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  • View in gallery
    Fig. 1.

    Image of Florida showing the flight region near Cocoa Beach and the area covered by the Landsat-5 satellite. Dashed lines indicate a subset that covers the coastal area (see section 4)

  • View in gallery
    Fig. 2.

    (left) Landsat-5 satellite image of the cumulus field observed near Vero Beach (FL) at 1453:16 UTC 10 Aug 1995. Observed region is about 65 km × 65 km and the satellite resolution is 30 m × 30 m. (right) Binary reproduction of left image after using a threshold of 27.3% of the maximum light intensity that can be measured by channel 2

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    Fig. 3.

    Size distributions obtained from four flight days during SCMS and a 4-day average: (a) cloud fraction density; (b) mass-flux density; (c) in-cloud buoyancy flux density. The bin size Δl = 270 m

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    Fig. 4.

    (left) Averaged in-cloud profiles of vertical velocity, virtual potential temperature, total water content, and liquid water potential temperature of flights RF12, RF13, RF16, and RF17 and (right) the average over the four flight days, where the effect of altitude on the measurements has been eliminated. The cloud sizes have been rescaled to unity. The bars denote the rms values of the deviations from the mean. These bars thus do not denote an error, but are a measure of the turbulence

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    Fig. 5.

    Schematic representation of a cloud with a descending shell of air just outside the cloud boundary and two mechanisms that can explain the descent: mechanical forcing and evaporative cooling. The consequences for the cross section profiles of virtual potential temperature and total water are also depicted schematically

  • View in gallery
    Fig. 6.

    (a) Normalized cloud cover density α(λ) of the Landsat image obtained by using a threshold of 27.3% of the maximum measurable light intensity of channel 2. The bin size Δλ = 270 m. (b) Threshold sensitivity analysis. Cloud cover σ, the cloud cover dominating size λm, and the characteristic length scale λc [defined as the first moment of α(λ)], as a function of the intensity threshold

  • View in gallery
    Fig. 7.

    Binary image of an artificial field of circular clouds. The clouds are distributed following the functional form n2(d) ∼ dβ exp(−d/D), with d the cloud diameter, β = −1.7, and D = 3.3 km. The cloud cover dominating size is then located at 1 km

  • View in gallery
    Fig. 8.

    Normalized cloud cover density α1(l) an aircraft would measure one-dimensionally in the cumulus field observed by the Landsat image with cloud cover density α2(λ). The bin sizes are 270 m

  • View in gallery
    Fig. 9.

    Four examples that cause the cloud cover and cloud fraction densities to diverge from each other: (a) nontrivial 3D shape; (b) wind shear; (c) multilevel clouds; (d) cloud frayness

  • View in gallery

    Fig. A1. Cloud fraction vs chosen threshold (droplet concentration of the FSSP-100). Seven droplets per cm−3 is the chosen threshold used for conditional sampling of the flight data

  • View in gallery

    Fig. B1. (left) A 2D image of several (shaded) structures. The area fraction (the sum of the shaded areas Ai divided by the total surface A) should equal the sum of all intercepts li (thick lines) divided by the total line length (sum of thick and dashed lines) in the case of many random lines. (right) Circular clouds with diameter d. The cloud intersection has a length l. In a single cloud, each point y is equally likely to be passed

  • View in gallery

    Fig. B2. Cloud cover density n1(l)l an aircraft would measure one-dimensionally for a prescribed circular cloud field n2(d) for Eq. (B12) with β = −1.7 and D = 3.3 km. The y axis is written in arbitrary units (a.u)

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Size Distributions and Dynamical Properties of Shallow Cumulus Clouds from Aircraft Observations and Satellite Data

Stefaan M. A. RodtsThermal and Fluids Sciences, Delft University of Technology, Delft, Netherlands

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Peter G. DuynkerkeInstitute of Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, Netherlands

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Harm J. J. JonkerThermal and Fluids Sciences, Delft University of Technology, Delft, Netherlands

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Abstract

In this paper aircraft observations of shallow cumulus over Florida during the Small Cumulus Microphysics Study (SCMS) are analyzed. Size distributions of cloud fraction, mass flux, and in-cloud buoyancy flux are derived. These distributions provide information on the specific contribution of clouds with a certain horizontal size and reveal, for example, which size has the largest effect on cloud fraction or vertical transport. The analysis of four flights shows that the mass flux and buoyancy flux are dominated by intermediate-sized clouds (horizontal dimension of about 1 km). The cloud fraction, on the other hand, is found to be dominated by the smallest clouds observed. These clouds are additionally found to have a negative contribution to the mass flux, yet a positive contribution to the buoyancy flux.

