What is Adiabatic Fraction in Cumulus Clouds: High-Resolution Simulations with Passive Tracer

The process of mixing in warm convective clouds and its effects on microphysics, is crucial for an accurate description of cloud fields, weather, and climate. Still, it remains an open question in the field of cloud physics. Adiabatic regions in the cloud could be considered as non-mixed areas and therefore serve as an important reference to mixing. Therefore, the adiabatic fraction (AF) is an important parameter that estimates the mixing level in the cloud in a simple way. Here, we test different methods of AF calculations using high-resolution (10 m) simulations of isolated warm Cumulus clouds. The calcu5 lated AFs are compared with a normalized concentration of a passive tracer, which is a measure of dilution by mixing. This comparison enables us to examine how well the AF parameter can determine mixing effects, and to estimate the accuracy of different approaches used to calculate it. The sensitivity of the calculated AF to the choice of different equations, vertical profiles, cloud base height, and its linearity with height are all tested. Moreover, the use of a detailed spectral bin microphysics scheme demonstrates that the accuracy of the saturation adjustment assumption depends on aerosol concentration, and leads to 10 an underestimation of AF in pristine environments.


Introduction
Warm convective clouds were found to have a major role in the high uncertainty that clouds exert on climate change research (Sherwood et al., 2014;Zelinka et al., 2020). Clouds' radiative forcing, defined as the change that anthropogenic aerosols impose on clouds' radiative properties and life-cycle (e.g. the aerosols indirect effect), is considered to be negative (i.e. cooling; 15 IPCC 2013; Boucher et al. 2013). On the other hand, the feedbacks of warm clouds on the changing climatic system were recently shown to be positive, due to reduction in cloud cover (Ceppi et al., 2017;Nuijens and Siebesma, 2019). A major drawback in understanding the effects of shallow convection on climate and their representation in models are the processes of entrainment and mixing. These processes have a major impact on cloud properties and hence on their radiative forcing and feedbacks. As an example, high aerosol loading conditions increase the number of droplets and their surface area to volume 20 ratio, which increases the diffusion efficiency. This increases the liquid water content in the core of the cloud (Albrecht, 1989) and the evaporation at the cloud's edge. Thus, the intensity of mixing plays an important role in the non-monotonic response of clouds to aerosol loading (Small et al., 2009;Dagan et al., 2017). Mixing also affects convection and its vertical fluxes, which are important for climate models (De Rooy et al., 2013). Mixing effects on microphysical cloud properties is still an open clouds are still under debate (Gerber, 2000;Khain et al., 2019). This has significant consequences for shallow clouds, since cloud top height, as well as microphysical cloud properties, depend on the existence/absence of an adiabatic core. Further, obtainment of the adiabaticity level is important for parameterizations of the vertical mass fluxes (De Rooy et al., 2013), and for remote sensing retrievals, in which the radiation transfer calculations depend on adiabatic microphysical profiles (Merk et al., 2016). Hence, the usage of a simple parameter that characterizes the mixing level can be very beneficial. The ideal 30 way to evaluate dilution by mixing is to use a passive tracer, which is a conservative variable in moist adiabatic processes, (i.e. does not change during evaporation/condensation). Sub-cloud tracers are preferable over natural conservative variables, such as total water mixing ratio or equivalent potential temperature, as they are absent from the clouds' surroundings. This eliminates the need for knowledge about the conservative variable's initial profile and assumptions on its mixing processes.
However, such fictitious tracers do not exist in in-situ measurements and remote sensing and are only being used in numerical 35 simulations, aiming for process-level understanding of mixing. The level of adiabaticity (i.e. deviation from a perfect adiabatic state) can also be a measure of mixing in cases where radiation and sedimentation are negligible, thus it is widely common to use adiabatic fraction (AF ) as a proxy for adiabaticity.
The AF is determined as: where LWC is the liquid water content (g/m 3 ) at a specific location, and LW C ad is the theoretical liquid water content that a parcel would have if it was lifted adiabatically from the cloud base to a specific height. The definition of LW C ad is not consistent in the literature; many studies define LW C ad using the moist adiabatic lapse rate as derived by Yau and Rogers (1996), with inherent saturation adjustments assumption (i.e., S(t, z) ≈ 0). This definition considers LW C ad as the maximal potential of LWC. This maximal value does not describe the true potential because it ignores the fact that the potential of LW C ad is 45 limited by the condensation efficiency. Saturation adjustment assumes that the total amount of water vapor that exceeds the concentration for saturation will condense instantaneously. Such assumption ignores the relaxation time for condensation that determines the condensation efficiency, and depends on the available surface area of the droplets. Cases of clouds with high supersaturation values can occur in clouds with low droplet concentrations (i.e. low surface area to volume ratio) or very strong updrafts. The various approaches for AF calculations differ in the way by which they calculate the reference LW C ad . The

