Theoretical analysis of mixing in liquid clouds – Part 3: Inhomogeneous mixing

An idealized diffusion–evaporation model of timedependent mixing between a cloud volume and a dropletfree 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 ofR 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 literature typically considers two types of turbulent mixing: homogeneous and inhomogeneous (e.g. Burner and Brenguier, 2006;Devenish et al., 2012;Kumar et al., 2012). The concept of inhomogeneous mixing in clouds was introduced by Baker and Latham (1979), Baker et al. (1980) and Blyth et al. (1980). In a laboratory exper- 5 iments by Latham and Reed (1977) showed that mixing of cloud environment and sub-saturated air resulted in complete evaporation of some droplets, whereas others remained unchanged. The detailed analysis of classical concepts of homogeneous and inhomogeneous mixing is given in the study by Korolev et al. (2015, hereafter Pt1).
The studies of inhomogeneous mixing were closely related to exploration of differ-10 ent mechanisms explaining enhanced growth of cloud droplets and warm precipitation formation (Baker et al., 1980;Baker and Latham, 1982). However, the concepts of homogeneous and inhomogeneous mixing have a much wider application in cloud physics than simply the formation of large-sized droplets assumed in cases of extreme inhomogeneous mixing. In fact, they are closely related to mechanisms involved in the 15 formation of DSDs in clouds and to the description of such formations in numerical cloud models. Pinsky et al. (2015) (hereafter referred to as Pt2) discussed the mechanisms of homogeneous mixing. It was shown that homogeneous mixing takes place at scales below about 0.5 m. In air volumes of larger scales, droplets located in different parts of the volume experience different subsaturations (saturation deficits). Such mix-Introduction when both the characteristic droplet radius and the droplet concentration change, the mixing was considered as "intermediate". The microphysical processes that describe intermediate mixing remain largely uncertain. It is clear that the application of the concepts of homogeneous and inhomogeneous mixing depends on the spatial scale of mixing volumes (Pt1). In the frame of 5 existing convention the mixing is considered as homogeneous if τ mix /τ pr 1, where τ mix ∼ L 2/3 ε −1/3 is the characteristic time scale of turbulent mixing of a volume with a characteristic linear scale L of an entrained volume, and τ pr = 4πDrN −1 is time of phase relaxation, characterizing response of the population of droplets to the changes of humidity. Here ε is dissipation rate of turbulent kinetic energy, N is a concentration of 10 droplets in cloud volume, r is a mean radius of droplets and D is the diffusivity of water vapor (a list of notations of variables used in the study is given in Table A1). In this case, the process of homogeneous mixing can be divided into two stages. During the first stage, the initial gradients of microphysical and thermodynamic variables rapidly decrease to zero. By the end of this stage the temperature, humidity (hence, supersat-15 uration) and droplet concentration fields are spatially homogenized. During the second stage, which duration is relatively long, droplets evaporate and increase the relative humidity in the volume. The process ends when the relative humidity (RH) becomes equal to 100 %, or when all droplets completely evaporate, so that the final RH ≤ 100 %. At scales larger than ∼ 0.5 m, τ mix /τ pr > 1 and the spatial gradients of RH remain for 20 a long time. Consequently, droplets within mixing volumes experience different subsaturations and the mixing should be considered as inhomogeneous. If τ mix /τ pr 1, the mixing is considered to be extremely inhomogeneous.
In this study, we analyze the process of inhomogeneous mixing in cases when the initial DSD in a cloud volume is monodisperse. The structure of the paper is changing of microphysical and thermodynamical variables: first is the turbulent diffusion resulting in mechanical stirring the environment and smoothening the gradients of temperature, water vapor and droplet concentration, whereas the second process is related to the reaction of the population of droplets on the undersaturated environment resulting in droplet evaporation and phase transformation. In the frame of this study 10 the process of inhomogeneous mixing is investigated basing on the analysis and solution of 1-D diffusion-evaporation equation. The conceptual cartoon presenting initial conditions used in the following discussion is shown in Fig. 1.
Let us consider mixing of two equal volumes: the cloud volume (left in Fig. 1) and the dry volume (right in Fig. 1), each has a linear size of L/2. The value of L is thought to 15 be the external turbulence scale, of several tens or a few hundred meters. The mixing starts at t = 0. The process of mixing of two volumes is considered to be adiabatic: i.e. at the domain of mixing the mass and energy are conserved. The cloud volume is initially saturated S 1 = 0, the initial droplet concentration is N 1 , and the initial liquid water mixing ratio is q w1 = 4πρ w 3ρ a N 1 r 3 0 . In the cloud-free (right) volume at the initial moment 20 RH 2 < 100 % (i.e. S 2 < 0), N 2 = 0, and q w2 = 0. Therefore, the initial profiles of these quantities along the x axis are step functions The initial profile of droplet concentration is shown in Fig. 1. This is the simplest inhomogeneous mixing scheme, wherein mixing takes place only in the direction x, and the vertical velocity is neglected. The process of turbulent diffusion (turbulent mixing) 5 is described by a 1-D equation of turbulent diffusion with a turbulent coefficient K . The mixing is assumed to be driven by isotropic turbulence 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) = Cε 1/3 L 4/3 (2) 10 In Eq.
(2) C is a constant. Equation (2) means that turbulent diffusion occurs at scales much larger than the Kolmogorov microscale, i.e. at scales where processes of molecular diffusion can be 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. 15

