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

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 ) = kl α exp (− l β ), 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 m g , 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.