## Abstract

Mixing is one of the most important processes associated with atmospheric moist convection. It determines the two-way interaction between clouds and their environment, thus having a direct impact on the time evolution of convection. The fractional entrainment rate *ε*—the main parameter related to mixing—is often parameterized in global circulation models as a function of updraft properties, and at the same time has a strong influence on how convection evolves. Within the framework of cumulus thermal vortices in large-eddy simulations of convection, here we first investigate the validity of some of the most common parameterizations of *ε*, and then investigate how relevant *ε* is for the fate of these thermals. We find that 1/*R*, where *R* is a measure of the thermal’s radius, best parameterizes *ε*, but it explains only about 20% of the total variance. On the other hand, we find that both *ε* and favorable initial conditions—including high initial saturated fraction of the thermals—are key factors that affect the thermals’ ascent rate, mean buoyancy, and distance traveled. The lifetimes of thermals, however, seem not to be affected significantly by either *ε* or initial conditions, which supports the view of cumulus convection as a succession of many short-lived thermals. Finally, our results suggest that for the majority of in-cloud cumulus thermals the important role of environmental moisture in the deepening of convection results mainly from providing the initial moisture for the short-lived thermals as they initiate at different altitudes above cloud base, rather than favoring their buoyancy as they rise through it.

## 1. Introduction

Atmospheric convection is one of the processes that shape the climate of our planet. It is the main vertical transport mechanism of mass and energy within the troposphere, and the main source of precipitation within the tropics. Understanding and being able to properly simulate—or parameterize—atmospheric convection is thus crucial for Earth system modeling. Although current global circulation models (GCMs) reproduce many of the observed features of atmospheric convection around the globe, many fundamental questions still remain unanswered. For example, we still do not fully understand how clouds, circulation, and climate interact (Bony et al. 2015). Processes such as the Madden–Julian oscillation, for instance, are still not satisfactorily simulated in GCMs (e.g., Ahn et al. 2017). Furthermore, current parameterizations of cumulus convection have become so complex that it is extremely hard to improve them or simply to understand the reasons for their achievements or their flaws. It seems necessary to go back a few steps in the conceptual development of such parameterizations in order to gain fundamental understanding and, by doing so, hopefully improve them as well.

The process of mixing is crucial in atmospheric convection, and perhaps one of the most complicated to deal with. How strongly cumulus clouds mix with their environment defines how the clouds themselves evolve, and how they impact their environment. The importance of cumulus cloud mixing for climate cannot be overemphasized. For example, Klocke et al. (2011) show that the value used for entrainment rate in a climate model is strongly related to the model’s climate sensitivity. Determining the “right” value for mixing rates is thus crucial for climate modelers, and quite often such parameters are sought after as functions of other known quantities. Some simple mathematical relationships are then used to estimate its value. On the other hand, mixing has an impact on how convection evolves. For example, mixing of cloudy air with its environment can potentially moisten the environment and dry the in-cloud air, affecting the cloud’s surroundings, hence the large scale, with a possible impact on the fate of the cloud itself. Thus, mixing is both a consequence of certain cloud properties and a cause for how these properties evolve. Here, we will study both aspects of mixing using LES of shallow to deep convection in tropical conditions. A thorough review of how mixing has been studied throughout the past decades can be found in de Rooy et al. (2013).

To study mixing in cumulus clouds, we must first define the two systems that exchange properties via this process. In other words, where is the boundary of a cloud? Cloud boundaries are *fuzzy* by nature, so a perfect answer to this question does not exist. Furthermore, the discrete nature of model grids makes the spatial and temporal evolution of a cloud boundary very hard to trace with the necessary precision to correctly estimate mixing directly. Sophisticated methods are then required in order to do this, for instance that by Romps (2010), who defines a binary “activity operator” that indicates if a given grid box is part of the cloud (active) or not (inactive), depending on thresholds of water content, buoyancy, and vertical velocity. Entrainment and detrainment rates are then calculated from the rates at which grid boxes change from inactive to active and vice versa. Another successful attempt was carried out by Dawe and Austin (2011a,b, 2013), who developed a complex interpolation method for the subgrid-scale surface of a cloud that allows them to compute the velocity of the cloud surface with high precision in large-eddy simulations (LESs). Entrainment and detrainment rates are then computed directly from the relative velocity of the air and the cloud surface. All these methods for estimating entrainment directly yield higher values, by about a factor of 2–3, than “bulk” estimates that typically use conserved tracers [see de Rooy et al. (2013) for details on such methods].

