The conceptual diagrams of homogeneous and extreme inhomogeneous mixing are shown in Fig. 1. During the first stage of extreme inhomogeneous mixing the subsaturated parcel is engulfed into the cloudy environment (Fig. 1a1). Then, the droplets at the interface of the sub-saturated parcel and the cloud environment undergo complete evaporation until the air within the engulfed volume reaches saturation (Fig. 1a2). After that the saturated but droplet-free parcel mixes with the rest of the cloud environment (Fig. 1a3). The result of inhomogeneous mixing is that the cloud parcel has reduced droplet concentration and the droplet sizes remain unchanged.

In the case of homogeneous mixing after entraining into a cloud (Fig. 1b1), the subsaturated parcel “instantly” mixes up with its cloud environment (Fig. 1b2) leading to undersaturation of the total volume. Then, all droplets throughout the mixed volume undergo simultaneous evaporation until the equilibrium state is reached. The result of homogeneous mixing is a cloud volume with reduced concentration of droplets and droplets with reduced sizes (Fig. 1b3).

Based on mass and energy conservation the final state of the bulk parameters (i.e., liquid water mixing fraction, humidity, temperature, etc.) is the same for both types of mixing. However, in the case of extreme inhomogeneous mixing saturation is reached through complete evaporation of some fraction of droplets, and their sizes remain constant. Whereas in case of homogeneous mixing saturation is reached through a uniform evaporation of droplets, and the total number of droplets remains unchanged. It should be noted that in both cases the droplet concentration decreases due to dilution by the mixed droplet-free sub-saturated parcel.

The following discussion will be specifically focused on the microphysical properties formed at the final stage of the homogeneous and extreme inhomogeneous mixing. The processes occurring during mixing state (i.e., transition a1 → a2 and b1 → b2 in Fig. 1) remain outside the frame of this work. Following the formalism of homogeneous and extreme inhomogeneous mixing, the process of mixing reaches the final stage when (1) the entrained and cloud environment are mixed up and the spatial gradients of the microphysical (N, β, q, etc.) and environmental (T, S, e, etc.) parameters approach zero; (2) the diffusional process related to droplet evaporation comes into equilibrium. The second condition is completed when (a) the environment reaches saturation state, or (b) the entire population of droplets is completely evaporated, if the entrained air is sufficiently dry.

The above description of homogeneous and extreme inhomogeneous mixing is highly idealized. Actual in-cloud mixing does not occur as a sequence of discrete events (Fig. 1) that individually come to equilibrium only to be followed by the next discrete mixing events. But rather it is occurring continuously on a cascade of different spatial and timescales. Broadwell and Breidenthal (1982) summarized the experimental evidence and proposed the following description of mixing in turbulent shear layers. Mixing takes place in a series of events. Two shear layers exchange mass by engulfing parcels from an opposite layer into localized zones. The initially large-scale filaments of the two gases break down towards smaller scales due to the action of turbulence. The turbulence stretches the interface between the gases and enhances the molecular diffusion across the increasing surface. The actual mixing of the engulfed volume is a molecular diffusion process that is most effective after the break down volumes reduce to the Kolmogorov viscosity scale. It is anticipated that the reaction of the ensemble of droplets is a combination of homogeneous and inhomogeneous mixing with domination of one type of mixing over the other depending on the characteristic spatial and timescales of the environment determined by turbulence, cloud microphysics, state parameters and stage of mixing.

The foregoing discussion will be focused on mixing between saturated cloud parcels and out-of-cloud sub-saturated air. The cloud parcel contains droplets with average diameter D¯1, liquid mixing ratio q1 and number concentration N1. The initial temperature in the cloud parcel is T1, relative humidity RH1=1, where RH = e/es(T) (the explanation of variable notations is provided in Table 1). The second parcel is droplet free (N2=0), sub-saturated with initial relative humidity RH2<1 and temperature T2. The mixing occurs isobarically, i.e., p=const. At the final stage of mixing the temperature and humidity formed in the resulting parcel are T and RH (Appendix A). The process of mixing is completed, when the mixed parcel reaches equilibrium due to the air saturation (i.e., RH = 1), or due to the complete evaporation of droplets. In the latter case the final humidity is RH ≤ 1. The effect of the vertical velocity and vertical travel on final T, RH, and q is not considered here, i.e., vertical velocity uz=0.

