Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
A SIMPLIFIED SLAB PULL MODEL:
AN EXAMINATION INTO THE OCCURRENCE AND
RECURRENCE OF GREAT INTERPLATE
SUBDUCTION EARTHQUAKES
Gwo-Shyh Song and Brain T. R. Lewis
ABSTRACT
In the current subduction paradigm the principal force is that due to the cold,
dense plate sinking into the less dense hot mantle. This force is resisted by viscous
forces in the mantle and by frictional forces in the "trench" region where large
subduction earthquakes occur. In our simplified model this process is represented by a
mass (m2) resting on a horizontal plate, with the mass (m2) being pulled by a spring (of
constant k) connected to another mass (m1) sinking into a fluid of viscosity P,. The
mass m2 on the plate has a static coefficient of friction f x and a dynamic coefficient fd .
Both physical and mathematical realizations of this model produce periodic slipping of
the mass m2, that represents the stick-slip process of subduction.
The Viscosity η is the dominant factor controlling the subduction rate, Young's
modulus E (k of the spring) determines the displacement of slipping, and to first order
the ratio η/E controls the recurrence rate of slipping.
Application of this model to a system with earth-like properties gives estimates of
earthquake recurrence times that are in reasonable agreement with actual subduction
zones. The model also gives insight into the critical factors controlling recurrence rates.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
INTRODUCTION
Large interplate thrust earthquakes occur due to accumulation of strain associated
with subduction processes in the trench region. That the stress associated with this
strain is released by large earthquakes with a statistically characterizable recurrence rate,
is itself an indication that subduction is episodic or stick-slip. It is also evidence that for
long periods during the subduction process the subducting slab can be viewed as
sinking into the mantle under its own weight while suspended from the locked zone at
the trench.
This research is motivated by a desire to understand the general phenomenon of
subduction and the recurrence rates of great subduction earthquakes. Great subduction
earthquakes are the cause of the largest natural disasters and this knowledge can be of
use in the Taiwan-arc "region" where has been classified as one of the highly potential
areas for these great earthquakes (McCann et al., 1979).
The negative buoyancy force of the cold and dense subducted slab is a key factor
in driving the plates. From the thermal model of McKenize (1969), this force, called
slab pull, can be estimated by integrating over the slab forces caused by the density
contrast between the denser subducting material and the density of "normal"
surrounding mantle material. Slab pull (as large as 1013 N/m) contributes the major
stresses inside the slab (Richardson, et al., 1979; Turcotte, et al. 1982; Spence, 1987,
1989). To maintain equilibrium of forces in the system, most of this pull is balanced by
drag forces from the viscous mantle. Tensional stresses in the downdip part of the slab
build up slowly, as a result of the viscous drag to cause the long recurrence time
between slipping events (earthquakes) along the "main thrust zone" (ITZ). The
existence of downdip tensional stress in the subducted slab is observed at depths above
300km from seismicity (Isacks and Molnar, 1969) and is evidence that motion of the
subducted slab is resisted at its upper end (main thrust zone is in this case).
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
A simplified one-dimensional mechanical analogy (Fig 1) of this process consists
of a sphere (subducted slab) sinking into a less dense viscous fluid (mantle) with the
sphere connected by a spring (elastic part of the slab) to a mass (oceanic lithosphere)
resting on a horizontal plate. It is to represent the process of subduction and the
recurrence of great subduction earthquakes. The spring force accumulates as the elastic
slab (spheric weight) sinks through the viscous asthenosphere (fluid).
In this paper the intention is to use this simplified model to determine and justify
parameters used in controlling the behavior of the subducted slab. Furthermore, the key
components that control periodicities of the stick-slip motion will be analysized to
understand the mechanisms acting in subduction zones.
With this model, viscosity of the mantle controls the subduction rate for a given
slab-pull and ridge-push force. The parameters controlling the recurrence rate of
interplate earthquakes are 1) ratio of viscosity and Young's modulus of the tensional
subducted slab, 2) tensional behavior of the slab, which relates to the geometry of
subduction (dip and thickness of slab), and other pull or resistance forces which are
related to deeper phase transitions at 450km and 650km depth.
k
↓y
Fig 1. A mechanical analog for the subduction process. The sphere represents the subducted lithosphere; the
spring is the elastic portion of the subducted slab; and the fluid represents the viscous mantle. The
interface between the block and horizontal plate is the interface thrust zone. Frictional resistance
across the interface is governed by the weight of the block, which represents the overlying mass of a
continental margin.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
A MATHEMATICAL REALIZATION
A conceptual diagram of the subduction process is shown in Fig 2 together with
the principal forces (after Spence, 1987). Fig 1 shows a simplified mechanical analog
of this process.
