Atmos. Chem. Phys., 9, 1289–1302, 2009
www.atmos-chem-phys.net/9/1289/2009/
© Author(s) 2009. This work is distributed under
the Creative Commons Attribution 3.0 License.
Atmospheric
Chemistry
and Physics
Turbulent dispersion in cloud-topped boundary layers
R. A. Verzijlbergh1 , H. J. J. Jonker1 , T. Heus1,* , and J. Vilà-Guerau de Arellano2
1 Department
of Multi-Scale Physics, Delft University of Technology, Delft, The Netherlands
and Air Quality Section, Wageningen University, The Netherlands
* current affiliation: Royal Netherlands Meteorological Institute, De Bilt, The Netherlands
2 Meteorology
Received: 22 September 2008 – Published in Atmos. Chem. Phys. Discuss.: 18 November 2008
Revised: 5 February 2009 – Accepted: 9 February 2009 – Published: 18 February 2009
Abstract. Compared to dry boundary layers, dispersion in
cloud-topped boundary layers has received less attention. In
this LES based numerical study we investigate the dispersion
of a passive tracer in the form of Lagrangian particles for four
kinds of atmospheric boundary layers: 1) a dry convective
boundary layer (for reference), 2) a “smoke” cloud boundary
layer in which the turbulence is driven by radiative cooling,
3) a stratocumulus topped boundary layer and 4) a shallow
cumulus topped boundary layer.
We show that the dispersion characteristics of the smoke
cloud boundary layer as well as the stratocumulus situation
can be well understood by borrowing concepts from previous
studies of dispersion in the dry convective boundary layer. A
general result is that the presence of clouds enhances mixing
and dispersion – a notion that is not always reflected well in
traditional parameterization models, in which clouds usually
suppress dispersion by diminishing solar irradiance.
The dispersion characteristics of a cumulus cloud layer
turn out to be markedly different from the other three cases
and the results can not be explained by only considering the
well-known top-hat velocity distribution. To understand the
surprising characteristics in the shallow cumulus layer, this
case has been examined in more detail by 1) determining
the velocity distribution conditioned on the distance to the
nearest cloud and 2) accounting for the wavelike behaviour
associated with the stratified dry environment.
Correspondence to: H. J. J. Jonker
([email protected])
1
Introduction
This paper describes the dispersion of a passive tracer in different types of atmospheric boundary layers with emphasis
on the dispersion in cloudy boundary layers. Understanding
the diffusion of pollutants in cloudy boundary layers is important for climate, atmospheric chemistry and air quality.
Clouds are known to transport pollutants from the boundary layer to higher regions in the atmosphere, a phenomenon
referred to as cloud venting (e.g., Cotton, 1995). An intricate coupling exists between particles in the atmosphere and
clouds: not only do clouds enhance the upward vertical transport of pollutants (gases, aerosols) they are also strongly influenced by them. The optical properties as well as the lifetime of a cloud are known to depend on the aerosol distribution in the cloud’s environment. In turn, both the optical properties of clouds as well as their lifetimes affect the
earth’s radiation budget and hence global climate.
Chemical processes in the atmosphere are also influenced
by clouds. First of all they affect transport of chemical compounds through the atmosphere and enhance turbulent mixing of different species. In addition clouds can alter the
photodissociation rates of chemical compounds around them
(Vilà-Guerau de Arellano et al., 2005).
Finally, next to the importance of dispersion on climate
and atmospheric chemistry, the quality of the air we live
in is also affected by meteorological conditions. Predicting ground level concentrations of possibly harmfull substances requires detailed knowledge about the relation between weather conditions and dispersion.
The classical work relating dispersion and turbulence was
done by Taylor (1921). This analysis, however, was based on
homogeneous turbulence, whereas atmospheric motions are
often very complex and characterized by non-homogeneous
turbulence. Pasquill (1961) proposed a Gaussian plume
model with a vertical dispersion coefficient depending on the
Published by Copernicus Publications on behalf of the European Geosciences Union.
1290
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
meteorological circumstances. Basically, the vertical dispersion coefficient is then related to the stability of the atmosphere, which is related to the amount of insolation. In this
view, clouds have a damping effect on dispersion in daytime
conditions.
The subject of atmospheric dispersion has been further extensively studied in the laboratory, in field experiments and
by numerical methods. The pioneering water tank experiments of Willis and Deardorff (1978) demonstrated the effects of the non-homogeneous turbulence of a convective
boundary layer (CBL) on the diffusion of particles. They
showed, rather surprisingly at the time, that a near-ground
release resulted in a quickly rising plume (in terms of the
peak concentration), but an elevated release resulted in a
descending plume that only rises after impinging on the
ground. The water tank results have later been verified by
full-scale atmospheric experiments (Briggs, 1993). Lamb
(1978) approached the problem numerically and used the
velocity fields from Large Eddy Simulations to investigate
the dispersion of particles. Among others, Nieuwstadt and
de Valk (1987) used the advection of a passive scalar in an
LES model to describe the dispersion of both buoyant and
non-buoyant plumes in the dry CBL. More recently Dosio
and Vilà-Guerau de Arellano (2006) gave a thorough statistical description of dispersion in the dry CBL in an LES
based study. By now, it is well understood that the skewed
velocity distribution is responsible for the observed descent
of the plume maximum for elevated releases of non-buoyant
plumes in the CBL.
In contrast with the number of studies on dispersion in
the CBL is the modest number of studies on dispersion in
other types of boundary layers, in particular cloudy conditions. Dispersion in the stable boundary layer was studied
by e.g. Hunt (1985), Kemp and Thomson (1996) and more
recently Weil et al. (2006). The effects of a stratocumulus
cloud deck on dispersion in the nocturnal boundary layer
have been explored by Sorbjan and Uliasz (1999). Evidence
was found that the vertical diffusion of pollutants in a stratocumulus topped boundary layer is non-Gaussian and depends on the location of the source in the boundary layer.
Concerning shallow cumulus clouds, some field experiments
have demonstrated the effect of cloud venting (e.g., Ching
et al., 1988; Angevine, 2005). Vilà-Guerau de Arellano et al.
(2005) have shown in a LES study how shallow cumulus enhance vertical transport of pollutants, thereby specifically focussing on the influence on chemical transformations. Weil
et al. (1993) used ice-crystals as a tracer to study relative dispersion in an ensemble of cumulus clouds. Chosson et al.
(2008) did a numerical study of the dispersion of ship tracks,
and the role of stratocumulus and cumulus clouds therin.
They focussed mainly on the first few turn-over times of the
cloud venting process.
Because a comprehensive study of dispersion in cloudy
boundary layers that includes dispersion over longer time
scales of pollution within the cloud layer appears to be missAtmos. Chem. Phys., 9, 1289–1302, 2009
ing, the objective of this detailed numerical study is to investigate and statistically describe turbulent dispersion in different types of cloudy boundary layers. To this end we perform large eddy simulations together with a Lagrangian particle module. Four types of boundary layers will be considered: 1) the clear convective boundary layer, 2) a boundary
layer filled with radiatively cooling smoke, 3) the stratocumulus topped boundary layer and finally 4) the shallow cumulus topped boundary layer. The differences and similarities between these four atmospheric situations offer a unique
opportunity to gain more insight in the observed dispersion
characteristics. The primary aim of this paper is to study
the influence of the clouds on the dispersion. Where necessary, the deeper mechanisms that control the dispersion in
and around clouds will be briefely treated. In Sect. 2 we describe the methodology consisting of the numerical setup, the
case characteristics and the definition of statistical quantities.
Section 3, in which the results are presented and discussed,
is divided into two parts: a phenomenological part with a
qualitative description of the dispersion characteristics in the
different boundary layers is followed by a more quantitative
part.
2
2.1
Methodology
LES model and Lagrangian particle dispersion model
The LES-code used in this research is version 3 of the Dutch
Atmospheric LES (DALES3) as described by Cuijpers and
Duynkerke (1993). In this study, Lagrangian particles rather
than a concentration field of a scalar are used as a representation of the pollutants. To this end, a Lagrangian Particle Dispersion Module (LPDM) as described in Heus et al. (2008) is
implemented in the LES. Tri-linear interpolation determines
the particle location within each grid cell; the LPDM takes
subgrid-scale motion into account through a model that is
largely based on the criteria for stochastic Lagrangian models formulated by Thomson (1987). The implementation of
these criteria in LES models described in Weil et al. (2004) is
followed in the present LPDM. Due to the parallelization of
the LES some extra attention was required to handle the case
of particles moving from one processor to another. A dynamical list structure in the form of a linked list was used to cope
with this issue: a particle moving from, say, the first processor to the second will be deleted from the list of the first and
added to the list of the second processor. Every record in the
list hence points to a particle and every record can contain as
many entries as wished, e.g., positions, velocities, temperature etc.