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values of LW C ad can be obtained using parcel modeling or direct calculations, which can be performed in a bulk approach (e.g. using conservation of energy or water mass of all phases (Brenguier, 1991), or by using analytical thermodynamic considerations (Khain and Pinsky, 2018;Pontikis, 1996). The different methods are detailed in section 2.3.
AF is commonly used to study the effects of mixing on clouds' microphysical structure. Observations (Freud et al., 2008) and numerical modeling (Zhang et al., 2011) used AF to show the effects of mixing on the effective radius profile in cloud fields.

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Conditioning aircraft measurements of cumulus and stratiform clouds according to AF were used to examine the effects of mixing on the width of the droplets size distribution (DSD; Pawlowska et al. 2006;Pandithurai et al. 2012;Kim et al. 2008;Bera 2021. AF is also commonly used in mixing diagram analyses for determination of mixing types (Gerber et al., 2008;Schmeissner et al., 2015). Additionally, it is used to calibrate in-situ aircraft measurements (Brenguier et al., 2013). Some studies approximated AF by normalizing LWC by the maximal measured value at a given height (LW C max ; Bera 2021).

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However, some clouds may not contain adiabatic regions, or their adiabatic pockets may not be sampled, thus, the normalization of in-situ measurement data by the maximal value might lead to an overestimation of AF , as LW C max ≤LW C ad . In this study, we evaluated the accuracy of different methods and assumptions for AF estimation, by simulating several single warm cumulus clouds in high-resolution (10 m), using different aerosol concentrations. The dynamic model is coupled to a spectral bin microphysics model for explicit representation of the microphysical processes and the resulted supersaturation 65 field. The high resolution allows to solve the turbulent fluxes in more detail, and reduces the model dependence on sub-grid parameterizations, which improves mixing representation. Moreover, the small grid spacing enables better detection of local maxima in the 3D field (e.g. LWC, supersaturation, updraft Khain et al. 2004;Fan et al. 2009). SAM is a non-hydrostatic, inelastic model with cyclic boundary conditions in the horizontal direction. Sub-grid turbulence parameterization was performed using a 1.5-closure scheme. Analysis by Pinsky et al. (2021) 80 shows that turbulent motions in this design obey the − 5 3 law. To avoid the effects of the cloud on itself via the cyclic boundaries, we chose the domain size to be 5.12 km, which is much larger than the cloud scale (∼800 m diameter). The horizontal resolution was set to 10 m, and the vertical resolution to 10 m up to 3 km, and 50 m for the last kilometer (maximal cloud top is 2 km).
The time resolution was 0.5 seconds. Initial vertical profiles of water vapor mixing ratio and potential temperature (inversion at 1500-2000 m), and constant large-scale forcing and surface fluxes were taken from the BOMEX case study (Siebesma 85 et al., 2003). The horizontal background wind was set to zero and aerosols were distributed only below cloud base (600 m).
The cloud was simulated for one hour, and was initialized by a perturbation of 0.1 K in the center of the domain, with a horizontal radius of 500 m and a vertical radius of 100 m (from the surface). The perturbation decays to zero as a Cosine square function of 0 < x < π 2 from the center to the edge of the radius, and random noise is added (perturbation type 7 in SAM manual). The SBM is based on solving kinetic equations for size distribution functions of water drops and aerosol. Both size 90 distributions are defined on a doubling mass grid containing 33 bins. The drops radii range between 2 µm and 3.2 mm. The size of aerosols serving as cloud condensational nuclei (CCN) ranges between 0.005 and 2 µm. Following Jaenicke (1988) and Altaratz et al. (2008), the size distribution of the aerosols was represented by a sum of three log-normal distributions describing fine, accumulation, and coarse mode aerosols, typical for the maritime boundary layer. Three clouds with different aerosol concentrations (Na) were simulated, with Na= 5, 50, and 500 cm −3 .

Passive tracer setup
For quantification of the dilution level of the cloud, we used a passive tracer that disperses in space and time by advection and turbulent diffusion that is set according to the sub-grid scheme. The tracer is uniformly distributed in the sub-cloud layer, from the surface up to 600 m (mean cloud base). Throughout the simulation, the measured concentration is normalized by the sub cloud initial concentration; therefore, a concentration equal to unity indicates no dilution. Fig. A1 in the appendix shows three 100 snapshots: the tracer's initial spatial distribution, its distribution and values at the time of the cloud's maximal development (33 minutes), and at the end of the simulation, after 55 minutes.