Without the loss of generality the masses of the cloudy and sub-saturated volumes prior to the mixing are assumed to have a unit masses, i.e., m1=1 and m2=1. The mixing process will be considered as mixing of μ fraction of the cloud parcel with (1-μ) fraction of the second (sub-saturated) parcel. The mixing cloud fraction may vary within the range of 0≤μ≤1. Therefore, the mass of the resulting mixed parcel is equal to m1μ+(1-μ)m2=1. This approach simplifies the consideration of mixing and allows us to consider all possible proportions of the mixing of two volumes.

The mixing ratio of liquid water q formed at the final stage of mixing is determined by the mass of the mixing cloud water μq1 and amount of evaporated water required to saturate the newly formed mixed volume δqm. The mass balance of liquid water for the mixing volume yields q=μq1-δqm, where δqm=cpaRvTm02L2ln⁡1+es(Tm0)RaL2pcpaRv2Tm021+RHm0es(Tm0)RaL2pcpaRv2Tm02≅-Sm0A2 is the mixing ratio of liquid water required to saturate 1 kg of volume with temperature Tm0 and humidity RHm0 (Appendix A); Tm0, RHm0 and Sm0, are the temperature, relative humidity and supersaturation formed in the volume after instantaneous air mixing, but before droplets start evaporating (Appendix A); es(Tm0) is the saturation vapor pressure at temperature Tm0.

Equation (1) is a non-linear function of μ, since Tm0, em0 and thus δqm depend on μ. Equation (1) can be simplified, if T1=T2. In this case Tm0=T1=T2, and es(Tm0)=es(T1)=es(T2). Given that, the expression under logarithm in Eq. (2) can be expanded in series resulting in (Appendix B) δqm=(1-μ)δq∗, where δq∗=cpaRvT22L2ln⁡1+es(T2)RaL2pcpaRv2T221+RH2es(T2)RaL2pcpaRv2T22≅-S2A2 is the mixing ratio of liquid water required to saturate 1 kg of the entrained dry air with temperature T2 and humidity RH2. Substituting Eq. (3) in Eq. (1) gives q=μq1-1-μδq∗. The value of δq∗ does not depend on μ, and Eq. (5) is a simple linear function of μ. The comparisons with numerical simulations showed that Eq. (5) provides accuracy within a few percent, when the temperature difference T1-T2<2∘. Although, in many cases T1-T2 may vary a wide range reaching 10 ∘C or higher, clouds with T1-T2<2∘ are quite common. Therefore, for the sake of simplicity, Eq. (5) and the assumption T1≈T2 will be used in the following consideration of mixing.

It should be noted that Eqs. (1) and (5) are valid for the cases when μ>μcr. Here μcr is critical mixing fraction, which separates partial and complete evaporation of cloud water in the mixing volume (Sect. 2.4). Cases when μ≤μcr correspond to complete evaporation of droplets, and q=0.

The temperature at the final stage of mixing can be estimated as (Appendix C) T=Tm0-1-μδq∗Lcpa,whenμ>μcrT=Tm0-μq1Lcpawhenμ≤μcr. Eqs. (1), (5), (6) were obtained based on mass and energy conservation, and they do not depend on how mixing proceeds. Therefore, Eqs. (1), (5), (6) are valid for both homogeneous and inhomogeneous mixing.

As mentioned in Sect. 2.2 the process of mixing is complete only after reaching equilibrium by saturating the mixed volume or by evaporating of all cloud droplets depending on the mixing fraction μ. The critical mixing fraction μcr, corresponding to evaporation of all droplets, can be found from Eq. (5) when q=0, i.e., μcr=δq∗q1+δq∗. Critical mixing fraction separates μ in two subranges: (a) 1≥μ>μcr where q is described by Eqs. (1) or (5) and RHm=1; (b) μcr≥μ≥0 where q=0 and RHm≤1.

