Atmos. Chem. Phys., 9, 7481–7490, 2009
www.atmos-chem-phys.net/9/7481/2009/
© Author(s) 2009. This work is distributed under
the Creative Commons Attribution 3.0 License.
Atmospheric
Chemistry
and Physics
Analytical treatment of ice sublimation and test of sublimation
parameterisations in two–moment ice microphysics models
K. Gierens and S. Bretl
Deutsches Zentrum für Luft- und Raumfahrt, Institut für Physik der Atmosphäre, Oberpfaffenhofen, Germany
Received: 18 February 2009 – Published in Atmos. Chem. Phys. Discuss.: 30 April 2009
Revised: 11 September 2009 – Accepted: 21 September 2009 – Published: 7 October 2009
Abstract. We derive an analytic solution to the spectral
growth/sublimation equation for ice crystals and apply it to
idealised cases. The results are used to test parameterisations of the ice sublimation process in two–moment bulk
microphysics models. Although it turns out that the relation between number loss fraction and mass loss fraction is
not a function since it is not unique, it seems that a functional parameterisation is the best that one can do in a bulk
model. Testing a more realistic case with humidity oscillations shows that artificial crystal loss can occur in simulations of mature cirrus clouds with relative humidity fluctuating about ice saturation.
1
Introduction
Cloud microphysical models can essentially be grouped into
bulk and bin models. Bulk models describe a cloud in terms
of gross quantities like total water or ice mass and total number concentration of hydrometeors. Although an assumption
of the underlying size or mass spectrum of the hydrometeors is usually made, its evolution is not completely described
since only the low-order moments are predicted in the model
equations. Bin models instead describe a cloud in much more
detail by explicitly accounting for the size or mass spectrum.
Here the mass spectrum is divided into a large number of bins
and the microphysical processes directly change the number
of particles in the various bins (some recent examples can be
found in Lin et al., 2002; Monier et al., 2006; Leroy et al.,
2007, 2009). While bulk microphysics models are designed
rather for computational speed than for an exact description
of the evolution of a cloud’s mass spectrum, it is vice versa
for the bin models which pay for more exact results with
Correspondence to: K. Gierens
([email protected])
higher computational costs, that is, longer computing time
and higher memory requirements. For many applications (for
instance in an operational weather forecast model) it is too
expensive to use a bin model, hence bulk models are needed.
Nonetheless, their behaviour should be tested under a large
variety of situations such that their strenghts and weaknesses
become understood as much as possible.
Cloud microphysical models that use a two–moment bulk
scheme predict both the temporal and spatial variations of the
total number and total mass of the droplets and ice crystals.
The temporal change of number and mass concentrations are
determined by process rates which are sometimes difficult to
formulate. One example is deposition. When ice crystals
grow by vapour deposition, the ice mass grows accordingly
while the number of crystals is constant. This is the easy
case. The difficult case arises in subsaturated conditions.
When ice crystals sublimate, the ice mass decreases accordingly, while the decrease rate of crystal number cannot be formulated in a straightforward way. The prognostic equations
for ice number and mass concentration (for recent examples
cf. Morrison and Grabowski, 2008; Spichtinger and Gierens,
2009) contain deposition terms, DEP (for mass concentration) and N DEP (for number concentration) in the nomenclature of Spichtinger and Gierens (2009).
The problem is now, that there is an equation for DEP ,
but not for N DEP . Harrington et al. (1995) have performed
a large series of simulations to find an appropriate parameterisation for N DEP . Spichtinger and Gierens (2009) roughly
followed their results and used the following simple approximation to calculate N DEP :
fn = fmα ,
(1)
where fm is the mass fraction sublimated in the current time
step, fn is the desired number fraction, and α is a constant parameter. Harrington et al. (1995) suggest a range
of 1≤α≤1.5 for this parameter, which is set to α = 1.1 by
Published by Copernicus Publications on behalf of the European Geosciences Union.
7482
K. Gierens and S. Bretl: Analytical treatment of sublimation
Spichtinger and Gierens (2009). Morrison and Grabowski
(2008) use α = 1, i.e. fn = fm .
The constancy of α is certainly an oversimplification, since
this would mean that ice crystals in fall streaks sublimate in
a similar way than ice crystals deep inside a cloud where
small humidity fluctuations around ice saturation might lead
to crystal loss. Hence, we deemed it worth to investigate
this process in more detail and to test potential alternative
parameterisations.
For this purpose we will use an analytic solution to
the problem, proceeding from the spectral form of the
growth/sublimation equation, to be derived next in Sect. 2.
From the analytical solution we compute number and mass
loss fractions (Sect. 3) and derive timescales (Sect. 4). Alternative parameterisations will be tested in Sect. 5. A discussion and conclusions are presented in the final Sects. 6 and
7.
but expressions like mb are always meant as (m/U M)b .
With the chosen units, a certain value of a (which is a number) means that the sublimation rate of a 1 ng ice crystal is
a ng/s. In this paper we use crystal masses and values of
a that are typical in cirrus clouds under subsaturated conditions, i.e. the temperature is below −38◦ C, the pressure
below 300 hPa and the relative humidity with respect to ice
ranges from say 10 to 98%. We used b = 0.37 and a width of
the mass distribution of σm = 2.85. Note that the exact values
are not important for the principle considerations that follow.
We introduce the following coordinate transformation:
2
Time dependent crystal mass distribution
As ice crystals in a cloud generally have a large variety of
sizes, bulk microphysics models take account of this variety by implicitly or explicitly assuming an underlying crystal
size distribution, which can be expressed as a crystal mass
distribution or mass spectrum, f (m). Crystal growth by
vapour deposition or crystal sublimation changes the mass
spectrum, so that it becomes also a function of time, i.e.
f (m, t). The temporal change of the mass spectrum due to
deposition/sublimation can be described by an equation that
has the form of a continuity equation in mass space (see, e.g.,
Wacker and Herbert, 1983):
∂
∂
f (m, t) = −
(ṁf ).
∂t
∂m
(2)
f (m, t) dm can equivalently be interpreted as the time dependent probability that an ice crystal selected randomly in
a cloud has a mass between m and m + dm. The probabilistic interpretation will turn out useful for the analytical derivations presented in the following. (f (m, t) has
to be distinguished carefully from fractions fk which are
marked by a lower index.) ṁ ( = dm/dt) is the depositional
mass growth/sublimation rate of a single crystal of mass m.
Koenig (1971) has shown that ṁ can be parameterized as a
function of m in the simple form:
m b
ṁ
=a
.
(3)
U M/U T
UM
Here, a and b depend on temperature and pressure, and
a depends additionally on the degree of ice supersaturation.
For subsaturated cases, a<0. Spichtinger and Gierens (2009,
their Fig. 5) show that this approximation is very good over
four orders of magnitude in m. U M and U T are unit mass
and unit time, for which we use 1 ng and 1 s, respectively. In
the following equations we drop the reference to the units,
Atmos. Chem. Phys., 9, 7481–7490, 2009
x=
m1−b
1−b
(4)
(= const).
ẋ = a
That is, in the new coordinates the growth/sublimation rate
is a simple constant. Let g(x) be the probability density function of x. Then the spectral growth equation obtains the simple form:
∂
∂g
g(x, t) = −a ,
∂t
∂x
(5)
which is solved by a characteristic:
(6)
g(x, t) = g̃(x − at).
g̃ can be any function in principle, but here we use probability density functions, of course. It is obvious that the time
development of g(x, t) is merely a shift of the function as a
whole along the x-axis. All central moments of order higher
than one are invariant, that is, the shape of g(x, t) does not
change over time.
We start (t = 0) from a certain initial mass distribution
(e.g., log-normal), normalized to one for the present paper:
f (m, 0) dm = √
m 2
1 [ln( m0 )]
exp −
2 (ln σm )2
(
1
2π ln σm
)
1
dm, (7)
m
substitute m with x (this yields another log-normal with
1−b
x0 = m1−b
0 /(1 − b) and ln σx = ln σm ):
x 2
1 [ln( x0 )]
g(x, 0) dx = √
exp −
2 (ln σx )2
2π ln σx
(
1
)
1
dx,
x
(8)
and then replace x with x −at, which represents the temporal
evolution of the mass distribution in the x-coordinates:
(9)
g(x, t) dx =
√
1
2π ln σx
2
[ln( x−at
x )]
1
exp − 12 (ln σ0 )2
x−at dx.
x
Finally, we have to substitute back x with m1−b /(1 − b),
to get the desired result:
www.atmos-chem-phys.net/9/7481/2009/
K.
Gierens
S.
Bretl:
Analytical
treatment
ofsublimation
sublimation
K. Gierens
Gierens
andand
Bretl:
Analytical
treatment
of
and
S.S.Bretl:
Analytical
treatment
of sublimation
3
1
1
1.4 1.4
3
7483
1.2 1.2
0.8 0.8
1
f(m,t)
0.8 0.8
0.6 0.6
φn
0.6 0.6
φn
f(m,t)
1
0.4 0.4
0.4 0.4
0.2 0.2
0 0
0.001
0.001
0.01 0.01
0.1 0.1
Crystal
massmass
(ng) (ng)
Crystal
1
10
1
Fig.Fig.
1.1. f1.(m,
t) for
various
times
t under
withwith
a =with
(m,
t) for
various
times
t under
sublimation
a =
Fig.
ff(m,
t)
for
various
times
t sublimation
under
sublimation
◦
(approximatly
for for
−50−50
C, ◦◦250
hPa,hPa,
RHRH
==
98%).
iRH
(approximatly
−50
C, 250
250
hPa,
= 98%).
i 98%).
a−0.004
= −0.004
−0.004
(approximatly
for
C,
t=0
i
t =t =
0 (solid
red,red,
the the
initial
log–normal
distribution),
and t = t =
0 the
(solid
initial
log–normal
distribution),
(solid
red,
initial log–normal
distribution),
and t = 10,and
30, 60,
10, 30, 30,
60, 120120
s (solid
green
and and
blue, dashed
red and
(solid
green
dashed
red blue).
and blue).
12010,
s (solid60,
green sand
blue,
dashed blue,
red and
blue).
Singularities
of this
formulation
can can
appear
at negative
m m
Singularities
of this
formulation
appear
at negative
which
are are
irrelevant.
TheThe
singularity
developing
at mat=m0= 0
which
irrelevant.
singularity
developing
is integrable.
TheThe
temporal
evolution
of the
initially
log–log–
is integrable.
temporal
evolution
of the
initially
 Fig.

