Box model simulations
In the following box simulations, the sensitivity of the LCM collection
process to the number of simulated superdroplets, different splitting
approaches, and the approaches' specific parameters is investigated.
Therefore, the box model simulation considers collection as the only
microphysical process.
Setup
Although zero-dimensional simulations do not have a spatial extent,
allocating a certain weighting factor requires a reference volume to
represent a defined droplet concentration. Therefore, the volume of a grid
box is 8×103 m3, which corresponds to an isotropically spaced
grid with Δx=Δy=Δz=20m. The simulation
time is 3600 s with a constant time step of 1 s. To ensure
adequate statistics, 25 344 boxes are calculated, and results are averaged
over this ensemble. (The number of ensemble members represents the maximum
amount of grid boxes which can be calculated on four computing nodes in an
appropriate time.) In the following this method is referred to as a single-box
model.
Besides the traditional single-box approach, a new multi-box approach is
introduced. In contrast to the calculation of independent grid boxes, the
multi-box approach allows superdroplets to move from one grid box to the next
by prescribing a stochastic velocity (but no mean motion) in
Eq. (), using 25 344 grid boxes, as in the single-box
ensemble above, with cyclic boundary conditions among which the superdroplets
are allowed to move. The stochastic velocity component is chosen in such a
way that it corresponds to a kinetic energy dissipation rate of
ϵbox=0.01 m2s-3, which is typical for
shallow cumulus clouds e.g.,.
This multi-box approach has one distinct advantage over the ensemble mean of
the same amount of individual box model simulations (single-box model), which
results from the difficulties to initialize a DSD with superdroplets of a
constant weighting factor, as it is done in many applications of LCMs in the
literature
e.g.,.
A single-box model simulation suffers crucially from this initialization
method due to a wrong representation of the largest and rarest superdroplets
their Fig. 17. In doing so, the rarest and
largest, and therefore most important superdroplets for the collection
process, are a priori over- or underestimated. An exchange of superdroplets
between the collection boxes helps to mitigate this problem. Moreover, this
new approach is closer to the representation of collection in
three-dimensional simulations, in which a superdroplet is not bound to a single grid box.
The impact of different numbers of superdroplets per grid box and the use of
splitting for the traditional single-box approach will be discussed first;
then, the new introduced multi-box approach will be presented for both
splitting and non-splitting cases. Box model simulations will be compared to
the results of , who used a high-resolution bin model. The
purpose of this study is, however, not the exact reproduction of these
results but a computationally efficient approximation to them using
splitting. Accordingly, the initialization of the box simulation follows
, using an exponential initial DSD:
n(r,t=t0)=3Ninitr03⋅r2exp-r3r03,
where Ninit=300 cm-3 is the droplet number
concentration. The initial mean radius is r0=9.3 µm, which
leads to a liquid water content of L0=1 gm-3. Following
, we set the minimum droplet radius to rmin=1.5µm. Superdroplet radii are then selected by a random
generator which follows the distribution given by Eq. ().
All superdroplets receive the same initial weighting factor:
Ainit=Ninit⋅ΔVNP,
which ensures the number concentration of 300cm-3. This method
is also described as νconst-init in
, which has been chosen in this study to resemble
the initialization of superdroplets in less-idealized applications but also
significantly hinders collisional growth.
As reference, the “singleSIP” initialization of
is also used for the single-box model. In contrast to the previously described
initialization, the initial DSD is discretized using logarithmically spaced
bins. The number of bins corresponds to the number of superdroplets. To each
bin, a superdroplet with a corresponding mean radius and weighting factor is
assigned. The maximum radius of the initial distribution is approximately
33µm, which corresponds to a number of concentrations of
1/ΔV. This avoids superdroplets with a weighting factor less than 1.
Note that this (not always applicable) initialization technique represents
the inherent variability in droplet radii and their abundance across the
initial spectrum much more accurately than the previously described method,
and therefore results in a much better agreement with literature references.
In addition to analyzing the DSD directly, the temporal development of the
zeroth and second moment of the mass density distributions is examined. Due
to mass conservation in all applied approaches, the first moment is constant
in time and will not be shown. The moments of the mass distribution fm are
defined as
Mk=∫mkfm(m)dm,
where m is the mass and fm(m) denotes the number concentration
distribution. Note that the zeroth moment M0 is the number concentration
and the second moment M2 is proportionate to the radar reflectivity, and
thus highly sensitive to the largest droplets in the DSD.
