Weakly nonlinear wave propagation in magma containing crystals and bubbles

A comprehensive understanding of seismic P-wave propagation characteristics in magma is essential for the estimation of the volume fraction of bubbles and viscosity of magma. Previous theoretical studies on wave propagation in magma have predominantly focused on linear regimes. In this study, we theoretically investigate P-wave propagation in magmas containing melts, bubbles, and crystals, with a focus on the weak nonlinearity of P-waves. We begin by developing two fundamental equations incorporating an apparent viscosity that simplifies the complex rheological effects induced by bubbles and crystals: the first equation describes the effective two-fluid model, which distinguishes between bubbles and an effective medium comprising melts and crystals, and the second describes the three-phase mixture model representing a combination of all three phases. Subsequently, we derive a nonlinear wave equation, specifically, the Korteweg–de Vries–Burgers (KdVB) equation for low-frequency long waves, from these foundational equations. We then analyze the influence of the volume fraction of bubbles and crystals in magma on the speed of P-waves, as well as the dependence of propagation characteristics on the angular frequency of P-waves. Finally, we numerically solve the KdVB equation and discuss the waveform change in the magma chambers.