The effect of 1-D and 3-D thermal radiation on cloud droplet growth in
shallow cumulus clouds is investigated using large eddy simulations
with size-resolved cloud microphysics. A two-step approach is used
for separating microphysical effects from dynamical feedbacks.
In step one, an offline parcel
model is used to describe the onset of rain. The growth of cloud droplets to raindrops is
simulated with bin-resolved microphysics
along previously recorded Lagrangian trajectories. It is shown that thermal
heating and cooling rates can enhance droplet growth and raindrop
production. Droplets grow to larger size bins in the 10–30 µm radius
range. The main effect in terms of raindrop production arises from recirculating parcels,
where a small number of droplets are
exposed to strong thermal cooling at cloud edge. These recirculating
parcels, comprising about 6 %–7 % of all parcels investigated, make up
45 % of the rain for the no-radiation simulation and up to 60 % when 3-D
radiative effects are considered. The effect of 3-D thermal radiation
on rain production is stronger than that of 1-D thermal
radiation. Three-dimensional thermal radiation can enhance the rain amount up to 40 %
compared to standard droplet growth without radiative effects in this
idealized framework.
In the second stage, fully coupled large eddy simulations show that
dynamical effects are stronger than microphysical effects, as far as the
production of rain is concerned. Three-dimensional thermal radiative effects again exceed
one-dimensional thermal radiative effects. Small amounts of rain are
produced in more clouds (over a larger area of the domain) when thermal
radiation is applied to microphysics. The dynamical feedback is shown to be an
enhanced cloud circulation with stronger subsiding shells at the cloud
edges due to thermal cooling and stronger updraft velocities in the
cloud center. It is shown that an evaporation–circulation feedback reduces the amount of rain
produced in simulations where 3-D thermal radiation is applied to
microphysics and dynamics, in comparison to where 3-D thermal radiation
is only applied to dynamics.
Introduction
Cloud droplets form in saturated environments by condensation of water vapor on
cloud condensation nuclei (CCN). In the first phase of its lifetime,
cloud droplet growth follows Köhler theory . If a certain critical radius is
reached a droplet can grow further, following diffusional droplet growth
theory. From a certain droplet size onward, rain formation processes such as
collision and coalescence dominate growth .
The droplet size distribution in clouds has important implications for the
Earth's atmosphere. The size distribution of droplets determines how much
solar radiation is reflected back to space. Smaller droplet sizes
reflect more radiation back to space (for constant liquid water), thus leading
to a cooling of the atmosphere, while larger droplets allow radiation to
penetrate more easily to the surface, thus allowing more radiation to
be absorbed .
Furthermore, the droplet size distribution determines the formation of rain in
clouds. Droplets that reach the 10–30 µm radius range can lead to rain
formation. Only very small numbers of droplets of this size (on the order of 1 per
liter) are necessary to initiate the process of collision and coalescence.
It is known that a broad droplet size spectrum is necessary for these processes
to start; however, cloud droplet growth in the diffusional growth theory slows
down when droplets reach 10 µm and collision and coalescence is not yet
effective (the so-called collision–coalescence bottleneck ). Different
processes can cause broadening of the droplet size spectra, e.g.,
turbulence , the associated supersaturation
fluctuations , giant CCN
and radiation
. As soon as rain is initiated, the
cloud system morphology and intrinsic properties can change as the
dynamics of the system change.
Radiative effects on cloud droplet growth have been studied in various ways in
the past. Among the earliest are the studies by and
. Both analyzed the growth of an individual droplet and showed
that droplets can grow to 20 µm and larger by radiative cooling, even in a
subsaturated environment.
and studied
the effect of radiation on the growth of a droplet population.
showed increased droplet growth in the diffusional
droplet growth regime, while also included
collision–coalescence and found earlier onset of rain by a factor of 4.
An important issue of the application of radiation to droplet
growth is the timescale of the temperature exchange.
estimated the time until
droplets reach a steady state in temperature exchange. For most droplet sizes,
the timescale was small enough to make the assumption of a steady state
system feasible. simulated radiative fog, thus including
microphysical and dynamical feedbacks. The inclusion of the radiative term in the droplet growth
equation had important consequences for the lifetime of fog.
The enhanced growth of larger droplets by radiation and associated gravitational settling
caused a reduction of liquid water in the fog. The oscillation of liquid water
(with a period of 15–20 min) could only be simulated by including radiative
effects. simulated stratocumulus clouds,
using a bin microphysical model including 1-D radiation. The stronger diffusional
growth in the simulations with radiative effects reduced supersaturation and
therefore the number of small droplets. The reduced number of
droplets and the larger droplet size resulted in more drizzle and therefore a lower
cloud optical thickness. Observations of nocturnal
stratocumulus were remodeled with Lagrangian parcels by . They stated that “a
simple Lagrangian model suggested that the larger drops grew within the zone
of high net radiative loss around cloud top”. used a large eddy simulation
(LES) and an independent parcel model, including bin microphysics
and radiative effects on droplet growth. They showed that only parcel trajectories
spending long periods of time at cloud top (10 min or more) can cause the
droplet size spectrum to broaden via radiative cooling. They also found an earlier onset of drizzle
production; however, this occurred along parcels that would produce drizzle
anyhow. They concluded that radiative cooling may reduce the time for drizzle onset.
The recent theoretical study of and direct numerical
simulations by re-emphasize the hypothesis that thermal radiation might
influence droplet growth significantly and lead to a broadening of the
droplet size spectra and thus enhance the formation of
precipitation. Similarly, investigated the effect of thermal
radiation on rain formation in a precipitating shallow cumulus case and found broadening of the
droplet size spectrum and earlier rain formation.
In this study, we investigate the role of thermal radiation on cloud droplet
growth in cumulus clouds. The limited lifetime of cumulus clouds changes
the radiative impact compared to former studies where stratiform
clouds where investigated. The finite size of the cumulus
clouds and the high local cooling rates of several
hundred kelvin per day (K d-1) at cloud top and at cloud sides (e.g., )
suggest that the investigation of 3-D thermal radiation effects
might have a significant effect on droplet growth.
