Forcing of seismic waves travelling through a bubbly magma

The idea of amplification of seismic waves in magma was first introduced by Lensky et al. (2002) who examined the compressibility and the effective bulk viscosity of a bubbly suspension that expands due to the growth of gas bubbles in a supersaturated melt. At the initial stages of growth, when diffusion is efficient and viscous resistance of the melt controls expansion, bulk viscosity is negative. Only later, when growth decelerates, it turns positive. Bulk viscosity is negative whenever the conversion rate of potential chemical energy of dissolved volatiles into expansion work is higher than the rate of dissipation of kinetic energy to heat. They suggested that when bulk viscosity is negative, part of the excess energy in the system may be converted into seismic energy resulting in the amplification of seismic waves. We have studied this possibility by examining the dynamics of pressure waves in an expanding bubbly magma. The integration of pressure oscillations with bubble growth dynamics was carried-out by following the path laid by Commander and Prosperetti (1989). They developed a wave equation for bubbly suspensions and showed that the introduction of bubbles can be accounted for by an additional term, the second time-derivative of the gas volume fraction. Combining their equation with the bubble growth model (Navon & Lyakhovsky, 1998), we now solve for the amplitude of pressure oscillations in the suspension. The expansion of the additional term leads to a wave equation which includes additional terms for damping and forcing. We solved this equation for the initial stage, following pressure drop, when growth is controlled by the viscous resistance of the melt. We used typical conditions and properties of magma in a conduit below a dome. The results show that in this case, forcing overcomes damping and the amplitude of the pressure waves increases with time. A numerical model is constructed to follow the evolution of the pressure waves during the full course of growth. We show for the first time that under appropriate conditions, pressure oscillations in an expanding, supersaturated bubbly suspension are amplified. This effect should be important for the generation of volcanic seismicity.