Abstract
Convective selfaggregation is a modelling paradigm for convective rain cell organisation over a constanttemperature tropical sea surface. This setup can give rise to cloud clusters developing over timescales of weeks. In reality, seasurface temperatures do oscillate diurnally, affecting the atmospheric state and influencing rain rates significantly. Over land, surface temperatures vary more strongly. Here, we carry out a suite of cloudresolving numerical experiments, and find that qualitatively different dynamics emerge from modest surface temperature oscillations: while the spatial distribution of rainfall is homogeneous during the first day, already on the second day, the rain field is firmly structured. In later days, this clustering becomes stronger and alternates from day to day. We show that these features are robust to changes in resolution, domain size and mean surface temperature, but can be removed by a reduction of the amplitude of diurnal surface temperature oscillation, suggesting a transition from a random to a clustered state. Maximal clustering occurs at a scale of \({l}_{\max }\approx 180\ {\rm{km}}\), which we relate to the emergence of mesoscale convective systems. At \({l}_{\max }\), rainfall is strongly enhanced and far exceeds the rainfall expected at random. Simple conceptual modelling helps interpret the transition to clustering, which is driven by the formation of mesoscale convective systems, and brings about daytoday moisture oscillations. Our results may help clarify how continental extremes build up, and how cloud clustering over the tropical ocean could emerge as an instance of spontaneous symmetry breaking at timescales much faster than in conventional radiative–convective equilibrium selfaggregation.
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Introduction
Due to their relatively low resolution, current general circulation models cannot simulate mesoscale convective organisation explicitly^{1}. Yet, the recent observed positive trend in regional tropical rainfall was stated to predominantly stem from changes in the frequency of organised deep convection^{2}. Similarly, in midlatitudes, the majority of floodproducing rain was attributed to mesoscale convective systems (MCSs)^{3,4}. In line with Moeng and LeMone^{5}, we here take MCSs to be longlived complexes of rain cells spanning ~100 km in diameter. Over continental regions, the clustering of convection in an MCS poses a risk to humans as severe storms can lead to intense downdraughts and flash flooding^{3,4,6,7}. Pronounced longlived clustering can be simulated in a process known as convective selfaggregation forming over the timescale of weeks^{8,9,10}. There, the radiative–convective equilibrium (RCE) experimental setup^{11,12} is usually employed, assuming spatially and temporally constant surface temperature (~300 K). In RCE selfaggregation, radiation feedbacks have emerged as the “smoking gun” for sustaining and increasing clustering^{8}. Still, factors such as seasurface temperature feedbacks^{13}, domain size, geometry and resolution^{10}, as well as periodically varying insolation^{14,15} or coldpool effects^{16,17} were also stated to influence RCE selfaggregation.
Prescribing constant boundary conditions is an elegant model simplification, but it is not always realistic. Diurnal seasurface temperature amplitudes can be as large as two^{18,19} to five^{20} Kelvin, especially under weak surface wind conditions and strong insolation. Diurnal surface temperature amplitudes of this magnitude change the properties of the atmosphere^{20}, resulting in diurnally varying precipitation intensity and affecting largescale phenomena, such as the Madden–Julian Oscillation (MJO) or the El Niño phenomenon. Satellite observations show a diurnal cycle in cloud height for the MJO^{21,22}. Diurnally varying precipitation intensity also emerges from numerical simulations that account for differences in insolation^{23}. Further, in contrast to RCE selfaggregation simulations, observed tropical cloud clusters typically are more transient, as they often persist for less than two days^{24}.
Mechanistically, cold pools (CPs), that is, denser air formed by rain evaporation under convective clouds, were long implicated in the organisation of convection^{25,26}, both by thermodynamic and mechanical effects^{27,28,29,30,31}, and were suggested to lead to clustering^{32,33}. In the following, when referring to MCS, we take this to encompass both the cluster of deep convective cells and the associated CPs, which are often longlived. The continental diurnal cycle of deep convection is characterised by an afternoon or evening peak in rainfall^{31,34,35,36,37,38}. Over the ocean, this daily period can further be modulated by bi–diurnal oscillations, a dynamics observed by Chen and Houze for local cloudiness in MCSs and referred to as “diurnal dancing”^{39}.
Previous studies of convective selforganisation under diurnally varying surface temperature either did not address the emergence of MCSs or simulated a relatively small domain^{37}. Current stateoftheart regional climate models are now capable of simulating much larger areas (beyond 1000 km horizontally at kilometre resolution)^{40,41,42,43}. Encompassing features, such as terrain variation and inhomogeneous external forcing, adds unique regional insight but makes mechanistic analysis of selforganised clustering more cumbersome.
Through an idealised setup, we analyse the multiday organisation in the convective rain field over land and sea. In our setup, surface heating is spatially homogeneous but oscillates diurnally. Our focus is on the spontaneous emergence of clustering. We show that a transition from a homogeneous to a strongly clustered state occurs spontaneously when the amplitude of the diurnal surface temperature oscillation is sufficiently large. We refer to the emergent convective clusters as MCSs because of their typical scale of ~100 km^{44} and term the clustering process diurnal selfaggregation due to the lack of imposed spatial scales. This clustering oscillates bidiurnally and continues to strengthen from day to day, features we back by a conceptual model. Finally, we discuss the relevance for the emergence of extreme continental rainfall and the implications for organised convection over the tropical ocean.
Results
We carry out a suite of numerical experiments on square mesoscale model domains (L × L) of up to L = 960 km horizontal length and horizontal grid resolutions of 1 km and finer. Diurnally oscillating, spatially homogeneous surface temperature, T_{s}(t), is prescribed as
where \(\overline{{T}_{s}}=298\ {\mathrm{K}}\), t_{0} ≡ 24 h is the duration of the simulated model day, and \(\overline{{T}_{s}}\) and T_{a} represent the temporal average and amplitude of T_{s}(t), respectively. The diurnal insolation cycle is chosen to be typical for the equator and peaks at noon (just like the surface temperature T_{s}). We provide sensitivity experiments exploring resolution, domain size, rain evaporation, surface conditions and insolation (see details in “Methods”). We contrast simulations with different diurnal surface temperature amplitudes, T_{a}, but equal average surface temperature, and refer to experiments with T_{a} = 2 K, 3.5 K and 5 K as A2, A3.5 and A5, respectively. Modifiers, such as A5b or A5sea (see Table 1), label sensitivity studies. Unless explicitly stated, our key results regarding clustering occurring under varying values of T_{a}, are qualitatively robust under the sensitivity experiments. Each numerical simulation is run for several days, allowing for a spinup and quasisteadystate period (Table 1; Supplementary Fig. 1).
