasing X is evident in Fig. 9, along with increasing values of the weaker concentrations, for x> 1.4, as ‘fillmg in’ occurs. The lack of more compfete regularity in the pattern of the curves for successive X is probably attributable to sampling error. An interesting feature of the figure is the presence of a kink in the distribution, at least for X > 1.4. The presence and direction of the kink is associated with a skewed distribution of log C such that small values of log C have a long ‘tail’ while large values have a small tail. Thus, the distribution of C found here can only be described as log-normal in a very rough sense. The cumulative distributions of log C for the
Groundlevel concentration fluctuattons from a buoyant and a non-buoyant source
1305
b
-O-
Y
0
*y
Y-
--Q
0
Y
my
Y-
Fig. 9. (a) Lateral profiles ofCfy) from the seven expertments with non-buoyant effluent, for X averaging 1.4. (b) Lateral profiles of C(y) from the six experiments with buoyant effluent, for X averaging 2.07.
case, F, = 0.11, are shown in Fig. 11. The number of samples within any one distribution at a particular value of X in this case increased from 345 at IE?= 1 to 3330 at x = 3.7. Except for much smaller values of C in the uppermost 20% of the population for AT< 1.8, the distribution is quite similar to that of buoyant-effluent
I
5 CUMULATIVE
FREOUENCY (%I
Fig. 10. Cumulative frequency distributions (abscissa) of log C from non-buoyant cases (FL = 0) for mean values of X ranging from 0.65 to 3.7.
IO
I
I
2030 50 70 00 90 95 CUMULATIVE FREQUENCY (%)
Fig. 11. Cumulative frequency distributions of log C from buoyant cases (F, = 0.11) for mean values of X ranging from 1.0 to 3.7.
J. W DEARWRFF and G.
1306
the non-buoyant source. It also exhibits the same kink, whose position shifts towards smaller cumulative frequencies as X increases. The effect of uncertainty in the fraction of zeroconcentration values upon these distributions is to cause noticeable uncertainty in the ordinate at very small frequencies of occurrence, but little uncertainty for cumulative frequencies exceeding about 303;. The ‘filling in’apparent in both Figs lOand 11, as the weaker concentrations become somewhat greater downwind and the distribution begins to approach a horizontal line of uniformity, suggests gross violation of assumption (a) of the log-normality plausibility argument in Section 5. That is the assumption that after each mixing event a parcel’s concentration becomes less, as if it always mixed with cleaner fluid. Instead, parcels with weak concentrations have increasing opportunity to mix with parcels of greater concentration as travel time increases, and as tluid entrained into the plume gradually adopts meanplume characteristics. A very similar kink in the cumulative probability distribution of vertical velocity and temperature within the convective atmospheric boundary layer has been reported by Lenschow (1970) and others. It is in the sense that upward motions and associated aboveaverage temperatures have a substantial tail to their distribution, while the more prevalent downdrafts and associated below-mean temperatures are more uniform. The present, similarly kinked distribution of log C might be a manifestation of the same convective skewness, if upward motions are associated with smaller log C values and downward motions with larger log C values. There is some indirect evidence of this. For 1.2 < X < 3 or 4 the mean vertical distribution of C for the non-buoyant source near the surface is one of mean C increasing somewhat with height (Willis and Deardorff, 197Q so that upward motions might then be correlated with below average
E. WILLIS
concentrations, and vice versa. For the buoyant effluent, the increase of mean C with height must be even more pronounced, as discussed in Section 3. However. we cannot offer this qualitative observation as an explanation of the kink, since the diffusion from an elevated point source is so different than that from a large-area surface source, and since negative eddy diffusivities of vertical diffusion have been reported by Deardorff and Willis (1975). Observed concentration variances
In the inner region, - o/2 < y -Z 012 and the outer region 3 ug < 1y 1-c uyr and at z = 0.08h. values of the ratio p/i?* were calculated from six of the seven experiments with non-buoyant emissions and from four of the six with buoyant emissions. (These results were obtained at a later date than the concentration distributions and not all the original data tapes were still available.) For this purpose zero and near-zero values were of course included in the analyzed data, which therefore include plume meander effects as well as in-plume fluctuations. In calculating the ratio, values of the numerator and of T in the denominator were separately averaged from all available data within a small range of X before forming the ratio. (~ was obtained from 2’”- T2. Results are shown for the non-buoyant releases in Fig. 12 and for the buoyant releases in Fig. 13. The pronounced decrease in p/i?’ with X is the main feature of interest, and is very definite despite relatively large sampling error. (The points at X = 1.4 in Fig. 12 and at X = 3.2 for the outer region in Fig. I3 were ignored in fairing the smooth curves through the remaining points, because in these three instances the anomalies were caused by single events within just one of the nine and five traverses, respectively, from which the ensemble means in these cases were formed.) The squared fluctuation-to-mean ratio for the outer region
-a
v
Fig. 12. Measured values of c’~/TZfor the non-buoyant case as a function of X. Measurements denoted by A represent an average eve-rthe region Iyl < 0,/2; those denoted by circles represent an average over the region a,/2 -Z lyl < uY.
Groundlevel
concentration
fluctuations
I
from a buoyant and a non-buoyant
I
I
1307
source
I
8 F, =Oll 7
6
3
2
0
Fig.
13.
Measured values of C-“/C’for
the buoyant as in Fig. 12.
of Fig. 13 exceeds that of the inner region by some ifs;’ did not vary over the entire region from - u, to u,, but T varied in Gaussian fashion. For Fig. 12 the scatter is worse but the outer values tend to exceed the inner. Taken together, the results support previous findings (Csanady, 1967,1973) that 7 decays considerably less rapidly with y than does %. 60x,
about what is to be expected
The greater values of c’2/T2 for the buoyant release are believed associated with the elevated plume axis in this case; consequently the near-surface sampling often encountered relative fluctuation values on the (lower) fringe of the mean plume, where relative intensities are known to be of greater magnitude (Hanna, 1984). Because of the finite sampling volume, the measured relative intensities are underestimates of point values [even in the presence of small but finite molecular diffusivity, D,where the ‘conductivity’ cutoff scale, 1 = (~D'/E)"~, was approximately 0.04 mm, v is the kinematic viscosity and E is the rate of dissipation. See Chatwin and Sullivan (1979) for a discussion of the role of A.] With a doubled sampling volume, obtained by doubling Ay in a reanalysis of the concentration variances, it was found that p was decreased by 24 y0 at X = 0.5, falling to an 18 y0 reduction at X = 4. The present estimates of the variance may therefore be too small by a fraction comparable to these. In this reanalysis, it may be noted that the intra-plume variance is unaffected by altering Ay. The ratio of the intra-plume variance to the total variance was found to be 0.27 f 0.04, where the latter variation is the standard deviation of the mean. The ratio 0.27 was found to be insensitive to the buoyancy of the source, location within the mean plume, and the downstream distance.
