12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

PROBABILITIES OF HIGH-CONCENTRATION EVENTS IN SCALAR
POINT-SOURCE PLUME IN A TURBULENT BOUNDARY LAYER

Miaoyan Pang and Kapil Chauhan∗Centre for Wind, Waves, and Water
School of Civil Engineering, The University of Sydney, Syndey, NSW 2006
∗ kapil.chauhan@sydney.edu.au

ABSTRACTsuggest the possible existence of self-similarity of concentration PDFs.
Various distributions are fitted to concentration data inpast studies, including but not limited to Weibull, Beta,
Gamma, log-normal, and Gaussian distributions. The results
are summarised by (Cassiani et al., 2020, See Table 2). Most
researchers have found agreement of the Gamma distribution(Nironi et al., 2015; Efthimiou et al., 2016; Oettl & Ferrero,2017) with their data, while Yee et al. (1993) concludes thata Weibull distribution describes the concentration PDFs better. Hence, there is no broad agreement on a representative
functional form. It is to be noted here that these studies attempt to fit the full-signal PDF, which includes near-zero andnon-zero concentrations. However, instantaneous concentration of scalar, particularly in a meandering plume, is intermittent, i.e. the concentration is zero for a certain duration andthen peaks to values much greater than the mean-concentration
at other times. Such concentration behaviour is not suitably
represented by a Gamma distribution or a Weibull distribution,
which prescribes p(C = 0) = 0. Here C is the instantaneousconcentration and p is the probability density function. Therefore when fitting the full range of the concentration PDF, one
needs to take into consideration both the possible peak at C = 0
due to intermittency and the overall shape at C > 0. Nironi
et al. (2015) verifies the effect of intermittency, π, by plotting
the PDF for signals with various intermittency. Hypothetically,
the PDF of a uniform source with no molecular diffusion is

In this paper, we study the probability density functions(PDFs) of concentration in a plume of passive scalar released
from a point-source in a turbulent boundary layer. The emphasis is on the high concentration events, i.e. the tail of the
PDF. Past studies on PDFs have focused on limited positionsin the plume, such as at the plume centreline. Peak concentration events are not well-documented either. We extend theanalysis of PDFs of the temporal concentration signal and theunderlying instantaneous peaks for many locations across thewhole plume width. It is found that a Gamma distribution canpartly describe the PDFs, i.e. for finite concentration magnitudes, a result that is in agreement with previous studies. A
mathematical model for the tail of PDFs for high concentration
magnitudes can be simplified with a pure exponential function
and the corresponding decay exponents are found to be similar
across the plume. A scaling factor is then used to collapse the
PDF tails cross the plume. The analysis show that the decay
exponents are proportional to the power of plume half-width.
On the other hand, the scaling factor has a Gaussian distribution and hence could be related to the mean concentration
profile. Therefore, mean concentration profiles could be used
to predict the high concentration tails. We also study the tailof the PDFs of peaks over certain threshold and maxima overa period. The tails of these PDFs also share the same decay
exponents as that of the full-signal PDFs.

INTRODUCTION

p(C; x,t) = π(x,t)δ (C −C1 ) + (1 − π(x,t)) δ (C)The spread of a plume in a turbulent boundary layer has
been extensively studied. However, due to the complexity ofthe turbulent flow, prediction of the instantaneous concentration downstream of the source cannot be realised. Hence, one
has to rely on statistics to predict occurrence of instantaneous
events with desired certainty, of which high-concentration
events are of practical interest. The approach is to use the
PDFs of concentration at different points in space as they contain information of all moments as well as the probability ofoccurrence of extreme events.
Chatwin & Sullivan (1990) highlight the scaling law between the mean concentration and the rms of the concentrationfrom both theory and experimental data for uniform source.
They extend their theory to similarity of the ratio of higherorder consecutive moments. However their predictions are notsuitable for near field where the concentration fluctuations arehighly dependent on the source condition. Their work is complemented by that of Sawford & Sullivan (1995) for near field
and background flow of homogeneous turbulence. Since the
PDF includes the information of all moments, these studies(1)

where C1 is the source concentration(O’Brien, 1978; Chatwin& Sullivan, 1990). Here the intermittency is accounted for by
the Dirac-delta function at C = 0. When molecular diffusion
is present, the first term becomes a probability functionp(C; x,t) = π(x,t) f (C; x,t) + (1 − π(x,t)) δ (C)(2)

