- Email: [email protected]
Ground-level concentration fluctuations from a buoyant and a non-buoyant source within a laboratory convectively mixed layer
1210
Discussions
tation of some o f the existing National Ambient Air Quality Standards.
Environmental Science Department General Motors Research Laboratories Warren, MI 48090-9055, U.S.A.
DAVID P. CHOCK
REFERENCES
Bharucha-Reid A. T. (1960) Elements of the Theory of Marker Processes and Their Applications. McGraw-Hill, New York. Chock D. P. (1985) Statistics of extreme values of air quality--a simulation study (to be submitted).
GROUND-LEVEL CONCENTRATION FLUCTUATIONS FROM A BUOYANT AND A NON-BUOYANT SOURCE WITHIN A LABORATORY CONVECTIVELY MIXED LAYER Deardorff and Willis (1984) and Willis and Deardorff (1983) report on mean plume rise and diffusion and concentration fluctuations observed in buoyant and non-buoyant plumes released in their laboratory's convective tank. Lamb (1982) and Moninger et al. (1983) have shown that properly-scaled mean plume observations in the convective tank are similar to observations in the real atmosphere. Appropriate scaling parameters are the mixing depth, h, and the convective velocity scale, w, = (ghH/cvp) 1/3, where H/cpp is the kinematic surface heat flux. In this note it will be shown that a simple analytical model for concentration fluctuations provides a good fit to the convective tank observations. Probability distribution functions of pollutant concentrations can usually be fitted by a two-parameter analytical formula, such as the clipped-normal (Lewellen and Sykes, 1983), the log-normal with intermittency (Wilson, 1982) or the exponential with intermittency (Barry, 1977; Hanna, 1984). Intermittency, I, is defined as the fraction of non-zero readings in the time-series of concentrations at a given monitor. In any of the three suggested formulas the full distribution is completely determined once any two of the three parameters (I, ac, ~ ) are specified, where a c and ~-" are the standard deviation and the mean of the concentration data. Wilson (1982) and Hanna (1984) show that the exponential distribution is more valid for atmospheric observations around a single point source, where strong variations in wind direction nearly always occur and the resulting concentration time series is highly intermittent. For an exponential distribution function, the following formula can be used to predict the ratio of the standard deviation of the concentration fluctuations to the mean: ac/C
a o = 0.0078 h, T L = 0.5h/w,, a v = 0.6w, h e = 0.15 h non-buoyant h e = 0.40 h buoyant (assumes a receptor height of 0.086 h) Lamb (1982) has derived the empirical formulas az/h = X/2 for X < 2 / 3 and a , / h = l / 3 for X > 2 / 3 from previous experiments in Deardorff's tank, where dimensionless downwind distance X equals xw,/uh. However, Lamb states that this formula is only approximate at X less than one because of the non-Gaussian shape of the vertical velocity distribution, and is uncertain for X greater than about one because the vertical distribution becomes nearly uniform beyond that point. (For a uniform distribution az/h = 0.3 from the definition of the second moment.) In order to assure that the exponential term in Equation (2)approaches unity, which will better parameterize the effects of a uniform distribution at large X on concentration fluctuations, the formula a,/h = X/2 is used for all X in our application. It is also assumed that the vertical and lateral turbulence and diffusion are homogeneous in the convective boundary layer, so that Equation (3) applies equally to ay and az. Resulting predictions of %/C and I are compared with the observations in Figs 1 and 2, respectively. The predicted lines tend to pass through the observed points on the figures. The biggest disagreements between observations and predictions occur at large X in Fig. 1, where observed ac/C drops to less than 0.5 while predicted ac/~ remains at about unity. This disagreement may be due to the constraining effects of the tank walls at large downwind distances, which limit the plume meandering and make the exponential p d f assumption invalid. After sufficient time has elapsed in the tank, ac/-~ would approach zero, similar to the fluctuations in cream concentrations in a cup of coffee. In the atmosphere, no such constraints exist (except in valleys) and Gifford (1982) has shown that meandering exists at time scales out to several days. These calculations have assumed that concentrations are observed in an infinitely small volume. Deardorff and Willis (1984) state that the finite averaging distance (about 0.01 h) for the concentration measurement in their tank may reduce the effective observed oc/C by about 10°o from what it would be for instantaneous measurements in an infinitesimal volume. This effect can be studied using an equation suggested by Venkatram (1979) and assuming that XA/X l equals tA/t I, where x I and t I are integral distance and time scales for the concentration fluctuations:
(1)
Oc(XA)/Oc(O) = (2 XI (I _ XI (I _ e x p ( _ XA/Xl) ) ))1,2.(4)
The intermittency at a receptor beneath the plume centerline can be estimated from a formula derived by Hanna (1984):
The parameter x A is the averaging distance. As Sykes (1984) has shown, x I for concentration fluctuations is not necessarily equal to the integral distance scale for velocity fluctuations (UTL), he suggests a simple formula for calculating the ratio of me integral scale of the concentration fluctuations, x I, to the integral scale of the velocity fluctuations, UTL, near the center of the plume:
=
((2/•)- 1) i:2.
