Observations and high‐resolution simulations of convective precipitation organization over the tropical Atlantic

High‐resolution simulations (grid spacing 2.5 km) are performed with ICON‐LEM to characterize convective organization in the Tropics during August 2016 over a large domain ranging from northeastern South America, along the tropical Atlantic to Africa (8,000×3,000 km). The degree of organization is measured by a refined version of the wavelet‐based organization index (WOI), which is able to characterize the scale, the intensity and anisotropy of convection based on rain rates alone. Exploiting the localization of wavelets both in space and time, we define a localized version of the convective organization index (LWOI). We compare convection observed in satellite‐derived rain rates with the corresponding processes simulated by ICON‐LEM. Model and observations indicate three regions with different kinds of convective organization. Continental convection over West Africa has a predominantly meridional orientation and is more organized than over South America, because it acts on larger scales and is more intense. Convection over the tropical Atlantic is zonally oriented along the ITCZ and less intense. ICON and observations agree on the number and intensity of the African easterly waves during the simulation period. The waves are associated with strong vorticity anomalies and are clearly visible in a spatiotemporal wavelet analysis. The central speed and the wavelength of the waves is simulated well. Both the scale and intensity components of LWOI in ICON are significantly correlated with environmental variables. The scale of precipitation is related to wind shear, CAPE and its tendency, while the intensity strongly correlates with column‐integrated humidity, upper‐level divergence and maximum vertical wind speed. This demonstrates that the LWOI components capture important characteristics of convective precipitation.


INTRODUCTION
Convection and its organization play an essential role in the dynamics of the climate system. For instance, around 70% of the total global precipitation over land is caused by deep convection (Xu and Zipser, 2012). Besides moisture fluxes through condensation and rainfall, it has crucial influence on the energy and momentum transport in the global circulation (Stevens and Bony, 2013). Convective processes interact with the global circulation and are driven by large-scale forcing due to waves, fronts and convergence lines (Duda and Gallus Jr. 2013). Different forcing mechanisms and environmental quantities such as wind shear or instability lead to weakly (e.g. pulse storms, scattered convection) or strongly (e.g. squall-lines, mesoscale convective systems) organized convection. Phenomena in the latter category live on larger spatial and temporal scales and are linked to strong forcing and increased wind shear (Moncrieff, 2010). Supercells, squall-lines and other mesoscale convective systems (MCSs) are associated with e.g. large hail, severe winds, tornadoes and/or heavy rainfalls (Groenemeijer et al., 2017). Thus, the correct representation of convection and its degree of organization in numerical weather and climate prediction systems is vitally important.
In recent years the resolution of numerical weather prediction models has increased rapidly (grid spacing ≈ 1-10 km) and many convective processes are resolved explicitly. However, in climate simulations with general circulation models (grid spacing ≈10-100 km), convection still needs to be parametrized to describe organized features like the Madden-Julian oscillation satisfactorily (Peters et al., 2017). Convective parametrizations are responsible for large uncertainties in climate prediction models and are still under vital debate (e.g., Birch et al., 2015;Badlan et al., 2017;Holloway et al., 2017;Moncrieff et al., 2017;Wing et al., 2017). One major aspect in current research is the question of whether and how convective organization should be parametrized (e.g., Mapes and Neale, 2011). Therefore different degrees of convective organization need to be quantified.
Especially in the Tropics, where convection acts on very different scales ranging from cumulus convection and pulse storms up to hurricanes or the Madden-Julian oscillation (Petersen and Rutledge, 2001), a correct representation of convective organization is important. Most tropical convection is driven by the intertropical convergence zone (ITCZ), but heterogeneous continental landmasses (e.g., Africa and South America) greatly influence its initiation and organization as well. Current general circulation models are not able to predict the exact position of the ITCZ correctly (Möbis and Stevens, 2012;Nolan et al., 2016), with the consequence that convection and its degree of organization are represented inadequately.
In our study, we focus on the tropical Atlantic including the adjacent parts of South America and Africa in August 2016 to investigate convective processes over three different regions: the northeastern coast of South America including the Amazon and Tocantins delta, the tropical Atlantic along the ITCZ, and West Africa including western parts of Sahel, Guinea Highlands and Gulf of Guinea. Earlier studies have shown that the types of convection, its organization and the forcing mechanisms are fundamentally different within these regions. Garstang et al. (1994), Rickenbach (2004) and Romatschke and Houze Jr. (2010) have shown that convection over northeastern South America is initiated by convergence lines along the coastline due to the easterlies. Intense long-living MCSs (mostly squall-lines) develop, move inland and benefit from the relatively flat terrain and high moisture in the Amazon and Tocantins delta. Over Africa, convection is driven by the African easterly waves (AEWs). AEWs modulate organized convection over Africa (Duvel, 1990;Mekonnen and Rossow, 2018;Tomassini, 2018) and initiate strong squall-lines with heavy rainfall (e.g., Carlson, 1969;Fink and Reiner, 2003;Mekonnen et al., 2006;Crétat et al., 2015). AEWs, defined as westward propagating lower-tropospheric disturbances over Africa, have a period of 2.5-5 days (Burpee, 1972;Lubis and Jacobi, 2015) and a phase speed of 8-10 m⋅s −1 (Reed et al., 1977;Price et al., 2007). The wavelength of AEWs ranges from 2,000 km (e.g., Burpee, 1974) up to 5,000 km (Diedhiou et al., 1999;Kiladis et al., 2006), which correspond to zonal wavenumbers between 8 and 19. Thus, strongly organized MCSs, frequently organized into squall-lines (Mathon and Laurent, 2001;Jackson et al., 2009;Rickenbach et al., 2009), move westward across the tropical Atlantic. The large-scale forcing over the tropical Atlantic is weaker than over Africa (Xie and Carton, 2004). In general there are fundamental differences between convection over land and ocean (Zipser et al., 2006;Janiga and Thorncroft, 2013). Convection over the tropical Atlantic shows lower cloud tops and less intense rain rates, but more stratiform rainfall than over Africa (Schumacher and Houze Jr. 2006;Futyan and Del Genio, 2007;Liu et al., 2007;Janiga and Thorncroft, 2014;. To characterize these regions of different convective organization, we use high-resolution convection-permitting simulations and compare them to two kinds of satellite-derived rain rate estimations, namely passive microwave radiometers and infrared measurements. Senf et al. (2018), using the same dataset as the present study, evaluated brightness temperatures simulated by the ICOsahedral Non-hydrostatic model (ICON) against satellite observations over the tropical Atlantic using a histogram matching technique. Their object-based analysis shows that ICON simulates convection more realistically over continental regions than over the ocean, but contains more small-scale features and overestimates shallow cumulus clouds.
Convective organization is generally measured by indices such as the simple convective aggregation index (SCAI; Tobin et al., 2012), the organization index (I org ; Tompkins and Semie, 2017) or the convective organization potential (COP; White et al., 2018). These conventional indices are based on cluster algorithms of brightness temperatures, vertical wind velocity and outgoing long-wave radiation, respectively. A comparison of the different convective organization indices is presented in Pscheidt et al. (2019).
Local convective organization has previously been investigated by Li et al. (2018) calculating the information entropy (Shannon, 1948) from satellite observations. They find that information entropy shows only small variances between weak and strong convective organization. Thus, defining convective organization locally is still challenging.
In meteorology, wavelets are used to verify precipitation or cloud fields (e.g., Casati et al., 2004;Yano and Jakubiak, 2016;Weniger et al., 2017;Kapp et al., 2018;Buschow et al., 2019) or to characterize the structure of convection (e.g., Yano et al., 2001a;Yano et al., 2001b;Klein et al., 2018). Brune et al. (2018) developed the wavelet-based organization index (WOI) to differentiate between non-organized and organized convection over Germany based on rain rates. Typically, convective organization measurements (SCAI, I org , COP) are directly applied to satellite measurements (e.g., Tobin et al., 2012;Stein et al., 2017;Senf et al. 2018). In recent studies the indices are also used for radar data (Moseley et al., 2019;Pscheidt et al., 2019) or even satellite-derived rainfall (Holloway, 2017). Because the rain rate-based WOI is not yet tested for other variables such as outgoing long-wave radiation or brightness temperature, and has performed well for midlatitude rain rates, we analyze the structure of rain rates over the tropical Atlantic to characterize convective organization in this study. We refine WOI and exploit the wavelet's inherent power of localization in order to answer the following questions: • How is convection organized over northeastern South America, tropical Atlantic and West Africa?
• Does ICON simulate convective organization correctly?
• How do environmental variables influence convective organization?
In order to explicitly quantify the role of AEWs with regard to these questions, we furthermore complement our spatial wavelet analysis with a spatiotemporal wavelet transform following Kikuchi and Wang (2010).
This article is structured as follows. Section 2 gives an overview of the ICON simulations and the observations. In Section 3 we present the WOI modifications including the extension to the local WOI (LWOI) and briefly introduce spatiotemporal wavelet transforms. After analyzing convective organization using the LWOI components, we turn to the temporal evolution of convection and link environmental quantities to the LWOI components in Section 4. We conclude the article with a summary in Section 5.