Adiabatic fraction calculations
Although AF was used in many studies over the years, there are different methods for calculation of LW C ad , which are often not well defined in the literature (details of the calculations are often missing). The value of LW C ad can be calculated in 105 different ways that differ by method, assumptions, and practical implementations. In this section, we present three commonly used methods for AF calculation and the following assumptions that can be made.
The equation for supersaturation (S) for an adiabatically ascending parcel is given by Korolev and Mazin (2003): where w is the updraft velocity and A 1 and A 2 are thermodynamics parameters, which depend on temperature and water vapor mixing ratio that vary with altitude.
Eq. 2 is obtained by differentiating S = (e−es) es , and using a quasi-hydrostatic approximation that is valid for updrafts weaker than 10 m s . e is the water vapor partial pressure, e s is the saturated water vapor partial pressure over liquid, g is the gravity 115 acceleration, T is temperature, L w is the latent heat of evaporation, c p is the heat capacity of air under constant pressure, R v and R a are the gas constants of water vapor and dry air, respectively, and ρ v and ρ d are the density of water vapor and dry air, respectively.
The first term in the RHS of Eq. 2 is the source of S by adiabatic cooling, and the second term is the sink of S due to water vapor loss and latent heat release by condensation. Since Eq. 2 does not include effects of mixing, the value of LWC equals LW C ad . Considering the changes in S in the vertical direction only, and transforming the time domain to vertical coordinates using w leads to: And the LWC in an adiabatic parcel is: where z=0 at cloud base.
One can see that LW C ad is not only a function of z, but also depends on temperature and humidity (via parameters A 1 and A 2 ), as well as on vertical velocity and aerosols through the supersaturation term. When S << 1, Eq. 4 can be simplified to: Eq. 5 shows that in regions where S increases with height (e.g. near cloud base, or in pristine environments), LW C ad will be smaller than its maximal value, because some amount of water vapor in excess of supersaturation remains in the gas phase.
At the exception of these cases, the S term is small compared to the first term on the RHS of Eq. 5. Neglecting the term which includes the supersaturation, we can write: Taking A1 A2 as a constant leads to the well-known linear LW C ad profile, which is the first assumption to be examined in this study (section 3.2.1).
Two alternative approaches can be used to calculate LW C ad . One is using the total water-mixing ratio q t = q l + q v ( g kg ), which is a conservative value in moist adiabatic processes. q v is the water vapor mixing ratio and q l is the liquid water mixing 140 ratio. At cloud base q l = q l0 ≈ 0, and so q t0 = q v0 . For undiluted parcels q t0 = q t (z), and at any altitude above cloud base q t (z) = q l (z)+q v (z). Assuming saturation adjustment (i.e. S(z) = 0) means that q v = q vs , where q vs is the water vapor mixing ratio in saturation that can be calculated according to the Clausius-Clapeyron equation. LW C ad can then be defined using q l , as: The third approach is to use the conservation of moist static energy (h), where h = L w q l + c p T + gz. Differentiating h with respect to z, conserving it with height ( dh dz = 0) and multiplying by ρ d gives (Schmeissner et al., 2015): Eq. 8 shows that the difference between the lapse rate of an adiabatic parcel and the dry adiabatic lapse rate is due to condensation, and thus can be translated into LW C ad . We note here that this method avoids the use of saturation adjustments.

