cess
Atmospheric
Chemistry
and Physics
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Atmospheric
Measurement
Techniques
Open Access
Atmos. Chem. Phys., 13, 9021–9037, 2013
www.atmos-chem-phys.net/13/9021/2013/
doi:10.5194/acp-13-9021-2013
© Author(s) 2013. CC Attribution 3.0 License.
Sciences
Biogeosciences
E. Kienast-Sjögren1 ,
P.
Spichtinger2 ,
and K.
Gierens3
Open Access
Formulation and test of an ice aggregation scheme for two-moment
bulk microphysics schemes
1 Institute
Climate
of the Past
Correspondence to: E. Kienast-Sjögren ([email protected])
Earth System
Open Access
Open Access
Cirrus clouds, in particular at temperatures higher than
−40◦ C, often contain very large ice crystals with maximum
dimensions exceeding 1 mm (Heymsfield and McFarquhar,
2002, Fig. 4.6). These large crystals generally have complex
shapes (Field and Heymsfield, 2003, Fig. 3), and many of
them seem to be aggregates of simpler crystals, although one
has to be careful in identifying irregular crystals with ag-
Open Access
Introduction
Open Access
1
gregates (Bailey and Hallett, 2009).
But also in cold cirrus
Dynamics
clouds (T < −40 ◦ C) aggregated ice crystals can be found
(e.g., Kajikawa and Heymsfield, 1989; Connolly et al., 2005;
Bailey and Hallett, 2009), indicating that aggregation might
also play a role for the cold Geoscientific
temperature regime.
Instrumentation
The process of ice aggregation
was already investigated in
the 19th century. From September
1842 on,
Faraday made a
Methods
and
series of experiments in order to investigate the ability of ice
Data Systems
to stick onto other ice particles (Faraday, 1859), which was
called “regelation” by Tyndall (1857). During this time, the
accepted explanation was the so-called pressure melting proposed by Thomson (1859,Geoscientific
1860); the main idea is that sufficient compressive
forces
exist
at the contact region, causing
Model Development
melting if the ice particles are brought together. However,
results by Nakaya and Matsumoto (1954) show that the required pressures are far to high to be realistic under atmoHydrology
andthe existence of
spheric conditions. Faraday
(1859) proposed
a so-called “liquid-like” Earth
layer at the
ice
surface, which soSystem
lidifies in case of contact with another piece of ice. This approach was supported about 100Sciences
yr later by Weyl (1951) and
Fletcher (1962). Additionally, measurements by Nakaya and
Matsumoto (1954) and Hosler and Hallgren (1960) indicate a
temperature dependence of the sticking ability, which could
be explained by the liquid
layer Science
on top of the ice crystals.
Ocean
Kingery (1960) proposed a different way to explain ice aggregation, namely ice sintering. Two (spherical) ice particles
attach at a single point, which is not a thermodynamically
stable state; in order to minimize surface free energy, a neck
between the spheres is formed, thus the two particles stick together. This process of ice sintering,
course, would be supSolidofEarth
ported by the “liquid-like” layer, as proposed. The details of
Open Access
Abstract. A simple formulation of aggregation for twomoment bulk microphysical models is derived. The solution
involves the evaluation of a double integral of the collection kernel weighted with the crystal size (or mass) distribution. This quantity is to be inserted into the differential equation for the crystal number concentration which has classical
Smoluchowski form. The double integrals are evaluated numerically for log-normal size distributions over a large range
of geometric mean masses. A polynomial fit of the results
is given that yields good accuracy. Various tests of the new
parameterisation are described: aggregation as stand-alone
process, in a box-model, and in 2-D simulations of a cirrostratus cloud. These tests suggest that aggregation can become important for warm cirrus, leading even to higher and
longer-lasting in-cloud supersaturation. Cold cirrus clouds
are hardly affected by aggregation. The collection efficiency
is taken from a parameterisation that assumes a dependence
on temperature, a situation that might be improved when reliable measurements from cloud chambers suggests the necessary constraints for the choice of this parameter.
Open Access
Received: 24 July 2012 – Published in Atmos. Chem. Phys. Discuss.: 13 September 2012
Revised: 20 May 2013 – Accepted: 18 July 2013 – Published: 9 September 2013
Open Access
for Atmospheric and Climate Science, ETH, Zurich, Switzerland
for Atmospheric Physics, Johannes Gutenberg-University, Mainz, Germany
3 Deutsches Zentrum für Luft- und Raumfahrt, Institut für Physik der Atmosphäre, Oberpfaffenhofen, Germany
2 Institute
The Cryosphere
Open Access
Published by Copernicus Publications on behalf of the European Geosciences Union.
M
9022
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
this process are discussed by Hobbs (1965). From measurements by Kumai (1964) (image reprinted in Hobbs, 1965)
and Hobbs and Mason (1964) the existence of small quasispherical ice crystals (diameter ∼ 10 µm) sticking together is
evident, even at cold temperatures down to −37 ◦ C or even
below. For large ice particles mechanical interlocking (Jiusto
and Weickmann, 1973) may play a role, especially for large
dendritic snow flakes. In summary, it is likely that ice sintering in combination with the “liquid-like” layer is the main
process for ice aggregation at small sizes; the modern term
for “liquid-like” layer is quasi-liquid-layer (QLL). At high
temperatures near melting point or even down to −15 ◦ C,
the existence of QLLs is quite evident (Kahan et al., 2007;
Sazaki et al., 2012); however, it is not clear if, for the cold
temperature regime T < −40 ◦ C, the QLL still exists; also
from a theoretical point of view, this question is still undecided (Ryzhkin and Petrenko, 2009).
Although details are still unclear, it is evident that aggregation can only occur when ice crystals collide. Collisions can
be caused by a variety of processes, for instance turbulent
motions and gravitational settling of crystals. For crystals
larger than a few µm, gravitational settling is the most efficient aggregation process (Jacobson, 2005, Fig. 15.7). Turbulent fluctuations in clouds can however enhance the gravitational aggregation process as a result of synergy between dynamics and microphysics (Sölch and Kärcher, 2011): cirrus
clouds developing in a steady uplift situation have a thin nucleation zone at their top. New crystals form there as soon as
the ongoing cooling drives the relative humidity over the nucleation threshold. The number of new ice crystals is a strong
function of dSi /dt, the rate at which the supersaturation increases at the threshold. Turbulent motions lead to variations
in dSi /dt, thus, in consequence to variations in the number
concentration of new crystals. If dSi /dt is by chance particularly low, only few crystals form and they grow subsequently
in highly supersaturated air with only weak competition for
the excess water vapour. Thus they first grow large by deposition, obtain large fall speeds, and can then collect many ice
crystals on their way from cloud top to base.
The dominance of gravitational collection has some consequences: (i) the importance of aggregation decreases with
altitude (thus with decreasing temperature) because the absolute humidity decreases (roughly exponentially) and therefore mean crystal dimensions decrease with altitude; (ii) aggregation is more important in deep than in shallow (ice)
clouds; (iii) aggregation is more important in well-developed
than in young (ice) clouds.
Since for atmospheric investigations the evolution of an
ensemble of ice particles must be evaluated, the stochastic
collection equation must be investigated. This type of equation was already investigated by Smoluchowski (1916, 1917)
in order to describe Brownian coagulation analytically. For
investigating the evolution of a size distribution, the stochastic collection equation must be solved or treated in a numerical way. In former studies, different treatments of modelling
Atmos. Chem. Phys., 13, 9021–9037, 2013
ice aggregation has been obtained. Some authors have simulated aggregation via spectral models (Khain and Sednev,
1996; Cardwell et al., 2003) or even by single particle tracking (Sölch and Kärcher, 2011).
Field and Heymsfield (2003) believe that size distributions of ice crystals are dominated by depositional growth
for small particles (e.g. up to 100 µm) and dominated by aggregation for larger particles which they underpin by demonstrating that the crystal size distributions for large crystals
display a scaling behaviour. “Scaling” is a modern expression for the attainment of a self-preserving size distribution
(SPD) (see Pruppacher and Klett, 1997, ch. 11.7.2, see also
Sect. 4.4): the SPD theory suggests that the process of coagulation makes a crystal population loose memory of its
initial size distribution and attaining asymptotically a size
distribution of a relatively simple form. The further evolution of the latter with time can be described simply by scaling transformations, that is, when the x (size) and y (number) axes are transformed with two simple functions of time
(x 0 (t) = x fx (t), y 0 (t) = y fy (t)), the size distribution is represented by a constant curve in this changing coordinate system. Such scaling behaviour in ice clouds has been demonstrated by several researchers and traced back to a dominance
of aggregation processes (e.g. Westbrook et al., 2007).
This behaviour is also a kind of justification for the
modelling aggregation processes via bulk models, i.e. using (fixed) size-distributions and predicting general moments
such as number and mass concentration, respectively. This
was done in some former studies (see, e.g., Passarelli, 1978;
Lin et al., 1983; Mitchell, 1988; Levkov et al., 1992; Ferrier, 1994; Lawson et al., 1998; Field and Heymsfield, 2003;
Morrison et al., 2005). However, the main focus in all these
model studies as well as in most observational studies (see,
e.g., Field et al., 2006; Connolly et al., 2012) is on the “high”
temperature regime, i.e. T > −30 ◦ C, where precipitation is
mainly formed via the ice phase. There are only few observations in the cold temperature regime, mostly in convective
outflow cirrus clouds (Connolly et al., 2005), indicating that
aggregation of ice crystals happens. However, even in cold
cirrus clouds formed in situ, aggregates are sometimes found
(Kajikawa and Heymsfield, 1989), thus aggregation might
occur at these temperature and might have an impact on microphysical properties.
The bulk ice microphysics scheme by Spichtinger and
Gierens (2009a) so far did not represent ice aggregation; as
only cold cirrus clouds have been simulated, this was tolerable. However, as seen above, aggregation can be an important process and a complete cirrus microphysics scheme must
have a representation of it. This new treatment is described
in this study. It has to be noted that although the scheme can
treat multiple classes of ice (e.g. ice formed by homogeneous
nucleation and ice formed by heterogeneous nucleation), it is
not yet possible to compute aggregation between these different classes. Therefore we describe here aggregation between
crystals of a single class of ice.
www.atmos-chem-phys.net/13/9021/2013/
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
2
As stated above, in order to find the prognostic equation
for N we have to integrate, i.e.
Mathematical formulation of aggregation for
two-moment bulk microphysics schemes
Bulk microphysics schemes do not explicitly resolve the size
spectrum of the modelled hydrometeors as spectral models
(also known as bin models) or even models following single
particles do. Instead, bulk models use only some low order
moments of the a priori assumed size distribution and predict their temporal evolution subject to microphysical processes as nucleation, depositional or condensational growth
(and evaporation or sublimation), sedimentation, and aggregation. All these processes can be formulated by first considering the process for a single particle (or two particles in the
case of aggregation) and then computing integrals over the
assumed size distribution. Two-moment schemes consider
the evolution of the zeroth and another low-order moment,
which are proportional to the particle number concentration
(N) and mass concentration (qc ). We use a crystal mass distribution f (m) instead of a size pdf, thus we use the zeroth
and first moment of f (m). In the following we present the
theory for an assumed mass distribution, which we use in
two versions, namely n(m)dm is the number concentration of
particles having masses between m and m+dm, and f (m)dm
is the normalised
version of this, namely f (m) = n(m)/N
R
where N = n(m)dm (the total numberR concentration irrespective of particle mass). This implies f (m)dm = 1.
Evidently, aggregation does not change the mass concentration, thus the prognostic equation for qc is simply
∂qc
= 0.
∂t agg
As this paper deals with aggregation alone, we will drop in
the following the lower index referring to the process. In order to formulate the differential equation for N we start by
writing down the following master-equation (e.g. Pruppacher
and Klett, 1997)
∂n(m, t) 1
=
∂t
2
Zm
9023
∂N (t)
∂
=
∂t
∂t
Z∞
Z∞
∂n(m, t)
n(m, t)dm =
dm
∂t
0
0
Z Z∞
1
=
dm dm0 K(m0 , m − m0 ) n(m0 , t) n(m − m0 , t)
2
0
|
{z
}
=:I1
Z
Z∞
dm dm0 K(m, m0 ) n(m, t) n(m0 , t) .
−
0
|
{z
=:I2
0
Z∞
− K(m, m0 ) n(m, t) n(m0 , t) dm0 .
(1)
0
Here, K(m, m0 ) is the so-called coagulation kernel (i.e. the
rate at which the crystal concentration changes due to aggregation per unit concentration of crystals of mass m and per
unit concentration of crystals of mass m0 ). The first rhs integral describes the formation of particles of mass m from
aggregation of two smaller particles, and the second rhs integral describes the aggregation of particles of mass m with
other crystals of arbitrary mass, which leads to a loss of particles of mass m. Note that we can extend the 1st integral upper
limit to infinity without changing its value because n(m−m0 )
is zero for negative arguments. This fact will be used below.
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}
It is easy to see that every combination of m and m0 in the
first integral occurs as well in the second one. Thus, both
integrals are equal (I1 = I2 ), apart from the factor 1/2, so
that the result is
Z Z∞
∂N (t)
1
=−
K(m, m0 ) n(m, t) n(m0 , t) dm0 dm.
(3)
∂t
2
0
The derivation of this result is as follows: first we interchange
the order of integration in the first integral (I1 ), resulting in
Z∞
Z∞
0
0
I1 = dm n(m , t) dm K(m0 , m − m0 ) n(m − m0 , t).
0
0
Now we substitute x for m − m0 in the inner integral, with
dx = dm. The integral limits can still be set to zero and infinity, because n vanishes for negative arguments. Therefore
Z∞
Z∞
0
0
I1 = dm n(m , t) dx K(m0 , x) n(x, t).
0
K(m0 , m − m0 ) n(m0 , t) n(m − m0 , t) dm0
(2)
0
Since it does not matter whether we write the integration
variable as x or as m and because the integrand is symmetric
in its two variables, we see that I1 equals the second integral
from above, and this completes our proof of Eq. (3).
