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Variability of Optical Depth and Effective Radius in Marine Stratocumulus Clouds Szczodrak, Malgorzata; Austin, Philip H.; Krummel, P. B. 2001

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2912 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESq 2001 American Meteorological SocietyVariability of Optical Depth and Effective Radius in Marine Stratocumulus CloudsM. SZCZODRAKRosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, FloridaP. H . A USTINAtmospheric Sciences Programme, University of British Columbia, Vancouver, British Columbia, CanadaP. B . K RUMMELCSIRO Atmospheric Research, Aspendale, Victoria, Australia(Manuscript received 31 January 2000, in final form 12 March 2001)ABSTRACTRadiance measurements made by the Advanced Very High Resolution Radiometer (AVHRR) at 1-km (nadir)spatial resolution were used to retrieve cloud optical depth (t) and cloud droplet effective radius (reff) for 31marine boundary layer clouds over the eastern Pacific Ocean and the Southern Ocean near Tasmania.In the majority of these scenes (each roughly 256 3 256 km2in extent) t and reffare strongly correlated,with linear least squares yielding a regression curve of the form reff} t1/5. This relationship is consistent withan idealized model of a nonprecipitating layer cloud in which 1) the average cloud liquid water content increaseslinearly with height at some fraction of the adiabatic lapse rate ina1km2vertical column, and 2) the normalizedhorizontal variability of the cloud liquid water path exceeds the variability of a scaled measure of the clouddroplet number concentration. In contrast, other scenes of similar horizontal extent show little or no correlationbetween retrieved values of t and reff. These scenes include thicker clouds in which precipitation may be occurring,as well as cloud layers with spatially distinct regions of varying reff.In situ aircraft measurements were made simultaneously with six AVHRR overpasses as part of the SouthernOcean Cloud Experiment. The clouds sampled by these flights were significantly thicker than the typically 200-m-thick eastern Pacific stratocumulus, with large vertical and horizontal variability. On five of the six flights,aircraft measurements of the cloud-top effective radius were well matched by the satellite retrievals, and in twoof these layers reff} t1/5.1. IntroductionMarine stratus and stratocumulus clouds play a sig-nificant role in the planetary energy budget. Globally,boundary layer clouds act to decrease the net radiativeforcing by 15 W m22due to their large reflectivity (Hart-mann et al. 1992). This reflectivity varies with cloudparameters such as cloud fraction, column-integratedliquid water, and the mean surface area of cloud drop-lets. Knowledge of the seasonal and spatial variabilityof these cloud parameters is a prerequisite for under-standing feedbacks between boundary layer cloud prop-erties and natural or anthropogenic climate change.Efforts to incorporate prognostic equations for cloudliquid water content and sulphate mass in global climatemodels have underscored the uncertainties inherent inCorresponding author address: Dr. Philip H. Austin, AtmosphericScience Programme, University of British Columbia, 6339 StoresRoad, Vancouver, BC V6T 1Z4, Canada.E-mail: paustin@eos.ubc.capredicting the impact of aerosol concentration on cloudreflectivity (Lohmann and Feichter 1997). At the sametime, a new generation of cloud-resolving models, runwith horizontal and vertical resolutions of tens of metersand domain sizes of kilometers, are making detailedpredictions about the distribution of fluctuations incloud liquid water and cloud droplet size. To better con-strain predictions at both ends of the modeling spectrum,observations of cloud liquid water path, particle size,and droplet number concentration across a range ofscales are needed.Several recent studies have shown that satellite-basedpassive remote sensing can provide information on thecloud optical depth t (related to the extinction of thedirect solar beam), the surface area-weighted mean ra-dius reff(or effective radius; see appendix A for defi-nitions), the liquid water path (lwp), and a measure ofthe droplet number concentration in the column (see,e.g., Han et al. 1994; Nakajima and Nakajima 1995;Platnick and Valero 1995; Han et al. 1995, 1998a;Greenwald and Christopher 1999; Han et al. 1998b).1OCTOBER 2001 2913SZCZODRAK ET AL.TABLE 1. The grid system for values of cloud base (z), cloud thickness (Dz), solar zenith angle (u0), satellite zenith angle (u), opticaldepth (t), and effective radius (reff) for the lookup tables used for the satellite t, reffretrievals.Quantity Grid pointsz (km)Dz (km)u0(8)u (8)f (8)treff(mm)1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.00.1, 0.2, 0.5, 1.0, 2.00, 5, 10, 20, 30, 35, 40, 45, 50, 55, 60, 65, 700, 5, 10, 20, 30, 35, 40, 45, 50, 55, 600–180 (every 108)1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 50, 704, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 30Below we use the algorithm of Nakajima and Nakajima(1995) to retrieve t and cloud-top