Comparison between the three methods
In this work, we solved the Lagrangian equations presented above from the outputs of the Eulerian model, using the assumption 155 that in a time scale of ∼ 5 minutes the thermodynamic profiles in the cloud are fixed during growth and mature stages. It implies that the profiles of temperature and humidity can be used to predict the conditions to which a parcel in the cloud base would be exposed as it ascends. The most accurate way to consider profiles of temperature (T(z)) and specific humidity (q v (z)) for LW C ad calculations is to obtain them from the undiluted core of the cloud, where q v is maximal and T is warmer due to release of latent heat. If there is a perfect undiluted adiabatic core, its AF value is equal to one, and it will coincide with the maximum 160 normalized value of the tracer (Tr). Fig. 1 shows cross-sections of Tr and the three different methods for calculating AF (see list below) when it reached its maximal height and mass (33 minutes). The sensitivity of the methods to the choice of profiles is tested in Fig. 1 by calculating each method twice, first with accurate "least diluted" profiles, and second, by approximating the adiabatic (undiluted) profiles, using the points with the highest updraft values at each level.
The methods are denoted as: 165 1. AF ref : calculated according to Eq. 6 using the in-cloud profiles of A 1 and A 2 . This method for AF calculation will be used as the reference method from herein (reasoning for this choice is provided below).
The accurate estimations of the adiabatic vertical profiles of T and q v were obtained here by averaging the values of those 170 parameters in the voxels containing the highest 1% Tr values at each altitude, and the results are presented in Fig. 1a-c. The cross-section of Tr is provided in Fig. 1d.
The vertical profiles of T and q v that were used in Fig. 1a-c (based on the simulated Tr) can also be calculated using the maximal values of LWC or updraft. It is hard to obtain these types of profiles from in-situ measurements that do not contain the theoretical tracer, nor the full 3D distribution of the cloud variables. Thus, for the sake of simplicity, and in order for the 175 methods presented in this paper to be comparable to measurements, we approximated the profiles by averaging the values in the voxels with the highest 5% updraft values at each altitude. This methodology was used to estimate the T and q v profiles throughout this study. It is shown that AF ref remains almost similar when using either the approximated or accurate profiles.
On the other hand, AF qt and AF dT dz exhibit some underestimations and overestimations compared to the accurate profiles, respectively. These differences are explained in detail below. Fig. 2 presents the differences between each AF method when 180 using the approximated profiles and the Tr values (as shown in Fig. 1d). The apparently good agreement between AF ref and Tr, as presented in Fig. 1e is more closely examined in Fig. 2a, where differences are detected. Close to cloud base, the AFs experience non-realistic, non-homogenous values due to the inhomogeneity of the cloud base. Moreover, the values of LWC and LW C ad near the cloud base are small, hence their ratio exhibits high sensitivity even when differences from the reference are minor (chosen according to the highest updraft). These differences are no longer observed around 100 m above cloud base.

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For the sake of comparison with Tr, the points near cloud base with AF>1 were set to one. Determination of the cloud base height was achieved using the vertical profile of the cloud horizontal cross-section area. All clouds exhibited a local maximum in the cross-section area around 600 m during their growing stage. Aiming to choose the cloud base as a level that can represent the cloud with "enough" cloudy voxels, we chose to define it as the height above the level of initial condensation, in which the area covers 90% of the local maximal area. Changing the criteria threshold from 90% to 33% can decrease the cloud base height by up to 30 m. This definition, using the 90% criteria, was found to be stable for all simulations during the growing stage of the clouds, and is considered optimal for AF calculations since it maintains an optimal agreement between AF ref and Tr in the regions of high values. When comparing the cross-sections of Tr with each AF presented in Fig. 1 (more than 100 m above cloud base), one can see that the AFs decrease toward the cloud edge faster than Tr (Tr>AF; see Fig. 2a). This is because AF is also affected by evaporation, and not only dilution, as in the case of Tr (i.e. mechanical mixing). The opposite is observed in 195 higher levels, at slightly diluted regions, where T r < AF ref < 1. These regions represent a more complex difference between AF and Tr, which is also caused by condensation and evaporation. Tr can change only due to mechanical mixing and hence, is almost a one-directional process; once the parcel is diluted, it has low probability to restore its initial Tr concentration.
This means that Tr has a memory of the mixing history, unlike AF that has a source/sink process. A parcel can regain liquid water after a mixing event, if supersaturation is reached again at a later time. This means that a parcel in the margins of the diluted, but then the flow pattern of the toroidal vortex inside the cloud forces them upward; Hence, they cool and re-condense some of the water they lost. The phenomenon of rapid growth of droplets in an updraft following an entrainment event was suggested as a mechanism for rain initiation (Baker et al., 1980;Yang et al., 2016). Correlation of the blue regions (where AF>Tr) with strong updrafts (as part of the toroidal vortex) was found for different time-steps and different cloud simulations. with AF ref is observed, although AF dT dz with the approximated profiles is slightly larger in the sub-adiabatic regions at higher levels (i.e. having smaller values of LW C ad (z)). This is explained by the fact that AF dT dz considers the difference between dT dz and the dry lapse rate as a consequence of condensation, and uses it to define LW C ad . Diluted parcels are colder than the adiabatic core because they were mixed with colder environmental air, and may have experienced evaporation. This difference between absolutely adiabatic and slightly diluted parcels increases with height, as the parcel is aging. For these 215 reasons, using diluted voxels to estimate the adiabatic profiles will lead to a larger temperature gradient (more negative) that is closer to the dry lapse rate, falsely inferring less condensation, and biasing LW C ad toward smaller values. The arguments above explain the difference between Fig. 1a-b where AF dT dz ≈ AF ref , and Fig. 1e-f, where AF dT dz >AF ref . These findings suggest that AF ref is less sensitive to the choice of adiabatic profiles, because it is constrained by two free parameters (T and q v ), rather than only T. The final method, AF qt , is shown to be very sensitive to the choice of profiles, as presented in Figs. 1c,g.

