ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-16-9273-2016Theoretical analysis of mixing in liquid clouds – Part 3: Inhomogeneous
mixingPinskyMarkKhainAlexanderalexander.khain@mail.huji.ac.ilhttps://orcid.org/0000-0002-8429-4127KorolevAlexeihttps://orcid.org/0000-0003-3877-8419Department of Atmospheric Sciences, The Hebrew University of
Jerusalem, Jerusalem, IsraelEnvironment Canada, Cloud Physics and Severe Weather Section,
Toronto, CanadaAlexander Khain (alexander.khain@mail.huji.ac.il)28July201616149273929726August20154November201530May20163June2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/16/9273/2016/acp-16-9273-2016.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/16/9273/2016/acp-16-9273-2016.pdf
An idealized diffusion–evaporation model of time-dependent mixing between a
cloud volume and a droplet-free volume is analyzed. The initial droplet size
distribution (DSD) in the cloud volume is assumed to be monodisperse. It is
shown that evolution of the microphysical variables and the final equilibrium
state are unambiguously determined by two non-dimensional parameters. The
first one is the potential evaporation parameter R, proportional to the
ratio of the saturation deficit to the liquid water content in the cloud
volume, that determines whether the equilibrium state is reached at 100 %
relative humidity, or is characterized by a complete evaporation of cloud
droplets. The second parameter Da is the Damkölher
number equal to the ratio of the characteristic mixing time to the phase
relaxation time. Parameters R and Da determine the type of mixing.
The results are analyzed within a wide range of values of R and
Da. It is shown that there is no pure homogeneous mixing, since the
first mixing stage is always inhomogeneous. The mixing type can change during
the mixing process. Any mixing type leads to formation of a tail of small
droplets in DSD and, therefore, to DSD broadening that depends on
Da. At large Da, the final DSD dispersion can be as large
as 0.2. The total duration of mixing varies from several to 100 phase
relaxation time periods, depending on R and Da.
The definitions of homogeneous and inhomogeneous types of mixing are
reconsidered and clarified, enabling a more precise delimitation between
them. The paper also compares the results obtained with those based on the
classic mixing concepts.
Introduction
Cloud physics typically investigates two types of turbulent mixing:
homogeneous and extremely inhomogeneous (e.g., Burnet and Brenguier, 2007;
Andrejczuk et al., 2009; Devenish et al., 2012; Kumar et al., 2012). The
concept of extremely inhomogeneous mixing in clouds was introduced by Latham
and Reed (1977), Baker and Latham (1979), Baker et al. (1980) and Blyth et
al. (1980). According to this concept, mixing of cloud air and sub-saturated
air from cloud surrounding results in complete evaporation of a fraction of
cloud droplets, whereas the size of other droplets remain unchanged. The studies
of extremely inhomogeneous mixing were closely related to investigation of
different mechanisms underlying enhanced growth of cloud droplets and warm
precipitation formation (Baker et al., 1980; Baker and Latham, 1982). The
concept of homogeneous mixing suggests that all the droplets partially
evaporate, so the liquid water content decreases while the droplet
concentration remains unchanged (Lehmann et al., 2009; Pt1). The significance
of the concepts of homogeneous and inhomogeneous mixing goes far beyond
formation of large-sized droplets. In fact, these concepts are closely
related to the mechanisms involved in formation of droplet size distributions
(DSD) in clouds and to the description of this formation in numerical cloud
models. A detailed analysis of the classical concepts of homogeneous and
extremely inhomogeneous mixing is given by Korolev et al. (2016, hereafter
Pt1).
Mixing in clouds includes two processes: mechanical mixing caused by
turbulent diffusion and droplet evaporation accompanied by increasing
relative humidity. The relative contribution of these processes can be
evaluated by comparison of two characteristic timescales: the characteristic
mixing timescale τmix∼L2/3ε-1/3 (where
L is the characteristic linear scale of an entrained volume and
ε is the dissipation rate of turbulent kinetic energy) and the
time of phase relaxation τpr=4πDr¯N-1 (where N is droplet concentration in a cloud volume,
r¯ is the mean droplet radius and D is the diffusivity of water
vapor) characterizing the response of the droplet population to changes in
humidity (the list of notations is given in Appendix). The choice of the
phase relaxation time as the characteristic timescale of mixing is discussed
by Pinsky et al. (2016) (hereafter referred to as Pt2) and will be further
elaborated below.
Mixing is considered homogeneous if τmix/τpr≪1.
At the first stage of mixing, the initial gradients of the microphysical and
thermodynamic variables rapidly decrease to zero. By the end of this stage,
the fields of temperature, humidity (hence, the relative humidity, RH) and
droplet concentration are spatially homogenized and all the droplets within
the mixing volume experience the same saturation deficit. During the
relatively lengthy second stage, droplets evaporate and increase the relative
humidity in the volume. It was shown that homogeneous mixing takes place at
scales below about 0.5 m (Pt2).
At spatial scales larger than ∼ 0.5 m, τmix/τpr>1 and the spatial gradients of RH remain for a long time. Consequently,
droplets within the mixing volume experience different subsaturations, thus
the mixing is considered inhomogeneous. At τmix/τpr≫1, the mixing is considered extremely inhomogeneous.
According to the classical conceptual scheme, during the first stage of
extremely inhomogeneous mixing a fraction of droplets is transported into the
droplet-free entrained volume and evaporates completely. The evaporation
continues until the evaporating droplets saturate the initially droplet-free
volume. At the second stage, turbulent mixing between the cloud volume and
the initially droplet-free (but already saturated) volume homogenizes the
gradients of droplet concentration and other quantities. Since both volumes
are saturated, mixing does not affect droplet sizes. As a result, the final
(equilibrium) state is characterized by the relative humidity
RH = 100 % and the DSD shape similar to that before mixing, but with
a lower droplet concentration. The same result (a decrease in droplet
concentration but unchanged droplet size) is expected in cases of both
monodisperse and polydisperse initial DSD. Since the DSD shape does not
change, the characteristic droplet sizes (i.e., the mean square radius, the
mean volume radius and the effective radius) do not change either in the
course of extremely inhomogeneous mixing.
Thus, according to the classical concepts, the final equilibrium state with
RH = 100 % is reached either by a partial evaporation of all droplets
(homogeneous mixing) or a total evaporation of a certain portion of droplets
that does not affect the remaining droplets (extremely inhomogeneous mixing)
(Lehmann et al., 2009; Pt1).
In analyses of in situ measurements, the observed data are usually compared
with those expected at the final state of mixing as assumed by the classical
mixing concepts. If droplet concentration decreases without a corresponding
change in the characteristic droplet radius, the mixing is considered
“extremely inhomogeneous”. If the characteristic droplet radius decreases
with an increase of the dilution level while droplet concentration decreases
insignificantly, the mixing is identified as “homogeneous”. If both the
characteristic droplet radius and the droplet concentration change, the
mixing is considered as “intermediate”. Quantitative evaluations of the
microphysical processes specific for intermediate mixing remain largely
uncertain.
As was discussed in Pt2, the final states of mixing suggested by the
classical concepts are only hypothetical. To understand the essence of the
final equilibrium states of mixing and evaluate the time needed to reach
them, it is necessary to consider the time evolution of DSD in the course of
mixing process. Time-dependent process of homogeneous mixing was analyzed in
Pt2. It was shown that in important cases of wide polydisperse initial DSDs,
the final state substantially differs from that hypothesized by the classical
concepts.
In this study, which is a Pt3 of the set of studies, we analyze the
time-dependent process of inhomogeneous mixing. The structure of the paper is
as follows. The main concept and the basic equations for time-dependent
inhomogeneous mixing are described in Sect. 2. Analysis of non-dimensional
diffusion–evaporation equations is presented in Sect. 3. The design and the
results of simulations of non-homogeneous mixing are outlined in Sects. 4 and
5. A discussion clarifying the concepts of homogeneous and inhomogeneous
mixing is presented in concluding Sect. 6.
The main concept and the basic equations
During mixing of cloud volume and entrained air volume, the following two
processes determine the change of the microphysical and thermodynamical
variables: turbulent diffusion resulting in mechanical smoothing of the
gradients of temperature, water vapor and droplet concentration, and droplet
evaporation accompanied by phase transformation. In this study, inhomogeneous
mixing is investigated based on the analysis and solution of a 1-D
diffusion–evaporation equation. To our knowledge, the idea of using a
diffusive model of turbulent mixing to describe the mixing process was first
proposed by Baker and Latham (1982). A diffusion–evaporation equation was
also analyzed by Jeffery and Reisner (2006). In order to get a more precise
understanding of the physics of the mixing process the analysis is performed
under the following main simplifying assumptions:
Turbulent mixing is analyzed neglecting vertical motions of mixing
volumes, droplet collisions and droplet sedimentation.
The total mixing volume is assumed adiabatic.
Mixing is assumed to take place only along the x-direction, i.e., a 1-D
task is considered;
The initial DSD in the cloud volume is assumed monodisperse.
Other assumptions and simplifications are discussed below.
A schematic illustration of the initial conditions used in the study is shown
in Fig. 1. Two air volumes are assumed to mix: a cloud volume (left) and a
droplet-free volume (right), each having the linear size of L/2. The value
of L is assumed within the range of several tens to a few hundred meters.