Instead of using sophisticated gridbox-based methods to define the extent of a cloud, we opt for going one level deeper and defining what the “basic convective elements” of clouds are. The assumption of this “basic convective element” is one of the fundamental aspects of cumulus cloud parameterization. The *steady entraining plume*, first advocated by Stommel (1947), has been the most popular choice due to its simplicity and success in representing the large-scale impacts of convection. In the simplest case, it assumes that a cloud (or even an ensemble of clouds) can be represented by one steady updraft that entrains environmental air as it rises up to the level of neutral buoyancy, where it detrains. However, except for intense and highly organized convective systems such as supercells, transient thermals—or bubbles—would be a more realistic choice (de Rooy et al. 2013; Yano 2014). Cumulus clouds have been shown to be made of a large number of small, transient thermals that account for their typical “cauliflower” aspect (e.g., Scorer and Ludlam 1953; Woodward 1959; Sánchez et al. 1989; Blyth et al. 2005; Damiani et al. 2006).

Significant effort was devoted to study thermals in the 1950s and 1960s, mostly from laboratory experiments and theoretical studies (e.g., Scorer and Ludlam 1953; Malkus and Scorer 1955; Scorer and Ronne 1956; Woodward 1959; Levine 1959; Turner 1963); a thorough review of such studies is given by Yano (2014). In recent years this topic has regained inertia, with the appearance of several numerical and theoretical studies aimed at understanding the properties of cumulus thermals (Sherwood et al. 2013; Romps and Charn 2015; Romps and Rusen 2015; Hernandez-Deckers and Sherwood 2016; Moser and Lasher-Trapp 2017; Morrison 2016, 2017; Morrison and Peters 2018). Notice that from the steady entraining plume view, a cloudy updraft is easily mapped with a gridbox-based approach by including grid boxes with vertical velocity, liquid water content, and buoyancy above certain thresholds. From the thermal—or bubble—point of view this must be done with dynamical arguments. A thermal is considered a coherent volume of air that rises as a whole, but may have internal circulations, inhomogeneities in thermodynamic properties, etc. For example, the typical structure of a thermal is close to that of Hill’s spherical vortex, with an axisymmetric recirculation around the center (e.g., Woodward 1959; Sherwood et al. 2013; Yano 2014). Such a structure contains regions with downward or stagnant motion, and may also have regions with unsaturated air due to mixing or evaporative heating. These regions still make part of the rising thermal because they are dynamically linked to the cloud, but would not be included from a gridbox-based point of view.

Here, we use as a starting point the thermal—or bubble—view of convection. Hernandez-Deckers and Sherwood (2016, hereafter HS2016) describe the main dynamical properties of cumulus thermals based on LES of shallow to deep convection, and here we extend those results by exploring the role of mixing for these thermals. In the future this could evolve into the development of new cumulus parameterizations, but even if this is not successful, most likely such an approach will improve our understanding of convective processes and have an indirect impact on new parameterization development.

We begin here by investigating whether the mixing rate of a thermal with its environment is determined by its physical properties such as size, altitude, buoyancy, or ascent rate, in analogy to what many simple convective models assume for steady plumes. Next, we explore the role of mixing in determining the physical and dynamical properties of a thermal. This leads us to another relevant question pointed out by Romps and Kuang (2010) and Böing et al. (2012): are cloudy updrafts affected more by their initial conditions at cloud base, or by their entrainment history above cloud base? We address this question here, but within the context of cumulus thermals: is the fate of a thermal (its ascent rate, distance traveled, buoyancy, etc.) more determined by its initial conditions or by its mixing history? Notice that a “convective air parcel” may be part of several thermals as it rises through the cloud, so the answers may be different in each case. Finally, we also investigate the importance of moisture; in particular, a thermal’s initial moisture content and its environment’s moisture content.

The paper is organized as follows. Section 2 describes the LES and how entrainment rates are diagnosed for the tracked thermals. Section 3 presents the results, where we first explore the possible dependence of entrainment rate on several expected quantities, then investigate the impact of different entrainment rates on the thermals’ fates, and finally explore the role of environmental moisture on cumulus thermals. Section 4 summarizes our findings.

## 2. Simulations and method

This study builds on the work by HS2016 and uses their methodology to identify and track cumulus thermals in LES of tropical convection, which we briefly summarize here. Cumulus thermals closely resemble spheres, so it is a reasonable approximation to assume them as such. The location of a thermal’s center is estimated by assuming it matches the location of its peak updraft velocity, and its trajectory by tracking this point in time. A thermal’s size is then estimated by finding the radius that encloses a volume with average vertical velocity equal to the ascent rate obtained from the trajectory of the peak updraft location. A thermal is then fully specified by the location of its center and the magnitude of its radius over time. For further details about this tracking method and its performance, please refer to HS2016.