Domainmean time series
Unsurprisingly, the differences in diurnal surface temperature amplitudes are reflected in larger diurnal amplitudes of atmospheric nearsurface temperatures (Fig. 1a; Supplementary Fig. 1a, b). Changes in the time series of domainmean rain rate are more profound (Fig. 1b): whereas A5 yields a relatively sharp midafternoon singlepeak structure, the curve transitions to a broader and doublepeaked structure for A2, where it approximates the diurnal cycle typical of oceanic convection^{45}. For T_{a} = 0, only a weak nocturnal maximum remains (Supplementary Fig. 2f, compare: Janowiak et al.)^{46}. Again, the differences in the temporal mean (horizontal lines) are minimal, reflecting radiation constraints on rainfall^{47}. Real surface temperatures normally peak at a delay relative to local solar noon. Such delays accordingly shift the onset of organisation; hence, radiative effects are not key in this timing (Supplementary Fig. 2e). Curves that vary in a similar manner to the rain rate are found for rain area fraction, which, by contrast, differs from those of rain rate immediately before the central peak (Fig. 1c). These differences are made more transparent when inspecting rain rates conditional on a threshold (Fig. 1d). Both mean and heavy precipitation show a pronounced evening peak for A5 and an earlymorning or nocturnal peak for A2. In summary, whereas time averages of rain rate are nearly identical for numerical experiments with varying diurnal surface temperature amplitudes, the time series differ markedly.
Quantifying clustering
Do the differences in domainmean time series hint at differences in spatial organisation? Consider the horizontal patterns formed by surface rainfall, visualised by temporally averaging the rainfall pattern during each model day (Fig. 1e, f). During the spinup from the initial condition (day 1), both A2 and A5 show modest and relatively homogeneous convective activity throughout the domain. During the subsequent model days, convection intensifies for both simulations because nearsurface temperatures gradually increase. In A2, the spatial pattern of events remains rather homogeneous (Fig. 1e). In contrast, for A5, an inhomogeneous pattern selforganises, with several subregions receiving pronounced average rainfall, whereas others are all but dry (Fig. 1f, days 4 and 5). Furthermore, for A5, temporal alternations in surface rainfall rate are apparent when comparing one day to the next (compare: Fig. 1f, day 4 vs 5): a cluster on 1 day leaves an almost rainfree area the next day.
To quantify such spatiotemporal inhomogeneities, we determine all surface rain event tracks and compute their centreofmass positions. Tracks are thereby defined as spatially and temporally contiguous rainy grid boxes (see “Methods”). We then break the horizontal domain area down into square boxes of side length l, yielding n(l) ≡ (L/l)^{2} such boxes, and determine the number of tracks located in each of the boxes. The probability p_{l} of a track occurring within one of the boxes at random would be p_{l} = n(l)^{−1} and the binomial
hence describes the probability of m of N randomly distributed tracks lying in one of these boxes during the model day. The variance of counts m, given a side length l, denoted as Var_{ran}(l; N), is^{48}
which we compare with the variance of the empirical data
where 〈m〉 ≡ N/n(l) is the average number of tracks per box, and the sum runs over all boxes i. The comparison of the two quantities in Eqs. (3) and (4) hence measures how much variation is obtained empirically when contrasted against completely uncorrelated data. This comparison can be expressed in a clustering coefficient, which we define as \({\mathcal{C}}(l)\equiv {\rm{V}}{{\rm{ar}}}_{emp}(l;N)/{\rm{V}}{{\rm{ar}}}_{ran}(l;N)\). \({\mathcal{C}}(l)\) is below unity when tracks are regularly spaced and above unity when tracks are clustered. The results show that, for small diurnal surface temperature amplitudes (experiment A2a, Fig. 2a) \({\mathcal{C}}(l)\,<\,1\) for all box sizes l; hence, the spacing of tracks is generally more regular than expected at random. For larger diurnal surface temperature amplitudes (A3.5 and A5, Fig. 2b, c) regular spacing with \({\mathcal{C}}(l)\,<\,1\) is found only for relatively small box sizes of l ≈ 20 km, whereas spacing at larger box sizes is strongly clustered, that is, \({\mathcal{C}}(l)\gg 1\). Moreover, this clustering increases from day to day (Fig. 2d).
We also define a box size l_{max}, at which \({\mathcal{C}}(l)\) is maximal. Despite some variation, l_{max} ≈ 180 km can be identified from A5 (Fig. 2c, e). In addition, we measure the autocorrelation c(τ) between day d and day d + τ by computing the Pearson correlation coefficient of daily mean precipitation rates from all grid boxes (Fig. 2f). We find rainfall to be anticorrelated from one day to the next for all box sizes (c(1) < 0), suggesting a local inhibitory effect of rain and positively correlated two days into the future (c(2) > 0). The magnitudes of c(1) and c(2) both increase with box size l, but appear to level off near l = 200 km (Fig. 2f).
Clustering as a result of celldensity difference
We now explore the mechanistic origin of the clustering. Differences in the areal number density, that is, the count of rain cells per unit area, between A2 and A5, are evident from the rainfall diurnal cycle (Fig. 1b, c). The peak in rain intensity and fractional area covered by rain cells is nearly twice as high for A5 compared with A2. Given that the area, intensity and duration of each individual rain cell are all similar for A2 and A5 (Supplementary Table 1), roughly twice the number of rain events will coincide during the peak of rainfall for A5 compared with the peak of A2. As mentioned, CPs are known to mediate interactions between convective rain cells^{32,49}. We expect interactions to become more relevant at increased rain cell density because CPs will more often collide as they spread along the surface. To quantify density effects on the clustering dynamics in A2 versus A5, we first perform a simple CP tracking. The method identifies spatially and temporally contiguous patches of low buoyancy, measured by a 1 K temperature depression (for detail, see “Methods”). In A2, CP areas typically do not exceed 500 km^{2}, have modest temperature depressions and lifetimes of generally less than 2 h (Fig. 3a–c). In A5, where CPs only occur during parts of the day, CP areas often exceed 10^{3} km^{2}, and such large CPs have much stronger temperature depressions and substantially longer lifetimes (Fig. 3d–f). The formation of very large and strongly negatively buoyant CPs suggests that CPs from distinct rain cells often bunch together before each CP has fully expanded. We employ the CP tracking to detect merging events, that is, instances where two previously separate CP areas combine. For A5b, CP merging is indeed ubiquitous, whereas it is all but absent in A2b (Supplementary Fig. 4c).
