Formation of propagation pattern in two‐phase flow systems with application to volcanic eruptions

Abstract

The hydrodynamic behaviour of a two‐phase system with different densities and viscosities is investigated, assuming that both phases are continuous phases and incompressible and that no phase change occurs. Mass and momentum conservation equations describe the relative motion of the two phases under the gravitational force when the inertia term is not negligible. Using the derived equations the stability of a spatially constant structure along gravity is examined in the one‐dimensional case for an infinite system without boundaries, where the basic state is taken as constant velocity and volume fractions. Two special cases are considered: the first is a gas‐liquid system where the density of one of the phases is neglected; the second is a deformable solid‐liquid system where the densities of the two phases are taken to be the same except for the calculation of the buoyancy force. The linear stability analysis shows that a small perturbation always generates a propagating wave which grows unstably with a characteristic wavelength. Since the spatially uniform structure of a two‐phase system is always unstable, the growth pattern should be observed in nature if the growth rate and spatial scale of the system are suitable. The dependence of the wavelength and growth rate on parameters is divided into two regimes. In one regime the wavelength, which equals the compaction length, and growth rate depend on the typical pore size. In the other regime, on the other hand, they do not depend on the typical pore size but only on the kinematic viscosity of the liquid phase and the volume fraction, given the frictional law. Which regime a real system chooses depends on a nondimensional parameter. The first case (gas—liquid system) is applied to magma effusion in an eruption‐style volcano. The system consisting of a silicate liquid phase and a gas phase can have a wavelength from metres to tens of metres and a growth rate from the order of less than a second to 1 week depending on viscosities, typical pore size and volume fraction. As a result, the rhythm and discreteness of magma effusion observed at volcanic eruptions may be explained by the mechanism presented here. The theory provides a relation between the observed period of the magma effusion rate, the mixture exit velocity of gas plus magma and the volume fraction of magma, given the frictional law. Using the relation, the mixture exit velocity is estimated from the observed period for the 1986 Izu‐Oshima eruption. The applicability to partially molten systems in the mantle and the core and basic assumptions are discussed. Copyright © 1988, Wiley Blackwell. All rights reserved