1. Introduction
Latent heat released during water phase transition is an important source of energy driving formation and evolution of cumulus convection. For the tropical shallow cumulus clouds considered in this study, the latent heat is released predominantly through condensation and evaporation. Their formulation in computer models depends on grid resolution [large-eddy simulation (LES)/cloud-resolving model (CRM)/numerical weather prediction (NWP)] and the method chosen to describe the microphysics (explicit/bin or parameterized/bulk). In an LES model, because of its fine grid, it is reasonable to assume the grid cell is either entirely saturated or undersaturated and to therefore calculate the phase transition rate (TR) in the whole grid cell. In the case of explicit microphysics, the TR can be quite accurately described by the condensation equation (Mordy 1959), while for bulk microphysics it is calculated using the moist saturation adjustment scheme (Soong and Ogura 1973). The same scheme is often used for CRMs.
In meso- and large-scale models, including NWP models, cloudy regions may occupy only a fraction of a grid cell; therefore, the grid-mean relative humidity may be less than saturation. Nevertheless, the threshold for saturation can be reached locally in some parts of the grid. Formulation of condensation thus requires subgrid information on the statistical distribution of cloud parameters. As the saturation vapor value is a function of temperature and pressure, the phase transition is explicitly linked to the thermodynamical and radiative processes affecting temperature and humidity. The close interaction between microphysics and thermodynamics makes parameterization of phase transitions in meso- and large-scale models much more challenging than parameterization of microphysical conversion processes (such as autoconversion and accretion), primarily because the latter are defined by self-contained closure relationships.
In our previous paper (Kogan 2021a) we described a linear relationship between phase TR and the upward plus mass flux (MFP). Both condensation rates (CR) and evaporation rates (ER) were found to have a remarkably high correlation coefficients with MFP (R = 0.99 and 0.98, respectively). These results should not be surprising, as mass flux has long been recognized as a major factor affecting cumulus convection (see, e.g., Arakawa and Schubert 1974; Geleyn et al. 1982; Tiedtke 1989; Bechtold et al. 2008; Suselj et al. 2019). It is also well known that vertical velocity has a strong effect on supersaturation and, therefore, cloud microphysics (see, e.g., Squires 1952; Paluch and Knight 1984; Politovich and Cooper 1988). In this study we focused on the direct effect of MFP on processes of phase transition, specifically, on evaluation of the sensitivity of phase transition rates to mass, but also to buoyancy flux and some other microphysical variables. The goal was to identify the parameters that have the strongest effect on phase transition rates.
Another important goal is establishing a theoretical basis for the LES-derived linear relationship between transition rates and MFP. We have developed a theoretical formulation based on the equation for supersaturation and the concept of “quasi-steady” supersaturation. The theory and LES results suggest that the TR–MFP relationship is valid for small local volumes and can therefore be integrated over larger areas, such as over a horizontal plane to obtain relationships for horizontal mean variables that vary in the vertical. Such vertically dependent relationships may serve as the basis for parameterization; this task will be the focus of Part II of our study. Here, as a first step, we analyzed variables integrated over the whole cloud. Integral cloud characteristics are of interest to studies of cloud contributions to regional and global processes (see, e.g., Haynes et al. 2011; Oreopoulos et al. 2017; Nelson and L’Ecuyer 2018; Tomassini 2020).
2. Model and dataset
The simulations were made using a version of the CIMMS LES model (Kogan et al. 1995; Khairoutdinov and Kogan 1999) called System for Atmospheric Modeling—Bulk Microphysics (SAMBM; Kogan et al. 2012). The dynamical core of SAMBM consists of the System for Atmospheric Modeling (SAM), developed by M. Khairoutdinov (Khairoutdinov and Randall 2003). SAM is based on nonhydrostatic, anelastic dynamics and uses a monotonic, positive-definite advection scheme for scalars (Smolarkiewicz and Grabowski 1990). The positive-definite property of the scheme is especially important for advection of microphysical variables, such as cloud-drop mixing ratios and concentrations. The bulk microphysical parameterization in SAMBM uses the formulation for cumulus clouds described in Kogan (2013). This parameterization has been tested against simulations using the CIMMS LES with explicit microphysics (SAMEX; Kogan et al. 2012) in case studies of northeast Atlantic marine stratocumulus [the Atlantic Stratocumulus Experiment (ASTEX); Albrecht et al. 1995) and marine trade cumulus based on the Rain in Cumulus over the Ocean (RICO; Rauber et al. 2007) field campaigns.