In addition, we compute the CP height, measured by detecting temperature anomalies in the vertical dimension. In A5b, when CPs first appear, they show heights comparable to those of A2b (Supplementary Figs. 4a, b and 5). Subsequently, a double peak forms, where groups of merged CPs obtain much larger heights, reaching close to the level of free convection (≈1100 m). Larger CP heights h and deeper temperature depressions \(T^{\prime}\) are consistent with larger mean surface wind speed \({v}_{cp} \sim {(hT^{\prime} )}^{1/2}\) under CPs^{50} (compare: panels in Supplementary Fig. 5) and surface fluxes (Supplementary Fig. 1e–h), resonating with markedly enhanced nearsurface moisture and convective available potential energy (CAPE) along CP boundaries in A5 (compare: Supplementary Fig. 6e, j).
We aim to formulate a simplified model that incorporates the observed numberdensity differences. To this end, consider first a detailed comparison of CPs formed in A2 and A5: in A2, CPs are spatially isolated from one another, and the area covered by each CP remains small (Fig. 4a). In A5, many CPs occur so close to each other, that their temperature anomalies inevitably merge (Fig. 4a; Supplementary Fig. 4c), forming a large patch of dense air. This combined CP, which we associate with the emergent MCS, reaches a greater height and develops a relatively cold and dry region in its interiour where convection is suppressed. This inhibitory effect in the MCS interiour can be ascribed to the divergence of the level of free convection (LFC) (Fig. 4b) and a depletion of CAPE (Supplementary Fig. 6e, j).
In contrast, as the combined CP spreads outward from the dense region of rain cells, new cells are often triggered at its front—further feeding the MCS. At this combined CP’s gust front, hence at the periphery of the emergent MCS, the greater CP height allows boundarylayer environmental air to be forced higher up and set off new rain cells. In addition, the CP’s surroundings benefit from the moisture transported by the combined CP’s front and the additional latent heat flux and increased CAPE along the combined CP’s edges (compare Supplementary Fig. 6e, j). Indeed, many subsequent rain cells do form near the perimeter of the combined CP, whereas this is rare for A2 (thin black contours in Fig. 4a). To quantify this moisture redistribution from within the MCS to its surroundings, we contrast domain subregions of A5, which receive intense versus weak precipitation during a given model day (Supplementary Fig. 6f–h). Regions of intense rainfall are characterised by enhanced moisture near the cloud base (z ~1 km) before precipitation onset, but marked depletion after rain has occurred. Conversely, areas of weak rainfall show nearly a “mirror image”, with depressed moisture before but enhanced values after precipitation. The bi–diurnal dynamics for A5 can hence be characterised as an alternation of cloudbase moisture, driven by the lateral expansion of MCSs, which leaves behind an inhibitory dry patch in their interior that is marked by low CAPE (Supplementary Fig. 6i, j). On the subsequent day, this drying in locations of previous MCSs leads to relatively enhanced nearsurface and cloudbase moisture in those regions not affected by these previous MCSs—in turn providing favourable conditions for convective activity there (for detail, see Supplementary Information Text and Supplementary Fig. 6). We further find a positive correlation between cloudbase moisture and precipitation, which could help promote the transition from shallow to deep convection^{29,51} and potentially reinforce the updraughts. In A2, daytoday moisture alternations are lacking (see Supplementary Fig. 6g, h for A5b to Supplementary Fig. 6b, c for A2b), a finding that falls in line with the absence of organised convection. As with rainfall, the spatial pattern of cloudbase moisture again reflects a coarsening of scales for A5 relative to A2 (Supplementary Fig. 6d, i).
In total, the analysis suggests that an MCS emerges, when several CPs combine, become deeper and excite new rain cells along their common gust front. The emergence of an MCS on one day causes a suppressed region the subsequent day. As a further check, we weaken CPs by decreasing the ventilation coefficients (a_{v} and b_{v} in Eq. (24) of ref. ^{52}) to a fraction of their default values. These coefficients control rain evaporation, hence temperature depression \(T^{\prime}\) and CP propagation (\(\sim {T}^{^{\prime} 1/2}\)) (for details, see “Methods”). Indeed, decreasing these coefficients systematically reduces the spatial extent of the rainfree regions (Supplementary Fig. 3).
Clustering from a twolevel atmosphere model
A key characteristic of A5 is that parts of the day see no rainfall at all. In contrast, immediately after the onset of rain (near midday, Fig. 1b), the area covered by precipitation is relatively large—corresponding to a high number density of rain events and CPs. Our simplified model describes the system domain as a square lattice of sites, where each site can reside in two different states. For simplicity, we consider each pair of a rain cell and its CP as one entity. We let this entity occupy one lattice site, termed an “active site”, and say that it fills the area of a rain cell a_{0} ≈25 km^{2} (Supplementary Table 1). Lattice sites not active are considered “vacant”. In practice, our model mimics the qualitative clustering dynamics by allowing MCSs to emerge, whenever a sufficient number of active sites occur close to each other.
Assume that, at a given time, the fraction of active sites is p_{0}, and sites are independently populated. That is, each site of a square lattice contains a rain event at probability p_{0}. To exemplify, in Fig. 4c, i, p_{0} is relatively small, and few sites are active. In Fig. 4c, ii, p_{0} is larger, more sites are therefore active and many of them are now neighbours. To incorporate the effect of an MCS, we simply let the remaining vacant sites be more likely to become active when they are immediate neighbours to a sufficiently large area A > A_{crit} of spatially contiguous active sites (compare: Fig. 4a). This is accomplished by assigning increased probabilities to the neighbourhood sites (Fig. 4c, ii, orange sites). When p_{0} is small (p_{0} ≪ 1), the system will, however, be unlikely to contain contiguous rain areas exceeding A_{crit} (compare: shaded box in Fig. 4a). To be explicit: the probability of finding two active sites on two neighbouring sites is proportional to \({p}_{0}^{2}\), and this probability will decay exponentially with the number of sites contained in the contiguous area^{53}.
But how does p_{0} emerge from the diurnal cycle dynamics, and how can bi–diurnal temporal correlations be captured (Fig. 2)? To selfconsistently incorporate these features, we describe the population of rain cells within a twolayer atmosphere model consisting of a prescribed boundary layer temperature T_{bl}, varying sinusoidally with a small diurnal surface temperature amplitude t_{a} (see Fig. 1a), and an interactive freetropospheric temperature T_{ft}. We compute the probability for a rain cell to become active by coupling it to the temperature difference between the boundary layer and the free troposphere ΔT = T_{bl} − T_{ft}, as an approximation for atmospheric stability. When T_{bl} increases during the model day, rain events will eventually be set off. Two key processes impact on the freetropospheric temperature: thermal radiation to space reduces T_{ft}, whereas latent heat transfer by rain events increases it. The boundarylayer temperature (T_{bl}) is not directly affected by rain cells. However, buoyancy depressions, arising mainly from sustained drying after rain events, are implemented by an inhibitory potential. In practice, once an active site transitions back to vacant, it needs to “wait” until it can become active again (for details, see “Methods”).