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The deviation from Tr in Fig. 2c demonstrates a substantial underestimation of AF qt , due to a similar argument as discussed above for AF dT dz . Using a slightly diluted parcel, with a smaller q v or q vs compared to the core, is expected to falsely infer more condensation and larger LW C ad . The bias is stronger for this method because it depends on q v (or its estimation as q vs (T )). The simplicity of this method is also its downfall; since q v is an order of magnitude larger than ql (liquid mixing Figure 2. Cross-sections of the differences between the various AF methods and the tracer. Vertical cross-sections for the differences between methods using approximated profiles ( Fig. 1e-g) and the tracer (Fig. 1d). (a) Difference between AFref and Tr ( Fig. 1e minus Fig.   1d). (b) Same as a, for AF dtdz . (c) Same as a, for AFqt. ratio), small errors can cause significant effects when estimating LW C ad using only q v . Another disadvantage of AF qt is that 225 it is commonly used with the saturation adjustments assumption (i.e. S ≈ 0), by estimating q v to be q vs . This assumption can lead to underestimation of AF qt in conditions of pristine environment (low aerosol concentrations) as explained in detail in section 3.2.3. The results of this section suggest that the analytical solution for AF ref using Eq. 5 is a more accurate and stable method to calculate AF, as it shows similarity to Tr and robustness for different choices of vertical profiles. We also note that profiles of T and q v are often obtained from the environment (see section 3.2.2). This will lead to substantial errors when using 230 AF dT dz or AF qt , which showed sensitivity to the choice of profiles. Furthermore, Eq. 5 allows isolating different assumptions such as linearity, with height and saturation adjustments.

Testing the effects of assumptions made when calculating AF
Next, we examine the effects of several commonly used assumptions on AF ref calculation, in order to estimate their impact.
Although the magnitude of the difference that should be considered as significant depends on the application, we define here a 235 considerable difference as 0.1, which is 10% of the maximal LW C ad .
The approaches that will be examined next are: 1. AF linear : Using Eq. 6 and keeping A1 A2 constant from the cloud base.
3. AF s : Including the supersaturation term using Eq. 5. 4. AF +50 : Using Eq. 6, but estimating the cloud base height to be 50 m higher.
5. AF −50 : Using Eq. 6, but estimating the cloud base height to be 50 m lower.

Linear LW C ad
AF linear is a very common method, based on the assumption that LW C ad is linear with height (i.e. neglecting the dependence of A1 A2 on temperature and humidity). This implies that A1 A2 can be used as a constant, based on the known values at the cloud 245 base (Pontikis, 1996). Indeed, small negative differences from to the non-linear AF ref (i.e., underestimation) are observed in Fig. 3b when using A1 A2 as a constant from the cloud base. Pontikis (1996) derived such a solution for stratocumulus clouds, and noted that the error using this method would increase for deeper clouds. On the same note, Brenguier (1991) argued that the linear assumption is valid for shallow clouds (depth of up to 200 hPa, ≈ 2km). This emphasizes that the usage of the AF linear method is restricted to shallow clouds, and should be used with care or avoided altogether for deeper clouds.