About 200 flight intersections of cumuli with horizontal sizes larger than 500 m are used to obtain average horizontal cross-section profiles of vertical velocity, liquid water content, liquid water potential temperature, and virtual potential temperature. A thin shell of descending air just around the cloud emerges as a conspicuous feature. Evidence is found that the descent is mainly caused by evaporative cooling, which results from lateral mixing at the cloud boundary.

A Landsat satellite image near the flight region is analyzed to compare the cloud size distributions with the aircraft data. The cloud cover in the image appears to be dominated by much larger clouds than the aircraft observations indicated. To account for the different measurement methodology (two-dimensional versus one-dimensional) an equation with which one can predict the cloud size distribution that results from performing line measurements in a prescribed two-dimensional cumulus field is derived. The equation reveals that the aircraft cloud size distributions are always biased toward smaller cloud sizes. This effect is nevertheless not enough to reconcile the aircraft and satellite data, presumably because the analysis neglects the variability of clouds in the vertical direction.

Deceased

Corresponding author address: Dr. Harm J. J. Jonker, Thermal and Fluids Sciences, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands. Email: h.jonker@ws.tn.tudelft.nl

Abstract

In this paper aircraft observations of shallow cumulus over Florida during the Small Cumulus Microphysics Study (SCMS) are analyzed. Size distributions of cloud fraction, mass flux, and in-cloud buoyancy flux are derived. These distributions provide information on the specific contribution of clouds with a certain horizontal size and reveal, for example, which size has the largest effect on cloud fraction or vertical transport. The analysis of four flights shows that the mass flux and buoyancy flux are dominated by intermediate-sized clouds (horizontal dimension of about 1 km). The cloud fraction, on the other hand, is found to be dominated by the smallest clouds observed. These clouds are additionally found to have a negative contribution to the mass flux, yet a positive contribution to the buoyancy flux.

About 200 flight intersections of cumuli with horizontal sizes larger than 500 m are used to obtain average horizontal cross-section profiles of vertical velocity, liquid water content, liquid water potential temperature, and virtual potential temperature. A thin shell of descending air just around the cloud emerges as a conspicuous feature. Evidence is found that the descent is mainly caused by evaporative cooling, which results from lateral mixing at the cloud boundary.

A Landsat satellite image near the flight region is analyzed to compare the cloud size distributions with the aircraft data. The cloud cover in the image appears to be dominated by much larger clouds than the aircraft observations indicated. To account for the different measurement methodology (two-dimensional versus one-dimensional) an equation with which one can predict the cloud size distribution that results from performing line measurements in a prescribed two-dimensional cumulus field is derived. The equation reveals that the aircraft cloud size distributions are always biased toward smaller cloud sizes. This effect is nevertheless not enough to reconcile the aircraft and satellite data, presumably because the analysis neglects the variability of clouds in the vertical direction.

Deceased

Corresponding author address: Dr. Harm J. J. Jonker, Thermal and Fluids Sciences, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, Netherlands. Email: h.jonker@ws.tn.tudelft.nl

1. Introduction

Clouds play a key role in climate change prediction (Wielicki et al. 1995) and in weather forecasting (Tiedtke 1989) through radiative effects, latent heat release, and precipitation. For example, cloud morphology affects the cloud radiative properties, which is one of the most important parameters controlling the earth's radiation budget (Sengupta et al. 1990). Besides these atmospheric applications, clouds have impact on remote sensing systems. For example the identification of clouds and their removal from a satellite image has been studied frequently (Reynolds et al. 1978; Coakley and Bretherton 1982; Wielicki and Welch 1986; Wielicki and Parker 1992; Koren and Joseph 2000).

