Concentration Fluctuations in Smoke
Plumes Released Near the Ground
Boundary-Layer Meteorology
An International Journal
of Physical, Chemical and
Biological Processes in the
Atmospheric Boundary Layer
ISSN 0006-8314
Volume 137
Number 3
Boundary-Layer Meteorol
(2010) 137:345-372
DOI 10.1007/s10546-010-9532x
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Boundary-Layer Meteorol (2010) 137:345–372
DOI 10.1007/s10546-010-9532-x
ARTICLE
Concentration Fluctuations in Smoke Plumes
Released Near the Ground
H. E. Jørgensen · T. Mikkelsen · H. L. Pécseli
Received: 3 July 2009 / Accepted: 28 July 2010 / Published online: 28 August 2010
© The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract Near-ground artificial cloud releases in the turbulent atmospheric boundary layer
were investigated experimentally by Lidar measurement techniques. Simple scaling relations between the average concentration and the lowest order moments are suggested by
simple analytical models, and the experimental results are tested against these hypotheses.
We find strong evidence for a simple scaling of the standard deviation, skewness and kurtosis
with the average concentrations at the downwind distances observed in our experiments.
Near-ground concentration fluctuations in fixed as well as moving frames of references are
investigated. The scaling is supported by data from several experimental sites and different atmospheric stability conditions. One conclusion of the study is that relatively accurate
estimates for the standard deviation, skewness and kurtosis can be obtained for the concentration fluctuations, given a reliable estimate of the space-time varying average concentration
field.
Keywords Concentration fluctuations · Lidar scattering techniques ·
Smoke diffusion experiments
1 Introduction
The present study concerns instantaneous puff and continuous smoke plume releases,
as investigated by backscatter Lidar techniques (Lewellen and Sykes 1986; Jørgensen
and Mikkelsen 1993; Mikkelsen et al. 1995, 2002; Munro et al. 2003a, b). Remote
H. E. Jørgensen · T. Mikkelsen
Risø National Laboratory, Technical University of Denmark, 4000 Roskilde, Denmark
e-mail: haej@risoe.dtu.dk
T. Mikkelsen
e-mail: tomi@risoe.dtu.dk
H. L. Pécseli (B)
Department of Physics, University of Oslo, Box 1048, Blindern, 0316 Oslo, Norway
e-mail: hans.pecseli@fys.uio.no
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H. E. Jørgensen et al.
sensing of concentration fluctuations by Lidar (Light detection and ranging) is a relatively novel technique and similar studies have previously been carried out by other means
(Fackrell and Robins 1982; Dinar et al. 1988). We are also aware of relatively early
experimental investigations based on light scattering techniques (Becker et al. 1957;
Shaughnessy and Morton 1977); a short summary of these and similar results was compiled by Chatwin and Sullivan (1990). The experimental part of the present study is
based on data analysis of the spatially and temporally resolved concentration profiles
obtained by the Lidar backscatter. We test some simple scaling laws for the variance,
skewness and kurtosis of the space-time varying concentration fluctuations. The analysis is carried out in the absolute fixed frame of reference, as well as in the moving frames of reference that follow the lateral centre-of-mass of the plumes or puffs.
We have taken the concentration fluctuation data for this analysis from a comprehensive database, originating from several independent experiments involving the controlled
release of contaminants in a turbulent atmosphere. We expect that the results obtained can
have practical value for estimating the basic statistical variability of concentration fluctuations of contaminants released (accidentally or deliberately) in the atmospheric surface
layer. Concentration (or density) fluctuations have received interest also in other contexts (Labit et al. 2007). Our results add support to an empirical relation between the
kurtosis and the skewness that has been reported in the literature. We also find empirical evidence for a similar relation between the normalized standard deviation and the
skewness.
The analysis of Lidar-averaged data deals with concentration estimates that are in general
different from the local concentration values. The difference could be large particularly in
assessing values of concentrations found in the high concentration range of the probability
density function of concentration. That is, the concentration found from a ∼1 m long spatial
average (ignoring the beam diameter) will generally be much less than that found for a sample averaged over ∼1 mm (the conduction cut-off length). We argue, however, that the basic
relations being analyzed in the present study have general applicability, and our results will
be relevant also for a wider range of experimental conditions.
The emphasis of the present study is on the analysis of experimental results, as outlined
in Sects. 4 and 5. Logically, it is however an advantage first to present the basic analytical
results that gave rise to the study in the form presented here. The relevant theoretical discussions are summarized in Sect. 2 with some additions in Sect. 3. The experimental results are
discussed in Sect. 6, while finally Sect. 7 contains our conclusions. Two appendices contain
some relevant details concerning inequalities for standard deviations, skewness and kurtosis
in Appendix A, and some simple models for spatial Lidar averaging in Appendix B. The
latter results are not directly applied to our data, but can have applications for data from large
downwind distances in other experiments.
2 Statistical Analysis of the Concentration Variations
The statistical properties of the space-time varying concentration, c(y, t), are best described
by the probability density P(c; y, t) for c at a given position y at a certain time t, i.e. each
spatial position has, at any time, an assigned concentration probability density. This probability can be specified in any frame of reference, including a moving one. The notation allows
for the possibility of anisotropy, but for the homogeneous isotropic cases the probability
densities depend on |y| only.
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2.1 Continuum Model
The probability density for a non-uniform concentration can be written as (Sawford and
Sullivan 1995)
P(c; y, t) = δ(c − Sc (y ))Q(y ; y, t)dy ,
(1)
V0
where Sc (y) is the initial concentration distribution and Q(y ; y, t)dy is the transition probability (Sawford 2001) for a fluid element from y at t = 0, reaching a position y at time t. The
initial concentration is assumed to be confined to a finite volume, V0 . For simplicity, we use
the shorthand notation dy = dy1 dy2 dy3 , here and in the following, in terms of the vector
components with indices 1, 2, 3. The space-time variation of the average concentration and
of other statistical averages will be different for the rest frame and for a moving frame. From
an experimental point of view, the results have different averaging characteristics in the two
cases. In the absolute frame, we find a horizontal “meandering” of the plume, which is caused
by the turbulence on scales larger than the instantaneous plume width. This results in larger
sampling uncertainties as compared to the analysis in the moving frame of reference as also
carried out in this study. In the moving frame the cloud expansion is predominantly caused
by eddy sizes smaller than or comparable to the instantaneous width or puff sizes (Mikkelsen
et al. 1987).
It has been suggested (Sawford and Sullivan 1995) that at later times after release (t >
y0 /u ∗ with y0 being the initial puff size at the source and u ∗ the turbulent friction velocity
in the shear flow, Panofsky et al. 1977; Schlichting and Gersten 2000) the initial locations
of the fluid particles within the source region are “forgotten”. Provided this limit exists, we
then have Q(y ; y, t) ≈ Q(0; y, t). In this case the integral in (1) can be decoupled (Sawford
and Sullivan 1995) to give
1
δ(c − Sc (y ))dy ,
(2)
P(c; y, t) = (1 − α(y, t))δ(c) + α(y, t)
V0
V0
with α(y, t) ≡ V0 Q(y ; y, t)dy ≈ V0 Q(0; y, t). The assumption giving (2) can evidently
only be justified for late times after release, but it is not evident how late this has to be. Also,
we note that (2) assumes the initial concentration to be restricted to a finite volume in space.
Physically, this is quite reasonable, but excludes Gaussian models, for instance, which in
principle extend to infinity.
Even (2) contains too much information in the sense that its full space-time variation
can be difficult to present. It is therefore an advantage to consider its lowest moments, and
consider only the first three moments, the mean square relative concentration fluctuations
σ 2 (y, t), the skewness S (y, t), and the kurtosis K (y, t). We have (Sawford and Sullivan
1995)
σ 2 ≡ (c − c)2 = α(λ2 − αλ21 ),
(3)
σ S ≡ (c − c)3 = α(λ3 + 2α 2 λ31 − 3αλ2 λ1 ),
(4)
σ 4 K ≡ (c − c)4 = α(λ4 − 3α 3 λ41 + 6α 2 λ2 λ21 − 4αλ3 λ1 ),
(5)
3
n (y )dy /V
with λn = V0 Sc
0 for n ≥ 2, and the average concentration c ≡ C (y, t) =
cm
0 c P(c; y, t)dc, where we here find c = αλ1 , again with α = α(y, t). As long as
molecular diffusion is ignored, the assumption of incompressible flows is consistent with
the assumption of a constant maximum concentration level Cm , i.e. regardless of molecular
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Fig. 1 a Variation of the
normalized standard deviation
σ 2 /C 2 shown as a function of
the intermittency factor α. Full
line shows the ideal uniform
concentration model, Sect. 2.1.1,
while the dashed line corresponds
to a radial concentration variation
cos(π y/2D), and cos2 (π y/2D)
is shown by a dotted line. b The
variation of the skewness (which
can assume also negative values)
on a semi-logarithmic scale for
varying intermittency factor α.
c Similarly we show variation of
the kurtosis for varying
intermittency factor α
H. E. Jørgensen et al.
(a)
(b)
(c)
diffusion there is a maximum value (the largest source value) of concentration that cannot
be exceeded.
