Optics of Fractal Clusters Such as Smoke

Abstract

Scalar or vector light of wavelength 2/k strikes N small refracting and absorbing spherules, each of radius a, which have coagulated into a sparse random cluster with fractal dimension D (for smoke, D 1 78). It is assumed that ka<< 1 but the cluster size R = aNlID may be larger than the wavelength. Using a mean field theory it is shown that multiple scattering is negligible for all N if D <2, and becomes important when N-(ka) -D(D -2) if D> 2. Cross-sections are calculated as functions of N, D and the complex refractive index of the spherules. If D < 2 the scattering cross-section per spherule rises with N and saturates when kR >> 1, at a value exceeding that of an isolated spherule by a factor of order (ka) -D; if D> 2 the same quantity increases as N l ' -2/D/(ka) 2 . For D<2 the absorption cross-section is N times that of a solitary spherule. These results are very different from those for spherules coagulating into compact solid spheres (D = 3), and are important for the optics of sooty smoke, with Dr1-78, which is used as an example throughout. IntroductionSmoke is formed as small soot spherules [1] which stick together when they touch and thereby coagulate. Further aggregation leads to sparse random clusters with a fractal dimension of about 78 2, 3]. This fractality has not been included in studies of the optics of smoke. The omission is important for the scattering and extinction of radiation by smoke lofted into the atmosphere by large fires, such as those produced by multiple nuclear explosions [4][5][6][7]. Our purpose here is to present a theory of the optics of smoke in which the importance of the fractal structure is fully brought out. Implications for the nuclear winter predictions will be discussed in a separate publication.The central simplifying principle will be that the wavelength of the illuminating radiation, which is about 05 for the visible and 10 p in the infrared, greatly exceeds the spherule radius a, which is about 0005 u-0 0 5 p for sooty smoke. In other words, the observation scale is much bigger than the inner fractal scale. However, the size R of a cluster (that is, the outer fractal scale) increases with aggregation ( § 2) as the number of spherules in a cluster increases, from a (when N= 1) to values which may considerably exceed (when N is several thousand). We shall calculate the optical cross-sections as functions of N and show that the optics of clusters with R<<A2 and R>>A can be very different. Moreover the differences depend strongly on whether the fractal dimension D is less than 2 or greater than 2. Downloaded by [Monash University Library] at 00:

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