where T' = t/TL, and T L is the Lagrangian time scale of the velocity fluctuations. The parameter 6o equals ao/x//2%TL , where ao is the source size. H a n n a (1984) presents a figure which contains graphical solutions to Equation (3) for several assumed values of 8-oz. In order to apply these formulas to the convective tank data, the following assumptions are made, based on information in the papers by Deardorffand Willis (1984)and Willis and Deardorff (1983):
\
1 = (ayi~Tzl/ffyT~TzT)exp ( --he/2azT), 2 2
(2)
where subscript I and subscript T refer to the instantaneous and time-mean plume spreads, and h e is the elevation of the mean plume centerline above the receptor. The ratio al/a T is given by a formula suggested by Gifford (1982) and Lee and Stone (1983) based on a solution to the Langevin equation: °2 = ° o 2 + T ' - ( 1 - - e - T ' ) - - 0 . 5 ( 1 _ e - r ' ) 2 a.~ 8o2+T ' - ( 1 - e - r ' )
(3)
Xl/UT L =
XA
XA
0.5 (~rc/C)- 2 In (1 + 2ac/C ).
(5)
Deardorff and Willis' data show that ac/~" is about 2 at X equal to 1.0, and Equation (5) predicts that xt/uT L would equal 0.20 at that distance. Consequently the ratio XA/X l is about 0.20 at X equal to 1.0, and the ratio of the standard deviations ac(XA)/ac(0) is predicted by Equation (4) to be
Discussions
1211
I I I • •
I I l l l
Gc
Nor)buoyant .... Buoyant .....
\
•
•
o
.,)1J2
Mod.,:
~
~
A
~
~ ~ ~
Fig. 1. Observed (points) and predicted (lines) ac/C for Dcardorff and Willis' (1984) tank data.
Model: ~ =
0 2 ~12ex p (-he2/2Oz2) aT
• •
•
1.o
JO
Nonbuoya nt Buoyant
•
/
Fig. 2. Observed (points) and predicted (lines) intermittency I for Deardorff and Willis' (1984) tank data.
about 0.97. Thus the theory predicts about a 3 ~ reduction in oc/~" due to the finite sampling volume, which is less than the reduction derived by Deardorff and Willis. The difference may be due to small scale fluctuations in the plume interior that are not accounted for by the simple models in Equations (4) and (5). In any case, an error on the order of 10% has little effect on the conclusions reached in this note.
Acknowledoements--This research was sponsored by the Ag 19 : 7-~1
Electric Power Research Institute, with Dr Glenn Hilst as project monitor, and the Army Research Office, with Dr Walter Bach as project monitor.
Environmental Research and Technology, Inc., 696 Viroinia Road, Concord, MA 01742, U.S.A.