We show illustrative results in Fig. 1a–c for three examples of Sc (y), for different initial
concentration distributions, an ideal uniform concentration model, for cos(πr/2D), and for
cos2 (πr/2D), with D being the radius in an initially spherical release, see also discussion
in Appendix B. The three distributions have different integrated concentrations, but a normalizing coefficient vanishes for the quantities σ 2 /C 2 , S and K . In particular in Fig. 2a
we show a parametric representation of S 2 (α) + 1 and K (α), while in Fig. 2b we have a
parametric representation of σ 2 (α)/C 2 (α) and S (α), which
will be useful later on.
The full
√
1
2
2
2
2
line in Fig. 2b has the analytical expression σ /C = 2 2 + S + S 4 + S , which is
later used as a reference when fitting data points.
For the idealized “two-level” model for the release, where the concentration is either
zero or Cm , the effects of molecular diffusivity are particularly important (Zimmerman and
Chatwin 1995; Mole et al. 2008). The skewness
(4) attains negative values in the case λ2 −
λ22 − 8λ3 λ1 /9 < (4/3) λ21 α < λ2 +
123
λ22 − 8λ3 λ1 /9, with λ22 ≥ 8λ3 λ1 /9, while also
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(a)
349
(b)
Fig. 2 a Parametric presentation of S 2 (α) + 1 versus K (α) with the intermittency factor α as the parameter.
See Fig. 1 for explanation of the symbols. b We show also a parametric presentation of the two dimensionless
quantities σ 2 (α)/C 2 (α) versus S (α) again with the intermittency factor α as the parameter
0 ≤ α ≤ 1. Such inequalities for skewness and kurtosis can be used as a simple test for the
validity of a simple model like (3)–(5) for an actual experiment.
2
By the Schwartz inequality we have λ2 ≡ V0 Sc 2 (y )dy /V0 ≥ V0 Sc (y )dy /V0 ≡
λ21 , and we have C Av = λ1 to be the average initial concentration in the cloud. Evidently, for
0 ≤ α ≤ 1 we always have σ 2 ≥ 0. In particular, the relation (3) has been compared with data
from several different experiments, and good agreement was found (Chatwin and Sullivan
1990). The initial shape of the cloud is not essential for the mathematical expressions for λi .
In the simplest model, the initial probability density of the position of a fluid element can
be taken to be uniformly distributed within the initial volume of the cloud. In this way, the
only information concerning the initial concentration distribution is retained in its size, and
the concentration Sc (y) in (2) is then taken to be uniform, giving λn = 1, for all n. It turned
out that even this modified model gave good agreement with numerical simulations (Krane
et al. 1996, 2003).
The set of expressions (3)–(5) represent a class of models, in the sense that they contain
the initial concentration distribution Sc as an input parameter. An important property of the
model is that it predicts a parametric relationship between σ, S and K , as well as higher
order normalized moments, these relations depending on Sc as indicated on Fig. 2a and b. In
particular it seems that the relation between S and K is rather robust, by depending only
weakly on Sc , see Fig. 2.
2.1.1 An Exact Result
When λ j = 1 for all j, the expressions (3)–(5) contain the exact results for the model with the
initial concentration being either Cm or zero (Csanady 1967; Mole and Clarke 1995; Yee and
Chan 1997), where the probability for a vanishing concentration is P(c = 0) = 1 − α(y, t),
while the probability for detecting a concentration Cm is P(c = Cm ) = α(y, t), giving
P(c) = (1 − α(y, t)) δ(c) + α(y, t)δ(c − Cm ).
(6)
The first term on the right-hand side of the probability density function (PDF) in (6) corresponds to positions outside the plume where 1 − α is the probability of being outside and δ(c)
is the corresponding PDF (i.e. zero) of those concentrations; the second term is the PDF for
positions in the plume where α is the probability for being inside the plume and its density is
1/V0 times the integral concentration. It is demonstrated in Appendix B that in certain limits
the result (6) can serve as a model also for a Lidar-averaged signal.
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For such a model system, all the information is contained in the intermittency, defined
as the probability α = α(y, t) for a selected spatial position, y, to be inside the cloud at
time t. For this simple case C (y, t) = α(y, t)Cm , and it is readily demonstrated that the
skewness attains negative values if the intermittency is in the interval 1/2 < α(y, t) < 1,
with S → −∞ for α → 1 and S → ∞ for α → 0. As a reference for the magnitude of
K it is
to use the value 3for a Gaussian
distribution. It is also found that K < 3
customary
√ √ 1
1
for 2 1 − 1/ 3 < α(y, t) < 2 1 + 1/ 3 , or 0.21 < α < 0.79. For this two-level
concentration model (Mole and Clarke 1995) we have
K = S 2 + 1.
(7)
The results for this simple reference model are illustrated in Fig. 1a–c with full lines (Krane
et al. 2003). The skewness can also attain negative values for very large concentrations,
though for the data to be described in the following we do not observe this. To observe C as
high as Cm we have to be in the small time regime of the release, where t < y0 /u ∗ , that is
we need to observe the concentrations at short times for the initial value problem, or, for the
constant source release, to be close to the source in the downstream direction.
Physically, the initial evolution of S in the simple model can be described by noting
that, inside the cloud, the concentration must retain its initial value or decrease just after
release, implying S < 0 there. Correspondingly, outside the initial cloud circumference, the
concentration must remain zero or increase, implying S > 0 there. The minimum value for
the kurtosis is K = 1, while K diverges for α → 0 and 1.
As a guideline, we may expect that the skewness is negative at early times after release,
in regions where the average concentration exceeds approximately 1/2 of its initial value.
The skewness is then expected to change sign at later times when the average concentration
becomes small everywhere. Similarly, the kurtosis is expected to start out exceeding 3, may
be reduced for a while, but ultimately becomes large at large times, to diverge as the average
concentration goes to zero everywhere.
With the assumption of fast-fading memory in the puff or the instant plume’s moving
centre-of-mass frame it is natural to extend the hypothesis by also assuming the position
of a volume element of marked fluid to be the result of many small independent displacements. In that case we can take C (y, t|R) to be a Gaussian centered at R, and characterized
by a single parameter, namely its standard deviation (Yee 1998). This standard deviation is
related to the two-particle mean square separation tensor (Csanady 1973). This assumption of
a Gaussian spatial distribution has the important consequence that the average concentration
can be described by a one-parameter family of functions, giving considerable simplification
to the analysis. The time it takes before the average crosswind concentration can be approximated by a Gaussian depends on the initial distribution, but it will generally be after a time
t > y0 /u ∗ .
Note that for the “single concentration level” model we have C (0, t) ≤ Cm , since the centre-of-mass need not be inside the volume of marked fluid, at least not in every realization.
Imagine, for instance, that an initially spherical cloud is deformed into a horseshoe shape by
the turbulent velocity field. Then the centre of mass is in the centre of the horseshoe, where
the concentration is zero (in that particular realization). This can happen in many realizations,
so when the ensemble average is taken the centre-of-mass concentration becomes smaller
than the peak concentration. Similarly, when considering the cloud evolution in the fixed
frame of reference, we need not find the origin of the release to be inside the cloud at later
times (Krane et al. 2003).
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The analysis outlined in the present and foregoing sections basically express higher order
moments of the statistical distributions of the space-time varying concentration fluctuations
by the local average of the concentration. The models can be tested in many ways, for instance
also by analyzing the concentration along a line of sight, which need not pass through the
centre-of-mass of the cloud.
The simple relationship (7) resulting from the single concentration level model has been
generalized (Mole and Clarke 1995; Schopflocher and Sullivan 2005; Mole et al. 2008) to
K = AS 2 + B,
(8)
and introducing two constants A and B that both turn out to be in a range 1–3 for a wide set
of data.
There are no restrictions on the spatial dimensionality of the system implied in (3)–(5).
Thus, the conjectured results (3)–(5) have been tested in numerical simulations of twodimensional turbulence by Krane et al. (2003). Their studies used a multi-step initial distribution as an approximation to a continuous “bell-shaped” initial concentration distribution.
The results gave good agreement with the simple model discussed here, making it worthwhile to attempt a similar study also for field experiments. In the numerical simulations of
Krane et al. (2003), the method for obtaining the local distribution of a test cloud avoids
“the Gibbs phenomenon” from Fourier transforming steep local concentration gradients that
would “contaminate” the results. These effects would be particularly conspicuous for the
estimates of the higher order moments of the local probability density of the concentration.
The simple model (6) has been generalized by introducing more parameters (Lewis and
Chatwin 1997). Here we restrict the analysis by relaxing the assumption of a “single concentration level” model by the continuous and differentiable source distributions, used also
in Fig. 1a–c.
3 A Model Flow
The discussions of (3)–(5) refer to ideal point measurements of the local concentrations, but
this is not what is obtained by the Lidar. In this subsection we give a short discussion of the
relation between the point measurements and the Lidar results. The following discussions
are restricted to cases where the Lidar pulse has the symmetry H (y) = H (−y) with respect
(k) = H
(−k). This is a valid
to its centre-of-mass, implying that its Fourier transform is H
assumption in our case as well as in many others. We denote Fourier transformed variables
by
·.
