a. Small Cumulus Microphysics Study

We use data collected by the C-130 aircraft operated by the National Center for Atmospheric Research (NCAR) that participated in the Small Cumulus Microphysics Study (SCMS). The study took place from 17 July to 13 August 1995 near Cocoa Beach, Florida. The field project focused on aircraft and radar studies of shallow cumulus clouds in which the very early initiation of warm rain was active. Knight and Miller (1998) and French et al. (1999) have presented observations of the geometrical and microphysical structure of cumulus clouds during the SCMS campaign. More information on the meteorological context to the observations, such as mean profiles, surface fluxes, etc., can be found in the paper by Neggers et al. (2002), who use the data to evaluate the results of large eddy simulation (LES) study based on SCMS.

Four flight days with the presence of shallow cumulus are used in our analysis: flights RF12, RF13, RF16, and RF17, on 5, 6, 10, and 11 August, respectively. The flight region was 20 km × 20 km around 28.7°N and 80.7°W (Fig. 1). The clouds were quite small with typical diameters of less than 3 km. The cloud base was typically at a height of about 500 m. Cloud top was found to be around 2500 m. Details of the four flight days are listed in Table 1. The C-130 carried instrumental installations that were primarily adapted for observations of turbulence, radiation, and cloud physics. A summary of the instrumentation used in the analysis is presented in appendix A.

Once the individual clouds have been identified, one can determine the cloud number density n₁(l): the number of clouds with a size in the interval l and l + dl. The subscript 1 denotes the one-dimensional character of the aircraft measurements. The total number of observed clouds N₁ is the integral of the cloud number density

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An important quantity describing the cumulus field is of course, the cloud fraction σ_f, here taken as the ratio of cloudy points to the total number of points. The cloud fraction σ_f can be decomposed analogous to N₁:

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where α_f(l) is the so called cloud fraction density, which is related to the number density by

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with L the total flight length. The cloud fraction density α_f(l) provides information of the effect of cloud size on the cloud fraction and combines the competing effects of cloud number and cloud size—bigger clouds occur less frequently but have a larger impact per cloud.

Parameterization of transport due to cumulus clouds is often performed using the mass-flux approach (Randall et al. 1992; Tiedtke 1987; Siebesma and Holtslag 1996). The mass flux M_c in a cloud population is proportional to the product of the cloud fraction σ_f and the cloud-average vertical velocity w′^c. So, apart from cloud fraction, we also study quantities directly relevant to cloud dynamics such as the mass flux M_c and the in-cloud buoyancy flux. In addition we consider size decompositions of these dynamical quantities in order to reveal which cloud size—if any—is most important.

The mass flux M_c = ρ₀σ_fw′^c can be decomposed into the mass-flux density μ(l)

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where μ(l) is given by

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ρ₀ is a reference density, and w′^l is the average vertical velocity of clouds of size l.

The in-cloud buoyancy flux B_c in a cloud population is proportional to the cloud average product of the virtual temperature excess θ^′_υ and in-cloud vertical velocity w′ and can be decomposed into the in-cloud buoyancy flux density β(l); that is,

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where g is the gravitational acceleration, θ₀ a reference virtual potential temperature, and where the density β(l) is defined as

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Here, (g/θ₀)w′θ^′_υ^l denotes the average in-cloud buoyancy flux of clouds with size l.

Cumulus clouds couple the boundary layer to the free troposphere by means of transport. Both the mass flux and the buoyancy flux are important terms in the quantification of this transport. Apart from the overall cloud averages, the cloud size decompositions provide additional information: they reveal the net contribution of clouds within a certain size range. Once it is known which clouds—small or big—dominate the transport, research can focuss on these cloud sizes. Note that both the mass-flux density μ(l) and the in-cloud buoyancy flux density β(l) depend on the cloud fraction density α_f(l).

b. Landsat satellite data

We used the Landsat Thematic Mapper (TM) images from the Landsat-5 satellite to obtain a cloud cover distribution as a comparison for the size distributions derived from the flight measurements. Only high-resolution images over Florida near the flight region of the SCMS campaign of 10 August were available (the day of flight RF16) from the Landsat-5 satellite. The observed cumulus field is shown in Fig. 2; it covers an area of 65 km × 65 km. The lower-left point in this picture is at 27.9°N, 80.9°W. The upper-right point is at 27.2°N, 80.4°W (see also Fig. 1). The passing-by time of the satellite was 1453:16 UTC at an altitude of 705 km. The TM data have a resolution of 30 m × 30 m in the visible and near-infrared reflected bands.