Initial profiles, large-scale vertical velocity, and tendencies of temperature and moisture corresponding to average conditions over the 3-week period from 16 December 2004 to 8 January 2005 of the RICO campaign were used for initialization and forcing of the LES simulation (described in detail in the LES model intercomparison study; vanZanten et al. 2011). Total cloud condensation nuclei (CCN) concentration was assumed to be 104.4 cm−3, a concentration similar to those measured by Passive Cavity Aerosol Spectrometer Probe (PCASP) during the RICO flight RF11 on 7 January 2005. The model horizontal and vertical grid spacings for the simulation were 100 and 40 m, respectively, with a total of 500 × 500 × 100 grid points (a 50.0 × 50.0 × 4 km3 domain). The total simulation time was 32 h, with the dynamical time step of 1 s and microphysical time step for condensation calculations of 0.1 s.
3. Results
a. Phase transition rates and macroparameters
The domain-averaged turbulent kinetic energy (TKE) and accumulated surface precipitation during the simulation are shown in Fig. 1.
Evolution of mean TKE and accumulated surface precipitation.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
After the initial 8-h spinup and early cloud formation, the cloud system evolved to a stage of mostly growing, weakly precipitating clouds (8–18 h). From 18 to 25 h it progressed to a quasi-stable stage of mature moderately precipitating clouds and then to the final stage of heavily precipitating decaying clouds lasting until the end of the simulation.
Over the course of the 32-h run, the simulation data were archived every half hour from 8 to 32 h. From this archived dataset we selected 2031 clouds by applying combined thresholds of a liquid water path > 20 g m−2, and a horizontal cross-section size > 90 and volume > 800 cloudy grid points. A cloudy grid point was defined as having a liquid water content (LWC) > 0.05 gm−3.
As cloud volumes are quite large (they vary in size from a fraction to tens of cubic kilometers), it is convenient to describe TR in units of tons per second (t s−1). Similar to (1), cloud water and rainwater (qc and qr) are also integrated over the whole cloud to yield total cloud and rainwater content (QC and QR),1 also measured in tons.
Establishing statistically robust relationships between TR and relevant thermodynamical variables is a challenging task in a system of clouds at various stages of their evolution. We found that the best approach to the problem is to separate the entire dataset into subsets stratified by the cloud size. A similar method was used in our previous work aimed at PDF parameterization development (Kogan and Mechem 2014, 2016); specifically, the dataset was arranged by ascending order of cloud depth and separated into four subsets, and the boundaries between them were chosen such that each subset yields about the same amount of condensate per second.2 The first two subsets (groups of clouds) consisted of small clouds that were mostly at the growing stage but also contained remnants of the larger clouds, which dissipated. The third group (G3) consisted mostly of mature clouds, and the fourth (G4) of a mixture of mature and decaying clouds.
Table 1 and Fig. 2a show the basic statistics of overall cloud characteristics in each group. On average, the G1 clouds had their tops at ∼2 km, surface footprints of ∼1.8 km2, and volumes of ∼0.9 km3.
Mean (top value for each group number) and standard deviation (bottom value) of cloud top, area, volume, and surface precipitation (precip) rates in each of four groups.
Mean and standard deviation of cloud physical parameters in each of the four groups. (a) Cloud top, surface area, volume (Ctop, Area, Vol; in km, km2, km3, respectively); (b), cloud and rainwater content (QC, QR; in tons); (c) per cloud condensation/evaporation rate (in ton s−1); (d) scatterplots of phase transformation (CR/ER) dependence on cloud volume for all groups. S denotes the slope of the best-fit line.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
The linear increase in cloud-top height from G1 to G2 of about 0.3 km, and then similar increases to G3 and G4, were accompanied by an exponential increase in cloud surface footprint and volume (Fig. 2a). The growth in cloud water content (QC) (Fig. 2b) trailed the growth in cloud volume; for example, the increase in cloud volume from G2 to G4 by a factor of 8.6 resulted in cloud water content growth by a factor of only 5.3 (see also Table 1). The rainwater content, however, grew at more than twice that rate (Fig. 2b), obviously because growth by condensation was combined with conversion of cloud water to rain by autoconversion and accretion.