Cumulus clouds are important as they dynamically couple the boundary layer to the free troposphere: cumulus clouds transport heat, moisture, momentum, and pollutants through the inversion. However, shallow cumulus has typical sizes (of about a kilometer) that cannot be resolved in global circulation models, mesoscale models, and even not in the current cloud-resolving models. Only collective effects of cumulus clouds can be represented by means of parameterization. Implementation of a cumulus parameterization scheme has lead to significant improvements in tropical forecasting at the European Centre for Medium-Range Weather Forecasts (ECMWF) (Tiedtke 1989). Recent research has proposed various parameterization schemes (e.g., Nordeng 1994; Siebesma and Holtslag 1996; Grant and Brown 1999; Lin 1999; Gregory 2001; Lappen and Randall 2001) and is still evolving.

From observations of shallow cumulus fields, it is known that cumulus clouds can be of many sizes, ranging from a few meters up to a few kilometers. The small-sized cumuli largely outnumber the large-sized cumuli, but do not necessarily contribute the most to area coverage properties like the cloud cover. Distributions of cumulus cloud size have been frequently studied. In the 1960s and 1970s high-flying aircraft were used for determining the size of cumulus clouds (Plank 1969; Hozumi et al. 1982), but the postprocessing of the photographs involved enormous labor. In the last few years remote sensing instruments, like satellite images, and radar came more into focus (Lopez 1977; Cahalan and Joseph 1989; Sengupta et al. 1990; Machado and Rossow 1993; Benner and Curry 1998). Due to increasing computer capacity, numerical studies on cloud size statistics (Brown 1999; Neggers et al. 2003) and on cloud geometry (Siebesma and Jonker 2000) have been performed as well. Other statistical properties that have been studied with remote sensing measurements are nearest neighbor spacing (Joseph and Cahalan 1990; Sengupta et al. 1990) and cloud fractal dimensions (Lovejoy 1982; Cahalan and Joseph 1989; Benner and Curry 1998).

Most of these studies have focused on the statistical properties of clouds. However, the cloud sizes that contribute the most to the cloud cover do not necessarily contribute the most to dynamical properties like mass flux or in-cloud buoyancy flux. Additional knowledge of the size decompositions of mass flux and buoyancy flux can contribute to a better insight in growth and decay processes of cumulus clouds and to improved parameterization schemes. Aircraft observations are very well suited to yield size distributions of dynamical quantities.

In this paper we first analyze aircraft observations of cumulus fields above Florida to obtain (horizontal) size distributions of cloud fraction, mass flux, and buoyancy flux. After having established which cloud sizes have the greatest impact on the vertical transport, we focus on these clouds and present average horizontal cloud cross sections to get more insight into the cloud dynamics. Finally we compare the cloud size distributions derived from the flight data with the cloud size distributions obtained from a Landsat satellite image near the flight region.

The data used in our analysis and necessary definitions are described in section 2. The results of the aircraft measurements are discussed in section 3. The results of the satellite image and the comparison with the flight data are presented in section 4. Conclusions and a discussion can be found in section 5.

2. Data description and definitions

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.