To obtain a cloud cover distribution from the TM images, the first step is to determine a cloud reflectivity threshold that identifies each pixel as cloudy or noncloudy resulting in a binary image. Since certain surface regions have albedos comparable to clouds (beaches, some agricultural lots), cloud detection is not as straightforward as it may seem and research in this field is still continuing (Wielicki et al. 1995). Numerous methods exist for detecting cloudy pixels in satellite imagery, for example, the spatial coherence method (Coakley and Bretherton 1982), the probability of cloudy pixel method (Joseph 1986), a technique that makes use of statistics or climatological assumptions (Wielicki and Welch 1986), the bispectral iterative method (Joseph and Cahalan 1990), and several other methods using spectral information (Wen et al. 2001; Benner and Curry 1998; Koren and Joseph 2000).

Since we consider only a single Landsat image in our study, we decided to use the simple detection method of thresholding with a single channel in the visible light region (band 2 with wavelength 0.52–0.60 μm). Relying on man's formidable pattern recognition capabilities, we determined the threshold value by visual inspection. This led to a threshold which is 27.3% of the maximum light intensity that can be measured by channel 2 (Fig. 2). The additional use of the IR band (i.e., an extra restriction of a temperature threshold), according to the thresholding technique followed by Benner and Curry (1998), made no improvement due to the presence of a (cold) cirrus layer. In section 4, we study the impact of the intensity threshold on the cloud size distribution for a range of threshold values.

Individual clouds are defined as the cluster of contiguous pixels exceeding the threshold. Of each individual cloud, its linear size λ is calculated from the surface A of the cloud in the image:

λA(9)

Next, we calculated the cloud number density n(λ)—the number of clouds with a linear size between λ and λ + dλ. The total number of clouds N from the satellite image is again the integral over the cloud number density:

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Similarly, the cloud cover can be decomposed into the cloud cover density α(λ):

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where the cloud cover density α is related to the number density according to

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where S is the total area of the image.

At the boundary of the image, the cloud sizes are underestimated because of the cutoff. This effect diminishes as the sampling area increases. Plank (1969) argued that the minimum area in which this effect can be neglected measured 27 km × 27 km, whereas the Landsat picture investigated was 65 km × 65 km. To further clarify the influence of the four boundary sides the area has been divided into nine equal subareas of about 22 km × 22 km. The cloud cover distributions of these nine individual regions were averaged and the result was compared to the distribution of the whole region. The difference in the total number of clouds was less than 2.5% and the general shape of the cloud cover distribution virtually identical. The boundary effects are therefore of minor importance.

c. Cloud fraction versus cloud cover

In the literature, different appellations are in use to describe the percentage of the area covered by clouds (cloud size, cloud cover, cloud fraction, sky cover, cloud amount, etc.). Previously, we have defined the cloud fraction and the cloud cover. We emphasize that these are two different types of “area covered by clouds.” In this paper, we adopt the following convention: 1) cloud cover σ is the ratio of the total area of vertical projections of the clouds to the total area; 2) cloud fraction σ_f is the ratio of the total area of the cross sections of the clouds in a horizontal plane to the total horizontal area. The cloud fraction is a function of height and is always less than or equal to the cloud cover. As the satellite images are vertical projections, only a cloud cover density can be obtained from the satellite data. Conversely, flight measurements can only provide a cloud fraction density.

a. Cloud fraction distribution

Figure 3a shows the distribution of the cloud fraction obtained from flights RF12, RF13, RF16, and RF17 during SCMS. Furthermore an average over these four days is shown. Because the aircraft flew at different heights, this density needs to be seen as an average over the cloud layer (600–2000 m). Table 2 summarizes the number of cloud intersections for each flight day and the cloud fraction.

