Introduction
Warm clouds play a crucial role in the water cycle and energy balance on
Earth . Understanding the whole life cycle of warm clouds,
including formation, development and precipitation, is important for better
prediction of local weather and global climate. Cloud droplet growth is
dominated by diffusion of water vapor at the early stage of cloud
development, while collisional growth is considered to be the most important
mechanism for drizzle formation and warm cloud precipitation
. The concept of a cloud parcel rising adiabatically in
the atmosphere has been used to study cloud microphysical properties for
decades. In a hypothetical initially subsaturated air parcel rising
adiabatically, cloud forms at the lifting condensation level and the growth
of cloud droplets due to diffusional growth can be accurately predicted if we
know the aerosol chemical composition. On one hand, because the growth rate of a cloud
droplet is inversely proportional to droplet size, diffusional growth is
inefficient when the droplet diameter is larger than 20 µm. On the
other hand, collisional growth is efficient when the droplet diameter is
larger than 38 µm . Meanwhile, the sizes of the
smaller cloud droplets will approach those of the larger droplets and narrow
the cloud droplet size distribution (CDSD), which is also unfavorable for
collisional growth . If only diffusional growth is
considered, the CDSD becomes narrower and several tens of minutes even up to
hours will be needed for a cloud droplet to reach efficient-collision size in
an ascending cloud parcel. However, the CDSD in a real cloud is usually wider
than predicted by an adiabatic cloud parcel model and drizzle-size cloud
droplets are frequently observed in warm clouds
e.g.,.
The broadening of the CDSD has a strong effect on precipitation and
radiation. A broader CDSD implies larger differences in the terminal velocity
of droplets. This is beneficial for collision coalescence and might cause the
fast-rain process in the atmosphere e.g.,. In addition, a
broader CDSD increases the relative dispersion, which is the ratio of
standard deviation to the mean CDSD. Previous studies show that an increase
in relative dispersion is relevant to the albedo effect and can either
increase or decrease albedo susceptibility depending on the broadening
mechanism . An interesting question is why
the CDSD is wider than predicted, in particular why large droplet sizes are
frequently observed in the clouds e.g.,. Several
mechanisms have been proposed that can be divided into two categories:
turbulence-induced spectra broadening and aerosol-induced spectra broadening.
A brief review is given next for each category.
Turbulence is ubiquitous in the clouds and can cause CDSD broadening in both
condensation and collision processes e.g.,.
Turbulence induces vertical oscillations of air parcels and causes
fluctuations in temperature, water vapor concentration and supersaturation
e.g.,. The effects of supersaturation
fluctuations on droplet condensational growth in turbulent environments have
been studied for several decades e.g.,. A
qualitative description of this mechanism is that some “lucky” cloud
droplets experience relatively larger supersaturation or stay a relatively
longer time in the cloud compared with the other cloud droplets; therefore
they can grow larger in size and broaden the CDSD. Recent theoretical and
experimental studies support this mechanism and provide ways to quantify the
resulting width of the droplet size distribution
e.g.,.
Turbulence can also modulate the condensational growth of cloud droplets
through mixing and entrainment
e.g.,. In addition, turbulence
can enhance the collision efficiency between droplets and produce “lucky”
cloud droplets through stochastic collisions, which has been confirmed by
direct numerical simulations and Lagrangian drop models
e.g.,.
Aerosols, which serve as condensation nuclei of cloud droplets, can also
cause CDSD broadening in turbulent environments through several mechanisms.
First, turbulence-induced mixing and entrainment can trigger in-cloud
activation of haze particles, which can broaden the left branch of the size
distribution e.g.,.
Secondly, giant cloud condensational nuclei (GCCN, usually defined as
aerosols with a dry diameter larger than a few µm) provides an embryo for
large droplets, which can broaden the right branch of the size distribution and
can be important for warm rain initiation
e.g.,. Recently,
investigated the effect of GCCN on droplet growth and rain
formation using a cloud parcel model. They found that GCCN provides an embryo
for big droplets at the activation stage and, more importantly, GCCN enhances
droplet growth after activation due to the solute effect. For example,
droplets formed on GCCN can still grow through the condensation of water
vapor in the downdraft region even though the environment is subsaturated
with respect to pure water . This, in fact, is an extreme
case of Ostwald ripening.
Ostwald ripening for cloud droplets is the phenomenon when larger droplets
grow and smaller droplets shrink due curvature and/or solute effects and,
thus, it can broaden the CDSD at both small and large ends of the
distribution. investigated the growth of cloud droplets
in a rising air parcel. Results show that the variance of the squared radius of
the CDSD was constant during the condensational growth process if both
curvature and solute effects were ignored, but it was increased if those
effects were considered. This “condensational broadening” is more
pronounced in clouds with high cloud droplet number concentration and low
vertical velocity. In turbulent clouds, droplets will experience
supersaturated/subsaturated conditions in updraft/downdraft regions.
studied the evolution of the CDSD driven by supersaturation
fluctuations in a vertically oscillating air parcel. Supersaturation
fluctuations in his study mean that air is supersaturated in the updraft and
subsaturated in the downdraft; however no spatial inhomogeneity of
supersaturation is considered in the parcel. Results show that the growth and
evaporation cycles during the CDSD evolution are irreversible if the solute
and curvature effects are considered. This “CDSD irreversibility”
(terminology used in his paper) will promote the growth of large cloud
droplets, lead to evaporation or even deactivation of small cloud droplets,
and thus broaden the CDSD. argued that stronger turbulent
fluctuations of supersaturation would result in a broader CDSD. This is
contrary to , who found that supersaturation fluctuations are
not responsible for CDSD broadening and the formation of large droplets. The
curvature and solute effects on Ostwald ripening, activation and deactivation
have been topics of study in recent years
e.g., but, to our knowledge, the
relative roles of the curvature effect and solute effect on CDSD broadening
have not been investigated.
