Let us consider a population of objects (that we will name “thermals” in the following, but they could be clouds or updrafts in general) randomly placed in space following a uniform distribution, characterized by an averaged length L0, a density D0, and an exponential size distribution: 2S0(x)=1L0e-xL0 For the reasons explained above and in Appendix , it is assumed that the merging takes place between objects of effective size βL0 with β≥ 1. It is possible to compute the size distribution of thermals after merging, by distinguishing the two types of thermals that emerge: those that have merged and those that have not merged yet. The analytical treatment (detailed in the Supplement) shows that after letting the thermals merge once or several times, the size distribution of the thermals that have not merged is written: 3Sunmerged(x)=1L1e-xL1withL1=L01+βD0L0 and the size distribution of the thermals that have merged is asymptotically exponential for large thermal sizes (x ≫ L0): 4Smerged(x)=1L2e-xL2withL2=L0eβD0L0

Moreover, analytical expressions are also derived for the density of unmerged thermals, D1, and the density of merged thermals, D2, as follows: 5D1=D0e-βD0L01+βD0L06D2=1-e-βD0L01+βD0L0(1+βD0L0)2βL0(1+eβD0L0)

After merging, the size distribution of thermals can thus be written as a sum of two exponential functions as written in Eq. (), with L1 and L2 defined as above and p1+p2=1. The calculations indicate that: 7p1=11+(1+βD0L0)e-βD0L08p2=(1+βD0L0)e-βD0L01+(1+βD0L0)e-βD0L0

These calculations, illustrated by Fig. , thus show that the merging of thermals that are characterized initially by an exponential size distribution of length scale L0 produces a second population of thermals, and that the size distribution of the thermal population after merging can be represented by the sum of two exponential functions, characterized by two length scales L1 and L2. In the absence of merging (βD0L0=0), L1=L2=L0 (Fig. a) and p1=p2=0.5 (Fig. b): the size distribution can be represented by a single exponential. However, as the merging efficiency βD0L0 increases, L1 decreases (because the smaller thermals are statistically less likely to be affected by the merging process) while L2 increases (because the merging produces larger thermals).

The total density of thermals after merging (D, which is always ≤ D0) is the sum of the densities of non-merged and merged thermals. Using Eqs. (5) and (), we obtain the following analytical expression: 9D=D1+D2≈D0e-βD0L01+βD0L0+1-e-βD0L01+βD0L0(1+βD0L0)2βL0(1+eβD0L0) Solving the equation ∂D1∂(βD0L0)=0 shows that there is an optimal βD0L0=1φ≈0.618 (where φ=1+52 is the golden number) which maximizes the total number of non merged thermals, and that D reaches a maximum value Dcrit≈0.34βL0 for βD0L0=βD0L0crit=0.83. There is therefore a critical merging efficiency of thermals beyond which the merging becomes so efficient in producing larger but fewer thermals that the densities of thermals before and after merging become anti-correlated (Fig. c). In other words, the D-βD0L0 curve is concave down, with a local maximum at βD0L0=0.83.

Although the physical meaning of L1 and L2 is clear (these length scales relate to the mean chord lengths of unmerged and merged thermals, respectively), the physical meaning of p1 and p2 is not so clear. When βD0L0→0 (i. e., no merging), Eq. () and Fig. b show that L1=L2 and p1=p2=0.5. However, when L1=L2, any values of p1 and p2 satisfying p1 + p2 = 1 (including p1=1 and p2=0) would describe the same (single) exponential size distribution. Therefore, p2 should not be interpreted as the proportion of thermals in the second population (it is just an asymptotic approximation) and the ratio D2D1+D2 is a better measure of the proportion of merged thermals than p2. In addition, the influence of merging on the size distribution is best described by L2-L1 or (as will be shown later, Fig. b) by the non-dimensional quantity L2-L12L0, and the absence of merging is best described by L2→L1 or by the density of merged thermals D2→0.

These calculations thus support our hypothesis that the second exponential of the size distribution results from the merging process. Reciprocally, they also show that it is possible to infer the properties of the thermal population before merging from the size distribution of thermals after merging (characterized by L1 and L2): by combining Eqs. () and () to eliminate L0, we obtain: 10βD0L0=WLeL2L1-1 where WL is the Lambert W function satisfying x=W(x)eW(x) and 11L0=L2e-βD0L0orL0≈L1L2 (the second expression for L0 is obtained after a multiplication of Eqs. and , followed by a first order Taylor expansion of the exponential function). Moreover, as shown in the Supplement, the coverage fraction of thermals can be expressed as: 12f=1β(1-e-βD0L0). Therefore it is possible to infer β in the observations from Eqs. () and ().

Although analytical calculations support the hypothesis that the merging process is sufficient to explain the presence of a mixture of exponential distributions, they are based on a number of mathematical simplifications that were needed to make the calculations tractable. Therefore we test the validity of the theory and further test the hypothesis that merging can explain the second population of chords in the size distribution of thermals, by developing a simple one dimensional statistical model.

We assume that initially the thermals are uniformly and randomly distributed along a domain of length Ldomain=1000 km with a mean spacing λ0=1/D0, following a Poisson process, and that they have an exponential size distribution (Eq. ) of characteristic size L0=100 m (this value is chosen to be close to the averaged thermal length measured in the surface layer, Fig. a). For the sake of simplicity, we assume β=1.