MODEL AND DATA
Simulations for August 2016 are performed with a limited-area high-resolution version of the ICON model  within the High Definition Clouds and Precipitation for Advancing Climate Prediction (HD(CP) 2 ) project, funded by the German Ministry for Education and Research. For detailed information on ICON we refer to Wan et al. (2013), Dipankar et al. (2015), Reinert et al. (2016), and Heinze et al. (2017).
In this study, we use ICON simulations over the tropical Atlantic with a horizontal grid spacing of Δ x ≈ 2.5 km and 75 terrain following full levels (76 half levels) up to a height of about 30 km. Simulations over 36 hr were initialized with analyses from the European Centre for Medium-Range Weather Forecasts (ECMWF) at 0000 UTC and were performed for each day in August 2016. Boundary data were provided by ECMWF every 3 hr. Because spin-up takes at least 6 hr (Heinze et al., 2017), we neglected the first 12 hr and chose the last 24 hr of each simulation to fully cover August 2016. This guarantees that the model physics can evolve and, although the simulations are not continuous, they are close to the observations. The simulations include lots of interesting features such as shallow cumulus clouds, hurricanes, squall-lines and vortex streets during August 2016 and cover the tropical Atlantic as well as parts of the South American and African continents . Due to the high resolution, the long simulation period and persistent convective activity over the domain, the simulations are a good basis to investigate convective organization in the Tropics. However, ICON overestimates shallow cumulus and low-level clouds, which produce weak small-scale drizzle and an increased number of small-scale features (Senf et al., 2018).
The ICON output is interpolated from the original triangular model grid on to a coarser regular grid with a longitudinal extent from 67.45 • W to 14.45 • E and 9.55 • S to 19.45 • N by steps of 0.10 • in both directions. This corresponds to a horizontal grid spacing of about 10 km, which is close to the effective resolution of the used ICON (7Δ x − 8Δ x ; Heinze et al., 2017). The temporal resolution of the 2D output variables is 30 min, and 3D output is available every hour.
In contrast to the object-based study with satellite data by Senf et al. (2018), we characterize convective organization on the basis of simulated rain rates. To compare convective organization in ICON with observations, high-resolution rain rate measurements in space and time are required. In the absence of a sufficiently dense rain-gauge network or radar observations, we rely on satellite-derived rain estimations from the Integrated Multi-satellite Retrievals for Global precipitation measurement (IMERG; Huffman et al., 2015) project 1 to assess the observed state of convective organization. Our primary data sources are passive microwave (MW) radiometers from low earth orbit satellites, which constitute the most direct measurements of precipitation available. Over water bodies, emissions from falling hydrometeors near the surface can easily be separated from surface emissions due to the difference in temperature. Over land, MW measurements are mainly determined by scattering from frozen-phase hydrometeors. Both kinds of data are intercalibrated using the global precipitation measurement core satellite's active MW radiometer -essentially a space-borne radar. Over land, the data undergo further calibration to rain-gauge-derived climatologies to create the final IMERG product.
While this procedure yields the overall most trustworthy available information, it may suffer from spatial and temporal inconsistencies. The satellites used to obtain the rain fields change perpetually and the measurement principle differs over land and sea. In addition, the temporal resolution is limited to the frequency of over-passing satellites. The Goddard profiling algorithm employed in IMERG therefore uses sophisticated time interpolation and auxiliary data from infrared (IR) images to create the half-hourly dataset we use. How strongly these processing steps alter the spatial structure of the resulting rain fields remains to be seen. In order to address this uncertainty in the observations, we therefore include the IR-derived rain rates as a secondary source of information. These images are based on a single instrument on a geostationary satellite, which always covers the complete domain and delivers half-hourly data. These data are less trustworthy since precipitation is inferred indirectly from cloud-top temperatures. For example, low warm precipitating clouds 1 https://pmm.nasa.gov/category/keywords/imerg; accessed 1 February 2020 may not be detected by the IR measurement, while high cirrus clouds can potentially be mis-interpreted as rain.
We aim at a more complete comparison using two observation datasets both to one another and to the simulations. Aside from model validation, the intercomparison of different satellite observations with respect to their spectral characteristics holds some interest in itself.