The growth in condensation and evaporation rates (CR, ER) followed more closely the cloud volume growth (Fig. 2c). For instance, Vol4 was 8.6 times Vol2 (Fig. 2b), corresponding to a 7.7-fold increase in condensation and 6.9-fold increase in evaporation rate. The strong sensitivity of CR and ER to cloud volume is also evident in Fig. 2d, which shows a scatterplot of these rates versus volume for clouds from all groups. The overall evaporation rates were, on average, about 42% of the condensation rates (the slope of the best fit line S = 0.56 for ER versus S = 1.34 for CR).
Figures 2a–c show the means of parameters of individual clouds in a group; Fig. 3 shows these parameters integrated over the entire group.
Total CR, ER, and PR integrated over the group (tons−1). Numbers over the ER columns denote the ratio of evaporation to condensation in percentages. Numbers over the precipitation rate (PR) columns show percentages of precipitation ratio in each group to the total precipitation from all groups.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
By design, the groups were selected to condense approximately equal sums of water vapor. While the overall fraction of evaporation to condensation rates was about 42%, it varied between clouds from different groups. These fractions are shown in Fig. 3 for each group above the ER green columns; they are smaller for G1/G4 clouds but larger for G2/G3 clouds (37.4% and 41.4% versus 45.9% and 44.0%, respectively). Naturally, in the G1 group the majority of clouds grew and therefore evaporation in them lagged condensation. On the other hand, the mature G3 clouds develop stronger downdrafts where more cloud drops can evaporate. In G4 clouds, raindrops are large enough to fall out and add more to precipitation than to evaporation; this will lower the fraction of evaporation compared with that in G3 clouds.
The precipitation in G4 was quite large, which is clear from the numbers over the precipitation rate (PR) columns (colored in purple in Fig. 3). These numbers show the percentages of precipitation in each group relative to the total precipitation from all groups. The G3 and G4 clouds combined contributed 75% of the total precipitation. G1 and G2 clouds contributed another quarter (8% and 17%, respectively). The G3 clouds precipitated about 70% more water than they condensed, and for G4 clouds this number was 100%. Clearly, both G3 and G4 clouds were losing water; this is another indication that these clouds were entering or already at the stage of decay.
While the contribution of G1/G2 clouds to total precipitation was small, the data in Table 1 show a quite large variation in precipitation from these clouds. Notably, the standard deviations of precipitation rates are larger than their mean values by a factor of 2.6 for G1 and a factor of 1.6 for G2 clouds. The explanation is that the G1/G2 groups contained not only growing clouds, but also remnants of larger dissipating clouds. This increased the variability of rainwater and precipitation. We will discuss the effects of dissipating clouds in more detail in section 3e.
Figure 4 shows the ratios of the volumes occupied by various cloud parameters to the volume occupied by condensation. The cloud parameters of interest are evaporation (EVP), updraft (UPD), downdraft (DND), upward buoyancy flux (BFP), cloud water (QC), and cloud liquid water content (QL). As the cloud boundaries are defined by the cloud liquid water content, the volume of QL is by definition the volume of the cloud itself. As expected, in the predominantly growing G1 clouds the evaporation volume is smaller than the condensation volume (mean ratio of about 90%). In the G2 clouds the volume of evaporation is about 120%, and in G3/G4 about 140% of the condensation volume.
Mean and standard deviation of the ratio of the volumes of parameters shown on the x axis to the volume occupied by condensation. EVP: evaporation; UPD: updraft; DND: downdraft; BFP: upward buoyancy flux; QC: cloud water; QL: total liquid water.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
Evidently the area of evaporation, which is expanding by detrainment as clouds grow, is larger than the more compact area of updrafts, where most condensation takes place. However, not all updrafts result in condensation, as some weak updrafts may be undersaturated and therefore contribute to evaporation. The sum of the volume of weak updrafts (part of the UPD column above the y = 1 red line in Fig. 4) and the downdraft volume equals the evaporation volume. Among other variables, the upward buoyancy flux volume is larger than the condensation volume by roughly 40% (with the exception of G1 clouds where the difference is 20%). The volume occupied by QC is about 60% larger than the condensation volume in all cloud groups. This indicates that in this “extra” volume where supersaturation is negative, the cloud water, nevertheless, does not evaporate immediately but forms a cooling envelope encompassing the condensational core.