normal
distribution
is shown
in Fig.
1 for
sublimation
con1 afor
2connormal
distribution
is shown
in
a sublimation
1−b −(1−b)at


m


dition
(a
<
0).
It
is
evident
that
in
the
mass
coordinates
the


dition (a < 0). It is evident that
in the
lnmass coordinates

the
1−b
m
1
1
distribution
does
not
retain
its
initial
shape;
deviations
from
0
distribution
its initial
from
− shape; deviations
exp
f (m,
t) dm = does
√ not retain
2 (ln σ )in2 
the the
initial
shape
are2π
very

2When
(1When
−web)consider
lnvery
σpronounced.
initial
shape
are
we
consider
m


m pronounced.
 initial
in
stead
a growth
situation
(a >
shape
is much
stead
a growth
situation
(a 0)
>the
0) the initial
shape
is much
better
conserved
(not(not
shown).
A more
general
solution
of the
better
conserved
shown).
A more
general
solution
of the
1 equation
spectral
growth
is
presented
in
Appendix
A.
× spectral growth equation
dm
(10)
is
presented
in
Appendix
A.
b
m − (1 − b)atm
3 3Mass
lossloss
andand
number
lossloss
fractions
Mass
number
fractions
Singularities of this formulation can appear at negative m
WeWe
assume
thatthat
ice ice
crystals
havehave
a minimum
massmass
of
which
are
irrelevant.
The
singularity
developing
at m
mofthr
=m0thr
is
assume
crystals
a minimum
−3
(e.g.
10
ng).
Then
we
define
for
k
∈
{0,
1}
the
following
−3
integrable.
The
temporal
evolution
of
the
initially
log-normal
(e.g. 10 ng). Then we define for k ∈ {0, 1} the following
integrals which
give theintotal
number
mass fractions
of
distribution
is shown
for and
a sublimation
condition
integrals which
give theFig.
total1 number
and mass fractions
of
the
ice
mass
distribution
exceeding
the
threshold:
(a<0).
is evident
that inexceeding
the massthe
coordinates
the iceIt mass
distribution
threshold: the distriZ∞ not
bution does
retain its initial shape; deviations from the
Z∞k
initial
shape are
pronounced.
When we consider(11)
instead
Ik (t) :=
m very
f (m,
t)dm.
Ik (t) :=
mk f (m, t)dm.
(11)
a growth
situation
(a>0)
the
initial
shape
is
much
better
conmthr
mthr
served (not shown). A more general solution of the spectral
Since f depends on time, so does Ik . Harrington et al. (1995)
growth
is on
presented
Appendix
A.
Sinceequation
f depends
time, soin
does
I . Harrington
et al. (1995)
consider the fractional total mass lossk(φ1 ) and number loss
consider the fractional total mass loss (φ1 ) and number loss
(φ0 ) at time t which can be written as
(φ0 ) at time t which can be written as
Ik (0) − Ik (t)
= loss
Ik (0)
Ik.(t) loss fractions
3φk (t)
and−number
φMass
.
k (t) = Ik (0)
Ik (0)
(12)
(12)
Note that we use the notation with numerical indices (since
We assume
that
havewith
a minimum mass
of(since
mthr
Note that
weice
usecrystals
the
notation
indices
the expressions
moments
suggestnumerical
it) interchangeably
−3 ng). using
(e.g.
10
Then
we
define
for
k
∈
{0,
1}
the
following
thetheexpressions
using
suggest
interchangeably
with
notation using
(m,moments
n) as indices
(the it)
notation
that is
with
the
notation
using
(m,
as
indices
notation
thatof
is
integrals
which
give
the total
number
and(the
mass
fractions
used
in the
quoted
papers),
i.e. n)
the
following
identities
hold:
used
in
the
quoted
papers),
i.e.
the
following
identities
hold:
the
ice
mass
distribution
exceeding
the
threshold:
φ0 ≡ φn , φ1 ≡ φm , I0 ≡ In , I1 ≡ Im (and correspondingly
, φ1 ≡ φmbelow).
, I0 ≡ Inφ,nI(φ
I is (and
correspondingly
0 ≡fkφn
1 ≡
for φthe
introduced
plotted
in Fig. 2.
m) m
for the fk introduced below). φn (φm ) is plotted in Fig. 2.
∞
Z
Ik (t) :=
mk f (m, t)dm.
(11)
mthr
www.atmos-chem-phys.net/9/7481/2009/
10
0.2 0.2
0
0
0
0
0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8
φm φm
1
1
Fig. Fig.
2. Total
fraction
of number
loss, loss,
φn ,loss,
vs.
fraction
of
2.
of of
number
φn ,φtotal
vs.
total
fraction
of of
Fig.
2. Total
Totalfraction
fraction
number
total
fraction
n , vs.
massmass
loss, loss,
φ
for
initialinitial
geometric
mean mean
masses
from
1from
m, φ
masses
1 1
mass
loss,
φmm,various
,for
forvarious
various
initialgeometric
geometric
mean
masses
from
to 1000
ng. Note
that the
curves
are almost
congruent.
Time Time
runs runs
to
ng.
the
areare
almost
congruent.
to 1000
1000
ng. Note
Notethat
that
thecurves
curves
almost
congruent.
Time runs
alongalong
the curves
from (0, 0)(0,
to (1, 1). 1).
alongthe
thecurves
curvesfrom
from (0,0)0)toto(1,(1,
1).
TimeTime
runsruns
alongalong
the curves
fromfrom
(0, 0) to 0)
(1,to1).(1,We
see
the curves
Wethat
see that
Since
fis depends
on
time, so(0,
doesnegligible
Ik . 1).
Harrington
et al.
initially
there
mass
loss
combined
with
num-numinitially
there
is
mass
loss
combined
with
negligible
(1995)
consider
the
fractional
total
mass
loss
(φ
)
and
num1
ber loss,
but the
loss overhauls
in theinlater
phases
of of
ber loss,
butnumber
loss overhauls
the later
ber losssublimation
(φ0 )the
at number
timeprocess.
t which
canisbewhat
written
as phases
the ongoing
This
we
expect
for
the ongoing sublimation process. This is what we expect for
massmass
distributions
masses
exceeding
the threshIk (0)with
− Iwith
(t) mode
distributions
masses
exceeding
the threshkmode
φk (t) Different
=
. can be expected for exponen(12)
old mass.
behaviour
old mass. Different
behaviour
can
be
expected
for
exponenIk (0)
tial distributions.
We
note
that
Harrington
et
al.
(1995)
also
tial distributions. We note that Harrington et al. (1995) also
Note that
we
usea the
notation
with numerical indices (since
had examples
with
a different
behaviour.
had examples
with
different
behaviour.
Inthe
a In
cloud
model,
we
generally
do
neither
haveit)
knowledge
expressions
using
moments
suggest
interchangeably
a cloud model, we generally do
neither
have
knowledge
of the
initial
values
I
(0)
nor
of
the
time
t
which
must
be in-be that
with
the
notation
using
(m,
n)
as
indices
(the
k
of the initial values Ik (0) nor of the time t whichnotation
must
in- is
terpreted
here
as
the
time
passed
by
since
sublimation
started.
used in the
quoted
i.e.by
the
following
identities
hold:
terpreted
here
as the papers),
time passed
since
sublimation
started.
(Since
act inna, cloud
model simultaneously
it for the
φ0many
≡φnmany
, processes
φ1 ≡φ
Iin1 ≡I
correspondingly
m , I0 ≡Iact
m (and
(Since
processes
a cloud
model
simultaneously
it
would
even make
sense toφnintroduce
such a time
variable
fknot
introduced
(φ
) is plotted
in Fig.
2.variable
Time runs
would
not even below).
make sense
tomintroduce
such
a time
or toalong
track the
values).
in 1).
a model
we that
usu- initially
the “initial”
curves
from
(0, Hence,
0) toHence,
(1,
see
or to track
the “initial”
values).
inWe
a model
we usually can only consider the fractional mass and number loss
therecan
is mass
loss combined
with negligible
ally
only consider
the fractional
mass andnumber
numberloss,
loss but
per time step, which can be written as
the time
number
overhauls
in the later
per
step,loss
which
can be written
as phases of the ongoing
Ik (t)process.
− I (t + This
∆t) is what we expect for mass dissublimation
Ik (t)k− Ik (t +.∆t)
fk (t, ∆t) =
(13)
ftributions
(t,
∆t)
=
.
withImode
masses exceeding
the threshold(13)
mass.
k
k (t)
Ik (t)
Different
behaviour
can
be
expected
for
exponential
distriThese fractions depend on the size of the time step ∆t, but
These
fractionsnote
depend
the size of
the
step ∆t,had
but exbutions.
that on
Harrington
al. time
(1995)
still on
the timeWe
since sublimation
started.etExamples
foralso
varstill
on
the
time
since
sublimation
started.
Examples
for
varwith
a different
behaviour.
ious amples
time step
values
∆t are plotted
in Fig. 3: These curves
stepmodel,
values we
∆t generally
are plotteddo
inneither
Fig. 3: have
Theseknowledge
curves
Intime
a{f
cloud
link ious
pairs
m (ti , ∆t), fn (ti , ∆t)} for times ti = i · ∆t
link
pairs
{f
(t
,
∆t),
f
(t
,
∆t)}
for
times
t
=
m
i
n
i
i
of
the
initial
values
I
(0)
nor
of
the
time
t
which
must
be in(i = 1, 2, 3, . . .) running kuntil the ice is completely sub-i · ∆t
(i
=
1,
2,
3,
.
.
.)
running
until
the
ice
is
completely
subterpreted
here
as
the
time
passed
by
since
sublimation
started.
limated. (The yellow dot at (1, 1) was computed with a
(The
yellow
atice
1) was
computed
with a it
(Since
many
processes
