Verification of Near Surface Wind Patterns in Germany using Clear Air Radar Echoes

The verification of high-resolution meteorological models requires highly resolved validation data and appropriate tools of analysis. While much progress has been made in the case of precipitation, wind fields have received less attention, largely due to a lack of spatial measurements. Clear-sky radar echoes could be an unexpected part of the solution by affording us an indirect look at horizontal wind patterns: Regions of horizontal convergence attract non-meteorological scatterers such as insects; their concentration visualizes the structure of the convergence field. Using a two-dimensional wavelet transform, this 5 study demonstrates how divergences and reflectivities can be quantitatively compared in terms of their spatial scale, (horizontal) anisotropy and direction. A long-term validation of the highly resolved regional reanalysis COSMO-REA2 against the German radar composite RADOLAN shows surprisingly close agreement. Despite theoretically predicted problems with simulations in or near the ‘grey-zone’ of turbulence, COSMO-REA2 is shown to produce a realistic diurnal cycle of the spatial scales larger than 8km. In agreement with the literature, the orientation of the patterns in both data-sets closely follows the mean 10 wind direction. Conversely, an analysis of the horizontal anisotropy reveals that the model has an unrealistic tendency towards highly linear, roll like patterns early in the day.

of well defined objects is not guaranteed? Skinner et al. (2016), who are interested in tornado forming mesocyclones, chose to focus on the rotational component of the wind field by verifying only the horizontal vorticity. Model and observations are subjected to several spatial filters and then thresholded at manually selected values before the object based MODE technique (Davis et al., 2009) and the image-morphing DAS of Keil and Craig (2009) are applied. Their approach is justified because well-defined objects, i.e., tornadic supercells, clearly exist in the specific case study under consideration. Bousquet et al. 45 (2008) find a similar answer to the vector-problem by verifying horizontal divergences against the corresponding values from the French Multi-Doppler network. Besides point-wise measures, these authors apply a simple scale-separation approach based on a Haar wavelet decomposition of the wind fields. Other recent attempts at spatial wind verification include Zschenderlein et al. (2019) who apply the object-based SAL technique (Wernli et al., 2008) to tresholded predictions of gusts (i.e. absolute wind speed), and Skok and Hladnik (2018) who sort wind vectors into classes based on their speed and direction and use the 50 popular fractions skill score (Roberts and Lean, 2008, FSS) to find the scales on which the predicted classes agree with the observations.
In this study, we take a similar route as Skinner et al. (2016) but instead of the rotational component we focus on the horizontal divergence of the near surface wind field. Under the right environmental conditions, the spatial pattern of this divergence field can be observed in widely available radar reflectivity data: On warm, rain-free days, convergent boundary 55 layer circulations attract swarms of insects which are drawn in and actively attempt to resist the vertical motion of updraughts (Wilson et al., 1994). The resulting increased concentration of biological scatterers within the radar beam reflects the pattern of convergence and divergence. Numerous studies including Weckwerth et al. (1997Weckwerth et al. ( , 1999; Thurston et al. (2016) ;Banghoff et al. (2020) have used this kind of data to study the dominant patterns of boundary layer organization. Atkinson and Wu Zhang those patterns. Numerous studies have used radar data to observe these phenomena (see references in Banghoff et al. (2020)); Banghoff et al. (2020) also present a first long-time climatology using ten years of reflectivities and Doppler velocities from a single radar station in Oklahoma. They manually classified radar images from over 1000 days into cells, rolls and unorganized patterns, reporting organized features on 92 % of summer days without rain. Santellanes et al. (2021) exploited this data-set to study the environmental conditions that favor the different modes of organization. 65 In the present investigation, we aim to study a similarly large data-base of reflectivities from the German RADOLAN-RX composite and compare it to divergence structures from the regional reanalysis COSMO-REA2 (Wahl et al., 2017), covering the time-span from 2007 to 2013. We limit our analysis to small environments around each radar station and consider both the entire COSMO-REA2 time-series (for an overall model climatology) and the subset where clear air radar echoes are available (for verification).