∂N(0, t) ∂x
where q v is the water vapor mixing ratio. Since during mixing different droplets experience different supersaturations, the initially monodisperse DSD will turn into polydisperse DSD. The droplets that were transported into an initially dry volume will undergo either partial or complete evaporation. Introduction Now we will derive the basic system of equation that describes the processes of diffusion and evaporation which occur simultaneously. The first equation is written for value Γ defined as This value is conservative in a moist adiabatic process, i.e. its value is insensitive to 5 phase transitions (Pinsky et al., 2013(Pinsky et al., , 2014. In Eq. (4) the coefficient A 2 = 1 q v + L 2 w c p R v T 2 is weak function of temperature, and it changes by ∼ 10 % when temperatures change by ∼ 10 • C (Pinsky et al., 2013). In this study, it is assumed that A 2 = constant. In Eq. (4) is the liquid water mixing ratio and g(r) is the DSD. The quantity Γ obeys the diffusion equation with the boundary conditions ∂Γ(0,t) ∂x = ∂Γ(L,t) ∂x = 0 and the initial profile at t = 0 Therefore, in the left volume, function Γ(x, 0) is positive, and in the right volume it is negative.
Since Γ does not depend on phase transitions, Eq. (5) can be solved independently of other equations. The solution of Eq. (5) with initial condition Eq. (6) is (Polyanin and Zaitsev, 2004) Γ(x, t) = ∞ n=0 a n exp − K n 2 π 2 t L 2 cos nπx L = where the Fourier coefficients of expanding the step function Eq. (6) are An example of the spatial dependencies of Γ(x, t) at different time instances during 5 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 a formation of the constant limit value of function Γ Γ(x, ∞) = 1 2 (Γ(0, 0) + Γ(L, 0)) . 10 The second basic equation is the equation for diffusional droplet growth taken in the following form (Pruppacher and Klett, 2007) where σ = r 2 is the square radius of droplets and F =  The third main equation describes the evolution of DSD. In the following discussion the DSD will be described in a form f (σ), which is the distribution of the square of the radius. Such presentation directly utilizes the property of the diffusion growth Eq. (9) according to which changes made to the square radius during DSD evolution do not depend on the drop radius. The time changes of DSD are reduced to shifting distributions, while the shape of the distribution is assumed to remain the same. The standard DSD g(r) is related to f (σ) as g(r) = 2r · f (σ).
The normalized condition for f (σ) is where N is the droplet concentration. Using DSD f (σ), the liquid water mixing ratio can 10 be represented as integral The 1-D diffusion-evaporation equation for non-conservative function f (σ) can be written in the form (Rogers and Yau, 1989) ∂f where the first term on the right-hand side of Eq. (13) describes changes in the DSD due to the process of 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 where q w (x, t) is calculated according to Eq. (12). Equations (12), (14), and (15) represent a closed set of equations allowing the calculation of f (x, t, σ). It is interesting to consider equations for moments of DSD. Let us define a moment Multiplying Eq. (14) by σ α , integrating within limits [0, . . ., ∞] and assuming that σ α f (σ) → 0 when σ → ∞ yield a recurrent formula for DSD moments 10 Equation (17) provides a recurrent relationship between moments of different orders. Such a relationship was discussed in Pinsky et al.'s (2014) analysis of diffusion growth in an ascending adiabatic parcel. In particular, the equation for liquid water mixing ratio that is a moment of the order of α = 3 2 can be written as where the mean radius r(x, t) = The characteristic time of the process of evaporation and the change in supersaturation is the phase relaxation time (Korolev and Mazin, 2003;Pt1) Using Eq. (19), Eq. (18) can be rewritten as From Eqs. (20) and (15), the equation for supersaturation can be written in the following simple form Equations (20) and (20a) show that changes in microphysical variables are determined by the rate of spatial diffusion (the first term on the right-hand side of these equations) and evaporation (the second term on the right-hand side). Equation similar to Eq. (20a) was used by Jeffery and Reisner (2006).

Analysis of non-dimensional equations
Spatial diffusion and evaporation depend on many parameters. An analysis best begins 15 with the basic equation system sketched out in non-dimensional form. Let us define a time scale corresponding to the initial phase relaxation time in a cloud volume and non-dimensional time t = t/τ 0 . We also use the following non-dimensional parameters: Non-dimensional phase relaxation time normalized liquid water mixing ratio non-dimensional conservative function 10 normalized square of droplet radius normalized droplet concentration The definition Eq. (22g) means that the integral of a non-dimensional initial size distribution over the normalized square radius is equal to unity. Non-dimensional distance and non-dimensional time are defined as The most widely used non-dimensional parameter showing comparative rates of diffu-5 sion and evaporation is the Damkölher number: is characteristic time of mixing.

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Using these non-dimensional parameters, Eq. (20) can be rewritten in nondimensional form The initial and boundary conditions should be re-written in non-dimensional form as well. For instance, the normalized initial condition for the non-dimensional function q( x, 0) can be derived from Eqs. (1c) and (22b) The solution for Γ( x, t) can be obtained by a normalization of solution Eq. (7) where is a non-dimensional parameter. The solution Eq. (28) at t → ∞ depends only on this parameter R Non-dimensional parameter R is one of the important parameters that controls the process of mixing. In this study, we consider cases when R < 0 since S 2 < 0, i.e. droplets can only evaporate in the course of mixing. The R yields the ratio of the amount of water vapour that need to be evaporated in order to saturate the mixing environment 15 (that is determined by S 2 ) and the initial liquid water mixing ratio q w1 .
The type of mixing and its result depend on the relationship between the magnitudes of two terms: diffusion and evaporation.
< −1 then Γ( x, ∞) < 0. It means that the droplet-free volume V 2 is too dry and all droplets completely evaporate. At the final equilibrium stage RH < 100 %, i.e. S(x, ∞) < 0. If R = S 2 A 2 q w1 > −1, then Γ( x, ∞) > 0. This means that the mixed volume in the final stage contains droplets, i.e. the mixing leads to an increase in cloud volume. At the final equilibrium stage, RH = 100 % (i.e. S(x, ∞) = 0). The case |R| = S 2 A 2 q w1 = 5 S 2 1 corresponds to either RH ≈ 100 % (i.e. S 2 ≈ 0) and/or the liquid water content in the cloud volume is large. In this case, the second term on the right-hand side of Eq. (25) is much smaller than the first term. In this case results of mixing will be driven by the turbulent diffusion only. The value of Da plays the most critical role, since it determines mixing type. In the 10 case Da → 0 (often referred to as homogeneous mixing), the diffusion term at the beginning of the mixing process is much larger than the evaporation (second) term on the right-hand side of Eq. (25). Then, within a short period, the total homogenization of all variables in the mixing volume takes place and all spatial gradients become equal to zero. At this point, the first term on the right-hand side becomes equal to zero, and 15 the second term on the right-hand side of Eq. (25), describing droplet evaporation, becomes dominant. Thus, the process of mixing consists of two stages: a short stage of inhomogeneous mixing and a longer stage of homogeneous mixing. The evolution of the microphysical variables at the stage of homogeneous mixing is described in detail in Pt2.