Now we go on using the normalised mass distribution. In
this form, Eq. (3) reads
1
∂N (t)
= − N2
∂t
2
Z Z∞
K(m, m0 ) f (m, t) f (m0 , t) dm0 dm. (4)
0
In a mathematical sense, the double integral over the kernel function is nothing else than its expectation value for the
given distribution f (m, t). This is usually notated as hKi(t)
where we have retained the time dependence for clarity. The
prognostic equation for N (t) is therefore
N (t)2
∂N (t)
=−
hKi(t)
∂t
2
(5)
Atmos. Chem. Phys., 13, 9021–9037, 2013
9024
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
A similar formulation was already given by Murakami
(1990), however for the conversion from ice to snow and
without the statistical interpretation of hKi. Assuming that
hKi(t) is a constant during a single time step 1t in the bulk
scheme, there is a formal analytical solution of the form
N(t + 1t) =
N(t)
,
(1/2)N(t)hKi(t ∗ )1t + 1
(6)
where t ∗ is any appropriate time within the time step. The
form of this solution has already been obtained by Smoluchowski (1916, 1917) for Brownian coagulation (see also
Pruppacher and Klett, 1997, Sect. 11.5). It is seen that for
long timesteps such that 1t 1/[N(t)hKi(t ∗ )] the final
N(t + 1t) becomes independent of the initial N(t). With the
new value of N and the (here unchanged) value of qc we can
compute the updated mean mass of f (m, t + 1t), i.e. we can
compute the updated mass distribution.
The prognostic equation for N is the desired result, and the
solution can in principle be computed for arbitrary forms of
the mass distribution and for arbitrary coagulation kernels.
Nevertheless, the necessary integrations are tedious and it
may be justified to construct a look-up table where the required values of hKi(t) can be read off. The computation of
the integrals for special choices of f (m) and the kernel is
demonstrated next.
3
3.1
Computation of the double integrals
Choice of a mass distribution
In principle we could use any probability density function on
R+ for f (m). Following Spichtinger and Gierens (2009a),
we use here a log-normal distribution, i.e.
"
#
1
1 log(m/mm ) 2 1
.
(7)
f (m) = √
exp −
2
log σm
m
2π log σm
Here, log denotes natural logarithm. The normalised f (m)
has two parameters. The first parameter is the modal mass
(or geometric mean) mm , which is updated after every time
step by the prognostic values of number and mass concentration, respectively; the second parameter is the geometric
standard deviation σm , which is usually fixed or formulated
as a function of the mean mass. The mass distribution is
then given by normalisation with number concentration, i.e.
n(m) = N ·f (m). The general kth moment of the mass distribution is denoted by µk [m] and for log-normal distributions
we obtain
Z∞
1
k
k
2
µk [m] := m n(m)dm = N · mm exp
(k log σm ) . (8)
2
0
Number and mass concentration are prognostic variables
in our scheme, represented by the general moments N =
Atmos. Chem. Phys., 13, 9021–9037, 2013
µ0 [m], qc = µ1 [m],
respectively,
with a mean mass of m̄ =
1
2
qc /N = mm exp 2 (log σm ) .
Aggregation would tend to lead to a deviation from the
log-normal distribution towards an exponential one. This effect cannot be taken into account in our model and would require some development, for instance introduction of an ice
class “aggregates” with exponential distribution and development of an aggregation scheme between small ice crystals
(log-normally distributed) and aggregates. All this is future
work.
3.2
Choice of an aggregation kernel
We assume that ice crystals aggregate in particular when
large crystals fall through an ensemble of small crystals,
when they collide and stick together. This particular mechanism is called gravitational collection and can be described
by the following form of a collection kernel (Pruppacher and
Klett, 1997, p. 569)
K (R, r) := π (R + r)2 |v(R) − v(r)|E(R, r).
(9)
Here, R and r are the “radii” (see below) of the larger and
smaller colliding ice crystals, respectively, such that the first
factor on the rhs is the geometric cross section for the collision. The second factor is the absolute difference of the fall
speeds of the two crystals, that is, the speed of their relative
motion. Because of hydrodynamic (and potentially other) effects it is not just the geometric cross section that determines
whether two crystals collide, and even if they collide they
do not need to stick together. Therefore a correction factor
E(R, r) is applied which accounts for these effects. E is usually called collision or collection efficiency. Choices of E
will be presented below.
The next problem to solve is to formulate the collection
kernel for non-spherical ice crystals instead of spheres. Here
we do this for hexagonal cylinders, a common shape for
ice crystals as used for instance in Spichtinger and Gierens
(2009a). For convenience, the size is replaced by the particle mass. This can be done since mass (m) and size (L) are
related (see e.g. Heymsfield and Iaquinta, 2000), usually expressed via power laws (e.g. m = αLβ ). Using these relations, we obtain the following expression for the surface of a
hexagonal ice crystal of mass m
v
u
u 2α β1
β+1
β−1
1
2
1
t
A(m) =
·αβ m β +6· 1
(10)
· m 2β
√
ρb
α β 3 3ρb
where ρb = bulk density of ice = 0.81 × 103 kg m−3 .
Assuming randomly oriented columns (analogous to the
usual approximation for radiation parameterisation in Ebert
and Curry, 1992) we obtain r 2 = A(m)
4π by replacing the ice
crystals surface by the surface of a sphere, such that
1 p
(R + r)2 =
2 A(M)A(m) + A(M) + A(m) .
(11)
4π
www.atmos-chem-phys.net/13/9021/2013/
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
E.Kienast–Sjögren et al.: Ice aggregation in 2-moment schemes
The
terminal
be
v(m)
= γ mδ velocity
· c(T , p).of each of the falling particles can(13)
described using the following power law
The parameters α, β, γ and δ needed to translate K(R, r) into
= γmδ · c(T,p)
(13)
K(M, m) are givenv(m)
in Spichtinger
and Gierens (2009a). Note
that the parameters usually are constants for values of m in a
The
parameters
α, β, Thus,
γ andthis
δ needed
translatesplitting
K(R,r)
certain
mass interval.
leads totoa generic
of
into
arecan
given
in Spichtinger
Gierens
(2009a).
the K(M,m)
integrals, as
be seen
in the nextand
section.
A correction
Note
usuallyisare
constants
fortovalues
of
factorthat
forthe
theparameters
terminal velocity
added
in order
consider
m
in a certain
mass
interval.
Thus, this
leads to adrag.
generic
density
changes,
leading
to different
aerodynamic
The
splitting
of factor
the integrals,
as can be
in the
next section.
and
correction
is represented
as seen
follows
(Heymsfield
AIaquinta,
correction
factor for the terminal velocity is added in or2000)
der to consider
leading to different aerody density
a changes,
p correction
T b factor is represented as follows
namic drag. The
c(T , p) =
,
(14)
(Heymsfield and
p0 Iaquinta,
T0 2000)
hPa,
a T =233
b K, a = −0.178,
with constants p0 = 300
0T
p
c(T,p)
b = −0.394. Since
the= correction factor, depends only(14)
on
p0
T0
temperature and pressure, respectively, it can be treated as
a constant
for the
integrals.
with
constants
p0calculations
= 300hPa,ofTthe
0 = 233K, a = −0.178,
b = −0.394. Since the correction factor depends only on
3.3 Computation of the integrals
temperature and pressure, respectively, it can be treated as
aThe
constant
forvalues
the calculations
the) were
integrals.
integral
(hKi in m3ofs−1
calculated using an
adaptive Simpson quadrature (e.g. Lyness, 1969) for different
3.3 Computation of the integrals
values of the geometrical standard deviation σm . The calculated integral results for σ = 2.85 are indicated as squares in
The integral values (hKi inm m3 s−1 ) were calculated using an
Fig. 1. The calculation was done for T = 233 K and 300 hPa,
adaptive Simpson quadrature (e.g. Lyness, 1969) for different
i.e. c(T , p) = 1 in Eq. (14). For other choices of atmospheric
values of the geometrical standard deviation σm . The calcuparameters the values change moderately. For reasons given
lated integral results for σm = 2.85 are indicated as squares
below we compute the integrals with a collision efficiency of
in Figure 1. The calculation was done for T = 233K and
unity. The integration was conducted up to a modal mass of
300hPa,
i.e. c(T,p) = 1 in eq. (14). For other choices of
106 ng = 10−6 kg (equivalent to a length ∼ 8 mm) as this is
atmospheric parameters the values change moderately. For
the upper limit for an aggregated particle, which will be used
reasons given below we compute the integrals with a colliin the later parameterisation. The calculated integral values
sion efficiency of unity. The integration was conducted up
were divided into four ranges because of mass dependent coto a modal mass of 106 ng = 10−6 kg (equivalent to a length
efficients α, β, γ and δ. For each range a polynomial was fit∼ 8mm) as this is the upper limit for an aggregated partited through the calculated values. The combination of these
cle, which will be used in the later parameterization. The
polynomials is displayed as solid line in Fig. 1. The four
calculated integral values were divided into four ranges bepolynomial ranges correspond to changes in the growth and
cause of mass dependent coefficients α, β, γ and δ. For each
sedimentation behaviour of ice crystals (e.g. changes of the
range a polynomial was fitted through the calculated values.
parameters α, . . . , δ) as indicated in Spichtinger and Gierens
The combination of these polynomials is displayed as solid
(2009a):
line in Figure 1. The four polynomial ranges correspond to
−4 ng
changes
growth
sedimentation
behaviour
of ice
1. 1 ×in10the
< mmand
≤ 2.5
× 10−3 ng: mainly
hexagonal
crystals
changes
the parameters
ice(e.g.
crystals
with of
aspect
ratio 1. α,...,δ) as indicated
in Spichtinger−3
and Gierens (2009a)
2. 2.5 × 10 ng < mm ≤ 4 × 102 ng: columns with aspect
ratio−4
larger
1. 1·10
ng <than
mm1.
≤ 2.5·10−3 ng: Mainly hexagonal ice
crystals with aspect ratio 1.
www.atmos-chem-phys.net/13/9021/2013/
0.0001
1e-06
aggregation kernel <K> (m3 s-1)
We replace the radii (R, r) with the corresponding masses
We
radii (R,r) with the corresponding masses
(M,replace
m) andthe
obtain
(M,m) and obtain
1 p
2 A(M)A(m) + A(M) + A(m) ·
K(M, m) =
K(M,m) = 4
p− v(m)|E(M, m).
(12)
1|v(M)
2 A(M )A(m) + A(M ) + A(m) ·
The terminal4velocity of each of the falling particles can be
|v(M
v(m)|E(M,m).
(12)
described using
the) −
following
power law
9025
5
numeric integral value
polynomial interpolation
1e-08
1e-10
1e-12
1e-14
1e-16
1e-18
1e-20
1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06
modal mass (kg)
3 −1 as a function of the modal
Fig. 1.
1. hKi
hKi integral
Fig.
integral values
values (in
(in m
m3 ss−1 )) as
a function of the modal
mass
(in
kg)
of
the
crystal
mass
distribution and
and for
for σσm
= 2.85.
2.85.
m=
mass (in kg) of the crystal mass distribution
The
corresponding
sizes
range
between
a
modal
length
of
0.6
and
The corresponding sizes range between a modal length of 0.6 and
8000 µm.
µm. Exact
shown as
as
8000
Exact values
values from
from aa numerical
numerical integration
integration are
are shown
squares. The
The solid
solid lines
lines represent
represent polynomial
polynomial fits
fits in
in 44 different
different mass
mass
squares.
ranges, indicated
indicated by
by the
the black
black lines.
lines.
ranges,
2 ng < m ≤ 1×104 ng:
3. 2.5
4×10
theColumns
sedimentation
veloc2.
· 10−3
ng <mmm ≤ 4 · 102 ng:
with aspect
ity
gets
larger.
ratio larger than 1.
2 4 ng < mm : even 4larger columns.
4. 41·×
3.
1010
ng < mm ≤ 1 · 10 ng: The sedimentation velocity gets larger.
As can be seen in Fig. 1, the polynomial fits to the exact
4
integral
values
good; the maximum deviation is less
4. 1 · 10
ng <are
mvery
m : Even larger columns.
than 10 % and the mean deviation even less than 2 %. Thus
As
be seen incan
Figure
1, theaspolynomial
to the while
exact
thecan
polynomials
be used
an accuratefits
solution
integral
values are time.
very good; the maximum deviation is less
saving computing
than
10%fits
andwere
the calculated
mean deviation
even less
than
2%. Thus
These
for a whole
range
of log-normal
the
used as an
accurate
solution in
while
sizepolynomials
distributionscan
withbegeometric
standard
deviations
the
saving
time. The coefficients for these fits are
range computing
σm = 1.90–5.29.
These
calculated
a whole
lognormal
given
in fits
thewere
appendix,
TableforA1.
In Fig.range
2 theofkernels
for
size
with geometric
standard
deviations
in disthe
thesedistributions
vales are displayed
against modal
mass
mm of the
tribution
wellcoefficients
as against mean
massfits
of are
the
range
σm (upper
= 1.90panel)
− 5.29.as The
for these
2 ) (lower
distribution
= mm exp(0.5
· (log
)
panel).
As
given
in the m̄
appendix,
table 2.
Inσfigure
2
the
kernels
for
m
expected
theory,
the kernels
larger
form
am
wider
distriof the
disthese
valesbyare
displayed
againstare
modal
mass
bution; this
behaviour
be seen
clearly mean
in the mass
upperof
panel
tribution
(upper
panel)can
as well
as against
the
2
of Fig. 2. Inm̄fact,
kernels
for
a
fixed
modal
mass
vary
distribution
= mthe
exp(0.5
·
(logσ
)
)
(lower
panel).
As
m
m
over morebythan
one order
of magnitude
forfor
different
geometexpected
theory,
the kernels
are larger
a wider
distriric standard
deviations.
