a. The truncated gamma distribution

The full expressions for the droplet concentration N_T, the liquid water mixing ratio l, and the reflectivity Z for nondrizzling clouds (assuming Rayleigh scattering, which is still valid at the proposed radar frequency of 94 GHz) can be written as

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where ρ_w is the density of liquid water, ρ_o is the density of dry air, Γ(α) is the gamma function, and the P factor is defined as

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The three shape parameters α, a, and b can be determined from (2) to (4) using individually measured size spectra by calculating N_T, l, and Z while integrating over the spectral bins and solving the three transcendental equations for the three unknowns. The effective radius r_e is a derived parameter, and is defined as the ratio of the third moment and the second moment of the droplet distribution:

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while the optical depth is given by

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where z_b is cloud-base height and Q_ext is the extinction efficiency (∼2).

In the case of drizzle Eq. (2)–(6) are used as well, but the P-correction factors are disregarded, which means that we resort to the complete gamma function.

b. Size distribution moments and universal links to cloud depth and cloud optical depth

Using the simplifying assumption that the total droplet concentration N_T and α are constant with height, then the substitution of (2), (3), and (8) into (7) yields an optical depth of

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where Γ(α) has been replaced by (α − 1)! and ψ₁ = {[(α + 1)(α + 2)]/[(α + 3)²]}^1/3.

Similarly, the expressions for the effective radius and the reflectivity can also be written in terms of l(z) and N_T:

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where ψ₂ = [(α + 6)(α + 4)(α + 5)]/[(α + 3)(α + 2)(α + 1)]. Z has the dimension m³, but is usually expressed in mm⁶ m⁻³ or its logarithmic equivalent dBZ; 0 dBZ corresponds to 1 mm⁶ m⁻³. The dimension of r_e(z) is m; because of the small size of cloud droplets the unit μm is used in this paper. The mixing ratio l(z) is dimensionless. Here N_T is the number of particles per unit of volume; in this paper cm⁻³. The optical thickness is dimensionless. In this paper, r_max is set to 23 μm. The reflectivity is thus a quadratic function of height, and inversely proportional to N_T. This implies that, all other factors being equal, the reflectivity in the Northern Hemisphere will generally be lower than in the Southern Hemisphere because industrial sources of air pollution and thus cloud condensation nuclei are more abundant in the Northern Hemisphere than in the Southern Hemisphere. During the SOCEX in situ observations were made of marine stratiform clouds located west of Tasmania (Boers et al. 1996, 1998). This region is subjected to an extremely low concentration of anthropogenic cloud condensation nuclei, resulting in droplet concentrations that are among the lowest in the world. Equation (11) indicates that the data from SOCEX would generate the highest reflectivities, all other factors being equal. Equation (11) also suggests the well-known possibility to retrieve liquid water content from cloud reflectivity, provided N_T is derived from other sources.

c. Application to in situ data

As mentioned, the three parameters describing the gamma function may be estimated using Eq. (2)–(4), where the values of N_T, l, and Z are those computed from the in situ measured droplet concentrations. Table 1 contains the mean values of the three parameters of the truncated gamma distribution. The values are obtained using the in situ data collected during SOCEX 1 and SOCEX 2. All data were taken using the same probes. Data reflect level flight legs at various altitudes inside clouds and correspond to scales from 5 up to 20 km. So, the smallest scales are comparible to those of the volume-averaged radar reflectivity but can, for longer averaging time, be larger by a factor of three to four as we will see below.

For comparison, the results for CLARE98, which was held over Chilbolton (United Kingdom), and some research flights conducted off the Namibian coast, are given as well (courtesy P. Francis, Met Office). For all campaigns, the in situ measured particle size distributions are averaged over the length of an entire flight, which is about 10–30 min. Table 1 contains the average value of the cloud parameters encountered during the four experiments. For the total droplet concentration in the nondrizzle case, the range of measured values is given as well. When compared to the truncated gamma distribution, the nonexistent tail of the complete gamma distribution contributes up to 30% to the total reflectivity. Thus, it is essential to use the truncated gamma function in order to make proper estimates of the distribution's parameters.