We implement reasonable coefficients for these processes and find the simulations to reach a repetitive diurnal cycle (Fig. 4e, inset). Indeed, for small t_{a}, modest rainfall is present during the entire day, whereas for larger t_{a}, rainfall is either intense or absent. Timeaveraged rain areas for large and small t_{a} match (compare: Fig. 1b), a result of the radiative constraint and in agreement with the numerical experiments. The earlier phase in Fig. 4e, inset versus Fig. 1b, can be attributed to the fact that we directly prescribed atmospheric boundarylayer temperature T_{bl}, rather than surface temperature itself, causing an immediate precipitation response.
Considering the variance of the spatial pattern for large t_{a}, the simplified model indeed produces increased clustering over time and larger spatial scales (Fig. 4d, colours from red to green, and Supplementary Fig. 7). Conversely, clustering is absent for small t_{a} (Fig. 4e). This can be explained intuitively: when the thermal forcing caused by the boundarylayer temperature T_{bl} increases rapidly during the day, many rain events will be set off during a short time—leading to large p_{0} during those times. The negative feedback on the freetropospheric temperature T_{ft} will then rapidly cause the “budget” of rainfall to be used up. MCS will form as long as the increased probability at the edges of the contiguous CP patches counteracts the ongoing increase of T_{ft}. Hence, MCSs will be able to spread, as long as this is the case, thus setting the time (≈6 h) and space (≈100 km) scale for MCS, which is significantly larger than the scale of a single rain event (≈1 h and ≈5 km).
Conceptualising further
The two scales of individual rain cells (~5 km) and MCS (~100 km) allow for a further simplified conceptual view, which builds on domain total moisture being conserved from daytoday. In this view, the occurrence of rain cells is directly proportional to the available moisture and rain cells act to redistribute moisture, once they form an MCS. Take the model domain to be broken down into a square lattice consisting of blocks. Each “block” consists of 20 × 20 sites for single rain cells of area a_{0}, together yielding a block size of 100 km × 100 km, which provides space for an entire MCS. In each block, an MCS is only set off if a sufficient number of rain cells are present. We find that a simple set of rules can model the MCS dynamics: (1) first, initialise each block in this square lattice, by assigning an integer, encoding the number of rain cells, drawn from a binomial distribution; (2) update the system once per day, by letting all blocks above a particular threshold (an MCS forms) hand over their content (the moisture transported by the MCS) at equal parts to their four neighbouring blocks; after this distribution step, all sites collect the spilt content, hence conserving total moisture.
When the mean of the binomial distribution is low compared with the threshold, which is the case for small t_{a}, no MCS will form and no redistribution will take place (Fig. 4f, blue points, and lower row of squares). In contrast, when t_{a} is sufficiently large, a long sequence of reallocations will occur, leading to a checkerboardlike clustering, which strengthens in time (red curve and upper row of squares). This example is sufficient to capture the increase of normalised variance for A5 and the lack of it for A2 (Fig. 4f).
Discussion
Our study analyses the spontaneous emergence of convective clustering, when the amplitude of the diurnal cycle surface temperature is varied. This “diurnal selfaggregation” occurs without imposing spatial scales by external forcing, as with radiative–convective equilibrium (RCE) selfaggregation^{8,9,10}. Clearly, the two cases differ regarding the surface boundary conditions. However, also the spatial and temporal scales of the clustering, and in particular the physical processes causing it, differ. In RCE selfaggregation, clusters are temporally coherent over multiple days, whereas in diurnal selfaggregation, the spatial patterns of convection are almost opposite from one day to the next. The findings here suggest that, when the diurnal surface temperature variation is sufficiently large, the clustering, and thereby the potential for flash floods, increases from daytoday. Increasing model horizontal resolution to 0.5 or 0.2 km (A5c and A5d in Supplementary Fig. 3) leads to even stronger clustering effects, in agreement with recent findings for continental surface conditions^{42}. It should be tested if higher resolution leads to stronger coldpool interaction, hence more activation in areas with dense rain cell occurrence. The current findings suggest that observational studies on extremes^{54,55,56} should additionally focus on diurnal surface temperature amplitude, not just the mean.
In RCE selfaggregation, the standard explanation for clustering invokes arguments based on differential cooling and circulation changes induced by this cooling. Remarkably, the memory enabling clustering in diurnal selfaggregation is purely thermodynamically driven: we do not detect any sustained changes of circulation that harbour a memory from one day to the next (Supplementary Fig. 1i, j). The findings hence suggest two possible “shortcuts” to observed mesoscale clustering over the ocean: (i) at times, seasurface temperature oscillations may be sufficient to kickstart clustering; (ii) clustering may emerge over land surfaces, where a strong diurnal cycle prevails, and may then be advected over the sea, where it could grow further under more RCElike conditions. Practically, it could be checked through simulations, whether a state of persistent, spatially and temporally coherent selfaggregation is reached more quickly when seasurface temperatures transiently oscillate—thereby exciting initial clustering on the scale of l_{max}. Both “shortcuts” require that clustering can persist or grow once formed—a requirement consistent with hysteresis effects^{57}. In our conceptual model, hysteresis is, in fact, easy to achieve. Once an imbalance between neighbouring grid cells is established, sustained redistributions will be possible, even when the overall event number density is then lowered. It should further be explored, if the daytoday variation we find can be amplified by an interactive surface, such as by “cloud shading”^{8,13}.
The specific feature of the MCS, formed by higher cell and thus coldpool density at larger diurnal surface temperature amplitude, is to trigger new convective cells at its periphery. These new cells form cold pools, feeding the emergent MCS and further forcing updraughts near its boundary. Notably, new convective cells are also formed within the interior of the MCS (Fig. 4a), a finding in line with collision effects of multiple coldpool gust fronts^{17,32}. These interior cells could further act to deepen and cool the combined cold pool driving the MCS expansion. Together, MCSs hence act to excite new convection both within and around the combined coldpool area. In conventional selfaggregation studies, strong cold pools were found to inhibit clustering^{16}. However, under the diurnal cycle conditions studied here, they combine to promote the formation of MCSs due to the high numbers of cold pools occurring in spatially confined subregions within a relatively short time.