To obtain size distributions from the aircraft measurements, first a useful definition of “cloudy” is needed. Usually a cloudy region is defined by the presence of liquid water. However, throughout the whole flight, water droplet measurements indicate the presence of droplets. We used a concentration threshold of seven droplets per cubic centimeter (see appendix A) to distinguish between cloudy and noncloudy regions. In our analysis an individual cloud is defined as consisting of successive points exceeding the droplet threshold and its horizontal size follows simply from the time Δt the aircraft spent in the cloud multiplied by the aircraft velocity υ:
lυt.
With a flight speed of 100 m s−1 and 10 Hz liquid water measurements, individual clouds larger than 10 m could be detected during SCMS. Throughout this paper, when we refer to cloud size, we always mean the horizontal size of the cloud. We use the following (arbitrary) terminology regarding size: small clouds have size less than 500 m, intermediate have size between 500 and 1500 m, and large clouds have size larger than 1500 m.
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
i1520-0469-60-16-1895-e2
An important quantity describing the cumulus field is of course, the cloud fraction σf, here taken as the ratio of cloudy points to the total number of points. The cloud fraction σf can be decomposed analogous to N1:
i1520-0469-60-16-1895-e3
where αf(l) is the so called cloud fraction density, which is related to the number density by
i1520-0469-60-16-1895-e4
with L the total flight length. The cloud fraction density αf(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 Mc in a cloud population is proportional to the product of the cloud fraction σf and the cloud-average vertical velocity wc. So, apart from cloud fraction, we also study quantities directly relevant to cloud dynamics such as the mass flux Mc 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 Mc = ρ0σfwc can be decomposed into the mass-flux density μ(l)
i1520-0469-60-16-1895-e5
where μ(l) is given by
i1520-0469-60-16-1895-e6
ρ0 is a reference density, and wl is the average vertical velocity of clouds of size l.
The in-cloud buoyancy flux Bc in a cloud population is proportional to the cloud average product of the virtual temperature excess θυ and in-cloud vertical velocity w′ and can be decomposed into the in-cloud buoyancy flux density β(l); that is,
i1520-0469-60-16-1895-e7
where g is the gravitational acceleration, θ0 a reference virtual potential temperature, and where the density β(l) is defined as
i1520-0469-60-16-1895-e8
Here, (g/θ0)wθυl 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 αf(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
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:
i1520-0469-60-16-1895-e10
Similarly, the cloud cover can be decomposed into the cloud cover density α(λ):
i1520-0469-60-16-1895-e11
where the cloud cover density α is related to the number density according to
i1520-0469-60-16-1895-e12
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 σf 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.

3. Size distributions and dynamical properties from aircraft observations

All the distributions depicted in this paper are obtained by sorting into equidistant bins (Δl = Δλ = 270 m) on the linear size (l or λ) axis. All densities are normalized to show percentages of the total value. In the rest of the paper much attention will be payed to the peak size, that is, the size at which a peak occurs in the density. We will refer to this peak size as the dominating size.

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 θυ, the liquid water potential temperature θl, and the total water content qt 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): ρ0wφMc(φcφe), where φc and φe 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 Mc(φcφe)/(ρ0wφ) ≈ 0.5 for the conserved variables φ = qt and φ = θl. 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 qt than the environment; that is, a dip in the cross section profile of qt is expected. The other view, evaporative cooling resulting from entrainment, predicts a value of qt 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 θυ profile corroborates this viewpoint. However, the profiles of qt and θl 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.

4. Cloud cover distribution from satellite data

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 λm, and its first moment λc:
i1520-0469-60-16-1895-e13
[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 λm and the first moment λc are virtually insensitive to threshold variations.

Since the dominating size λm 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, λm = 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
i1520-0469-60-16-1895-e14
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
i1520-0469-60-16-1895-e15
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 dβ (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) ∼ dβ 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) ∼ d2+β exp(−d/D). We took β = −1.7 (best fit to our data) and D = 3.3 km, which ensures the dominating size dm to be located at 1 km [because dm = 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 lm that dominates α1(l). In this example we find the peak in α1(l) to be located at lm = 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 dm translates into a 1D dominating size lm 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 lm. 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 lc and dc, which are defined as the first moments of α1(l) and α2(d), respectively. As derived in appendix B, we find
i1520-0469-60-16-1895-e16
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 λm = 700 ± 100 m; for the 1D cloud cover density it was found at lm = 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 lm ≈ 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.

5. Conclusions

In this paper we studied in situ (flight) measurements of cumulus fields above Florida during the SCMS campaign in August 1995. Four flights in the presence of cumulus enabled us to obtain size distributions of the cloud fraction, mass flux, and in-cloud buoyancy flux. We analyzed a high-resolution Landsat-5 satellite image of the cumulus field of one of the flight days to obtain a cloud cover distribution with which we could compare the flight measurements.