(their Fig. 11) showed that local peak differences in cooling rates between 1-D and 3-D
thermal radiation in cumulus cloud fields can reach 20 %–120 %,
depending on the cloud field resolution. But the differences between
1-D and 3-D thermal radiation are not only focused on local grid
boxes. and showed that layer
averaged 1-D and 3-D heating and cooling differences can be up to 1 K d-1,
which is the same order of magnitude as clear-sky cooling. Whether the stronger local 3-D cooling affects droplet growth compared
to 1-D thermal cooling and if cooling in general causes changes in
droplet growth in cumulus clouds are questions addressed in this
study. The focus of this work is on thermal radiative effects on
droplet growth. At the end of the study, we will briefly investigate
thermal radiative effects on dynamics as shown by
, or .
The paper is structured as follows: Sect. provides
the necessary theory and Sect. the model setup. Section
analyzes the results of our study. Summary,
conclusion and outlook are provided in Sect. .
Theory
The energy budget at a droplet surface is described by Eq. (), which combines
water vapor diffusion to the droplet and latent heat release, where
lv is the latent heat, dm dt-1 the change in mass (m) over time (t),
r the droplet radius, K the thermal diffusivity, and Td
and Tinf the droplet temperature and the temperature of the
surrounding air:
lvdmdt=4πrK(Td-Tinf).
Following , the equation can be extended by a radiative term
(Eq. ), where HRλ(r)=4πr2qabs,λ(r)Fnet,λ(r) is the emitted or absorbed power of an
individual droplet. qabs,λ(r) is the absorption efficiency per droplet
radius and wavelength (λ), and Fnet,λ(r) is the net radiative gain or loss of a
droplet per radius and wavelength in watt per square meters (W m-2).
lvdmdt=4πrK(Td-Tinf)+∫λHRλ(r)dλ transformed the equation to the
notation of the bin microphysical model of . We will
follow their notation in the following, as we use the same bin microphysical
model. Thus, Eq. () becomes
dmdt=C(P,T)m2/3m1/3+l0[η(t)+J(P,T)m1/3HR(m)],
where l0 is a length scale representing gas kinetic effects,
η(t) the excess specific humidity (qv-qs(T)) and
C(P,T)=4πCrs; C=RvTinfDes(Tinf)+lvTinfK(lvRvTinf-1),
where D is the diffusion coefficient, Rv is the specific gas constant
for moist air and es is the saturation vapor pressure.
J(P,T) summarizes constants concerning the radiative term J(P,T)=rslvαcKRvT with rs the
saturation mixing ratio, αc=[34πρl]13 and ρl the liquid water density.
We note that the radiative cooling is an increasing function of droplet mass.
HR(m) of a droplet is the wavelength band (i) integrated radiative
gain or loss, weighted by the absorption efficiency for a
mass size bin (k) for the bin microphysical
model. showed that the radiative term HR(m)
can be approximated with the mean mass (m‾k) of a drop
size bin k:
HR(m)=∑iNbandsqabs,i(m)Fnet,i4≈∑iNbandsq‾abs,i(m‾k)Fnet,i=HR(m‾k).
This radiative term must be included in the equation for
supersaturation and for droplet growth. The equation for the
supersaturation, in our case water vapor excess η, is
dηdt=D-A(P,T)dMdt,
where the function A(P,T) connects the integrated mass growth rate dMdt-1
to changes in η.
Including the integrated radiative terms of the mass growth rate R, Eq. () becomes
η(t)={[η(t0)-DG]e-G(t-t0)+DG}-RG[1-e-G(t-t0)],
where D represents the increase or decrease in η due to dynamics, G is the contribution to η from the standard droplet growth and
R the contribution to η from radiatively driven
droplet growth. Here it can be seen that the additional radiative term
can increase or decrease η due to radiative heating or cooling. For a more detailed explanation the reader is referred
to , their Eqs. (6)–(10).
For solving the condensation equation in the two-moment framework of ,
where both mass and number in a bin k are predicted,
Eq. () has to be integrated over one time step from t0
until tf. Again, we follow
to calculate the forcing τ (the gain or
loss of mass of a droplet) of the droplet growth equation:
∫momfm1/3+lom2/3=C(P,T)∫t0tfη(t)dt+m‾k(1/3)C(P,T)7J(P,T)HR(m‾k)Δt=τd+τr=τ,
where τ is the combined dynamic (τd) and radiative
(τr) forcing of the droplet growth equation, and m0 and mf are
the initial and final mass of the droplet before and after condensation or evaporation.
What remains now is to derive the radiative term Fnet,λ(r) in
Eq. () from the radiation scheme in the LES model. Heating rates in
LES models are calculated spectrally from bulk water. These heating
rates include contributions from liquid water (cloud water) as well as
water vapor and other atmospheric gases. Former studies, e.g., and , used a 1-D radiative
transfer approximation and calculated the individual droplet absorption and emission from the upwelling
and downwelling fluxes. We, however, include 3-D radiative effects. Our 3-D radiative transfer approximation is designed to provide 3-D
heating rates. We estimate the individual droplet emission or absorption from a
volume heating rate and therefore have to separate the heating or cooling from the liquid water
phase (HRliquid) from the total heating or cooling (from liquid water and
atmospheric gases, HRtot). We follow the approach of which
showed the relationship between heating or cooling rates and the actinic
flux F0 to be
8HRtot,λ=-kabs,λF0,9HRliquid,λ=-kabs,liquid,λF0,
where kabs is the total absorption coefficient and kabs,liquid the absorption coefficient of liquid water.
Combining these two equations it follows that the heating or cooling rate resulting from the
liquid water absorption is
HRliquid,λ=-kabs,liquid,λkabs,λHRtot,λ.
This total heating rate now has to be distributed among all droplets
in the volume. The total heating or cooling from the liquid water of a grid box (for
a single wavelength λ
or wavelength band i) is the sum of all
droplet contributions to the heating or cooling:
HRliquid,λ=∫n(r)hλ(r)dr,
where n(r) is the number of droplets of radius r per radius interval dr,
and hλ(r)=4πr2qabs,λ(r)Fnet,λ(r) is the heating or
cooling rate of each droplet at radius r with the absorption efficiency qabs,λ(r)
and the net heating of each droplet Fnet,λ(r).
