Simulated and observed horizontal inhomogeneities of optical thickness of Arctic stratus

2.1 Vertical Distribution of Ice in Arctic clouds (VERDI) campaign

Cloud remote sensing data and atmospheric profile measurements by dropsondes from the airborne VERDI campaign (Bierwirth et al., 2013; Schäfer et al., 2015, 2017a) are exploited in this study. VERDI was based in Inuvik, Canada, and was conducted in April/May 2012. The data were collected aboard the Polar 5 research aircraft of the Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research (AWI). The measurement flights were carried out in the region over the Beaufort Sea, which was mostly covered by sea ice, but also included sea-ice-free areas (polynyas). Mostly stratiform, low-level liquid and mixed-phase clouds within a temperature range of −19 to 0 ^∘C were investigated (Costa et al., 2017). Here, the analysis is focused on a persistent cloud layer probed on 4 consecutive days from 14 to 17 May 2012. The applied measurements were performed in close vicinity (less than 50 km) over constant surface conditions (open ocean; polynyas). The persistent cloud cover in the respective area decreased continuously from day to day with cloud top altitude decreasing from about 880 m on 14 May to around 200 m on 17 May (Klingebiel et al., 2015; Schäfer et al., 2015, 2017a).

The Polar 5 research aircraft was equipped with a set of cloud and aerosol in situ and remote sensing instruments (Bierwirth et al., 2013; Schäfer et al., 2015; Klingebiel et al., 2015). Atmospheric profiles of temperature, humidity, wind speed, and direction were derived from dropsonde measurements, which were regularly released during all flights.

2.2 Horizontal fields of cloud optical thickness

The qualitative and quantitative description of the cloud inhomogeneities is performed using 2-D fields of cloud optical thickness τ. Marshak et al. (1995), Oreopoulos et al. (2000), and Schröder (2004) proposed to study horizontal cloud inhomogeneities using cloud top reflectances. However, Schäfer et al. (2017a) pointed out that radiance measurements include the information of the scattering phase function (e.g., forward/backward scattering peak, halo features). To avoid artifacts in the inhomogeneity analysis from such features, parameters that are independent of the directional scattering of the cloud particles have to be analyzed. Therefore, to characterize the observed and simulated cloud fields regarding their horizontal cloud inhomogeneities, the cloud optical thickness is applied, which does not include the fingerprint of the scattering phase function.

Figure 2(a, e) Potential temperature, (b, f) relative humidity, (c, g) wind speed, and (d, h) wind direction for the four investigated cases. The dropsonde data are shown in the first row (a–d) and the 2 h domain-averaged profiles after spin-up time of the simulations are shown in the second row (e–h). Dropsondes were released closest to the imaging spectrometer measurements.

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The 2-D fields of τ are retrieved from 2-D fields of reflected solar spectral radiance, which were collected with the imaging spectrometer AisaEAGLE (Schäfer et al., 2013, 2015). Using those data, Schäfer et al. (2017a) retrieved 10 cases of 2-D fields of cloud optical thickness τ (Schäfer et al., 2017b). From those 10 cases, four are selected for the comparison with the LES results obtained from COSMO. Figure 1 illustrates selected sections (1.2 km by 3.0 km) of four cases. The full widths and lengths of the available 2-D fields of τ range from 1.7 to 26.8 km. Their spatial resolution is between 2.6 and 3.6 m (depending on the vertical distance between aircraft and cloud).

During the time period from 14 to 17 May 2012, the areal average of τ of the observed clouds decreased from 8.1 to 4.3 (compare Table 2,  Schäfer et al., 2017a). The selected sections in Fig. 1 illustrate the temporal evolution of τ. In particular, from 15 to 17 May 2012, a reduction of the horizontal cloud inhomogeneity is visible, which is confirmed by Schäfer et al. (2017a), who also found a continuous reduction of cloud inhomogeneity during those 4 consecutive days. Furthermore, directional features, which are prominent on 14 May, seem to be reduced, which is also confirmed by Schäfer et al. (2017a).

2.3 Vertical profiles of atmospheric parameters

During each measurement flight, Vaisala dropsondes (type RD94) were used together with the Vaisala AVAPS (Airborne Vertical Atmosphere Profiling System) dropsonde receiving system (Hock and Franklin, 1999; Coleman, 2003). The dropsondes were released to sample profiles of meteorological parameters – air pressure (p), air temperature (T), relative humidity (RH), wind speed (v), and wind direction (WD) – below the aircraft, which typically operated at about 3 km altitude to sample the entire cloud and ABL structure. The accuracy of the dropsonde measurements is specified by the manufacturer as ±0.4 hPa for the air pressure, ±0.2 ^∘C for the air temperature, ±2 % for the relative humidity, and ±0.5 m s⁻¹ for the wind speed. The potential temperature (Θ), RH, v, and WD profiles for the four investigated cases are displayed in Fig. 2.