act
a(1,
cloud
model
simultaneously
verylimated.
large time
step,
such thatdot
thein
sublimated
completely
very
large
time
step,
such
that
ice sublimated
completely
within
this
single
time
step).
Thetothe
curves
fn (fsuch
)
all
have
would
not
even
make
sense
introduce
a
time
variable
m
within
thisThey
single
time
curves
fmeans
all have
n (fm ) that
similar
nearstep).
the fmThe
–axis
whichin
or shape.
to track
the start
“initial”
values).
Hence,
a model
we ususimilar
shape.
start
near the fwithout
which means that
m –axis number
the initial
mass
lossThey
occurs
alone
ally can
only
consider
thealmost
fractional
mass and loss,
number loss
the initial mass loss occurs alone almost without number loss,
per time step, which can be written as
fk (t, 1t) =
Ik (t) − Ik (t + 1t)
.
Ik (t)
(13)
Atmos. Chem. Phys., 9, 7481–7490, 2009
4
7484 4
K. Gierens and S. Bretl: Analytical treatment of sublimation
K. Gierens
Gierensand
andS.S.Bretl:
Bretl:
Analytical
treatment
of sublimation
Analytical
treatment
of sublimation
0.0035
1
0.0035
1
0.003
0.003
0.8
0.0025
0.0025
0.8
f’f’nn
0.002
0.002
0.0015
0.0015
fn
fn
0.6 0.6
0.001
0.001
0.4 0.4
0.0005
0.0005
0
0.2
00
0.2
0.0005
0.0005
0.001
0.001
0.0015
0.002
0.0025
0.003
0.0015
0.002
0.0025
0.003
f’m
f’m
fn0 n00vs.
0
0
0
0
0.2
0.2
0.4
0.4
0.6
fm0.6
0.8
0.8
1
1
fm
Fig. 3. Mass loss (fm ) and number loss (fn ) fractions for series of
time steps.
are 100,
200, 300,
Mass loss
losssublimation
(fm
and number
number
(fnn)steps
of
Fig. 3.subsequent
Mass
(f
lossTime
(f
fractions
for series
m)) and
400 s (solid
red, green,
blue,
dashed
red),
and time
runs 200,
along300,
each
steps
are 100,
subsequent
sublimation
time
steps.
Time
curve from
the point
neares
to thered),
fm –axis
to the
point
nearest
to
red, green,
blue,
dashed
and time
runs
along
each
400 s (solid
the f –axis. The individual time steps are marked on each curve.
the point
point neares
neares to
to the
the ffmm -axis
curve fromnthe
–axis to
to the
the point
point nearest to
Arrows indicate the direction of time. The yellow point at (1, 1) was
-axis.
The
individual
time steps are marked on each curve.
the fnnthe
–axis.
resultThe
of aindividual
test with a very long time step that should guarantee
Arrows
indicate
the
direction
of time. The
yellow
pointThe
at (1,
1) was
that the ice sublimates completely
within
that step.
black
line
0.89
should
guarantee
the result
a test with
very
is theofattempt
of a fita of
thelong
later time
parts step
of thethat
curves,
fn =f
m . The
that step.is m
The
black
that the
ice geometric
sublimatesmean
completely
initial
mass for within
the calculations
ng. line
0 =1
0.89
fm
is the attempt of aa fit
fit of
of the
thelater
laterparts
partsof
ofthe
thecurves,
curves,ffnn==f
m . The
= 1 ng.
ng.
initial geometric mean mass for the calculations is m
m00 =1
and after about 600 s they reach an “attractor” that can be apα̃
proximately be fitted by fn = fm
with α̃ = 0.89. α̃ < 1
again
signifies
that
later
in
the
sublimation
thebenumandThese
after about
600depend
s they reach
an size
“attractor”
that
apfractions
on the
of theprocess
timecan
step
1t,
α̃ of the curves shows that fn
ber
loss
catches
up.
The
shape
proximately
be fitted
fn sublimation
= fm with started.
α̃ = 0.89.
α̃ < 1
but still on the
time by
since
Examples
issignifies
not reallythat
function
the
mathematical
of fnumm beagain
later
in(in
the
sublimation
the
for various
time astep
values
1t
are
plottedprocess
insense)
Fig. 3:
These
cause
it
is
not
unique
even
if
the
timestep
is
fixed.
This
makes
ber
losslink
catches
The
shape
ofi ,the
curves
shows
that
fn
curves
pairs up.
{f (t
fn (t
1t)}
for times
ti =
i · 1t
i , 1t),
it questionable m
whether
a functional
dependence for
parameis
not1,really
a. function
(inuntil
the mathematical
sense) of fsubli(i =
2,
3,
.
.)
running
the
ice
is
completely
m beterisation of sublimation should be used in bulk models.
cause
is(The
not
unique
even
the(1,
timestep
is computed
fixed.
This makes
mated.it The
yellow
dotifat
a
timestep
dependence
of 1)
fk was
suggests
a Taylorwith
expanit
questionable
whether
a
functional
dependence
for
paramevery
large
time
step,
such
that
the
ice
sublimated
completely
sion which yields
terisation
sublimation
should
be used in bulk models.
within thisofsingle
time step).
The
2 curves fn (fm ) all have simfktimestep
(t, ∆t) =dependence
fk0 ∆t + O(∆t
). suggests a Taylor expan(14)
ilarThe
shape.
They start
near theoffmfk–axis
which means that the
As
Fig.
3
shows,
the
higher
orders
cannot
be
neglected
at
sion
initialwhich
massyields
loss occurs alone almost without number loss,
time
steps
of
the
order
100
s
and
longer.
Hence
the
following
after
about
600
they 2reach
an “attractor” that can(14)
be
fand
∆t) =
fisk0 ∆t
+ sO(∆t
).
k (t, analysis
strictly
valid
models
with
approximately
be
fitted
by only
fn =for
fmα̃cloud
withresolving
α̃ = 0.89.
α̃ <
1
small
time steps
of higher
the orderorders
seconds
(wherebetheneglected
order
As
Fig.
3 shows,
cannot
at
again
signifies
thatthe
later
in the sublimation
processhigher
the numterms of
arethe
negligible),
but
it might
still
give some
guidance
time
steps
order
100shape
s and
longer.
Hence
the following
ber loss
catches
up. The
of
the curves
shows
that fn
for
the
treatment
of
sublimation
in
large–scale
models.
analysis
is strictly
valid(in
only
for
cloud resolving
models
is not really
a
function
the
mathematical
sense)
of
fmwith
beFirst, we have
small
time
steps
of
the
order
seconds
(where
the
higher
order
cause it is not unique even if the timestep is fixed. This makes
dIkstill
(t) give some guidance
∂f
∆t)
k (t,
terms
are
negligible),
it−1
might
it questionable
whether
a=functional
dependence
for parame= fk0 (t)but
,
(15)
∂∆t
Ik (t) in dt
for
the treatment
of sublimation
terisation
of sublimation
should
be large–scale
used in bulkmodels.
models.
0
First,
we still
haveretains
the timeof
dependence.
Weanote
that 1/f
Thewhich
timestep
dependence
fk suggests
Taylor
expank (t)
can
be yields
interpreted as−1
a timescale
sion
which
dIk (t) for the change of the cor∂fk (t,
∆t)
= fk0 (t) =
,
(15)
∂∆t
Ik (t)
2
f (t,
1t) = f 0 1t + O(1t
). dt
(14)
k
0
k
which still retains the time dependence. We note that 1/fk0 (t)
3 shows,asthea higher
orders
cannot
be neglected
at
canAsbeFig.
interpreted
timescale
for the
change
of the cortime steps of the order 100 s and longer. Hence the following
Atmos. Chem. Phys., 9, 7481–7490, 2009
0 0
for
fm
m
0
initial
geometric
mean
masses
from from
1
Fig.
Fig. 4.
4. f vs. f forvarious
various
initial
geometric
mean
masses
1
for
various
initial
geometric
mean
masses
from 1
vs.
fm red,
Fig.
4. ng
fng
n (dashed
to
1000
solid
green,
blue,
red).red).
NoteNote
that the
inverse
to
1000
(dashed
red,
solid
green,
blue,
that
the inverse
to 1000of
(dashed
red, solidtogreen,
blue,
red).forNote
inverse
values
the
axis
thethe
timescales
the
change
of
values
ofng
the
axiscorrespond
correspond
to
timescales
for that
the the
change
of
values
of
the
axis
correspond
to
the
timescales
for
the
change
of
the
corresponding
integrals
I
(t)
in
seconds.
Arrows
indicate
the
the corresponding integralsk Ik (t) in seconds. Arrows indicate the
direction
of time. integrals Ik (t) in seconds. Arrows indicate the
the corresponding
direction of time.
direction of time.
responding integral. The derivatives of the integrals can be
analysis is strictly valid only for cloud resolving models with
computed
in the
following
responding
integral.
Theway:
derivatives of the integrals can be
small time steps of the order seconds (where the higher order
computed
Z∞ in the following
terms
but itway:
might still give some guidance
dIk are negligible),
k ∂f
=
m
·
dm,
(16)
for
the
treatment
of
sublimation
in large-scale models.
∞
Z
dt
∂t
dI
∂f
k
First,
we have
=mthr
mk ·
dm,
(16)
dt (t, 1t)
∂t −1 dI (t)
∂f
andk the m
partial
derivative
of
the
mass
distribution
function
is:
k
0
thr
= fk (t) =
,
(15)
∂1t
Ik (t) dt
∂f
and =
thef ×
partial derivative of the mass distribution function
(17) is:
which
still retains the time dependence. We note that 1/fk0 (t)
∂t
∂f be interpreted as a timescale for the change of the corcan
= f×
(17)
∂t
³integral.
responding
The ´derivatives of the integrals can
be


