1. Mathematical models of conduit flows during explosive eruptions (Kamchatka steady, transient, phreatomagmatic) Oleg Melnik †‡, Alexander Starostin †, Alexey Barmin †, Stephen Sparks ‡, Rob Mason ‡ † Institute of Mechanics, Moscow State University ‡ Earth Science Department, University of Bristol

2. Conduit flow during explosive eruption Schematic view of the system  xt Flow regimes and boundaries. Homogeneous from magma chamber until pressure > saturation pressure. Constant density, viscosity and velocity, laminar. Vesiculated magma from homogeneous till magma fragmentation.  Bubbles grow due to exsolution of the gas and decompression.  Velocity and viscosity increases.  Flow is laminar with sharp gradients before fragmentation due to viscous friction. Fragmentation zone or surface (?).  Fragmentation criteria. Gas-particle dispersion from fragmentation till the vent.  Turbulent, high, nonequilibrium velocities.  subsonic in steady case, supersonic in transient.

3. Kamchatka steady Barmin, Melnik (2002) Magma - 3-phase system - melt, crystals and gas. Viscous liquid  (concentrations of dissolved gas and crystals). Permeable flow through the magma. Account for pressure disequilibria between melt and bubbles. Fragmentation due to critical overpressure. 2 particle sizes after fragmentation.

4. Mass conservation equations (bubbly zone)  - volume concentration of gas (1-  ) - of condensed phase  - volume concentration of crystals in condensed phase=const  - densities, “m”- melt, “c”- crystals, “g” - gas c - mass fraction of dissolved gas = k p g 1/2 V - velocities, Q - discharge rates for “m”- magma, “g” - gas

5. Momentum and bubble growth  - mixture density - resistance coefficient (32 - pipe, 12 -dyke) k(  ) - permeability  g - gas viscosity p- pressure “s”- mixture, “m”- condensed phase, “g”- gas

6. Equations in gas-particle dispersion F - interaction forces: ”sb” - between small and big particles “gb” - between gas and big particles  s,  b - volume fractions of particles  - volume fraction of gas in big particles

7. Fragmentation wave

9. Ascent velocity vs. chamber pressure

10. Model of vulcanian explosion generated by lava dome collapse (Kamchatka transient)

11. Assumptions Flow is 1D, transient Velocity of gas and condensed phase are equal Initial condition - V = 0, pressure at the top of the conduit > p atm, drops down to p atm at t =0 Two cases of mass transfer: equilibrium (fast diffusion), no mass transfer (slow diffusion) Pressure disequilibria between bubbles and magma

12. Mechanical model

13. Results of calculation (eq. case)

14. Discharge rate and fragmentation depth equilibrium mt. no mass transfer

15. Model of phreatomagmatic eruption Model of phreatomagmatic eruption Model of the magma flow in the conduit with influx from the porous layer Model of magma flow in the conduit Model of water flow in the porous layer

16. Transient Problem Transient Problem __ magma discharge __ water influx __ fragmentation front

17. Conclusions Set of models for steady-state and transient conduit flows. Realistic physical properties of magma. New fragmentation criteria. Explanation of transition between explosive and extrusive eruptions, intensity of volcanic blasts, cyclic variations of discharge rate during phreatomagmatic eruptions.