From 14 to 15 May, the cloud top inversion increased from 810 to 880 m, while for the subsequent 2 days, the inversion layer decreased to 440 m on 16 May and to 200 m on 17 May 2012 (Fig. 2a). In conjunction with the decrease of the cloud top altitude, the cloud base decreased as well until it almost reached the surface on 17 May. The relative humidity (Fig. 2b) confirms the initial increase and consecutive decrease of the cloud top and base altitudes. The inversion strength decreased over the time period from about 5 to 1 K mainly because the temperature of the surface layer continuously decreased; the ABL became more stable.

Figure 2c illustrates that the near-surface wind increased during the 4 days from about 1 to 10 m s⁻¹. Except for the case on 14 May, the daily increase of the near-surface wind speed is observed as well in higher altitudes to up to 1 km. Following Jacobson et al. (2013), this is related to low-level jets (LLJs) for the days from 15 to 17 May.

3.1 COSMO: general setup

COSMO is a non-hydrostatic, limited-area atmospheric forecast model (Schättler et al., 2015). Here, it is used in a semi-idealized LES setup, which follows the description by Loewe et al. (2017), based on Ovchinnikov et al. (2014), and Paukert and Hoose (2014). The two-moment cloud microphysics scheme by Seifert and Beheng (2006) predicts the number densities and masses of six hydrometeor types. The different ice-phase hydrometeor growth processes are parameterized. The radiative transfer is described by a two-stream radiation scheme (Ritter and Geleyn, 1992). It is calculated every 2 s and has a direct cloud radiative feedback. A 3-D prognostic turbulence scheme describes the turbulent fluxes of heat, momentum, and mass by a first-order closure after Smagorinsky and Lilly (Herzog et al., 2002; Langhans et al., 2012). The horizontal size of the model domain used by Loewe et al. (2017) was 6.4×6.4 km with a spatial resolution of 100 m. Here, this setup is applied as well. However, analyzing cloud inhomogeneities requires a fine horizontal spatial resolution of the model simulations. Therefore, for the comparison with the imaging spectrometer measurements analyzed here, the spatial resolution is also increased to 50 m for addition model runs. In those cases, the domain size is reduced to 32 by 32 grid points (1.6 km × 1.6 km) for computational constraints. A further reduction of the spatial resolution was not possible due to numerical instabilities, probably caused by the propagation and growth of perturbations stemming from imbalanced initial and/or boundary conditions (Duran, 2010). The vertical height range of 22 km is divided into 166 levels, which are more dense for the ABL with a typical vertical resolution of around 15 m up to the inversion height of the different days of investigation. The initialization profiles of temperature, humidity, wind speed, and wind direction are based on the dropsonde data, which are partly affected by horizontal variability, when slowly passing the cloud and drifting horizontally. Therefore, parts of the original profiles (Fig. 2) are smoothed vertically for initialization of the model. The surface of the model is open ocean and the surface fluxes depend on the surface temperature, which is 273.5 K for the seawater surface. Moreover, ERA-Interim reanalysis data from the European Centre for Medium-Range Weather Forecasts (Dee et al., 2011) have been used to complete the profiles above the altitude where the dropsondes were released. Other model parameters include the description of the large-scale subsidence, which is adjusted to the temperature inversion height, the relaxation to fixed cloud droplet number concentration (CDNC), and ice crystal number concentration (ICNC). The spin-up time was set to 2 h following Ovchinnikov et al. (2014). The CDNCs are based on measurements of the Small Ice Detector (SID3;  Vochezer et al., 2016). During the 4 investigated days, CDNCs of 90 to 100 cm⁻³ were observed as summarized in Table 1. The concentration of ice crystals was below or at the detection limit of the SID3. Therefore, the ICNC were assumed to be one particle per liter according to observations of mixed-phase Arctic stratus during the Indirect and Semi-Direct Aerosol Campaign (McFarquhar et al., 2011; Ovchinnikov et al., 2014). The inversion height of the temperature, z(T_in), is necessary for the description of the large-scale subsidence in the model and is represented by the inversion height of the dropsonde profiles, which are used for initialization of the simulations (Table 1).

