# Advances on the Links Between Turbulent and Submeso- to Mesoscales During EUREC4 A

E. Gauvrit, M.-n. Bouin, J.-m. Delouis, F. Boulanger

► To cite this version:

E. Gauvrit, M.-n. Bouin, J.-m. Delouis, F. Boulanger. Advances on the Links Between Turbulent and Submeso- to Mesoscales During EUREC4 A. Earth and Space Science, 2025, 12 (2), 10.1029/2024EA003865. https://hal-05016815

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# Earth and Space Science

## RESEARCH ARTICLE

10.1029/2024EA003865

**Special Collection:**

Recent progress in atmospheric boundary layer turbulence and implications to surface-atmosphere exchange

**Key Points:**

* Wavelet scattering transform (WST) is a new promising tool to reveal scale modulations in airborne observations of atmospheric turbulence

* In trade winds, modulating scales are very different in roll vortices and convective cells environments

* WST reveals organization of the turbulence along rolls and the impact of mesoscales at the cloud base in convective cell conditions

**Correspondence to:**

M.-N. Bouin,
marie-noelle.bouin@meteo.fr

**Citation:**

Gauvrit, E., Bouin, M.-N., Delouis, J.-M., & Boulanger, F. (2025). Advances on the links between turbulent and submeso- to mesoscales during EUREC⁴A. *Earth and Space Science*, 12, e2024EA003865.
https://doi.org/10.1029/2024EA003865

Received 8 JUL 2024
Accepted 11 NOV 2024

**Author Contributions:**

**Conceptualization:** M.-N. Bouin, J.-M. Delouis, F. Boulanger

**Data curation:** E. Gauvrit, J.-M. Delouis

**Formal analysis:** E. Gauvrit, M.-N. Bouin, J.-M. Delouis, F. Boulanger

**Funding acquisition:** J.-M. Delouis, F. Boulanger

**Investigation:** M.-N. Bouin

**Methodology:** E. Gauvrit, M.-N. Bouin, J.-M. Delouis, F. Boulanger

**Project administration:** J.-M. Delouis, F. Boulanger

**Resources:** E. Gauvrit, F. Boulanger

**Software:** E. Gauvrit, J.-M. Delouis

**Supervision:** M.-N. Bouin, J.-M. Delouis, F. Boulanger

© 2025. The Author(s).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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## Advances on the Links Between Turbulent and Submeso- to Mesoscales During EUREC⁴A

E. Gauvrit¹,² ID, M.-N. Bouin¹,³ ID, J.-M. Delouis¹ ID, and F. Boulanger² ID

¹Laboratoire d'Océanographie Physique et Spatiale (LOPS), Univ. Brest, CNRS, Ifremer, IRD, Brest, France, ²Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France, ³CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France

**Abstract** Turbulent processes in the atmospheric boundary layer (ABL) are parameterized in numerical weather prediction and climate models. Better understanding their modulation by larger-scale organized structures, some of them being represented explicitly, is thus of great interest. In this study, we test an innovative statistical tool, the Wavelet Scattering Transform (WST) on turbulent measurements of 3D velocities at different levels in the ABL during the EUREC⁴A campaign near Barbados. The measurements were done in trade wind environment over the sea. They encompass two categories of ABL convection, roll vortices (RV) or convective cells (CC) whose organization depends on cold pool and density currents. Statistical tools such as Fourier transforms or moment analysis give access to levels of energy, characteristic sizes and variances in CC and RV conditions. The WST provides further information, with strong modulations of the turbulent scales by submeso- to mesoscales in CC, whereas modulating scales are smaller than the horizontal scale of rolls in RV. Penetrating dry tongues from the ABL top are revealed by modulation scales varying with height. In RV conditions, modulations differ in cross-wind and along-wind observations confirming that along-wind measurements do not provide good sampling of the ABL turbulence. WST are a valuable tool for investigating ABL turbulence and its interactions with coherent structures, and could be used on scalar variables and surface fluxes.

**Plain Language Summary** Turbulence, which involves movements at a scale smaller than 300 m, cannot be directly modeled for weather forecasts or climate, although it significantly affects the atmospheric layer close to the ground. Understanding how it is influenced by larger organized motions allows for indirect evaluation, since these motions can be reproduced in models. In this study, we use a new statistical tool to explore the connections between organized motions and turbulence from airborne observations in two types of atmospheric conditions: horizontal rolls and convective cells with rain. We demonstrate that this tool can capture all the information about atmospheric structures, similar to traditional statistical methods. In addition, we find that the sizes of the structures that influence turbulence differ greatly between the rolls and the cells. In cells, there are descents of dry air from the layers above. In rolls, turbulence is stronger between rolls than within them. This promising tool will be used to study temperature and humidity turbulence, as well as turbulence closer to the surface.

## 1. Introduction

Turbulence is not represented explicitly by atmospheric numerical models of kilometric or coarser resolution. Current turbulence parameterizations are based on a set of assumptions, including horizontal homogeneity at the scale of the model grid cell and stationarity at the corresponding advective time scale. Also, strong hypotheses are made about the -5/3 frequency slope of the inertial part of the velocity spectrum (e.g., Kaimal et al., 1976). These assumptions have been increasingly challenged with the detection from in situ observations or Large Eddy Simulations (LES) of boundary layer processes responsible for larger (sub-mesoscale) coherent transport (LeMone et al., 2019, for a review). Resulting horizontal inhomogeneities clearly violate the homogeneity assumption thought to organize the turbulent scales close to the surface. Most of these coherent structures are related to the so called shallow convection, with submesoscale characteristic length scales. With recent or expected improvements in model resolution, explicit representation of coherent structures is becoming feasible, while explicit representation of turbulence is out of reach. More precisely, boundary layer processes may be usefully categorized according to their scales into organized 2D structures at submeso- to mesoscales, and classical law-of-the-wall 3D turbulence at smaller scales. The cutoff length scale
**Validation:** E. Gauvrit, J.-M. Delouis, F. Boulanger

**Visualization:** E. Gauvrit

**Writing – original draft:** E. Gauvrit, M.-N. Bouin

**Writing – review & editing:** E. Gauvrit, M.-N. Bouin, J.-M. Delouis, F. Boulanger

between the two corresponds to a spectral gap (e.g., Smedman-Högström & Högström, 1975; Stull, 2012; Van der Hoven, 1957). The existence of this spectral gap is still the subject of active research, but it is used to justify considering coherent, resolved processes and parameterized turbulence as dynamically independent in numerical modeling. Investigating the link between submeso- or mesoscale coherent structures and turbulent scales is therefore an important step toward more accurately representing fluxes and buoyancy in kilometer-scale, operational simulations.

Shallow convection and the associated horizontal inhomogeneities have motivated numerous studies for several reasons (Atkinson & Wu Zhang, 1996; Etling & Brown, 1993): (a) they correspond to the presence of cumulus or stratocumulus clouds, which cover a large part of global ocean and reduce the shortwave radiative budget; this impacts the thermal balance of the atmosphere at global scale and climatic time scales (Bony & Emanuel, 2005); (b) they are associated with a significant share of the global transport of heat, moisture and momentum, which impacts the location and strength of Hadley-Walker cells, intraseasonal oscillations and mid-latitude jets (Bony et al., 2015; Slingo & Slingo, 1988); (c) within the atmospheric boundary layer (ABL), they influence the transport and associated processes: strengthening of mixing by rolls, transport of heat and moisture from the surface to the top of the ABL, surface gusts due to downward transport of momentum (Lorsolo et al., 2008; Morrison et al., 2005; Rivière et al., 2020); (d) they are related to the turbulent intensity near the surface and modulate the surface fluxes at submeso or meso scales. Unlike NWP models, climate simulations are unable to represent them explicitly and rely on parameterizations. To improve these representations several research programs of the past decades aimed at better understanding the dynamics of the trade cumuli and their link with the large scale conditions.

Most of these programs were based on airborne observations over land (e.g., LeMone, 1976; Lohou et al., 1998; Weckwerth et al., 2004) or at sea (e.g., Holland, 1970; LeMone & Pennell, 1976; Nicholls & Lemone, 1980). Several campaigns focused on the trade cumulus region, which plays a key role in the radiation budget because of its large extent (Medeiros et al., 2008). Steady winds result in larger evaporative fluxes there and the moisture end up feeding the tropical convergence zones (Tiedtke et al., 1988; Zuidema et al., 2012). By condensing or precipitating, trade cumuli also change the thermodynamics of the surrounding layers (Bony et al., 2017).

In the last 50 years, measurement campaigns probed the atmospheric conditions and cloud organizations near the Barbados or Puerto-Rico (e.g., LeMone & Pennell, 1976; Nair et al., 1998; Rauber et al., 2007). They especially targeted the presence of ABL roll vortices (RV) and convective cells of various sizes and organizations (CC). Recently, the EUcidating the RoI of Cloud Circulation Coupling in ClimAte (EUREC⁴A) field experiment aimed at characterizing the Marine Atmospheric Boundary Layer (MABL) properties and their link with the cloud organization from turbulent to mesoscales (Stevens et al., 2021). Some of the airborne instruments targeted the cloud fraction and parameters close to the cloud base. Others provided high-frequency measurements of the ABL properties between the surface and the bottom of the cloud layer. These observations confirmed that RV and CC largely prevail in this region of trade cumuli (e.g., Brilouet et al., 2023; Touzé-Peiffer et al., 2022).