Here we consider an adiabatic cloud parcel that experiences vertical
oscillations, with cloud droplets that are formed on polydisperse,
submicrometer aerosols. Results confirm that the CDSD is broadened during
diffusional growth due to Ostwald ripening and associated droplet
deactivation and reactivation, which is consistent with previous studies
e.g.,. In this study, we investigate (1) what
are the relative roles of the solute and curvature effects on CDSD
broadening, and (2) what other factors can affect this broadening? This paper
is organized as follows. Section introduces the basic setup
for the cloud parcel model, which is similar to except that
there are no GCCN. Results related to CDSD broadening and the associated
sensitivity studies are detailed in Sect. . Conclusions are
summarized in Sect. , including a discussion of implications
in cloud observations and modeling.
Initial dry aerosol radii for different grids.
Grid number
rdry (nm)
Grid number
rdry (nm)
Grid number
rdry (nm)
Grid number
rdry (nm)
1
503
26
191
51
72.4
76
27.5
2
484
27
184
52
69.7
77
26.4
3
466
28
177
53
67.0
78
25.4
4
448
29
170
54
64.5
79
24.5
5
431
30
163
55
62.0
80
23.5
6
414
31
157
56
59.7
81
22.6
7
399
32
151
57
57.4
82
21.8
8
384
33
146
58
55.2
83
20.9
9
369
34
140
59
53.1
84
20.2
10
355
35
135
60
51.1
85
19.4
11
341
36
130
61
49.2
86
18.6
12
328
37
125
62
47.3
87
17.9
13
316
38
120
63
45.5
88
17.3
14
304
39
115
64
43.8
89
16.6
15
292
40
111
65
42.1
90
16.0
16
281
41
107
66
40.5
91
15.4
17
271
42
103
67
39.0
92
14.8
18
260
43
98.8
68
37.5
93
14.2
19
250
44
95.0
69
36.0
94
13.7
20
241
45
91.4
70
34.7
95
13.2
21
232
46
87.9
71
33.4
96
12.7
22
223
47
84.6
72
32.1
97
12.2
23
214
48
81.4
73
30.9
98
11.7
24
206
49
78.3
74
29.7
99
11.3
25
198
50
75.3
75
28.6
100
10.8
Methods
Historically there are two types of bin microphysics: fixed bin scheme and
moving-size-grid scheme (see Sect. 4.2.1 in and references
therein). The advantage of the moving-size-grid method is that it can avoid
artificial CDSD broadening. In this study, we use a cloud parcel model with a
moving-size-grid microphysics scheme, where discrete particle sizes on a 1-D
grid (initially the radii of dry aerosols; e.g., Table ) each
grow/shrink according to the environmental conditions to modify the
“moving size” of the grid element. The original version of the model was
designed to study cirrus clouds by , and then warm
clouds . In recent years, this model has been
modified and applied to investigate various microphysical problems
e.g.,. In the current
version of the parcel model, air pressure (p), parcel height (h), air
temperature (T), water vapor mixing ratio (qv) and radii of haze and
cloud droplets (ri) are prognostic variables, which are calculated using
the variable-coefficient ordinary differential equation solver (VODE)
. Specifically, p is calculated from the hydrostatic equation
and h depends on the vertical velocity (w). Similarly to Eq. (11) in
, T is calculated from
dTdt=-gcp,airw+lvcp,airdqwdt,
where g is the gravitational acceleration, cp,air is the heat capacity
of air, lv is the latent heat of water vaporization and qw is the
liquid water mixing ratio. The first term in Eq. () is the cooling due
to dry adiabatic ascent, and the second term is the microphysical
contribution due to the release of latent heat of condensation. Because the
total water mixing ratio is conserved in the parcel, a decrease in water
vapor mixing ratio (-dqv) equals an increase in liquid water mixing
ratio (dqw). Air supersaturation (Se), which controls the growth of
haze and cloud droplets, is calculated from T, p and qv. A brief
introduction of the model setup and the main mathematical formulations used
for cloud microphysical processes are described below.
In this study, the parcel starts rising at about 300 m below cloud base
and starts descending at about 300 m above cloud base, which is similar
to , except that our cloud parcel then experiences upward and
downward oscillations between 50 m above cloud base and 300 m above
cloud base (see Fig. a). The ascending and descending velocities
are set to be 0.5 and -0.5 m s-1 for the control case.
At the parcel's initial altitude of 600 m, the initial air temperature is
284.3 K, pressure is 938.5 hPa and the saturation ratio is 0.856,
which are as same as .
(a) Trajectory of cloud parcel with upward and downward oscillations. Velocity is constant and is 0.5
for the ascending parcel and -0.5 m s-1 for the descending parcel. The dashed line is the cloud base, and the red and
blue lines represent ascending and descending parcels, respectively. (b) Initial dry aerosol size distribution. The total aerosol number
concentration is 1000 cm-3.
The initial dry aerosols are ammonium sulfate with a log-normal size
distribution range of 10 to 500 nm in radius. The submicrometer
aerosols are parsed into 100 grids (discrete droplet size in each grid
detailed in Table ), where the median radius is 50 nm and the
geometric standard deviation is 1.4. The total number mixing ratio is
1000 mg-1 for the control case, which is about 1000 cm-3 (see
Fig. b). The model first calculates the equilibrium size of haze
droplets for each grid at 85.6% relative humidity, as does
. The equilibrium size of haze particles for the ith grid
(ri) at initial relative humidity is obtained by solving the equation
Ssat(ri)=RH(t=0) iteratively, where Ssat is the saturation ratio
for a solution droplet, calculated from the Köhler equation
p. 172:
Ssat≡ees(T)=as(rd,i,ri)exp2σsρwRvTri,
where e is the water vapor pressure in air, es is the saturated water
vapor pressure over a solution droplet at T, ρw is the density of
water and Rv is the gas constant for water vapor. σs is the
water activity of the haze droplets, which is a function of temperature and
solute p. 133. as is the water activity of haze
droplets, which depends on the composition of aerosol, size of dry aerosol
(rd) and size of haze droplets (r). In this study, as for cloud
droplets is calculated from laboratory-based parameterizations (Eq. 2 in
).
Only diffusional growths of haze and cloud droplets are considered in our
model. Collision coalescence, sedimentation, mixing and entrainment are
ignored. The growth of haze or cloud droplet for the ith grid is calculated
from
dridt=1riSe-SsatG,
where G is the growth parameter given by,
G=ρwRvTDv′es(T)+ρwlvkT′TlvRvT-1.