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The case Da → ∞ corresponds to the extremely inhomogeneous mixing, according to the conventional view. In this case the diffusion term is much smaller than the evaporation term, so microphysical processes take place under significant spatial gradients of RH. In the limit of Da = ∞ the adjacent volumes do not mix at all, and remain isolated one from another. This case is equivalent to the existence of two independent Da the mixing is inhomogeneous, but both factors, turbulent diffusion and evaporation, contribute simultaneously to the formation of the DSD. Using Eq. (14) and normalization Eq. (22f), the equations for non-dimensional size distribution can be written as where δ( σ − 1) is a delta function. Table 1 presents a summary of the non-dimensional variables used in this study and the range of their changes. It is shown that six parameters determining the geometrical 10 and microphysical properties of mixing can be reduced to two non-dimensional parameters. Such representation allows for a more efficient analysis of the mixing process. The ranges of the changes of the variables in Table 1 correspond to the simplifications used in the study: the initial DSD is monodisperse and RH ≤ 100 %.

Damkölher number Da in clouds
The characteristic mixing time τ mix can be evaluated using Eqs. (2) and (24) There is significant uncertainty regarding the evaluation of τ mix and Da in clouds. This uncertainty is largely related to the choice of the coefficient C in the expression Eq. (33).

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ACPD 15,2015 Theoretical analysis of mixing in liquid clouds -Part 3  Monin and Yaglom (1975) and Boffetta and Sokolov (2002), C ≈ 0.2. 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 cloud may vary in a wide range reaching few orders of magnitude. It is interesting that the values of Da 5 in stratocumulus clouds can be similar or even higher than those in cumulus clouds, since both τ mix and τ pr in stratiform clouds are larger than in cumulus clouds.
In our simulations, we compare the evolution of microphysical parameters within a wide range of Da (from 1 up to 500) and R (from −1.5 up to −0.1). Da = 1 is considered the case closest to homogeneous mixing, while Da = 500 represents extremely 10 inhomogeneous mixing.

Numerical method
Calculations were performed using MATLAB solver PDEPE. We solve the equation system Eq. (31) for normalized DSD f ( x, t, σ j ) with initial condition Eq. (32) and Neumann boundary conditions where j = 1, . . ., 24 are the bin numbers on a linear grid of square radii. The number of grid points along the x axis was set equal to 81. In calculation of the last term on the right-hand side of Eq. (31), the normalized supersaturation S was calculated first using normalized conservative equation 30338 ACPD 15,2015 Theoretical analysis of mixing in liquid clouds -Part 3 where Γ( x, t) is calculated using Eq. (28). Then, this term was represented using Eq. (9) as Therefore, at each time step the DSD f first was shifted left to the value 2 3 S∆ t, where ∆ t is a small time increment chosen such that 2 3 S max ∆ t ≤ ∆ σ 2 . Then, the shifted DSD 5 was remapped onto the fixed square radius grid σ j . We used Kovetz and Olund's (1969) remapping method, which conserves droplet concentration and LWC. After remapping, the differences between the new and old DSDs were recalculated. The new values of LWC within the DSD were determined using Eq. (26). MATLAB utility PDEPE automatically chooses the time step needed to provide stability of calculations.

Full evaporation case
First, we consider the case R = −1.5, when all cloud water should evaporate. This process corresponds to the cloud dissipation caused by mixing with the entrained dry air. At the final stage, RH is expected to be uniform and negative in the entire volume.
15 Figure 3 shows spatial and time changes of S for Da = 1, 50 and 500. As seen from Fig. 3 at the final stage for all three cases S = −0.25, which is in agreement with the analytical solution of Eq. (30). The final negative value indicates that all droplets were completely evaporated during mixing. In the case Da = 1 ( Fig. 3a and b), two stages of supersaturation evolution can be identified. The first short stage, t < 0.4τ pr , is the 20 period of inhomogeneous mixing during, when the gradients of RH persist. By end of the second stage of about 14τ pr the equilibrium state is reached. Thus, at small Da ACPD 15,2015  both types of mixing take place during mixing between two volumes till the reaching the final equilibrium stage. Delimitation between mixing types in this case is somehow arbitrary and will be discussed in Sect. 5.3. In the cases of Da = 50 and Da = 500, spatial gradients exit during the entire period until reaching the equilibrium stage (approximately 50τ pr and 300τ pr , respectively) ( Fig. 3c-f). Therefore during these periods inhomogeneous mixing takes place. Figure 4 shows spatial (upper row) and x − t (lower row) changes of LWC for the same case as in Fig. 3. These diagrams demonstrate significant difference in the rates of evaporation at different Da numbers. Complete evaporation (LWC = 0) is reached at Da = 1, 50 and 500 at about 12, 22 and 120 relaxation times, respectively. 10 Analysis of Figs. 3 and 4 allows segregation of two characteristic times periods: (1) T mix during this period spatial gradients of microphysical parameters persist and mixing is inhomogeneous, and (2) T ev , during which droplet evaporation takes place. Both times are dimensionless and normalized using τ 0 . Time T ev is equal either to the time of total droplet evaporation (in the case of R < −1.0) or the time when the saturation 15 deficit in the mixing volume becomes equal to zero (or close to zero if R > −1.0). Quantitative evaluations of T mix and T ev will be given in Sect. 5.3. At t < T mix , droplets in the mixing volume experience different saturation deficit. Toward the end of this stage, the saturation deficit becomes uniform over the entire mixing volume. At Da = 1, the homogenization of the saturation deficit and all microphysical variables takes place during 20 a very short time of about 0.5τ pr , and then the evaporation of droplets is assumed to take place under the same supersaturation conditions, so T mix T ev . Figure 4a and b shows that at t ≈ 0.35, normalized LWC drops down from 1 to 0.4. Since the average value of LWC in the initial volume is equal to 0.5 (see initial condition Eq. 27), 20 % of droplet mass evaporates during this short inhomogeneous period.