The use of Lagrangian particles has the advantage of being
able to track individual particles in time, thereby allowing the
calculation of Lagrangian statistics. Contrary to Nieuwstadt
and de Valk (1987), an instantaneous plane source rather
than an instantaneous line source is used in this study. We
www.atmos-chem-phys.net/9/1289/2009/
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
can view the plane source of 10242 ≈106 particles homogeneously distributed over the domain in both horizontal directions as an ensemble of 1024 linesources, that are released instantaneously. For every case, 3 simulations are run,
with the release height close to the boundary layer height
zi (z=0.2zi ), in the center of the boundary layer (z=0.5zi ),
and near the surface (z=0.2zi ), respectively. For all cloudy
cases, we define the boundary layer height as the height of
the maximum gradient in θv . This definition ensures that the
cloud layer is incorporated within the boundary layer. Details about the numerics of the simulations vary between the
different cases and will be discussed in the next section.
2.2
1291
p(w)
w
(a) CBL
p(w)
Case descriptions
w
Hereafter we describe briefly the four different atmospheric
situations that have been under consideration. In particular
the velocity distributions (depicted schematically in Fig. 1)
are discussed, since they are important for the dispersion
characteristics. Numerical values of the case characteristics
are listed in Table 1. Figure 2 shows the profiles of the virtual potential temperature flux hw0 θv0 i. These profiles give an
indication about the dynamics and the structure of the boundary layer.
2.2.1
Dry convective boundary layer
(b) Smoke
p(w)
w
(c) Stratocumulus
p(w)
The CBL is characterized by a well mixed layer, a strong
surface heat flux and a capping inversion. This gives rise to a
positively skewed velocity distribution: strong localized upCBL
w
drafts surrounded by moderate compensating downdrafts, as
(d)
Cumulus
depicted schematically in Fig. 1a. Figure 2a shows the buoyancy flux profile for the CBL. The boundary layer height,
Fig. 1. Conceptual
representation
of the velocity
distributions
in thelayers.
averaged over the first hour, was 900 m high,
in representation
Fig.as
1. given
Conceptual
of the velocity
distributions
in the different
boundary
different boundary layers.
Table 1, and increased slightly during the simulation. No
large scale horizontal wind was imposed. The simulation
2.2.1 Dry
convective boundary layer
was run on a grid of 2563 points, with a horizontal
resolution of 1x=1y=25 m and a vertical of 1z=6m, resulting in
a domain of 6.4 km×6.4 km×1.5 km. A120
timestep
of 1t=1
s
The CBL
is characterized
by a well
mixed
layer,
heat
and a capping in
top (Smoke).
This
results
in a strong
mirror surface
image of
theflux
vertical
and a 5th order advection scheme (Wicker and Skamarock,
velocity
distribution
from
the
CBL,
as
depicted
in
Fig.
1b
This gives rise to a positively skewed velocity distribution: strong localized updrafts sur
2002) have been used, except for the advection of momenFigure 2b shows the buoyancy flux in the smoke cloud.
by moderate
compensating
downdrafts,
as adepicted
in Fig.
1a. Figure 2a sh
tum, for which a second order central-difference
scheme has
Next to the
absence of
surface schematically
heat flux, we also
observe
been used. The particles were released after buoyancy
three hours
of
entrainment
at The
the top
of the smoke
cloud.averaged over the first hour, wa
flux profile
for the CBL.
boundary
layer height,
simulation.
In the smoke case the domain measured
high, as given in table 1, and increased slightly during the simulation. No large scale ho
3.2 km×3.2 km×1.2 km.
Horizontal and vertical reso3
2.2.2 Smoke cloud boundary layer 125 wind was imposed. lutions
The simulation
was run on am
gridand
of 256
points,mwith
resol
are 1x=1y=12.5
1z=6.25
anda horizontal
the
number
of
grid
points
is
256
in
the
horizontal
and
200
in
the
∆x = ∆y = 25m and a vertical of ∆z = 6m, resulting in a domain of 6.4km × 6.4km ×
The smoke case used in this study is described in an intervertical direction. For the scalar variables the monotonous
A timestep
of ∆t =kappa
1s andadvection
a 5th order
advection
scheme (Wicker
and Skamarock,
The smoke
comparison study by Bretherton et al. (1999).
et al., 1995)
and for 2002) ha
scheme
(Hundsdorfer
case is particularly useful to gain understanding
of
the
strasecond order
central-differences
used, except for themomentum
advection ofthemomentum,
for which
a second orderscheme
central-difference
tocumulus case, for it has similar radiation characteristics,
with a timestep of 1t=0.5 s have been used. The particles
has been
used.
particles were released after three hours of simulation.
but there are no condensation processes or surface
fluxes
to The were
released after two hours of simulation.
additionally drive convection. Making the analogy with the
CBL, instead of heating at the bottom (CBL) we have radiative cooling due to the divergence of the radiative flux at the
5
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Atmos. Chem. Phys., 9, 1289–1302, 2009
1292
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
Table 1. Characteristics of the different cases: the approximate boundary layer height zi at the moment of particle release, the cloud base
height zcb at the moment of particle release, the surface moisture and heat fluxes hw 0 qt0 i0 and hw 0 θl0 i0 , the convective velocity scale w∗ as
defined in Eq. (2), the characteristic timescale t∗ .
dry CBL
smoke
stratocumulus
shallow cumulus
zi
[m]
zcb
[m]
hw 0 qt0 i0
[kg kg−1 m s−1 ]
hw 0 θl0 i0
[K m s−1 ]
w∗
[m s−1 ]
t∗
[s]
900
700
700
2000
–
0
350
500
0
0
1×10−5
1.2×10−4
9.4×10−2
0
1×10−2
1.5×10−2
1.42
0.92
0.87
1.66
633
760
804
1204
1400
1000
800
1000
height (m)
height (m)
1200
800
600
600
400
400
200
200
0
-0.02
0
0
0.02
0.04 0.06
<w’ θ’v>
0.08
0.1
-0.01
(a) CBL
0.02
3000
800
2500
height (m)
height (m)
0.01
<w’ θ’v>
(b) Smoke
1000
600
400
200
0
-0.01
0
2000
1500
1000
500
-0.005
0
0.005
<w’ θ’v>
0.01
0.015
0
-0.01
(c) Stratocumulus
0
0.01
0.02
<w’ θ’v>
0.03
0.04
(d) Cumulus
Fig. 2. Virtual potential temperature flux hw 0 θv0 i in the four′ different
boundary layers. The profiles are an average over the first hour after
Fig. 2. Virtual potential temperature flux hw θv′ i in the
four different boundary layers. The profiles are an
particle-release.
average over the first hour after particle-release.
smoke case, thus giving rise to a more symmetric velocity
2.2.3 Stratocumulus topped boundary layer
distribution, as depicted in Fig. 1c.
130 2.2.2 Smoke cloud boundary layer
Figure 2c shows the virtual potential temperature flux of
The stratocumulus case under consideration is the Atlantic
the stratocumulus case. We already stated that the stratocuStratocumulus
Transition
Experiment
(ASTEX,
de
Roode
The smoke case used in this study is described in mulus
an intercomparison
study by Bretherton et al.
case is a combination of the CBL and the smoke case,
and Duynkerke, 1997). Data from flight 2, A209, has been
butunderstanding
here we specifyofthat
is especially in case,
the cloud
used. (1999).
Convection
in smoke
a stratocumulus
topped boundary
layerto gain
The
case is particularly
useful
theit stratocumulus
for layer
(starting at approximately 350 m) that the smoke cloud charis driven by a combination of processes: radiative cooling
it has
characteristics,
but latent
there areacteristics
no condensation
or surface
fluxes
to looks
are found.processes
In the subcloud
layer, the
profile
at cloud
top,similar
surface radiation
fluxes of heat
and moisture and
more like that in the CBL.
heat release
due to condensation.
This is the
only case
is
additionally
drive convection.
Making
thethat
analogy
with the CBL, instead of heating at the bottom
subject to a mean horizontal wind of (0.5,−10) m s−1 , that is
The numerical grid in the stratocumulus case consists of
3 points
135
(CBL)
wethroughout
have radiative
cooling
due to the
divergence
the radiative
flux at the
top (Smoke).
This m
roughly
constant
the bulk
of the domain.
