A Dynamical Model of Seismogenic Volcanic
Extrusion, Mount St. Helens, 2004-2005
Richard Iverson
U.S. Geological Survey
Cascades Volcano Observatory
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
Feb. 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
Extruded volume (m3)
80x106
60x106
40x106
3 /s
5m
.
1
at 05
n
i o . 20
s
u
r Dec
t
x
e 4y
d
a
00
e
2
t
.
s ec
D
20x106
0
10/1/04 12/1/04 2/1/05 4/1/05
6/1/05
8/1/05 10/1/05 12/1/05
Time (month/day/year)
Fact 2: striated fault gouge that coats the
surface of the newly extruded dacite plug
exhibits rate-weakening frictional strength
S. Schilling photo
Shear stress (kPa)
72
0.475
slower slip (1.5 x 10
71
-6
0.470
m/s)
0.465
70
faster slip (3 x 10
-6
m/s)
0.460
69
0.455
constant normal stress = 159 kPa
0.450
68
0
2
4
Displacement (mm)
6
8
Approximate friction coefficient
Fact 2: striated fault gouge that coats the
surface of the newly extruded dacite plug
exhibits rate-weakening frictional strength
Example
of 24 hours
of seismicity,earthquakes
Dec. 1, 2005
Fact
3: repetitive
“drumbeat”
occurred almost periodically (T ~ 100 s), had
magnitudes ≤ 2, hypocenters < 1 km
directly beneath the new dome, and mostly
“hybrid” waveforms with impulsive onsets.
Constants
Parameters that evolve
as prescribed functions
of dependent variables
or time
Dependent variables
that evolve with time
Rock density ρr
Magma density ρ
Magma compressibility α1
Conduit compliance α2
1-D
“SPASM”
model
1-D conservation of mass and momentum leads to
du
1
 g 
 pA   u  F 
dt
m0   t
dp
1/ V

 Au  RB  Q 
dt 1   2
1
dV

dt 1   2
 A u  RB  Q   Q  B
where
0

R  1
 1  exp[1 ( p  p0 )]
r
r
and

u 
1
F  sgn(u ) mg   0 1  c sinh

uref 

Obtain equation for damped, forced oscillations
of normalized extrusion velocity
Kgt0
u
du u
1
d
(1  Kt )
 2D



2
dt  V 
V  (Q  RB ) /A
dt 
2
where
u   u /[(Q  RB) / A]
t  = t /t0
[m0 (1 +  2 ) V0 ]
t0 
A
 t0
1/ 2
K 
m0
V   V / V0
1 t0 dF
D  K
2 m0 du
Find exact solutions, steady or oscillatory, if V´ =1
and D is constant, but behavior is unstable for D < 0
Predicted free oscillation period of u' (linear theory)
[m0V0 (1  2 )]
T  2 t0  2
 2 [(1   2 ) r H con H plug ]
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 friction coefficient,  / 0

u 
1
F  sgn(u ) mg   0 1  c sinh

uref 

1.00
c = - 0.005
for u/uref <1,
approximates linear
rate dependence
c = - 0.01
0.95
c = - 0.015
c = - 0.02
0.90
for u/uref >1,
approximates
logarithmic rate
dependence
c = - 0.025
c = - 0.03
0.85
0.80
0
20
40
60
Relative velocity, u / uref
80
100
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:
1 t0
D
2 m0
 dF 
 du 
u u0

gt0 u0
1
  c0
2
u0 uref

  u
1   0
  uref

which simplifies to
gt0 
1
D   c0

2
u0 
if u0/uref >> 1



2




1/ 2





Computed
start-up
behavior with
T =10 s, D =−0.01
and initial
conditions
u = Q/A, p = p0,
V = V0
Phase-plane representation of start-up behavior with
D = −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
Details for D = −2
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
Results for D = −2
For fixed D,
sensitivity of
limit cycles to
choices of c and λ
is nil. That is,
static friction and
rate weakening
have counterbalancing effects
on dynamics.
Commensurate with
7 × 107 N force drop
during slip event
Effect of disequilibrium initial condition
(0.005% initial excess magma pressure)
Conclusions
1. 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.