For the general case when T1≠T2, μcr, can be found by solving the non-linear equation μcrq1-δqm(μcr)=0. Figure 2 shows comparisons of dependences of μcr vs. q1 calculated from Eq. (7) and those deduced from a numerical model (Sect. 3). Critical mixing fraction μcr is also shown by black stars in Fig. 4. The locations of the stars in Fig. 4 coincide well with the locations, where the modeled microphysical moments become zero. The obtained agreement between analytical and modeled μcr in Figs. 2 and 4 validates the developed approach.

Within the framework of extreme inhomogeneous mixing some fraction of droplets undergo complete evaporation, whereas the rest of the droplets remain unchanged. Therefore, such a process results in scaling the droplet size distribution f(D), i.e., f(D)=kf1(D), where k is some coefficient dependent on μ and the initial environmental parameters of the mixing volumes, f1(D) is the droplet size distribution before mixing. Equation (9) yields relationships between pairs nth and mth moments MnMn1=MmMm1, where Mn is the nth moment of f(D). Therefore, it is anticipated that for extreme inhomogeneous mixing droplet number concentration N (0th moment), extinction coefficient β (2nd moment), liquid water mixing ratio q (3rd moment), along with other moments, will correlate with each other, i.e., NN1=ββ1=qq1. One of the consequences of Eqs. (9)–(10) is that the characteristic droplet sizes D¯, D2, Dv, Deff will remain constant during inhomogeneous mixing.

For the case T1=T2 and μ>μcr Eqs. (5) and (11) yield the dependence of N vs. μ N=N1μ-(1-μ)δq∗q1β=β1μ-(1-μ)δq∗q1. For a general case when T1≠T2 the term (1-μ)δq∗ in Eqs. (12) and (13) should be replaced by δqm(μ) (Eq. 2).

For homogeneous mixing, when μ>μcr, the droplet number concentration changes only due to dilution by the entrained air, i.e., NN1=μ. Assuming T1=T2, and substituting Eq. (5) in Eq. (14) yields NN1=q+δq∗q1+δq∗. Following Eq. (15) N and q are linearly related for homogeneous mixing. However, no linear relationships exist between other moments. Thus, substituting the definition of the liquid water mixing ratio q=πρwNDv3/6ρa in Eq. (15) yields the relationship between mean volume droplet size, concentration and liquid water mixing fraction Dv3Dv13=1+1-N1Nδq∗q1Dv3Dv13=qq1q1+δq∗q+δq∗. In a similar way the relationship between the extinction coefficient β=QπND22/4, N and q can be written as ββ1=NN11+1-N1Nδq∗q12/3ββ1=qq12/3q+δq∗q1+δq∗1/3. In Eqs. (17a) and (17b) it is assumed that D2≈Dv.

Substituting in Eq. (16) the expression for the time of phase relaxation τp=1/bND¯ (e.g., Squires, 1952; Korolev and Mazin, 2003) and assuming D¯≈Dv yields ττ1=N1N1+1-N1Nδq∗q1-1/3. For the cases when the temperature difference T1-T2 exceeds a few degrees, the effect of μ on Tm and Sm should be taken into consideration in the calculations of evaporated water. For such cases δqm (Eq. 2) should be used instead of δq∗. Using Eq. (14) δqm can be presented as a function of NN1, i.e., δqmμ=δqmNN1. Replacing Eq. (5) by Eq. (1) in the above consideration, the equations Eqs. (15)–(18) can be rewritten as NN1=q+δqmNN1q1Dv3Dv13=1-δqmNN1q1N1N=qq+δqmqq1ββ1=NN11-δqmNN1q1NN12/3=q2/3q+δqmββ11/3q1τpτp1=N1N1-δqmNN0q1N1N-1/3. Equations (19)–(22) can be solved numerically.

Following Eq. (5), if (1-μ)μδq∗q1≪1 then q1≥q≫δq∗. If the condition in Eq. (23) is satisfied, then the terms associated with δq∗ in Eqs. (15)–(18) can be neglected. This results in correlation of all moments, i.e., N/N1=β/β1=q/q1 (compare with Eq. 11). This corresponds to the degenerate case, when the difference between the homogeneous and inhomogeneous mixing vanishes. Thus, the dimensionless parameter ξ=1-μμδq∗q1 in Eq. (23) can be used for characterization of proximity of the homogeneous mixing moments to those formed during extreme inhomogeneous mixing.