refffrom 25 relativelyshallow stratocumulus cloud layers located off the Cal-ifornia coast and six deeper and more variable layersover the Southern Ocean. These scenes are represen-tative of a larger set of 404 retrievals that we have doneon 63 days of satellite data. In this paper we describethe distributions of optical depth and effective radiusand their correlations, and propose a simple cloud modelto account for the correlations between t and reffob-served in 19 of the 31 cloud layers.In section 2 we briefly review the Nakajima and Na-kajima (1995) algorithm and discuss the uncertaintiesin the retrievals of t and reff, while in section 3 wedescribe the datasets used in this paper. In section 4 wepresent the resulting retrievals for thin and thick clouds,with joint probability densities and mean layer statistics.In section 4b we use in situ aircraft measurements tovalidate reffretrievals in thicker clouds. Section 5 con-tains a summary and discussion of the results.2. Retrieval algorithm and uncertaintiesa. Retrieval algorithmWe use the algorithm of Nakajima and Nakajima(1995) to retrieve cloud optical depth and cloud dropleteffective radius given radiance measurements in Ad-vanced Very High Resolution Radiometer (AVHRR)channels 1 (centered at 0.63 mm), 3 (centered at 3.74mm), and 4 (centered at 10.8 mm). The algorithm firstcalculates model radiances in the same wavelengthrange as AVHRR channels 1 and 3 given a three layeratmosphere with specified cloud-base height, cloudthickness, visible optical depth, and droplet effectiveradius. The absorption coefficient for the overlying at-mosphere is specified using LOWTRAN-7, with tem-perature and vapor soundings taken from the midlatitudesummer profile (Kneizys et al. 1988). The satellite-mea-sured radiances are then used to choose the best-fit setof model parameters, minimizing the difference betweenobserved and model radiances while iteratively cor-recting for the emitted radiation in channel 3, as de-scribed in Nakajima and Nakajima (1995). The param-eter set used in the model and the view and solar zenithangles included in the calculation are listed in Table 1.b. Retrieval error estimatesUncertainty in the retrieved values of cloud opticaldepth and cloud droplet effective radius arises fromthree sources of error.1) Approximation error: the error introduced by as-sumptions made in the forward radiative transfermodel, including the assumption of a vertically uni-form profile of reffwith height and of climatologicalvapor and temperature profiles. Nakajima et al.(1991) used radiances calculated from simulatedclouds for which reffincreased linearly with heightto test their t,reffretrieval and found the retrievedreffto be roughly 10% less than the cloud-top reffofthe vertically inhomgeneous cloud. The same retriev-al overestimated t by 1%–5%, with uncertainty inthe atmospheric water vapor profile adding 1%–3%to the t and reffuncertainties. Recently Brenguier etal. (2000) performed a similar calculation with arange of cloud profiles and found their retrieved reffto decrease from between 100% to 80% of the mod-eled cloud-top value as the cloud-top reffis increasedfrom 6 to 18 mm.2) Independent pixel approximation (IPA) error: thiserror is introduced by neglecting the horizontal ra-diative transfer between pixels. For overcast scenes,Chambers et al. (1997) estimate that this error forthe retrieval of t can range between 10% for 30-mLandsat pixels to less then 5% for 5.7-km pixels ata solar zenith angle of 638. As Table B1 indicates,solar zenith angles for the AVHRR scenes were gen-erally between 508 and 608, with satellite view anglestypically less than 308. Given the 1-km AVHRR na-dir pixel size we anticipate that the IPA error shouldfall within this range of values. The IPA introducesa negative bias in retrievals of t. The effect of theIPA on reffhas not been investigated for cloud re-trievals, but we anticpate that smaller channel 3 ra-diances due to inhomgeneous clouds would result inlarger values of retrieved reff. The Southern Oceanclouds, which were more irregular and had lowercloud fractions, may suffer from larger IPA errors.3) Measurements (retrieval) error: this is dominated bythe uncertainty of sensor calibration and digitization,estimated for t at about 15% for our range of t andsolar angle values (Pincus et al. 1995). The AVHRR2914 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESFIG. 