Evolution of droplet size distribution (DSD) due to mixing between cloudy and
dry volumes is investigated for different values of the cloud fraction and
for different initial DSD shapes. The analysis is performed using
a diffusion–evaporation model which describes time-dependent processes of
turbulent diffusion and droplet evaporation within a mixing volume. Time
evolution of the DSD characteristics such as droplet concentration, LWC and
mean volume radii is analyzed. The mixing diagrams are plotted for the final
mixing stages. It is shown that the difference between the mixing diagrams
for homogeneous and inhomogeneous mixing is insignificant and decreases with
an increase in the DSD width. The dependencies of the normalized cube of the mean
volume radius on the cloud fraction were compared with those on normalized
droplet concentration and found to be quite different. If the normalized
droplet concentration is used, mixing diagrams do not show any significant
dependence on relative humidity in the dry volume.
The main conclusion of the study is that traditional mixing diagrams cannot
serve as a reliable tool for analysis of mixing type.
Introduction
The effects of mixing of cloudy air with surrounding dry air on cloud
microphysics are still the focus of many studies (see overview by Devenish
et al., 2012). Processes of mixing are investigated in observations (Yum
et al., 2015; Bera at al., 2016a, b), large eddy simulations (Andrejczuk
et al., 2009; Khain et al., 2018) and direct numerical simulations (Kumar
et al., 2014, 2017). Processes of mixing and their effects on droplet size
distributions were recently investigated in a set of theoretical studies
(Yang et al., 2016; Korolev et al., 2016 (hereafter, Pt1); Pinsky et al.,
2016a, b).
Pt1 presented analysis of conventional (classical) concept of mixing and
introduced the main parameters characterizing homogeneous and extremely
inhomogeneous mixing. In the classical concept two volumes, one cloudy and
one droplet-free, mix within an unmovable adiabatic mixing volume. At
a monodisperse initial droplet size distribution (DSD), homogeneous mixing
leads to a decrease in droplet size and droplet mass content, while the
number of droplets remains unchanged. Extremely inhomogeneous mixing is
characterized by decreasing the number of droplets due to full evaporation of
some fraction of droplets penetrating the initially dry air volume while the
DSD shape in the cloud volume remains unchanged. As a result of extremely
inhomogeneous mixing, droplet number decreases while the mean volume radii
remain unchanged. At a polydisperse DSD, the extreme homogeneous mixing is
characterized by proportional changes in DSD for all droplet radii (Pt1).
Since widely used mixing diagrams describe the final equilibrium stage of
mixing within the mixing volume, they do not contain information about changes
in microphysical quantities in the course of mixing.
Pinsky et al. (2016a, hereafter Pt2) analyzed the time evolution of initially
monodisperse and polydisperse DSD during homogeneous mixing. It was shown
that the result of mixing strongly depends on the shape of the initial DSD. At
a wide DSD, evaporation of droplets (first of all, of the smallest ones) is
not accompanied by a decrease in the mean volume or effective radius.
Moreover, the values of the radii may even increase over time. This result
indicates that the widely used criterion of separation of mixing types based
on the behavior of the mean volume radius during mixing is not generally
relevant and may be wrong in application to real clouds.
Pinsky et al. (2016b, hereafter Pt3) introduced a diffusion–evaporation model
which describes evolution DSDs and all the microphysical variables due to two
simultaneously occurring processes: turbulent diffusion and droplet
evaporation. Mixing between two equal volumes of subsaturated and cloudy air
was analyzed; i.e., it was assumed that the cloud volume fraction μ=1/2.
The initial DSD in the cloudy volume was assumed monodisperse. These
simplified assumptions allowed to reduce the turbulent mixing equations to
two-parameter ones. The first parameter is the Damkölher number,
Da, which is the ratio of the characteristic mixing time to the
characteristic phase relaxation time. The second parameter is the potential
evaporation parameter R characterizing the ratio between the amount of
water vapor needed to saturate the initially dry volume and the amount of
available liquid water in the cloudy volume.
Within the Da-R space, in addition to the two extreme mixing types
defined in the classical concept, two more mixing regimes were distinguished,
namely intermediate and inhomogeneous mixing. It was shown that any type of
mixing leads to formation of a tail of small droplets, i.e., to DSD
broadening. It was also shown that the relative humidity in the initially dry
volume rapidly increases due to both water vapor diffusion and evaporation of
penetrating droplets. As a result, the mean volume and effective radii in the
initially dry volume rapidly approach the values typical of cloudy volume. At
the same time, the liquid water content (LWC) remains significantly lower
than that in the cloudy volume over a much longer time than required for the
effective droplet radius to grow.
In the present study (Pt4) we continue investigating the turbulent mixing
between an initially droplet-free volume (referred to as dry volume) and
a cloudy volume. The focus of the study is investigation of DSD temporal
evolution and analysis of the final equilibrium DSD. In comparison to Pt3,
the problem analyzed in this study is more sophisticated in several aspects:
The dependences of different mixing characteristics on cloud volume fraction 0≤μ≤1 are analyzed. In
this
case the equations of turbulent mixing cannot be reduced to the
two-parameter problem as was done in Pt3.
The initial DSDs in cloud volume are polydisperse. We use both narrow and wide initial DSD described by
gamma
distributions with different sets of parameters. The DSDs are the same as
those used in Pt2. Mechanisms of formation of wide DSDs in clouds including
DSDs in undiluted cloud cores have been investigated in several studies (e.g.,
Khain et al., 2000; Pinky and Khain, 2002; Segal et al., 2003; Prabha
et al., 2011). These studies show that the DSD broadening is caused by in-cloud
nucleation of droplets within clouds as well as by collisions between cloud
droplets. It was shown that DSDs in adiabatic volumes can be wide and first
raindrops or drizzle drop arise namely in non-diluted adiabatic cloud parcels
(Khain et al., 2013; Magaritz-Ronen et al., 2016). We use both narrow and
wide DSDs in the form of gamma distribution with typical parameters used in
different cloud-resolving models. The DSDs that are used as initial ones in
cloudy volumes could also be formed under influence of mixing during their
previous history. The mechanisms of the formation of initial DSD are not of
interest in the study since they do not affect the analysis.
