Theoretical study of mixing in liquid clouds . Part 1 : classical concepts

13 The present study considers final stages of in-cloud mixing in the framework of classical 14 concept of homogeneous and extreme inhomogeneous mixing. Simple analytical relationships 15 between basic microphysical parameters were obtained for homogeneous and extreme 16 inhomogeneous mixing based on the adiabatic consideration. It was demonstrated that during 17 homogeneous mixing the functional relationships between the moments of the droplets size 18 distribution hold only during primary stage of mixing. Subsequent random mixing between already 19 mixed parcels and undiluted cloud parcels breaks these relationships. However, during extreme 20 inhomogeneous mixing the functional relationships between the microphysical parameters hold 21 both for primary and subsequent mixing. The obtained relationships can be used to identify the 22 type of mixing from in situ observations. The effectiveness of the developed method was 23 demonstrated using in-situ data collected in convective clouds. It was found that for the specific set 24 of in-situ measurements the interaction between cloudy and entrained environments was dominated 25 by extreme inhomogeneous mixing. 26 27 28 29 30 31

2 present novel techniques of investigating the effect of mixing both from a theoretical standpoint and through in-situ observations. Second, in contrast to the reviewer, we support the common practice of using idealized models of complex cloud processes, in order to investigate physical mechanisms without being bogged down by the multitude of other processes involved. Idealized considerations (e.g. adiabatic assumptions) are widely used in cloud physics as well as in physics in general. The assumptions are clearly articulated at the beginning of each paper in order to let a reader judge about the level of idealization of the utilized approaches.
Third, as regards to novelty, the following new results have been obtained: a) The first paper suggests a new technique for identifying type of mixing (homogeneous or inhomogeneous) based of the analysis of the moments of droplet size distributions. It was shown that homogeneous mixing breaks functional relationships between the moments. Nothing like that has been done before. A novel approach for identifying mixing from in-situ observations was proposed. The comments obtained by the authors from their colleagues showed that the proposed technique start to be utilized by other research groups.
b) The second paper considers homogeneous mixing. One of the important finding of this paper is an analytical universal solution describing the rate of evolution microphysical parameters as well as the final equilibrium state (mixing diagram). It is shown that in case of polydisperse droplet size distributions evolution of droplet spectra can lead to increase in characteristic size of droplets in contrast to widely accepted "classical" view, when the characteristic droplet size is decreasing. It was shown that evaporation time can be expressed in terms of time of phase relaxation. This is important for definition of reaction time in Damkoller number.
c) The third paper is dedicated to inhomogeneous mixing. A theoretical framework for a time dependent mixing of two volumes that accompanies by cloud droplet evaporation is developed. A new turbulence-evaporation model of time evolution of ensemble of droplets under different environmental parameters is proposed. In contrast to previous studies the Damkoller number is introduced as a result of re-normalization of mixing-evaporation equation, rather than empirically. It is shown that any mixing leads to droplet spectrum broadening. For the first time the scientifically grounded demarcation between homogeneous and inhomogeneous mixing in the space of environmental parameters is performed.
The authors regret that Referee 1 overlooked all these novelties. The authors also believe it is impossible to follow the recommendation of Referee 1, to combine all papers into one single, summary paper. While the papers all consider the same subject, they perform completely different functions with regard to investigating the issues of mixing.

Comments:
A small technical comment: I think the terminology the papers use is not correct. The limiting cases should be referred to as homogeneous and extremely inhomogeneous mixing. Everything between the two is the inhomogeneous mixing. 6 A brief description of the model is provided at the beginning of section 3 "The simulations have been performed with the help of a parcel model similar to that in Korolev (1995). The ensemble of droplets in the simulation was assumed to be monodisperse. For the case of extreme inhomogeneous mixing the amount of evaporated water ∆ required to saturate the mixed volume was calculated first. If ∆ < 1 , then the concentration of evaporated droplets was calculated as . Then, the concentration of the remaining droplets = 1 − was recalculated based of the calculation on the volume formed after mixing. If ∆ ≥ 1 , then all droplets evaporate, and = 0. For the case of homogeneous mixing in the first step the engulfed parcel instantly mixes with the cloud parcel resulting in a new humidity 0 , temperature 0 and volume 0 . After that the droplets start evaporating until either their complete evaporation or saturation over liquid is reached. The calculations stopped when, either < 0.2μm or ( − )/ < 0.001, respectively." This description is sufficient for cloud physicist to reproduce the results in section 3.
Section 3 were shortened and rearranged (see revised manuscript with ALL marked-ups). Figs.4-6 were converted into one figure as proposed by Reviewer. 11. Section 3.5 is perhaps a good start to a follow-up investigation. At the moment, it does not belong to this paper.