source as a function
of X. Symbols
However, the intra-plume variance defined here may not coincide with the meandering component as usually defined; here the total variance is represented by c’zJ__Tz
= (;rr_F)+(p_T2) A
(5) B
where c^is the mean from a particular realization over either the inner region or outer region. Term A is the within-plume variance and term B the intra-plume variance. Our results suggest that the within-plume variance dominates in the convective boundary layer, at least for 0.5 < X < 4. The pronounced decrease of c’1/r2 with distance associated with the effects of downstream diffusion from the continuous source and, at later times, with vertical diffusion after the presence of the mixed-layer ‘lid’, as well as the surface, had been felt. It may be noted that an opposite result of Sawford (1983) was obtained with a model which assumed onedimensional mixing and may not have taken into account these two homogenizing influences. seems
Intermittency
Values of the intermittency, y, in the inner and outer regions of the mean plume, for the buoyant and nonbuoyant releases, are shown in Fig. 14. (A value of y = 1 means that the contaminant was always present.) The threshold level chosen for the presence or absence of C was 0.05 (c’2)“’ or 0.05, whichever was larger. Results show the expected smaller values at smaller X, indicative of plume meanderings, especially near the buoyant source where plume rise caused y = 0.
J. W
1308
DFARDORFF and G.
E. WILLIS
B
Si-
,
2-
X Rg. 14. Measured values of intermittency, y, over the inner region (A)and the outer region (circles) as a function of X. Open symbols denote non-buoyant source. closed symbols denote the buoyant source.
Quantitative values are, however, believed to be rather sensitive to both the threshold level and the size of the sampling volume.
6. SUMMARY
Acknowledgements-This work was supported by the Environmental Protection Agency under Grant R808919. P. Stockton provided expert technical support. We thank three anonymous reviewers for providing valuable criticisms of earlier versions of this paper.
AND CONCLUSIONS
Using mixed-layer scaling, the near-surface concentration at moderate distances downwind of a stack emitting a buoyant efauent characterized by F, = 0.11 has been obtained and on the average compared quantitatively with the glc from a non-buoyant source that was otherwise identical and subject to identical environmental conditions. Although results for the buoyant effluent will depend somewhat upon conditions just above the top of the mixed layer, as well as upon stack height, those obtained are believed representative of typical atmospheric mixed-layer conditions when the mixed-layer depth is much greater than the stack height. As would be expected from the effect of plume rise, the buoyant e&tent has a much smaller mean maximum gic, with delayed appearance downstream, in comparison with the non-buoyant eIIluent. It also evidences an extremely slow decay of centerline glc downstream of its (faint) surface maximum, in comparison with the very rapid decay in the non-buoyant case immediately downstream of this (highly peaked) maximum. The cumulative frequency distribution of sampled concentrations was presented for the same two cases for various dimensionless downwind distances between 0.6 and 4. From these, the expected maximum concentration to be exceeded near the ground under convective conditions 2% or less of the time, for example, may be obtained. In both cases, for X > 1.2, the distributions are found to be log-normal in only a bimodal sense, with the weaker concentrations having a greater spread of log C than the stronger concentrations. The concentration variance relative to the square of the mean, near they = 0 axis, is found to decrease quite strongly with downwind distance.
REFERENCES
Bierly E. W. and Hewson E. W. (1962) Some restricttve meteordogi&l conditions to be considered in the-design of stacks. J. appl. Met. 1, 383-390. Bowne N. E. (1981) Preiiminary results from the EPRI plume model validation project-Plains site. Electric Power Research Institute, Palo Alto, CA, 94304, April, 1981, EPRI EA-178~SY. Caughey S. J. (1982) Observed characteristics of the atmospheric boundary layer. In Atmospkeric Turbukwe and Air Pollution Modelling (edited by Nieuwstadt R. T. M. and van Dop H.). D. Reidel, Dordrecht. Caughey S. J. and Patmer S. G+ (1979) Some aspects of turbulence structure through the depth of the convective boundary layer. Q. JI R. met. Sot. l@!, 811-827. Chatwin P. C. and Sullivan P. J. (1979) The basic structure of clouds of dhTusing contaminant. In Mathemutical Modellina ofTurbulent Difiusion in the Environmentkdited by Harris 6. J.k pp. 3-32: Academic Press, New York. Csanady G. T. (1967) Concentration guctuations in turbuient diiTusion. J. atmos. Sri. 24, 21-28. Csanady G. T. (1973)7’urbu/[email protected] the Enurronmenr. D. Reidel, Boston. DeardorfFJ. W. (1970) Convective veio&ty and temperature s&es for the unstable planetary boundary layer and for Ray&h convection. J.‘atmos. ki. 27, 12il-i213. DeardorfFJ. W. and Willis G. E. (1975) A narameterization of diffusion into the mixed htyer.‘J.ap&. ket. 14, Mil-i458. Hanna S. R. (1984) Concentration fluctuations in a smoke plume. Atmospheric Eltoirortlwnzlg. 1091-l 1%. Irwin J. S. and Cope A. M. (1979) Maximum surface concentration of SO2 from a moderate-size stream-electric power plant asa function of power plant load. Atr&spheric Environment13, 195-197. Kaimai J. C., WyngttardJ. C, Haugen D. A.,Cottf 0. R, bmmi Y., CaunhGy S. J., and Ret&inns C. J. (1976) Turbulence structt& in-theconvective boundary layer.J. &os. Sci. 33, 2152-2169. Kerman B. R. (1981) A note on the maximum ground level concentration for inversion-iimited chimney plumes. Atmospheric Environment 15, 119-I 22. Lamb R. G. (1982) Diffusion in the convecttve boundary
Groundlevel concentration fluctuations firom a buoyant and a non-buoyant source layer. In Atmospheric Turbulence and Air Pollution Modelling (edited by Nieuwstadt F. T. M. and van Dop H.). D. Reidel, Dordrecht. Lenschow D. H. (1970) Airplane measurements of planetary boundary layer structure. J. appl. Met. 9, 874-884. Lenschow D. H., Wyngaard J. C. and Pennell W. T. (1980) Mean-field and second-moment budgets in a baroclimc, convective boundary layer. J. otmos. Sci. 37, 1313-1326. Liu H. T., Lin J. T., Delisi D. F. and Robben F. A. (1977) Apphcation of a fluorescence technique to dyeconcentration measurements ma turbulent jet. Proc. Symp. Flow Open Channels and Closed Conduits, Nat. Bur. Std. Pub. 484.423446. Manins P. C. (1979) Partial penetration of an elevated inversion layer by chimney plumes. Atmospherz Environment 13, 733-741.