For a given superdroplet ensemble the moments for each grid box are
calculated with
Mk=∑n=1NPAnmnk/ΔV,
where mn is the single droplet mass (mn=4/3πρlrn3) of a
superdroplet.
Box model results
First, the sensitivity of the collision algorithm to the number of
superdroplets is examined using the LCM as a single-box model. Second, the
improvements by the splitting method on collisional growth is evaluated.
Subsequently, those investigations are repeated for the multi-box approach.
Single-box approach
Figures and show the mass density distribution
after 3600s and the temporal development of the moments for the
LCM applied as a single-box model using the singleSIP initialization by
. Each grid box is initialized with a different
number of superdroplets (colored lines). The reference solution of
is shown as a black solid line. Figure shows
that even with 87 superdroplets the solution of can be
reproduced well and a further increase in the number of superdroplets only
leads to minor improvements. The small deviations between the bin model
solution and the LCM can be traced back to the different solution of the
collection equation e.g.,. Overall, it can be seen
that the solution of the LCM converges with an increasing number of
superdroplets. The moments of mass distribution (Fig. ) also show
convergence with an increasing number of superdroplets. This good
representation of collision growth is in line with the results with
.
Mass density distribution for the single-box approach after
3600s for the “singleSIP” initialization. The black solid line
denotes the solution of . The colored dashed curves show
the solution of the LCM with different numbers of superdroplets per grid
box.
Moments of the mass density distribution as a function of time
obtained from the single-box simulations for the “singleSIP” initialization.
The black solid line denotes the solution of . The colored
dashed curves show the solution of the LCM with different numbers of
superdroplets per grid box.
Mass density distribution for the single-box approach after
3600 s. The black solid line denotes the solution of
. The colored dashed curves show the solution of the LCM
with different numbers of superdroplets per grid box.
Moments of the mass density distribution as a function of time
obtained from the single-box simulations. The black solid line denotes the
solution of . The colored dashed curves show the solution
of the LCM with different numbers of superdroplets per grid box.
Now, Figs. and show the same quantities but for the
initialization with identical weighting factors. In Fig. , a
significant deviation of the mass density distribution of the reference
solution can be seen for all configurations. An excessively pronounced first
maximum is found for all superdroplet concentrations, while the second
maximum is at too small droplet sizes. Also, fluctuations occur for radii
larger than 100 µm, resulting from insufficient superdroplet
statistics in this range. However, as the initial number of superdroplets
increases, the depletion of the first maximum and the development of the
second maximum is reproduced better. Figure a shows that in all
cases the decrease in the number concentration is underestimated. Also for
the second moment (Fig. b), values are predicted too low in nearly
all cases. All in all, it can be observed that an increase in the number of
superdroplets leads to a better agreement of the results with the bin model
even though difference are still significant for 1000 superdroplets per grid
box.
Mass density distribution for the single-box approach after
3600s. The black solid line denotes the solution of
, the black dashed curve the reference case (without
splitting). The colored dashed curves show solution for splitting simulation
with different configurations.
Moments of the mass density distribution as a function of time
obtained from single-box simulations. The black solid line denotes the
solution of , the black dashed curve for the reference
simulation (without splitting). The colored dashed curves show the solutions
for different splitting configurations.
In Figs. and the mass density distribution after
3600s and the temporal development of the moments applying the
splitting algorithm in different configurations are shown. Again, the
splitting modes are abbreviated with S for the simple splitting method and
G for using the splitting method based on a gamma distribution. The number
following S or G indicates the splitting radius in microns. For all
simulations, the maximum permissible number of superdroplets per grid box is
limited to NP,max=1000. The maximum splitting factor is
ηmax=20. By selecting these limits, which are chosen to
represent the upper limit of computationally feasible three-dimensional
simulations, it is possible to obtain an estimate of the quality of the
individual splitting methods. The influence of the choice of these parameters
is discussed below. All simulations are initialized with NP=87
superdroplets per grid box.
The black dashed line (Const.) shows the reference LCM case in which no
splitting is applied. Comparing the non-splitting case to splitting cases, the
results are significantly improved with respect to the reference solution.