The equation for supersaturation that was used in this study is valid at low humidity in the initially dry volume and
is more general and compared with that used in Pt3, which makes the DSD
calculations more accurate.
At the same time, some simplifications used in Pt3 are retained in this
study. The vertical movement of the entire mixing volume is neglected;
collisions between droplets and droplet sedimentation are not allowed. Also,
we consider a 1-D diffusion–evaporation problem. We neglect the
changes of temperature in the course of mixing, which is possibly a less
significant simplification. All these simplifications allow for revealing the
effects of turbulent mixing and evaporation on DSD evolution.
Formulation of the problem and model design
In this study, the process of mixing is investigated based on the solution
of 1-D diffusion–evaporation equation (see also Pt3). According to this
equation, evaporation of droplets due to negative supersaturation in the
mixing volume takes place simultaneously with turbulent mixing. Since
droplets within the volume are under different negative supersaturation
values until the final equilibrium is reached, the modeled mixing is
inhomogeneous. The droplets can evaporate either partially or totally. The
evaporation leads to a decrease in droplet sizes and droplet
concentration.
The initial state at t=0. The left
volume is a saturated cloudy volume; the right volume is an undersaturated
dry air volume.
Like in Pt3, the process of turbulent diffusion is described by a 1-D
equation of turbulent diffusion. The equation does not describe formation of
separate turbulent filaments. Instead, it describes averaged effects of
turbulent vortices of different scales by modeling of turbulent diffusion,
characterized by a typical value of turbulent diffusion coefficient K. The
mixing is assumed to be driven by isotropic turbulence at scales within the
inertial sub-range where Richardson's law is valid. In this case, turbulent
coefficient is evaluated as in Monin and Yaglom (1975):
K(L)=Cε1/3L4/3.
In Eq. () ε is the turbulent kinetic energy dissipation
rate and C=0.2 is a constant (Monin and Yaglom, 1975), Boffetta and Sokolov
(2002). Equation () means that we consider the effects of turbulent
diffusion at scales much larger than the Kolmogorov microscale; i.e., the
effects of molecular diffusion are neglected. In the simulations, we use
L=40m and ε=20cm2s-3. This means that
in the present study mixing is performed by vortices smaller than several
tens of meters, which agrees with measurements in warm cumulus (Cu) (Gerber
et al., 2008). The value of the turbulent kinetic energy dissipation rate chosen
is also typical for small Cu (e.g., Gerber et al., 2008). These parameters
correspond to Da values of several hundred. The model allows
utilization of other values of L and ε typical of other cloud
type (say, deep convective clouds), which can change results quantitatively,
but not qualitatively.
Geometry of mixing and the initial conditions
The conceptual scheme presenting mixing geometry and the initial conditions
used in the following analysis are shown in Fig. 1.
At t=0 the mixing volume of length L is divided into two volumes: the
cloud volume of length μL and the dry volume of length
(1-μ)L, where 0≤μ≤1 is the cloud volume
fraction. The entire volume is assumed closed, i.e., adiabatic. At t=0 the
cloud volume is assumed saturated, so the supersaturation S1=0. This
volume is also characterized by the initial distribution of the square of the
droplet radii g1(σ), where σ=r2. The initial liquid
water mixing ratio in the cloudy volume is equal to
qw1=4πρw3ρa∫0∞σ3/2g1(σ)dσ. The integral of
g1(σ) over σ is equal to the initial droplet concentration
in the cloud volume N1=∫0∞g1(σ)dσ.
The initial droplet concentration in the dry volume is N2=0, the initial
negative supersaturation in this volume is S2<0 and the initial liquid
water mixing ratio qw2=0. Therefore, the initial profiles of these
quantities along the x axis are step functions:
N(x,0)=N1if0≤x<μL0ifμL≤x<L,S(x,0)=0if0≤x<μLS2ifμL≤x<L,qw(x,0)=qw1if0≤x<μL0ifμL≤x<L.
The initial profile of droplet concentration is shown in Fig. 1 (bottom). This is the
simplest inhomogeneous mixing scheme, wherein mixing takes place only in the
x direction, and the vertical velocity is neglected.
Since the total volume is adiabatic, the fluxes of different quantities
through the left and right boundaries at any time instance are equal to zero,
i.e.,
∂N(0,t)∂x=∂N(L,t)∂x=0;∂qw(0,t)∂x=∂qw(L,t)∂x=0;∂qv(0,t)∂x=∂qv(L,t)∂x=0,
where qv is the water vapor mixing ratio.
To investigate of mixing process for different initial DSD, we assume that
DSD in the cloud volume can be represented by a gamma distribution:
f(r,t=0)=N0Γ(α)βrβα-1exp-rβ,
where N0 is an intercept parameter, α is a shape parameter
and β is a slope parameter of distribution. The DSD f(r) relates to
distribution g1(σ) as f(r)=2rg1(σ). We performed
simulations with both initially wide and narrow DSDs. The width of DSD is
determined by a set of parameters. The parameters of the initial gamma
distributions used in this study are presented in Table 1. Parameters of the
distributions are chosen in such a way that the modal radii of DSD and the
values of LWC are the same for both distributions. These distributions were
used in Pt2 for analysis of homogeneous mixing.