Discrete wavelet transforms
To describe spatial structures in the precipitation fields, we decompose the rain rates with a redundant discrete wavelet transform. For a comprehensive overview of wavelets and their mathematical background, we refer to Daubechies (1992). We compute the estimated locally stationary wavelet spectrum following Eckley et al. (2010) with help of the R Statistical Programming Language package LS2W (Eckley and Nason, 2011). The decomposition of a 2 J × 2 J field leads to a set of 2 J × 2 J × J × D wavelet coefficients with J scales (1,…, J) and D = 3 directions (east-west, north-south, diagonal). Here, the jth scale corresponds to features with a spatial extent of approximately 2 j grid-points. Since the wavelet coefficients are not independent, the spectral energy is biased towards large scales where most of the information in the local spectra is redundant. The correction of this bias following Eckley and Nason (2011) may lead to negative energy values at some locations. We discuss the ramifications of this phenomenon in Section 3.3.
The wavelet analysis requires a quadratic domain with 2 n × 2 n , n ∈ ℕ, grid points and periodic boundaries. Thus, we pad our original output field (820 × 291) with zeros to obtain a field of size 1024 × 1024. To reduce the gradients between the original precipitation field and the synthetic zero-precipitation regions, we follow Weniger et al. (2017) and Kapp et al. (2018) and smooth the outer 25 grid points with a linearly decreasing filter. Both largest scales (9-10; 2,560-5,120 km) are strongly influenced by the formulation of the boundary conditions. Scale 8 (1,280 km) is smaller than the domain size, but the support length of the D4 wavelet exceeded the domain size. Due to these technical issues, we focus on scales 1-7 (10-640 km) in the 2D wavelet analysis, although convection may be organized into AEWs, which act on even larger scales.
In contrast to Brune et al. (2018), who only studied the spatially averaged wavelet spectra, we now intend to analyze the local spectra at every grid point. The location at which we store the respective wavelet coefficient therefore becomes relevant. Since the Daubechies basis functions used by Eckley et al. (2010) can exhibit complicated structures within a large support area, it is not entirely obvious to which grid point should be attributed an individual projection. As an ad hoc solution, we store each coefficient at the corresponding centre of mass of the wavelet. This guarantees that spectral energy corresponding to any individual precipitation feature is located close to the feature itself, and leads to local spectra which are relatively straightforward to interpret.