where f is the PDF (similar to equation 14 by Chatwin & Sullivan (1989)). In a follow-up study, Yee & Chan (1997) discuss
fitting a shifted and clipped-Gamma distribution to the full signal with assumptions that simplify the role of molecular diffusion. By examining the exceedance probability distribution of
the data and the model, the higher concentrations are found to
be under-predicted.
The tail of PDFs are of interest because high concentration of pollutants are more concerning, whereas low concentration measurements are affected by the instrumentation (Cassiani et al., 2020). High concentration statistics e.g. 90th and112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)98th percentile are used in regulations to quantify odour impact(Oettl & Ferrero, 2017; Invernizzi et al., 2020, e.g.). The challenge of modelling the tail of concentration is compoundedby unavailability of reliable data as long sampling times areneeded. Only a few studies focus on fitting certain distributions to the tail of the PDFs, although the extent of fits varyin each study. Schopflocher (2001) and Mole et al. (2008)
used Extreme Value Theory to investigate the ‘goodness’ of fitof a Generalized Pareto Distribution (GPD) for concentration
larger than the local mean. The variation of the shape parameter in the GPD cross-plume is found to be relatively constant in
Schopflocher (2001). Efthimiou et al. (2016) used a Gamma
distribution to fit the 75th to 99th percentile of the concentration data whereas Oettl & Ferrero (2017) found that a Weibull
distribution describes the PDF well when the concentration ishigher than the 90th percentile. Among these three fits, both
Weibull and Gamma have a exponential tail while GPD doesnot. Hence, our goal is to: a) evaluate the appropriate distribution to fit the PDF tails of concentration, and b) assess thevariation of the distribution parameters across the plume.
This conference paper briefly describes our initial findings on the parameterisation of the high concentration tails using data from plume released at different source-heights in aturbulent boundary layer.

10-2

p(C)
10-4

z
10-6

0

50

100

150

200

150

200

C, ppm
(b)

10-2

z

p(C)
10-4

10-6

0

50

100

C, ppm

EXPERIMENTAL DETAILSA neutrally buoyant gas mixture (1.5% iso-Butylene
mixed in 98.5% Nitrogen) is released from a point-source located at different heights in a turbulent boundary layer. Further
details of the facility and the experiments can be found in Talluru et al. (2017). Simultaneous measurements of velocity andconcentration measurements are acquired, however here weare only concerned with the concentration measurements. The
instantaneous concentration downstream of the plume sourceis measured using a Photo-Ionisation Detector (PID) that issensitive to the iso-Butylene present in the release mixture andhas a frequency response of 400 Hz. Multiple acquisitions are
made across the plume with a sampling time of 300 sec foreach position. The PID is calibrated before and after each experiment to account for background concentration and instrument drift. After applying the calibration technique outlined in
Talluru et al. (2019a), the mean- and rms-concentration profiles are found to agree well with the Gaussian (or ReflectedGaussian) behaviour that is well-known (Talluru et al. (2017,2018)). However, there is still a ±3% systematic error and±2% repeatability error (Talluru et al., 2017). Here we will
examine the concentration signals when the plume is released
at Sz /δ = 32/δ + , 350/δ + , 0.1, and 0.25, Sz being the sourceheight and δ = 0.31 m being the boundary layer thickness.
The measurements are acquired at a distance Sx =0.5δ , δ , 2δ ,and 4δ downstream of the source.

Figure 1. PDFs as a function of the instantaneous concentration, C, for Sx = 0.1δ , Sz = 1δ . The solid black line represents the PDF on the plume centreline and the red lines are theones above the centreline. The blue lines are the PDFs below
the centreline. The exponential behaviour is indicated by thedashed black line.
tion PDFs better. The Gamma distribution is given as,p(C) =



1
C
k−1
C
exp
−
,
θ
Γ(k)θ k(3)

where k is the shape parameter and θ is the scale parameter.
Typically, PDFs of the instantaneous concentration, C,
andpinstantaneous concentration normalised by its r.m.s.,