The cloud of contaminants is characterized by a space-time varying concentration c(y, t)
assigned to each spatial position at time t. This scalar variation is subsequently monitored
by the Lidar. The spatial averaging
by the Lidar pulse at any given time t can be described
by a convolution c(y, t) ≡ H (y − y)c(y , t)dy = H (y − y )c(y , t)dy ≡ H ⊗ c,
with the symbol ⊗ denoting the convolution integral and y is the centre of the Lidar pulse
at time t. The spatial variation of the Lidar pulse intensity is given as H (r) ≥ 0, where, for
brevity, we omit a temporal variable, that should be included in order to account for the pulse
propagation. The spatial Lidar averaging indicated by the “overline” should not be confused
with ensemble averaging. If the cloud c is singly connected, the same will hold for c, while
the converse statement is not necessarily true. A statistical distribution of c implies that c is
statistically distributed as well, albeit with a different distribution.
Although it is not measured here, it is of interest to note some properties of the averaged velocity u(y, t) ≡ H ⊗ u(y, t). By the assumption of incompressibility we have
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k · ∇ · u(y, t) = 0. Fourier transforming this expression gives i H
u = 0, indicating that
when u is incompressible, then so is u. By these arguments we also have that ∇ · g = ∇ · g
for any function g(y) where the Fourier transform exists since the Fourier transforms of the
right-hand and left-hand sides of the expression are equal.
We let the concentration c follow an ideal continuity equation ∂c/∂t + u · ∇c = 0, where
u is the space-time varying fluid velocity vector, where incompressibility has been assumed,
∇ · u = 0. We here assumed ideal conditions by ignoring losses and diffusive motions,
for instance. We can introduce a model concentration field c(y, t), which is again a scalar
assigned to each spatial position y at time t. The scalar c will also follow a continuity equation, which we attempt to determine. Thus, from the continuity of c we find trivially for
spatial Lidar averaging
∂c
+ ∇ · (uc) = 0,
∂t
(9)
which is an exact relation with the given assumptions.
The model continuity equation for c becomes
∂c
+ ∇ · (V c) = 0,
∂t
(10)
where V(y, t) is the postulated effective model fluid velocity. In order for the two continuity
equations to agree, we require
∇ · (V c) = ∇ · (uc) = ∇ · (u c).
(11)
Apart from an arbitrary rotational additive field, ∇ × A, this equation has the solution
V(y, t) = u(y, t) c(y, t)/c(y, t) demonstrating the realizability of the model flow, given
the actual flow determined by c(y, t) and u(y, t). Note in particular that V = u, where, as
it turns out, the explicit averaged velocity u does not enter the expressions at all. Note also
that ∇ × V = 0 in general.
If the Lidar pulse is very narrow, we can approximate H (y) by a δ(y)-function, apart
from a constant. In this case c ≈ c and V ≈ u, as expected. More generally we note that
c(y, t) is likely to vary continuously with the spatial variable, even for the case where the
“source” concentration c has a discontinuous “top-hat” distribution. The continuity equation for c is consistent with the conservation of c(y, t)dy. While we
have ∇ · u = 0 by assumption, we do not have any a priori guarantee that also the model
flow V(y, t) is incompressible. The spectral characteristics of u will also be different from
the characteristics of V, in general.
It is possible to add some statements concerning the incompressibility of the model flow.
Writing (11) in a Fourier transformed version, again with · denoting the Fourier transformed
function, we have
k · k·
V ⊗
c=H
u ⊗
c,
or
k·
(k ) − (k) V(k − k ) H
u(k − k ) H
c(k )dk = 0.
(12)
(13)
This relation serves also to determine V from c for given u, given also a spatial Lidar
intensity variation, H , although the result may have little practical value in this respect.
However, for large times we may again argue for a principle of fading memory, and expect
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that V(k) becomes only weakly dependent on the actual space-time evolution of the concentration in the given realization. Consequently, we expect the result to be valid also for
a uniformly distributed concentration at the initial time t = 0, implying the approximation
(k)/H
(0). In this limit
c(k ) ≈ c0 δ(k ) in the integral. This, in turn, implies V(k) ≈ u(k)H
we thus find that V ≈ u, and that V is incompressible if u and thereby u are incompressible. The limit considered here corresponds physically to the case where the initial cloud is
stretched and folded so many times that it covers the volume of the Lidar pulse uniformly, to
a good approximation. (The other extreme limit, where we can set c(k ) = constant may be
interesting, but of little relevance here. It corresponds to the case of an initial point source
release, with infinite memory in the flow.)
The basic results (3)–(5) were derived under relatively mild assumptions: the central and
essential assumption is that the cloud behaves as a continuous fluid, so that a small volume
element at (y0 , t0 ) is connected to another at (y1 , t1 ), with t1 > t0 , by a single path, so that a
transition probability can be defined. The argument then builds
on the additional assumption
of “fading memory”, allowing expressions such as λn = V0 Sc n (y )dy /V0 and c = αλ1
to be written solely in terms of the source concentration Sc (y), irrespective of the details of
the time evolution. This argument applies to the model flow as well. In addition is an implicit,
or auxiliary, assumption of incompressibility, by taking the maximum cloud concentration
to be constant. In the case of the ideal “exact” top-hat model, this assumption enters by having the maximum concentration constant at all times. Note that no assumption of universal
subranges, or similar, enters the arguments.
We can attempt to apply the arguments for obtaining the previous basic results on the
model flow. We have demonstrated that also c(y, t) follows a continuity equation, as appropriate for a continuum. As mentioned before, we have no reason to expect that the model
flow is incompressible, but so far as V(y, t) is approximately of the same magnitude as
u(y, t), we might anticipate that the difference in this respect is not large. For a cloud with a
continuously varying concentration the compressibility is supposedly of minor consequence,
since a small compression merely introduces a concentration level at a somewhat different
position as compared to where it would have been found otherwise. Consequently, we expect
the previous basic results to be representative for the model flow, which is that obtained
by the Lidar beam along the line of sight. Results deduced for the ideal or physical scalar
concentration field c(y, t) are thus likely to be representative of the effective concentrations
c(y, t) detected by the Lidar as well.
In Appendix B we mention some solvable limiting cases for the Lidar return signal probability density. In Fig. 3 we show schematically an illustration of a Lidar beam crossing a cloud
of scatterers at a later time t1 , where the initial cloud has been deformed by the turbulence.
Each of the points in the volume covered by the Lidar beam can be traced back to the time
of release t0 , as also shown in the figure. Given a position of the beam centre (y1 , t1 ), we can
define a fraction of the beam f i = f i (y1 , t1 ) that is inside the cloud, and correspondingly the
fraction 1− f i that is outside. If t1 t0 we can argue that the convoluted line shown at t = t0
covers the initial cloud uniformly: basically, this is the same assumption used to obtain (2)
where we here use the argument for each point contributing to the Lidar backscatter. In the
limit where molecular diffusion is ignored, the fluid volume V L that is covered by the Lidar
beam at t1 is the same as shown at t0 , see Fig. 3. By these assumptions, the Lidar backscatter
gives a signal s
s ∼ fi
VL
V0
Sc (t = 0, y)dy.
(14)
V0
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Fig. 3 Schematic illustration of a Lidar beam crossing a cloud of scatterers, shown at t = t1 . Also illustrated
is the spatial distribution at the time of release t = t0 of those particles that contribute to the Lidar signal at
t = t1 > t0
The probability density of s is consequently determined via the probability density for f i ,
within this model. In particular we have the Lidar estimate for the average concentration at
(y1 , t1 ) to be f i (y1 , t1 ) (V L /V0 ) V0 Sc (t = 0, y)dy.
We can estimate the time where the approximation giving (14) becomes applicable by
recalling that the elongation of a contour due to turbulent motions is exponential to a good
approximation, as shown for instance by two different realizations of two-dimensional turbulence (Krane et al. 1996, 2003). Analytical results (Batchelor 1952) refer to a line element,
giving the same results.
√ For this latter case an exponential elongation ∼exp(ζ t) was anticipated, with ζ = c ε/ν where c is a universal constant of order unity, implying that ζ is
proportional to and of the same order of magnitude as the root-mean-square of the average
Eulerian spatial velocity derivatives (Batchelor 1952). We argue that the approximation (14)
becomes useful at a time when a contour having the initial length equal to the Lidar’s spatial
pulse length L L has expanded to a length larger than the initial diameter of the cloud. For
short Lidar pulses this is a significant restriction. Implicit in the argument is the assumption
that the elongation is at least approximately exponential also in the backward tracing of the
turbulent velocity field.
4 Data Analysis
4.1 Comments on the Scaling of the Diffusion Experiments in Terms
of Atmospheric Surface-Layer Turbulence
The presented measurements of concentration and concentration fluctuations have all been
obtained near the ground within the atmospheric surface layer (with near-logarithmic vertical
wind profiles, and constant shear stress −u w = u ∗ 2 ).