Dv′ and kT′ are the modified physical diffusion
coefficient of water vapor
and the modified thermal diffusion coefficient, respectively p. 337-338:
Dv′=Dvriri+λ+4Dvαmc‾airri,
and
kT′=kTriri+λ+4kTαTc‾airnaircp,airri.
Here Dv is the physical diffusion coefficient, kT is the thermal
diffusion coefficient, λ is the mean free path of air,
c‾air is the mean molecular speed of air and nair is the
number concentration of air. αm is the mass accommodation
coefficient and αT is the thermal accommodation coefficient; in
this study, we choose αm=1.0 and αT=1.0.
Ssat in the growth equation (Eq. ) is calculated from the
Köhler equation (Eq. ). Therefore, the curvature effect
(exponential part in Eq. ) and the solute effect (as in Eq. ) are considered during the growth process for each grid. It should
be noted that there are several methods to calculate the solute effect with
the relative deviations for activation ranging up to 20%, but the
differences are small for droplet growth . In addition,
different choices of parameters – such as σs, αm and
αT – can also cause differences in droplet growth
. How the choices of different parameters would affect
our results is worth studying in the future. The total simulation time is 3 h, and variables including temperature,
pressure, height, water vapor mixing ratio, as well as droplet size and
number concentration for each grid are recorded every 1 s.
Results and discussions
Cloud droplet size distribution broadening
For the control case, the liquid water mixing ratio increases linearly with
height in the ascending branches and decreases in the descending branches as
shown in Fig. a. Liquid water mixing ratio in the ascending
branch is slightly smaller than that in the descending branch at the same
height due to the kinetic effect (or hysteresis effect), which is consistent
with . The saturation ratio has an increasing trend in the
ascending branch after each cycle, but has a decreasing trend in the
descending branch (indicated by red and blue arrows in Fig. b).
Droplet size for two moving size grids is shown in Fig. c.
Droplet size in the grid monotonically increases with the dry aerosol mass
associated with the grid. The solid line is for the cloud droplet that formed
on a dry aerosol of 503 nm and represents the largest droplet in our
simulation. It grows in the ascending branch but it evaporates in the
descending branch. Also, the droplet size for this grid increases after each
cycle. The dashed line in Fig. c is for the cloud droplet that
formed on a dry aerosol of 51 nm. For this cloud droplet, the changes in
radius with height are similar for the initial few cycles, after which the
cloud droplet deactivates and becomes a haze particle. Ultimately, the
aerosol is reactivated again as a cloud droplet by the end of the simulation
(green dashed line). Also notice that a second mode appears in the CDSD due
to the reactivation of aerosols after about 2 h (see Fig. d). It
should be mentioned that the critical radius, where the Köhler curve
peaks and a droplet is activated, is 3.6 µm for a cloud droplet formed
on a dry aerosol of 503 nm, and 0.44 µm when formed on a dry
aerosol of 51 nm. Figure d shows that all droplet radii are
larger than 4 µm at the end of updraft cycle, indicating that all
cloud droplets are activated at that point. Because GCCN do not exist in our
simulation and the oscillation frequency is low, all cloud droplets have
enough time to grow and to be activated in the updraft region. In this study, we
focus on the CDSD at the end of the updraft cycle so the growth and
evaporation of unactivated cloud droplets e.g., will
not affect the final CDSD. The CDSD broadens after each cycle as the larger
droplets become larger and the smaller droplets either remain similarly sized
or become smaller. All these features are consistent with
(see Fig. 5 in his paper).
Thermodynamical and microphysical properties of an adiabatic cloud parcel with upward and downward oscillations.
(a) Liquid water mixing ratio changes with height. (b) Cloud parcel saturation ratio changes with height. Arrows
in (b) represent the evolution of the saturation ratio profile with time. (c) Radii changes of two selected cloud
droplets with height. The solid line is for the largest cloud droplet that formed on a dry aerosol with a radius
of 503 nm, and the dashed line is for a droplet that formed on an aerosol of 51 nm. The red and blue lines
in (a)–(c) represent ascending and descending parcels, and the black dashed line indicates cloud base height. The
green dashed line indicates the reactivation of that grid. The black and green circles are referred to in the
text. (d) Cloud droplet size distribution changes with time. The black line represents the mean cloud droplet
radius change with time. The yellow dashed line is the change in mean droplet size for the ascending-only
cloud parcel with a constant velocity of 0.5 m s-1, and the upper and lower dashed gray lines represent
the largest and smallest cloud droplets in the ascending-only parcel.
analytically investigates the narrowing and broadening of
cloud droplet size distribution during condensation when solute and curvature
effects are considered. He considers a cloud parcel oscillating vertically in
simple harmonic motion. Results show that the CDSD evolution is irreversible
if solute and curvature effects are considered. Irreversibility of the CDSD
will not only promote the growth of large droplets, but it will also lead to
the evaporation, or even deactivation of small cloud droplets, and thus
broaden the CDSD. However, the relative roles of the solute effect, curvature
effect, deactivation and reactivation on the broadening of droplet size
distributions have not been investigated.