Two simulations on LES mode (no cumulus or boundary layer schemes are used) are performed with version 3.8 of the Weather Research and Forecasting (WRF) Model created by the National Center for Atmospheric Research (Skamarock et al. 2008). We run the simulations at 65-m resolution in the horizontal and 400 vertical levels with a vertical spacing that varies from ~20 m in the lower levels up to ~200 m near the top of the model domain, which is 20 km. We use eight acoustic time steps per model time step, which is 0.25 s. We run the daytime convective development over the land TRMM-LBA case (Grabowski et al. 2006), which starts at 0730 local time from a moist initial sounding from the amazon (23 February 1999 in Rondonia, Brazil) and has imposed diurnally varying surface fluxes and diabatic cooling rates. This case generates convection that gradually deepens throughout the day, reaching an altitude of about 12 km by the end of the simulation around 1330 local time. We use periodic boundary conditions, a domain size of 20 km × 20 km × 20 km, and a 1.5-order turbulent kinetic energy closure. The two simulations differ in the microphysics schemes used: LBA1 uses Thompson’s scheme (Thompson et al. 2008) and LBA2 uses Morrison’s (Morrison et al. 2009) scheme. For more details on these model numerics and settings, please refer to Skamarock et al. (2008). Simulation data are available upon request.

Thermals are tracked offline using 1-min output within a 10 km × 10 km × 20 km column throughout three different time periods that roughly correspond to shallow, congestus, and deep convective regimes. These three periods are from 0930 to 1030, 1100 to 1200, and 1200 to 1300 LT in LBA1, and from 1000 to 1100, 1100 to 1200, and 1200 to 1300 LT in LBA2. The choice of these three periods in each simulation is aimed at matching qualitatively three different stages of convection (shallow, congestus, deep), which occur at slightly different times in each case, hence the different times. Nevertheless, for this study we consider the three periods altogether for the different analyses performed, so this becomes irrelevant. Figure 1 shows the time evolution of the two simulations in terms of cloud water content and surface precipitation rate. The aim of these simulations is to generate realistic—but idealized—convective clouds in which we can track a large number of thermals and study their properties during all stages of convection. Both simulations reproduce qualitatively the expected behavior in terms of the gradual deepening of convection and the increase in precipitation rate throughout the day (Grabowski et al. 2006), but with some differences regarding the cloud water content distribution, which of course result from the different microphysics schemes. For example, notice the two layers between 6–8 and 11–14 km that have cloud water content of 0–0.04 g kg^{−1} in LBA2 since the beginning of the simulation. This is due to a weak homogeneous layer of ice generated by Morrison’s scheme from the initial sounding, but has no clear no impact on the convective development of the simulation. The only significant impact would be through radiation processes, but these are not present since we use imposed diabatic cooling rates and surface fluxes. It is not the aim of this paper to investigate these differences, but rather to increase the robustness of our results against the possible impact of the microphysics parameterization used.

Physical properties of thermals such as size, ascent rate *W*, buoyancy *B*, lifetime, and distance traveled Δ*Z* are computed as in HS2016. Additionally, since lifetime has a discrete value (in minutes), we add to each thermal’s lifetime a random number between −0.5 and 0.5 min in order to smooth the corresponding plots; notice that this has no effect on the corresponding statistics and, hence, on the conclusions we reach. We estimate the rate of mixing of thermals with their environment through three different quantities. First is the instantaneous fractional entrainment rate, a direct measurement of entrainment that we denote here as . It is obtained by computing the integrated mass flux across the thermal’s surface in the thermal’s frame of reference and dividing by the thermal’s vertical mass flux, as presented by HS2016 in their Eq. (7). We reproduce this formula here for convenience:

where “ is the step function (only air coming into the thermal counts), is the velocity relative to the thermal, is the unit vector pointing inward from the thermal’s surface’” (HS2016), *ρ* is density, *dS* is a surface element of the thermal, *w* is the absolute vertical speed, and *dV* is a volume element of the thermal. We compute this for each output time step in which a thermal is tracked, and then obtain an average value for each thermal. Notice that since this result is obtained from instantaneous values at each time step of a thermal’s lifetime, it does not include any contribution from the change in size or morphology of the thermal, and thus detrainment equals entrainment within this context. In our identification method we assume that thermals are spherical, which as HS2016 show, turns out to be true on average, and individual thermals do not deviate much or in any systematic way from this shape. Thus, we expect that the contribution from changes in morphology are not significant, and will be mostly canceled out when averaging over a large sample of thermals.

To consider the impact of changes in the size of the thermal, our second measure is the fractional *net* entrainment rate, which is also used following HS2016 and is obtained from the change in size—or, rather, mass—of a thermal: , where *M* is the mass of the thermal and *Z* the altitude of its center. We compute this for every two consecutive output time steps in which a thermal is tracked and then obtain an average value per tracked thermal. Notice that this gives additional information to that given by regarding the relative importance of entrainment and detrainment throughout the lifetime of a thermal. But for example, being close to zero does not necessarily imply there is little mixing. It means that the thermal has entrained nearly as much as it has detrained throughout its lifetime.