The parameter $z_{INV}/L$ with $z_{INV}$ the inversion height and $L$ the Obukhov length is often used as an estimate of the ABL structure (Obukhov, 1971). For $z_{INV}/L > -5$, convection is preferentially organized along rolls (Figure 1a), whereas three-dimensional convective cells will dominate for $z_{INV}/L < -25$ (Figure 1b) with some mixed type of organized convection in between (Etling & Brown, 1993). However, in the case of the Barbados trade cumuli, recent investigations based on 10 years of satellite imagery evidenced a more diverse range of organizations with four categories of cloud patterns labeled flower, fish, gravel or sugar, depending on their size, vertical extent, organization and capacity of precipitation (Bony et al., 2020). In the case of CC, in situ observations and LES experiments show that generation of cold pools (CP) by evaporating precipitation plays a key role in the development of new convective cells and in their organization in mesoscale arcs (Seifert & Heus, 2013; Tompkins, 2001; Zuidema et al., 2012). Cooled air downdrafts generate density currents, which spread at the surface up to 200–400 m in the trade wind conditions. They modify the boundary layer properties by creating a cold dome (~-1 K with respect to their environment), and a dry core and/or moisture ring due to enhanced evaporation flux at their edge (Figure 1b). As a feedback to cloud occurrence and organization, cold pools trigger convective updrafts through mechanical or thermodynamic forcing (Schlemmer & Hohenegger, 2016). In fact, sensitivity tests using LES showed that cloud structures and organization are strongly impacted by the CP formation (Seifert & Heus, 2013).
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Mixing ratio fluctuations in the surface layer ($z = 5$ m) from LES simulations in (a) convective rolls and (b) large convective cell conditions. The LES simulations have been designed using the model and configuration described in Dauhut et al. (2023), except that the subgrid condensation scheme has been activated. The surface mixing ratio is used here as a proxy of low-level moist convergence and cloud fraction.</figcaption></figure>

Roll vortices (Figure 1a) span the height of the ABL and are generally oriented from 10° to 20° to the left of the wind at the ABL top, with a maximum vertical velocity at 0.33 $z_{INV}$ (LeMone, 1973) and an aspect ratio typically between 2 and 10 (Etling & Brown, 1993). In particular, rolls have been observed in the trade wind cumulus region chosen for the EUREC⁴A campaign (e.g., Grossman, 1982; LeMone, 1976; Pennell & LeMone, 1974). As for CC, strong links exist between the type and structure of clouds and the ABL organization: clouds form above the updraft part of the RV circulation, corresponding to low-level convergence (Etling & Brown, 1993). Consistently, enhanced moisture flux and gusts are observed under the cloudy part of the RV organization, with 50% of the momentum flux concentrated in 14% of the area (LeMone, 1976). Cloud-free areas correspond to the sinking branch of the RV, transporting dry air from above and reinforcing the contrast of moisture between updrafts and downdrafts. Several studies based on observations and LES underline the strong anisotropy of turbulent fields in the MABL due to the RV organization, with impacts on their distribution and spectral characteristics (Lohou et al., 2000; Nicholls & Readings, 1981).

All the mesoscale to submesoscale structures listed above emphasize the strong links between the boundary layer dynamics and the cloudy organization in the trades. In addition to these feedback mechanisms, the ABL is frequently impacted by intrusions of air masses from above, possibly in link with larger scale dynamics. As a result, the high-frequency specific humidity field shows a higher variance and a pronounced negative skewness. This has been observed in cumulus cloud conditions, for instance in the Barbados (Nicholls & Lemone, 1980) or over land with confirmation of the mechanisms from LES (Couvreux et al., 2007). Depending on the conditions, dry tongues may concern a few hundred meters below the cloud base or the whole ABL down to the ground.

Coherent structures in the ABL and intrusions of upper air masses have been characterized using statistical methods like power spectra analysis, integral length scales, conditional sampling or quadrant analysis (LeMone, 1973; LeMone, 1976; Lohou et al., 1998, 2000; Couvreux et al., 2007; Denby et al., 2022, among others). Power spectrum is based on the Fourier transform and provides a statistical representation of the signal variance at each scale. It has desirable properties such as translation invariance, scale separation, and asymptotic normality (Cheng & Ménard, 2021). These characteristics make it a valuable tool for identifying characteristic scales in measurements, such as the integral length scale ($L_α$, with α the variable of interest) or the size of the most energetic eddies ($λ_α$) (Brilouet et al., 2017; Darbieu et al., 2015; Kaimal & Finnigan, 1994; Lenschow & Stankov, 1986; Lohou et al., 2010). However, the power spectrum has limitations in terms of extracting additional information. For example, the delocalized Fourier kernel ($e^{-ikx}$ with k the wave number) leads to the
loss of location information in real space (phase). Furthermore, calculating second- and higher-order moments (such as spectra, bispectra, and structure functions) involves repeated multiplications of the input signal, increasing its variance and resulting in a less robust statistical description. Consequently, these higher-order statistics are ineffective for characterizing non-Gaussian spatial structures, which are linked to interactions between scales. To understand how turbulence is modulated by larger organized motions, a low variance statistical tool is needed.

In this study, we investigate the relationships between turbulence and larger scales present in the MABL utilizing the Wavelet Scattering Transform (WST) (Mallat, 2012). The WST is a compact statistical representation that has low variance and preserves important characteristics similar to the power spectrum, such as translation invariance, energy extraction, and scale separation. Moreover, the WST is capable of extracting information on non-Gaussian behaviors like filaments or vortices (in 2D). Initially developed in computer vision, it has found applications in various fields such as image and audio signal processing (Andén & Mallat, 2011, 2014; Bruna & Mallat, 2013; Oyallon et al., 2019; Sifre & Mallat, 2013). It has also been used in quantum chemistry, medicine, geography, and astrophysics for tasks including classification (Cheng & Ménard, 2021; Kavalerov et al., 2020; Liu et al., 2020), parameter inference (Cheng et al., 2020; Eickenberg et al., 2022; Hirn et al., 2017; Valogiannis & Dvorkin, 2022), noise separation (Delouis et al., 2022; Régaldo-Saint Blancard & Eickenberg, 2023), and exploratory data analysis (Allys et al., 2019; Regaldo-Saint Blancard et al., 2020). We present here a first attempt to use WST in the field of atmospheric turbulence, to focus on the scale interactions among identified structures in wind measurements collected in different MABL organizations during the EUREC⁴A field campaign. Section 2 briefly presents the data used for analysis and classification of atmospheric conditions. Section 3 takes advantage of classical statistical tools to characterize the turbulent and mean atmospheric conditions in the MABL. Section 4 presents the WST principles and its application to the data set. Section 5 gives the results of our analysis, which are discussed in Section 6 with some perspectives.

## 2. Data

The EUREC⁴A campaign took place in January-February 2020 over the tropical Atlantic Ocean in the downstream winter trades, east of Barbados (Bony et al., 2017, 2022; Stevens et al., 2021). Many instruments were deployed from several measurement platforms including four research aircrafts, four research vessels and the Barbados Cloud Observatory (BCO). Among the instrumented aircrafts, two were dedicated to measure the lower atmosphere properties, the ATR-42 (ATR) sampling the turbulence in the subcloud layer (Brilouet et al., 2021), and the High Altitude and Long Range Research Aircraft (HALO) characterizing the dynamical and thermodynamical environment at larger scale (Konow et al., 2021) from photographs and dropsondes. In this section, we first provide a succinct description of the data employed in our analyses.

### 2.1. Turbulence Data

During EUREC⁴A the ATR made 20 flights equipped with meteorological sensors measuring wind, temperature and humidity at 25 Hz corresponding to a mean spatial (horizontal) resolution of $l_s = 3.9$ m. Mean wind conditions in the MABL during the campaign correspond to easterlies with an average speed of $10.47 \pm 2.44$ m.s⁻¹ (Figure 2) and a slight positive trend from January to March. Flight patterns were designed according to these mean wind conditions and include both along- and cross-wind transects to sample MABL structures in a comprehensive way. On each flight, the aircraft followed the same trajectories along R and L patterns (Figure 2) inside the cloud and the subcloud layers respectively, providing measurements between 800 and 2,000 m, and between 250 and 800 m above sea level. The observations have been partitioned into horizontal continuous segments of ≈60 km (or 10 min; the segments are only approximately 60 km long, without any impact on the quality of our results). The data set paper by Brilouet et al. (2021) provides more details about the processing strategy, the calibration and quality controls applied to the data. The data set includes two types of data, turbulent moments calculated over each segment and fluctuating time series. We use the latter to investigate the turbulent scale interactions of the MABL by analyzing the 3D components of the wind ($u', v', w'$), with $u$ corresponding to the eastward component and $v$ to the northward component. As this study focuses on the dynamical aspects of the MABL, we restrain our analyses to $u', v', w'$.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** ATR trajectory (red line) for the RF11 flight (5 February 2020) showing R and L patterns at *z* = 749 *m* (around cloud base) and *z* = 276 *m* (close to the middle of the MABL), respectively. HALO circular trajectory where dropsondes are released (violet dots). ATR mean wind distribution (wind rose), corresponding to easterlies.</figcaption></figure>

## 2.2. Auxiliary Data

### 2.2.1. HALO

Environmental conditions associated with each ATR flight are given by Bony et al. (2022). Meteorological parameters such as the trade inversion height ($z_{INV}$), the subcloud-layer top height ($z_{SC}$) and the lifting condensation level height ($z_{LCL}$) have been computed from the Joint dropsonde Observations of the Atmosphere in tropical North atlaNtic mesoscale Environments (JOANNE) data set (George et al., 2021). These dropsondes were launched from the HALO during circular flights at 10,000 m above sea level around the ATR trajectories (Figure 2). The corresponding values for the flights considered in this study are $z_{LCL}$ = 673 m, [605, 758] m, $z_{SC}$ = 773 m, [700, 870] m, and $z_{INV}$ = 2746 m, [1,960, 3,050] m.