Table 1Model setup specifications of the different mixed-phase cloud simulations of 4 VERDI campaign days.

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3.2 Domain-averaged cloud properties and temporal evolution

Time series of simulated liquid water content (LWC) and ice water content (IWC) for the four selected cases are shown in Fig. 3. During the four considered flights, only few ice crystals were observed. The simulated clouds consisted mostly of liquid water droplets except for 15 May, in which more IWC was built at about 4 h after model initialization (Fig. 3b). Furthermore, the cloud top is around 1000 m for 14 and 15 May (Fig. 3a, b). However, the cloud top height increases over time in all four simulations because of entrainment of air through the top of the ABL. This is evident in the temporal evolution of LWC, which has a maximum between 0.25 and 0.35 g kg⁻¹ near the cloud top. The Arctic clouds on 16 and 17 May are the lowest simulated clouds with a cloud top initially around 450 and 350 m, respectively (Fig. 3c, d).

Figure 3Domain averages of LWC (blue color scale) and IWC (red–yellow color scale) of the four simulations during the VERDI campaign. Please note the different color scale for the IWC in panel (d).

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The four simulations show differences in the temperature, relative humidity, and wind speed profiles (Fig. 2e–g), which in general still agree with the initial dropsonde profiles after the spin-up time (Fig. 2a–c). The height of the ABLs and the strength of the inversions are lower in the simulations of 16 and 17 May. Furthermore, for the simulation on 17 May, a second inversion develops in the ABL near the surface around 60 to 150 m. The ABL structure is well mixed in the simulation of 16 May although no second temperature inversion is built near the surface. The simulation of 16 May shows a wind shear from around 150 to around 100^∘ (Fig. 2g) and a decrease of v with height above the cloud top height, which is also seen in the dropsonde profiles (Fig. 2c). The other simulations do not show a turning of the wind directly above the inversion height.

The simulated mixed-phase clouds of the four VERDI flights show a liquid water path (LWP) around 35 to 50 g m⁻². The highest LWP is seen in the simulation of 14 May, which increases towards 50 g m⁻² at the end of the simulation. The simulation of 15 May has the lowest LWP values. Furthermore, the LWP remains very stable until the end of the simulation. The ice water path (IWP) and the snow water path (SWP) of all four simulations are small, especially for the simulated clouds on 14, 16, and 17 May, which fits well with observations.

For the comparison of the simulated and observed horizontal cloud structures (cloud inhomogeneities), fields of simulated cloud optical thickness (τ_sim) are compared to retrieved fields of cloud optical thickness from the measurements (τ_meas). The τ_sim is calculated within the COSMO model considering the amount of liquid water and the solar spectrum. However, it cannot be expected that COSMO is capable of reproducing the detailed spatial and temporal cloud evolutions, which are captured by the observed fields of τ, accurately (inhomogeneity features and directional structures). Therefore, besides the comparison of observed and simulated clouds with regard to macrophysical cloud features (cloud vertical extent, cloud optical thickness) of the individual cases, instead of point-by-point comparisons of cloud parameters, statistical bulk parameters describing the horizontal cloud inhomogeneities, their directional structures, and the temporal evolution of both will be compared.

4.1 One-dimensional statistical bulk parameters

For the quantitative description of the cloud inhomogeneities from the simulated fields of cloud optical thickness (τ_sim) obtained from COSMO and measurement-based retrieved fields of cloud optical thickness (τ_meas) collected during the VERDI campaign, statistical techniques are applied. Following Schäfer et al. (2017a), different statistical quantitative measures of the cloud inhomogeneities are derived using the mean and standard deviation of the particular τ field and three 1-D inhomogeneity parameters: ρ_τ (Davis et al., 1999; Szczap et al., 2000), S_τ (Davis et al., 1999; Szczap et al., 2000), and χ_τ (Cahalan, 1994; Oreopoulos and Cahalan, 2005). They are given by

A homogeneous cloud is characterized by ρ_τ=0 and S_τ=0. Higher values of ρ_τ and S_τ indicate more pronounced cloud inhomogeneity. However, both of them have no predefined upper limit. Therefore, ρ_τ and S_τ only sustain a quantitatively significance, when their values for different cases are compared to each other. The 1-D inhomogeneity parameter χ_τ ranges between 0 and 1, with values close to unity indicating horizontal homogeneity and values approaching zero characterizing high horizontal inhomogeneity. Due to the limited range between 0 and 1, χ_τ is not only a qualitative but also quantitative measure.