1. The geographical location of 25 northeastern Pacific scenesused in section 4. Markers used for each scene are explained in moredetail in section 4. N: scenes where for which t } reff; C: scenes withbimodal distribution of Nsat(see section 4 for the definition); n: sceneswith thick clouds.channel 1 radiance measurements for the five sat-ellites were calibrated following Kaufman and Hol-ben (1993).Both Han et al. (1994) and Platnick and Valero (1995)also considered the errors listed above. Platnick andValero (1995) estimated the worse-case net uncertaintyin an AVHRR retrieval of reffto be 620%, accountingfor the channel 3 measurement error, an unknown in-cloud water vapor absorption and droplet size distri-bution uncertainties. Han et al. (1994) similarly esti-mated retrieval errors for their study to be between 1and 2 mm. This is in agreement with validation resultsusing collocated aircraft measurements. Nakajima andNakajima (1995) and Platnick and Valero (1995)showed differences between satellite and in situ esti-mates of the effective radius differ by roughly 10%,with good qualitative agreement in regions of varyingreff. In a set of comparisons with a ground-based mi-crowave radiometer and a pyranometer, satellite retriev-als of reffand liquid water path lwp by Han et al. (1995)show lwp agreement within ł20% and reffagreementto within approximately 10%.3. DataWe have selected 25 scenes of 1 km 3 1 km localarea coverage AVHRR data with dimensions of ap-proximately 256 km 3 256 km that are representativeof a set of roughly 400 similarly sized retrievals, drawnfrom 67 different orbital swaths. The 1987 data werearchived as part of the First International Satellite CloudClimatology Program (ISCCP) Regional Experiment(FIRE) (July 1987), while data from June and July 1994and 1995 were archived and processed at the Universityof British Columbia. The location for these scenes isshown in Fig. 1 and Table B1; we will also discuss sixSouthern Hemisphere summer scenes from the SouthernOcean near Tasmania, acquired during the SouthernOcean Cloud Experiment II (SOCEX II, February 1995;see Table B2).For each scene, completely overcast and clear pixelsare identified using the spatial coherence analysis tech-nique of Coakley and Bretherton (1982). We limited ourretrievals to the cloudy pixels for which the standarddeviation of 2 3 2 arrays of the AVHRR channel 4brightness temperature was below 0.5. For the north-eastern Pacific scenes the fraction of cloudy pixels meet-ing this threshold exceeds 80%; for the more variableSouthern Ocean scenes the fully cloudy fraction falls tobetween 30% and 50%. Failure to exclude pixels withlarger spatial variances results in the retrieval of anom-alously large droplets. This is due to a combination oftwo effects: 1) lower reflectivity due to cloud inho-mogeneity and 2) an overestimate of the channel 3 ther-mal emission due to sea surface contamination of thepixel. When the erroneous thermal emission is subtract-ed from the channel 3 radiance, the resulting low re-flectivity produces a mistaken estimate of larger clouddroplets.4. Resultsa. Distributions of optical depth and effective radiusFigure 2 shows log–log contour plots of reffversus tfor scenes 1, 4, 5, and 24 of Fig. 1. Shown in the figureis a mean regression line (solid) of the formlog(r ) 5 b log(t) 1 a.eff(1)The mean regression coefficients b and a for each 2563 256 pixel scene are found by averaging 100 individualregressions of sets of independent pixels, subsampledto remove spatial correlation. We determine the corre-lation length for the t and refffields using the two-dimensional semivariogram (second-order structurefunction):2S (Dx) 5^[t(x 1Dx) 2 t(x)] &,2(2)where Dx is the (x, y) separation vector between twopixels, and ^&represents an average over all pixels sep-arated by Dx.The autocorrelation length scale for the scene is thentaken as the distance over which S2reaches its maxi-mum; typically this is 7–10 km for both the t and refffields (Isaaks and Srivastava 1989; Szczodrak 1998).The random subsets are selected with pixels separatedby the autocorrelation length, and a linear regression isthen performed on each of these subsets of roughly 3001OCTOBER 2001 2915SZCZODRAK ET AL.FIG. 2. Log–log contour plots of reffand t from four cloudy scenes. The contours are of the frequency densityh, where hD logtD logreffdenotes the number of pixels with optical depth and effective radii in the range (logt, logt9,logt 1Dlogt; logreff, log , logreff1Dlogreff). The dot–dashed, labeled lines are isolines of lwp9reffgiven by (3c); the dotted labeled lines are isolines of Nsatgiven by (3b). Also shown on each panel is the linearleast squares regression line (solid), with the slope b and its uncertainty. Clouds with both high (t,reff) correlationsand b ł 0.2 are denoted by a N in Fig. 1.pixels yielding the individual regression coefficients thatare averged. Subsampling in this way minimizes theeffect of local spatial correlations on the fit, so that theresulting correlation reflects scenewide behavior. Theresulting mean slope and intercept a and b with uncer-tainties are shown in Fig. 2. We also calculate the Pear-son’s linear correlation coefficient for each subsampledfit, assuming uncorrelated errors of 15% for t and 20%for reffas discussed in section 2b. The correction for thebias introduced into the regression by the log–log trans-formation is negligible for these scenes (Jansson 1985).As the solid line in each of the four panels of Fig. 2indicates, the t,reffscatterplots in each of these fourpanels are well fit by a relationship of the form reff}t1/5. This is also true of 13 other scenes of the 25 shownin Fig. 1 (marked with square boxes), which all yieldmean log–log regression slopes b 5 1/5 within the sam-pling uncertainty, with correlation coefficients greaterthan 70%. Of the 404 images analyzed