Finally, we compute a third measure of mixing, which is a “bulk” estimate that uses a conserved tracer, in this case moist static energy (MSE). Below the freezing level, where MSE is approximately conserved, is estimated assuming that any change in the MSE of a thermal must be due to the entrainment of environmental air that has a different MSE:

where mse_{th} is the MSE inside the thermal—which we take as the average MSE inside the thermal; mse_{env} is the MSE of the thermal’s environment—which we take as the average MSE in a region that extends 10 times the thermal’s radius in all horizontal directions; and is the corresponding fractional entrainment rate. We do not compute for thermals above the freezing level. This type of conserved-tracer estimate of *ε* is widely used in LES studies of shallow convection (e.g., Siebesma and Cuijpers 1995; Siebesma et al. 2003), but as de Rooy et al. (2013) point out, this measure is tracer dependent. It also has strong assumptions; for instance, it assumes that the tracer—in this case MSE—is homogeneously distributed with constant value mse_{th} inside the thermal, and the same is true for the thermal’s environment, which is assumed to have MSE = mse_{env} everywhere in the thermal’s environment. Hence, detrainment is not expected to change mse_{th}, and any entrained air is expected to have MSE = mse_{env}.

Comparing the histograms of the probability distributions and the vertical profiles of *ε* using the three different methods (Fig. 2), we see that the instantaneous method has the highest values, and the average net entrainment is close to zero, just as HS2016 found. The fact that is so close to zero on average suggests that changes in the size of the thermals are not an important component of entrainment, and thus the instantaneous estimate is a reasonable approximation, because thermals are on average entraining as much as they are detraining. However, this instantaneous method may be taking into account fast recirculations of air (e.g., entraining of air that has been recently detrained), which could or could not be considered important for mixing. We argue that within the context of cumulus thermals these recirculations are important for mixing since they are indicative of turbulence and as such they also reflect the intensity of the mixing. However, we are aware that there could still be some overestimation because the actual shapes of the thermals may deviate from the spherical shape we assume, which could cause false recirculations. Although we cannot precisely quantify this overestimation, should nevertheless serve as an upper limit for the *true *

On the other hand, we expect to underestimate *ε* (i.e., the true *ε*). The main reason for this is the assumption that MSE has a constant value everywhere inside the thermal and everywhere in its environment. The fact that this may not be true has two impacts that lead to an underestimation of . First, environmental air with higher MSE than mse_{env} is more likely to be entrained, so the actual MSE contrast in Eq. (2) should be smaller than that considered, which implies that . Second, air inside the thermal with lower MSE than mse_{th} is more likely to be detrained, which would cause mse_{th} to decrease less with height than expected from only entrainment (or even increase with height if no environmental air is entrained!), resulting once more in an underestimation of *ε*. Thus, as an estimate for *ε* has several deficiencies, but it does provide a lower limit for *ε*. Taking the mean values of and as the lower and upper limits for the average *ε*, we obtain for LBA1 and for LBA2.

To study the sensitivities and impacts of *ε*, here we will use . We do not use because as we have pointed out above, this estimate not only underestimates *ε*, but it also underestimates it because of detrainment. In fact, turns out to be highly correlated to : we find correlation coefficients between and of 0.74 and 0.75 in LBA1 and LBA2, respectively, and of 0.27 and 0.26 between and . Thus, is useful for providing a lower limit for *ε*, but it is not a good estimate for *ε* because it is the result of a mixture of *ε* and . From here on, unless otherwise stated, *ε* refers to .

## 3. Results

### a. Common parameterizations of entrainment

Several relationships have been proposed by different authors that express entrainment in terms of other fundamental quantities in order to parameterize it. Thus, these relationships depict entrainment as a consequence of other known properties. Here, we test a few of the most frequently used relationships. Notice that the relationships we test here have in general been proposed in a different context than ours. Usually, these have been brought up with the steady entraining plume in mind and, often, are related to shallow convection. Therefore, a negative result here does not imply that these relationships are *wrong* in the original setting they were proposed. However, fundamental justifications for these relationships are often also applicable to cumulus thermals, but perhaps with different parameters, for instance. Thus, it is still valid to test these relationships here, but having in mind that our results do not necessarily validate or invalidate previous studies that are not in the context of cumulus thermals.

#### 1)

The relationship, where *R* is a measure of the linear horizontal extent of the “cloud,” was first proposed for cumulus clouds by Levine (1959), and further advocated by Simpson (1971). Based on the simple conceptual plume model proposed by Stommel (1947), cumulus clouds are treated primarily as cumulus *towers*, with entrainment into these towers considered mainly through the sides. Assuming turbulent homogeneous mixing along the sides would imply that the fractional entrainment rate should vary as due to the surface-to-volume ratio of such towers. The same relationship would hold for mixing along the entire surface of a sphere, since it has the same surface-to-volume ratio, with *R* its radius. The simplicity of this relationship is very attractive, and has been widely used until the present day, particularly to parameterize turbulent entrainment rates (e.g., Arakawa and Schubert 1974; Tiedtke 1989). To test this relationship with our tracked thermals, we plot *ε* against for individual thermals (black dots in the left panels in Fig. 3), and bin the data by (blue line). This is indeed the best relationship we find. It explains about 21% of the variance, with a slope of ~0.3, slightly higher than the commonly used value of 0.2 (Simpson 1971; Tiedtke 1989; de Rooy et al. 2013). However, notice that for individual thermals there is still a large fraction of the variance that is unexplained.