In the majority of the flights the observed cloud fraction density did not reveal a peak at an intermediate size; instead the cloud fraction was dominated by the smallest clouds in the ensemble. On the other hand, the mass flux and in-cloud buoyancy flux were found to be dominated by intermediate-sized clouds. The less frequent occurrence of clouds with these sizes is apparently compensated by their, on average, higher vertical velocity and therefore higher impact on the overall transport.

We observed a negative contribution to the mass flux by the smallest clouds, which implies that on average these small-sized clouds are descending; their contribution to the buoyancy flux was, however, positive. As the intermediate-sized clouds contribute the most to the mass flux and the buoyancy flux, we studied in greater detail the thermodynamic structure of these clouds. We presented averaged (horizontal) cross section profiles of the vertical velocity, the virtual potential temperature, the liquid water potential temperature, and the total water content, of cumulus clouds with sizes larger than 500 m. A striking feature is the presence of a thin shell of descending air just outside the cloud boundary. Based on our data, in particular on the conspicuous dip in the virtual potential temperature at the cloud edges, we attributed this descending shell mainly to evaporative cooling resulting from mixing, rather than to mechanical forcing (Jonas 1990).

The cloud cover density derived from the satellite image is in good agreement with other observational studies—intermediate cloud sizes dominate the cloud cover. At first sight, this finding seemed to contradict the aircraft observations which, as mentioned, revealed the cloud fraction to be dominated by the smallest clouds observed. However, due to the difference in measurement methodology, the densities cannot be directly compared. We analytically derived an equation that can translate 2D cloud number distributions into 1D distributions. Given a 2D cloud number density, this equation enables one to predict the corresponding 1D cloud number density, that is, the density as a (hypothetical) aircraft would have observed it. The analysis revealed that 1D densities are always biased toward smaller sizes, irrespective of the 2D cloud number distribution.

However, the predicted bias was not enough to reconcile the aircraft data with the satellite data, presumably because we neglected in our analysis the variability of clouds in the vertical direction. The nontrivial 3D shape of a cloud, the presence of wind shear, the possibility of multiple stacked clouds, and cloud frayness are four effects that will lead to opposing biases—a bias toward larger scales for satellite cloud cover densities, and at the same time a bias to smaller scales for aircraft cloud fraction densities. Unfortunately, we cannot push the conclusions on the discrepancy between satellite and aircraft observations too far, because the observations were not well collocated. Hopefully, future observational studies will pay extra attention to matching aircraft (or other) observations and satellite observations both in time and place.

One of the challenges of atmospheric LES studies is to simulate representative 3D cumulus fields in great detail. Thus far LES studies have proven to accurately predict the average thermodynamic state of the cumulus layer and more or less reasonably the turbulent state. The measured cumulus cloud cross section profiles presented in this paper may constitute some additional test to check whether or not individual cumuli are well represented. This can give further insight into the mixing mechanisms at the cloud boundary. Additionally, 3D LES cloud fields could nicely serve to mimic the aircraft measurements, in order to test whether the shift in dominating cloud size is supported.

Acknowledgments

The data collected by means of the C-130 of NCAR during SCMS were kindly supplied by Dr. C. A. Knight. We acknowledge NCAR and its sponsor, the National Science Foundation, for the use of the observational data. We thank Piet Jonker for his computer programming support and Stephan de Roode for many discussions. We are grateful to three anonymous referees for their useful comments and suggestions.

This article would not have seen daylight without the support, suggestions, positive criticism, and ceaseless enthusiasm of Peter Duynkerke. The sudden loss of our colleague and friend has been a great shock. We hope that his ideas are clearly articulated in this paper.

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APPENDIX A

Instruments

The C-130 instrumental installations were adapted primarily for observations of turbulence, radiation, and cloud physics. The instruments used for this study are listed in Table A1.

The total water content was calculated from the liquid water content and the specific humidity (qt = ql + qυ). A Gerber optical liquid water content device (Gerber et al. 1984) was used to determine the liquid water content. A Lyman–Alpha absorption hygrometer was used to measure the specific humidity, and a Rosemount platinum resistance thermometer was used to observe the temperature.