This
indicates
thatinthe
the
bution;
this behaviour
can
be seen
clearly
the width
upper of
panel
size
verykernels
important
the strength
of thevary
agof
fig.distribution
2. In fact,is the
for for
a fixed
modal mass
gregation
over
more process.
than one order of magnitude for different geometric standard deviations. This indicates that the width of the
3.4 distribution
Comparison
withimportant
other model
parameterisations
size
is very
for the
strength of the aggregation process.
For testing our new parameterisation, we first compare
the derived
kernelswith
withother
already
existing
parameterisations.
3.4
Comparison
model
parameterisations
There were some former attempts in deriving aggregation
parameterisations
for parameterisation,
hydrometeors, especially
and
For
testing our new
we firstsnow
compare
ice
crystals.
Passarelli
(1978)
derived
an
analytical
kernel
the derived kernels with already existing parameterisations.
for aggregating
crystals,
leading
to an unhandy
expresThere
were someice
former
attempts
in deriving
aggregation
pasion
of
hypergeometric
functions.
However,
he
assumed
an
rameterisations for hydrometeors, especially snow and ice
crystals. Passarelli (1978) derived an analytical kernel for
Atmos. Chem. Phys., 13, 9021–9037, 2013
9026
6
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
E.Kienast–Sjögren et al.: Ice aggregation in 2-moment schemes
0.0001
aggregation kernel <K> (m3 s-1)
1e-06
1e-08
1e-10
1e-12
σm=1.90
σm=2.23
σm=2.85
σm=3.25
σm=3.81
σm=4.23
σm=5.29
1e-14
1e-16
1e-18
4πρbs
1e-20
1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06
modal mass (kg)
0.0001
aggregation kernel <K> (m3 s-1)
1e-06
1e-08
1e-10
1e-12
1e-14
scribed parameterisations (e.g., Morrison et al., 2005, using
scribed
parameterisations
(e.g.,maybe
Morrison
al., 2005, using
Passarelli’s
parameterisation),
withetmodifications.
Passarelli’s
parameterisation),
maybe
with
modifications.
For the low temperature regime (T < −40 ◦ C) only one
For
low temperature
(T <(2012)
−40C◦is) only
one
Schumann
available.
recent the
parameterisation
by regime
recent
parameterisation
by
Schumann
(2012)
is
available.
Schumann (2012) estimated the aggregation kernel (his
Schumann
(2012) estimated
the aggregation
kernel (his equaEq. 52, translated
into our formulation)
to be
tion 52, translated into our formulation) to be
K = E · 16 · π r̄ 2 v(r̄)
(15)
K = E · 16 · πr̄2 v(r̄)
(15)
3m̄ 1331
m̄
usingthe
thevolume
volumemean
meanradius
radiusr̄r̄== 34πρ
withaabulk
bulkice
ice
using
with
bs
σm=1.90
σm=2.85
σm=3.81
σm=5.29
Schumann (2012)
1e-16
1e-18
1e-20
1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06
mean mass (kg)
Fig. 2.
kernels
forfor
different
geometFig.
2. Fits
Fitsononcalculated
calculatedaggregation
aggregation
kernels
different
georic
standard
deviations.
Upper
panel:
aggregation
kernels
vs.
modal
metric standard deviations. Upper panel: Aggregation kernels
vs.
mass. The
change
due to the
the distribution
can be seen
modal
mass.
The change
duewidth
to theofwidth
of the distribution
can
clearly.
kernels vs.
mean vs.
mass.
Since
the
be
seen Lower
clearly.panel:
Loweraggregation
panel: Aggregation
kernels
mean
mass.
mean
mass
depends
on
the
geometric
width
of
the
distribution,
the
Since the mean mass depends on the geometric width of the distrivariation
the kernels
to different
width iswidth
smaller.
Addibution,
theofvariation
of thedue
kernels
due to different
is smaller.
Schumann
(2012)
tionally, the aggregation
kernel
as derived
by by
Additionally,
the aggregation
kernel
as derived
Schumann
(2012)is
shown
between aa
is
shownforforcomparison.
comparison.The
Thecorresponding
corresponding sizes
sizes range
range between
modal
length
of
0.6
and
8000
µm.
modal length of 0.6 and 8000 µm.
aggregating ice crystals, leading to an unhandy expression
of hypergeometric functions. However, he assumed an exexponential size distribution, because he was interested in
ponential size distribution, because he was interested in agaggregation of snow, i.e. aggregation at warm temperatures.
gregation of snow, i.e. aggregation at warm temperatures.
The assumption of exponential distributions simplified the
The assumption of exponential distributions simplified the
expressions. This procedure is not viable in our case beexpressions. This procedure is not viable in our case because we use log-normal distributions for ice crystals (as
cause we use lognormal distributions for ice crystals (as jusjustified in Spichtinger and Gierens, 2009a). Mitchell (1988)
tified in Spichtinger and Gierens, 2009a). Mitchell (1988)
derived an aggregation kernel in a similar way as Passarelli
derived an aggregation kernel in a similar way as Passarelli
(1978), using exponential distributions and, similar to our
(1978), using exponential distributions and, similar to our
treatment, power laws for terminal velocities. Again, the agtreatment, power laws for terminal velocities. Again, the
gregation parameterisation was derived for the warm temaggregation parameterisation was derived for the warm temperature range, with the special assumption of an exponenperature range, with the special assumption of an exponential distribution. Ferrier (1994) used an approach similarly to
tial distribution. Ferrier (1994) used an approach similarly to
ours, using gamma size distribution. He evaluated the douours, using gamma size distribution. He evaluated the double integrals numerically in order to create a look-up table.
ble integrals numerically in order to create a look-up table.
However, again this parameterisation was made for the warm
However, again this parameterisation was made for the warm
temperature regime.
regime. Many
models rely
rely on
on these
these dedetemperature
Many other
other models
Atmos. Chem. Phys., 13, 9021–9037, 2013
−3
density of
of917
917kg
kgmm−3
For comparison
comparison we
we used
usedour
ourtermitermidensity
. . For
nal
velocity
formulation,
the
volume
mean
radius
was
denal velocity formulation, the volume mean radius was derived from
from the
the mean
mean mass
mass ofofan
anice
icecrystal;
crystal; the
theefficiency
efficiency
rived
(lower panel)
panel) some
some kernels
kernels
set toto be
be EE≡≡1.1. In
isis set
In Fig.
fig. 22 (lower
for our
ourformulation
formulation(σ
(σmm==1.9/2.85/5.29,
1.9/2.85/5.29,representing
representingnarnarfor
row/medium/broadsize
sizedistributions)
distributions)are
areshown
showninincomparcomparrow/medium/broad
ison with
with the
thekernel
kernelasasderived
derivedbybySchumann
Schumann(2012).
(2012).The
The
ison
kernelsare
areplotted
plottedagainst
againstthe
themean
meanmass
massm̄.
m̄.At
Atleast
leastininthe
the
kernels
−14 ≤ m̄ ≤ 10−9
−9 kg, the qualitative agreement
massrange
range10
10−14
mass
≤ m̄ ≤ 10 kg,
the qualitative agreement
quite good,
good, although
although the
the kernel
kernel by
by Schumann
Schumann(2012)
(2012)isis
isis quite
about55times
timeshigher
higherthan
thanour
ourparameterisations.
parameterisations.InInthe
themass
mass
about
−14 kg and m̄ ≥ 10−9
−9 kg there is a larger overrangesm̄
m̄≤
≤10
10−14
ranges
kg and m̄ ≥ 10 kg
there is a larger overestimation compared
compared to our parameterisations.
estimation
parameterisations. These
Theseoveresovertimations are
larger masses
massesdo
donot
not
estimations
arenot
notcrucial,
crucial, since
since (1) the larger
applyfor
forthe
theparameterisation
parameterisationby
bySchumann
Schumann(2012)
(2012)which
whichisis
apply
used for
for contrails,
contrails, where
where small
small toto moderate
moderatecrystal
crystalmasses
masses
used
prevail, and
and (2)
(2) for
for the
the very
very small
small masses
masses the
theaggregation
aggregation
prevail,
timescales
much larger
larger than
than cloud
cloudand
andcontrail
contraillifetimes
lifetimes
time
scales are much
(seebelow).
below).
(see
3.5 Time
Timescale
3.5
scale analysis
In
Inorder
orderto
toestimate
estimatethe
thepossible
possibleimpact
impactofofaggregation
aggregationon
onice
ice
crystal
estimate the
thetimescales
time scalesof
crystal number
number concentrations, we estimate
of
aggregation
aggregation
1 2
∂N ! N
2
1
2=
− =N∂N
hKi=! =N ⇔ =
⇔ −τ
(16)
− N 2 hKi
−τ
=
(16)
2 ∂t
τ N · hKi N · hKi
2
τ ∂t
In fig. 3 the timescales of aggregation are displayed for a typIn Fig. 3 the timescales of aggregation are displayed for
ical size distribution with geometric standard deviation value
a typical size distribution with geometric standard deviaof σm = 2.85 and for typical ice crystal number concentration value of σm = 2.85 and for typical ice crystal numtions in cirrus clouds
(see, e.g., Krämer et al., 2009) in the
et al.,
ber concentrations in cirrus clouds (see, e.g., Krämer
range between N = 104 m−3 = 10L4−1 −3
and N = 107 m−3 =
2009)−3
in the range between N = 10 m = 10 L−1 and N =
10 cm
, respectively. Since aggregation is a pure sink for
107 m−3 = 10 cm−3 , respectively. Since aggregation is a
ice crystal number concentrations, the time scale is negative;
pure sink for ice crystal number concentrations, the timescale
however, in fig. 3 we show absolute values of τ for a better
is negative; however, in Fig. 3 we show absolute values of τ
representation.
for a better representation.
It is evident, that only in a very narrow range set by paramIt is evident that only in a very narrow range set by parameters number concentration and ice crystal size, respectively,
eter number concentration and ice crystal size, aggregation
aggregation might play a role. Since the life time of cirrus
might play a role. Since the life time of cirrus clouds might
clouds might be in the order of one day (e.g. Spichtinger et
be in the order of one day (e.g. Spichtinger et al., 2005), this
al., 2005), this time interval might serve as an upper limit for
time interval might serve as an upper limit for the impact
the impact of aggregation on cirrus clouds. Since in clouds
of aggregation on cirrus clouds. N
Since10in4 m
clouds
with only a
−3
with only a few ice crystals (e.g.
) the ice crys4 m−3 )=the
few
ice
crystals
(e.g.
N
=
10
ice
crystals
can grow
tals can growth to large sizes - at least at high temperatures
www.atmos-chem-phys.net/13/9021/2013/
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
E.Kienast–Sjögren et al.: Ice aggregation in 2-moment schemes
4.1 Test of aggregation only
4.1 Test of aggregation only
1e+18
aggregation time scale |τ| (s)
1e+16
N=1045 m-3
N=106 m-3
N=107 m-3
N=10 m-3
1e+14
1e+12
9027
7
4.1.1 Maximum
Maximum
aggregation
4.1.1
aggregation
σm=2.85
1e+10
1e+08
1e+06
1 day
10000
100
1 hour
1 min
1
0.01
1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06
mean mass (kg)
Fig.
medium width
widthofofthe
theice
ice
Fig. 3. Timescales
Time scalesof
of aggregation
aggregation for a medium
crystal
distribution (σ
(σm
= 2.85)
2.85)and
andfor
fortypical
typicalice
icecrystal
crystalnumnumcrystal size distribution
m=
ber
as found
found in
in cirrus
cirrus clouds
cloudsatatcold
coldtemperatures
temperatures
ber concentrations as
(see,
et al.,
al., 2009).
2009). The
The corresponding
correspondingsizes
sizesrange
rangebebe(see, e.g., Krämer et
tween
length of
of 0.6
0.6 and
and 8000
8000µm.
µm.
tween a modal length
to
large
sizes can
– atbeleast
at high influenced
temperatures
this regime
- this
regime
effectively
by –aggregation.
can
be clouds
effectively
influenced
Cirrus clouds
Cirrus
containing
manyby
iceaggregation.
crystals are usually
domcontaining
manycrystals.
ice crystals
arealthough
usually dominated
by small
inated by small
Thus,
ice crystal number
crystals.
Thus,isalthough
ice crystal
number concentration
concentration
able to reduce
the aggregation
time scalesis
able
to reduce
the aggregation
timescales
orderssize
of magby orders
of magnitudes,
an upper
limit inbycrystal
due
to thermodynamic
constraints
leads
to less
aggreganitudes,
an upper limit
in crystal
size
due effective
to thermodynamic
tion. We will
see later,
that
this veryaggregation.
simple estimation
from
constraints
leads
to less
effective
We will
see
time that
scalethis
analysis
will beestimation
coroborrated
by timescale
detailed tests.
later
very simple
from
analysis
will be corroborated by detailed tests.
4 Various tests
4 Various tests
The change in particle number density per timestep is described
in (5)inasparticle number density per time step is deThe
change
scribed in Eq. (5) as
∂N (t)
N (t)2
=−
hKi(t)
2
∂N(t)
N (t) ∂t
2
=−
hKi(t).
∂t solution of2 this can either be achieved through an exact
The
solution
involving
variables
with
the following
The
solution
of thisseparation
can eitherofbe
achieved
through
an exact
result involving separation of variables with the following
solution
result
N (t)
N (t + ∆t) = N(t)
(17)
(1/2)N (t)hKi(t∗ )∆t + 1
(17)
N(t + 1t) =
(1/2)N(t)hKi(t ∗ )1t + 1
or through the following Euler approximation
or through the following Euler approximation
∂N (t)
N (t + ∆t) = N (t) + ∂N(t) · ∆t
(18)
N(t + 1t) = N (t) + ∂t · 1t
(18)
∂t
N (t)22
(19)
= N (t) − N (t) hKi(t) · ∆t
= N (t) − 2 hKi(t) · 1t.
(19)
2
Tests have shown that both methods give practically identiTests
have shown
that both
methods
giveperformed
practically
identical results.
The following
tests
have been
with
the
cal
results.