All observed flights indicate that there is no intermediate dominating size; the cloud fraction is dominated by the smallest observed clouds (l up to 300 m). Apparently the large number of small cumuli have more effect on the cloud fraction than the size of larger clouds. Only one flight (RF17) shows a peak in the cloud fraction density. For all other flights the peak is not visible or falls into the first bin.

The domination of the cloud fraction by the smallest-sized clouds is remarkable when compared to various remote sensing studies of the cloud cover. For example, the photographs taken from high-flying aircraft (e.g., Plank 1969) and other remotely sensed images like radar and satellite pictures (e.g., Sengupta et al. 1990; Wielicki and Welch 1986) indicate a cloud cover dominated by sizes of approximately half of the maximum observed cloud size, although the individual peak sizes can differ from a few hundred meters to about 1.5 km. However, it should be noted that the in situ measurements of the aircraft during SCMS only yield cloud fraction densities, which are intrinsically different than the cloud cover densities, which result from a projection.

Despite the frequently presented cloud cover densities, we have not found literature on cloud fraction decompositions from field observations. Only Neggers et al. (2003) show a cloud fraction distribution of an LES study based on the Barbados Oceanographic and Meteorological Experiment (BOMEX). They present a cloud fraction density at an altitude of 900 m (approximately 400 m above cloud base) dominated by intermediate-sized clouds. The largest clouds had linear sizes of about 750 m. The cloud fraction dominating clouds in the LES study had sizes of about 350 m.

In section 4 we make a direct comparison of our obtained cloud fraction density with the cloud cover density obtained from a satellite image of the cumulus field near the same region of the flight measurements and discuss the differences between the observed cloud fraction density and cloud cover density in more detail.

b. Mass-flux and in-cloud buoyancy flux distribution

The mass-flux density μ(l) and in-cloud buoyancy flux density β(l), defined in Eqs. (6) and (8), are shown in Figs. 3b and 3c. Whereas the cloud fraction is dominated by the smallest-sized clouds, it is interesting that the mass flux and the in-cloud buoyancy flux—both functions of the cloud fraction—are dominated by intermediate-sized clouds. Both μ(l) and β(l) reveal a peak near 1 km. Apparently, the intermediate-sized clouds, although small in number, are the main contributors to the vertical transport. Despite the fact that the largest-sized clouds are individually highly dynamic, they occur too seldom to play an important role.

A mass-flux decomposition based on observations has been presented earlier by Nitta (1975). This decomposition was extracted from observed large-scale conditions by the use of the Arakawa and Schubert (1974) scheme. Nitta (1975) deduced that, during BOMEX, the smallest clouds with a diameter up to 100 m contributed most to the total mass flux. Neggers et al. (2003) presented mass-flux decompositions based on LES studies of the Atmospheric Radiative Measurement program (ARM), BOMEX, and SCMS, and also found intermediate cloud sizes dominating the mass flux.

A surprising observation is the negative contribution to the mass flux by the smallest clouds (Fig. 3b). This implies that, on average, these small-sized clouds are descending. While there is significant scatter between the flights, the negative contribution to the mass flux consistently emerges for all four flights. Negative mass fluxes have also been reported by, for example, Grinnell et al. (1996), who employed ground-based Doppler radar together with aircraft observations. They emphasized the dependence of the vertical mass-flux profile on the cloud's life cycle: especially the decaying phase was associated with a negative mass-flux profile.

c. Average horizontal cloud cross sections

The presented flux densities revealed that intermediate-sized clouds are most important to the vertical transport. We therefore investigated these clouds in more detail, focussing on the in-cloud thermodynamical structure. We produced averaged (horizontal) cross section profiles of the vertical velocity w, the virtual potential temperature θ_υ, the liquid water potential temperature θ_l, and the total water content q_t in cumulus clouds with linear sizes larger than 500 m during the four flights. The restriction to 500 m and larger ensures the dominating clouds to be included in the average and also provides sufficient data points per cloud. As the airplane flew at a constant speed of 100 m s⁻¹, it spent about 5 s in a 500-m-sized cloud, where it collected 50 data points at a sampling rate of 10 Hz. As the aircraft pilot tried to intersect the large clouds in the middle, the cross sections did not say anything about cloud-top or cloud-base profiles. With the restriction to sizes of 500 m and larger, we observed 72, 48, 58, and 22 clouds during the four flights RF12 through RF17, respectively. Comparable studies have been performed by, for example, Nicholls (1989), for downdrafts in stratocumulus, and by Jonas (1990), for cumulus.