for this region,40% satisfy these two the criteria.In appendix A we review relationships between theoptical depth t, the effective radius at cloud top reff, theliquid water path lwp, and the number concentration Nthat hold for an idealized layer cloud in which cloudliquid water content (lwc) increases linearly with heightand cloud droplet number concentration N is constantwith height:1/5 22/5 1/5r 5 aN t (3a)eff 1 sat1/2 1/2 25/2N 5 a t r ł kN/ˇb, (3b)sat 1 efflwp 5 a tr . (3c)0efThe definitions for the coefficients a0and a1and theirexpected range of variability are given in appendix A,along with that of k, an empirical parameter that relatesreffto the volume mean radius rvoland b, which describesthe rate of increase of lwc with height. The very simplecloud model described by (3) predicts that t,reffretriev-als for clouds with low variability in the quantityshould exhibit the power-law relationship be-1/5 21/2aN1sattween t and reffobserved in Fig. 2. Below we will focuson relative variability of Nsatand lwp, and use satellite-retrieved values of reffas estimates of cloud-top reffin2916 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESFIG. 3. Histograms of t,reff, Nsat, and lwp for the four scenes of Fig. 2.(3). As mentioned in section 2b, this produces an un-derestimates of reffin clouds for which reffis increasingwith height. Assuming uncorrelated errors in t and reff,the impact on the derived quantities in (3) would be a10% underestimate lwp and a 25% overestimate in Nsatgiven a 10% underestimate in reff.The assumptions in (3) that number concentration isroughly constant with height in the cloud and that liquidwater content increases linearly with height are consis-tent with aircraft and balloon observations of both pre-cipitating and nonprecipitating stratocumulus clouds inthe north Atlantic, northeastern Pacific, and mid-Atlan-tic (e.g., Nicholls 1984; Caughey and Kitchen 1984;Austin et al. 1995; Khairoutdinov and Kogan 1999;Brenguier et al. 2000). A linear increase of liquid watercontent is also found in the mean lwc profiles of manynumerical simulations of layer clouds, including modelsbased on higher-order closure (e.g., Wang and Wang1994) and two-dimensional large eddy simulations withbin-resolved microphysics (e.g., Khairoutdinov and Ko-gan 1999). There are also cases, however, in which boththe assumptions of approximately constant droplet num-ber concentration and a near-linear vertical profile ofliquid water content are invalid. For example, decoupledcloud layers, and clouds in which droplets at cloud topare being removed by evaporation, violate both theseassumptions and break the connection between Nsatandcloud-top number concentration assumed in (3).To help characterize these cloud layers we have in-cluded isolines of Nsat5 t1/2and lwp 5 a0t reff1/2 25/2ar1efin Fig. 2 and in the histograms for these scenes (Fig.3), with values for a0and a1given in appendix A. Sev-eral studies have shown that the retrieval of lwp via (3c)from both AVHRR and Geostationary Operational En-vironmental Satellite (GOES) imager measurements oft and reffcompares well with the surface microwavemeasurements of stratocumulus clouds of Han et al.(1995) and Greenwald et al. (1999). For example,Greenwald et al. (1999) report a root-mean-squared dif-ference of 17 g m22between GOES imager and surfacemicrowave data in a cloud layer with lwp values be-tween 0 and 200 g m22.In situ and satellite comparisons for Nsatare moredifficult, because they require simultaneous knowledgeof the cloud microphysics (N, k) and the bulk thermo-dynamics (b,a1). Han et al. (1998b) compare satelliteretrievals of a related quantity (the column number con-centration Ncor product of the droplet number N andlayer thickness H), for four aircraft flights in marinestratocumulus, and show 2 factor of 2 agreement be-1OCTOBER 2001 2917SZCZODRAK ET AL.tween in situ and retrieved Ncvalues. In light of theuncertainties itemized in section 2b and appendix A andthe assumptions underlying the relation between t,reffand the droplet concentration N in (3), we will focushere on contrasts between Nsatand lwp variability, rec-ognizing that large variations in Nsatcould originate fromfluctuations in cloud microphysics or departures fromthe assumptions underlying (3c), while small Nsatvari-ance could arise from compensating fluctuations in someor all of N, k, and b.In Fig. 3 we compare the t,reff, Nsat, and lwp distri-butions for the scenes shown in Fig. 2. The mean liquidwater paths for scenes 1, 4, 5, and 24 range from 80 to116gm22, which are typical of thinner clouds observedby ground-based and satellite microwave radiometermeasurements for this region (Han et al. 1995; Zuidemaand Hartmann 1995). The Nsatdistributions for all fourscenes are nearly normal, with values of skewness de-fined as3(N 2 N )sat satB 5 , (4)3sranging from 1023to 1022, or 5–10 times smaller thanthat for the lwp distributions. The lwp and optical depthdistributions for scenes 1, 4, and 5 are positivelyskewed, and are well fit with either gamma or lognormaldistributions. In contrast, scene 24 is characterized bya unimodal Nsatdistribution