Moninger W. R., Eberhard W. L., Griggs G. A., Kropfli R. A. and Kaimal J. C. (1983) Simultaneous radar and lidar observations of plumes from continuous point sources. Proc. 21st Radar Meteorology Conf., Edmonton, Alberta, Am. Met. Sot. Nicholls S. and LeMone M. A. (1980) The fair weather boundary layer m GATE The relationship of subcloud fluxes and structure to the distribution and enhancement of cumulus clouds. J. atmos. SCL 37, 2051-2067. Nieuwstadt F. T. M. (1980) Application of mixed-layer similarity to the observed dispersion from a ground level source. J. appl. Met. 19, 157-162. Sawford B. L. (1983) The effect of Gaussian particle-pair distribution functions in the statistical theory of concen-
1309
tration fluctuations in homogeneous turbulence. Q. JI R. met. sot. 109, 339-354.
Venkatram A. (1980) The relationship between the convective boundary layer and dtspersion from tall stacks. Atmospheric Enwonment
14, 763-767.
Weil J. C. and Jepsen A. F. (1977) Evaluation of the Guassian plume model at the Dickerson power plant. Armospherrc Environment 11, 901-910. Wilhs G. E. and Deardorff J. W. (1974) A laboratory model of the unstable planetary boundary layer. J afmos SCI. 31, 1297-1307. Willis G. E. and Deardorff J. W. (1976) A laboratory model of diffusion into the convective planetary boundary layer. Q. Jl R. met. Sot. 102.427445. Willis G. E. and DeardortTJ. W. (1978) A laboratory study of dispersion from an elevated source within a modeled convective planetary boundary layer. Atmospheric Environment 12, 1305-1311. Willis G. E. and Deardorff J. W. (1979) Laboratory observations of turbulent penetrativeconvection planforms. J geophys. Res. 84, 295-302. Willis G. E. and Deardorff J. W. (1983) On plume rise withm a convective boundary layer. Atmospheric Encwonment 17. 2435-2447.
Wyngaard J. C., Cot6 0. R. and Izumi Y. (1971) Local free convection, similarity, and the budgets of shear stress and heat flux. J. armos. Sci. 28, 1171-1182. Wyngaard J. C. and LeMone M. A. (1980) Behavior of the refractive index structure in the entraining convective boundary layer. J. armos. SCL 37, 1573-1585
Enc,rownen,
Vol IS.
No
7. pp
1297-1309.
1984
ooo4-6981184 $3 00 + 0.00 Q 1984 Pcrgamon Press Ltd
Pnntcd I” Great Bntam
GROUNDLEVEL CONCENTRATION FLUCTUATIONS FROM A BUOYANT AND A NON-BUOYANT SOURCE WITHIN A LABORATORY CONVECTIVELY MIXED LAYER J. W. DEARDORFF and G. E. WILLIS Department of Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, U.S.A. (Ftrsr receitied 18 July 1983 and infinolfirm
13 February 1984)
Abstract-With the use of mixed-layer scaling, the near-surface mean concentration and the concentration fluctuations at moderate distances downwind of a stack emitting a highly buoyant efRuent are obtained and compared quantitatively with those from a non-buoyant source that is otherwise Identical. The environment ISa laboratory mixed layer in a state of free convection with mean wind simulated by towing the stack along a horizontal line. Although the buoyant effluent has a much smaller maximum mean groundlevel concentration occurring at a greater downwind &stance than the non-buoyant eflluent, the decay of mean concentration from that pomt downwind is found to be exceedingly slow in the buoyant case. The cumulative frequency distributions of sampled concentrations greater than zero for these same two cases are found to be log-normal in only a bimodal sense, with the weaker concentrations having a greater logarithmic range in amplitude than the stronger concentrations. The ratio of the root-mean-square (r.m.s.) concentration fluctuation to the mean concentration, in the central portion of the mean plume, is found to decay with downwind distance. Homogenization associated with both downstream and vertical mixing IS believed responsible for the rapid decay observed. Key word index: Concentration fluctuations, ground level concentrations from buoyant and non-buoyant sources, mixed-layer dispersion, relative concentration intensity.
1. INTRODUCTION The effluent concentrations observed at ground level downwind of tall stacks under conditions of thermal convection are of primary concern to researchers of diffusion. Highest concentrations are believed to occur under these conditions (Bierly and Hewson, 1962; Irwin and Cope, 1979; Venkatram, 1980) for a given wind speed and source strength. The efficiency of the convective motions in carrying portions of elevated stack plumes downwards is responsible for their early appearance at the ground. The thermal-convection tank is proving to be a valuable laboratory apparatus for studying dispersion within a convectively mixed boundary layer (Lamb, 1982) under conditions when the lower region of shear induced turbulence (z < -L) occupies only a small fraction of the total boundary-layer depth (h). In that case h + - L, where L is the Monin-Obukhov length, and thermal convection dominates the turbulence production at heights exceeding -L (Wyngaard er al., 1971). The condition h % - L occurs frequently in the daytime over land under sunny or partially sunny conditions, and it is also common for the stack height itself to exceed -L. For these reasons, a laboratory convection tank is a useful tool even though no mean flow exists within its convective boundary layer (L = 0).
A re-analysis of Prairie Grass data using mixed-layer
scaling (Nieuwstadt, 1980) has provided supportive evidence for the laboratory results using mixed-layer scaling in the case of the mean dispersion from a nearsurface non-buoyant source. Also, recent measurements at the Boulder Atmospheric Observatory for a non-buoyant source located at the height z = h/2 have largely verified the laboratory results for this source height (Moninger et al., 1983). Of special interest in both cases is the lack of mean-plume Gaussianity in the vertical direction. The concentration fluctuations are also of great interest, since the mean concentration distribution (from data averaged over a time period with relatively uniform surface conditions and relatively constant wind velocity) gives little information about the nearpeak values of concentration and associated toxicity which may occur over short time periods and in small sample volumes. The present study therefore emphasizes the concentration fluctuations (near ground level) measured at various scaled distances downwind of the stack. Although the non-buoyant stack plume is a most important reference case for study, primary interest resides in the dispersion of buoyant plumes (e.g. Weil and Jepsen, 1977; Manins, 1979; Venkatram, 1980; Kerman, 1981) since the majority of elevated point sources are smoke stacks emitting warm gases. Therefore, in this study we compare near-surface concentration data (the mean and fluctuation statis-
1297
I W. DEARDOWI
1298
and G.