A passive contaminant (smoke in this case) was released near the ground at an effective
height of 1–2 m. Lidar measurements in the form of a time series of near-ground, crosswind
instantaneous concentration profiles were obtained at downwind distances ranging between
50 and 300 m. For neutral conditions (no buoyancy effects) the mean vertical wind profile
is logarithmic (Panofsky et al. 1977; Schlichting and Gersten 2000) and determined by the
surface roughness length z 0 and the friction velocity u ∗ , i.e. U (z) ≈ (u ∗ /0.4) ln(z/z 0 ).
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The puffs and instant plumes diffused consequently in very anisotropic turbulence, with
vertical scales limited by the measurement height 1–2 m above ground and by bigger lateral
(outer) horizontal turbulent scales in the range of 500–1000 m as determined by the boundary-layer height.
In terms of scaling subranges including velocity and length scales the ground-released
diffusion experiments were characterized by near-ground surface-layer scaling, with anisotropic turbulence scales characterized by velocity anisotropy, σu > σv > σw , where all three
wind-component standard deviations are proportional to u ∗ .
4.2 Time and Scale Limitations in the Experiment
The instantaneous crosswind concentration profiles have all been obtained within the downwind range interval characterized as follows. The experiments took place in the intermediate
time limit, characterized by: y0 /u ∗ < t < t L , where t L is the Lagrangian time scale of
the near-ground lateral wind fluctuations. Vertical limits are 0 < σz pu f f (x) ≤ L W and
σ y pu f f (x) L v , where L w (z) is the vertical mixing length scale at the measurement height
z (1–2 m) and L v is the horizontal/crosswind outer scale of the surface-layer horizontal turbulence; σz pu f f (x) is the mean instantaneous puff height and σ y pu f f (x) is the mean instantaneous
cross-wind puff size at the downwind measurement distance x.
4.3 Mean Puff Concentration Scaling
Based on results from the relative diffusion scaling literature we expect (Cionco et al. 1999)
that:
(1) The mean vertical puff diffusion (starting from a surface release), scales with: σz pu f f ∼
g(u ∗ t, z/L), where L is the Obukhov length scale (Chaudhry and Meroney 1973).
(2) The mean crosswind diffusion scales with σ y pu f f ∼ u ∗ t (surface-layer scaling,
Chatwin 1968; Sawford 2001).
(3) The crosswind distance neighbour function D(l) for concentration fluctuations
c (y)c (y + l) have been found to scale as c 2 exp(−l/u ∗ t) (Sawford 2001;
Mikkelsen et al. 2002).
4.4 Observations
Essential parts of the experimental set-up and conditions in general are discussed in detail
elsewhere (Jørgensen and Mikkelsen 1993; Cionco et al. 1999; Nielsen et al. 2002; Mikkelsen
et al. 2002) so here a short summary will suffice. The temporal half-width of the aerosol backscatter Lidar’s impulse response function is 10 ns, yielding an effective 1.5 m spatial resolution
(−3 dB) along the beam direction, within the measured instantaneous cloud concentration.
The Lidar transmitted radiation (λ = 1064 nm) beam diameter is approximately 1 mm at
the Lidar, and expands to approximately 1 m at a distance of 103 m, so at relevant distances,
we can assume a beam diameter of 0.05–0.3 m. Data were stored for processing using a
300 MHz bandwidth 250 Mega-samples s−1 (single shot) digital oscilloscope. The recorded
backscatter is, after a Klett correction for extinction within the smoke plumes, interpreted
as a measure for the relative variation of the instantaneous concentration along the Lidar
beam, convoluted with the Lidar pulse. The Nd:YAG laser has a 10 mJ output per pulse at
55 Hz pulse repetition rate. Backscatter profiles from aerosols in the beam path is detected
at up to 50 Hz by a silicon photo diode, and stored for processing in a digital oscilloscope.
The Lecroy digital oscilloscope has effectively 8 bit resolution at 400 Mega-samples s−1
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(effective bandwidth 300 MHz at −3 dB) and hence = 1/256 in our case. More technical
data are given in Jørgensen and Nielsen (1999) and Mikkelsen et al. (2002).
Artificially generated smoke plumes (aerosol plumes) provided the experimental releases.
A smoke generator produced continuous releases of sub-micron aerosol particles, by mixing
liquid SiCl4 and a 25% water solution of NH4 OH in their neutralizing, stoichiometric ratio
(1:3.2) into a strong atomizing air stream vapourizing nozzle, that produced a well-mixed
initial plume of diameter (standard deviation) about 1 m at a few metres downwind of the
source. Beyond this point, say at most a 10 m downwind distance from the source point, the
plume disperses in equilibrium with the surrounding atmospheric turbulence. We measured
the effective plume width at this equilibrium position to be approximately 3–5 m, which is
taken to be the effective initial width.
After an initial mixing phase, the resulting white “smoke plumes” consisted of conglomerate SiO2 and NH4 Cl aerosols of mean particle sizes a little less than 1 µm mean particle size.
The statistical spread of the particle sizes is large, and the forms of the particles are irregular.
In-situ aerosol measurements taken on the smoke plume particle size distributions, obtained
from both the plume’s centreline and from its sides, were steady both in time and shape.
Stable and dense artificial smokes plumes were maintained over the duration (1/2–1 h) of the
experiments. Details of the data handling and analysis are given in Jørgensen and Mikkelsen
(1993) and Mikkelsen et al. (2002).
The database analyzed in the present work consists of results from five different experiments at various sites, in the following denoted “Borex”, Meppen, Germany (9), “Borris”,
western Jutland, Denmark (3), “Fladis”, Landskrona, Sweden (1), and “Porton Down Madona”, UK (12), where the number of independent datasets from each individual experiment is
given in the parentheses. The “Trial” experiment, also at Landskrona, Sweden (1), involved
a heavy gas. Details concerning distances from the source are given in Nielsen et al. (2002)
and Mikkelsen et al. (2002).
The parameters differed somewhat for the different experiments. Thus, at the Borex,
Meppen, Fladis, Porton (Madona) campaigns we had 0.33 Hz sampling for the profile measurements with approximately 1.5-m spatial resolution. Borris95 was sampled at 8 Hz and a
spatial resolution of approximately 3 m.
Table 1 describes the experimental sites and Table 2 gives an overview of meteorological
conditions and the plume dimensions at the measurement location. The meteorological and
turbulence parameters are deduced from measurements at 7.5 m height, except for the Fladis
experiments, where wind speeds were measured at 10 m and turbulence at 4 m (Nielsen et al.
1997). For the Borex89 experiment the measured turbulence is only available as 10- min
averages of the shear stress u ∗ , Obukhov length L, heat flux, wind speed and temperature.
In the other experiments time series of turbulence are also available.