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The changes in the growth rate of LW C ad ( A1 A2 in Eq. 6) with height, which can lead to deviation from AF linear (or AF env , discussed next), occur mostly due to changes in A 2 . This is because A 1 (Eq. 2a), which is a parameter in the term for cooling by ascent, depends only on temperature, and exhibits a negligible change in the case of our shallow clouds. A 2 (Eq. 2b), which relates to the S sink term (by condensation), depends on T and q v , and increases with height. Fig. 4 demonstrates the sensitivity of A 2 to q v and T, by presenting the differences in the profiles of A 2 from the true profile (calculated using the original q v and 255 T profiles), and when keeping T or q v as a constant (with the value set at cloud base). The true profile of A 2 (in blue) shows an increase with height, as mentioned earlier. One can see that the A 2 profile with a constant T (red curve) is very similar to the true profile. On the other hand, the profile's gradient decreases significantly when using q v as a constant (yellow line). This demonstrates that depletion of water vapor in higher levels of the cloud is the major factor that impacts A 2 values (see the inverse relation to water vapor mixing ratio in Eq. 2b) and the deviation of LW C ad from its linear relations. Figure 4. The sensitivity of A2 profile to temperature and humidity. The growth rate of LW C ad depends on the ratio of A1 A2 (see Eq. 6). LW C ad changes with height depend mostly on A2, since A1 exhibits little sensitivity. The blue curve is the true profile of A2. Red and yellow A2 profiles are calculated using constant temperature and humidity, respectively, and the dashed black line is the profile obtained using the environmental sounding.
3.2.2 LW C ad using sounding profiles (environmental profiles) The advantage of using environmental profiles is that they can be obtained from sounding data, and can be considered as constant reference values for an ensemble of clouds (the whole cloud field). Such application can have large errors in cases where the T and q v profiles in the cloud's core and the environment exhibit large differences (e.g. penetration of cloud into the inversion layer, or into higher levels of the atmosphere). The profile of A 2 when calculated using the environmental profiles 265 is given in dashed black line in Fig. 4. It shows that the environmental A 2 profile values are larger than the profiles in the core of the cloud (especially in the inversion layer above 1500 m, where q v decreases fast). The larger A 2 values lead to a smaller LW C ad (A 2 is in the denominator in Eq. 6), and hence, to an overestimation of LW C env , as seen in Fig. 3c. The small overestimation witnessed in our case (trade wind cumulus in Barbados), will probably be greater for deeper clouds, where the gradients between the core and the environment are larger.

The role of the supersaturation term in conditions of low aerosol concentrations
Considering the profiles of T and q v is necessary if one wishes to dismiss the saturation adjustment assumption (i.e. dS dz ≈ 0). This assumption is almost inherent in most previous works that we know of. The supersaturation can be significantly greater than zero in regions of high updrafts and/or small droplet concentrations (for example, in the first tens of meters above cloud base, and in pristine environments). Thus, if one wants to achieve accuracy near cloud base, or compare different clouds under 275 different aerosol loading conditions (e.g. studying aerosol effects on cloud mixing), one has to use AF s as calculated by Eq. 5.
The second term in Eq. 5, referred to here as the S term, depends on the vertical profile of S. It is worth noting that S profiles are available only in modeling studies, since in-situ measurements of S cannot reach the desired accuracy, as far as we know. Fig. 3d demonstrates that for cases where the cloud develops under conditions of high aerosol concentrations (Na=500 cm −3 ), neglecting the S term introduces a negligible underestimation of AF near the cloud base (AF s > AF ref ). However, this is not 280 the case for cleaner environments (lower Na). Neglecting the S term means assuming that the parcel condenses all of the excess water vapor that forms as it ascends and cools. This overlooks the limited condensation efficiency in pristine environments that prevents a full consumption of water vapor by the drops, and lets S increase (i.e., dS dz and the second term are larger than zero). To evaluate this effect, we simulated two additional clouds in a cleaner environment, with lower aerosol concentrations (Na=50 cm −3 and 5 cm −3 ). Before analyzing these simulations, we first had to make sure that there is no significant sedimentation 285 in the clouds at this stage. Sedimentation creates liquid water loss and downdrafts, which violate the adiabatic assumption, and lead to a deviation of AF from Tr. For the example examined in this section, we used Tr to assure that sedimentation can be neglected in the time-steps we chose for comparison (timing of maximal cloud top height of each cloud). Fig. A2, in the appendix, shows vertical cross-sections of AF ref and Tr for the cloud simulations with Na=50 cm −3 and 5 cm −3 , at the time steps used for this example. There is a good agreement between AF ref and Tr for Na=50 cm −3 at 33 minutes, and Na=5 cm −3 290 at 31 minutes. This is not the case for Na=5 cm −3 after 40 minutes, when the cloud precipitates, and sedimentation can no longer be neglected. In Fig. A2 we observe regions in the clouds with large differences between Tr and AF ref values.
In Fig. 5, we present the deviation of AF s (when including the saturation term) from AF ref for the three different simulations. Fig. 5a (same as Fig. 3d) shows that the deviation is negligible for high Na. The deviation increases and spreads to higher levels when Na is smaller (Fig. 5b-c). Under pristine conditions (Na=5 cm −3 , Fig. 5c), the underestimation of AF ref spreads 295 throughout the entire cloud column. The profiles of S are depicted in Fig. 5d for all cases, presented as the mean S of the voxels with the highest 5% updraft in each level. Fig. 5d shows that saturation adjustment is a reasonable approximation for AF calculations in polluted clouds, because the S profile is smaller, and more importantly-almost constant. The gradient of the S profile of the Na=50 cm −3 case is positive also far from the cloud base (at 1000-1400 m). It doesn't introduce very large errors to AF ref (as can be seen in Fig. 5b) because the gradient is relatively small, and thus the second term in Eq. 5 is 300 negligible compared to the first term. Near cloud base, the first term in Eq. 5 is small, and thus, even a relatively small gradient in S can be significant, and lead to a large difference between AF ref and AF s .