The range of μ in ξ is limited by μcr<μ≤1, so that 0<1-μμ≤q1δq∗. This gives the range of changes of ξ, i.e., 0≤ξ≤1 for the mixing without complete evaporation of all droplets. The degenerate case corresponds to ξ→0, whereas ξ→1 corresponds to a maximum difference of the moments for homogeneous and extreme inhomogeneous mixing.

As follows from Eqs. (4) and (23) approaching to the degenerate case (ξ→0) occurs, when one of the following conditions or their combination is satisfied: (a) RH2→1; (b) es(T)→0 at low temperatures; (c) q1≫δq∗; (d) μ→1. The effect of RH, T, q1 and μ on mixing will be demonstrated in Sect. 3.

Figure 3 shows dependence of ξ vs. μ. The grey area in Fig. 3 indicates the region where identification of type of mixing from in situ measurements (Sect. 5) may be hindered due to proximity of the moments for homogeneous and extreme inhomogeneous mixing. Thus for δq∗/q1=0.01 identification of type of mixing is ambiguous for nearly the entire range of μ.

For the general case, when T1≠T2, it should be ξ=|δqmμ|μq1. An absolute value |δqmμ| should be used in ξ since δqmμ can be negative (Appendix A, Fig. A1) if mixing results in supersaturation Sect. 3.4).

The coefficient ξ may be useful for identification type of mixing from in situ observations.

In conclusion of Sect. 2 it is worth noting, that for both homogeneous and extremely inhomogeneous mixing most of the relationships between moments include the ratio δq∗q1. As follows from Eq. (A9) (in Appendix A) δq∗q1≅S2A2q1, which is equal to the so-called “potential evaporation parameter” R (Pinsky et al., 2016a, b). The potential evaporation parameter R plays an important role in determining scenarios of droplet evaporation in a turbulent environment.

Figure 4 shows the results of the simulation of different moments and state parameters vs. μ. The calculations were performed for different relative humidity of the entrained parcel RH2=0.2, 0.5, 0.8 and 0.95. As seen from Fig. 4 for the case of homogeneous mixing only N and q are linearly related with μ, the rest of the variables have non-linear dependences on μ. For the case of extreme inhomogeneous mixing all f(D) moments and droplet sizes linearly depend on μ. Note, for μ≤μcr all moments are equal to zero.

Since the amount of the evaporated liquid water does not depend on the type of mixing, the dependences of q(μ) are the same for both homogeneous and inhomogeneous mixing (Fig. 4a). The type of mixing has the most pronounced effect on the droplet concentration (Fig. 4b) and droplet sizes (Fig. 4e).

Figure 4(g) shows the dependences RHm0 and RH vs. μ. Here RHm0 is the relative humidity at the initial stage of homogeneous mixing before droplets start evaporating (Fig. 1b2). Figure 4h presents comparisons of modeled T(μ) and those calculated from Eqs. (6a), (6b) and (C4). The independence of q(μ), RH(μ) and T(μ) on type of mixing (Fig. 4a, g, h) is the consequence of the mass and energy conservation, which are not contingent on type of mixing.

The diagrams in Fig. 5a–c show the dependences of normalized β, q and Dv vs. N/N1 calculated from numerical simulations and analytical equations from Sect. 2. The calculations were performed for different humidity of the entrained air RH2. As seen from Fig. 5a–c, the normalized dependences for homogeneous mixing q(N), β(N) and Dv(N) tend to approach the line of extreme inhomogeneous mixing when relative humidity RH2 approaches 1. This is consistent with the degenerate case, when ξ→0 (Sect. 2.7). In this case droplets behave as a passive admixture, and they do not interact with the environment.

Figure 5d–f demonstrate the sensitivity of q(N), β(N) and Dv(N) to liquid water mixing ratio q1. It is seen that the increase of q1 results in q(N), β(N) and Dv(N) (calculated for homogeneous mixing) approaching towards q(N), β(N) and DvN for the inhomogeneous mixing. In other words, the sensitivity of the microphysical parameters to the type of mixing increases with the decrease of q1. From a practical viewpoint it means that from in situ observations the difference between homogeneous and inhomogeneous mixing is anticipated to be more pronounced for the cases with a relatively low liquid water mixing ratio (e.g., q1<1 g kg-1). Such behavior is consistent with the consideration in Sect. 2.7.