### 2.2.2. Satellite Observations

In addition to in situ measurements, satellite observations provide a coverage of the atmosphere at high spatial resolution. The Geostationary Operational Environmental Satellite GOES-16 collected images every minute between 19°N–5°S and 49–66°W from 14 January to 14 February. The Advanced Baseline Imager onboard GOES-16 (GOES-R Calibration Working Group, 2017) provides observations of the cloud organizations in the vicinity of the ATR trajectories through two distinct spectral bands: Band 2 (visible at 0.64 µm) corresponds to reflectance measurements and Band 13 (infrared at 10.35 µm) to the brightness temperature. The visible channel with a 500 m spatial resolution reveals fine scale structures but works only during daytime. The infrared channel provides observations during both day- and nighttime with a lower resolution of 2 km and was used for a few flights at night.

### 2.2.3. Radiative Measurements

Upwelling and downwelling radiative fluxes have been measured by two sets of Kipp and Zonen pyranometers (shortwave, 0.2 – 3.6 µm range) and pyrgeometers (longwave, 4.5 – 42 µm range) placed at the top and at the bottom of the ATR (Stevens et al., 2021).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** GOES-16 images showing examples of ATR flights in CC (a–c) and RV conditions (d–f). Blue (resp. orange) arrows indicate flight segments.</figcaption></figure>

All these data have been used to categorize the turbulent data sets from the ATR flights into the two types of organizations (CC or RV as presented in Section 1).

## 2.3. Prior Classification

To compute the scale modulations in CC and RV conditions, we first classify each ATR segment using the corresponding environmental cloud structures observed by GOES-16 images and the radiative fluxes as a proxy of the cloud cover. Figure 3 illustrates 6 different scenes where ATR segments are either in CC (Figures 3a–3c) or RV environments (Figures 3d–3f). Convective cells associated with CP are marked by arc-shaped cumuli at their front edge (Droegemeier & Wilhelmson, 1985; Moncrieff & Liu, 1999; Rotunno et al., 1988). At their center, there is either a central cloud indicating precipitation for the active case (Figures 3a and 3b, Langhans and Romps (2015)), or clear sky for the passive case (Figure 3c). Longitudinal RV can be identified by cloud streets aligned in the direction of the mean wind (Figures 3d and 3e) (Etling & Brown, 1993; Young et al., 2002). Sometimes the lack of cloud hinders the ability to discern the underlying physical processes, but it does not imply the absence of organization in the MABL. Figure 3f gives an example in which the ATR flew under clear sky conditions, while convective rolls have been identified and are well documented (Brilouet et al., 2023).

We retain only segments that remain entirely within a single type of coherent structure, either CC or RV, disregarding those that cross multiple cloud organizations. Consequently, we obtained 22 and 26 segments, respectively, for the CC and RV categories (see Tables A2 and A3).

# 3. Preliminary Results

To investigate the distribution of energy and fluctuations within the ABL under CC and RV conditions, preliminary investigations focused on analyzing the energy spectrum, turbulent kinetic energy (TKE), and probability density functions (PDFs) of wind speed measurements. The energy spectrum was calculated across 12 logarithmic bins ranging from 10 m to 20 km. Turbulent kinetic energy was evaluated over sections of 10 km,
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Average log-bin energy spectra of (a) *u*'̇, (b) *v*'̇ and (c) *w*'̇ for segments in CC (blue) and RV (orange) conditions. Vertical colored dashed lines represent the largest energetic scale $\lambda_\alpha$. Shaded areas and error bars represent the standard deviation and the standard error, respectively. Dotted lines correspond to the $k^{-2/3}$ turbulent cascade.</figcaption></figure>

producing six TKE measurements per segment, from which a mean value was calculated. Subsequently, wind spectra and TKE values were averaged over segments to determine the ensemble averages for both CC and RV conditions.

## 3.1. Velocity Energy Distribution

Figure 4 shows the energy spectra $kE_\alpha(k)$, with $k$ the wave number, expressed with respect to the wavelength $\lambda$ for the three velocity components averaged over each segment. Within the inertial range ($10 \text{ m} \le \lambda \le 1000 \text{ m}$), the energy densities of $u'$, $v'$, and $w'$ are higher in RV than in CC conditions. Conversely, variability in the inertial subrange of horizontal velocity components (Figures 4a and 4b) is much lower in RV conditions than in CC conditions. This difference in energy levels is supported by the mean TKE, which is higher in RV conditions ($0.27 \pm 0.08 \text{ m}^2 \text{s}^{-2}$) than in CC conditions ($0.08 \pm 0.04 \text{ m}^2 \text{s}^{-2}$), even if this difference is not significant at a level of uncertainty of 95%. We also investigate whether this level of energy is dependent on the height of measurements. Dividing the measurements in three groups corresponding to the lower half of the ABL, upper half of the ABL and cloud layer does not show any difference in energy (significant at 95% level of uncertainty) between these groups.

The peak of energy is reached at shorter wavelengths for RV than for CC conditions. The largest energetic scale ($\lambda_\alpha$, with $\alpha = u'$ or $v'$) for horizontal velocities in RV conditions is close to 2,500 m, likely corresponding to the horizontal extension of the rolls. For the CC conditions, the peak of energy is much higher and close to 10 km, almost at the upper edge of the sampling of the observations. A weak spectral gap is observed for wavelengths between 1000 and 5000 m, in agreement with observations of long time series of horizontal turbulence (Kang & Won, 2016; Larsén et al., 2016). Note that Larsén et al. (2016) observed a clear decrease with height of the spectral gap, and that the lowest level of the measurements used here is much above their highest level of 100 m. Energetic horizontal processes are present and likely dominant up to scales of at least 10 km in CC conditions.

The vertical velocity spectra also show a shift between the peak wavelength $\lambda_{w'}$ in RV conditions, slightly below 1,000 m and likely corresponding to the maximum height of the rolls and its counterpart in CC conditions around 2,500 m (Tables A2 and A3). We interpret this as a discrepancy between the height of the rolls constrained by $z_{SC}$ in the first case, and the height reached by the shallow convection close to $z_{JNV}$ in the second case. Note that it is not possible to check whether the subcloud level $z_{SC}$ is different in RV and CC conditions, because the only available information is the mean $z_{SC}$ value for the whole flight and not for every segment. These results are consistent with previous studies making the link between spectral peaks of energy and constraints provided either by the height of the MABL or by the size or coherent structures organizing the transport in the MABL (Lohou et al., 2000; Nicholls & Readings, 1981). Typically, in RV conditions at sea, airborne measurements performed cross-wind and along-wind showed different peak length scales, with different levels of sharpness defined as the
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Probability density function computed for time-series of *u*'', *v*' and *w*' for the two different categories, CC (a–c) and RV (d–f). Each PDF (colored solid line) is represented in log-scale against its respective normal law N(μ = 0, σ) (solid line) as well as its opposite distribution (dashed line). Standard deviation (σ), skewness (μ̃₃) and kurtosis values (μ̃₄) are given for each case.</figcaption></figure>

curvature of the transition region of the spectrum around the peak (Brilouet et al., 2017). The same shift of the spectral peak constrained by the subcloud height zsc is observed here for the vertical velocity spectrum (Figure 4c).

## 3.2. Probability Density Functions

We then analyze the PDFs for each velocity component (*u*'', *v*'', *w*'') for the two groups of MABL conditions (Figure 5). Each distribution is compared to a normal law N(μ, σ), where μ and σ correspond to the average and standard deviation of the variable (with μ = 0 here). All components strongly deviate from normal distributions, meaning that fluctuations are not only governed by a Gaussian process. This deviation is characterized by two key aspects: asymmetry (normalized skewness μ̃₃) and a pronounced increase at high fluctuations (normalized kurtosis μ̃₄). The latter indicates the presence of intermittent processes, which occur more frequently in CC than in RV conditions (see Figures 5a–5c). The positive heavy tail in the distribution of *u*' in CC conditions (Figure 5a, μ̃₃ = 0.64) indicates gusts blowing mainly against the mean wind. We relate them to the gust fronts associated with convergence at the edge of the CPs, which are ubiquitous in CC conditions. In RV conditions, both the skewness and kurtosis of the horizontal velocities are lower than in CC conditions, indicating that the flow is more symmetrical and more regular.

The distribution of the vertical velocities in CC (Figure 5c) is almost symmetrical, with as many intermittencies in the downdrafts as in the updrafts (strong normalized kurtosis μ̃₄ = 4.59, low normalized skewness μ̃₃ = 0.37). In the RV case, the vertical velocities are skewed toward positive values (Figure 5f), indicating more intermittent processes in updrafts (no similar tail is associated with downdrafts). As already noted by LeMone (1976), rolls concentrate turbulence and turbulent fluxes in the convergence and updraft regions by advecting horizontally and transporting upwards the elements generating turbulence at the surface. The standard deviations of the vertical velocities are higher in RV than in CC conditions, but this is largely due to much larger variability on the largest length scales, whereas, in the inertial range, the variability is much lower (Figure 4c).
This preliminary analysis is useful for identifying characteristics of the MABL associated with coherent struc-
tures. At the indiscriminate flight levels between 250 and 2,000 m, significant differences are observed between
RV and CC conditions. For the three velocity components, the energy level at all length scales (excluding the
longest ones for the horizontal components) is significantly higher in RV than in CC conditions. In RV conditions,
the most energetic scales correspond to the characteristic scales of the rolls. In striking contrast, the horizontal
energetic scales in CC conditions show no limitation in the large length scales, showing that coherent patterns of
10 km or more are present in the MABL. The variability (standard deviations) is also much higher in the length
scales between ≈10 and ≈1,000 m for CC than for RV, for all three velocity components. From what is known
about convective cells associated with trade cumuli and the examples shown Figure 3, segments in CC conditions
likely encompass a larger variety of conditions: in the MABL, this may include downdrafts beneath precipitating
clouds (Tompkins, 2001), stably stratified conditions close to CP centers (Zuidema et al., 2012), mixed layers
with enhanced moisture surface fluxes close to the edge of CPs (Schlemmer & Hohenegger, 2016), gusts and
enhanced turbulence with updrafts at the edge of CPs or between CPs. In RV environment, the flow is more
organized with likely more small eddies in the ascending than descending branches, but less variety of conditions
(LeMone, 1973). Finally, gusts opposed to the main wind are detected in CC conditions only.