Assuming steady state (e.g., ), HRliquid,λ is equally distributed
among all droplets, and the individual heating or
cooling (Fnet,λ(r)) of a droplet of size r in
Eq. () is therefore
Fnet,λ(r)=HRliquid,λ∫4πr2n(r)qabs,λ(r)dr.
Methodology
To estimate the effect of 1-D and 3-D thermal radiation on cloud droplet
growth we use a two-stage approach. First, to estimate the impact
of thermal radiation on droplet growth and to gain insight into physical
processes, we use Lagrangian parcels recorded during
a LES (System for Atmospheric Modeling, SAM;
) with the bin-emulating
two-moment bulk scheme of .
These parcel trajectories are then used to
drive an independent (offline) parcel model including a bin-microphysics scheme .
We separate
between 1-D (RRTMG, , 1DR) and 3-D thermal (, 3DR)
radiative effects and compare both results
to the droplet growth without radiative impacts (NR) and to each other. This approach allows
us to focus on the effect of thermal radiation on droplet growth, without the
interaction of changing dynamics that would occur in a fully coupled
LES.
Second, we run a fully coupled LES with the
Tel Aviv University (TAU) bin-microphysics scheme , where
1-D and 3-D thermal radiative effects (heating rates) are applied to the
droplet growth and to the dynamics, or to just one
of the two. We chose a shallow cumulus case with weak precipitation
(BOMEX, Barbados Oceanographic and Meteorological EXperiment) where we expect the effects of thermal
radiation on cloud droplet growth to be tangible and not overwhelmed by the rapid development
of precipitation encountered in deeper trade-wind cumulus environments.
We expect that it would be harder to discern these effects in a more strongly precipitating case.
In both cases the simulations were run with 75 m horizontal and 50 m vertical
resolution for a 45km×45km domain.
The simulations for the trajectories were run for 6 h in
total; the last 2 h are used for evaluation. The coupled LES cases
were run for 8 h in total.
For the first part of the study 2.7 million Lagrangian air parcel trajectories were recorded
in the last 2 h of the BOMEX simulation with a 2 s time
step. The simulation was driven by 1-D thermal radiation, but we recorded 3-D thermal
radiation along the same parcels. This allows us to compare the same parcels,
driven by the same variables (liquid water potential temperature,
pressure, vertical velocity) in the later part of the study. The
difference in the results of the independent parcel model ensemble is therefore only
due to the difference in the 1-D and 3-D thermal heating or cooling rates
and their impact on cloud droplet growth. (Changes to the approach of
, were explained in Sect. .) The
total number of aerosol particles (assumed to be ammonium sulfate) is
100 cm-3 with a median radius of 0.1 µm and a geometric
standard deviation of 1.5 (assuming a log-normal distribution).
The bin model includes diffusional growth and the growth by
collision and coalescence and covers 33 size bins with a mass doubling
from one bin to the next. The radius of the first bin (lower bound) is 1.56 µm.
Aerosol particles are activated based on the locally calculated supersaturation
and placed in the first bin.
We neglect the solute and kelvin effect in this framework, because they have a minor
impact for r>1.56µm. Kinetic and ventilation effects are taken into account.
A few comments are in order regarding our approach.
With the parcel model, we focus on the effects of thermal radiation
on microphysics, neglecting any changes in cloud development that would occur
due to feedbacks within a LES framework. A further advantage of this
method is that spurious
spectral broadening due to advection is avoided .
A key limitation of this method is that drop sedimentation is not represented;
drops do not fall off a trajectory of interest, and drops from other trajectories
do not fall onto that trajectory. Because all droplets follow
the parcel trajectory, the liquid water content (qc) is not reduced as the parcels do not
“rain out” and radiation does not change along the parcel trajectories
when the size distribution (or qc) changes. The
method is thus mostly useful for examining the onset of drizzle.
One can consider the trajectory approach to be an imperfect
but useful model (as documented in )
for examining the combined effect of droplet growth and thermal
radiation with and without the radiative effects in a framework that
allows for realistic and quantifiable exposure to strong radiative
cooling at cloud edges.
The analysis of characteristic timescales of important
processes for a droplet radius of 20 µm, such as diffusional
droplet growth (χgrowth), diffusional droplet growth with
radiation (χgrowth,rad) and sedimentation (χsed),
supports our argument about the usefulness of the
approach, despite the fact that sedimentation is not represented in the
parcel model. The characteristic timescales for the three processes are on
the order of minutes for the diffusional droplet growth
(χgrowth=6 min 40 s and χgrowth,rad=5 min 30 s) and on the
order of an hour for sedimentation (χsed=1 h 23 min). For the full calculation of the
timescales, the reader is referred to the appendix (Appendix ).
This clear signal lends credence to the use of the parcel model.
In contrast, LES allows for a more faithful treatment of
these processes because of the coupling of interactive components but at the
expense of transparency of the radiative effects on droplet growth.
In combination the two modeling approaches allow insights that
neither could have produced by themselves.
As the microphysical schemes in our LES and in the offline
parcel model are different (two-moment bulk vs. bin), small differences in
the predicted liquid water can occur. Therefore, the calculated
heating or cooling rates of the LES might occasionally be too high for
the application in the parcel model, thus causing unrealistic
droplet growth. We therefore applied a threshold to the
cooling. Whenever the distributed droplet cooling (Fnet) was
larger than the black body emission (σT4/6; the factor 1/6
accounting for the window regions and emission to only one hemispheric
dimension) the cooling of the droplet was set to the black body
emission value. Tests showed that the discrepancy between the liquid
water content of the parcel model and the LES occurs most often at the
edges of clouds where qc is very small. In this area, droplet cooling can be regarded as
“black”, because droplets are exposed to clear sky.
The coupled LESs have a similar setup, but we used the
bin-microphysics scheme from the beginning of the simulation. We
restarted after 4 h from a base simulation with 1-D thermal radiation
passed to dynamics only. We separate five cases:
1-D thermal radiation applied to dynamics only (1DD),
1-D thermal radiation applied to dynamics and droplet growth (1DD_1DM),
3-D thermal radiation applied to dynamics only (3DD),
3-D thermal radiation applied to dynamics and droplet growth (3DD_3DM), and
1-D thermal radiation applied to dynamics and 3-D radiation
applied to droplet growth (1DD_3DM).