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Thus, despite a very short inhomogeneous mixing stage, evaporation plays an important role even at Da = 1.
Since the initial states are not homogeneous, there is always some period during which spatial inhomogeneities are present. With an increase in Da, the duration of ACPD 15,2015  Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the inhomogeneous stage increases and the duration of the homogeneous stage decreases. In the case of Da = 500, homogenization of the saturation deficit requires 250τ pr , which is twice as long as the total droplet evaporation time, i.e. T mix ≈ 2T ev . This means that at Da = 500, the entire process of droplet evaporation takes place in the presence of spatial gradients of supersaturation. After full evaporation, spatial 5 gradients of the water vapour mixing ratios remain. Such mixing can be regarded as inhomogeneous.
At Da = 50, the time of complete evaporation is approximately equal to the time of supersaturation homogenization, i.e. T mix ≈ T ev . In this case, as in the case of Da = 500, the process of droplet evaporation occurs in an environment of a non-uniformly 10 distributed saturation deficit and also can be regarded as inhomogeneous.
The differences in the process of droplet evaporation at different Da can be seen in Fig. 5. Figure 5 shows the relationships between N and q plotted with a certain time increment, so that each symbol on the diagrams corresponds to a particular moment of time. The set of these symbols forms curves. Each panel of Fig. 5 shows three 15 such curves corresponding to different x: the centre of an initially cloud volume ( x = 1/4); the centre of the mixing volume ( x = 1/2); and the centre of an initially dropletfree volume ( x = 3/4). The directions of time increase are shown by arrows along the corresponding curves in the Fig. 5. The initial points of the curves corresponding to t = 0 are characterized by values q = 1 and N = 1 at x = 1/4, and by values q = 0 and 20 N = 0 at x = 3/4. The behaviour of the N-q relationships provides important information about the mixing process. At t < T mix , there are spatial gradients of N and q, i.e. N and q are different at different x. This means that the three curves at t < T mix are different and do not coincide. At t > T mix , the spatial gradients of N and q disappear and the three 25 curves coincide. So, when the curves do not coincide, mixing is inhomogeneous, and the coincidence of the curves indicates that mixing becomes homogeneous. In Fig. 5a and b (Da = 1 and 5 respectively), the curves coincide at point A, corresponding to time t = T mix . 15,2015 Figure 5a and b shows that at Da = 1 and Da = 5 the process of mixing consists of two stages: inhomogeneous and homogeneous. The time instance t = T mix separates these two stages. In turn, the period of homogeneous evaporation can be separated into two parts. In the first part, droplets evaporate only partially and q decreases under the same droplet concentration. This stage is very pronounced at Da = 1, when q 5 decreases from about 0.4 to 0.1 at the unchanged droplet concentration. At the later stage, when q < 0.1, droplets evaporate totally, beginning with smaller ones, so both droplet concentration and q rapidly decrease to zero. At Da = 5 (Fig. 5b), at the stage of homogeneous evaporation (that begins at point "A") a decrease in q is accompanied by a decrease in N. 10 At Da = 50 (Fig. 5c), curves corresponding to different values of x do not coincide, except at the final point "F", where N = 0 and q = 0. This means that horizontal gradients exist and mixing is inhomogeneous till the final equilibrium state is reached. Droplets penetrating into the dry volume begin evaporating, so only a small fraction of droplets reaches the centre of the dry volume, as seen in Fig. 5c, x = 3/4 (black curve). 15 Accordingly, at x = 3/4 droplet concentrations and q reach their maxima (of 0.1 and 0.05, respectively) and then decrease to zero. In the case of Da = 500 (Fig. 5d), all droplets evaporate before reaching the centre of the dry volume, indicating an extreme spatial inhomogeneity of droplet evaporation. Hence, only two curves for x = 1/4 and x = 1/2 are seen in Fig. 5d. 20 Figure 5 also shows that the slopes of the curves describing the N-q relationships are different at different values of x, and that they change over time. At large Da, the slopes of the curves describing the dependencies N-q in the initially cloud volume are close to linear. However, the slope at a high value of q is still lower than that at a low value of q. This can be attributed to the fact that when q is large, it decreases faster Introduction It was discussed in Pt1 that according to the classical concept, for extremely inhomogeneous mixing the ratio of different DSD moments (e.g. N/q) remains constant. For the dimensionless N and q the scattering points should be aligned along the 1 : 1 line. Therefore, the closeness of particular cases to the classical inhomogeneous mixing can be determined from deviation of N-q curve from 1 : 1 line. One can see that in 5 case Da = 500 the N-q relationship is closer to linear.

ACPD
Despite the fact that at R < −1 all 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 DSD evolutions at different Da. In the case of Da = 1, different DSDs very 10 rapidly form at different values of x (panel a). The widest DSD occurs at x = 1, i.e. at the outer boundary of the initially non-cloud volume. This is natural, because the supersaturation deficit is greatest at x = 1. At t > T mix ≈ 0.4 DSDs become similar at all values of x (Fig. 6b). The DSD width continues to increase due to partial droplet evaporation. This time corresponds to the horizontal segment of the N-q relationship 15 in Fig. 5a. Figure 6c shows the DSD at the stage when a decrease in LWC is accompanied by a decrease in number concentration. The corresponding point at the N-q diagram at this time instance is quite close to the point "F" at which N = 0 and q = 0.
In the case of Da = 50, DSDs are different at different x during the entire period of mixing. One can see that while DSDs at x > 0.5 are wide and droplet evaporation is 20 accompanied by a shift of DSD maximum to smaller droplet radii (a feature typically attributed to homogeneous mixing), the DSD maximum at x < 0.5 (originally cloud volume) shifts toward smaller radii only slightly until t = 3.17 (Fig. 6e). Further droplet evaporation leads either to total droplet evaporation (at x ≥ 0.5) or to a shift of DSDs to small droplet sizes (panel f). The maximum droplet concentration remains at x = 0. The 25 difference in DSDs at different x occurs because of a lack of correspondence between curves representing different N-q relationships, as shown in Fig. 5c. Figure 6 shows that DSD shapes evolve substantially over time, although the final state is characterized by total cloud droplet evaporation.

