Particle size distributions and the sequential fragmentation/transport theory applied to volcanic ash

Abstract

The assumption that distributions of mass versus size interval for fragmented materials fit the log normal distribution is empirically based and has historical roots in the late 19th century. Other often used distributions (e.g., Rosin‐Rammler, Weibull) are also empirical and have the general form for mass per size interval: n(l) = klα exp (−lβ), where n(l) represents the number of particles of diameter l, l is the normalized particle diameter, and k, α, and β are constants. We describe and extend the sequential fragmentation distribution to include transport effects upon observed volcanic ash size distributions. The sequential fragmentation/transport (SFT) distribution is also of the above mathematical form, but it has a physical basis rather than empirical. The SFT model applies to a particle‐mass distribution formed by a sequence of fragmentation (comminution) and transport (size sorting) events acting upon an initial mass m′: n(x, m) = C ∫∫ n(x′, m′)p(ξ)dx′ dm′, where x′ denotes spatial location along a linear axis, C is a constant, and integration is performed over distance from an origin to the sample location and mass limits from 0 to m. We show that the probability function that models the production of particles of different size from an initial mass and sorts that distribution, p(ξ), is related to mg, where g (noted as γ for fragmentation processes) is a free parameter that determines the location, breadth, and skewness of the distribution; g(γ) must be greater than −1, and it increases from that value as the distribution matures with greater number of sequential steps in the fragmentation or transport process; γ is expected to be near −1 for “sudden” fragmentation mechanisms such as single‐event explosions and transport mechanisms that are functionally dependent upon particle mass. This free parameter will be more positive for evolved fragmentation mechanisms such as ball milling and complex transport processes such as saltation. The SFT provides better fits to many types of volcanic ash samples than does the log normal curve. Modeling of the SFT shows its similarity to the log normal curve on size frequency histograms; it differs by its variable skewness controlled by γ. Skewed distributions are typical of many volcanic ash samples, and characterization of them by the SFT allows interpretation of eruptive and transport mechanisms.

An example of enhanced tephra deposition driven by topographically induced atmospheric turbulence

“…the portion deposited by normal single-particle sedimentation and the portion experiencing accelerated deposition) are each described by two sub-populations, rather than one. Although log-normal distributions are not necessarily the most appropriate description of a population of fragmented grains (Wohletz et al 1989), it is clear from the variable asymmetry of the Cha1 grain-size distributions that a single statistical distribution cannot describe every sample. Thus, the occurrence of distinct grain-size populations, even at sites upwind of Yate, appears to be real.…”

“…). We used the SFT software (Wohletz et al 1989) to fit a spline curve to the sieved mass-intervals and then fitted lognormal distributions, aiming to achieve a fit to the data with a residual of <10 % (Fig. 3).…”

A review of volcanic ash aggregation

“…7) through deconvolution of measured bulk sample polymodal particle size distributions (Wohletz et al 1989). Aggregate fallout during the 18 May 1980 Mount St. Helens eruption was linked to the formation of a distal mass deposition maximum centred on ~ 330 km from the volcano (Carey and Sigurdsson, 1982).…”

Dynamics and energetics of bubble growth in magmas: Analytical formulation and numerical modeling

“…If temperatures approach the vitrification point, then brittle fragmentation into angular pyroclasts may occur instead of disruption into liquid spray [Alidibirov, 1994;Wohletz et al, 1989]. Even if gas leaves the foam through interpore channels without bursting it into fragments [Cashman and Mangan, 1994], vaporization cooling of the melt can lead to adiabatic vitrification without the necessity for radiative cooling to the atmosphere.…”