Reply: This section 3.5 was turned into section 4 in the revised manuscript. This section has a direct link to the subject of the paper, which might not be well articulated in the original text. The text of section 4 underwent significant modification to make it more clear. The purpose of this section is to demonstrate a breakup of functional relationships between the microphysical moments during progressive homogeneous mixing. This has a direct link to the subject of the paper, i.e. how microphysical moments are related to each other. A physical explanation of this phenomenon is also provided in the section 4 (new Fig.10). The results of this section help interpretation of in-situ observation (conceptual diagram in Fig.12) and explain broad scattering of data points in case of homogeneous mixing. This is specifically relevant to the past studies of mixing from in-situ observations. 12. Section 3.7. This is really not a summary.
Reply: The tittle of this section was renamed to "Expected relationships between the moments" (line 492). The text of the former section 3.7 was rewritten and moved into Sections 5.1 in the revised manuscript.
13. Section 4 is long and does not bring anything new in my view. What is the point of having it here? I was not able to follow detailed discussion in section 4.1 and references to the specific figures. Section 4.2 can be omitted. I question the link between in-cloud observations and the results of theoretical analysis that the previous sections provide.
Reply: Section 4 ("In-situ observations") in the original manuscript is changed to Section 5 in the revised manuscript. This section demonstrates how the results obtained in sections 2, 3 and 4 can be utilized for identification of mixing type from in-situ observations. This is a logical continuation of the theoretical study started at the beginning of the manuscript, which 7 ended by demonstration of its application to cloud measurements. The novel results in this section are: (1) the scattering diagrams of homogeneous and inhomogeneous mixing in Fig.12; (2) demonstration of utilizing the new approach for identification of type of mixing. Most of the previous studies to identify homogeneous mixing were based on the comparisons of measurements with the − calculated for the first stage of mixing. Such attempts have a limited success and in many ways may be misleading. This section demonstrates utilization of other moments, which makes identification of type of mixing more robust.
Section 4.2 was shortened and some of it parts moved to section 6 "Discussion". It bring up a warning that utilization of the developed approach for identification of type of mixing has limited capability and that it should not be blindly applied to a random cloud. 9 Reply: The authors shortened several pages of the text in order to reduce the size of the manuscript and make it concise. A number of cross references were added in all three parts in order to link them together. As it is seen now, part 1 is closely related to part 2 and it uses the same approach. Part 3 utilizes the results of part 2. The first part uses experimental data to demonstrate the how the theoretical outcomes could be verified from in-situ measurement. In our opinion such comparisons with experimental results are natural, and if it is not there, it probably might be requested by reviewers.
We also checked Jeffery's works on mixing. However, no discussions of the effect of mixing on the DSD second moment were found. We appreciate, if this reference could be provided.
2. After a long preliminary discussion, the most important paragraph in the introduction is on page 30214 starting at Line 26: "Besides the effect on N and r the type of mixing is anticipated to manifest itself in relationships between other moments of the droplet size distribution…" It should be further explained in that paragraph why it is valuable to analyze different moments. Are they expected to be more insightful than the traditional mixing diagram methodology; is it making applications of mixing to other fields clearer; etc?
Reply: The paragraph explaining importance of the effect of mixing on the DSD moments was added in the introduction following the Reviewer's comment (lines 102-106): "It is shown that the newly obtained relationships between the moments provide a more robust identification of type of mixing from in-situ measurements as compared to conventional − 3 relationships used in mixing diagrams. Relationships between moments may be useful for parameterization of mixing in numerical simulations of clouds and climate, interpretations of remote sensing measurements." 3. In Fig. 9 and after, a multiple-step mixing process is envisioned. The approach is to consider mixing between a cloud and the dry environment, and then to consider subsequent mixing events between that parcel and the cloud again. Why did the authors choose to take this view instead of considering a cloud parcel progressively mixed with clear air? Some motivation for that choice is needed and some discussion of how the results would be expected to differ. For example, if one were to focus on the dry air first, dots should be concentrated at lower end in Figure 10.
Reply: The modeling of the progressive mixing presented in the paper corresponds to the case when the entrained dry air is interacting with the cloudy environment. The final state of this interaction is a diluted cloud. The progressive mixing of the cloud environment with the environmental dry air corresponds to detrainment, which ultimate state is dry cloud free air. It can be show that during detrainment the relationships between moments will be the same as during primary mixing. The authors consider that the case of detrainment is less interesting, and left it outside the frame of the manuscript in order to keep it concise. However, following the reviewers suggestion a paragraph was added in the revise manuscript in order to explain the motivation of our choice (lines 455-460): "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.
where ( ) is the mixing fraction at the -th stage of mixing. Eqs. (15)-(24) also remain valid for the progressive mixing with the dry air only. " 4. There are many mistakes in the paper, including errors in the equations, at least according to the derivations as I am able to follow them. Again, the physics is difficult enough by itself, without having to make corrections. Please thoroughly check all results and the typesetting.