The mixing starts at t=0. The cloud volume is initially saturated S1=0, the initial droplet concentration is N1 and the initial liquid
water mixing ratio is q1=4πρw3ρaN1r03. In the droplet-free volume the initial conditions are
RH2<100 % (i.e., S2<0), N2=0 and q2=0. Therefore,
the initial profiles of these quantities along the x axis are step
functions
The schematic illustration of the 1-D mixing problem considered in
the study. The initial state at t=0 is illustrated. The left volume of
length L/2 is a saturated cloudy volume; the right volume is a
non-saturated air volume from the cloud environment.
N(x,0)=N1if0≤x<L/20ifL/2≤x<LS(x,0)=0if0≤x<L/2S2ifL/2≤x<Lq(x,0)=q1if0≤x<L/20ifL/2≤x<L
The initial profile of droplet concentration is shown in Fig. 1. In this
study, averaged equations are used. We do not consider mixing at scales below
several millimeters. At the scales of averaging, there exist clear
definitions of droplet concentration, supersaturation and other “macro
scale” quantities. The mixing is assumed to be driven by isotropic
turbulence within the inertial sub-range where the Richardson's law is valid.
Accordingly, turbulent diffusion (turbulent mixing) is described by a 1-D
equation of turbulent diffusion with a turbulent coefficient K. The
turbulent coefficient is evaluated as proposed by Monin and Yaglom (1975)
K(L)=Cε1/3L4/3.
In Eq. (2), C is a constant. Equation (2) is valid in case turbulent diffusion
is considered, i.e., at scales where molecular diffusion can be neglected.
Since the total mixing volume is adiabatic, the fluxes of different
quantities through the left and right boundaries of the volume are equal to
zero at any time instance, i.e.,
∂N(0,t)∂x=∂N(L,t)∂x=0;∂q(0,t)∂x=∂q(L,t)∂x=0;∂qv(0,t)∂x=∂qv(L,t)∂x=0,
where qv is the water vapor mixing ratio.
During mixing, droplets in the mixing volume experience different
subsaturations, therefore, the initially monodisperse DSD will become
polydisperse. The droplets that were transported into the initially
droplet-free volume will undergo either partial or complete evaporation. The
evaporation leads to a decrease in both droplet size and droplet
concentration.
The basic system of equations that describes the processes of diffusion and
of evaporation which occur simultaneously is to be derived. The first
equation is written for value Γ defined as
Γ=S+A2q.
This value is conservative in a moist adiabatic process, i.e., it does not
change during phase transitions (Pinsky et al., 2013, 2014). In Eq. (4), the
coefficient A2=1qv+Lw2cpRvT2 is a weak function of temperature that changes by
∼ 10 % when temperatures change by ∼ 10 ∘C (Pinsky
et al., 2013). In this study, it is assumed that A2= constant. In
Eq. (4), q=4πρw3ρa∫0∞r3f(r)dr is the liquid water mixing ratio and f(r) is the
DSD. The quantity Γ obeys the diffusion equation
∂Γ(x,t)∂t=K∂2Γ(x,t)∂x2
with the boundary conditions ∂Γ(0,t)∂x=∂Γ(L,t)∂x=0 and the initial profile at
t=0Γ(x,0)=A2q1if0≤x<L/2S2ifL/2≤x<L
Therefore, function Γ(x,0) is positive in the left volume, and
negative in the right volume.
Since Γ does not depend on phase transitions, Eq. (5) can be solved
independently of other equations. The solution of Eq. (5) with initial
conditions (Eq. 6) is (Polyanin and Zaitsev, 2004)
Γ(x,t)=∑n=0∞anexp-Kn2π2tL2cosnπxL=12S2+A2q1+(A2q1-S2)∑n=1∞sin(nπ/2)nπ/2exp-Kn2π2tL2cosnπxL,
where the Fourier coefficients of expanding the step function (Eq. 6) are
a0=12A2q1+S2an=(A2q1-S2)sin(nπ/2)nπ/2,n=1,2,…
An example of spatial dependencies of Γ(x,t) at different time
instances during the mixing is shown in Fig. 2. One can see a decrease in the
initial gradients and a tendency to establish a horizontally uniform value
of Γ. Since the initial volume was divided into two equal parts, the
diffusion leads to formation of a constant limit value of function Γ(x,∞)=12Γ(0,0)+Γ(L,0).
An example of Γ(x,t) evolution during mixing.
The second basic equation is the equation for diffusional droplet growth,
taken in the following form (Pruppacher and Klett, 2007):
dσdt=2SF,
where σ=r2 is the square of droplet radius and
F=ρwLw2kaRvT2+ρwRvTes(T)D. The value of
coefficient F is considered constant in this study. The solution of Eq. (9)
is
σ(t)=2F∫0tS(t′)dt′+σ0.
The third main equation describes the evolution of DSD. In the following
discussion, the DSD will be presented in the form g(σ) which is the
distribution of the square of the radius. This formulation directly utilizes
the property of the diffusion growth equation (Eq. 9) according to which the
time changes of DSD are reduced to shifting the distributions in the space of
square radii, while the shape of the distribution remains unchanged. The
standard DSD f(r) is related to g(σ) as f(r)=2r⋅g(r2).
The normalized condition for g(σ) is
N=∫0∞g(σ)dσ,
where N is the droplet concentration. Using DSD g(σ), the liquid
water mixing ratio can be presented as integral
q=4πρw3ρa∫0∞σ3/2g(σ)dσ.
The 1-D diffusion–evaporation equation for the non-conservative function
gσ can be written in the form (Rogers and Yau, 1989)
∂g(σ)∂t=K∂2g(σ)∂x2-∂∂σdσdtg(σ)
where the first term on the right-hand side of Eq. (13) describes changes in
the DSD due to spatial diffusion, while the second term on the right-hand
side describes changes in the DSD due to evaporation. Substitution of Eq. (9)
into Eq. (13) leads to the following equation
∂g(x,t,σ)∂t=K∂2g(x,t,σ)∂x2-2S(x,t)F∂g(x,t,σ)∂σ.
To close Eq. (14), Eq. (4) should be used in the form
S(x,t)=Γ(x,t)-A2q(x,t),
where q(x,t) is calculated according to Eq. (12). Equations (12, 14, 15)
constitute a closed set of equations allowing calculation of g(x,t,σ).
To proceed to the equations for DSD moments, let us define a moment of DSD
g(σ) of order α as
mα=σα‾=∫0∞σαg(σ)dσ.
Multiplying Eq. (14) by σα, integrating within limits 0…∞ and assuming that σαg(σ)→0 when σ→∞, yield a recurrent formula for the DSD moments
∂mα(x,t)∂t=K∂2mα(x,t)∂x2+α2SFmα-1(x,t).
Eq. (17) provides a recurrent relationship between the DSD moments of
different orders. A similar relationship was discussed by Pinsky et al. (2014)
while analyzing diffusion growth in an ascending adiabatic parcel.
In particular, the equation for the liquid water mixing ratio that is a
moment of the order of α=32 can be written as
∂q(x,t)∂t=K∂2q(x,t)∂x2+4πρwN(x,t)r¯(x,t)FρaS(x,t),
where the mean radius r¯(x,t)=m1/2m0.
In the general case, Eq. (18) is not closed, since concentration N(x,t) and
r¯(x,t) are unknown functions of time and spatial coordinates.
The characteristic time of evaporation and of supersaturation change is the
phase relaxation time (Korolev and Mazin, 2003)
τpr=ρaF4πρwA2Nr¯.
Using Eq. (19), Eq. (18) can be rewritten as
∂q(x,t)∂t=K∂2q(x,t)∂x2+1τpr(x,t)1A2Γ(x,t)-q(x,t)=K∂2q(x,t)∂x2+1A2τpr(x,t)S(x,t).
From Eqs. (20) and (15), the equation for supersaturation can be written in
the following simple form
∂S(x,t)∂t=K∂2S(x,t)∂x2-S(x,t)τpr(x,t).
Eqs. (20) and (20a) show that changes in the microphysical variables are
determined by the rate of spatial diffusion (the first term on the right-hand
side of these equations) and of evaporation (the second term on the
right-hand side).
Analysis of non-dimensional equations
Spatial diffusion and evaporation depend on many parameters. It is best
to start the analysis from the basic equation system presented in a
non-dimensional form. A timescale corresponding to the initial phase
relaxation time in a cloud volume can be defined as
τ0=ρaF4πρwA2N1r0
and the non-dimensional time is t̃=t/τ0. Other
non-dimensional parameters to be used are: the
non-dimensional phase relaxation time
τ̃pr=τpr/τ0=N1r0N(x̃,t̃)r¯(x̃,t̃),
the normalized liquid water mixing ratio which is equal to the normalized
liquid water content
q̃=qq1,
the normalized supersaturation
S̃=SA2q1,
the non-dimensional conservative function
Γ̃=ΓA2q1,
the normalized square of droplet radius
σ̃=σr02,
the normalized droplet concentration
Ñ=N/N1
and the non-dimensional DSD
g̃(σ̃)=r02N1g(σ)
with normalization Ñ=∫01g̃(σ̃)dσ̃. The definition (Eq. 22g) means that the integral of a
non-dimensional initial size distribution over the normalized square radius
is equal to unity.