To explore the relative roles of different factors in this CDSD broadening
mechanism, three more cases are tested here. For the first case, we turn off
both the solute and curvature effects for all cloud droplets after 700 s;
this is the time when the cloud parcel first reaches 50 m above cloud
base and is just below the oscillation layer. Specifically, we set
Ssat=1 for all droplets. The result is shown in Fig. a. The CDSD repeats for each cycle in this particular case, consistent with
, and the total cloud droplet number concentration (n) is
constant (red solid line in Fig. d). For the second case, we only
turn off the curvature effect but retain the solute effect. Specifically, we
ignore the exponential term in Eq. () such that Ssat=as. The
result in Fig. b shows that the largest droplet (with the most
solute) can grow after each cycle while the smallest droplet size (with the
least solute amount) associated with a moving size grid does not change much
after each cycle. However the largest droplet size that a grid can reach is
much smaller than that in the control case. Because the saturated water vapor
pressure over a droplet formed on larger aerosol is lower than that formed on
smaller aerosol due to the solute effect, the larger droplet grows faster
than the smaller droplet in the updraft region, and it evaporates slower in
the downdraft region. For this case, the solute effect alone cannot explain
the larger cloud droplets in the control case. In addition, n is also a
constant and droplet deactivation does not occur (green dashed line in Fig. d). In the third case, we consider both curvature and solute
effects, but we do not allow droplet reactivation. This means that once the
droplet deactivates it cannot be activated again. The result in Fig. c shows that the growth of the largest cloud droplet is similar to
the control case, but the size of the smallest cloud droplet associated with
a grid also increases after each cycle. The reason for this CDSD broadening
is the Ostwald ripening effect, where large droplets grow at the expense of
small ones. Past studies have concluded that the ripening effect is typically
slow and inefficient for droplet growth . The vertical
oscillations near cloud base that are considered here, however, allow for droplet
deactivation and result in the decrease of n with time (see Fig. d), as in the control case. Thus, the typically inefficient Ostwald
ripening is amplified through the resulting deactivation of the smallest
droplets. An early suggestion of this behavior is shown in Fig. 8 of
. The only difference between the control and this
simulation is that n for the control case increases near the end of the
simulation because of droplet reactivation (see Fig. d). It
should be mentioned that the step changes in n in Fig. d are a
result of using a discretized grid method to represent the continuous
spectrum. A downward step in n means droplet deactivation, and an upwards
step in n means droplet reactivation. Deactivation and reactivation can
also be seen from the CDSD qualitatively: droplet deactivation occurs when
the peak value of CDSD decreases (from red to blue as shown in Fig. d), while droplet reactivation occurs when a subset of smaller
cloud droplets appears.
(a) Cloud droplet size distribution (CDSD) changes with time without solute or curvature effects.
(b) CDSC changes with time with the solute effect but without the curvature effect. (c) CDSD changes with time
including both solute and curvature effects but where droplet reactivation is not considered. (d) Total cloud
droplet number concentration (n) changes with time for the different cases. The gray region in (a)–(c) represents
the range of the droplet size spectrum for the control case, and the black lines represent the mean cloud droplet radius change with time.
From Fig. a and b, we can see that the solute effect
contributes part of the CDSD broadening compared with the control case. But
the solute effect alone is not enough to explain the growth of the largest
cloud droplet. Droplet deactivation, which is related to the curvature
effect, plays a crucial role here (see Fig. c). Because the
oscillations occur within the cloud region, 50 m above cloud base,
droplet deactivation is surprising to us. There are two related questions:
(1) why do some cloud droplets deactivate in the cloud region while others do
not? (2) Why is droplet deactivation related to the CDSD broadening?
The reason for the droplet deactivation is mainly because the cloud parcel
experiences upwards and downwards oscillations. In the downdraft region, the
air is subsaturated, which supports droplet evaporation. In addition, the
saturated water vapor pressures over polydisperse droplets are different via
both the solute and curvature effects. Smaller droplets with less solute and
larger radii of curvature have higher saturated water vapor pressures, and
thus evaporate faster than larger droplets in the downdraft region.
Therefore, smaller droplets will evaporate first in the downdraft region.
The reason why droplet deactivation is related to the CDSD broadening can be
explained in two ways. From the thermodynamic point of view, the liquid water
mixing ratio is roughly a constant at a given height for each cycle (see
Fig. a). As the n decreases due to the droplet deactivation, we
can expect that on average droplet size will be larger because the same
amount of water will be redistributed on fewer cloud droplets. From the
kinetic point of view, quasi-steady state supersaturation (sqs) will
become larger after each cycle due to droplet deactivation, as shown in
Fig. b. sqs, the environmental supersaturation in
quasi-steady state, is inversely proportional to the integral of the mean droplet
size r‾ and the droplet number concentration (n), sqs∝(r‾n)-1
e.g.,. Here the decrease
in n due to droplet deactivation is much greater than the change of
r‾; therefore, sqs will increase with decreasing n. This
means that larger droplets grow even faster in the updraft region, and
smaller droplets evaporate even faster in the downdraft region – beyond the
solute effect alone. Conversely, an increase in sqs will enhance droplet
deactivation for smaller droplets, and it will also reinforce the growth of
larger droplets in a positive feedback.
Microphysical properties at cloud top for different cases: rmax is the largest cloud droplet radius
in a moving size grid, rmin is the smallest cloud droplet radius in a grid, r‾ is the mean cloud droplet
size, σ is the standard deviation of the droplet radius, σ/r‾ is the relative dispersion and n is
the cloud droplet number concentration. Case 0 is when the cloud parcel reaches the cloud top for the first time
with the same setup as the control case (shown as black circle in Fig. ). For other cases, results
represent the parcel at cloud top for the last time after 3 h simulation; an example of the control case
is shown as the green circle in Fig. . Bold values represent cases of broader cloud droplet size distribution with relative dispersions larger than 0.15.
rmax (µm)
rmin (µm)
r‾ (µm)
σ (µm)
σr‾
n (cm-3)
deactivation
reactivation
Case 0
9.1
4.2
5.8
0.5
0.088
654
no
no
Ascending only
17
12
13
0.55
0.041
654
no
no
Control
17
6.1
7.5
1.6
0.22
260
yes
yes
αm=0.06
17
5.1
7.0
1.9
0.27
299
yes
yes
Ngrid=200
17
5.9
7.5
1.6
0.22
260
yes
yes
Pure water
7.8
5.9
6.0
0.086
0.014
654
no
no
Only solute effect
13
5.8
6.0
0.21
0.035
654
no
no
Without reactivation
18
7.9
10
1.1
0.11
111
yes
no
Low Na
16
9.6
11
0.40
0.036
92
no
no
High Na
17
3.1
4.7
1.5
0.32
913
yes
yes
Low w
13
7.7
8.8
0.60
0.068
191
yes
no
High w
17
4.6
5.3
1.0
0.19
695
yes
yes
Thin ΔH
17
6.2
8.5
1.4
0.16
192
yes
yes
Thick ΔH
9.0
4.1
5.8
0.50
0.087
654
no
yes
One question relevant to precipitation initiation is how fast can the largest
cloud droplet grow in an oscillating parcel compared with droplets in an
ascending-only parcel? For the latter case, the cloud parcel ascends at a
vertical velocity of 0.5 m s-1 for 3 h with the same initial
condition as the control case. At the end of the simulation, the cloud parcel
reaches about 6000 m and cloud droplets are supercooled (around 248 K),
but we ignore ice nucleation in this study. The mean (yellow dashed line) and
largest/smallest (upper/lower gray dashed lines) cloud droplets in an
ascending-only cloud parcel are also shown in Fig. d. It can be
seen that the size of the largest cloud droplet in a moving size grid at the
cloud top in each cycle of the oscillating parcel (blue color bar) is similar
to that in the ascending-only parcel (upper gray line). This is quite
surprising because when the parcel reaches 1200 m for the first time
(i.e., the top of the oscillation cycle), the largest cloud droplet radius is
9.07 µm (see Table and Fig. c); however after
several cycles, the largest cloud droplet radius at 1200 m is 17.3 µm. The size is similar to the largest droplet size associated with a
moving size grid in an ascending-only parcel at a height of about 6000 m.