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For a fair, quantitative validation of the model, the spatial patterns must be analyzed objectively. Here, we rely on the wavelet-based SAD verification methodology of Buschow and Friederichs (2021) which applies a series of directed filters to objectively determine the dominant spatial Scale, Anisotropy and Direction in an image. A closely related approach was used to define a wavelet-based index of convective organization in radar and satellite images by Brune et al. (2021).
To what extent a model at O(1km) horizontal resolution can be expected to realistically represent boundary layer circulations 75 in the so-called 'Grey-Zone' regime (Wyngaard, 2004) between parametrized and resolved turbulence is a difficult question which poses further theoretical challenges to the verification process. Section 2 therefore briefly summarizes some of the relevant theoretical and experimental results from the literature. Data and methodology are described in sections 3 and 4.
Section 5 presents the results of our analysis, including the model-based climatology of divergence structures and its validation against RADOLAN. Some discussion of our findings is given in section 6, section 7 examines what conclusions can be drawn 80 and identifies avenues for future research.
2 Theory and modelling of mesoscale shallow convection Zhou et al. (2014) have demonstrated how occurrence and basic properties of shallow convective circulation in the atmospheric boundary layer can be understood in analogy to Rayleigh Bénard thermal instability. In the classic framework, the circulation regime of a fluid between two heated plates is determined by the Rayleigh number where d is the distance between the plates, β = dT /dz is the temperature gradient, and the coefficients g, α, k, ν denote gravitational acceleration, thermal expansion coefficient, thermal conductivity and kinematic viscosity, respectively. Eddies of wavelength λ start to grow when Ra exceeds a critical value Ra c (λ). The qualitative sketch in figure 1 shows that this marginal stability curve has a global minimum near λ = 2d. For Ra<Ra c (2d), the flow is laminar and heat is exchanged via conduction.  (2014) argue that an analogous stability curve applies to the atmospheric boundary layer. In this case, Ra is replaced by a turbulent Rayleigh number of similar form as Eq. 1 wherein the depth d is replaced by the boundary layer height H. On a sunny day, the earth's surface is heated and the vertical temperature gradient, as well as the height of the boundary layer increase.
The theory predicts that, once a critical Ra is crossed, the initial wavelength of the circulation should be near λ = 2H ≈ 3 km; 95 both smaller and larger eddies begin to develop later.
The simulation of this process is challenging because a model with grid-spacing δ can never resolve eddies with λ < 2δ.
In large eddy simulations with δ << 2H, convection will correctly be initiated at the natural critical Ra c with a wavelength of ∼ 2H. Current NWP models, on the other hand, have δ 2H. In this case, the first eddies to form as Ra increases have λ ≈ δ and initiate at a grid-spacing dependent value Ra c (δ). For global or regional models with δ 10 km, the critical value 100 is so large that such circulations will never form under realistic conditions. Modern mesoscale models, however, operate at δ = O(1km) and Ra c (δ) becomes attainable. The result is a potentially unrealistic model circulation, the scale and initiation time of which depends on δ. This is one example of the so-called Terra Incognita or Grey-Zone of turbulence (Wyngaard, 2004;Honnert et al., 2020), where the highest energy vortices are too large to be adequately represented by the boundary layer parametrization but too small to be explicitly resolved by the dynamical core of the model. Ching et al. (2014) observed this 105 phenomenon in nested WRF simulations, Poll et al. (2017) detected it in TerrSysMP, the atmospheric component of which is COMSO. Using LES runs of the same models as a reference, both of these studies found that simulations with grid spacing on the kilometer scale initiate turbulence too late and too energetically. In the present study, we will investigate how frequently such small-scaled circulations occur in the climatology of COSMO-REA2 and how they compare to radar observations.