These five simulations allow us to (a) look at the effect of thermal
radiation on droplet growth, (b) separate between 1-D and 3-D thermal
radiative effects and (c) to separate the droplet growth effect from
dynamical effects.
Simulations resultsParcel model – cloud field statistics and properties
Figure shows a time snapshot
of the cumulus field and selected time-dependent trajectories (red). From our
2.7 million parcel trajectories we selected about 340 000 that make
contact with a cloud for further investigation. This number was chosen as it
provides us with a statistically representative result and a number of parcels that could
still be handled in a finite amount of time in the post-processing.
Time snapshot of the BOMEX shallow cumulus cloud
field. Displayed are qc and selected parcel trajectories (red).
The effect of 1-D and 3-D thermal radiation on the growth of cloud
droplets depends (among other factors) on the length of time that a droplet is exposed to
thermal cooling (in other words, that a droplet is located close to
cloud edges or cloud top) and the strength of the cooling. found that droplets
have to spend about 10 min in a cooling area to experience a noticeable effect on the
droplet size distribution. We therefore first investigated different
properties of our trajectories:
in-cloud residence time and
time spent in the vicinity of cloud edges or cloud tops.
Histogram of the time that parcels spend in a cloud. For
the sampling of the data, a threshold of 0.01 g kg-1 of the
qc was used to separate cloudy from
non-cloudy regions.
Histogram of the time that parcels spend at cloud top or
cloud side. For the sampling of the data, a threshold of 0.01 g kg-1 of the
qc was used to separate cloudy from
non-cloudy regions. To separate cloud edge regions from the cloud
interior, four different thresholds of the cooling rates were used (4,
10, 20, 100 K d-1).
For the cloud residence time, we used a threshold of 0.01 g kg-1 to
separate between cloudy and cloud-free areas. We then traced among
our 340 000 parcel trajectories the time periods during which a parcel stays in a
cloud. Inevitably, a parcel can contact a cloud more than once, in which
case the hits were counted as multiple
events. Figure shows a histogram of the time that
our parcels spend in clouds. Most of the parcels spend less than 15 min
in a cloud, but we also find some rather long periods of more than
25 min. This is in agreement with former results
(e.g., ).
The time at cloud side was estimated by setting the same threshold
for qc (0.01 g kg-1) and additionally
setting four different thresholds in terms of heating rates (-4,
-10, -20, -100 K d-1) for 1-D and 3-D thermal radiation. Again, multiple hits were possible for
each parcel trajectory. The histograms are shown in
Fig. . One-dimensional (blue) and three-dimensional
(orange) thermal radiative transfer simulations show that most of the parcels spend less than
5 min in a certain cloud volume encompassing a cooling threshold.
For the 100 K d-1 threshold, no parcel
exceeds 3 min. However, there are some parcels which spend 10 min or
longer, especially when 3-D radiative effects are considered in volumes experiencing cooling of 10–20 K d-1.
This is simply due to the fact that a larger volume of each cloud experienced
cooling rates in 3-D radiative transfer. The possibility that thermal
radiation can affect cloud droplet growth is therefore given. In the
following, we will take a closer look at individual parcel
trajectories and the overall statistics of the 340 000 parcel
trajectory ensemble.
Parcel model – cloud droplet growth including thermal radiative
effectsIndividual parcels
We now focus on individual parcels. An example of a parcel
trajectory is given in Fig. . The qc is shown in
color (Fig. a). This illustration of the trajectory includes a temporal dimension. Each data
point is recorded at a different time step. The parcel rises in the
beginning, enters an area of high qc (red, arrow (i)),
followed by a decrease in qc (blue area, arrow
(ii)), but never drops to zero, before entering again an area of high
qc (red, arrow (iii)). Finally, qc decreases again and the
parcel leaves the cloud.
The three-dimensional visualization of qc, the parcel
position and liquid water path (lwp) of the selected scene. (a)
shows the parcel trajectory. The qc at each time step of the selected parcel is
colored. Three time intervals were selected for the following
figures: time intervals where the parcel stays in high-qc areas (i, iii) and one
where the qc drops substantially but does not reduce to zero
(i). (b), (c) and (d) show the parcel trajectory (gray, again time
dependent). For the time interval in focus, the qc is again colored. The displayed clouds are chosen at the
center time of the interval, as is the lwp. The red
marker displayed in the lwp field shows the projected
location of the center time step.
The other three panels of Fig. combine time snapshots of
the cloud field and the temporal development of the parcel trajectory. The
cloud field is shown at the time marked by the red dot on the trajectory. The
surface shows the liquid water path, lwp, of the selected cloud field at that
specific time. The red dot on the surface is the vertical projection of the
location of the parcel at the time. Figure b shows the
updraft area where the parcel first enters an area of high qc.
The parcel (at that time) is located in the upper part of the cloud where it
experiences cooling. In the following, the cloud grows, and at the next shown
time step (Fig. 4c) a significantly larger cloud with more qc is
encountered. The parcel is now located at the outer edge of the cloud
(especially visible in the lwp field, red dot). qc has dropped
below 0.01 g kg-1 but does not decrease to zero in the following,
meaning the parcel never leaves the cloud. The cloud grows further
(Fig. d)) and the parcel is located again in an area of
high qc. We will see later that this “recirculation” of parcels
occurs occasionally and can cause a broadening in the droplet size spectrum.
It is likely that radiative effects become more important in this case,
because parcels pass cloud edges where thermal cooling per droplet is strong.
Time series of different properties of the first selected
parcel. Shown are height, vertical velocity, qc, heating or cooling rate, predominant radius and the
heating or cooling rate per droplet. Gray areas show time intervals
where the qc is below 0.01 g kg-1. The red
dotted lines show selected time steps used in the following
analysis.
Parcel trajectory 1
We now take a more detailed look at the same parcel
trajectory shown in
Fig. . Figure shows this
selected parcel, which is characterized by moderate vertical velocities (peaking at about
6 m s-1 in the beginning but not exceeding 2 m s-1 later on). The parcel
stays in the cloud for about 20 min and twice experiences radiative cooling
(for about 8 and 2 min). We chose four
different time steps for further investigation (red dotted lines
at 14, 16, 21 and 25 min). The first time step was chosen shortly
after the parcel passes the first volume of strong cooling and is
recirculating. Here, we defined “recirculation” loosely as an event
where the qc along a parcel trajectory becomes very
low (in this case 0.007 g kg-1).