More precisely, the fluctuations that occur for large droplet radii are
successfully removed by splitting. Furthermore, a better representation of
the second maximum is also achieved by splitting. Independent of the
splitting mode, simulations with the same splitting radius provide similar
results. The only exception is between the simulations G10 and
S10, in which the assumed gamma distribution enables effective
splitting at slightly larger radii in G10 compared to S10.
This results in a better agreement of S10 with the bin reference. In
general, a reduction of the splitting radius leads to an improved
representation of the mass density distribution. However, for all splitting
simulations the reduction of the first maximum is underestimated, while the
second maximum is only inadequately represented.
Similar conclusions are possible from Fig. , in which the
time series of the zeroth and second moment of the DSD are shown. The best
agreement for the number concentration is achieved by S10, where
many superdroplets are cloned at a very early stage. For all splitting
configurations, the second moment shows a strong improvement in comparison to
the LCM reference case without splitting (Const.) where this value is largely
underestimated. Accordingly, splitting leads to an improved representation of
the collisional growth in LCMs but there are still very large deviations from
the bin reference.
These results exhibit how strongly collisional growth suffers from the
initialization with a constant weighting factor, consistent with
. Since large superdroplets are initialized only in
a few grid boxes, collisional growth is subject to a great variability in the
different realizations among the ensemble. Due to that, the following
subsection will repeat this investigations using the multi-box approach,
which reflects the collisional growth in 3-D applications more appropriately.
Same as Fig. but for the multi-box approach, i.e.,
interactions between the grid boxes are possible.
Same as Fig. but for the multi-box approach, i.e.,
interactions between the grid boxes are possible.
Same as Fig. but for the multi-box approach, i.e.,
interactions between the grid boxes are possible.
Multi-box approach
Figure shows the mass density distribution after 3600s
time for different numbers of superdroplets (colored lines) using the
multi-box approach without splitting. One can see that as the number of
superdroplets increases, a better agreement with the bin model is achieved.
Especially the simulations with 512 and 1000 superdroplets per grid box can
reproduce the mass density distribution well. However, for these cases, a
stronger decrease in the first maximum is observed. This can be attributed to
accelerated accretion, which is favored by the combination of a few large
droplets with an overestimated weighting factor and a large number of
superdroplets with radii of about 10 µm. In contrast, a
decelerated depletion of the first maximum and a weaker second peak are
detected for simulations with a lower number of superdroplets. This results
from the insufficient representation of the initial DSD, especially that of
large droplets, which are crucial for effective collisional growth.
In Fig. , the temporal evolution of the number concentration and
the second moment are shown. In simulations with a high number of
superdroplets, a too strong reduction of the number concentration is
predicted; and contrarily, the decrease in the zeroth moment is underestimated
in cases with only 15 and 37 superdroplets. This tendency is also observed
for the second moment. Simulations with a high number of superdroplets
overestimate the reference, whereas simulations with only a few superdroplets
result in too low values. However, comparing the results of the non-splitting
cases (Const.) in the single- and multi-box simulations, the latter
already provides improved results with respect to the bin model. The results
show that this initialization artifact can be successfully mitigated by the
newly introduced stochastic exchange between the grid boxes. For typical
applications, however, the required amount of at least 512 superdroplets per
grid box, necessary to derive satisfying results without splitting, is
computationally unfeasible.
To maintain a reasonable amount of superdroplets, these box simulations will
be repeated now, using the splitting approach. Here all parameters
(initializing all simulations with 87 superdroplets per grid box) and
splitting thresholds are identical as for the single-box approach described
above but the superdroplets are now allowed to move between grid boxes.
Figure shows the mass density distribution after 3600 s
for different splitting configurations. Clear differences in the consistency
with the bin reference solution can be seen. In particular, the simulations
S10 and G10 show a good agreement with the results of
. In both cases, the bimodal shape of the spectrum is
represented well. However, for the other simulations, the deviation from the
reference solution increases with increasing splitting radius, but less with
the splitting mode. Both simulations with a splitting radius of
40 µm show no improvements in comparison to a simulation without
splitting (Const., black dashed line), except in the right tail of the
distribution. Figure shows the moments for the different
splitting configurations. The two plots indicate a slightly faster
precipitation process than in the bin model, but the general agreement with
the reference is much higher than without splitting (Fig. ).