Integral length scales $L_\alpha$ have not been used in this analysis. Indeed, they give information on the size of the coherent structures, which is not the subject of this study. Tables A2 and A3 nevertheless list the $L_w'$ values as provided in Brilouet et al. (2021). As seen here, the PDF approach, although useful to highlight certain properties, offers a limited perspective. For instance, it is not possible to determine if the observed heavy tails are associated with intermittency at small or large scales. Moreover, it fails to retain the localization of the structure. A detailed study of intermittency requires localized information, which is why the current study employs WST (presented in the next section) to enhance the statistical analysis.

4. Method: Wavelet Scattering Transform

The WST, first proposed by Mallat (2012), is a signal processing technique that produces a concise and informative set of statistical features from complex data. It achieves this by using a series of wavelet convolutions, modulus non linearities, and averaging. This process mimics the hierarchical iterations found in convolutional neural networks (CNNs). However, unlike CNNs, the WST does not require any training phase, as it employs pre-defined convolution kernels. In addition, the use of a known transform enhances the interpretability of the results. According to Bruna and Mallat (2013, their Table 4), the use of WST for classification improves the challenge of image pattern recognition described in their article, achieving better results than methods relying on the power spectrum or convolutional neural networks.

In this section, we present a simplified version of the WST formalism, specifically in the context of the one-
dimensional case. For a more comprehensive mathematical derivation and detailed properties of the scattering
transform, we recommend referring to Mallat (2012) and Bruna and Mallat (2013).

4.1. A Sparse Representation

High-frequency samples of turbulent flows longer than 10–15 min capture a large set of structures of different
length scales and amplitudes, ranging from large eddies to the dissipative scale. The resulting signals can be seen
as a combination of regular and intermittent patterns at specific points. The WST is used to analyze these localized
features by decomposing the signal into different scales using an orthogonal wavelet basis. A wavelet is a
localized sine wave in real space and a band-pass filter in Fourier space, enabling the examination of both time
and frequency information together. Wavelets are particularly effective in analyzing signals that have varying
frequencies and transient characteristics.

Wavelets are employed to decompose a signal into distinct frequency components. This collection comprises
wavelets that share the same shape but vary in size, which is achieved by scaling a mother wavelet $\psi$. Typically,
the scaling factor is selected as $2^j$, where the scale index $j$ is an integer ranging from 0 to $J-1$, and $J$ denotes the
maximum number of scales considered. Consequently, the value of $2^j$ must not exceed the length of the input
signal.

In the one-dimensional case, the dyadic wavelet family $\psi_j(x)$ can be expressed as follows
$$ \psi_j(x) = 2^{-j/2} \psi(2^{-j}x), \qquad (1) $$

where $\psi(x)$ represents the mother wavelet. In the framework of the WST, a commonly used choice for the mother wavelet is the Morlet wavelet. The Morlet wavelet is defined as the product of a plane wave with a unit wave-number $k_0$ and a Gaussian window with a width of $\sigma$:

$$ \psi(x) = \alpha(e^{ik_0x} - \beta) e^{-x^2/2\sigma^2}, \qquad (2) $$

where $\alpha$ and $\beta$ are adjusted to ensure that the wavelet has a unit $L^2$-norm and satisfies the admissibility condition. The admissibility condition states that wavelets should be localized filters, which can be expressed as $\int \psi(x) dx = 0$ or equivalently $\hat{\psi}(0) = 0$, where $\hat{\psi}$ corresponds to the Fourier transform of $\psi$ (Ashmead, 2012).

Then $\psi_j(x)$ defines a set of orthogonal basis functions that satisfy the Littlewood-Paley condition $(\sum_j \hat{\psi}_j(k)^2 = 1)$ (Bruna et al., 2015). This condition ensures that $\psi_j(x)$ uniformly covers the Fourier space, except for the zero frequency, thereby conserving energy. The full details of the Fourier spectrum include the amplitude at zero frequency, revealing the mean value throughout the domain. The Fourier transform of $\psi_j(x)$, $\hat{\psi}_j(k)$, corresponds to a bandpass filter centered on wavenumber $k_j = 2^{-j}k_0$ and has a bandwidth proportional to $(2^j\sigma)^{-1}$. This property makes the wavelet family stable with respect to small changes of the signal, but the resolution decreases with increasing $j$. In this paper, we use $k_0 = \frac{3\pi}{4l_s}$ and $\sigma = 0.8$, as prescribed in similar cases by Andreux et al. (2020), leading to wavelets centered on scales $\lambda_j = \frac{8}{3} 2^j l_s$, with $l_s$ the sampling length ($\approx 4$ m in our case).

## 4.2. WST Formalism

The WST involves successive layers of convolution applied to an input signal $X_0(x)$ using a wavelet family $\psi_j(x)$ followed by a modulus operation. In each layer, a new set of signals is generated, characterizing the intensity of fluctuations observed in the previous layer as a function of $x$. This process can be expressed as

$$ X_n = |X_{n-1} \star \psi_j|, \qquad (3) $$

where $n$ is the layer index. Subsequently, an averaging operation is applied at each order, providing translation-invariant WST coefficients $S_n \equiv \langle X_n \rangle$, where $\langle \cdot \rangle$ represents the average operator over the $x$-axis. Since each iteration of wavelet convolution and modulus propagates the signal energy to lower frequencies, the total signal variance $\sigma^2(X_0)$ is partitioned into scattering coefficients through an infinite expansion series (Mallat, 2012),

$$ \sigma^2(X_0) = \sum_{n=0}^{\infty} \sum_{(j_1, \dots, j_n) \in \mathbb{Z}^n} S_n^2(j_1, \dots, j_n). \qquad (4) $$

Bruna and Mallat (2013) have shown that the energy carried by the scattering coefficients decreases to 0 as $n$ increases, becoming negligible for orders $n \ge 3$. Therefore, only the first three coefficients are necessary to statistically characterize the input signal. Each coefficient provides unique information about the signal properties, which will be described in detail in the following.

### 4.2.1. Zero- and First-Order

The representation of the initial three-order ($n = 0, 1, 2$) WST operations is illustrated in Figure 6. Initially, the input signal $X_0(x)$ is averaged, resulting in the first WST coefficient $S_0 \equiv \langle X_0 \rangle$, which represents the average amplitude of the signal across $x$ (Figure 6b). Next, $X_0$ is convoluted with a first wavelet family $\psi_{j_1}$ (represented in Figure 6a). Each convolution decomposes the signal at a distinct scale $\lambda_{j_1}$, with $j_1 = 0, \dots, J - 1$. Afterward, the modulus operator is applied, leading to $X_1(x, j_1)$ (Figure 6c) which represents for each selected scale their localized fluctuation amplitudes. This step increases the dimensionality of $X_0$ by introducing a dependency on both $x$ and $j_1$. The first-order coefficients are obtained by averaging $X_1$ over $x$, and read as
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Diagram illustrating the initial three orders of the Wavelet Scattering Transform (WST), constituting a multi-layer network. (a) Wavelet family labeled by the scale index *j*. (b) Input signal $X_0$ averaged to $S_0$. (c) Scale decomposition of $X_0$ after convolution with $\psi_{j_1}$, (d) Scale decomposition of the first-order subsignals $X_1$. (e)–(f) Average operator applied on both $X_1$ and $X_2$ over *x*, resulting in WST coefficients $S_1$ and $S_2$, respectively.</figcaption></figure>

$$S_1(j_1) \equiv \langle X_1 \rangle = \langle |X_0 \star \psi_{j_1}| \rangle. \qquad (5)$$

$S_1$ measures the fluctuation strength in $X_0$ at different scales $j_1$ (Figure 6e). It is similar to the power spectrum $E(k_{j_1})$, but differs in two main ways. First, the WST uses a non-linear modulus operator instead of the squared modulus, which prevents the amplification of fluctuations. This results in coefficients with low variance that are less affected by outliers compared to the power spectrum. As a result, the WST coefficients are expressed in the $L^1$ norm, which cannot be directly interpreted as energy levels. Second, the WST uses wider frequency bins to capture fluctuations, while the power spectrum uses narrower bins. These two properties ensure that the WST is robust to small perturbations (noise) and deformations. It has been shown that, assuming self-similarity, the first-order WST coefficients and the power spectrum are related as

$$S_1(k_{j_1})^2 \approx k_{j_1} E(k_{j_1}), \qquad (6)$$

(Bruna et al., 2015). In the inertial range, where the energy distribution of turbulence follows the $-5/3$ power law ($E \sim k_{j_1}^{-5/3}$), this translates to $S_1 \sim k_{j_1}^{-1/3}$. For simplicity, in the following discussion, we refer to $k_{j_n}$ ($\lambda_{j_n}$) as $k_n$ ($\lambda_n$).