The second time step was chosen after qc has risen again, the third
time step shortly before the second cooling phase and the fourth when
the parcel leaves the cloud.
The drop distribution at these four time steps is shown in
Fig. . Figure a–d show the drop size spectra
(dm/dr) themselves, and Fig. e–h show the
ratio of the spectra of the 1DR and 3DR simulations and the NR simulation of
the parcel.
Drop size distribution dm/dr plot for the four selected time steps displayed by
the red dotted line in Fig. .
(a)–(d)
show dm/dr. The lower
row shows the ratio between the NR case and the 1DR and 3DR
case. Gray areas in (e)–(h) display
size bins where mass occurs in the 1DR and 3DR case but not
the 1DR case and therefore no ratio could be calculated.
In the beginning, hardly any differences can be seen in the drop
spectra between the NR, 1DR and 3DR simulations. The spectrum broadens over time.
Looking at the ratio of dm/dr of the 1DR/3DR and
the NR simulation reveals a decrease in mass in
the small bins and an increase in the larger bins for the radiation
simulations. This changes later in the simulation when the
simulations with thermal radiation increase mass over the entire drop spectrum.
dm/dr for the 1DR and 3DR simulations exceeds the
NR simulations up to 1.5 times for r>10µm. The
size spectrum broadens and a drizzle mode develops. The peak of the
spectra remains at about 15 µm. The ratio for 3DR simulations
always exceeds that for
1DR simulations. A factor of more than 2 is reached for dm/dr of the 3-D
radiation simulation compared to the NR simulation. The droplet concentration in
the 20 µm bin (not shown) increases by up to 15 % for the 3DR case along this trajectory.
The possible increase in droplet growth by thermal radiation does not
only depend on the time that a droplet is exposed to cooling, but also
on the magnitude of the cooling and the size of the droplet. Recall that the
larger the droplet, the more effectively radiation can act on
it as the droplet absorbs and emits radiation more
effectively. Radiative effects become stronger from a radius of about
10 µm on. Additionally, the radiative impact competes
with the dynamical effects, which depend on the vertical velocity (see
Eq. ). It
therefore follows that the larger the droplet and the weaker the
updraft, the more radiation can affect droplet growth. Figure shows the temporal development of the individual
droplet heating or cooling rate for three different sizes (left
column). This heating or cooling rate is the fraction of the spectrally
and bin-integrated cooling rate per bin (Eq. ), integrated over all wavelengths. The center
radius of the corresponding bin is given in each figure. Gray shaded areas and the red lines
are identical to those shown in Fig. for comparison. Note the change in
the y axis for these figures.
We find that the cooling per droplet increases
with increasing radius and that 3-D cooling is stronger than 1-D cooling.
Wavelength integrated, bin-resolved heating or cooling
rates and forcing τd and τr.
The right column (Fig. 7) shows the forcing τ (Eq. ), which is the
total driving force for condensation in each bin.
The gray line shows the dynamical forcing (τd), which if compared to
Fig. follows the vertical velocity trend. The radiative
forcing (τr) is shown in yellow (3-D) and blue (1-D) for the
same four size bins. Note that a cooling per droplet (left side) causes a
positive contribution to the droplet growth and therefore a positive
forcing (right side). The radiative forcing
is smaller than the dynamical one but has the same order of
magnitude. An additional boost is given to the
droplet growth shortly before 15 min, after which the dynamical forcing rises
again. This small radiative perturbation is sufficient to cause the
increase in dm/dr seen in Fig. b and f. The radiative
forcing becomes strongest towards the end of the parcel trajectory,
counteracting the negative dynamical forcing, especially for the
larger size bins.
Parcel trajectory 2
This second parcel experiences stronger dynamical forcing. Vertical
velocity rises and falls throughout the parcel's lifetime and peaks at
more than ±6 m s-1. The parcel recirculates twice. During these
two periods the parcel experiences radiative cooling, which causes a
broadening of the droplet size spectrum (Figs. and
). Due to the strong dynamical forcing, the parcel shows broader spectra than the first trajectory. As before, a
substantial increase in condensed water is found for the 1DR and 3DR
simulations, peaking for the 3-D thermal radiation
simulation.
Similar to Fig. but for the second selected
parcel.
Similar to Fig. but for the second selected
parcel.
These examples illustrate the variety of ways in which radiative
effects can act on a cloud droplet. The time that a parcel spends in
a certain cooling area, the magnitude of the cooling, the size of the
droplets at the time of cooling and the dynamical forcing contribute
to droplet growth with different magnitudes. The ideal situation
for the radiative effects to enhance droplet growth would be
droplets of size of about 10 µm or more, a cooling period of more than
5 min in a cooling of 20 K d-1 or more, and vertical velocities close to
zero. Because these effects usually do not occur together, the overall effect on the droplet growth from these factors is small.
A strong effect is found when parcels recirculate. Whenever a
parcel reaches cloud edge, the number of droplets is small. Yet, these
droplets are exposed to cloud top or cloud edge cooling which, due to
the limited number of droplets, is close to the maximum cooling that a
droplet can experience (2 in Fig. ). Additionally, these parcels already include
larger droplets, where radiative effects are stronger. The droplets
experience additional growth by radiative cooling during the
recirculation time and return into the cloud with a slightly broadened size
distribution (3 in Fig. ). The droplet size distribution subsequently continues to broaden (4 in Fig. ).
Schematic figure of the droplet growth for a
recirculating parcel.
Summarizing, these analyses of individual parcel trajectories have shown that radiatively
enhanced droplet growth can occur in “lucky situations” or when
recirculation occurs. The increased droplet growth for recirculating
parcels agrees well with prior results of enhanced droplet growth in areas of net
radiative loss (see, e.g., ). The radiative cooling does not seem to cause droplet growth in individual
parcels beyond the NR case (as also found by ), but thermal radiative effects enhance
the mass per bin and occasionally allow droplets to grow into larger bins.
In the following, we will take a more general look at the effects in our
parcel trajectory ensemble.