The fraction of the cloud occupied by condensation is shown by the inverse ratio of QL/CND volume (last columns in Fig. 4 panels). This fraction decreases steadily (52.6%, 44.8%, 42.4%) for G1, G2, and G3 clouds, respectively, and finally to the lowest fraction of 41.7% for the decaying G4 clouds. The difference between the QC and QL columns shows the volume occupied by QR only; its ratio to total cloud volume increases as clouds grow bigger: 13.2%, 24.7%, 30.5%, and 32.5% for G1, G2, G3, and G4 clouds, respectively.
So far we have described the general parameters of simulated shallow cumulus clouds. These LES model data derived from different stages of cloud system evolution may, among other applications, be useful in developing and evaluating simpler but computationally efficient cloud-topped boundary layer models (see, e.g., Lilly 1968; Schubert 1976; Albrecht 1979).
b. Condensation rate correlations
In this section we describe the relationships between CR and cloud variables relevant to the condensation process. Figure 5 shows the degree of correlation between CR and these variables for G1 clouds. Obviously, they include the upward mass and buoyancy fluxes, as well as cloud water variables, such as cloud water content and drop concentration. For completeness, we also show CR correlations with rain variables.
Scatterplots of parameters shown in plot legends together with the correlation coefficient R. Group G1. Here NC and NR denote the integral cloud drop and raindrop concentration, respectively, and QC and QR denote the integral cloud and rain liquid water content, respectively.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
The very high correlation between condensation and buoyancy flux is not surprising, as most of the time condensation indeed happens in areas of positive buoyancy. CR also correlates well with cloud water content and cloud drop concentration but, as one might expect, correlates poorly with rain parameters. The most remarkable result, however, is the nearly perfect correlation of CR with the upward mass flux (R = 0.994).
Clouds in groups G2–G4 reveal similar results. For brevity, we show them in Fig. 6 only for mass and buoyancy fluxes, as well as cloud water parameters, omitting less relevant rainwater variables. The scatterplots show the same perfect CR correlation with MFP (R = 0.994), and a slightly weaker but still high correlation with BFP (R = 0.93–0.95). The correlation with drop concentration is worse than that with cloud water content, which is understandable because relatively large variations in the number of small cloud droplets as a result of fluctuations in vertical velocity may lead to large variations in total drop concentration, but insignificantly affect CR.
As in Fig. 5, but for groups G2–G4.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
In summary, the correlation between CR and MFP is remarkably high, and valid for clouds in all groups. The physical basis of the high CR–MFP correlation will be described in section 3d below.3
c. Evaporation rate correlations
Figure 7 shows correlations between ER and buoyancy flux. For G1/G2 clouds, ER correlates better with the downward buoyancy flux. As already mentioned, the G1/G2 clouds contain remnants of the larger, decaying clouds in which undersaturated downdrafts prevail and contribute directly to evaporation. In dissipating clouds, downward fluxes are affected by the falling rain to a significant degree and not completely connected to the upward motion. For G3/G4 clouds, in contrast, the correlation coefficient is slightly higher for upward buoyancy flux. In growing clouds, it is likely that downward motion is less affected by microphysics but is more of a dynamical feature of the mass conservation law; as a result, both up and down fluxes are interconnected, and ER can be well defined by either of them. A more definite conclusion requires a comprehensive analysis of data that is beyond the scope of the current study.
As in Fig. 6, but for correlation between buoyancy flux and evaporation rate. Groups G1–G4.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
The mass flux (Fig. 8) results are somewhat more surprising. First, with the exception of G1 clouds, the ER–MFP correlation is the same or even slightly higher than the ER–MFM correlation (cf. the left and right panels in Fig. 8). Second, ER correlates better with MFP than with BFP. This is consequential, given the fact that CR is also strongly correlated with the upward mass flux; as a result, both CR and ER can be expressed as a function of a single variable, MFP.
As in Fig. 7, but for correlation between mass flux and evaporation rate.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
Analysis of the slopes of the linear fits for the CR–MFP scatterplots in Figs. 5 and 6 showed that they all have values that differ by only a few percent. Similar results are true for the ER–MFP scatterplots in Fig. 8. This suggests that the data from all groups can be combined and approximated by a single linear fit.