The estimated gamma parameters can now be used to construct a diagram of l vs N_T, with isolines of the effective radius, reflectivity, and optical depth [Eq. (9)–(11)]. To obtain the optical depth and height above cloud base the additional two assumptions are that β = 0.4 and that N_T and α are constant with height. The parameter β reflects the fact that clouds are rarely adiabatic; the value of 0.4 follows from a ground-based remote sensing experiment during the Clouds And Radiation experiment (CLARA) campaigns in 1996 in the Netherlands (Boers et al. 2000). In that experiment it was shown that β may vary between 0.2 and 0.6 for a closed stratocumulus field.

The purpose of Fig. 1 is to visualize the link between the cloud parameters and their vertical variation in the nondrizzle case, and provides a clear picture of the expected reflectivity, mean particle size, and optical depth given that the liquid water mixing ratio and the total droplet concentration are known. In other words, the diagram provides an estimate of three cloud parameters given that two other parameters are known. Clearly, this figure is only an estimate based on only a few experiments and is therefore valid only as a guide to expected behavior. The right-hand side of Fig. 1 represents the nondrizzle component of the particle spectra where α = 9.93 (average α over SOCEX1 and SOCEX2), whereas the left-hand side is the drizzle component where α = 1.39. Notice that the optical depth, as represented by Eq. (9), is only valid for the nondrizzle case, where it may be assumed that the liquid water mixing ratio varies linearly with height. Therefore, its isolines are omitted from the left side of the figure.

To illustrate the universal character of this diagram, if N_T = 300 cm⁻³ and assuming a cloud depth of 200 m, then the radar reflectivity is expected to be around −30 dBZ at the cloud top (and decreasing below cloud top), while the cloud optical depth would be about 8. When N_T = 50 cm⁻³, however, a cloud of the same depth would have a reflectivity of −22 dBZ near cloud top and an optical depth of roughly 5. This is an important result as it directly visualizes the restrictions on cloud detection by the spaceborne radar. It implies that a typical sensitivity of −30 dBZ for cloud radar would not be sufficient to detect many continental clouds. On the other hand, increasing the sensitivity to −40 dBZ would greatly improve the detectability of continental clouds. The in situ–based values are in line with the actual radar observations described in Clothiaux et al. 1995.

The in situ probe data from SOCEX1, SOCEX2, CLARE98, and Namibia (diamonds, asterisks, squares, and crosses, respectively, plotted as l vs N_T) are superimposed on the figure to illustrate the contribution of the drizzle component to the total reflectivity. The scatter in the diagram for a given N_T is not in disagreement with what could be expected due to adiabatic variation, assuming that β varies between 0.2 and 0.6. Two examples are considered. In the first example the nondrizzle component contains a total number of droplets of 33 cm⁻³, while in the second example this number is 87 cm⁻³. First, consider the observation O₁ marked with a triangle. The nondrizzle component (N_T = 33 cm⁻³, l = 0.066 g kg⁻¹, r_e = 9.84 μm, as computed from the in situ measured particle spectra) has a reflectivity of −27.23 dBZ. Its associated drizzle component (N_T = 0.053 cm⁻³, l = 0.099 g kg⁻¹, r_e = 105.51 μm) has a reflectivity of 8.11 dBZ and thus dominates the total reflectivity. In this case, the contribution of the drizzle component to the total liquid water mixing ratio is also significant (60.19%). For observation O₂, the total concentration of the nondrizzle component is 87 cm⁻³, with a liquid water mixing ratio and effective radius of 0.098 g kg⁻¹ and 8.19 μm, respectively. The reflectivity is of the same order as O₁, namely, −27.35 dBZ, but this observation contains far less drizzle (N_T = 0.0088 cm⁻³, l = 0.011 g kg⁻¹, r_e = 90.12 μm), leading to a drizzle contribution to the total liquid water mixing ratio of only 10.24%. However, since the reflectivity is proportional to the sixth moment of the droplet spectrum, the contribution of the drizzle component (−4.40 dBZ) to the total reflectivity is still predominant (99.5%).

Practically all observations during the SOCEX campaigns are drizzle contaminated, leading to reflectivities that are dominated by the drizzle component of the particle spectrum. The contribution of the drizzle component to the total liquid water mixing ratio is significant (contribution larger than 10%) for 78% of all observations. The situation is similar but less serious for CLARE98. No drizzle was reported over Namibia.