A mechanical realization of this model was achieved by allowing a brass mass to
sink into a beaker of honey, with the mass connected via a spring to a block of plastic
resting on an aluminum plate. This realization clearly shows stick-slip behavior and the
period between slipping events (earthquakes) can be readily increased (or decreased) by
increasing (or decreasing) the viscosity of the honey by cooling (or heating) or by
decreasing (or increasing) the stiffness of the spring. The stick-slip behavior is fairly
irregular because of asperities on the aluminum surface. A diagrammatic representation
of the forces, displacements and velocities in this system are shown in Fig 3
The one-dimensional negative buoyancy force Fp exerted by a spherical weight,
of density ñs, and radius a, sinking into a fluid of density ρw . with viscosity η under a
gravitational acceleration g, is (ñs-ρw )g(4/3)ða3. If this weight is hung by a spring, with
spring constant k, and the upper end of the spring is fixed, the motion of the sinking
sphere is described by the following equation:
m1 (d 2 y/dt2 )= Fp -6πηa (dy/dt)-ky
(1)
where m1 is the mass of the sphere (ie. (4/3)πa3 ñs) and 6πηa (dy/dt) is the viscous force
on the surface of the sphere with sinking rate dy/dt (Batchelor, 1967). y is the vertical
distance coordinate, t is time, ky is the force transmitted by the spring, assuming that y
is equal to zero when the spring is unstretched, and dy/dt is the instantaneous sinking
rate of the sphere.
Assuming the system is over-damped, that is, η2 >> m1 k/9π2a2 and η>>k, and
with initial conditions y=0, dy/dt=0 at t=0, the solution of (1) is:
y(t)=Fp /k[1-exp(-kt/q)]
(1a)
where q=6πηa.
If the spring force ky(t) is not greater than the frictional force of the block on the
plate the block will not move and remains at rest. From (1a), the sinking distance may
approach Fp /k as t goes to infinity. However, as soon as the spring force is greater than
the frictional force the block will begin to move. Equation (1) is not going to be valid
and needs modifications, as follows.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Fig 2. Schematic diagram, adapted from Spence (1987), of driving and resisting forces acting on descending
lithosphere. General features observed at subduction zones are shown. The major plate bending
occurrs just downdip of the ITZ. The slab pull force, Fp, shown by the insert diagram, and ridge push
force is resisted by the viscous force of the asthenospbere and by the frictional force in the ITZ.
Corner flow in the asthenospbere may balance the vertical component of the slab pull force, FN, and
resist the plate sinking and in controlling the rollback of the trench. Unlabeled arrows within the plate
show expected character of extension (open arrows) and compression (meeting arrows) due to
subduction. The stars indicate the seismicity, with the big star representing great thrust earthquakes
on the ITZ. Marginal basins are observed in subduction zones with extensional environments in the
back-arc "region", or with active trench-rollback (Ruff & Kanamori, 1980; Carlson, 1983; Garfunkel
et al., 1986; Jarrard, 1986).
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Suppose that the friction resistance between the block (of mass m2 ) and surface of
the plate is characterized by the static frictional coefficient fs and the dynamic frictional
coefficient fd . Then, to initiate sliding from the state of rest, a horizontal force fsm2g
must be applied to the block. A horizontal force fd m2g will resist the motion of the
block when the block is moving. Let us assume that sliding is initiated (at t0 =0) when
fsm2g =ky0, or when y0 = fsm2g/k. The subsequent motion of the sphere(of displacement
y*, with y*=y-y0 ) and of the block (of displacement x) during the time period of the
sliding of the block (t0 to t1, t1 is at the time the sliding ended) are described by the
following equations:
m1 (d 2 y*/dt2 ) =Fp -q(dy*/dt)-k(y*+y0 -x)
(2)
m2 (d 2 y*/dt2 ) =k(y*+ y0 -x) -fd m2g
(3a)
=k(y*-x) + (fs-fd ) m2g
(3b)
with initial conditions:
x=0
dx/dt=0
y*=0
dy*/dt=R at t0
(4)
R is the sinking rate of the mass ml when the spring is stretched an amount of y0 .