E.
WILLIS
,-_.
tics) from a highly buoyant source with those from an
,c-’
...__-..-.-._-. _‘__._~-?‘_
_
.:.
-_
.:_-
_..
_
/ ,
otherwise identical non-buoyant source. Our emphasis here is on near-surface concentrations. For the effects of plume buoyancy upon plume rise and upon the mean plume envelope aloft as measured within the laboratory convectively mixed layer, the reader is directed to a recent paper by Willis and Deardorff (1983). The degree of plume buoyancy chosen for study here IS relatively large, so that most observed buoyant plumes from tall stacks will undergo behavior intermediate to the non-buoyant and buoyant cases studies. It would obviously be desirable to study the quantitative behavior of plumes of intermediate buoyancy also, but this must await future multi-year investigations.
/
_
,
)1
Jcv___
STACK \
i’J’
_j / j
/
TARK WALL
2. EXPERIMENTAL METHODS The laboratory model is descnbcd in detail elsewhere (Wittis and Deardoti, 1974) Briefly, it consists of a water tank, 1.24m on a side, with a convectively mixed layer of depth, It, of 0.232 m mean height, and an overlying stable layer with stratitI&on of 0.05 C mm-’ in the experiments to be reported. This m&Mayer turbulence is driven by a Iarge surface heating rate of about 5 kW m- ‘. The model includes no mean flow and thus no mean shear-generated turbulence. Favorable comparisons of atmospheric mixed-layer turbufence statistics with those from the model appear in Willis and Deardozff (1974), Caughey and Palmer ( 1979),Wyngaard and LeMone (f980), Lenschow et ai. (1980). Nichoils and LeMone (1980) and Caughey (1982). The effects of a uniform mean wind m bending over a vertically emitted plume are simulated in the water tank by towing a mode1 stack of height 2, = 0.031 m at a horizontal speed of U = 0.2 w,, where w, is the stack-exit velocity of 0.1 m s - I. Thedimensionfess release height is zJh = 0. I34. A dimensionless ‘stack buoyancy’ parameter, F, . is defined as F, = w,rf (S/PO)(PO-P&W:
uh),
(1)
where r, is the stack radius of I.0 mm, (p. - pS) is the initial effiuent density deficit relative to the ambient fluid, equal to O&41p0 for the buoyant plumes to be treated and zero for the non-buoyant plumes; and w+ is the free-convection velocity scale (Deardorff, 1970)whose value avera8ed 9 mm s- ’ in the present experiments. Values of F, used in this study are 0 for the non-buoyant releases and 0.11 for the buoyant releases. The latter value is typical for a large industrial plant stack, such as that at Sudbury, Ontario, ona summer day with winds near 3-4 m s-l. Willis and Deardorff (1983) found F, to be the most appropriate dimensionle~ stack-buoyancy parameter, under convective conditions, in governing plume behavior for X exceeding a few tenths. In (I), the numerator relative to U determines the potential for stack e!Iluent to rise due to its buoyancy, while the denominator, w: h, is a measure of the ability of convective ~un~r~layer turbulence to break up and dilute the plume over the height h. A planform top view of the tank and expertmental arrangement appears In Fig. 1. A high intensity argon-ion (blue) laser beam extended the w&h of the tank in the J direction at a height of 0.020 m. Due to index of refraction variations near the heated lower boundary the beam could not be positioned iower without encountering excessive vertical displacements. This height of 0.086 h, where concentration measurements were made, is here referred to as ‘ground-level”. After the lower-surface heating had been imtiated and the convective motions had become fuily established, the plume emission and stack translation were begun. When the stack
Fig. 1. Top planform depiction showing the towed stock, diffusing stack phime, and the tiscr beam intersecting the plume @a&cd line) where fluorescence occurs. Tests by Willis and Deardorff (1983) showed that there was no significant effect upon the phune dispersion or ambient turbulence associated with the moving towing tube (to left of stack) which rested on the bottom surface of the convection tank. had been towed some distance in Y beyond the locatmn of the laser beam the turbulent convective motions will have carried portions of the plume down to a height where they can intercept the blue beam {dashed line of Fig. 1). The stack plume (water-alcohol mixture) contained a small amount of RhodamineS dye which fluoresces when energized by the laser beam. A photo-multiplier detector, equip@ with an optical fitter whose window passed light in the fluorescent wavelengths, viewed a narrow section of the laser beam along y. This detector was located a small height above the water level in the tank, and was periodically traversed along the axis of the beam (see Fig. 2). The intensity of the emitted light is linearly proportional to the efftuent concentration, c. The minimum concentration which couki be measured, determined by the system noise teve1, was about 59; of the vertically well mixed value at y = 0 near the end of the experiment. Details of the laser-dye-fluorescence technique have been reported by Liu et ni. (1977). Traverses across the dispersing plume by the remote detector were made every 7-10s giving a series of concenTroverstng track
P,hotomultlpller
II
tf
II
Stab@
2 ..
I Stock’
2 I L Loser beam
Fig. 2. End view (in y-z plane) of convection tank containing mixed layer and stable water hyer above, and schematically showing buoyant e&ent within the m&d layer. The photomuItiplier detector and its traversing track at top are mounted dire&y above the laser be&n, while thr stack moves along -x (directed into the paper).
Groundlevel concentration fluctuations from a buoyant and a non-buoyant source tration profdes in yand t (time). The plume-fluctuation pattern is assumed frozen at the traverse midpoint time, since the detector translation speed was 20 times greater than the r.m.s. turbulence velocity. Through the use of mixed-layer scaling (e.g. see Willis and Deardorff, 1976) the measured variables were converted into the dimensionless distance downwind from the stack X = (w*/V) (x/h) and dimensionless concentration C = ch= U/S
(2)
where S is the source emission rate. The concentration measurements were averaged over an ensemble of seven experiments for the buoyant case and six experiments for the non-buoyant case, to yield the mean values. From these measurements the lateral spread of the plume at ground level, the cross-wind integrated concentrations, and concentration fluctuation statistics were also obtained. In five additional sets of experiments, both with buoyant and non-buoyant effluent, y-integrated concentrations at the same near-surface height were obtained through continuous measurement of the net in-tank absorption of a He-Ne (red) laser beam directed along y at this height when the stack eflluent contained some added blue due. The dimensionless y-integrated concentration is defined by m CY = s -m
C dylh,
(3)
1299
which can be seen to have unit value if the effluent IS uniformly distributed in the vertica! up to the height (h) utilized in the definition of C, with zero value above. The respective calibration constants (to relate sensor voltages to dimensionless concentrations or to their y-integrated value) made use of this condition, upon thoroughly mixing all the water ofdepth H in the tank at the end of each experiment and substituting H in place of h in (2) or (31.With this method of calibration, the source emission rate, 5, need not be known since only the dimensionless concentrations (involving c normalized by S) were obtained. However, for large values of X in the non-buoyant case (6 < X < 8) it was found that TY from the calibration lacked l&15”,, of reaching unity. Therefore, the calibration constants were increased by lo”, in both the non-buoyant and buoyant cases. There is consequently an absolute uncertainty of about this amount in the values of results to be presented, presumably associated with uncertainty in the effective value of h. The tank side walls begin to influence the diffusion after a particle travel time corresponding to X L 4; hence results will not be presented here for values of X appreciably greater than this. 3.