Table 1 Description of the experimental sites
Campaign Site description
Borex89 A German military proving ground (Meppen) with flat terrain and surface roughness z 0 = 0.01 m
Borris94 A Danish military proving ground (Borris) with flat terrain and surface roughness z 0 = 0.01 m
Fladis
Madona
A Swedish safety exercise ground (Landskrona) with flat area (z 0 = 0.04 m) and some
upstream buildings
A British military proving ground (Porton Down) with rolling hills and surface roughness
z 0 = 0.02 m
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Table 2 Ground-release experiments and associated meteorology
ID
Date
x
u
T
v v
u∗
H
E
L
σtot
σp
Bor4c
10/08-89
110
5.6
24.3
1.436
0.42
159
1.27
−35
26.3
12.0
Bor5b
10/08-89
170
5.5
27.2
0.950
0.43
28
0.93
−220
21.7
11.1
Bor8a
11/08-89
290
5.6
19.6
1.357
0.43
87
1.03
−67
33.6
15.8
Bor8b
15/08-89
290
7.4
19.8
0.989
0.53
84
1.43
−132
20.9
14.5
Bor11a
15/08-89
370
8.0
26.8
2.187
0.62
211
2.52
−86
40.8
19.3
Bor11b
17/08-89
370
8.9
27.2
3.714
0.65
116
3.07
−181
41.8
18.3
Bor17a
17/08-89
160
6.2
22.3
2.067
0.50
137
1.64
−68
29.4
9.8
Bor17b
17/08-89
160
6.0
22.2
1.861
0.47
93
1.54
−84
29.1
10.2
Bor17c
17/08-89
160
5.5
21.9
0.998
0.36
3
1.09
−999
21.9
8.9
Bor17e
17/08-89
160
3.3
20.8
0.348
0.18
−17
0.32
25
27.3
8.1
borr04b
04/07-94
200
7.7
24.3
1.116
0.539
2.798
−216
20.5
13.0
12.5
65
borr04c
04/07-94
200
7.0
23.8
2.037
0.485
33
2.656
−305
35.0
borr15d
15/07-94
350
5.4
21.1
1.235
0.463
151
2.476
−59
65.1
25.0
b15d_00
15/07-94
350
5.3
21.3
1.328
0.446
152
2.567
−52
56.1
28.1
b15d_30
15/07-94
350
5.4
21.1
0.991
0.469
143
2.564
−64
62.9
25.8
b15d_60
15/07-94
350
5.6
21.0
1.207
0.477
155
2.280
−62
65.3
23.1
mad14h
14/10-92
230
6.9
16.8
1.017
0.497
30
1.587
−365
30.3
15.9
mad14i
14/10-92
230
6.7
16.5
1.197
0.506
11
1.541
−1095
28.8
13.5
mad14j
14/10-92
230
7.1
16.2
0.817
0.515
6
1.386
−2046
21.9
14.1
mad14k
14/10-92
230
6.3
15.5
0.648
0.426
−16
1.203
435
22.0
13.2
mad15h
15/10-92
340
7.8
17.3
0.906
0.491
51
1.507
−209
27.0
15.8
mad15i
15/10-92
340
7.6
17.0
0.790
0.421
34
1.279
−198
28.8
15.4
mad15j
15/10-92
340
7.4
16.5
0.827
0.425
20
1.284
−348
25.1
16.0
mad15k
15/10-92
340
5.6
16.5
0.529
0.319
7
1.158
−428
28.6
18.1
mad18a
18/10-92
560
2.1
14.8
0.089
0.159
mad19e
19/10-92
400
3.2
16.6
0.330
0.225
0.2
25
0.142
−1522
78.4
29.2
0.430
−41
58.2
30.3
mad21g
21/10-92
600
4.4
16.1
0.304
0.245
3
0.593
−420
39.1
22.8
mad21k
21/10-92
310
2.3
15.2
0.287
0.109
−0.8
0.380
155
77.7
34.6
trial23
30/08-94
222
6.6b
17
1.220
0.53
106
3.463
−112
41.6
17.5
trial25
30/08-94
222
4.5b
17
0.884
0.43
33
2.091
−201
31.1
21.0
trialn
30/08-94
222
4.2b
17.3
0.506
0.37
25
1.636
−155
32.8
19.6
Exp., run identifier (bor: Borex89; borr: Borris94; mad: Madona; tri: Fladis); x, distance [m] from the source
to the measuring path; u, wind speed [m s−1 ], normally at a reference height of 7 m above terrain; T , atmospheric temperature [◦ C]; v v , variance [m2 s−2 ] of wind-speed perturbations perpendicular to the average
wind direction; u ∗ , friction velocity [m s−1 ]; H , upward heat flux [W m−2 ] estimated by eddy correlation of
sonic anemometer measurements; E, turbulent kinetic energy (1/2)(u u + v v + w w )[m2 s−2 ] (estimated
by sonic anemometer); L, Obukhov length [m]; σtot , time-averaged plume width [m] in a fixed frame of
reference; σ p , time-averaged plume width [m] in a frame of reference moving with the plume centreline; date,
day/month-year
a Wind speed measured at a reference height of 2 m above terrain
b Wind speed measured at a reference height of 10 m above terrain
In Fig. 4 we show an experimental site (here: Porton Down, UK), while Fig. 5 shows
an illustrative example of raw data as obtained from the detector. A possible interference
from natural dust/aerosols creating a background concentration can be estimated from the
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Fig. 4 Experimental set-up
showing the smoke release from
the “Madona” experiment. The
Lidar will typically be at a
position in the middle of the
figure
Fig. 5 Example of raw data for
the concentration c, as obtained
from the detector output after
post-processing, including a Klett
correction for 1/r 2 range,
extinction and threshold
corrections. Concentrations are
given in arbitrary units, while the
time delay for the received pulse
is expressed in terms of an
effective distance
backscattered signal seen in front of the plume (on this figure distances less than 50 m). We
find that an uncertainty due to these effects can be ignored here.
Figures showing average concentrations in the fixed frame as well as moving frames are
shown by Mikkelsen et al. (2002). The range gates of the Lidar concentration measurements
are “binned” into 500 range gates, here denoted by an index j . A spatial position is deduced
from the time of flight of the Lidar return, and used as the horizontal axis in Fig. 5. The
Lidar backscatter provides a signal as illustrated in Fig. 5, and the concentration of released
material is deduced from this signal, apart from a numerical constant which can vary from
one experiment to the other. The Lidar signal thus measures the relative cross-wind concentration variation at selected downwind distances from the source. The information of the
absolute concentration requires an independent calibration, which is not available from aerosol backscatter alone. The raw data from the present experiment can be compared to, for
instance, those by Mylne and Mason (1991), Zimmerman and Chatwin (1995) and Munro
et al. (2003a). The line-of-sight averaging over the effective 1.5 m spatial extent of the Lidar
pulse results here in less spiky signals in comparison.
To illustrate differences between the two frames mentioned before (fixed and moving
frames), we show in Fig. 6a and b some basic results for illustrating the characteristics of
the concentrations fluctuations. We thus show cn 1/n for n = 1, 2, 3, 4 for the fixed and
for the moving frames, respectively. It is evident that the uncertainties on the estimates are
significantly larger in the fixed frame, as compared to the moving frame. Ideally, we would
expect a distribution close to a Maxwellian also in Fig. 6a. For the moving frame, where
the meandering motion of the plume is removed, we find much less smearing out during the
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(b)
0.0008
0.0006
1
c1
2
c2
3
c3
4
c4
0.001
0.0008
0.0006
1
c1
2
c2
3
c3
4
c4
cn
0.0004
0.0002
0.0002
n
n
cn \
0.0004
arb. units
arb. units
(a)
0
0
50
100
150
200
250
300
100
Distance m
50
0
50
100
150
Distance m
Fig. 6 a Fixed-frame and b moving-frame concentration statistics, showing cn 1/n for n = 1, 2, 3, 4
averaging process, so that the ensemble averages determined from measurements are more
closely related to the values of the concentration in the individual realizations (Chatwin
and Sullivan 1979), and consequently the differences between the four curves cn 1/n with
n = 1, 2, 3, 4 are smaller.
From the Lidar returns, we thus obtain time series of the cross-wind instantaneous concentration profiles c(x j ) spatially smoothed only by the effective Lidar-pulse length of 1.5 m.
The measurements are repeated M times (i.e. with M number of Lidar pulses), and averages
are obtained as
cn (x j ) =
M
1 n
c (x j ),
M
(15)
=1
where M (typically of the order of several thousands) can vary from one experiment to the
other. We collect, store and process typically several thousand instantaneous profiles into the
dataset of a single Lidar experiment, lasting 10–30 min.
The expression (15) serves to illustrate the importance of one practical limiting case
caused by the bit-resolution of the signal. We can have all c = 0 except one: this will
typically happen at a position x j at the very edge of the plume, say j = j0 . In this case,
we evidently have cn (x j0 ) = n /M, with being the basic unit in the bit-resolution
of the Lidar signal. Calculating σ 2 /c2 = (c2 − c2 )/c2 , we find at this position
σ 2 /c2 = M − 1 ≈ M, the approximations applying for large M.
√ For the skewness in this
particular case we find S = (M 2 − 3M + 2)/(M − 1)3/2 ≈ M, and for the kurtosis
K = (M 3 − 4M 2 + 6M − 3)/(M − 1)2 ≈ M, giving the exact relation K = S 2 + 1
for this case. These expressions can be seen as limiting values to be found at very small
concentrations.
5 Results
The results of Sect. 2 demonstrate that a physically non-trivial model exists, where there are
simple relations between the average concentration and the corresponding standard deviations, the skewness and the kurtosis at any given position and at any time. We are not aiming
at a verification of the simple results summarized in Sect. 2, but attempt rather to generalize the question as to whether some universal functional relation exists between C , σ, S
and K ? If there is such a relation, it can be made conspicuous by plotting for instance σ
vs. C , and similarly for S and K . In case such a relation exists, we expect that for each
experiment all the data points will lie on a curve, which can subsequently be determined
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empirically from the experimental data. If there is nothing like such a relation, the points will
scatter throughout the plane. As an intermediate case, we can have the data points lying in a
“band”, and with a width representing the “goodness” of the approximation by a continuous
curve.
We distinguish results for the absolute and the centre-of-mass frames. We emphasize again
that the large-scale meandering of the cloud in the absolute frame gives a larger uncertainty
of our estimates as compared to the centre-of-mass frames.
5.1 Fixed Frame
We consider it to be an advantage to work with dimensionless quantities, so rather than
using c2 vs. C , we show the normalized concentration fluctuation intensity σ 2 /C 2 ,
with σ 2 ≡ c2 − C 2 vs. C instead, see for instance Fig. 7. Similarly, we
use the
standard expressions for skewness and kurtosis, S ≡ c3 + 2C 3 − 3c2 C /σ 3 , and
K ≡ c4 − 3C 4 + 6c2 C 2 − 4c3 C /σ 4 , respectively. The use of C /Cm as the main
independent variable can be seen as a replacement for the physical location within the plume.
Thus, the lower values of σ 2 /C 2 , S , and K in Fig. 7, and similar, should be near the centreline (where C /Cm ≈ 1) and the high values of these variables should be in the tails of the
plume where C /Cm is small. In each case (i.e. data from each site) we have Cm to be the
maximum concentration of the average over the M number of Lidar pulses. In Fig. 7 we
show the full range of data obtained so that the uncertainty for different levels of C /Cm can
be estimated.