The supersaturation equation for an adiabatic immovable volume can be written
in the form 1S+1dSdt=-A2dqwdt, where S is supersaturation
over water, and the coefficient A2=1qv+Lw2cpRvT2 is slightly
dependent on temperature (Korolev and Mazin, 2003) (notations of other
variables are presented in Appendix A). In our analysis we consider A2 to be a constant. As follows
from the supersaturation equation, the quantity
Γ(x,t)=ln[S(x,t)+1]+A2qw(x,t)
is a conservative quantity, i.e., it is invariant with respect to phase
transitions. In Eq. (), S(x,t) can be comparable
with unity by the order of magnitude. The conservative quantity Γ(x,t) obeys the following equation for turbulent diffusion,
∂Γ(x,t)∂t=K∂2Γ(x,t)∂x2,
with the adiabatic (no flux) condition at the left and right boundaries
∂Γ(0,t)∂x=∂Γ(L,t)∂x=0 and the initial profile at t=0Γ(x,0)=A2qw1if0≤x<μLln[S2+1]ifμL≤x<L.
From Eq. () it follows that Γ(x,0) is positive in the cloud
volume and negative in the initially dry volume. The mean value of function
Γ(x,0) can be written as follows:
Γ‾=1L∫0LΓ(x,0)dx=A2qw1L∫0μLdx+ln[S2+1]L∫μLLdx=μA2qw1+(1-μ)ln[S2+1].Γ‾ can be either positive or negative. In the latter case
a complete evaporation of droplets in the course of mixing takes place.
The solution of Eq. () with the initial condition Eq. () is
(Polyanin et al., 2004):
Γ(x,t)=∑n=0∞anexp-Kn2π2tL2cosnπxL=μA2qw1+(1-μ)ln[S2+1]-2(ln[S2+1]-A2qw1)∑n=1∞sin(nπμ)nπ×exp-Kn2π2tL2cosnπxL.
One can see that function Γ(x,t) depends on three independent
parameters A2qw1, S2 and μ. This function does not
depend on the shape of the initial DSD in the cloud volume. In the final
state when t→∞, Γ(x,t) is
Γ(t=∞)=μA2qw1+(1-μ)ln[S2+1].
Therefore, Γ(t=∞) depends on the cloud fraction and the initial
values of the liquid water mixing ratio in the cloud volume and the relative
humidity in the initially dry volume.
The final equilibrium values of supersaturation S(x,∞) and liquid
water mixing ratio qw(x,∞) can be calculated using
Eq. (). The case Γ(t=∞)>0 corresponds to the
equilibrium state with S(x,∞)=0 and qw(x,∞)=μqw1+(1-μ)ln[S2+1]A2, when droplets
remain but do not evaporate any longer.
The case Γ(t=∞)<0 corresponds to the equilibrium state with
qw(x,∞)=0 and S(x,∞)=(1+S2)1-μexpμA2qw1-1. In this equilibrium state
droplets are totally evaporated, and volume remains subsaturated
S(x,∞)<0. At given qw1 and S2, there is a critical
value of the cloud fraction μcr which separates these two
possible final equilibrium states. This critical value corresponds to Γ(t=∞)=0 and can be calculated from Eq. () as
μcr=ln[S2+1]ln[S2+1]-A2qw1.
Another expression for μcr was formulated in Pt1.
Spatial–temporal variations of
conservative function 100 ×Γ(x,t) for different cloud
fractions μ and initial RH2= 80 %.
The examples of spatial–temporal variations of function Γ(x,t) for
different cloud fractions and initial RH = 80 % are shown in Fig. 2.
The upper panels (μ=0.1) correspond to the case of final total droplet
evaporation and negative final function Γ, whereas the middle and
bottom rows (μ=0.5 and μ=0.9) illustrate partial evaporation cases
when the total mixing volume reaches saturation. It is interesting that the
time required for the final equilibrium state to be reached practically does
not depend on the cloud fraction, which is ∼ 180 s for the illustrated
cases. The cases μ=0.1 and μ=0.9 demonstrate a strong non-symmetric
spatial variability of Γ(x) function during the first 50 s. At μ=0.5, a nearly full compensation between saturation deficit in the dry
volume and available liquid water in the cloud volume takes place if S(x,∞)=qw(x,∞)=Γ(x,∞)=0 at the
equilibrium state. However, the compensation at μ=0.5 is not full because of
the nonlinearity of Γ in Eq. ().
Diffusion–evaporation equation for DSD
To formulate the diffusion–evaporation equation we use a simplified equation
for droplet evaporation (Pruppacher and Klett, 1997), in which the curvature
term and the chemical composition term are omitted
dσdt=2SF,
where F=ρwLw2kaRvT2+ρwRvTew(T)D=const (notations of other variables are
presented in Appendix A).
The solution of Eq. () is
σ(t)=2F∫0tS(t′)dt′+σ0.
Equation () means that, in the course of evaporation, distribution
g(σ) shifts to the left without changing its shape. The
diffusion–evaporation equation for function g(x,t,σ) can be written
in the form
∂g∂t=K∂2g∂x2+∂∂σdσdtg.
Combining Eqs. () and () yields
∂g(x,t,σ)∂t=K∂2g(x,t,σ)∂x2+2SF∂g(x,t,σ)∂σ.
Equation () is similar to the diffusion–evaporation equation for
size distribution function used in Pt3. The first term on the right-hand side
of Eq. () describes the effect of turbulent diffusion, while the
second term describes the changes of size distribution due to droplet
evaporation. To close this equation, one can use Eq. () written as
S(x,t)=exp[Γ(x,t)-A2qw(x,t)]-1
and the equation for liquid water mixing ratio
qw(x,t)=4πρw3ρa∫0∞σ3/2g(x,t,σ)dσ.