The
following
tests
have
been
performed
with
the
Euler approximation. Further tests have shown that the polyEuler
approximation.
Further
tests
have
shown
that
the
polynomial approximation of the hKi integrals was a sufficient
nomial
approximation
of the
integrals tests
was ahave
sufficient
approximation
(see above),
so hKi
the following
been
approximation
(see
above),
so
the
following
tests
have
been
performed using the polynomial approximation.
performed using the polynomial approximation.
www.atmos-chem-phys.net/13/9021/2013/
Thenew
newformulation
formulationofofthethe
aggregation
process
tested
The
aggregation
process
waswas
tested
ininMATLAB
start
values
forfor
thethe
particle
numMATLABwith
withdifferent
different
start
values
particle
num4
4 to
4 particles
ber
10410to
5 · 10
particles
per cubic
berdensity
densityranging
rangingfrom
from
5×
10
per cumetre.
We let
occur as
the as
only
We
bic metre.
Weaggregation
let aggregation
occur
theprocess.
only process.
set
E
≡
1
for
these
tests,
that
is,
the
following
results
show
We set E ≡ 1 for these tests, that is, the following results
a show
maximum
effect ofeffect
aggregation.
The aggregation
was runwas
a maximum
of aggregation.
The aggregation
for
1000
s,
i.e.
approximately
17
minutes.
If
aggregation
run for 1000 s, i.e. approximately 17 min. If aggregation ococcurs,
the particle
particle number
andand
thethe
curs, the
number density
densityNNwill
willdecrease
decrease
mean
mass
(
m̄
=
q
/N
)
increase.
Using
the
expressions
forfor
c
mean mass (m̄ = qc /N) increase. Using the expressions
the
compute
then
a new
modal
thelog-normal
log-normaldistribution
distributionwewe
compute
then
a new
modal
mass
f (m)
is is
used
in in
thethe
next
massand
andthe
thecorresponding
correspondingnew
new
f (m)
used
next
timestep.
For
thethe
calculations,
wewe
setset
an an
upper
boundary
forfor
time step.
For
calculations,
upper
boundary
−6
the
themass
massofofthe
theaggregated
aggregatedparticles
particlesatat1010−6kg,
kg,which
whichcorcorreresponds
to aa particle
particle size
size of
of 88mm.
mm.Larger
Largerice
icecrystals
crystalswill
willnot
sponds to
not
occur
model.
occur
in in
thethe
model.
Figure
4
shows
results
of of
these
tests.
ForFor
small
initial
results
these
tests.
small
initial
Figure 4 showsthethe
−11
modal
masses
(e.g.
10
kg,
green
line)
and
starting
with,
modal masses5(e.g. 10−11 kg,
green
line)
and
starting
with,
for example, 10 particles
m−3−3
, after simulating 1000 s we
5 particles
for example,
10
m
,
after
simulating
1000
s we
still have 105 particles
m−3 . Thus almost no aggregation ocstill have 105 particles m−3 . Thus almost no aggregation
occurs. If the initial modal mass is instead increased to 10−9 −9
kg
curs. If the initial modal mass is instead
increased
to
10
kg
−3
(blue line) while still starting with 105 m
particles, we
(blue line) while 4still−3
starting with 105 m−3 particles, we
only have about 10 m
particles left after 1000 s. Thus
only have about 104 m−3 particles left after 1000 s. Thus
about 90% of the particles have aggregated.
about 90 % of the particles have aggregated.
As expected, when small crystals (i.e. small modal
As expected, when small crystals (i.e. small modal
masses) predominate, nothing happens. The probability for
masses)
nothing
happens.
probability
collision ispredominate,
negligible and
the relative
fallThe
speeds
are low.for
collision
is
negligible
and
the
relative
fall
speeds
The larger the particles get, the more they aggregate.areForlow.
The
largerinitial
the particles
get, the(i.e.
more10they
−9 aggregate. For the
the
largest
modal masses
kg) the particles
−9 kg) the particles stick
largest
initial
modal
masses
(i.e.
10
stick together very fast so that the iteration has to be stopped
together
very fast
before
reaching
1000sos.that the iteration has to be stopped before
reaching
1000
Timesteps from 0.1s.s (timescale for microphysics) to 10 s
Timesteps
from 0.1 sgave
(timescale
for microphysics)
(timescale
for dynamics)
very similar
results, that is,to
the10 s
(timescale
for
dynamics)
gave
very
similar
results,
that
is, the
parameterization is consistent and convergent with different
parameterisation
is
consistent
and
convergent
with
different
timesteps. We chose to use a timestep of 1 s.
timesteps. We chose to use a time step of 1 s.
4.1.2 Introducing temperature dependency
4.1.2 Introducing temperature dependency
Experience from field measurements suggests that aggregation occurs more efficiently in warmer than in colder air (e.g.
Experience from field measurements suggests that aggregaKajikawa and Heymsfield, 1989). This behaviour is also contion occurs more efficiently in warmer than in colder air (e.g.
sistent with the possible existence of a QLL on top of ice
Kajikawa
and Heymsfield, 1989). This behaviour is also concrystals, even at low temperatures. The temperature depensistent
with
the possible
existenceparameterization
of a QLL on top
dence is expressed
by the following
for of
theice
crystals,
even
at
low
temperatures.
The
temperature
depencollision efficiency of ice crystals (Lin et al., 1983; Levkov
is expressed
the following
parameterisation
etdence
al., 1992)
which isby
independent
of the
crystal masses for
andthe
collision
efficiency
of
ice
crystals
(Lin
et
al.,
1983;
Levkov
dependent only on temperature (the original papers do not
et al., 1992)
which
is independent ofisthe
crystal
masses and
mention
whether
the parameterisation
based
on measuredependent
only
on
temperature
(the
original
papers
do not
ments)
mention whether
the
parameterisation
is
based
on
measureE(T ) = exp(0.025 · (T − 273.16))
(20)
ments)
With this parameterization, E can be taken out from the integral calculation and treated as a prefactor. Note here, that
E(T ) = exp(0.025 · (T − 273.16)).
(20)
Atmos. Chem. Phys., 13, 9021–9037, 2013
89028
8
E. Kienast-Sjögren
et al.:
Ice
aggregation
inin
two-moment
schemes
E.Kienast–Sjögren
al.:Ice
Iceaggregation
aggregation
in2-moment
2-momentschemes
schemes
E.Kienast–Sjögren
etetal.:
−9
Number
Number density
density after
after 1000 s
6 6
1010
66
10
10
−13
mm
=10
=10−13kg
kg
mm
T=210KK
T=210
T=220KK
T=220
T=230
T=230KK
T=240
T=240KK
No
Notemperature
temperaturedependence
dependence
−11
−11
mm
=10
=10
mm
kg
kg
−10
=10−10
kg
mm
=10
kg
mm
5
=10−9−9kg
kg
mm
=10
mm
5
5
10
10
−3
[m−3
NN [m
]]
N [m−3
]
N [m−3]
5
1010
−9
Numberdensity
densityafter
after1000
1000s,s,mm=10
=10
Number
kgkg
mm
4
4
4
1010
4
10
10
3
3
10
4
10 10
4
10
3
5
105
10 −3
N [m ]
N 0 [m−3]
6
10 6
10
0
10 3 4
10
10 4
10
5
10 5
10−3
N0 [m −3
]
N [m ]
6
10
6
10
0
Fig.
the start
start of
ofthe
theiteration
iteration(x(x
Fig. 4.
4. Particle
Particle number
number density N at the
Fig. 4. Particle number density N000 at the start of the iteration (xAxis)and
and after
after 1000
1000 ss of
of iteration (N
axis)
(N,, y-Axis).
y axis). For small
small modal
modal
Axis) and after 1000 s of iteration (N , y-Axis). For small modal
masses, the particle number
thethe
itermasses,
numberdensity
densityhardly
hardlychanges
changesduring
during
itmasses, the particle number density hardly changes during the iteration and
eration
andthe
theplot
plotisisalmost
almostaastraight
straightline.
line.As
Asreference
referencefor
forno
noagagation and the plot is almost a straight line. As reference for no aggregation, aa black
black line is plotted. As expected,
gregation,
expected, larger
larger modal
modalmasses
masses
gregation, a black line is plotted. As expected, larger modal masses
result in
in increased
increased aggregation.
result
aggregation. Thus the particle number
number density
density
result in increased aggregation. Thus the particle number density
decreases during
during integration, which is shown
decreases
shown in
in the
the graph.
graph. The
Thecorcordecreases
during
integration,
which is shown in
the graph. The
corresponding size
size to
to the modal masses plotted
responding
plotted ranges
ranges between
between 66 and
and
responding
size that
to the
masses
plotted rangeswith
between
6 and
345µm.
µm.Note
Note
themodal
tests have
have
been
345
that the
tests
been
performed with E
E≡
≡1,
1, that
that
345
µm.maximum
Note thateffect
the tests
have been performed
with Ered
≡ 1, that
is, the
the
of aggregation
is,
maximum effect
of
aggregation is
is seen
seen here.
here. The
The red line
lineisis
is,almost
the maximum
effect
of
aggregation
is
seen
here.
The
red
line is
equal to
to the
the black
black line,
line, thus
almost equal
thus can
can hardly
hardly be
be seen
seen in
in the
theplot.
plot.
almost equal to the black line, thus can hardly be seen in the plot.
Fig. 5. Particle number density N0 at the beginning of the iteraFig.
Particlenumber
numberdensity
densityNN
beginning
iteraFig. 5. Particle
thethe
beginning
of of
thethe
iteration
0 0at at
tion (x-Axis) and after 1000 s of iteration
(N, y-Axis) for a modal
tion
(x-Axis)
and
after
1000
s
of
iteration
(N,
y-Axis)
for
a
modal
(x axis)
and
after
1000
s
of
iteration
(N
,
y
axis)
for
a
modal
mass
of
mass
of 10−9
−9kg, which corresponds to a particle with modal length
mass
kg,corresponds
which corresponds
to awith
particle
with
modal
length
10−9 of
kg,10which
to a particle
modal
length
345
µm.
345 µm. As a reference for no aggregation, a black line is plotted.
345
As a reference
for no aggregation,
a black
line isAdding
plotted.
As aµm.
reference
for no aggregation,
a black line
is plotted.
a
Adding a temperature dependency shows a weakened aggregation
Adding
a temperature
dependency
shows a aggregation
weakened aggregation
temperature
dependency
shows
a
weakened
effect
with
effect with decreasing temperatures. The lower the tempertures the
effect
with decreasing
temperatures.
Thetemperatures
lower the tempertures the
decreasing
temperatures.
The
more
particles
are present at
thelower
end ofthe
the simulation. the more parmore
areatpresent
of the simulation.
ticles particles
are present
the endatofthe
theend
simulation.
With this parameterisation, E can be taken out from the inteexperimental evidence of the exact form of the temperature
gral
calculation and treated
asexact
a prefactor.
here
that exexperimental
of the
formthis
ofNote
thethe
temperature
dependence isevidence
not given.
Nevertheless,
is
only temperimental
evidence
of
the
exact
form
of
the
temperature
dedependence
is not given.
this is thewhich
only temperature dependence
we Nevertheless,
found from literature,
also
pendence
is
not
given.
Nevertheless,
this
is
the
only
temperaperature
we found from literature, which also
seems todependence
be reasonable.
ture dependence
we found from literature, which also seems
seems
to
be
reasonable.
The temperature dependent collision efficiency is 0.21 ≤
toThe
be reasonable. dependent collision efficiency is 0.21 ≤
E(T ) temperature
≤ 0.44 for typical temperatures in ice clouds (210 ≤
The≤temperature
dependent
collisioninefficiency
is (210
0.21 ≤
≤
E(T
0.44 Thus
for typical
temperatures
ice clouds
T ≤)240K).
we expect
reduced aggregation
effects
on
E(T
)
≤
0.44
for
typical
temperatures
in
ice
clouds
(210
≤
TN≤ compared
240K). Thus
we previous
expect reduced
aggregation
effectsthe
on
to the
tests when
we introduce
T
≤
240
K).
Thus
we
expect
reduced
aggregation
effects
on
Nnew
compared
to the 5previous
tests
when
weconcentrations
introduce the
factor. Figure
shows the
final
crystal
N
compared
to
the5 previous
when
weconcentrations
introduce
the
new
factor. for
Figure
showswithout
thetests
final
crystal
as before
aggregation
temperature
dependency
new
factor.
Figure
5
shows
the
final
crystal
concentrations
as(Ebefore
for for
aggregation
temperature
dependency
= 1) and
different without
temperatures.
As expected,
with
as
before
for
aggregation
without temperature
decreasing
temperature
becomes
lessdependency
important.
(E
=
1) and
for
differentaggregation
temperatures.
As expected,
with
(E
= 1)
and
for different
temperatures.
As (240
expected,
Even
at the
highest
considered
temperature
K) thewith
redecreasing
temperature
aggregation
becomes
less
important.
decreasing
temperature
aggregation
becomes(240
less
important.
duction
of the
aggregation
effecttemperature
is considerable,
inK)
particular
Even
at the
highest
considered
the reEven
atofthe
considered
temperature
K)
the
refor high
initial
number
densities.
This
finding (240
is aingood
arguduction
thehighest
aggregation
effect is
considerable,
particular
duction
of
the
aggregation
effect
is
considerable,
in
particular
ment
ignoring
aggregation
simulations
for
highfor
initial
number
densities.inThis
finding isofa cold
goodcirrus
argufor
high
initial
number
densities.
finding
isofa cold
good
arguclouds
not
only
E
is small
but
also
the median
crystal
ment
forwhere
ignoring
aggregation
inThis
simulations
cirrus
ment
for
ignoring
aggregation
in
simulations
of
cold
cirrus
masseswhere
are smaller
than
clouds
not only
E in
is warm
small cirrus.
but also the median crystal
clouds where not only E is small but also the median crystal
masses
are smaller than in warm cirrus.
masses
are of
smaller
than in within
warm cirrus.