Figure 5g–i shows the effect of temperature on the normalized q(N), β(N) and DvN for T1=T2. Figure 5g–i indicate that the difference between the moments becomes most pronounced at warm temperatures, whereas at cold temperatures (e.g., T=-30 ∘C), q(N), β(N) and Dv(N) for homogeneous mixing are approaching those for the extreme inhomogeneous mixing limit.

Such behavior is explained by the fact that liquid water deficit δqm decreases with decreasing temperature (Appendix A, Fig. A1). At low temperatures (T=-30 ∘C) the amount of evaporated water δqm is so small that homogeneous mixing with dry out-of-cloud air will have approximately the same effect as mixing with saturated air (i.e., degenerate case, Sect. 2.7).

Overall, as follows from Fig. 5 the results from the analytical predictions (Sect. 2) turned out to be in a good agreement with numerical simulations.

Isobaric mixing of two nearly saturated volumes with T1≠T2 may result in supersaturated environment (e.g., Rogers, 1976; Bohren and Albrecht, 1998). Mixing resulting in supersaturation is different in principle from the mixing with evaporating droplets. In this case the meaning of homogeneous and inhomogeneous mixing becomes ambiguous. Formation of supersaturation leads to different dependences between ND¯, β, q, D¯ and N as compared to those shown in Figs. 4–5, when T1=T2.

Figure 6 presents a set of diagrams similar to those in Fig. 4, but calculated for the cases, when T1≤T2. It turns out that for the case of extreme inhomogeneous mixing the temperature difference between T1 and T2 breaks down linear dependences of the microphysical moments (e.g., ND¯, β, q Fig. 6a, c, d) vs. μ.

Figure 7 presents the effect of the temperature difference ΔT on the normalized dependences q(N), β(N) and Dv(N). In clouds, high supersaturation resulting from isobaric mixing may lead to activation of interstitial CCN, which may increase N and decrease Dv (Korolev and Isaac, 2000). However, no activation of new droplets during isobaric mixing was allowed in this study. For the cases when RHm0>1 (Fig. 7, AB on line 1) the condensed water was uniformly distributed between available droplets. Therefore, q(N), β(N) and Dv(N) calculated for homogeneous and extreme inhomogeneous mixing coincide with each other on this interval.

Numerical simulations also showed that the effect of temperature on mixing is more pronounced for the cases when the cloud temperature is warmer than that of the entrained air, i.e., T1>T2, as compared to the cases with T1<T2.

In the previous sections the mixing was considered as a single event, i.e., μ fraction of the cloudy air mixed up with (1-μ) fraction of entrained dry air. Such mixing will be referred to as “primary” mixing. Primary mixing results in an ensemble of elementary volumes characterized by a set of microphysical and state parameters, i.e., Dv(μ), N(μ), RH(μ) and T(μ). Each of these parameters has a functional dependence on μ, and what is important, these parameters have functional relationships between each other.

In reality mixing is a continuous process. It does not stop after the primary mixing. The elementary volumes formed after primary mixing continue to progressively mix with each other.

The second stage of mixing will result in an ensemble of elementary volumes characterized by a set of parameters Dv(2), N(2), RH(2), T(2), etc. Here the superscript (2) indicates the stage of mixing. After the second stage the mixed volumes undergo subsequent stages of mixing.

The idealized conceptual diagram of the progressive mixing is shown in Fig. 8. Each stage of mixing results in an ensemble of elementary parcels formed for different μj(n). The elementary parcels continue mixing with each other and cloud environment during the subsequent mixing stages. As mentioned in Sect. 2.1, the actual process of mixing is indeed much more complex than the sequence of discrete events portrayed in Fig. 8. However, as it will be shown below, this simplified consideration of progressive mixing allows for the establishment of the main features of evolution of relationships between the microphysical moments affected by mixing. The obtained results facilitate identification of a type of mixing from in situ measurements.

