“The Erupt Model”, or “Some of what we don’t know about the physics of conduit flow”

1. “The Erupt Model”, or “Some of what we don’t know about the physics of conduit flow” Karl L. Mitchell and Lionel Wilson Planetary Science Research Group, Environmental Science Department, I. E. N. S., Lancaster University, Lancaster LA1 4YQ, U. K. Volcanic Eruption Mechanism Modelling Workshop University of New Hampshire, Durham, NH, November 2002

2. Introduction • Early work of Lionel Wilson: terrestrial v extraterrestrial volcanism. • Inspired by desire to understand volcanic eruptions on other planets. • Feedback between planetary science and terrestrial volcanology. • Figure below shows the (effectively) unique position of the Earth in styles of activity.

3. Introduction (2) • The Erupt model • Very old model, dating back to the 1970. • Allows important physics to be implemented easily. • Similarity with results from e.g. Valentine and Wohletz. • Implementation • Simultaneous solution of conservation of mass, momentum & energy. • 4th order Runge-Kutta iterative method.

4. Conservation laws • Momentum • Inertia + driving pressure + friction + gravity = 0 • Poor for non-Newtonian or inhomogeneous flow. • Doesn’t work well for Strombolian eruptions, foamy flows, high crystal content magmas (rhyolites) or eruptions through inclined conduits. • Mass • What happens to magma densities when volatiles are dissolved? • Exsolved volatiles seem to appear out of nowhere.

5. Conservation laws (2) • Energy • Adiabatic treatment (e.g. Buresti and Casarosa). • Enthalpy + K. E. + P. E. = 0, therefore a net loss of heat. • Assumes no heat transfer across conduit walls, so all frictional losses become heat within the magma/volatiles. • Massive temperature loss (often 100 K or more) near vent for magmas with high volatile content over a few hundred metres, typically taking a few seconds. • Fragmentation of clasts might possible due to sudden cooling? • Also, potential for frictional heating. • Perhaps evidence in field (e.g. Taupo Ignimbrite – ca. A. D. 186) or lab (G. R. O. S. S. experiments at Lancaster or Bristol).

6. Governing equations • Two solutions • (1) Give area profile as a function of height to get pressure and temperature. • Either a parallel-sided fissure or cylinder, or based on a dyke shape. • Normally either “choked” to Mach 1 at the surface, or exit pressure equals surface atmospheric pressure. • Often results in ridiculous wall stresses. • (2) Give pressure profile as a function of height to get temperature and pressure. • Used for pressure-balanced solution -> approximates de Laval nozzle. • Numerator and denominator must equal unity at transonic point. • Can only occur if erosion and/or deformation results in zero wall stress. • Reality? • Systems may evolve to nearly pressure balanced.

7. Supersonic compressible flow • Problem: vent cross section becomes massive. • Reason: • Velocity of eruption orthogonal to direction of flow exceeds Mach 1. • Expansion wave is limited to the speed of sound, so > Mach 1 lateral expansion is impossible. • Solution: limit dr/dz < 1/Mach.

8. Supersonic compressible flow (2) • Problem: shocks and rarefaction waves appear. • Prandtl-Meyer expansion. Flow is refracted inward. • Extremely difficult to solve or simplify. • Multiple Prandtl-Meyer cells. • Applies both above and below the surface.

9. Supersonic compressible flow (3)

10. Supersonic compressible flow (2)

11. Conduit inclination a b c d • Not all eruptions vertical eruptions from the top of a magma chamber. • Plinian style activity is generally central vent. • Flank eruptions are common and and tend to result in more effusive activity. • Hence we have designed erupt so that it can be run in an inclined conduit. • Bubblerise and erupt. • We have developed an iterative method using erupt to model flow properties and bubblerise to model bubble nucleation, growth and accumulation. • Result: get some indication of regimes in which “slug flow” occurs. • Indication of when Hawaiian, Strombolian and Effusive activity occur. Results from sugar-solution analogue experiments. From left to right: (a) vertical bubbly flow, (b) vertical slug flow, (c) bubbles show tendency to rise at 6o, (d) bubbles accumulate at 29o.

12. Conclusions • Conduit inclination is a first-order influence on eruptive style. • Even just a few degrees can drastically affect flow properties. • It is likely that many effusive, Strombolian and Hawaiian style eruptions are through inclined conduits. • This explains how effusive activity is possible on Mars, given the diagram at the start of the talk, without speculating on mechanisms of almost complete volatile depletion. • Further work is needed in the study of supersonic jets. • Prandtl-Meyer expansion must be incorporated into models of magma ascent if we are to understand evolved explosive eruptions. • Further work on the feedback between conduit flow and conduit shape is also necessary.