AF sensitivity to cloud base heights
Last, we tested how sensitive AF calculations are to the errors in the estimation of the cloud base height. Fig. 3e-f shows the deviation from AF ref when having an error of ±50 m in cloud base height. When overestimating (underestimating) cloud 305 base height, as in Fig. 3e (3f), the estimated LW C ad is smaller (larger), and AF is larger (smaller). These results demonstrate that such small errors in a parameter that is often taken for granted can introduce large errors in adiabaticity estimation. For example, the Lifting Condensation Level (LCL), which is often used to approximate cloud base height, can be obtained from a tephigram, or calculated by several proposed analytical equations. Calculating LCL from surface conditions, as suggested by Bolton (1980), Lawrence (2005), or Romps (2017), approximates cloud base height to be 515, 550, or 525 m, respectively. These approximations are lower than the height that was found as optimal for AF calculations using the sub-cloud layer tracer (≈600 m, depending on simulation time and cloud properties). Additionally, LCL is known to be an underestimation of cloud base height when the convective parcel is driven by perturbation in temperature. In such a case, the perturbation reduces the parcels' relative humidity, and therefore the parcel starts the condensation at a higher altitude (above the LCL).
This can cause an overestimation of LW C ad and an underestimation of AF. The opposite will occur when the convection is 315 driven by a humidity fluctuation (Hirsch et al., 2017). We note that most in-situ measurements and space-borne remote sensing observations lack the tools required to measure the cloud base height, and usually estimate it based on LCL.

Mean differences between the assumptions with time
So far, we examined the AF calculations at one time-step the time of maximum development of the clouds (∼33 min). The robustness of the results tested by estimating the deviation of each method from AF ref over height, and as a function of time, 320 along the cloud's lifetime. Since the deviations are more pronounced in regions of high AF, we chose to consider for this analysis only the cloudy regions with AF ref > 0.5. Note that these sub-adiabatic regions are important, and highly debated. Fig. 6 shows the mean deviation for a cloud with Na=500 cm −3 . Here, we observe that the statistics along the cloud's lifetime agree with the instantaneous qualitative pictures presented in Fig. 3. As demonstrated in Fig. 6a, the linear assumption (AF linear ) underestimates AF for altitudes above cloud base. AF env , using the environmental profiles (Fig. 6b), exhibits a small over-325 estimation, which becomes significant near the cloud top, at the inversion layer. Here, we define a significant difference as larger than 0.1 in absolute value (marked by the black contour). Considering the S term in the polluted case does not have a considerable effect (Fig. 6c). Errors in the estimations of cloud base by 50 m lead to relatively large errors in AF-up to 1000 m ( Fig. 6d-e). Repeating the same analysis for the clean case (with Na=5 cm −3 ) gives similar results for all methods but AF s (see Fig. 7c) which supports the argument made in section 3.2.3. The time series is shorter in this case because sedimentation starts 330 after 33 minutes. It seems that the observed differences between the AF calculation methods don't change with time during the growth stage of a particular cloud, for both pristine and polluted conditions.  An accurate calculation of the adiabatic fraction (AF) is crucial in two main aspects: 1. An accurate AF can promote a high-resolution measure of the mixing state of sampled parcels. This may advance the 335 research of mixing processes in shallow clouds and their effects, which remain open questions in the field of cloud physics.
2. To Allow mapping of the occurrence and extent of adiabatic regions in shallow clouds, which are still under debate.
Adiabatic processes are simpler to predict, hence adiabaticity is assumed in remote sensing retrieval algorithms and cloud parametrizations in weather and climate models.