The thermals are placed onto the domain one at a time. Everytime one is placed, it is checked whether the new thermal overlaps with an already existing thermal. If so, then these thermals are merged such that the edges of the new thermal is the leftmost extent of the leftmost old thermal and the rightmost extent of the rightmost old thermal (Fig.  in the Appendix), as assumed also in the mathematical calculations. After the merging processes takes place, the coverage fraction is counted. If this coverage fraction is less than a pre-specified value fTH, then the simulation proceeds by placing a new thermal, checking for overlap, merging if there is overlap, then computing the coverage fraction again. This continues until the coverage fraction in the simulation equals fTH. Once they are equal, the simulation is ended. The simulation is then repeated 10 times to generate more statistics. The thermal positions and lengths are recorded both before and after the merging process takes place.

Figure a shows the chord length distributions obtained through this process for a range of fTH values, which (from Eq. ) amounts to a range of βD0L0 values and thus of merging efficiencies. In the case of weak merging efficiency, the final distribution is close to the initial exponential distribution. However, for stronger merging efficiencies we note the formation of larger chords and an increasing deviation from the initial distribution, with the formation of a long tail. Each final distribution turns out to be well fitted by a sum of two exponentials. As merging is the only process represented in this model, it shows that if the initial size distribution of thermals is exponential, merging is a sufficient process to explain the formation of a second population of larger thermals and produce a final chord length distribution that is well fitted by a double exponential.

This is further confirmed by Fig. b that shows the decomposition of the size distribution into merged and unmerged thermals for a given merging efficiency. Although the size distribution of the thermals that have not merged yet is exponential and associated with a shorter lengthscale than the initial distribution (L1<L0), the size distribution of the thermals that have merged tends, for large chord lengths, to an exponential distribution. This is in line with the theory that predicts that the distribution of merged thermals is only asymptotically exponential (that is, for lengths much larger than L0).

We then use the simple model to assess the ability of the analytical calculations to predict L1, L2 and the thermal densities. Although not perfect, we note a fairly good agreement between simulations and theoretical predictions, both for the length scales (Fig. c) and for the evolution of the densities of merged and unmerged thermals with the merging efficiency (Fig. d). These results give us confidence in the validity of the analytical treatment, and encourage us to use this theory to interpret the observations.

Given that trade wind clouds are rooted in subcloud layer thermals , we now investigate how the merging of boundary layer thermals imprints the size distribution of clouds near their base. For this purpose, we first assess the extent to which the physical framework presented in Sect.  can help interpret EUREC⁴A observations (summarized in Tables  and ). From Eqs. () and () and the length scales L1 and L2 inferred from the observed chord length distributions, we infer L0 and D0. From Eq. () and the fractional coverage of thermals or clouds measured for each flight, we infer β. Then, from the values of (D0, L0, β) associated with each flight, we compute the density of thermals expected from the theory (Eq. ) and compare it with the density that was actually measured during the campaign.

For most of the flights there is a good agreement, both in the subcloud layer and near cloud base (Fig. a). Since the measured thermal density was not used to diagnose (D0, L0, β), this can be considered as an independent consistency test of the theory with the observations. Moreover, since the theoretical prediction of the density is based only on the effect of the merging process, it suggests that the variability of the thermal density over the course of the campaign primarily reflects the effect of the variability of the merging process on the thermals field. Nevertheless, there are a few discrepancies at the lowest density values, where observations report a higher thermal density than predicted by the theory. In these cases, the thermal density seems to depend not only on the merging process, but also on other factors. These factors probably include the influence of the low-level convergence associated with the circulations created by cloud updrafts or shallow mesoscale circulations such as those revealed by , which can increase the thermal density below the clouds but are not included in the merging theory, nor in the simple statistical simulations.

In the analytical calculations, the strength of the merging process is quantified by βD0L0. As the merging of objects of initial length scale L0 results in a size distribution of objects characterized by length scales L1≤L0 and L2≥L0, we expect L2-L1 to vary together with βD0L0. This is indeed what we find (Fig. b), with (L2-L1) varying linearly with βD0L0 and (L2-L1)/(2L0)≈βD0L0 for both thermals and clouds. It suggests that the metric (L2-L1)/(2L0)≈(L2-L1)/(2L1L2), which is derived directly from the fit of the observed chord length distributions, can be used as a simple proxy for the merging efficiency of thermals or clouds.

The variation of the thermal density with the merging efficiency of thermals is shown on Fig. c. As predicted by the theory and the simple model (Sect. ), the correlation between these two variables is positive for weak merging efficiencies and negative for stronger merging efficiencies. This anti-correlation is explained by the fact that the merging process produces larger but fewer thermals, which reduces the thermal density. However, we note that in observations the anti-correlation starts at a lower value of the merging efficiency than in Figs. c or d. This is because in Nature the effective area of influence of a thermal is larger than the thermal size itself (β>1) owing to the circulation induced by the thermal around it , and there is a positive correlation between the merging efficiency and β (Fig. d). This makes the merging even more efficient in reducing the thermal density than in the absence of such a circulation.

Figure d suggests that the flight-to-flight variability in merging efficiency is primarily governed by variations in β. Therefore it is important to clarify the physical interpretation of this parameter.