The data in Fig. 7 seemingly consist of two close “traces” representing the instantaneous
concentration profiles. The almost continuous one originates from the near side of the plume,
the other part from the rear. In this latter part, the Lidar beam intensity is somewhat depleted
in all individual realizations, and the signal-to-noise ratio is consequently reduced. Ideally
these two traces should be identical, and we can use the difference as an estimate for the
uncertainty of the measurements. For C /Cm > 5 × 10−2 we find this uncertainty to be
of the order of 10–15%. For smaller values of C /Cm we have a significant increase in the
uncertainty, and these points will not be used in the parametric presentations of S , and K
to be shown later.
We note that the observed variations of normalized fluctuations, skewness and kurtosis
follow the simple model from Sect. 2.1.1 well, and in particular we note that also the differences in numerical values found in Fig. 1a–c are recovered here, i.e. the numerical value
of the skewness is close to an order of magnitude smaller than the kurtosis and the normalized mean square fluctuations. For a one-to-one comparison with the results from the
simple analytical model in Fig. 1a–c, we would need the Lidar signal corresponding to the
maximum concentration in the plume. This is not available, since we cannot make certain
that the Lidar beam is crossing the spatial region of maximum concentration. In addition, due
to molecular diffusion (which is ignored in the simple model discussed in Sect. 2.1.1) the
maximum concentration level at a given distance from the source will, in general, decrease
in the downstream direction. For this reason we normalized the abscissa with the maximum
value of c at the relevant position, giving the maximum abscissa value to be unity.
A possible relation between the mean square concentration fluctuation and the average concentration has been investigated previously. That study was based on data from a rough-walled
boundary layer, with propane as the contaminant in neutrally buoyant mixtures, using a fast
response flame ionization detector system (Wilson et al. 1982).
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BORR04B.fxm
(a)1000
100
50
C
2
C
2
500
C2
Fig. 7 a Example of a
parametric
representation
of
c2 − C 2 /C 2 vs. C /Cm .
The different points correspond
to different positions in the plume
in the fixed frame. Corresponding
variations of b the skewness, S
and c kurtosis, K . Data points
are normalized by the maximum
value of C ≡ c, to give the
largest abscissa value as unity.
These data originate from the
“Borris” experiment (Nielsen
et al. 2002; Mikkelsen et al. 2002)
361
10
5
1
5
10
4
10
3
10
2
10
1
10
1
C Cm
BORR04B.fxm
(b)
20
10
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1
10
5
10
4
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3
10
2
10
1
1
C Cm
BORR04B.fxm
(c)
100
50
10
5
1
10
5
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4
10
3
10
2
10
1
1
C Cm
We obtained experimental results for the fixed frame normalized concentration variations,
skewness and kurtosis, respectively. We show here the parametric presentations corresponding to Fig. 2a and b, and are given in Fig. 8a and b. These results are consistent with (3)–(5)
for a cosn profile as discussed for the models in Fig. 1a–c, when we take an exponent n ≈ 2–4.
In particular we note a near-linear relationship between K and S 2 + 1 as indicated in the
figure caption. The scatter in data appears from Fig. 8.
The experimental results shown in this section have a relatively large statistical scatter, in
spite of the averaging over many Lidar pulses. In this fixed frame, we find that the large-scale
motions of the plume often give low average concentrations over large spatial regions, and
the uncertainty on the statistical estimates becomes large. Averages over very long time series
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All experiments
(a) 10
6
2
1
8
4
2
0
2
4
6
8
10
All experiments
8
6
C
2
C
2
(b)
C2
Fig. 8 a Experimental test of the
relative variations of the
skewness, S and kurtosis, K .
The simple analytical model (6)
gives K = S 2 + 1, as shown by
a red line for reference. The
figure includes all our data from
the experiments, here presented
for the fixed frame of reference.
The data can be fitted by
K = 1.3 S 2 + 2.1. b
Parametric plot of σ 2 /C 2 and the
skewness S , using the average
concentration C as a parameter.
Also this figure includes data
from all the experiments referring
to the fixed frame of reference.
The red curve gives the analytical
model result obtained for a “flat
topped” initial density
2
distribution.
The curve
σ 2 /C
=
1 2+S2 +S 4+S2
5
gives an adequate approximation
to the data points
H. E. Jørgensen et al.
4
2
0
1
2
3
would be necessary to improve the reliability of the estimates, i.e. a large time between Lidar
pulses would be needed to justify the assumption of each pulse representing independent
realizations. The signal-to-noise ratio is significantly improved by an analysis that refers to a
moving centre-of-mass frame, where the slow, large-scale, meandering motion of the plume
is removed, to be discussed in the following.
5.2 Moving Frame
The analysis discussed in Sect. 5.1 has a counterpart referring to a moving frame of reference.
To discuss this formulation of the problem, we take each of the traces obtained experimentally, and calculate the centre-of-mass for that dataset, and organize a set of realizations,
where this centre-of-mass is at the origin. The ensuing averages are made on this set. Again,
we characterize the probability distributions of the concentrations with an average, a standard
deviation, a skewness and a kurtosis. Results corresponding to those in Fig. 7 are shown in
Fig. 9, but now in the centre-of-mass, or the moving, frame of reference. We emphasize that
the present definition of a centre-of-mass refers to the line-of-sight of the Lidar beam, and
need not correspond to the mass centre of the entire cloud. The analytical expressions (3)–(5)
do not require the Lidar beam to cross the centre of the cloud. The present choice of frame
has the purpose of removing the effects of the slow large-scale motions in the flow. Again
we note that ideally the two traces representing the near and distant parts of the cloud should
be identical. Also here we can use the difference as an estimate for the uncertainty of the
measurements. For C /Cm > 5 × 10−2 we find this uncertainty to be of the order of 10%,
slightly better than for the absolute frame results in Fig. 7, and also the scatter in data for
C /Cm < 5 × 10−2 is reduced, although it is still too large to allow these data to be included
in the ensuing parametric representation of skewness and kurtosis.
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C
100
2
C
C 2 vs. C . The different points
correspond to different positions
in the plume in the moving frame.
b Shows the corresponding
variations of the skewness, S
and c the corresponding kurtosis,
K . We have normalized all data
points by the maximum value of
C , to give the largest abscissa
value as unity. The present data
originate from the “Borris”
experiment (Nielsen et al. 2002;
Mikkelsen et al. 2002)
BORR04B.moc
(a) 1000
C2
Fig. 9 a Example of parametric
representation of c2 − C 2 /
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In Fig. 10a–c we show the variation of the normalized variance of the concentration
(c − c)2 /c2 for varying c/Cm , where we also here normalized all curves to have the
maximum abscissa at unity. The most significant test of the model (6) is presumably given
by comparison with the result K versus S 2 + 1, see Fig. 11a. This test has the advantage
of not requiring any knowledge of the absolute concentration level. We find a close to linear
relationship for K < 6, followed by an increasing scatter. We note that the scatter in the data
is somewhat larger here than in the fixed-frame presentation, Fig. 8. In Fig. 11b we show a
parametric presentation of σ 2 /C 2 versus the skewness S , with the average concentration as
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100
C
2
C
2
(a)1000
10
C2
Fig. 10 a c2 − C 2 /C 2 for
varying average concentrations
C . b Shows the corresponding
variations of skewness. Note that
the skewness is limited in
magnitude due to the data
sampling, and consequently the
average saturates for low
concentrations. c Shows the
similar kurtosis variation. The
figure includes data from all the
experiments referring to the
moving frame of reference. The
red curves give the averages
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parameter; also this figure contains data from all the experiments. The scatter in the data is
approximately the same as in Figs. 8b and 11b.
The results summarized in Fig. 10a–c have a significantly reduced scatter, as compared
to the corresponding fixed-frame counterparts, demonstrating the advantage of the moving,
or centre-of-mass, frame of reference, but we also note that this conclusion does not hold
for the parametric representations of Fig. 11a and b, where the scatter is comparable to or
slightly larger than for the fixed-frame presentation.
We obtained empirical evidence for a good data fit by relations such as (8), with properly
chosen coefficients A and B in both frames, see Figs. 8 and 11. This result is consistent
with other results (Mole and Clarke 1995; Schopflocher and Sullivan 2005), but they are
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C
2
C
2
(b) 8
4
C2
Fig. 11 a Experimental test of
the relative variations of the
skewness, S and kurtosis, K .
The simple analytical model (6)
gives K = S 2 + 1, see red line.
The data can be fitted by
K = 1.5S 2 + 2.3. b Shows the
parametric plot of σ 2 /C 2 and the
skewness S , using the average
concentration C as a parameter.
The figure includes data from all
the experiments referring to the
moving frame of reference. The
red curve gives the simplest
analytical model result. For this
2
case the curve σ 2 /C
=
1 2+S2 +S 4+S2
25
gives an adequate approximation
to the data points
365
2
0
1
2
3
here found for Lidar-averaged data with two types of analysis, the fixed and centre-of-mass
systems, as defined here. We find it interesting that also an empirical relation between σ 2 /C 2
and S can be found (see Figs. 8b, 11b) with an experimental scatter that is not significantly
different from that found for the relation between S and K . For the latter case we used as
reference the functional form found for the “single concentration level” model.