#### 2)

Based on an dependence of *ε* and the steady entraining plume suggested by Stommel (1947), *ε* should scale as , where *Z* is the height of the plume, or in our case the altitude reached by the thermal. This obeys the expected linear increase of a plume’s radius with height. From a different perspective, if *ε* is proportional to the inverse of the largest eddies, which in turn scale with *Z* (Romps 2010), then *ε* should scale with . This dependence of *ε* has been widely used, particularly in parameterizations for shallow convection (e.g., Siebesma 1996; Jakob and Siebesma 2003; Bretherton et al. 2004), supported by the fact that bulk plume estimates from observations in shallow cumulus tend to show such dependence. Binning our data, we can fit this dependence with a slope of 0.3–0.4 (see second column in Fig. 3), but the explained variance is only 5%–7%. This clearly does not explain the largest variance in *ε*, and we thus agree with Romps (2010) in arguing that no significant dependence of *ε* is found in our results. This is also consistent with the average profiles from Fig. 2, which show no significant height dependence.

#### 3)

Another relationship was suggested by Gregory (2001) arguing that the kinetic energy that entrained air must acquire in order to become part of the updraft must eventually come from the updraft buoyancy. Based on this assumption, the kinetic energy of entrained air would scale with updraft buoyancy, , so , where *B* is buoyancy and is the in-cloud updraft velocity. In other words, from this point of view more buoyancy implies more available energy for ambient air to be entrained into the updraft. This expression is used by the Goddard Institute for Space Studies (GISS) GCM to estimate *ε* within their moist convection scheme (Kim et al. 2011). Taking as a thermal’s ascent rate and *B* its average buoyancy during its tracked period, we can fit a straight line to the binned data and obtain a slope of about 0.3–0.4. This explains close to 15% of the variance, which would make it the second best relationship after . However, once more the largest variance is clearly not captured by this relationship (third column in Fig. 3).

#### 4)

The ECMWF operational model uses an entrainment formulation based on Bechtold et al. (2008), which is relative humidity (RH) dependent. As opposed to the previous relationships presented here, this formulation is not based on any physical mechanism that is easily described, but rather with the aim of constructing a parameterization that reproduces an expected behavior of convection, regardless of the physical mechanism, which is totally valid from a parameterization point of view. The main motivation behind this formulation is to represent the observed positive impact of midtropospheric humidity in convection, which is done in such a way that high environmental relative humidity reduces entrainment, which in turn tends to intensify convection. The formulation is (de Rooy et al. 2013)

where , , is the saturation specific humidity, and is the cloud-base level.

To verify this relationship, we compute the mean profile of relative humidity and at each time step of the model, and assign a value for to each thermal according to its average altitude. The result (right panels in Fig. 3) yields a slope of about 1.7 × 10^{−3} m^{−1} for LBA1, or 0.8 × 10^{−3} m^{−1} for LBA2. The explained variance is around 6%, again a very low value that suggests that this relationship is not useful within the context of cumulus thermals. The difference in slopes between the two simulations is due to the last bin in LBA1, which with only a few thermals causes the slope to increase. This also points out the small amount of robustness of this relationship within the context of cumulus thermals.

The four relationships we have presented here seem to show some truth behind the corresponding dependence, but none of them is able to explain more than ~20% of the variance for individual thermals. Certainly, the relationship has the best results, followed by . However, there is still a large fraction of the variance that remains unexplained. To determine if these entrainment formulations are useful or not, we would need to investigate whether the unexplained variability is due to random errors in the calculation of *ε*, or if it is real in such a way that it has some clear impact on the evolution of the thermals. We will return to this after section 3b, since we need to evaluate the impacts of entrainment in order to do this.