As explained in Sect. , β was introduced in the merging framework to capture the ability of convective objects to interact and attract each other without direct contact, thereby facilitating merging. For thermals or clouds, which transport air upward in an updraft, such interactions can arise from the circulations induced around them as a consequence of mass conservation (, Appendix ). In this context, β can be interpreted as the radius of influence (or basin of attraction) that an object exerts on its surroundings through the circulation it generates. In other words, β corresponds to the region where a given thermal can capture its neighbours through the circulation it creates. Since objects probably move to achieve the merging process, the amount of movement depends on β. However, β may also encapsulate other mechanisms, such as the effects of imposed mass convergence in the subcloud layer (for instance, induced by an overlying cloud or associated with an external circulation), which increases thermal density and thereby enhances merging.

Several other interpretations can be inferred from the definition of β (Appendix ): β=1+HhTlifeTtransit, where Tlife is the lifetime of the updraft and Ttransit the time necessary for an air parcel to travel from the bottom to the top of the updraft. The ratio Tlife/Ttransit can be viewed as the number of successive warm bubbles (Nbubbles) that rise over the course of the life of an updraft. Therefore, a persistent, long-lived convective system will be associated with a large β.

Finally, for clouds β can also be expressed in terms of the ratio between the area of the anvil of the cloud and its core size. Indeed, if the cloud core size is Lcore=LA, and uA is the horizontal velocity of the outflow layer, the cloud anvil size is given by Lanvil=2uATlife. Therefore we get: LanvilLcore=2uATlifeLA=Hh′TlifeTtransit where h′ is the depth of the outflow layer at the cloud top and H is the depth of the updraft. β is thus directly related to the (aspect) ratio between the size of the anvil and the size of the cloud core.

To summarize, β quantifies the effect (in space and time) of convective-scale circulations on the merging of thermals or clouds. It increases with the radius of influence that a convective object exerts on its surroundings through the circulation it generates, with the lifetime of the convective object and, in the case of clouds, with the aspect ratio of the cloud field: 13β=1+HhTlifeTtransit14=1+HhNbubbles15=1+h′hLanvilLcore

Because the life time of an updraft is usually at least as long as the transit time, β is always larger than 1, and is typically of a few units (it actually ranges between 1 and 5 in the case of thermals, Fig. d). However, as shown later (Fig. d), it can reach much larger values (5 to 30) for long-lived updrafts that typically produce extensive anvil clouds, as observed in Flowers.

Having checked the consistency of the observations with the theory, we can now further interpret the observations in the light of the merging theory. Using Eqs. () and () we can estimate the length scale of objects that, after merging, would lead to size distribution length scales (L1 and L2) similar to those observed, and thus obtain clues as to the origin of the merged objects. This is done using the thermal chord length distributions measured near the ocean surface, in the subcloud layer and near cloud base and using the cloud base width distributions.

Figure a shows that in the surface layer, L0TH estimates (97 m ± 24 m) are very close to the mean size LTH of thermals (97 m ± 22 m) and to L1TH in that layer (Table ). It suggests that at that level, thermals experience very little merging and remain largely independent of each other. Within the subcloud layer and near cloud base, on the other hand, L0TH estimates (159 ± 27 and 172 ± 26 m, respectively) are close but smaller than the mean thermal sizes found at the same level (LTH=170±45 m in the subcloud layer and 204 ± 42 m near cloud base). The thermal size distributions measured within the subcloud layer and near cloud base are thus consistent with those expected from the merging of thermals through the depth of the subcloud layer.

Figure b shows the L0CLD values inferred for each flight from L1CLD and L2CLD (Fig. ). In this case, L0CLD is bimodal: in the presence of a single cloud population, L0CLD≈L1TH (measured near cloud base or in the subcloud layer) ≈L1THsat, while in the presence of two cloud populations, L0CLD≈L2TH (measured near cloud base or in the subcloud layer) ≈L2THsat (the close relationship between the thermal length scales and L0CLD is further illustrated in Fig. S2). Moreover, the density of clouds prior to merging D0CLD correlates well with the density of cloudy thermals DTHsat (Fig. c) or updrafts DTHup and is of a similar order of magnitude. In fact, D0CLD is slightly higher than DTHsat, suggesting that the merging may involve not only cloudy thermals but also, to a lesser extent, clouds that are not – or no longer – rooted in active thermals.

It thus appears that unmerged thermals that overshoot the LCL form the first population of (very shallow) clouds, and merged thermals that overshoot the LCL generate cloud shoots which, after merging with each other and/or with unmerged saturated thermals, form the second population of clouds, that are on average wider and deeper. The merging of thermals and cloud shoots thus exerts a strong control on the type of clouds present.