### 4.2.2. Second-Order

After generating the first-order subsignals $X_1$, the same process is applied to each extracted scale $j_1$. This results in the second-order operation $X_2(x, j_1, j_2)$ (Figure 6d), which decomposes $X_1$ into scales $\lambda_2$ using a second wavelet
family $\psi_{j_2}$, with $j_2 = 0, \dots, J-1$. The average operator is applied to $X_2$, producing the second-order coefficients $S_2$ defined as

$$S_2(j_1, j_2) \equiv \langle X_2 \rangle = \langle \|X_0 \star \psi_{j_1} | \star \psi_{j_2} \| \rangle. \quad (7)$$

Then, $S_2$ measures the amplitude at which a fluctuation observed at the scale $j_1$ is modulated at the scale $j_2$ (Figure 6f). Although all combinations of the scales $j_1$ and $j_2$ are mathematically feasible, quantifying modulations smaller than the previously filtered scale $j_1$ is physically meaningless, as no new information can be obtained only on the scale $j_2$ if $j_2 < j_1$. Therefore, only the modulation of the scale $j_2 > j_1$ is relevant, which provides information on the amplitude ratio of the physical cascade between the scales. As a result, only the coefficients where $j_2 > j_1$ are retained, reducing the total number of coefficients $S_2$ to $J(J-1)/2$. Furthermore, modulations that provide amplitude ratios between scales do not give information on whether the cascade is direct or inverse. To compare scale interactions between different measurements taken at different times and under varying meteorological conditions, it is useful to consider descriptors that are independent of the intensity of fluctuations. The coefficients $S_2$ capture fluctuations in the previous order signals $X_1$ and carry a portion of their energy. To remove the dependency of $S_2$ on $S_1$, it is common to use normalized second-order scattering coefficients $\tilde{S}_2$, which are defined as

$$\tilde{S}_2(j_1, j_2) = \frac{S_2(j_1, j_2)}{S_1(j_1)}. \quad (8)$$

$\tilde{S}_2$ is a descriptor that remains unaffected by multiplication of the input signal by a constant factor, and is independent on the level of energy. This is useful for comparing modulation amplitudes across different scales $j_1$ as well as in measurements taken under varying environmental conditions. Bruna et al. (2015) have found characteristic decays in $\tilde{S}_2$ for Poisson processes, fractional Brownian motions and Lévy processes. They proved mathematically that for stationary Gaussian processes $\tilde{S}_2$ follows a power law of $2^{-j_2/2} (\approx \lambda_2^{-1/2})$. In their study, they also observed a modulation cascade within the inertial subrange for a laboratory-controlled turbulent jet flow. This modulation cascade follows a relationship of $\tilde{S}_2 \propto 2^{-j_2/5} (\approx \lambda_2^{-1/5})$. These two expected behaviors provide valuable insights into characterizing the structures found in complex signals from the MABL, where $\tilde{S}_2$ offers a set of coefficients that characterize intermittent and self-similar processes. As a result, processes that are invariant to scaling should display the same $\tilde{S}_2$ coefficients no matter of $j_1$. Within the context of the interstellar medium, the terms $\tilde{S}_2$ have never shown scale invariance, as noted by Allys et al. (2019); Delouis et al. (2022); Auclair et al. (2024). Thus, we focus our analysis on the coefficients $S_1$ and $\tilde{S}_2$, where the latter will be compared to the predicted power laws.

# 5. Results

The WST analysis is applied to all segments identified as CC (22 segments, Table A2) or RV (26 segments, Table A3). Since the length of segments is $L \approx 60$ km, we select $J = 12$ to obtain wavelets ranging from $\lambda_1 \approx 10$ m (or $\frac{8}{3}2^0 l_s$ m, with $l_s \approx 4$ m the sampling) to $\approx 44$ km (or $\frac{8}{3}2^{12} l_s \approx \frac{8}{3} \times 4096 \times 4$ m). We then analyze our signals through 12 different scales $\lambda_1$ ranging from 10 m to 21 km (rounded values of the $\lambda_1$ are given in Table A1 and used in the following, for instance $\lambda_1 = 100$ m corresponds actually to 83 m). For each segment and wind speed component, we compute the first- and second-order WST moments. We then calculate the ensemble average and standard deviation for the two subgroups, CC and RV.

## 5.1. First-Order Coefficients

The first-order WST coefficients $S_1$ for variables $u'$, $v'$, and $w'$ (Figure 7) indicate the average magnitude of velocity fluctuations at different scales $j_1$ ($\lambda_1$). We qualitatively recover the power spectra for the three wind components, as well as the Kolmogorov power law, which describes the inertial subrange where $S_1$ is proportional to $k_1^{-1/3}$. We observe that RV conditions have higher levels of fluctuations at almost all wavelengths compared to CC conditions, for both the horizontal and vertical components. Similarly to the largest energetic scales ($\lambda_\alpha$) obtained from power spectra, we have identified the maximum fluctuation wavelengths ($\Lambda_\alpha$) for each curve $S_1$.
<figure><img src="image_9.png" /><figcaption>**Figure 7.** First-order scattering coefficients for the three wind speed components (a) *u*′, (b) *v*′, and (c) *w*′, averaged for CC (blue) and RV (orange) conditions. The vertical blue and orange dashed lines indicate the maximum fluctuation wavelengths Λ<sub>α</sub>.</figcaption></figure>

(indicated by vertical dashed lines in Figure 7). For RV conditions, peaks in the horizontal and vertical velocity at scales around 2,500 and 1,500 m respectively align with the values obtained from power spectrum analysis. In contrast, CC conditions show no peak and the maximum values are reached at the upper wavelength boundary (Λ<sub>u</sub>′,Λ<sub>v</sub>′ ≥ 20 km) at the same order of magnitude as λ<sub>u</sub>′,λ<sub>v</sub>′ ≈ 10 km. For the vertical component, the correspondence is even better since Λ<sub>w</sub>′ ≈ λ<sub>w</sub>′ (≈2,500 m). After checking the good consistency between these S<sub>1</sub> coefficients and the analysis of the power spectra, the second-order coefficients are used to access scale modulations.

**5.2. Second-Order Coefficients**

Normalized second-order scattering coefficients (\tilde{S}_2) for the three velocity components measure how fluctuations at a certain scale $\lambda_1$ are modulated by larger scales $\lambda_2$ (Figure 8). For the sake of clarity, we represent $\tilde{S}_2$ only for $\lambda_1 \in [10, 100, 300, 1000]$ m, even if intermediate wavelengths are available. The parameter $\tilde{S}_2$ is given in logarithmic scale for each fluctuation size $\lambda_1$ as a function of the modulation wavelength $\lambda_2$ that ranges from 20 m to 20 km. The -1/5 and -1/2 slopes as identified by Bruna et al. (2015) correspond to the expected behaviors of a turbulent energy cascade and a Gaussian process, respectively.

Regardless of the two subgroups, $\tilde{S}_2$ does not exhibit significant variability between $u'$, $v'$ and $w'$, with coefficients nearly equal at all scale modulations. At each length scale $\lambda_1$, the highest modulation occurs when $\lambda_2 = \lambda_{j+1}$. This indicates that fluctuations of a particular scale are strongly influenced by the scale immediately above it. Furthermore, some $\tilde{S}_2$ coefficients have the same values for different length scales $\lambda_1$, for $\lambda_1 < 100$ m. These coefficients come together along the -1/5 slope. This is a sign of scale-invariant processes occurring in the inertial subrange for both CC and RV conditions. As detailed below, there are strikingly different behaviors in the two subgroups beyond a cut-off wavelength, where $\tilde{S}_2$ diverge between CC and RV.

**5.2.1. RV Conditions**

For the smallest length scale at $\lambda_1 \approx 10$ m (Figures 8a, 8e, and 8i) $\tilde{S}_2$ decrease until a cut-off length scale $\lambda_2^{c1}$ (300 m) according to the -1/5 power law, corresponding to the expected modulation cascade for turbulent flows. For $\lambda_2$ between $\lambda_2^{c1}$ and $\lambda_2^{c2} \approx 2,500$ m, the modulation amplitudes of the 10 m length scale are the same and equal to ~0.12. Above $\lambda_2^{c2}$ the modulation amplitudes follow the -1/2 slope indicating that 10 m length scales are randomly modulated by mesoscale motions. For larger $\lambda_1$ up to 1,000 m (Figures 8b–8d, 8f–8h, and 8j–8l) we observe the same behavior with a -1/5 slope until $\lambda_2^{c1}$ and a -1/2 slope above $\lambda_2^{c2}$. While $\lambda_2^{c2}$ remains equal to 2,500 m, $\lambda_2^{c1}$ gradually increases from 300 m to $\lambda_2^{c2}$. As a result, this transitional regime disappears for $\lambda_1$ larger than 100 m.
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Mean normalized 2^nd^-order scattering coefficients ($\tilde{S}_2$) for u', v', w' (top, center, bottom) for CC (blue) and RV (orange) categories, for scales $\lambda_{j1}$ of 10 m (a, e, i), 100 m (b, f, j), 300 m (c, g, k), and 1000 m (d, h, l). The -1/5 slope (solid) and -1/2 slope (dashed line) correspond to modulations obtained for turbulent jet flow in laboratory and random Gaussian process, respectively (Bruna et al., 2015). The vertical green and red lines indicate the $\lambda_2^{c1}$ and $\lambda_2^{c2}$ thresholds. Error-bars give the standard error of the mean while shaded areas represent the standard deviation in each groups. The top x-axis scale, which is common to all subplots corresponds to $\lambda_1$ whereas the bottom x-axis scale corresponds to $\lambda_2$ for each graph.</figcaption></figure>