Interestingly, the observations on the right-hand side of Fig. 1 appear to be constrained by the −20 dBZ isoline, not by their apparent height above cloud base, as would be expected if it were a random set of spectra. While it cannot be ruled out that the spectra were taken under highly selected, nonrandom conditions, it is more probable that it reflects the effect of drizzle on cloud depth. Several studies (Pincus and Baker 1994; Boers 1995) demonstrate that drizzle formation restricts cloud depth. In other words, deep clouds cannot be sustained in an environment of few droplets, as the few droplets would grow large and initiate the collision–coalescence process.

From a remote sensing perspective, it is useful to know a priori whether a cloud contains a significant amount of drizzle. As SOCEX data yield among the highest reflectivities expected of boundary layer clouds, Fig. 1 shows directly that the maximum reflectivity due to the nondrizzle component is around −20 dBZ, whereas the minimum reflectivity due to the drizzle component is about −10 dBZ. Thus, recalling that each data point represents one in situ observation averaged over an entire flight leg, on average there appears to be an evident jump of approximately 10 dBZ in reflectivity between a drizzle-contaminated and a drizzle-free droplet spectrum. Furthermore, inspection of the SOCEX datasets has shown that if the total reflectivity is below −5 dBZ, then the contribution to the total liquid water mixing ratio due to drizzle is negligible (5.4%, 0.0047 g kg⁻¹ on average), and the total concentration of drizzle is low (0.0052 cm⁻³ on average). This delineation can be used to discrimate drizzling and nondrizzling clouds. It is interesting to note that Frisch et al. (1995) observed a similar delineation between drizzle and cloud droplets, albeit in their observations during the Atlantic Stratocumulus Transition Experiment (ASTEX) campaign −16 dBZ deliniation was more appropriate.

Clearly, the reflectivities isolines in Fig. 1 are only estimates based on the average of the gamma parameters (Table 1). To indicate the accuracy in the estimates of the reflectivity and the effective radius based on Fig. 1, given the liquid water mixing ratio and the total particle concentration, the standard deviations in the estimates of Z and r_e are computed as

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where X_i is the theoretically derived value of either Z or r_e [Eq. (10) and (11)] at coordinate (N_T,i, l_i), where N_T,i and l_i are the in situ–measured values. Here O_i is the calculated Z or r_e, based on the in situ observations, and T is the total number of observations. The resulting standard deviations are given in Table 2.

It shows that Fig. 1 can be used to predict r_e with an accuracy of approximately 5% in the nondrizzle case and roughly 30% in the drizzle case. The error of Z varies then from approximately 1.5 dB for the nondrizzle case to approximately 6 dB for drizzle. This error estimation is based on in situ observations done under different conditions and will likely also represent a realistic range of values of the β value for adiabatic conditions. This result demonstrates that this type of plot can be used as a universal guideline for reflectivity prediction.

a. Cloud depth

The estimated parameters describing the gamma distribution are used in this section to construct a model of the reflectivity field and to simulate the expected spaceborne radar return. First, the performance of spaceborne radar with regards to the vertical structure of boundary layer clouds is considered. The impact of the large radar volume on the horizontal variation is discussed next with the aid of a fractal model, which simulates a cloud with a horizontal extent of 25 km × 25 km based on a mean total concentration, cloud thickness, and cloud top.

Inserting (8) into (11) yields

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Thus, as mentioned before, the radar reflectivity varies quadratically with the height above cloud base for prescribed values of β and N_T. To simulate the expected return for a spaceborne radar, the actual reflectivity field is convoluted with the radar signal:

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where S_i is the radar pulse shape function, which is assumed to be either a boxcar function or a Gaussian function; z′ is a running parameter.

Figure 2 shows a vertical profile of Z (solid line), in accordance with Eq. (13) where N_T = 50 cm⁻³ (Figs. 2a,b) or 250 cm⁻³ (Figs. 2c,d), β = 0.4 and the cloud depth is 400 m. The dashed lines in the figures are the convoluted signals, and the dotted lines are the convoluted signals sampled at an interval of 500 m, which is the typical height resolution of a spaceborne radar, taking into account a radar sensitivity of −30 dBZ (WMO–WCPR 1994). Figures 2a and 2c show the convolution of the reflectivity field with a boxcar function and Figs. 2b and 2d result from convolution with a Gaussian function. Both radar signals have a pulse length of 500 m.

