ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-18-7251-2018Bridging the condensation–collision size gap: a direct numerical simulation of continuous droplet growth in turbulent cloudsBridging the condensation–collision size gapChenSisisisi.chen@mail.mcgill.cahttps://orcid.org/0000-0002-9598-8255YauMan-KongBartelloPeterXueLulinMcGill University, Montréal, Québec, CanadaNational Center for Atmospheric Research, Boulder, Colorado, USASisi Chen (sisi.chen@mail.mcgill.ca)25May201818107251726210January20188February201830April20187May2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://acp.copernicus.org/articles/18/7251/2018/acp-18-7251-2018.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/18/7251/2018/acp-18-7251-2018.pdf
In most previous direct numerical simulation (DNS) studies on droplet growth in
turbulence, condensational growth and collisional growth were treated
separately. Studies in recent decades have postulated that small-scale
turbulence may accelerate droplet collisions when droplets are still small
when condensational growth is effective. This implies that both processes
should be considered simultaneously to unveil the full history of droplet
growth and rain formation. This paper introduces the first direct
numerical simulation approach to explicitly study the continuous droplet growth by condensation and collisions
inside an adiabatic ascending cloud parcel. Results from the
condensation-only, collision-only, and condensation–collision experiments are
compared to examine the contribution to the broadening of droplet size
distribution (DSD) by the individual process and by the combined processes.
Simulations of different turbulent intensities are conducted to investigate
the impact of turbulence on each process and on the condensation-induced
collisions. The results show that the condensational process promotes the
collisions in a turbulent environment and reduces the collisions when in
still air, indicating a positive impact of condensation on turbulent
collisions. This work suggests the necessity of including both processes
simultaneously when studying droplet–turbulence interaction to quantify the
turbulence effect on the evolution of cloud droplet spectrum and rain
formation.
Introduction
Theoretical studies indicate that for droplets in the size
range of 15–30 µm in radius, referred to as the condensation–collision
size gap, neither condensational growth nor collisional growth is effective
in producing precipitation. Classical parcel models
generally yielded very narrow droplet size distributions (DSDs) and took a
rather long time to form rain . In nature, wide DSDs and
large droplets are frequently observed in cumulus and even stratocumulus
clouds e.g.,. This size gap
problem represents a longstanding challenge in the ongoing quest to
understand the warm-rain initiation process. In the literature, various
mechanisms have been proposed to accelerate rain development, such as
small-scale turbulence , the presence of giant
aerosols , entrainment of unsaturated
air , and large-eddy hopping
. This study focuses on the effect of
small-scale turbulence-containing eddies in the inertial and
dissipation range with length scales ≪ 10 m as shown in Fig. 1 of
, which can be resolved by the technique of direct
numerical simulation (DNS).
Several mechanisms related to turbulence have been proposed to explain the
fast growth of droplets in the condensation–collision size gap
. As a result of the response of droplet
inertia to turbulent eddies of different scales, turbulent flow creates two
effects: the non-uniform distribution of cloud droplets (clustering effect)
and the increase in the relative velocities between droplets (transport
effect). A number of DNS studies have reported that the geometric collision
rate of droplets increases as turbulence intensifies . Concomitantly, turbulence modifies the response of a
droplet to the local disturbance flow induced by other droplets through
hydrodynamic interactions to increase the collision efficiency
. In particular,
demonstrated that the turbulence enhancement of collisions became most
significant among droplet pairs of similar sizes, suggesting that turbulence
may efficiently broaden the narrow DSD generated from condensational growth.