Again, general differences between the bin model and the LCM are caused by
the initialization, which cannot be fixed by splitting. The initialization
with constant weighting factors will always deviate from the exponential
initialization used by . Therefore, subsequent collisions,
which are improved by splitting, cannot agree with the bin solution by
whatsoever. In general, the decreasing difference among
the different LCM simulations, as it is occurring due to splitting, needs to
be seen as a proof of concept, and not the comparison with the bin results.
All in all, it is shown that collisional growth is better represented by
using the splitting method in both the single- and multi-box simulations.
Furthermore, the choice of the splitting mode is secondary, but the splitting
radius is identified as the most crucial parameter. The multi-box simulations
exhibit a distinct advantage over the single-box simulations. Due to the
presence or absence of sufficiently large droplets that might initiate
collision and coalescence, as a result of the initialization, collisional
growth can be overestimated in certain grid boxes while it is
underestimated in others. Splitting and the subsequent stochastic exchange
are able to distribute these so-called precipitation embryos among the entire
ensemble where they are able to initiate collision and coalescence as
sketched in Fig. , which would not be possible in the single-box
approach.
Same as Fig. but for the multi-box approach, i.e.,
interactions between the grid boxes are possible.
Schematic representation on how splitting affects the spatial
distribution of large superdroplets. The squares outline the different grid
boxes with superdroplets of the size of cloud droplets (blue) and
superdroplets representing rain drops (dark red). Without splitting
(a), the rain drop is represented by only one superdroplet. In the
splitting case with the multi-box approach (b), this superdroplet is
cloned into several superdroplets, which are able to move in other grid boxes
(due to their individual subgrid-scale velocities) where they
initiate or affect collisional growth.
Sensitivity to splitting thresholds
The limiting parameters of the splitting algorithm are now examined in
sensitivity studies using the multi-box approach. For this purpose, the
parameters of the maximum possible number of superdroplets per grid box
NP,max, the maximum splitting factor ηmax, and the
splitting radius rspl are varied for the splitting mode
G; the base state of this mode is defined as rspl=10µm, ηmax=20, and NP,max=1000.
This base state is varied by individually changing the parameters
rspl, ηmax, and NP,max. Furthermore, all
simulation are identically initialized with Ninit=87
superdroplets per grid box.
Figure a shows the mass density distributions after
3600s for different values for NP,max. We find that a
value of NP,max=150 is sufficient to reach convergence for this
setup. Since the initial superdroplet concentration is Ninit=87
for all configurations of NP,max, it can be concluded that
NP,max is a necessary but not crucial parameter as long as
NP,max⩾150. This reduction of the maximum number of
superdroplets per grid box results in a reduction of the computational time
by a factor of 15 compared to the simulation with NP,max=1000.
The sensitivity studies for the maximum splitting factor show that this has
no influence on the results (Fig. b). An explanation for this is
that the algorithm is executed at every time step and thus only the clone
rate but not the absolute number of the clones is affected. More precisely, a
low value of ηmax may reduce how many clones are produced at a
time step. However, results show that this effect is negligible since a
superdroplet will be cloned sufficiently fast at the subsequent time steps as
long as NP≤NP,max.
As shown before, the development of the spectrum is highly sensitive to the
choice of the splitting radius. Figure c shows that the results
converge with decreasing splitting radius, with no significant deviations for
configurations with rspl≤15µm. This can be
attributed to the fact that especially the largest droplets (in this case
with radii of approximately 15µm) are crucial for initiating
the collisional growth. Accordingly, an improved representation of these
droplets leads to an improved representation of the whole collisional growth
process.
Single cloud
Setup
In this case, we are simulating an idealized shallow cumulus cloud in the form of
a rising warm air bubble as in . The model domain is
1920m×7680m×3840m in the x-,
y-, and z-direction, respectively. An isotropic grid spacing of 20 m is used.
The simulation time is 3000 s using a constant time step of 0.1 s.
The warm air bubble is triggered by a Gaussian-shaped potential temperature
perturbation θ*
θ*(y,z)=θ0⋅exp-12⋅y-ycσy+z-zcσz,
where θ0=0.4K is the maximum temperature difference, which
decreases with a standard deviation of σy=200m and
σz=150m in the y- and z-direction, respectively. The
center of the bubble is set to yc=3840m and
zc=170m. Due to the two-dimensional character of the
temperature excess, the initial temperature perturbation is elongated
homogeneously along the x axis.
