Two-phase dynamics of volcanic eruptions: compaction, compression and the conditions for choking
Résumé
Volcanic eruptions involve high-speed turbulent flows of gas and magma mixtures in which peak gas velocities often approach 600 m s−1. Such speeds are well in excess of the slow mixture sound speeds of approximately 150 m s−1 predicted by pseudo-gas theory, in which the gas and magma phases are assumed to be locked to each other; eruptions thus appear to be highly super-sonic. Indeed, flow in the volcanic conduit will choke when it reaches the sound speed and to attain supersonic velocities one must invoke special conduit nozzle shapes. However, the pseudo-gas approximation is a long-wavelength assumption, while choking and acoustic shocks are a short-wavelength effect; thus, the approximation is likely inapplicable for the choking phenomenon. To allow for non-pseudo-gas effects, such as phase separation, in the short-wavelength limit requires a more complete treatment of two-phase eruptions. We therefore develop and explore a model for two-phase, high Reynolds number flow of a compacting suspension of magma particles in a compressible gas. Flow properties, such as mixture density, are controlled both by gas content as well as gas compressibility, both of which vary according to different processes of compaction and compression, respectively. The two phases of the mixture separate because of their different densities, and the interaction forces (turbulent drag and inertial exchange) can be complex. The model is used to examine acoustic-porosity wave propagation and the development of shocks or choking structures in a volcanic conduit. Sound waves in separable mixtures are highly dispersive with fast waves propagating at the pure gas sound speed at small wavelengths, slow waves travelling at the pseudo-gas speed at long wavelengths, and pure attenuation and sound blocking at intermediate wavelengths. As shock fronts in gas density develop they become increasingly short wavelength features and thus will only cause choking if the maximum speed in the flow reaches the pure gas sound speed. Non-linear, finite-amplitude steady-state models of eruptions in a volcanic conduit show that compaction occurs over the magma particle gravitational deceleration height, and either suppresses gas expansion for fast eruptions, or isopycnally collapses the gas volume near the base of the erupting column for slow eruptions. Once compaction ceases, the gas expands toward a shock structure or choking point, which is coincident with a rapid gas acceleration and a high-speed vent eruption. Increased turbulent drag between the gas and particles suppresses compaction effects but greatly sharpens the shock front at the choking point. Although the standard pseudo-gas models predict such choking to occur at low velocities, the full two-phase theory always has choking occur when the gas reaches the pure gas sound speed, in keeping with the sound-speed dispersion analysis. Therefore, the full two-phase theory predicts choking to occur at the pure gas sound speed, which (for water vapour at the relevant high temperatures) is about 700 m s−1. Eruption velocities of 600 m s−1 are therefore fully consistent with the limit imposed by this choking condition, and no special conditions to obtain supersonic eruptions, such as nozzled conduit geometries, are necessary.
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