In anal256of
with a horizontal
resolution
of 1x=1y=25
ogy with the previous cases, in terms of the driving mechaand a vertical resolution of 1z=6.25 m, spanning a domain
results in a mirror image of the vertical velocity distribution from the CBL, as depicted in Fig. 1b
nisms, the stratocumulus case can be regarded a combination
of 6.4 km×6.4 km×1.6 km. Like the smoke case, the kappa
of the CBL
and
the
smoke
cloud.
This
translates
into
a
veadvection
scheme
scalars and
for moFigure 2b shows the buoyancy flux in the smoke cloud.
Next
to theforabsence
of acentral-differences
surface heat flux,
locity distribution that is a combination of the CBL and the
mentum with a timestep of 1t=0.5 s have been used. After
we also observe entrainment at the top of the smoke cloud.
In the Phys.,
smoke9,case
the domain
Horizontal and vertical resoluAtmos. Chem.
1289–1302,
2009 measured 3.2km × 3.2km × 1.2km. www.atmos-chem-phys.net/9/1289/2009/
140
tions are ∆x = ∆y = 12.5m and ∆z = 6.25m and the number of grid points is 256 in the horizontal
and 200 in the vertical direction. For the scalar variables the monotonous kappa advection scheme
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
two hours, regarded as spin-up period, the particles were released.
2.2.4
Shallow cumulus topped boundary layer
The shallow cumulus (in the remainder of the article referred
to as cumulus) case used in this study is derived from the
Small Cumulus Microphysics Study (SCMS) as described in
Neggers et al. (2003). The cumulus topped boundary layer
can be considered as two layers on top of each other. The
subcloud layer has the characteristics of a dry CBL. The velocity distribution in the cloud layer is often thought and also
parametrized (e.g., Siebesma and Cuijpers, 1995) as positively skewed: strong localized updrafts in the cloudy regions and homogeneously distributed compensating downdrafts elsewhere, see Fig. 1d. Recent studies by Heus and
Jonker (2008) and Jonker et al. (2008) have however shown
that downward mass transport occurs mainly near the edge
of a cloud, a mechanism referred to as the subsiding shell.
From the buoyancy flux profile, Fig. 2d, it can be seen that
cloud base is located at approximately 500 m and the cloud
layer extends to 2500 m. The subcloud layer has a profile
similar to the CBL.
Numerical resolutions in the cumulus case are
1x=1y=25 m in the horizontal and 1z=20 m in the
vertical. With 2563 points, this amounts to a domain of
6.4 km×6.4 km×5.2 km. A centered-difference integration
scheme with a timestep of 1t=1 s has been used. The
particles were released after three hours of spin-up, allowing
for a fully developed cumulus field.
1293
in the definition by Deardorff (1980) the integration is up to
the inversion height zi , we integrate over the entire domain
because in the cumulus case the definition of the inversion
height is not so clear. Concerning the other cases, since there
is hardly any buoyancy flux above the inversion height, replacing Lz by zi in Eq. (2) leads only to a very small difference.
The typical timescale is defined as
t∗ =
zi
.
w∗
2.4
Statistics
(3)
In this section we introduce the statistical variables necessary
to describe the dispersion characteristics. The instantaneous
local concentration c(x, y, z, t) of particles is computed by
counting the number of particles Np in a small control volume 1V =1x1y1z centered at (x, y, z). This value is divided by the total number of particles Ntot so that we have a
normalized concentration:
c(x, y, z, t) =
Np (x, y, z, t)
Ntot 1x1y1z
(4)
Z
c dx dy dz = 1
(5)
V
The various statistical parameters can now be defined. The
first statistical moment or mean plume height is given by
Z
zc dx dy dz
z=
(6)
V
2.3
Scaling parameters
In order to compare the results of the different boundary layers, we introduce the following dimensionless velocity and
timescales. For the CBL, it would be natural to use the convective velocity scale based on the surface flux:
w∗ =
g 0 0
w θv 0 zi
θ0
1/3
(1)
However in situations where the surface-flux is not the
main driving mechanism of convection, an alternative velocity scale can be defined based on Deardorff (1980):
1/3
Z
g Lz 0
w∗ = c 1
w θv dz
θ0 0
(2)
where Lz is the domain height. The factor c1 =2.5 ensures
that integrating a linear flux profile from a value of w0 θv0 0
at the surface to 0.2 w 0 θv0 0 at the inversion height yields a
velocity scale that is consistent with Eq. (1) . Equation (2)
makes sense from a physical point of view, since the integral
represents the production of turbulent kinetic energy. Furthermore, it is the most consistent choice, since we can now
use Eq. (2) for all the cases under consideration. Although
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The vertical dispersion coefficient is defined by
Z
σz2 = (z − z)2 c dx dy dz
(7)
V
where the difference with the mean plume height is used
rather than the source height. For the skewness of the plume
we get
Z
1
Sz = 3
(z − z)3 c dx dy dz
(8)
σz V
Another useful quantity is the horizontally integrated concentration, that we will call the vertical concentration profile
Z
1
c dx dy
Cz (z, t) =
(9)
Axy A
were Axy =Lx Ly is the horizontal domain size. Equivalently
we define a horizontal concentration profile according to
Z
1
Cy (y, t) =
c dx dz
(10)
Axz Axz
were Axz =Lx Lz is a vertical cross-section of the domain, Lz
denoting the domain height.
Atmos. Chem. Phys., 9, 1289–1302, 2009
1294
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
0
1
t/t*
2
0
Cz(z,t)
2.5
1200
1000
800
2
1.5
600
1.5
400
1
200
0.5
z (m)
z (m)
0.5
200
0
0
0
0
500
1000
t (s)
1500
0
0
500
(a)
0
1
t/t*
2
0
Cz(z,t)
2.5
2000
1.5
400
1
200
6
t/t*
8
10 12
Cz(z,t)
2.5
2
2000
1.5
1500
1
1000
0.5
0
4
2500
2
600
2
3000
z (m)
z (m)
800
0.5
500
0
0
500
1000 1500
t (s)
(b)
1000
0
Cz(z,t)
2.5
2
1
400
2
1000
800
600
t/t*
1
1000 1500 2000
t (s)
0
0
(c)
3600 7200 10800 14400 18000
t (s)
(d)
Fig. 3. Horizontally integrated plume evolution in four different boundary layers; (a) for the CBL, (b) for the smoke case, (c) for stratocu3. cumulus.
Horizontally
integrated
plume
evolution
in four
different boundary
a) for
theheight
CBL,and
b)the
forevolution
the
mulus, andFig.
d) for
A plane
of particles
has been
released
instantaneously
at half thelayers;
boundary
layer
of the
vertical concentration
profile
C
(z,
t)
as
defined
in
Eq.
(9)
has
been
plotted.
The
concentration
profile
has
been
multiplied
by
1000
to
obtain
z
smoke case, c) for stratocumulus,
and d) for cumulus. A plane of particles has been released instantaneously
a convenient scale. The points represent the location of the maximum concentration.
at half the boundary layer height and the evolution of the vertical concentration profile Cz (z, t) as defined in
9 has been plotted. The concentration profile has been multiplied by 1000 to obtain a convenient scale. The
Velocity statistics
for the spin-up. We briefly discuss the general features of
points represent the location of the maximum concentration.
Fig. 3. In the next section we will go into more detail for
The probability density function (PDF) of velocity that we
each case individually.
will consider is based on the velocities of the Lagrangian
The
evolution
of explained
the horizontally
integrated
plume in the
throughout the boundary layer. The initial descent of the
plume
can be
from the
skewness
particles. Next to this PDF, we shall consider the Lagrangian
CBL, Fig. 2a, has the familiar shape that was first described
velocity
by as shown in Fig. by
240 autocorrelation
of the vertical function,
velocity defined
distribution,
1a.Willis and Deardorff (1978): The plume concentration
maximum
initially isatransported
to the ground
plume
layer resembles
reversed version
of whatby the subu0The
(t)u0 (t
+ τ ) evolution in the smoke cloud boundary
L
siding
motions,
then
rises
by
the
thermals,
until eventually
Ru (τ ) =
(11)
σu2 in the dry CBL. The plume maximum rises
happens
it is
reaches
inversion,
theuntil
plume
entirelythe
mixed.
After remains
3 turnoverthere
times the particles
are
almost
homogeneously
distributed
throughout
the
some time and
again. The
with u0 (t)=u(t)−u(t)
the descends
velocity fluctuation
of observed
a particle,plume can again be understood by considering the
boundary layer. The initial descent of the plume can be exthe overbar
represents
thevertical
averagevelocity
over all distribution
particles at as
allschematically depicted in Fig. 1b.
skewness
of the
plained from the skewness of the vertical velocity distributimes, and u can be u,v,w.
tion, as shown
1a. for short times, much
245
The plume evolution in the stratocumulus topped boundary
layerinis,Fig.
at least
The plume evolution in the smoke cloud boundary layer
more symmetric than in the CBL and the smoke case.