A schematic of the impact of the merging process on thermals and clouds is represented in the lower half of Fig. : turbulence near the surface produces a large density of thermals. As they rise across the depth of the subcloud layer, some of them merge and become wider. This results in two thermal populations coexisting in the subcloud layer and near cloud base: those that have merged (of length scale L2TH), and those that have not merged yet (of length scale L1TH). As a result of merging, the thermal density decreases with height. The thermals that overshoot the lifting condensation level (about one out of five on average during EUREC⁴A) saturate at their top and form “cloud shoots” whose base has initially the same size as the saturated thermals that produced them (L0CLD≈L1TH≈L1THsat or L0CLD≈L2TH≈L2THsat, Fig. b). As will be shown later (Fig. c), a higher density of thermals (DTH) is associated with a higher density of saturated thermals DsatTH (Fig. S5, consistent with the fact that when the density of thermals is high, the boundary layer is moister and the LCL is lower) and thus a higher density of “cloud shoots” (D0CLD, Fig. c). When cloud shoots form close to each other (which occurs more easily when thermal merging is weak and thus the thermal density around cloud base is high), they can merge. It forms larger bases and leads to the formation of wider and deeper clouds.

Each ATR flight was typically associated with two to three hours of in-situ and remote sensing measurements around the cloud base level. During this time, HALO was dropping 3 × 12 dropsondes along three consecutive, 200 km diameter circles . From these dropsondes, a horizontal wind divergence and then an area-averaged mesoscale vertical velocity could be estimated . From the vertical velocity measured around cloud base (Wb) and an analysis of the subcloud layer mass budget , an area-averaged mass flux Mb could also be estimated . In addition, by using high frequency (25 Hz) in-situ measurements of vertical velocity and humidity from the ATR , we could estimate a linear cloud-base mass flux along the ATR trajectory as MbATR=Σiρiaiwi, with ai=H(rhi-0.98)/N, where H is the Heaviside function, rhi and wi are the relative humidity and vertical velocity (assuming zero mean vertical velocity over each 30 km segment) measured in each point i of the trajectory, N is the total number of measurements made at the cloud base level for each flight, and ρi is the air density assumed to be 1 kg m⁻³ for simplicity (see for a justification of these approximations).

Despite differences during flights where the spatial scale of the cloud organization was larger than the region sampled by the ATR (e.g. RF14), the two independent Mb estimates exhibit the same large flight-to-flight variability, and both correlate positively with the density of thermals near the cloud base level (Fig. a). From we know that Mb co-varies with the mesoscale vertical motion around cloud base (Wb), and indeed ascending branches of mesoscale circulations (Wb>0) tend to be associated with stronger Mb than subsiding branches (Fig. b). We also note that shallow mesoscale circulations tend to be associated with a heterogeneous distribution of thermals, as thermals are more concentrated in regions of mesoscale ascent and low-level convergence and more sparse in regions of low-level divergence. However, the relationship between thermal density and Wb exhibits some outliers. They may be due to the presence of cold pools (e.g. during RF17 and RF18 that were associated with a strong precipitation), which affect the low-level divergence and therefore the measurement of mesoscale vertical motions , and likely modulate the distribution of thermals. The relationship between the thermal density inferred from ATR turbulence measurements and the Mb or Wb estimates inferred from HALO dropsondes might also be affected by the different area and time samplings of the two aircraft (e.g. during RF17).

What controls the magnitude of Mb? Since the cloud-base mass flux is known to be more strongly modulated by the cloud size than by the in-cloud vertical velocity , we expect higher values of Mb to be related to the presence of wider clouds. Indeed, when two cloud populations are present, Mb increases with the length scale of the second cloud population (Fig. c), which increases with the merging efficiency of clouds (Fig. d, Sect. ). The flight-to-flight variations in Mb can also be interpreted as a result of variations in the thermal population. Noting that Mb can be well approximated by the product of the mean density, length and vertical velocity of cloudy thermals Mb≈wsatTH⋅DsatTH⋅LsatTH (Fig. S6), it appears that Mb variations are primarily governed by DsatTH variations (and to a lesser extent by wsatTH variations), which are roughly proportional to variations of total density of thermals DTH (Fig. S5). In other words, a weaker thermal merging is associated with a higher density of thermals (DTH) and saturated thermals (DsatTH); this leads to a higher density of cloud shoots (D0CLD, Fig. c) and thus promotes cloud merging and the formation of wider cloud bases (L2CLD increases), which eventually leads to a stronger mass flux.

However, we note that sometimes a strong mass flux can occur in the absence of cloud merging (Fig. c): in RF15 we observe only one cloud population, and the strong mass flux comes from the many small clouds that form on top of a very high density of thermals. In fact, the comparison of RF15 with the following flight (RF16, which occurred a few hours later on the same day) shows that the many small clouds of RF15 later began to merge and form a second population of clouds with wider cloud bases.

What role does the thermal-cloud coupling play in mesoscale circulations? The theoretical study of showed that cumulus mass fluxes favor the development of mesoscale ascents, and the modeling study of showed that the low-level mass convergence associated with mesoscale ascents increases the density of thermals in the subcloud layer. EUREC⁴A observations support these results, but also suggest that a high density of thermals will eventually favor thermal merging, resulting in fewer and wider but more widely spaced thermals. This will reduce cloud merging, and thus Mb, potentially to the point where Wb will become less ascending or even descending. In this way, thermal merging is likely to temper, or act as a negative feedback, on the growth of shallow mesoscale circulations.