VISUAL APPEARANCE OF THE PLUMES
Side-view photographs of the buoyant and nonbuoyant stack plumes are shown in Figs. 3(a) and (b), respectively. In looking for effects of stack plume
a
b
no
Fig. 3. Side-view photographs of a buoyant plume (a) and a non-buoyant plume (b). The dark line appearing in the background near height z/h = 0.1 is the intersection of the tank rear wall and bottom plate.
J W DEARDORFFand G. E. WILLIS
1300
buoyancy it is most instructive to view the photographs as a unit. For both cases the stack variables, r,, w,, h, and U were the same, and on a statistical basis both plumes encountered similar environmental turbulence fields. The only quantity that varied for the two cases is (pO- p,), so that F, = 0.11 in (a) and 0 in (b). Immediately after stack exit the plumes assume the familiar bent-over appearance associated with real stack plumes in the atmosphere. Plume looping, characteristic of plumes released into an unstable atmospheric boundary layer, is evident. The plume rise due to the added buoyancy is clearly evident in (a). A quantitative estimate of its average magnitude, as well as that of plumes of lesser buoyancy, is reported in Willis and Deardorff (1983). Perhaps the most striking comparative feature of the photographs is the effect of added plume buoyancy in inhibiting large-scale downward motions of the plume. Clearly, large portions of the non-buoyant plume are carried to ground level at distances near X = 0.5 before encountering plume lift-off due to the surface sweep-out effect of vigorous large-scale updrafts. Initial plume descent to ground level (in the mean) followed by sweep-out is absent for the buoyant plume. With F, = 0.11 the plume buoyancy and associated vertical momentum are still sufficiently great at distances out to X = 1 to counteract largely the downdraft motions of the convection. Later, of course, the stack plume becomes more dilute and can then be carried to ground level by the turbulent motions. For the buoyant case, large portions of the plume apparently become trapped in the upper reaches of the boundary layer (near the inversion level), and then serve for some time as a continuous source of material to be carried downward, as in the process of fumigation. This effect is quite distinct from the elevated concentration maximum occurring in a convectively 1.0
I
mixed layer with a non-buoyant ehluent (see Willis and Deardorff, 1976, 1978). In that case the eievated maximum resulting from the sweep-out effect is often too weak to be very noticeable from individual photographs, and occurs at z/h = 0.75. For the highly buoyant release, the stronger elevated maximum is estimated to lie near z/h = 1.0. when the quantity (gadtJ/&)“’ (we/h) = 8, where the numerator 1s the Brunt-Vaisala frequency within the inversion, 2 IS the coefficient of thermal expansion. and /Z/M+ lb the convective time scale within the mixed layer. A typical value for this ratio is 10 for atmospheric convective conditions. In our laboratory model this ratio can be varied within the range l-20. The smaller this ratio, the greater is the expected penetration of the buoyant effluent above the average top of the mixed layer, all else being equal. The existence of a distinct, identifiable stack plume obviously breaks down for unstable conditions at relatively short downstreamdistances when the turbulent motions break up the plume so effectively that the concept of a plume entity becomes nebulous except in the statistical sense.
4.~m
MEANCONCENTRATIONANDITSLATERAL SPREAD
Smoothed lateral plume spreads, obtained from m OJ
--m
-m
using measured values of c, are shown in Fig. 4 vs dimensionless downwind distance. The r.m.s. standard deviations of the mean (error bars) indicate that the difference between the buoyant and non-buoyant cases is statistically negligible, at least for smaii X, suggesting that the cross-wind spread of the plume is essen-
I
I
_F*=O 0.8
_
__
F,
= 0.11
0.6
Fig. 4. The dimensionless lateral plume spread [square-root of Equation (4) J vs dimensionless downwind distance for the buoyent and non-buoyant stack phrmes. Error bars represent f one standard deviation of the mean from the seven experiments in each ease..
Groundlevel concentration fluctuations from a buoyant and a non-buoyant source tially independent of F,. This conclusion would appear to be counter to the usual assumption of crosswind plume spread being proportional to plume rise which is dependent on F,. However, initial plumespread enhancement due to buoyancy is presumably present at the elevated plume close to the source but appears to be negligible compared to the total spread after the plume reaches ground level much later (X = 0.7). With a tow speed of Cl = 2.2 w* the stack moves a distance of about 1.5 h over the increment AX = 0.7. This distance is roughly equivalent to the characteristic dominant horizontal convection scale (Kaimal et al., 1976; Willis and Deardorff, 1979). Thus, any portion of the plume appearing at ground level within the distance X = 0.7 can be expected on the average to have been under the influence of and rapidly dispersed by a single large-scale (and horizontally diverging) downdraft. Its effects on the plume spread could dominate and mask any spread due to internally generated turbulence. The r.m.s. plume spreads are systematically about 10% smaller, at all values of X, than those we previously measured in the non-buoyant plume studies using the oil-droplet-in-water technique (Deardorff and Willis, 1975). Possibly, the present measurement method slightly underestimates plume spreads by missing extremities of the plume where the dye may be very dilute and consequent light emission weak and undetectable. The measured groundlevel, laterally integrated, dimensionless concentration, F, is shown in Fig. 5. The pronounced effect of plume buoyancy (F, = 0.11) in 4
I
0 r\ /
3-
I
/c : I :
‘\
delaying and hindering the appearance of groundlevel concentrations (glc) is evident. The added plume buoyancy is effective in reducing the observed maximum glc by an order of magnitude. Only for X exceeding 3 or 4 does the laterally integrated glc in the buoyant case reach levels comparable to those attained in the non-buoyant case. Mean centerline (p = 0) concentrations were obtained from the smoothed curves of Figs. 4 and 5 using the Gaussian assumption; i.e. by dividmg cy by (2n)“’ a,/h. They are shown in Fig. 6. The effect of plume buoyancy in reducing the maximum centerline glc is even more pronounced than in Fig. 5 for cY due to the earlier appearance of glc in the non-buoyant case when aYis smaller. Specifically, for F, = 0.11 the peak mean concentration is reduced by a factor of about 12 and the appearance of the maximum is delayed by AX = 0.4 at least, as compared with the non-buoyant case. In fact, this faint maximum in Fig. 6 at X = 1.0 may not be statistically significant, and could reflect small sampling errors or small departures of the assumed lateral distribution from the Gaussian. Also, quantitative comparisons such as this can depend sensitively upon the stack height and initial momentum jet of the stack effluent in the non-buoyant case. Nevertheless, it seems safe to conclude that the short-range benefits of effluent buoyancy of this magnitude in reducing the glc are mostly lost for X > 3. For typical values of mixedlayer scaling variables, this dimensionless distance corresponds to a distance downwind from the stack of from 5 to 20 km. I
I ,--F*=
0
x-F*=
0.11
co\, \ \ 0
1301
\G lJ \ 0 '\ \
Fig. 5. Mean values of ?? near ground level (r/h = 0.086; r,/h = 0.14) vs dimensionless downwind distance. X, for the buoyant case (x’s are data points through which dotted curve is faired) and the non-buoyant case (circles are the data points through which dashed curve is faired). p is defined in Equations (2) and (3).