To obtain the averaged cumulus cloud cross section profiles of the four flights, we rescaled all clouds (larger than 500 m) to unit length. All in-cloud measurement points of an observed quantity were divided into 10 equidistant bins. In order to compare the cumulus cloud cross section with its environment, the same procedure was followed for the out-cloud regions: an equal number of in-cloud measurement points was taken before the airplane flew into a cloud and after the airplane exited the cloud. Also these observations were rescaled to unit length and divided into 10 equidistant bins. The results are plotted on a scale ranging from −1 to 2, where the interval [0, 1] pertains to the cloudy region. Evidently, since the airplane penetrated clouds randomly, the results should be symmetric around 0.5, provided there are enough data points.

The results for the vertical velocity, the virtual potential temperature, the liquid water potential temperature, and the total water content of the four flights are shown on the left in Fig. 4. All profiles have a quite similar shape, although the absolute values differ from day to day. This is due to the fact that the boundary layer is slightly different each day.

On the right in Fig. 4, we have eliminated the height effect and have averaged the four flight days into one profile; the average value of the region before the aircraft penetrated the cloud was subtracted from all the measured values before binning and averaging. The bars in the picture indicate the root-mean-square deviations from the mean and are a measure for the turbulence; they thus do not indicate the error in the measurements. The deviations are higher inside the cloud than outside the cloud, as clouds are more turbulent than the environment.

Note that the magnitude of the turbulent fluctuations is typically as large as the difference between the average in- and out-cloud values. This observation is important with respect to the validity of the mass-flux approximation, also referred to as the “top-hat” approximation (e.g., Wyngaard and Moeng 1992; Siebesma and Cuijpers 1995; de Laat and Duynkerke 1998): ρ₀w′φ′ ≈ M_c(φ_c − φ_e), where φ_c and φ_e represent the cloud average and the environmental average of φ, respectively. In passing we therefore mention that on the basis of the SCMS data we typically find M_c(φ_c − φ_e)/(ρ₀w′φ′) ≈ 0.5 for the conserved variables φ = q_t and φ = θ_l. This value is close to the theoretical value of ≈0.6 derived by Wyngaard and Moeng (1992) based on joint Gaussian distributions, it is also close to the value found by de Laat and Duynkerke (1998) for stratocumulus, but it is significantly lower than the value of 0.8–0.9 derived from LES for shallow cumulus (BOMEX) by Siebesma and Cuijpers (1995).

Returning to Fig. 4, one observes that the vertical velocity in the cloud is on average between 1 and 2 m s⁻¹; outside the cloud, slightly negative values should have been observed. Our results are slightly polluted for the following reason. For the out-cloud averaging we took a region, the size of the cloud, before and after the cloud intersection. If a neighboring cloud was in that region it was also averaged as environment, which explains the detected upward motion in this region. As a few more neighboring clouds were in the sample region on the right side than on the left side, the variances to the right of the cloud are larger than on the left side.

On average the cloudy air parcels move upward, but just outside the cloud boundary a thin shell of air moving downward is observed. This descending shell of air was also observed by Jonas (1990). Jonas argued that around active (growing) clouds, a descending shell of air is due mainly to mechanical forcing and not to evaporative cooling following from entrainment. Lateral entrainment into such an active cloud thus originates at much higher levels. No such shell was found around decaying clouds, but near the cloud edge, in the cloud interior, downdrafts were found to be driven by evaporative cooling.