and bimodal distributionsof t,reff, and lwp. Images of these retrieved values (notshown) show that the scene contains two distinct regionswith lwp maxima of 33 and 151 g m22but similar Nsatdistributions with modes of 78 and 83 cm23, respec-tively.While the power-law relationship between t and reffshown in Fig. 2 occurs frequently, the majority of the400 scenes for which retrievals have been performedhave more complex t,reffvariability. Figures 4–7 showrepresentative examples of these more complex clouds.Figures 4a and 4b shows two scenes (9 and 10) that aresimilar to the bimodal scene 24 of Fig. 2d, with spatiallydistinct regions with differing reffdistributions. To high-light this spatial separation we have shown images ofthe two scenes in Figs. 4c and 4d, with the pixel colorsassigned according to whether the Nsatvalues are greateror less than 85 and 30 cm23, respectively. These twoscenes differ from scene 24 in that the Nsatstatistics aredistinct in each subregion, as indicated by the bimodalNsatdistributions in Fig. 5c. The contrasting joint t,reffdistributions are consistent with other AVHRR retrievalsshowing sharp contrasts between cloud properties across100-km regions (e.g., Nakajima and Nakajima 1995).Scenes 7 and 8 (not shown) also exhibit this kind ofspatially distinct variation in Nsat.Figures 6 and 7 show a third type of t,Nsatvariabilityfound in thicker clouds, defined as layers with modeoptical depths larger than 20. In these four layers themean liquid water path exceeds 100 g m22, while themean refffor each scene is between 8.5 and 12.5 mm.Scene 14, from 16 July 1987, is also shown as a scat-terplot in Nakajima and Nakajima (1995; their Fig. 17,panel B-3). This scene has pixels with reff. 15 mm,for t , 20, but smaller reffvalues at larger t. As Na-kajima and Nakajima (1995) remark, the absence ofpixels with both large lwp and large reffmay be due tothe removal of water by precipitation in thick cloud withlow number concentrations, where droplets are largeenough to permit collision–coalescence. In contrast,however, scene 10 (Fig. 4) shows retrieved pixels witheffective radii larger than 20 mm at optical depths great-er than 20 and lwp . 270gm22.The means of the distributions of t,reff, lwp, and Nsatfor all 25 scenes are summarized in Fig. 8. We distin-guish in the figure between the 17 scenes for which thelinear regression of reff, t, as described for Fig. 2, yieldsb 5 0.2 within the uncertainty of the fit, and other clouds(circled scene numbers in Fig. 8). The criterion that b5 0.2 within the regression uncertainty is met by 17 ofthe 25 scenes in Fig. 1. Mean optical depths for thesescenes range from 7.8 to 20, while mean effective radiivary from 6.6 to 13 mm, with larger mean optical depthsgenerally corresponding with larger mean reff. In con-trast, mean values of t and reffare uncorrelated for theeight (circled) scenes which for which b – 0.2.Figure 9 shows the means of the two derived quan-tities Nsatand lwp together with the standard deviationfor each quantity. As expected, for all scenes in whichregression yields reff} t1/5, the Nsatstandard deviation(sNsat) normalized by the mean is more than three timesthe size of the normalized standard deviation of lwp.The trend for these scenes is for higher values of Nsatto occur in thinner cloud. There is little systematic geo-graphical variation in Nsatvalue with distance offshore;the location of scenes with Nsat. 200 cm23ranges from100 to more than 1000 km offshore. There is also noobvious retrieval bias with view angle or optical depth:both large and small normalized Nsatdeviations occuracross the entire range of t and lwp, and across a meanview angle range of 58–488 (see Table B1).b. In situ measurementsIn this section we present coincident aircraft and sat-ellite measurements of stratocumulus cloud layers ob-served during six flights flown to the west of Cape Grim,Tasmania, as part of SOCEX II. Boundary layer cloudsduring SOCEX II were typically cumulus penetratinginto stratocumulus, with the solid cloud layer being ap-proximately 400–600 m thick, and overall cloud layerbeing approximately 1 km thick. The six analyzed SO-CEX II flights are listed in Table B2, together with thetime of the matching satellite overpass. Thermodynamicprofiles on most flights show discontinuities indicatingdecoupling between cloud and subcloud layers. Allflights with the exception of flight 7 reported drizzle,with the largest rainfall rates reaching 50–60 mm day212918 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESFIG. 