submitted to NSF/EAR December 200

Introduction

After more than a century of research, we still do not understand why volcanoes erupt!

The situation is quite different than for earthquakes. We understand both the plate motion that causes the steady increase in stress on locked faults and the nature of the frictional instability that causes the sudden fault rupture and that produces seismic motion.

For volcanoes, we understand something about the mantle melting process that supplies the magma, although magma flux estimates are much poorer than the corresponding tectonic loading rates for faults. We understand almost nothing about the eruptive instability. Some people have hypothesized that eruptions are caused by an instability within the magma chamber that suddenly drives up the pressure (e.g. Linde et al. 1994, 1994; Brodsky et al. 1998). Others have suggested that it is caused by the sudden failure of the magma chamber containment, such as faulting or cracking in its lid (e.g. Einarsson & Brandsdottir, 1980). Still others have suggested that it is caused by an sudden increase in supply from below (e.g. Aki, 1981). These are all good ideas. They all have some corrobatory evidence suggesting that they have occured in a few instances. What is lacking is a methodology for routinely descriminating between them, to learn which are active in particular eruptions.

Seismically-triggered eruptions arguably provide a good starting place for studies of the cause of eruptions. The fact of the triggering has been rigourously established by the statistical correlation of eruptions and nearby large earthquakes (Linde and Sacks 1998). Furthermore, the triggering seems to be dynamic - that is, caused by the shaking itself - rather than by the static stress change associated with the fault displacement. If the link between the shaking and the volcanic instability can be established, light might be shed both on the specific mechanism of seismic triggering, and also on magma chamber instabilities in a more general sense.

"Rectified Diffusion" (Sturtevant-Bradford et al. 1996; Brodsky et al. 1998) is a mechanism that has recently been hypothesized to occur in a magma chamber when a volcanic eruption is triggered by a regional earthquake. It is a non-linear process in which the seismic vibrations rachet-up the pressure in pre-existing volatile (e.g. water) bubbles within the magma, causing an overall rise in pressure that leads to diking, and hence to an eruption. The racheting is caused by a change in thickness of the diffusive boundary layer that surrounds a bubble, with a compressed bubble having a thicker boundary layer than a dilated one. The passage of an oscillatory seismic wave through the magma chamber results in faster diffusion into the bubble from the magma (assumed saturated or oversaturated with volatile) during the dilation than diffusion out during the compression. A a net transport of volatile from the magma into the bubble results. Brodsky et al.'s (1998) analysis of the underlying physics of Rectified Diffusion indicates that it can plausibly occur in realistic magmatic systems.

The key idea behind this proposal - which I will justify below - is that the Rectified Diffusion mechanism causes a loss of energy of the incident seismic waves. This loss of energy can broadly be called "seismic attenuation", although its physics is somewhat different than internal friction that is normally associated with that term. Measurements of the seismic attenuation of magma chambers are therefore a potential tool for monitoring the eruptive potential of a volcano. Hypothetically, we might imagine that if seismic attenuation were monitored over decadal timescales, one would observe a sudden increase at a time when conditions, such as state of saturation and bubble concentration and size, favored Rectified Diffusion. Furthermore, it may provide a means of monitoring the amount of bubbles in the magma chamber even when the Rectified Diffusion effect is not strong enough to permit a seismically triggered eruption. It may therefore provide a means of detecting other magma chamber instability mechanisms, such as Linde et al's (1994) hypothesis that a pressure flucuation is caused by a sudden rise of bubbles. Attenuation tomography of the magma chamber might even provide evidence that the distribution of bubbles in the magma chamber has changed.

This proposal is to analyze this effect in detail: to develop meand for calculating the properties of the attenuation (magnitude, frequency-dependence, dependence on magma and bubble properties).

A Thought Experiment

Imagine a pendulum consisting of a ball on a string, where the string is anchored to an object that can slide horizontally (Figure 1). If the slider is held fixed, normal oscillatory motion of the pendulum can occur.

Imagine, however, that the slider is made to rachet slowly to the right whenever the pendulum is to the right of the slider and swinging to the right. The oscillation will rapidly damp out. (I've tried this with a string and a weight - it really does). The reason is that in this configuration the string is exerting a rightward force on the slider, and if the slider moves to the right, net work (equals force time distance) is done on it. The work is supplied by the energy of motion of the pendulum.

The same sort of attenuation mechanism is routinely used by automobile drivers, when they prevent a car from oscillating horizontally on its supsension after a short stop, by letting it roll forward a little.

The point is that a nonlinear coupling between oscillatory and linear motion can result in a net transfer of energy out of the oscillatory part of the system. As I show below, a broadly similar effect occurs during Rectified Diffusion: energy is transferred out of the seismic ocsillation as the ambient pressure and volumetric strain of the magma chamber increases.