The non-dimensional distance and the non-dimensional time are defined as
x̃=x/L;t̃=t/τ0.
A widely used non-dimensional parameter showing the comparative rates of
diffusion and evaporation is the Damkölher number:
Da=τmixτ0=L2Kτ0,
where
τmix=L2K
is the characteristic timescale of mixing. Using the non-dimensional
parameters listed above, Eq. (20) can be rewritten in a non-dimensional form
as
∂q̃(x̃,t̃)∂t̃=1Da∂2q̃(x̃,t̃)∂x̃2+1τ̃pr(x̃,t̃)Γ̃(x̃,t̃)-q̃(x̃,t̃)=1Da∂2q̃(x̃,t̃)∂x̃2+1τ̃pr(x̃,t̃)S̃(x̃,t̃)
where
q̃(x̃,t̃)=N(x̃,t̃)σ3/2‾N1r03=∫0∞σ̃3/2g̃(x̃,t̃,σ̃)dσ̃.
The initial conditions and the boundary conditions should be rewritten in a
non-dimensional form as well. For instance, the normalized initial condition
for the non-dimensional function q̃(x̃,0) can be derived from
Eqs. (1c) and (22b)
q̃(x̃,0)=1if0≤x̃<1/20if1/2≤x̃<1
The solution for Γ̃(x̃,t̃) obtained by a
normalization of solution (Eq. 7) is
Γ̃(x̃,t̃)=121+R+1-R∑n=1∞sin(nπ/2)nπ/2exp-n2π2t̃Dacosnπx̃,
where
R=S2A2q1
is a non-dimensional parameter referred to, hereafter, as a potential
evaporation parameter (PEP). The PEP is proportional to the ratio of the
amount of water vapor that should evaporate in order to saturate the
initially droplet-free volume (that is determined by S2) to the initial
available liquid water q1 in the cloud volume. The solution of Eq. (28)
at t̃→∞ depends only on parameter R.
Γ̃(x̃,∞)=121+R
The importance of PEP that determines a possible final state was illustrated
in Pt1. PEP is also the sole parameter enabling calculation of the normalized
mixing diagram for homogeneous mixing (Pt2). In this study, we consider cases
when R<0 since S2<0, i.e., when droplets can only evaporate in the
course of mixing.
The solution of Eq. (25) and the type of mixing depends on the values of two
non-dimensional parameters, namely, Da and R. Thus, when
R=S2A2q1<-1, Γ̃(x̃,∞)<0. It
means that the initially droplet-free volume V2 is too dry and all the
droplets in the mixing volume evaporate completely. At the final equilibrium
state RH < 100 %, i.e., S(x,∞)<0. If R=S2A2q1>-1, Γ̃(x̃,∞)>0. This means that the mixed
volume in the final state contains droplets, i.e., the mixing leads to an
increase of the volume with droplets, i.e., the cloud volume. At the final equilibrium
state, RH = 100 % (i.e., S(x,∞)=0). The case when R=S2A2q1=S̃2≪1 corresponds to either RH close to 100 % (i.e., S2 is
close to zero) (this case corresponds to the degenerated case considered in
Pt1), and/or to the case when the liquid water mixing ratio in the cloud
volume is large. In case R≪1, the second term on the
right-hand side of Eq. (25) is much smaller than the first term, and the
mixing is driven by turbulent diffusion only.
In case Da→0 (often considered as homogeneous mixing), at the
beginning of the mixing the diffusion term is much larger than the
evaporation term, the second term on the right-hand side of Eq. (25). As
mixing proceeds, within a short time period the total homogenization of all
the variables in the mixing volume is established and all the spatial
gradients become equal to zero. At this time instance, the first term on the
right-hand side becomes equal to zero, and the second term on the right-hand
side of Eq. (25), describing droplet evaporation, becomes dominant. Thus, the
analysis of the Eq. (25) shows that mixing consists of two stages. The first
mixing stage is a short stage of inhomogeneous mixing and the longer second
stage of homogeneous mixing. The evolution of the microphysical variables
during homogeneous mixing is described in detail in Pt2.
Da→∞ corresponds to extremely inhomogeneous mixing, according to
the classic concept. In this case, the diffusion term is much smaller than
the evaporation term, so evaporation takes place under significant spatial
gradients of RH. At Da=∞, the adjacent volumes do not mix
at all and remain separated. This is equivalent to the existence of two independent
adiabatic volumes. Another interpretation of the limiting case Da=∞ is an infinite fast droplet evaporation. Both scenarios at
Da→∞ indicate simplifications in the definition of the
extremely inhomogeneous mixing. At intermediate values of Da, mixing
is inhomogeneous, when both turbulent diffusion and evaporation contribute
simultaneously to formation of the DSD.
Main parameters of the problem and their non-dimensional forms*.
QuantitySymbolNon-dimensional formRange of normalized valuesTimett̃=tτ00…∞Distancexx̃=xL0…1Square of drop radiusσσ̃=σr020…1Droplet concentrationNÑ=NN10…1Liquid water mixing ratioqq̃=qq10…1Distribution of square of drop radiusg(σ)g̃(σ̃)=r02N1g(σ)Conservative functionΓΓ̃=ΓA2q1-∞…1SupersaturationSS̃=SA2q1-∞…0Relaxation timeτprτ̃pr=τprτ01…∞Damkölher numberDaDa=τmixτ0=L2Kτ00…∞Potential evaporation parameter (PEP)RR=S2A2q1-∞…0
* All normalized values depend on the initially given values
of L, N1, r0, A2, S2 and K.
Using Eqs. (14) and normalization (22f), the equations for the
non-dimensional size distribution can be written as
∂g̃(x̃,t̃,σ̃)∂t̃=1Da∂2g̃(x̃,t̃,σ̃)∂x̃2+23Γ̃(x̃,t̃)-q̃(x̃,t̃)∂g̃(x̃,t̃,σ̃)∂σ̃.
Eq. (31) is solved with the following initial conditions
g̃(x̃,0,σ̃)=δ(σ̃-1)if0≤x̃<1/20if1/2≤x̃≤1
where δ(σ̃-1) is a delta function.
Table 1 presents the list of all the non-dimensional variables used in this
study and the ranges of their variation. It is shown that six parameters
determining the geometrical and microphysical properties of mixing can be
reduced to two non-dimensional parameters, which enables a more efficient
analysis of mixing. The ranges of parameter variations in Table 1 correspond
to the simplifications used in the study (the initial DSD is monodisperse and
RH ≤ 100 %).
Design of simulationsDamköhler number in clouds
The characteristic mixing
time τmix can be evaluated using Eqs. (2) and (24)
τmix=1Cε-1/3L2/3.
There is significant uncertainty regarding the evaluation of τmix and Damköhler number, Da, in clouds, which is largely related to the choice of
coefficient C in expression (33). These values differ in different studies:
C=10 (Jeffery and Reisner, 2006); C=1 (Lehmann et al., 2009) and
C≈0.2 (Monin and Yaglom, 1975) and Boffetta and Sokolov (2002).
According to Lehmann et al. (2009), the values of Da in clouds of
different types range from to 0.1 to several hundred. Thus, estimation of
Da in clouds may vary within a wide range up to a few orders of
magnitude. Da values in stratocumulus clouds can be similar or
even higher than those in cumulus clouds, since both τmix and
τpr in stratiform clouds are larger than in cumulus clouds.
In our simulations, we compare the evolution of the microphysical parameters
within a wide range of Da (from 1 up to 500) and of R (from -1.5
up to -0.1). Da= 1 represents the case closest to homogeneous
mixing, while Da=500 indicates extremely inhomogeneous mixing.
Numerical method
Calculations were performed using MATLAB solver PDEPE. We solve the equation
system (Eq. 31) for normalized DSD
g̃(x̃,t̃,σ̃j) with the initial
condition (Eq. 32) and the Neumann boundary conditions
∂g̃(0,t̃,σ̃j)∂x̃=∂g̃(1,t̃,σ̃j)∂x̃=0,
where j=1…24 are the bin numbers on a linear grid of square radii.
The number of grid points along the x̃ axis was set equal to 81.
In calculation of the last term on the right-hand side of Eq. (31), the
normalized supersaturation S̃ was calculated first using the
normalized conservative equation
S̃(x̃,t̃)=Γ̃(x̃,t̃)-q̃(x̃,t̃),
where Γ̃(x̃,t̃) is calculated using Eq. (28).
Then, this term was formulated using Eq. (9) as
Horizontal dependencies (upper row) and x̃-t̃
dependencies (lower row) of normalized supersaturation at Da=1,
Da= 50 and Da= 500 and at R=-1.5.
Panel (b) is plotted in semi-log scale.
23S̃(x̃,t̃)∂g̃(x̃,t̃,σ̃j)∂σ̃j≈g̃x̃,t̃,σ̃j+23S̃Δt̃-g̃x̃,t̃,σ̃jΔt̃.
Therefore, at each time step, the DSD g̃ first was shifted to the
left of the value 23S̃Δt̃, where Δt̃ is a small time increment chosen so that23S̃maxΔt̃≤Δσ̃2. Next, the shifted DSD was remapped onto the fixed
square radius grid σ̃j. We used the remapping method
proposed by Kovetz and Olund (1969), which conserves droplet concentration
and LWC. After remapping, the differences between the new and old DSDs were
recalculated. The new values of LWC were then determined using new values of
DSD and Eq. (26). MATLAB utility PDEPE automatically chooses the time step
needed to provide stability of calculations.
