Expansion dynamics of volatile-supersaturated liquids and bulk viscosity of bubbly magmas

Abstract

We derive expressions for the bulk viscosity of suspension of gas bubbles in an incompressible Newtonian liquid that exsolves volatiles. The suspension is modelled as close packed spherical cells and is represented by a single cell (‘cell model’). A cell, consisting of a gas bubble centred in a spherical shell of a volatile-bearing liquid, is subjected to decompression that is applied at the cell boundary, and the resulting dilatational boundary motion and driving pressure are obtained. The dilatational motion and the driving pressure are used to define the bulk viscosity of the cell, as if it were composed of a homogeneously compressible fluid. By definition, the bulk viscosity is the relation between changes of the driving pressure and changes in the resulting expansion strain rate. The bulk viscosity of the suspension is obtained in terms of two-phase parameters, i.e. bubble radius, gas pressure and the properties of the incompressible continuous liquid phase. The resulting bulk viscosity is highly nonlinear. At the beginning of the expansion process, when gas exsolution is efficient, the expansion rate grows exponentially while the driving pressure decreases slightly, which means that the bulk viscosity is formally negative. This negative value reflects the release of the energy stored in the supersaturated liquid and its transfer to mechanical work during exsolution. Later, when bubbles are large and the gas influx decreases significantly, the strain rate decelerates and the bulk viscosity becomes positive as expected in a dissipative system. We demonstrate that amplification of seismic waves travelling through a volcanic conduit filled with a volatile saturated magma may be attributed to the negative bulk viscosity of the compressible magma. Amplification of an expansion wave may, at some level in the conduit, damage the conduit walls and initiate the opening of a new pathway for magma eruption. We also consider the energy related to positive and negative bulk viscosities.