C/ c2 (c = C − C, concentration fluctuations), are used asa variable (Nironi et al., 2015; Talluru et al., 2017). Since,
c2 varies across the plume, to compare the PDFs at two distinct locations within the plume, here we study the PDF of theinstantaneous concentration, C.
Figure 1 plots the experimentally determined PDFs for
Sx = 0.1δ , Sz = 1δ . The PDF at the plume centreline is shownby the solid black line. PDFs for measurements below the centreline are shown by red lines in figure 1a whereas those formeasurements above the centreline are shown by blue lines infigure 1b. It is evident that the high-concentration tails havean exponential behaviour (a straight line on linear – log axes).
Moreover, the indicative slope of the tail of each PDF appears
to be similar too, i.e. they have similar decay exponent. Also,
it is seen that in the vicinity of C = 0, the PDFs exhibit a sharp
peak due to the presence of near-zero values in the signal resulting from the intermittent nature of concentration in a meandering plume. This peak behaviour near C = 0 cannot bemodelled by the Gamma distribution, even though it is suitablefor higher concentration values. Since the Gamma distribution

PROBABILITY DISTRIBUTION FUNCTIONS
Although various distributions are used in literature (Cassiani et al., 2020, e.g.) to fit the full-signal PDF of measured
concentration, none is appropriate for describing the full rangeof concentration signal. The quality of the fit, typically assessed by some measure such as the R-square, also depends onthe fidelity of the experimental data and the range over which
fitting is performed. Most experimental observations indicatethat Weibull and Gamma distributions describe the concentra-

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) 0.02(a)
Data
Exponential
GPD

0.015

0.15

0.15

z
/

0.005

z
/

0.1

0

50

100

150

200

0
0

250

10

20

3

C, ppm

0.1

0.05

0.05

(b)

30

0
0

0.1

0.2

0.3

A

Figure 3. For Sx = 0.1δ , Sz = 1δ , (a) θ , the decay exponent,from the exponential fit to the high concentration tail of the
PDFs. The filled circles are for full-signal PDFs and the empty
circles are from PDFs of peaks. (b) Coefficient A for fullsignal PDFs.

10-2

p(C)

0.2

0.2

p(C) 0.01

0

(b)

10-4

0

50

100

150

200

mean concentration at plume centreline. It is seen that the decay exponent is almost constant across the plume, indicativeof a statistical similarity for high-concentration events. Thisbehaviour of the concentration-tail is analogous to the similarity of concentration-spectra observed by Talluru et al. (2019b).
The median of decay exponents in figure 3(a) is used to plot thedashed line in figures 1a and 1b, indicating that all tails could
be modelled using a single θ value. Schopflocher (2001) andMole et al. (2008) also found the shape parameter that theyused to fit a GPD to the PDF tail to be constant. Mole et al.
(2008) argued that the physical reason why the tails has a universal character is due to the velocity field and the molecular
diffusion. Further, we note that the distribution of the coefficient A has a Gaussian shape about the plume centreline, as
illustrated in figure 3b.

250

C, ppmFigure 2. Comparison of Generalized Pareto Distribution
and Exponential fits for the PDF tails at the plume centreline
for Sx = 0.1δ , Sz = 1δ , on a (a) linear – linear axes (b) linear– log axes.

is characterised by the exponential term in equation 3 at higher
concentrations, consistent with the experimental observations,
we simplify the fitting to the tail using a purely exponentialfunction given as,

SPATIAL VARIATION OF PARAMETERSC
p(C) = A · exp −, C > C′θIn this section we extend the analysis to other source distances and source heights. Hereafter, Sz /δ = 32/δ + is referred
to as Ground Level Source (GLS) and the other three Sz as Elevated Sources (ES). We find that PDFs of all ES consideredhere exhibit a near constant behaviour of the decay exponent,θ (as observed in figure 1b, although θ varies for each Sz andSx combination). This implies that the PDFs of C can be multiplied by a factor such that the tails of the PDFs for each source
condition will collapse onto single curve. Here we denote thisfactor as 1/µ. It is obvious that µ ≡ A/Acentreline from equation 4 and µ = 1 at the plume centreline. Figure 4a shows theresulting PDFs for Sx = 0.5δ and Sz = 0.1δ and 4b plots thoseof Sx = 4δ and Sz = 0.1δ . Similar to figure 1, the solid blackline is the PDF at plume centreline and the dashed black line
represents the average slope of tail. Figure 4a and 4b demonstrate the self-similarity of PDF tails and the possibility to develop scaling laws for modelling. A representative decay exponent, θ is determined for plume released from four source
distances and four source heights. We then examine the trendof the decay exponent by plotting θ and θmax , where θ is themean decay exponent and θmax is the decay exponent at theplume centreline. Figure 5 illustrates the relationship between
θ and plume half-width calculated from the Crms profile, δz ,for all ES. Firstly, the values of θ and θmax for a given Sz
and Sx combination are close, except for those of Sx = 4δ andSz /δ = 350/δ + (black diamond symbol). A possible reason(4)