Parcel model – ensemble results
As a next step, we evaluated our 340 000 parcel trajectory ensemble to
see if we find changes in droplet size and rain amount, as represented by the local volume flux of water, or rain rate.
Joint histogram of integrated water and maximum mean radius. The first
figure shows all data from the NR simulation. The other
figures show the difference of the number of occurrences of 1DR
vs. NR, 3DR vs. NR and 3DR vs. 1DR.
Figure shows a histogram of maximum mean radius along
a parcel trajectory versus the integrated qc along the parcels.
The first figure shows the number of occurrences for the NR case. Integrated
qc mostly occurs in a range of 0–10 g kg-1 min-1
with maximum mean radii up to 20 µm. Larger droplets and integrated
qc amounts exist but are comparatively small in number. When
comparing the number of occurrences of the 1DR case to the NR case, we find
an increase in the number of larger droplets for small qc amounts
and for those between 5 and 10 g kg-1 min-1 and a decrease in
the directly smaller bin. Radiation thus enhances the growth of droplets for
a specific qc for very small droplets and for droplets in the
10–25 or 30 µm range. There is also a tendency for the larger
drops to grow to larger sizes in the 1DR case. For 3-D thermal radiation we
see a similar picture. The number of droplets growing to larger sizes is even
higher than in the 1DR case. Comparing the results of the 3-D thermal
radiative transfer simulation to the 1DR simulation shows the additional
increase in the 3DR case. We confirm here that due to thermal radiation
droplets in the critical range tend to grow to larger sizes.
Next, we calculate the total rain rate at each time step
(accounting for drop radii >20µm) for the
entire trajectory ensemble. In the first hour, the absolute differences in the rain rate
between the three setups is small. Absolute differences become larger
over time and are clearly visible during the last 40 min of the analyzed
time period (Fig. ). Looking at the
relative differences between either the 1DR simulation and the NR case, or 3-D thermal radiation
simulation and the NR case, we find differences of 10 %
for the 1DR case in the first hour and 20 % for
the 3-D thermal radiation case. Relative differences increase commensurately with the
absolute differences towards the end of the 2 h simulation and
reach as high as 40 % for the 3-D thermal radiation case.
Total parcel rain rate calculated from the trajectory
ensemble simulation (accounting for drop radii >20µm).
(a) shows the absolute rain rate for the
NR, the 1DR and the 3DR cases. (b) shows the relative differences of 1DR simulation
and the NR case, as well as 3DR
simulation and the NR cases.
We then separated the rain rates according to different factors that
could affect droplet growth on our
trajectories. Following on the results of our investigation of the
individual trajectories, we calculated rain rates for parcels with
certain thresholds of updraft speeds, cumulative cooling, or time
spent at cloud side. About 50 % of the rain rate arises from parcels that are
in an updraft region of 3 m s-1 or more (regions typically associated
with higher qc), but differences between the
NR and 1DR and 3DR cases are small. The largest
radiative effect emerges from parcels that recirculate (see Fig. ). We define
“recirculation” by setting a threshold in terms of qc of
0.01 g kg-1.
The time periods for recirculation events are shown in
Fig. . Most of the parcels spend a few
minutes outside a cloud. More than 90 % of the recirculation events
are shorter than 5 min.
Up to 58 % of the parcel rain rate of the 3DR simulation
arises from recirculating parcels, while in the case of NR
about 45 % of the parcel rain rate arises from recirculating parcels. The
largest increase is found within the last 20 min of the
investigated timeframe. Differences of 5–10 % are found in the first
20 to 50 min, while in between there is no difference between the
three simulation types. Setting an upper limit in time (e.g., 5 min,
which includes more than 90 % of our recirculating parcels), the
changes in our results are very small. The maximum contribution of the
rain rate reduces to 56 %. For a time threshold of 2 min, the
maximum reduces to 45 %. In this context it should be noted
that only 6 %–7 % of our 340 000 parcel trajectories are classified as
recirculating according to our definition.
Remarkably, these 6 %–7 % can contribute up
to 60 % of the total parcel rain rate. The parcel rain rate due to
recirculation (when normalized to each of the corresponding simulations and
therefore without considering radiative effects) is about 30 % in
our study. found a
similar magnitude in their study. About 50 % of the rain rate emerged from
recirculating parcels. These zones might be considered
the birth place of precipitation embryos, which subsequently become
important for accelerating collision and coalescence.
Histogram of the time period of recirculation
events. Recirculation events are defined by threshold in terms
of qc of 0.01 g kg-1.
Total rain rate calculated from the trajectory ensemble
simulation from recirculating parcels with a threshold of
0.01 g kg-1. The dashed lines show the total rain rate, and the
solid lines show the rain rate from recirculating parcels as well as
the relative difference between the cumulated sum rain rates from
recirculating parcels compared to the cumulated sum total NR rain rate.
Coupled LES – cloud droplet growth under the impact of thermal radiative effects
Next, we investigate the effect of 1-D and 3-D thermal radiation in
a fully coupled system. To this end, we ran a set of BOMEX simulations as
described in Sect. . Here we (a) look at the effects
of thermal radiation on microphysics in a coupled system and (b) compare
it to the effect on dynamics.
The effect of thermal radiative transfer on
microphysics
We start with variables concerning rain and first
focus on the microphysical effect. Figure shows
rain water path, surface precipitation fraction, domain-averaged surface precipitation
rate and the cumulative surface precipitation rate of four of the five
simulations. We take the simulation with 1-D radiation on dynamics (1DD) as
our reference case. We focus first on the discussion of the rain water path. We find a small increase
in rain water path an hour after restart in the case of 3-D radiation acting on
microphysics and dynamics (3DD_3DM); however, differences among
the four simulations never become significant.
Surface precipitation fraction (over the total domain) shows an increase
for all simulations where radiation is coupled to the diffusional
droplet growth. The strongest increase is found for the 3DD_3DM
case. This suggests that more clouds produce rain when radiation is
coupled to the droplet growth, but the total amount of rain water
produced does not change substantially.