#### 5) Other relationships

Emanuel and Zivkovic-Rothman (1999) and von Salzen and McFarlane (2002) parameterize (lateral) entrainment as proportional to the vertical gradient of in-cloud buoyancy. The reasoning behind this is that an increase in buoyancy with height induces lateral entrainment due to mass conservation. Within the context of spherical cumulus thermals we can estimate the buoyancy gradient as the change in time of the mean buoyancy of the thermal as it rises throughout its lifetime. However, if anything, our results (not shown here) suggest a weak inverse relation of *ε* and . Thus, at least for cumulus thermals, this reasoning does not work. Another similar example is the case of the relationship suggested by Lin (1999), which states that . Based on several cloud resolving model experiments, Lin (1999) obtained a value for *α* close to −1, which has been interpreted or approximated as an inverse dependence of *ε* on buoyancy, (Romps 2010). Lin’s interpretation of this relationship is that high in-cloud buoyancy results from low dilution rates that imply low *ε* values, and low in-cloud buoyancy must result from strong dilution rates (i.e., high values of *ε*). Notice that these two relationships are conceptually built upon an impact of entrainment on buoyancy, rather than the opposite. This approach is dealt with in the next subsection.

### b. The impact of entrainment on the fate of thermals

Can we predict other thermal properties from entrainment? In other words, does entrainment determine to some extent the dynamical and physical properties of thermals? At first sight, Fig. 4 suggests that the answer is yes, but certainly not 100%. The binned values of *B*, *W*, and have a clear tendency of increasing for smaller values of *ε*, whereas *lifetime* does not show this tendency. Not surprisingly, *W* is the most clearly impacted variable, with the binned values explaining more than 40% of the total variance. Much more spread can be seen in —whose binned values explain about 20% of the variance, *B*—whose binned values explain 10%–13% of the variance, and even more in lifetime—whose binned values only explain 5%–7% of the variance. The larger spread in compared to *W* can be understood from the fact that lifetime does not increase for small values of *ε*. Rather, it shows a slight decrease for , so although thermals are faster, they live slightly less so that does not increase as steeply and consistently as *W*. The fact that lifetime does not further increase for smaller *ε* below 2 × 10^{−3} m^{−1}, but rather decreases, is slightly puzzling. We hypothesize that the reason could be that these very fast thermals are more difficult to follow by our tracking algorithm, and/or have a higher chance of encountering obstacles such as turbulence or other thermals as they rise, so that they simply break sooner. Nevertheless, the little sensitivity of lifetime to *ε* seems quite robust, and supports HS2016’s view of convection as a succession of many small, short-lived bubbles instead of a few long-lived plumes that extend throughout the entire depth of the clouds.

Romps and Kuang (2010) pose the question of “nature versus nurture,” which investigates whether cloud-base properties or entrainment history are more relevant in determining the fate of a convecting air parcel. They find that entrainment history is a much better predictor than cloud-base properties, favoring nurture over nature. Böing et al. (2012) reach the same conclusion using Lagrangian particle trajectories in LES of shallow to deep convection. Within our framework the question scales down to individual thermals, which are much shorter lived than convective air parcels and, thus, may behave differently. In fact, a single convective air parcel may travel within several thermals throughout its ascent from cloud base to cloud top. Thus, we pose here the question of whether the entrainment history or initial conditions are more important for the fate of individual cumulus thermals. Figure 4 suggests that entrainment definitely has an impact, particularly on *W* and , to a lesser extent on *B*, and very little or not at all on a thermal’s lifetime.

Figure 5 investigates the impact of initial buoyancy and initial ascent rate on several thermal properties. The clearest impact of is on *W* with the binned averages explaining more than 30% of the variance. On the other hand, has a stronger impact on (30% of variance explained) than on *B* (~20% of the variance explained), and in both cases lifetime is the least sensitive, with an explained variance of 10% or less, but still seems to respond slightly to . Thus, Figs. 4 and 5 suggest that both entrainment history and initial conditions play an important role in determining the fate of a cumulus thermal, with lifetime only slightly being affected by the initial buoyancy. This can be clearly seen from Fig. 6, where we show *W*, , and lifetime associated with the initial buoyancy for three categories of thermals based on their *ε* values: low *ε *, mid-*ε *, and high *ε *. These three categories roughly split the cases at the 15th and 85th percentiles. Notice how these relationships shift upward as *ε* decreases, except for lifetime. A similar result is found when these quantities are plotted against instead of (not shown here), but with slightly smaller differences between *ε* categories, and even less dependence of lifetime on than on .

We now turn back to the question that remained open from section 3a: is the unexplained variability from the formulations of entrainment as a consequence real, or simply due to noise or errors in the computation of *ε*? We test this by taking a subsample of thermals in which the quantity to test (say, ) does not vary much, and verify if the remaining spread in *ε* from this population has an impact on the fate of the thermals, now that we know that *ε* does have such impacts. In other words, if the spread is caused by errors or noise in the computation of *ε*, we would expect that the thermals from the subsample with similar properties (say, ), would not show strong impacts on *W*, , or lifetime due to the differences in *ε*. On the contrary, if we find similar impacts on these properties as we found in the entire population of thermals due to the remaining scatter in *ε*, then that would suggest that the unexplained variability of *ε* is not simply due to errors in the computation of *ε* or noise.