where C′ is arbitrary. Two other studies that particularly focus
on the tail of the concentration PDF are by Mole et al. (2008)
and Schopflocher (2001). They use GPD instead of an exponential function. A GPD fitted to our data as an example isplotted in figure 2. It can be observed that, though the prediction of a GPD seems to be acceptable on a linear – linear plot,
it over-predicts in the range of approximately 75 < C < 175
on a linear – log scale and under-predicts beyond that. Thus
its prediction appears to be reasonably good on a linear plot,
more stringent examination reveal that the present data is notsuitably represented by the GPD. The parameters A and θ inequation 4 are determined using an iterative non-linear leastsquares method for each PDF and their variation across theplume are shown in figure 1. The decay exponent, θ , is plotted in figure 3a as filled symbols, whereas the coefficient, A
is plotted in figure 3(b). It was found that the tail of PDF
for wall-normal locations near the edge of the plume on either side have a slightly different decay exponent. This indicates a decreased statistical convergence towards the plumeedge (Chatwin & Sullivan, 1990; Yee & Chan, 1997). Basedon the above observation, figures 1 and 2 only includes the locations in the plume where C > 0.1Cmax , where Cmax is the312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)

100
10-2

p(C)=7

10-1

7

10-4

10-2
10-6
0

100

200

300

400

500

10-3

600

100

C, ppm(b)

C=C max

100

Figure 6. The factor used to collapse p(C), µ, plotted against
C/Cmax for all ES. For legend refer to figure 5.

10

(a)
10-2

-4

0

5

10

15

20

25

p(C)=7p(C)=710-2

30

C, ppmFigure 4. Collapsed PDFs of C for (a) ES Sx = 0.5δ , Sz =0.1δ . (b) ES Sx = 4δ , Sz = 0.1δ . The collapsed PDFs are
obtained by dividing a the original PDF p(C) by µ.

10-4

10-6
0

100

200

300

400

500

C, ppm0:5/; 350=/ +/; 350=/ +2/; 350=/ +4/; 350=/ +

0:5/; 0:1/; 0:1
2/; 0:1
4/; 0:1(b) 0.07

0:5/; 0:25/; 0:25
2/; 0:25
4/; 0:25

0.06
0.05

102

z=/

0.04
0.03
0.02

3

1

10

0.01
0
16

18

20

22

24

26

28

30

3
0

10

0.05

0.1

0.15

0.2

Figure 7. a) Collapsed PDFs of C for a GLS, Sx = 0.5δ ,Sz /δ = 32/δ + . b)θ calculated for each PDF in the crossplume direction.

0.25

/z =/
Figure 5. θ vs δz (half-plume width) for all ES. The firstnumber in the legend is the source distance, Sx , and the second is source height, Sz /δ . Solid symbols are centreline PDFs
and hollow ones are the average slopes for all PDFs.

D
log-log scale for all ES. It is observed that µ ∝ C/Cmax ,where D is some exponent. Again, decreased statistical convergence is observed for Sx = 4δ and Sz /δ = 350/δ + here
(black diamond symbol). In contrast to the above observation,
the decay exponent θ varies significantly across the plume fora GLS. The change in slope for each PDF is evident in figure
7a and consequently the PDF tails do not collapse to a single
curve after being scaled. Instead the profile of θ for GLS in
figure 7b resemble the mean concentration profile for the GLS.
We infer that this behaviour of changing decay exponent might
be due to wall effect and further analysis is need to characterise
this behaviour.

is that at farther downstream distances from the source, theprofile of a lower ES evolves to be similar to that of a GLS.
Secondly, it is apparent that θ decreases in the streamwise direction and is proportional to δzB , where B is some exponent.
The other parameter in equation 4, A, shows a Gaussian-likebehaviour, as plotted in figure 3b. Thus µ is likely to be related
to the mean concentration. Figure 6 plots C/Cmax vs µ in a4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) 35(a) 120

