On the fractal dimension of the fallout deposits: A case study of the 79 A.D. Plinian eruption at Mt. Vesuvius

We investigated the existence of a fractal law (power law) distribution of size pyroclastic fragments erupted
during the fallout phase of the 79 A.D. Plinian eruption at Mt. Vesuvius. In particular, we performed a particle
size distribution analysis on 18 white and grey pumice samples collected in six sites distributed in the SW
sector of Mt. Vesuvius. Our measurements show that the fragmentation of samples in the investigated range
(from 32 mm to 850 μm) follows a power law, guaranteeing the scale invariance of the process. The
relationship frequency-size distribution of the fragments is verified independently from the nature (i.e.,
pumices and lithics) and stratigraphic height of the considered samples in the pyroclastic deposit. Therefore,
the fractal fragmentation theory can be indicated for evaluating the relationship between the intensity of
fragmentation (fractal dimension D) and eruption energy. In this way the apparent chaotic distribution of the
particles in the fallout deposits hides a self-organized complexity revealed by the retrieved power law
distribution. We further remark that a key aspect of our analysis is the founded evidence that the fractal
dimension of the lithics is systematically greater than that of the pumices.

Bak, P., Tang, C., Wiesenfeld, K., 1988. Self-organized criticality. Physical Review, A 38,
364–374.
Bennet, J.G., 1936. Broken coal. Journal of the Institute of Fuel 10, 22–39.
Bindeman, I.N., 2005. Fragmentation phenomena in populations of magmatic crystals.
American Mineralogist 90, 1801–1815.
Borgia, A., Tizzani, P., Solaro, G., Manzo, M., Casu, F., Luongo, G., Pepe, A., Berardino, P.,
Fornaro, G., Sansosti, E., Ricciardi, G.P., Fusi, N., Di Donna, G., Lanari, R., 2005.
Volcanic spreading of Vesuvius, a new paradigm for interpreting its volcanic
activity. Geophysical Research Letters 32, L03303. doi:10.1029/2004GL022155.
Brown,W.K.,Wohletz, K., 1995. Derivation of theWeibull distribution based on physical
principles and its connection to the Rossin–Rammler and lognormal distribution.
Journal of Applied Physics 78, 2758–2763. Carey, S., Sigurdsson, H., 1987. Temporal variations in column height and magma
discharge rate during the 79 A.D. eruption of Vesuvius. Geological Society of
America Bulletin 99, 303–314.
Cioni, R., Marianelli, P., Sbrana, A., 1992. Dynamics of the A.D. 79 eruption: stratigraphic,
sedimentological and geochemical on the successions from the Somma–Vesuvius
southern and eastern sectors. Acta Vulcanologia 2, 109–123.
Cioni, R., Marianelli, P., Sbrana, A., 1999. Pyroclastic deposits as a guide for
reconstructing the multi-stage evolution of the Somma–Vesuvius caldera. Bulletin
of Volcanology 60, 207.
Cioni, R., Marianelli, P., Santacroce, R., Sbrana, A., 2000. Plinian eruptions. In: Sigurdsson,
H. (Ed.), Encyclopaedia of Volcanoes. Academic Press.
Fujiwara, A., Kamimoto, G., Tsukamoto, A., 1977. Destruction of basaltic bodies by high
velocity impact. Icarus 31, 277–288.
Gurioli, L., Cioni, R., Sbrana, A., Zanella, E., 2002. Transport and deposition of pyroclastic
density currents over an inhabited area: the deposits of the A.D. 79 eruption of
Vesuvius at Herculaneum, Italy. Sedimentology 49, 929–953.
Hartmann, W.K., 1969. Terrestrial, lunar, and interplanetary rock fragmentation. Icarus
10, 201–213.
Kaminski, E., Jaupart, C., 1998. The size distribution of pyroclasts and the fragmentation
sequence in explosive volcanic eruptions. Journal of Geophysical Research 103,
29759–29779.
