# The Effect of Droplet Size Distribution on the Determination of Cloud Droplet Effective Radius

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2 Eleventh ARM Science Team Meeting Proceedings, Atlanta, Georgia, March 9-, Also, the effects of the droplet size distribution on the retrievals of r e are compared among retrievals made from using different near-infrared (NIR) channels, i.e.,.4,.65,.5, and.75 µm, respectively. Theorectical Droplet Size Distributions In cloud modeling and radiative transfer calculations, cloud droplet size spectra are usually characterized by the lognormal or gamma distributions. The two theoretical distributions are chosen because they adhere more closely to the droplet size spectra measured by the in situ probes during many stratus and stratocumulus observation campaigns. The lognormal size distribution is defined by n log N (r) = exp [ (ln r ln r) / σlog ], () πσ r log where is the logarithmic width of the distribution as it characterizes the radius spectral dispersion, N = n(r) dr is the total droplet concentration per unit volume, and r is the median radius. The r e for the lognormal distribution is given by The standard gamma distribution used here is defined by n gam r e log = r exp (5σ / ), (4) ( b)/ b r /(ab) N r e (r) =, Γ[( b) / b] (5) ( b) / b (ab) where a = r e denotes the r e, b = σ gam denotes the spectral dispersion, and Γ is the gamma function. Relationship Between r e, w, and N In many cloud and climate studies, droplet r e is determined based on the parameterization relationship given by d e c(w / N) r = (6) where c and d are constants, generally determined by empirical fitting to the in situ observations at local cloud experiments (e.g., Martin et al. 994; Liu and Daum ). In principle, the relationship between r e and w/n can be derived from Eqs. () and (), which is given by r e 4 = πρ w r n(r)dr w (7) n(r)dr N

4 Eleventh ARM Science Team Meeting Proceedings, Atlanta, Georgia, March 9-, (a) 6 r eff (µm) 8 4 Log-normal Std. Gamma =.7 σ gam =.7 =.5 σ gam =.5 =.5 σ gam =.5 (b) W/N (g m - cm - ) 6 r eff (µm) 8 4 Observations (W/N) / Figure. Theoretical relationship between (a) re and w/n and b) re and (w/n)/ obtained based on the lognormal (solid curves) and standard gamma (dashed curves) distributions for various size spectral dispersions. Some observations as described in the text are also plotted in (b).

5 Eleventh ARM Science Team Meeting Proceedings, Atlanta, Georgia, March 9-, Reflectance (%) (a) Nadir View: (θ, θ, φ φ ) = (6o, 7 o, o ).4 µm.65 µm.5 µm µm r eff 6 µm µm 6 µm 4 µm µm Reflectance (%) (b) Forward View: (θ, θ, φ φ ) = (6o, 6o, o ).4 µm.65 µm.5 µm.75 µm Reflectance (%) (c) Backward View (θ, θ, φ φ ) = (6o, 6o, 8o ).4 µm.65 µm.5 µm.75 µm Figure. The dependence of NIR reflectances at.4,.65,.5, and.75 mm on slog for (a) nadir, (b) forward, and (c) backward viewing directions. Results are shown for constant re =, 6,, 6, and 4 mm with a cloud optical depth of. 5

6 Eleventh ARM Science Team Meeting Proceedings, Atlanta, Georgia, March 9-, (a).4-µm (b).65-µm,σ gam =.7,σ gam =.6 Difference in r e (Retrieved Input) (µm) (c).5-µm ,σ gam =.44,σ gam = (d).75-µm Input r e (µm) Figure. The dependence of the difference between the retrieved and input re on the input spectral dispersion (s) plotted as a function of the input re for (a).4, (b).65, (c).5, and (d).75 mm. The solid curves were derived based on the lognormal distribution with various input of slog; likewise, the dashed curves were derived based on the standard gamma distribution. 6

7 Eleventh ARM Science Team Meeting Proceedings, Atlanta, Georgia, March 9-, Summary This study examines the impact of droplet size distribution on the relationship between r e, w, and N and the sensitivity of remote sensing retrieved r e to different a priori assumptions of the droplet size spectral dispersion. The determination of r e, based on w and N, is found to be dependent on the droplet size distribution and its size spectral dispersion. Even with constant w and N, the determination of r e may vary by a few microns with changes of from.7 to.5 for a lognormal distribution. The remote sensing determination of r e, based on NIR reflectance measurements, is also dependent on the droplet size distribution. The dependence is generally smaller at near nadir viewing angles than at forward or backward scattering directions. With the lognormal size distribution, a change of ±.5 in spectral dispersion from =.5 may lead to a change of about ± µm in the mean of the r e retrievals at around µm. The change increases as r e increases. Further studies, based on observational spectral dispersions of the droplet size distributions, are needed to quantify their effects on the determination of r e. Acknowledgement This work was supported by the U.S. Department of Energy Grant DE-FG-97ER66 under the Atmospheric Radiation Measurement (ARM) Program. Corresponding Author Dr. Zhanquing Li, () References Charlson, R. J., S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, Jr., J. E. Hansen, and D. J. Hofmann, 99: Climate forcing by anthropogenic aerosols. Science, 55, 4-4. Han, Q., W. B. Rossow, and A. A. Lacis, 994: Near-global survey of effective droplet radii in liquid water clouds using ISCCP data. J. Climate, 7, Liu, Y. and P. H. Daum, : Spectral dispersion of cloud droplet size distributions and the parameterization of cloud droplet effective radius. Geophy. Res. Lett., 7, Liu, Y. and J. Hallett, 997: The / power-law between effective radius and liquid water content. Quart. J. Roy. Meteor. Soc.,, Martin, G. M., D. W. Johnson, and A. Spice, 994: The measurement and parameterization of effective radius of drops in warm stratocumulus clouds. J. Atmos. Sci., 5,

8 Eleventh ARM Science Team Meeting Proceedings, Atlanta, Georgia, March 9-, Miles, N. L., J. Verlinde, and E. E. Clothiaux, : Cloud droplet size distributions in low-level stratiform clouds. J. Atmos. Sci., 57, 95-. Twomey, S., M. Piepgrass, and T. L. Wolfe, 984: An assessment of the impact of pollution on global cloud albedo. Tellus, 6B,

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