STEVEN R. HANNA
1212
Discussions REFERENCES
Barry P. J. (1977) Stochastic properties of atmospheric diffusivity. In Sulfur and Its Inorganic Derivatives in the Canadian Environment, pp. 313-358. National Research Council of Canada. Deardorff J. W. and Willis G. E. (1984) Ground-level concentration fluctuations from a buoyant and nonbuoyant source within a laboratory conveetively mixed layer. Atmospheric Environment 18, 1297 1310. Gifford F. A. (1982) Horizontal diffusion in the atmosphere: a Lagrangian-dynamical theory. Atmospheric Environment 16, 505-512. Hanna S. R. (1984) The exponential probability density function and concentration fluctuations in smoke plumes. Boundary-Layer Met. 29, 361-375. Lamb R. G. (1982) Diffusion in the convective boundary layer. In Atmospheric Turbulence and Air Pollution Modellin# (edited by Nieuwstadt F. T. M. and van Dop H.), pp. 159-230. D. Reidel, Dordrecht. Lee J. T. and Stone J. L. (1983) The use of Eulerian initial conditions in a Lagrangian model of turbulent diffusion. Atmospheric Environment 17, 2477-2481. Lewellen W. S. and Sykes R. I. (1983) On the use of concentration variance predictions as a measure of natural uncertainty in observed concentration samples. Proc. 6th Syrup. of Turb. and Diffl Am. Met. Soc, Boston, MA. Moninger W. R., Eberhard W. L., Briggs 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, Soc. Sykes R. I. (1984) The variance in time-averaged samples from an intermittent plume. Atmospheric Environment 18, 121-123. Willis G. E. and Deardorff J. W. (1983) On plume rise within a convective boundary layer. Atmospheric Environment 17, 2435-2447. Wilson D. J. (1982) Predicting risk of exposure to peak concentrations in fluctuating plumes. Alberta Environment. Venkatram A. (1979) The expected deviation of observed concentrations from predicted ensemble means. Atmospheric Environment 13, 1547--1549.
AUTHORS'
REPLY
Hanna (1985) shows that the exponential distribution with assumed intermittency, as applied to centerline concentrations downwind from a compact source, yields predictions for ~rc/C which agree satisfactorily with our laboratory mixed-layer results, if X does not exceed about 3. Although this result is encouraging, it should be kept in mind that the concept of intermittency is not satisfactory unless the dependency on threshold level, e¢ is included; i.e. the selected measurement level below which the concentration is considered to be zero or background level. As a passive plume disperses, the turbulence within the plume is no different than that outside. Therefore, the plume detrains some of its material into the 'environment', just as it entrains environmental air into itself. This means that detraining blobs or threads of pollutant are stretched out ever finer near the edges of the instantaneous plume, leading to exceedingly small (but non-zero) concentrations there. For X not too large, this same fluid can reappear at the mean-plume centerline position either due to plume meanders or reentrainment into the interior of the plume. Hence, what may seem like perfectly pure fluid intermittently observed at the mean-plume centerline will actually often contain trace a m o u n t s of emitted material, with concentrations ranging
from undetectable on up. The fraction of the time such fluid is present at the sensor will depend upon the arbitrary threshold level. We are not convinced that an optimum threshold level exists to minimize this uncertainty. Similar arguments exist for the sampling volume and/or the individual-sample duration period. For a given concentration threshold level, a greater sample size means less likelihood that relatively pure fluid with subthreshold concentration will be observed. On this point, Hanna (Equation 4) refers to Venkatram (1979) for a formula for estimating the reduction of observed concentration-fluctuation variance due to finite sampling volume. Using this same formula, and our finding (Deardorffand Willis, 1984) that the variance was reduced by some 20" 0 when x n was increased from 0.02h to 0.04h, we calculate that XA/X I = 0.65. This value is much larger than Hanna's deduced value xa/x I = 0.20 at X = 1. This emphasizes that the uncertainties here are large. A careful calculation ofx~ should be made from the original data; however, its value will be too large if the data samples are "smoothed' due to finite sample size. Chatwin and Sullivan (1979) have pointed out the basis of this potential problem: were it not for molecular diffusion or brownian diffusion, an ideal instrument with vanishingly small sample volume would always sense either pure uncontaminated