This means that the largest cloud droplet size for a grid in an oscillating
parcel at 1200 m is much larger than calculated from a traditional cloud
parcel model (ascent only), and hence shows “superadiabatic” growth. In
addition, the size of the smallest cloud droplet for a grid and the mean
droplet size are larger in an ascending-only parcel. Differences between the
mean droplet sizes increases after each cycle, especially at the end of the
simulation due to the reactivation of numerous small droplets. Therefore, the
relative dispersion, which is the ratio of the standard deviation to the mean
of a droplet size distribution, also increases after each cycle, and is much
larger than in an ascending-only cloud parcel.
Sensitivity studies
In this subsection, we investigate the effects of several factors on the CDSD in
the adiabatic parcel model with vertical oscillations. Previous studies show
that aerosol number concentration and vertical velocity are the two most
important factors controlling cloud properties in an adiabatic cloud parcel
model e.g.,. Two regimes are
frequently considered: an aerosol-limited regime exists when there is an ample
supply of water, and the cloud droplet number concentration is limited by the
aerosol number concentration; and an updraft-limited regime exists when
supersaturation is starved, and the cloud droplet number concentration is
limited by the updraft velocity. In the updraft-limited region, cloud
droplets will compete with each other for the limited available water, and
the larger aerosols will suppress the activation of smaller aerosols
. Based on , the
aerosol-limited regime exists when the ratio of the vertical velocity to
droplet number concentration, w/n, is larger than 10-3 m s-1 cm3 and the updraft-limited region occurs when the w/n ratio is smaller
than 10-4 m s-1 cm3. For the control case, the w/n ratio is
7×10-4 m s-1 cm3, which is in the transitional regime. In
this subsection, we choose several values of aerosol number concentration and
vertical velocity to investigate the CDSD in the aerosol-limited and
updraft-limited regimes. In addition, we also test
the effect of the recirculation layer thickness on the CDSD broadening.
Effect of total aerosol number concentration
We test two other aerosol number concentrations, 102 and
104 cm-3, and keep the median radius and geometric standard
deviation the same as the control case (see Fig. a and c).
These values are chosen to represent the conditions for clean clouds
(102 cm-3) and polluted clouds (104 cm-3), which are
consistent with previous studies e.g.,. Considering a
vertical velocity of 0.5 m s-1, they also represent the
aerosol-limited regime (the 102 cm-3 case leads to a w/n ratio of
5×10-3 m s-1 cm3) and the transition regime (the 104 cm-3 case leads to a w/n ratio of
4×10-4 m s-1 cm3). The results show that the CDSD for the relatively clean case
(102 cm-3) behaves similarly to the solute effect alone (compare
Figs. b and b) – there is neither droplet deactivation
nor reactivation. The CDSD broadening is due to the ripening effect alone,
which is not as efficient as when it is accompanied by deactivation as in the
control case. For the relatively polluted case (104 cm-3), both
droplet deactivation and reactivation occur (see Fig. d). The
largest cloud droplet acts similarly to that in the control case, while the
smallest cloud droplet is larger 1.5 h into the simulation but then begins
to become smaller compared with the control case. We interpret these
observations as follows. For the clean case, all aerosols are activated, and
all droplets are able to grow to a relatively large size, making them
unlikely to deactivate. However for the polluted case, not all CCN are activated;
there are consequently some smaller droplets that cannot grow very large and will evaporate first in the downdraft region. Another explanation from
is that the CDSD broadening occurs when air supersaturation
(Se) is smaller than the critical supersaturation for the smallest cloud
droplets (Ssat(rsmall)). For this condition, the smallest cloud
droplets evaporate and the largest cloud droplets might grow slightly if
Se>Ssat(rlarge) or evaporate slightly if
Se<Ssat(rlarge), thus leading to broadening. If the water vapor
mixing ratio in air is much larger on average than the saturated water vapor
mixing ratio over droplet, only narrowing of the CDSD occurs. Because
in-cloud supersaturation decreases with increased aerosol concentration, it
is expected that the Ostwald ripening is more efficient in polluted cloud,
which is also consistent with .
(a) Aerosol size distribution for a low number concentration of 102 cm-3. (b) Cloud droplet
size distribution changes with time for the low aerosol number concentration case. (c) Aerosol size distribution
for the high number concentration of 104 cm-3. (d) Cloud droplet size distribution changes with time
for the high aerosol number concentration case. Gray lines in (a) and (c) represent the control case with a total
aerosol number concentration of 103 cm3, and gray regions in (b) and (d) are the range of the cloud
droplet size spectrum for the control case.
Effect of vertical velocity
Two vertical velocities (0.1 and 1.0 m s-1) are used to
test their influence on CDSD broadening. These values are chosen based on
observations that updrafts in stratocumulus clouds are in the order of 0.1 m s-1 and in cumulus clouds are in the order of 1.0 m s-1
. Results also show that they correspond to the
aerosol-limited regime (the 1.0 m s-1 case leads to a w/n ratio of
10-3 m s-1 cm3) and the transitional regime (the 0.1 m s-1 case leads to a w/n ratio of 5×10-4 m s-1 cm3).