In-cloud temperature measurements are not trivial. In the presence of water droplets, the measurements can be biased. For example Raga et al. (1990) mention temperature errors caused by wetting of the sensor. Since initially we found most in-cloud parcels to be negatively buoyant despite the substantial upward velocities, we checked the influence of liquid water on the platinum resistance thermometer. We found that the difference between the platinum resistance thermometer and the dew point thermometer showed a linear correlation of 0.4 K per 1 g kg−1 liquid water (checked for several flights). In this article we only use temperatures from the resistance thermometer that have been corrected (subscript c) for this linear behavior:
TcTql
The (virtual) potential temperature θ(θυ) was calculated with the corrected temperatures [Eq. (A1)] and the pressure measurements (Rosemount Model 1201F1). A linear expression for the liquid water potential temperature θl is used:
i1520-0469-60-16-1895-ea2
in which lυ is the specific latent heat for vaporization, and cpd the specific heat capacity for dry air at constant pressure.

The inertial reference system (IRS) of the aircraft is normally used to determine the position and speed of the airplane. It measures the acceleration of the aircraft and calculates its speed and position by integrating the accelerations. However, this system can slowly accumulate large errors over time. The global positioning system (GPS) is used to correct for these errors. On the basis of the statement by the Research Aviation Facility (RAF) of NCAR that these corrected variables are the best available when GPS is functioning, we used the corrected values.

It is easy to indicate clouds or cloudy regions looking at the sky; however, objective determination of cloud boundaries from aircraft observations are less straightforward. Ideally, the presence of water droplets indicates a cloudy region, whereas the absence of droplets indicates the environment. However, in practice, a small amount of water droplets is measured throughout the whole flight; a threshold needs to be chosen. The Forward Scattering Spectrometer Probe (FSSP) was used to measure water droplets with radii in the range of 3.7 to 50.5 μm. Figure A1 shows the cloud fraction as a function of the chosen threshold value for the four flight days. The strong decline in cloud fraction stops at concentrations of six droplets per cubic centimeter. Varying the threshold between 7 and 50 droplets per cubic centimeter did not effect the results significantly. Therefore we used a threshold of seven droplets per cubic centimeter to distinguish individual clouds.

APPENDIX B

Analytical Relation between One- and Two-Dimensional Cloud Number and Cover Densities

The aircraft data are 1D (measurements along a line), while the satellite data are 2D (vertical projections). If we assume the aircraft to have flown randomly, it is known from stereology (Russ 1986) that the measured cloud fraction should equal the cloud fraction based on a 2D measurement method: if infinite random lines are drawn on a 2D image with a total surface S, the length li that is within the structures to be measured, divided by the total line length L, equals the area fraction (Fig. B1):
i1520-0469-60-16-1895-eb1
However, we are also interested in size decompositions. To see how the two different methods relate we will attempt to deduce an analytical relation between the “flight” cloud number density and the number density of a given cloud field. Hereby, we assume a field of circular clouds with a prescribed cloud number density n2(d) (d being the diameter of the cloud; subscript 2 denoting the 2D character). The derivation will be in three steps—1)–3).
  1. If the aircraft intersects a cloud, the probability of intersecting a cloud with diameter d is given by
    i1520-0469-60-16-1895-eb2
    which reflects the fact that smaller clouds have less chance to be intersected than larger clouds. Upon introducing the average diameter
    i1520-0469-60-16-1895-eb3
    with N2 the total number of clouds in the 2D field, we can rewrite Pa(d) into
    i1520-0469-60-16-1895-eb4
  2. Let us now look at a single circular cloud with a diameter d (Fig. B1). If we assume the probability density Pb(y | d) of flying through a cloud with a certain diameter d at a position y to be uniform (each point y is equally likely to be passed), we get
    i1520-0469-60-16-1895-eb5
  3. As is evident, the length l of a line cut through the circle will vary, depending on the level at which the circle is cut (Fig. B1). It will be largest around its center and minimal at the top.