J. W DEARDORPFand G. E WILLIS
1302
\
_F,= _
F,=
0
1
0.11
6
Fig. 6. Mean values of dimensionless centerline (J’ = 0) groundlevel concentration vs X for the buoyant (dashed curve) and non-buoyant (solid curve) cases.
Of special interest in Fig. 6 is the prolonged downwind region 0.9 < X < 4 or greater over which C(X, 0, 0) varies only slightly. A similarly broad maximum in c(x, 40) at analogous distances from a buoyant source from the Kincaid power plant in Illinois was observed by Bowne (1981). In that case, F, is estimated to have averaged 0.1 (Bowne, personal communication). We attribute this extremely slow decay to the fumigation discussed in Section 3 as well as to residual plume buoyancy within the mixed layer. It is interesting that it can occur in such a manner as to produce a nearly constant glc over such a relatively great centerline distance. The 8atness of the non-buoyant glc curve of Fig. 6 for 2 < X c 4 is also interesting. For the smaller source height of 0.08 h the glc along y = 0 was found to increase slightly with X in this region by Willis and Deardorff (1976)due to the convective-eddy sweep-out effect which operates at smaller X. This effect diminishes with increasing source height.
5. CONCENTRATION
DISTRIBUTIONS
An example of the visual appearance of the fluorescing dye within the laser beam of the laboratory tank is shown in Fig. 7 (bottom), along with the remotely sensed crosswind proliIecentered at thesame time. The correspondence between the two is seen to be close but imperfect largely because of the finite time that was
required by the photomultiplier detector for each traverse. The averaging volume seen by this detector is estimated to have been defined by Ay = O.O2h,and Ax = AZ = 0.007h (for h = 0.23 m). The latter two dimensions were supplied by the effective diameter ofthe laser beam itself. Thus, the spatial resolution was rather high and extreme irregularities in individual lateral concentration profiles were typically observed, as in Fig. 7. The expected log-normal distribution
The concentration distribution could be expected to be log-normal, to first approximation, by thefollowing argument (Csanady, 1973, p. 225). After exiting the stack where the concentration is cO,a parcel of efIluent diffuses into increasingly numerous sub-parcels each undergoing repeated numbers (tat) of mixing events. After each mixing event each sub-parcel’s concentration is assumed to be diminished by a random factor p,(O -z p, < 1). At some later time, a particular subparcel which had undergone n diluting impulses would have a concentration of c = c,p,p,p, pm,so that In(c/c0)=lnp,+Inp2+lnp,+ ... + In p.. Although the distribution Of pi may not be normal, by the Central Limit Theorem the distribution of the sums of a large number of independent pi values, or their logarithms, is normal. Hence In c/c0 from a large collection of sub-parcels would be normally distributed, giving a log-normal distribution of c/co. It may be noted that this plausibility argument refers
Groundlevel concentration fluctuations from a buoyant and a non-buoyant source
-1.5
-1.0
0
1.0
1303
1.5
y/h Fig. 7. Upper: Profile of C m an expenment with non-buoyant effluent, at X = 1.33. Lower: photograph fluorescing dye wtthm the laser beam at the same time.
only to sub-parcels inal undilute
deriving ultimately from the origefIiuent, and therefore having a concen-
tration greater than zero or background level. Note also that since the above argument involves following particular parcels as they meander and dilute, there is no explicit requirement that y or z be fixed, although a fixed value of travel time, or of X, is implied. This simple theory, besides having nothing to say about the degree of plume meandering or pollutant intermittency within the plume, has at least two other drawbacks: (a) the assumption that pt < 1 requires that a weakly polluted parcel never mix with more polluted neighboring fluid and (b) allowing p, to vary randomly means that when p1 is small, for example, a mixing event creates a much larger sub-parcel that has the much smaller resultant concentration. When pr is close to unity, on the other hand, little additional volume is
of
Implied to be involved in the mixing event which leaves the concentration close to its previous value. The final sample drawn by the theory does not take this bias into account. Nevertheless, Csanady (1973) presents evidence in support of the log-normal distribution in flows not involving thermal convection. Effect offinite sampling volume One of several questions this argument does not address involves the averaging volume, or averaging period, of the individual con~ntration m~surement. If non-zero values of concentration on a relatively fine scale are distributed log-normally, what will be the distribution of concentrations measured with coarser resolution for which adjacent zero values may have been included in the average? To explore this question, we set up a two-dimensional numerical grid of 900
1304
_i W. DEARDORF~ and G. E. WILLIS
values of finite concentration prescribed randomly In space, with values satisfying a log-normal distribution. Then, non-overlapping compact sets of four adjacent concentrations were averaged together over the whole grid, and the distribution of the resulting values (225 of them) obtained. The result is shown in Fig. 8 by the dashed curves. The effect of the averaging when no zeroes were present is to reduce extreme values, decreasing the standard deviation of log C, but leaving the distribution still log-normal. If, however, a substantial fraction of zero values are present on the fine scale, namely 540 out of 900 in this case (with finite values again possessing a log-normal distribution), after averaging over sets of four the resulting smoothed values have the distribution shown by the solid curve labelled “after”. (There are then 200 finite values out of the 225 total.) The most dilute 20 3, of the finite population then no longer shares the log-normal distribution which the rest retain. In results to be presented, the fraction of the sampled population having zero concentration (in the region -by < y < oY)is estimated to have been about 40 y0 for X = 1.4, decreasing with X and being greater at small X due to lateral meandering of plume material. Thus, the fraction of zero values was not excessive, and the finite averaging volume used in this study is not cause for expecting log-normality to have been violated if present on a finer scale.