1−b
m
−(1−b)at


a
b
1−b
computedlnin the following
way:
(1 − b)am
m0
+
.
2 [m1−b − (1´
³

σ
)
−
b)at]
m
−
(1 − b)atmb  
(1 − b)(ln
m
∞
1−b
Z
m
−(1−b)at

a
ln
dIk
∂f 1−b
(1 − b)amb(16) 
0
= 0 mk · mdm,
+
.
0
dtf(1
∂t
2
1−b
fm isσplotted
in Fig.
4. −
These

n vs.
−mb)(ln
− (1
b)at]curves
mlook
− (1similar
− b)atmb 
m ) [m
to those inthrFig. 3, and indeed they are equivalent to those for
aand
unitthe
time
step of
1 s.
partial
derivative
of the mass distribution function is:
0
fn0 vs. fm
is plotted in Fig. 4. These curves look similar
to those in Fig. 3, and indeed they are equivalent to those for
4a∂funit
Time
scales
time
step of 1 s.
= f×
(17)
∂t
 curves φ

The
(φ
)
in
Fig.
2
look
very
similar
for
different
n m
m1−b −(1−b)at


initial
mean
masses,
but
if
we
would
include
tick
marks
for


ln
a


4 Time scales m1−b
b
(1
−
b)am
0
time along the curves they would differ for
different
curves.
+
.

m −with
(1 −respect
b)atmb 
(1can
− b)(ln
)2try
[m1−b
− (1 these
− b)at]
We
therefore
to unify
functions