Makingaagain
the analogy
with
the happens
previousin the dry
resembles
reversed
version of
what
3 Results and discussion
CBL.
The
plume
maximum
rises
until
it
reaches
cases, this can be explained by recalling that the stratocumulus topped boundary layer can be
seen the inversion,
remains
there
some
time
and
descends
again.
The ob3.1 Plume phenomenology for different types of boundary
as
a
combination
of
the
CBL
and
the
smoke
case.
served
plume
can
again
be
understood
by
considering
the
layers.
skewness
of
the
vertical
velocity
distribution
as
schematiPerhaps the most striking observation in Fig. 3 is the extremely slow plume spread in the shallow
cally depicted in Fig. 1b.
To give a first general impression of the dispersion character250of the
cumulus
case, where
the layers,
particles
been released atThe
1200m,
i.e.,
in the middle
of the cloud layer.
istics
four different
boundary
thehave
time evolution
plume
evolution
in the stratocumulus
topped boundary
layer
is,
at
least
for
short
times,
much
more
of the concentration
profiles,
integrated
over
the
two
horiThe slow plume spreading is especially surprising given that the dispersion in the cloud layer, (Fig.symmetric
than in the CBL and the smoke case. Making again the analzontal directions, are depicted in Fig. 3. For all the cases,
6 particles
is a composite
of vigorously
turbulent
inside
thethe
clouds,
andcases,
compensating
in by recall10242 ≈103d),
were released
instantaneously
in a updrafts
horogy
with
previous
this can bemotion
explained
izontal plane
at
half
the
boundary
layer
height,
2
or
3
h,
deing
that
the
stratocumulus
topped
boundary
layer
the environment. This configuration of the cloud layer results in a top-hat velocity distribution thatcan be seen
pending on the case, after the start of the simulation to allow
as a combination of the CBL and the smoke case.
2.5
is strongly skewed (Fig. 1d; Heus et al., 2008a). On the basis of this velocity distribution, one would
Atmos. Chem. Phys., 9, 1289–1302, 2009
www.atmos-chem-phys.net/9/1289/2009/
10
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
Sz
1
2
t/t*
3
4
5
1
800
600
z/zi
pw
0.75
0.5
200
0.25
–
400
1
-4
0.75
400
0.5
200
0.25
0.5
0.4
0.3
0.2
0.1
0.8
0.6
0.4
0.2
0
-0.2
0.8
0.6
0.4
0.2
0
-0.2
0
1000
2000
t (s)
-2
-1
0
1
w(m/s)
2
3
4
τ/ t*
w
400
300
200
100
-3
0
R L(τ)
600
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
zmax/zi
800
σz/zi
σz (m)
zmax (m)
–
z (m)
0
1295
1
2
3
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
4
5
0.2zi
0.5zi
0.8zi
0
1000
2000
τ (s)
3000
3000
Fig. 5. Vertical velocity
based upon
particles released
the entire boundary lay
Fig. 5.distribution
Vertical velocity
distribution
based homogeneously
upon particlesinreleased
homogeneously
in the
entire
boundary
and autocorrelation
for different release
levels
in the
CBL layer and autocorrelation for
different release levels in the CBL
Fig. 4. Plume characteristics for the dry convective boundary layer.
√
secondly the vertical dispersion coefficient approaches the limit σz /zi = 1/ 12 ≈ 0.3 and final
concentration
dispersion
coefficientfor
for release
release from
(solid
mum concentration
level,level,
dispersion
coefficient
from
(solid line)
z = 0.2z
i ,(dashed
the skewness ofbeen
the plume
should
zero: Szsince
≈ 0. this
The results
are becomes
in satisfactory agreeme
plotted
for approach
the full range,
quantity
line) z=0.2zi ,(dashed line) z=0.5zi ,and (dotted line) z=0.8zi ; and
irrelevant
when
we
approach
a
vertically
homogeneous
,and (dottedskewness,
line) z =
0.8z
;
and
skewness,
for
release
from
z
=
0.5z
.
i from z=0.5zi .
i
with other numerical
studies by Nieuwstadt (1992) and Dosio and Vilà-Guerau parde Arellano (2006
for release
characteristics
for the dry convective boundary layer. From top to bottom: mean plume height,
From top to bottom: mean plume height, height of maximum
ticle distribution. In the mean plume height and especially
the location of the maximum we see the characteristics as deBriggs (1993).
scribed
the issue
introduction:
a near-ground release results in a
sion characteristics
similar
thoseobservation
in the CBL.
We3 is
will
to in
this
in
Perhaps the
most to
striking
in Fig.
thecome
ex- back
steeply
rising
plume,
whereas
elevated release results in a detremely slow plume spread in the shallow cumulus case,
3.2.2 Vertical velocity statistics
scending
plume.
Initially,
the
skewness
of the plume reflects
where the particles have been released at 1200 m, i.e., in
the
skewness
in
the
vertical
velocity
distribution
(only shown
the middle of the cloud layer. The slow plume spreading
In Fig. 5(top), the PDF of vertical velocity of the Lagrangian particles has been plotted and h
for
release
at
0.5z
).
A
vertically
homogeneous
distribution
i
is especially
given that the dispersion in the cloud
s of dispersion
in the surprising
CBL
indeed
the
positively
skewed
distribution.
This
PDF
is
based
on
particles
released
is
reached
for
all
releases
after
approximately
t=4t
∗ . Thishomogeneous
layer, (Fig. 3d), is a composite of vigorously turbulent upwell
mixed
situation
is
characterized
by
three
conditions:
280 in
in the entire
drafts inside the clouds, and compensating motion
en- CBL, so it represents the velocity distribution in the entire CBL rather than at a specifi
al dispersion
the mean
plume heightofisvertical
approximately
the boundary
vironment. This configuration of the cloud layer results
a Lagrangian
height.inThe
autocorrelations
velocity forhalf
particles
released at three differe
layer
height
z≈0.5z
,
secondly
the
vertical
dispersion
coeffii
top-hat velocity distribution that is strongly skewed heights
(Fig. 1d;
√
has been plotted in Fig. 5(bottom). The autocorrelations are in agreement with the on
cient
approaches
the
limit
σ
/z
=1/
12≈0.3
and
finally
the
z i
Heusorder
et al., statistical
2009). On the
basis of and
this velocity
distribution,
ond and third
moments
the height
of maximum
concentration
as behaviour
found
by
Dosio
et
al.
(2005).
The
oscillating
in
the
autocorrelation
reflects
the large sca
skewness
of
the
plume
should
approach
zero:
S
≈0.
The
z
one would expect dispersion characteristics similar to those
results
are
in
satisfactory
agreement
with
other
numerical
coherent
vertical
motions.
previous in
section
have
plotted
function
the3.5.
dimensionless time in Fig. 4
the CBL.
Webeen
will come
backastoathis
issue in of
Sect.
Nieuwstadt
Dosiostatistics
and Vilà-Guerau
285
We conclude studies
that the by
dispersion
results(1992)
and theand
velocity
of the CBL de
are in satisfacto
rent release
heights.
Arellano
(2006),
who
in
turn
validated
their
results
with ex3.2 Statistics of dispersion in the CBL
agreement with the literature. We shall therefore treat it as a reference case in understanding t
perimental
data from
different release heights (0.2zi , 0.5zi and 0.8zi ) have been
chosen to
cover
a large
parte.g. Willis and Deardorff (1978) and
results of the other
cases.(1993).
Briggs
3.2.1 Vertical dispersion
275
who in turn validated their results with experimental data from e.g. Willis and Deardorff (1978) an
ry layer, in order to observe how the dispersion characteristics change with height. The
The first, second and third order statistical moments and the
3.2.2
Vertical velocity statistics
imum concentration
has not concentration
been plottedasfor
the full
range,
since this quantity becomes
height of maximum
defined
in the
previous
12
section have
been plotted
as a function particle
of the dimensionless
In Fig.
5 (top),plume
the PDF of vertical velocity of the Laen we approach
a vertically
homogeneous
distribution. In
the mean
time in Fig. 4 for three different release heights.
grangian particles has been plotted and has indeed the pos-
pecially the The
location
of the maximum
we(0.2z
seei ,the
as described
in the
three different
release heights
0.5zcharacteristics
itively
skewed distribution.
This PDF is based on particles
i and 0.8zi )
have been chosen to cover a large part of the boundary layer,
released homogeneously in the entire CBL, so it represents
a near-ground
release results in a steeply rising plume, whereas elevated
release results
in order to observe how the dispersion characteristics change
the velocity distribution in the entire CBL rather than at a
with
height.