    With the probability density Pb(y | d), we can deduce the probability density Pc(l | d) of intersecting the cloud with a certain intersection length l, given a cloud with diameter d as
    PclddlPbyddy.
    From this equation and the theorem of Pythagoras
    l2y2d2
    we can find Pc(l | d):
    i1520-0469-60-16-1895-eb8
    This equation tells us what the probability will be of measuring an in-cloud path of length l provided that the aircraft intersects a cloud with diameter d(l < d ≤ ∞).
Now if we intersect a cloud we can calculate the probability P1(l) of measuring an in-cloud path of length l in a whole field of circular clouds:
i1520-0469-60-16-1895-eb9
With this we find for the 1D cloud number density n1(l):
i1520-0469-60-16-1895-eb10
where N1 is the total number of intersected clouds. The 1D cloud cover density α1(l) = n1(l)l/L [see Eq. (4)] becomes
i1520-0469-60-16-1895-eb11
with α2(d) = n2(d)d2/S the 2D cloud cover density [see Eq. (12)].
If we prescribe the cloud number density of the cumulus field [n2(d)] (e.g., from a satellite image), we can calculate the number density and cloud cover density that the aircraft will measure. As an example we will use a number density based on a combination of an exponential term and a power-law term:
n2ddβed/D
The parameter D has a dimension of length and is used in this calculation to adjust the maximum of the cloud cover density [n2(d)d2] at a peak diameter dm = 1000 m. Equation (B10) then has an analytical solution (Prudnikov et al. 1992),B1 where we use p = 1/D and γ = 1 + β:
i1520-0469-60-16-1895-eb13
With, for example, β = −1.7 and D = 3.3 km to ensure a peak in the cloud cover density at dm = 1000 m [since dm = D(β + 2)], we can plot the 2D cloud cover density n2(d)d2 and the calculated 1D cloud cover density n1(l)l, which an aircraft would measure in this case (Fig. B2).

The effect of 1D measurements is immediately seen: whereas we set the peak in the cloud cover density n2(d)d2 at dm = 1000 m, the (aircraft) 1D cloud cover density reveals a peak at a cloud size l = 700 m.

We verified this effect and the analytical relation [Eq. (B10)] with a flight simulation. First of all, an imaginary field of circular clouds, randomly placed, with the number density n2(d) according to Eq. (B12) (with β2 = −1.7 and D = 3.3 km) was made (Fig. 7). An aircraft flying randomly through this field was simulated and the corresponding number density n1(l) was measured. In total, about 30 000 clouds were put in a field of 115 km × 115 km, which resulted in a cloud cover of 20%. In the flight simulation 4000 lines were drawn. The resulting cloud cover was 19.8%, in conformity with the 2D cloud cover of 20%. The peak in the cloud cover density shifted from dm = 1000 m in the 2D case to lm = 700 ± 100 m in the 1D case, in agreement with the analytical solution [Eq. (B13), see Fig. B2].

An analytical relation denoting the shift in the peak sizes (2D compared to 1D) cannot be deduced without prescribing the number density n2(d). However, we would like to have a single parameter that signifies the shift in peak size independent of the prescribed cloud number density. We therefore use the first moments of the 1D and 2D cloud cover density, denoted lc and dc respectively, as characteristic sizes. Using Eq. (15), the characteristic size dc becomes
i1520-0469-60-16-1895-eb14
which is the third moment divided by the second moment of the 2D number density. Similarly, the characteristic size lc becomes
i1520-0469-60-16-1895-eb15
After substituting Eq. (B10) and rearranging the integral, we obtain
i1520-0469-60-16-1895-eb16
Combining Eqs. (B14) and (B16) yields the relation between the two characteristic sizes:
i1520-0469-60-16-1895-eb17
This relation implies that a 1D obtained cloud cover distribution is dominated by scales that are smaller than the 2D peak sizes, whatever 2D cloud number distribution.

Fig. 1.
Fig. 1.

Image of Florida showing the flight region near Cocoa Beach and the area covered by the Landsat-5 satellite. Dashed lines indicate a subset that covers the coastal area (see section 4)

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 2.
Fig. 2.

