Abstract. If climate models produced clouds having liquid water amounts close to those observed, they would compute a mean albedo that is often much too large, due to the treatment of clouds as plane-parallel. An approximate lower-bound for this "plane-parallel albedo bias" may be obtained from a fractal model having a range of optical thicknesses similar to those observed in marine stratocumulus, since they are more nearly plane-parallel than most other cloud types. We review and extend results from a model which produces a distribution of liquid water path having a lognormal-like probability density and a power-law wavenumber spectrum, with parameters determined by stratocumulus observations. As the spectral exponent approaches -1, the simulated cloud approaches a well-known multifractal, referred to as the "singular model", but when the exponent is -5/3, similar to what is observed, the cloud exhibits qualitatively different scaling properties, the socalled "bounded model". The mean albedo for bounded cascade clouds is a function of a fractal parameter, 0 << 1, as well as the usual plane-parallel parameters such as single scattering albedo, asymmetry, solar zenith angle, and mean vertical optical thickness. A simple expression is derived to determine from the variance of the logarithm of the vertically-integrated liquid water. The albedo is shown to be approximated well by the plane-parallel albedo of a cloud having an "effective" vertical optical thickness, smaller than the mean thickness by a factor χ(f), which is given as an analytic function of f. California stratocumulus have a mean fractal parameter (f) ≈ 0.5, relative albedo bias of 15%, and an effective thickness 30% smaller than the mean thickness (χ ≈ 0.7). For typical observed values of mean liquid water and (f), the effective thickness approximation gives a plane-parallel albedo within 3% of the mean albedo.
Scaling Properties of the Mean Multiplicity and Pseudorapidity Density in e−+e+, e±+p, p(
p
¯
)+p, p+A and A+A(B) Collisions