Mass density distribution for the box simulation after
3600 s. The black solid line denotes the solution of
. In (a), sensitivity studies for different
values of NP,max are presented. In (b), simulations for
different values of ηmax are shown. In (c), results
for different splitting radii are displayed. All sensitivity studies are
conducted using the splitting mode G.
The initial profiles for temperature and specific humidity are based on the
shallow cumulus case by . Note that no background winds,
large-scale forcings, or surface fluxes are considered. The superdroplets are
released at the beginning of the simulation and are uniformly distributed in
the entire model domain. For all three directions in space, the average
distance of the superdroplets is initially 4.5 m. This results in a
superdroplet concentration of approximately 87 superdroplets per grid box and
roughly 4.55×108 superdroplets in total. Using a weighting factor of
Ainit=9.0×109, an initial cloud condensation nuclei (CCN)
concentration of 100 cm-3 is represented. Additionally,
simulations with 15 and 186 superdroplets per grid box are carried out, in
which the weighting factor is adjusted such that the CCN concentration of
100 cm-3 is retained. If merging is applied, only superdroplets
with a radius smaller than rmer=0.1µm and with a
weighting factor smaller than Amer=Ainit/2 are allowed to
merge.
At the surface, superdroplets are absorbed if their radius is larger than
1.0 µm. For smaller particles, a reflection boundary condition
is assumed to avoid the change that the surface acts as a CCN sink. Horizontal
boundaries are prescribed with cyclic conditions. Moreover, for collision and
coalescence, the kernel by is used. An overview of all
conducted simulations is given in Table 1.
Summary of the main parameters for the single-cloud simulations.
Simulation
NP
Initial weighting factor
Splitting
rspl
NP,max
ηspl/max
Merging
Const.NP15
15
5.0×1010
no
–
-
–
no
Const.NP87
87
9.0×109
no
–
-
–
no
Const.NP186
186
4.3×109
no
–
-
–
no
S10
87
9.0×109
yes
10 µm
150
20
no
S20
87
9.0×109
yes
20 µm
150
20
no
S20 merging
87
9.0×109
yes
20 µm
150
20
yes
G20
87
9.0×109
yes
20 µm
150
20
no
G20 merging
87
9.0×109
yes
20 µm
150
20
yes
Mass density distribution after 1800 s for the idealized
single-cloud simulations using parameters described in
Table .
Single-cloud results
Microphysical properties
Figure shows the cloud-averaged mass density distribution at
t=1800 s for the configurations listed in
Table . The left part of the spectrum is reproduced
quantitatively consistent in all cases. This implies that both the splitting
and the merging processes have no artificial impact on the diffusional growth
process, which prevails in this region of the spectrum. However, the right
tail of the DSDs differs significantly when the splitting algorithm is
applied. The biggest drops are almost 350 µm smaller for the
reference case (black lines) compared to simulations with splitting.
Furthermore, splitting effectively reduces the fluctuations which occur in
the reference cases for radii above 100 µm. The mass density
distributions imply that the choice of the splitting mode does not affect
cloud microphysical results. Likewise, the simulation S10, in which
the splitting radius is reduced to rspl=10µm, shows
almost no difference in the mass density distribution compared to cases with
rspl=20µm. Thus, it can be deduced that a
splitting radius of rspl=20µm is sufficient for
this cloud. Further investigations (not shown) in which rspl is
successively increased to 30µm show that a larger splitting
radius leads to strong deviations from simulations with smaller splitting
radii. This indicates that droplets with radii larger than
20µm need to be represented in a statistically sufficient way
to initiate the precipitation process correctly. It should be emphasized,
however, that these results are only valid for a cloud with a relatively
strong diffusional radius growth. A reduction of the splitting radius might
be required for settings in which collisions dominate the droplet's growth at
smaller radii as it is the case in the previously presented box simulations.
This behavior can be ascribed to different requirements on the superdroplet
number for the convergence of different growth processes. The left part of
the spectrum is dominated by diffusional growth which can be sufficiently
represented by just a couple of superdroplets per grid box. By contrast,
collisional growth is highly sensitive to the superdroplet number and the
correct representation of large droplets. An improved representation of these
droplets is ensured by the splitting algorithm, no matter what splitting mode
is used.