The
curves
φσnm(φ


m ) in Fig. 2 look very similar for different
to time as well by introducing a dimensionless time τ . The
initial mean masses, but if we would include tick marks for
corresponding
τ –tick marks would then become nearly contime
the
curves they
would
differcurves
for different
curves.
fn0 along
vs.For
fm0this
is plotted
4. These
look
similar
gruent.
we can in
firstFig.
compute
the time T it
needs
to to
We
can
therefore
try
to
unify
these
functions
with
respect
those
in
Fig.
3,
and
indeed
they
are
equivalent
to
those
sublimate completely (i.e. to mass zero) a crystal having ex- for a
to time
wellm
a dimensionless
time τof. The
unit
time
step
ofby
s.introducing
actly
theas
mass
, which
can be any
characteristic mass
01
corresponding
τ
–tick
marks
would
then
become
nearly
the chosen distribution, (for instance the median, mean,
or congruent. For this we can first compute the time T it needs to
4sublimate
Time scales
completely (i.e. to mass zero) a crystal having exactly the mass m0 , which can be any characteristic mass of
look verythe
similar
for mean,
different
The
curves distribution,
φn (φm ) in Fig.
the chosen
(for2 instance
median,
or
initial mean masses, but if we would include tick marks for
www.atmos-chem-phys.net/9/7481/2009/
K.K.Gierens
Gierensand
andS.S.Bretl:
Bretl:Analytical
Analyticaltreatment
treatment of
of sublimation
sublimation
www.atmos-chem-phys.net/9/7481/2009/
1.4
1.2
1
0.8
h(τ)
mode
mass).
wethey
takewould
for m0differ
the geometric
mean
mass
time
along
the Here
curves
for different
curves.
of can
the chosen
log–normal
distribution.
This timewith
is respect
We
therefore
try to unify
these functions
to time as well
by introducing a dimensionless time τ . The
m1−b
0
corresponding
τ -tick
marks would then become nearly conT =
.
(18)
− b)we can first compute the time T it needs to
gruent.|a|(1
For this
sublimate
(i.e. to τmass
zero)
With thiscompletely
time we introduce
as t/T
, i.e.a crystal having exactly the mass m0 , which can be any characteristic mass of
(b−1)
the
chosen
or
τ=
t|a|(1distribution,
− b)m0
. (for instance the median, mean,(19)
mode mass). Here we take for m0 the geometric mean mass
the dimensionless
time variable
ofHence,
the chosen
log-normal distribution.
Thistakes
timeinto
is account
the chosen initial characteristic mass of the mass distribution
1−b
and themcurrent
sublimation rate. It does not take into account
0
T other
= quantities
. like the width of the mass distribution. (18)
We
|a|(1 − b)
define
With this
introduce
t/T , i.e.
f 0 time
(a, mwe
d ln Iτnas
(τ )/dτ
0, τ )
h(τ ) = n0
=
.
(20)
f (a, m(b−1)
, τ)
d ln Im (τ )/dτ
τ = t|a|(1 m
− b)m00 .
(19)
h(τ ) can be interpreted as the ratio of two timescales for
Hence,of
thethe
dimensionless
variable
takes and
into their
account
change
Ik integrals. time
These
timescales
rathe
massinofFig.
the 5mass
tiochosen
vary ininitial
time.characteristic
h(τ ) is plotted
for distribution
various iniand
current sublimation
It does
not take
tialthe
geometric
mean massesrate.
(from
1 to 1000
ng)into
andaccount
various
other
quantities
likeWe
thecan
width
of thethemass
distribution.
We
sublimation
rates.
observe
following
behaviour:
define
up to a normalised time τ ≈ 0.1 sublimation leads almost
only tofa0 (a,
mass
m0loss
, τ ) and dthe
ln number
In (τ )/dτdensity of the ice crysn
h(τ
)=
(20)
= Number loss. commences at about
tals
stays0 nearly constant.
f (a, m , τ )
d ln Im (τ )/dτ
τ ≈ 0.1 mand the0fractional number
loss rate exceeds the fractional
massbeloss
rate (h >as1)the
at about
τ ≈
and thereafter,
h(τ ) can
interpreted
ratio of
two1 timescales
for
that is,ofonce
characteristic
mass
had
sufficient
time
to
change
the Ithe
integrals.
These
timescales
and
their
rak
completely.
shapeinofFig.
a curve
h(τvarious
) depends
on
tiosublimate
vary in time.
h(τ ) isThe
plotted
5 for
initial
the widthmean
(and shape)
the initial
massng)
spectrum.
The trangeometric
massesof(from
1 to 1000
and various
subsition
from
mass
loss
to
number
loss
regime
is
sudden
limation rates. We can observe the following behaviour:forupa
distribution
gradual
for a broad
one
(not shown).
tonarrow
a normalised
time and
τ ≈0.1
sublimation
leads
almost
only to
However, as long as we chose a fixed value for σm (as in the
a mass loss and the number density of the ice crystals stays
figure), all these functions are very similar and they can be
nearly constant. Number loss commences at about τ ≈0.1
fitted with a generalised log–logistic function:
and the fractional number loss rate exceeds the fractional
mass loss rate (h>1)
ψ at about τ ≈1 and thereafter, that is,
³
´¡ ¢ .
h̃(τ ) =
(21)
p had sufficient time to sublimate
once the characteristic
p+1
0
1 + p−1 τmass
τ
completely. The shape of a curve h(τ ) depends on the width
(and
of the is
initial
The transition from
Theshape)
formulation
suchmass
that τspectrum.
0 is the inflexion point of the
mass
loss
to
number
loss
regime
is
sudden
forwell
a narrow
fit function. The latter is plotted in Fig. 5 as
with ψdis=
tribution
and
gradual
for
a
broad
one
(not
shown).
1.24, p = 2.4, τ0 = 0.3. ψ is the asymptotic valueHowever,
of the fit
asfor
long
as we
chose
value unity
for σm
(as inisthe
figure), all
large
values
of τa.fixed
It exceeds
which
an expression
these
functions
are
very
similar
and
they
can
be
fitted
with a
for our earlier finding that in the end the number loss exceeds
generalised
log–logistic
the mass loss
rate. The function:
parameter p controls the steepness of
the fit around the inflexion point, i.e. it measures how fast
ψ
sublimation
loss
changes
from
. the mass loss to the number (21)
h̃(τ
)=
p+1the inflexion
τ0 p
regime.1 Finally,
point
is
found
here
at
0.3
which
+ p−1 τ
means that the
transition into the number loss regime occurs
at a considerably shorter time than is needed to sublimate a
The formulation is such that τ0 is the inflexion point of
crystal with the chosen characteristic
mass.
the fit function. The latter is plotted in Fig. 5 as well with
We see that it is possible to find a certain universal beψ = 1.24, p = 2.4, τ0 = 0.3. ψ is the asymptotic value of
haviour of the sublimation curves that do not depend on inithe
fit for large values of τ . It exceeds unity which is an
tial mean mass or sublimation rate. The function does howexpression
forwhen
our earlier
that
in the
enddistributions,
the number
ever change
we usefinding
different
initial
mass
loss exceeds the mass loss rate. The parameter p controls the
steepness of the fit around the inflexion point, i.e. it measures
7485
5
0.6
0.4
0.2
0
0.01
0.1
1
10
τ
Fig. 5.
5. h(τ
h(τ) )vs.
vs.dimensionless
dimensionlesstime
timeτ τfor
forvarious
variousinitial
initial
geometric
Fig.
geometric
mean masses
massesand
andsublimation
sublimationrates:
rates:
1 ng,
a=
−0.04(red);
(red);
mean
m0m=1
a=
−0.04
0 =ng,
m00 =
10 ng, aa =
= −0.09 (green);
(green);mm
= 100ng,
ng,a =
a =−0.2
−0.2(blue);
(blue);
m
=10
0 0=100
m
ng, aa==−0.5
−0.5
(violet).These
These
curves
similar
m00 ==1000
1000 ng,
(violet).
curves
areare
so so
similar
that
that
be seen.
dashed
black
a generalized
onlyonly
one one
can can
be seen.
TheThe
dashed
black
lineline
is aisgeneralized
log–
log–logistic
function
logistic function
thatthat
fits fits
h(τh(τ
). ).
e.g.
to achanges
differentfrom
σm (not
shown).
how by
fastchanging
sublimation
the mass
loss The
to themain
numproblem
still thatFinally,
the valuethe
of inflexion
h dependspoint
on theistime
since
ber loss isregime.
found
here
sublimation
not have into
this the
information
at 0.3 whichstarted.
means As
thatwe
thedotransition
number in
loss
cloud
know easily
whether
process
is
regimemodels
occurswe
at cannot
a considerably
shorter
timethe
than
is needed
still
in the phase
where
mass
is much
larger than
the
to sublimate
a crystal
with
the loss
chosen
characteristic
mass.
number
lossthat
(τ <
or already
theaphase
where
numberbeWe see
it 0.1)
is possible
to in
find
certain
universal
loss
catches
up sublimation
(τ > 1) or somewhere
between
(τ close
haviour
of the
curves thatindo
not depend
on to
inithe
point).
tial inflexion
mean mass
or sublimation rate. The function does howIt is
clear that
τ0 ordistributions,
τ = 1 is
ever
change
whenthe
wetime
use corresponding
different initialtomass
ae.g.
characteristic
timetoofa the
sublimation
process.
It might
be
by changing
different
σm (not
shown).
The main
useful
to
know
it
for
practical
applications
that
we
consider
problem is still that the value of h depends on the time since
next.
For the started.
parameters
in Fig.
5 the this
timeinformation
scales range in
sublimation
As used
we do
not have
from
40
to
250
s,
that
is,
time
steps
of
cloud
resolving
models is
cloud models we cannot know easily whether the process
are
mostly
smaller
than
the
characteristic
times,
while
thosethe
still in the phase where mass loss is much larger than
of
climate
models
are