fluid or concentrations having full source strength. Thus, close to a source where molecular diffusion has not had much time to act, the intensity of concentration 'spikes' may be greatly underestimated and x I overestimated. On the other hand, Equation (4) may be in error, as, for example, if the spectrum of the concentration fluctuations should turn out to be more complicated than that associated with a simple exponential autocorrelation. The spectrum might possibly be bimodal, with one peak at the scale Hanna has in mind and a second peak at the much smaller.scales of the mean separation between pollutant strands or filaments. For these reasons, we have reported values of our sampling volume and threshold level. However, it appears that the dependencies of intermittency, 1, and of x I upon sampling volume and downwind distance, and of I upon ec as well, need to be determined if theories making use of the intermittency concept are to be tested comprehensively. The theories should take into account both xA and cc" Regarding our measurements Ofac/'C < 1 at the larger X vs the exponential distribution prediction, we mentioned in our paper how continued entrainment of fresh fluid into the top of the boundary layer will prevent this value from approaching zero too closely. A m i n i m u m value of 0.1, for example, would not be inconsistent with our data in Hanna's Fig. 1. The convective mixed layer, even in the laboratory, is in this and other respects unlike the mixed volume in a cup of coffee with cream stirred in. In our convection tank, a period corresponding to X = 5 must be exceeded before wall reflections (back into the region in which we made fluctuation measurementsl first begin to occur. One must wait a period corresponding to X _~ 35 before lateral mixing within the tank is nearly complete. Our data in question extend out only to X = 4. Concerning plume meanders in the convectively mixed layer of the lower atmosphere vs that in the laboratory apparatus, the root mean square (r.m.s.) angle of such meandering is, in both cases, measured by ao ~- at,/u, where o,, is the r.m.s lateral wind fluctuation. In both cases, this value is about 0.7w,/u, although in experiments prior to 1977 we utilized a deeper mixed layer for which wall effects were less minimal and av/w, near the surface was only 0.55 instead of 0.7. Our value of w,/u utilized in the concentrationfluctuation measurements was about 0.44, also a typical atmospheric value. The plume-meandering angles we observed are thus expected to have been typical of those in the atmospheric mixed layer over flat, uniform terrain in the absence of appreciable mesoscale motions. We are therefore interested in whether field measurements o f a c / C at X > 3 or 4 under conditions of thermal convection will also show small ac/(" values.
Discussions
tation of some o f the existing National Ambient Air Quality Standards.
Environmental Science Department General Motors Research Laboratories Warren, MI 48090-9055, U.S.A.
DAVID P. CHOCK
REFERENCES
Bharucha-Reid A. T. (1960) Elements of the Theory of Marker Processes and Their Applications. McGraw-Hill, New York. Chock D. P. (1985) Statistics of extreme values of air quality--a simulation study (to be submitted).
GROUND-LEVEL CONCENTRATION FLUCTUATIONS FROM A BUOYANT AND A NON-BUOYANT SOURCE WITHIN A LABORATORY CONVECTIVELY MIXED LAYER Deardorff and Willis (1984) and Willis and Deardorff (1983) report on mean plume rise and diffusion and concentration fluctuations observed in buoyant and non-buoyant plumes released in their laboratory's convective tank. Lamb (1982) and Moninger et al. (1983) have shown that properly-scaled mean plume observations in the convective tank are similar to observations in the real atmosphere. Appropriate scaling parameters are the mixing depth, h, and the convective velocity scale, w, = (ghH/cvp) 1/3, where H/cpp is the kinematic surface heat flux. In this note it will be shown that a simple analytical model for concentration fluctuations provides a good fit to the convective tank observations. Probability distribution functions of pollutant concentrations can usually be fitted by a two-parameter analytical formula, such as the clipped-normal (Lewellen and Sykes, 1983), the log-normal with intermittency (Wilson, 1982) or the exponential with intermittency (Barry, 1977; Hanna, 1984). Intermittency, I, is defined as the fraction of non-zero readings in the time-series of concentrations at a given monitor. In any of the three suggested formulas the full distribution is completely determined once any two of the three parameters (I, ac, ~ ) are specified, where a c and ~-" are the standard deviation and the mean of the concentration data. Wilson (1982) and Hanna (1984) show that the exponential distribution is more valid for atmospheric observations around a single point source, where strong variations in wind direction nearly always occur and the resulting concentration time series is highly intermittent. For an exponential distribution function, the following formula can be used to predict the ratio of the standard deviation of the concentration fluctuations to the mean: ac/C