The improved statistics of large superdroplets are also shown in
Fig. , where the absolute number of superdroplets per logarithmic
radius (log(r)) bin is presented. It is noticeable that in the reference
simulations, this number decreases significantly for larger droplets
(starting from a radius of approximately r=20µm). In
simulations in which no splitting operations are carried out, the largest
droplets are represented by only a few tens of superdroplets in the whole
model domain. For the S mode, the superdroplet concentration is kept almost
constant (except in the right tail) for all splitting cases. For the
G mode, a second maximum at 100 µm can be observed. This can be related
to the calculation of the splitting criterion. The approximation of the mass
density distribution by a gamma distribution results in a somewhat lower
splitting factor for superdroplets close to the splitting radius in
comparison to the S mode, which shifts the superdroplet production to
larger radii in the G mode.
Macrophysical properties
In Fig. , the development of the cloud is shown in a time series of
several macroscopic properties. The behavior of the different splitting
configurations can be clearly seen in Fig. a, which depicts the
ratio of the current superdroplet number to its initial value. In simulations
without splitting, the superdroplet number remains nearly constant. A clear
increase in the superdroplet number can be observed when splitting is used,
with maximum increase of about 15 % for S10. In all other
splitting cases, the increase in superdroplet number is notably lower and
starts approximately 500 s later, which corresponds to the larger
splitting radius of rspl=20µm. The lowest increase
in superdroplet number is observed in the merging cases in which the maximum
number of superdroplets is reached during the growing phase of the cloud and
decreases in the dissipation stage.
Total number of superdroplets per logarithmic radius bin after
t=1800s for the idealized single-cloud simulations using
parameters described in Table .
Figure b and c show the temporal evolution of the liquid water
path (LWP) and the rain water path (RWP). The RWP is defined as the integral
mass of all droplets with r≥40µm. It is notable that the
LWP is the same for all simulations, which emphasizes the mass-conserving
character of the splitting algorithm and its negligible impact on the general
development of the cloud. In the reference simulations (represented as a mean
of five ensembles for each case) of Figs. c and d, one can see an
increase in the precipitation parameters (RWP, radar reflectivity, and
precipitation sum) for an increased number of superdroplets. However, the
differences among the ensemble members are quite large, which is shown by
the range (gray area) and the band of plus–minus one standard deviation from
the mean (light blue area) derived from all 15 ensemble members. Overall, the
splitting simulations have a slight tendency to compare better with the
reference cases using 87 and 186 superdroplets. Admittedly, since the results
are (for the most part) within one standard deviation, it can be concluded
that splitting has no significant influence on the global precipitation
parameters.
Figure e and f display the precipitation rate and the total
precipitation reaching the ground. The precipitation rate in the reference
simulations without splitting exhibit high temporal variances (black lines).
Those variances are successfully reduced in all splitting simulations. This
can be explained by the better representation of precipitation in the
splitting simulations by a larger number of superdroplets, resulting in a
more uniform removal of liquid water by precipitation. As expected from the
RWP, splitting slightly increases the total precipitation.
Figure shows the effect of splitting on the spatial distribution
of rain after 2100 s simulated time for the NP87
simulation (Fig. a) and the S20 splitting simulation (Fig. b).
Similar to the reduced temporal variance in the time series of the
precipitation rate (Fig. e), the spatial variance is also
significantly reduced using splitting. Again, the precipitation is
represented by only a few superdroplets in the simulation without splitting,
which leads to very high, localized precipitation rates. Due to splitting,
raindrops with large weighting factors are split into several superdroplets
with smaller weighting factors, resulting in the more realistic spatial
representation of the precipitation.
All in all, the splitting of large droplets, which results in an improved
representation of the collision process and thus the DSD, also partly
influences the macroscopic properties of the cloud. In particular, rain water
content, radar reflectivity, and precipitation rate are represented in a more
realistic manner. Due to the improved statistics, the temporal and spatial
variance of these parameters is significantly reduced. However, the whole
cloud life cycle, which is driven by the general dynamics and thermodynamics,
is largely unaffected by splitting. Additionally, the merging shows no
influence on the physical outcomes.
To estimate the increase in computing time due to splitting, we conducted
three simulations (Const. NP87, S20,
and S20 merging) with comparable time measurements. Here we observe
that the splitting simulation S20 requires 19.2 % more computing
time than the reference simulation Const. NP87.
If applied, merging allows a massive reduction of the number of
superdroplets, reducing the computing time by 18 % and the storage demand
(which is proportional to the number of superdroplets) by at least 7 %
compared to simulations applying only splitting (Fig. a). All in
all, the simulation applying both splitting and merging is only 1.2% slower
than the reference simulation Const. NP87.