often
larger.
number loss (τ <0.1) or already in the phase where number
loss catches up (τ >1) or somewhere in between (τ close to
the inflexion point).
5 ItTests
of parameterisations
in two–moment
models
τ = 1 is a
is clear
that the time corresponding
to τ0 or
characteristic time of the sublimation process. It might be
In the foregoing sections we have seen that it is necessary
useful to know it for practical applications that we consider
to know the time since sublimation started when its effect on
next. For the parameters used in Fig. 5 the time scales range
the number and mass concentrations is to be treated correctly,
from 40 to 250 s, that is, time steps of cloud resolving models
and the the problem is just that this time cannot be defined
are mostlyonce
smaller
the characteristic
while those
generally
otherthan
processes
act in a cloudtimes,
as well.
of climate models are often larger.
In a two–moment model one needs both quantities, fn and
fm . It is easier to formulate an equation for fm than for fn
because
is an analytic equation
for the mass change
5 Teststhere
of parameterisations
in two-moment
modelsof
single crystals, while there is none such for the number (it is
either
or one,sections
which iswe
non–analytic
in the
In thezero
foregoing
have seen that
it mathematis necessary
ical
sense).
A formulation
for fm can
be when
made its
in effect
a rela-on
to know
the time
since sublimation
started
tively
straightforward
once the current
mass
spectrum
is
the number
and mass way
concentrations
is to be
treated
correctly,
known
assumed.
Weishave
sublimation
and theorthe
problem
just seen
that in
thisSect.
time2 how
cannot
be defined
generally once other processes act in a cloud as well.
Atmos. Chem. Phys., 9, 7481–7490, 2009
changes
the shape of
the mass
spectrum.
is ignored
in
In a two-moment
model
one needs
both This
quantities,
fn and
such that
errors arise
mass
ftwo–moment
to formulate
an equation
forinfmthethan
forloss
fn
m . It is easiermodels,
computation
well
as in the
number
(as we
because
there as
is an
analytic
equation
forloss
the fractions
mass change
of
will
see
below).
single crystals, while there is none such for the number (it
Therezero
are or
twoone,
principle
to parameterise
subis either
which possibilities
is non-analytic
in the mathematlimation
in
two–moment
models,
either
one
defines
f
as a
ical sense). A formulation for fm can be made in an reladirect
function
of
f
or
one
formulates
f
independently
m way once the current
n mass spectrumof
tively straightforward
is
f
.
The
second
approach
is
a
bit
risky,
it does not
m
known
or assumed. We have seen in Sect. 2 since
how sublimation
guarantee
fn =
for spectrum.
fm = (0, 1),
so is
that
a mixed
changes
thethat
shape
of (0,
the 1)
mass
This
ignored
in
approach
that
gives
this
guarantee
would
perhaps
be
prefertwo-moment models, such that errors arise in the mass
loss
able. Whenasfnwell
is aasdirect
fm ,fractions
the power
computation
in thefunction
numberofloss
(as law
we
α
is
the
simplest
formulation
that
fulfills
f
=
(0, 1)
f
=
f
n
n
will see m
below).
for
fm =are
(0,two
1). principle possibilities to parameterise subThere
In
the
following
we test
a number
potential
limation in two–moment
models,
eitherofone
definesformulafn as a
tions.
The
tests
are
all
based
on
the
corresponding
“benchdirect function of fm or one formulates fn independently
of
functions φn (φm ) obtained from the analytical solufmark”
m . The second approach is a bit risky, since it does not
tion derived
in fSect.
2, see Fig. 2. That is, we take a formulaguarantee
that
n = (0, 1) for fm = (0, 1), so that a mixed
tion
of
f
,
and
compute
the corresponding
functionbeφprefern
n (φm )
approach that gives this guarantee
would perhaps
up to complete sublimation and compare the result to the
able. When fn is a direct function of fm , the power law
benchmark φ (φm ). The tests use the following algorithm:
fn = fmα is then simplest
formulation that fulfills fn = (0, 1)
for1.fmStart
= (0,loop
1). over time steps;
In the following we test a number of potential formulations.
The testsmass
are change:
all baseddM
on ∝
theaµ
corresponding
“bench2. compute
b dt;
mark” functions φn (φm ) obtained from the analytical solucompute
change:
fmis,=we
−dM/M
;
2. That
take a formulation3.derived
in relative
Sect. 2, mass
see Fig.
tion of fn , and compute the corresponding function φn (φm )
4. compute number change (various methods, see below:
up to complete sublimation
and compare the result to the
e.g. fn = f α → dN = −fn N or dN ∝ aµb−1 dt);
benchmark φn (φmm). The tests use the following algorithm:
5. update mass and number, update mode mass for f (m),
1. Start
loop
over time
steps;
output
of various
quantities;
2.6. compute
change: 1M ∝ aµb 1t;
end timemass
step loop.
3. compute relative mass change: fm = −1M/M;
5.1 Power laws
4. compute number change (various methods,
see below:
α
A first exampleαshowing a test of fn = fm
α = 1.1
e.g. fn = fm → 1N = −fn N or 1N∝aµwith
b−1 1t);
(Spichtinger and Gierens, 2009) is given in Fig. 6. We see
that
the parameterisation
produces
too high
lossf frac5. update
mass and number,
update
modenumber
mass for
(m),
tionsoutput
in theofearly
phases
of sublimation while in the final
various
quantities;
phase the mass loss fractions are overestimated relative to
6. number
end timeloss.
stepObviously,
loop.
the
a choice of α < 1 would deteriorate the situation, and a much larger value of α would only
5.1
Power
improve
the laws
agreement for the initial phase of sublimation at
the price of much worse results in the later phases.
α
A first
example
a test
n = fphase
m with
One could
try toshowing
use a large
α in of
the finitial
andαα=<1.1
1
(Spichtinger
and
Gierens,
2009)
is
given
in
Fig.
6.
We see
later. The problem is, however, that one cannot decide
in
that
produces
toothe
high
number loss
fracthe the
bulkparameterisation
cloud model in which
phase
sublimation
process
tions
in
the
early
phases
of
sublimation
while
in
the
finala
is, as we have seen in the foregoing sections. Thus, such
phase
the
mass
loss
fractions
are
overestimated
relative
to
possibility is not given.
the number loss. Obviously, a choice of α<1 would deteriorate
situation,
and a much larger value of α would only
5.2 the
Other
functions
improve the agreement for the initial phase of sublimation at
Weprice
tested
relations
fn and fm as
the
of other
muchfunctional
worse results
in thebetween
later phases.
well,
particular
give
a zero
derivative
fm α<1
=0
Oneincould
try to such
use athat
large
α in
the p
initial
phaseatand
α with α > 1
1−
fm
(not shown).
Examples
are fn = that
1 − one
later.
The problem
is, however,
cannot
decide in
the bulk cloud model in which phase the sublimation process
Atmos. Chem. Phys., 9, 7481–7490, 2009
K.
K. Gierens
Gierens and
and S.
S.Bretl:
Bretl: Analytical
Analyticaltreatment
treatmentofofsublimation
sublimation
1
0.9
0.8
0.7
0.6
φn
6
7486
0.5
0.4
0.3
0.2
0.1
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
φm
1
Fig.
Fig. 6.
6. φφnn vs.
vs. φ
φm
as parameterized
parameterized inin Spichtinger
Spichtingerand
andGierens
Gierens
m ,, as
(2009)
(2009) (red
(redlines,
lines,for
forvarious
variouschoices
choicesofofsublimation
sublimationrates
ratesand
andinitial
initial
geometric
geometric mean
meanmasses),
masses),compared
comparedwith
withthe
theanalytical
analyticalsolution
solutionfor
for
various
various initial
initial geometric
geometricmean
meanmasses
massesand
andsublimation
sublimationrates
rates(black
(black
line).
line). Only
Only one
one black
blackline
linecan
canbe
beseen
seenbecause
becausethe
thedifferences
differencesbebetween
tween the
the analytical
analyticalsolutions
solutionsare
aresmaller
smallerthan
thanthe
thethickness
thicknessofofthe
the
curve.
curve.
and
= have
[cos((f
+ 1]/2. This
did not
yieldsuch
real a
m −
is, asfnwe
seen
in 1)π)
the foregoing
sections.
Thus,
improvements
over
the
simple
power
law.
In
these
cases,
possibility is not given.
φn ≈ 0 until most of the ice mass is sublimated (unless α is
very
to functions
one).
5.2 close
Other
5.3
Usingother
maximum
sublimating
crystal
massfn and fm as
We tested
functional
relations
between
well, in particular such that give a zero
at fm = 0
p derivative
Alternatively
can compute
the maxα
1−fmodel
(not shown). one
Examples
are fnin
= the
1− cloud
m with α>1 and
imum
crystal mass
mmax that sublimates within a timestep.
fn = [cos((f
m −1)π )+1]/2. This did not yield real improveAssuming
the
Koenig
is cases,
given as
ments over the simpleapproximation
power law. In this
these
φn ≈0 until
1/(1−b)
most
of
the
ice
mass
is
sublimated
(unless
α
is
very
close
m
=
(|a|∆t)
.
(22)to
max
one).
With this quantity at hand we can compute
R mmaxmaximum sublimating crystal mass
5.3 Using
f (m) dm
fn = 0R ∞
,
(23)
Alternatively
onedm
can compute in the cloud model the maxf
(m)
0