In our observations most of the clouds observed were active clouds. However, two findings suggest that the thin shell of descending air is not due to mechanical forcing, but to evaporative cooling. To substantiate this conclusion, we show in Fig. 5, in a schematic way, the two opposing views—mechanical forcing versus evaporative cooling—and also indicate the consequences for total water and buoyancy. Briefly, if the descending air results from mechanical forcing it will possess the properties of the air mass at a higher level. As the total water content decreases with height, the shell is likely to have a lower value of q_t than the environment; that is, a dip in the cross section profile of q_t is expected. The other view, evaporative cooling resulting from entrainment, predicts a value of q_t between the environment and the cloud value, since the air in the shell is subject to strong mixing. But note that this view also predicts a significant dip in the buoyancy cross section, because negative buoyancy is assumed to be the major driving force behind the downward motion. Such a dip is not likely for mechanical forcing, where buoyancy plays a subordinate role, and a value between cloud and environment is expected.

If we compare our observations in Fig. 4 with the schematic picture in Fig. 5, it appears the evidence in our case points to evaporative cooling due to mixing rather than to mechanical forcing as the major driving mechanism of the observed descending shell of air. In particular the conspicuous dip in the θ_υ profile corroborates this viewpoint. However, the profiles of q_t and θ_l are clearly “sharper” than depicted in Fig. 5; apparently the effect of mixing is being compensated by the downward transport caused by the descending shell of air.

a. Cloud cover density

Figure 6a shows the distribution of the cloud cover for the cumulus field analyzed from the Landsat image. Using an intensity threshold of 27.3% (see section 2b), 8400 clouds in the field were observed with a cloud cover of 19.1%. This is more than twice the cloud fraction observed on the same day by flight RF16 (8.2%). However, if one bears in mind that the satellite image covers a different region (Fig. 1) and that the cloud cover, which involves a projection, is always larger than the cloud fraction (section 2), the difference in absolute values between cloud cover and cloud fraction is not surprising.

What is surprising, however, is the finding that the cloud cover density is dominated by an intermediate cloud size, as revealed by the peak at 700 m in Fig. 6a, whereas the aircraft observations of section 3 showed that the cloud fraction density was dominated by the smallest clouds in the ensemble (Fig. 3a). To rule out the possibility that the chosen threshold influences the results of the satellite analysis, we have determined for a range of threshold values the cloud cover density α(λ), its peak size λ_m, and its first moment λ_c:

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[see also Eq. (B14)]. The results, presented in Fig. 6b, reveal that the threshold has the expected strong effect on the cloud cover, but a minor effect on the size distribution, since the dominating size λ_m and the first moment λ_c are virtually insensitive to threshold variations.

Since the dominating size λ_m emerges as a robust feature, the relevant question that arises is whether the results of the satellite and aircraft observations are in mutual disagreement or not. Prior to answering this question we have to establish to what extent the satellite data and aircraft data can be compared. First of all, the aircraft and satellite observations were not well collocated. Figure 1 shows that the flight paths were concentrated near the coast, whereas the satellite image covers part of the coastal area, part of the sea and a relatively big part of the mainland. Furthermore, Fig. 2 indicates that cumulus convection was not present over sea, feeding the idea that the vicinity of the sea strongly influences the aircraft measurements, and possibly contributes to the discrepancy between the size distributions inferred from the satellite and aircraft data. However, if we calculate the cloud cover density in a subset of the satellite image, chosen roughly as big as the flight area and oriented in similar way along the coast (see the dashed rectangle in Fig. 1), we obtain a cloud cover density very similar to Fig. 6a, with the peak located at the same position, λ_m = 700 m. Furthermore, as mentioned in section 2b, dividing the image into nine subsets, also yields cloud cover densities that are very similar. So the satellite analyses consistently yield an intermediate dominating size. On the other hand, the aircraft measurements, performed on four different days, consistently yield a much smaller dominating size.

The second important issue regarding the validity of a direct comparison, is the difference in measurement methodology, the most important difference being that the aircraft data are one-dimensional and the satellite data two-dimensional. The latter issue will be addressed in the next section, where we analytically derive an equation that couples the number densities with each other, and enables one to translate a number density obtained from satellite data into the one-dimensional analogue, that is, the corresponding number density as the aircraft would have observed it.