6 Discussion
The results summarized in Sects. 2.1 and 2.1.1 invite the following basic postulate: the lowest
order averages, i.e. standard deviation, skewness and kurtosis depend on the spatial and temporal variables, but only through a functional dependence on C (y, t). We hereby postulate,
for instance, that σ = σ (C (y, t)), and similarly S = S (C (y, t)) and K = K (C (y, t)).
From the continuity equation, where we for the moment ignore molecular diffusion, we have
(Seinfeld 1983; Falkovich et al. 2001)
∂
C (y, t) + ∇ · u(y, t)c(y, t) = 0,
∂t
(16)
∂
c(y, t)2 + ∇ · u(y, t)c2 (y, t) = 0,
∂t
(17)
and
where u(y, t) is the incompressible turbulent velocity field. We assumed that a mean flow
can be removed by a change in frame of reference.
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With ∂c2 /∂t = (dc2 /d C )∂C /∂t, the basic postulate can be reformulated by these
equations to give
∇ · u(y, t)c2 (y, t)
dc2 =−
.
(18)
dC
∇ · u(y, t)c(y, t)
Similar expressions can be obtained for the skewness and kurtosis.
In the case dc2 /d C = constant, the relation (18) implies that, in an average sense,
the local concentration is convected by the random flow in the same manner as the squared
concentration. In case we have a “flat-top” initial concentration distribution, being either
zero or C0 , we evidently have in the absence of diffusivity, a constant of proportionality
c2 (y, t)/c(y, t) = C0 in each realization, and a relation such as c2 /C = Cm follows
trivially.
6.1 Consequences of Diffusivity
In actual physical viscous flows, it is not possible to have conditions where the effect of
molecular diffusion is entirely absent. Molecular diffusion with a constant diffusion coefficient D can be introduced using a model equation (Seinfeld 1983; Yee and Chan 1997;
Falkovich et al. 2001)
∂
c(y, t) + u(y, t) · ∇c(y, t) = D∇ 2 c(y, t),
(19)
∂t
where c(y, t) is the concentration of the contaminant, and u(y, t) is the velocity vector of
the transporting flow in the given realization. We assume also here the velocity field to be
independent of c, for a passive contaminant. Equation 19 is to be supplemented
by an initial
condition c(y). With D = 0 in (19)it is trivially demonstrated that cn (y, t)dy = constant
for alln. For D = 0, we still have c(y, t)dy = constant, but now d c2 (y, t)dy/dt < 0,
and d c3 (y, t)dr/dt < 0, for instance.
With the assumption of a vanishing mean flow, u(y, t) = 0 we find by Lidar and ensemble averaging of (19) the relation
∂
c(y, t) + ∇ · u(y, t)c(y, t) = ∇ · (D∇c(y, t)).
∂t
(20)
As shown before, we have ∇ · u(y, t)c(y, t) = ∇ · u(y, t)c(y, t), etc. for a relevant class
of Lidar pulses. The last term on the left-hand side of (20) represents a turbulent flux and
the term on the right-hand side a diffusion flux. In this particular case, molecular diffusion is
usually considered to be of minor importance, since the gradient operators on the right-hand
side act on the average contaminant concentration, i.e. the diffusive flux is in general small
compared to turbulent fluxes.
We also have
∂ 2
c (y, t) + ∇ · u(y, t)c2 (y, t) = ∇ · D∇ c2 (y, t) − 2D (∇c(y, t))2 ,
(21)
∂t
and similar expressions involving cn (y, t), etc (Falkovich et al. 2001).
The important difference between (20) and (21) is that, in the last term of the latter, the
gradient operator acts directly on the small scales in the individual realizations. Without the
Lidar averaging, the last term on the right-hand side in (21) and its higher order counterparts
will eventually dominate the first one in magnitude, as finer and finer scales develop. In such
a limit we expect ∂c2 (y, t)/∂t < 0 and therefore c2 (y, t) → 0 as t → ∞, and similarly
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for all cn (y, t). In (21) we have the first term on the right-hand side to represent diffusive
fluxes, which are usually considered negligible.
We define L M (t) as the length scale associated with the average concentration, and m (t)
as the length scale characterizing the thickness of filaments in individual realizations, both
quantities evidently varying with time. Ignoring at first the Lidar averaging, the relative magnitude of the second term on the left-hand side and the two terms on the right-hand side of
(21) is then
u 2 1/2
LM
:
D
2
LM
:
D
2m
(22)
where u 2 ≡ u · u. We use u 2 1/2 c as an estimate for the average flux on the left side
of (21), while ∇c2 ∼ c2 /L M and (∇c)2 ∼ c2 /m . In the present notation, the Péclet
number is u 2 1/2 L M /D. In general, the first term on the right-hand side of (21) is smaller
2 ≤ 1. We can ignore the effects of molecular diffusion
than the last one by the ratio 2m /L M
provided D u 2 1/2 2m /L M .
We now include also the Lidar averaging as given in (21). The Fourier transformed version
. For small k this term is negligible because of the k 2
of the term D∇ 2 c gives −Dk 2 cH
falls off rapidly for increasing k in realistic cases, we can argue that
multiplier. Since H
for the Lidar-averaged equation, the diffusion term in (20) is of minor importance. (For the
present experiment it was thus found that the form of the Lidar pulse was well approximated
by a Gaussian, implying that its Fourier transform is also a Gaussian, thus decaying exponentially as |k| → ∞.) The smooth concentration variation found in, for instance Fig. 5
(as compared to results from other studies), is thus most likely to be a consequence of the
Lidar averaging. The pulse length of the Lidar here sets a lower limit of 1.5 m to the fine
structure we are able to observe.
The discussion here made reference to an initial value problem of a contaminant release.
The basic conclusions will, however, apply to constant source releases as well.
7 Conclusions
In the present study we have described results from a series of Lidar measurements, with
particular attention to the lowest order statistical moments of the space-time varying concentrations. Our analysis support previous results by for instance Chatwin and Sullivan (1990)
concerning the scaling of the standard deviation with average concentration, although those
experiments were based on different methods. In addition, we analyzed also the variation
of skewness and kurtosis, and in particular, we addressed the problem in fixed as well as
moving frames of reference. We demonstrated that a simple model for the concentration statistics gives a good account of the variations of the mean-square concentration fluctuations,
the skewness and the kurtosis, σ 2 , S and K . It is evident that if a concentration model
containing few parameters, as used in Figs. 1a–c, 2a and b is proposed, it is then possible to
obtain a parameter fit by results as those shown in Fig. 11a and b, for instance. The interesting
observation here is, in our opinion, that the scatter is relatively modest, and in reasonable
agreement with the simplest generalization of a flat-top model distribution.
One interesting aspect of the present study is the apparent similarity between the lowest
order concentrations statistics in both fixed and moving frames. The possible differences are
below the noise level, which, as stated, is larger for the fixed frame.
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In spite of the spatial averaging by the Lidar pulse, the simple model (8) fits the results in
Figs. 8 and 11 quite well by adjusting the constants A and B, consistent with results given
by, for instance, Mole and Clarke (1995) and Schopflocher and Sullivan (2005). In our case
the fit is performed for several datasets as outlined in Sect. 4. Our results thus add support
to an empirical relation between the kurtosis and the skewness that has been reported in
the literature. We also find empirical evidence for a similar relation between the normalized
standard deviation and the skewness. The form of (8) seems to be universal, and the A and
B differ only modestly from the fixed and moving frame representations, see Figs. 8 and 11.
Comparison of the results in Figs. 8 and 11 also indicate that σ 2 /C is largest in the fixed
frame of reference, as expected.
An obvious, but nonetheless important, conclusion from the present analysis is that provided the space-time evolution of the average concentrations is known, we are able to give
a rather accurate estimate for the space-time evolution of the corresponding concentration
fluctuations, as well as their skewness and kurtosis. The average concentration is relatively
easy to model: often a Gaussian form can be postulated with a time varying width (for the
initial value problem) (Mikkelsen et al. 1987; Krane et al. 2003). In such cases we have
C a V0
1 2 2
C (y, t) ≈ (23)
√ 3 exp − 2 y / (t)
(t) π
in three spatial dimensions, with Ca being the average concentration in the initial volume
V0 , i.e. at t = 0, and expressions for the standard deviation, skewness S (y, t) and kurtosis
K (y, t) then follow trivially from (3)–(5). The actual model then determines (t), where
different results are found for the absolute and for the moving frames, since in the absolute
frame any motion in the flow disperses particles, while in the moving frames only the relative
motions are relevant. In particular, we note that Fig. 6 indicates that a Gaussian model for
the average concentration is indeed appropriate, at least in the moving frame of reference.
Several analytical models have been developed for the expansion of a contaminant cloud
released in a turbulent environment (Smith 1959; Smith and Hay 1961; Sawford 1982;
Mikkelsen et al. 1987; Krane et al. 2003) and many seem to be in very good agreement with
observations from experiments (Mikkelsen et al. 1987), or numerical simulations (Krane et al.