Altogether, the present work interpolates between the established RCE setup for oceanic convection and typical boundary conditions for continental convection. Over the ocean, the origin of initial clustering could be found in modest seasurface temperature fluctuations—in line with those observed^{18,19,20}. Over continents, fluctuating surface temperatures are wellknown to imprint pronounced diurnal rainfall variations^{31,34,35,36,37,38}, and observed MCS rainfall was reported to increase^{2,58}, elevating the risk of flash floods. In summary, the present results suggest that explanations for tropical and summertime midlatitude flash floods might also be found in temperature variations.
Methods
Largeeddy model, boundary and initial conditions
We simulate the convective atmosphere using the University of California, Los Angeles (UCLA) Large Eddy Simulator (LES) with subgridscale turbulence parameterised after Smagorinsky^{59}. This is combined with a delta fourstream radiation scheme^{60} and a twomoment cloud microphysics scheme^{61}. Radiation interacts with the atmosphere including clouds, but it does not impact the surface temperature and hence surface fluxes. The diurnal insolation cycle is chosen typical for the equator (Supplementary Fig. 2d). Rain evaporation is implemented after Seifert and Beheng^{52}. Diurnally oscillating, spatially homogeneous, surface temperature, T_{s}(t), is prescribed, as
where \(\overline{{T}_{s}}=298\ {\mathrm{K}}\), t_{0} = 24 h is the duration of the simulated model day, \(\overline{{T}_{s}}\) represents the temporal average and T_{a} the amplitude of T_{s}(t). Surface heat fluxes are computed interactively and depend on the vertical temperature and humidity gradients as well as horizontal wind speed. Horizontal wind speed is approximated using the Monin–Obukhov similarity theory, which applies scaling arguments to assume logarithmically increasing horizontal wind speed with height^{62}. Temperature and humidity were initialised using observed profiles that potentially represent convective conditions^{31}. However, due to the repeated diurnal cycle forcing, the system eventually establishes a selfconsistent vertical temperature and moisture profile.
Model grid, dynamics and output
The model integrates the anelastic equations of motion on a regular horizontal domain with varying horizontal grid spacing dx and periodic boundary conditions (Table 1). The vertical model resolution is 100 m below 1 km, stretches to 200 m near 6 km and reaches 400 m in the upper layers, with the model top located at 16.5 km. A sponge layer is employed between 12.3 km and the model top. Horizontal resolution dx and domain size vary (Table 1). The Coriolis force and the mean wind were set to zero with weak random initial perturbations added as noise to break complete spatial symmetry. For all two and threedimensional model variables, the output timestep varies between experiments between Δt_{out} = 5 min and 15 min. At each output timestep, instantaneous surface precipitation intensity, as well as the threedimensional moisture and velocity fields, is recorded for the entire model domain. In addition, at 30s and 5min intervals, respectively, spatially as well as horizontally averaged time series were extracted from the numerical experiments.
Sensitivity experiments
The principal focus of this study is the response to different values of diurnal surface temperature amplitude, T_{a}. The main experiments (A5a, A2a, A5b and A2b) were carried out at T_{s} = 298 K, contrasted T_{a} = 2 K and T_{a} = 5 K (Table 1) and used dx = 1 km horizontal model resolution. One intermediate value of T_{a} = 3.5 K was tested to constrain the transition to clustering further. The control case of T_{a} = 0 K was also tested as a benchmark. These simulations assumed that surface latent heat fluxes occurred at 70% of their potential value. This mimics a land surface, which is reasonable given the mean surface latent and sensible heat fluxes of LHF ≈ 57 W/m^{2} and SHF ≈18 W/m^{2}, respectively, yielding a Bowen ratio of B ≈ 0.30, realistic for forested land. These main experiments already explore domainsize effects, as the clustering observed could be influenced by the finite system size. We find that the results vary little between these domain sizes. We conducted additional experiments to test the sensitivity to a range of modifications: horizontal model resolution (A5c, A5d, Supplementary Fig. 3), where dx = 0.5 km and dx = 0.2 km were used; coldpool strength (A5vent05, A5vent001, Supplementary Fig. 3), where the ventilation coefficients a_{v} and b_{v} in Eq. (24) of Seifert and Beheng^{52} were respectively reduced to 0.5 and 0.01 of their default values; surface conditions, where the surface evaporation was that of a sea surface (Supplementary Fig. 2a–c, as well as A5sea and A2sea in Supplementary Fig. 3, here B ≈ 0.15); the phase of the surface temperature diurnal cycle, described by a term τ, when generalising Eq. (5) to
relative to that of the diurnal insolation cycle. τ = 3.6 h and τ = 6 h (A5ph3.6h and A5ph6h, respectively) were applied, accounting for a possible lag of the surface temperature peak towards the mid or late afternoon, as well as a constant solar radiative forcing (A0constrad and A5constrad). The dominant effect of nonzero τ is to shift the diurnal precipitation cycle accordingly (Supplementary Fig. 2e), whereas clustering still occurs (Supplementary Fig. 3). In contrast, setting insolation to the temporal average (dashed line in Supplementary Fig. 2d) has the effect of shifting the diurnal cycle of precipitation to earlier times (Supplementary Fig. 2e, dotted vs solid red line). However, also for constant insolation, clustering is still observed for A5 (Supplementary Fig. 3) (for further details, see Table 1 and Supplementary Table 1).
Rain cell and coldpool tracking
In all experiments, we track rain cells using the Iterative Rain Cell Tracking Method (IRT^{63}). In the twodimensional surface precipitation field corresponding to any output timestep, the IRT first detects all spatially contiguous patches (using connectivity 4) of rain intensities exceeding a threshold I_{0} = 0.5 mm h^{−1}, termed rain objects. Tracks are then identified, by determining any rain objects that overlap from one output timestep to the next. In the case of merging of tracks, the track of a larger area is continued under the same track ID (the threshold value θ was set to unity). In our variance analysis, we use the set of coordinates formed by the initial positions of each track, which is defined here as the precipitationweighted centre of mass of the first rain object belonging to each track. The locations of all subsequent rain objects of the same track are discarded in our variance analysis, to avoid artefacts from double counting of the positions of the nearly collocated objects.
To account for possible changes in the CP extent between the different simulations, CPs were tracked by a simple temperaturedepression method. In this method, the IRT was modified, such that at any timestep, any grid box within the lowest model level with a temperature depression, measured relative to the spatial mean at the same time, exceeding a threshold of one Kelvin, was recorded. As for rainfall, spatially contiguous patches of temperature depression were collected into objects and assigned a unique track identifier. To increase the signal, only objects exceeding an area threshold of 10 km^{2} were considered. Going forward in time, overlapping objects were identified and considered to be the same CP track. Again, the track of the larger CP area is continued under the same track identifier^{63}.