4.2 Test
aggregation
a box model
4.2 Test of aggregation within a box model
4.2
Test of aggregation
within
a box modelidealized box
For investigating
the impact
of aggregation
model
simulationsthewere
carried
out. Here, we
use a maxFor
investigating
impact
of
idealized
box
For
investigating
the
impact
ofasaggregation
aggregation
idealised
box
imum
efficiency
E
≡
1
as
well
a
temperature
dependent
model
simulations
were
carried
out.
Here,
we
use
a
maxmodel simulations were carried out. Here, we use a maxiimum efficiency E ≡ 1 as well as a temperature dependent
4.2.1 Model description and setup
4.2.1 Model
Model description
description and
and setup
set-up
4.2.1
Atmos. Chem. Phys., 13, 9021–9037, 2013
mum efficiency
≡ 1 to
as see
wella realistic
as a temperature
dependent
efficiency
E(T ) inEorder
impact of aggregaefficiency
E(T
)
in
order
to
see
a
realistic
impact
of
aggregation on cirrus clouds.
tion on cirrus clouds.
In
In this
thissection
sectionwe
wetest
testthe
theeffect
effectofofaggregation
aggregationononthetheiceice
In this number
section we
test the effect
of aggregation
on the ice
crystal
concentration
in
the
crystal number concentration in theframework
frameworkofofa abox
box
crystal
number
concentration
in
the
framework
of a box
model
model (Spichtinger
(Spichtinger and
and Gierens,
Gierens,2009a)
2009a)which
whichweweconconmodel
(Spichtinger
and inGierens,
2009a)
which weproconsider
realistic
test
microphysical
sidera amore
more
realistic
testthat
in various
that various
microphysical
sider acan
more
realistic
test in thatThe
various
microphysical
processes
act
simultaneously.
box
is
thought
to
repprocesses can act simultaneously. The box is thought to
cessesan
can
act simultaneously.
The box
is thought
to
represent
initially
cloud
free
air
parcel
which
is
lifted
with
represent an initially cloud free air parcel which is lifted
an initially
free air
parcelthe
which
is lifted
with
aresent
constant
vertical cloud
velocity.
During
cooling
procewith a constant vertical velocity. During the cooling proa constant
vertical freezing
velocity.of During
cooling
procedure,
homogeneous
aqueous the
solution
droplets
cedure, homogeneous freezing of aqueous solution droplets
dure, “homogeneous
homogeneous nucleation”,
freezing of parameterized
aqueous solution
(short
afterdroplets
Koop
(short “homogeneous nucleation”, parameterised after Koop
(short
“homogeneous
nucleation”,
parameterized
after Koop
et
al., 2000)
will occur,
i.e. ice crystals
are formed.
In
et al., 2000) will occur, i.e. ice crystals are formed. In
the
supersaturated
environment
thecrystals
ice crystals
grow byIn
et al.,
2000) will occur,
i.e. ice
are formed.
the supersaturated environment the ice crystals grow by
diffusional
growth (based
on approximations
by Koenig,
the supersaturated
environment
the ice crystals
grow by
diffusional growth (based on approximations by Koenig,
1971)
to larger
sizes.(based
The parameterisations
for by
bothKoenig,
prodiffusional
growth
on approximations
1971) to larger sizes. The parameterisations for both processes
described
detail
in Spichtinger and
1971) are
to larger
sizes.in The
parameterisations
for Gierens
both processes are described in detail in Spichtinger and Gierens
(2009a).
determining
the impact
of aggregation
for
cesses areFor
described
in detail
in Spichtinger
and Gierens
(2009a). For determining
the impact
of aggregation
for difdifferent
and velocity
regimes,
idealizedfor
(2009a). temperature
For determining
the impact
ofmany
aggregation
ferent
temperature
velocity
many idealised
box
simulations
wereand
carried
out. regimes,
Each simulation
starts
different
temperature
and
velocity
regimes,
many idealized
box
simulations
were
carried
out.
Each
simulation
at
p=
300hPa. The
temperature
given by Tstarts
=
box
simulations
wereinitial
carried
out. Eachis simulation
starts
at p = 300 hPa. The initial
temperature
is given
by by
T =
210/220/230/240K,
vertical
velocity range
is given
at p = 300hPa. Thetheinitial
temperature
is given
by T =
210/220/230/240
K, the vertical velocity
range
is total
given
w
= 0.01/0.02/0.05/0.1/0.2/0.5/1/2/5
m range
s−1
. −1The
210/220/230/240K,
the vertical velocity
is given
by
by
w
=
0.01/0.02/0.05/0.1/0.2/0.5/1/2/5
m
s
.
The
total
simulation
time is calculated by tsim = 720m/w;
w = 0.01/0.02/0.05/0.1/0.2/0.5/1/2/5
m s−1 . this
Theprototal
simulation
timethat
is calculated
=simulation
720m/w; the
this same
procecedure
ensures
at the endby
oftsim
simulation
time
isatcalculated
by
tthe
= 720m/w;
this prosim
dure
ensures
that
the
end
of
the
simulation
the
same
enenvironmental
conditions
(T,p) are
reached
for each
cedure
ensures
that at the
of the
simulation
theinital
same
vironmental
conditions
(T ,end
p) are
reached
for each
initial
environmental conditions (T,p) are reached for each inital
www.atmos-chem-phys.net/13/9021/2013/
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
temperature run (or equivalently, an altitude difference of
1zsim = 720 m is reached). For each set-up three scenarios were calculated: without aggregation, with temperaturedependent aggregation and with maximum aggregation. For
deriving the maximum impact of aggregation, the box is a
closed system, i.e. ice crystals stay in the box. However, for
more realistic treatment, we have to consider sedimentation.
The process of sedimentation is treated in the box model as
described in Spichtinger and Cziczo (2010). Here, we repeat
some key features of this treatment. Time and space is discretized by time steps 1t and space increments 1z, respectively; t n := n·1t. Generally, the sedimentation process for a
quantity ψ (e.g. ice mass mixing ratio qc or ice crystal number concentration N) can be described in the following way
top
ψ(t n+1 ) = ψ(t n ) · exp(−αψ ) +
Fψ
ρvψ
· 1 − exp(−αψ ) (21)
v 1t
with the terminal velocity vψ for the quantity ψ; αψ = ψ1z
top
denotes the respective Courant number and Fψ is the flux
through the top of the layer. For our purpose, we use fluxes
for quantities ice mass mixing ratio qc and ice crystal number
density N; the mass and number weighted terminal velocities
are then given by the following expressions
v qc
1
=
qc
vN =
1
N
Z∞
n(m)m v(m)dm
(22)
0
Z∞
n(m)v(m)dm
(23)
0
top
whereas n(m) denotes the mass distribution. Since Fψ is
usually unknown for a single box, we can assume, that the
flux through the top is given by a fraction of the flux through
top
the bottom, i.e., Fψ = fsed Fψbottom with the sedimentation
parameter fsed . Using this ansatz, it is possible to categorise
different sedimentation scenarios:
– fsed = 1.0 corresponds to no sedimentation as net effect. The flux exiting the lower part of the box has the
same magnitude as the flux entering the top of the box.
– fsed = 0.9 corresponds to regions in the middle and
lower part of the cloud. The flux leaving the region is
almost but not completely balanced by the flux entering
that region from above.
– fsed = 0.5 corresponds to the top region of a cloud. The
flux leaving that region is only half balanced by a flux
from above.
9029
In the following we discuss a typical scenario of a steady
updraught of w = 0.05 m s−1 at high temperatures (i.e. initial
temperature T = 240 K).
4.2.2
Results for an initial temperature of 240 K
As a baseline experiment we consider first a case without the
effect of sedimentation, that is, with fsed = 1. The temporal
variation of the crystal number density in the box is shown
in Fig. 6. As the box is cooled down from 240 K, the threshold supersaturation for homogeneous nucleation is reached
about 120 min later, and the nucleation burst leads to a high
crystal concentration (≈ 2 × 104 m−3 ). The further development depends strongly on whether we include aggregation or
not. Without aggregation, the crystal concentration stays essentially constant (a weak reduction is caused by the expansion of the lifting box). With aggregation, however, the crystal concentration decreases strongly (by about 95 %) within
two hours. In the scenario with a temperature-dependent aggregation the reduction of the number concentration is less
drastic, but still of the order 85 %. This happens in this academic case because the large crystals effectively stay within
the box (the crystals leaving the box are replaced by identical
crystals entering from above) and aggregate over the whole
simulation time.
Now we turn sedimentation on, allowing the large crystals to leave the box without complete replacement from
above. For the top region of the clouds (fsed = 0.5), the results are displayed in Fig. 6. Again, nucleation occurs after
120 min and a large number of ice crystals appear. These
grow by vapour deposition and RHi decreases. Sedimentation is a much more important process now than aggregation,
which can be seen from the timescale of the decrease of N
which occurs much faster than in the previous case where
sedimentation was switched off. The two curves representing the cases with and without aggregation are almost identical, also after further nucleation bursts occur. This is possible
because RHi starts to increase again because of the ongoing
cooling when N is sufficiently diminished.
In the middle of the cloud, where a good deal of falling ice
crystals are replaced by crystals falling from above (fsed =
0.9), aggregation is a bit more important than at the top of
the cloud. This is shown in Fig. 6. We see that the reduction of N after the nucleation burst is slower than at the
top of the cloud because more falling crystals are replaced.
Aggregation accelerates the reduction of N such that a certain level of N is now reached some ten minutes earlier in
the case with aggregation than without. The scenario with
temperature-dependent aggregation lies between the two others. Secondary nucleation bursts do not occur in either case
in the middle of the cloud until the end of the simulation.
These three scenarios are used in the simulations in order
to investigate the impact of aggregation under different, but
realistic conditions.
www.atmos-chem-phys.net/13/9021/2013/
Atmos. Chem. Phys., 13, 9021–9037, 2013
10
9030
E.Kienast–Sjögren et al.: Ice aggregation in 2-moment schemes
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
ice crystal number concentration (m-3)
20000
15000
no aggregation
with aggregation
temp. dep. aggregation
10000
5000
0
0
30
60
90
ice crystal number concentration (m-3)
20000
15000
120
150
time (min)
180
210
240
120
150
time (min)
180
210
240
120
150
time (min)
180
210
240
no aggregation
with aggregation
temp. dep. aggregation
10000
5000
0
0
30
60
90
ice crystal number concentration (m-3)
20000
15000
no aggregation
with aggregation
temp. dep. aggregation
10000
5000
0
0
30
60
90
Fig. 6. Ice crystal number concentration as a function of time for different
factors.concentration
Top panel: fsed
1.0, i.e. no
Fig.
6. sedimentation
Ice crystal number
as a=function
of sedimentime for
tation, leading
to the strongest
of aggregation;
different
sedimentation
factors.impact
Top panel:
fsed = 1.0,middle
i.e. nopanel:
sedfsed = 0.5, leading
representing
upper impact
part of of
a cloud;
bottommiddle
panel:
imentation,
to thethe
strongest
aggregation;
fsed =f0.9,
representing
the lower
of a part
cloud.
withpanel:
0.5, representing
thepart
upper
of Simulations
a cloud; bottom
sed =
out aggregation
indicated bythe
redlower
lines,part
simulations
including
agpanel:f
representing
of a cloud.
Simulased = 0.9,are
gregation
withaggregation
maximum strength
are represented
by blue
lines and
tions
without
are indicated
by red lines,
simulations
simulations
with temperature-dependent
aggregation
are indicated
including
aggregation
with maximum strength
are represented
by
by green
All simulations
at T = 240 K, p =aggregation
300 hPa and
blue
lineslines.
and simulations
with starts
temperature-dependent
are indicated
driven by by
a constant
vertical
of w
= 0.05
.
are
gree lines.
Allupdraught
simulations
starts
at Tm=s−1
240K,
p = 300hPa and are driven by a constant vertical updraught of
w = 0.05 m s−1 .
Atmos. Chem. Phys., 13, 9021–9037, 2013
peratures, for all choices of fsed and both aggregation sce4.2.3 This
Lower
narios.
is initial
indeed temperatures
what we found from our box model
simulations: the curves for the cases with and without agThe box model
simulations
at thus
240 K
shown
aggregregation
differ not
drastically,
wehave
refrain
fromthat
showing
gation
generally
has
a
weak
effect
on
the
evolution
of
crysthe simulations in detail.
tal number concentrations. Since the collision efficiency decreasesMaximum
exponentially
with of
decreasing
temperature,
we from
there4.2.4
impact
aggregation
as derived
fore expect
a negligible
effect of aggregation at lower tembox model
simulations
peratures, for all choices of fsed and both aggregation scenarios.
This is indeed
what we
foundoffrom
our boxformodel
For determining
the maximum
impact
aggregation
difsimulations:
the
curves
for
the
cases
with
and
without aggreferent temperature and velocity regimes, we investigate
the
gation
do not
differ drastically,
thus we refrain
fromfsed
showing
box
model
simulations
with sedimentation
factor
= 1,
the no
simulations
in detail.
i.e.
sedimentation
is allowed. In order to determine the
maximum impact of aggregation we define an aggregation
4.2.4 f Maximum
impact of aggregation as derived from
factor
agg : The factor is given by the ratio of ice crystal
box
model
simulations
number concentrations of the aggregation run and the reference run (without aggregation):
For determining the maximum impact of aggregation for different temperature and velocity
regimes, we investigate the
n(t)aggregation
fagg = with sedimentation factor fsed(24)
box model simulations
= 1,
n(t)no aggregation
i.e. no sedimentation is allowed. In order to determine the
maximum
of aggregation
define an aggregation
with
t =endimpact
of simulation,
and forwetemperature-dependent
factor fagg : the
the factor
is given
by the ratio
of ice
crystal
numsimulations
analogon
is formed.
In fig.