The surface precipitation
rate shows no clear changes, but in accumulation the simulations
with the radiative–microphysical coupling produce more rain (10 %
for 3DD_3DM). There is a subtle increase in the accumulated rain rate and rain
water path when thermal radiation is coupled to the diffusional droplet growth. However, when
comparing 1DD_3DM to 1DD_1DM, no difference can be found. Hence, the
small increase in rain in the case of 3DD_3DM must arise from the
effects of 3-D thermal radiation on dynamics, not on microphysics. We
will investigate the 3-D effect further in the following.
Temporal development of rain water path (a), surface
precipitation fraction (b), domain-averaged surface precipitation
rate (c) and the accumulated domain-averaged surface
precipitation rate (d). The time series is shown from the
restart time of 4 h onward. We compare four of our five
simulations here where thermal radiation was coupled to the
diffusional droplet growth.
3-D thermal radiative effects
Figure shows the same variables as
Fig. but now comparing the results of
the 3DD_3DM and 3DD. In the beginning, rain water path and rain
rate show no noticeable difference. After 7 h of the simulation,
rain water increases for 3DD. Prior to 7 h, as also discussed
in Sect. , the fraction of surface rain rate is slightly enhanced in
3DD_3DM compared to 3DD, which again suggests that the
coupling to microphysics does not produce more rain but that more
clouds produce small amounts of rain, while in the 3DD case changes
in the dynamics cause somewhat stronger rain in fewer clouds.
Figure d shows again the accumulated surface rain rate over
time. The 3DD simulation produces more rain. This was counterintuitive at
first, because it was expected that thermal radiative effects enhance
droplet growth.
Similar to Fig. but for 3DD and
3DD_3DM. The gray shaded area shows the time period between
6.7 and 7.3 h which is investigated in the further analysis.
We pose the following hypothesis to explain this behavior:
enhanced droplet growth, due to 3-D thermal radiation at the cloud edges, decreases the evaporation at the cloud edges, causing weaker evaporative cooling and weaker downward motion, representing an evaporation–circulation feedback.
This hypothesis constitutes a negative feedback to changes in the cloud
circulation found by and will be explained in the
following. It is analogous to the previously documented evaporation–circulation feedback
due to changing aerosol concentrations in cumulus clouds
and earlier studies that identified the relationship between the
horizontal buoyancy gradient and the vortical circulation around a cloud;
stronger cloud edge evaporation generates stronger horizontal buoyancy gradients,
increased turbulence kinetic energy (TKE), and
enhanced mixing and entrainment .
found an enhanced cloud circulation due to thermal
radiative effects. It was shown that cloud top cooling
caused stronger updraft velocities in the clouds and, due to
the side cooling, stronger subsiding shells at the cloud edge. Due to
the stronger updrafts, clouds were deeper, more turbulent and contained more qc. The results from
and the above posed hypothesis, can, if correct,
explain the differences in surface rain fraction and rain rate between
the two simulations. We will therefore investigate the profiles of cloud water mixing
ratio, precipitation flux, evaporation rate and buoyancy production of
TKE (Fig. ), averaged over half an hour marked by the gray shading in
Fig. . This time period is chosen as it is the period
shortly before and at the beginning of the increase in rain
production. All four variables show higher values for 3DD
compared to 3DD_3DM.
Time-averaged profiles of cloud water mixing ratio,
precipitation flux, evaporation rate and buoyancy production of
the TKE averaged from 6.7 to 7.3 h.
Finally, we look at the temporally averaged profiles of updraft and
downdraft vertical velocities in saturated areas. Here we also find
stronger downdraft and stronger updrafts in the 3DD case. These
analyses lend credence to our hypothesis and (Fig. ).
Time-averaged profiles of updraft and downdraft vertical
velocity in saturated areas averaged from 6.7 to 7.3 h.
1-D vs. 3-D thermal radiative effects
Finally, we compare the results of 1DD and 3DD to examine the
effect of 3-D thermal radiation on dynamics. We focus again on
rain production as this was not included in . As expected,
3-D radiation causes an increase in
all the rain-related variables (shown in
Fig. ). To prove that a change in the cloud
circulation is also causing the increase in rain, we look at the
profiles of the updraft and downdraft vertical velocity in saturated
areas, averaged over two time periods marked in gray in
Fig. . The time periods were again chosen
because they include the beginning of rain enhancement by 3-D thermal
radiation compared to 1-D thermal radiation. For both time periods,
Fig. shows enhanced downdrafts and updrafts for
the 3DD simulation compared to 1DD.
Similar to Fig. but for 1DD and
3DD. The gray shaded area shows the time period between
4.6 and 5.3 h as well as 6.7 and 7.3 h, which are investigated in the further analysis.
Similar to Fig. but for 1DD and
3DD and for the time period between
4.6 and 5.3 h as well as 6.7 and 7.3 h.
Summary
The coupling of thermal radiative effects to microphysics can lead to the formation of larger cloud droplets and drizzle
droplets. For 3-D thermal radiative effects, additional cooling occurs
at cloud edges, which can strengthen the effect. The coupled LESs showed the following.
When thermal radiation is coupled to microphysics, there is a
small increase in rain production for 1-D radiative effects (1DD_1DM). This could be due to
recirculation of droplets, as shown in the parcel model
study. However, the change in rain in the coupled simulations is
very small. When coupling radiation to droplet growth it matters little whether 3-D or 1-D thermal radiation
is applied. The increase in the surface precipitation fraction
suggests that rain is produced in more clouds distributed over the domain.
When 3-D thermal radiative effects are considered we find
(counterintuitively) overall more rain in the simulation with
dynamics only.
The fact that more rain is produced by the simulation coupled to
dynamics only is hypothesized to be due to an evaporation–circulation feedback caused
by the larger droplets in the 3DD_3DM simulation.
When comparing the 1-D and 3-D thermal radiative effects on
dynamics we find an increase in the rain production for 3-D thermal
radiation.
The dynamical effect caused by 1-D and 3-D thermal radiation is a
change in the cloud circulation as already found by
where thermal radiation increases upward and
downward vertical velocities in and in the near-cloud environment,
which causes deepening and more rain.
Finally, we note that the overall differences concerning
precipitation are small and might not be detectable relative to
differences associated with perturbations to thermodynamical inputs in
an ensemble of simulations (see, e.g., ).
Conclusions
In this study we investigated the effect of thermal radiation on cloud
droplet growth and rain formation in shallow cumulus clouds. We used a two-stage approach, which
allowed us to separate microphysical from dynamical effects.