30

100
80

C, ppmC, ppm

25
20
15

60
40

10

20

5
0

=c

0

0.05

0.1

0.15

0

0.2

0

2

4

t, sec

6

8

10

150

200

250

t==c

(b)

(b)10-110-2p(Cp;2 )p(Cp;1 )

10-2
10-3

10-4

10-4

0

50

100

150

200

250

0

Cp;1 , ppm

50

100

Cp;2 , ppm

Figure 8. (a) Instantaneous concentration measured for a 0.2sec period with identified peaks. (b) Probability density functions for peak values only for Sx = 0.1δ , Sz = 1δ .

Figure 9. a) Instantaneous concentration measured for a period of 10 time scale for concentration, τc , with identified
peaks. b) Probability density functions for peak values in eachbin for Sx = 0.1δ , Sz = 1δ . The dashed line represents the
decay exponent calculated from p(C), in this case, the green
triangle in figure 5.

EXTREME EVENT ANALYSIS
In this section we examine the PDFs for the extreme values in the concentration signal, for which the extreme values
are considered to be the local maxima. Two approaches are
adopted. First, all local maxima that are above a threshold are
examined and second, local maximum over multiple segmentsof the signal are analysed.

signal at any location is acquired for 300 seconds and thereafter this sampling time is divided into segments of pre-definedduration. We have adopted the pre-defined duration as thetime-scale for concentration, τc , which is found from the autocorrelation function. The signal is then divided in to segments,
each of duration τc , and the maximum C p,2 from each segment
is collected. This method is demonstrated in figure 9a. The
PDFs of C p,2 are plotted in figure 9b. As anticipated, these
PDFs also have an exponential tail, with a decay exponent
similar to that for p(C) and p(C p,1 ) in figures 1 and 8b, respectively.

Peak over threshold
Here we determine the local maxima (see figure 8a) in
the instantaneous signal at a height z such that the peak values(C p,1 ) are larger than the local mean, C(z). The PDFs for thesepeak values only are plotted in figure 8b and similar to thefull-signal PDFs, the PDFs for the peaks also exhibit an exponential behaviour at high concentrations. The decay exponent
is evaluated for these PDFs, plotted as hollow symbols in figure 3a, and is close to the values for the full-signal PDFs (solid
symbols in figure 3a). We also find that decay exponent for the
PDFs of the peaks is nearly constant, evidencing similarity.

Summary
The results presented here arise from extensive measurements and thus bring in new information to the already extensive literature on PDFs for scalar concentration. The PDFs ofconcentration measured for varying source heights and source
distances exhibit statistical similarity across the plume. We
have presented evidence for the decay exponents to be invariant across the plume through the PDFs for p(C), p(C p,1 ), andp(C p,2 ). A novel finding is in the fact that the decay exponent
is related to the plume half-width.
The spread of passive scalar is dominated by two mechanisms: turbulent mixing and molecular diffusion (Gifford,
1959). Large concentration events in the plume are thought
to reside on thin strands or sheets and caused by local turbulence (Yee & Chan, 1997). Molecular diffusion is also signif-