a o = 0.0078 h, T L = 0.5h/w,, a v = 0.6w, h e = 0.15 h non-buoyant h e = 0.40 h buoyant (assumes a receptor height of 0.086 h) Lamb (1982) has derived the empirical formulas az/h = X/2 for X < 2 / 3 and a , / h = l / 3 for X > 2 / 3 from previous experiments in Deardorff's tank, where dimensionless downwind distance X equals xw,/uh. However, Lamb states that this formula is only approximate at X less than one because of the non-Gaussian shape of the vertical velocity distribution, and is uncertain for X greater than about one because the vertical distribution becomes nearly uniform beyond that point. (For a uniform distribution az/h = 0.3 from the definition of the second moment.) In order to assure that the exponential term in Equation (2)approaches unity, which will better parameterize the effects of a uniform distribution at large X on concentration fluctuations, the formula a,/h = X/2 is used for all X in our application. It is also assumed that the vertical and lateral turbulence and diffusion are homogeneous in the convective boundary layer, so that Equation (3) applies equally to ay and az. Resulting predictions of %/C and I are compared with the observations in Figs 1 and 2, respectively. The predicted lines tend to pass through the observed points on the figures. The biggest disagreements between observations and predictions occur at large X in Fig. 1, where observed ac/C drops to less than 0.5 while predicted ac/~ remains at about unity. This disagreement may be due to the constraining effects of the tank walls at large downwind distances, which limit the plume meandering and make the exponential p d f assumption invalid. After sufficient time has elapsed in the tank, ac/-~ would approach zero, similar to the fluctuations in cream concentrations in a cup of coffee. In the atmosphere, no such constraints exist (except in valleys) and Gifford (1982) has shown that meandering exists at time scales out to several days. These calculations have assumed that concentrations are observed in an infinitely small volume. Deardorff and Willis (1984) state that the finite averaging distance (about 0.01 h) for the concentration measurement in their tank may reduce the effective observed oc/C by about 10°o from what it would be for instantaneous measurements in an infinitesimal volume. This effect can be studied using an equation suggested by Venkatram (1979) and assuming that XA/X l equals tA/t I, where x I and t I are integral distance and time scales for the concentration fluctuations:
(1)
Oc(XA)/Oc(O) = (2 XI (I _ XI (I _ e x p ( _ XA/Xl) ) ))1,2.(4)
The intermittency at a receptor beneath the plume centerline can be estimated from a formula derived by Hanna (1984):
The parameter x A is the averaging distance. As Sykes (1984) has shown, x I for concentration fluctuations is not necessarily equal to the integral distance scale for velocity fluctuations (UTL), he suggests a simple formula for calculating the ratio of me integral scale of the concentration fluctuations, x I, to the integral scale of the velocity fluctuations, UTL, near the center of the plume:
=
((2/•)- 1) i:2.
where T' = t/TL, and T L is the Lagrangian time scale of the velocity fluctuations. The parameter 6o equals ao/x//2%TL , where ao is the source size. H a n n a (1984) presents a figure which contains graphical solutions to Equation (3) for several assumed values of 8-oz. In order to apply these formulas to the convective tank data, the following assumptions are made, based on information in the papers by Deardorffand Willis (1984)and Willis and Deardorff (1983):
\
1 = (ayi~Tzl/ffyT~TzT)exp ( --he/2azT), 2 2
(2)
where subscript I and subscript T refer to the instantaneous and time-mean plume spreads, and h e is the elevation of the mean plume centerline above the receptor. The ratio al/a T is given by a formula suggested by Gifford (1982) and Lee and Stone (1983) based on a solution to the Langevin equation: °2 = ° o 2 + T ' - ( 1 - - e - T ' ) - - 0 . 5 ( 1 _ e - r ' ) 2 a.~ 8o2+T ' - ( 1 - e - r ' )
(3)
Xl/UT L =
XA
XA
0.5 (~rc/C)- 2 In (1 + 2ac/C ).