7 the
aggregaber concentrations
themaximum
aggregation
run and the
tion
factor is shownoffor
aggregation
(topreference
panel)
run temperature-dependent
(without aggregation): aggregation (bottom panel).
and
Figure 7 can be interpreted as follows:
n(t)aggregation
(24)
fagg =
– Forn(t)
lowno
vertical
velocities at warm temperatures, homoaggregation
geneous nucleation does produce just low number conwithcentrations
t = end of(see,
simulation,
temperature-dependent
e.g., fig. and
12 infor
Spichtinger
and Gierens,
simulations
theaanalogon
is formed.
aggregation
2009a, for
relationship
betweenInwFig.
and7Nthe
). Since
there
factor
is shown
for maximum
panel)
and
is only
few competition
on aggregation
the available(top
water
vapour,
temperature-dependent
aggregation
these few ice crystals
can grow (bottom
to large panel).
sizes; thus agFigure
7 canisbe
interpreted
gregation
quite
effectiveasasfollows:
we know from time scale
analysis in sec. 3.5. This can be seen in an aggregation
– factor
For low
vertical velocities
warm
temperatures,
hoapproaching
values at fataggr
∼ 0.01
− 0.1
mogeneous nucleation does produce just low number
concentrations
(see, e.g.,
Fig. 12temperatures
in Spichtinger
and
– As
soon as we proceed
to colder
and/or
Gierens,
2009a,velocities
for a relationship
w and
N).
higher
vertical
the picturebetween
changes.
Under
Since
there is only
fewmore
competition
on the
such
conditions,
much
ice crystals
are available
producedwain
ter vapour, these
few ice crystals
can grow
to large
homogeneous
nucleation
events. Thus,
although
thesizes;
ice
thus aggregation
quite supersaturated
effective as we
know from
crystals
still live in aishighly
environment,
timescale
analysis
in Sect.
3.5.available
This canwater
be seen
in an
they
have to
compete
for the
vapour.
aggregation
factorbut
approaching
valuessizes.
at faggr
∼ 0.01–
Thus,
they grow,
only to smaller
Since
for
0.1efficiency of aggregation the size is much more imthe
portant than the number concentration, in this regimes
– aggregation
As soon as iswe
proceed
to colder
temperatures
and/or
less
efficient.
This leads
to larger aggrehigher
vertical
velocities
the
picture
changes.
Under
gation factors. For very cold temperatures, when
ice
such
conditions,
many
more
ice
crystals
are
produced
crystals stay really at small sizes, aggregation is not imin homogeneous
nucleation
portant
anymore, i.e.
faggr ∼ events.
0.9 − 1.Thus, although the
ice crystals still live in a highly supersaturated environment,
they have
to compete
the available
wa– The
temperature
dependence
of theforcollision
efficiency
ter
vapour.
Thus,
they
grow,
but
only
to
smaller
sizes.
reduces the values but not the qualitative result. Thus,
Since
for the efficiency
of aggregation
the size
is much
the
aggregation
factors are
closer to 1 than
in the
case
more
important
than
the
number
concentration,
in not
this
of E ≡ 1, however, the structure of the curves does
regimes
aggregation
is
less
efficient.
This
leads
to
larger
change
aggregation factors. For very cold temperatures, when
www.atmos-chem-phys.net/13/9021/2013/
In the next subsection we present the setup of the simulations. Then we will present and discuss the results.
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
9031
E.Kienast–Sjögren
et
al.:
Ice
aggregation
in
2-moment
schemes
11
0.8
1
aggregation factor fagg
aggregation factor fagg
T=240K
T=230K
T=220K
T=210K
0.9
0.5
T=240K
T=230K
T=220K
p T=210K
= 300 hPa
0.6
0.2
0.5
0.1
0.4
0
0.01
0.3
1
0.2
0.9
0.1
0.8
0
0.7 0.01
0.61
0.9
0.5
altitude (km)
0.8
0.4
0.7
0.3
0.1
1
vertical velocity (m/s)
10
p = 300 hPa
0.1
1
T=240K
vertical velocity (m/s) T=230K
T=220K
T=210K
10
0.5
0.1
0.4
0
0.01
0.3
0.1
1
vertical velocity (m/s)
11
10
914
T
813
10
914
813
712
712
611
611
510
49
200
8
210
220 230 240 250
temperature (K)
260
270
low
middle
high
510
4 9
0
8
20
40
60
80 100
relative humidity wrt ice (%)
120
7
6
Fig. 8. 6Initial vertical profiles for the simulations;
left: temperature,
Fig.
8. 5Initial
vertical
the5simulations;
left:
temperamiddle:
pressure,
right:profiles
relativefor
humidity
with respect
to ice
for dif4
4
ture,
middle:
pressure,
right:
relative
humidity
wrt
ice
for
different
ferent
set-ups
middle
and
200 210 (low,
220 230
240 250
260high
270 altitude
0
20 range,
40
60corresponding
80 100 120
temperature
(K)
relative
humidity wrt
setups
(low,
middle
and
high
altitude range,
to high,
medium
and
low
temperature
range,corresponding
see Table
1).iceto(%)high,
medium and low temperature range, see tab. 1).
setups (low,
high
range,
to high,
lowmiddle and
7.5 ≤
z ≤altitude
9
235.3
≥corresponding
T ≥ 222.3
Layer
altitude
(km)
temperature
medium and
low temperature
range,
see tab.
middle
8.5
≤z≤
10 226.8
≥ 1).
T ≥(K)
213.5
10
0.2
p = 300 hPa
0.1
Fig. 7. Maximum
impact of aggregation for box simulations using
0
0.1
1
10
a constant 0.01
updraught (i.e. cooling
rate) at different
initial temperavertical velocity (m/s)
tures. The impact of aggregation is determined by the aggregation
factor defined in eq. (24). All simulations are without sedimentation
7. Maximum
of aggregation
box Aggregation
simulations using
inFig.
order
to give the impact
maximum
impact. Topfor
panel:
fac−1
Fig.
Maximum
impact
of
aggregation
box simulations
a constant
updraught
(i.e.vertical
cooling
rate) atfor
different
tor
for7.different
realistic
velocities
(0.01
≤initial
w ≤ 5temperam susing
)
constant
(i.e. cooling
at different
initial
The updraught
impact
of aggregation
israte)
determined
by the
aggregation
atatures.
different
inital temperatures.
The
collision
efficiency
istemperaset to
The
impact
aggregation
determined
the
aggregation
All simulations
are without
sedimentation
factor
inpanel:
Eq.of(24).
Etures.
≡ 1. defined
Bottom
same
as inistop
panel,
butbyfor
temperature
factor
defined
eq.efficiency
(24). All simulations
are
without
sedimentation
in order
tocollision
giveinthe
maximum
impact.
panel:
aggregation
dependent
E
= E(TTop
) as
given
by eq.
(20).factor
in
to give
the maximum
impact. (0.01
Top panel:
facfororder
different
realistic
vertical velocities
≤ w ≤Aggregation
5 m s−1 ) at diftor
forinitial
different
realistic vertical
velocities
(0.01 ≤iswset≤to5 m
s−11.)
ferent
temperatures.
The collision
efficiency
E≡
at
different
inital
temperatures.
Thebutcollision
efficiencydependent
is set to
Bottom
panel:
same
as in top panel,
for temperature
Note, that the shape of the curves for different temperature
E
≡ 1. Bottom
panel:
in topby
panel,
but for temperature
collision
efficiency
E = same
E(T ) as given
Eq. (20).
regimes
is nearly
same; actually,
dependent
collisionthe
efficiency
E = E(Tthe
) ascurve
givenof
byaggregation
eq. (20).
factor for temperature Tinit = 240K could be shifted to the
left in order to represent the curves for other temperatures
ice crystals stay really at small sizes, aggregation is not
even quantitatively.
Note,important
that the shape
of the
different temperature
anymore,
i.e.curves
faggr ∼for
0.9–1.
regimes is nearly the same; actually, the curve of aggregation
4.3
aggregation
within
a 2Dcould
model:
factorTest
for of
temperature
Tinit
= 240K
be Simulations
shifted to the
of
synoptically
driven
cirrostratus
left– inThe
order
to
represent
the
curves
for
other
temperatures
temperature dependence of the collision
efficiency
even reduces
quantitatively.
the values but not the qualitative result. Thus,
In order to investigate the impact of aggregation in a more
the aggregation factors are closer to 1 than in the case
realistic situation we implemented the new aggregation pa4.3 of
Test
of1,aggregation
within
a 2Dofmodel:
Simulations
E≡
however, the
structure
the curves
does not
rameterization into the EULAG model including the alof synoptically driven cirrostratus
change.
ready mentioned bulk microphysics scheme (Spichtinger and
Gierens, 2009a). We investigate typical formation conditions
In
order
investigate
of
in a more
Note
thattothe
shapeclouds,
of the
the impact
curves
foraggregation
different
for
stratiform
cirrus
i.e.
a synoptic
scale temperature
updraught.
realistic
situation
wesame;
implemented
paregimes is
nearly the
actually, the
the new
curveaggregation
of aggregation
rameterization
into theTinit
EULAG
alfactor for temperature
= 240 Kmodel
could including
be shiftedthe
to the
ready
bulk microphysics
(Spichtinger
and
left inmentioned
order to represent
the curvesscheme
for other
temperatures
Gierens,
2009a). We investigate typical formation conditions
even quantitatively.
for stratiform cirrus clouds, i.e. a synoptic scale updraught.
www.atmos-chem-phys.net/13/9021/2013/
11
Table
Initialvertical
vertical profiles
positions
for iceFig. 8.1.Initial
forand
thetemperature
simulations;ranges
left: temperasupersaturated
layers
(low/middle/high).
Layerpressure,
altitude
temperature
ture, middle:
right:(km)
relative humidity
wrt(K)
ice for different
T=240K
T=230K
T=220K
p T=210K
= 300 hPa
0.6
0.2
14
7
0.8
0.4
0.7
0.3
14
low simulaIn the
next subsection Twe present
the setup ofmiddle
the
13
13
high
tions.
Then
we
will
present
and
discuss
the
results.
12
12
altitude (km)
0.61
altitude (km)
0.7
altitude (km)
aggregation factor fagg
aggregation factor fagg
0.9
high
9.5
217.9
low
7.5 ≤
≤ zz ≤
≤ 11
9
235.3 ≥
≥T
T≥
≥ 204.6
222.3
middle
8.5
≤
z
≤
10
226.8
≥
T
≥
213.5
Layer
altitude (km)and temperature
temperature
(K) for iceTable 1. Initial
ranges
high vertical
9.5 positions
≤ z ≤ 11 217.9 ≥ T ≥ 204.6
lowlayers (low/middle/high)
7.5 ≤ z ≤ 9
235.3 ≥ T ≥ 222.3
supersaturated
middle
8.5 ≤ z ≤ 10 226.8 ≥ T ≥ 213.5
high
9.5 ≤ z ≤ 11 217.9 ≥ T ≥ 204.6
4.3 Test
of aggregation within a 2-D model: simulations
4.3.1
Setup
Tableof
1. synoptically
Initial verticaldriven
positions
and temperature ranges for icecirrostratus
supersaturated
(low/middle/high)
For
simulatinglayers
stratiform
cirrus clouds as typical for midlatitudes
vertical
profiles of
of aggregation
temperature and
In orderwe
to specify
investigate
the impact
in a presmore
sure
as shown
in fig.
respectively;theadditionally,
we prerealistic
situation,
we 8,
implemented
new aggregation
pa4.3.1 anSetup
scribe
ice–supersaturated
layer with
vertical
extension
rameterisation
into the EULAG
model
including
the of
al∆z
= 1.5km
at different
altitudes (top
of (Spichtinger
layer at ztopand
=
ready
mentioned
bulk microphysics
scheme
For simulating stratiform cirrus clouds as typical for mid9/10/11km,
i.e. We
low/middle/high).
The
vertical conditions
extension
Gierens, 2009a).
investigate typical
formation
latitudes we specify vertical profiles of temperature and presand
the corresponding
temperature
rangesscale
are updraught.
presented in
for stratiform
cirrus clouds,
i.e. a synoptic
In
sure as shown in fig. 8, respectively; additionally, we pretable
1. We
use a 2Dwe
domain
the next
subsection
present(x-z-plane)
the set-up in
of the
the troposphere
simulations.
scribe an ice–supersaturated layer with vertical extension of
(∆x = 100m) and
with
horizontal
extension
Lx = 12.7km
Thena we
will present
and discuss
the results.
∆z = 1.5km at different altitudes (top of layer at ztop =
a vertical extension 4 ≤ z ≤ 14km (∆x = 50m). At initialisa9/10/11km,
low/middle/high). The vertical extension
4.3.1
Set-upi.e.temperature
tion
the potential
field is superimposed by Gausand the corresponding temperature ranges are presented in
sian noise with standard deviation σθ = 0.025K. We choose
tablesimulating
1. We use a 2D domain
(x-z-plane)
the troposphere
cloudswind,
asintypical
for midaFor
moderate windstratiform
shear forcirrus
horizontal
i.e. du/dz
=
= 12.7km
(∆x
= 100m) and
and
with
extension
Lx profiles
−3 a
−1horizontal
−1
latitudes,
we
specify
vertical
of
temperature
10 s with u(z = 0) = 0 m s , leading to a maximum
a verticalasextension
4 ≤−1
z ≤ 14km (∆x =we
50m). At initialisapressure
shown
an doicewind
of umax
≈ 10in
mFig.
s 8;atadditionally,
z = 14km. Theprescribe
whole 2D
tion the potential
temperature
fieldextension
is superimposed
by1.5
Gaussupersaturated
layer
with
vertical
of
1z
=
km
main is lifted with a constant vertical velocity. In order
sian
noise with
standard
deviation
σzθ top
= 0.025K.
Wekm,
choose
at
different
altitudes
(top
of
layer
at
=
9/10/11
i.e.
to investigate different synoptic conditions we choose two
a moderate wind−1
shearvertical
for horizontal
i.e.
du/dz =
−1 wind,
low/middle/high).