The
height
of
maximum
concentration
has
not
specific
The Lagrangian autocorrelations of vertical
ng plume. Initially, the skewness of the plume reflects the skewness
in height.
the vertical
ibution (only shown for release at 0.5zi ). A vertically homogeneous distribution is
www.atmos-chem-phys.net/9/1289/2009/
Atmos. Chem. Phys., 9, 1289–1302, 2009
ll releases after approximately t = 4t∗ . This well mixed situation is characterized by
ns: the mean plume height is approximately half the boundary layer height z ≈ 0.5zi ,
1296
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
t/t*
0
1
2
3
4
z/zi
0.25
–
0.5
200
–
z (m)
0.75
400
pw
1
600
0.5
200
0.25
0.4
0.3
0.2
0.1
200
Sz
0
-0.2
-0.2
-0.4
-0.4
w
0
1000
2000
t (s)
-1
0
1
2
1
2
3
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
3
4
4
0.2zi
0.5zi
0.9zi
0
0
-2
t*
0
L
100
-3
w(m/s)
R (τ)
σz (m)
300
-4
zmax/zi
0.75
400
σz/zi
zmax (m)
1
600
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
3000
1000
2000
t (s)
3000
Fig. 7. Vertical velocity
autocorrelation
for the
smoke
case
Fig. 7.distribution
Verticaland
velocity
distribution
and
autocorrelation
for the
smoke case.
Fig. 6. Plume characteristics for the smoke case. From top to botcharacteristics for the smoke case. From top to bottom: mean plume height, height of maximum
tom: mean plume height, height of maximum concentration
level, already impinged to the ground while the majority of particles is still in a slow updr
downdrafts
dispersion
coefficient
releasefrom
from(solid
(solid line)
level, dispersion
coefficient
forfor
release
line)z=0.2z
z = 0.2z
i ,(dashed line) z = 0.5zi ,and
i ,(dashed
halfway
to the inversion.
fast moving
descending
particles
can thus onlyin‘compensate’
for t
opposite The
behaviour
as the
one found
for dispersion
the
line) z=0.5z ,and (dotted line) z=0.8z ; and skewness, for
release
i
i
i
= 0.8zi ; and
skewness,
from
z=0.5z . for release from z = 0.5zi .
CBL,
although
it is
smaller.
The maximum
slow moving rising
particles
as long
as somewhat
they have not
hit the ground
yet. Indeed,cona closer look sho
centration level for the low release height shows a small dip,
before starting to rise as expected. Presumably because of the
opposite
is observed
for horizontal
the mean plume
height
in the
cs of dispersion
in for
theparticles
smoke cloud
boundary
layer310 heights
velocity
released
at three different
hasdirection,
relatively
small
domain
size
of CBL
this case.
case, events
been plotted in Fig. 5 (bottom). The autocorrelations are in
like penetrative up- and downdrafts show up immediately af3.3.2 The
Velocity ter
statistics
agreement with the ones found by Dosio et al. (2005).
particle release.
cal dispersion
oscillating behaviour in the autocorrelation reflects the large
Another feature is the following: the mean plume height
Fig. 7 shows the vertical velocity distribution and the Lagrangian autocorrelation of vertical v
scale coherent vertical motions.
from the release at 0.5zi slightly ascends for a while, simidispersion characteristics
for
the
smoke
cloud
boundary
layer
As opposite
mentioned
inare
theshown.
smoke
boundaryto
layer.
As expected,
we observe
reversed
We conclude that the dispersion results and thelocity
velocity
larlycloud
but
the mean
plume height
in thetheCBL.
Thissymmetry of t
statistics
of the
CBL cloud
are in satisfactory
agreement
withregarded
the
seem
rather of
peculiar
at distribution
first sight,issince
a mass
bal- The fact th
case withmight
CBL,
although
the
velocity
somewhat
narrower.
the dynamics
of the
smoke
boundary layer
can smoke
be
athemirror-image
literature. We shall therefore treat it as a reference
case
in
ance
over
a
horizontal
plane
is
zero
by
conservation
of
mass,
315 we find here the same symmetry is not surprising, since the velocity statistics obviously determi
see this directly
in thethe
dispersion
characteristics.
release
at half
the the mean plume height to remain conunderstanding
results of the
other cases. Considering the thus
we would
expect
the dispersion statistics. Concerning the autocorrelations, we observe again close agreement w
stant. However, we must realise that the highest velocities
yer height, 3.3
where
in the CBL the plume initially descends
and impinges
to the ground,negative because of the coherent vertical motions that ma
the CBL.
The autocorrelation
Statistics of dispersion in the smoke cloud boundary
are found inbecomes
downdrafts and hence the particles that were
initially
in the
strongest
downdrafts
impinged
to the
particles inversion.
undergo:
theyFurthermore,
first
reach
the capping
inversionalready
where they
cannot go
any further and a
e case the plumelayer
maximum rises until it reaches the capping
ground
while
the
majority
of
particles
is
still
in
a
slow
updraft
caught inafter
a downdraft.
It is worth noting in Fig. 7 that the line from the particles releas
hat a well-mixed
distribution
of particles (σz ≈ 0.3) then
is reached
approximately
3.3.1 Vertical
dispersion
halfway
to the inversion. The fast moving descending parti320 closest to the ground that differs from the others, whereas in the CBL it is the line from the high
cles can thus
onlyone
“compensate” for the slow moving rising
e as in the CBL. The skewness of the plume follows the opposite behaviour
as the
release
that differs
most. It
that
the have
autocorrelations
of the velocity
fields far
In Fig. 6 the dispersion characteristics for the smoke
cloud
particles
asseems
long as
they
not hit the ground
yet. Indeed,
a away from t
boundary
layer
are
shown.
As
mentioned
before,
since
the
persion in the CBL, although it is somewhat smaller. Theturbulent
maximum
level
closer
look shows
that case,
for very
short times,
sourcesconcentration
(the bottom
in the
CBL
entrainment
zone inthe
themean
smokeplume
case) are closer to t
dynamics of the smoke cloud boundary layer can be regarded
height is constant. The same phenomenon, although in the
elease height
shows a small
dip,CBL,
before
rise asin expected.
because
a mirror-image
of the
westarting
see this to
directly
the dis- Presumably
opposite direction, is observed for the mean plume height in
persion characteristics.
release
half the
the CBL
case.
ely small horizontal
domain size Considering
of this case,the
events
likeatpenetrative
upand downdrafts
14
boundary layer height, where in the CBL the plume initially
mediately after
particle
descends
and release.
impinges to the ground, in the smoke case the
3.3.2 Velocity statistics
plume
maximum
rises
until
it
reaches
the
capping
inversion.
eature is the following: the mean plume height from the release at 0.5zi slightly ascends
Figure 7 shows the vertical velocity distribution and the LaFurthermore, we observe that a well-mixed distribution of
particles
(σz ≈0.3)
reachedplume
after approximately
sameThis grangian
autocorrelation
of vertical velocity in the smoke
similarly but
opposite
to theis mean
height in thethe
CBL.
might seem
rather
time as in the CBL. The skewness of the plume follows the
cloud boundary layer. As expected, we observe the reversed
that for very short times, the mean plume height is constant. The same phenomenon, although in t
rst sight, since a mass balance over a horizontal plane is zero by conservation of mass,
ld expect the
meanChem.
plume
height
to remain 2009
constant. However, we must realise that the
Atmos.
Phys.,
9, 1289–1302,
www.atmos-chem-phys.net/9/1289/2009/
cities are found in downdrafts and hence the particles that were initially in the strongest
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
3.4
0
1
t/t*
2
3
4
1
z/zi
0.75
0.5
200
0.25
–
400
–
z (m)
600
0.75
400
0.5
200
0.25
0.5
0.4
0.3
0.2
0.1
σz (m)
300
200
100
0.4
0.2
zmax/zi
600
σz/zi
zmax (m)
1
Sz
symmetry of the smoke case with the CBL, although the
velocity distribution is somewhat narrower. The fact that
we find here the same symmetry is not surprising, since the
velocity statistics obviously determine the dispersion statistics. Concerning the autocorrelations, we observe again close
agreement with the CBL. The autocorrelation becomes negative because of the coherent vertical motions that many particles undergo: they first reach the capping inversion where
they cannot go any further and are then caught in a downdraft. It is worth noting in Fig. 7 that the line from the particles released closest to the ground that differs from the others, whereas in the CBL it is the line from the highest release
that differs most. It seems that the autocorrelations of the velocity fields far away from the turbulent sources (the bottom
in the CBL case, entrainment zone in the smoke case) are
closer to the exponential function.
