A DYNAMICAL MODEL OF DOME-BUILDING SPASMS, MOUNT ST. HELENS, 2004-2005

A newly developed model can explain the relationship between two key aspects of the 2004-2005 eruption of Mount St. Helens: (1) the enduring, dome-building phase of the eruption was characterized by quasisteady extrusion (~1-2 m³/s) of a solid dacite plug surfaced with striated fault gouge; and (2) extrusion was accompanied by >10⁶ repetitive, small earthquakes (~M 1-2) that occurred at regular intervals (~30-300 s). The extraordinary persistence of these “drumbeat” earthquakes motivates the hypothesis that they resulted from discrete boundary slips (<1 cm each) that occurred as the plug was forced incrementally upward by a nearly steady influx of solidifying magma. This “Seismogenic Plug of Ascending, Solidifying Magma” (i.e., “SPASM”) hypothesis is formalized mathematically by considering conservation of mass and momentum of the solid plug and underlying magma, and by adopting simple constitutive postulates for the compressibilities of the magma and conduit wall rock and for the frictional resistance F of the plug sliding against the conduit walls. The resulting model reduces to a nonlinear system of three ordinary differential equations describing coevolution of the upward plug velocity, u, basal magma pressure, p, and volume of the liquid-filled portion of the conduit, V. In general these equations must be solved numerically, but they also yield analytical results showing that u will persistently oscillate, provided that F does not increase as u increases (i.e., friction is not rate-strengthening). Predicted oscillation periods correspond well with the observed interval between drumbeat earthquakes. Moreover, if F decreases nonlinearly as u increases (representing a decay from static to dynamic friction) oscillations of u sharpen into discrete stick-slip cycles. Even if the system begins far from equilibrium (presumably necessary to instigate a volcanic eruption), it is strongly attracted to this type of near-equilibrium, oscillatory state.