Wavelet-based organization index (WOI)
The wavelet-based organization index (WOI) was introduced by Brune et al. (2018) to differentiate between organized convective structures (e.g., squall-lines, supercells, clustered multicells, MCSs) and unorganized convection (e.g., pulse storms, scattered convection). The WOI consists of three components describing the horizontal scale of precipitation (WOI 1 ), its intensity (WOI 2 ) and anisotropy (WOI 3 ). They are defined as E s represents the small-scale energy on scales 1-3 (10-40 km), while E l represents the large-scale energy on scales 4-7 (80-1,280 km) in the wavelet spectrum, averaged over the domain and each direction. r is the number of grid points with rain rates above zero. E d s stands for the small-scale and E d l for the large-scale domain-averaged energy in the east-west (d = 1), north-south (d = 2) and diagonal (d = 3) directions.

Modified wavelet-based organization index
Negative energy induced by the bias correction of the spectral coefficients may result in a WOI 1 > 1 and a noisy WOI 3 . Previous studies Brune et al., 2018;Kapp et al., 2018) used the sharp Haar wavelet (Haar, 1910), which is also known as the first member of the Daubechies family (D1; Daubechies, 1992). Preliminary investigations have shown that negative energy is reduced drastically for higher orders of Daubechies wavelets. However, the support length of Daubechies wavelets increases with their order, resulting in less sharp localization. As a compromise between good localization and less negative energy, we choose the extremal phase D4 wavelet for all the following calculations. Remaining negative energy is set to 0 before calculating the indices.
In order to render all three components comparable, we scale WOI 3 with the factor of maximal anisotropy (3∕2

√
3). Strong convective precipitation results in high mean spectral energy E and is concentrated on a small number of grid points r of the total domain N x ⋅ N y . To ensure that intense rain rates lead to a WOI 2 → 1, we multiply the total energy by −1 and subtract the exponential term from 1. Thus, for less intense rain over larger parts of the domain, WOI 2 is 0, because The resulting WOI components describe scale (sc), intensity (in) and anisotropy (ai) of convection and are defined as

Local wavelet-based organization index
For the localized analysis of convective organization, we calculate WOI sc , WOI in and WOI ai at each rainy grid point (i, j) using with where E d k (i, j) describes the wavelet spectrum at grid point (i, j) for scale k = 1, 2, … , 7 and direction d = 1, 2, 3. Note that the LWOI in is not identical to the rain intensity, rather it denotes the total local variance integrated over all scales and directions. LWOI in becomes large if there are sharp gradients in the precipitation field, as is the case for squall-lines.
The code to calculate WOI and LWOI is available as an R package calcWOI (Brune et al., 2019), which computes the original WOI as defined in Brune et al. (2018), the modified WOI components defined in Section 3.3 and LWOI components of an arbitrary 2D array.

Theory of spatiotemporal wavelet transforms
Apart from its effect on the spatial patterns of precipitation, convective organization manifests itself in the movement and life cycle of thunderstorms. Since tropical weather systems are known to propagate predominantly in zonal directions, in our case westward across the Atlantic, we can infer their spatiotemporal characteristics from the meridional mean as a function of longitude and time. To quantify and compare the size and speed of the various moving systems present in this kind of data, we employ a spatiotemporal wavelet transform (STWT) which was first applied to the analysis of tropical waves by Kikuchi and Wang (2010). Following these authors, we use the complex Morlet mother wavelet given by This function of space (here longitude) x and time t is a plane wave with frequency 0 = 6 and wave number k 0 = ±6, localized by a Gaussian envelope. Here, the sign of k 0 determines the direction of propagation. The daughter wavelets of the STWT are given by where denotes shift in time and b shift in space; a > 0 is a common scaling parameter for both directions, and the parameter c > 0 corresponds to the phase speed of the daughter wavelet. The power P related to one of these basis functions is given by where the normalization 1∕(a 3 c) guarantees that the total power integrated over all daughters equals the variance of the original data (up to a further normalization dependent only on the choice of mother wavelet). Before applying the transform given by Equation (15) to the meridional means, we add zeros around the data to get a square field of 2 11 × 2 11 , thereby avoiding aliasing while speeding up the fast Fourier transformation used to perform the convolutions. Kikuchi and Wang (2010) demonstrate how the parameters a and c can be related to the frequency and wavenumber k of the corresponding Fourier transform. While c is exactly the usual phase speed ∕k, the interpretation of a, which scales time and space alike, is less straightforward. To facilitate the interpretation of our results, we therefore replace a by the wavelength For a derivation of this identity, as well as further technical details, the reader is referred to Kikuchi and Wang (2010) and references therein. Lastly, a natural way to represent the direction of the daughter wavelets, originally determined by k 0 , is to attach the sign of k 0 to the phase speed, meaning that the daughter denoted by c = −10 corresponds to k 0 = −6, c = 10. In principle, we could apply the same discrete two-dimensional wavelet transform to the longitude-time diagrams. For two reasons, we use the STWT basis described above. Firstly, we expect the meridional means to exhibit inherently wave-like behaviour (Figure 4). The complex Morlet wavelet, which is simply a localized plane wave, therefore seems appropriate. Secondly, the spatial F I G U R E 1 Snapshot of (a-c) rain rate in mm hr −1 , (d-f) LWOI sc , (g-i) LWOI in and (j-l) LWOI ai of a West African squall-line at 0000 UTC on 20 August 2016 represented in (a, d, g, j) microwave observations, (b, e, h, k) infrared observations, and (c, f, i, l) ICON simulations. Only grid points with rain rates ≥ 0.1 mm⋅hr −1 are displayed. In (a-c) the organization index (I org ; Tompkins and Semie, 2017), the simple convective aggregation index (SCAI; Tobin et al. 2012) and the convective organization potential (COP; White et al. 2018) are shown for the same threshold (0.1 mm⋅hr −1 ). In (d-l), the spatial average of LWOI sc , LWOI in and LWOI ai are shown at top right transform is limited to three directions, allows only scales which are powers of 2, and neglects the important larger scales. For spatiotemporal data, the basis functions correspond to exactly three distinct phase speeds, combined with periods of half a month, a quarter of a month and so on. The continuous STWT entails basis functions which are directly related to any desired combination of speed and scale, making it far more convenient for this purpose.