(5)
Deardorff and Willis' data show that ac/~" is about 2 at X equal to 1.0, and Equation (5) predicts that xt/uT L would equal 0.20 at that distance. Consequently the ratio XA/X l is about 0.20 at X equal to 1.0, and the ratio of the standard deviations ac(XA)/ac(0) is predicted by Equation (4) to be
Discussions
1211
I I I • •
I I l l l
Gc
Nor)buoyant .... Buoyant .....
\
•
•
o
.,)1J2
Mod.,:
~
~
A
~
~ ~ ~
Fig. 1. Observed (points) and predicted (lines) ac/C for Dcardorff and Willis' (1984) tank data.
Model: ~ =
0 2 ~12ex p (-he2/2Oz2) aT
• •
•
1.o
JO
Nonbuoya nt Buoyant
•
/
Fig. 2. Observed (points) and predicted (lines) intermittency I for Deardorff and Willis' (1984) tank data.
about 0.97. Thus the theory predicts about a 3 ~ reduction in oc/~" due to the finite sampling volume, which is less than the reduction derived by Deardorff and Willis. The difference may be due to small scale fluctuations in the plume interior that are not accounted for by the simple models in Equations (4) and (5). In any case, an error on the order of 10% has little effect on the conclusions reached in this note.
Acknowledoements--This research was sponsored by the Ag 19 : 7-~1
Electric Power Research Institute, with Dr Glenn Hilst as project monitor, and the Army Research Office, with Dr Walter Bach as project monitor.
Environmental Research and Technology, Inc., 696 Viroinia Road, Concord, MA 01742, U.S.A.
STEVEN R. HANNA
1212
Discussions REFERENCES
Barry P. J. (1977) Stochastic properties of atmospheric diffusivity. In Sulfur and Its Inorganic Derivatives in the Canadian Environment, pp. 313-358. National Research Council of Canada. Deardorff J. W. and Willis G. E. (1984) Ground-level concentration fluctuations from a buoyant and nonbuoyant source within a laboratory conveetively mixed layer. Atmospheric Environment 18, 1297 1310. Gifford F. A. (1982) Horizontal diffusion in the atmosphere: a Lagrangian-dynamical theory. Atmospheric Environment 16, 505-512. Hanna S. R. (1984) The exponential probability density function and concentration fluctuations in smoke plumes. Boundary-Layer Met. 29, 361-375. Lamb R. G. (1982) Diffusion in the convective boundary layer. In Atmospheric Turbulence and Air Pollution Modellin# (edited by Nieuwstadt F. T. M. and van Dop H.), pp. 159-230. D. Reidel, Dordrecht. Lee J. T. and Stone J. L. (1983) The use of Eulerian initial conditions in a Lagrangian model of turbulent diffusion. Atmospheric Environment 17, 2477-2481. Lewellen W. S. and Sykes R. I. (1983) On the use of concentration variance predictions as a measure of natural uncertainty in observed concentration samples. Proc. 6th Syrup. of Turb. and Diffl Am. Met. Soc, Boston, MA. Moninger W. R., Eberhard W. L., Briggs 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, Soc. Sykes R. I. (1984) The variance in time-averaged samples from an intermittent plume. Atmospheric Environment 18, 121-123. Willis G. E. and Deardorff J. W. (1983) On plume rise within a convective boundary layer. Atmospheric Environment 17, 2435-2447. Wilson D. J. (1982) Predicting risk of exposure to peak concentrations in fluctuating plumes. Alberta Environment. Venkatram A. (1979) The expected deviation of observed concentrations from predicted ensemble means. Atmospheric Environment 13, 1547--1549.