4.2 Two-dimensional autocorrelation analysis

Two-dimensional autocorrelation analysis is applied to quantify the typical scales of cloud inhomogeneities and to identify directional patterns of the cloud structures (Schäfer et al., 2017a). To derive the autocorrelation functions, each field of τ is correlated with itself, while it is shifted pixel by pixel (observations) or grid point by grid point (simulations) against itself. The values of the resulting correlation coefficients after each shift are in the range between −1 (perfect negative correlation) and 1 (perfect positive correlation). Correlation coefficients with values of 0 identify no correlation. Here, only the degree of correlation matters and not if it has a positive or negative sign. Similar to Schäfer et al. (2017a), squared autocorrelation functions $P_{\mathit{\tau }}^{\mathrm{2}}$ are used to avoid ambiguous interpretations. The $P_{\mathit{\tau }}^{\mathrm{2}}$ reach values between 0 (no correlation) and 1 (perfect correlation).

The particular correlation coefficients at the derived distances identify the similarity of the horizontal cloud structures. If the cloud is horizontally homogeneous, the correlation coefficients stay constant over large distances. If the cloud is rather inhomogeneous, the correlation coefficients already drop at closer distances. Therefore, $P_{\mathit{\tau }}^{\mathrm{2}}$ as a function of distances is a measure of the size of the dominant cloud structures.

A quantitative value for the distance, at which cloud structures are different from each other (namely decorrelated), is the decorrelation length ξ_τ (Schäfer et al., 2017a). It is the distance, at which $P_{\mathit{\tau }}^{\mathrm{2}}$ drops to

In a 2-D autocorrelation function, ξ_τ can differ depending on the orientation, if the cloud structures have a predominant orientation. To quantify this directionality, ξ_τ is calculated along (${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$) and across (${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$) the predominant direction. The larger the differences between ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$, the more cloud structures are orientated.

Figure 4(a–b) Horizontal fields of normalized τ_meas (VERDI) and τ_sim (COSMO) for the case on 15 May 2012. (c–d) Two-dimensional autocorrelation coefficients $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ and $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$, calculated for fields of τ displayed in panels (a) and (b). (e–f) One-dimensional autocorrelation coefficients along (straight white line marked in panels c and d) and across (dashed white line marked in panels c and d) predominant directional structure. The grey dotted line illustrates the threshold for the estimation of ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$.

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Figure 4a shows a section of an observed field of τ_meas, retrieved from the measurements on 15 May. The selected section has a swath of 1.3 km (oriented in y direction) and a length of 6 km (oriented in x direction). Figure 4b shows the corresponding field of τ_sim (6 km × 6 km, adapted to the selected length of the measurement case), which is simulated with COSMO 2 h after the spin-up time for the case of 15 May. For comparability reasons, both fields of τ are normalized by their maximum.

Although the swath (y direction) of the field of τ_meas is smaller by a factor of almost 5 compared to the field of τ_sim, larger cloud structures of similar size and shape are obvious in both fields of τ_meas and τ_sim. However, with 488 spatial pixels along the swath (spatial double binning was applied during measurements) and a field of view of 37^∘, AisaEAGLE's spatial resolution is approximately 1.3 m for a target in a distance of 1 km. Thus, the spatial resolution of AisaEAGLE is relatively high, compared to the spatial resolution of 100 m from COSMO. Thereby, the exact pixel size of AisaEAGLE depends on the distance between aircraft and cloud, which leads to pixel sizes between 2.6 and 3.6 m for the four investigated cases. Due to the 30 to 40 times higher spatial resolution of AisaEAGLE, compared to COSMO, the measurements show cloud features, which cannot be resolved by COSMO. Those features on a spatial scale below 100 m may have an effect on the statistical (1-D inhomogeneity parameters) and spatial comparison (autocorrelation analysis) of the particular fields of τ.

To quantify the size and orientation of the represented cloud structures in the observations and simulations, Fig. 4c and d show the calculated squared 2-D autocorrelation coefficients $P_{\mathit{\tau }}^{\mathrm{2}}$. To calculate them, different numbers of legs (shifts) have to be applied for $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ and $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$. The applied field of τ_meas consists of 2700 by 450 spatial pixels. Therefore, restricted by the shorter side, 225 by 225 (half of swath pixel number, calculated into x and y direction) legs are chosen for the calculation of the 2-D $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$. The COSMO results consist of 64 by 64 grid points. This allows 32 by 32 legs for the calculation of $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$.