For CP height, a criterion based on a fixed temperature deviation was found inappropriate, as temperature differences are smaller aloft. Therefore, we measure CP height by evaluating the average temperature of the domain and the standard deviation at each output timestep and vertical model level k and comparing the local temperature T_{0} at each grid cell (i, j, k) with this value. If T_{0} is two standard deviations below the domain mean at the same height level k, this grid box is considered part of a CP. Within each column (i, j), the CP height is determined as the highest level that fulfils this criterion.
A simplified mesoscale convective system (MCS) model
A square lattice of s × s sites is initialised, where the area of each site is taken as the average area a_{0} occupied by a single convective rain cell, a_{0} ≈ 5 km × 5 km (compare: Supplementary Table 1). The total domain area A hence is A ≡ a_{0}s^{2}. Each site can be in one of two convective states, namely active or inactive. Our model assumes boundarylayer processes to be local. In contrast, the free troposphere acts as a reservoir of large heat capacity, where heat is quickly redistributed in the entire free troposphere by gravity waves^{64}; thus, spatial homogeneity is assumed there. The model incorporates three fundamental processes affecting event formation: (1) spontaneous activation due to the moderate drive of the diurnal cycle temperature forcing, (2) refractory dynamics due to CPs and (3) strong activation due to combined CPs ahead of MCS fronts. Below, we discuss the estimation of model parameters for these processes (summarised in Table 2).
Spontaneous activation. To define a vertical temperature gradient, we consider quantities T_{bl} and T_{ft}, which represent the deviation of the boundary layer and freetropospheric temperature from their assumed steadystate values. T_{bl}(t) is prescribed and oscillates harmonically as
where t_{0} = 24 h. The freetroposphere temperature anomaly T_{ft} is initialised to zero. We also define the temperature difference ΔT ≡ T_{bl} − T_{ft}, and its normalised version \(\Delta T^{\prime} \equiv \Delta T/\Delta {T}_{0}\), where the reference scale ΔT_{0} ≈ 0.3 K is taken as two standard deviations of LESsimulated nearsurface temperature. \(\Delta T^{\prime}\) serves as a proxy for both convective available potential energy (CAPE) and convective inhibition (CIN). The basic dynamics proceeds in discrete model timesteps of 0.5 h, reflecting the typical timescale of convective cloud formation. At each timestep, the default probability p_{0} for a vacant site to become active is taken as
Equation (8) makes the qualitative assumption that there is no activity at all for stratified conditions (\(\Delta T^{\prime}\, <\,0\)), and events occur with certainty when the temperature gradient is much larger than typical fluctuations. Both of these limits could be softened, as could the assumption of linearity at intermediate \(\Delta T^{\prime}\). One could equally argue for a smooth function of \(\Delta T^{\prime}\), such as an error function. Nonetheless, Eq. (8) captures the observed fact that more initial activity occurs when surface temperature changes quickly. Without any further perturbations, the probability p_{ij} to initiate convection in a site (i, j) will be equal for all sites (p_{ij} = p_{0}).
Spatial structure. Notably, Eq. (8) does not depend on the position within the lattice. Local modifications cause spatial structure (compare: Fig. 4a). Cold pools have two effects: reduction of local temperature and reduction of local humidity. However, the recovery timescale for the former is fast (τ_{cp} ≈ 3 h), whereas that of the latter can be slower (τ_{inh} ≈ 24 h, compare: Supplementary Fig. 6)^{27}. To consider the former, we take active sites to persist for τ_{cp}, during which no further rain cell initiation is possible at the same site. Simultaneously, the local probability p_{ij} is reduced as \({p}_{ij}={p}_{0}{p}_{{\mathrm{inh}}}\exp (\delta t/{\tau }_{{\mathrm{inh}}})\), where δt is the time after the occurrence of the rain event and p_{inh} > 0. p_{inh} controls the fidelity of the anticorrelation from daytoday, and the results are qualitatively not affected by it.
An MCS is defined to occur when a 4connected contiguous patch of active sites exceeds the threshold area A_{crit} ≡ n_{0}a_{0}, with n_{0} = 20 (Fig. 4). While this is the case, and \(\Delta T^{\prime}\, >\,0\), the probability p at each of the surrounding sites (i, j) becomes p_{ij} → p_{ij} + p_{act}, which acts to lower the barrier presented by CIN. The increased CIN above CPs is reflected in the strongly elevated LFC above the CP or MCS (Fig. 4a, b). When p_{ij} is no longer a surrounding site, the term p_{act} is no longer applied for this site. The initiation probability p_{ij} is then defined analogously to the one in Eq. (8).
Radiative constraint. Generally, T_{bl} rises during the day as the surface is heated by insolation. p_{ij} will then eventually become positive at some sites, and rain cells can be produced there. When this occurs, latent heat is transferred to the free troposphere, increasing T_{ft}, thus subsequently lowering ΔT and thereby p_{ij}. Simultaneously, the site (i, j) is shut down by the adverse buoyancy effects through CP formation. T_{ft} will relax by heat loss (P_{out} ≈ 200 W m^{−2}) through outgoing thermal radiation. According to the Stefan–Boltzmann law, \({P}_{out}=\sigma {T}_{eff}^{4}\), with σ ≈ 5.7 × 10^{−8} W m^{−2} K^{−4}, the Stefan–Boltzmann constant—translating to an effective emission temperature T_{eff} ≈ 250 K. Through P_{out}, the free troposphere would hence cool by ~2 K d^{−1}. As a crude estimate of the change in T_{ft} through latent heat transfer, we estimate the freetropospheric heat capacity C_{ft} = M_{ft}c_{pd}, assuming dry air. c_{pd} ≈ 1 kJ kg^{−1} and the mass of the free troposphere
using z_{LCL} ≈ 1 km and z_{top} ≈ 16 km for the lifting condensation level (LCL) and top of the troposphere, respectively. Hence, C_{ft} ≈ 8 × 10^{6} Jm^{−2} K^{−1}. From rain cell tracking of our LES simulations, we obtain the average rain cell lifetime to be ≈1 h (Supplementary Table 1), the mean cell area a_{0} ≈ 20 km^{2} and the average cell rain rate of ≈4 mm h^{−1} (Supplementary Table 1); hence, each rain cell can be assumed to yield M_{event} = 4 kg m^{−2} of liquid water. Assuming that the corresponding latent heat was previously deposited in the free troposphere upon ascent, each rain event heats the free troposphere by Q_{event} = L_{v} M_{event}. We take this heat to increase T_{eff} accordingly by \(\delta \ {T}_{{\mathrm{eff}}}={Q}_{{\mathrm{event}}}{C}_{ft}^{1}{{\mathrm{s}}}^{2}\), where s^{−2} results from the ratio a_{0}/A, yielding δT_{eff}s^{2} ≈ 1 K: if one rain event occurred instantaneously at each site, the freetropospheric temperature would rise by 1 K.