We conclude from the dispersion results in the smoke case
that they can be understood in the light of the vertical velocity
distribution in analogy with the CBL. Furthermore, also in
the purely radiatively driven smoke case, i.e., in the absence
of insolation to generate a surface heat flux, rapid mixing
throughout the boundary layer is observed.
1297
0.4
0.2
0
0
-0.2
-0.2
-0.4
-0.4
0
1000
2000
t (s)
3000
Statistics of dispersion in the stratocumulus topped
Fig. 8. Plume characteristics for the stratocumulus case. From
Fig. 8. Plume characteristics
for the stratocumulus case. From top to bottom: mean plume
boundary layer
top to bottom: mean plume height, height of maximum concen-
3.4.1
Vertical dispersion
dispersion
coefficientforforrelease
release from
from (solid
(solid line)
of maximum concentrationtration
level,level,
dispersion
coefficient
line) z = 0.2z
z=0.2zi ,(dashed line) z=0.5zi ,and (dotted line) z=0.8zi ; and
z = 0.5zi ,and (dotted line)skewness,
z = 0.8z
and skewness,
fori .release from z = 0.5zi .
i ; release
for
from z=0.5z
The statistical moments of the plume evolution in the stratocumulus case have been plotted in Fig. 8. We repeat here
that to some extent the stratocumulus exponential
topped boundary
layer
function.
is not very significant, but for a stronger decoupled stratocucan be regarded as a combination of the CBL and the smoke
mulus case (i.e. the situation where there is a stably stratified
We conclude
dispersion results in the smoke case that they can be unde
case, for it has both the surface flux characteristic
for from
the the
layer just underneath cloud base, Duynkerke et al. (2004)) a
CBL and radiative cooling at cloud
like the
smoke
cloud. velocity
clear separation
between
the “smoke-like”
in the
325 top,light
of the
vertical
distribution
in analogy
with the dispersion
CBL. Furthermore,
also
This is reflected partially in the dispersion characteristics, escloud layer and the “CBL-like” dispersion in the subcloud
driveninismokelayer
case,
in the absence of insolation to generate a surface he
pecially in the skewness of the plume,radiatively
which is, although
cani.e.,
be expected.
tially positive, much closer to zero than in the previous two
After t≈3t
are spread homogeneously in the
∗ theisparticles
mixingevolution.
throughout
layer
observed.
cases, indicating a more symmetric plume
Thethe boundary
vertical direction, comparable to the CBL and smoke case almean plume height and height of maximum concentration are
beit a bit sooner than in the latter cases. This result confirms
more similar to the ones found in the smoke
case though.
and quantifies
the dispersion results
forboundary
stratocumulus
clouds
3.4 Statistics
ofIndispersion
in the stratocumulus
topped
layer
terestingly, a close inspection of Fig. 8 shows that for the reby Sorbjan and Uliasz (1999). We emphasize again that this
lease at z=0.2zi the plume maximum descends to the ground
strong dispersion is contradicting simple dispersion models
Vertical
dispersion
before rising again and the release in3.4.1
the cloud
layer does
in which the dispersion parameter depends on the amount
the opposite. This can possibly be explained by realising
of insolation. A stratocumulus topped boundary layer has
that the buoyancy flux profile in
the
subcloud
layer
is
domito unity,inand
have according
330 The statistical momentsaofcloud
the cover
plumeclose
evolution
thewould
stratocumulus
case to
have been p
nated by the positive surface flux (like the CBL), whereas the
these models a dispersion coefficient almost an order of magcloud layer looks more like the smoke
Roode
and that nitude
8. case.
We de
repeat
here
to some
extent
topped
boundary layer can
lower
than inthe
thestratocumulus
CBL. Apparently,
only considering
Duynkerke (1997) also noted that this leads to a skewness
the amount of insolation is insufficient to describe the disperas awith
combination
of the CBL and the smoke case, for it has both the surface flux
of the velocity distribution that changes
height: negsion in a cloudy boundary layer like the stratocumulus case.
atively skewed in the cloud layer andfor
positively
skewed
in
the CBL
and radiative
cooling at cloud top, like the smoke cloud. This is reflecte
the subcloud layer. This means that dispersion around cloud
3.4.2 Velocity statistics
dispersion
characteristics, especially in the skewness of the plume, which is, alth
base can be quite different dependingthe
on the
exact location.
Within the cloud the maximum concentration has a tendency
In Fig. 9, the velocity statistics from the stratocumulus case
335 positive, much closer to zero than in the previous two cases, indicating a more sym
to go up, but outside the cloud, the maximum will go down.
have been plotted. The velocity distribution is much more
In a well-coupled stratocumulus case evolution.
like this one, The
the effect
symmetric
thanand
in the
previous
cases, whichconcentration
is in agreement are more s
mean plume
height
height
of maximum
ones found in the smoke case though. Interestingly, a close inspection of Fig. 8 show
www.atmos-chem-phys.net/9/1289/2009/
Atmos. Chem. Phys., 9, 1289–1302, 2009
release at z = 0.2zi the plume maximum descends to the ground before rising again an
in the cloud layer does the opposite. This can possibly be explained by realising that
1298
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
3000
1
2500
height(m)
1.2
0.6
0.4
2000
1500
1000
500
0.2
0
0
-3
-2
-1
0
1
w(m/s)
2
3
1
2
1800s
3600s
7200s
14400s
28800s
600s
1800s
3600s
7200s
14400s
28800s
3000
3
4
0.2zi
0.5zi
0.8zi
2000
1500
1000
500
0
Fig. 10. Time evolution of the vertical concentration profile for the cumulus case. Bottom: release in the
Fig. 10. Time evolution of the vertical concentration profile for the
cumulus case. Bottom: release in the subcloud layer (200 m). Top:
t/t
release in the cloud layer0 (1250
m).
2
4
6
8
subcloud layer (200 m). Top: release in the cloud layer (1250 m).
*
2000
τ (s)
3000
2000
1600
1200
800
400
1
2000
1600
1200
800
400
0
1
0.75
z/zi
1000
z (m)
0
–
w
R L(τ)
0
600s
2500
τ/ t*
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
4
height(m)
-4
0.5
–
pw
0.8
0.25
0.5
0.25
600
0.3
400
0.2
300m
2000m
1250m
200
Sz
zmax/zi
0.75
σz/zi
σz (m)
zmax (m)
From Fig. 2d it was already clear that dispersion in the cumulus
cloud layer was very different than in the other cases.
g. 9. Vertical velocity
distribution
and Lagrangian
autocorrelation
for the
stratocumulus topped boundary
Fig. 9. Vertical
velocity
distribution
and Lagrangian
autocorrelation
It
is
instructive
to look at the plume evolution in another way
for the stratocumulus topped boundary layer.
yer
than the contourplot from Fig. 3. Figure 10 shows the vertical concentration profile as defined by Eq. (9) for releases in
5 Statistics of dispersion in the cumulus topped boundary layer
the sub-cloud layer (bottom) and in the cloud-layer (top) at
with the characteristics of the plume. The autocorrelations
different times. Figure 11 shows the statistical moments of
very much resemble the ones from the CBL and the smoke
5.1 Vertical dispersion
the plume evolution.
case. The autocorrelation of the particles released at the top
Fig. 10 (bottom), it can be seen from the release in
of the
cloud
layertohas
a shape between
similar to
the one
in sub-cloud
the smokelayer and In
the cumulus case
it is
important
distinguish
release
in the
release
the
sub-cloud
layer that dispersion in the sub-cloud layer
Fig.
11.
Plume
characteristics for the cumulus topped boundary layer. Note that cloudbase is approximately at
case, again confirming that the stratocumulus cloud layer has
the cloud layer. Unless specified otherwise, the presented graphs represent the dynamics
of the
is
initially
analogous
the
CBL:
theconcentration
plume level,
maximum
de0.25z
. From
top to bottom:
mean plumeto
height,
height
of maximum
dispersion coefficient
the same characteristics as the smoke cloud.
forscends
release fromto
(solid
line)ground
z = 0.2z ,(dashed
line)
z = rises
0.5z ,andagain.
(dotted line)We
z = 0.8z
; andobserve
skewness, for
the
and
then
also
tire ABL, i.e. theSome
composite
cloudyabout
and clear
finalover
remarks
the regions.
generality of the results in
release from z = 0.5z .
this in Fig. 11, although the location of the plume maximum
From Fig. 2d the
it was
already clear that
dispersion
in the cumulus
cloud
layer
was very
stratocumulus
case.