The equation system (15–17) for distribution g(x,t,σ) should be
solved under the following initial condition,
g(x,0,σ)=g1(σ)if0≤x<μL0ifμL≤x<L,
and using the Neumann boundary conditions
∂g(0,t,σ)∂x=∂g(L,t,σ)∂x=0.
These equations were solved numerically on a linear grid of droplet
radii rj within the range 0–50 µm, where
j=1…50 are the bin numbers. The number of grid points along the
x axis was set equal to 81. In numerical calculations, the “evaporation
term” in Eq. () was approximated as
2SF∂g(x,t,σ)∂σ≈gx,t,σ+2SFΔt-gx,t,σΔt.
A shift and subsequent remapping of DSD using the method proposed by Kovetz
and Olund (1969) were implemented to solve Eq. () with the help
of the MATLAB solver PDEPE. After calculation of the g(x,t,σj) function,
DSD f(x,t,rj) was calculated using the relationship f(x,t,rj)=2rjg(x,t,σj).
Spatial–temporal variations of DSD and of DSD parameters
Mixing may take a significant time. Cloud microphysical parameters measured
in in situ observations correspond to different stages of this transient
mixing process. During mixing, DSDs and its parameters change substantially,
which makes it reasonable to analyze these time changes.
Time evolution of DSD in the centers of
the initially cloudy volume (a) and of the initially dry air
volume (b) at initially narrow DSD. The initial mixing parameters
are RH2= 80 %, T= 10 ∘C,
p= 828.8 mb and L= 40 m.
Figure 3 shows time evolution of initially narrow DSD in the centers of the
cloudy volume and of the initially dry volume. The values of DSD in the
initially cloudy volume decrease, while there are no significant changes in
the DSD shape. At μ=0.7, the modal droplet radius remains unchanged
during mixing staying equal to 10 µm. At μ=0.3, the effect of
droplet diffusion on DSD is stronger, and mixing leads not only to a decrease
in the DSD values but also to a decrease in the modal droplet radius in the
cloudy volume. At both μ=0.3 and μ=0.7, mixing leads to broadening
of the initial DSD due to the appearance of the tail of small droplets. The
tail of small droplets is especially pronounced in the initially dry volume
due to maximum evaporation of penetrated droplets.
The rate of the DSD growth in the initially dry volume depends on the value
of the cloud fraction. At a low cloud fraction, the droplet concentration and
droplet mass remain substantially lower for the main period of mixing process
than that in the cloudy volume. At the same time, the modal DSD radius
increases reaching 80 % of its maximum value already within the first
5 s. This is due to the fast increase in the relative humidity during
mixing, so large droplets penetrating the initially dry volume do not decrease in size significantly determining only small changes of the values of modal, mean
volume and effective radii. Thus, we see two stages of DSD evolution within
the initially dry volumes: at the first stage penetrated droplets evaporate
totally or partially. The partially evaporating droplets form the tail of small droplets. The formation of the
tail of smallest droplets does not lead to significant changes in the size
of the largest droplets. Note that according to the equation of diffusion
growth/evaporation in sub-saturation conditions, the rate of droplet radii
decreases inversely proportional to the droplet radius. This means that if,
say, radius of a 2 µm droplet decreases twice during a certain
time instance, the radius of a 20 µm droplet will decrease by less
than 0.1 µm, i.e., remains approximately unchanged. At this
stage, diffusion of water vapor from a cloudy volume and evaporation of penetrating
droplets lead to a rapid growth of relative humidity RH. This growth of RH
decreases evaporation rate of droplets penetrating initially dry volume
later. At the second stage mixing leads to the increase in the droplet number
due to droplet diffusion from cloudy volume. Since RH is high, this
diffusion is not accompanied by significant change in droplet sizes, so DSD
grows similarly at all radii.
The same as in Fig. 3 but for the
initially wide DSD.
At the initially wide DSD (Fig. 4), the modal radii of the DSD do not change.
This means that at the initial RH of 80 %, mixing and evaporation lead
to a fast saturation of the initially dry volume, after which the peak radius
remains unchanged in this volume. In the initially cloud volume RH remains
close to 100 %, so the DSD decrease is related to dilution by initially
dry air.
It is interesting that at μ=0.3, the maximum value of the DSD maximum in
the initially dry volume is reached during the transition period (Fig. 4, at
t= 80 s) and then decreases toward the equilibrium state.
This behavior is caused by the competition between the diffusion and droplet
evaporation.
Spatial dependences of droplet
concentration, LWC and the mean volume radius within the mixing volume at
different time instances at a narrow initial DSD. The initial mixing parameters
are RH2= 80 %, T= 10 ∘C,
p= 828.8 mb and L= 40 m.
Figure 5 shows spatial dependences of droplet concentration, LWC and the mean
volume radius within the mixing volume at different time instances at narrow
initial DSD. At small values of the cloud fraction, diffusion of water vapor
and droplets, as well as droplet evaporation, leads to a fast decrease in
droplet concentration and LWC in the initially cloud volume. The mean
volume radius in this volume decreases by about 15 % in the course of
mixing. It is natural that, at a large cloud fraction, droplet concentration and
LWC in the initially cloudy volume decrease slowly, while these quantities in
the initially dry volume increase rapidly. At both small and large cloud
fractions, the mean volume radius in the initially dry volume grows rapidly
during the mixing toward its values in the initially cloudy volumes, even if
droplet concentration and LWC remain much lower than in the adjacent cloud
volume.
The same as in Fig. 5 but for wide
DSD.