Data availability
The rainfall simulation data that support the findings of this study are available in the online repository with the identifier 10.5281/zenodo.3839932.
Code availability
The computer code to generate the main analysis results is available in the online repository with the identifier 10.5281/zenodo.3841916.
References
Moncrieff, M. W., Liu, C. & Bogenschutz, P. Simulation, modeling, and dynamically based parameterization of organized tropical convection for global climate models. J. Atmos. Sci. 74, 1363–1380 (2017).
Tan, J., Jakob, C., Rossow, W. B. & Tselioudis, G. Increases in tropical rainfall driven by changes in frequency of organized deep convection. Nature 519, 451–454 (2015).
Stevenson, S. N. & Schumacher, R. S. A 10year survey of extreme rainfall events in the central and eastern united states using gridded multisensor precipitation analyses. Monthly Weather Rev. 142, 3147–3162 (2014).
Feng, Z. et al. Structure and evolution of mesoscale convective systems: Sensitivity to cloud microphysics in convectionpermitting simulations over the United States. J. Adv. Modeling Earth Syst. 10, 1470–1494 (2018).
Moeng, C.H. & LeMone, M. A. Atmospheric planetary boundarylayer research in the US: 19911994. Rev. Geophys. 33, 923–931 (1995).
Smith, J. A., Baeck, M. L., Zhang, Y. & Doswell, C. A. III Extreme rainfall and flooding from supercell thunderstorms. J. Hydrometeorol. 2, 469–489 (2001).
Cintineo, R. M. & Stensrud, D. J. On the predictability of supercell thunderstorm evolution. J. Atmos. Sci. 70, 1993–2011 (2013).
Bretherton, C. S., Blossey, P. N. & Khairoutdinov, M. An energybalance analysis of deep convective selfaggregation above uniform SST. J. Atmos. Sci. 62, 4273–4292 (2005).
Khairoutdinov, M. F. & Emanuel, K. A., Aggregated convection and the regulation of tropical climate. In 29th Conference on Hurricanes and Tropical Meteorology (American Meteorological Society, Tucson, AZ, 2010).
Muller, C. J. & Held, I. M. Detailed investigation of the selfaggregation of convection in cloudresolving simulations. J. Atmos. Sci. 69, 2551–2565 (2012).
Held, I. M., Hemler, R. S. & Ramaswamy, V. Radiativeconvective equilibrium with explicit twodimensional moist convection. J. Atmos. Sci. 50, 3909–3927 (1993).
Tompkins, A. M. & Craig, G. C. Radiativeconvective equilibrium in a threedimensional cloudensemble model. Q. J. R. Meteorological Soc. 124, 2073–2097 (1998).
Hohenegger, C. & Stevens, B. Coupled radiative convective equilibrium simulations with explicit and parameterized convection. J. Adv. Modeling Earth Syst. 8, 1468–1482 (2016).
Ruppert, J. H. Jr. & Hohenegger, C. Diurnal circulation adjustment and organized deep convection. J. Clim. 31, 4899–4916 (2018).
Ruppert, J. H. Jr. & O’Neill, M. E. Diurnal cloud and circulation changes in simulated tropical cyclones. Geophys. Res. Lett. 46, 502–511 (2019).
Jeevanjee, N. & Romps, D. M. Convective selfaggregation, cold pools, and domain size. Geophys. Res. Lett. 40, 994–998 (2013).
Haerter, J. O. Convective selfaggregation as a cold pooldriven critical phenomenon. Geophys. Res. Lett. 46, 4017–4028 (2019).
Weller, R. A. & Anderson, S. P. Surface meteorology and airsea fluxes in the western equatorial pacific warm pool during the toga coupled oceanatmosphere response experiment. J. Clim. 9, 1959–1990 (1996).
Johnson, R. H., Rickenbach, T. M., Rutledge, S. A., Ciesielski, P. E. & Schubert, W. H. Trimodal characteristics of tropical convection. J. Clim. 12, 2397–2418 (1999).
Kawai, Y. & Wada, A. Diurnal sea surface temperature variation and its impact on the atmosphere and ocean: a review. J. Oceanogr. 63, 721–744 (2007).
Tian, B., Waliser, D. E. & Fetzer, E. J. Modulation of the diurnal cycle of tropical deep convective clouds by the MJO. Geophys. Res. Lett. 33, L20704 (2006).
Suzuki, T. Diurnal cycle of deep convection in super clusters embedded in the MaddenJulian Oscillation. J. Geophys. Res.: Atmos. 114(D22), 1–14 (2009).
Liu, C. & Moncrieff, M. W. A numerical study of the diurnal cycle of tropical oceanic convection. J. Atmos. Sci. 55, 2329–2344 (1998).
Chen, S. S., Houze, R. A. Jr. & Mapes, B. E. Multiscale variability of deep convection in realation to largescale circulation in TOGA COARE. J. Atmos. Sci. 53, 1380–1409 (1996).
Droegemeier, K. K. & Wilhelmson, R. B. Threedimensional numerical modeling of convection produced by interacting thunderstorm outflows. Part I: control simulation and lowlevel moisture variations. J. Atmos. Sci. 42, 2381–2403 (1985).
Rotunno, R., Klemp, J. B. & Weisman, M. L. A theory for strong, longlived squall lines. J. Atmos. Sci. 45, 463–485 (1988).
Tompkins, A. M. Organization of tropical convection in low vertical wind shears: the role of cold pools. J. Atmos. Sci. 58, 1650–1672 (2001).
Böing, S. J., Jonker, H. J. J., Siebesma, A. P. & Grabowski, W. W. Influence of the subcloud layer on the development of a deep convective ensemble. J. Atmos. Sci. 69, 2682–2698 (2012).
Schlemmer, L. & Hohenegger, C. Modifications of the atmospheric moisture field as a result of coldpool dynamics. Q. J. R. Meteorological Soc. 142, 30–42 (2016).
Feng, Z. et al. Mechanisms of convective cloud organization by cold pools over tropical warm ocean during the amie/dynamo field campaign. J. Adv. Modeling Earth Syst. 7, 357–381 (2015).
Moseley, C., Hohenegger, C., Berg, P. & Haerter, J. O. Intensification of convective extremes driven by cloudcloud interaction. Nat. Geosci. 9, 748 (2016).
