Dynamics of seismogenic volcanic extrusion resisted by a solid surface plug, Mount St. Helens, 2004-2005

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 <10⁸
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.