Here’s a paper by Liu et al. deriving a theoretical relative dispersion based on the aerosol concentration and updraft velocity.

Analytical expression for the relative dispersion of the cloud droplet size distribution (Subscription required)

Abstract: An analytical expression that relates the relative dispersion (ratio of standard deviation to mean radius) of the cloud droplet size distribution to CCN spectra and updraft velocity is derived from adiabatic growth theory of cloud droplets. Coupled with the Twomey expression for droplet concentration, the analytical expression is used to examine the relationship of relative dispersion to droplet concentration under different combinations of CCN spectra and updraft velocities. These analytical results compare favorably with the corresponding simulations of an adiabatic parcel model. The analytical expression theoretically demonstrates that an increase in aerosol loading (CCN concentration) leads to concurrent increases in the droplet concentration and relative dispersion whereas a larger updraft velocity leads to a higher droplet concentration but a smaller relative dispersion.

Liu, Y., P. H. Daum, and S. S. Yum (2006), Analytical expression for the relative dispersion of the cloud droplet size distribution, Geophys. Res. Lett., 33, L02810, doi:10.1029/2005GL024052.

This paper has been in the queue for awhile now, and it doesn’t look like I’m going to get around to critiquing it. So here’s the abstract.

A modeling study of the effect of drizzle on cloud optical depth and susceptibility
Graham Feingold, Reinout Boers, Bjorn Stevens, and William R. Cotton

This paper examines the impact of drop spectral broadening, generated by the collection process, on the optical depth, cloud albedo, and susceptibility of marine stratocumulus clouds. The results are arrived at using (1) the output from a simple box model calculation of collection and (2) the output from an eddy-resolving model of stratocumulus clouds that explicitly represents the size distribution of the drops. It is shown that commonly used relationships for cloud optical properties developed for narrow spectra do not generally apply to spectra undergoing spectral broadening. The optical depth dependence on the drop number concentration to the one-third power is shown to be an overestimate of the optical depth when spectra broaden through collection. In addition, the cloud susceptibility dependence on drop number is shown to be larger for spectra experiencing broadening than for narrow spectra.

This is the fourth article in the spectral dispersion series. I’ll be summarizing / reviewing the 2006 paper by Lu and Seinfeld in the Journal of Geophysical Research, Effect of aerosol number concentration on cloud droplet dispersion: A large-eddy simulation study and implications for aerosol indirect forcing.

Abstract: Through three-dimensional large-eddy simulations of marine stratocumulus we explore the factors that control the cloud spectral relative dispersion (ratio of cloud droplet spectral width to the mean radius of the distribution) as a function of aerosol number concentration and the extent to which the relative dispersion either enhances or mitigates the Twomey effect. We find that relative dispersion decreases with increasing aerosol number concentration (for aerosol number concentrations less than about 1000 cm^−3) because smaller droplets resulting from higher aerosol number concentrations inhibit precipitation and lead to (1) less spectral broadening by suppressed collision and coalescence processes and (2) more spectral narrowing by droplet condensational growth at higher updraft velocity because reduced drizzle latent heating at cloud top results in increased boundary layer turbulent kinetic energy production by buoyancy and thereby stronger turbulence. Increased spectral broadening owing to increased cloud-top entrainment mixing, also as a result of increased boundary layer turbulence, is relatively insignificant compared with outcomes 1 and 2. The coefficient k, an important parameter that relates cloud droplet effective radius and volume mean radius in large-scale models, is a function of skewness and relative dispersion of the distribution and is negatively correlated with relative dispersion. Increasing k with increasing aerosol number concentration leads to maximum enhancement of the cloud susceptibility (the change of cloud optical depth due to change of cloud droplet number concentration) over that attributable to the Twomey effect alone by about 4.2% and 39% for simulated FIRE and ASTEX cases, respectively.

Continue Reading »

This is the third in a series of articles on spectral dispersion. This paper is a little off-topic, but it’s important for understanding the next post.

Lu and Seinfeld in the Journal of the Atmospheric Sciences, Study of the Aerosol Indirect Effect by Large-Eddy Simulation of Marine Stratocumulus [PDF].

Abbreviated Abstract: A total of 98 three-dimensional large-eddy simulations (LESs) of marine stratocumulus clouds covering both nighttime and daytime conditions were performed to explore the response of cloud optical depth (Ï„) to various aerosol number concentrations (Na = 50–2500 cm−3) and the covarying meteorological conditions (large-scale divergence rate and SST)… The second indirect effect is found to enhance (reduce) the overall aerosol indirect effect for heavily (lightly) drizzling clouds; that is, Ï„ is larger (smaller) for the same relative change in Na than considering the Twomey (first indirect) effect alone. The aerosol indirect effect (on Ï„) is lessened in the daytime afternoon conditions and is dominated by the Twomey effect; however, the effect in the early morning is close but slightly smaller than that in the nocturnal run.

Continue Reading »