Convective organization of an African squall-line
To illustrate the different representations of convective organization in MW observations, IR observations, and ICON simulations, we show rain rates and the LWOI components on 20 August 2016 at 0000 UTC in Figure 1. A westward-moving squall-line developed during 19 August 2016 and travelled to West Africa. The position of the squall-line is similar in both observations and ICON, but the intensity and its east-west extent differ. In the observations there are two local precipitation maxima in the squall-line, while ICON simulates a more uniform, meridionally oriented precipitation line. The horizontal extent of the squall-lines is largest in the MW observations and slightly smaller in the IR observations. ICON simulates much finer structures and a narrow squall-line. In addition to the heavy precipitation event in West Africa, the MW observations include lots of weak and large-scale precipitation over the Eastern Atlantic, which is almost completely missing in the IR observations. ICON simulates more maritime rainfall than IR, but the precipitation features are smaller and more intense than in MW observations. ICON fails to simulate correctly the location of the single-cell storms over the African continent, but their structure and intensity is represented more satisfactorily.
LWOI sc indicates that most precipitation features in the MW observations act on large scales. In IR only the squall-line is characterized as a large-scale feature (LWOI sc > 0.9), while other continental cells are of small scale (LWOI sc < 0.2). In the ICON simulation LWOI sc is larger than 0.5 only for some pixels within the squall-line and over Central Africa, while otherwise rainfall in ICON acts mostly on small scales. Both observations and ICON agree that the squall-line is intense (LWOI in ≈ 1). The continental cells are less intense in MW than in IR or ICON. Although LWOI ai is generally noisy, anisotropy along the squall-line seems slightly increased. Especially in ICON, the north-south oriented squall-line is characterized as a linearly organized structure with LWOI ai > 0.9.
In Figure 1a-c we provide values of I org , SCAI and COP based on rain rates ≥ 0.1 mm⋅hr −1 for MW, IR and ICON. Note that all three indices are not tuned for rain rates and were originally based on vertical wind velocity (I org ), brightness temperatures (SCAI) and outgoing long-wave radiation (COP). I org suggests that organization is higher in ICON than in MW and IR, while both COP and SCAI (lower value for SCAI means higher degree of organization) indicate that MW and IR rainfall is more organized than in the ICON simulations. The spatial average of LWOI sc is consistent with SCAI and COP and suggests that the scale of precipitation structures in ICON is much smaller than in both observations, which results in a high number of small objects. Note that SCAI and COP are sensitive to the number of cells and the area . The reason why I org characterizes ICON rain rates as most organized is because the small-scale objects are close together. Intensity (LWOI in ) and anisotropy (LWOI ai ) are not assessed by I org , SCAI or COP. The LWOI components provide additional information on the spatial structure of convection. None of the other convective organization indices provide localized maps of convective organization.