The same as in Fig. 3, but for normalized LWC. Left bottom panel is
plotted in semi-log scale.
Results of simulationsFull evaporation case
First, we consider the case R=-1.5, when all the cloud water evaporates
completely. This process corresponds to the cloud dissipation caused by
mixing with the entrained dry air. At the final state, RH is expected to be
uniform and negative over the entire mixing volume.
Dependencies of normalized values of droplet concentration on
normalized LWC at different Da and R=-1.5. Blue symbols mark the
center of the cloudy volume (x̃=1/4), red symbols mark the interface
between the cloudy volume and the dry volume (x̃=1/2), and black
crosses mark the center of the initially droplet-free volume
(x̃=3/4). Symbols are plotted at different time instances. Symbols
at t=0 show initial values of droplet concentration and LWC at the three
values of x̃. Arrows show the direction of movement of the points at
the diagram with time. Point “A” marks the beginning of the spatially
homogeneous stage, t̃=Tmix. Point “F” marks the final
state. The dashed line indicates the relationship between Ñ and
q̃ in extremely inhomogeneous mixing (according to the classical
concept).
Figure 3 shows spatial and time changes of S̃ for Da=1,
50 and 500. At the final state for all three cases S̃=-0.25,
which is in agreement with the analytical solution of Eq. (30). The final
negative value indicates that all the droplets completely evaporated during
mixing. At Da=1 (Fig. 3a, b), two stages of supersaturation
evolution can be identified. The first short stage with t<0.4τpr is the period of inhomogeneous mixing, when the gradients of RH persist. By
end of the second stage of about 14τpr, the equilibrium state
is reached. Thus, at small Da both types of mixing take place. In
the cases of Da=50 and Da=500, the spatial gradients
exit during the entire period of mixing until the equilibrium state is
reached (approximately 50τpr and 300τpr,
respectively) (Fig. 3c, d, e, f). Therefore, at these Da mixing is
inhomogeneous during entire mixing.
Figure 4 shows spatial changes (upper row) and changes in
x̃-t̃ coordinates (lower row) of normalized LWC for the same
case as in Fig. 3. These diagrams demonstrate a significant difference in the
evaporation rates at different Da values. Complete evaporation
(LWC = 0) is reached at Da=1, 50 and 500 by about 12, 22 and
120 relaxation time periods, respectively.
Analysis of Figs. 3 and 4 allows one to introduce two characteristic time
periods: (1) period Tmix during which the spatial gradients of the
microphysical parameters persist, and mixing is inhomogeneous, and (2) period
Tev during which droplet evaporation takes place. Both time
periods are dimensionless and normalized using τ0. Time period
Tev is equal either to the time of complete droplet evaporation
(when R<-1.0) or to the time period during which the saturation deficit in
the mixing volume becomes equal to zero (or close to zero if R>-1.0), i.e.,
evaporation is actually terminated. Quantitative evaluations of Tmix and Tev will be given in Sect. 5.3. At t̃<Tmix, droplets in the mixing volume experience different saturation deficits.
Toward the end of time Tmix the saturation deficit becomes uniform
over the entire mixing volume because of mechanic mixing. At Da=1, the homogenization of the saturation deficit and all the microphysical
variables takes place during a very short time of about 0.5τpr,
and then the evaporation of droplets is assumed to take place under the same
subsaturation conditions, so Tmix≪Tev.
Time evolution of DSD during droplet evaporation at Da=1
(upper row) and Da=50 (bottom row). In each panel, the normalized
DSD are shown at different values of horizontal coordinate x̃.
Different panels show DSD at different time instances.
Figure 4a, b show that at t̃≈0.35, normalized LWC drops down
from 1 to 0.4. Since the average value of the normalized LWC in the mixing
volume is equal to 0.5 (see the initial condition in Eq. 27), 20 % of the
droplet mass evaporates during this short inhomogeneous period. Thus, despite
being quite short, inhomogeneous mixing stage plays an important role even at
Da=1.
Since at t=0 the mixing volume is not spatially homogeneous by definition,
there is always a period while spatial inhomogeneity exists. With increasing
Da, the duration of the inhomogeneous stage increases and the
duration of the homogeneous stage decreases. At Da=500,
homogenization of the saturation deficit requires 250τpr, which
is twice as long as the time of complete droplet evaporation, i.e.,
Tmix≈2Tev. This means that at Da=500,
droplet evaporation takes place in the presence of the spatial gradients of
supersaturation. After complete evaporation of droplets, spatial gradients of
the water vapor mixing ratios remain. This kind of mixing is regarded as
inhomogeneous.
At Da=50, the time of complete evaporation is approximately equal
to the time of supersaturation homogenization, i.e., Tmix≈Tev. In this case, as at Da=500, the droplets experience
different saturation deficit within the mixing volume, so mixing is
inhomogeneous at Da=50.
The differences in droplet evaporation at different Da can be seen
in Fig. 5, showing the relationships between Ñ and q̃
plotted with a certain time increment, so that each symbol in the diagrams
corresponds to a particular time instance. These symbols form curves. Each
panel of Fig. 5 shows three curves corresponding to different x̃:
the center of the initially cloud volume (x̃=1/4); the center of the
mixing volume (x̃=1/2) and the center of the initially droplet-free
volume (x̃=3/4). The directions of the time increase are shown by
arrows along the corresponding curves. The initial points of the curves
corresponding to t̃=0 are characterized by values q̃=1 and
Ñ=1 at x̃=1/4, and by values q̃=0 and
Ñ=0 at x̃=3/4 .
The behavior of the Ñ-q̃ relationship provides important
information about mixing process. At t̃<Tmix, there are
spatial gradients of Ñ and q̃, i.e., Ñ and
q̃ are different at different x̃. This means that the three
curves at t̃<Tmix do not coincide. At
t̃>Tmix, the spatial gradients of Ñ and
q̃ disappear and the three curves coincide. When the curves do not
coincide, mixing is inhomogeneous, and the coincidence of the curves
indicates that the mixing becomes homogeneous. In Fig. 5a and b
(Da=1 and Da=5, respectively), the curves coincide at
point A corresponding to time t̃=Tmix.
Profiles of normalized supersaturation at different Da
and different R>-1.
Figure 5a, b show that at Da=1 and Da=5, mixing
consists of two stages: inhomogeneous and homogeneous. The time instance
t̃=Tmix separates these two stages. In turn, the period of
homogeneous mixing (when evaporation is spatially homogeneous) can be
separated into two sub-periods. During the first sub-period, droplets
evaporate only partially and q̃ decreases at the same droplet
concentration. This sub-period is very pronounced at Da=1, when
q̃ decreases from about 0.4 to 0.1 at the unchanged droplet
concentration. At the second sub-period, when q̃<0.1, droplets
evaporate completely, beginning with smaller ones, so both the droplet
concentration and q̃ rapidly drop to zero. At Da=5
(Fig. 5b), at the stage of homogeneous evaporation (that begins at point
“A”) the decrease in q̃ is accompanied by a decrease in
Ñ.
At Da=50 (Fig. 5c), curves corresponding to different values of
x̃ do not coincide, except at the final point “F”, where
Ñ=0 and q̃=0. This means that horizontal gradients exist
during the entire mixing process and mixing is inhomogeneous till the final
equilibrium state is reached. Droplets penetrating into the initially
droplet-free volume begin evaporating, so only a small fraction of droplets
reaches the center of the droplet-free volume, as seen in Fig. 5c,
x̃=3/4 (black curve). Accordingly, at x̃=3/4 the droplet
concentrations and q̃ reach their maxima (of 0.1 and 0.05,
respectively) and then decrease to zero. At Da=500 (Fig. 5d), all
the droplets evaporate before reaching the center of the dry volume,
indicating an extremely high spatial inhomogeneity of droplet evaporation.
Hence, only two curves for x̃=1/4 and x̃=1/2 are seen in
Fig. 5d.
Profiles of normalized LWC at different Da and at
different R>-1.
Figure 5 also shows that the slopes of the curves describing the
Ñ-q̃ relationships are different at different values of
x̃ and change over time. At large Da, the slopes of the
curves describing the dependencies Ñ-q̃ in the initially
cloud volume are close to linear. However, the slope at a high value of
q̃ is still flatter than that at a low value of q̃. This
can be attributed to the fact that when q̃ is large, it decreases
faster than the concentration Ñ because some fraction of droplets
evaporate only partially. At the end of the mixing when q̃ is small,
Ñ decreases faster than q̃, because the droplet
concentration is determined by the smallest droplets, while q̃ is
determined by larger droplets.
As was discussed in Pt1, according to the classical concept of extremely
inhomogeneous mixing, the ratio q/N remains constant. For dimensionless
Ñ and q̃, the scattering points should be aligned along
the 1 : 1 line. Therefore, the closeness of particular cases to the
classical extremely inhomogeneous mixing can be evaluated by the deviation of
the Ñ-q̃ curve from the 1 : 1 line. One can see that at
Da=500 the Ñ-q̃ relationship is closer to linear.
Despite the fact that at R<-1 all the droplets within the mixing volume
evaporate, it is interesting to follow the DSD evolution during this process.