The
extension
and
the
correspondvalues
w = 5cm s and w = 8cm
s . In order to obtain
10−3
s−1
with u(z
0)are
=
0presented
mofs−1
, leading
to 1.
a (i.e.
maximum
ing
temperature
ranges
in Table
We use
similar
conditions
at=the
end
the
simulations
thea
−1
wind
of
u
≈
10
m
s
at
z
=
14km.
The
whole
2D
domax
2-D
domain
(x-z-plane)
in
the
troposphere
with
a
horizonsame vertical distance of lifting ∆zlif t = 720m or equivamain
is
lifted
with
a
constant
vertical
velocity.
In
order
tal
extension
L
=
12.7
km
(1x
=
100
m)
and
a
vertical
exlently a cooling xof ∆T ≈ 7.04K), the simulation time is adto investigate
different
synoptic
conditions
we choose
two
tension
≤ z ≤of
14wkm
(1x
m).
At initialisation
the pojusted;
in4 case
= 5cm
s=−150the
total
simulation
time
is
−1
−1
values
w
=
5cm
s
and
w
=
8cm
s
In
order
to
obtain
−1 . by
tential
temperature
field
is
superimposed
Gaussian
noise
∆t = 240min, whereas for w = 8 cm s the total simulation
similar
conditions
at the
end
of the
simulations
(i.e. the
with
standard
deviation
σθ =
0.025
K. We
choose a moderate
equivasame shear
vertical
of wind,
liftingi.e.
∆zdu/dz
lif t = 720m
wind
fordistance
horizontal
= 10−3ors−1
with
lently
a cooling
of ,∆T
≈ 7.04K),
the simulation
is adu(z
= 0)
= 0 m s−1
leading
to a maximum
wind time
of umax
≈
justed;
= The
5cmwhole
s−1 the
total
simulation
time
10
m s−1inatcase
z = of
14 w
km.
2-D
domain
is lifted
withis
∆t = 240min, whereas for w = 8 cm s−1 the total simulation
Atmos. Chem. Phys., 13, 9021–9037, 2013
9032
12
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
E.Kienast–Sjögren et al.: Ice aggregation in 2-moment schemes
a constant
velocity.
to turned
investigate
time isvertical
∆t = 150min.
AsIntheorder
results
out todifferent
be similar
synoptic
conditions
we
choose
two
values
w
=
5 cm s−1
and
for the chosen vertical velocities in terms of impact
of aggrew = 8gation,
cm s−1we
. Inwill
order
to
obtain
similar
conditions
at
the
endthe
concentrate on a detailed investigation of
of thecase
simulations
verticalthe
distance
lifting
w = 5 cm (i.e.
s−1 . the
For same
investigating
impact of
of aggrega1zlifttion
= 720
equivalently
a cooling
of reference
1T ≈ 7.04
K),we
we m
useorthree
different setups:
In the
case,
the simulation
time is adjusted;
in case
of w = 5 cm s−1
switch off aggregation;
in scenario
“temperature-dependent”
the total
simulation
is 1t =parameterization
240 min, whereas
for w =the
we use
the full time
aggregation
including
−1 the totaldependency,
described
4.1.2.As
In order
8 cm stemperature
simulationastime
is 1t in
= sec.
150 min.
the
toturned
see theout
maximum
effectfor
of the
aggregation,
we usevelocithe sceresults
to be similar
chosen vertical
i.e. here
aggregation
hasonefties innario
terms“maximum
of impact impact”,
of aggregation,
wethe
will
concentrate
−1
ficiency
E
≡
1.
We
assume
that
ice
forms
by
homogeneous
a detailed investigation of the case w = 5 cm s . For invesnucleation
only,ofparameterized
afteruse
Koop
et different
al. (2000).setThe
tigating
the impact
aggregation we
three
background
aerosol
(sulphuric
acid)
is
prescribed
with
a
logups: in the reference case, we switch off aggregation; in sce= 25nm and
distribution with (dry)
radius
nario normal
“temperature-dependent”
wemodal
use the
fullrmaggregation
geometrical standard deviation of σr = 1.5, respectively.
parameterisation including the temperature dependency, as
described
in Sect. 4.1.2. In order to see the maximum effect
4.3.2 Results and discussion
of aggregation, we use the scenario “maximum impact”, i.e.
here the
aggregation
has efficiency
E≡
1. Wetoassume
that
In general
the simulations
behave
similar
those carried
ice forms
homogeneous
nucleation
only,
out byby
Spichtinger
and Gierens
(2009b)
forparameterised
the case of pure
Koop et al. (2000).
The background
aerosol
(sulfuric
after homogeneous
nucleation:
As the domain
is lifted
it cools
acid) by
is prescribed
with a log-normal
distribution
withincreases
(dry)
adiabatic expansion
and the relative
humidity
modaluntil
radius
rm =are
25 nm
and at
geometrical
standard
deviation
crystals
formed
the threshold
for homogeneous
Fig. 9 shows part of the temporal evolution of
of σr nucleation.
= 1.5.
the reference simulation in time steps of 30 min, starting at
60min, and
i.e. discussion
the state for t = 60/90/120/150min is dis4.3.2 t =Results
played. The formation of a quite homogeneous cirrostratus
can bethe
seen
to occur after
about
2 h simulation
time by
In general
simulations
behave
similar
to those carried
outincreasing
ice
water
content
IWC
=
q
·
ρ
(black
isolines).
c
air
by Spichtinger and Gierens (2009b) for the case of pure hoSome structure
is formed
the horizontal
driving
mogeneous
nucleation:
as the by
domain
is lifted itwind
cools
by
small circulations inside the layer. In the further evolution,
adiabatic expansion and the relative humidity increases unthe vertical depth of the cloud layer is extended due to seditil crystals are formed at the threshold for homogeneous numenting ice crystals, reaching about ∆z ∼ 3km at the end of
cleation.
Figure 9 shows part of the temporal evolution of
the simulation at t = 240min. Fig. 10 shows the results at the
the reference
in Mean
time steps
end of thesimulation
simulations.
valuesof
of 30
icemin,
waterstarting
content,atice
t = 60
min, i.e.
the state
for t = 60/90/120/150
min iswrt
dis-ice
crystal
number
concentration
and relative humidity
played.
formation
a quite
homogeneous
cirrostratus
areThe
shown,
averagedofover
the domain.
As expected,
the imcan be
seen
to occur after
about 2with
h simulation
time
byinin-the
pact
of aggregation
increases
temperature,
even
creasing
water
= qc · ρair efficiency
(black isolines).
casesice
with
E ≡content
1 whereIWC
the aggregation
itself has
Somenostructure
is formed
by the
horizontal
wind driving
temperature
dependence.
Obviously
the remaining
factors
the collisioninside
kernelthe
contribute
significantly
the tempersmallincirculations
layer. In
the furthertoevolution,
ature dependence
andcloud
this islayer
due to
fact that
crystals
can
the vertical
depth of the
is the
extended
due
to sedgrowice
larger
in the reaching
higher absolute
humidity
environment
imenting
crystals,
about 1z
∼ 3 km
at the endat
temperatures.
Thus,
geometrical
factor
of thehigher
simulation
at t = 240
min.the
Figure
10 shows
the evidently
results
factor
depending
on the condifferat thegrows
end ofwith
thetemperature.
simulations.The
Mean
values
of ice water
encecrystal
of terminal
fall concentration
speeds increases
averagehumidity
with mean
tent, ice
number
andonrelative
crystal size
if the
of the
size distribution
so. InAsthe
with respect
to ice
arewidth
shown,
averaged
over the does
domain.
formulation of Spichtinger and Gierens (2009a) the width of
expected, the impact of aggregation increases with temperthe mass distribution (i.e. the square root of the second cenature, even in the cases with E ≡ 1 where the aggregation
tral moment) is proportional to the mean mass. Therefore
efficiency
itself has no temperature dependence. Obviously
higher temperatures lead to more aggregation also via the terthe remaining
factorsinin
themodel.
collision kernel contribute sigminal velocities
this
nificantly
to
the
temperature
dependence
andofthis
is due to
Aggregation increases the
average mass
ice crystals
and
the fact
that
crystals
can
grow
larger
in
the
higher
so leads to stronger sedimentation which has anabsolute
effect on
humidity
environment
at higher
temperatures.
Thus, theclouds.
geoice mass
and number
concentration
in the simulated
metrical factor evidently grows with temperature. The factor
depending on the difference of terminal fall speeds increases
on average with mean crystal size if the width of the size
Atmos. Chem. Phys., 13, 9021–9037, 2013
Fig.9.9. Time
Time evolution
evolution of
withwith
a time
incre-inFig.
ofreference
referencesimulation
simulation
a time
ment ofof∆t1t
==
30min,
starting
at t at
= 60min
(shown
are states
for
crement
30 min,
starting
t = 60 min
(shown
are states
t =t 60/90/120/150min,
respectively).
Black
isolines
indicate
ice
for
= 60/90/120/150 min,
respectively).
Black
isolines
indicate
water
content
(IWC
qc q· ρair
, increment ∆IWC = 2mgm−3−3
),
ice
water
content
(IWC= =
c ·ρair , increment 1IWC = 2mg m ),
grey lines indicate isentropes (increment ∆θ = 4K).
grey lines indicate isentropes (increment 1θ = 4 K).
Mean crystal number concentrations and ice water contents
distribution does so. In the formulation of Spichtinger and
are strongly (up to and partly exceeding a factor 2) reduced
Gierens
(2009a)inthe
of theusing
massthe
distribution
(i.e. the
by aggregation
thewidth
simulation
highest temperasquare
root
of
the
second
central
moment)
is
proportional
ture. The effects are of similar quality in the colder cases, but to
the
mean Effects
mass. Therefore
higher
temperatures
lead to more
smaller.
on the mean
profiles
of relative humidity
are
aggregation also via the terminal velocities in this model.
Aggregation increases the average mass of ice crystals
and therefore leads to stronger sedimentation which has an
www.atmos-chem-phys.net/13/9021/2013/
12
11.5
11
11
10.5
10.5
10
10
9.5
9
8.5
12
reference
aggregation, temperature dependent
aggregation, maximum impact
11
10.5
10
9
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8
8
7.5
7
7
6.5
6.5
25000
50000
75000
100000
125000
6.5
1
2
3
4
5
6
7
8
9
10
6
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
mean relative humidity wrt ice (%)
3
mean ice water content (mg/m )
12
12
reference
aggregation, temperature dependent
aggregation, maximum impact
10
altitude (km)
10.5
10
9.5
9
8.5
7.5
7
7
6.5
6.5
6
6
20000
30000
40000
50000
60000
10.5
10
9
8
10000
11
8.5
7.5
7
6.5
1
2
3
4
5
6
7
8
9
10
6
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
mean relative humidity wrt ice (%)
mean ice water content (mg/m3)
12
12
reference
aggregation, temperature dependent
aggregation, maximum impact
11
10
altitude (km)
10.5
10
9.5
9
8.5
7.5
7
7
6.5
6.5
6
6
15000
10.5
10
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10000
9.5
9
8.5
8
7.5
7
6.5
0
20000
11
9
8
reference
aggregation, temperature dependent
aggregation, maximum impact
11.5
9.5
8
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reference
aggregation, temperature dependent
aggregation, maximum impact
11.5
10.5
0
12
altitude (km)
11
9
8.5
8
mean ice crystal number concentration (m-3)
11.5
9.5
7.5
0
70000
reference
aggregation, temperature dependent
aggregation, maximum impact
11.5
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8
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reference
aggregation, temperature dependent
aggregation, maximum impact
11
altitude (km)
11
12
11.5
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reference
aggregation, temperature dependent
aggregation, maximum impact
7
mean ice crystal number concentration (m-3)
11.5
9
8.5
8
0
150000
9.5
7.5
6
0
altitude (km)
11.5
9.5
7.5
6
altitude (km)
9033
13
altitude (km)
12
11.5
altitude (km)
altitude (km)
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
E.Kienast–Sjögren et al.: Ice aggregation in 2-moment schemes
1
2
3
4
5
6
7
8
9
10
6
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
mean relative humidity wrt ice (%)
3
mean ice crystal number concentration (m-3)
mean ice water content (mg/m )
Fig.
ice respect
(right) at
Fig.10.
10.Vertical
Verticalprofiles
profilesofofmean
meaniceicecrystal
crystalnumber
numberconcentration
concentration(left),
(left),iceicewater
watercontent
content(middle)
(middle)and
andrelative
relativehumidity
humiditywrt
with
to
the
of the
(t =simulation
240min, w
cmmin,
s−1 )wfor
Top row:
low Top
temperature
row:
iceend
(right)
at simulation
the end of the
(t =
=5240
= different
5 cm s−1 )temperature
for differentregimes.
temperature
regimes.
row: lowconditions,
temperaturemiddle
conditions,
medium
bottom
row: high
temperature
middle temperature
row: mediumconditions,
temperature
conditions,
bottom
row: highconditions
temperature conditions.
1
1
1
0.1
0.1
0.1
www.atmos-chem-phys.net/13/9021/2013/
-3
frequency of occurrence
-3
frequency of occurrence
frequency of occurrence
-3
frequency of occurrence
frequency of occurrence
frequency of occurrence
effect on ice mass and number concentration in the simulated
Also the total surface area of the ice crystals decreases by
0.01
0.01
clouds.0.01Mean crystal number concentrations and
ice water
aggregation. This and the
above mentioned increase of sed0.001
0.001
0.001
contents
are strongly (up to and partly exceeding
a factor 2)
imentation fluxes diminish
the sink for supersaturation and
0.0001
0.0001
0.0001
reduced
by aggregation in the simulation using
the highest
higher relative humidities
are maintained over longer periods
1e-05
temperature.
The effects are of similar quality 1e-05
in the colder
of time compared to the1e-05
reference cases without aggregation.
reference
reference
reference
1e-06
1e-06
cases,1e-06
but aggregation,
smaller.