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Answering these questions can improve our process-level understanding, in-situ measurements, and remote sensing retrievals. This will improve models' representation of shallow convection, and may reduce the magnitude of the shallow clouds' contribution to the uncertainty in weather and climate models.
In this study, we tested the accuracy of different approaches that are commonly used to calculate adiabatic fraction (AF). We used high-resolution (10 m) simulations of isolated trade wind cumulus clouds, that solve the turbulent flow down to scales 345 that are rarely achieved. This enables a better representation of mixing, and relaxes the dependency on sub-grid parameterization schemes. A sub-cloud layer's passive tracer (Tr), which is an accurate measure of mechanical mixing, was added to the simulations and was used as a reference point. The calculation of AF using Eq. 4-6 (which is based on a Lagrangian model) from a snapshot of an Eulerian model, demanded bridging the gap between these two different perspectives. This was achieved by assuming stationarity of the thermodynamic profiles in the cloud core. Through the validation of AF calculations by Tr 350 values, we found that this assumption holds for the temperature profile, but not for the specific humidity (q v ). The method that is based only on q v (Eq. 7) exhibited a weaker agreement with Tr. Some regions in the cloud emphasized the important differences between AF and Tr. While Tr follows the complex flow in the cloud and records all mixing events, AF is based on a one-dimensional model whose reference lies in the core. For that reason, AF cannot describe processes that occur in the margins. Moreover, condensation that occurs after a mixing event can delete records of earlier evaporation/dilution events.

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As an example, the toroidal vortex drives entrainment events followed by updrafts, which cause some parcels to experience dilution and evaporation (decrease in Tr and AF), followed by condensation that increases AF.
Three different calculation methods of AF (Eq. 5,7,8) were compared with the tracer. The most robust method (Eq. 5) is an analytical solution that allows the isolation of different assumptions and evaluation of their accuracy. The important findings and their implications are as follows:

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Assuming a linear profile of LW C ad , or using the sounding profiles of temperature and humidity instead of the in-cloud profile, produces small errors at higher levels of shallow clouds (∼ 2 km). The small error in AF for shallow clouds obtained using environmental profiles suggests that it can be used as a constant reference for all clouds in the field.
The saturation adjustment assumption was integrated into the calculation of AF in most previous studies. Testing this assumption on clouds that develop in different environmental conditions (with different aerosol concentrations), revealed that it can lead to underestimation of AF. A simulation of a cloud in a pristine environment (Na=5 cm −3 ) yielded high supersaturation values (compared to the polluted case), and led to an underestimation of AF when assuming saturation adjustment (i.e. dS dz ≈ 0). This means that comparing clouds' mixing under different aerosol loading when using the saturation adjustment assumption may neglect some of the microphysical effects on clouds' dynamics, and mixing in particular.
AF was found to be sensitive to errors of ±50 m in the estimated cloud base height; especially in the first few hundred meters 370 above cloud base. Determining cloud base height is challenging for aircraft in-situ measurements, and is often obtained by estimating the lifting condensation level (LCL) from a tephigram or analytical solutions. Three analytical solutions that were tested here (Bolton, 1980;Lawrence, 2005;Romps, 2017) differed by 45 m, and underestimated the cloud base height that was optimal for AF calculations. Underestimation of the cloud base height can lead to a larger LW C ad , and thus to the underestimation of AF. This can lead to an underestimation of the extent of adiabatic regions in shallow clouds. Accurate estimation 375 of AF near the cloud base is challenging because these levels include a ratio of two small numbers (LWC and LW C ad ), and are not homogenous as mostly assumed. Moreover, the calculation of LW C ad in these levels exhibit high sensitivity to the determination of cloud base and the representative supersaturation profile.
All simulations demonstrated the existence of an adiabatic core (i.e., high values that are close to one for both AF and Tr) up until the cloud top. While the core is wide at the lower parts of the cloud, it narrows and breaks down to smaller fragments at 380 higher levels. The extent and frequency of adiabatic parcels in different levels of the cloud will be assessed in a subsequent study.
Finally, we point out the limitations of using AF in clouds deeper than the shallow convective clouds of the boundary layer: 1. AF is based on a quasi-hydrostatic equation, which is valid for updrafts smaller than 10 m s .
2. Supersaturation in clouds with strong updrafts can increase, leading to underestimations of AF when the S term is 385 neglected.
3. Sedimentation of particles from higher levels of deep clouds can increase LWC in their lower levels and lead to an overestimation of AF.
4. The rate of change in LW C ad is dominated by the parameter A 2 , which changes as water vapor is depleted in clouds, meaning that LW C ad ceases to be linear. The large differences expected in deep clouds between the in-cloud and 390 environmental profiles suggest that the latter is prone to large biases when used to predict AF.