For a relative low velocity of ±0.1 m s-1, the cloud parcel only
experiences one and a half cycles within 3 h (see Fig. a). The parcel reaches the cloud base in around 1 h, which is significantly
later than the control case due to the small velocity (see Fig. a). However, the largest cloud droplet size ultimately becomes
similar to that in the control case, and we also see the cloud droplet number
concentration decrease due to droplet deactivation. No droplet reactivation
occurs because the small velocity generates a low supersaturation in the
updraft region, which is unfavorable for droplet reactivation. For a relative
high velocity of ±1.0 m s-1, the cloud parcel can cycle more times
within 3 h (see Fig. c). The parcel reaches cloud base
faster than the control case (see Fig. c). Here we keep the
thickness of the recirculation layer constant. Therefore, larger vertical
velocity results in a higher oscillation frequency. Both droplet deactivation
and reactivation occur in this case, and the largest and smallest cloud
droplets behave similarly to the control case.
(a) The height of cloud parcel changes with time for the low velocity case of ± 0.1 m s-1.
(b) Cloud droplet size distribution changes with time for the low velocity case. (c) The height of the cloud
parcel changes with time for the velocity of ±1.0 m s-1. (d) Cloud droplet size distribution
changes with time for the high velocity case. Red and blue lines in (a) and (c) represent ascending and descending parcels, and gray lines represent the control case with
velocity of ±0.5 m s-1. The gray regions in (b) and (d) are the range of cloud droplet spectrum for the control case.
Effect of the thickness of the recirculation layer
Turbulence driven by cloud-top radiative cooling can result in various eddy
sizes in the stratocumulus-topped boundary layer . Two
different recirculation layer depths are tested, 150 and 350 m, to
investigate the effect of eddy size on CDSD broadening. For a recirculation
layer of 150 m, which is 100 m thinner than the control case, the
parcel experiences more cycles within 3 h (see Fig. a).
The total cloud droplet number concentration decreases with time due to
droplet deactivation, but no droplet reactivation occurs (see Fig. b). Therefore the largest cloud droplet is similar to the control
case, but the smaller cloud droplet is larger than in the control case. For a
recirculation layer of 350 m, the parcel can penetrate the cloud base
each cycle (see Fig. c). In this case, all cloud droplets are
deactivated below cloud base and reactivated again when the cloud parcel is
supersaturated in the next ascending branch. Therefore the CDSD is repeated
and no broadening occurs.
Discussion
We have studied the effects of total aerosol number concentration, updraft
velocity and the thickness of the recirculation layer on CDSD broadening.
However we note that there are other parameters used in this study that can
lead to the uncertainties in the results. For example, found
that using an insufficient number of model grids will lead to the narrow CDSD
reported by . found that both the
spectral discretization and the uncertainty in the value of the mass
accommodation coefficient can lead to uncertainty in the results. To test the
effects of the mass accommodation coefficient and spectrum discretization on the
CDSD, two more sensitivity studies are conducted. One case is to set the mass
accommodation coefficient (αm) to 0.06 based on .
It is expected that a smaller value of αm might suppress the growth
of cloud droplets. The other case is to change the number of grids from 100
to 200, while keeping other parameters the same as in the control case.
(a) The height of cloud parcel changes with time for the thin recirculation layer of 150 m.
(b) Cloud droplet size distribution changes with time for the thin recirculation layer case. (c) Aerosol
size distribution for the thick recirculation layer of 350 m. (d) Cloud droplet size distribution
changes with time for the thick recirculation layer case. Red and blue lines in (a) and (c) represent ascending and descending parcels, and gray lines in represent the control
case with recirculation layer of 250 m. The gray regions in (b) and (d) are the range of cloud droplet
size spectrum for the control case.
Table summarizes the microphysical properties at cloud top for
different cases. When the cloud parcel first reaches about 1200 m, the
largest cloud droplet radius associated with a moving size grid (rmax)
is 9.1 µm (case 0). If the cloud parcel continues rising for 3 h as for the ascending-only case, rmax=17 µm at 6000 m.
However if the parcel experiences recirculation within cloud region,
rmax can also be around 17 µm as long as deactivation occurs,
except for the low Na case (see Table ). If reactivation also
occurs, the smallest cloud droplet radius associated with a moving size grid
rmin is around 5 µm and the relative dispersion is larger than
0.1. It is interesting to note that low mass accommodation has a negligible
effect on rmax, but it has a stronger impact on rmin. This will
result in a broader CDSD compared with the control case. In addition, a low
mass accommodation coefficient inhibits the growth of cloud droplets and
leads to more activated cloud droplets . Results for 200 grids
are similar to those from the control case, which means that the 100 grids used
in this study are enough to limit the uncertainty due to spectrum
discretization.
From the above, we see that droplet deactivation and droplet reactivation
play crucially important roles in CDSD broadening in this study. Deactivation
of smaller droplets is important for the growth of larger cloud droplets
(e.g., see Figs. d, c, d, b, d and b). Droplet deactivation occurs in the descending branch for
smaller droplets due to both the curvature and solute effects (Ostwald
ripening). The evaporation of smaller cloud droplets with less solute makes
water vapor available for the growth of other larger cloud droplets. On
average, the largest cloud droplet size for a moving size grid increases with
time after each cycle.
Results from the sensitivity studies show that the relative dispersion is
larger than 1.5 for relatively polluted conditions when both deactivation
and reactivation occur (see Table ), which is consistent with the
values from observations and simulations
e.g.,. However the relative dispersion
has also been found to be larger than 1.5 for relatively clean conditions
e.g.,. This might be due to other
mechanisms, such as supersaturation fluctuations in a turbulent environment
or the collision coalescence process. It should be mentioned that the CDSD
observed in previous studies might have the problem of instrumental
broadening due to low instrument resolution or long-distance averaging of the
sampling volume . A broad CDSD is also observed
by recent holographic measurements, which limit the effect of instrument
broadening and have much higher temporal and spatial resolution than other
instruments, such as particle-counting probes
.
We note that deactivation is suppressed for a thin recirculation layer
ΔH=150 m as shown in Fig. b, and therefore the CDSD
broadening is not as efficient as the control case. However, the vertical
oscillations of an air parcel due to turbulence might be much smaller than
150 m. did not observe the enhanced CDSD broadening by
deactivation and reactivation with a shallower recirculation layer. One
interesting question is whether deactivation or reactivation can be inhibited for
a very thin recirculation layer. To answer this question, three more cases
are carried out with recirculation layers of 50, 5 and 1 m.
