A mathematical model of vulcanian eruptions

Abstract

The problem of an expansion wave propagating into a saturated magma is solved and used as a model of a vulcanian eruption. In our model for an explosive eruption we assume that a drop in pressure leads to the exsolution of magmatic volatiles. Initially the exsolved vapours create bubbles in the magma. We model the subsequent part of the exsolution process in which a foam is created; this is believed to be an essential feature of explosive volcanic eruptions. The foam also has the advantage that it can be modelled as a mixture or ‘pseudo‐gas’ without slippage between the phases. Eventually the foam breaks up and becomes volcanic ash. Assuming that the exsolution of vapour is given by Henry's law, that the temperature is constant, that the magma and vapour have equal velocities, and neglecting wall friction and gravitational effects, an analytic solution for pressure, velocity, and vapour fraction is obtained for the expanding mixture in a constant area duct. The exit velocity u for the mixture is u= (ø0RT0)1/2In(P0/p), where ø0 is the original mass fraction of dissolved vapour, R is the gas constant for the vapour, T0 the constant temperature, and p0/p the pressure ratio across the expansion. With ø0= 1 per cent, T0= 1000°K, and p0/p= 100 we find u0= 300ms‐1, consistent with observations for vulcanian eruptions. Copyright © 1990, Wiley Blackwell. All rights reserved