2003). We expect these and similar results (Chatwin and Sullivan 1990) can contribute to
practical applications in predictions of the consequences of, for instance, accidental releases
of chemical substances.
Acknowledgements We thank Søren Ott, Jan Trulsen and Bård Krane for valuable discussions, and Liv
Larssen from the Auroral Observatory in Tromsø for her kind assistance in preparing Fig. 3. We also thank
Dr. Leif Kristensen for his help in preparing the manuscript. One of the authors (HLP) was in part supported
by the “Effects of North Atlantic Climate Variability on the Barents Sea Ecosystem” (ECOBE) project.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided
the original author(s) and source are credited.
Appendix A: Some General Basic Results
We have the trivial relations (xx 4 − x 2 x 3 )2 ≥ 0 or (xx 3 − x 2 x 2 )2 ≥ 0, giving
x 4 ≥ x 3 2 /x 2 . The equality sign applies for x 2 = 0, where we have the “null identity”. More generally we have K ≥ S 2 + 1 when σ = 0. For completeness we give the
simplest proof here by taking the dimensionless vector X ≡ {1, x, x 2 } and then obtaining
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the expectation value of the tensor XX† , where † denotes “transpose”. With x = 0 we then
have
⎞
⎛
1
0 x 2 XX† = ⎝ 0 x 2 x 3 ⎠ .
x 2 x 3 x 4 Using a vector Y ≡ {x 2 2 , x 3 , −x 2 }, we construct the scalar Y† · XX† · Y, which
is identical to (Y† · X)(X† · Y) ≡ (Y† · X)(Y† · X)† ≥ 0. Consequently we have
x 4 x 2 − x 3 2 − x 2 6 ≥ 0. If x 2 = 0, we have trivially that x 3 = 0. When x 2 > 0
this gives the desired result K ≥ S 2 + 1, which remains correct also when x = 0. A more
general, but also more lengthy, analysis containing the foregoing results is given by Mole
and Clarke (1995).
The results summarized here have general validity. They have been quoted elsewhere
(Labit et al. 2007) with particular reference to general Beta distributions.
Appendix B: Models for Lidar Return Probability Densities
Some simple models for the Lidar return can be proposed in limiting, yet physically relevant,
cases. We can thus assume that the Lidar beam is crossed by one filament of the cloud where
the thickness of this filament is larger than the diameter of the Lidar pulse, but much smaller
than its length, i.e. the Lidar pulse is considered to be a “pencil of light” given by H (y0 , t0 ),
with centre-of-mass being at the position r0 at t = t0 . This case can be representative for
later times after release at the larger distances from the cloud’s centre-of-mass.
Let the concentration of the filament be c0 . With the assumed geometry we only need
to consider the coordinate z along the Lidar pulse, with the
filament at position z 1 . We
consider the filament to be thin to allow the approximation H (z − z 0 , t0 )c(z − z 1 )dz ≈
c0 w H (z 1 − z 0 , t0 ) for the return signal, where w H (z 1 − z 0 , t0 ) is the small common volume of the filament at position z 1 and the Lidar pulse. If the Lidar beam
is crossed by
many independent filaments we have the signal s at t = t0 being s ∼ c0 Nj H (z j − z 0 ),
where, for simplicity, we have assumed all filament concentrations and common volumes to
be the same. A coefficient containing scattering cross-section etc. is omitted for simplicity.
These assumptions can be relaxed, at the expense of a more lengthy notation. When, for
instance, two filaments occupy different spatial positions they will be integrated by the Lidar
pulse to produce a backscatter that can be up to twice the intensity corresponding to one
filament.
The subsequent ensemble averaging is over all positions and numbers N of crossing filaments. For the particularly simple case, where we have one and only one filament crossing
at a random uniformly distributed position in different realizations, N = 1. We can take
H (z) to be a Gaussian (which is appropriate for our case). Given P(s)ds = P(z 1 )dz 1 and
P(z 1 ) = 1/L L for 21 L L < z 1 < 21 L L and P(z 1 ) = 0 otherwise, it is easy to obtain the
probability density of the backscattered signal s to be
P(s) =
A
,
√
(s/c0 ) −ln(s/c0 )
(24)
with 0 < s < c0 , and with
singularities
√ for s = 0 and s = c√0 . We introduce A to be a normalizing constant. Since ds (s/c0 ) − ln(s/c0 ) = 2c0 ln(s/c0 ) diverges at s/c0 → 0,
we here restrict the effective length L of the Lidar pulse to some finite value. The singularity
at s = 0 is due to filaments being very far from z 0 , while that at s = c0 comes from filaments
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close to z 0 . The two singularities correspond to the two δ-functions in (6). Assuming only
one filament crossing of the Lidar beam, the distribution (24) is conditional by assuming
c0 and m to be given. If the statistical distribution of these quantities is known, we can use
Baye’s theorem to obtain the unconditional distribution. In general, we expect the singularity
for s = c0 to disappear, while that at s = 0 remains.
For the general case with N > 1 filaments crossing the Lidar beam, we assume the
positions of them to be independent and find for a given N (Rice 1944),
PN (s) =
1
2π
∞
e−isξ Nj=1 Ch j (ξ ))dξ,
(25)
−∞
in terms of the characteristic functions Ch(ξ ) for P(s), with
Ch 1 (ξ ) ≡ exp(isξ )
≡ eisξ P(s)ds = eiξ c0 wH (z 1 −z 0 ) P(z 1 ) dz 1 .
(26)
It is implicitly assumed that each filament contributes with c0 w. We assume that all characteristic functions of the probability densities for all H (z j − z 0 ) are the same. The density
of filaments μ = N /L L was introduced in terms
of the average number, N , of filaments
contributing in the Lidar pulse. We have P(s) = ∞
N =0 P(N )PN (s). The averaging over N
is performed assuming the probability
density
P(N
) = e−μL L (μL L ) N /N !, i.e. a Poisson
∞
μX
N
distribution for N . Using e = N =0 (μX ) /N !, we find
P(s) =
1
2π
∞
e
−isξ +μ
L L /2 iξ c wH (z−z )) 0 −1 dz
e 0
−L /2
L
dξ,
(27)
−∞
where it may not always be permissible to let L L → ∞, as mentioned in relation to (24).
The analytical expression (27) can be quite difficult to solve, even numerically, except in a
few special cases: we can, for instance, assume H (ξ ) to be a “box-function”. For this special
case we find (Pécseli 2000)
P(s) = (μ N /N !) e−μ δ(s − N c0 ) for N = 0, 1, 2, . . .,
the result
N /N ! = 1. For very low densities of μ, where it is unlikely
recalling the identity e−μ ∞
μ
N =0
to find more than one filament within the range of the Lidar pulse, we can expand this result by
including only terms up to first order in μ giving P(s) ≈ (1−μ)δ(s)+μδ(s−c0 ), i.e. the result
(6), although it here describes some slightly different physical conditions. This result can also
be used to model (24). Position and time after release enters through μ = μ(y, t). We expect
μ to decrease with distance from the instantaneous centre-of-mass of the cloud at any time.
The foregoing analysis assumes that the concentrations in all filaments are the same,
with the same common volume of Lidar beam and filament. The analysis can be extended
with little additional effort to account for statistically distributed filament concentrations but
assuming all common volumes to be at least approximately the same. We obtain
P(s) =
1
2π
∞
e
−isξ +μ
∞
0
P(c)
L L /2 −L L /2
eiξ cwH (z−z 0 )) −1 dzdc
dξ.
(28)
−∞
We will need the PDF of the concentrations c, and as a working hypothesis it can be assumed
to be given by the normalized concentration distribution in the original cloud at t = t0 in
Sect. 2.1. For one of the cases consideredpreviously, with Sc (y) = Cm cos(π y/2D), we
find P(c) = (24/Cm π 3 ) arccos2 (c/Cm )
1 − c2 /Cm2 . For Sc (y) = Cm cos2 (π y/2D),
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we find on the other hand P(c) = (12/Cm π 3 ) arccos2 (c/Cm )
c(Cm − c)/Cm2 as used
in Figs. 1 and 2. Now μ is the concentration of filaments, irrespective of their contaminant
concentration. We can determine the semi-invariants (or cumulants) as
∞
λn = μc w
n
H n (z)dz,
n
(29)
−∞
where we took L L → ∞. The result (29) does not need the Lidar signal PDF from (28), but
a result that follows more easily from Campbell’s theorem (Rice 1944; Pécseli 2000). For
the lowest order moments, the relations between the semi-invariants and the basic moments
are simple: λ1 = s, λ2 = s 2 − s2 ≡ σ 2 , λ3 = s 3 − 3ss 2 + 2s3 , etc. We find,
for instance, the scaling σ 2 /s2 ∼ 1/μ, i.e. a quantity we expect to increase with distance
from the average centre-of-mass of the cloud. The assumption of the common volumes of
the filaments and Lidar beams being approximately the same for all filaments can be relaxed
only when an appropriate PDF for w can be argued. The analytical procedure will, however,
be the same as outlined in the present Appendix.
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