Convective organization during August 2016
Figure 2a-c display the averaged rain rates in August 2016 observed by the MW observations, the IR observations and simulated by ICON. All three datasets show intense rainfall along the ITCZ, over northern parts of South America and over West Africa. ICON underestimates precipitation along the ITCZ compared to observations and fails to reproduce strong precipitation over the mountains of northern South America and West Africa. IR and ICON precipitation show finer structures than the MW observations. The analysis of snapshots (Figure 1) demonstrates that precipitation derived from MW observations includes frequent large-scale stratiform precipitation over the Atlantic, which is completely missing in IR and underestimated in ICON. These huge discrepancies between both observations are based on the different measuring methods. Note that the IR and MW rain rates are estimated from satellite images and may include large uncertainties because of the issues discussed in Section 3.
We calculate temporal averages of LWOI sc , LWOI in and LWOI ai at each location to provide an averaged characterization of local convective organization as displayed in Figure 2d-l. For this climatology we use the same mask for both observations and ICON. Only grid points with rain rates ≥0.1 mm⋅hr −1 for at least 2% of all 31 × 48 timesteps in MW, IR and ICON were taken into account to focus on the main precipitation regions and to exclude the incorrectly simulated drizzle in ICON. ICON reveals much smaller scales (LWOI sc in Figure 2d-f) compared to both observations due to the missing large-scale stratiform precipitation. In MW observations, the stratiform rainfall over the tropical Atlantic seems much larger in scale. In ICON and IR observations, African convection acts on the largest scales. This mismatch may be caused by the different satellite measurement method over ocean and land. However, ICON and both observations suggest that convection over the western tropical Atlantic and South America is of smaller scale than over Africa and eastern parts of the tropical Atlantic. Convection over South America is very localized. The large-scale features over Africa are associated with westward-propagating MCSs.
ICON and IR observations agree on the existence of intense West African rainfall caused by MCSs, but show differences over the ocean and South America, where ICON simulates weaker precipitation (Figure 2g-i). There are five local LWOI in maxima in MW observations, namely over the Guiana Shield, central tropical Atlantic, West African coast along the Guinea Highlands, Ghana and Gulf of Guinea west of Mount Cameroon. Especially over the eastern half of the tropical Atlantic, the LWOI in is low, although MW-derived precipitation is high ( Figure 2a). Thus, in the MW observations rainfall from the large-scaled systems is less intense and more persistent. In contrast, rainfall estimated from IR satellite images is very intense over the tropical Atlantic (high LWOI in ), but smaller in scale as seen before. These contrary results are due to the different measurement methods discussed in Section 2.
Anisotropy (Figure 2j-k) is slightly lower over South America and parts of West Africa in the MW observations than in ICON simulations and IR observations, but generally the ICON simulations and observations agree well in this respect. The variability of the LWOI ai over the complete domain is relatively small, mainly due to two reasons. Firstly, due to a missing rotational invariance of the two-dimensional wavelets used here, the LWOI ai is not completely invariant with respect to rotation, that is LWOI ai changes when a squall-lines changes F I G U R E 2 Monthly averaged (a-c) rain rates, (d-f) scale LWOI sc , (g-i) intensity LWOI in and (j-l) anisotropy LWOI ai of precipitation during August 2016 in (a, d, g, j) MW, (b, e, h, k) IR and (c, f, i, l) ICON. LWOI sc , LWOI in and LWOI ai are only shown for grid points with rain rates ≥0.1 mm⋅hr −1 for at least 2% of all timesteps in both observations and ICON. Dashed lines in (a-c) indicate latitude 2 • N its orientation. This may be relevant since meridionally oriented squall-lines over northern West Africa often propagate southwestwards at the end of their life cycle and change their orientation from north-south to diagonal. Secondly, convection over the tropical Atlantic may also organize into highly isotropic structures such as hurricanes. Thus although LWOI in is large, this results in a low LWOI ai . LWOI ai as a single indicator thus may miss highly organized structures, but can be useful to differentiate a squall-line from a highly organized isotropic cluster, which both have a large LWOI in . Because of the rotation issue and the noise in LWOI ai , we concentrate on LWOI sc and LWOI in in the remainder of the article.
Nevertheless, to investigate the preferred orientation of precipitation patterns, we compare the scale-averaged spectral energy for each direction and show the percentage of east-west and north-south oriented objects during August 2016 in Figure 3. The orientation of each rain pixel is given as the direction of maximal energy. East-west (Figure 3a-c), north-south (Figure 3d-f) and diagonal (not shown) sum up to 1. East-west and north-south oriented objects are dominating. Only less than 10% of the objects have a diagonal orientation, which is mainly due to the missing rotational invariance of the wavelet transform. Observations and ICON both show that convection is more zonally (east-west) oriented over the tropical Atlantic along and south of the ITCZ, while continental convection over Africa is mostly organized into meridionally (north-south) oriented squall-lines. Interestingly, MW, IR and ICON indicate that convection is also north-south oriented north of the ITCZ over the tropical Atlantic. This signal in orientation may be caused by eastward-moving squall-lines. Convection over South America shows no preferred direction. Wavelets may be used to describe convective organization not only in space but also in time. The two-dimensional discrete wavelet investigates the time-averaged degree of convective organization in space, but ignores the longevity and propagation speed of the precipitating systems. Figure 4 has shown us that the central part of the domain is traversed by a number of large, westward-moving clusters with elevated degrees of spatial organization. In the light of the analyses presented by Schlueter et al. (2019), we expect that these large, intense, anisotropic features should be related to AEWs. To verify this hypothesis and to quantify the spatiotemporal variability of our three datasets, we apply the STWT as described in Section 3.5 to the meridionally averaged rain fields. Phase speeds are varied within a range of 10 ≤ |c| ≤100 km⋅hr −1 and the considered wavelengths range from 150 to 4,000 km. Figure 5 shows the resulting wavelet spectra averaged over space and time. As expected, almost all spectral power is concentrated on the westward-moving side (c < 0). The largest values correspond to daily and sub-daily time-scales (i.e., small wavelengths and slow to moderate speeds). IR has the overall greatest variability in this regime, while spectral energy in ICON is concentrated on the smallest scales ( < 250 km, k > 20). The two satellite datasets feature a notable increase in power at T = 1 day, which is not as pronounced in ICON. This decreased prominence of a diurnal cycle in the model is likely related to the overall weak simulated precipitation over South America which varies mostly on daily time-scales (cf. Figure 4).