For each of the four formulations of entrainment as a consequence described in section 3a, we split the thermals into six groups of equal numbers of thermals organized according to one particular variable (e.g., ). From these six groups we choose the one in which this variable varies the least, and then produce scatterplots as in Fig. 4 but for this subset of thermals instead of the entire sample. Figure 7 shows this for LBA1 (for LBA2 the results are very similar). The resulting range of the corresponding formulation in each case is shown in the top panel, as well as the percentile values of the threshold values. Despite the much smaller range of variations of each formulation in each case, we see that *W*, , and lifetime show very similar features to the case with the entire population of thermals. Perhaps the only exception is in the third column regarding the impact on *B* and *W*. However, this is expected since by narrowing the range in we of course also narrow the ranges of *B* and *W*. The fact that the impacts on and lifetime are similar to those with the entire sample of thermals confirms this. Thus, we argue that the unexplained variability of the different formulations of *ε* from section 3a is still mostly real, and not only due to noise or errors in the computation of *ε*.

### c. Entrainment and the role of environmental moisture

From the preceding sections we can conclude that both entrainment and the initial conditions are relevant for the fate of a thermal. However, a large fraction of the variability in quantities like *W*, , or lifetime remains unexplained by variations in *ε* or the initial conditions (e.g., or ). A third factor that can affect both the initial conditions of a rising thermal and the possible impact of *ε* as the thermal rises is its saturated fraction, and that of the environment through which it rises. It is well known that the presence of free-tropospheric moisture is important for the development of cumulus clouds (e.g., Sherwood et al. 2010; Takemi 2015), because the entrainment of moist air into the cloud is expected to contribute to the development of convection. But how does this operate at the scale of individual thermals? Do thermals benefit from relatively moister surroundings, or do they benefit more from their initial internal moisture? Note that this is tightly connected to mixing: if little or no mixing takes place, a thermal that is moister than its environment would have an advantage against less moist thermals due to the release of latent heat as condensation takes place. But if strong mixing takes place, this relative advantage will decrease since mixing will tend to homogenize any temperature and moisture contrasts.

Here, we explore how the initial average relative humidity inside a thermal RH_{th} and that of its environment RH_{env} impact the thermal’s buoyancy averaged throughout its tracked lifetime *B*. For this we consider that the environment through which a thermal rises is a rectangular column of 500-m height with a width of 6 times the thermal’s mean radius, which starts just above the thermal as it is first identified. To explore the impact of RH_{th} on *B*, we “fix” RH_{env} by selecting a subsample of 30% of the total population of tracked thermals that have the closest values of RH_{env} to the median of the population. This gives us a large enough population of thermals with the smallest variation in RH_{env}. We divide this subsample of thermals into two, according to their value of *ε*, and plot *B* as a function of RH_{th} (left panels in Fig. 8). Next, to explore the impact of RH_{env} on *B*, we do the same but by extracting a subsample of thermals with little variation around the median of RH_{th} (right panels in Fig. 8).

Notice that the median of RH_{th} is higher than the median of RH_{env}, so the cases we consider here are of thermals that rise through a relatively drier environmental air, which is of course the most common situation in cumulus convection. Notice that our definition of cumulus thermals allows for noncloudy air to make part of a thermal, as opposed to the classical view in which all air inside a cloud thermal should be cloudy (i.e., with RH_{th} = 100%). We find that RH_{th} has a clear impact on the mean buoyancy of a thermal (left panels in Fig. 8). Thermals initially burdened with more noncloudy air (lower RH_{th}) lose more buoyancy than those with a lower fraction of noncloudy air (higher RH_{th}). Also, the higher the value of *ε*, the more noncloudy air will be incorporated into the thermal as it rises—because the environment is drier, leading to a further reduction in *B*. This constitutes further evidence that our novel approach of including noncloudy air in cumulus thermals is correct, since we do detect an impact of this noncloudy fraction of air in the subsequent dynamics of thermals.

On the other hand, a moist environment is expected to be less detrimental for a rising thermal than a dry environment, because mixing with a dry environment will tend to dilute buoyancy faster than mixing with a moist environment. Our tracked thermals suggest that the dependence of *B* on RH_{env} is weaker than its dependence on RH_{th}. It is hard to identify any impact from RH_{env} on the low entraining thermals, whereas for the high entraining thermals there is a slight positive impact from moister environments. This is evident for the lower values of RH_{env} (88%–92%), but above this range the impact disappears. This suggests that if we were to test this for much drier environments (perhaps for relative humidities below 80%), the impact might be much stronger. This could be important for the leading thermals at the top of the cloud and those near the edges, but for most of the thermals that are embedded within the cloud their environment will not be that dry, and hence their internal relative humidity will be more important than that of their immediate environment.