Figure 6 shows the time evolution of a normalized DSD at Da=1 and
Da=50. One can see a substantial difference in the DSD evolutions
at different Da. At Da=1, different DSDs are formed
very rapidly at different values of x̃ (panel a). The widest DSD
occurs at x̃=1, i.e., at the outer boundary of the initially
droplet-free volume. This is natural, because the supersaturation deficit is
the highest at x̃=1. At t̃>Tmix≈0.4, DSD
become similar at all values of x̃ (Fig. 6b). The DSD width
continues to increase due to partial droplet evaporation. This time period
corresponds to the horizontal segment of the Ñ-q̃
relationship in Fig. 5a. Figure 6c shows the DSD at the stage when a decrease
in LWC is accompanied by a decrease in number droplet concentration. The
corresponding point in the Ñ-q̃ diagram at this time
instance is quite close to the point “F” at which Ñ=0 and
q̃=0.
Profiles of normalized droplet concentration at different
Da and at different R>-1.
At Da=50, DSD are different at different x̃ during the
entire period of mixing. While DSD at x̃>0.5 are wide and droplet
evaporation is accompanied by a shift of DSD maximum to smaller droplet radii
(this feature is typically attributed to homogeneous mixing), the DSD maximum
at x̃<0.5 (the initially cloud volume) shifts toward smaller radii
only slightly until t̃=3.17 (Fig. 6e). Further droplet evaporation
either leads to a complete evaporation (at x̃≥0.5) or shifts the
DSDs to smaller droplet sizes (panel f). The maximum droplet concentration
takes place at x̃=0. Figure 6 shows that DSD shapes evolve
substantially over time, although the final state is characterized by
complete droplet evaporation.
Partial evaporation caseEvolution of the microphysical parameters at different values of
Da and R
Here we consider the process of mixing at R>-1, i.e., when not all the
droplets evaporate completely. Figure 7 shows the horizontal profiles of a
normalized supersaturation at different Da and R. One can see
that in all cases, the final state occurs when the equilibrium
supersaturation S̃=0 (RH = 100 %). However, this final value
is reached quite differently depending on Da. At Da=1,
rapid mixing leads to the formation of spatially homogeneous humidity and
supersaturation during a time period of a fraction of τpr.
Then, supersaturation within the mixing volume grows by evaporation of
droplets, which are uniformly distributed over the entire mixing volume. This
process of homogeneous mixing was analyzed in detail in Pt2.
At Da=500, changes in supersaturation take place largely within
the initially droplet-free volume. RH in the initially cloud volume undergoes
only small changes. This process agrees well with the classical concept of
extremely inhomogeneous mixing. However, a strong gradient of supersaturation
remains within the initially drop-free volume for a long time (tens of
τpr). At Da=50, the situation is intermediate. Mixing
is intensive enough to decrease RH in the initially cloud volume, but
spatially uniform RH is established within about 5–10τpr,
increasing with an increase in R. After this time instance,
mixing takes place according to the homogeneous scenario.
Figure 8 shows the horizontal profiles of normalized LWC at different
Da and R. At the same R, the final equilibrium values of LWC
are identical, as follows from Eq. (30); LWC decreases with an increase in
R. At any Da, the decrease in the LWC in the
cloud volume is caused largely by diffusion of droplets from the cloud volume
into the initially droplet-free volume.
At Da=500, evaporation in the cloud volume is small because
S̃ in these volumes is high during mixing
(Fig. 7). At Da=1, the process of spatial homogenization takes
place during fractions of τpr, i.e., Tmix<1. Then,
during a relatively lengthy period of 10τpr, evaporation
decreases LWC over the entire mixing volume, which is characteristic of
homogeneous mixing. At Da=50, spatial homogenization takes place
during about Tmix≈15. This is a slightly shorter time than
it takes to establish the final equilibrium stage Ttot. Different
Da cases reach equilibrium at different times. The process of
reaching a final uniform LWC lasts for 100τpr at
Da=500 and for about τpr at Da=1.
Figure 9 shows the profiles of the normalized droplet concentrations at
different Da and R. In contrast to LWC, the final concentration
depends both on Da and R. Hence, profiles at different
Da can have different shapes at the same value of R. At R=-0.1
(which corresponds to high RH in the initially dry volume) none of the
droplets evaporate, so the final normalized droplet concentration is equal to
Ñ=1/2. This means that all the droplets in the initially cloud
volume are now uniformly distributed between both mixing volumes. At larger
R, i.e., at lower RH in an initially droplet-free volume,
some droplets evaporate completely. The final concentration decreases with an
increase in Da.
Dependencies of normalized values of droplet concentration on
normalized LWC at different Da and at R=-0.5. Blue circles mark
the center of the cloudy volume (x̃=1/4), red symbols mark the
initial interface (x̃=1/2) and black crosses mark the center of the
initially dry volume (x̃=3/4). Arrows show the direction of movement
of the points with time. Point “F” marks the final stationary state of the
system. The dashed line indicates the relationship between Ñ and
q̃ in extremely inhomogeneous mixing (according to the classical
concept).
The physical interpretation of this dependence is clear. At low
Da, fast mixing leads to formation of a uniform RH throughout the
entire mixing volume, and this affects all the droplets. At high
Da, RH in the initially droplet-free volume remains low for a
long time, and droplets that penetrate can evaporate. Therefore, the fraction
of completely evaporated droplets increases with Da: at R=-0.1
there are no completely evaporated droplets at any Da. At R=-0.3 a
decrease in the droplet concentration takes place only at Da=500,
and at R=-0.5 the droplet concentration decreases already at
Da≥50.
Examples of DSD evolution in the initially cloudy volume
(x̃=1/4) (upper row) and in the initially dry volume
(x̃=3/4) (lower row) at R=-0.5 and at different values of
Da.
The comparative contributions of different factors in establishing the final
states of mixing are well seen in Fig. 10 presenting the relationships
between normalized concentration and normalized LWC at three values of
x̃: 1/4 (center of the cloudy volume), 1/2 and 3/4 (center of the
initially dry volume) at R=-0.5 and different values of Da.
Figure 10 is analogous to Fig. 5, but plotted for R>-1.
DSD at different x̃ at the beginning of the mixing process
for Da=1 and R=-0.5.
At Da=1 the mixing is very fast, which leads to a rapid decrease
in LWC and in the droplet concentration in the initially cloud volume and to
an increase of these quantities in the initially droplet-free volume. As a
result of the rapid mixing and homogenization, all the curves coincide at
point “A” (left panel). After this time instance, spatial homogeneous
evaporation takes place. Since at Da=1 only partial, but not
total, droplet evaporation occurs, the droplet concentration remains
unchanged even while LWC decreases. At Da=50 and
Da=500, the three curves coincide at the final point “F” only.
At Da=500, the relationship between the droplet concentration and
the mass becomes more linear (blue curve). The linear dependence is
consistent with the concept of extremely inhomogeneous mixing (see Pt1).
Considerations regarding the closeness of the Ñ-q̃
relationship to the line 1 : 1 as a measure of inhomogeneity of mixing made
at R<-1 are also valid for R>-1.
Evolution of DSDs and the DSD parameters
Figure 11 presents examples of the DSD evolution at the center of the
initially cloud volume (x̃=1/4) (upper row) and of the initially
droplet-free volume (x̃=3/4) at R=-0.5 and different values of
Da. Several specific features of the DSD are notable. As a result
of the rapid mixing at Da=1 (left column), DSD become similar in
both volumes already at t=0.317τpr (black lines). Further
evolution is similar in both volumes and is characterized by broadening of
the DSD and its shifting toward smaller droplet sizes. This
shift means a decrease in the mass at constant droplet concentration, which
is typical of homogeneous mixing.
Spatial dependencies of the relative DSD dispersion at different
time instances and at different values of Da and different
R>-1.
The initially monodisperse DSDs become polydisperse. The mechanism of the DSD
broadening at Da=1 is illustrated in Fig. 12, showing the DSD at
the earlier, inhomogeneous stage at different x̃. One can see that
within very short periods when the spatial gradient of saturation deficit
exists, droplets entering the initially droplet-free volume partially
evaporate, reaching their minimal size at x̃=1. In this way, a
polydisperse DSD forms. As the mixing proceeds, DSD become spatially
homogenized, as seen in the right panel of Fig. 12.
At Da=50 and Da=500, the DSD shapes substantially
differ from those at Da=1. There are two main differences: the peak
of the distribution shifts only slightly (at Da=50) or does not
shift at all (at Da=500). At the same time, the DSD develops a
long tail of small droplets. Since the mixing rate at these values of
Da is slow, droplets penetrating deeper into the initially dry
volume remain there for a long time and get smaller. As a result, at moderate
and large Da, a polydisperse DSDs form with droplet sizes ranging
from zero to 1. Formation of a long tail of small droplets in case of
inhomogeneous mixing was simulated in direct numerical simulation (DNS) by
Kumar et al. (2012), as well as by means of “the explicit-mixing parcel
model” (EMPM) (Krueger et al., 1997; Su et al., 1998; Schlüter, 2006).
Spatial dependencies of effective radius at different time instances
and at different values of Da and different R>-1.