COSMO-REA2
For a systematic investigation of low-level divergence structures, we ideally need a long, homogeneous time series of high resolution model data. The regional reanalysis COSMO-REA2 is uniquely suited for our need as it provides seven years (2007)(2008)(2009)(2010)(2011)(2012)(2013) of hourly output from the mesoscale model COSMO (Baldauf et al., 2011) at a horizontal resolution of 0.018 • or roughly 2 km. The model was run with 50 vertical levels over a domain covering Germany and the neighbouring countries. For a full 115 description of the used physics parametrizations, we refer to Wahl et al. (2017) and references therein. For our purposes, it is important to note that boundary layer fluxes are handled by a level-2.5 TKE-closure, shallow convection is parametrized via a modified Tiedtke mass-flux scheme while deep moist convection is left to the dynamic core. The data assimilation uses a continuous nudging scheme to relax the prognostic temperature, wind speed, pressure and relative humidity towards observations from stations, radiosondes, aircraft, ships and buoys. In addition, rain rates from radar observations are assimilated 120 via latent heat nudging (Stephan et al., 2008, LHN). Thus, on clear air days, the only source of mesoscale information (LHN) is inactive, meaning that while data assimilation can help create realistic environmental conditions, the fine-scale structure of the fields is a product of the dynamics and physics of the model. Horizontal divergences were calculated from the hourly 10 m wind vector fields as a simple finite difference approximation.

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RADOLAN (Radar online adjustment, 'RADar OnLine ANeichung') RX is the operational radar reflectivity composite of the 16 C-band radars operated by the German weather service. The output has a spatio-temporal resolution of 1 km×1 km×5 min and covers Germany and parts of its neighbours. The underlying radar scans are performed at an orography following elevation angle (∼ 1 • ) with an azimuthal resolution of 1 • and a range resolution of 250 m. Due to the beam geometry, the true native resolution of the reflectivity composite, as well as the height for which it is representative, depends heavily on the distance to 130 the radar station. Pejcic et al. (2020) show that the beams reach typical boundary layer heights of 1 − 1.5 km at about 100 km from the radar location. Therefore, relevant clear-air echoes caused by insects that cannot survive at low temperatures are expected to be found only in the immediate vicinity of the radars.
To get an idea of the type of data we rely on for our model validation, it is instructive to consider an example case. Figure 2   and Wu Zhang, 1996). A visual comparison of the reflectivities around, for example, the Berlin radar with the simulated divergences and the clouds in that region leads us to hypothesize that the boundary layer processes in COSMO-REA2 are not entirely unrealistic.

Data availability
As mentioned above, clear-air echoes typically only occur in a small environment around each radar. We therefore limit our 150 study to circular regions with 64 km radius, centred at the 16 radar station which were active throughout the COSMO-REA2 period. While simulated divergences are readily available at every such grid point for each hour between 2007 and 2013, the availability of clear-sky echoes depends on many factors including local topography, technical details of the radars, radar processing at DWD and the life-cycle of the biological scatterers. We consider an individual radar image incomplete if less than 50 % of pixels within our 64 km radius around the radar are above −10 dBZ (visual analysis of many example images 155 has shown that no significant signals exist between roughly −10 dBZ and the smallest stored value of −32.5 dBZ). From the remaining data, we must filter out rainy episodes, defined here somewhat arbitrarily as cases where at least 100 pixels exceed +10 dBZ. We will refer to all remaining images as complete. Table 1 shows that such complete clear air echoes are overall rare (well below 5 % of all hourly images) and their frequency varies considerably between radars. For this study, we neglect the four radar stations with the fewest data, thereby removing two urban (Essen, Frankfurt) and two coastal locations (Emden, Rostock). The twelve remaining radars give us roughly 20 thousand individual hourly images for comparison with COSMO-REA2. When studying the diurnal cycles below, we will furthermore include radar data at the full 5 min resolution which gives us over 200 thousand images.
In table 2, we see that the vast majority of clear sky echoes occurs during summer, particularly June and July, with considerable variability between the years. The preference for the warm season is expected since both insect activity and boundary layer height are increased by higher temperatures. Consequently, the daytime frequency of available data follows a diurnal cycle as well ( figure 3). In addition, there is a large second population of night time cases. The sudden increase in clear air echoes at dusk, as well as their absence in winter, hints at migrating swarms of insects as a likely explanation (Drake and Reynolds, 2012). We exclude these data because (1) the weaker nighttime convergences are less likely to influence the pattern of the insect cloud and (2) migrating swarms tend to inhabit thin layers near an atmospheric inversion which only partly intersect the 170 radar beam (cf. p.237 f. in Drake and Reynolds (2012)).