Moreover, it has also been argued that the supersaturation perturbation field
can arise from the fluctuation of temperature and water vapor in turbulence
and the differential local water vapor consumption
which is enhanced by droplet clustering. This may lead to a distinct growth
history by condensation for each droplet as it is transported in a turbulent
flow . However, several DNS studies found that small-scale
turbulence can only create small, if not insignificant, drop size broadening
through condensation . The
reason is that the average time that droplets are exposed to supersaturation
perturbations shortens as the turbulence intensifies and as droplets grow
larger and begin to sediment .
reported a wider size distribution when the Reynolds number of the flow,
which was calculated based on the computational domain size, increased from
40 to 185 and proposed a simple scaling to extrapolate the DNS result to the
typical size of a adiabatic cloud core (approximately 100 m wide or Reynolds
number ≈5000). However, caution should be exercised in applying this
scaling as DNS is not able to capture the spatiotemporal complexity of the
turbulence at scales larger than the size of the domain.
also used a similar model to but extended the simulation
time to 20 min to be comparable to the formation time of rain revealed in
real observations. They found that the variance of the droplet size
distribution was mainly determined by the large-scale flow, i.e., the
large-hopping effect suggested by and studied by
. Nevertheless, it should be noted that their conclusion
was based on the simplified assumption that both the mean updraft speed and
the mean supersaturation were zero. On the other hand, the DNS model of
considered a time-dependent and buoyancy-driven mean
vertical motion calculated from a given environmental sounding. In their
study of the effect of turbulence and entrainment on the evolution of cloud
droplets, it was found that the thermodynamic fluctuations caused by
turbulent advection prevented the buildup of the buoyancy force, leading to
an even slower evolution of the mean droplet size and the vertical velocity
as compared to those predicted by a parcel model.
A common limitation shared by most, if not all, previous DNS studies is that
the condensation process and collision–coalescence process were studied
separately. This assumption may be justifiable in a parcel model due to the
non-overlapping droplet size regimes of the two growth processes in
still air. However, this assumption is questionable in DNS studies which
reveal substantial turbulent enhancement of collisions among droplets in the
condensation–collision size gap. As there is an absence of DNS work on
continuous droplet growth incorporating both processes, it is the goal of
this study to unveil the full history of droplet growth and the DSD
broadening by condensational and collisional growth in a turbulent,
supersaturated environment undergoing an adiabatic ascent.
The purpose of this study is (1) to introduce the first DNS approach to
explicitly resolve the continuous droplet growth by condensation and
collision in shallow, turbulent clouds and (2) to answer the following two
questions. How does the droplet collisional process interact with the
droplet condensational growth process and what is the role of
turbulence in this interaction?
Our approach is to incorporate the droplet hydrodynamic collision and
condensation processes into a single DNS modeling framework. Arguably, this
model provides a first direct approach to bridge the condensation–collision
gap that has puzzled the cloud physics community for decades. The paper is
organized as follows. In Sect. we describe the sets of
equations adopted from and and the
accompanying modification. The simulated results from three sets of
experiments (condensation-only, collision-only, condensation–collision) in
various turbulent environments are given in Sect. ,
followed by a conclusion and remarks on the limitation of this study in Sect. .
Model description and experimental setup
This paper represents a sequel to as part of our ongoing
exploration of the evolution of cloud DSD affected by turbulence. The DNS
model adopted was originally developed by in perhaps
one of the earliest DNS approaches to simulate droplet growth in turbulence.
focused on the impact of turbulence on droplet
condensation, and thus collisions were not considered. A number of extensions
followed. resolved the droplet collisions using an
efficient collision detection technique. made changes to
allow simulation in larger domain sizes and introduced a new forcing scheme
to achieve a statistically steady turbulent dissipation rate.
added the local disturbance flow field induced by droplets
to obtain accurate turbulent collision efficiencies and droplet collisional
growth affected by both the disturbance flow and the turbulence
flow.
In the present study, the model from is further extended to
restore the thermodynamical framework of to include
condensational growth. Specifically, the whole DNS box is regarded as a
parcel ascending adiabatically from near the cloud base with a constant mean
updraft. Two sets of equations are used to solve for (1) the macroscopic
variables that describe the time evolution of the parcel mean state
properties and (2) the microscopic variables that describe the turbulent flow,
as well as the temperature and the water vapor mixing ratio fluctuation
fields. Furthermore, equations pertaining to the
thermodynamics are modified to improve the accuracy of
droplet condensational growth. For convenience of reference, a list of
constants is given in Appendix and the detailed equations
are provided in Appendix .
