Effects of relative motion between gas and liquid on 1-dimensional steady flow in silicic volcanic conduits: 1. An analytical method

We investigate the effects of vertical relative motion between gas and liquid on eruption styles by formulating a model for 1-dimensional steady flow in volcanic conduits. As magma ascends and decompresses, volatiles exsolve and volume fraction of gas increases. As a result, magma fragmentation occurs and the flow changes from bubbly flow to gas-pyroclast flow. In our model, a transitional region (‘permeable flow region’) is introduced between the bubbly flow region and the gas-pyroclast flow region. In this region, both the gas and the liquid are continuous phases, allowing the efficient vertical escape of gas through the permeable structure. We describe the features of conduit flow with relative motion of gas and liquid using non-dimensional numbers α, γ and ε. The parameter α represents the ratio of effects of wall friction to gravitational load, and is proportional to magma flow rate. The parameter γ represents the degree of decompression for the gas-pyroclast flow to reach the sound velocity at α = 1, and is proportional to r_c²/μ for given magma temperature and initial volatile content, where r_c is conduit radius and μ is liquid viscosity. The parameter ε is defined as the ratio of liquid–wall friction force to liquid–gas interaction force in the permeable flow region, and represents the efficiency of gas escape from magma. The values of γ and ε are determined only by magma properties and geological conditions such as liquid viscosity, magma permeability and conduit radius. We formulate a 1-dimensional steady-state conduit flow model to find non-dimensional magma flow rate α as a function of magma properties and geological conditions (e.g., γ and ε) under given boundary conditions. When the relative motion is taken into account with the assumption that magma fragmentation occurs when the gas volume fraction reaches some critical values, the pressure at the fragmentation level (P_f) decreases as the magma flow rate (α) decreases or the efficiency of gas escape (ε) increases, because gas escape suppresses the increase in the gas volume fraction accompanying magma ascent. When ε is so large that P_f is below the atmospheric pressure (P_a), the flow reaches the vent before fragmentation at low α. On the other hand, when ε is so small that P_f is greater than P_a, the flow reaches the vent after fragmentation at high α. These steady-state solutions of the flow at low and high α correspond to effusive and explosive eruptions, respectively. We present a graphical method to systematically find α. On the basis of the graphical method, a simple regime map showing the relationship between the assemblage of the solutions of conduit flow and the magma properties or the geological conditions is obtained.