Wavelet analysis
The idea of this study is to compare the correlation structures of the radar reflectivities and divergence fields, summarized in terms of scale, anisotropy and direction. To extract these properties from divergence and reflectivity images, we use the 175 SAD methodology of Buschow and Friederichs (2021): The image to be analyzed is convolved with a series of localized 2D wave-forms with varying scale and orientation. The analyzing filters are the so-called daughter wavelets which are generated by shifting, scaling and rotating a single, carefully designed wave function, the mother wavelet. The square of one wavelet coefficient, i.e., the result convolving the image with one of the daughters, represents the amount of variance present at a particular location for a particular combination of spatial scale and orientation. The dual-tree complex wavelet transform 180 (Selesnick et al., 2005) used in this study provides daughter wavelets with six orientations and up to J scales for an image of size 2 J × 2 J . Following Buschow and Friederichs (2021), the largest three scales are removed because their support is larger than the image, rendering their interpretation ambiguous. After spatial averaging, a radar image with 128 × 128 pixels is thus summarized by 4 × 6 values, the so-called wavelet spectrum. To extract the scale, anisotropy and direction from this spectrum, we treat the J × 6 values as point-masses arranged in a 3D space such that the six directions for one scale are at the vertices of -From the angular coordinate, we can determine the dominant orientation angle ϕ ∈ [0 • , 180 • ]. Note that ϕ is only meaningful if the anisotropy ρ is non-zero.
For a detailed description of the calculation of these properties, as well as the details of the wavelet transform itself, we refer to Buschow and Friederichs (2021) and references therein. The software for this analysis is freely available in the open source 195 dualtrees R-package (Buschow et al., 2020).
The central scale z is a dimensionless quantity which cannot be analytically transformed into an equivalent Fourier wavelength. Since the actual physical size of the patterns is of some interest in the present study, we derive an empirical relationship based on test images with fixed wavelength in appendix A. We find that, in the range of 1.5 < z < 2.5, the relationship is well described by a linear fit with It is important to note that this relationship is only approximately valid for the specific wavelets, scales and wave-like test images used in the present study. This equivalent wavelength is furthermore not identical to the spacing between wave-crests used as the measure of horizontal scale by Banghoff et al. (2020) because our λ includes also the scale perpendicular to the orientation of the features.

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To make the distribution of angles ϕ interpretable, we compute the angles of intersection between ϕ and the model wind direction (averaged over the regions around each radar). A relative angle ∆ϕ = 0 • thus means that the patterns align with the wind direction whereas ∆ϕ = 90 • indicates an orthogonal orientation.

Boundary conditions and pre-processing
The wavelet analysis described above requires data on a regular grid, ideally of size 2 n × 2 n to ensure fast computation times, 210 discontinuities at the boundaries must be avoided. This is only a minor factor for intermittent fields like rain but very important for data with non-zero values along each border. To achieve periodic boundaries, we cut out a 128 km × 128 km region (128 and 64 pixels for RADOLAN and COSMO-REA2, respectively) around each radar location and apply a circular Tukey window to smoothly reduce the field to zero (for divergences) or −10 dBz (for reflectivities) towards each side. A rectangular boundary would introduce spurious horizontal and vertical directions to the wavelet spectra.