imum crystal mass mmax that sublimates within a timestep.
while
fm isthe
computed
by the model in
theis usual
Assuming
Koenig approximation
this
given way.
as For
the calculation of the numerator integral we may use expres1/(1−b)
sions
truncated
moments
when analytical expressions for
mmax for
= (|a|1t)
.
(22)
them are available. For our special choice of a log–normal
With this quantity
hand
we can
compute
distribution
we haveat(cf.
Jawitz,
2004):
R
¸
Z mmax mmax f (m) dm ·
ln(mmax /m0 )
1
1
fn = R0f∞
,
(23)
√
+
(24)
(m) dm = erf
2
ln σm 2
0
0 f (m) dm2
It
turnsfout
that
this method
produces
for too
while
computed
by the
modelbad
in results
the usual
way.large
For
m is
and
too small time
steps.
For toointegral
large time
stepsuse
(even
1s
the calculation
of the
numerator
we may
expressions for truncated moments when analytical expressions for
www.atmos-chem-phys.net/9/7481/2009/
K. Gierens and S. Bretl: Analytical treatment of sublimation
7487
them are available. For our special choice of a log–normal
distribution we have (cf. Jawitz, 2004):
Z mmax
1
1
ln(mmax /m0 )
f (m) dm = erf
+
(24)
√
2
2
ln σm 2
0
sinusoidal oscillation. The sublimated ice is assumed to add
to the water vapour, i.e. to increase the relative humidity.
The initial conditions are chosen such that we start with ice
saturation and if we would completely sublimate the ice we
would reach RHi = 105%. The oscillation amplitude is assumed to be ±5%, as well. Hence we have:
It turns out that this method produces bad results for too
large and too small time steps. For too large time steps (even
1 s turned out too large for m0 = 1 ng and a = − 0.04) fn is
non-zero at the beginning, and for too short time steps (0.1 s)
we have again the problem that φn ≈ 0 until most of the ice
mass is sublimated.
5.4
Using moments
Another possibility one could think about is to formulate fn
in a cloud model via the moments of the mass distribution.
These are defined as
Z ∞
µk =
mk f (m) dm.
(25)
0
Spichtinger and Gierens (2009) use dqc /dt = aµb (corrections neglected) to calculate the tendency of the ice mixing ratio from which fm is computed. In analogy, we may
set dN/dt = aµb−1 to calculate the tendency of the number
mixing ratio and compute fn from it. (Although µb−1 is a
moment of negative order, it works since the corresponding integral is finite). Although this sounds a logical approach, it is not. In this case we have fm0 = − aµb /µ1 and
fn0 = − aµb−1 /µ0 , which shows that h(τ ) turns out as a constant (since the time dependent mode mass cancels out in the
division fn0 /fm0 and σm is constant). This is not only in contradiction to the analytical behaviour of h(τ ) (Fig. 5), but –
even worse – it also implies fn = cfm (c a constant) which violates the boundary conditions at (1, 1), unless c = 1, which
it is generally not since it depends on b (pressure and temperature dependent) and σm .
5.5
Effect of humidity fluctuations around ice saturation
Finally we test the effect of small–scale humidity fluctuations
around ice saturation (caused by random fluctuations or atmospheric waves) on sublimation. Such a situation can arise
in old contrails or cirrus clouds (where relative humidity had
enough time to equilibrate with the ice crystal ensemble) under certain synoptic conditions, namely when the large–scale
vertical wind is very weak (less than a few cm s−1 ). Since
a is a non–explicit function of time in such a case, the analytical solution is no longer valid and also the more general
solution of Appendix A is not applicable. Hence we choose a
simple numerical procedure. We divide the lognormal mass
distribution in 1000 discrete initial masses, compute the mass
growth/sublimation rates for each mass with 1 s timesteps.
Once a mass gets smaller than a threshold of 0.001 ng, it is
assumed to be lost. The fluctuations are represented with a
www.atmos-chem-phys.net/9/7481/2009/
RHi = 100 − 5 sin(2π ωt) + 5φm
(%).
(26)
Note that φm can become negative in the growth phases.
We use a geometric mean mass of 100 ng. At RHi = 95%
(220 K, 250 hPa) we get a = − 9.1×10−4 , the characteristic
time (i.e. τ = 1) is 5800 s. The oscillation frequency chosen was ω = 1/250 s−1 , which is a typical value for a BruntVäisälä oscillation in stably stratified air. A first test of the
numerical method with constant subsaturation gave φn (φm )
as in Fig. 2 which shows that the numerical procedure works
well. When we switch on the oscillation and add the sublimated ice to the relative humidity both mass and number
loss are reduced strongly; after 30000 s simulation time only
a few percent of the mass and less than one per mille of the
number are lost. With a lower frequency of ω = 1/2500 s−1
(gravity wave) these numbers are larger, because the initial
sublimation period lasts much longer, but still less than 10
percent of the number is lost after 30 000 s. The results can
be explained as follows: Initially the mean relative humidity
is ice saturation, but the sublimating ice adds to this mean,
such that quickly a new mean value is achieved that exceeds
saturation. Hence, in spite of the oscillations that always
lead transiently through subsaturated states, on the average
the sublimation process is halted. After the first sublimation
period all crystals that were small enough for sublimation
are lost, but in the following sublimation phases the remaining crystals are large enough to survive. Unfortunately, a
different behaviour is displayed by a model employing the
simple parameterisation fn = fmα . Here, more than 20% of
the ice is lost within 30 000 s and with ω = 1/250 s−1 , while
the ice mass stays approximatly constant. This is obvioulsy
the effect of adjusting the mass distribution after each time
step to a new log-normal distribution (with changed geometric mean mass). Hence although the small crystals are gone
within the first sublimation phase, there are new small crystals in the next, merely because the fixed distribution type
enforces this. This leads to crystal loss in every new sublimation phase. This means that under conditions of a cloud
in equilibrium (i.e. RHi ≈100%) small scale fluctuations in
the cloud model can lead to an artificial crystal loss with corresponding consequences on further microphysical evolution
and optical properties.
6
Discussion
Having performed the simple tests of the various possible parameterisations we see now clearer the difficulties inherent in
modelling ice sublimation in the framework of two-moment
Atmos. Chem. Phys., 9, 7481–7490, 2009
7488
K. Gierens and S. Bretl: Analytical treatment of sublimation
models. These difficulties arise in the same way when another mass distribution is chosen (we have also tested a generalised gamma distribution). An exponential distribution
which is typical for cloud droplets could give different test
results; this has not been tested in this paper because we are
mainly interested in improvement of cirrus and contrail models. But such tests could be done in an analogous manner.
As a general rule of thumb the analytical solutions showed
that over a large fraction of the sublimation phase the number
loss fraction equals roughly the mass loss fraction, as seen for
instance in Figs. 2 and 3. In particular for large-scale models
with long timesteps of the order of the sublimation timescale
it seems therefore reasonable to set fn = fm . For cloud resolving models which need timesteps much shorter than the
sublimation timescale problems can arise, in particular in the
beginning of the process, when the number concentration is
hardly affected although the mass concentration is already
diminishing. Hence it is not easy to find a better parameterisation for a small-scale model. The simple parameterisation
fn = fmα with α & 1 seems to obey the simple rule, but in
the oscillation case it went wrong although the more exact
simulation showed that the rule is valid in this case as well.
Unfortunately, the oscillation case is not so academic like
the other tests we have performed; we even took into account
that the sublimating ice mass contributes to the vapour phase
and thus to the relative humidity in the cloud. The oscillation
case is typical for mature clouds where crystals have consumed the excess vapour completely and where random fluctuations (turbulence) and atmospheric waves of various kind
affect the cloud evolution. In order to avoid artificial number
concentration loss in such quasi-equilibrium cases one could
either employ a larger value of α, such that larger mass loss
is needed before crystal loss commences. Or one could introduce a sublimation humidity threshold of several percent
below saturation. Both strategies have their disadvantages. A
sublimation threshold several percent below 100% is unphysical and may have unforeseeable effects. Larger α underestimates crystal loss in situations of steady substantial sublimation (as do other functional relations, as we have seen);