Time series of different variables for the idealized single-cloud
simulation for different initial numbers of superdroplets and splitting
configurations. In (a), the ratio of the actual and initialized
number of superdroplets in the whole model domain is shown. The liquid water
path (LWP) and rainwater path (RWP) are displayed in panels (b)
and (c), respectively. In (d), the total radar reflectivity
is shown. Panels (e) and (f) show the precipitation rate
and total precipitation, respectively. The reference simulations (runs
without splitting) are presented as a mean of five ensembles for each case.
Moreover, the light blue areas show the mean plus–minus one standard
deviation and the gray areas show the range derived from all 15 ensemble
members.
Vertical cross sections of the precipitation rate for the reference
case (a) and the splitting case S20 (b).
Cloud field
Setup
The setup for simulating a shallow cumulus field is based on the LES
intercomparison study by , using their initial profiles
for potential temperature, water vapor mixing ratio, large-scale forcings,
and surface fluxes. As in the original, the model domain covers an area of
about 12.8km×12.8km×4.0km in the
x-, y-, and z-direction, respectively. The grid spacing is Δx=Δy=100m in the horizontal, and Δz=40m in
the vertical. Moreover, the calculation of the domain-averaged quantities
follows the descriptions given in the original case.
Time series of (a) the liquid water path (LWP),
(b) rainwater path (RWP), (c) ratio of the actual number of
superdroplets to the initial number of superdroplets, (d) cloud
cover, (e) precipitation rate, and (f) total precipitation
for different initial numbers of superdroplets and splitting configurations.
The gray areas in (a), (b), and (d) indicate the
documented model variability in the simulated shallow cumulus case
.
Probability density function of precipitation rates for different
initial numbers of superdroplets and splitting configurations.
Three different simulations will be presented. In the cases LCM
NP87 and LCM NP400, the
number of superdroplets per grid box are 87 and 400, respectively. With
initial weighting factors of Ainit=1.89×1012 and
Ainit=7.0×1012, respectively; these represent a CCN
concentration of 100cm-3 in each case. Moreover, one more
simulation with splitting and merging is carried out. For this configuration,
in which the general settings of LCM NP87 are
adopted, the splitting mode S with rspl=20µm,
ηspl=20, and Aspl=Δx×Δy×Δz×1m-3=4.0×105 is used.
Aspl is chosen to allow number concentrations as small as
1m-3 to be represent by a single superdroplet.
Based on the previously presented results, the maximum number of particles
per grid box is set to NP,max=150. Merging is applied in
non-cloudy grid boxes for superdroplets with a radius smaller than
rmer=0.1µm and a weighting factor smaller than
Amer=Ainit/2.
Cloud field results
The analysis is focused on the influence of splitting on the macroscopic
properties of the shallow cumulus field. Figure shows time
series of (a) the LWP, (b) RWP, (c) ratio of the current superdroplet number
to its initial value, (d) cloud cover, (e) precipitation rate, and
(f) total precipitation. Despite the superdroplet number, all these
parameters agree in a statistical sense. In the cases without splitting, the
total superdroplet number decreases slightly in the course of the simulation
due to precipitation (Fig. c), while the simulation with
splitting increases the total superdroplet number by about 15 %. Note,
however, that both LWP and RWP are at the top of model variability documented
in (gray areas), which is in line with the results of
, who also used an LCM for the simulation of this
shallow cumulus case.
Considering the temporal variability in the precipitation rate and total
precipitation (Fig. e and f), no significant changes are
detectable using splitting or a very high number of superdroplets.
Nonetheless, a positive impact of splitting on the representation of
precipitation can be seen in the probability density function of the surface
precipitation rate (Fig. ). For the simulation with 400
superdroplets per grid box and the splitting simulation, the probability for
very high precipitation rates is smaller by about 1 order of magnitude
compared to the simulation LCM NP87. This
clearly shows that extremely high precipitation rates, resulting from
individual superdroplets with large weighting factors, are mitigated when
splitting is applied. Accordingly, splitting is important for a statistically
appropriate representation of individual rain events and necessary for the
process-level understanding of the precipitation process, but the general
features of the cloud field, as it was the case for the single cloud, are
largely unaffected.