Scatterplots of phase transition rates (CR: condensation; ER: evaporation) as a function of upward mass flux (MFP) for combined dataset of clouds from all groups. R is the correlation coefficient; S is the slope of the linear fit.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
Obviously, using the specific latent heat constant, one can directly relate formulation (3) to the latent heat released during phase transitions.
d. Theoretical foundation and implications for cloud supersaturation
We start by evaluating αqs, which is a function of temperature T and pressure P. We first calculate at each vertical level the values of T and P as horizontal averages over the cloudy areas, and then obtain αqs using (8), (10), and (15). The values of αles are determined as slopes of linear fits, either to the full LES dataset αles = 2.06 or to each individual group subset (2.17, 2.13, 2.052, and 2.049 for groups G1–G4, respectively).
The vertical variation of αqs is shown in Fig. 10a for the temperature profile
Profiles of cloud parameters as a function of the height above the cloud base: (a) the temperature and “quasi-steady” slope in the formulation (14) and (b) the vertical profile of the ratio of supersaturation to its quasi-steady value.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
In calculating αqs we made the assumption that
Comparison of the coefficient α based on the LES data and the theoretical value from Eq. (15) using the quasi-steady (QS) assumption. Columns show cases from four groups (G1–G4) separately and the full dataset (all).
While Table 2 compares the integral values of αqs and αles, a significantly more informative comparison would be made between their vertical profiles, or by showing the profile of κ factor. Figure 10b shows the profile of κ that was calculated using the αqs profile shown in Fig. 10a and the integral values of αles for the full dataset and, separately, for clouds in groups G1–G4.
As Fig. 10b shows, the supersaturation in the lower half of the clouds is, on average, smaller by 3%–5% than the quasi-steady value (κ < 1). This may be because of the small values of integral radius Ri once cloud drops begin to grow after being activated at the cloud base. The small values of Ri combined with intensifying updrafts lead to large quasi-steady supersaturations [see Eq. (12)] in the lower half of the cloud. At the upper levels Sqs is decreasing, because Ri is large, while updrafts decrease as they near the stable layer at the cloud top. The cloud supersaturations, on the other hand, may increase in the upper regions of the cloud, because of entrainment and mixing with cold and dry environmental air.
While all these factors may contribute to the increase of κ with height, the κ profiles in Fig. 11b will change quantitatively when the constant value of αles averaged over the whole cloud is replaced by the more accurate vertically dependent profiles of αles. Our preliminary results using the vertically dependent CR and MFP variables show that αles increases steadily with height; therefore, the pattern of κ factor increasing with height will remain qualitatively the same. The detailed analysis of this case will be presented in Part II of the paper.
Cumulative distribution of (a) the rainwater fraction QR/QC and (b) the evaporation fraction ER/CR for all four groups. (c),(d) As in (a) and (b), but for all clouds in the G1 group (G1_All) and for the subset of small clouds in the G1 group formed only between 8 and 9 h into simulation (G1_SC).
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
While the average values of α or κ factors (Table 2 and Fig. 10) may be useful for model validation, their physical interpretation has to be taken with a grain of salt. First, nonlinearity of the averaging procedure may introduce additional errors. Second and more important, the average variables may conceal significant local variations of, for example, supersaturation, depending on the degree of mixing and entrainment in a particular cloud location.
Therefore, (14) can be integrated over a cloud area larger than
Finally, we note that while the current study was focused on shallow Cu clouds, the relationship (3) has a more general nature and is valid for a wide variety of cloud types, including deep convective clouds as demonstrated by Grant et al. (2022).
e. Specifics of evaporation affecting its correlation with mass flux
Compared with condensation, the ER–MFP correlation is weaker (see Fig. 8), especially for clouds in the G1 group. The reasons are as follows. First, cloud and raindrops contribute differently to condensation and evaporation. Notably, condensation on cloud droplets is significantly larger than on raindrops, obviously, because it is proportional to Ri [Eq. (4)], and the contribution to Ri from raindrops is orders of magnitude smaller due to their much smaller concentrations.
Raindrops, however, contribute much more significantly to evaporation. At the earlier stage of clouds evolution, the updrafts and condensation prevail over downdrafts and evaporation (see, e.g., the G1 panel in Fig. 4). However, as clouds mature, the raindrops grow in size and numbers (predominantly by coagulation), and qr in areas of evaporation becomes comparable to qc.