A large variability of cloud mesoscale patterns was observed during the EUREC⁴A campaign , with the occurrence of each of the four known prominent patterns of tradewind cloudiness . However, most flights were associated with a mixture of cloud mesoscale organizations, and sometimes the ATR was sampling an area smaller than the scale of the cloud pattern itself (e.g. during RF09 the ATR spent most of its flight time in between the cloud systems that constitute the Flower pattern, ). In Fig. , we highlight the five flights associated with only one cloud population (in green), and six flights (out of 14) associated with two cloud populations and either high or low thermal densities (in orange and red, respectively). An additional flight is highlighted (RF08), which is associated with only one population of (large) thermals (Table ). The cloud patterns present on these different days are illustrated with satellite images (bottom of Fig. ). How do they differ in terms of thermal and cloud merging?

Figure a–b show that the measured thermal density is anti-correlated with the strength of thermal merging, and that cloud merging is anti-correlated with thermal merging. These features can be explained as following: when there is little thermal merging, the thermals are small but numerous (DTH is large), and therefore the cloud shoots rooted in these thermals form close to each other. Since βCLD>βTH, the clouds merge more easily than thermals, forming wider cloud bases (Fig. d). In contrast, a strong merging of thermals leads to wider but sparser thermals (DTH is small); the cloud shoots forming on top of these thermals are thus initially wider (because L0CLD=L2TH and L2TH increases with thermal merging) but are more widely spaced and therefore they merge less easily.

Figure b suggests that the Gravel pattern corresponds to a minimized thermal merging but maximized cloud merging, and Flowers to a maximized thermal merging and minimal cloud merging. Therefore, when two cloud populations are present: the Gravel pattern maximizes cloud base widths (and in-cloud vertical velocities at cloud base) and thus the convective mass flux, while the Flower patterns minimizes the cloud base widths (and in-cloud vertical velocities) and the convective mass flux (Fig. c–d). It is consistent with the observation that the clouds embedded in the Gravel pattern are often deeper and associated with a higher rain rate than those embedded in Flowers .

Figure a–b show that the situations with only one cloud population (and thus no cloud merging by definition) occur for a wide range of thermal densities and merging efficiencies. The clouds that form in these cases are very small and shallow because they are rooted in small, unmerged thermals (Sect. , Fig. b). In the absence of other cloud types (such as in RF06), this corresponds to a Sugar pattern . However, even in the presence of other cloud types, such clouds are also found because merged and unmerged thermals often co-exist. Therefore, Sugar-like clouds are present in all cloud mesoscale patterns, albeit in a varying proportion that depends on the thermal merging efficiency. When the thermal merging efficiency increases, the effective factor of thermals increases more quickly than their own lengthscale (i.e. βTH≫1, Fig. d). It makes the merging more and more efficient and thus the very shallow clouds more and more sparse. This explains why, in satellite images, the areas between the deepest clouds are less filled with very shallow clouds (and thus appear darker) in the case of Flowers (that are associated with strong thermal merging) than in the case of Gravel (Fig. c). It also explains why on a given day associated with a Flower pattern (e.g. on 2 February 2020), the ATR sampled one cloud population on one flight (RF09) when flying in-between the deep clouds, and two cloud populations on the other one (RF10) when flying across the deep clouds.

Interestingly, we note that the cloud merging efficiency of the different flights (0.85 ± 0.10) is always close to 0.83 (only the Gravel patterns are associated with higher efficiencies). It means that the coupling between thermals and clouds is such that it maximizes the cloud density (Sect.  and Fig. c). Since the Gravel patterns are associated with a high thermal density but low cloud densities, they are more likely to evolve until the cloud density maximizes, while the Flower patterns (which are fed by wide and longer-lived thermals) are likely to be more stable and persistent. It is consistent with βCLD, which increases with the lifetime of clouds and is larger for Flowers than for Gravel (Fig d).

Vertical and plan views of the interplay between thermals and clouds are represented schematically in Figs.  and . The left-hand side of the cartoons correspond to a case of weak thermal merging (and thus high thermal density), and the right-hand side to a case of strong thermal merging and low thermal density. The two sides thus correspond to Gravel- and Flowers-types of mesoscale organization, taking into account that the very shallow clouds topping unmerged thermals (represented in the middle of Fig.  or around deep clouds in Fig. ) are also part of these patterns. In the Flower case, deep clouds are represented with an extended cloud coverage at their top (a shallow anvil): it results from the water detrained from the convective core during the lifetime of the convective clouds, which can be particularly long in situations of strong thermal merging and large βCLD (Fig. d, Sect. ).

The response of trade cumulus clouds to global warming has long been an important contributor to the uncertainty in low-cloud feedback and climate sensitivity . In climate models, this uncertainty is primarily related to changes in cloud fraction near the cloud base level . EUREC⁴A observations allowed us to show that the climate models that predict the largest trade-cumulus feedbacks overestimate the cloud-base cloud fraction in the current climate, simulate a dependence of this cloud fraction on convection that is at odds with observations, and exhibit difficulties in simulating daily transitions between shallow and deeper trade cumuli . This calls for investigating the influence of the merging process on the cloud fraction near cloud base.

As discussed in Sect. , the final coverage of merging objects depends on their merging efficiency and β. According to Eq. (), the coverage increases as the initial density D0 or size L0 increases, but it is bounded by the maximum value fmax⁡=1/β. Therefore, for a given merging efficiency, the final coverage decreases as β increases. During EUREC⁴A, βCLD≥5 (Fig. d) and therefore 0.2 appears to be an upper bound for the cloud fraction around cloud base. Furthermore, since βD0L0CLD is never far from 0.83 (Fig. b), the cloud base cloud fraction is well approximated by (1-e-0.83)/βCLD (Fig. a). It highlights the important role of βCLD in modulating, and limiting, the cloud fraction around cloud base.