Kapteyn, J.C., van Uven, M.J., 1903. Skew Frequency Curves in Biology and Statistics.
Groningen. Astronomical laboratory, Noordhoft.
Kittleman, L.R., 1964. Application of Rosin's distribution in size-frequency analysis of
clastic rocks. Journal of Sedimentary Research 34, 483–502.
Kueppers, U., Perugini, D., Dingwell, D.B., 2006. “Explosive energy” during volcanic
eruptions from fractal analysis of pyroclasts. Earth and Planetary Science Letters
248, 800–807.
Lirer, L., Munno, R., Petrosino, P., Vinci, A., 1993. Tephrostratigraphy of the 79 A.D.
pyroclastic deposits in perivolcanic areas of Mt. Vesuvio (Italy). Journal of
Volcanology and Geothermal Research 58, 133–149. Mandelbrot, B.B., 1983. The Fractal Geometry of Nature. Freeman, New York.
Papale, P., Dobran, F.,1993. Modelling of the ascent of magma during the Plinian eruption
of Vesuvius in A.D. 79. Journal of Volcanology and Geothermal Research 58,101–132.
Pepe, S., 2002. Analisi del processo di frammentazione delle eruzioni Pliniane. Il caso del
Vesuvio. Master Degree Thesis. University of Naples Federico II. pp. 51. Sammis, C.G., Osborne, R.H., Anderson, J.L., Banerdt, M., White, P., 1986. Self-similar
cataclasis in the formation on fault gouge. Pure and Applied Geophysics 123, 53–78.
Santacroce, R. (Ed.), 1987. Somma–Vesuvius. CNR Quaderni della Ricerca Scientifica, vol.
8, p. 114.
Schoutens, J.E., 1979. Empirical analysis of nuclear and high explosive cratering and
ejecta. Nuclear Geophysics Sourcebook.
Sheridan, M.F., 1971. Particle-size characteristics of pyroclastic tuffs. Journal of
Geophysical Research 76 (23), 5627–5634.
Sheridan, M.F., Barberi, F., Rosi, M., Santacroce, R., 1981. A model for Plinian eruptions of
Vesuvius. Nature 289, 282–285. Sheridan, M.F.,Wohletz, K.H., Dehn, J.,1987. Discrimination of grain-size subpopulations
in pyroclastic deposits. Geology 15 (4), 367–370.
Sigurdsson, H., Cashdollar, S., Sparks, R.S.J., 1982. The eruption of Vesuvius in A.D. 79:
reconstruction from historical and volcanological evidence. American Journal of
Archeology 86, 39–51.
Sigurdsson, H., Carey, S., Cornell,W., Pescatore, T., 1985. The eruption of Vesuvius in A.D.
79. National Geographic 1 (3), 332–387.
Sigurdsson, H., Cornell, W., Carey, S., 1990. Influence of magma withdrawal on
compositional gradients during the A.D. Vesuvius eruption. Nature 345, 519–521.
Sparks, R.S.J., Bursik, M.I., Carey, S., Gilbert, J.S., Glaze, L.S., Sigurdsson, H., Woods, A.W.,
1997. Volcanic plumes. Text Book. Wiley and Sons, Chichester, England. Turcotte, D.L., 1986a. A fractal model of crustal deformation. Tectonophysics 132,
261–269.
Turcotte, D.L., 1986b. Fractals and fragmentation. Journal of Geophysical Research 91,
1921–1926.
Turcotte, D.L., 1992. Fractals and Chaos in Geology and Geophysics. Cambridge
University Press, U.K. 216 pp.
Turcotte, D.L., 1997. Fractals and Chaos in Geology and Geophysics, 2nd edition.
Cambridge University Press, U.K. 398 pp.
Wohletz, K.H., Sheridan, M.F., Brown, W.K., 1989. Particle-size distribution and the
sequential fragmentation transport-theory applied to volcanic ash. Journal of
Geophysical Research 94, 15703–15721.