Temporal evolution of convection
As mentioned in Section 1, AEWs are characterized by periods of 2.5 to 5 days and zonal wavenumbers of 8 to 19, which roughly corresponds to wave numbers 1 to 5 in our domain. We recognize that all three datasets contain a distinct increase in power for 2.5< T < 5 days, with a common local maximum at the small-scale edge of the AEW regime (c ≈ −25 km⋅hr −1 , ≈ 1, 900 km).
Our wavelet approach makes it very straightforward to localize this AEW pattern in space and time. Instead of averaging over these two dimensions, we simply average the full spectra over all speeds and wavelengths corresponding to the features of interest (thick lines in Figure 5). In Figure 6, we have superimposed the resulting contours of AEW-related power on the original Hovmöller diagrams of rain (repeated from Figure 4). As expected, all local maxima correspond to the large, long-lived systems crossing the Atlantic with most power lying close to the African coast (20 • W). Besides three dominant features persisting between 13 and 25 August 2016, we recognize one large system ending its life cycle near the beginning of the month and one beginning its westward journey near the end. While ICON has less power overall, related to generally lower intensities, the timing and locations are very similar to those of the satellite data. To enable a more direct comparison between model and observations, we summarize the time-averaged STWT spectra by calculating their barycentres in the -c plane, giving us one central speed and wavelength for every longitude. The resulting profiles (Figure 7) reveal the expected structure. Small, fast features dominate over the African continent and gain spatial extent while slowly losing speed over the ocean -the latter can be observed as a slight upwards bend of the diagonal stripes in Figure 4. The characteristics over South America are similar to West Africa, albeit with a tendency towards even smaller scales. The three datasets are in fairly good agreement, particularly over the ocean where AEWs are most prevalent. Differences between model and observations are more pronounced over the continents. ICON simulates lower speeds over northwestern South America and smaller

Link to environmental variables
The previous analysis suggests that tropical convection is organized into westward-moving squall-lines over West Africa, while convection over the tropical Atlantic is more persistent and zonally oriented. Now we investigate convective organization over the tropical domain from a dynamical perspective using the ICON-simulated 2D and 3D variables.
To this end, we select a set of 30 environmental variables and additionally calculate hourly tendencies of nine variables such as the consumption of CAPE or changes in surface pressure within an hour before the onset of convection and during convection. Because most of the precipitation along the tropical Atlantic is convective, we assume convection if rain rates are above 0.1 mm⋅hr −1 . We calculate the environmental variables for all rain pixels during August 2016. The temporal resolution of the 2D and 3D variables is 1 hr. To assess sampling uncertainty, we perform a five-day block bootstrapping. Six blocks of five running days are randomly chosen with replacement to generate 200 bootstrap samples each of 30 × 24 = 720 hr. We calculate meridional averages of the environmental variables and the LWOI components for each sample, and compute the temporal correlations between the environmental variables and LWOI sc and LWOI in as shown in Table 1. The spread of the 95% sampling interval in Table 1 and Figure 8 indicates the null hypothesis that a correlation may be due to random chance. Correlations outside these intervals are significant. LWOI sc is positively correlated with the tendency of CAPE (0.27) and the 6 km wind shear (0.24). CAPE consumption and high wind shear leads to more large-scale rain rates such as are caused by squall-lines. Surprisingly, CAPE itself is negatively correlated with LWOI sc (−0.25), meaning that precipitation acts on smaller scales when CAPE is high. However, this negative correlation is reversed over the central Tropical Atlantic, as seen in Figure 8e and discussed below. Correlations between LWOI in and environmental quantities are higher than for LWOI sc . Strong correlations between LWOI in and maximum vertical wind velocity (0.64), mean vertical wind velocity (0.60), upper-level divergence (0.58) and column-integrated relative humidity (0.53) indicate that rain rates are particularly intense with high LWOI in in the case of powerful updraughts in a humid environment.
We describe the six best correlating environmental variables (Table 1) in more detail and discuss their regional variations. Figure 8c shows that the temporal correlations between meridionally averaged LWOI sc and CAPE tendency are slightly positive (0.2) for all longitudes. The scale of convection gets larger when more CAPE is consumed by convection. The temporal average of CAPE tendency shows that the large-scaled African squall-lines consume up to 500-800 J kg −1 hr −1 . Over the Atlantic and along the African coast, rainfall reduces CAPE by 100 J kg −1 hr −1 or less, although LWOI sc is large. CAPE is also strongly reduced over South America and the Caribbean, where temporal averages of LWOI sc indicate less organized rainfall. Thus, the tendency of CAPE does not directly determine the scale of convection.
Correlations between LWOI sc and CAPE (Figure 8e) are even negative over both continents (−0.25) and maximal over the central tropical Atlantic (0.35). We conclude that high CAPE over the tropical Atlantic leads to large-scale precipitation, while the degree of organization over the South American and African continents does not depend on the amount of CAPE. The monthly averages show that CAPE is high over the West Atlantic (1,500 J kg −1 ) and below 1,000 J kg −1 over central tropical Atlantic and Africa, although LWOI sc indicates large-scale precipitation. This confirms that the influence of CAPE on LWOI sc is relatively small over both continents.
Wind shear over 6 km is slightly positively correlated with LWOI sc (Figure 8g). High wind shear fosters the evolution of large-scale systems such as squall-lines characterized by a large LWOI sc . The averaged wind shear is large over most parts of Africa (15 m⋅s −1 ), moderate over the West and Central Atlantic (5-10 m⋅s −1 ) and low over the East Atlantic (below 5 m⋅s −1 ) and fits well to the LWOI sc pattern shown in Figure 8a. Increased CAPE values in combination with moderate wind shear north of the ITCZ may be responsible for the north-south orientated convection in this region as observed in Figure 3.
Temporal correlations between LWOI in and the maximum vertical wind, upper-level divergence and column-integrated relative humidity show a similar pattern. Between 65 • W and 50 • W, correlations are nearly constant, increase over the tropical Atlantic and maximize at 30 • W, decrease slightly towards the West African coast and remain constant over West Africa. Correlations between the environmental variables and LWOI in are much larger than for LWOI sc and indicate that LWOI in is directly linked to the updraught velocity, especially over the Atlantic and the African continent, and the column-integrated humidity. The maximum of the TA B L E 1 Correlations over time of LWOI sc and LWOI in with the 39 environmental variables at calculation height z and time t (B denotes the value 1 hr before onset of convection, D the value at the onset of convection, and BD the difference between values at onset of convection and at 1 hr before)

Variables
Description z (km) t LWOI sc LWOI in   Figure 8d, f show these strong updraughts over the northern parts of West Africa, where LWOI in is maximal. Also over the Central Atlantic and over the South American continent, increased updraught velocities are in accordance with an increased LWOI in . Our analysis demonstrates that LWOI sc and LWOI in , which are calculated only on the basis of rain rates, provide useful information on convective organization. The LWOI components detect precipitation patterns of different kinds of convective storms.