In the presence of the thermodynamic fluctuation fields and the turbulence
flow field, droplets grow in two distinct ways simultaneously.
Droplets grow by condensation with its growth rate directly proportional
to the instantaneous supersaturation (see Eq. in
Appendix ). When a droplet moves relative to the air, the
water vapor field is not spherically symmetric around the droplet surface but
is modified depending on the direction of motion (the so-called ventilation
effect). This effect becomes important when droplets are greater than 30 µm in radius . In , all droplets
were smaller than 20 µm and this effect was not considered. However, the
present study allows droplets to grow larger and thus the ventilation
coefficient is added to the droplet growth equation. Following
, the curvature term and the solute term are
neglected in the equation, and the droplets are treated as pure water drops
since all droplets in this study are greater than 5 µm
.
Simultaneously, droplets grow through the collision–coalescence process.
The droplet motion and collisional growth are treated in the same manner as
in . Each droplet is tracked in the Lagrangian framework,
with its motion determined by gravity and the local fluid drag force
Eq. 1 in. Once two droplets collide, they coalesce to
become a bigger entity with its mass equal to the sum of the masses of the
collided droplets and its location being the barycenter of the binary system
before the collision. The velocity of the coalesced droplet is calculated
based on the conservation of momentum. Since we are particularly interested
in the condensation–collision size range, i.e., droplets smaller than drizzle
drops, defined as drops with a radius equal to larger than 100 µm, our
study only consider radius r≪ 100 µm. In addition, solving the
motion of large drops requires more complex consideration such as induced
turbulent wakes and drop deformation which are beyond the scope of this
study. Therefore, droplets reaching 100 µm are considered as fallouts
and are not allowed to grow further, i.e., they neither interact with other
droplets nor affect the local disturbance flows. It should be
noted that this assumption bears certain caveats. The Stokes' law assumption
for the disturbance flow becomes less accurate for droplets larger than 50 µm,
because droplets over 50 µm (and smaller than 100 µm) have
a particle Reynolds number of order one. However, since the collision
efficiency for droplets larger than 50 µm is close to unity due to the
large Stokes number, it is argued that the impact of the disturbance flow on
the collision statistics of those large particles would be secondary.
Furthermore, in all the simulations the calculated total number of collisions
remains below 10 % of the total number of droplets. Specifically, the number
of collisions is within 9 % in strong turbulence, below 3 % in weak
turbulence, and below 2 % in still air. It follows that the impact of reducing
the droplet number concentration due to collisions on the resulting DSD can
be assumed small. One alternative post-collision treatment maybe to introduce
a new, randomly located droplet into the domain once a collision happens so
that the droplet number concentration remains constant. However, the size of
droplets that should be introduced remains contentious and needs further
justification.
Three sets of experiments are conducted to evaluate the DSD broadening due to
the turbulence effect on different droplet growth processes: (1) droplet
growth by condensation only (referred to as the condensation-only
experiment), (2) droplet growth by collision–coalescence only (referred to as
the collision-only experiment), and (3) droplet growth by condensation and
collision–coalescence together (referred to as the condensation–collision
experiment). All experiments use the same initial DSD shape
adopted from an aircraft measurement in non-precipitating cumulus clouds
. The initial droplet number concentration is set as 80 cm-3 and a constant updraft of 2.5 m s-1
is used to represent the
condition of pristine maritime cumulus clouds.
