The hyperbolic distribution of particle sizes

Abstract

Many natural collections of small particles, such as cosmic and terrestrial dust, mineral and organic particles suspended in sea water, and fine sediments, have a size distribution well formulated by the equation N = Kx−c, where N is the number of particles larger than size x and K and c are constants. A number of equations are derived for calculating the interesting parameters characterizing hyperbolic distribution, which are illustrated by new Coulter Counter measurements of dust and sea water. An appendix lists useful equations, generalized for different size parameters.

A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters

“…Of these approximations, the power law is the most widely used for optical and ecological purposes [Jonasz, 1983;Briucaud et al, 1981;Stramski and Kiefer, 1991;Boss et al, 2001;Twardowski et al, 2001]. Many processes and laws in Nature conform to the power law PSD, over many orders of magnitude, such as dust, sea spray, bubble concentrations, sediments [Monahan and Zietlow, 1969;Bader, 1970] and organismal energetics [Chisholm, 1992;Seibel, 2007;Packard and Birchard, 2008]. While even a cursory examination of PSDs indicates a complexity that cannot be fully captured by a simple approximation, the power law has proven to be a good first-order approximation over several orders of magnitude for oceanic particles.…”

“…[6] The PSD slope estimated from the power law model provides information on the relative concentration of small to large particles: the steeper the slope, the greater proportion of smaller particles and the flatter the slope, the greater proportion of larger particles [Bader, 1970;Jackson et al, 1997]. For natural waters, PSD slopes generally vary from 3 to 5 with most values between 3.5 and 4 [Jonasz, 1983] and can be up to 7 for very small submicron particles [Loisel et al, 2006].…”

“…[5] Several mathematical descriptions of the PSD have been used over the years, including power law (or "Junge type"), sum of lognormal functions, Weibull distribution, gamma function, and characteristic vectors (principal components) [Junge, 1963;Bader, 1970;Kitchen et al, 1975;Risovic, 1993;Jonasz and Fournier, 1996]. Of these approximations, the power law is the most widely used for optical and ecological purposes [Jonasz, 1983;Briucaud et al, 1981;Stramski and Kiefer, 1991;Boss et al, 2001;Twardowski et al, 2001].…”

“…[2] The particle size distribution (PSD) represents the relationship between the size of particles and their concentrations in a polydispersion of particles and has been widely used to characterize marine particles [Bader, 1970]. Marine PSDs have relevance to diverse fields within oceanography, from biology to sedimentology, and play an important role in climate studies.…”

A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters

“…PSDs are commonly described using a Junge-type model, where the distribution is fit with a hyperbolic curve with the slope parameter s c describing the shape (i.e., steep or flat). For marine particle populations, s c typically varies between 2.5 and 5 for a differential distribution (as opposed to a cumulative distribution), with most values falling within 3.5-4.0 [Bader, 1970;Kitchen, 1977]. PSDs have also been described using an exponential model [Carder, 1970], a multisegmented hyperbolic distribution [Brown and Gordon, 1974;Harris, 1977;Kitchen et al, 1982], a Gaussian function superimposed on a multisegmented hyperbolic distribution [Jonasz, 1983], and a two-component distribution proposed by Risovic [1993], where the populations of small and large particles are each represented by a twoparameter gamma function.…”