University of Colorado
Boulder, CO 80309
Plane-parallel radiative transfer modeling of clouds in GCMs is thought to be an inadequate representation of the effects of real cloudiness. A promising new approach for studying the effects of cloud horizontal inhomogeneity is stochastic radiative transfer, which computes the radiative effects of ensembles of cloud structures described by probability distributions. This approach is appropriate because cloud information is inherently statistical, and it is the mean radiative effect of complex 3D cloud structure that is desired.
The simplest general way to derive a stochastic solution to the radiative transfer equation is to use the order of scattering solution of the deterministic system, because this expresses the radiative response explicitly in terms of the medium optical properties. The equation can be then multiplied by the joint probability density function (pdf) and integrated to obtain the ensemble mean radiative response. The deterministic solution has integrals over transmission and two angles for each order of scattering. These high-dimensional integrals are computed with Monte Carlo integration in a procedure called backward Monte Carlo radiative transfer. The stochastic solution for the ensemble mean has in addition integrals over the distances between successive scatterings, which are the stochastic random variables. The general joint pdf would include all of these path distances and also depend on the transmission and angular variables as well. The approximation used here is that the general joint pdf can be expressed in terms of conditional pdfs involving just two successive paths. The form of these path pdfs used here is f(k|T) and f(k2|T,k1), where k is the mean path extinction and T is the path transmission. Preliminary testing (Evans 1993) with 3D log-normal multifractal fields compared with many deterministic runs indicated that pdfs describing two paths are adequate, whereas those describing single paths are not.
Path pdfs were computed from Gerber cloud probe data from the Atlantic Stratocumumulus Transition Experiment (ASTEX) for all cloudy segments longer than 2 km. Approximate two path pdfs were derived from 1D extinction traces (made from the particle surface area channel). Assuming isotropic homogeneous cloud structure, the ensemble mean albedo for four stochastic approximations was computed. The independent pixel approximation (IPA) tended to agree with the Monte Carlo stochastic method (though the approximate nature of this stochastic method is evident from lack of flux conservation). The plane-parallel method had 8-13% higher albedo, while the source closure method of Gabriel and Evans (1995) had 6-8% higher albedo.
The good agreement for ASTEX cloud data between the Monte Carlo stochastic radiative transfer method and the IPA is explained by the characteristics of the path pdfs, which are close to those corresponding to the IPA: f(k|T)=f(k), f(k2|T,k1)=d(k2-k1). In contrast, path pdfs corresponding to the often modeled array of rectangular clouds are completely different, with the pdf of the second path mean path extinction (k2) almost independent of the first path (k1) (Figure 3.1).
An appealing approach, based on the fundamental physics of radiative transfer, can thereby be formulated for future studies of radiation and inhomogeneous clouds. The spatial variability of clouds can be characterized with path pdfÕs obtained from cloud probes, mm-wave radars, and microwave radiometers. Those aspects of real cloud variability that are most important for radiative transfer should be expressed with a few key parameters (cloud fraction, etc.) In principle, fast radiative transfer methods based on these parameters could be developed to include the effect of cloud inhomogeneity in cloud radiation GCM parameterizations. Finally, there would be the need to develop methods of remote sensing cloud inhomogeneity parameters on a global basis.
References
Evans, K. F., 1993: A general solution for stochastic radiative transfer, Geophys. Research Letters, 20, 2075-2078.
Gabriel, P. M. and K. F. Evans, 1995: Domain averaged solar radiative fluxes calculated by first order closures in two dimensional media, Submitted to J. Atmos. Sci.