All these cases have the same setup as the control case except for the
thickness of recirculation layer. The CDSD and total cloud droplet number
concentration for each case are shown in Fig. . It can be seen
that reactivation is inhibited for all cases, but deactivation always occurs.
More interestingly, the CDSD for all these three cases are similar, and the
decrease of total cloud droplet number concentration due to deactivation is
also similar. The evolution of the CDSD for a thin recirculation layer is
independent of air motion and degrades to a steady state where the CDSD
broadening is due to Ostwald ripening in a still environment.
Cloud droplet size distribution (CDSD) changes with time for different thicknesses of recirculation
layers: (a) ΔH=50 m, (b) ΔH=5 m and (c) ΔH=1 m. (d) Total cloud droplet number
concentration (n) changes with time for the different cases. The gray region in (a)–(c) represents the range
of the droplet size spectrum for the control case, and the black lines represent the mean cloud droplet
radius change with time.
One interesting result is that the size of the largest cloud droplet
associated with a moving size grid within each cycle is similar to that in
the ascending-only parcel (i.e., approximately within one micrometer), as
shown in Fig. . The general trends approximately follow the
growth rate that is independent of aerosol number concentration, vertical
velocity and the thickness of the oscillation layer, as long as deactivation
occurs. This suggests that the growth of the largest cloud droplets strongly
depends on the amount of time such droplets remain in the cloud (residence
time of cloud droplets), rather than the temporal variability of
supersaturation in updrafts and downdrafts. The reason for this is that the
environmental (i.e., the in-cloud) saturation ratio (Se) is buffered by
the equilibrium saturation ratio (Ssat) over smaller droplets. Figure shows the changes of Se and
Ssat over two droplets (same
used as in Fig. c) in the control case. Instead of being
symmetric around one for the pure water case (ignoring solute and curvature
effects), Se in the oscillating parcel is symmetric around Ssat
over the small cloud droplets. For example before 1.5 h, droplets formed
on ra=51 nm are the smallest cloud droplets in the population, and the
average Se (gray line) during one oscillation is roughly symmetric
around the blue line (Fig. ). The fact that Se is buffered
by Ssat over small cloud droplets is mainly because the number
concentration of the smallest cloud droplet (36 cm-3 in the control
case) is much larger than that of large cloud droplet (1.8×10-9 cm-3). When those small droplets deactivate
(between 1.5 and 2.5 h), Ssat (blue line) for those deactivated droplets is the same as
Se (gray line). During this period, Se is symmetric around
Ssat over the remaining small droplets (larger than the droplets formed
on ra=51 nm but smaller than for ra=503 nm). When the droplets
formed on ra=51 nm are reactivated (after 2.5 h), Se is
symmetric around Ssat(ra=51 nm) again until they are deactivated.
It should be mentioned that the number concentration of those reactivated
droplets increases steadily after each cycle after 2.0 h (See Fig. d). By the end of the simulation, the number concentration of the
reactivated droplets is similar to that of the remaining large droplets
(about 150 cm-3). Therefore, the effect of those reactivated droplets
on the environmental saturation ratio becomes stronger after 2.0 h (see
Fig. ).
The largest cloud droplet size after each cycle is plotted for the different previously discussed cases:
blue dots, control case; red dots, no reactivation case; pink dots, high number concentration case; green
dots, high vertical velocity case; and black dots, thin oscillation layer case. The gray line is for the
ascending-only case from Fig. , and the red line represents the growth of a droplet.
This symmetric property of Se can be also explained using the
quasi-steady supersaturation sqs which for pure water droplets is expressed as sqs∼wnr . This can be obtained from the analytical
expression of supersaturation in an adiabatic cloud parcel:
dSedt=Aw-Bnr(Se-1), where A and B are parameters depending
on thermodynamic properties . A symmetric distribution of
w around zero will generate a symmetric distribution of sqs around zero
(i.e., Se around one). If curvature and solute effects are considered,
sqs will be symmetric around sk given the same condition of w,
because dSedt=Aw-Bnr(Se-Ssat) and thus sqs∼wnr+sk, where sk=Ssat-1 is the equilibrium
supersaturation ratio over a monodisperse droplet. In the updraft region,
all droplets grow and the effect of sk is negligible. In the downdraft
region and for polydisperse cloud droplets, the large number of small cloud
droplets buffers the environmental conditions. Therefore Se is symmetric
around Ssat over smaller droplets before they deactivate in the
oscillating parcel. Se‾-Ssat controls the growth of a large
droplet and it is positive on average. That is why the large droplets can
grow after each cycle. In addition, the influence of Se fluctuations on
droplet growth is small if Ssat over a large droplet is much lower than
Se and its fluctuations. The extreme examples of this phenomenon are
when droplets form on GCCN in warm clouds or ice particles
form in mixed phase clouds. Therefore, the growth of the large droplet here
is dominated by its in-cloud lifetime. Previous studies show that although
the mean lifetime of cloud droplets is usually less than half an hour, the
residence time for some lucky cloud droplets can be longer than 1 h
e.g.,. Those long-lifetime cloud
droplets might contribute to large droplets in the cloud, similar to
long-lifetime ice particles in mixed-phase clouds .
Changes of the environmental saturation ratio (gray) and the equilibrium saturation ratios over two
droplets (red and blue) with time in an oscillating parcel. The blue line is for a droplet formed on
a dry aerosol with radius of 53 nm and the red line is for a droplet formed on a dry aerosol with
radius of 503 nm. The smaller cloud droplet (formed on a dry aerosol with radius of 53 nm)
deactivates at approximately 1.5 h and reactivates at approximately 2.5 h.
However if all cloud droplets are deactivated, CDSD broadening does not occur
(see Fig. d). Without droplet deactivation, the CDSD can also
broaden due only to the solute effect, as is the case when the curvature
effect is ignored (Fig. b) or when the total aerosol number
concentration is low (Fig. b). CDSD broadening due to the
ripening effect without droplet deactivation is not as significant as it is
with droplet deactivation, but it might also be important after several hours
as suggested by .
Droplet reactivation usually occurs in the updraft region after several
cycles, and those reactivated droplets will be deactivated again in the
downdraft region. Formation of smaller cloud droplets can broaden the CDSD at
smaller sizes, decrease the mean cloud droplet size, and increase the
relative dispersion. Meanwhile, the generation of new cloud droplets also
suppresses the growth of larger cloud droplets (see Fig. d).