AUTHORS'
REPLY
Hanna (1985) shows that the exponential distribution with assumed intermittency, as applied to centerline concentrations downwind from a compact source, yields predictions for ~rc/C which agree satisfactorily with our laboratory mixed-layer results, if X does not exceed about 3. Although this result is encouraging, it should be kept in mind that the concept of intermittency is not satisfactory unless the dependency on threshold level, e¢ is included; i.e. the selected measurement level below which the concentration is considered to be zero or background level. As a passive plume disperses, the turbulence within the plume is no different than that outside. Therefore, the plume detrains some of its material into the 'environment', just as it entrains environmental air into itself. This means that detraining blobs or threads of pollutant are stretched out ever finer near the edges of the instantaneous plume, leading to exceedingly small (but non-zero) concentrations there. For X not too large, this same fluid can reappear at the mean-plume centerline position either due to plume meanders or reentrainment into the interior of the plume. Hence, what may seem like perfectly pure fluid intermittently observed at the mean-plume centerline will actually often contain trace a m o u n t s of emitted material, with concentrations ranging
from undetectable on up. The fraction of the time such fluid is present at the sensor will depend upon the arbitrary threshold level. We are not convinced that an optimum threshold level exists to minimize this uncertainty. Similar arguments exist for the sampling volume and/or the individual-sample duration period. For a given concentration threshold level, a greater sample size means less likelihood that relatively pure fluid with subthreshold concentration will be observed. On this point, Hanna (Equation 4) refers to Venkatram (1979) for a formula for estimating the reduction of observed concentration-fluctuation variance due to finite sampling volume. Using this same formula, and our finding (Deardorffand Willis, 1984) that the variance was reduced by some 20" 0 when x n was increased from 0.02h to 0.04h, we calculate that XA/X I = 0.65. This value is much larger than Hanna's deduced value xa/x I = 0.20 at X = 1. This emphasizes that the uncertainties here are large. A careful calculation ofx~ should be made from the original data; however, its value will be too large if the data samples are "smoothed' due to finite sample size. Chatwin and Sullivan (1979) have pointed out the basis of this potential problem: were it not for molecular diffusion or brownian diffusion, an ideal instrument with vanishingly small sample volume would always sense either pure uncontaminated fluid or concentrations having full source strength. Thus, close to a source where molecular diffusion has not had much time to act, the intensity of concentration 'spikes' may be greatly underestimated and x I overestimated. On the other hand, Equation (4) may be in error, as, for example, if the spectrum of the concentration fluctuations should turn out to be more complicated than that associated with a simple exponential autocorrelation. The spectrum might possibly be bimodal, with one peak at the scale Hanna has in mind and a second peak at the much smaller.scales of the mean separation between pollutant strands or filaments. For these reasons, we have reported values of our sampling volume and threshold level. However, it appears that the dependencies of intermittency, 1, and of x I upon sampling volume and downwind distance, and of I upon ec as well, need to be determined if theories making use of the intermittency concept are to be tested comprehensively. The theories should take into account both xA and cc" Regarding our measurements Ofac/'C < 1 at the larger X vs the exponential distribution prediction, we mentioned in our paper how continued entrainment of fresh fluid into the top of the boundary layer will prevent this value from approaching zero too closely. A m i n i m u m value of 0.1, for example, would not be inconsistent with our data in Hanna's Fig. 1. The convective mixed layer, even in the laboratory, is in this and other respects unlike the mixed volume in a cup of coffee with cream stirred in. In our convection tank, a period corresponding to X = 5 must be exceeded before wall reflections (back into the region in which we made fluctuation measurementsl first begin to occur. One must wait a period corresponding to X _~ 35 before lateral mixing within the tank is nearly complete. Our data in question extend out only to X = 4. Concerning plume meanders in the convectively mixed layer of the lower atmosphere vs that in the laboratory apparatus, the root mean square (r.m.s.) angle of such meandering is, in both cases, measured by ao ~- at,/u, where o,, is the r.m.s lateral wind fluctuation. In both cases, this value is about 0.7w,/u, although in experiments prior to 1977 we utilized a deeper mixed layer for which wall effects were less minimal and av/w, near the surface was only 0.55 instead of 0.7. Our value of w,/u utilized in the concentrationfluctuation measurements was about 0.44, also a typical atmospheric value. The plume-meandering angles we observed are thus expected to have been typical of those in the atmospheric mixed layer over flat, uniform terrain in the absence of appreciable mesoscale motions. We are therefore interested in whether field measurements o f a c / C at X > 3 or 4 under conditions of thermal convection will also show small ac/(" values.