5.2.2. CC Conditions

In the CC case and for the length scale $\lambda_1 \approx 10$ m the $\tilde{S}_2$ coefficients show the same behavior as RV for $\lambda_2 < \lambda_2^{c1}$ (Figures 8a, 8e, and 8i). There is also the transitional regime between $\lambda_2^{c1} \approx 300$ m and $\lambda_2^{c2} \approx 2500$ m. For $\lambda_2 > \lambda_2^{c2}$ the $\tilde{S}_2$ coefficients increase with $\lambda_2$ in contrast to the RV case. This indicates an increased modulation between smallest and larger scales in CC conditions: conversely to rolls, convective cells in trades can have horizontal characteristic sizes up the mesoscales (50–100 km) and a vertical extent up to 2.5–3 km, with enhanced continuity between subcloud and cloudy layers (e.g., Schulz et al., 2021; Stevens et al., 2020). This behavior is the same for every $\lambda_1$ between 10 and 100 m, with, as for the RV case, a progressive decrease of the transition zone between $\lambda_2^{c1}$ and $\lambda_2^{c2}$. On the other hand, for length scale $\lambda_1 > 300$ m the increase of $\tilde{S}_2$ for $\lambda_2 > \lambda_2^{c2}$ is replaced by a decrease. For $\lambda_1 \approx 1000$ m (Figures 8d, h, and 8i) the slope of this decrease is compatible (taking into account the uncertainties) with the -1/5 slope.
### 5.2.3. Scale Interaction in MABL

The second-order coefficients of the WST analysis highlight more pronounced differences between the two groups of atmospheric conditions than the spectral analysis (or WST first-order coefficients). The short scales (up to 300 m) are mostly modulated by the scale immediately above, and are organized following the classical turbulent cascade until $\lambda_2^{c1}$.

The shortest scales (10 m, and, to some extent, scales up to 100 m) are also strongly modulated by scales between 300 and 2500 m, both in RV and CC conditions. In CC conditions, the strongest modulation of these shortest scales is due to the mesoscales, from 2500 m to 10 km. This indicates that energy is injected from mesoscale organized structures into turbulent scales in the MABL. In RV conditions, modulation of the turbulent scales by mesoscale organization is random, since the modulation coefficients $\tilde{S}_2$ follow the $-1/2$ power law. We interpret this as a strong limitation of the influence of coherent structures larger than 2500 m on the turbulent scales, 100 m and below, in link with the organization of MABL in horizontal rolls of characteristic size around 2500 m. Interestingly, the vertical velocity is also modulated by scales up to 10 km/2500 m in CC/RV conditions, which correspond to maximum horizontal (and not vertical) length scales present in the MABL.

Intermediate length scales (300–1000 m) are also influenced by larger scales, with the same difference between the CC and RV conditions. In the CC conditions, these intermediate scales are modulated by all coherent structures of larger scales, up to the maximum scale sampled by the method, 20 km. In RV conditions, the modulation is present but weak up to the same cut-off length scale of 2,500 m. This scale of 2,500 m appears to a strong limitation of the size of organized structures able to modulate both turbulent and submesoscales in the case of the RV MABL. A strong decrease of the energy level in the MABL for scales longer than a few kilometers was already observed by Etling and Brown (1993) or Brilouet et al. (2017), but the limitation of the size of the modulating structures is a new result, to our knowledge.

### 5.3. Variation With Height

In this section, we investigate whether the results obtained from the $\tilde{S}_2$ coefficients depend on the height of the measurements within the MABL. MABL dynamics at turbulent scales can be influenced both by turbulence generated by elements at the surface (related to friction and thermal instability) and transported upwards, and by larger scale, coherent structures, which can transport down turbulence and momentum from the top of the MABL (Couvreux et al., 2007; Etling & Brown, 1993). Thus, modulation of turbulent scales by submeso- to mesoscales can strongly vary between the lower half of the MABL and the bottom of the cloud layer. We present here the results for the $w'$ component and the lowest fluctuation length scale $\lambda_1$ (≈10 m). They can be generalized to the other components. We calculated the average value of $\tilde{S}_2$ for both CC and RV conditions in the different groups defined in Section 3.1, sampling the lower half, upper half of the MABL and bottom of the cloud layer. Figure 9 illustrates the evolution of $\tilde{S}_2^w$ for small-scale fluctuations ($\lambda_1$ ≈ 10 m) within these groups.

For Group 2 ($z \in [0.5, 1[z_{SC}];$ Figures 9b, 15 segments in RV conditions, 9 in CC conditions), the results are similar to those of the general case (Section 5.2). The modulation intensity cascades following the $-1/5$ slope until $\lambda_2^{c1}$, then a plateau until $\lambda_2^{c2} \approx 2500$ m. Above $\lambda_2^{c2}$ the CC and RV segments diverge with an increase of the modulation intensity for CC and a decrease with $-1/2$ slope for RV. Group 2 is the category with most samples, which can explain the similarity between these results and those of Section 5.2.

Group 3 ($z \ge z_{SC}$; Figures 9c, 4 segments under RV conditions, 11 under CC conditions) shows roughly the same behavior with two differences. First $\lambda_2^{c2}$ is higher (reaching ≈5000 m), which indicates likely the influence of larger structures at those heights. Second, the plateau effect between $\lambda_2^{c1}$ and $\lambda_2^{c2}$ is replaced by a slow increase for RV and a higher increase for CC. Due to the small number of samples none of these trends is significant. As a result, the intensity of the interaction between the smallest length scales and mesoscales (length scales above $\lambda_2^{c2}$) is higher than for Group 2.

Finally, the modulation amplitude is strikingly different for Group 1 ($z/z_{SC} \in [0.35, 0.5]$; Figure 9a and 7 segments in RV conditions, 2 in CC conditions) with no difference between CC and RV samples. The $\tilde{S}_2$ coefficient decreases following a $-1/5$ slope until $\lambda_2^{c2}$ and following a $-1/2$ slope after $\lambda_2^{c2}$, for both CC and RV segments. Conversely to the other groups, the smallest length scale is only randomly modulated by the length scales of 2500 m and above, for both RV and CC conditions. The CC conditions represent only two samples, but
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Vertical velocity modulation coefficients ($\tilde{S}_2^{n'}$) for small scale fluctuations ($\lambda_1$ = 10 m) for CC (blue) and RV (orange) conditions through three normalized height intervals: (a) $z/z_{sc} \in [0.35, 0.5]$; (b) $z/z_{sc} \in [0.5, 1]$; (c) $z > z_{sc}$.</figcaption></figure>

they are very consistent with each other (as shown by the standard deviation for $\lambda_2 \approx 10$ km) and significantly depart from those in CC conditions at other measurement heights.

The modulations of smaller scales by longer scales strongly depend on the level of measurements. Most samples are in the upper half of the MABL and the modulations there define the average pattern of the MABL. In the lower half of the MABL, turbulent scales are not modulated by submesoscales or mesoscales whatever the conditions. In CC conditions the scale modulations above the cloud base are very close to the average pattern. In RV conditions the turbulent scales are modulated by structures up to 5,000 m in size rather than 2,500 m. This indicates some transfer of energy from structures larger than the roll vortices, present above the MABL. Our understanding is that in the RV case the very organized MABL limits the maximum size of modulating length scales to the horizontal footprint of the rolls. Close to the MABL top, larger length scales brought by detrainment from the cloud layer can modulate the turbulent scales.

To test the hypothesis of intrusion of drier air from above into the top or upper half of the MABL we look at the water vapor distribution (mixing ratio $r_v$) at different levels (Figure 10). Mean values increase and standard deviations decrease when going from the top of the MABL to its lower half, indicating less variability. Around the bottom of the cloud layer, the very strong negative skewness indicates strong entrainment/detrainment between the clouds and their drier environment. Note that the secondary peak of the distribution close to $r_v'$ = 2.5 g kg$^{-1}$ can indicate either dry intrusion or variation of the flying height. Finally, the skewness decreases in the upper half of the MABL to reach -0.39 g kg$^{-1}$ in its lower half, corresponding to dry intrusions penetrating down to half the MABL, around 300 m. Dry tongues penetrating the MABL down to variable heights have been observed from airborne measurements in various types of environment (e.g., Mahrt, 1991; Weckwerth et al., 1996). They can contribute to the moisture heterogeneity of the MABL down to the surface and drive the shallow convection with the surface buoyant fluxes (Couvreux et al., 2007; Lohou et al., 2010).

## 5.4. Effect of Anisotropy

Previous studies have emphasized the impact of coherent structures, especially RV, in organizing the turbulence in the MABL along the mean wind. Analyses of airborne turbulent measurements for instance show more energy in the largest length scales in the cross-wind than along-wind samples, because eddies are stretched along the mean wind direction (Brilouet et al., 2017; Nicholls & Readings, 1981). The WST second-order coefficients are used to check the anisotropy of the scale modulations. Segments in RV conditions are split into parallel ($\|\|$, 8 segments) and perpendicular (⊥, 18 segments) to the mean wind, Figure 11.
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Histograms of the water vapor mixing ratios $r_v$ for all samples in the lower half (red), upper half of the MABL (blue) and bottom of cloud layer (green). The first three moments are indicated.</figcaption></figure>

The modulations of small-scale fluctuations at $\lambda_1 \approx 10$ m (Figure 11a) are different for along- and cross-wind segments. Cross-wind, the modulation coefficients correspond to those of the general RV case (Section 5.2), with two cut-off length scales separating the -1/5 slope, slight increase and -1/2 slope. Along-wind, the $\tilde{S}_2^w$ coefficients decrease continuously along the -1/5 slope until $\lambda_2^{c2} \approx 10$ km, then along the -1/2 slope above $\lambda_2^{c2}$. For intermediate scale fluctuations at $\lambda_1 \approx 100$ m (Figure 11b), the modulation coefficients in both directions are closer, with cross-wind coefficients larger than along-wind ones. The cut-off length scale are also different, respectively $\lambda_2^{c2} \approx 2.5$ km and 5 km. For larger length scales ($\lambda_1 \approx 1000$ m, Figure 11c) modulation coefficients are identical, with a -1/2 slope.