Stratocumulus
clouds
are
found
in different than
has only been plotted for short times for release in the submany
different forms;
fortheexample
the decoupled
the other cases.
It is instructive
to look at
plume evolution
in anothersituation,
way than the contourplot
cloud layer, because in a well18 mixed situation this quantity
as
mentioned
above,
can
be
expected
to
display
different
disom Fig. 3. Figure 10 shows the vertical concentration profile as defined by Eq. 9 for releases
in relevance. After 30 min we observe a vertically
looses its
persion characteristics than the case we considered. The relawell
mixed
profile in the sub-cloud layer, but clouds have
e sub-cloud layer
(bottom)
and
in thelayer
cloud-layer
(top) atofdifferent
times. layer
Figure 11 shows the
tive size
of the
cloud
to the height
the boundary
transported
a
small part of the particles into the cloud layer.
atistical moments
the be
plume
evolution.
can of
also
expected
to be of importance. Illustrative of the
Clouds continue to bring particles upwards, so the concenlatter point
the results
ofrelease
Sorbjan
Uliasz (1999),
whodispersion in the
In Fig. 10(bottom),
it can are
be seen
from the
in and
the sub-cloud
layer that
tration in the sub-cloud layer slowly decreases in time (cloud
considered
a
stratocumulus
case
where
the
cloud
layer
covb-cloud layer is initially analogous to the CBL: the plume maximum descends to the ground
and whereas the number of particles in the cloud layer
venting),
ered almost the entire boundary layer and found dispersion
grows.
These numerical results are in agreement with Vilàen rises again. We also observe this in Fig. 11, although the location of the plume maximum has
results that were closer to the results of the smoke case from
Guerau
de Arellano et al. (2005), although they did not obly been plotted
forstudy.
short times for release in the sub-cloud layer, because in a well mixed situation
this
serve the diminishing concentration in the subcloud layer
is quantity looses its relevance. After 30 minutes we observe a vertically well mixed profile in the
since they prescribed a continuous surface flux of pollutants.
Statistics
of dispersion
in the
cumulus
toppedinto
boundary
b-cloud layer,3.5
but clouds
have transported
a small
part
of the particles
the cloud layer.
TheClouds
effect of cloud venting can also be seen in Fig. 11, where
layer
we see a steadily increasing dispersion coefficient for the release in the sub-cloud layer. One could speculate that this
3.5.1 Vertical dispersion 17
dispersion coefficient has two components: one from the dispersion in the sub-cloud layer, which is constant after approximately 30 min and one from the effect of cloud venting,
In the cumulus case it is important to distinguish between rewhich has a much larger time-scale.
lease in the sub-cloud layer and release in the cloud layer.
Unless specified otherwise, the presented graphs represent
Next we consider the release in the cloud layer. The plume
the dynamics of the entire ABL, i.e. the composite over
is positively skewed, also observed in the bottom graph of
cloudy and clear regions.
Fig. 11, which reflects the skewed velocity distribution in
0.1
3
3
2
2
1
1
0
0
0
3000
6000
t (s)
9000
i
i
i
i
i
Atmos. Chem. Phys., 9, 1289–1302, 2009
www.atmos-chem-phys.net/9/1289/2009/
evolution of the vertical concentration profile for the cumulus case. Bottom: release in the
(200 m). Top:
release
in the cloud
(1250dispersion
m).
R. A.
Verzijlbergh
et al.: layer
Turbulent
in cloud-topped boundary layers
1.2
t/t*
4
6
1
8
1
2000
1600
1200
800
400
0
1
0.6
z/zi
0.75
0.4
–
0.5
0.2
0.25
0
0.3
400
0.2
300m
2000m
1250m
200
0.1
3
3
2
2
1
1
0
σz/zi
0.25
0
w(m/s)
2
4
Pwr
w (m/s)
0.5
-2
zmax/zi
-4
0.75
600
Sz
0.8
pw
2
2000
1600
1200
800
400
σz (m)
zmax (m)
–
z (m)
0
1299
10
8
6
4
2
0
-2
-4
-6
-8
-400
0.001
1e-04
1e-05
1e-06
1e-07
0
400
800
r (m)
1200
1600
0
0
3000
6000
t (s)
9000
Fig. 12. VerticalFig.
velocity
in the cumulus
cloud layer
(top)
and vertical
velocity
12. distribution
Vertical velocity
distribution
in the
cumulus
cloud
layerdistribution as a
function of distance
nearest
cloud edge
(bottom).
(top)to and
vertical
velocity
distribution as a function of distance to
nearest cloud edge (bottom).
Fig. 11. Plume characteristics for the cumulus topped boundary layer. Note that cloudbase is approximately at 0.25zi . From
topmean
to bottom:
plume
height,
height of maximum
concenop to bottom:
plumemean
height,
height
of maximum
concentration
level, dispersion coefficient
3.5.2 Velocity statistics
tration level, dispersion coefficient for release from (solid line)
m (solid line)
z = i0.2z
line) zi ,and
= 0.5z
(dotted
line)
z
=
0.8z
i ,(dashed
i ,andline)
i ; and skewness, for
z=0.2z
,(dashed
line) z=0.5z
(dotted
z=0.8z
;
and
i
The
distribution of vertical velocity in the cumulus cloud
skewness,
for
release
from
z=0.5z
.
i
= 0.5z .
e characteristics for the cumulus topped boundary layer. Note that cloudbase is approximately at
layer is shown in
1 Fig. 12 together with a twodimensional vecloudlayer
(1100m) r represents the
locity distribution,
new coordinate
0.8 where the
cloud layer (2000m)
subcloud
layer
(300m)
distance to the0.6nearest cloud edge, following
the cloud layer. The value of the skewness is about 3 times
Jonker et al.
0.4
higher than in the dry CBL, which
(2008). Negative values for r are locations inside the cloud.
18 is due to the rare but very
0.2
strong in-cloud updrafts in the cumulus layer (Heus et al.,
Jonker et al. (2008)
and Heus and Jonker (2008) have shown
0
2009). Nonetheless, looking at the height of the maximum
that cumulus -0.2
clouds are surrounded by a sheeth of descendconcentration, we observe that it descends only very slowly,
ing air, resulting
-0.4 from evaporative cooling due to mixing with
0 cloud.1000
unlike we would expect from the analogy with the CBL.
dry air outside the
Figure 2000
12 shows3000
that indeed parτ
Moreover, the dispersion parameter shows that the plume
ticles in a cloud move mostly upward, particles near a cloud
spreads much slower in the cumulus cloud layer than in all
move mostly downward. In the far environment particles
Fig. 13. Vertical velocity autocorrelation in the cumulus case
have zero average velocity, although the variance of the veother cases, not only in absolute sense (with the dispersion
coefficient measured in meters), but also in dimensionless
locity is clearly not zero. This velocity distribution explains
units. In the other cases, a vertically well mixed concentrawhy the plume maximum remains approximately constant,
tion profile was reached after t∼3t∗ ; in the cumulus cloud
since the majority of the particles find themselves far away
layer this is not even the case after t∼8t∗ . The combination
from clouds. However, this still does not explain why parof both graphs in Fig. 10 provides the following conceptual
ticles spread so slowly in the21cumulus cloud layer, because,
picture: Clouds transport particles (pollutants) from ground
after all, the velocity distribution in the stratocumulus topped
level to the cloud layer, where they detrain from the cloud
boundary layer is not so much different than in the cumulus
into the stable environment where in turn they hardly move
cloud layer. To understand this we invoke the autocorrelation
anymore – a phenomenon sometimes referred to as plume
functions from Fig. 13. The autocorrelation of the particles
trapping. We emphasize that this result is rather surprising
released in the sub-cloud layer very much resembles the ones
and cannot be understood using the classical view on vertiin the CBL, which is expected because the subcloud layer has
all the characteristics of a CBL. For release in the cloudlayer,
cal transport by cumulus clouds, i.e., strong narrow updrafts
there seems to be a wavelike component in the autocorrelain cloudy regions surrounded by homogeneously distributed
downdrafts. The latter motions would transport the pollution function. Recalling that the cloud layer in a cumulus
tants to cloud-base in a steady pace. The velocity statistics
field is stably stratified for dry air, we can expect the presence of buoyancy waves, as is also shown in a theoretical
should clarify this issue.
Rw (τ)
i
www.atmos-chem-phys.net/9/1289/2009/
Atmos. Chem. Phys., 9, 1289–1302, 2009
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
Brunt-Vaisala
ω from fit
3000
cloudlayer (1100m)
cloud layer (2000m)
subcloud layer (300m)
2500
height (m)
Rw (τ)
1300
2000
1500
1000
0
1000
2000
τ
3000
500
0
0.001
0.002
0.003
0.004
frequency (1/s)
13. Vertical velocity
autocorrelation
in theautocorrelation
cumulus case in the cumulus case.