Haerter, J. O., Böing, S. J., Henneberg, O. & Nissen, S. B. Circling in on convective organization. Geophys. Res. Lett. 46, 7024–7034 (2019).
Lochbihler, K., Lenderink, G. & Siebesma, A. P. Response of extreme precipitating cell structures to atmospheric warming. J. Geophys. Res.: Atmos. 124, 6904–6918 (2019).
Petch, J. C., Brown, A. R. & Gray, M. E. B. The impact of horizontal resolution on the simulations of convective development over land. Q. J. R. Meteorological Soc. 128, 2031–2044 (2002).
Guichard, F. et al. Modelling the diurnal cycle of deep precipitating convection over land with cloudresolving models and singlecolumn models. Q. J. R. Meteorological Soc. 130, 3139–3172 (2004).
Sato, T., Miura, H., Satoh, M., Takayabu, Y. N. & Wang, Y. Diurnal cycle of precipitation in the tropics simulated in a global cloudresolving model. J. Clim. 22, 4809–4826 (2009).
Schlemmer, L. The Diurnal Cycle of Midlatitude, Summertime Moist Convection Over Land in an Idealized Cloudresolving Model. PhD thesis, ETH Zurich (2011).
Haerter, J. O. & Schlemmer, L. Intensified cold pool dynamics under stronger surface heating. Geophys. Res. Lett. 45, 6299–6310 (2018).
Chen, S. S. & Houze, R. A. Diurnal variation and lifecycle of deep convective systems over the tropical pacific warm pool. Q. J. R. Meteorological Soc. 123, 357–388 (1997).
Ban, N., Schmidli, J. & Schär, C. Heavy precipitation in a changing climate: does shortterm summer precipitation increase faster? Geophys. Res. Lett. 42, 1165–1172 (2015).
Prein, A. F. et al. Simulating North American mesoscale convective systems with a convectionpermitting climate model. Clim. Dyn. 55, 1–16 (2017).
Rasp, S., Selz, T. & Craig, G. C. Variability and clustering of midlatitude summertime convection: testing the craig and cohen theory in a convectionpermitting ensemble with stochastic boundary layer perturbations. J. Atmos. Sci. 75, 691–706 (2018).
Satoh, M. et al. Global cloudresolving models. Curr. Clim. Change Rep. 5, 1–13 (2019).
Houze, R. A.Jr. Mesoscale convective systems. Rev. Geophys. 42, 1–43 (2004).
Yang, G.Y. & Slingo, J. The diurnal cycle in the tropics. Monthly Weather Rev. 129, 784–801 (2001).
Janowiak, J. E., Arkin, P. A. & Morrissey, M. An examination of the diurnal cycle in oceanic tropical rainfall using satellite and in situ data. Monthly weather Rev. 122, 2296–2311 (1994).
Held, I. M. & Soden, B. J. Robust responses of the hydrological cycle to global warming. J. Clim. 19, 5686–5699 (2006).
Feller, W. An Introduction to Probability Theory and Its Applications (John Wiley & Sons, 1957).
Böing, S. J. An objectbased model for convective cold pool dynamics. Math. Clim. Weather Forecast. 2, 43–60 (2016).
Etling, D. Theoretische Meteorologie: Eine Einführung (SpringerVerlag, 2008).
Zhang, Y. & Klein, S. A. Mechanisms affecting the transition from shallow to deep convection over land: inferences from observations of the diurnal cycle collected at the ARM southern great plains site. J. Atmos. Sci. 67, 2943–2959 (2010).
Seifert, A. & Beheng, K. D. A twomoment cloud microphysics parameterization for mixedphase clouds. Part 1: Model description. Meteorol. Atmos. Phys. 92, 45–66 (2006).
Christensen, K. & Moloney, N. R. Complexity and Criticality, Vol. 1 (World Scientific Publishing Company, 2005).
Lenderink, G. & Van Meijgaard, E. Increase in hourly precipitation extremes beyond expectations from temperature changes. Nat. Geosci. 1, 511–514 (2008).
Berg, P., Moseley, C. & Haerter, J. O. Strong increase in convective precipitation in response to higher temperatures. Nat. Geosci. 6, 181–185 (2013).
Coumou, D. & Rahmstorf, S. A decade of weather extremes. Nat. Clim. Change 2, 491 (2012).
Muller, C. & Bony, S. What favors convective aggregation and why? Geophys. Res. Lett. 42, 5626–5634 (2015).
Hu, H., Leung, L. R. & Feng, Z. Observed warmseason characteristics of mcs and nonmcs rainfall and their recent changes in the central united states. Geophys. Res. Lett. 47, e2019GL086783 (2020).
Smagorinsky, J. General circulation experiments with the primitive equations: I. the basic experiment. Monthly Weather Rev. 91, 99–164 (1963).
Pincus, R. & Stevens, B. Monte Carlo spectral integration: a consistent approximation for radiative transfer in large eddy simulations. J. Adv. Modeling Earth Syst 1, 1–9 (2009).
Stevens, B. et al. Evaluation of largeeddy simulations via observations of nocturnal marine stratocumulus. Monthly Weather Rev. 133, 1443–1462 (2005).
Stull, R. B. An Introduction to Boundary Layer Meteorology, Vol. 13 (Springer Science & Business Media, 2012).
Moseley, C., Henneberg, O. & Haerter, J. O. A statistical model for isolated convective precipitation events. J. Adv. Modeling Earth Syst. 11, 360–375 (2019).
Bretherton, C. S. & Smolarkiewicz, P. K. Gravity waves, compensating subsidence and detrainment around cumulus clouds. J. Atmos. Sci. 46, 740–759 (1989).
Acknowledgements
We thank Kim Sneppen for fruitful discussions on the simplified modelling. J.O.H. and B.M. gratefully acknowledge funding by a grant from the VILLUM Foundation (grant number: 13168) and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant number: 771859). S.B.N. acknowledges funding through the Danish National Research Foundation (grant number: DNRF116). The authors acknowledge the Danish Climate Computing Center (DC3) and the German Climate Computing Center (DKRZ).
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J.O.H. ran and processed the largeeddy simulations (LES), developed the conceptual models and wrote the paper. B.M. produced the case study and revised the paper. S.B.N. contributed to model development and implementation and revised the paper.
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Haerter, J.O., Meyer, B. & Nissen, S.B. Diurnal selfaggregation. npj Clim Atmos Sci 3, 30 (2020). https://doi.org/10.1038/s4161202000132z
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DOI: https://doi.org/10.1038/s4161202000132z
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