A Discrete Model of Rectified Diffusion

I have assembled a simple, discrete, numerical model of Rectified Diffusion. The magma chamber is modeled by a cylindrial pipe containing pistons, most of which are free to slide along the axis of the pipe (the two end pistons are held fixed) (Figure 2, top). Some of these pistons are separated by springs. These piston-spring-piston combinations represent the magma, which is assumed to be linearly elastic. Some of the pistons are separated by an isothermal ideal gas. These gaseous chambers represent volatile bubbles in the magma. Only the pistons have mass, and their motion is given by Newton's law, with a net force that is the sum of the force due to a spring and to gas pressure. Initially the system is motionless, with the force of the springs exactly balancing the force of the gas. The system can be made to oscillate by giving one of the pistons a sudden displacement. This displacement models a triggering earthquake. Since the system is perfectly (though not linearly) elastic, it oscillates indefinitely, without any long-term change in the total kinetic energy of the pistons (Figure 3, top left). Since gas is not allowed to enter or exit the chambers, the total amount of gas does not change with time (Figure 3, bottom left).

Now we attach volatile reservoirs to the pipe (Figure 2, bottom), so that volatiles can enter and exit the gas chambers through a valve. The reservoirs represent volatile dissolved in the magma. We assume that the vapor pressure of the volatile in the reservoirs is given by Henry's Law, that is, it varies lineary with the concentration of dissolved volatile. The valve allows flow of the volatile from the reservoir into the chamber when the vapor pressure of the reservoir is higher than the pressure in the chamber, and flow from the chamber into the reservoir when it is lower. Rectified Diffusion is implemented by having the flow rate depend linearly upon the pressure difference (i.e. diffusion) but with a diffusion coefficient that is higher for flow into the chamber than out of it. When the system is perturbed from its equilibrium by the sudden displacement of a piston, it begins to oscillate. However, the total kinetic energy of the pistons rapidly decrease with time (Figure 2, top right). Seismic attenuation occurs. The amount of gas in the chamber initally increase (Figure 2, bottom right), as is expected from the Rectified Diffusion mechanism, and then slowly decreases as the system returns to equilibrium. A similar change in the mean pressure in the chambers also occurs (not shown).

I have performed many numerical integrations of this system, experimenting with various choices for the numerous parameters. The link between the transfer of gas from the reservoirs (that produces the "desirable" pressure pulse) and the loss of kinetic energy in the pistons (the "seismic attenuation") seems very strong. In all cases considered a large pressure increase was correlated with strong attenuation.

Research Plan

While the discrete model discussed above indicates attenuation can occur, it does not enable us to calculate the quality factor ("Q") of a realistic magma chamber, or to predict its dependence on the relevant parameters (e.g. frequency, bubble size, state of saturation, etc.). I therefore propose to:

1. Derive the partial differential equations that are the continuum mechanical version of the system, solve them numerically for an incident plane compressional wave, and derive an expression for the rate of attenuation of that wave and the effective Q of the medium.

2. Modify the continuum model so that it has more realistic physics. In particular, use more precise equations of state for the gas pressure (Burnham et al. 1969) and the vapor pressure of the dissolved volatile (Dixon et al. 1995).

3. Develop a sets of parameters that together comprise a good model of a magma chamber, and use them to make quantitative predictions about the properties of the attenuation that might be observed in realistic circumstances.

4. Design an experiment would be able to detect this attenuation. Several possible senarios might include: an active-source experiment in which the Q of P-waves transmitted through the magma chamber is monitored; a passive experiment in which the Q of telseismic P waves is monitored; a passive experiment based on the monitoring of P-wave coda Q for local earthquakes. The choice of experiment will be determined by the predicted properties of Q (its amplitude, frequency dependence, etc.).

Benefits of this Research

The overall theme of this research is developing methodology for testing the various hypotheses that have been made about the causes for volcanic eruptions. The case is made here that Rectified Diffusion, which has recently been put forward as the cause of seismically triggered eruptions, can be possibly detected through its effect on seismic attenuation. Seismic measurements are, of course, only one of the many parameters that could be potentially monitored around a volcano. Geodetical, geochemical and geothermal measurements are also of potential application, though are not treated here. We note, however, that a wealth of seismic data has already been collected as a routine component of the monitoring of many volcanos. Methodology developed by the proposed research would have direct application to these datasets.

Management Plan

William Menke will be responsible for the overall management and timely completion of the project. Menke, assisted by Graduate Research Assistant Kristina Rodriguez, will perform the research.

Timetable

The research is expected to take one year.

Dissemination of Results

We will maintain archives of preliminary results on our institutional web sites (as we now do for previous studies, see for example http://www.ldeo.columbia.edu/user/menke). We will present results at scientific national meetings, such as the Fall AGU, and make a best-faith effort to publish them rapidly in a peer-reviewed journal.

Miscellaneous

  1. Figure 1 (EPS Format) Pendulum on a racheting slider.
  2. Figure 2 (EPS Format) Discrete model of magma chamber appropriate for studying interaction of Rectified Diffusion and seismic attenuation. (Top) Model without Rectified Diffusion, consisting of pistons, springs and gas-filled chambers. (Bottom) Model with Rectfied Diffusion, with gas reserviors and valves.
  3. Figure 3 (EPS Format) Results of numerical integration of discrete magma chamber model. (Left) Without Rectified Diffusion. (Right) With Rectified Diffusion.
  4. Summary
  5. Budget Justification
  6. References
  7. Facilities
  8. Prior Results
  9. Menke's Vita