As discussed in Sect.  and Appendix , simple physical arguments suggest that β encapsulates the influence that the circulation produced by convective objects exerts on neighbouring objects. Then, how to physically interpret the fact that βCLD constrains the cloud fraction? The fmax⁡ limit corresponds to the maximum cloud fraction for which the clouds' basins of attraction remain non-overlapping. In a cloud field with an area fraction 1/βCLD, then any new clouds born in the domain would necessarily be within an existing cloud's “basin of attraction” and would therefore merge with that cloud (in the simplest case where βCLD=1, a new cloud born in a region with a cloud fraction of unity would necessarily imply overlap and merging with existing clouds and no further increase in cloud fraction). Another interpretation is that the circulation induced by clouds likely promotes a mass convergence around their base level that favors the merging of thermals and thus decreases the cloud base fraction (Fig. ).

Moreover, the circulation induced by clouds facilitates the merging process all the more that the clouds live longer. How may thermal merging influence the cloud lifetime? When thermal merging increases, the thermals become wider, and therefore they are more likely associated with positive buoyancy and stronger vertical velocities , and thus with stronger circulations. Clouds are likely to live longer when they are fed by such active thermals, and therefore associated with a larger βCLD.

Consistently, the situations with weak thermal merging, that predominantly correspond to the Gravel type of organization (Sect. ), are associated with short-lived clouds, βCLD values ranging from 6 to 10 and a measured cloud fraction that ranges from 0.07 to 0.1 at cloud base. In contrast, the situations with strong thermal merging, that correspond to Flowers, embbed clouds that have much longer lifetimes (as shown by , on 2 February 2020 the cloud flowers followed along their Lagrangian trajectory seemed almost motionless for more than 12 h), βCLD ranges from 10 to 30 and the cloud base cloud fraction does not exceed 0.05. By enhancing the lifetime of clouds, thermal merging thus exerts a strong control on the cloud-base cloud fraction.

As explained in Sect. , βCLD can also be related to the aspect ratio of clouds, i.e. the ratio between cloud length scales at cloud top and at cloud base: β=1+γftopCLDfbaseCLD, where γ=h′/h is the ratio between the outflow and inflow layer depths of the air transported by the cloud circulation. Since fbaseCLD is bounded by 1/βCLD, the cloud cover measured from top is also bounded by 1γ(1-1/βCLD). Figure b shows the relationship between βCLD (inferred from ATR measurements as explained in Sect. ) and the ratio ftopCLDfbaseCLD, using ftopCLD measurements from the downward-looking instruments on board HALO .

Indeed, the two quantities are actually strongly correlated (Pearson correlation equals 0.84), and the relationship is reasonably reproduced using γ=3/2, supporting our hypothesis that the effective factor βCLD arises from the presence of cloud-induced circulations. We thus expect ftopCLD to be bounded by 1/γ=2/3. The histogram of ftopCLD values inferred from HALO measurements (considering the maximum cloud fraction estimates across the different instruments) during the whole EUREC⁴A field campaign shows that this value actually represents an upper bound for the measurements (Fig. c). Over the 86 circles flown by HALO during January–February 2020, the cloud fraction exceeded this value only twice, on 15 February 2020, when HALO was flying above a persistent layer of altostratus that has no reason to depend on boundary layer processes and thermal merging.

These findings are the result of analyzing and interpreting the interplay between thermals, clouds, mesoscale circulations and cloud patterns that was observed over the tropical Atlantic Ocean during the EUREC⁴A field campaign. During the campaign, the atmosphere was statistically sampled over a four-week period with two research aircraft that followed a repeated flight pattern. The ATR aircraft flying in the lower troposphere characterized boundary-layer thermal and cloud base chords using high-frequency humidity measurements and horizontal lidar-radar remote sensing while the HALO aircraft observed clouds from above and measured mesoscale circulations using dropsondes.

Airborne observations taken at several heights throughout the subcloud layer show that the density of thermal chords decreases with height while their average size increases. The observations also reveal that the distribution of thermal chord lengths is exponential in the surface layer, and that it is well fitted by a sum of two exponential functions higher up in the subcloud layer and near cloud base. Measurements of cloud chord lengths around the cloud base level also exhibit two populations of chords. Similar to thermals, the size distribution of cloud chords is well fitted by a sum of two exponentials, when considering either the whole campaign or individual flights. The length scale of the first cloud exponential is similar to the average size of individual thermals, while the length scale of the second cloud exponential is several times larger. Then, the detailed analysis and interpretation of these observations addresses three main questions: (1) What physical process explains the double exponential distributions? (2) How do the thermal and cloud size distributions relate to each other? (3) How do these distributions inform our physical understanding of the mesoscale organization of convection?

Physical insight and mathematical calculations, supported by simple statistical simulations, show that the merging of objects with initially exponentially distributed chord lengths leads to a sum of two exponential distributions. One exponential corresponds to objects that have merged, and the other corresponds to objects that have not yet merged. Furthermore, physical arguments suggest that the circulation created by convective objects influences the surrounding objects in a way that facilitates the merging process. This influence is formally similar to assuming that the objects have an effective length greater (by a factor β, the effective factor) than their actual length. The merging efficiency and effective factor of objects can be inferred from their chord length distribution and total coverage after merging.