Figure 13 shows the spatial dependencies of the DSD dispersion (ratio of DSD
r.m.s. width and the mean radius) at different time instances and different
values of Da and R. One can see that the dispersion increases
with an increase in Da and in R. This behavior
can be accounted for by the fact that the DSD broadening toward smallest
droplet size increases with the increase in Da and in R. The DSD dispersion increases with time and with an increase in
x̃, i.e., further into the initially droplet-free volume. At the same
time, spatial homogenization takes place, so at the final state at R=-0.5
the DSD dispersion reaches 0.11 at Da=1 and about 0.2 at
Da=50 and Da=500.
Observed DSD dispersion in different clouds typically ranges from 0.1 to 0.4
(Khain et al., 2000; Martin et al., 1994; Prabha et al., 2012) and can be
caused by the following factors: in-cloud nucleation (e.g., Khain et al., 2000;
Pinsky and Khain, 2002), spatial averaging along aircraft traverses (Korolev,
1995) and non-symmetry in droplet nucleation/denucleation (Korolev, 1995). As
seen in Fig. 13, this dispersion may be also caused by mixing at cloud edges
at moderate and large Da. Hence, inhomogeneous mixing leads to DSD
broadening.
The effective radius, reff, is an important DSD characteristic.
According to the classical concept, reff remains unchanged during
extremely inhomogeneous mixing, whereas decreases during homogeneous mixing.
Figure 14 shows spatial dependencies of reff at different time
instances and different values of Da and R. At R=-0.1 (high RH
in the surrounding volume) reff is similar for all values of
Da. So, at high R (i.e., close to zero), the behavior of
reff does not allow to distinguish between mixing types.
At a given R, the final reff increases with increasing
Da. For instance, at R=-0.5, reff at the final state
differs from the initial reff value by less than 6 % at
Da=500, while at Da=1reff decreases by
20 %. At moderate and high Da, large gradients of
reff exist during the mixing process. However, the gradient is
high only in the initially droplet-free volume where reff
decreases significantly due to the intense evaporation of droplets. Besides,
reff grows very rapidly in the initially droplet free volume, so
at high Da during most of the mixing time reff within
the mixing volume becomes close to the initial reff value in the
cloudy volume.
Delimitation between mixing types
Typically, the Da value is used as a criterion for delimitation
between mixing types. Da=1 is usually used as a boundary value
separating homogeneous and inhomogeneous mixing. As shown in Sect. 4, mixing
always starts as inhomogeneous. In the course of mixing, the initial spatial
gradients decrease and the air volumes either become identical or remain
different. In the former case, the second mixing stage is homogeneous. If
inhomogeneity persists until the equilibrium state is established, mixing
remains inhomogeneous during the entire period. Both mixing stages can be
characterized by duration, change in the droplet concentrations or LWCs, and
other quantitative characteristics. These characteristics are functions of
two non-dimensional parameters R and Da, which can be calculated
and used for delimitation between mixing types. Since mixing between volumes
may turn from inhomogeneous into homogeneous before reaching the equilibrium
state, it is necessary to use some quantitative criteria to delimit mixing
types. Below, delimitation is performed for R>-1 which corresponds to
partial evaporation of droplets by the end of mixing.
Characteristic time periods of mixing
Three characteristic time periods of mixing are distinguished: (a) mixing
period Tmix, during which spatial gradients are smoothing (may
be also called the homogenization period); (b) period Tev during
which S<0 and droplets evaporate until saturation is reached and (c) the
total mixing period Ttot that lasts until the final equilibrium
stage is reached. In our analysis, all three periods are assumed to be
dimensionless quantities.
We use solution (28) for conservative function Γ̃(x̃,t̃) to define quantitatively time period Tmix.
The deviation of the solution from its final value ΔΓ̃=Γ̃(x̃,t̃)-Γ̃(x̃,∞) at
t̃→∞ can be approximately estimated using the first term of
the series expansion as
ΔΓ̃max≈1-Rsin(π/2)π/2exp-π2t̃Dacosπx̃max=1-R2πexp-π2t̃Da.
From Eq. (37) the estimation of Tmix can be written as
Tmix=-Daπ2lnπ21-RΔΓ̃max.
Suppose the value of the maximum deviation is ΔΓ̃max=0.02. This is a small value compared to the initial leap
of function Γ̃, which is equal to 1-R. At ΔΓ̃max=0.02 the duration of the non-homogeneous
stage is evaluated as
Tmix=-Daπ2ln0.01π1-R.
Several studies evaluate the evaporation time for droplets of a particular
size using the equation for diffusion growth (e.g., Lehmann et al., 2009). In
our study, the evaporation time duration Tev is defined as the
period during which the maximum deviation of supersaturation from zero
exceeds the small value chosen as ΔS̃max=0.02:
S̃(x̃,Tev)≤ΔS̃max=0.02.
Although criterion (Eq. 39) is rather subjective, it has an advantage over
the criterion used by Lehmann et al. (2009), as Eq. (32) characterizes
evaporation of the droplet population taking into account the simultaneous
increase in supersaturation, but not of individual droplets of particular
size at constant S as in Lehmann et al. (2009).
At the end of the mixing, both the thermodynamic equilibrium and the
diffusion equilibrium are reached. Accordingly, the total time of mixing
Ttot is evaluated as the maximum of the two time periods needed to
achieve equilibrium Ttot=maxTmix,Tev. All the three characteristic time periods are normalized on the
phase relaxation time, and, therefore, depend on the two non-dimensional
parameters R and Da. The contours of the characteristic time
durations Tmix, Tev and Ttot in the
Da-R diagrams are shown in Fig. 15.
Contours of normalized mixing duration times on Da-R
plane. (a) Mixing time Tmix, (b) evaporation
time Tev, and (c) the total duration mixing time
Ttot.
(a) The boundaries between mixing types on the
Da-R plane designed according to criteria λ1=TmixTtot; (b) the boundaries between
mixing types on the Da-R plane designed according to criterion
λ2=2q̃(Tmix)-1R (Eq. 41). Dashed lines indicate the line corresponding to 2 %
deviation from the initial mean volume radius.
As follows from Eq. (38b), Tmix is proportional to Da.
The dependence of Tmix on R is not very strong, so Tmix slightly decreases with increasing R. This can be attributed to the fact
that the lower the R, the smaller the initial inhomogeneity of function
Γ̃ and the shorter the time to align this inhomogeneity is. At
small Da (high rate of homogenization of the volume), Tev depends largely on R. At large Da, Tev depends
substantially on Da, since the evaporation rate depends on the
number of droplets that diffuse to drier parts of the mixing volume. A
comparison of Fig. 15c with Fig. 15a and b shows that at small Da,
time Ttot is determined by Tev , while at large
Da, Ttot is determined by Tmix.
Determination of boundaries between the mixing types on the
R-Da plane
Several criteria can be proposed for delimitation between mixing types. We
consider these criteria for R>-1. As discussed above, mixing always starts
as inhomogeneous and late either become homogeneous or remains inhomogeneous
till the final equilibrium state is established. At small Da, the
homogenization takes place during Tmix<Ttot. The value of
time fraction λ1 of the inhomogeneous stage can serve as a
criterion for definition of homogeneous mixing. This formula for the fraction
can be written as
λ1=TmixTtot.
The case λ1≤0.5, most of the time the mixing takes place according
the homogeneous scenario and such a regime is reasonable to regard as
homogeneous mixing. If λ1(R,Da) changes within the range
of 0.5<λ1≤1, mixing appears to be intermediate. The criterion
(Eq. 40) depends on the non-dimensional parameters R and Da.
Figure 16a shows the boundaries separating mixing types on the Da-R plane. These boundaries separate all plane into
several zones. At very small R, the duration of the phase transition is
negligibly small. According to criterion (Eq. 40), in this case mixing should
be considered inhomogeneous, irrespective of the Da value.
Another criterion of delimitation between mixing types can be determined from
a comparison of LWC variation rates due to different mechanisms. The mean
normalized LWC (which is equal to the mean normalized liquid water mixing
ratio) can be written as integral q̃(t̃)=∫01q̃(x̃,t̃)dx̃.
The initial mean LWC is equal to q̃(t=0)=12. The final equilibrium LWC is equal to q̃(t=∞)=12(1+R) (Eq. 30). The total
amount of liquid water that evaporates in the course of mixing can be
quantified by the difference between these two values q̃(t=0)-q̃(t=∞=-12R. The amount of liquid water evaporated in the
course of the first inhomogeneous mixing stage is calculated by the equation
q̃(t=0)-q̃(Tmix)=12-q̃(Tmix). Hence, parameter λ2
which is a ratio of
λ2=q̃(t=0)-q̃(Tmix)q̃(t=0)-q̃(t=∞)=2q̃(Tmix)-1R
can serve as another possible criterion for delimitation between mixing
types. This ratio characterizes the fraction of liquid water that evaporates
at the initial inhomogeneous stage. Condition λ2<0.5 in this case
corresponds to homogeneous mixing, while condition 0.5≤λ2<1
corresponds to intermediate mixing. We regard the case λ2=1 as
inhomogeneous mixing. Certainly, criterion λ2 depends on the
non-dimensional parameters R and Da. Figure 16b illustrates
delimitation between mixing types on the Da-R plane according to
criterion λ2.