however, one can let α depend on the degree of subsaturation, with larger values at small and smaller values at larger
saturation deficits. This has been tested as well and it produces better results than the parameterisation with constant
α.
The question arises whether there are other pieces of information available at timestep level in the two-moment model.
We have: two independent moments, namely number and
mass concentrations, and direct functions of them (all other
moments, effective sizes, and so on). More information can
only be provided by the thermodynamic state at the timestep,
i.e. the relative humidity and the temperature. These quantities allow to additionally compute the maximum mass that
can evaporate within one timestep. However, we have seen
that even with this additional information it is difficult to construct a better parameterisation of sublimation; at least our
approach in the previous section was not successful although
it sounded plausible.
Our analytical tests were admittedly a bit academic. The
analytical solution is only valid for constant a, which would
mean that sublimated ice would disappear from the system.
In reality, the sublimating ice increases the vapour concentration which generally will lead to increasing a. Also the sublimation process was considered in isolation, which is reasonable for testing, but in reality other processes act simultaneously. The sublimation timescales can be very long especially when the subsaturation is low. In such a case, however,
there may well be other processes with shorter timescales
that then dominate the cloud evolution rendering the problems with sublimation less important. When subsaturation
is larger or the mean mass smaller, sublimation time scales
get shorter; then the sublimation process will quickly evolve
into that regime where fn ≈fm , such that the problems with
the initial phases lasts for a shorter period of time. Here it
would turn out disadvantagous to have a larger α or one of
the other approaches discussed in the previous section, because they lead to underestimation of the number loss over a
large fraction of the whole sublimation process.
Considering these arguments we think that, in spite of its
weaknesses, the simple approach fn = fmα with α slightly exceeding unity or dependent on the degree of subsaturation (or
unity for large-scale models) is still one of the best choices
one can make.
Atmos. Chem. Phys., 9, 7481–7490, 2009
7
Conclusions
The analytic solution of the spectral form of the equation for
deposition/sublimation and its application to ice sublimation
gave the following insights:
– The relation between fractional number loss and fractional mass loss is non-unique, that is, it is no function. However it seems advantageous to formulate it as
a function in parameterisations of ice microphysics for
both large-scale and small-scale models.
– Sublimation timescales are usually longer than
timesteps of cloud resolving models, and often but not
generally shorter than timesteps of large-scale models.
The latter case is easier to parameterise than the first
one.
– As a rule of thumb the analytical solutions showed that
over most of the sublimation process the number loss
rate equals approximatly the mass loss rate. It is therefore a good idea to set them equal in large-scale models
with long timesteps (say, 15 min. and more).
– For small-scale models (e.g. cloud–resolving and large–
eddy models) we have tested a number of alternatives to
the standard power law, but the power law turned out
www.atmos-chem-phys.net/9/7481/2009/
K. Gierens and S. Bretl: Analytical treatment of sublimation
7489
to produce the best results compared to the analytical
solution.
being the antiderivative of a(t) (for constant a, A = at as
in the main part of the paper). Now take the following viewpoint: Let m(0) be a random variable with a given probability
distribution f0 [m(0)], e.g. the log–normal. Then the solution
above (Eq. A2) effects a mapping from the random variable
m(0) to the random variable m(t), that is unique (and invertible) over a certain interval in time, [0, T ], depending on the
time dependence of a. Over this intervall it is possible to
compute the probability distribution of m(t), ft [m(t)], using
the substitution rule, i.e.
– Care has to be taken with the power law formulation
when a cirrus cloud or condensation trail at ice saturation undergoes small–scale humidity fluctuations (probably irrelevant for large–scale models). Without any
counter–measure the power law with α close to one
leads to severe overestimation of crystal loss. It might
be better to let α be a function of saturation deficit with
larger values at low deficit and values closer to one at
larger subsaturations.
The academic test cases of the present paper can only give
insight about the problems inherent in parameterisations of
ice sublimation. The success of an alternative parameterisation in “every day work” should, however, be checked against
spectrally resolving microphysics models where sublimation
acts in combination and in competition with all other processes.
The analytical solution points to the very core of the problem: the sublimation process requires an additional parameter for the complete description of the mass spectrum. The
additional parameter must be obtained from an additional
equation which is difficult to formulate in general once a
variety of other processes affect the shape of the spectrum
as well. To introduce such new equations is probably not
the best way to overcome the problems since this would add
a significant amount of complexity to the bulk models. If
exact solutions are required one should resort to bin models
which do not suffer from the mentioned problems. Nevertheless, bulk models are useful and needed whenever there are
computer time and memory constraints, which is a common
situation. Therefore, in spite of the problems, we encourage
their use, provided that their limits are thoroughly checked.
Appendix A
Solution of the spectral growth equation
A more general analytic solution of the spectral growth equation that handles for instances cases with time dependent a(t)
(but with constant b) is possible proceeding from the following point of view. Let
dm
= a(t) mb ,
dt
(A1)
then the solution of this nonlinear differential equation is
m(t)1−b − m(0)1−b = (1 − b)A(t)
(A2)
with
Z
A(t) =
t
a(t 0 ) dt 0
0
www.atmos-chem-phys.net/9/7481/2009/
(A3)
ft [m(t)] = f0 {m(0)[m(t)]}
dm(0)
dm(t)
and the derivative is
b
dm(0)
(1 − b)A(t) 1−b
.
= 1−
dm(t)
m(t)1−b
(A4)
(A5)
ft [m(t)] is the desired solution of the spectral growth equation. For the special case b = −1 (radius growth equation
for liquid droplets) formally identical solutions have been
obtained using another method (seemingly going back to Sedunov, 1974) by Brenguier (1991, eq. 2.5).
Appendix B
Notation
parameters in Koenig’s crystal growth
parameterisation
A
time integral of a (Appendix A)
f (m, t) time dependent ice crystal mass spectrum
f0 ≡ fn number loss fraction per timestep
f1 ≡ fm mass loss fraction per timestep
g(x, t)
transformed time dependent ice crystal mass
sprectrum
g̃(x − at) characteristic solution for g(x, t)
h(τ )
time dependent ratio of time scales
for number and mass loss
I0 ≡ In total integrated number loss
I1 ≡ Im total integrated mass loss
m
mass of a single ice crystal
ṁ
mass growth rate of a single ice crystal
m0
characteristic mass of the spectrum
or geometric mean mass
N
ice number mixing ratio
p
fit parameter (steepness of the transition
from the mass loss to the number loss regime)
qc
ice mass mixing ratio
RHi
relative humidity with respect to ice
t
time since sublimation started
T
time required to completely sublimate
crystal of characteristic mass;
length of time interval (Appendix A)
x
transformed crystal mass
a, b
Atmos. Chem. Phys., 9, 7481–7490, 2009
7490
ẋ
x0
α
α̃
1t
µk
φ0 ≡ φn
φ1 ≡ φm
σm
σx
ω
K. Gierens and S. Bretl: Analytical treatment of sublimation
transformed mass growth rate
transformed geometric mean mass
power law exponent for the sublimation
parameterisation
fit parameter
model time step
moment of order k
total fractional number loss
total fractional mass loss
geometric standard deviation of mass spectrum
geometric standard deviation of
transformed mass spectrum
oscillation frequency
Acknowledgements. We are grateful to Johannes Hendricks for
critically reading a preliminary version of the manuscript and to
Peter Spichtinger for providing a piece of software that we could
use for the numerical exercise. The comments of two reviewers
helped to improve the paper. The results of the work contribute
to the development of two-moment ice physics schemes in the
framework of the DLR project CATS.
Edited by: J. Quaas
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