A model for cyclic extrusion of a lava dome based on a stick-slip mechanism Antonio Costa 1,2 Geoff Wadge 1 & Oleg Melnik 3 (1) Environmental Systems Science Centre, University of Reading, Reading RG6 6AL, UK. (2) Istituto Nazionale di Geofisica e Vulcanologia, Naples, Italy. (3) Institute of Mechanics, Moscow State University, Moscow, Russia. 2012 EGU Meeting 22-27 April 2012 Vienna, Austria Poster session EGU2012-8619 ABSTRACT !" !" ! ! ! ! ! ! ! !" ! !! ! ! !! ! ! ! ! ! !! ! !" !"#$ !" !" ! !! ! ! !! ! ! ! !! ! ! ! ! !"#$ !" !" !" ! !"#$ ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !" ! ! ! ! ! !" where ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! Acknowledgements. This work was supported by NERC research grant reference NE/H019928/1. OEM acknowledges the support from the ERC grant no 228064 (VOLDIES), and the grant of the Russian Foundation for Basic Research (12-01-00465). 01-00465). Fig. 4: Dimensional oscillation period (scaled by multiplying for tc = 2300s) as a function of the initial conditions for the dimensionless pressure X(0) for differ- ent values of !. The dependence of the period from ξ is inversely proportional and can be calculated analytically. Lava dome eruptions are sometimes characterized by large periodic fluctuations in extrusion rate over periods of hours that may be accompanied by Vulcanian explosions and pyroclastic flows. Here we present a simple system of nonlinear equations describing a 1D flow of lava extrusion through a deep elastic dyke feeding a shallower cylindrical conduit. Stick-slip conditions depending on a critical shear stress are assumed at the wall boundary of the cylindrical conduit. By analogy with the behaviour of industrial polymers, the elastic dyke acts like a barrel and the shallower cylindrical portion of the conduit as a die for the flow of magma acting as a polymer. The model is able to reproduce some features of the observed short-period cyclicity. When we applied the model to the Soufrière Hills volcano, Montserrat, for which the key parameters have been evaluated from previous studies, cyclic extrusions with periods from 3 to 30 hours were readily simulated, matching observations. The model also explains the reduced period of cycles observed when a major unloading event occurs due to lava dome collapse. . Here R denotes the cylindrical conduit radius, g gravita- tional acceleration, ρ is the magma density, v is the verti- cal magma velocity, P is the pressure, v slip is the slip veloc- ity at the wall, Q is the mass flow rate,V dyke is the dyke volume, and γ is an effective compressibility-rigidity modulus that, in principle, can be expressed as a function of the magma bulk modulus, K, and the rigidity modulus of rocks surrounding the dyke, G. For the meaning of other symbols see Table 1. For further details on the model see Costa et al. (2012). Y X A B τ X (a) τ Y (b) 30 hours 3 hours 100 10 1 α=0.1 α=10. α=1.0 ξ= 0.15, m = 3 Period (hours) 1 0.1 0.6 0.5 0.4 0.3 0.2 0.7 0.8 0.9 X(0) Fig. 1: Simplified sketch of the investigated system consisting of an elastic dyke of height Ld, and width Wd (where Wd = 2a) that feeds a cylindrical conduit of length Lc and radius R and is filled with magma. This system is similar to an industrial polymer extruder, where the dyke acts as the barrel, the conduit as the die and the magma as a polymer. Modified after Costa et al. (2007). By analogy with the behaviour of industrial polymers, the elastic dyke acts like a barrel and the shallower cylindrical portion of the conduit as a die for the flow of magma acting as a polymer (Denliger and Hobblit, 1999). ~12 hours ~7 hours (slip law) (flow rate evolution) (pressure evolution) Dynamical system in dimensionless form Fig. 5 (left): Deformation (tilt) cycles (continuous curve) and associated seismic activity (histograms) from June 23rd to June 28th, 1997 on SHV, with the relative seismic am- plitude (RSAM) calculated for 20-minute windows using the seismic record at station MBGA. The sudden change on June 25th corresponds to the collapse of the lava dome. After Green and Neuberg (2006). Fig. 6 (right): Figure showing that the simple stick- slip mechanism we model is able to reproduce a change in the periodicity of the system similar to that showed abovr. We found that considering some values of the controlling parameters within their typical ranges it is possible to change the period of the system from ~12 hours to ~7 hours by changing the initial pressure condition by ∆X ≈ 0.15. Fig. 3: Phase diagram of equation system pre- sentend on the left. One oscillation corre- sponding to the initial conditions X(0)=0.3, Y(0)=0, is exemplified by the thick line with two turning points, A and B. Table 1. Parameters used in the simulations. Notation Description Value L Total conduit length 5000 m L c Cylindrical conduit length 1000 m R Cylindrical conduit radius 15 m V d Volume of the dyke region 10 7 m 3 ! Average overall magma density 2100 kg m -3 µ Average viscosity in the cylindrical region 3!10 10 Pa s G Rock rigidity modulus 1-6 GPa " Compressibility-rigidity parameter 0.3 GPa ! !" Flow rate at the dyke base 7350 kg s -1 (3.5 m 3 s -1 ) ! ! Characteristic pressure drop (in the cylinder) 13 MPa ! ! ! ! ! ! Characteristic time scale 2300 s ! ! ! !! ! ! Characteristic velocity scale 0.0065 m s -1 ! ! ! ! ! Cylinder aspect ratio 0.015 ! ! !"! ! !!! ! ! ! ! Dimensionless pressure parameter 0.0019 ! ! !"! ! ! ! ! ! Dimensionless compressibility parameter 0.0024-0.24 ! ! !! ! ! ! ! Dimensionless slip parameter 0.1-10 ! !" ! !" !" !! ! !! ! Dimensionless flow rate 0.76 Fig 2: Example of the time evolution of X(τ) (a) and Y(τ) (b) for the system illustrated in Fig. 3. We show only one example here because, typically, time evolution plots look similar with only the time scale changing. All the cases we explored clearly showed that the oscillation period is inversely proportional to ξ and, also decreases as α increases - Costa A., Wadge G., Melnik O. (2012) Cyclic extrusion of a lava dome based on a stick- slip mechanism, Earth Planet. Sci. Lett., in press. -Den Doelder, C. F. J., Koopmans, R. J., Molenaar, J., and Van de Ven, A. A. F., 1998, Comparing the wall slip and the constitutive approach for modelling spurt instabilities in polymer melt flows. J. Non-Newtonian Fluid Mech., 75, p. 25–41. - Denlinger, R., and R. P. Hoblitt (1999), Cyclic eruptive behavior of silicic volcanoes, Ge- ology, 27, 459–462. -Green, D.N., J. Neuberg, 2006. Waveform classification of volcanic low-frequency earth- quake swarms and its implication at Soufriere Hills Volcano, Montserrat, J. Volcanol. Geo- therm. Res.,153, 51-63. SYSTEM INVESTIGATED References PHYSICAL MODEL COMPARISON WITH OBSERVATIONS OSCILLATION PERIOD L L c L d P(t) Q in Q(t)