Evolution of microphysical parameters at different Da and R
Now we shall consider the process of mixing when R > −1, i.e. in cases when not all the droplets evaporate. Figure 7 shows horizontal profiles of a normalized supersaturation at different Da and R. One can see that in all cases, the final state occurs when the 5 equilibrium supersaturation S = 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 less than a fraction of τ pr . Then, supersaturation within the entire volume grows by the evaporation of droplets, which are uniformly distributed within the total volume. This process 10 (homogeneous mixing) was considered in detail in Pt2. At Da = 500, changes in supersaturation take place largely within an initially dry volume. RH in an initially cloud volume undergoes only small changes. This process agrees well with the classical concept of extreme inhomogeneous mixing. Note, however, that a strong gradient of supersaturation remains for a long time (tens of τ pr ) 15 within an initially drop-free volume. At Da = 50, the situation is intermediate. Mixing is strong enough to decrease RH in an initially cloud volume, but spatially uniform RH is established in about 5-10τ pr , increasing with an increase in |R|. After this time instance, mixing takes place according to the homogeneous scenario. Figure 8 shows horizontal profiles of normalized LWC at different Da and R. At the 20 same R, the final equilibrium values of LWC are the same, as follows from Eq. (30); LWC decreases with an increase in |R|. A decrease in the LWC in a cloud volume is caused largely by the process of the diffusion of droplets from a cloud volume to an initially dry volume for any Da. Evaporation in a cloud volume at Da = 500 is small because S is high in cloud vol-25 umes during mixing (Fig. 7). At Da = 1, the process of spatial homogenization takes place for fractions of τ pr , i.e. mixing. At Da = 50, spatial homogenization takes place during about T mix ≈ 15. This is slightly shorter time than it takes to establish the final equilibrium stage T tot . Different Da's reach equilibrium at different times. The process of reaching a final uniform LWC lasts 100τ pr in the case of Da = 500 and about τ pr in the case of Da = 1. Figure 9 shows profiles of normalized droplet concentrations at different Da and R. In 5 contrast to LWC, the final concentration depends jointly 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 an initially dry volume) there is no total droplet evaporation, so the final normalized droplet concentration is equal to N = 1/2. This means that all droplets in the initially cloud volume are now uniformly distributed between cloud and dry volumes. At larger |R| (lower RH in an initially dry volume), some droplets totally evaporate. The final concentration decreases with an increase in the Da.
The physical implication of such dependence is clear. At low Da, fast mixing leads to the formation of a uniform RH throughout the entire volume, and this affects all droplets. At high Da, RH in an initially dry volume remains low for a long time, and droplets that 15 penetrate can evaporate. Therefore, the fraction of fully evaporated droplets increases with Da. In the example presented in Fig. 9, at R = −0.1 there is no Da at which a full evaporation of individual droplets takes place. At R = −0.3 a decrease in droplet concentration takes place only at Da = 500. At R = −0.5 a decrease in droplet concentration takes place already at Da ≥ 50.

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The comparative contributions of different factors in establishing the final states of mixing is well seen in Fig. 10, which shows the relationships between normalized concentration and normalized LWC at three values of x: 1/4 (centre of cloudy volume), 1/2, and 3/4 (centre of initially dry volume) at R = −0.5 and different values of Da. Figure 10 is analogous to Fig. 5, but plotted for R > −1.

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In case of Da = 1 the mixing is very fast, which leads to a rapid decrease in LWC and concentration in the initially cloud volume and an increase of these quantities in the initially dry volume. As a result of the rapid mixing and homogenization, all curves coincide at point "A" (left panel). After this time instance, spatial homogeneous evaporation ACPD 15,2015  Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | takes place. Since at Da = 1 only partial, but not total, evaporation takes place, droplet concentration remains unchanged even while LWC decreases. In cases Da = 50 and 500, three curves coincide in one final point "F" only. In cases of Da = 500 the relationship between droplet concentration and mass becomes more linear (blue curve).
Recall that linear dependence is associated with concept of extreme inhomogeneous 5 mixing (see Pt1). Comments made about closeness of the N-q relationship to the line 1 : 1 as a measure of the inhomogeneity of mixing made for the case R < −1 are valid also in the case R > −1. Note that initially monodisperse DSDs become polydisperse. The mechanism of the DSD broadening at Da = 1 is illustrated in Fig. 12, where the DSD at the earlier, inhomogeneous stage at different x is shown. One can see in Fig. 12 that in the very 20 short periods when the spatial gradient of saturation deficit exists, droplets moving by diffusion to initially dry volume partially evaporate, becoming smallest at x = 1. So, a polydisperse DSD forms. The mixing leads then to a spatial homogenization of DSDs, as seen in the right panel of Fig. 12.

Evolution of DSDs and their parameters
In the cases Da = 50 and 500, the shapes of DSD substantially differ from those at penetrating to a greater distance into the initially dry volume remain there for long time and decrease to smaller sizes. As a result, at moderate and large Da, polydisperse DSDs form with droplet sizes ranging from zero to 1. The formation of a long tail of small droplets in the case of inhomogeneous mixing was simulated in direct numerical simulation (DNS) by Kumar et al. (2012), as well as by using the "explicit-mixing parcel 5 model" (EMPM) (Krueger et al., 1997;Su et al., 1998;Schlüter, 2006). Figure 13 shows the spatial dependencies of the DSD dispersion (ratio of DSD r.m.s. width and mean radius) at different time instances, and the values of Da and R. One can see that the dispersion increases with an increase in Da and |R|. The reason for such behavior is related to that the DSD broadening toward small size end increases 10 with an increase in Da and |R|. The DSD dispersion increases with time and with an increase in x (i.e. further into the dry 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 range from 0.1 to 0.4 (Khain 15 et al., 2000;Martin et al., 2004;Prabha et al., 2012). Such DSD dispersion may be explained by 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 from Fig. 13, such dispersion may be caused by mixing at cloud edges at moderate and large Da. So, inhomogeneous 20 mixing leads to DSD broadening. The effective radius is an important DSD characteristic. According to the classical concept, the effective radius is expected to remain unchanged during extreme inhomogeneous mixing, whereas during homogeneous mixing the effective radius is anticipated decreasing. Figure 14 shows the spatial dependencies of the effective radius at   15,2015  At a given R, the final effective radii increases with increasing Da. For instance if R = −0.5, the effective radius in the final state differs from the initial one by less than 6 % at Da = 500, while at Da = 1 it decreases by 20 %. Note that in cases of moderate and high Da, large gradients of the effective radius exist during the mixing process. Note, however, that the gradient is high only in an initially droplet-free volume, where 5 the effective radius decreases significantly due to the evaporation of droplets. Besides, effective droplet radius in the initially dry volume growth very rapidly, so at high Da during most of the mixing time effective radius within mixing volume is close to the initial value in the cloudy volume. This result agrees in general with the classical concept of extreme inhomogeneous mixing. 10