For each set of experiments (except for the condensation-only experiment),
three flow configurations are considered: purely gravitational case (i.e.,
still air), a weak turbulence case (with eddy dissipation rate ϵ= 50 cm2 s-3),
and a strong turbulence case (with ϵ= 500 cm2 s-3). The domain size of each simulation is about 10 cm in each
direction, with grid space ≈0.1 cm determined by the dissipation
rate as explained in . It is recognized that droplet
condensation in still air leads to a narrow DSD and the DSD broadening by
condensation impacted by small-scale turbulence is insignificant. Therefore,
during the condensation-only experiment, only the strong turbulence
simulation is performed to serve as an upper bound of the DSD broadening
among the three flow conditions. As a result, seven simulations are
performed. Each simulation lasts 6.5 min in real time, which is the
approximate duration required for the whole parcel to ascend from cloud base to 1000 m
above the base, which is representative of a typical cumulus development.
Results and discussion
We first compare the results from the three experiments to scrutinize the
contributions of the different droplet growth processes under the effect of
turbulence. Figure shows the DSDs at the end of each experiment
in strong turbulence. As a reference, the initial DSD is displayed with a
gray area. It should be noted that droplet number
concentrations below 0.001 cm-3 will be treated as statistical
uncertainty throughout the discussion, since they correspond to less than 2–4
droplets in the domain. Consistent with past findings, the turbulence effect
on droplet condensational growth is small. The condensation-only process
produces the narrowest size distribution among the three experiments and
droplets grow no larger than 20 µm at the end of the simulation. On the
other hand, in both the collision-only experiment (blue curve) and the
condensation–collision experiment (yellow curve), a substantial number of
large droplets are found. Furthermore, compared to the collision-only
simulation, the condensation–collision experiment generates more large
droplets and substantially larger droplets. The largest r reaches 100 µm at the end of the simulation compared to less than 65 µm in the
collision-only case. Meanwhile, the number concentration of r> 30 µm
droplets in the condensation–collision case increases by a factor of 2.3
(0.35 cm-3 compared to 0.15 cm-3 in the collision-only case).
Droplet size distributions at the
6.5th minute for the condensation-only case (red), collision-only case (blue) and
condensation–collision case (yellow). Dissipation rate is 500 cm2 s-3
for all cases with the initial size distribution shown as a dashed grey line.
Droplet number concentrations below 0.001 cm-3 are treated as
statistical uncertainty.
To examine whether this enhanced broadening due to the inclusion of the
condensational process depends on the flow, detailed comparisons between the
collision-only and the condensation–collision experiments are made under
three flow conditions: purely gravitational, weak turbulence, and strong
turbulence. Figure demonstrates the time evolution of the DSD in
the two sets of experiments under the three different flows. It is found
that
In the purely gravitational case (Fig. a and b), despite the condensation–collision experiment producing
larger maximum droplets at the end of the simulation relative to the
collision-only experiment (black outline in Fig. ), the number
concentration of large droplets is still negligible (as r> 35 µm
droplets stay below 0.001 cm-3 as seen from the expansion of the purple
edge with time).
In the turbulent cases, we find more large droplets and
much larger maximum droplet sizes in the domain when condensational growth is
considered. With weak turbulence, droplets larger than 35 µm (over 0.001 cm-3)
can be seen as early as 3.5 min in the condensation–collision
experiment but 6 min in the collision-only run. With strong turbulence,
large droplets were found in the third minute in the condensation–collision
simulation compared to the fourth minute without condensation. It is evidence
that both experiments experience earlier formation of large droplets as
turbulence intensifies while the inclusion of condensation further accelerate
the droplet growth. This result evinces that an effective
condensation–collision broadening mechanism that strengthened with
increasing turbulence intensity exists.
The time evolution of the DSD in the collision-only (Coll) experiments
(a, c, e) and the collision–condensation (Coll + Cond) experiments (b, d, f).
Results from the purely gravitational case (first row), weak turbulence
(ϵ= 50 cm2 s-3, c, d), and strong turbulence
(ϵ= 500 cm2 s-3, e, f) are demonstrated. The solid black
curve indicates the largest droplet of the entire domain. The droplet number
concentration (cm-3) on each size bin (bin width = 1 µm) is
displayed in color using a logarithmic scaling shown in the color bar.