Progressive homogeneous mixing was simulated with the help of a numerical model, where parcels were randomly mixed with each other and with the cloud environment. The mixing fraction μ was also set to be random during each mixing event. Models of stochastic mixing have been used in a number of studies (e.g., Krueger et al., 1997; Su et al., 1998; Burnet and Brenguier, 2007). In the present work the analysis of progressive mixing is expanded to examine its effect on the relationship between moments of the droplet size distribution.

The results of the progressive homogeneous mixing for the first four stages are presented in Fig. 9. As seen from Fig. 9 the functional relationship between the pairs of microphysical and state parameters exists only for the primary stage. For higher mixing stages these functional relationships break down. Thus, cloud volumes with the same N(2) may have different Dv(2). Figure 9 also shows that the regions of scattering of q(N), β(N) and Dv(N) for stages 2, 3 and 4 are limited from above by the inhomogeneous mixing (red dashed lines) and from below by primary homogeneous mixing (red solid lines).

Figure 10 presents a conceptual N-q diagram explaining breaking the functional relationships during progressive homogeneous mixing. After the first stage of mixing the N-q points will be scattered along the line OB and point C. The line OB corresponds to the ensemble of points with RH = 1. Therefore, the result of mixing between two saturated volumes randomly selected on AB, will remain on the same line. Point C corresponds to the ensemble of points with N=0, RH2≤RHCμ1≤1, where 0≤μ1<μcr. Therefore, mixing between point A (Fig. 10) and point C, when RH = 1 will result in scattering along the line AC (degenerate case). Points resulted from mixing between A (RH = 1) and point C when RH2≤RHC<1, will scattered over the ensemble of dashed lines shown in Fig. 10. These lines will fill the sector CAB. Random mixing between points on the line OB and C will eventually fill the entire sector COB. The same consideration can be applied to progressive mixing between other moments.

During the progressive mixing N(n), β(n), q(n) and Dv(n) formed in the elementary parcels tend to approach those in the undiluted cloud, i.e., N1, β1, q1 and Dv1. This process can be considered as a surrogate to the diffusion process between the cloud and sub-saturated out-of-cloud environment. The convergence of βn, q(n) and Dv(n) during the progressive mixing can be seen in Fig. 9, where the scattering of normalized qn(N), βnN and Dv(n)(N) becomes denser towards the top-right corner (1,1) with the increase of the stage of mixing.

It is worth noting that progressive mixing with the dry air does not break the functional relationships between the moments. This case is equivalent to detrainment of cloudy environment into dry air. It can be shown that Eq. (14) remain valid at any stage of progressive homogeneous mixing with dry air only, i.e., Nj/N1=μ(1)⋯μ(n-1)μ(n), where μ(n) is the mixing fraction at the nth stage of mixing. Equations (15)–(18) also remains valid for the progressive mixing with the dry air only.

As follows from Eq. (8) for the case of extreme inhomogeneous mixing the progressive mixing does not affect the functional relations between N(n), βn, q(n) and Dv(n) and other microphysical parameters. These relations remain the same regardless of the actual stage of mixing. This is one of the fundamental differences between homogeneous and extreme inhomogeneous mixing, which can be used for identification of type of mixing from in situ measurements.

Figure 11 shows modeled droplet size distributions averaged over the ensembles of elementary volumes corresponding to the first four stages of homogeneous mixing. As seen from Fig. 11a–d for the case with T1=T2 the droplet size distributions are broadened towards small sizes. Depending on the stage of mixing and mixing fraction μ the size distributions formed in each elementary volume may be unimodal or multimodal. However, due to the random nature of the modal sizes formed during mixing, the average size distributions become smooth and unimodal (Fig. 11a–d).

Broadening of droplet size distributions towards small sizes during homogeneous mixing is well known and it was demonstrated in a number of studies (e.g., Baker and Latham, 1982; Jensen and Baker, 1989; Jeffery, 2007; Kumar et al., 2013). However, if mixing results in supersaturation (Sect. 3.4), then the droplet size distribution may broaden towards larger sizes (Fig. 11e–h). For this to occur, both the temperature difference between the cloud and the environment T1-T2 and the relative humidity of the environment RH2 must be sufficiently large. Such conditions are inherently unstable, however, this might occur in regions that have been moistened through prior cloud detrainment. Thus homogeneous mixing may result in broadening of droplet size distributions towards either smaller or larger sizes (Fig. 11).