Reply: The authors highly appreciate the Reviewers efforts to improve our msnuscript and pointing out numerous typos. All specific comments listed below were addressed and the text of the manuscript was thoroughly checked.

Specific comments
1. Eq. 1, page 30218: As monodisperse cloud droplets are used in this part of the study, the droplet size distribution f(r) will confuse people. Especially Equations 2 and 3 only work for monodisperse droplets theoretically. Please explain and be consistent.
Reply: The relationships between moments are valid for relatively narrow polydisperse droplet size distributions. However, the modeling was performed for monodisperse size distributions. The confusion about assumption of monodisperse droplets during deriving relationships between the moments is probably coming from mentioning monodisperse size distributions in section 2.2. The statement about the assumption of monodispesity was removed from section 2.2 to avoid confusion.
3. It is difficult to connect Eq. 8 to Eq. 5. How do you prove Eq. 5 is (1-) Eq.8, when T1=T2=Tmo? Reply: The term (1-) appears as a result of expansion in series. Appendix B was added to clarify the derivation of this equation.
5. Line 6, page 30220: The neglect of latent heat is a strong assumption that removes possible important factors such as negative buoyancy production. It is valid in the range specified by the authors, but the limitation should be discussed. Does it restrict the results to certain environments or cloud types (e.g., shallow convection)? Reply: If fact the latent heat was accounted during derivation of Eq.3 (old Eq.8) (see Eq.A7 in Appendix A). The confusion regarding disregarding the latent heat is coming from inaccurate statement on page 30220 as indicated by Reviewer. The original purpose of this statement was to indicate that the temperature is included as a coefficient and it remains constant. In order to address the Reviewer's concern the calculation of temperature during mixing was added in the text (line 217-219): "The temperature at the final stage of mixing can be estimated as (appendix C) In order to demonstrate that * and allow accurate depiction of the temperature depression during mixing-evaporation process, the air temperature formed after mixing calculated from Eq. 6a,b was compared with the modelled temperature in Figs. 4h and 6h.
Reply: Corrected: line 208 in the revised manuscript. 7. Line 13, page 30220: missing space between "on" and "delta_q" Reply: This sentence was deleted in the revised manuscript.
8. Line 17, page 30220: the volume change due to temperature change should not affect liquid water mixing ratio, because it's connected to mass not volume as mentioned in point 4.
Reply: This paragraph was deleted.
Reply: The prefactor was corrected in the revised manuscript in Eq.(4) (former Eq.8).
12. Eq. 16: I believe the exponent should be -1/3, and inside the parentheses should be N_0/N. Reply: Corrected: Eq.18 in the revised manuscript.
14. Fig. 3: it looks like panels a and b are mixed up. Also the caption refers to liquid water mixing ratio but the axis label states LWC; needs to be consistent. Figure 3 labeling was corrected as per Reviewer comment. 15. Figs. 3 and 4: should use same format for S through the whole paper (e.g. 20% as in Fig.4 or 0.2 as in Fig. 3) Reply: Corrected. In the revised manuscript is replaced by in order to address the earlier Reviewer's comment regarding consistency of notations with part 2 and 3.

Reply:
is determined as a saturation ratio ans the units were adjusted throughout the text.  Figure A1 in the revised manuscript.
19. Fig. 7: why changes from r0=10um ( Fig. 4,5,6) to r0=5 um. And also changes the S from 50% to 90%? Reply: The sizes 10m and 5m were selected to demonstrate mixing for the cases 1 = 2 and 1 ≠ 2 in a most pronounced way. For the case RH2=50% no supersaturation will be formed. Positive supersaturation may occur only at RH2>80% and T<15C. Larger T seems to be uncommon for the tropospheric clouds.
20. Fig. 8 Fig.7 due to it proximity to point (1,1). In order to clarify this issue the following text was added (lines 398-401): "However, no activation of new droplets during isobaric mixing was allowed in this study. For the cases when 0 > 1 (Fig.  7, on line 1) the condensed water was uniformly distributed between available droplets. Therefore, ( ), ( ) and ( ) calculated for homogeneous and extremely inhomogeneous mixing coincide with each other on this interval.". 21. Line 5, page 30228: in Fig. 8, Delta_T is negative, here it's positive.