Delimiting between mixing types
Typically, the value of the Da number is used as a criterion to distinguish between types of mixing. Da = 1 is usually used as a boundary value separating homogeneous and inhomogeneous types of mixing. As shown in the previous section, the process of mixing may consist of two stages. Mixing always begins at an inhomogeneous stage. In 15 the course of mixing, the initial spatial gradients decrease and the air volumes either become identical or remain different. In the case of the former, the inhomogeneous stage is replaced by a homogeneous one. In cases in which inhomogeneity exists until the equilibrium state is established, the entire period of mixing is inhomogeneous. Both mixing stages can be characterized by duration, change in droplet concentrations or 20 LWCs, or other quantitative characteristics. These characteristics are some function of two non-dimensional parameters R and Da, which can be calculated and used for distinguishing between mixing types. Since mixing between volumes may change from inhomogeneous to homogeneous before reaching the equilibrium state, there is a necessity to use some quantitative criteria to delimit mixing types. Below, we carry out 25 such a delimitation for the important case when R > −1, which correspond to partial evaporation of droplets by the end of mixing. 15,2015

Characteristic time periods of the mixing process
One can define three characteristic periods: (a) mixing period T mix , during which spatial gradients are smoothened. This period also can be referred to as the period of homogenization, (b) evaporation period T ev , during which S < 0 and droplets evaporate. As soon as the saturation is reached, the evaporation is terminated, (c) total mixing period 5 T tot , which is the period till the final equilibrium stage is reached. All three times are dimensionless quantities in the study. We use solution Eq. (28) for conservative function Γ( x, t) to define quantitatively the mixing duration time T mix . The deviation of solution from its final value ∆ Γ = Γ( x, t) − Γ( x, ∞) when t → ∞ can be approximately estimated using the first term of the series 10 expansion as From Eq. (37) the estimation of T mix can be written as Let us set the value of maximal deviation ∆ Γ max = 0.02. This is a small value as compared to the initial jump of function Γ, which is equal to 1 − R. In this case, the duration of the non-homogeneous stage is evaluated as Several studies evaluate 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 T ev is defined as the period during which maximal supersaturation deviation from zero exceeds some small value ∆ S max = 0.02 Criterion Eq. (39) is somehow subjective. An advantage of this criterion as compared to that used by Lehmann et al. (2009) is that it characterizes evaporation of the droplets population taking into account simultaneous increase in supersaturation, but not of individual droplets of particular size under constant S. At the end of the mixing process both thermodynamic and diffusion equilibriums are 10 reached. Accordingly, the total time of mixing T tot is evaluated as the maximum of the two times needed to achieve equilibrium T tot = max {T mix , T ev }. Note that all three characteristic time periods are normalized on the phase relaxation time, and therefore depend on two non-dimensional parameters R and Da. The contours of the characteristic time durations T mix , T ev and T tot , on Da-R diagrams are shown in Fig. 15.

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As follows from Eq. (38b), T mix is proportional to Da. The dependence of T mix on R is not very strong, so T mix slightly decreases with an increase in R. This can be attributed to the fact that the lower R, the smaller the initial inhomogeneity of function Γ and the smaller the time to align this inhomogeneity. At small Da (high rate of homogenization of the volume), T ev depends largely on R. At large Da, T ev depends substantially on 20 Da, since the rate of evaporation depends on the number of droplets that diffuse to the drier parts of the mixing volume. A comparison of Fig. 15c with Fig. 15a and b shows that at small Da, time T tot is determined by T ev , while at large Da, T tot is determined by T mix .

Delimitation between the types of mixing
One can propose several criteria of delimitation between the types of mixing. We consider these criteria for the case R > −1. As discussed above, mixing always starts as inhomogeneous. Then it can convert to homogeneous or remain inhomogeneous till the establishing the final state equilibrium state. It is reasonable to refer the latter case 5 as inhomogeneous mixing. At small Da, the process of homogenization takes place during T mix < T tot . The time fraction λ 1 of the inhomogeneous stage can serve as a criterion of the definition of homogeneous mixing. This fraction can be defined as The case λ 1 ≤ 0.5, i.e. when most time the mixing takes place according the homogeneous scenario is reasonable to regard as homogeneous. In case λ 1 = 1, the mixing is inhomogeneous, as mentioned above. If λ 1 (R, Da) changes within the range 0.5 < λ 1 ≤ 1, the mixing is reasonably be refer to as intermediate. The criteria Eq. (40) depends on the non-dimensional parameters R and Da. Figure 16a shows these three zones on the Da-R plane. Note that at very small R, the duration time of phase transi- 15 tion is negligibly small. According to criteria Eq. (40), the mixing in this case should be considered as inhomogeneous, irrespective of the value of Da. Another criterion of delimitation between mixing types can be determined from a comparison of the rates of LWC change during different mechanisms. Let us define the mean normalized LWC as integral q( t) = 1 0 q( x, t)D x. The initial mean LWC is 20 equal to q(t = 0) = 1 2 . The final equilibrium LWC is equal to q(t = ∞) = 1 2 (1+R) (see 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 q(t = 0) − q(t = ∞) = − 1 2 R. The amount of liquid water evaporated in the course of the first, inhomogeneous stage of mixing is calculated from the equation can define another possible criterion of the delimitation between the types of mixing, the parameter λ 2 which is a ratio of This ratio characterizes the fraction of liquid water that evaporates at the initial inhomogeneous stage. Condition λ 2 < 0.5 in this case can be associated to homogeneous 5 mixing, while condition 0.5 ≤ λ 2 < 1 corresponds to intermediate mixing. We regard the case λ 2 = 1 as inhomogeneous mixing. Of course, criterion λ 2 depends on both nondimensional parameters R and Da. Figure 16b illustrates the delimitation between the types of mixing on the Da-R plane according to the criterion λ 2 . A comparison of Fig. 16a and b shows a similar separation of the Da-R plane into 10 three zones corresponding to homogeneous, intermediate and inhomogeneous mixing. At the same time, the boundaries separating these three zones are different for different criteria. However, one can conclude that at Da smaller than 4-10 and R < −0.1, mixing can be considered as homogeneous, and at Da larger than several tens mixing can be considered inhomogeneous. 15 In literature terms "inhomogeneous mixing" (Burner 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 normalized values N and q is represented by a straight 1 : 1 line, which is equivalent to the constant mean volume radius (in some studies effective radius is used instead of mean volume one).