Figure 6 shows the spatial dependences of droplet concentration, LWC and the
mean volume radius within the mixing volume at different time instances at
wide initial DSD.
A specific feature of mixing at a wide DSD is the increase in the mean volume
radius, so the ratio rvrv0>1. In the
course of mixing, the mean volume radius maximum is reached in the initially
dry volumes. This result can be attributed to the fact that in this volume
smaller droplets fully evaporate, so the concentration of large droplets
increases with respect to concentration of smaller droplets (Fig. 4, right
column). Scattering diagrams plotted using in situ observations often contain
points or groups of points with rvrv0>1
(or rere0>1, where re is
effective radius) within a wide range of normalized droplet concentration
(e.g., Burnet and Brenguier, 2007; Krueger et al., 2006; Gerber et al.,
2008). The result obtained in the present study shows that the behavior of
rvrv0 with time in the course of mixing
may depend on the DSD shape in the initially cloud volume that determines the
ratio between concentrations of small and large droplets in the course of
mixing. Of course, the DSD shape is only one possible reason for the appearance of
points with rvrv0>1 on the scattering
diagram.
We see that the transition to the final equilibrium state within the volume
with the spatial scale of 40 m is about 5 min (Fig. 7), which
is a comparatively long period of time compared to the characteristic times
of other microphysical processes, including droplet evaporation. During this
time the DSD changes substantially, especially at small cloud fraction. The
mean volume radius in the initially dry volume increases much faster than
LWC. As a result, mean volume radius in such a volume rapidly reaches the
values typical of cloudy air, while LWC still remains substantially lower
than in the cloudy volume. Despite some DSD broadening, the final DSDs in the
mixing volume resemble those in the initially cloud volumes. The main effect
of mixing is lowering the DSD values as the cloud fraction decreases.
Equilibrium state and mixing diagram
This study reconsiders the classical theory of mixing diagrams. In the
classical theory two volumes (cloudy and droplet-free) mix with each other
within a given unmovable mixing volume (see review by Korolev et al., 2016).
Mixing diagrams are typically plotted for times when all variables become
uniform within the mixing volume, i.e., when the equilibrium state is reached.
We plot the mixing diagram using the same simplifications used in the
plotting classical mixing diagrams, namely no vertical motions and no
collisions are assumed. These assumptions allow for better revealing the
microphysical effects of turbulent mixing. It is widely assumed that the
mixing type is determined by the Damköhler number, which depends only on drop
relaxation time and mixing time. No averaged vertical velocity and no
collision rate are included into this criterion.
We extend the theory, however, in several important aspects concerning
microphysical effects: (a) we consider the time-dependent process of mixing and
(b) initial droplet size distributions are assumed polydisperse.
Mixing considered in the present study always leads to the equilibrium state.
As was explained above, two equilibrium states are possible. The first one is
characterized by the total evaporation of cloud droplets
qw(x,∞)=0, whereas the second one occurs if the air in the
mixing volume becomes saturated, i.e., when S(x,∞)=0. At the given
initial values of qw1 in the cloud volume and of S2 in the
initially dry volume, there always exists the cloud fraction
μcr (Eq. 11) separating these two states.
Time required to reach the equilibrium
state vs. the cloud fraction at different initial RH for the initially narrow
DSD (a) and the initially wide DSD (b). Parameters of DSD
are given in Table 1. The initial mixing parameters are
T= 10 ∘C, p= 828.8 mb and
L= 40 m.
Dependences of the normalized cube of the
mean volume radius on the cloud fraction at different time instances for
x=0 (solid lines) corresponding to the initially cloud volume, and x=L
(dash line) corresponding to the initially dry volume. The time instances in
seconds are marked by numbers. The figure is plotted for the narrow initial
DSD for two values of RH2: 60 % (a) and
95 % (b). Parameters of DSD are given in Table 1. The initial
mixing parameters are T= 10 ∘C,
p= 828.8 mb and L= 40 m. Calculations performed
within the range of 0.1<μ<0.95.
The process of achieving the equilibrium state
Figure 7 shows the dependences of the time required to reach the equilibrium
on the cloud fraction, at different initial relative humidity values in the
dry volume and two initial DSDs (the parameters are presented in Table 1).
The characteristic time is defined here as the time from the beginning of
mixing to the time instance when inequality δ=N‾(t)-N‾(∞)N‾(0)-N‾(∞)<0.01 becomes
valid. The mean droplet concentration is calculated by averaging along
x axes (N‾(t)=1L∫0LN(x,t)dx). In the case of a total evaporation, N‾(∞)=0.
Each curve in Fig. 7 consists of two branches. The left branches correspond
to the total evaporation regime, while the right branches correspond to the
partial evaporation at equilibrium. The maximum time corresponds to the
situation when the available amount of liquid water is approximately equal to
the saturation deficit. A similar result was obtained in Pt1 and Pt2 for
homogeneous mixing. The maximum values of the characteristic time are about
4 min for a mixing volume of 40 m in length. The right
branches show that the characteristic time decreases with increasing cloud
fraction. Despite some differences in the curve slopes, the characteristic
times for wide and narrow DSD are quite similar.
Figure 8 shows dependences of the normalized cube of the mean volume radius on
the cloud fraction at different time instances for two values of x: x=0
(solid lines) corresponds to the initially cloudy volume, and x=L (dashed
line) corresponds to the initially dry volume. The figure is plotted for the
narrow DSD for two values of RH2: 60 and 95 %. Despite the
fact that the diffusion–evaporation equation allows simulating using any
initial RH, we do not consider in our examples the cases of very low RH of
dry volume. This is because at very low RH, say RH = 20 %, the cloud
fraction should exceed 0.8 to prevent total droplet evaporation in the
equilibrium state (at LWC = 1 gkg-1). At the same time, we
are interested in the equilibrium state at which droplets exist. Note that at
the lateral edges of warm Cu a shell of humid air arises, so RH
of the entrained air should be high enough (e.g., Gerber et al., 2008).