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For the reflectivity data, further pre-processing steps are required. Firstly, some radar images contain erroneous isolated pixels with unusual intensities which would artificially reduce the analyzed spatial scales. Following Lagrange et al. (2018), we therefore compare each pixel to the average over its eight nearest neighbours. If the difference is greater than 10 dBZ, the pixel value is replaced by the neighbourhood average. Secondly, the reflectivities around each radar often contain gaps of very small reflectivities (< −10 dBZ), caused for example by buildings, mountains or water bodies without scattering insects. These 220 arbitrarily shaped holes introduce an artificial pattern which is unrelated to the wind field and needs to be removed. Here, we fill in the gaps with a simple algorithm which iteratively replaces values below -10 dBZ with an average over the neighbouring non-missing pixels. The details of the gap-filling algorithm, as well as a demonstration of its effectiveness are given in appendix B.
Lastly, a comparison between the wavelet spectra of two images would normally require that both data sets be given on the 225 same grid. In our case, we can avoid re-gridding either field since the spatial resolutions differ by a factor of two. The second scale in RADOLAN thus corresponds to the first scale of COSMO-REA2. We can therefore simply remove the smallest scale from the radar image to make the spectra comparable. We have checked that the results are virtually identical when the radar images are bilinearly re-mapped instead. The largest daughter wavelet that fits into our domain is j = 4 for RADOLAN and j = 3 for COSMO-REA2, giving us three comparable scales with six directions each.

Climatology of divergence structures in COSMO-REA2
Based on section 2, we can expect that small-scaled, cellular circulations will form on warm sunny days, favored by high pressure and low wind speeds. Following the diurnal cycle of the boundary layer depth, these circulations start out small and become larger over the course of the day. According to Poll et al. (2017), Banghoff et al. (2020) and references therein, we 235 furthermore expect to see a more linear mode of organization on windier days. The orientation of these roll-like structures will generally follow the mean wind direction (Weckwerth et al., 1997). Both cells and rolls should leave an imprint on the scale and anisotropy and direction of the horizontal divergence fields. We therefore cut out square regions of 64 × 64 pixels around the twelve selected radar stations (table 1)  For the spatial scales in panel (a), we find that the prominent minimum around noon is indeed a common occurrence in all seasons except winter, indicated by bi-modal distributions between 9 and 15 UTC. During summer in particular, the smallerscaled mode, centred near z ≈ 1.75 or λ ≈ 10 km, is more likely than z > 2. Two modes can be seen with similar clarity in and 13 UTC, averaged around the selected radar locations. "small and round" cases have z < 1.86, ρ < 0.12, "small and linear" is z < 1.86, ρ > 0.32. The boxplot labeled "rx" contains all instances where at least one clear air radar echo is available.
anisotropy ( figure 5 b), on the other hand, is far weaker: A clearly increased likelihood for anisotropic features is only evident in summer between 9 and 12 UTC and the change in the distribution is far less pronounced than for z. While the formation of exceptionally small structures, oriented along the mean wind, is thus a common occurrence, the increased linearity around 260 noon seen in figure 4 b can only occasionally be observed.
Next, we are interested in the typical weather situation associated with the occurrence of these small and / or linear patterns.
To this end, we focus on the three hours around noon during the summer season and search for cases where both ρ and z are in the bottom 5 % of their climatological distribution ("small and round" mode). For the "small and linear" mode, we select those cases where z is in the bottom 5 % whereas ρ is in the top 5 % of its distribution. At these time-steps, as well as the 265 remaining "reference" cases, we compute spatial averages around the selected radar stations for several relevant variables from COSMO-REA2. Figure 6 shows that boundary layer height, 2 m temperature and surface pressure see a moderate increase during timesteps with small and linear patterns and a stronger increase if the pattern is small and round. In the latter cases, the median temperature is close to 25 • C and the boundary layer rarely falls below 2 km. Simultaneously, the average 10 m wind speed 270 is strongly reduced. Conversely the small and linear mode is associated with a significantly increased wind speed. Hence the boundary layer circulation in COSMO-REA2 qualitatively resembles Rayleigh Bénard convection.
In preparation for the quantitative comparison with radar data, figure 6 also includes the environmental conditions for days where at least one clear-sky RADOLAN image is available. We find that the radar echoes occur mostly on very warm days with moderately increased boundary layer depth and decreased wind speeds. This is consistent with the assumption of insects