The sampling region Data samples were collected from individual remote-sensor traverses within the region - aY < y < a,,, where br is the usual mean-plume lateral spread obtained from the ensemble of six or seven experiments using the traversing remote detector. Measurements outside this region were excluded from this analysis in order to minimize the uncertainty
between a zero value and a finite value iymg below the threshold of measurement. The observations were not restricted to discrete values of y, such that a detailed _Vdependence could be examined, because insufficient sampling would have resulted. Using the convection tank, one cannot simply accumulate concentration samples over a period of time at a particular downstream distance at J I_-0, for example, during any one experiment. Instead, the elapsed time after passage of the stack by a particular point converts to x/U using Taylor’s translation hypothesis, and hundreds of separate experiments would then be required to collect the necessary data for given values of x and y. Hence, off-centerline values, out to y = +a,, will be included in our analyses for various X. although in examining the concentration variance this region will be subdivided into two equal portions: inner and outer. Observed lateral profiles of instantaneous concentration, C(y), from the seven experiments with F, = 0 for X = 1.4 f 0.1 are shown in Fig. 9(a); similar profiles from the six experiments with F, = 0.11 for X = 2.07 f 0.1 are shown in Fig. 9(b). As might be expected, the peak values often lie outside the analyzed range of - gY < y < oY;in Fig. 9(a) they do so four out of seven times, and in Fig. 9(b) two or three out of six times. Thus, the fluctuation statistics appear to be rather uniform within the analyzed region and can therefore be expected to be fairly representative of what occurs at y = 0 itself. It may also be noted that the concentration centrotd usually lies considerably off to one side of the y = 0 axis due to plume meandering. Thus, it is a definite advantage to know in advance the location of the true _V= 0 axis (along the stack towing line in the experiments) independently of what might be estimated from the limited supply of data samples at any given X. Observed cumulative distributions
T
C
05
02 I
1 5
IO 2030 CUMULATIVE
50
70 80
90 95
PERCENTAGE
Fig. 8. Cumulative frequency distribution (abscissa) of a numekally generated random distribution of log C values (see ordinate, arbitrary scale) before small-scale spatial averaging (steeper dashed line) and after averaging (less steep dashed line). Solid curves show same comparison when initial field contains 607, zero values. Distributions pertain only to finite C values.
Measured cumulative distributions for the case of the non-buoyant effluent for - u,, c y < uYare shown in Fig. 10 on a log-normal probability diagram. The number of samples entering each distribution at a particular mean value of X (= m increased from 470 for x = 0.65 to 3525 for x = 3.7. The rapid fall-off of the larger concentrations with Increasing X is evident in Fig. 9, along with increasing values of the weaker concentrations, for x> 1.4, as ‘fillmg in’ occurs. The lack of more compfete regularity in the pattern of the curves for successive X is probably attributable to sampling error. An interesting feature of the figure is the presence of a kink in the distribution, at least for X > 1.4. The presence and direction of the kink is associated with a skewed distribution of log C such that small values of log C have a long ‘tail’ while large values have a small tail. Thus, the distribution of C found here can only be described as log-normal in a very rough sense. The cumulative distributions of log C for the
Groundlevel concentration fluctuattons from a buoyant and a non-buoyant source
1305
b
-O-
Y
0
*y
Y-
--Q
0
Y
my
Y-
Fig. 9. (a) Lateral profiles ofCfy) from the seven expertments with non-buoyant effluent, for X averaging 1.4. (b) Lateral profiles of C(y) from the six experiments with buoyant effluent, for X averaging 2.07.
case, F, = 0.11, are shown in Fig. 11. The number of samples within any one distribution at a particular value of X in this case increased from 345 at IE?= 1 to 3330 at x = 3.7. Except for much smaller values of C in the uppermost 20% of the population for AT< 1.8, the distribution is quite similar to that of buoyant-effluent
I
5 CUMULATIVE
FREOUENCY (%I
Fig. 10. Cumulative frequency distributions (abscissa) of log C from non-buoyant cases (FL = 0) for mean values of X ranging from 0.65 to 3.7.
IO
I
I
2030 50 70 00 90 95 CUMULATIVE FREQUENCY (%)
Fig. 11. Cumulative frequency distributions of log C from buoyant cases (F, = 0.11) for mean values of X ranging from 1.0 to 3.7.
J. W DEARWRFF and G.
1306
the non-buoyant source. It also exhibits the same kink, whose position shifts towards smaller cumulative frequencies as X increases. The effect of uncertainty in the fraction of zeroconcentration values upon these distributions is to cause noticeable uncertainty in the ordinate at very small frequencies of occurrence, but little uncertainty for cumulative frequencies exceeding about 303;. The ‘filling in’apparent in both Figs lOand 11, as the weaker concentrations become somewhat greater downwind and the distribution begins to approach a horizontal line of uniformity, suggests gross violation of assumption (a) of the log-normality plausibility argument in Section 5. That is the assumption that after each mixing event a parcel’s concentration becomes less, as if it always mixed with cleaner fluid. Instead, parcels with weak concentrations have increasing opportunity to mix with parcels of greater concentration as travel time increases, and as tluid entrained into the plume gradually adopts meanplume characteristics. A very similar kink in the cumulative probability distribution of vertical velocity and temperature within the convective atmospheric boundary layer has been reported by Lenschow (1970) and others. It is in the sense that upward motions and associated aboveaverage temperatures have a substantial tail to their distribution, while the more prevalent downdrafts and associated below-mean temperatures are more uniform. The present, similarly kinked distribution of log C might be a manifestation of the same convective skewness, if upward motions are associated with smaller log C values and downward motions with larger log C values. There is some indirect evidence of this. For 1.2 < X < 3 or 4 the mean vertical distribution of C for the non-buoyant source near the surface is one of mean C increasing somewhat with height (Willis and Deardorff, 197Q so that upward motions might then be correlated with below average
E. WILLIS
concentrations, and vice versa. For the buoyant effluent, the increase of mean C with height must be even more pronounced, as discussed in Section 3. However. we cannot offer this qualitative observation as an explanation of the kink, since the diffusion from an elevated point source is so different than that from a large-area surface source, and since negative eddy diffusivities of vertical diffusion have been reported by Deardorff and Willis (1975). Observed concentration variances
In the inner region, - o/2 < y -Z 012 and the outer region 3 ug < 1y 1-c uyr and at z = 0.08h. values of the ratio p/i?* were calculated from six of the seven experiments with non-buoyant emissions and from four of the six with buoyant emissions. (These results were obtained at a later date than the concentration distributions and not all the original data tapes were still available.) For this purpose zero and near-zero values were of course included in the analyzed data, which therefore include plume meander effects as well as in-plume fluctuations. In calculating the ratio, values of the numerator and of T in the denominator were separately averaged from all available data within a small range of X before forming the ratio. (~ was obtained from 2’”- T2. Results are shown for the non-buoyant releases in Fig. 12 and for the buoyant releases in Fig. 13. The pronounced decrease in p/i?’ with X is the main feature of interest, and is very definite despite relatively large sampling error. (The points at X = 1.4 in Fig. 12 and at X = 3.2 for the outer region in Fig. I3 were ignored in fairing the smooth curves through the remaining points, because in these three instances the anomalies were caused by single events within just one of the nine and five traverses, respectively, from which the ensemble means in these cases were formed.) The squared fluctuation-to-mean ratio for the outer region
-a
v
Fig. 12. Measured values of c’~/TZfor the non-buoyant case as a function of X. Measurements denoted by A represent an average eve-rthe region Iyl < 0,/2; those denoted by circles represent an average over the region a,/2 -Z lyl < uY.