Droplet number concentrations below 0.001 cm-3 are treated as
statistical uncertainty and thus are given no color in the
plot.
A condensation-induced broadening has been found in all three flow
conditions, though it seems that it is negligible in the case of still air.
This phenomenon can be explained by two main mechanisms:
The condensational growth process effectively produces droplets of small
sizes (r< 10 µm) to medium size (10–20 µm) due to the fast
growth rate of small droplets. This conjecture is supported by the result on
the right column of Fig. showing that, among the three
condensational cases, all droplets smaller than 15 µm become greater
than 15 µm within 4 min. As bigger droplets have higher collision
rates, the average collision rate in the domain is expected to increase
progressively as more medium-sized droplets are formed through condensation,
and they become more likely to be collected by other droplets.
Condensational growth narrows the DSD and provides a great number of
similar-sized droplets (i.e., the radius ratio between the small droplet and
large droplet, r/R, is close to unity). found that
turbulence enhancement of the collision rate is most significant in similar-sized
droplets and stays relatively weak for 0.2<r/R<0.8 (Fig. in
their paper). In an environment with similar-sized droplets created by
condensation, the turbulence-enhanced collisions are enhanced to accelerate
the production of large droplets.
Probability distribution function (PDF) of collisions with respect
to r/R at three different flow conditions (a, b: pure gravity;
c, d: weak turbulence; e, f: strong turbulence). Results from the
collision-only experiments (a, c, e) and the condensation–collision
experiments (b, d, f) are shown for comparison.
Panels (a–f) are the same as in Fig. but for the
collision frequency (cm-3 s-1) from different r/R pairs. Panels (g–i)
are the enhancement of collision frequency for different r/R pairs
due to the inclusion of condensation. The enhancement is calculated by taking
the ratio of the collision frequency from the condensation–collision
experiment and the collision-only experiment. Results from three different
flow conditions are demonstrated.
The first mechanism of enhanced collision rate due to larger mean droplet
sizes can also happen in the purely gravitational case but will be offset by
the inefficient gravitational collection process due to the DSD narrowing by
condensation. In turbulent cases, the condensational DSD produce
similar-sized droplets to allow the turbulence-enhanced similar-sized
collision process to act, leading to a positive feedback mechanism. Evidence
for this hypothesis can be found by comparing the probability distribution
function (PDF) of collisions with respect to r/R (the radius ratio between
the small droplet and big droplet in a droplet pair) in the collision-only
and the collision–condensation experiments. As seen in Fig. , the
PDF of collisions in either the weak turbulence or the strong turbulence
become more flattened when condensation is included. In particular, the
chance of similar-sized collisions (r/R>0.9) is substantially greater. On
the contrary, a narrower PDF is found in the purely gravitational case (Fig. a and b). Figure demonstrates the
distributions of collision frequency and the collision enhancement due to
condensation. It is found that in the purely gravitational case the number of
similar-sized collisions doubles in the condensation–collision experiment,
which results from the increased number of similar-sized droplets introduced
by condensation. It is obvious that increasing the intensity of turbulence
further enhances these collisions. The similar-sized collisions increase by a
factor of 3.5 in weak turbulence and a factor of 4.5 in strong turbulence.
Time evolution of the number of pair combinations
for (a) the different-sized droplets (r/R≤0.7) and (c) the
similar-sized droplets (r/R>0.7) in the collision-only experiments, and
(b) the different-size droplets and (d) the similar-sized droplets in the
condensation–collision experiments. The pair combination is computed using
the droplet number concentration (cm-3); therefore, the unit is in
cm-6. The color denotes the three different flow conditions which are
shown in the legend.
In the small r ratio range (r/R<0.7), the total collisions in the
purely gravitational case are lowered by more than half due to a reduced
number of those droplet pairs (Fig. g). However, in the
turbulent cases, the collision frequency instead experiences a mild increase
compared to the collision-only experiment. This increase is due to the fact
that condensation increases the population of medium-sized droplets (r= 10–20 µm) and
turbulence continues to enhance the collisions of these droplets.