Verification against radar reflectivities
In this section, we attempt to assess the realism of our model-based climatology using the clear-sky radar reflectivity data from RADOLAN. Besides cases with too many missing or rainy pixels, we also exclude all nighttime images. The remaining data 280 is subjected to the wavelet analysis as described in section 4.2.
Before analysing the statistics of the entire dataset, we briefly consider a few individual examples. Figure 7 shows five randomly selected cases from the Flechtdorf radar station.  with both data-sets agreeing that 2009-07-29, i.e., our example from figure 2, has the smallest and most anisotropic structure.
Overall, the decent visual similarity between COSMO-REA2 and RADOLAN is reflected in small to moderate differences in  Good agreement between model and observations can be seen in the distribution of the angle ϕ as well. In figure 9, we have pooled all radars together and consider only full hours where the model wind direction is known. Cases with small observed 305 anisotropy (ρ ≤ 0.1), i.e., ambiguous orientation, were removed as well. We find that, between 10 and 17 UTC, both sets of images are usually oriented within ±15 • of the mean model wind direction; the distributions of RADOLAN and COSMO-REA2 match almost perfectly. Before and after this interval, which coincides with the small-scale phase of the diurnal cycle, a wider variety of orientations is possible.
While the scale and orientation are thus in reasonably good agreement, the same can not be said for the anisotropy. Figure   310 10 shows that the observations are almost universally more isotropic than the model fields. The pattern of increasing linearity towards a maximum before noon seen in figure 4 b is clearly present in this sample of the model data. The observations, on the other hand, hardly contain this pattern at all with only a very weak maximum at 11 UTC and nearly constant values during the afternoon.
Aside from the climatological distribution and diurnal cycle, we are interested in the model's ability to represent the day-315 to-day variability of the spatial divergence patterns. For z and ρ, we can eliminate the overall bias and diurnal behaviour by subtracting the long-time mean for every daytime hour from the respective time series. To avoid residual effects of the annual cycle, we limit this analysis to the summer season. Timing errors within each day are furthermore removed by taking the daily minimum of z and maximum of ρ. Figure 11 a reveals that the remaining scale anomalies in COSMO-REA2 and RADOLAN are slightly correlated with many remaining errors below 0.1 and almost all below 0.2 (outer lines). As expected, the correlation 320 is even lower for ρ (figure 11 b) and the typical errors are relatively large even after the bias has been removed.

Discussion
The wavelet-based analysis for the task at hand? Concerning the trustworthiness of COSMO-REA2, it must be remembered that the 325 local divergence patterns are primarily the product of the model dynamics and parametrized turbulence, not the data assimilation. The environmental conditions which drive the formation of a particular mode of small-scale organization, however, can be expected to have good accuracy due to the continuous input of wind speed, humidity and pressure from weather stations. It is therefore not surprising that the model can accurately represent diurnal and annual cycles and differentiate between days with organized and unorganized situations. Consequently, the model climatology as described in section 5.1 qualitatively agrees 330 with our expectations from the literature. Whether or not the simulated small-scale structure can itself be trusted is questionable in light of the theory discussed in section 2. Our comparison with RADOLAN clear-air data suggests that, despite the proximity to the Grey-Zone, the modelled structures are not overall unrealistic. In interpreting this result, we must recall that the difference in native resolution between RADOLAN and COSMO-REA2 was handled by deleting the smallest scale from RADOLAN. We have thereby filtered out any variability below the model's effective resolution. Figure  indicate that the mesoscale model successfully simulates the spatial scales present in the real atmosphere. We can merely see that the remaining variability (upwards of λ ≈ 8 km), which both data-sets can represent, matches the observed diurnal cycle decently, especially at the rural stations.
As predicted by Zhou et al. (2014), the wavelengths of the simulated eddies are near the smallest scale resolved by the model. We note, however, that the underlying resolution of RADOLAN is 1 • in azimuth-and 250 m in range-direction. Inside 340 our 64 km radius, and particularly close to the radar, the internal resolution of the measurements is considerably finer than the used 1 km × 1 km grid. There is thus no obvious technical reason why, after filtering, RADOLAN should have to exhibit increased variability on the same scale as the model. We have experimentally re-calculated the central scales of the radar images including the previously removed smallest scales and found a slight shift in the cycle towards earlier hours. Conversely, if we remove the second smallest scale as well, a shift in the opposite direction emerges. This supports our interpretation that the 345 model simulates the patterns seen in the observations with an approximately correct diurnal cycle, on the scales we included; smaller-scaled variability, which would initiate earlier in the day, is resolved by neither COSMO nor RADOLAN. It should furthermore be noted that we make no direct statements about the intensity (variance) of the circulations. Such information cannot easily be inferred because the absolute radar reflectivities depend on the technical details of the radar, applied preprocessing and the unknown overall concentration of biological scatterers.