As Fig. 11a shows, in the G1 and G2 groups, respectively, only 2% and 7% of clouds have QR/QC > 1; in the G3 and G4 groups, respectively, these numbers are 30% and 43%. Larger QR/QC fractions in general correspond to larger fractions of ER/CR (Fig. 11b). For example, about 60% of G1 clouds have ER/CR > 0.3; this condition, however, is satisfied by 92% of G2 and almost 100% of G3/G4 clouds. In the latter case, the ratio of ER/CR varies in a narrow range from 0.3 to 0.9 with average ER/CR values of 44.0% and 41.4% for G3 and G4 clouds, respectively (see Fig. 3). Surprisingly, the ER/CR values for the smaller G1/G2 clouds vary in a much wider range, with 24% of G1 clouds and 40% of G2 clouds exhibiting a value of ER/CR exceeding 0.5. Notable also is the long-tail distribution in about 8%–10% of G1/G2 clouds with ER/CR > 0.8 (Fig. 11b). According to our data, the “long-tail” clouds with ER/CR > 0.8 account for 7.6% of total G1 cloud evaporation; for clouds with ER/CR > 0.5 the fraction is 29.5%. The long-tail phenomenon is explained by the fact that G1/G2 groups contain not only young growing clouds, but also remnants of the mature decaying clouds, which decrease in size as they dissipate. Figures 11c and 11d show the difference in results for G1 clouds when they are separated by the stage of their development. For this we selected from all G1 clouds a subset of growing small clouds (referred to as G1_SC). This subset includes only clouds formed in the G1 group at the very early stage of the simulation; that is, from 8 to 9 h into the simulation. The G3/G4 clouds were not yet formed at that time, and therefore no dissipating remnants of big clouds could be present in the G1_SC subset. As Figs. 12c and 12d show, clouds in the G1_SC subset have smaller, more “normal” QR/QC and ER/CR ratios compared with those in the full G1 dataset.
Evaporation fraction ER/CR as a function of integrated cloud liquid water color coded by the fraction of rainwater QR/QC for clouds in group G1.
Citation: Journal of the Atmospheric Sciences 79, 11; 10.1175/JAS-D-22-0060.1
However, even in the G1 group, not all clouds are equally affected by the QR/QC fraction. This is evident from Fig. 12 where the fraction ER/CR is shown as a function of total liquid water,5 QL, and color coded by the value of QR/QC. Clearly, increasing the fraction of rainwater in clouds of similar size leads to an increase in the evaporation fraction in the cloud; however, the effect is most pronounced in smaller clouds.
As a result, the correlation coefficient is lower in the G1/G2 groups, but remains quite high for datasets containing larger clouds. Even for the G1/G2 groups, the “contamination” of these groups with remnants of decaying big clouds has a limited effect because of the small number of G3/G4 clouds (202 clouds, or ∼10%, see Table 1) and consequently small number of their remnants. As a result, while it is important to note that evaporation could be most accurately defined by a 2D function of the dynamical variable MFP, as well as the microphysical variable QR or QR/QC, nevertheless, for the considered case of integral variables in shallow Cu clouds, evaporation dependence on the upward mass flux alone provides a reasonably good approximation.
In the previous section we mentioned that the CR–MFP relationship may be applied locally. For evaporation, the ER–MFP relationship may not necessarily hold for variables integrated over parts of the cloud. In this case, an additional complication arises because the evaporation of raindrops is much slower than the evaporation of cloud droplets. Larger phase relaxation times for raindrops lead to evaporation depending not only on downdraft, but also on microphysical parameters such as the mean size of raindrops and, correspondingly, their fall velocity. The latter affects the time raindrops fall through the undersaturated environment, consequently adding the ambient humidity as another variable shaping evaporation. The detailed investigation of the many factors affecting evaporation merits a separate investigation.
4. Conclusions
We analyzed parameters of condensation/evaporation/precipitation processes in a system of shallow Cu clouds selected from the 32-h LES under conditions observed during the RICO field project. The dataset consisted of 2031 clouds, which were stratified by cloud top and divided into four groups (G1–G4) with approximately equal amounts of condensation in each group. In our analysis we considered parameters characterizing cloud as a whole, that is, parameters integrated over the cloud volume.