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According to the definition 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 some idealization, which is never can realize in reality, our approach allows to evaluate to what extent the mixing can be regarded to as extremely inhomogeneous. The measure of inhomogenity of mixing is the closeness of the N-q relationship to the 1 : 1 straight line (see discussion related to Figs. 5 and 10).  Figure. 17a shows r.m.s. distance between the N-q relationship and the 1 : 1 straight line depending on Da and R. These dependences were calculated from set of points N i , q i uniformly distributed in spatial interval 0/1 and time interval 0/T tot . The equation where M is the total number of points. This distance corresponds to r.m.s. deviation of normalized mean volume radius from 1. The 5 dependences of last deviation on Da and R, which were estimated as δ/3 are shown in the Fig. 17b. This estimation is based on the fact that mass of drops is proportional to the cube of the mean volume radius. As expected, the distance decreases with an increase in Da. At large R all curves coincide indicating degenerative case when type of mixing becomes indistinguishable. 10 We conventionally assume that the value of the r.m.s deviation of normalized mean volume radius from the initial radius equal to 0.02 is reasonably small to determine the boundary of the extremely inhomogeneous mixing zone. The value 0.02 corresponds to droplet radii deviation of a few tenths of a micron, which is very small, and in insitu measurements this case would always be attributed to extremely inhomogeneous 15 mixing. This boundary is plotted in the Fig. 16 by broken line. The line shows that the mixing characterized by Da exceeding several hundred can be attributed to extreme inhomogeneous mixing. Note that between the boundary separating inhomogeneous mixing from the intermediate one and the zone of extremely inhomogeneous mixing there exists a wide zone of inhomogeneous mixing where the mean volume (or effec-20 tive) radius may drop by 10 % and more (Fig. 14), dispersion of DSD is substantial and the tail of small droplets is significant (Fig. 11). Mixing diagrams widely used for analysis of observed data (N − q dependences in final equilibrium state of mixing) do not contain this zone, so this zone have not been designed and studied yet.

Summary and conclusions
In this study, the process of inhomogeneous turbulent mixing is investigated using a simple a 1-D model of mixing between saturated cloud volume and undersaturated droplet-free volume. The process of mixing is simulated by solving a diffusionevaporation equation written in non-dimensional form. For simplicity, the initial volumes 5 of cloudy and environmental air were assumed to be equal, and the initial DSD in the cloudy volume was assumed to be monodisperse. The analysis of the diffusion-evaporation equation shows that the process of mixing and the final equilibrium state depend on two non-dimensional parameters. The first parameter R is proportional to the ratio between the saturation deficit in an initially dry volume and the initial liquid water content in a cloudy volume. At R < −1, the final state is characterized by total droplet evaporation and a spatially homogeneous saturation deficit. Such a case corresponds to the dissipation of cloudy volume that mixed with dry out-of-cloud air. In the case of R > −1, the final state is characterized by the existence of droplets and a zero saturation deficit (RH = 100 %). In this case the cloud volume 15 is increased after mixing with the entrained air. At small values of parameter |R| (e.g., when RH in the entrained volume is close to 100 %), the effect of droplet evaporation on microphysics is small, and, formally, the mixing should be regarded as extremely inhomogeneous. Strictly speaking this is a degenerate case, when there is no difference between homogeneous and inhomogeneous mixing. At R = 0, droplets turn into a pas-20 sive admixture and their turbulent diffusion will be the same as other thermodynamic parameters.
The second parameter is the Damkölher number, Da, which is the ratio between characteristic times of mixing and phase relaxation. This parameter compares the rates of spatial diffusion and evaporation processes. The analysis was performed within 25 a wide range of Da (from 1 to 500) and R (from −1.5 to −0.1). The final LWC and humidity in the volume are determined by the mass conservation and they do not depend on Da (see also Pt1). At the same time, the droplet concentration, as well as the shape and parameter of DSDs, strongly depend on Da.
It is shown that the process of mixing of initially different volumes consists of two stages and can be characterized by two times: the time during which microphysical characteristics become uniform over the total mixing volume T mix , and the time during 5 which a zero saturation deficit is reached (in cases of R > −1), T ev . At times t < T mix , the spatial gradients of microphysical values remain and the mixing regime can be regarded as inhomogeneous. At times t > T mix , evaporation takes place (if it does at all) within a spatially homogeneous medium, so all droplets experience the same saturation deficit. Such a regime can be regarded as homogeneous. It is shown, therefore, that at small Da mixing between two volumes that starts as inhomogeneous can turn out to be homogeneous to the end of mixing process.
This finding made desirable (and possible quantitatively) to perform delimiting between mixing types. We presented two quantitative criteria on the Da-R plane that allow us to delimit three mixing regimes: homogeneous, intermediate and inhomo-15 geneous. These criteria are based on comparison of characteristic duration times of mixing and the evaporation rates. These criteria showed that at Da less than about 5, mixing can be regarded as homogeneous, i.e. the main microphysical changes take place during the homogeneous stage. At 5 < Da < 30/50 the changes in microphysical parameters are larger at the inhomogeneous stage than that at the homogeneous 20 stage. In this case, the mixing can be regarded as intermediate. Last, at Da larger than several tens, the spatial microphysical gradients remain till the reaching of the final equilibrium stage. In this case, the mixing can be regarded as inhomogeneous. At Da larger than a few hundred the deviations from predictions of classical concept become relatively small. Mixing at such high Da can be attributed to extremely inhomogeneous.