4. (a) and (b) As in Fig. 2, but for two scenes with strongly bimodal distributions of Nsat. The heavy dashed Nsatisoline in eachcontour plot separates large reffpixels from small reff. These pixels are then colored black (small reff) and gray (large reff) in the maps beloweach contour plot [(c) and (d)]. Clouds with bimodal Nsatdistributions are denoted by a C in Fig. 1.and 20–30 mm day21on flights 8 and 11, respectively.Effective radii reported here are calculated using drop-size distributions measured using the forward scatteringspectrometer probe (FSSP) and 2DC probes on boardthe F27 aircraft as described in Boers et al. (1996).Sampling and sizing errors in the FSSP introduce un-certainties of at least 10% in droplet radius and 10% innumber concentration (Baumgardner and Spowart1990).Figure 10 shows a vertical profile of the flight pathof a typical SOCEX II flight mission (flight 10); SOCEXflight missions consisted of several horizontal stacksflown at different levels within a cloud layer. Flightslasted 3–4 h, with the horizontal in-cloud legs covering30–35 km over 7 min. The aircraft drifted with the wind,so that during a typical flight of 150 min a vertical plane30–35 km long was repeatedly sampled. As Fig. 10illustrates, there were typically six to eight encounterswithin 100–200 m of cloud top, ranging from samplesof several seconds on ascent/descent to 7-min horizontalsamples on level legs.The flight pattern shown in Fig. 10 allows us to obtaina picture of the vertical structure of the cloud layer overthe 150-min period of a stack set. Based on this, we areable to make a rough estimate of the cloud adiabaticityb for these clouds by taking the composite vertical liquidwater profile from the stacks and comparing it to anadiabatic profile taken from the mean cloud base. Figure11 shows these estimates of b from the six SOCEX IIflights plotted against total droplet number concentra-tion N (with its standard deviation), along with singlesounding estimates for the FIRE region of section 4ataken from Austin et al. (1995). The error bars for theSOCEX II flights are calculated from the uncertainty indlwc/dz given by the least squares fit to the stack profiledata, neglecting the FSSP sizing errors and the uncer-tainty associated with the variable cloud base charac-teristic of the SOCEX clouds. The figure shows that the1OCTOBER 2001 2919SZCZODRAK ET AL.FIG. 5. Histograms of t,reff, Nsat, and lwp for the two scenes of Fig. 4.FIG. 6. As in Fig. 2, but for four scenes with thicker cloud layers. Thick cloud scenes are denotedwith a n in Fig. 1.2920 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESFIG. 7. Histograms of t,reff, Nsat, and lwp for the four scenes of Fig. 6.FIG. 8. Scatterplots of the mean values of reff, t,Nsat, and lwp for all 25 northeastern Pacific scenesof Fig. 1. The eight scenes with circled numbers correspond to scenes with bimodal Nsatdistributionsor mean t . 20.1OCTOBER 2001 2921SZCZODRAK ET AL.FIG. 9. As in Fig. 8, but for lwp vs Nsat. Solid lines indicate 6 one standard deviation about themean value.FIG. 10. Aircraft vertical cloud sampling pattern (horizontal stacks) in SOCEX II flight 10.Time is in min from the start of the flight. Flight average cloud top and cloud base are indicatedby the dashed lines.2922 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESFIG. 11. Scatterplot of cloud adiabaticity b 5 (dlwc/dz)/(dlwcadiab/dz) vs cloud droplet number con-centration for the six SOCEX II flights (numbers) and for FIRE soundings (C) [using FIRE data fromAustin et al. (1995)]. Vertical error bars from the least squares fit to the stack profile of lwc, horizontalerror bars are 6 one standard deviation.SOCEX II b values tend to be slightly lower than thosetaken from FIRE, which is consistent with the obser-vations of both penetrative cumulus convection and de-coupling in the thermodynamic profiles, and drizzle re-sulting in the removal of cloud liquid water for theseclouds. These values can be compared with measure-ments of 0.4 , b , 0.7 for wintertime stratocumulusin the SOCEX II region (Boers et al. 1996).Figure 12 shows the aircraft estimates of reffand ttogether with log–log contour plots of the coincidentAVHRR retrievals. In each panel, the values of reffator near cloud top have been denoted with circles, whileaircraft measurements of reffbelow cloud top are shownas asterisks. We have assigned an approximate opticaldepth to each aircraft reffvalue by integrating the meansounding produced by all the level legs in a stack. Inclouds with large horizontal inhomogeneities such asthese, this will be a poor estimate of t in any particularpixel. Given the irregular cloud base, penetrating cu-mulus turrets, and convective lines observed in theselayers, we expect that the actual column-integrated op-tical depths might be significantly higher than those in-dicated by the time-averaged mean values presentedhere.Given this limitation in the aircraft t estimates, Fig.12 does show good agreement between satellite and insitu estimates of cloud-top reff. For flights 6 and 11 thesatellite retrieves large values of t with little variabilityin reff. For flight 6 the mean values for the in situ andsatellite-estimated reffare within 0.7 mm (13.9 mm forAVHRR vs 13.2 for the aircraft). Similar agreement isfound in flight 11 (14.0 vs 13.5 mm). In flights 7 and8 the cloud-top values of reffvaried between 5–10 (flight7) and 9–12 mm (flight 8), a range that is encompassedin the satellite retrievals for these layers, although theaircraft sampled significantly thinner cloud than the sat-ellite retrievals for flight 7. For flight 9 the aircraft-measured values of reffare smaller than 10 mm, in con-trast to 12-mm radii found by the satellite in the