Vesiculation process and bubble size distributions in ascending magmas with constant velocities

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Abstract
The vesiculation of magmas is the most important process which controls eruption style on terrestrial planets and physical and geological characteristics of volcanoes. This paper, focusing on the evolution of bubble size distribution, investigates the vesiculation behavior in ascending magmas with constant velocities. Taking into account homogeneous nucleation, growth by diffusion of volatile components, expansion by depressurization, and depletion of volatile components in a silicate melt by progressive vesiculation, the governing equations describing the vesiculation of magma and the evolution of bubble size distribution are formulated. The behavior of solution is controlled by three nondimensional parameters which represent the nucleation facility (related to the gas/silicate melt interfacial tension), the effective diffusivity (diffusivity divided by ascent velocity) of volatile components in a magma, and the initial saturation pressure. The governing equations are numerically solved for the limited range of variable parameters. Consequently, the bubble size distribution, the nucleation rate, and the volatile concentration in magma are obtained as functions of time (depth). The result provides the following qualitative understanding of the vesiculation process. The nucleation takes place localized in a narrow depth interval. The nucleation depth (pressure) where the nucleation rate reaches a maximum value increases with the nucleation facility and the effective diffusivity and almost linearly increases with the initial saturation pressure. The nucleation interval, the maximum nucleation rate, and the total number of bubbles nucleated decrease with the effective diffusivity. The volatile concentration in magma monotonically decreases with the progressive vesiculation, not by the nucleation of bubbles but by the growth of bubbles nucleated. The depletion factor (the ratio of the volatile concentration in magma at surface to the initial saturation) decreases with the nucleation facility and the effective diffusivity. The quantitative understanding is made by describing moments of the bubble size distribution function, the nucleation depth and interval, the maximum nucleation rate, and the depletion factor as functions of parameters representing an eruption. The results of numerical solutions are simply interpreted as the single nucleation event and the subsequent growth process using two independent quantities: the total number of bubbles per unit volume and the mean bubble radius. The results are also extrapolated to realistic values of parameters corresponding to the Plinian and sub‐Plinian eruptions. Under the condition of a constant depressurization and no interaction among bubbles, the total number of bubbles per unit volume is strongly reduced with increasing effective diffusivity, and bubbles grow according to the law in which the mean bubble radius is proportional to τ2/3, where τ represents the effective time of diffusion. These results suggest, through the silica content dependence of the diffusivity of volatile (water), that a silisic vesiculated rock generally includes a larger number of bubbles than a basic. Quantitative relationships are derived that can be used to determine two eruption parameters from moments of bubble size distribution function observed in vesiculated rocks quenched at the surface. The shape of the bubble size distribution function observed in vesiculated rocks is determined by the detail of the growth process of bubbles for the case where nucleation takes place at depth, whereas it is determined by the nucleation event for the case where nucleation near the surface is dominant. It is emphasized that derived relationships provide the important basis for estimating eruption parameters such as the nucleation depth or the hypothetical ascent velocity as well as the vesiculation history of erupted magma from the morphological analysis (determination of bubble size distribution) of vesiculated rocks.