Reply: The associated sentence was deleted in the revised manuscript.
22. Line16, page 30228: could you explain why "the effect is more pronounced when T1>T2 compared with T1<T2." Reply: When the entrained air is colder (T1>T2), it results in additional condensation of the cloudy air due to its cooling compared to the case when the dry air is warmer (T1<T2). This 13 statement is supported by the results of numerical simulations. This explanation was not included in the text for the sake of conciseness.
23. Line 27, page 30229: "becomes denser towards the top right corner" Is it because the mixed volume is mixed with cloud volume, not environmental volume? Reply: Yes. The mixing with the cloud environment results in approaching of the properties of mixing environment to the cloud properties. Eventually the entrained air is dissolved in the cloudy environment. Again, for the sake of brevity we did not expand this explanation in the manuscript.
24. Fig. 11: why use r0=5 um, not 10 um. It's better to use the same radius through the paper, except you want to do the sensitivity test. Reply: During the paper preparation the authors tried different r0. Unfortunately is does not work well for the same r0. Different r0 (5m and 10m) were used in order to demonstrate the most pronounced effect of mixing on microstructure. A relevant comment was embedded in the text to address this issue.
Reply: This is a good question. It was debated over years: how the averaging scale affects identification of the type of mixing, i.e. homogeneous versus inhomogeneous? The single instrument approach used in this and the majority of previous studies does not allow judgement about type of mixing at scales smaller than the averaging scale Lav. In part 2 it was shown that for typical cloud environmental conditions the upper spatial scale of homogeneous mixing is limited by few m. Inhomogeneous mixing depending on the conditions may cover a wide range of scales from cm to km. A discussion of spatial scales of homogeneous and inhomogeneous mixing is provided in parts 2 and 3. Another question related to in-situ observations is whether the mixing reached equilibrium state at the moment of measurement.
30. Fig. 14a: y axis unit (g/m3) not (km-1) Reply: The y-axis label in Fig.14a the mixing ratio of liquid water required to evaporate in order to saturate 1kg of the cloud volume formed after mixing with the entrained air, but before droplet start evaporating. Here 43. Line 14, page 30244: "is hold" should be "holds"?
Reply: Corrected following the reviewer comment: line 708 in the revised manuscript 15 44. Line 15, page 30244: Figure B1 is Figure 17. Reply: Figure A1 numbering was corrected to address the Reviewer comment 45. All marked up modifications can be viewed from a separately submitted file. into the cloudy environment (Fig. 1a1). Then, the droplets at the interface of the sub-saturated parcel 125 and the cloud environment undergo complete evaporation until the air within the engulfed volume 126 reaches saturation (Fig. 1a2). After that the saturated but droplet free parcel mixes with the rest of 127 the cloud environment (Fig. 1a3). The result of inhomogeneous mixing is that the cloud parcel has 128 reduced droplet concentration and the droplet sizes remain unchanged. 129 In the case of homogeneous mixing after entraining into a cloud (Fig. 1b1), the subsaturated 130 parcel "instantly" mixes up with its cloud environment (Fig. 1b2)  to dilution by the mixed droplet free sub-saturated parcel. 141 The following discussion will be specifically focused on the microphysical properties formed The foregoing discussion will be focused on mixing between saturated cloud parcels and out-168 of-cloud sub-saturated air. The cloud parcel contains droplets with average diameter ̅ 1 , liquid 169 mixing ratio 1 and number concentration 1 . The initial temperature in the cloud parcel is 1 , (the explanation of variable notations is provided 171 in Table 1). The second parcel is droplet free ( 2 = 0), sub-saturated with initial relative humidity is the nth moment of ( ). Therefore, it is anticipated that 249 for extreme inhomogeneous mixing droplet number concentration N (0th moment), extinction 250 coefficient (2nd moment), liquid water mixing ratio q (3rd moment), along with other moments, 251 will correlate with each other, i.e. One of the consequences of Eqs. (9)-(11) is that the characteristic droplet sizes ̅ , 2 , , eff 254 will remain constant during inhomogeneous mixing. 255 For the case 1 = 2 and > Eqs. (5) and (11) yield the dependence of vs.  The range of in is limited by < ≤ 1, so that 0 < 1− ≤ 1 * . This gives the range of 302 changes of , i.e. 0 ≤ ≤ 1 for the mixing without complete evaporation of droplets. The 303 degenerate case corresponds to → 0, whereas → 1 corresponds to maximum difference of the 304 moments for homogeneous and extremely inhomogeneous mixing. 