Nonlinear visco-elastic volcanic model and its application to the recent eruption of Mt. Unzen

A volcanic model is proposed. The model consists of a magma reservoir, which is a spherical hole in an elastic medium and is being replenished from below, and a conduit which is a circular cylinder in a visco-elastic medium. The model is expressed by a system of two coupled nonlinear differential equations. We solve the system as an initial value problem with a source term corresponding to the magma supply. The supply rate is expressed by a bell shape function with a constant part. This model works as follows: Supplied magma to the reservoir at a given rate increases pressure inside the reservoir. The increased pressure expands the conduit, through which the magma is discharged with the rate determined by Poiseuille's flow law. Although the system eventually goes to a steady state in which the in- and out-flows are balanced, the difference between response times of the reservoir and conduit causes the oscillation of the discharge rate. This means that a volcanic activity oscillates. For a pulse input expressed by a bell shape function, the number of oscillations increases when thr amount of inflow magma increases. This model is applied to the recent volcanic activities of Mt. Unzen. The model well simulates the semi-long term (few years) variation of the discharge rate of lava with a plausible replenishment rare of magma. Geometrical factors of the volcanic system (the volume and radius of the reservoir, the maximum and average radii of the conduit) and some physical constants (the maximum excess pressure, viscosity of the country rock and Mogi's coefficient) are estimated under assumptions on the following quantities; density difference between the country rock and magma, rigidity of the country rock, bulk modulus and viscosity of magma. All quantities calculated except the viscosity agrees with corresponding values determined by independent methods within a factor of two or three. We discussed physical background of these results together with the model itself. (C) 2000 Elsevier Science B.V. All rights reserved.