First an offline parcel model was used to investigate the effect of 1-D and 3-D
thermal radiation on cloud droplet growth. It was found that thermal
radiation in general has the potential to enhance droplet growth and rain formation. Three-dimensional thermal radiation
enhances droplet growth and rain formation more than one-dimensional thermal
radiation. It was shown that thermal radiation enhances the formation of precipitation
embryos in the 10–30 µm radius range.
These embryos have the potential to enhance rain formation in real clouds.
Thermal radiation can affect cloud droplet growth when one or more of the
following conditions are fulfilled.
Droplets have already grown to a size of about 10 µm or
more when being exposed to thermal cooling. A cooling period of more than 5 min
at a cooling rate of 20 K d-1 or more and vertical velocities close to
zero are favorable for radiative effects. If one or more of these factors occur, radiative cooling can
enhance droplet growth. The main effect was found in recirculating
parcels, which fulfill parts of the above-mentioned
criteria. Recirculating parcels include droplets that have already
grown to a certain size when passing a cloud edge area. At the cloud
edge area cloud droplets are exposed to large cooling (close to black
body emission), and this small number of droplets grows by thermal
cooling, which counteracts evaporation. When reentering a cloud,
droplet growth continues, generating a broader spectrum, which can enhance
rain formation. Only 6 %–7 % of our simulated parcels are classified
as recirculating, yet they can contribute up to 45 %–60 % of the
local rain rate.
Second, in a more realistic framework we investigated large eddy simulations where
thermal radiative effects were applied to droplet growth and
dynamics. It was shown that the effect on droplet growth is small. However more clouds produce small amounts of rain when
radiative effects are applied to the diffusional droplet growth; thus rain
covers a larger area of the simulation domain. Three-dimensional thermal radiative
effects exceed one-dimensional thermal radiative effects. The largest amount of
rain is produced when 3-D thermal radiation is applied to dynamics
only. This was initially considered to be counterintuitive since both microphysical and radiative effects tend to enhance rain.
We hypothesize that an evaporation–circulation feedback is responsible
for less rain in the simulation where radiation is also applied to
droplet growth: 3-D thermal cooling rates at the cloud edges enhance
droplet growth locally at the cloud edge, thus leading to weaker
evaporation rates, which in turn reduces the strength of the subsiding shell and the
horizontal buoyancy gradient, all leading to weaker cloud turbulence and lower rain
production. In simulations with 3-D thermal cooling only applied
to the dynamics, the enhanced cloud circulation causes stronger
updrafts in the cloud center, a cloud deepening and more condensation or rain formation.
These results could have implications in terms of cloud field
organization. As shown by , thermal radiation can
cause mesoscale organization of shallow cumulus clouds by changing
cloud circulation. It was shown that this change in cloud
circulation also occurs in the simulation in this
study. Furthermore, more rain produced by thermal radiation changes
the dynamics of the system as a whole. The larger area of the domain
covered by rain when radiative effects are applied to microphysics
could also lead to a feedback in terms of dynamics and cloud field
organization. Longer simulations are necessary to investigate the
organization feedbacks. Finally, a trade-wind cumulus case that tends to
deepen and generate more precipitation and organization would be worth
investigating.
Code availability
Input files and the model code for reproducing
the simulations and data of this study are available from the
corresponding author upon request.
Timescale calculation
The analysis of timescales supports our argument about the usefulness of
the parcel model, even though it does not represent certain
physical processes. The characteristic timescales for a
number of processes involved in our study (diffusional droplet growth, diffusional
droplet growth with radiation and sedimentation) are calculated in the
following. The analysis is performed for a droplet of 20 µm
radius. In order to avoid confusion, we use χ to represent timescales,
because the standard abbreviation τ is already used as the
forcing in the diffusional droplet growth equation (Eq. ).
The analysis shows a clear signal: sedimentation occurs on much longer
timescales than all the other processes. The timescale of
sedimentation is on the order of 1 h, while the
diffusional droplet growth timescales, with and without radiation, are on the order
of a few minutes.
Diffusional droplet growth
Diffusional droplet growth (which follows from Eq. ) is defined as
drdt=1ρC⋅Sr.
It therefore follows that the characteristic timescale χgrowth is
χgrowth=r2⋅CS=6min40s|,
with a droplet radius r of 20 µm, C=1e10 s m-2 and a
supersaturation S=0.01.
Diffusional droplet growth including thermal radiative
effects
The diffusional droplet growth including radiative effects (see also
Eq. , but defined in mass space) is
drdt=1ρC(Sr-F⋅Enet).
Therefore χgrowth,rad becomes (when assuming that r is
constant over time in the radiative term)
χgrowth,rad=r2⋅CS-r⋅F⋅Enet=5min30s,
with F≈1 and Enet=σT4⋅0.33≈-100 Wm-2 (one-third of
the black body radiation occurring in the window region).
Sedimentation
The timescale for sedimentation follows
χsed=Lv=1h23min,
with a typical length scale of L=100 m and a fall velocity at 20 µm
of v(20µm)=0.02 m s-1.
Competing interests
The authors declare that they have no conflict of interest.
Author contributions
CK implemented the 3-D radiative transfer scheme
into the LES, ran the simulations and performed the analysis. TY
implemented the bin-microphysics scheme into the LES. All authors
contributed to developing the basic ideas, discussing the results
and preparing the manuscript.
Acknowledgements
The authors acknowledge Bernhard Mayer and Jerry Harrington for useful discussion and Jan
Kazil for help with SAM. We gratefully acknowledge Marat Khairoutdinov (Stony Brook University) for developing and
making the System for Atmospheric Modeling (SAM) available.
The authors acknowledge the NOAA Research and Development
High Performance Computing Program for providing computing and storage resources that have
contributed to the research results of this paper.
Financial support
This research has been supported by the Deutsche
Forschungsgemeinschaft (DFG) (grant no. KL-3035/1) and the Bundesministerium
für Bildung und Forschung through the High Definition Clouds and
Precipitation for advancing Climate Prediction project (HD(CP)2) phase 2
(grant no. 01LK1504D).
Review statement
This paper was edited by Timothy Garrett and reviewed by two anonymous referees.
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