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Overall, the results of the present study are in line with the classical concepts regarding homogeneous and inhomogeneous mixing types. However, several important points emerge from our work that complement and clarify these assumptions. A com-ACPD 15,2015  parison of the classical concepts (see Pt1) and the present study is presented in Table 2. We comment on Table 2 as follows.
a. In contrast to many studies that analyze or assume the final, equilibrium state of mixing (Barnet and Brenguier, 2007;Gerber et al., 2008;Morrison and Grabowski, 5 2008;Hill et al., 2009) we consider the time-dependent processes of mixing and evaporation. The duration of the mixing process can last several minutes at moderate and high Da. In observations, we see mostly non-equilibrium stages, which may help to account for a quite-wide scattering of mixing diagrams even under the same values of Da (e.g., Lehmann et al., 2009). Note that time dependent mixing 10 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).
b. It is also shown that the slopes of the N-q (droplet concentration-LWC) relationship tends to the 1 : 1 line with increase in Da. The closeness of the relationship to this straight line can be considered as closeness to the extreme inhomoge- 15 neous mixing in terms of classical concept (see Pt1). It is found that the slope of the N-q relationship depends on the LWC and, accordingly, on time. At large LWC, q changes with time faster than N, while at low LWC, the concentration changes faster. Such differences were found in a numerical simulation of stratiform clouds in the vicinity of a cloud top . Types of 20 mixing are typically separate into homogeneous and extremely inhomogeneous. In this study it is shown that there are the wide ranges of Da and R, when mixing can be attributed to the intermediate and to inhomogeneous (but not extremely inhomogeneous). Within this zone effective radius can change by more than by 10-15 %. Standard mixing diagrams do not include this zone. To our knowledge, ACPD 15,2015 Theoretical analysis of mixing in liquid clouds -Part 3 c. Many studies assume existence of pure homogeneous mixing, when an initially monodisperse DSD would remain monodisperse. Our study shows that at the very beginning mixing is always inhomogeneous. This inhomogeneous stage leads to the formation of a polydisperse DSD, the width of which increases in the course of droplet evaporation. So, even at Da = 1, a monodisperse spectrum becomes 5 polydisperse.
d. It is shown that at small Da, process of mixing includes inhomogeneous and homogeneous stages. So, mixing changes its type during the mixing process.
e. The classical concept assumes that the effective radius always decreases during homogeneous mixing. In the present study, where we considered an initially monodisperse DSD, this conclusion proved largely valid, with the exception of the cases of small R. However, in Pt2, it was shown that depending of the shape of DSD, the effective radius can decrease, remain constant or increase during homogeneous mixing. Thus, a decrease in the effective radius during the mixing process cannot always be considered an indication of homogeneous mixing. 15 f. It is generally assumed that droplet concentration remains unchanged during homogeneous mixing. In the present study, as well as in Pt2, it is shown that since mixing leads to a polydisperse DSD, the smallest droplets may totally evaporate.
In the case of R < −1, the DSD becomes very wide and all droplets, beginning with the smaller ones, evaporate. 20 g. It is widely assumed that inhomogeneous mixing does not alter DSD shape, but rather only decreases droplet concentration. The present study showed that inhomogeneous mixing may significantly change the DSD shape. DSDs turned out to be quite different in different regions of mixing volumes. The main feature is broadening of DSD toward small droplets with relative dispersion up to 0.2-0.3. These values are quite close to those observed in atmospheric clouds (Khain et al., 2000). Such elongated tails of small droplets have been simulated by Schlüter ACPD 15,2015 Theoretical analysis of mixing in liquid clouds -Part 3  2006), who used the EMPM to describe turbulent diffusion (Kruger et al., 1997;Su et al., 1998) and by Kumar et al. (2012) using DNS. So, we see that the formation of a polydisperse DSD is a natural result of inhomogeneous mixing. Inhomogeneous mixing, then, is an important mechanism in the DSD broadening. The significant role of mixing on the DSD shape has been identified in many studies, 5 beginning with Warner (1973).
h. It has been long thought that the effective radius remains constant in inhomogeneous mixing. Our results indicate that, indeed, in the final equilibrium stage at comparatively high RH in a non-cloudy air volume, the effective radius is close to that found in an initially cloudy volume (especially at high Da). At the same time, the results show that during mixing, the effective radius varies in size, and is smaller in initially non-cloudy volumes. The effective radius also changes substantially in cases of low RH in the entrained volume at which most (or all) droplets evaporate by mixing.   Classical concept The present study

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The final equilibrium state is typically analyzed; in-situ observations are interpreted assuming the equilibrium state.
The mixing period can last several minutes. The microphysical structure during this period can differ substantially from that at the final stage. Types of mixing are typically separated into homogeneous and extremely inhomogeneous.
There are the wide ranges of Da and R, when mixing can be attributed to the intermediate and to inhomogeneous (but not extremely inhomogeneous). Mixing can start as purely homogeneous.
Any mixing between different volumes starts with the inhomogeneous stage. Homogeneous mixing leads to a DSD shift to small droplet sizes.
Homogeneous mixing does not always lead to a DSD shift to small droplet sizes (Pt2). Mixing can be analyzed within the frame of a monodisperse DSD.
Mixing always leads to formation of polydisperse DSD. In the course of homogeneous mixing, droplet concentration remains constant.
In the course of homogeneous mixing, droplet concentration does not always remain constant (Pt2). Extremely inhomogeneous mixing does not change the DSD shape.
Inhomogeneous mixing, including extremely inhomogeneous) leads to broadening of DSD towards small sizes. In the course of inhomogeneous mixing, the effective radius remains constant.
The effective radius varies only slightly (5-20 %) in an initially cloudy volume. The effective radius rapidly increases in an initially non-cloudy volume, approaching the size found in cloudy volumes. With increase in Da the difference between values of effective radius in the initially cloud volume and that in the final state decreases in agreement with the classical concept. 15,2015

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characteristic spatial scale of mixing m L w latent heat for liquid water J kg −1 m α moment of DSD of order α N droplet concentration m −3 N normalized droplet concentration -N 1 initial droplet concentration in cloud volume m −3 p pressure of moist air N m −2 q v water vapour mixing ratio (mass of water vapour per 1 kg of dry air)q w liquid water mixing ratio (mass of liquid water per 1 kg of dry air) q w1 liquid water mixing ratio in cloud volume q normalised liquid water mixing ratio -R ACPD 15,2015 15,2015   30372 ACPD 15,2015 15,2015