Based on this conceptual framework, we diagnose the merging efficiency and effective factor of thermals and clouds using EUREC⁴A observations, and we predict the thermal density that results from the merging process. The good agreement between this prediction and the measured thermal density (which was not used to infer the merging diagnostics) provides an independent test of the consistency between the theory and the observations. We then analyze the ensemble of EUREC⁴A observations in the light of this interpretation framework.

This analysis suggests that the thermals formed in the surface layer progressively merge as they rise through the subcloud layer. This decreases the thermal density and creates two populations: one of small, unmerged thermals averaging 100–120 m, and another of larger thermals averaging about 300 m. The cloud chord length distributions are closely related to these thermal populations (Fig. ). Merged and unmerged thermals constitute the roots of cloud-base widths: upon reaching the LCL, they saturate and give rise to cloud shoots (incipient cloud bases) which in turn merge and produce two cloud populations. The clouds that results from unmerged cloud shoots are horizontally small (they have roughly the same size as individual thermals), very shallow and non drizzling. On the other hand, the clouds that result from merged cloud shoots have a wider base, develop deeper and produce drizzle. The circulation they create around them (encapsulated by the effective factor) is also stronger, which likely reinforces the concentration and merging of underlying thermals, and reduces the presence of very shallow clouds in their vicinity. The interplay between thermals and clouds, combined with the merging process, has several significant implications.

First, the merging efficiencies of thermals and clouds are negatively correlated: when thermal merging is weak, the thermal density is high, which results in a high density of cloud shoots. This facilitates cloud merging, forms large cloud bases, increases the mesoscale mass flux and strengthens mesoscale circulations. However, the convergence of thermals below large clouds eventually strengthens thermal merging, which produces wider but more isolated thermals and a lower density of cloud shoots. This hinders cloud merging and reduces the mesoscale mass flux. The interplay between thermal and cloud merging thus represents a negative feedback on the growth of mesoscale circulations, thus regulating the intrinsically unstable growth of shallow mesoscale circulations .

Second, we observe a correspondence between the degree of thermal merging and the type of prominent mesoscale cloud pattern: situations of weak thermal merging tend to be associated with Gravel-type organization, while situations of strong thermal merging tend to be associated with Flower patterns. On the other hand, the very shallow clouds that cap single thermals can be found in all situations, either alone (thus forming a Sugar-type organization) or in association with other cloud types. Moreover, since cloud merging promotes the formation of larger cloud bases, deeper clouds and stronger mass fluxes, it contributes to the development of the shallow mesoscale circulations that have been shown to accompany the transitions from Sugar-Gravel to Flower types of organization in Large-Eddy Simulations . The analysis of these simulations, that will be presented in a separate paper, confirms this inferrence.

Finally, physical arguments suggest that the maximum cloud fraction that can be achieved at cloud base is inversely proportional to the cloud effective factor, which depends on the cloud lifetime. Since clouds presumably live longer when they are fed by wide, isolated thermals than when they are fed by small thermals, thermal merging reduces the cloud fraction near cloud base. This is consistent with the minimal cloud fraction measured at cloud base in Flower-type organizations, and with the positive relationship between cloud fraction and mesoscale mass flux pointed out by . Physical arguments also suggest that the cloud top coverage is limited by the lifetime of clouds, and EUREC⁴A measurements are consistent with this suggestion.

A number of observed features remain to be interpreted. For instance, a surprising observation is that the cloud merging efficiency is never far from 0.83 (Fig. b), which is the theoretical value that maximizes the cloud density after merging (Fig. ). Whether or not this is a general feature constrained by some physical process remains to be understood. Another interesting feature is the underestimate, compared to observations, of the thermal density predicted by theory in situations of maximal thermal merging or minimal thermal density after merging (Fig. a). This discrepancy suggests the influence of additional processes in the control of the thermal density. These processes might include the influence of mesoscale circulations, which concentrate thermals in ascending branches (as shown by Fig. b at the scale of a 200 km circle), or the presence of cold pools, which may concentrate thermals and thus favor thermal merging at their edge. The discrepancy may also result from the mass convergence induced by clouds in the subcloud layer, which may influence the distribution of thermals beneath clouds and thus thermal merging but is not adequately accounted for by the effective factor of thermals (because it arises from clouds). These influences will need to be studied. Finally, the sensitivity of thermal merging to factors such as the strength of surface turbulent fluxes, the Bowen ratio or environmental conditions, and the sensitivity of cloud merging to humidity and wind shear will have to be investigated.

More importantly, this study emphasizes the role of thermal- and cloud-induced circulations in shaping mesoscale organization and cloud patterns. Within the analysis framework presented here, these circulations are conceptualized by the effective factor β, which quantifies the basin of attraction exerted by a convective object on its surroundings and influences the merging process as if convective objects had an effective size β times their actual size. The factors that influence β remain to be clarified. For instance, how should we interpret the fact that inter-flight variability of βTH is larger than its intra-flight variability (Fig. d)? A deeper investigation into the dependence of β on convective object properties and environmental conditions would help answer this question. In addition, this study shows the role of merging in shaping the size distribution of thermals and clouds, but does not show how exactly the merging – or the contact between two adjacent objects – actually occurs. The influence on the size, orientation and movement of rising thermals and clouds, and consequently on merging efficiency, of processes such as entrainment at thermal and cloud edges, ambient horizontal flow or wind shear, or buoyancy-induced pressure gradients, should be further investigated with additional observations and/or simulations.