(left) Landsat-5 satellite image of the cumulus field observed near Vero Beach (FL) at 1453:16 UTC 10 Aug 1995. Observed region is about 65 km × 65 km and the satellite resolution is 30 m × 30 m. (right) Binary reproduction of left image after using a threshold of 27.3% of the maximum light intensity that can be measured by channel 2

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 3.
Fig. 3.

Size distributions obtained from four flight days during SCMS and a 4-day average: (a) cloud fraction density; (b) mass-flux density; (c) in-cloud buoyancy flux density. The bin size Δl = 270 m

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 4.
Fig. 4.

(left) Averaged in-cloud profiles of vertical velocity, virtual potential temperature, total water content, and liquid water potential temperature of flights RF12, RF13, RF16, and RF17 and (right) the average over the four flight days, where the effect of altitude on the measurements has been eliminated. The cloud sizes have been rescaled to unity. The bars denote the rms values of the deviations from the mean. These bars thus do not denote an error, but are a measure of the turbulence

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 5.
Fig. 5.

Schematic representation of a cloud with a descending shell of air just outside the cloud boundary and two mechanisms that can explain the descent: mechanical forcing and evaporative cooling. The consequences for the cross section profiles of virtual potential temperature and total water are also depicted schematically

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 6.
Fig. 6.

(a) Normalized cloud cover density α(λ) of the Landsat image obtained by using a threshold of 27.3% of the maximum measurable light intensity of channel 2. The bin size Δλ = 270 m. (b) Threshold sensitivity analysis. Cloud cover σ, the cloud cover dominating size λm, and the characteristic length scale λc [defined as the first moment of α(λ)], as a function of the intensity threshold

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 7.
Fig. 7.

Binary image of an artificial field of circular clouds. The clouds are distributed following the functional form n2(d) ∼ dβ exp(−d/D), with d the cloud diameter, β = −1.7, and D = 3.3 km. The cloud cover dominating size is then located at 1 km

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 8.
Fig. 8.

Normalized cloud cover density α1(l) an aircraft would measure one-dimensionally in the cumulus field observed by the Landsat image with cloud cover density α2(λ). The bin sizes are 270 m

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Fig. 9.
Fig. 9.

Four examples that cause the cloud cover and cloud fraction densities to diverge from each other: (a) nontrivial 3D shape; (b) wind shear; (c) multilevel clouds; (d) cloud frayness

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

i1520-0469-60-16-1895-fa01

Fig. A1. Cloud fraction vs chosen threshold (droplet concentration of the FSSP-100). Seven droplets per cm−3 is the chosen threshold used for conditional sampling of the flight data

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

i1520-0469-60-16-1895-fb01

Fig. B1. (left) A 2D image of several (shaded) structures. The area fraction (the sum of the shaded areas Ai divided by the total surface A) should equal the sum of all intercepts li (thick lines) divided by the total line length (sum of thick and dashed lines) in the case of many random lines. (right) Circular clouds with diameter d. The cloud intersection has a length l. In a single cloud, each point y is equally likely to be passed

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

i1520-0469-60-16-1895-fb02

Fig. B2. Cloud cover density n1(l)l an aircraft would measure one-dimensionally for a prescribed circular cloud field n2(d) for Eq. (B12) with β = −1.7 and D = 3.3 km. The y axis is written in arbitrary units (a.u)

Citation: Journal of the Atmospheric Sciences 60, 16; 10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2

Table 1.

Characteristics of flights RF12, RF13, RF16, and RF17 during SCMS

Table 1.
Table 2.

Characteristics of flights RF12, RF13, RF16, and RF17

Table 2.
Table 3.

Summary of different functional forms describing the number density proposed in the literature; a, b, and c are constants

Table 3.

Table A1. List of quantities measured by the instruments of the C-130

i1520-0469-60-16-1895-ta01

The solution of the integral in Prudnikov's book probably contains an error. The third term is written “+(2p)2−α−2β…”; however the factor 2 before p should be erased in our view. This has been verified by numerical integration of the integral.

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