After convolution, the “true” reflectivity is “smeared” out in the vertical direction, consequently synthetically increasing the cloud thickness by 500 m. The signal for a boxcar function (Fig. 2a) produces a thicker cloud than for a Gaussian function (Fig. 2b). Similar results apply for N_T = 250 cm⁻³ except that the sampled cloud appears thinner (Fig. 2c,d) due to the lower reflectivities.

If the convoluted signal is next sampled at 500-m intervals, which would complete radar signal processing, the radar-detected cloud depth is greatly affected by the digitization level and the radar sensitivity. Let us assume a −30-dBZ sensitivity. Figures 2a–d each have a left side and a right side. The right-hand figures are similar to the left-side figure, with the mere difference that the digitization levels have been changed by an altitude of 250 m. The reason for showing this option is that the exact distance from the radar to the cloud is always unknown. The digitization level affects the returned radar signal in a complex manner. Focusing on Fig. 2a, it appears that the apparent cloud size after digitization in the right-hand figure is 500 m thinner than on the left, while the vertical location of the cloud is approximately equal in both figures. The reason is that digitization of the convoluted signal on the left will detect two points exceeding a sensitivity of −30 dBZ, but only one on the right side. Consequently, the apparent cloud thickness on the left is 1000 m, but only 500 m on the right. Compared to the actual thickness of 400 m, it is clear that large errors in sizing may occur in this manner. From Figs. 2b and 2c it is clear that the digitization level may also modify the apparent vertical position of the cloud by about 250 m. As is the case for Fig. 2d, the sampling level may even cause the cloud not to be seen at all.

b. Horizontal variations in cloud depth and microphysics

Figure 2 considers purely the impact of the convolution and the digitization on the vertical structure of clouds. However, to simulate the expected radar return, an appropriate radar horizontal resolution has to be taken into account as well, which will have a great impact on the performance of the spaceborne radar to map the cloud stratification. To this end, a cloud with a horizontal extent of approximately 25 km × 25 km is simulated. This is accomplished with the aid of a bounded cascade model (Cahalan et al. 1994) and by considering the cloud top, the cloud thickness, and the total droplet concentration as scaling fractals. The purpose of using these fractals is not to construct a perfectly realistic cloud field, but to illustrate the effect of quasi-realistic fluctuations in controllable parameters.

Eight cascades are used, resulting in a grid of 256 × 256 pixels in the (x, y) plane. The radar footprint is taken as 1.4 km and one pixel is chosen to represent 100 m. Thus only a grid of 256 × 14 pixels is considered, yielding a cloud field of about 25 km × 1.4 km. The additional assumption that N_T is independent on height produces a 3D cloud model in the (x, y, z) plane. The effective radar footprint is assumed to be 4.0 km × 1.4 km, which results from averaging the footprint over 0.3 s. It is assumed that the satellite's speed is 8 km s⁻¹, so that the effective radar footprint is obtained by averaging over an additional 26 pixels, after having averaged over the radar footprint (14 pixels). Numerous simulations were performed; only some examples are given.

In Fig. 3a the resulting reflectivity field in the (x, z) plane (averaged across-track) is given for N_T = 50 cm⁻³. The large jumps in cloud height are included to simulate a worst-case scenario. However, observations from SOCEX confirm that the shown discontinuities do exist. Averaging the reflectivity across-track causes a smearing effect, similar to that produced by the convolution operation. Recalling that the mean cloud depth was originally 400 m, the reflectivity is now smeared over a height of roughly 600 m. The solid line represents the associated cloud top and cloud base, averaged across-track as well. Figure 3b displays the reflectivity after convolution with the boxcar function and shows that the amount of smearing caused by the convolution is approximately 400 m (Fig. 2). Notice that, although the reflectivity profile will be smeared out over a great vertical extent by the averaging and convolution operations, the mean value of the true cloud depth remains virtually unchanged (see also Fig. 4). The convoluted radar reflectivity, after averaging over the effective radar footprint, is given in Fig. 3c. A radar sensitivity of −30 dBZ is assumed. The solid line in Fig. 3c is the true cloud top and cloud base averaged over the same effective footprint, and the dashed line is the cloud top/base as “seen” by the radar. The along-track averaging modifies the reflectivity scene considerably, where the apparent cloud thickness reaches a maximum value of about 1.9 km. Figure 3d depicts the associated profiles of the optical depth and the droplet concentration (after averaging across-track). The optical depth ranges from 6 to 21, while the droplet concentration ranges from 30 to 60 cm⁻³.