In summary, the results of this study show that the CDSD can be broadened in
a vertically oscillating cloud parcel if both solute and curvature effects
are considered, consistent with the findings of previous studies
e.g.,. Although our model uses an idealized setup, the
sensitivity studies help explore the conditions under which this mechanism
may be important in the real clouds. The results show that CDSD broadening
due to Ostwald ripening can be enhanced in relatively polluted conditions
when deactivation and reactivation occur, such as typically exists for
continental clouds. For relatively clean conditions like marine clouds, other
CDSD broadening mechanisms might be more relevant, such as the collision
coalescence process or supersaturation fluctuations due to turbulence. When
deactivation and reactivation occur, the simulation results show that the
smallest cloud droplets do not change significantly after each oscillation
cycle, while the largest cloud droplets grow on average after each cycle. The
growth of the largest cloud droplet depends on its in-cloud lifetime. This is
because, due to the solute effect, the saturation water vapor pressure over
larger cloud droplets is smaller than the environmental water vapor pressure
that is buffered by numerous smaller cloud droplets with smaller amounts of
solute. It should be mentioned that the system is buffered by smaller cloud
droplets formed on smaller CCN when the number concentration of those
droplets is much higher than that for the largest cloud droplets formed on the
largest CCN. This may not be true under relatively clean conditions, where
the environmental supersaturation can be affected by droplets formed on the
largest CCN.
Conclusions and atmospheric implications
In this study, we investigate the condensation growth of cloud droplets in an
adiabatic parcel with vertical oscillations based on a moving-size-grid cloud
parcel model where cloud droplets are formed on polydisperse, submicrometer
aerosol particles. Both the solute and curvature effects are considered for
all cloud droplets before and after activation during the whole simulation.
The CDSD can also broaden by condensation growth due to Ostwald ripening
together with droplet deactivation and reactivation, which is consistent with
the results of . Droplet deactivation occurs in the
descending branch due to the combination of the solute and curvature effects.
Deactivation of smaller droplets makes water vapor available for other larger
droplets, and thus broadens the CDSD at larger sizes. The growth of the
largest cloud droplet in a vertically oscillating cloud parcel approximately
follows the growth rate in an ascending-only cloud parcel after each cycle,
and it is independent of aerosol number concentration, vertical velocity and
the thickness of the oscillation layer, as long as deactivation occurs. The
size of the largest cloud droplet strongly depends on the time that droplet
remains in the cloud rather than on the variability of the in-cloud
supersaturation. This is because the large number of smaller cloud droplets
buffers the environmental air: the environmental saturation ratio in an
oscillating parcel is symmetric around the equilibrium saturation ratio over
smaller cloud droplets. The growth rate for the largest cloud droplets can be
used to roughly estimate the large-size upper boundary of the CDSD, at least
in this study. Droplet reactivation usually occurs after a few cycles. These
cloud droplets are activated in the ascending branch, and deactivated in the
descending branch. They are usually very small (less than 5 µm) and
thus broaden the CDSD at smaller sizes. The mean cloud droplet size
significantly decreases when reactivation occurs, which leads to an increase
in relative dispersion. Conversely, the newly formed cloud droplets
compete against other cloud droplets for water vapor, thus suppressing the
growth of larger cloud droplets.
We note that there are additional factors that might affect droplet growth
that are not treated in this study. For example, we do not consider the
sedimentation of cloud droplets in this study, similar to
and . This is a reasonable assumption for an updraft velocity
of 0.5 m s-1 or above, but ignoring sedimentation in the low velocity
case (0.1 m s-1) will limit the accuracy of our results. In addition,
we do not consider the collision coalescence between droplets. Although CDSD
broadening is favorable for collision processes, it might be interesting to
determine how this broadening will accelerate rain formation.
We have used idealized simulations to analyze the CDSD broadening in a
vertically oscillating cloud parcel due to Ostwald ripening. There are three
necessary conditions for this CDSD broadening mechanism. The first condition
is that droplets form on polydisperse aerosol particles where larger cloud
droplets contain more solute. This is a very general occurrence in the
atmosphere due to the complexity of aerosol size and composition
. The second condition is that a cloud experiences
upward and downward oscillations. This is also a general occurrence in
natural clouds due to turbulence and circulations that can become established
within a cloud layer . The third condition is that cloud
droplets have a long in-cloud residence time, e.g., longer than 1 h. This
is consistent with previous studies that cloud droplet residence time plays
an important role in CDSD broadening due to the Ostwald ripening effect
. We expect that this mechanism of CDSD
broadening is possible in the real clouds under those specific conditions.
It should be mentioned that one limitation of this study arises from the use
of the adiabatic assumption for 3 h simulations. Turbulence can result
in not only upward and downward oscillations but also in entrainment and
mixing . The latter can cause cloud droplet
evaporation, deactivation and reactivation . In
addition, the lifetime of the cloud parcel is usually less than 1 h
. Therefore, one should be aware that results in this
study are based on a very idealized state. More realistic studies should
consider mixing processes where for example a trajectory ensemble model would
be a suitable tool . How important this
mechanism is to CDSD broadening in real clouds compared with other mechanisms
is worth future investigation, but is beyond the scope of this study.
There is an implication of this mechanism for the cloud modeling community.
Most of the bulk and bin microphysical schemes only consider the curvature
and solute effects during the activation process based on Köhler theory,
and cloud droplets are assumed to be pure water after they are activated.
Tracking the solute distribution for each bin of cloud droplet is possible
using a joint 2-D bin aerosol-cloud microphysical scheme, but it is very
computationally expensive e.g.,.
The mechanism of CDSD broadening in this study requires the model to consider
both solute and curvature effects all the time (i.e., before and after
activation, deactivation and reactivation). Our results suggest the
importance of solute and curvature effects to the deactivation and
reactivation processes, which are consistent with previous studies
e.g.,. However the
results are counter to some other studies where details of activation and
deactivation are argued to be unimportant in the cloud simulation
e.g.,. Large eddy simulations
with a similar microphysical treatment would be useful to investigate how
important this mechanism is to CDSD broadening in more realistic clouds.