The abundant number of those medium-sized droplets boosts the number of
similar-sized collisions by turbulence to produce larger droplets. Meanwhile,
the larger size from growth by condensation substantially increases the
chance of those droplets being collected by other larger droplets.
Furthermore, the formation of large droplets due to the turbulence-enhanced
collisions in turn contributes to the growing collector droplet population,
thus further increasing the chance of these medium-sized droplets being
collected. In the purely gravitational case, this process is inhibited by the
insignificant similar-sized collisions in spite of their number being doubled
from the collisional-only case.
To further illustrate the influence of turbulence on the
enhancement of the condensation-induced collisions, we have studied the
impact of condensation on the evolution of the number of droplet pairs. We
divided the droplet pairs into two groups: the similar-sized pair group with
r/R>0.7 and the different-sized pair group with r/R≤0.7. We then
calculated the total number of droplet pairs within the two groups. By
comparing the results from the collision-only experiments and the
collision–condensation experiments, we are able to separate the enhancement
of the collision rate solely due to turbulence and the enhancement directly
associated with the inclusion of condensational growth.
Figure shows the time evolution of the total number
of droplet pairs for the two groups in the domain. For the convenience of
comparison among the three turbulent cases, the pair numbers are calculated
based on the droplet number concentration (cm-3). For example, for the
droplet pair of r1 and r2 with concentrations of nd1 and nd2,
the pair number is nd1nd2 if r1≠r2 and
nd1nd22 if r1=r2. Therefore the unit of the pair
number is in cm-6. It is found that in the collision-only experiment the
number of different-sized droplet pairs stays relatively constant (Fig. a). Meanwhile, the number of similar-sized droplets undergoes
only a weak decay (Fig. c). Compared to the pure gravity case,
turbulence effectively accelerates similar-sized collisions, while the
enhancement of different-sized collisions is relatively small. On the one
hand, the turbulence enhancement of similar-sized collisions is due to the
fact that turbulence has a stronger effect on the similar-sized collision
efficiency . On the other hand, it has been found that the
turbulence clustering effects are more significant for droplets of similar
sizes . They tend to cluster in the same regions of the flow
because of similar droplet inertia and terminal velocities. This effect has
been confirmed previously in a number of studies e.g., and is especially pronounced for large droplets. The reason is
that small droplets have small Stokes numbers, and they adjust very quickly
to changes in the flow and therefore behave more like fluid tracers than
inertial droplets. Consequently, with growing droplet size, turbulence
clustering of similar-sized droplets becomes more significant and the number
of similar-sized pairs undergo an accelerated decline. This can be seen in
Fig. c where the curve of the turbulent case deviates from the
purely gravitational case.
By contrast, in the condensation–collision experiment the
trend of the number of droplet pairs behaves in a more complex fashion due to
the inclusion of condensation. As illustrated by Fig. b and d,
the droplet growth experiences two different stages. The number of
different-sized pairs significantly decreases in the first 2 min mainly
due to the rapid condensational growth of droplets with r< 15 µm. This
is demonstrated in Fig. where the droplet number concentration for
r< 15 µm quickly reduces from larger than 1 cm-3 to below
0.001 cm-3 in the first 2 min, while the production of large
droplets is still negligible. Concurrently, the number of similar-sized
droplets significantly increases during the first 2 min and steadily
decreases thereafter (Fig. d). The large increase in
similar-sized pairs in the collision–condensation experiments during the
first 2 min significantly increases the number of turbulence-enhanced
similar-sized collisions. After 2 min, the condensational effect
diminishes and the collision–coalescence process takes over in modulating the
droplet pair population. The subsequent decline of the number of
similar-sized pairs and the increase in the number of the different-sized
pairs mainly arise from the collision–coalescence process.
Conclusions
This work provides the first DNS study to explicitly resolve continuous
droplet growth by condensation and collision in a turbulent environment. The
results are expected to contribute toward resolving the warm-rain initiation
problem.