Figure 5Illustrated are sections of one and the same field of τ_meas from 14 May 2012 with a spatial resolutions of (a) ≈3 m (original resolution), (b) 50 m (COSMO resolution), (c) 100 m (COSMO resolution), (d) 150 m, and (e) 300 m. (f–j) Squared 2-D autocorrelation coefficients $P_{\mathit{\tau }}^{\mathrm{2}}$ calculated for the fields of τ_meas displayed in panels (a) to (e). (k–o) Squared 1-D autocorrelation coefficients $P_{\mathit{\tau }}^{\mathrm{2}}$ calculated along straight red line in panels (f) to (j). Estimated decorrelation length ξ_τ is marked by horizontal and vertical black lines and labeled by its value. Red dot marks ξ_τ as derived from the case with the original spatial resolution of 3 m.

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The resolved domains and spatial resolutions, which are displayed in Fig. 4c and d, show significant differences, which reveals that a direct comparison is difficult. Applying the 2-D autocorrelation analysis to the observations allows to resolve small-scale cloud structures with high spatial resolution (approximately 2.7 m) but only within a narrow spatial range below 1 km. Contrarily, the same analysis for COSMO delivers $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ with lower spatial resolution of 100 m but over a larger spatial range (about 3.2 km, in Fig. 4d only displayed until 2 km). Thus, also large-scale cloud structures are covered by COSMO (purple stripes in Fig. 4d) but not in the observations. Therefore, the large-scale structures cannot be compared between observations and simulations. With respect to a comparison of the small-scale structures, the spatial sizes (spatial resolution, domain size) of both data sets need to be conformed to make a direct comparison possible.

Furthermore, both Fig. 4c and d show predominant directional features of the cloud structures. Their lengths and widths are derived from 1-D autocorrelation functions along (straight white line in Fig. 4c and d) and across (dashed white line in Fig. 4c and d) those predominant directional structures and a subsequent estimation of ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$. The derived ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$ show an overall agreement but still differ from each other. For the observations, ${\mathit{\xi }}_{\mathit{\tau },\text{meas}}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau },\text{meas}}^{\leftrightarrow }$ reach distances of approximately 500 and 250 m, respectively. Contrarily, for the simulations, ${\mathit{\xi }}_{\mathit{\tau },\text{sim}}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau },\text{sim}}^{\leftrightarrow }$ reach distances of approximately 800 and 400 m, respectively. This is a further indication that it is necessary to make the fields of τ_meas and τ_sim conform with respect to their spatial resolution and domain. In the following, this is done by (i) averaging the observed fields of τ_meas to the spatial resolution of the simulated fields of τ_sim and (ii) improving the spatial resolution of the simulations themselves.

Figure 4e and f further illustrate that it is not possible to compare the large-scale structures between observations and simulations. The large-scale structures, which are covered by the COSMO simulations, are identified by a second increase of the $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ at distances (at approximately 1 km in Fig. 4f) larger than ξ_τ. The width of the measured fields is too narrow to cover such a second increase in the $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ (compare Fig. 4e). Therefore, the further comparison of the cloud structures, which are identified in the observations and simulations, is restricted to the small-scale cloud structures with sizes below 1 km only.

4.3 Final data preparation – adjustment of spatial resolution and domain

To compare both data sets, the fields of τ_meas, which are retrieved from the imaging spectrometer measurements, are averaged to the spatial resolution of the COSMO τ_sim fields. The investigations of the single cases during VERDI are performed for spatial resolutions of 50 m (32 by 32 grid points) and 100 m (64 by 64 grid points). All other model parameters are kept constant with respect to the analysis performed by Loewe et al. (2017).

In order to average the observed fields of τ_meas to the spatial resolution of 50 and 100 m, the τ_meas values of distinct numbers of neighboring pixels are averaged. The number depends on the single pixel size of the particular cases, which is a function of the vertical distance between aircraft and cloud. For the four investigated cases, this number varies between 13 (26) and 18 (36) pixels, which are needed to generate pixel sizes of τ_meas comparable to the 50 m (100 m) spatial resolution of COSMO.