Comparison of Fig. 16a and b shows that both criteria lead to nearly similar
separation of the Da-R plane into three zones corresponding to
homogeneous, intermediate and inhomogeneous mixing. At the same time, the
boundaries separating these zones are different depending on the delimitation
criterion used. Nevertheless, it can be concluded that mixing can be
considered homogeneous at Da below 4–10 and R<-0.1 and
inhomogeneous at Da exceeding several tens.
(a) Dependencies of the r.m.s. distance of the
Ñ-q̃ relationship curve from straight line 1 : 1 suggested
by classical concept of extremely inhomogeneous mixing. The dependencies are
plotted for different values of Da and R. (b) The same
as to the left panel but for r.m.s. deviations of the mean volume radius
curve from that initial constant value assumed in the classical concept.
Terms “inhomogeneous mixing” (Burnet and Brenguier, 2007) and “extremely
inhomogeneous mixing” (Lehmann et al., 2009; Gerber et al., 2008; Pt1) are
used to denote the mixing regime when the relationship between the normalized
values Ñ and q̃ is represented by a straight 1 : 1 line,
which is equivalent to the constant mean volume radius (in some studies, the
effective radius is used instead of the mean volume radius. According to the
definition used in the present study, extremely inhomogeneous mixing is the
limiting case of inhomogeneous mixing when Da→∞. Despite
the fact that the extremely inhomogeneous mixing is only an idealization our
approach allows to determine to what extent mixing can be considered to be
close to this limiting case. The measure of inhomogeneity of mixing is the
closeness of the Ñ-q̃ relationship to the 1 : 1 straight
line (see discussion above related to Figs. 5 and 10).
Figure 17a shows rms distance between the Ñ-q̃
relationship and the 1 : 1 straight line, depending on Da and
R. These dependences were calculated using the set of points Ñi,q̃i uniformly distributed over spatial interval 01 and time interval 0Ttot. The equation
for estimation is δ=12M∑i=1MÑi-q̃i2, where M is the total
number of points. This distance corresponds to r.m.s. deviation of the
normalized mean volume radius from 1. The dependences of the last deviation
on Da and R and estimated as δ/3 are shown in Fig. 17b.
This estimation is based on the fact that the total mass of droplets is
proportional to the cube of the mean volume radius. As expected, the distance
decreases with increasing Da. At large R, all the curves
coincide indicating a degenerative case when type of mixing becomes
indistinguishable.
We choose the value δ/3 equal to 0.02 to determine the boundary of
the extremely inhomogeneous mixing zone. The value of 0.02 corresponds to
droplet radii deviation of a few tenths of a micron, which is so low that in
in situ measurements this case would always be attributed to extremely
inhomogeneous mixing. In Fig. 16 this boundary is marked by the broken line. The
boundary shows that the mixing at Da exceeding several hundred can
be attributed to the extremely inhomogeneous. Between the boundary separating
inhomogeneous mixing from the intermediate one and the boundary separated
inhomogeneous mixing from extremely inhomogeneous there exists a wide zone of
inhomogeneous mixing where the mean volume (or the effective) radius may drop
by 10 % and more (Fig. 14), and where the DSD dispersion is substantial
and the tail of small droplets is long enough (Fig. 11). Mixing diagrams
currently used for analysis of observed data (N-q dependences in the final
equilibrium state of mixing) do not contain this zone which, therefore, has
remained unrecognized and uninvestigated.
Summary and conclusions
In this study, inhomogeneous turbulent mixing is investigated using a simple 1-D model of mixing between a saturated cloud volume and an undersaturated
droplet-free volume. The mixing is simulated by solving a
diffusion–evaporation equation written in the non-dimensional form. For
simplicity, the initial volumes of cloudy and droplet-free air were assumed
to be equal, and the initial DSD in the cloudy volume was assumed
monodisperse.
Analysis of the diffusion–evaporation equation shows that the time-dependent
process of mixing and the final equilibrium state depend on two
non-dimensional parameters. The first parameter R, referred in this paper
as potential evaporation parameter (PEP) is proportional to the ratio between
the saturation deficit in the initially droplet-free volume and the initial
liquid water content in the cloudy volume. At R<-1, the final state is
characterized by complete droplet evaporation and a spatially homogeneous
saturation deficit, which indicates dissipation of the cloudy volume. At
R>-1, the final state is characterized by existence of droplets and zero
saturation deficit (RH = 100 %). In this case, the cloud volume
expands after mixing with the entrained air. At small values of R (e.g., when RH in the entrained volume is close to 100 %), the
effect of droplet evaporation on microphysics is small, and, formally, this
kind of mixing should be regarded as extremely inhomogeneous. Strictly
speaking, this is a degenerate case, when homogeneous and inhomogeneous
mixing cannot be distinguished (see also Pt1). At R=0, the droplet
population turns into a passive admixture and its turbulent diffusion will be
the same at different thermodynamic parameters.
The second parameter is the Damkölher number (Da),
which is the ratio between the characteristic mixing time and the phase
relaxation time. This parameter compares the rates of spatial diffusion and
evaporation. Parameter Da (Eq. 23) logically appears in the
non-dimensional form of the diffusion–evaporation equation showing that
Da is the ratio of the mixing time defined as τmix=L2K , to the initial drop relaxation time. The expression for
this non-dimensional parameter clearly shows that since we consider an
ensemble of evaporating droplets, the drop relaxation time evaluated just
before the mixing is the characteristic timescale of inhomogeneous mixing
process. In several studies (e.g., Baker and Latham, 1979; Burnet and
Brenguier, 2007; Andejchuk et al., 2009) a question was raised as to which
timescale should be used in formulation of the Damkölher
number: the time of an individual droplet evaporation at constant saturation
deficit, or the phase relaxation time. This study, as well Pt2 show that the
phase relaxation time is the answer. The mixing time is introduced via the
turbulent diffusion coefficient which is a natural measure characterizing the
diffusion rate and, in particular, determines the propagation rate of the
fronts in the fields of droplet concentration and other microphysical
parameters. The turbulent diffusion coefficient is widely used to describe
mixing in cloud models at resolved scales.
The analysis was performed within a wide range of Da (from 1 to
500) and of R (from -1.5 to -0.1). The final LWC and the humidity in
the mixing volume are determined by the mass conservation and do not depend
on Da (see also Pt1 and Pt2). At the same time, the droplet
concentration, as well as the shape of DSD and their parameters strongly
depend on Da.
It is shown that the mixing of air volumes with initially different
thermodynamical and microphysical parameters consists of two stages
characterized by two time periods: the time during which microphysical
characteristics become uniform over the total mixing volume Tmix,
and the time during which zero saturation deficit is reached (at R>-1),
Tev. At t̃<Tmix, the spatial gradients of the
microphysical values remain and the mixing regime can be regarded as
inhomogeneous. At t̃>Tmix, droplet evaporation, if it
occurs at all, takes place within a spatially homogeneous medium, so all the
droplets in the mixing volume experience equal saturation deficit. This
regime can be regarded as homogeneous. It is shown, therefore, that at small
Da mixing between two volumes that starts as inhomogeneous can
become homogeneous towards the end of mixing.
Comparison of analysis based on the classic concepts of mixing and
the results of the present study.
Classical conceptThe present studyOnly the final equilibrium state is typically analyzed; results of in situ observations are interpreted assuming the equilibrium state.The mixing period can last several minutes and more. The microphysical structure of the mixing volumes during this period can differ substantially from that at the final stateTypes of mixing are separated into homogeneous and extremely inhomogeneous.There are the wide ranges of Da and R values, at which mixing can be regarded as intermediate or inhomogeneous (but not extremely inhomogeneous).Mixing can start as purely homogeneousAny mixing starts with the inhomogeneous stageHomogeneous mixing leads to a DSD shift to small droplet sizesHomogeneous mixing does not always lead to the DSD shift to small droplet sizes (Pt2). The shift depends on the DSD shape.Mixing can be analyzed within the framework of a monodisperse DSDMixing always leads to formation of polydisperse DSDIn the course of homogeneous mixing, droplet concentration remains constantIn the course of homogeneous mixing, droplet concentration does not always remain constant (Pt2)Extremely inhomogeneous mixing does not change the DSD shapeInhomogeneous mixing (including extremely inhomogeneous) leads to broadening of the DSD towards small sizesIn the course of inhomogeneous mixing, the effective radius remains constantThe effective radius varies only slightly (5–20 %) in the initially cloud volume. The effective radius rapidly increases in the initially droplet-free volume, approaching the value of effective radius in the cloud volume. With increasing Da, the difference between the values of the effective radius in the initially cloud volume and that at the final state decreases in agreement with the classic concept.
This finding allows to delimit between mixing types. We presented two
quantitative criteria on the Da-R plane that allow to delimit
three mixing regimes: homogeneous, intermediate and inhomogeneous. These
criteria are based on comparison of the characteristic duration mixing and
the evaporation rates. According to the criteria, at Da below
about 5, mixing can be regarded as homogeneous, i.e., the main microphysical
changes take place during the homogeneous stage. At 5<Da<50, the
changes in the microphysical parameters are more significant at the
inhomogeneous stage than at the homogeneous stage. In this case, the mixing
can be regarded as intermediate. Finally, at Da exceeding several
tens, the spatial microphysical gradients remain until the final equilibrium
stage is reached. In this case, the mixing can be regarded as inhomogeneous.
At Da exceeding a few hundred the deviations from predictions
based on the classical concept of extremely inhomogeneous become relatively
small, which justifies regarding this mixing to as extremely
inhomogeneous.