Results from the condensation-only, collision-only, and
condensation–collision experiments are compared to examine the contribution
to the DSD broadening by the individual process and by the combined processes
acting in concert. Three different flow environments (still air, weak
turbulence, and strong turbulence) are investigated to scrutinize the impact
of turbulence in the condensation-induced collisions. By comparing the
collision frequencies of the collision-only experiment and the
condensation–collision experiment, it is found that condensational growth
boosts the collisions when the flow is turbulent and slows down the
collisions for the case of still air.
In the purely gravitational experiment, the abundant similar-sized droplets
generated by condensation inhibit the gravitational collection process, and
the collision frequency of r/R<0.7 reduces by half. As a result, the number
concentration of droplets larger than 35 µm remains lower than 0.001 cm-3 throughout the simulation.
In the turbulence experiments, a greater number of large droplets are
produced, and their appearance occurs faster as turbulence intensifies,
implying an effective turbulence impact on droplet size broadening.
Furthermore, droplets larger than 35 µm form 1–2 min earlier in the
collision–condensation experiments. It follows that these droplets appear as
early as the third minute in the strong turbulence situation. This result suggests
that a sophisticated model that takes into account both the
turbulence-enhanced collisions and condensation-induced collisions under
the effect of turbulence should be used to study the cloud droplet spectrum
broadening and rain formation.
Finally we remark on the limitation of this study and some suggestions for
future work. It has been found that the evolution of the DSD and the rain
formation time highly depend on the initial shape of the DSD and the droplet
number concentration. Therefore, simulations of different initial DSDs are to
be conducted to better understand its dependency. In addition, the initial
DSD used in this study is taken from flight observations, which represents an
average over a long sampling time and a wide sampling volume. In this case,
the initial DSD is not guaranteed to be representative of the steady-state
DSD from aerosol activation and condensational growth in adiabatic cloud
cores. However, with the continuous advancement of in situ
and laboratory measurement technology such as HOLODEC
and the PI
chamber , representative sampling of the DSD near the cloud
base inside adiabatic cores may be possible in the near future. It is also
desirable to include the aerosol activation process to enable cloud particles
to grow from the very beginning (i.e., dry aerosols in sub-cloud regions). We
strive to explore this approach in a future study. Besides, the model can
also be modified to study other microphysics processes such as ice nucleation
which is poorly parameterized for deep convective clouds and cirrus clouds
as well as particle electrification which is potentially important in aerosol
scavenging and droplet collisions.
The data in this study were produced by the direct
numerical simulation (DNS) model and are available upon request.
List of constants
Dv=2.55×10-5water vapor diffusivity (m2 s-1)Dt=2.22×10-5thermal diffusivity (m2 s-1)ν=1.6×10-5air kinematic viscosity (m2 s-1)Ka=2.48×10-2thermal conductivity of air (J m-1 s-1 K-1)Rv=461.5Individual gas constant for water vapor (J kg-1 K-1)Ra=287Individual gas constant for dry air (J kg-1 K-1)L=2.477×106specific latent heat for water (J kg-1 K-1)Cp=1005specific heat for air (J kg-1 K-1)Γd=-g/Cpdry adiabatic lapse rateρw=1000.0density of water (Kg m-3)Sch=ν/DvSchmidt number
Equations for the DNS modelMicroscopic equations
The condensational growth rate of an individual droplet with radius Ri is
as follows:
dRi2dt=2KfvS.