Note that this does not imply that environmental moisture is unimportant for thermals, since their initial saturated fraction will be determined by the environment anyway. What we find here is that once a thermal is rising, the environmental moisture through which it rises is of less importance than its initial internal moisture, at least within the relative humidity range that we can span in our simulations, which is indeed narrow due to the high environmental humidity in our simulations. A larger range of relative humidity might reveal a stronger impact, but from our results we can argue that the initial relative humidity of a thermal is the main driver, not the environmental moisture with which the thermal mixes as it rises. However, since thermals are so short lived and they are initiated at various heights, environmental moisture becomes relevant as the input for internal moisture of thermals as they initiate.

## 4. Summary and discussion

Based on the methodology of HS2016, we track thousands of cumulus thermals in two large-eddy simulations (LESs) of convection based on the TRMM-LBA case of daytime convective development over land. We use these thermals to investigate how entrainment can be expressed in terms of known properties of the thermals, and to study how entrainment impacts thermal properties as well as its relative importance compared to the initial conditions. Our aim is to gain a fundamental understanding about the role of mixing at the scale of the building blocks of atmospheric moist convection: cumulus thermals.

We explore commonly used expressions that predict entrainment from other updraft properties. Although such relationships have been often devised with a different conceptual model in mind—for example, the steady entraining plume—the physical reasoning behind them can usually also be applied to cumulus thermals, so we explore them knowing that our results are not necessarily directly comparable to previous studies based on different conceptual models. The best skill for predicting average fractional entrainment rate *ε* is provided by the relationship , which explains about 20% of the variance, followed by , which explains about 15% of the variance. We do not find any significant dependence of *ε*, consistent with the fact that thermals do not grow or shrink as much as they rise (HS2016), as opposed to what would be expected from plumes of constant buoyancy sources (e.g., Stommel 1947, 1951).

The impact of entrainment on thermal properties is investigated by exploring how certain quantities like ascent rate *W*, mean buoyancy *B*, vertical distance traveled , and lifetime are affected by *ε*. We find that low values of *ε* effectively translate into high values of *W*, with more than 40% of the variance explained by this. In addition, and *B* are impacted by *ε*, but less consistently, and with less explained variance than *W*. On the other hand, lifetime shows little sensitivity to *ε*.

Initial conditions, in particular initial buoyancy and initial ascent rate , are found to strongly influence *W*, *B*, and , while only very weakly lifetime. This, together with the results on the impact of *ε*, suggest that both entrainment history and initial conditions are relevant for the fate of cumulus thermals, except regarding their lifetime. Thus, as opposed to what Romps and Kuang (2010) and Böing et al. (2012) find for convective air parcels, the fate of cumulus thermals is not only determined by their entrainment history, but also by their initial conditions as they start to rise. This of course has to do with the fact that cumulus thermals have short lifetimes [~4–5 min on average (HS2016)], compared to convective air parcels, which may remain in a cloud for several tens of minutes. The fact that thermal lifetime seems not to respond in any significant way to either *ε* or initial conditions supports the view of HS2016 of convection as a succession of many short-lived thermals instead of a few organized and extended updrafts.

Another factor that is important for the fate of thermals is moisture. We find that a thermal’s initial saturated fraction has an important impact on the thermal’s buoyancy, which is reduced with stronger mixing, as expected. On the other hand, the environmental moisture through which thermals rise does not impact the thermals’ buoyancy as much as the thermals’ internal moisture does. However, the range of environmental relative humidity that can be tested in our simulations is relatively narrow (88%–96%), and we do find a slight negative impact at the lower end of this range. Thus, for thermals that would be exposed to much drier environments, this could become an important factor. On the other hand, our result does not imply that environmental moisture is unimportant for cumulus thermals, since these thermals have short lifetimes and initiate throughout the cloud depth; the way convection grows is not through thermals traveling farther, but through more thermals initiating higher up (HS2016). Thus, environmental moisture has an impact by feeding moisture into these newly initiating thermals at higher altitudes, but not so much by affecting their individual ascents as they rise through it. This provides a simple explanation for the relevance of midtropospheric moisture for convection (e.g., Sherwood et al. 2010; Takemi 2015), and perhaps a hint on how to implement it in parameterizations: if we imagine deep convective clouds as a large population of small, short-lived thermals that initiate at various levels, a moist layer in the midtroposphere is the perfect fuel for initiating more vigorous thermals from higher altitudes.

## Acknowledgments

This work was funded by the Australian Research Council Centre of Excellence for Climate System Science (ARCCSS), and DHD was also supported through Agreement FP44842-134-2015 with the Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José de Caldas and the Universidad Nacional de Colombia (UNAL). The authors thank three anonymous reviewers who provided valuable feedback that has significantly improved this paper.

## REFERENCES

*Workshop on New Insights and Approaches to Convective Parametrization*, Reading, United Kingdom, ECMWF, 25–57, https://www.ecmwf.int/sites/default/files/elibrary/1996/12223-mass-flux-approach-atmospheric-convection.pdf.

## Footnotes

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