These results were obtained in the frame of the formalism of homogeneous and inhomogeneous mixing. The following two works in this series (Pinsky et al., 2016a, b) will discuss the broadening of polydisperse and monodisperse f(D) during both homogeneous and inhomogeneous mixing in greater detail.

Prior to proceeding with the analysis of in situ data we summarize the results of the previous consideration on how homogeneous and extreme inhomogeneous mixing is expected to manifest itself in relationships between basic microphysical parameters, such as N, β, q and Dv.

For extreme inhomogeneous mixing the relationship between the pairs of N, β and q are determined by linear dependencies Mn=αnmMm (Eq. 10) at any stage of mixing. As follows from Eq. (11) the slopes αnm for q(N), β(N) and q(β) are equal to the ratios q1/N1, β1/N1, and q1/β1, respectively, where N1, β1 and q1 correspond to undiluted adiabatic values. The values of N1, β1 and q1 may vary depending on the location inside the cloud and environmental conditions at the cloud base. Thus, the adiabatic value of q1 is a function of elevation above the cloud base ΔZ, whereas N1 depends on the vertical velocity at the cloud base uz and the aerosol load. Therefore, the scattering of q-N points will be aligned along an ensemble of different lines determined by q1/N1, which are specific to different cloud volumes. The conceptual diagram of the scattering of q-N measurements in a cloud with extreme inhomogeneous mixing is shown in Fig. 12a. The scatter diagrams for other moments (e.g., q-β, N-β) will have similar patterns as that in Fig. 12a.

For the case of homogeneous mixing the functional relationship between the pairs of N, β, q and Dv are disrupted by a progressive mixing. As shown in Sect. 4.1 the ensemble of points of N, β and q will be scattered within a sector, which is limited by lines determined by Eq. (11) (extreme inhomogeneous mixing) and Eqs. (15)–(17) (primary homogeneous), respectively (Fig. 9). What is important is that the top of the sectors for q(N) and β(N) correspond to adiabatic points [N1,q1] and [N1,β1], respectively. Since N1, β1 and q1 may vary within the same cloud, it is anticipated that the N, β and q measurements will be scattered within an ensemble of sectors as shown in Fig. 12b.

It is important to note that during the first stage of homogeneous mixing prior to reaching equilibrium, functional relationships between the microphysical moments do not exist either. After the instant mixing of cloud fraction μ with entrained air (Fig. 1b2), qm0=μq0 and Nm0=μN0. This state corresponds to point D in Fig. 10. After that droplets start evaporating until liquid mixing ratio reaches point A (Fig. 10), which corresponds to the equilibrium state (RH = 1). Therefore, during evaporation time q-N points will be scattered along the line AD. Since, point D can be located anywhere on OC, the ensemble of q-N points corresponding to non-equilibrium state will fill the COB area.

Thus, the absence of the functional relationships between the moments during homogeneous mixing may occur both during progressive mixing and during primary mixing prior to reaching the equilibrium state. The evaporation time required to reach equilibrium during homogeneous mixing is discussed in detail in Pinsky et al. (2016b), and it is usually limited by a few tens of seconds. However, progressive mixing is not limited in time. Therefore, it is very likely that no functional relationship between microphysical parameters will be observed within a set of in situ measurements.

Figure 12 demonstrated a fundamental difference in scattering of q-N for homogeneous and extreme inhomogeneous mixing, which will be used to facilitate identification of type of mixing in the following section.

The measurements were obtained on the University of Wyoming King Air aircraft during the COPE-MED project in South-Western part of the UK during July–August 2013 (Leon et al., 2016). The UW King Air was equipped with a suite of microphysical instruments, including a DMT Cloud Droplet probe (CDP), designed for measurements of droplet sizes and their concentrations in the nominal size ranges 1–50 µm.