Effects
on
the
mean
profiles
of
relative
huThe
statistics
of
relative
humidities
typically
temperature dependent
aggregation, temperature dependent
aggregation, temperature
dependent peak at values
aggregation, maximum impact
aggregation, maximum impact
aggregation, maximum impact
1e-07 to look at
1e-07 is best seen in the cold case where
midity1e-07
are
present,
but10000
here it100000
is more1e+06
instructive
slightly
above1e+06
100 %. This
100
1000
100
1000
10000
100000
100
1000
10000
100000
1e+06
ice crystal
number Fig.11).
concentration (mThe
)
ice crystal number aggregation
concentration (m )
ice crystal
) i . At the
the statistics (see
below,
pdfs of number concenhas hardly any effect
on number
the concentration
pdf of (m
RH
1
1
tration of1 ice crystals displayreference
in our simulations broad
maxhigher
temperatures, where
aggregation becomes
more effireference
reference
aggregation, temperature
aggregation, temperature dependent
aggregation, temperature dependent
−1dependent
aggregation,
maximum
impact
aggregation,
maximum
impact
aggregation,
maximum
impact
ima at around
N
∼
100
L
.
Whereas
there
is
hardly
any
efcient
we
see
the
peaks
shifted
to
higher
values
(i.e.
more sigi
0.1
0.1
0.1
fect on the statistics of number concentration in the low temnificant “quasi equilibrium” supersaturation), clearly an ef0.01 case, aggregation shifts the peaks to lower
0.01
perature
values
fect of the less effective0.01sink for water vapour in a cloud afand broadens them. As expected, this effect becomes more
fected by aggregation (see also the discussion in Spichtinger
0.001
0.001
0.001
prominent at higher temperatures. This is clearly a signature
and Cziczo, 2010). This effect is most pronounced in those
of the0.0001
aggregation-enhanced sedimentation (see
also discusregions of a cloud that
0.0001
0.0001otherwise approach ice saturation
most quickly, typically the middle part of the cloud. Thus agsion in Spichtinger and Gierens, 2009a). The high number
1e-05
1e-05
concentration
distributions
are merely
gregation
to1e-05
ice
within
30 40 50 tails
60 70 of
80 90the
100 110
120 130 140 150 160 170
30 little
40 50 60af70 80 90 100
110 120 130 140contributes
150 160 170
30 supersaturation
40 50 60 70 80 90 100 110
120 130 140relatively
150 160 170
relative humidity wrt ice (%)
relative humidity wrt ice (%)
humidity wrt ice (%)
fected by aggregation; this is quite plausible because high
warm cirrus clouds. Cold cirrus is relative
hardly
affected by aggrenumber concentrations are usually coupled with small ice
gation according to our simulations (and under the condition
Fig. 11. Statistics of ice crystal number concentrations (top row) and relative humidity wrt ice (bottom row) for different temperature regimes
crystals,
thus
aggregation
is
weakly
effective
in
this
range.
(left: low temperature conditions, middle: medium temperature conditions, right: high temperature conditions).
Atmos. Chem. Phys., 13, 9021–9037, 2013
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
3
mean ice crystal number concentration (m-3)
mean ice water content (mg/m )
mean relative humidity wrt ice (%)
1
1
0.1
0.1
0.01
0.001
0.0001
1e-05
reference
aggregation, temperature dependent
aggregation, maximum impact
1e-06
1e-07
100
1000
10000
0.01
0.001
0.0001
1e-05
1e-06
100000
reference
aggregation, temperature dependent
aggregation, maximum impact
1e-07
100
1e+06
ice crystal number concentration (m-3)
1000
10000
1e-05
100000
100000
1e+06
reference
aggregation, temperature dependent
aggregation, maximum impact
0.1
frequency of occurrence
frequency of occurrence
10000
1
0.01
0.001
0.0001
1e-05
1000
ice crystal number concentration (m-3)
0.1
0.0001
reference
aggregation, temperature dependent
aggregation, maximum impact
1e-07
100
1e+06
reference
aggregation, temperature dependent
aggregation, maximum impact
0.1
0.001
0.0001
1e-06
1
reference
aggregation, temperature dependent
aggregation, maximum impact
0.01
0.01
0.001
ice crystal number concentration (m-3)
1
frequency of occurrence
frequency of occurrence
1
0.1
frequency of occurrence
frequency of occurrence
Fig. 10. Vertical profiles of mean ice crystal number concentration (left), ice water content (middle) and relative humidity wrt ice (right) at
the
end of the simulation (t = 240min, w = 5 cm s−1 ) for different
temperature regimes.
Top Ice
row:aggregation
low temperature
conditions, middle
row:
9034
E. Kienast-Sjögren
et al.:
in two-moment
schemes
medium temperature conditions, bottom row: high temperature conditions
0.01
0.001
0.0001
1e-05
1e-05
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
relative humidity wrt ice (%)
relative humidity wrt ice (%)
relative humidity wrt ice (%)
Fig.
icerespect
(bottomtorow)
for different
Fig.11.
11.Statistics
Statisticsofofice
icecrystal
crystalnumber
numberconcentrations
concentrations(top
(toprow)
row)and
andrelative
relativehumidity
humiditywrt
with
ice (bottom
row)temperature
for differentregimes
temper(left:
temperature
conditions,
middle:
medium
temperature
conditions,
right:
high temperature
conditions).
aturelow
regimes
(left: low
temperature
conditions,
middle:
medium
temperature
conditions,
right: high
temperature conditions).
that the gravitational collection kernel is the only relevant
one for cirrus clouds).
In the simulations with stronger updraught we can see differences in details, but qualitatively they behave similar to
the simulations shown. Therefore, we do not deem it necessary to present them here.
4.4
Scaling size distribution
The log-normal crystal mass distribution is used in the
scheme of Spichtinger and Gierens (2009a) and the question arises whether this is an appropriate choice in situations
where aggregation dominates ice growth. In such cases the
evolving crystal size distribution apparently can be mapped
onto a universal shape:
f (m, t) =
mm (t)
m0
−θ
m
ψ
,
mm (t)
where m0 is a mass unit (to make the prefactor dimensionless) and ψ(x) is a function that depends on time only via
the ratio m/mm (t) (Field and Heymsfield, 2003; Westbrook
et al., 2004, 2007; Sölch and Kärcher, 2011). The temporal evolution of the size distribution can thus be captured by
scaling both axes with a function of the time-varying modal
mass, the m axis with its inverse and the f axis with its θ-th
power. Although the log-normal distribution can be treated
in this way (with θ = 1 for constant geometric width), it is
not a true solution for gravitational aggregation. Researchers,
guided by numerical simulations, tend to use for ψ(x) functions with an exponential upper tail (e.g. Field and Heymsfield, 2003, use a gamma distribution), however, it is not
Atmos. Chem. Phys., 13, 9021–9037, 2013
mathematically proven that the gravitational collection kernel has a scaling solution at all (Aldous, 1999). One condition
for this is the homogeneity of the kernel function K(m, M).
From the mass-length and length-fall speed relations we can
see that the kernel can only be a homogeneous function if the
exponents in these relations are constant over all sizes. This
is only so for spheres. The exponents change, however, with
size for ice crystals and the shape of the crystals, for instance
expressed by the aspect ratio, changes with size (see, for
instance, Heymsfield and Iaquinta, 2000). In our case with
hexagonal columns it is the function A(m), the surface of
the crystal, where m appears with two different exponents
(for the basal and prism faces, respectively); A(m) is nonhomogeneous. Hence, for ice the collection kernel is not a
homogeneous function and therefore there is no true scaling.
If it is nevertheless possible to match observed size distributions to a universal scaling function, this suggests that there is
an approximate scaling for large ( 100 µm) crystals, which
requires that the mentioned exponents are mainly constant
in this range. The upper tail of a log-normal distribution is
slightly bent upward on a semilog plot, that is, it does not
have a perfect exponential; but it might be possible to fit the
observed size distributions of large crystals with log-normals
as well as with gamma-distributions (in particular, given the
noise in the data). Unless there is a mathematical proof that
ice aggregation leads to a specific size distribution other than
the log-normal there is no need to abandon it.
www.atmos-chem-phys.net/13/9021/2013/
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
5
Conclusions
We have derived from the master-equation for coagulation a
simple formulation of aggregation for two-moment bulk microphysical models. So far we developed the formulation for
aggregation of crystals belonging to the same class only (the
microphysics scheme of Spichtinger and Gierens (2009a) allows more than one class of ice). A more general formulation with aggregation of crystals from different classes is
rather a numerical than a mathematical problem, but a difficult one; thus, it is beyond the scope of our study. The core of
the present formulation is a double integral of the collection
kernel weighted with the crystal size (or mass) distribution,
which is the expectation value of the kernel. This quantity
is to be inserted into the differential equation for the crystal
number concentration which is of a form that was already derived by Smoluchowski (1916, 1917). The double integrals
are evaluated numerically for log-normal size distributions
over a large range of geometric mean masses. The direct evaluation of the integrals within a cloud simulation run takes a
lot of computing time and is not recommended. Instead the
pre-calculated results can either be read from a look-up table
or – even better – a polynomial fit of the results can be used
that yields good accuracy.
We have tested the new parameterisation in various environments: stand-alone (to see how the solution of the differential equation behaves and to test the polynomial fits), in
a box-model (where aggregation occurs simultaneously with
other microphysical processes and where a first check can be
made, whether and when aggregation is important), and in
a 2-D simulation of a cirrostratus cloud (where additionally,
cloud dynamics can enhance or dampen the effects of aggregation). Overall these tests suggest that aggregation can
become important at (relatively) warmer cirrus temperatures,
affecting not only ice number and mass concentrations, but
leading also to higher and longer-lasting in-cloud supersaturation. Sedimentation fluxes are increased when aggregation is switched on. Cold cirrus clouds are hardly affected
by aggregation. The temperature dependence originates not
only from an assumed temperature dependence of the collection efficiency but also from the other factors in the collision kernel: Higher temperatures imply larger ice crystals
and larger spread in terminal velocities (if the assumed type
of size distribution is such that the width of it increases with
increasing mean size). From timescale analysis the importance of aggregation can be derived depending on number
concentration and size of the ice crystals. For cold clouds it
is often justified to ignore aggregation when the research focus is on ice mass and number densities. However, when the
focus of research is crystal habits and their effect on radiation, aggregation should not be ignored since cirrus clouds
usually contain complex, irregular and imperfect ice crystals
as reported by Bailey and Hallett (2009). The authors have
shown that even cold cirrus contains complex ice crystals that
may be the result of aggregation. They occur predominantly
www.atmos-chem-phys.net/13/9021/2013/
9035
at high supersaturation, but supersaturation does not directly
appear in the formulation of the kernel function. One might
test whether an extension of the formulation of the collection
efficiency (i.e. E = E(T , S)) yields better results; however,
this kind of further development as well as sensitivity studies is far beyond the scope of this study and is left for future
work. Rather it is desirable to measure collection efficiencies
in big cloud chambers. By doing so, one should simultaneously test whether aggregation is the only process that leads
to complex forms of ice crystals. This is not probable since
rosette shaped crystals are too regular to be formed by random collisions. There is obviously a gap in our understanding and much occasion remains for further experimental research before we should develop our numerical formulations
into unjustified detail.
Appendix A
Fit parameters
In Table A1 the coefficients for the fitting polynomials are
given. For each log-normal distribution with a certain width,
as given by the geometric standard deviation, the numerical
values are fitted by four polynomials for the four mass intervals. The polynomials Pi (i = 1 . . . 4) are of the form
Pi (x) =
3
X
ak x k .
(A1)
k=0
Atmos. Chem. Phys., 13, 9021–9037, 2013
9036
E. Kienast-Sjögren et al.: Ice aggregation in two-moment schemes
Table A1. Coefficients ak , k = 1, . . . , 4 for the fitting polynomials
as given by Eq. (A1) for different geometric standard deviations σm
of the underlying ice crystal mass distribution of log-normal type.
σm = 1.9
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.161743
0
0.063897
0.018204
1.944540
0.006629
−1.500559
−0.583488
7.225331
1.250592
12.517525
6.744979
−25.574820
−26.587122
−53.800911
−42.659445
Acknowledgements. K. Gierens contributes with the model improvement to the DLR project CATS. The numerical simulations
with the 2-D model were carried out at the European Centre for
Medium-Range Weather Forecasts (special project “Ice supersaturation and cirrus clouds”). We thank Ulrich Schumann for
comments on a previous version of this study leading to significant
improvements.
Edited by: A. Nenes
σm = 2.23
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.093471
0
0.039604
0.011304
1.062465
0.004782
−0.937555
−0.346037
4.185680
1.255352
8.213213
4.069962
−26.554164
−26.213817
−42.654474
−32.510841
σm = 2.85
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.047997
0
0.011625
−0.000954
0.508658
0.001689
−0.301539
0.055933
2.484682
1.261169
3.433380
−0.268175
−26.254489
−25.66479
−30.376550
−16.747973
σm = 3.25
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.035623
0
0.003945
−0.007756
0.366723
−0.000604
−0.130459
0.275873
2.099524
1.262784
2.165867
−2.610739
−25.904090
−25.352383
−27.047191
−8.349591
σm = 3.81
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.026046
0
0.000967
−0.011432
0.264208
−0.003465
−0.065920
0.378891
1.875774
1.262705
1.690159
−3.540836
−25.334124
−24.968373
−25.608373
−5.453425
σm = 4.23
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.021865
0
0.003797
−0.014204
0.224846
−0.005474
−0.125940
0.454091
1.839392
1.261214
2.100785
−4.193066
−24.833040
−24.705781
−26.349780
−3.503260
σm = 5.29
polynomial
P1 (x)
P2 (x)
P3 (x)
P4 (x)
a3
a2
a1
a0
0.016174
0
0.005342
−0.012141
0.182345
−0.010140
−0.151146
0.357770
1.927033
1.253076
2.193905
−2.867059
−23.518787
−24.113388
−25.950771
−8.842166
Atmos. Chem. Phys., 13, 9021–9037, 2013
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