Dynamics of seismogenic volcanic extrusion resisted by a solid surface plug, Mount St. Helens, 2004-2005: Chapter 21 in A volcano rekindled: the renewed eruption of Mount St. Helens, 2004-2006

The 2004-5 eruption of Mount St. Helens exhibited sustained, near-equilibrium behavior characterized by nearly steady extrusion of a solid dacite plug and nearly periodic occurrence of shallow earthquakes. Diverse data support the hypothesis that these earthquakes resulted from stick-slip motion along the margins of the plug as it was forced incrementally upward by ascending, solidifying, gas-poor magma. I formalize this hypothesis with a mathematical model derived by assuming that magma enters the base of the eruption conduit at a steady rate, invoking conservation of mass and momentum of the magma and plug, and postulating simple constitutive equations that describe magma and conduit compressibilities and friction along the plug margins. Reduction of the model equations reveals a strong mathematical analogy between the dynamics of the magma-plug system and those of a variably damped oscillator. Oscillations in extrusion velocity result from the interaction of plug inertia, a variable upward force due to magma pressure, and a downward force due to the plug weight. Damping of oscillations depends mostly on plug-boundary friction, and oscillations grow unstably if friction exhibits rate weakening similar to that observed in experiments. When growth of oscillations causes the extrusion rate to reach zero, however, gravity causes friction to reverse direction, and this reversal instigates a transition from unstable oscillations to self-regulating stick-slip cycles. The transition occurs irrespective of the details of rate-weakening behavior, and repetitive stick-slip cycles are, therefore, robust features of the system's dynamics. The presence of a highly compressible elastic driving element (that is, magma containing bubbles) appears crucial for enabling seismogenic slip events to occur repeatedly at the shallow earthquake focal depths (<1 km) observed during the 2004-5 eruption. Computations show that fluctuations in magma pressure accompanying such slip events are <3 kPa, indicating that deviations from mechanical equilibrium are slight and that coseismic force drops are <10 8 N. These results imply that the system's self-regulating behavior is not susceptible to dramatic change-provided that the rate of magma ascent remains similar to the rate of magma accretion at the base of the plug, that plug surface erosion more or less compensates for mass gain due to basal accretion, and that magma and rock properties do not change significantly. Even if disequilibrium initial conditions are imposed, the dynamics of the magma-plug system are strongly attracted to self-regulating stick-slip cycles, although this self-regulating behavior can be bypassed on the way to runaway behavior if the initial state is too far from equilibrium.