The focus of the study was to explore dependencies of the phase transition rates on the most appropriate thermodynamical variables, in particular the upward and downward mass and buoyancy fluxes. The highest correlation found was between the condensation rate and the upward mass flux (MFP) (correlation coefficient R = 0.99); the correlation with the upward buoyancy flux was weaker (R ∼ 0.93–0.96, depending on cloud group). The LES data also showed that CR is not only highly correlated with MFP, but also that the relationship between them can be expressed as a linear function (CR = αlesMFP).
While a strong correlation between the upward mass flux and condensation rate was not surprising (obviously due to strong dependence of supersaturation on the updraft strength), the very high correlation coefficient (R = 0.99) is quite remarkable. This LES model result has been examined theoretically by expanding the solution of the supersaturation equation in a rising Lagrangian air parcel model to air volumes in the LES model updrafts. The theory reveals the same linear relationship CR = καqs MFP, where αqs is a function of temperature and pressure, and κ is a ratio of S/Sqs (S and Sqs are supersaturation and its quasi-steady value, respectively). By equating the LES and theoretically derived expressions for CR, we can calculate κ = αles/αqs. The LES data show that κ is equal to 1, with an error of less than 5%. This fact supports the generally accepted notion that supersaturation in many areas of shallow Cu clouds is close to its quasi-steady value. Another conclusion that can be made from the fact that κ ≈ 1 is that the condensation rate may be defined by only a single dynamical parameter, the upward mass flux. The condensation rate, however, will depend on microphysical parameters (e.g., the integral radius) in areas of mixing and entrainment where supersaturation differs from its quasi-steady value. It was also noted that the theoretical CR–MFP relationship was derived based on the equation for cloud drop condensational growth and thus is valid for small local volumes. It can therefore be applied to study local variations of S in the cloud.
In addition, the TR–MFP relationship can be integrated over larger areas, such as over a horizontal plane to obtain relationships for horizontal mean variables that vary in the vertical. Such vertically dependent relationships may serve as the basis for parameterization; this task will be the focus of Part II of our study.
The process of evaporation is more complex, as it is affected not only by dynamical but also by microphysical parameters, specifically by the ratio of the rain to cloud water. Because of their larger phase relaxation times, raindrops evaporate more slowly. The rate of evaporation depends on the mean size of raindrops, and so does their fall velocity, which determines the time rainwater spends in the unsaturated ambient air. As a result, the correlation of evaporation rates with a single dynamical parameter, MFP, is weaker than that for condensation; this is especially pronounced for small clouds in groups G1/G2. The reason for this is that our cloud dataset was stratified only by cloud top, and therefore the G1/G2 groups contain not only newly formed growing clouds, but also remnants of larger clouds that decrease in size as they dissipate. These dissipating remnants have a much higher ratio of rain to cloud water, and therefore correlate less with the updraft. Our results suggest that a more accurate formulation of evaporation, especially in small clouds, could be provided by a two-parameter function that depends on both the dynamical variable MFP and the microphysical variable QR or QR/QC. Nevertheless, especially when considering the overall latent heat effect of shallow cumulus clouds, the results demonstrate that phase transition rates in shallow Cu clouds can be quite accurately estimated using a single variable, the upward mass flux.
We use capital letters to denote the integral cloud parameters (e.g., QC, QR, QL), while using lowercase notation (qc, qr, ql) for local variables defined at a grid point (x, y, z).
The exact division is limited because of finite discretization of the cloud-top height.
The theoretical description of the CR–MFP linear relationship is also given in Kogan (2021b).
Exact calculation of TR according to (5) eliminates the need for the condensation parameterization used in conventional two-moment bulk microphysics models. These condensation parameterizations assume that all water vapor above saturation is immediately condensed, and the residual supersaturation at the end of each time step is zero (“moist saturation adjustment” by Soong and Ogura 1973).
The total liquid water QL is well correlated with the cloud volume and may serve as its proxy.
Acknowledgments.
This investigation was supported by ONR Grant N00014-20-1-2050. The author is grateful to David Mechem for helpful comments on the draft of the manuscript, and to the three reviewers for many constructive comments. The computing for this project was performed at the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma.
Data availability statement.
Data are available upon request from the author.
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