The findings of this study offer new opportunities to understand and predict the mesoscale organization of convection, as well as its role in climate.

Given the importance of the transition between shallow and deeper trade cumuli in cloud feedback , and the uncertainty surrounding the role of cloud mesoscale organization in climate sensitivity , it will be important to verify that the models used to study convective organization and cloud feedbacks realistically represent this essential piece of atmospheric physics. By relating the statistical distribution of thermal and cloud chords to the processes that control the cloud's geometry, convective mass fluxes and mesoscale circulations, this study paves the way towards a better interpretation of the ability of numerical models to predict the different forms of mesoscale convective organization: Why do Large Eddy Simulation (LES) models or Cloud Resolving Models exhibit more success or difficulty predicting Flower-type of organization than Gravel? Why do they predict clouds that may be too scattered or on the contrary excessively wide, deep and clustered? How does the representation of thermal merging and the thermal-cloud interplay depend on the spatial resolution of models? These questions may be addressed through model inter-comparisons such as EUREC⁴A-MIP (https://eurec4a.eu/mip, last access: 14 June 2025) or large ensembles of simulations such as the Cloud Botany dataset . Analyzing these simulations would also allow us to investigate how environmental conditions influence thermal merging, which will advance understanding of how the mesoscale convective organization might respond to climate change.

In turn, the conceptualization of thermal and cloud populations as a mixture of two populations that interact and evolve through merging could help develop conceptual models that aim at representing the spectrum of cumulus clouds, their dynamics and their mesoscale organization. Such conceptual models could also be used to parameterize the mesoscale organization of convection in coarse general circulation models. With the exception of , pioneering studies in this direction have often described the statistical distribution of cloud base widths using power laws or other heavy-tail distribution functions . Studies also noticed the frequent presence of a scale break dependent on the spatial organization of convection (e.g., ), and suggested that the merging of cloud cores could influence the scale behavior of the cloud size distributions. The mathematical arguments presented in Sect.  suggest that a double exponential would be a more natural description of these distributions. It will need to be confirmed.

Finally, since EUREC⁴A took place in a regime of shallow convection, the question arises as to whether the findings of this study are specific to shallow convection, or could apply to a broader range of convective regimes. Exponential distributions of updraft chord lengths and mass fluxes have been pointed out in various contexts, ranging from observations of cloud-free continental convection to idealized simulations of deep convection . It remains to be clarified whether the absence of a second exponential may be due to the absence of merging (as may occur in simulations without mesoscale organization), and/or to a spatial resolution of simulations or observations that is too coarse to detect the smallest thermals or clouds. In any event, several elements suggest a certain universality of our results. First, the mathematical calculations and simple statistical simulations presented in Sect.  are not specific to shallow convection and would apply equally to deep convection. Second, the interplay between thermals and clouds that has been characterized here for different organizations of shallow clouds resembles that at work during the transition from shallow to deep convection over land or ocean , including in the presence of mesoscale circulations . However, this important question deserves further investigation.

Another question to be addressed is how much the merging process contributes to the self-organization of convection. Thermal merging has been suggested to play a role in the inverse energy cascade from the smaller to larger scales . In addition, modelling studies suggest that cold pools or radiatively-driven circulations are not necessary to organize shallow convection , and this is consistent with our results. Whether the merging process can also lead to the spontaneous organization of deep convection, and should be added to the list of physical processes that have already been identified , will have to be explored.

The recent MAESTRO (Mesoscale organisation of tropical convection, https://maestro.aeris-data.fr, last access: 14 June 2025) field campaign, which took place in August–September 2024 near Cape Verde as part of ORCESTRA (Organized Convection and EarthCARE Studies over the Tropical Atlantic, https://orcestra-campaign.org, last access: 14 June 2025), is an opportunity to explore the universality of the merging process across convective regimes. During MAESTRO, the ATR measured again humidity at a fast rate at different levels of the subcloud layer and at cloud base, in a wide range of meteorological situations ranging from shallow to deep convection . Using the same methodology as described in Sect. , we analyzed the thermals sampled during this campaign. Figure  shows that the thermal chord length distributions derived from MAESTRO observations resemble those from EUREC⁴A, further supporting the idea of a certain universality in the processes revealed by EUREC⁴A observations. However, further studies will be needed to confirm this conclusion, and to investigate how the interplay between thermal, clouds and circulations varies across a large diversity of convective organizations.

Finally, this study emphasizes the importance of considering the circulations induced by convective objects to understand the mesoscale organization of convection. This insight harkens back to a school of thought considering convective objects not just as thermodynamic entities (e.g. ) but also as geometric and dynamic entities (e.g. ). As fine-scale atmospheric models are now being used on large domains and multi-scale observations of convection, such as those from EUREC⁴A or ORCESTRA, are becoming available, a more complete understanding of the dynamical nature of thermal and clouds, and its implications for mesoscale oganization, is not only becoming warranted, but also possible.