Abstract
As the second attempt at unifying treatment of atmospheric particle systems, this paper further examines shape characterization of atmospheric particles. First, to support the theoretical framework developed in Part I, methods for studying non-spherical particles are reviewed. It is argued that these different methods can be unified under fractal geometry through the generalized power laws given in Part I. Empirical power-laws for hydrometeors scattered in literature since 1935 are summarized and reevaluated in terms of fractals. Second, generalization from self-similar to self-afflne particles is discussed. Self-affinity of atmospheric particles is exemplified by examining the exponents in the power laws between the length along a- and c-axis of ice crystals. It is argued that unlike Euclidean and self-similar particles, self-affine particles do not have a simple dimensional relation between original particles and their projections; the relation for projection of self-similar particles and Mandelbrot’ thumb rules for intersection respectively set the lower and upper bound. Using published data, self-affine particles are shown to exist in the atmosphere. The existence of self-affine particles in turn calls for instruments that can simultaneously measure mass, area and maximum diameter (or their equivalents).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arntt, W. P., et al. (1995), Direct airborne sampling of small ice crystals and the concentration and phase of haze particles, 9th symp. on Met., Observ, & Instr., 415–418.
Auer, A., and D. Veal (1970), The dimensions of ice crystals in natural clouds, J. Amos. Sci., 27: 919–926.
Baron, P. A. and K. Willeke (1993), Aerosol fundamentals, In Aerosol Measurement, Principles, Techniques and Applications. (Eds) K. Willeke and Baron, 8–22 pp.
Bashkirova, G., and T. Pershina (1964), On the mass of snow crystals and their fall velocities, Tr. Gl. Geo fiz. Observ., 165: 83–100.
Berry, M. V. (1989), Falling fractal flakes, Physica D 38, 29–31.
Bruintjies, R. et al. (1987), An examination of double-plate ice crystals and the initiation of precipitation in continental cumulus clouds, J. Atmos. Sci., 44: 1331–1349.
Cai, J. et al. (1993), Comparision of size and morphology of soot aggregates as determined by light scattering and electron microscope analysis, Langmuir, 9: 2861–2867.
Colbeck, L. et al. (1989), The dynamics and structure of smoke aerosols, J. Aerosol Sct., 20: 875–878.
Davis, C. L. and A. H. Auer (1974), Use of isolated orographie clouds to estimate the accuracy of diffusional ice growth equations, Proc. Cloud Phys. Conf. 1974, Tucson, USA, 141–147.
Family, F. and T. Vicsek (1991), Dynamics of Fractal Surfaces, World Scientific, 480 pp.
Falconer, K. (1990), Fractal geometry: mathematical foundations and applications, John Wiley & Sons, Chichester.
Forrest, S. R. and Witten (1979), Long-range correlations in smoke-particle aggregates, J. Phys. A12, L109-L117.
Herdan, G. et al. (1960), Small Particle Statistics, Butterworths, London.
Heymsfield, A. J. (1972), Ice crystal terminal velocities, J. Atmos. Sci., 29: 1348–1357.
Heymsfield, A. J. and M. Kajikawa (1987), An improved approach to calculating terminal velocities of plate-like crystals and graupel, J. Atmos. Sci., 44: 1088–1099.
Kajikawa, M., and A. J. Heymsfield (1989), Aggregation of ice crystals in cirrus, J. Atmos. Sci., 46: 3108–3121.
Katrinak, K. A. et al. (1993), Fractal geometry of carbonceous aggregates from urban aerosol, Environ. Sci. Technol., 27: 539–547.
Kaye, B. H. (1978), Specification of the ruggedness and/or texture of a fine particle profile by its fractal dimension, Powder Technol., 21: 207–213.
Kaye, B. H. (1981), Direct Characterization of Finaparticles, John Wiley & Sons. New York.
Klingen, H. J. and P. Roth (1989), Size analysis and fractal dimension of diesel particles based on REM measurements with an automatic imaging system, J. Aerosol Sci., 20: 861–864.
Knight, N. C., and A. J. Heymsfield (1983), Measurement and interpretation of hail density and terminal velocity, J. Atmos. Sci., 40: 1510–1516.
Korvin, G. (1992), Fractal Models in Earth Science, Elsevier, Amsterdam.
Langleben, M. P. (1954), The terminal velocity of snowflakes, Q. J. Met. Soc., 80: 174–184.
Lesaffre, F. (1989), Characterization of aerosol aggregates through fractal parameters, Effects due to humidity, J. Aerosol Sci., 20: 857–860.
Liu, Y. (1995), On the generalized theory of atmospheric particle systems, Adv. Atmos. Sci., 12: 419–438.
Lovejoy, S. and D. Schertzer and A. A. Tsonis (1987), Functional box-counting and multiple elliptical dimensions in rain, Science, 235: 1036–1038.