The curve plotted for the time instance of 300 s corresponds to the
equilibrium state (hereafter the equilibrium curve). The curves above the
equilibrium curve correspond to the initially cloudy volume, and the curves
below the equilibrium curve correspond to the initially dry volume. One can
see how curves of both types approach the same final state. During the mixing
the curves move over the (rvrv0)3-μ plane toward the equilibrium curve. As a result, the curves plotted in
Fig. 8, corresponding to different time instances of the mixing, together
cover the entire area of the panels.
During this movement the distance from the curves to the horizontal line
(rvrv0)3=1 changes, and the
curves'
slopes increase. In our case of L= 40 m, the mixing remains
inhomogeneous the during entire mixing process, so the change in the distance
from the curves to the horizontal line (rvrv0)3=1 characterizes the temporal changes over the
mixing process, but not a change in mixing type.
It is noteworthy in this relation that scattering diagrams plotted using
in situ observations reflect mixing between different multiple volumes at
different stages of the mixing process. Accordingly, points in the scattering
diagrams can be far from the equilibrium location. Figure 8 indicates,
therefore, that scattering diagrams show snapshots of transient mixing
process when the distance from points in the diagrams to line
(rvrv0)3=1 characterizes the stage of
the mixing process, but not the mixing type.
The dependences of the normalized cube of the mean volume radius on the cloud
fraction at different time instances at wide DSD also indicate approaching to
the equilibrium curve, while all the curves correspond to
(rvrv0)3>1 (not shown).
Note that in several studies normalized effective radius is used for plotting
scattering and mixing diagrams, but not mean volume radius (Gerber
et al., 2008; Freud et al., 2011). Comparison of scattering and mixing
diagrams in the study plotted using mean volume and effective radii did not
reveal any significant differences (not shown).
Mixing diagrams
Using the diffusion–evaporation equations (Eqs. 15–17) we calculated the
equilibrium DSD for different initial relative humidity values and different
cloud fractions. Each calculation was performed for both narrow and wide
initial DSD (parameters shown in Table 1). These equilibrium DSDs were used to
calculate mixing diagrams showing dependences of the normalized cube of the
effective radius on the cloud fraction.
Mixing diagrams. Normalized cube of the
mean volume radius vs. the cloud fraction for initial narrow DSD (a)
and initial wide DSD (b). The dependencies correspond to the
equilibrium state. Parameters of initial DSD are presented in Table 1. Solid
and dashed lines show the mixing diagrams for inhomogeneous and homogeneous
mixing, respectively. The initial mixing parameters are
T= 10 ∘C, p= 828.8 mb and
L= 40 m.
The corresponding mixing diagrams for homogeneous mixing case were also
calculated for comparison. To this effect, the supersaturation and DSD in
both the cloud and the dry volumes were aligned, taking into account the
cloud fraction value μ. The alignment led to the following initial values
of supersaturation and DSD within the mixing volume:
S0=(1-μ)S2;g0(σ)=μg1(σ).
Upon the alignment, time evolution values of DSD under homogeneous
evaporation in an adiabatic immovable parcel were calculated until the
equilibrium state was reached. These equilibrium DSDs were used to calculate
mixing diagrams for homogeneous mixing. To do this, we used the parcel model
proposed by Korolev (1995) that describes evaporation by means of equations
with temperature-dependent parameters. Figure 9 shows the mixing diagrams
plotted for initial narrow and wide DSD cases.
While all the curves in the mixing diagram for narrow DSD are below the
straight line (rvrv0)3=1, the
curves for wide DSD are above this line. The explanation of this effect is
given in Sect. (Fig. 6). The curves plotted for homogeneous and
inhomogeneous mixing demonstrate an important feature, namely that, at given
values of RH and qw1 in the initially dry volume, the values
μcr of the cloud fraction at which all the droplets evaporate
are approximately the same for any type of mixing. This condition is the
consequence of the mass conservation law determined by Eq. () and
does not depend on the initial DSD shape. In standard mixing diagrams (e.g.,
Lehmann et al., 2009; Gerber et al., 2008; Freud et al., 2011), the
horizontal straight line (rvrv0)3=1
(or (rere0)3=1) is typically
plotted for the entire range of the cloud fraction [0…1], while the
curves corresponding to homogeneous mixing are plotted for different RH
within the range [μcr(RH2)…1]. As a result,
the high difference between extremely inhomogeneous and homogeneous mixing
types is clearly seen at low RH and at small cloud fractions. The condition
that μcr is the same for different mixing types indicates that
the mixing diagrams may look nearly similar for μ>μcr. This
means that the range of the cloud fractions required for comparison of
diagrams aimed at determination of a mixing type shortens as RH2
values in the surrounding air decrease.
The comparison of panels (a) and (b) in Fig. 9 shows that the
differences between the diagrams for homogeneous and inhomogeneous mixing
types are more pronounced for initially narrow DSD. The maximum difference
should take place for monodisperse DSD considered in Pt1, Pt2 and Pt3.
Within the range of μ>μcr, the distance between the curves
corresponding to different mixing regimes is small even for narrow DSD and
low RH2. The lower difference is related to the fact that at high
RH2 the curves in the mixing diagrams are close to the horizontal
straight line in both regimes, while at low RH2,
μcr is small and both curves should drop to zero in the
vicinity of μ=μcr.