Figure 3e is the equivalent of Fig. 3c, but now for the larger droplet concentration of 300 cm⁻³. Theoretically, the reflectivity decreases by about 7.8 dBZ when the total concentration increases by a factor of 6 [all other factors being equal, see Eq. (13) or Fig. 1]. According to Fig. 1 the reflectivity will exceed −30 dBZ when the height above cloud base is larger than 200 m (which was only about 70 m for N_T = 50 cm⁻³). Consequently, the apparent cloud thickness is about 200 m thinner than the true thickness. Averaging the reflectivity across-track and by the convolution procedure, however, will compensate this effect by smearing out the reflectivity so as to obtain a cloud depth that is comparable to the true depth. Note however that there is now a sizable portion of the cloud that will not be detected by the radar at all.

For comparison with the estimated cloud thickness (h_est), the actual cloud thickness is defined in two ways. One method is to average the cloud thickness of the model over the effective footprint of the spaceborne radar, which generates the true cloud depth (h). The second method is to average the reflectivity of the model over the same effective footprint, and then define the cloud top and cloud base as the heights at which the reflectivity crosses the −30 dBZ threshold. The cloud depth that results from this method (h₂) is greatly affected by the averaging of the reflectivity over the effective radar footprint and is used to compare with h_est to gauge the impact of the convolution and the sampling operations on the retrieved cloud depth. The former method is only slightly affected by the averaging procedure, thus resulting in a much smaller value of the cloud depth when compared to the cloud thickness obtained using the latter method. This is illustrated in Fig. 4, which shows a vertical profile of reflectivity as the solid line (at the 11-km distance indicator in Fig. 3) after averaging over the effective radar footprint. The dashed line is the expected radar return and the total concentration is 50 cm⁻³. The cloud thickness using the second method is 1630 m (top and base are at 2820 and 1190 m, respectively), while the true cloud thickness resulting from the first method is only 439 m (top and base are at 2086 m and 1647 m, respectively). Clearly, in Figs. 3c and 3e, h₂ will follow exactly the contours of the reflectivity profiles, and it is seen that the difference between h₂ and h_est (indicating the impact of the convolution and the sampling operation together) reaches a maximum value of approximately 500 m. When the averaging is taken into account as well (i.e., comparison with true cloud depth), the inaccuracy is much larger. The mean error is roughly 600 m, with a maximum error of about 1500 m. Although it should be recognized that the simulation shows a “break” in the cloud at 13 km over which the smoothing takes place (and thus represents a worse case scenario), these are very large errors and imply that the cloud-filled volume as estimated by the radar can on some occasions be vastly larger than present in reality.

The smearing effect depends on the radar sensitivity; the higher the sensitivity, the larger the smearing effect. If we take the cloud volume of the fractal model as reference, then a doubling occurs for a −40 dBZ threshold and 70% increase in apparant cloud volume for a −30 dBZ threshold. After the next processing step, namely sampling (digitization), the increase in volume referenced from the base value ranged from 20% to 100%. This large range is entirely dependent on the actual distance from the cloud to the radar, as explained in Fig. 2. Clearly, the error made in estimating the cloud top and cloud base is smaller for higher droplet concentrations. However, this does not imply that the radar performance is better for higher droplet concentrations. It only means that two sources of error, namely low radar sensitivity and convolution, partly cancel each other out because, as mentioned, the reflectivity is inversely proportional to N_T. Consequently, for higher N_T, the amount of smearing caused by the convolution, averaging, and sampling is reduced.

