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Malvagi, F, Byrne RN, Pomraning GC, Somerville RCJ.  1993.  Stochastic Radiative Transfer in a Partially Cloudy Atmosphere. Journal of the Atmospheric Sciences. 50:2146-2158.   10.1175/1520-0469(1993)050<2146:srtipc>2.0.co;2   AbstractWebsite

A radiation treatment of the broken-cloud problem is presented, based upon various stochastic models of the equation of radiative transfer that consider the clouds and clear sky as a two-component random mixture. These models, recently introduced in the kinetic theory literature, allow for non-Markovian statistics as well as both vertical and lateral variations in the cloudiness. Numerical results are given that compare different models of stochastic radiative transport and that point out the importance of treating the broken-cloud problem as a stochastic process. It is also shown that an integral Markovian model proposed within the atmospheric radiation community by Titov is entirely equivalent to a special case of a simple low-order differential model. The differential form of Titov's result should be easier than the integral form to implement in any general circulation model.

McFarquhar, GM, Iacobellis S, Somerville RCJ.  2003.  SCM simulations of tropical ice clouds using observationally based parameterizations of microphysics. Journal of Climate. 16:1643-1664.   10.1175/1520-0442(2003)016<1643:ssotic>2.0.co;2   AbstractWebsite

A new bulk parameterization of the dependence of ice cloud effective radius (r(e)) on ice water content (IWC) is developed using in situ observations of the size and shape of ice crystals in tropical anvils. This work extends previous parameterizations because information about the number, size, and shape of ice crystals with diameters smaller than 100 m m is included and in that a range of possible fit coefficients, rather than single values, is given to reflect the fact that r(e) can vary significantly about its mean parameterized value. The parameterization is implemented in the Scripps single column model (SCM), and simulations of tropical clouds over the Atmospheric Radiation Measurement ( ARM) program's tropical western Pacific (TWP) site and over the Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE) domain are conducted. Sensitivity studies determine how the range of possible fit coefficients, which reflects the uncertainty in the parameterization of r(e), relates to uncertainties in modeled cloud radiative forcings (CRFs). When r(e) is chosen one or two standard deviations higher or lower than the mean parameterized value, temporally averaged shortwave CRFs can differ by up to 17.7 W m(-2) from that value estimated from the mean parameterized r(e), the difference depending on the time period and location; differences in longwave CRFs are substantially less. When other uncertainties in the parameterization are accounted for, such as those based on the observed numbers of smaller crystals, CRFs can differ by up to 25 W m(-2) from that determined by the base parameterization. When r(e) is randomly chosen for each simulation time within one or two standard deviations of the most likely r(e) for that IWC, shortwave CRFs can still differ from that of the base simulation by up to 13.9 W m(-2), with an enhancement of shortwave reflection of up to 4.9 W m(-2) observed on average. Therefore, the average of a series of such simulations may not equal a simulation of average conditions, a finding that may have important ramifications. Both interactive simulations, where changes in cloud heating rates feed back upon predicted cloud masses, and noninteractive simulations, where changes in heating rate do not feed back upon cloud mass, are performed in order to determine how and why different parameterizations affect the CRFs. It is shown that differences in longwave heating rates, associated with different versions of the parameterization, alter the mass of ice and liquid water produced at various levels, this change in cloud mass in turn affects the CRF. This change can either amplify or reduce the change in CRF associated with the more direct effect of varying the r(e) parameterization, namely, that smaller particles reflect more shortwave radiation given the same mass content. The amount of liquid water present in low clouds is an important indicator of whether changing ice cloud microphysical properties will have an important effect on CRF.