Hence, the turbulent scales in the RV group are significantly more modulated cross-wind than along-wind, indicating an anisotropy of the modulation at short scales. Along-wind, samples simply follow the turbulent cascade up to $\lambda_2^{c2} \approx 10$ km and $\tilde{S}_2 = 0.074$. Cross-wind, the relative deviation is about 50% higher than along-wind for 10 m length scales, and 25% higher for 100 m length scales (Figure 11b). The cross-wind cut-off length scale $\lambda_2^{c2}$ corresponds to the width of the rolls whereas along-wind it matches the roll lengths elongated by the mean flow. The roll vortices observed during the campaign can be described as coherent 5-to-10-km elongated structures about 2.5 km apart. These results compare well with those of Brilouet et al. (2017) in other atmospheric conditions with significantly more sharpness around the peak of the spectrum in the cross-wind than along-wind cases and a shift of the spectral peak toward larger length scales in the along-wind cases. Here, cross-wind turbulence scales are influenced by coherent scales near 2,500 m. Along-wind turbulence scales are only slightly affected by larger scales with a turbulent cascade up to approximately 5,000–10,000 m (Figures 11a and 11b).
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Roll Vortices (RV) vertical velocity modulation coefficients ($\tilde{s}_2^{v'}$) for fluctuations at $\lambda_1 = 10$ m (a), 100 m (b) and 1,000 m (c). Along- (||) and cross-wind (⊥) segments are represented in blue and red lines, respectively. The cut-off length scale $\lambda_2^{c2}$ is given for each subgroup.</figcaption></figure>

## 6. Discussion and Perspectives

Using WST on airborne measurements of turbulent 3D velocities within the trade cumulus MABL provides information on the turbulence, the submeso- to mesoscale structures in the MABL and their modulations. The ATR turbulent data of EUREC⁴A are a well-suited data set for testing the capabilities of the method: it covers length scales from 4 m to more than 10 km; it includes almost 50 samples equally divided in two distinct categories of roll vortices and convective cells; and it covers several heights within the MABL and close to the bottom of the cloud layer, with cross-wind and along-wind samples in RV conditions. Thanks to previous work and well advanced exploitation of these data, plenty of information on the dynamics of the trade winds MABL is available to interpret our results. Part of information given by spectral analysis and distributions is also present in WST analysis: turbulent cascade of energy between 10 and 300 m for all components and both CC and RV conditions, peak of energy around 2,500 m (1,000 m) for horizontal (vertical) components in the RV cases, no limitation in the horizontal sizes of structures in CC cases. It also gives information about the modulations of turbulent scales (between 10 and 300 m) by submeso- to mesoscales (1,000 m and longer scales). Turbulence is always modulated by longer scales, which are limited to 2,500 m in the RV case. Interestingly, the vertical velocity turbulence is also modulated by scales up to 2,500 m, even if maximum length scales of the vertical velocities culminate at ≈1,000 m. We interpret this first as a limitation of penetration of larger scales within the subcloud layer (decoupled from the upper layer at the inversion level), second as a confirmation of transport modulation of turbulence by the roll dynamics. In CC conditions, modulating scales reach the 10-km upper limit of the sample on the vertical component also. Previously published results about the effects of entrainment/detainment at the top of the MABL and the influence of drier air masses penetrating down to the middle of the MABL are also confirmed by the WST analysis at different heights. In RV conditions, the modulation is much more marked in cross-wind than along-wind samples. Using WST on along-wind measurements give information on the size of the modulating structures, 5–10 km here. This also confirms that along-wind measurements should be used cautiously to derive general properties of the MABL in the presence of organized structures (Brilouet et al., 2017). Finally, both in CC and RV conditions, the largest modulating scales of the turbulence out of the inertial range correspond to the horizontal maximum fluctuation wavelengths $\Lambda_{u',v'}$ even for the vertical components.

The two types of well-defined environments sampled by the present data set can be easily separated using WST. The method seems promising for exploring other conditions without a priori knowledge, or at the boundary between different types of conditions. Also, investigating the impact of the flight height and orientation with
respect to mean wind would have been more accomplished with more samples better spanning the whole height of the MABL. Similarly, exploring in more detail the variations with height or the anisotropy within the CC MABL was not feasible due to the limited number of samples and the expected complexity of the interpretation in the CC conditions.

The WST method also has limitations. WST captures scale modulation but does not inform about the relative positions of scales. It cannot be used to search for patterns such as the alignment of large- and small-scale structures. This issue can be addressed by using covariance between scales (Cheng et al., 2024). In the current study, WST was applied independently to the three velocity components. Future research will use statistics to cross-reference components to understand for example, how the small scales of one variable are modulated by the large scales of another variable. Another limitation of the WST is due to the dyadic property of the wavelet used to reduce the variance of the extracted coefficients. As a drawback, the obtained frequency accuracy is poor. This could be improved by modifying the wavelet shape to enhance frequency resolution at the expense of the variance. Finally, it should be noted that while the WST identifies the modulation between scales it does not provide any information about the direction of this cascade.

Some similarities may be expected between integral length scales and WST second-order coefficients but these methods are based on different principles and give unrelated results. The integral length quantifies the autocorrelation of structures, while the second order WST evaluates the modulation of specific, smaller scales $j_1$ by larger ones $j_2$. Integral length scale can be seen as the length of phase coherence. In contrast, the second-order WST only verifies the presence of scales without needing phase coherence thanks to the modulus usage. Values of the integral length scales of vertical velocities and corresponding characteristic second-order scattering coefficients $\lambda_2^{c1}$ and $\lambda_2^{c2}$ are given in Tables A2 and A3 and show no apparent relationship, even at the order of magnitude. Within the scattering covariance framework (Cheng et al., 2024), the second-order term determines the covariance between scales, incorporating phase information into the modulation between scales.

In the present study aiming at evaluating the method and its possibilities in probing the MABL, only velocity components were used. Temperature and humidity turbulent fluctuations play a very strong role in the MABL organization and evolution and its links with the cloud layer. Applying the same methodology to these parameters is certainly an interesting perspective. Especially, it could help to decipher the respective roles of dynamics and buoyancy in organizing the MABL, and characterize the relative scale modulations using cospectra. We expect strong discrepancies between the dynamic and thermal characteristics of the MABL and between the processes controlling its organization. More specifically, humidity and temperature fields have been observed to be strongly anisotropic and governed by processes at the surface or at the top of the MABL, at the scales of organized structures and of the turbulence (Lohou et al., 2000). The WST method could provide useful information about these characteristic scales and the processes from which they stem.

## Appendix A: Appendix

See Table A1.

See Table A2.

See Table A3.
<table><caption>**Table A1**
List of $\lambda_1$ Values, in m</caption><thead><tr><th>J</th><th>λ<sub>1</sub></th><th>Min</th><th>Max</th><th>λ<sub>1</sub><sup>a</sup></th></tr></thead><tbody><tr><td>1</td><td>10.4</td><td>7.8</td><td>15.6</td><td>10</td></tr><tr><td>2</td><td>20.8</td><td>15.6</td><td>31.2</td><td></td></tr><tr><td>3</td><td>41.6</td><td>31.2</td><td>62.4</td><td></td></tr><tr><td>4</td><td>83.2</td><td>62.4</td><td>124.8</td><td>100</td></tr><tr><td>5</td><td>166.4</td><td>124.8</td><td>249.6</td><td></td></tr><tr><td>6</td><td>332.8</td><td>249.6</td><td>499.2</td><td>300</td></tr><tr><td>7</td><td>665.6</td><td>499.2</td><td>998.4</td><td></td></tr><tr><td>8</td><td>1,331.2</td><td>998.4</td><td>1,996.8</td><td>1,000</td></tr><tr><td>9</td><td>2,662.4</td><td>1,996.8</td><td>3,993.6</td><td>2,500</td></tr><tr><td>10</td><td>5,324.8</td><td>3,993.6</td><td>7,987.2</td><td>5,000</td></tr><tr><td>11</td><td>10,649.6</td><td>7,987.2</td><td>15,974.4</td><td>10,000</td></tr><tr><td>12</td><td>21,299.2</td><td>15,974.4</td><td>31,948.8</td><td></td></tr></tbody></table>