Groundlevel
concentration
fluctuations
I
from a buoyant and a non-buoyant
I
I
1307
source
I
8 F, =Oll 7
6
3
2
0
Fig.
13.
Measured values of C-“/C’for
the buoyant as in Fig. 12.
of Fig. 13 exceeds that of the inner region by some ifs;’ did not vary over the entire region from - u, to u,, but T varied in Gaussian fashion. For Fig. 12 the scatter is worse but the outer values tend to exceed the inner. Taken together, the results support previous findings (Csanady, 1967,1973) that 7 decays considerably less rapidly with y than does %. 60x,
about what is to be expected
The greater values of c’2/T2 for the buoyant release are believed associated with the elevated plume axis in this case; consequently the near-surface sampling often encountered relative fluctuation values on the (lower) fringe of the mean plume, where relative intensities are known to be of greater magnitude (Hanna, 1984). Because of the finite sampling volume, the measured relative intensities are underestimates of point values [even in the presence of small but finite molecular diffusivity, D,where the ‘conductivity’ cutoff scale, 1 = (~D'/E)"~, was approximately 0.04 mm, v is the kinematic viscosity and E is the rate of dissipation. See Chatwin and Sullivan (1979) for a discussion of the role of A.] With a doubled sampling volume, obtained by doubling Ay in a reanalysis of the concentration variances, it was found that p was decreased by 24 y0 at X = 0.5, falling to an 18 y0 reduction at X = 4. The present estimates of the variance may therefore be too small by a fraction comparable to these. In this reanalysis, it may be noted that the intra-plume variance is unaffected by altering Ay. The ratio of the intra-plume variance to the total variance was found to be 0.27 f 0.04, where the latter variation is the standard deviation of the mean. The ratio 0.27 was found to be insensitive to the buoyancy of the source, location within the mean plume, and the downstream distance.
source as a function
of X. Symbols
However, the intra-plume variance defined here may not coincide with the meandering component as usually defined; here the total variance is represented by c’zJ__Tz
= (;rr_F)+(p_T2) A
(5) B
where c^is the mean from a particular realization over either the inner region or outer region. Term A is the within-plume variance and term B the intra-plume variance. Our results suggest that the within-plume variance dominates in the convective boundary layer, at least for 0.5 < X < 4. The pronounced decrease of c’1/r2 with distance associated with the effects of downstream diffusion from the continuous source and, at later times, with vertical diffusion after the presence of the mixed-layer ‘lid’, as well as the surface, had been felt. It may be noted that an opposite result of Sawford (1983) was obtained with a model which assumed onedimensional mixing and may not have taken into account these two homogenizing influences. seems
Intermittency
Values of the intermittency, y, in the inner and outer regions of the mean plume, for the buoyant and nonbuoyant releases, are shown in Fig. 14. (A value of y = 1 means that the contaminant was always present.) The threshold level chosen for the presence or absence of C was 0.05 (c’2)“’ or 0.05, whichever was larger. Results show the expected smaller values at smaller X, indicative of plume meanderings, especially near the buoyant source where plume rise caused y = 0.
J. W
1308
DFARDORFF and G.
E. WILLIS
B
Si-
,
2-
X Rg. 14. Measured values of intermittency, y, over the inner region (A)and the outer region (circles) as a function of X. Open symbols denote non-buoyant source. closed symbols denote the buoyant source.
Quantitative values are, however, believed to be rather sensitive to both the threshold level and the size of the sampling volume.
6. SUMMARY
Acknowledgements-This work was supported by the Environmental Protection Agency under Grant R808919. P. Stockton provided expert technical support. We thank three anonymous reviewers for providing valuable criticisms of earlier versions of this paper.
AND CONCLUSIONS
Using mixed-layer scaling, the near-surface concentration at moderate distances downwind of a stack emitting a buoyant efauent characterized by F, = 0.11 has been obtained and on the average compared quantitatively with the glc from a non-buoyant source that was otherwise identical and subject to identical environmental conditions. Although results for the buoyant effluent will depend somewhat upon conditions just above the top of the mixed layer, as well as upon stack height, those obtained are believed representative of typical atmospheric mixed-layer conditions when the mixed-layer depth is much greater than the stack height. As would be expected from the effect of plume rise, the buoyant e&tent has a much smaller mean maximum gic, with delayed appearance downstream, in comparison with the non-buoyant eIIluent. It also evidences an extremely slow decay of centerline glc downstream of its (faint) surface maximum, in comparison with the very rapid decay in the non-buoyant case immediately downstream of this (highly peaked) maximum. The cumulative frequency distribution of sampled concentrations was presented for the same two cases for various dimensionless downwind distances between 0.6 and 4. From these, the expected maximum concentration to be exceeded near the ground under convective conditions 2% or less of the time, for example, may be obtained. In both cases, for X > 1.2, the distributions are found to be log-normal in only a bimodal sense, with the weaker concentrations having a greater spread of log C than the stronger concentrations. The concentration variance relative to the square of the mean, near they = 0 axis, is found to decrease quite strongly with downwind distance.
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