Fig. 13.
Vertical velocity
Fig. 14. Comparison
profiles of thebetween
Brunt-Väisälä
frequency
calculated by Eq.fre12 and the values
Fig.between
14. Comparison
profiles
of theasBrunt-Väisälä
quency
astocalculated
by Eq.
(12) and thefrom
values
of ωrelease
obtained
by
ω obtained by fitting
Eq. 13
the Lagrangian
autocorrelations
different
heights.
fitting Eq. (13) to the Lagrangian autocorrelations from different
analysis for stable boundary layers by Csanady (1973). The
release heights.
frequency of these oscillations, the Brunt-Väisälä frequency,
4 Conclusions
is given by
21
s
460 We investigated the dispersion characteristics in different types of boundary layers in an LES bas
g dθ
resemble the cloud venting results as discussed by Chosson
ω=
study (12)
together with
a Lagrangian
particle
dispersion module.
Comparison
with the
et al.
(2008); after
transporting
the pollutants
upward,
theextensively do
20 dz
cloud
the pollutants
in the
stable
cloud
layer. The
umented dispersion
in disposes
the dry convective
boundary
layer
showed
satisfactory
agreement with t
One could argue that the autocorrelation of the vertical velocrather
surprising
conclusion
from
this
view
is
that
particles
literature.
ity in a stratified medium is made of two components: one
not only need a cloud to go up, but also one to go down in its
Theone
vertical dispersion results from the smoke case and the stratocumulus case are well und
from stochastic motion associated with turbulence and
surrounding shell.
465 stood
in terms of the vertical velocity distribution in the boundary layer. With the CBL as a ref
from the wavelike motion associated with buoyancy
waves.
Mathematically this would then translate to
ence case, the smoke case has the opposite velocity distribution and hence also mirrored dispersi
R L (τ ) = e
−
τ
TL
cos(ωt)
characteristics. 4Radiative
cooling at the top of the smoke cloud leads to strong narrow downdra
Conclusions
(13)
and compensating updrafts. This negatively skewed velocity distribution translates into an initia
We investigated the dispersion characteristics in different
rising
plume maximum.
where T L is then some characteristic timescale over
which
types of boundary layers in an LES based study together with
the turbulent velocity is correlated. If the suggestion
buoy470 ofThe
stratocumulus case that we considered can best be viewed as combination of the smoke ca
a Lagrangian particle dispersion module. Comparison with
ancy waves is indeed true, then ω in Eq. (13) should
andcorrethe CBL. The
layer, where
radiative dispersion
cooling is dominant,
the smoke case t
thecloud
extensively
documented
in the dryresembles
convective
spond with the one from Eq. (12). To verify this we commost, whereas the
sub-cloud
layer,
where the
surface fluxes
dominate,
more
the CBL.
boundary
layer
showed
satisfactory
agreement
with
theresembles
literputed the autocorrelations for many different release heights,
ature.
study proof dispersion in other types of stratocumulus situations, such as the decoupled stratocumulu
fitted the results with Eq. (13) and so obtained a vertical
L
The
file of T and ω. From the LES fields we have the
virtual
would
be interesting
to vertical
pursue. dispersion results from the smoke case and the
stratocumulus
case are well understood in terms of the vertitemperature profiles, thereby allowing us to make
475 a vertical
Future parametrizations of vertical dispersion in the smoke case and the stratocumulus case mig
cal
velocity
distribution
in the boundary layer. With the CBL
profile of Eq. (12). Figure 14 shows the results. The values
benefit from exploiting
the symmetry
with
respect case
to thehas
CBL,
parametrizations
as
a
reference
case,
the
smoke
thewhere
opposite
velocity have alrea
of ω as fitted from the autocorrelations correspond reasondistribution
been proposed and
validated. and hence also mirrored dispersion characterisably well with the ones calculated from the Brunt-Väisälä
tics.
Radiative
the top
ofand
thethe
smoke
cloud leads
frequency. The difference between the two curves It
can
be be emphasized
should
that incooling
both theatsmoke
case
stratocumulus
casetowe observe rap
strong narrow downdrafts and compensating updrafts. This
explained partially by realising that the autocorrelations
repmixing throughout the boundary layer. This observation contradicts simple dispersion models
negatively skewed velocity distribution translates into an iniresent ensembles of particles released at a certain height, but
480 which clouds have
a damping
effect maximum.
on dispersion by blocking insolation. In addition, as alrea
tially
rising plume
after a short while a part of these particles are not at their
pointed out by Sorbjan
Uliasz (1999),
stratocumulus
can dramatically
change
The and
stratocumulus
case
that we considered
can best
bethe dynamics
release height anymore.
viewed
as
combination
of
the
smoke
case
and
the
CBL.
The
The conceptual picture that emanates from the above
conthe nocturnal boundary layer and corresponding dispersion characteristics.
cloud layer, where radiative cooling is dominant, resembles
siderations is that the vast majority of particles in the The
cumushallow cumulus case shows markedly different dispersion characteristics. Apart from t
the smoke case the most, whereas the sub-cloud layer, where
lus cloud layer is just lingering in the environment, every
the surface fluxes dominate, more resembles the CBL. A
now and then get disturbed from their vertical position when
study of dispersion in other types of stratocumulus situations,
a cloud perturbs the whole system and will then start to os22
such as the decoupled stratocumulus, would be interesting to
cillate for a while around its original position. This explains
pursue.
why we do find a velocity distribution with a reasonable veFuture parametrizations of vertical dispersion in the smoke
locity variance, but yet hardly observe any plume spreadcase and the stratocumulus case might benefit from exing. The effective transport of pollutants is done solely by
ploiting the symmetry with respect to the CBL, where
the cloud and the surrounding subsiding shell. Dispersion of
parametrizations have already been proposed and validated.
pollutants released within a single cloud would first closely
Atmos. Chem. Phys., 9, 1289–1302, 2009
www.atmos-chem-phys.net/9/1289/2009/
R. A. Verzijlbergh et al.: Turbulent dispersion in cloud-topped boundary layers
It should be emphasized that in both the smoke case and
the stratocumulus case we observe rapid mixing throughout
the boundary layer. This observation contradicts simple dispersion models in which clouds have a damping effect on
dispersion by blocking insolation. In addition, as already
pointed out by Sorbjan and Uliasz (1999), stratocumulus can
dramatically change the dynamics of the nocturnal boundary
layer and corresponding dispersion characteristics.
The shallow cumulus case shows markedly different dispersion characteristics. Apart from the expected cloud venting (clouds transporting particles from the sub-cloud layer
upwards), we observed that the dispersion in the cloud layer
is much slower than in all other cases and as anticipated
solely on the basis of the velocity distribution. It turns out
that the velocity distribution as a function of distance to cloud
and the vertical velocity autocorrelations need to be invoked
to understand the observations. By doing so, the view that
emanates is that particles far away from clouds display oscillating behaviour associated with buoyancy waves. This
results in a velocity distribution with a variance comparable
to the other three cases, but yet a very slow spreading of the
plume. The overall picture that emerges is that clouds deposit
pollutants emitted at ground level in the cloud layer, where
they remain for very long times.
This view has consequences for the concentration variance in the cloud layer. Since pollutants in the environment far away from clouds mix very slowly throughout the
cloud layer, there will remain areas with high concentrations
for relatively long times. This might have consequences for
chemical processes that are often non-linear with respect to
the concentrations of the reactants. Additionally, this can be
an important mechanism in establishing the initial conditions
of the residual nocturnal boundary layer.
At a more detailed level than could be given in this study,
many cloud characteristics have potentially significant influence on the dispersion in cloudy boundary layers. For instance, decoupling between the cloud layer and the subcloud
layer, precipitation, shear and interaction with the free troposphere (especially for the stratocumulus case), and the precise configuration of the subsiding shell in the cumulus case,
can alter the dispersion significantly. Such studies require
in-depth studies of the respective cases, and could be interesting for future work. However, we believe that, like in the
dry CBL, a reliable first order view has been given here by
considering the strength, amount and location of the convective updrafts and downdrafts.
Acknowledgements. This work was sponsored by the Stichting
Nationale Computerfaciliteiten (National Computing Facilities
Foundation, NCF), with financial support from the Nederlandse
Organisatie voor Wetenschappelijk Onderzoek (Netherlands
Organization for Scientific Research, NWO).
Edited by: S. Galmarini
www.atmos-chem-phys.net/9/1289/2009/
1301
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