<table><caption>**Table A2**
Details of the ATR Flight Segments in CC Conditions</caption><thead><tr><th>Segment</th><th>z</th><th>z<sub>SC</sub><sup>a</sup></th><th>z<sub>INV</sub><sup>a</sup></th><th>λ<sub>w</sub><sup>b</sup></th><th>L<sub>w</sub><sup>b</sup></th><th>λ<sub>2</sub><sup>c</sup></th><th>λ<sub>2</sub><sup>c</sup></th></tr></thead><tbody><tr><td>RF09-L4B</td><td>1,100</td><td>730</td><td>2,760</td><td>1,296</td><td>444</td><td>166 ± 50</td><td>2,662 ± 799</td></tr><tr><td>RF10-R1A</td><td>776</td><td>700</td><td>2,910</td><td>3,255</td><td>609</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF10-R1B</td><td>783</td><td>700</td><td>2,910</td><td>1,654</td><td>515</td><td>333 ± 100</td><td>2,662 ± 799</td></tr><tr><td>RF10-R2A</td><td>781</td><td>700</td><td>2,910</td><td>1,989</td><td>357</td><td>333 ± 100</td><td>2,662 ± 799</td></tr><tr><td>RF10-R2B</td><td>780</td><td>700</td><td>2,910</td><td>1,213</td><td>582</td><td>166 ± 50</td><td>1,331 ± 399</td></tr><tr><td>RF10-R3A</td><td>778</td><td>700</td><td>2,910</td><td>2,072</td><td>1109</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF10-R3B</td><td>778</td><td>700</td><td>2,910</td><td>1,480</td><td>229</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF10-L1B</td><td>608</td><td>700</td><td>2,910</td><td>1,188</td><td>578</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF10-L2A</td><td>307</td><td>700</td><td>2,910</td><td>776</td><td>134</td><td>83 ± 25</td><td>2,662 ± 799</td></tr><tr><td>RF13-L2A</td><td>316</td><td>790</td><td>2,850</td><td>435</td><td>86</td><td>1,331 ± 399</td><td>10,650 ± 3,195</td></tr><tr><td>RF13-L3A</td><td>627</td><td>790</td><td>2,850</td><td>978</td><td>237</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R1A</td><td>995</td><td>870</td><td>2,810</td><td>1,471</td><td>263</td><td>333 ± 100</td><td>2,662 ± 799</td></tr><tr><td>RF14-R1B</td><td>915</td><td>870</td><td>2,810</td><td>1,364</td><td>623</td><td>83 ± 25</td><td>666 ± 200</td></tr><tr><td>RF14-R1D</td><td>917</td><td>870</td><td>2,810</td><td>3,358</td><td>780</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R2A</td><td>913</td><td>870</td><td>2,810</td><td>3,153</td><td>341</td><td>83 ± 25</td><td>1,331 ± 399</td></tr><tr><td>RF14-R2B</td><td>791</td><td>870</td><td>2,810</td><td>4,473</td><td>392</td><td>83 ± 25</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R2C</td><td>796</td><td>870</td><td>2,810</td><td>2,548</td><td>411</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF14-R2D</td><td>796</td><td>870</td><td>2,810</td><td>1,205</td><td>267</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R3A</td><td>791</td><td>870</td><td>2,810</td><td>1,382</td><td>242</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R3B</td><td>792</td><td>870</td><td>2,810</td><td>1,627</td><td>246</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R3C</td><td>795</td><td>870</td><td>2,810</td><td>2,156</td><td>332</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-R3D</td><td>793</td><td>870</td><td>2,810</td><td>1,347</td><td>371</td><td>333 ± 100</td><td>10,650 ± 3,195</td></tr></tbody></table>

Note. All values are in m. <sup>a</sup>The $z_{SC}$ and $z_{INV}$ values (Brilouet et al., 2021) are given for each flight (not each segment) and are the same for CC and RV segments within a same flight. <sup>b</sup>The $\lambda_w'$ and $L_w'$ values correspond to $\lambda_\alpha$ and $L_\alpha$ with $\alpha = w'$ (vertical velocity). <sup>c</sup>The standard deviation on the length scales is computed according to the discrete, dyadic scale.

<footnote><sup>a</sup>$\lambda_1^*$ corresponds to the equivalent (approximate) length scale used in the text.</footnote>
<table><caption>**Table A3**
Details of the ATR Flight Segments in RV Conditions</caption><thead><tr><th>Segment</th><th>z</th><th>z<sub>SC</sub></th><th>z<sub>INV</sub></th><th>λ<sub>w</sub>'</th><th>L<sub>w</sub>'</th><th>λ<sup>c1</sup><sub>2</sub></th><th>λ<sup>c2</sup><sub>2</sub></th></tr></thead><tbody><tr><td>RF14-L1A</td><td>659</td><td>870</td><td>2810</td><td>972</td><td>284</td><td>5,325 ± 1,597</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-L1B</td><td>659</td><td>870</td><td>2810</td><td>1027</td><td>176</td><td>1,331 ± 399</td><td>2,662 ± 799</td></tr><tr><td>RF14-L2A</td><td>323</td><td>870</td><td>2810</td><td>710</td><td>109</td><td>5,325 ± 1,597</td><td>5,325 ± 1,597</td></tr><tr><td>RF14-L2B</td><td>324</td><td>870</td><td>2810</td><td>1672</td><td>298</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF15-R3A</td><td>825</td><td>770</td><td>2770</td><td>788</td><td>147</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF15-R3B</td><td>826</td><td>770</td><td>2770</td><td>710</td><td>196</td><td>666 ± 200</td><td>10,650 ± 3,195</td></tr><tr><td>RF15-R3C</td><td>825</td><td>770</td><td>2770</td><td>1231</td><td>393</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF15-R3D</td><td>825</td><td>770</td><td>2770</td><td>775</td><td>140</td><td>333 ± 100</td><td>5,325 ± 1,597</td></tr><tr><td>RF15-L1A</td><td>620</td><td>770</td><td>2770</td><td>1376</td><td>276</td><td>2,662 ± 799</td><td>10,650 ± 3,195</td></tr><tr><td>RF15-L1B</td><td>620</td><td>770</td><td>2770</td><td>782</td><td>159</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF15-L2A</td><td>302</td><td>770</td><td>2770</td><td>686</td><td>118</td><td>2,662 ± 799</td><td>2,662 ± 799</td></tr><tr><td>RF16-R2C</td><td>791</td><td>820</td><td>3050</td><td>1882</td><td>413</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF16-R3A</td><td>793</td><td>820</td><td>3050</td><td>1078</td><td>240</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF16-R3B</td><td>792</td><td>820</td><td>3050</td><td>966</td><td>193</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF16-R3C</td><td>788</td><td>820</td><td>3050</td><td>727</td><td>129</td><td>666 ± 200</td><td>10,650 ± 3,195</td></tr><tr><td>RF16-R3D</td><td>786</td><td>820</td><td>3050</td><td>1078</td><td>182</td><td>333 ± 100</td><td>2,662 ± 799</td></tr><tr><td>RF16-R4C</td><td>782</td><td>820</td><td>3050</td><td>2544</td><td>345</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF16-R4D</td><td>780</td><td>820</td><td>3050</td><td>1356</td><td>250</td><td>666 ± 200</td><td>5,325 ± 1,597</td></tr><tr><td>RF18-L1A</td><td>552</td><td>760</td><td>2860</td><td>1059</td><td>631</td><td>2,662 ± 799</td><td>10,650 ± 3,195</td></tr><tr><td>RF18-L1B</td><td>549</td><td>760</td><td>2860</td><td>782</td><td>137</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF18-L2A</td><td>277</td><td>760</td><td>2860</td><td>632</td><td>110</td><td>333 ± 100</td><td>1,331 ± 399</td></tr><tr><td>RF18-L2B</td><td>279</td><td>760</td><td>2860</td><td>1405</td><td>493</td><td>10,650 ± 3,195</td><td>10,650 ± 3,195</td></tr><tr><td>RF19-L1A</td><td>600</td><td>740</td><td>1960</td><td>1256</td><td>484</td><td>1,331 ± 399</td><td>5,325 ± 1,597</td></tr><tr><td>RF19-L1B</td><td>598</td><td>740</td><td>1960</td><td>884</td><td>157</td><td>666 ± 200</td><td>2,662 ± 799</td></tr><tr><td>RF19-L2A</td><td>303</td><td>740</td><td>1960</td><td>535</td><td>117</td><td>666 ± 200</td><td>1,331 ± 399</td></tr><tr><td>RF19-L2B</td><td>306</td><td>740</td><td>1960</td><td>1705</td><td>361</td><td>1,331 ± 399</td><td>2,662 ± 799</td></tr></tbody></table>

Note. All values are in m.

## Data Availability Statement

The turbulence data used for the calculation of WST coefficients in this study are available in the AERIS data platform via https://doi.org/10.25326/128 (Brilouet et al., 2021). The GOES-16 ABI data are publicly available online at https://doi.org/10.7289/V5BV7DSR (GOES-R Calibration Working Group and GOES-R Series Program, 2017). The satellite images in Figure 2 are retrieved from the imagery of the Earth Observing System Data and Information System (EOSDIS) Worldview application (https://wvs.earthdata.nasa.gov) and from the NASA ATOMIC-EUREC4A GOES-16 ABI imagery (https://satcorps.larc.nasa.gov/cg-ibin/site/showdoc?doc-id=4&cmd=fiel-dexperimen-thomepage&exp=ATOMC-EUREC4A). WST coefficients produced in our analysis are stored in a data set in netCDF format. This data set as well as codes to generate it are openly available in a Zenodo repository at https://zenodo.org/doi/10.5281/zenodo.11479386 (Gauvrit, 2024). Figures were made with Proplot version 0.9.7 (Davis, 2021), available under MIT license on GitHub https://github.com/proplot-dev/proplot?tab=readme-ov-file.
Acknowledgments

The authors thank Pierre-Etienne Brilouet for the discussions on ATR data and his invaluable explanations regarding turbulence data processing. Erwan Allys, Pablo Richard, and Constant Auclair are acknowledged for their essential help in understanding and adapting the WST for atmospheric and oceanographic sciences. The authors acknowledge the comments and suggestions of two reviewers that helped to improve this paper. This research has been funded by the CNRS for the AstrOcean project under the 80Prime initiative. We acknowledge the use of imagery from the Worldview Snapshots Application (https://vws.earthdata.nasa.gov) part of the Earth Observing System Data and Information System (EOSDIS).

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