Furthermore, for the simulations with 100 m spatial resolution, the domain sizes of the measurements and simulations need to be adapted. The applied COSMO domain size of 6.4 km by 6.4 km is about 3 to 4 times larger than the domain size of the measurements. Therefore, to compare both data sets, the COSMO domain size is also reduced to the width and length of the corresponding τ_meas field from the measurements. Therefore, for the comparison, only a squared domain in the center of COSMO's τ_sim field is used, whose size corresponds to the size of the particular field from the measurement. For the four investigated cases, this results in COSMO domains composed out of 12 by 12 to 16 by 16 grid points (1.2 km by 1.2 km to 1.6 km by 1.6 km). Longer stripes of τ_meas fields and stripes according to their lengths across the COSMO domain are not used, because the investigations are focused on small-scale cloud inhomogeneities, which are already covered by the smaller squared domain size given by the swath of the τ_meas fields.

For the COSMO simulations, which use 50 m spatial resolution, the domain size is reduced to 32 by 32 grid points, resulting in a total domain of 1.6 km by 1.6 km, which is comparable to the observations. Therefore, the domain of those simulations is not adapted for the comparisons.

However, to increase the statistics, which might be otherwise too small because of the finally applied small domain but large pixel sizes, for COSMO, averages of the resulting $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ over all output time steps after spin-up are used. For the measured fields, whose lengths are much longer than their widths, squared domains (size determined by swath of τ_meas) are cut along the measured stripe and the resulting $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ values are averaged accordingly. Increasing the number of available $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$, which can be used for averaging, is a further reason to use squared domains instead of stripes.

To test possible effects arising from the change of spatial resolution and to check if the relevant scales of cloud inhomogeneity are lost, when reducing the resolution of the measurements, Fig. 5a–e show sections of one and the same field of τ_meas from the case of 14 May but displayed with a different spatial resolution of 3 m (original resolution), 50 m (COSMO fine resolution), 100 m (COSMO original resolution), 150 m, and 300 m resolution. Figure 5f–j show the corresponding squared 2-D autocorrelation coefficients. The red line illustrates the direction which is used to calculate the squared 1-D autocorrelation functions and decorrelation lengths ξ_τ, displayed in Fig. 5k–o. The fields from the 2-D autocorrelation analysis show that, except for the spatial resolution of 300 m, the directional structure of the cloud inhomogeneities is still captured, when the spatial resolution is reduced. However, the decorrelation lengths, derived from the 1-D autocorrelation analysis, increase with decreasing spatial resolution from ξ_τ=327 m at 3 m spatial resolution to ξ_τ=600 m at 300 m spatial resolution. Therefore, decreasing spatial resolution leads to larger ξ_τ, which indicates larger cloud structures. This means that reduced spatial resolution will generate fields of τ with larger spatial scales.

Figure 6Comparison of (a) mean and standard deviation and (b) inhomogeneity parameters (ρ and S) as a function of spatial resolution for the fields of τ_meas illustrated in Fig. 5a–e.

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To test the influence of the spatial resolution on the overall inhomogeneity, Fig. 6a shows the results for the mean and standard deviation of the fields of τ, illustrated in Fig. 5. Figure 6b shows the corresponding 1-D inhomogeneity parameters (ρ_τ and S_τ). While the mean value of τ stays constant for all spatial resolutions, its standard deviation decreases with increasing pixel size. This indicates that the fields of τ become more homogeneous the larger the pixel size is. Similarly, the values of both 1-D inhomogeneity parameters (ρ_τ and S_τ) decrease with increasing pixel size.

Therefore, in the following analysis, comparing the simulated against observed fields of τ, the simulations with the finer spatial resolution of 50 m are used. The simulations with 100 m spatial resolution are used to discuss the model sensitivity with respect to the spatial resolution.

5.1 Magnitude of inhomogeneity

The fields of τ obtained from the spectral imaging remote sensing (τ_meas) are compared to the fields of τ derived from the COSMO simulations (τ_sim). To validate the cloud inhomogeneity in the simulated fields, the statistical techniques from Sect. 4.1, including the averaging of the measured fields to 50 and 100 m pixel size, are applied. Table 2 lists the mean values of τ, standard deviation σ_τ, and the three 1-D inhomogeneity parameters (ρ_τ, S_τ, and χ_τ) for the observations and the simulations with the two different spatial resolutions of 50 and 100 m.

Table 2Mean value of τ, standard deviation σ_τ, and the three 1-D inhomogeneity parameters (ρ_τ, S_τ, and χ_τ) calculated for all four cases from the observations and the simulations with the two different spatial resolutions of 50 and 100 m.