CONCLUSION
Especially along the Tropics, where intense and long-lived convective systems develop, numerical weather prediction and climate models often fail to represent the degree of organization successfully. To assess the degree of convective organization and to analyze the relevant processes in detail, we use half-hourly output of high-resolution ICON simulations (grid spacing 2.5 km) over the tropical Atlantic (domain size 8,000 × 3,000 km) and parts of South America and West Africa during August 2016.
Comparisons are made with passive microwave radiometer observations (MW) and infrared measurements (IR) from the Integrated Multi-satellite Retrievals for Global precipitation measurement project (IMERG). ICON represents convective rainfall satisfactorily; the model contains even finer structures and produces small-scale drizzle over northern and southern parts of the tropical Atlantic. The IR measurements include strong precipitation inside deep convective cores and have potentially higher spatiotemporal consistency, but miss precipitation like stratiform rain outside the updraught regions.
To identify the degree of convective organization, we modified the wavelet-based organization index (WOI). We use the Daubechies 4 instead of the Daubechies 1 (Haar) wavelet, which results in less negative spectral energy. We furthermore normalize all three WOI components between 0 (non-organized convection) and 1 (organized convection). The resulting improved WOI allows us to study convective organization on scales ranging from the complete domain to individual grid-points without any further adjustments.
There are discrepancies between convective organization simulated by ICON and observed in MW-and IR-derived rain rates. Both observations and ICON show that convection over West Africa is more organized than over South America, because the scale of convection (LWOI sc ), its intensity (LWOI in ) and anisotropy (LWOI ai ) are higher. Due to the different measurement principles, the scale of rain rates in MW observations is higher than in the IR observations and ICON. IR observations show by far the greatest intensities (LWOI in ≈ 1). LWOI ai indicates that convection over northern parts of West Africa is slightly more linearly organized than over South America. LWOI ai is not completely invariant for similar but differently oriented squall-lines, and as a single index would not identify highly organized isotropic structures over the tropical Atlantic such as hurricanes. But in combination with LWOI sc and LWOI in , the LWOI ai index gives useful additional information on convective clusters. However, the issue of non-invariance with respect to rotation needs to be solved to fully exploit the anisotropy information. An analysis of the predominant orientation of the wavelet spectrum shows that convection over Africa is organized into meridional squall-lines while maritime convection is preferably zonally oriented.
Studying the temporal evolution using spatiotemporal wavelet transforms (STWT), we find that in ICON the small-scale variability is higher, but the diurnal cycle of convection, especially over South America, is less pronounced. However, the large-scale AEWs (c ≈ 25 km⋅hr −1 , ≈ 1, 900 km, 2.5< T < 5 days) are simulated satisfactorily in space and time. ICON and the observations agree on the central speed and wavelength of the AEWs. They include the fast westward-moving squall-lines (|c| > 45 km⋅hr −1 , < 1, 000 km) and more persistent, slow-moving meridionally oriented convection (|c| < 40 km⋅hr −1 , ≈ 1, 200 km) over the Atlantic.
We use a set of 3D variables provided by ICON to relate convective organization to convective characteristics and the environmental forcing. The pattern of scale and intensity of convection measured by LWOI sc and LWOI in correlates with different environmental variables. It turns out that LWOI sc is mainly modulated by wind shear. Large CAPE is not essential to simulate large-scale features, but the overlap of convective instability and strong wind shear is important. LWOI in is controlled by vertical wind speed and upper-level divergence, which indicate strong updraughts.
In this study, we provide a promising approach to characterize convective organization locally. Compared to the entropy analysis by Li et al. (2018), the LWOI components vary over the domain and are in agreement with convective and environmental quantities. The LWOI components are tested over a large domain, but only for roughly 1,500 time steps. To get a long-lasting climatology of organization over the tropical Atlantic and its temporal variability during a year, this analysis could be extended to several years. It is also possible to study convection over other regions (e.g., North America, Europe or tropical Pacific) to improve the representation of convective organization in climate and weather prediction models.
This wavelet analysis could be directly applied to IR measurements such as outgoing long-wave radiation or other variables. However, the wavelet spectra may be more sensitive for low-variation cloud structures than for small-scale highly variable rain rates.

ACKNOWLEDGMENTS
We gratefully acknowledge financial support by the project High Definition Clouds and Precipitation for Advancing Climate Prediction HD(CP) 2 , funded by the German Ministry for Education and Research (BMBF) under grant FKZ01LK1507B (Sebastian Brune). Special thanks go to Velibor Pejcic for his advice on the IMERG satellite data. We are also very grateful to two anonymous reviewers for their constructive comments on an earlier version of the article.