Figure 13 shows a time series of droplet concentration, extinction coefficient, liquid water content (LWC) and mean volume droplet diameter measured by the CDP during transit through a convective cell on 18 July 2013. The CDP data were sampled at 10 Hz, which corresponds to approximately 10 m spatial averaging. Visual examination of the spatial changes of N, β and LWC shows strong correlation. The amplitude of changes of these parameters reaches nearly 100 percent with respect to their maximum. Contrary to that, the spatial variations of D¯ and Dv are quite conservative and their values remain nearly constant. With the exception of two cloud holes between 13:50:42 and 13:50:44, the amplitude of fluctuations of Dv does not exceed 8 % with standard deviation of 2.2 %.

Figure 14 shows scatter diagrams of LWC(N), β(N), LWC(β) and Dv(N) measured by the CDP during seven consecutive penetrations of the same convective cell extended over a period of approximately 19 min. One of these penetrations is shown in Fig. 13. The measurements were conducted at H=5500 m and T=-12 ∘C. The relative humidity of the ambient air was approximately 20 %. At the beginning of the sampling no precipitation size particles were observed in the cloud. However, by the end of the sampling period some raindrops and ice crystals were present in the cloud. Despite the presence of some precipitation size particles, the scatter diagrams in Fig. 14a, b and d demonstrate high correlation between pairs N, β and LWC. The mean volume diameter in Fig. 14c shows very little changes from 19 to 17 µm, when concentration changes from 1100 to 500 cm-3. However, for N<200 cm-3, the volume diameter decreases to 12–15 µm.

Red lines in Fig. 14 indicate q(N), β(N), LWC(β) and Dv(N) calculated for the 1st stage of homogeneous mixing. The calculations were performed for a monodisperse f(D) with D1=18.5 µm, N1=1100 cm-3, and state parameters as during the measurements. Comparisons of dependences q(N), β(N), LWC(β) and Dv(N) based on in situ measurements with those obtained from numerical simulations of homogeneous mixing show minor difference for high concentrations 700 cm-3<N<1100 cm-3 (Fig. 14a–c). The simulation also shows that for this specific case the differences between homogeneous and inhomogeneous mixing does not exceed 10 % when 700 cm-3<N<1100 cm-3. Such difference remains within the errors of measurements. Therefore, in this specific cloud for the regions with N>700 cm-3 the type of mixing cannot be unambiguously identified from the analysis of the dependences LWC(N), β(N), LWC(β) and Dv(N). This is consistent with the assessment of feasibility of segregation of homogeneous and inhomogeneous mixing in Fig. 3 (dashed line). Since for homogeneous mixing μ=NN1 (Eq. 14), then assuming N1=1100 cm-3, Fig. 3 suggests good separation of the moments for N<700 cm-3.

For the regions with N<500 cm-3 the deviation between homogeneous mixing simulations and in situ measurements in Fig. 14 becomes well pronounced, and it extends beyond possible errors of measurements. This suggests that the mixing in these regions is dominated by the extreme inhomogeneous type.

Figure 15 shows the same type of diagrams as in Fig. 14, which were measured during 45 consecutive traverses through an ensemble of deep convective cells. The sampling altitude varied in the range 3000 m < H < 4500 m, temperature -11 ∘C < T < 0 ∘C, relative humidity in the vicinity of clouds 15 % < RH < 65 %. The cloud measurements were extended over a period of 2 h 13 min, which is suggestive that the convective cells were sampled at different stages of their lifetime. At the sampling level the concentration of raindrops varied from zero to a few per liter, and their diameter did not exceed 2 mm.

What is interesting is that the scattering of LWC(N), β(N) and LWC(β) (Fig. 15a, b and d) is limited by the sector, which originates from the zero point as in Fig. 12a. Analysis of the measurements showed that the data points LWC(N), β(N), LWC(β) in each individual cloud traverse are well aligned along the lines with different slopes (e.g., Fig. 14). After averaging over the ensemble of clouds, the area of the scattered points turned out to be located inside a sector limited by the lines with smallest and largest slopes.

Comparisons of the scatter diagrams LWC(N), β(N) and LWC(β) in Figs. 14 and 15 with the conceptual diagrams in Fig. 12 unambiguously suggest that interaction between cloud and environment in the studied clouds was dominated by inhomogeneous mixing. It should be emphasized that analysis of a stand-alone mixing diagram N-Dv would not allow one to unambiguously draw such a conclusion.