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The greater disagreement at the urban radar locations has two main explanations. On the one hand, it is likely that buildings and unrelated radio signals introduce excessive noise into the images, overshadowing the natural signal. This explanation is supported by the lack of complete images at the Essen and Frankfurt stations, both of which are located in highly urbanized regions (Frankfurt is the city with the most skyscrapers in Germany). On the other hand, the urban landscape itself can influence Aside from spatial scales, the anisotropy of the divergence pattern, i.e., the difference between linear and cellular organization, is of interest. Here, the model's tendency towards more linear patterns earlier in the day could not be confirmed observationally. On the one hand, it is plausible that the lack of finer-scale variability leads to the simulation of unnaturally regular stripes. On the other hand, gaps and noise have a larger impact on the anisotropy than the scale (cf. appendix B), making 360 these results somewhat less robust.
Lastly, it should again be emphasized that our clear air data-set provides no information on nighttime and winter and is biased towards cases with high temperatures where small-scale circulations are likely to occur. Our validation is therefore mostly conditional on the occurrence of these phenomena; whether or not the model correctly differentiates between days with and without organized shallow convection could only partly be judged (cf. figure 11).

7 Conclusions and outlook
The main goal of this study was to explore the use of clear-sky radar data for the evaluation of simulated low-level divergence structures. A wavelet-based verification methodology, developed and extensively tested for precipitation data, was used to summarize the spatial patterns in terms of scale, anisotropy and direction. We have demonstrated that model-based divergences and radar reflectivities are comparable at this level of abstraction. Our investigation of the German radar network has shown 370 that usable clear sky echoes are rare overall and almost non-existent in winter. This supports the assumption that such daytime echoes are caused by small insects, the life cycle and habitats of which may also explain the substantial differences between radars as well as strong year to year variations. The relatively long time-span from 2007 to 2013 nonetheless resulted in a robust data set of over 20.000 individual images, mostly during summer, where the modelled patterns could be verified against spatial observations. At most radar locations, both data sets show a very similar diurnal cycle in the spatial scales and orientations 375 with a strong preference for small-scaled (λ ≈ 10 km) features around noon. The orientation during the small-scaled phase of the cycle is almost always within 15 • of the mean wind. The fact that this observation holds for both data sets also implicitly confirms that the model adequately represents the mean wind direction. COSMO-REA2 furthermore simulated a trend towards increasingly linear features at the start of the small-scaled phase which could not be found in the observations. As discussed above, a more complete set of observations might be able to clarify whether this indicates deficiencies of the model or the 380 observations or (likely) both.
Based on the overall decent agreement with the radar observations, we may put some trust in the model's behaviour at the unobserved parts of the time series as well. If COSMO-REA2 is thus to be believed, mesoscale shallow convection, favored by high pressure (clear skies) and temperatures, as well as weak winds, is a common occurrence in Germany in all seasons except winter; during JJA, the small-scale mode is more likely than the larger-scaled configuration. Its onset a few hours after 385 sunrise is characterized by a transition phase with larger scaled, isotropic divergence patterns, the orientation of which switches from ∼ 70 • to ∼ 0 • with respect to the mean wind direction. While most patterns are isotropic, i.e., cellular in nature, there is also a weaker signal of linear organization. This more roll-like mode is most often simulated during JJA between 9 and 12 UTC and preferably occurs when winds are unusually strong and the boundary layer is shallower than in the cellular cases.
These simulated features are qualitatively consistent with the theory, as well as previous observations of mesoscale shallow 390 convection.
Concerning future prospects, it must be emphasized that we have relied on only the most widely available kind of radar observations. Modern dual-polarization Doppler radars produce a wealth of further information, which would for example allow us to confidently separate insect-related echoes from unhelpful noise and clear up the nature of the night-time echoes (Zrnic and Ryzhkov, 1998;Melnikov et al., 2015). Additionally, parameters like mean wind speed and direction, and even 395 the boundary layer height (Banghoff et al., 01 Aug. 2018) could be inferred directly from the radar instead of relying on the model (Banghoff et al., 2020). Lastly, we re-iterate that small scales below ∼ 8 km were filtered out in this study in order to fairly evaluate the mesoscale model. Depending on their frequency, weather radars can observe much finer details of the turbulent boundary layer. A similar strategy to ours could therefore also provide useful information for the objective validation Code and data availability. Software for the dual-tree wavelet transformation is available in the dualtrees R-package (Buschow et al., 2020). In addition, the specific version (0.1.4) used for this manuscript has been permanently archived at https://doi.org/10.5281/zenodo. 5027277 (Buschow, 2021a). COSMO-REA2 is currently available from the website of the Hans Ertel centre (reanalysis.meteo.uni-bonn.de).
RADOLAN is available via the DWD OpenData portal (opendata.dwd.de). The cropped reflectivity and divergence fields around the used 405 radar station have been archived at https://doi.org/10.5281/zenodo.5036447, together with all auxiliary data and software needed to fully reproduce the figures in this manuscript from scratch (Buschow, 2021b).