thinestpixels subject to retrieval. In this broken cloud, as inflight 7, it is likely that some flight-leg pixels have beenrejected because of failure to meet the spatial coherencecriteria. Subvisible cirrus contamination is also a pos-sibility for this cloud; there are patches of cirrus awayfrom the flight line in this image, although no highclouds are discernible in the Coakley–Bretherton dia-gram for flight 9.Overlying each of the contour plots in Fig. 12 is anNsatisoline equal to the mean Nsatfor the scene. Forflights 9 and 10 this isoline is also within 1%–2% ofthe value of the log–log regression, which gives an av-erage slope of b 5 0.2 6 0.01 in each case. Althoughwe expect these complicated, precipitating cloud layersto both have larger uncertainties in the retrieved valuesof t and reffand larger variability in b and k, we canqualitatively compare the distributions of Nsat51OCTOBER 2001 2923SZCZODRAK ET AL.FIG. 12. Log–log contour plots of the retrieved t,refffrequency density h (as in Fig. 2) for the sixSOCEX II flights. Open circles denote aircraft-measured reffat or near cloud top, while asterisks denotereffmeasurements below cloud top. Nsatisolines for the mean Nsatfor each scene are also shown with themean Nsatvalues indicated in each panel.t1/2shown in Fig. 12 with the in situ measure-1/2 25/2ar1efments of N and b given in Fig. 11, assuming that N 5Nsat/k. Taking k 5 0.8, using the mean b for eachˇbflight gives satellite-retrieved values of N for flights 6,10, and 11 of N 5 77, 176, and 76 cm23, respectively.These are within the range of N 6 sNgiven in Fig. 11,although the maximum in situ number concentrationsobserved in flights 10 and 11 (250 and 110 cm23) arelower than the maximum inferred from Nsat(350 and181 cm23, respectively). As noted in section 4a we ex-pect satellite-retrieved reffto underestimate the cloud-top reff, which would produce overestimates of Nsat.There is little agreement between the in situ N and theNsatestimates for the other flights, again possibly dueto a sample mismatch between the Lagrangian aircrafttrajectories and the location of successfully retrievedpixels.5. DiscussionSeveral authors [e.g., Han et al. (1994, p. 493); Na-kajima and Nakajima (1995, p. 4057); Han et al.(1998a)] have noted the tendency for satellite-retrievedreffto increase with increasing t in thinner clouds, anddecrease with increasing t in optically thicker clouds.In light of Figs. 2, 4, and 6, we can be more quantitativeabout the variability of t and reffin thinner clouds (t ,20): we find that a power law of the form reff} t1/5captures much of the variability between t and reffin19 of the 31 layers we presented. Twenty-five of thesescenes are drawn from a larger sample of 404 imagestaken from 67 satellite swaths during several summersin the northeastern Pacific. Of these 404 scenes, 40%also show a power-law relationship with an exponentequal to 1/5 within the uncertainty of the log t–log reffregression. Other scenes in this ensemble with similarvalues of optical depth, cloud fraction, satellite view,and solar zenith angles show larger reffvariability andlittle correlation between t and reff.Recently, Brenguier et al. (2000) have inferred dis-tributions of cloud droplet number concentration fromremotely sensed t,reffretrievals. Using an aircraft-mounted two-channel radiometer flown in the mid-At-lantic during the Second Aerosol Characterization Ex-periment, they retrieved t and reff, while simultaneouslymeasuring cloud thickness and droplet size spectra froma second in situ aircraft. Figure 8 of their paper mapsthe two-channel reflectances to droplet number concen-tration assuming an adiabatic liquid water content pro-file and constant number concentration in the columnbeneath the radiometer; there is good agreement be-tween the number concentration inferred from this sim-2924 VOLUME 58JOURNAL OF THE ATMOSPHERIC SCIENCESple model and the in situ samples, and a strong corre-lation of the two-channel radiance along isolines of con-stant number concentration.It is possible to make a similar interpretation of thesatellite retrievals presented here, provided that the as-sumptions of a vertically uniform mean droplet con-centration and linearly increasing mean liquid watercontent hold on the 1-km horizontal scale of the satellitepixels. In that case isolines of Nsat} t1/2map to25/2reffconstant values of kN/ . We expect both precipitationˇband entrainment to produce a negative correlation be-tween N and b (cf. Fig. 11), while Martin et al. (1994)show evidence for negative correlation between N andk, so that if the simple model holds, Nsatvariability likelyplaces only a lower bound on the variability of N inthese thinner layer clouds.In the survey of northeastern Pacific scenes fromwhich the 25 retrievals are taken we find that cloudswith mean optical depths greater than 20 typically ex-hibit lower (t,reff) correlations. As SOCEX flights 9and 10 show, however, a power law of the form reff}t1/5can at least occasionally describe variability in thick-er clouds with significant precipitation. Simultaneousmeasurements of lwp, cloud-top reff, and optical depth,combined