As regards the wide DSD case, the difference between the curves corresponding
to different mixing type is negligible (Fig. 9b)
Final normalized droplet concentration
vs. cloud fraction for initially narrow DSD (a) and initially wide
DSD (b). Parameters of initial DSD are shown in Table 1. Dashed line
shows the results of equivalent homogeneous mixing. The initial mixing
parameters are T= 10 ∘C, p= 828.8 mb
and L= 40 m.
Dependencies of the normalized cube of the
mean volume radius on normalized droplet concentration for different initial
relative humidity values. (a) Narrow initial DSD. (b) Wide
initial DSD. The initial mixing parameters are
T= 10 ∘C, p= 828.8 mb and
L= 40 m.
Effect of the relative humidity
In measurements carried out at cloud boundaries and in cloud simulations, the
cloud fraction is not known; therefore, it is widely accepted to use
normalized droplet concentration instead of the cloud fraction (Burnet and
Brenguier, 2007; Gerber et al., 2008: Lehmann et al., 2009). Droplet
concentration is normalized by the maximum value along the airplane traverse.
The difference between the cloud fraction and normalized droplet
concentration is obvious: the cloud fraction is a parameter given as the
initial condition. At the same time, normalized droplet concentration changes
with time and space due to complete evaporation of some droplet fraction.
Figure 10 shows dependencies of normalized droplet concentration on the cloud
fraction at the equilibrium final state of mixing. One can see a substantial
deviation from 1:1 linear dependence, especially at low RH.
Note that
droplet concentration decreases in the course of both homogeneous and
inhomogeneous mixing if the initial DSDs are polydisperse. The fraction of
totally evaporating droplets increases with decreasing RH2. As
expected, droplet concentration in homogeneous mixing is higher than that in
inhomogeneous mixing. The difference between droplet concentrations at wide
DSD is lower than at narrow DSD.
Figure 11 shows the dependencies (rvrv0)3 on normalized droplet concentration for narrow and wide DSD in
inhomogeneous mixing. The normalization by droplet concentration in the
initially cloud volume at t=0 was used. Taking into account the dependences
of normalized droplet concentration on the cloud fraction μ (Fig. 10),
one can get the curves shown in Fig. 11 which actually coincide at different
RH2. The lack of the sensitivity to RH2 can be
attributed to the fact that a decrease in RH leads to a decrease in
normalized droplet concentration, so the curves corresponding to low RH in
Fig. 9 shift to the left when the normalized droplet concentration is used
instead of μ. The shape of the dependences in Fig. 11b is explained by an
increase in the mean volume radius with decreasing droplet concentration.
Thus, the mixing diagrams plotted in the plane (rvrv0)3 vs. normalized droplet concentration do not depend
on the relative humidity of the surrounding dry air. This result indicates an
additional difficulty in distinguishing between mixing types based on
scattering diagrams plotted using in situ data in these axes. The
concentration of observed points in these scattering diagrams close to the
line (rvrv0)3=1 is often
interpreted as an indication of homogeneous mixing, but at high RH in the
surrounding air (Gerber et al., 2008; Lehmann et al., 2009). High values of
RH in the penetrating air volumes are usually explained by formation of
a layer of moist air around the cloud boundary (Gerber et al., 2008; Knight
and Miller, 1998).
The reference values of droplet concentration and the effective radius used
for normalization in the present study are taken as the initial values in the
cloud volume before it mixes with the neighboring dry volume. In real
in situ measurements the reference values of these quantities are typically
chosen in a less diluted cloud volume along the airplane traverse. This
reference volume may be quite remote from the particular mixing volume. It
can lead to a shift of the mixing diagram with respect to the
(rvrv0)3=1 line, as well as to
a large variation in mixing diagram shapes, which are not related, however, to the mixing
type (e.g., Lehmann et al., 2009).
Discussion and conclusion
This study extends the analysis of mixing performed in Pt3, where the
diffusion–evaporation equation served as the basis, the initial DSD were
assumed monodisperse and the cloud fraction was chosen as μ=1/2. In the
present study, the analysis focuses on the temporal and spatial evolution of
initially polydisperse DSD and investigates mixing diagrams obtained for
narrow and wide initial DSD within a wide range of the cloud fraction values
(0.1–0.95). It is shown that results of mixing and the structure of mixing
diagrams depend on the initial DSD shape. This finding indicates that mixing is a problem that cannot be
determined by a single parameter (e.g., the Damkölher number as often assumed) or even by two
parameters (the Damkölher number and the potential evaporation parameters
as assumed in Pt3). The temporal changes of DSD and their moments during
mixing are calculated. Although DSD broaden, they tend to remain similar to
the original DSD. The main changes come from the cloud air dilution by the
dry air, which leads to a decrease in droplet concentration for all droplet
sizes. The changes of DSD and its shape are minimum in the initially cloud
volumes, especially at significant cloud fractions. The droplet radii
corresponding to the DSD peak do not change significantly in any case. In the
initially dry volumes, mixing and evaporation of penetrated droplets leads to
a rapid increase in RH. Consequently, large droplets penetrating these
volumes do not change their sizes significantly. As a result, the mean volume
radius in these volumes rapidly increases and reaches the values typical of
cloud volumes, while LWC remains lower than in the cloud volume for most of
the mixing time. At narrow DSD, the mean volume (and effective) radius
remains smaller than that in the initially cloud volume. At wide DSD, the
mean volume (and effective) radius may become larger than that in the initial
DSD. This increase in the effective radius is attributed to the fact that
evaporation of smaller droplets leads to the increase in the fraction of
larger droplets in the DSD. In this study, and in Pt3, it is shown that mixing
leads to DSD broadening. This contrasts with the classical theory, in which
initially monodisperse DSDs remain monodisperse in the course of mixing. This
problem is analyzed in detail in Pt 3. Note that in real clouds there are
many mechanisms leading to DSDs broadening (e.g., Pinsky and Khain, 2002).
