An Optimised Model for the Terminal Velocities of Isolated Bubbles in Magma

Abstract

The rise of single, uncoupled bubbles in a Newtonian liquid medium is a well constrained phenomenon that occurs in a variety of scenarios. Their velocity is an important variable in volcanic eruption modelling, in which overpressures and fragmentation driven by bubble nucleation, growth, and degassing have been proposed as possible eruption mechanisms. Recent work has synthesised a parameterisation for rising rates across all bubble shape regimes. This thesis tests the empirical model against a larger experimental dataset and specifically within a magmatic regime, to produce an improved parameterisation for volcanic systems. This will aid in the creation of a validated quantitative model for bubble ascent in magma.

Data have been collated from ninety-one experimental studies carried out on rising bubbles. Experiments have primarily been carried out using low viscosity (<1 Pa.s) and relatively low density (<2000 kg m-3) fluids. These are poor analogues for magmatic conditions. To constrain the dataset to relevant magmatic conditions, several dimensionless numbers were used. The Reynolds number (Re), the ratio of inertial to viscous forces in the fluid, defines whether the flow will be laminar or turbulent. The Eötvös number (Eö), describing the effect of surface tension, and the Morton number (M), characterising the shape of the rising bubble. The motion of isolated bubbles rising can then be described within three distinct regimes: the spherical, ellipsoidal and spherical-cap regimes, with boundary conditions between each of these regimes identified by the dimensionless parameters. Scaling analysis of magma rheology shows that bubbles remain in the spherical regime within volcanic conduits. This is visualised on an Eö-Re plot, with bubbles occurring within silicate melts having a Re number of less than 1. Consequently, the following parameterisations are suggested for the terminal velocity of individual bubbles rising in magma:

v_b= 1/√((144〖μ_f〗^2)/(g^2 〖ρ_f〗^2 d^4 )+〖μ_f〗^□(4/3)/(〖〖0.202〗^2 g〗^□(5/3) 〖ρ_f〗^□(4/3) d^3 ))

v_b is the bubble terminal velocity, d is the diameter the bubble, ρ_f and μ_f are the fluid density and absolute viscosity respectively, and g is the acceleration due to gravity.

Constraining the experimental dataset collected to magmatic analogues within the relevant bubble rising regimes shows that highly viscous magmas, such as low temperature rhyolites, are poorly represented. Viscoelastic analogues could be used; however, these produce experimental difficulties. Further studies of isolated bubbles rising in highly viscous magmatic analogues would help optimise this equation further for use in eruption modelling.