305 As follows from Eqs. (4) and (23)   Since the amount of the evaporated liquid water does not depend on the type of mixing, the 343 dependences of ( ) are the same for both homogeneous and inhomogeneous mixing (Fig.4a). The 344 type of mixing has the most pronounced effect on the droplet concentration (Fig.4b) and droplet 345 sizes (Fig.4e). 346 Figure 4g shows the dependences 0 and vs. , Here 0 is the relative humidity at 347 the initial stage of homogeneous mixing before droplets start evaporating (Fig. 1b2).  activation of interstitial CCN, which may increase and decrease (Korolev and Isaac, 2000). 397 However, no activation of new droplets during isobaric mixing was allowed in this study. For the 398 cases when 0 > 1 (Fig. 7, on line 1) the condensed water was uniformly distributed 399 between available droplets. Therefore, ( ), ( ) and ( ) calculated for homogeneous and 400 extremely inhomogeneous mixing coincide with each other on this interval. 401 Numerical simulations also showed, that the effect of temperature on mixing is more 402 pronounced for the cases when the cloud temperature is warmer than that of the entrained air, i.e. After the second stage the mixed volumes undergo subsequent stages of mixing. 418 The idealised conceptual diagram of the progressive mixing is shown in Fig. 8. As mentioned 419 in Sect. 2.1, the actual process of mixing is indeed much more complex than the sequence of discrete 420 events portrayed in Fig.8. However, as it will be shown below, this simplified consideration of and q will be scattered within a sector, which is limited by lines determined by Eq. (11) (extreme 510 inhomogeneous mixing) and Eqs. (15)-(17) (primary homogeneous), respectively (Fig. 9). What is 511 important, is that the top of the sectors for ( ) and ( ) correspond to points [ 1 , 1 ] and [ 1 , 1 ], 512 respectively. Since 1 , 1 and 1 may vary within the same cloud, it is anticipated that the , and 513 q measurements will be scattered within an ensemble of sectors as shown in Fig. 12b.
It is important to note that that during homogeneous mixing prior reaching equilibrium, 515 functional relationships between the microphysical moments do not exist either. After the instant 516 mixing of cloud fraction with entrained air (Fig. 1b(2)), 0 = 0 and 0 = 0 . This state 517 corresponds to point in Fig.10. After that droplets start evaporating until liquid mixing ratio 518 reaches point (Fig.10), which corresponds to the equilibrium state ( = 1). Therefore, during 519 evaporation time − points will be scattered along the line . Since, point can be located 520 anywhere on , the ensemble of − points corresponding to non-equilibrium state will fill the 521 area. previous studies, including this one, identification of type of mixing was based on the assumption 608 that the sampled cloud volume is in equilibrium state ( = 1), and that it reached the final stage 609 of mixing (Fig.1 a2, a3, b3). It is possible that at the moment of measurement the process of mixing 610 is not complete and the droplet free filaments remained undersaturated (Fig.1 a1, b1, b2). In this 611 case the relationship between different moments may be well described as  (Fig.12a). However, for homogeneous mixing the scattering data points will be limited by a 652 sector originating at ( 1 , 1 ) and ( 1 , 1 ) (Fig.12b). Utilizing a stand-alone conventional − 653 mixing diagram may not provide unambiguous answer about type of mixing. The objective of this section is to find the amount of liquid water, which is required to be 672 evaporated in order to saturate the parcel formed after mixing. Assume that 1 , 2 are the mixing 673 vapor ratios in the cloudy and entrained parcels, respectively, and 1 , 2 are their respective initial 674 temperatures. First, we find the saturation ratio 0 formed after instant mixing of the cloud and 675 entrained before the cloud droplets start evaporating. 676 The vapor mixing ratio formed in the mixed volume will be where ( 0 ) is the saturated vapor pressure at temperature 0 . 693 The process of evaporation is accompanied by changing humidity and temperature due to latent 694 heat of vaporization. This process is described by the Eq. (C2) in Korolev and Mazin (2003). 695 Assuming the process to be isobaric (i.e. vertical velocity = 0) and absence of ice ( = 0), 696 Eq. (C2) (Korolev and Mazin, 2003) yields   719 As follows from Eq.(A4) for the case 1 = 2 with high accuracy 0 = 1 = 2 . Therefore, 720 ( 0 ) = ( 1 ) = ( 2 ). Dividing Eq.(B1) by yields In most liquid clouds 1 = 1 (Korolev and Mazin 2003). Therefore, Eq.B2 turns into The expression under logarithm can be presented as the first two terms of the series expansion 727 of (1 + The energy conservation for evaporating droplets can be written as is the specific heat capacity of the moist air