The finding of an intermediate cloud size that dominates the cloud cover is in agreement with other observational studies (e.g., Plank 1969; Sengupta et al. 1990; Wielicki and Welch 1986), and with numerical studies (e.g., Neggers et al. 2003). Neggers et al. (2003) also reveal the strong universality of cloud cover densities: when rescaled by the dominating cloud size, almost identical cloud cover densities were obtained for three different LES cases.

Many more studies have focussed on the cloud number density in search of a universal functional form. Several proposed functional forms are listed in Table 3: exponential (Plank 1969; Hozumi et al. 1982), lognormal (Lopez 1977), a single power law (Machado and Rossow 1993), and, more recently proposed, a double power law with a scale break (Cahalan and Joseph 1989; Sengupta et al. 1990; Benner and Curry 1998). We stress here that all proposed functional forms (except for the single power law) reveal an intermediate dominating size if one determines the corresponding cloud cover density [cloud number density multiplied by l²; see Eq. (12)]. It remains yet unclear which physical processes influence this dominating size, which varies from case to case (typically from 100 m to a few kilometers). Cahalan and Joseph (1989) assumed it might be related to the largest individual convective cells that exist in the boundary layer. Jonker et al. (1999) stressed the importance of fluctuations in the subcloud specific humidity field.

b. Relation between satellite and aircraft number densities

In this section we study how to translate a number density obtained from satellite data into the corresponding number density as the aircraft would have observed it. To confine our analysis we neglect the third dimension (the vertical) and assume the aircraft to traverse the two-dimensional cloud field in random directions.

From stereology (e.g., Russ 1986; see also appendix B) it is known that the total cloud fraction (or cloud cover) is independent of the dimensionality (1D or 2D) of the method, provided that there have been an infinite number of random line measurements. If we assume the aircraft to have flown randomly and for a long period of time, we can conclude that the observed cloud fraction is representative for the real (2D) cloud fraction. However, the flight-based cloud number density will generally differ from the the satellite-based number density. For instance, the aircraft will hardly ever penetrate the clouds along their full diameters, and therefore always observe smaller sizes. This effect causes the 1D number density to be biased toward smaller sizes. On the other hand, the probability of intersecting a large cloud is much higher than intersecting a small cloud. This effect results in a bias to larger sizes. Both effects are not exactly opposing and therefore do not cancel each other. In appendix B we take these and other effects into account and for the simplified case of circular clouds we analytically derive a relation that couples the “satellite” number density to the “aircraft” number density. If n₂(d) denotes the number density of a given field of circular clouds with cloud diameters d, and n₁(l) denotes the (aircraft) number density of the observed intersection lengths l, we then have

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where N₁ is the total number of clouds observed by the aircraft and N₂ the total number of clouds in the 2D field. Furthermore, the symbol d represents the average cloud diameter based on the density n₂(d) [see also Eq. (B3)]. From Eq. (14) we can immediately derive a relation between the corresponding cover densities. Employing Eqs. (4) and (12), we obtain

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where L is the total flight length, and S the total area of the image. With this relation, one can calculate the cloud cover density obtained from 1D measurements in 2D cloud fields. In other words, given the satellite observation n₂(d), Eq. (15) enables one to predict the cloud cover/fraction density the aircraft will observe. In addition, if the 2D cloud cover is dominated by a certain size, the effect of 1D measurements on this size can also be calculated.

To show the procedure, we will present an example with a number density based on a combination of an exponential term, as proposed by, for example, Plank (1969), and a power-law d^β (e.g., Cahalan and Joseph 1989; Machado and Rossow 1993; Sengupta et al. 1990; Benner and Curry 1998; see also Table 3); that is, n₂(d) ∼ d^β exp(−d/D). Figure 7 shows an artificial cloud field consisting of circular clouds with this number density. The cloud cover density is given by α₂(d) ∼ d^2+β exp(−d/D). We took β = −1.7 (best fit to our data) and D = 3.3 km, which ensures the dominating size d_m to be located at 1 km [because d_m = D(β + 2)]. In appendix B we calculate the analytical solution to Eq. (15) for this particular choice of α₂(d), which enables us to extract the size l_m that dominates α₁(l). In this example we find the peak in α₁(l) to be located at l_m = 700 m, that is, a reduction of the dominating size by a factor 1.4.

