Post on 25-Dec-2015
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A Dynamical Model of Seismogenic Volcanic Extrusion, Mount St. Helens, 2004-2005
Richard IversonU.S. Geological SurveyCascades Volcano Observatory
S. Schilling photoFeb. 22, 2005
Fact 1: extrusion rate of solid dacite plug is nearly constant when measured over timescales ranging from a few minutes to a few months
Time (month/day/year)
10/1/04 12/1/04 2/1/05 4/1/05 6/1/05 8/1/05 10/1/05 12/1/05
Ex
tru
de
d v
olu
me
(m3 )
0
20x106
40x106
60x106
80x106
steady extrusion at 1.5 m
3 /s
Dec. 2004 - D
ec. 2005
Fact 1: extrusion rate of solid dacite plug is nearly constant when measured over timescales ranging from a few minutes to a few months
S. Schilling photo
Fact 2: striated fault gouge that coats thesurface of the newly extruded dacite plugexhibits rate-weakening frictional strength
Displacement (mm)0 2 4 6 8
Sh
ear
stre
ss (
kPa)
68
69
70
71
72
Ap
pro
xim
ate
fric
tio
n c
oef
fici
ent
0.450
0.455
0.460
0.465
0.470
0.475
faster slip (3 x 10-6
m/s)
slower slip (1.5 x 10-6
m/s)
constant normal stress = 159 kPa
Fact 2: striated fault gouge that coats thesurface of the newly extruded dacite plugexhibits rate-weakening frictional strength
Example of 24 hours of seismicity, Dec. 1, 2005Fact 3: repetitive “drumbeat” earthquakesoccurred almost periodically (T ~ 100 s), hadmagnitudes ≤ 2, hypocenters < 1 km directly beneath the new dome, and mostly“hybrid” waveforms with impulsive onsets.
Magma compressibility α1
Conduit compliance α2
Magma density ρ
Constants
Parameters that evolveas prescribed functionsof dependent variablesor time
Dependent variablesthat evolve with time
Rock density ρr
1-D“SPASM” model
0
1du g pA u F
dt m t
1 2
1/dp V Au RB Q
dt
1
1 2
dV A u RB Q Q B
dt
01 01 1 exp[ ( )]
r r
R p p
10sgn( ) 1 sinh
ref
uF u mg c
u
where
and
1-D conservation of mass and momentum leads to
01(1 ) 2
( )
2
2
Kgtu du ud Kt D dt V V Q RB /Adt
/[( ) / ]u u Q RB A 0t = t /t0/V V V
1 20 1 2 0 0 0
00 0
[ ( ) ] 1
2
/m + V t t dFt K D K
A m m du
where
Obtain equation for damped, forced oscillations of normalized extrusion velocity
Find exact solutions, steady or oscillatory, if V´ =1and D is constant, but behavior is unstable for D < 0
Predicted free oscillation period of u' (linear theory)
0 0 1 20 1 2
[ ( )]2 2 2 [( ) ]r con plug
m VT t H H
A
Results for ρr=2000 kg/m3
Hcon = 8 km
Variable damping D arises from use of nonlinear rate-weakening friction rule for sliding at plug margins:
Relative velocity, u / uref
0 20 40 60 80 100
Rel
ativ
e fr
ictio
n co
effic
ient
, /
0
0.80
0.85
0.90
0.95
1.00
c = - 0.005
c = - 0.025
c = - 0.02
c = - 0.015
c = - 0.01
c = - 0.03
for u/uref <1, approximates linear rate dependence
for u/uref >1, approximates logarithmic rate dependence
10sgn( ) 1 sinh
ref
uF u mg c
u
If κ = 0, B = Q, and t0 is constant, behavior of numerical solutions depends almost entirely on D evaluated at the equilibrium slip rate u = u0= Q/A:
0
1/ 22
0 0 0 00
0 0
1 11
2 2u u ref ref
t gt u udFD c
m du u u u
00
0
1
2
gtD c
u
which simplifies to
if u0/uref >> 1
Phase-plane representation of start-up behavior withD = −0.01 and initial conditions u=Q/A, p = p0, V = V0
Time series and phase-plane representations of stick-slip limit cycles computed for T =10 s and various values of D, with initial conditions u = 0, p = p0, V = V0
With D = -2, work done against friction during a slip cycle is 2×108 J, similar to energy release in a M 2.3 earthquake
For fixed D, sensitivity of limit cycles to choice of u0/uref in the friction rule is slight, provided that u0/uref ≥ 1
Results for D = −2
For fixed D,sensitivity of limit cycles tochoices of c and λis nil. That is,static friction andrate weakeninghave counter-balancing effects on dynamics.
Results for D = −2
Commensurate with7 × 107 N force dropduring slip event
Conclusions1. Stick-slip oscillations are inevitable as a consequence of momentum
conservation, driving force supplied by compressible magma, restoring force supplied by gravity, and rate-weakening plug boundary friction.
2. Use of realistic (i.e. best-guess) parameter values produces stick-slip oscillations with roughly the correct period, amplitude, and force drop to
produce repetitive “drumbeat” earthquakes at MSH.
3. Fluctuations in magma pressure during stick-slip cycles are very small, a few kPa, implying that departures from equilibrium are very slight.
4. Long-term, oscillatory behavior of the system is remarkably stable unless magma influx or composition changes or friction evolves.
5. Initial conditions far from equilibrium probably didn’t exist at MSH. If they had, a large pulse of motion would have occurred initially, irrespective of the type of frictional resistance.