with a quantitative description of t,reffre-trieval uncertainties using three-dimensional radiativetransfer models, would be very useful in determiningthe underlying physical constraints on the variability inlwp and cloud droplet number concentration for theselayer clouds.Acknowledgments. We are grateful to Takashi Na-kajima and Teruyuki Nakajima for providing the sourcecode for the retrieval algorithm used in this paper, alongwith numerous helpful discussions, and to Jorgen Jen-sen, whose comments improved this paper. The paperwas improved by comments from Teruyuki Nakajimaand two anonymous referees. Our research was sup-ported by grants from the Canadian Atmospheric En-vironment Service, the National Science and Engineer-ing Research Council of Canada, and the Common-wealth Scientific and Industrial Research Organisationof Australia (CSIRO). SOCEX was partly funded by theCSIRO Office for Space Science and Applications(COSSA), the Cooperative Research Centre for South-ern Hemisphere Meteorology, and CSIRO’s Multi-Di-visional Climate Change Research Program. Thanks forSOCEX work are due to Jorgen Jensen, Reinout Boers,Sunhee Lee, Dan Gogoasa, Bernard Petraitis, and otherstaff at CSIRO Atmospheric Research.APPENDIX ASimple Relations among t,reff, Nsat, and lwpSeveral authors have derived the relationship betweenthe retrieved variables t and reff, the number concen-tration N, and the liquid water path lwp for a simpleadiabatic cloud layer (e.g., Pontikis 1993; Boers andMitchell 1994). Below we adopt the notation of Bren-guier et al. (2000), using an additional term (b) to rep-resent the departure of the liquid water content profilefrom adiabatic.Denoting the number concentration at height z withradii between r, r 1 dr as n(r, z)dr, the definitions forthe total droplet concentration N, effective radius reff,the volume mean radius rvol, the liquid water contentw, liquid water path lwp, and the optical depth t aregiven by‘3n(r, z)rdrE0r (z) 5 , (A1a)eff‘2n(r, z)rdrE0‘N(z) 5 n(r, z) dr, (A1b)E0‘43w(z) 5 r n(r, z)prdr, (A1c)w E301/33w(z)r (z) 5 , (A1d)vol124pN(z)rwHlwp 5 w(z) dz, (A1e)E0H ‘2t 5 Qn(r, z)prdrdz, (A1f)ext EE[]00where rwis the density of water, z is the height abovecloud base, H is the cloud thickness, and Qextis themean scattering efficiency.We will assume that the mean number concentrationis roughly constant with height in the 1 km2AVHRRpixel (N(z) 5 N), and that the mean liquid water contentincreases linearly with height at some fraction b of itsadiabatic valuew(z) 5 bCz,w(A2)where Cw, termed the moist adiabatic condensate co-efficient by Brenguier et al. (2000), is a weak functionof temperature and pressure, varying between 1.8 and2.25 3 1023gm23m21in the temperature range 280–290 K at 980 hPa. Substituting (A2) into (A1f), inte-grating, and using the empirical relationship betweenand found by Martin et al. (1994) gives a slightly33rreff volmodified version of Eq. (12) of Brenguier et al. (2000):32/3 2/3 1/3 5/3t 5 pQAb (kN) H , (A3)ext5where A 5 Cw/(4/3)prw, and the coefficient k was foundby Martin et al. (1994) to vary from 0.67 to 0.80 60.07with higher values associated with maritime air masses.1OCTOBER 2001 2925SZCZODRAK ET AL.Sensitivity tests using simple cloud models as wellas comparisons with in situ aircraft measurements in-dicate that the retrieved value of reffis typically within85%–90% of its value at cloud top (e.g., Nakajima andKing 1990; Nakajima and Nakajima 1995). Making theapproximation that this retrieved value, reff,sat, is equalto reff(H) and substituting k1/3rvol(H) for reffin (A3) gives212 5t 5 aNr, (A4a)1 sat eff5Aa 5 , (A4b)13pQextwith Nsat5 kN/.ˇbThe relationship for lwp is calculated similarly, sub-stituting (A2) into (A1e), integrating, and using the re-lationship between rvol, reff, and w at cloud top to obtainlwp 5 a tr , (A5a)0ef10rwa 5 . (A5b)09QextValues for Nsatand lwp in this paper are calculatedassuming Qext5 2 and Cw5 Cw(temperature 5 285 K)5 2 3 1023gm23m21; 285 K is a representative cloud-base temperature for both the northeastern Pacific andSOCEX clouds.APPENDIX BSatellite Scene LocationsTABLE B1. Year/month/day and satellite number and orbit for the images of Fig. 1.Scene No. Date/orbit ID Lat (8N) Long (8W) Zenith (8) View (8)12345678910111213141516171819202187/07/16 N10-429487/07/09 N10-419487/07/12 N10-423794/07/17 N11-2994387/06/23 N10-395194/07/16 N11-2992987/06/23 N10-395187/07/14 N10-426595/06/15 N12-2122087/07/14 N09-1332494/07/17 N11-2994394/07/17 N11-2994387/07/14 N09-1332487/07/10 N10-420894/07/16 N11-2992987/07/12 N10-423787/07/16 N10-429494/07/12 N11-2980287/07/31 N10-450794/06/23 N11-2980294/07/16 N11-2983035.0328.6634.8935.3830.6124.4038.4028.1924.8629.2338.2737.9333.8430.6023.6333.5332.1134.8929.2925.5440.322132.462124.622129.882127.932131.162121.112128.622117.652131.892126.942124.202124.932123.842121.002126.022124.712133.302129.882126.122128.832132.1254.650.854.951.264.864.850.154.363. N11-2971795/07/14 N14-2026687/07/07 N10-416687/07/04 N10-412338.2034.1129.6231.522132.302124.342124.812121.4259.951.348.352.948.045.511.910.0TABLE B2. 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