Dependences of the normalized cube of the mean volume radius on the cloud
fraction (rv/rv0)3 as a function of μ at
different time instances form the set of curves filling the entire
(rv/rv0)3-μ plane. Therefore, both the slope
and the distance of these curves with respect to the horizontal line
(rv/rv0)3=1 change with time. This means that
this distance characterizes the temporal changes in the course of mixing, but
not the mixing type (which remains inhomogeneous during the entire mixing
time). The mixing process is comparatively long (several minutes), so the
final equilibrium stage is hardly achievable in real clouds.
It is highly important that the critical values of the cloud fraction
μcr corresponding to total droplet evaporation are the same for
any mixing type. This means that the curves in a mixing diagram corresponding
to homogeneous and inhomogeneous mixing types should be compared only within
the range of μ>μcr. The range width of μ>μcr decreases with decreasing relative humidity in the
initially dry volume. Taking into account significant scattering of observed
points, this condition greatly hampers the problem of how to distinguish
between mixing types.
Another important result of the study is that mixing diagrams for homogeneous
and inhomogeneous mixing plotted for polydisperse DSD do not differ much. The
largest difference occurs for initially narrow DSD (the maximum
difference takes place for initially monodisperse DSD), but even in this case
the difference is not large enough to reliably distinguish mixing type, owing
to the significant scatter of observed data. At wide DSD, this difference
between mixing diagrams for homogeneous and inhomogeneous becomes negligibly
small.
The cloud fraction μ is a predefined parameter and is not determined from
observations. Consequently, in the analysis of in situ measurements the
normalized droplet concentration is typically used instead of the cloud
fraction. However, there is a significant difference between the cloud
fraction prescribed a priori and the normalized droplet concentration that
changes due to total evaporation of some fraction of droplets. We have shown
that the utilization of normalized droplet concentration in mixing diagrams
is not equivalent to the utilization of the cloud fraction. The important
conclusion is that when mixing diagrams are plotted using the normalized
concentration, the sensitivity to RH disappears. This conclusion is valid
even when RH in the initially dry volume is as low as 60 %. This
conclusion clearly contradicts the widespread assumption that mixing types
can be easily distinguished in mixing diagrams in case of low relative
humidity of the surrounding air.
In the present study as well as in Pt3 and modeling studies performed by
Andrejczuk et al. (2006, 2009) and Khain et al. (2018), it is shown that time
needed to establish equilibrium is either quite long or even never
reached. This means that the scattering diagrams observed in situ are just
snapshots of the transient mixing process. In order to show how different
the equilibrium and intermediate states are, we investigate the transition to such
equilibrium assuming that the mixing volume remains adiabatic (i.e., isolated)
during the entire period of mixing. This is, of course, a serious
simplification made to compare the results with those predicted by the classical
concept. Another simplification of the model is the neglecting the
intermittency in the process of mixing that takes place in real clouds.
To summarize, our general conclusion is that the simplifications underlying the
classical concept of mixing are too crude, making it impossible to use
scattering diagrams for comprehensive analysis of mixing and especially for
determination of mixing types. At the same time, scattering diagrams may
contain useful information concerning intensity of mixing, the DSD width and
other parameters of DSDs (see Khain et al., 2018).
List of symbolsSymbolDescriptionUnitsA21qv+Lw2cpRvT2, coefficient–anFourier series coefficients–CRichardson's law constant–cpspecific heat capacity of moist air at constant pressureJkg-1K-1Dcoefficient of water vapor diffusion in airm2s-1DaDamkölher number–ewater vapor pressureNm-2ewsaturation vapor pressure above flat surface of waterNm-2F(ρwLw2kaRvT2+ρwRvTew(T)D), coefficientm-2sf(r)droplet size distributionm-4g(r)droplet size distributionm-5g0(σ)initial distribution of square radius in homogeneous mixingm-5g1(σ)initial distribution of square radiusm-5kacoefficient of air heat conductivityJm-1s-1K-1Kturbulent diffusion coefficientm2s-1Lcharacteristic spatial scale of mixingmLwlatent heat for liquid waterJkg-1Ndroplet concentrationm-3N0parameter of gamma distributionm-3N‾mean droplet concentrationm-3N1initial droplet concentration in cloud volumem-3ppressure of moist airNm-2qvwater vapor mixing ratio (mass of water vapor per 1 kg of dry air)–qwliquid water mixing ratio (mass of liquid water per 1 kg of dry air)–qw1liquid water mixing ratio in cloud volume–RS2A2qw1, non-dimensional parameter–Rvspecific gas constant of water vaporJkg-1K-1rdroplet radiusmr1initial droplet radiusmreeffective radiusmre0initial effective radiusmSe/ew-1, supersaturation over water–S2initial supersaturation in the dry volume–S0initial supersaturation in homogeneous mixing–TtemperatureKttimesxdistancemαparameter of gamma distribution–βparameter of gamma distributionm-1Δttime stepsμcloud fraction–μcrcritical cloud fraction–εturbulent dissipation ratem2s-3Γ(x,t)conservative function–ρaair densitykgm-3ρwliquid water densitykgm-3σsquare of droplet radiusm2
Codes of the diffusional–evaporation
model are available upon request.
The authors declare that they have no conflict of
interest.
Acknowledgements
This research was supported by the Israel Science Foundation (grants 1393/14,
2027/17) and the Office of Science (BER) of the US Department of Energy
(award DE-SC0006788, DE-FOA-0001638). Edited by:
Timothy Garrett Reviewed by: two anonymous referees
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