Maxima over a period
The probability of maxima in a fixed time-period is commonly studied in meteorology and hydrology, e.g. monthly
maxima and annual maxima. Typically, Generalised Extreme
Value (GEV) distributions are then fitted for the prediction of
extreme events when limited data are available. For example, in wind engineering, the Weibull distribution is more appropriate for the probability distribution of annual maximum
wind speed(Holmes & Bekele, 2021). Hence we apply a similar approach to the concentration measurements, where the512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Fackrell, J. E. & Robins, A. G. 1982 Concentration fluctuations and fluxes in plumes from point sources in a turbulent
boundary layer. Journal of Fluid Mechanics 117 (1), 26.
Gifford, F. 1959 Statistical roperties of a fluctuating plume dispersion model. Advances in Geophysics 6, 117–137.
Holmes, J. D. & Bekele, S. 2021 Prediction of design windspeeds and structural safety. In Wind Loading of Structures,chap. 2, pp. 40–64. Milton: Taylor & Francis Group.
Invernizzi, M., Brancher, M., Sironi, S., Capelli, L., Piringer,
M. & Schauberger, G. 2020 Odour impact assessment byconsidering short-term ambient concentrations: A multimodel and two-site comparison. Environment International
144 (June), 105990.
Mole, N., Schopflocher, T. P. & Sullivan, P. J. 2008 High concentrations of a passive scalar in turbulent dispersion. Journal of Fluid Mechanics 604, 447–474.
Nironi, C., Salizzoni, P., Marro, M., Mejean, P., Grosjean, N.
& Soulhac, L. 2015 Dispersion of a passive acalar fluctuating plume in a turbulent boundary layer. Part I: velocity andconcentration measurements. Boundary-Layer Meteorology
156 (3), 415–446.
O’Brien, E. E. 1978 Stochastic properties of scalar quantities
advected by a non-buoyant plume. Journal of Fluid Mechanics 89 (2), 209–222.
Oettl, D. & Ferrero, E. 2017 A simple model to assess odour
hours for regulatory purposes. Atmospheric Environment
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Fluid Mechanics 289, 141–157.
Schopflocher, T. P. 2001 An examination of the right-tail of the
PDF of a diffusing scalar in a turbulent flow. Environmetrics
12, 131–145.
Talluru, K. M., Hernandez-Silva, C. & Chauhan, K. A. 2019aA robust calibration technique for concentration measurements using ionisation detectors. Measurement Science andTechnology 30 (7).
Talluru, K. M., Hernandez-Silva, C., Philip, J. & Chauhan,
K A. 2017 Measrements of scalar released from point
sources in a turbulent boundary layer. Measurement Scienceand Technology 28, 1–13.
Talluru, K. M., Philip, J. & Chauhan, K. A. 2018 Local transport of passive scalar released from a point source in aturbulent boundary layer. Journal of Fluid Mechanics 846,
292–317.
Talluru, K. M., Philip, J. & Chauhan, K. A. 2019b Self-similar
spectra of point-source scalar plumes in a turbulent boundary layer. Journal of Fluid Mechanics 870, 698–717.
Yee, E. & Chan, R. 1997 A simple model for the probability density function of concentration fluctuations in atmospheric plumes. Atmospheric Environment 31 (7), 991–
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icant for these high concentration patches due to locally highspatial gradient. Given that the plume width relative to theboundary layer thickness is small, the influence of the local
turbulence would be for the whole plume. Thus the two mechanisms for the spread of concentration are persistent across theplume resulting in high concentration events having the same
decay exponent in the corresponding PDFs.
Here we have only studied the high-concentration tail of
the PDFs and thus the behaviour of low-level concentration
is not examined. The analysis is not aimed towards describing the full range of concentration fluctuations, as typicallydone on past studies. The exponential tail of the PDFs is undoubtedly evident in our measurements, possible due to thelong sampling times. This behaviour can thus suggest theappropriate function form for the full-scale PDF, e.g. theGamma distribution asymptotes to an exponential behaviour
for large concentrations. Similar observations have been madeby Schopflocher (2001). Although it should be noted that theexponential distribution also results in finite probability, nomater how small, for concentration magnitudes that are above
the source concentration. This is unrealistic due to the everpresent molecular diffusion. However a possible application
is to model the 90th and 99th percentile, C90 and C99 , which
are studied in Fackrell & Robins (1982) and Efthimiou et al.
(2016).
Efforts are also ongoing to reconcile the observations
made for concentration PDFs in this paper with the results forthe concentration spectra by Talluru et al. (2019b) and the phenomenological model for the role of large-scale velocity structures by Talluru et al. (2018). As such, we have not yet parameterised these statistical variations (particularly for figures
5 and 6) to model the probability or for predictions.

Acknowledgements
The first author would like to thank Dr Murali Krishna
Talluru for valuable insights and suggestions. The authors arealso thankful to the reviewer(s) who suggested the inclusion of
PDFs for extreme events in the analysis presented here.

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physical interpretation of scalar fluctuation measurementsfrom many turbulent shear flows. Journal of Fluid Mechanics 212, 533–556.
Efthimiou, G. C., Andronopoulos, S., Tolias, I. & Venetsanos,
A. 2016 Prediction of the upper tail of concentration distributions of a continuous point source release in urban environments. Environmental Fluid Mechanics 16 (5), 899–921.

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