Appendix A: Empirical relationship between scale and wavelength
To approximately translate the central scale into an equivalent Fourier wavelength λ, we apply the exact method described in section 4.2 to synthetic test images of pure sine-waves, given by 410 f (x, y) = sin (2π(k x x + k y y)) + where is a Gaussian white noise term with zero mean and variance 0.04. Figure A1 shows that the relationship between z and λ is nearly linear for this idealized signal. For z < 1.5 and z > 2.5, the curve becomes non-linear because most variance is outside the range of scales covered by our wavelet transform. The linear fit yields λ/∆ x = 4.464 · z − 2.765. Since we are merely interested in a rough approximation with round numbers, we simplify the result for ∆ x ≈ 2 km to obtain equation 2. For this study, we are not interested in the radar reflectivities themselves, or even their full spatial correlation function, but only the estimates structural characteristics ρ, ϕ, z. To mitigate the effects of holes, i.e., regions with Z ≤ −10 dBz, in the radar images, we implement a simple iterative algorithm to smoothly fill in the gaps: (1) Find missing points with at least one non-missing neighbour, (2) replace values of those points with an average over the up to eight adjacent non-missing 420 values and (3) repeat from (1) until all gaps are filled. The result is similar to inverse distance interpolation but (at least in our implementations of the two algorithms) considerably faster. To test the success of our approach, we select 300 nearly complete (less than 3 % missing data) clear-air radar echoes from our data-set and artificially add the gaps form 300 other randomly selected incomplete images. In figure B1, we compare ρ, ϕ, z, estimated with and without the gap-filling algorithm.
As expected, the impact of the gaps is massive but our algorithm mostly mitigates the effects. We have repeated the experiment 425 with inverse distance interpolation (not shown) and found no substantial improvement over the iterative procedure.   Figure B1. Anisotropy ρ (a), angle ϕ (b) and scale z (c) estimated from nearly complete images (x-axis) and images with added holes (y-axis). Black dots show the results of the iterative gap-filling algorithms; values obtained without gap filling are shown in grey.
Competing interests. The authors declare that they have no conflict of interest.