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Both measurements and simulations show the highest areal averaged cloud optical thickness on 14 May with ${\stackrel{\mathrm{‾}}{\mathit{\tau }}}_{\text{meas}}=\mathrm{7.8}\pm \mathrm{1.5}$ and ${\stackrel{\mathrm{‾}}{\mathit{\tau }}}_{\text{sim}}=\mathrm{7.9}\pm \mathrm{0.6}$ at 50 m spatial resolution and ${\stackrel{\mathrm{‾}}{\mathit{\tau }}}_{\text{meas}}=\mathrm{8.1}\pm \mathrm{1.2}$ and ${\stackrel{\mathrm{‾}}{\mathit{\tau }}}_{\text{sim}}=\mathrm{6.9}\pm \mathrm{0.5}$ at 100 m spatial resolution, which indicate an overall agreement. During the course of the following days, the large-scale subsidence led to a decrease of the cloud top altitude and cloud geometrical thickness and corresponding lower values of τ and σ_τ. For these days, the model and observations are still in agreement. However, compared to the spatial resolution of 100 m, it is obvious that the finer resolved simulations lead to better agreement between measurements and simulations.

Regarding the cloud inhomogeneity, the absolute values of the 1-D inhomogeneity parameters (ρ_τ, S_τ, and χ_τ) do not compare well for the simulations with 100 m spatial resolution. The results for the COSMO simulations show lower 1-D inhomogeneity parameters (more homogeneous) by a factor of 2 and higher, compared to the results from the measurements. The agreement between the observations and simulations increase with the finer spatial resolution of 50 m but still does not match perfectly. The reason might be that the comparably lower inhomogeneity derived from COSMO for both spatial resolutions is caused by its effective spatial resolution, which is approximately three times 50 m or accordingly three times 100 m (Skamarock, 2004). Although the pixel size of AisaEAGLE is adapted to the COSMO spatial resolution by averaging over neighboring pixels, COSMO's effective spatial resolution is larger, which might lead to larger homogeneity of the simulations compared to the observations. Furthermore, COSMO simulates the cloud at the same location where it is initialized. Contrarily, the AisaEAGLE measurements took place along a stripe of several kilometers. The simulated clouds may not change in between the time steps as much as the measurements of the clouds along the measurement stripe do. Therefore, averaging over COSMO's time steps might further produce more homogeneous results than averaging over AisaEAGLE's squared domains along the flight track.

However, the observations show that the cloud field became more homogeneous from 14 to 15 May as indicated by lower values of ρ_τ, which reduce from 0.209 to 0.115. From 15 to 16 May, ρ_τ increases to 0.145, which indicates a cloud field with slightly higher inhomogeneity. Then, on 17 May, ρ_τ reduced to 0.132, showing that the cloud field became more homogeneous again. These different cases with high and low ρ_τ are reproduced by COSMO independent of the chosen spatial resolution. Larger discrepancy between modeled and observed inhomogeneity parameters only occurred on 14 May, when the observations were influenced by large-scale cloud structures.

Nevertheless, the lower/higher inhomogeneity is also imprinted in the inhomogeneity parameters (S_τ and χ_τ), which are smaller/larger in both measurements and simulations, indicating that COSMO performs well with regard to the 1-D inhomogeneity parameters.

Figure 7(a–d) Exemplary selected sections of fields of τ_meas observed during VERDI from 14 to 17 May 2012. (e–h) Mean 2-D autocorrelation coefficients $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ derived for fields of τ_meas from VERDI. (i–l) Exemplary selected fields of τ_sim simulated with COSMO (50 m spatial resolution) for the VERDI cases from 14 to 17 May 2012. (m–p) Mean 2-D autocorrelation coefficients $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ derived for fields of τ_sim. (q–t) Decorrelation length ξ_τ along strongest (straight blue and red lines) and weakest (dashed blue and red lines) extents of 2-D autocorrelation coefficients derived from $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ in panels (e)–(h) and $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ in panels (m)–(p), respectively.