Figure 5 displays the accuracy in the estimates of the cloud depths as a function of N_T. Four situations have been plotted. The asterisks and the diamonds represent the situations where the mean cloud thickness is 400 m, assuming a sensitivity of −40 and −30 dBZ, respectively. The cases where the mean cloud thickness is 200 m assuming a sensitivity of −40 and −30 dBZ are denoted by the triangles and the squares, respectively. The accuracy is computed as the standard deviation in the estimates of the cloud depth using Eq. (12), where X_i is the estimated cloud depth, and O_i is the true cloud depth. The percentages in the diagram reflect the amount of the actual cloud that is detected by the spaceborne radar, when different from 100%. It shows that a radar with a sensitivity of −30 dBZ will be unsuitable to detect clouds thinner than 200 m. When the cloud is at least 400 m, virtually all of the cloud will be detected using the same sensitivity, up to a total concentration of roughly 250 cm⁻³. Radar that is capable of detecting a minimum reflectivity of −40 dBZ will be able to detect almost all clouds, which confirms the conclusions by Fox and Illingworth (1997a).

c. Retrieval of cloud liquid water and effective radius

Equations (10) and (11) suggest a straightforward way to obtain cloud droplet effective radius and liquid water in case the droplet concentration is known from other sources, because r_e is dependent on the 1/3 power of l, while Z is quadratic in l (see also Baedi et al. 2000). It is instructive to see the distribution of cloud droplet effective radius and liquid water retrieved after convolution of the radar pulse with the cloud reflectivity. Figure 6 shows the result that, while the range of liquid water and effective radius values sampled from the volume-averaged, -digitized, and -convoluted reflectivity is much smaller than the actually observed range of values, the mean values are not all that far from the truth. For N_T = 50 cm⁻³, the model/retrieved values of Z, l, and r_e are −18.09/−23.08 dBZ, 0.20/0.18 g m⁻³, and 11.75/10.58 μm, while for N_T = 300 cm⁻³ they are −25.88/−28.27 dBZ, 0.20/0.17 g m⁻³, and 6.47/6.58 μm. However, the modal values of these parameters are substantially different from each other.

d. Drizzle

Thus far the contribution of the drizzle component of the particle spectrum has been neglected. Figure 7 shows how the drizzle component affects the vertical profiles of the effective radius, liquid water mixing ratio, and the reflectivity. The data are measured on 16 July 1993 during the SOCEX 1 campaign. On this day a uniform stratocumulus cloud over the Southern Ocean approximately 400 m thick was encountered, with drizzle showers below the cloud deck. The total droplet concentration was about 35 cm⁻³. The asterisks in the figure contain only FSSP data, and the diamonds represent the merged FSSP and 2DC spectra. It shows that the drizzle contribution to the radar reflectivity completely overshadows the nondrizzle contribution. In the presence of drizzle, clouds will seem to extend to the lowest level where drizzle would be observed, which in many cases will almost be to the surface. Furthermore, the reflectivity will not be related at all to the liquid water mixing ratio.

Boers et al. (1998) and Gerber (1996) demonstrated from observations that there is a threshold of the cloud droplet effective radius beyond which clouds will drizzle, although there appears to be some disagreement as to the exact value of the threshold. If we take a compromise value based on both studies (r_e = 12 μm) and compare this value against the value of r_e at cloud top for the model as shown in Figs. 3 and 4, then we find the entire cloud would be susceptible to drizzle if the droplet concentration was 50 cm⁻³, and practically none of the cloud would be if the droplet concentration was 300 cm⁻³. This means that in the drizzle regions the cloud radar reflectivity would be overwhelmed by reflectivity from droplets in and below cloud base, thus destroying the link between reflectivity, cloud depth, and liquid water content. Furthermore, it would mean that for the situation of N_T = 50 cm⁻³, the cloud volume would be even more overestimated than it already is, as the drizzle signal from below cloud base would make it appear that clouds were present there as well.

Figure 1 can be used to broadly demarcate the clouds susceptible to drizzle. If we take a 12-μm isoline of effective radius as the boundary between clouds that drizzle and clouds that do not, then we find that this line follows approximately the following droplet concentration–cloud depth (i.e., height above cloud base) coordinates: (10 cm⁻³, 60 m), (50 cm⁻³, 300 m), (300 cm⁻³, 1000 m). These numbers represent clouds with a varying optical thickness: from very thin to thick. Clearly, only very shallow marine nondrizzling clouds can be sustained for small concentrations, while at the high concentrations representing continental conditions, deep nondrizzling boundary layer clouds can be sustained. Of course, the precise location of the boundary is vague due to the fluctuations of the microphysical properties of individual cloud fields, and the uncertainty in precision of the threshold value of effective radius.