It is not directly clear from Eq. (15) how a 2D dominating size d_m translates into a 1D dominating size l_m for arbitrary densities n₂. For each choice of n₂(d) one would have to evaluate the integral in Eq. (15) and differentiate it with respect to l to find the 1D dominating size l_m. However, we would like to have a single parameter that signifies the shift in peak size independent of the prescribed cloud number density. Additional insight in the generic case can indeed be obtained by studying the so-called characteristic length scales l_c and d_c, which are defined as the first moments of α₁(l) and α₂(d), respectively. As derived in appendix B, we find

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independent of the prescribed functional form of the number density n₂(d). This relation implies that, compared to α₂, the 1D cloud cover distribution α₁ will always be biased to smaller sizes, irrespective of the 2D cloud number distribution n₂.

We validated our analytical predictions based on Eq. (15) by simulating an imaginary aircraft flying randomly through the particular 2D cloud field (Fig. 7) and measuring the 1D cloud cover density. The results of this simulation were in excellent agreement with the analytical solution [see Eq. (B13) and Fig. B2].

But since clouds are not circular, a more realistic simulation could also be performed by using the binary Landsat image as the 2D cloud field (Fig. 2b). Figure 8 shows the 1D cloud cover density as measured by a hypothetical aircraft flying randomly through the Landsat image. In the flight simulation the aircraft flew 5000 times in random directions through the image. The resulting cloud cover is about 18.9%, close to the 19.1% observed by the satellite. The peak in the 2D cloud cover density is at λ_m = 700 ± 100 m; for the 1D cloud cover density it was found at l_m = 430 ± 50 m—a reduction by a factor 1.4.

So, also for noncircular, real-life cloud fields, it has been shown that 1) the 2D cloud cover equals the 1D cloud cover, and 2) the 1D cloud cover density peaks at a smaller size than the 2D cloud cover density.

c. Cloud cover versus cloud fraction—Intrinsic difference

Previously we have seen that a 1D measured cloud cover density will differ from a 2D measured cloud cover density, while the cloud cover itself is the same in both cases. Both the “flight simulations,” the analytical example (appendix B), and Eq. (16) reveal a bias toward smaller scales when measuring one-dimensionally. Typically a reduction of the dominating size by a factor of 1.4 is anticipated.

However, the (true) aircraft observations of section 3 revealed a cloud fraction density that was dominated by clouds of size l_m ≈ 150 m, whereas the cloud cover density obtained from the Landsat image exhibited a peak at 700 m (see Fig. 6 for comparison)—a reduction by a factor of at least 4.5. Clearly, our analysis only partially explains the shift. In the derivation of the 1D and 2D cloud cover relation we eliminated the intrinsic difference between cloud cover and cloud fraction; by confining ourselves to only two dimensions, a similarity between cloud cover and cloud fraction was assumed. Hence, we did not account for variability in the vertical direction. This difference between cloud cover and cloud fraction is very important, as clarified in Fig. 9. For instance, as clouds are not cylindrical, the 3D shape of the cloud (Fig. 9a) reduces the aircraft intersection length l relative to the (linear) size observed by the satellite. A similar effect occurs due to wind shear, which tilts the clouds (Fig. 9b). If there are two clouds one overlapping the other (Fig. 9c), it is observed as one big cloud by the satellite, but the aircraft detects two small clouds. Similarly, if an aircraft intersects a frayed cloud it will detect two or more smaller clouds, even though these parts relate to the same cloud (Fig. 9d).

These examples show the importance of the variability in the vertical direction, and clearly illustrate the intrinsic difference between cloud fraction and cloud cover. In all four examples the net effect constitute biases in opposing directions: the 2D cloud cover density is biased toward larger sizes, whereas at the same time the 1D cloud fraction density is biased toward smaller sizes.