Locatelli, J. D., and P. V. Hobbs (1974), Fall speeds and masses of solid precipitation particles, J. Geophy. Res. 79(15): 2185–2197.
Magill, J. (1991), Fractal dimension and aerosol particle dynamics, J. Aerosol Sci., 22: S165–168.
Magono, C., and C. W. Lee (1966), Meteorological classification of natural snow crystals, J. Fac. Sci., Hokkaido Univ., Ser 7 II, 321–335.
Malinowski, S. P. and L., Zawadzki (1993), On the surface of clouds, J. Atmos. Sci., 50: 5–13.
Maloney, D. J. et al. (1995), A new approach to determine external surface and volume of irregularly shaped particles, Aerosol Sci. and Technol., 22: 60–72.
Mandelbrot, B. B. (1967), How long is the coast of Britain? Statistical self-similarity and the fractal dimension, Science, 165: 636–638.
Mandelbrot, B. B. (1983), The fractal geometry of nature, W. H. Freeman and Company, New York.
Mitchell, D. L. (1996), Use of mass- and area-dimensional power-laws for determining precipitation particle terminal velocities, J. Atmos. Sci., 53: 1710–1723.
Mitchell, D. L. et al. (1990), Mass-dimensional relationships for ice particles and influence of riming on snowfall rates, J. Appl. Meteor., 29: 153–163.
Mitchell, D. L. et al. (1996a), Modeling cirrus clouds, Part I: Treatment of bimodal size spectra and case study analysis, J. Atmos. Sci., 53: 2952–2966.
Mitchell, D. L. et al. (1996b), Modeling cirrus clouds. Part II: Treatment of radiative properties, J. Atmos. Sci., 53: 2967–2988.
Nakaya, U., and T. Terada (1935), Simultaneous observations of the mass, falling velocity and form of individual snow crystals, J. Fac. Sci. Hokkaido Univ., Ser. II, 1, 191–201.
Naumann, K. H. and H. Bunz (1991), Aerodynamic properties of fractal aerosol particles, J. Aerosol Sci., 22: S161–164.
Nyeki, S. and Colbeck, I. (1994), The measurement of the fractal dimensions of individual in situ soot agglomerates using a modified Millikan cell technique, J. Aerosol Sci., 25: 75–90.
Pruppacher, H. R., and J. D. Klett (1978), Microphysics of Clouds and Precipitation, D. Reidel.
Rogak, S. N. and Flagan, R. C. (1993), The mobility and structure of aerosol agglomerates, Aerosol Sci. and Technol., 18: 25–47.
Samson, R.J. et al. (1987), Structural analysis of soot agglomerates, Langmuir, 3: 272–281.
Sander, L. M. (1985), Growth by particle aggregation, in Scaling Phenomena in Disordered System, (Eds.) R. Pynn and A. Skjeltorp, 31–48 pp.
Scheoeder, M. (1991), Fractals, Chaos, Power Lawa. W. H. Freeman and Company, New York.
Schmidt-Ott, A. (1988a), In situ measurement of the fractal dimensionality of ultrafine aerosol particles, Appli. Phys. Lett., 52: 954–956.
Schmidt-Ott, A. (1988b), New approaches to in situ characterization of ultrafine agglomerates, J. Aerosol Sci., 19: 553–563.
Schmidt-Ott, A. et al. (1990), Scaling behaviour of physical parameters describing agglomerates, J. Aerosol Sci., 21: 711–717.
Takayasu, H. (1990), Fractals in the Physical Sciences, Joun Wiley & Sons, Chichester.
Tohno, S. and K. Takahashi (1990), Morphological and dynamic characterization of Pb fume particles undergoing Brownian coagulation, J. Aerosol Sci., 21: 719–732.
Wu, Z. and Colbeck, I. (1991), Measurement of the dynamic shape factor of fractal clusters, J. Aerosol Sci., 22: S239–240.
You, L., Y. Chen and T. Wang (1987), A study on the relationship between the masses and dimension of snow crystals, J. Academy Meteor. Sci., 2: 197–201.
Zhang, H. X. et al. (1988), In situ optical structure factor measurements of an aggregating soot aerosol, Langmuir, 4: 867–871.
Zikmunda, J. and G. Vali (1972), Fall patterns and fall velocities of rimed ice crystals, J. Atmos. Sci., 29: 1334–1347.
Zikmunda, J. (1972), Fall velocities of spatial crystals and aggregates, J. Atmos. Sci., 29: 1511–1515.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yangang, L. On the unified theory of atmospheric particle systems part ii: self–affine particles. Adv. Atmos. Sci. 14, 369–388 (1997). https://doi.org/10.1007/s00376-997-0057-2
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s00376-997-0057-2