On the whole, the results of the present study are in line with the classic
concepts defining homogeneous and inhomogeneous mixing types. However,
several important points emerge from our work show serious limitations of
classical concepts. A comparison of the classical concepts and the present
study is presented in Table 2. Analysis of Table 2 shows the following.
In contrast to many studies that analyze only the hypothetical final
(equilibrium) state of mixing (Burnet and Brenguier, 2007; Gerber et al.,
2008; Morrison and Grabowski, 2008; Hill et al., 2009), we consider the
entire time-dependent processes of mixing and evaporation. At moderate and
high Da, the mixing can last several minutes. In in situ
observations, we see mostly non-equilibrium stages which may account for a
rather wide scattering of mixing diagrams even at the same values of
Da (e.g., Lehmann et al., 2009).
Note that time-dependent mixing was also considered in several studies (e.g.,
Baker et al., 1980; Baker and Latham, 1982; Jeffery and Reisner, 2006;
Krueger et al., 1997; Kumar et al., 2012) using different approaches and
numerical models. These studies, however, do not contain analysis on
non-dimensional diffusion–evaporation equation.
It is also shown in the study that the slopes of the
Ñ-q̃ relationship (between the normalized droplet
concentration and LWC) tends to the 1 : 1 line with increasing
Da. The closeness can be considered as a measure of extremely
inhomogeneous mixing in terms of the classical concept (see Pt1). It has been
found that the slope of the Ñ-q̃ relationship depends on the
LWC and, accordingly, on time. At large LWC, q̃ changes with time
faster than Ñ, while at low LWC the concentration changes faster.
Although mixing types are usually separated into homogeneous and extremely
inhomogeneous, we have shown that there are wide ranges of Da and
R at which mixing should be considered intermediate or inhomogeneous, but
not extremely inhomogeneous. Within these ranges the effective radius can
change by more than 10–15 %. Standard mixing diagrams do not include
this range that, to our knowledge, has never been investigated despite the
fact that multiple in situ measurements indicate its existence (e.g., Lu et
al., 2014).
Many studies assume the existence of pure homogeneous mixing during
which the initially monodisperse DSD remains monodisperse. Our study shows
that at the very beginning, mixing is always inhomogeneous. This
inhomogeneous stage leads to the formation of a polydisperse DSD that broadens in
the course of droplet evaporation. Hence, even at Da=1 the
initially monodisperse spectrum becomes polydisperse.
It is shown that at small Da, mixing includes both
inhomogeneous and homogeneous stages, which means that type of mixing can
change during the mixing process.
The classical concept assumes that the effective radius always decreases
during homogeneous mixing. Assuming an initially monodisperse DSD, we have
found this conclusion largely valid, with the exception of small R. At the
same time, it was shown in Pt2 that during homogeneous mixing, the effective
radius can decrease, remain constant or increase depending of the initial DSD
shape. Thus, a decrease in the effective radius during mixing cannot always
be considered an indication of homogeneous mixing. Similarly, the
invariability of the effective radius during mixing in the process cannot always
be considered an indication of extremely inhomogeneous mixing.
It is generally assumed that during homogeneous mixing droplet
concentration remains unchanged. In the present study, as well as in Pt2, it
is shown that since mixing leads to a polydisperse DSD, the smallest droplets
may completely evaporate. At R<-1, the DSD becomes very wide and all the
droplets, the smallest ones first, evaporate.
It is generally assumed that inhomogeneous mixing does not alter DSD
shape, but only decreases droplet concentration. The present study showed
that inhomogeneous mixing significantly changes the DSD shape. DSD were found
to be quite different in different regions of mixing volumes. The main
feature is the DSD broadening toward small droplet size, so the relative
dispersion grows up to 0.2–0.3. These values are quite close to those
observed in atmospheric clouds (Khain et al., 2000). Elongated tails of small
droplets during mixing were simulated by Schlüter (2006) who described
turbulent diffusion following Krueger et al. (1997) and Su et al. (1998) as
well as Kumar et al. (2012) using DNS. We see that formation of a
polydisperse DSD is a natural result of inhomogeneous mixing and, therefore,
inhomogeneous mixing is an important mechanism of DSD broadening. A
significant impact of mixing on DSD shape was found identified in multiple
studies, beginning with Warner (1973).
The effective radius has been assumed to remain constant during
extremely inhomogeneous mixing. Our results indicate that, indeed, at the
final equilibrium stage at comparatively high RH the effective radius is
close to that in the initially cloudy volume (especially at high
Da). At the same time, we found that the effective radius varies
in size and is smaller in the initially droplet-free volumes.
The results obtained in parts Pt1 and Pt2, and especially in the current
study (Pt3) dedicated to analysis of turbulent mixing mechanisms in clouds
determine the directions for future work. Since the widely used mixing
diagrams show only a hypothetical equilibrium state, but not the
instantaneous state of mixing that likely correspond to transition periods,
the efficiency of the standard mixing diagrams is questionable. Moreover, the
standard diagrams miss a very important mixing regime, namely, inhomogeneous
mixing that occurs between two limiting cases of homogeneous and extremely
inhomogeneous mixing (Fig. 16).
We believe that the results obtained will help to improve understanding and
interpretation of mixing process both in in situ measurements and modeling.
The approach allows us to investigate the relationship between the main
microphysical parameters typical of inhomogeneous mixing, that differ from
those in the limiting cases of extremely inhomogeneous mixing. In addition,
utilization of polydisperse DSD when solving diffusion–evaporation equation
allows to investigate the role of the initial DSD shape in mixing. In situ
measurements (e.g., Burnet and Brenguier, 2007; Gerber et al., 2008; Lehmann
et al., 2009) and numerical models (Magaritz-Ronen et al., 2016) show a wide
scattering of data on the scattering diagrams. We expect that the location of various
points on the diagrams (e.g., rv3 vs. dilution rates) depends on
the shape of the initial DSDs and characterizes the stage of mixing. The
method applied in the study allows for the investigation of evolution of DSD moments
over space and time.
Recently, there has been vigorous discussions concerning the possible
existence of a high humidity layer near cloud edges that might affect mixing
of cloud with its surroundings (Gerber et al., 2008; Lehmann et al., 2009).
In our opinion, this layer does exist and forms as a result of turbulent
mixing of cloud with surrounding dry air, accompanied by complete droplet
evaporation. The approach developed in the present paper allows one to analyze
formation of such humid layers.
We believe that the results obtained in this study will foster the
development of physically grounded parameterization of mixing in cloud
models.
Data availability
Numerical codes of the model are available upon request.
List of symbols
List of symbols.
SymbolDescriptionUnitsA21qv+Lw2cpRvT2, coefficientnda0, anthe Fourier series coefficientsndCthe Richardson's law constantndcpspecific heat capacity of moist air at constant pressureJ kg-1 K-1Dcoefficient of water vapor diffusion in the airm2 s-1Dathe Damkölher numberndewater vapor pressureN m-2essaturation vapor pressure above a flat water surfaceN m-2FF=ρwLw2kaRvT2+ρwRvTes(T)D, coefficientm-2 sf(r)droplet size distributionm-4g(σ)distribution of square radiusm-5g̃(σ̃)normalized distribution of square radiusndkacoefficient of air heat conductivityJ m-1 s-1 K-1Kturbulent diffusion coefficientm2s-1Lcharacteristic spatial scale of mixingmLwlatent heat for liquid waterJ kg-1mαmoment of DSD of order αm-3Ndroplet concentrationÑnormalized droplet concentrationndN1Initial droplet concentration in a cloud volumem-3ppressure of moist airN m-2qliquid water mixing ratiokg kg-1q1Initial liquid water mixing ratio in a cloudy volumekg kg-1qvwater vapor mixing ratiokg kg-1q̃normalized liquid water mixing ratio equal to normalized LWCndrdroplet radiusmr0initial droplet radiusmr0mean droplet radiusmrvmean volume radiusmRS2A2q1, potential evaporation parameter (PEP)ndRaspecific gas constant of moist airJ kg-1 K-1Rvspecific gas constant of water vaporJ kg-1 K-1Se/ew-1, supersaturation over waterndS̃normalized supersaturationndS2Initial supersaturation in a dry volumendS̃maxmaximal normalized supersaturationndTtemperatureKTmixnormalized duration of inhomogeneous stagendTevnormalized duration of evaporationndTtotnormalized duration of mixingndttimest̃non-dimensional timendxdistancemx̃non-dimensional distancendλ1,λ2criteria of delimitation between the types of mixingndεturbulent dissipation ratem2 s-3Γ(x,t)conservative functionndΓ̃normalized conservative functionndρaair densitykg m-3ρwdensity of liquid waterkg m-3σsquare of droplet radiusm2σ̃normalized square of droplet radiusndτprphase relaxation timesτ̃prnormalized phase relaxation timendτmixcharacteristic time of mixingsτ0Initial timescales
“nd” denotes non-dimensional.
Acknowledgements
This research was supported by the Israel Science Foundation (grant 1393/14),
the Office of Science (BER), the US Department of Energy Award DE-SC0006788
and the Binational US-Israel Science Foundation (grant 2010446). A. Korolev's
participation was supported by Environment Canada. Edited by: T. Garrett
Reviewed by: two anonymous referees
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