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5.2 Spatial inhomogeneity scale

The 2-D autocorrelation functions are calculated to compare the typical spatial scales and the directional character of the small-scale cloud inhomogeneities (no large-scale inhomogeneities like roll convection) of observations and simulations. The 2-D autocorrelation coefficients ($P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$; $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$) for each case are shown in Fig. 7e–h for the measurements and in Fig. 7m–p for the simulations (50 m spatial resolution). Additionally, representative fields of normalized τ_meas (Fig. 7a–d) and τ_sim (Fig. 7i–l) are added. The 2-D autocorrelation analysis was applied to the simulated fields of τ_sim orientated in a north–south and west–east grid. The orientation of the observations is determined by the flight direction. Therefore, the orientation of the fields of τ_meas and $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ are rotated into the direction of the COSMO grid. One-dimensional $P_{\mathit{\tau }}^{\mathrm{2}}$ values are calculated manually along the dominant direction (straight red and blue lines in Fig. 7e–h and m–p) and across it (dashed red and blue lines in Fig. 7e–h and m–p). For $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ (red) and $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ (blue), the results are displayed in Fig. 7i–l. The dotted black line illustrates the threshold for the estimation of ξ_τ.

The observations on 14 May are influenced by a large-scale cloud structure, which is caused by large-scale dynamic forcing and leads to an increase of the autocorrelation coefficients for distances larger than 800 m. Furthermore, during this day, a significant directional structure from northwest to southeast is observed. Along this direction, the cloud field stays homogeneous over a wide range (ξ_τ=800 m). Across this predominant structure, the small-scale cloud structures reach a decorrelation length of ξ_τ=300 m. During the following days, the orientation of the directional structure turns eastwards in the observations. The differences between ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$ decrease. This characterizes a weakening of the directional structure of the cloud field.

Table 3Calculated decorrelation lengths ξ_τ,meas and ξ_τ,sim for the two different spatial resolutions of 50 and 100 m along (${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$) and across (${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$) the observed and simulated predominant directions (compare Fig. 7e–h and m–p for 50 m spatial resolution).

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Comparing the results for $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ with $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ reveals that the large-scale cloud structure is not well simulated for the case of 14 May. This results most probably from the small domain size of COSMO, which is fixed over the same location, when averaging the $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ over a set of time steps. Contrarily, the averages of $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ from the measurements are performed over a set of squared domains along the flight track. Thus, the chance to cover also larger structures is higher for the measurements compared to the simulations. However, the overall small-scale directional structures are well simulated. On 14 May, a significant directional structure from northwest to southeast is observed, which then turns eastwards for 15 to 17 May. Except on 16 May, the predominant simulated directions of the cloud fields are almost identically to the observations.

Furthermore, the results for $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ and $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ show that COSMO simulations using a spatial resolution of 50 m produce similar sizes of the small-scale cloud structures compared to the measurements. In Fig. 7m–p, the covered areas of $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ are of similar sizes compared to the areas covered by $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$ in Fig. 7e–h. Table 3 lists the resulting ξ_τ,meas and ξ_τ,sim calculated along (${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$) and across (${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$) the predominant structures found in Fig. 7e–h and m–p. A comparison reveals only minor differences between ξ_τ,meas and ξ_τ,sim. The best agreement is achieved on 15 and 17 May, when ξ_τ,meas and ξ_τ,sim show almost identically results. On 16 May, the differences are slightly larger, while on 14 May the differences are significantly larger, which might result from the insufficient simulated large-scale cloud structure. For the simulations with 100 m spatial resolution (graph not shown), the directional features still compare well between observations and simulations. Like for the measurements on 14 May, a predominant northwest to southeast direction is simulated, which then turns eastwards. Thereby, the cases on 14 May and 16 May show the strongest directional features (largest differences between ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ and ${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$, compare Table 3) with ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ on 14 May larger than the width of the observed field of τ_meas. Although on 17 May COSMO simulates a more isotropic structure (${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }\approx {\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }\approx \mathrm{400}$ m) of the cloud inhomogeneities compared to the measurements (${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }=\mathrm{370}\text{m}\ne {\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }=\mathrm{260}\text{m}$), it captures the reduction of the overall directionality. Therefore, the overall results with regard to the directional structure provided by COSMO are acceptable. However, the covered areas of the 2-D autocorrelation functions, where the values of $P_{\mathit{\tau },\text{sim}}^{\mathrm{2}}$ are higher than e⁻² are larger compared to the areas covered by the particular $P_{\mathit{\tau },\text{meas}}^{\mathrm{2}}$. Therefore, the ξ_τ,meas and ξ_τ,sim calculated along (${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$) and across (${\mathit{\xi }}_{\mathit{\tau }}^{\leftrightarrow }$) the predominant structures do not compare well (compare Table 3). Like expected from Fig. 5, the values from the simulations (except for ${\mathit{\xi }}_{\mathit{\tau }}^{\updownarrow }$ on 14 May) are larger compared to the values from the observations by 20 % to 30 %.