Here K-1=ρwRvTesat(T)Dv+LρwKaT(LRvT-1),
where esat is the saturated water vapor pressure. fv refers to the droplet ventilation
coefficient. The value of fv is determined by the empirical formulas from the
laboratory experiment of Beard and Pruppacher (1971):
fv=1.0+0.108(NSc1/3Rep1/2)2,forNSc1/3Rep1/2<1.4,fv=0.78+0.308(NSc1/3Rep1/2),for51.4>NSc1/3Rep1/2≥1.4,
where Rep=2Ri|V|ν is the droplet Reynolds
number and V is the velocity of droplet i. S is the supersaturation in the
grid cell where droplet i is located, defined as
S=qvqvs-1,
where qv is the water vapor mixing ratio, with its corresponding saturated
value qvs determined by temperature ((2.17)–(2.18) in Rogers and
Yau, 1989). We assume that all droplets residing in the same grid cell are exposed
to the same supersaturation environment. The scaler fields of qv and
temperature T can be decomposed into the parcel mean state and the
perturbation state. The parcel mean state is calculated via the macroscopic
set of equations shown in Sect. and the perturbations are
calculated as follows:
∂T′∂t=-∇⋅(UT′)-W′Γd+LCpCd′+Dt∇2T′,∂qv′∂t=-∇⋅(Uqv′)-Cd′+Dv∇2qv′,
where W′ is the vertical perturbation velocity. Cd′=Cd-CdM is the differential condensation rate between the grid cell and the
whole parcel. Given Eq. (), the condensation rate inside the
grid cell can be simplified as
Cd=1ma∑in43πρwdRi3dt=4maπρwKfv∑inRiS.
The turbulent velocity field U is governed by the incompressible
Navier–Stokes equations:
∂U∂t+(U⋅∇)U=-1ρa∇P+ν∇2U+F,∇⋅U=0,
where P is the perturbation pressure deviation from the hydrostatic pressure
PM (Eq. B14). The pressure term can be dropped when the equations
are solved in vorticity form. F is the external forcing. We used
the forcing method of to maintain the turbulence. The
droplet motion is governed by fluid drag force and gravity:
dV(t)dt=V(t)-Ũ(X(t),t)τp+g,
where τp denotes the droplet response time. For r< 40 µm, Stokes drag
force is applied and τp=(2ρw9νρa)r2. Droplet
terminal velocity can be obtained using VT=gτp. For r≥ 40µm,
the terminal velocity derived from the experimental data is applied to those
big droplets: VT=k2r; here k2=8×103s-1p. 126. Ũ is the flow velocity at the
droplet center, contributed by the turbulent flow field U and the
disturbance flow Udist caused by neighboring droplets
. The superposition method by is used to
calculate the disturbance flow.
Macroscopic equations
The time evolution of the parcel mean temperature TM, water vapor mixing
ratio qvM, pressure PM, and density ρaM are described as below.
All variables of parcel mean are denoted with a subscript M.
dTMdt=-WMΓd+LCpCdMdqvMdt=-CdMCdM=1Ma∑iNddt(43πρwRi3)=4MaπρwKfv∑iNRiS,PMdt=-ρaMgWMρaM=PMRaTM
The total fields of T and qv are calculated by adding the macroscopic
variables and the perturbation variables.
The authors declare that they have no conflict of interest.
Acknowledgements
We would like to acknowledge Paul A. Vaillancourt from Environment and
Climate Change Canada for providing the original DNS code and offering
constant help in this project. We also thank Jorgen Jensen and Hugh Morrison from NCAR for their valuable discussions.
Special thanks to Yeti Li for his insightful comments in shaping the first draft of this paper. We
would also like to thank Brian Dobbins and Jeremy Sauer from NCAR for
their generous help in improving the code performance. Luxlin Xue appreciates the
support of the Beijing Weather Modification Office through the Beijing Municipal
Science and Technology Commission (grant no. D171100000717001).
We also acknowledge the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC). Computations were made on
the Cheyenne supercomputer (10.5065/D6RX99HX) provided by NCAR's
Computational and Information Systems Laboratory, sponsored by the National
Science Foundation and on supercomputer Cedar provided by WestGrid
(www.westgrid.ca) and Compute Canada (Calcul Canada; www.computecanada.ca).
Edited by: Eric Jensen
Reviewed by: Wojciech Grabowski and one anonymous referee
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