At this instant, the viscous force on the sphere is qR which is equal to the residual
forces in the system, Fp - fsm2g; resulting in R being the value of (Fp - fsm2g)/q.
From (2) we obtain:
x(t)={m1 (d 2 y*/dt2 )+q(dy*/dt)+k(y*+y0 )-Fp }/k
(5)
Substituting this value of x(t) in (3) we obtain the following fourth-order differential
equation in dy*/dt.
d4 y*/dt4 + (q/m 1 )d 3 y*/dt3 +k(1/m1 +1/m2 )d 2 y*/dt2
+(qk/m 1 m2 )dy*/dt=(F p -fd m2 g)k/m 1 m2
(6)
The initial conditions are given by equation (4), and are converted to the following
expressions by using equation (2):
y* = 0
dy*/dt =R
d2 y*/dt2 = 0
d3 y*/dt3 = -kR/m 1
(7)
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Under the assumptions that the system is overdamped, that is; η>>k, and that
Fp >fsm2 g, the solution of equation (6) is:
y*(t) = - [(fs-fd ) m2 g/q] ( m2 /k)1/2sin[ t(k/ m2 ) 1/2]+(FP - fd m2 g) t/q
(8)
The first derivative of y*(t) is:
dy*/dt = - [(fs-fd ) m2 g/q]cos[ t(k/m2 )1/2]+ (FP - fd m2 g)t/q
(9)
The second term of equation (8) is linear in t and greater than the first term
because of the assumed properties. Therefore we use y*(t)=(Fp -fdm2g)tlq in equation
(3b), giving:
m1 (d2 x/dt2 )+kx=( FP - fd m2 g) tk/q+(fs-fd ) m2 g
(10)
The first term of the right hand side of equation (10) is much smaller than the
second term and can be neglected. As a result, equation (10) is y-independant and
m1 (d2 x/dt2)+kx=(fs-fd ) m2 g
(11)
The solution is:
x(t)=[( fs-fd ) m2 g/k][1-cos(t (k/m2 )1/2)
(12)
and its first derivative is:
dxldt=[(fs-fd ) m2 g/( m2 k) 1/2][sin(t (k/ m2 ) 1/2)
(13)
The force drop is defined as k[x(t1 )-x(t0)], where t0 =0 and t1=π /(k/ m2 ) 1/2. From
(12) the force drop is obtained to be 2(fs-fd )m2 g and shown in Fig 3(a).
The time interval between sliding periods is easily found as follows:
At the start of sliding:
fs m2 g = Fp (l-e-kt'/q )
At the end of sliding:
fd m2 g = (fs-fd ) m2 g +Fp (l - e-kt”/q )
The term (fs-fd )m2g occurs because this force is available to impart momentum
when sliding has started, and q=6ηπa. Therefore
∆t = t’-t’’ = (6ηπa/k) ln{1+2(fs-fd )m2 g / (FP - fsm2 g)}
(14)
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
We note the following properties of this mechanical system, as derived from the
mathematical realization.
[1] The viscosity of the fluid and the negative buoyancy force of the sphere
determine (to first order) the magnitude of the sinking rate of the sphere.
[2] The motion of the block is stick-slip. The time interval between slipping
events is inversely proportional to the spring constant(k) and proportional to the
coefficient of viscosity, and also dependent on the negative buoyancy force and the
static and dynamic frictional forces (equation (14)).
[3] In the case of η>>k, the detailed motion of the block while sliding, that is it's
acceleration, deceleration, and displacement, is controlled primarily by the spring
constant(k) and the static and dynamical friction forces. Displacement of the block is
independent of the initial velocity, of the sinking sphere as soon as the block starts to
slide. The time period of the slipping event of the block is inversely proportional to the
ratio of (k/m2 ) 1/2.
Fig 3. A diagram which shows the forces (a), the displacements (b) and the speeds (c) in the
mechanical system shown by Figure 1. In (a) the trace represents the historv of the pull force
to block by the sinking sphere, and the force drop after the slide is 2(f s-f d )mg with m being the
mass of the block. The duration from time t' to t'', is a cycle of rupture event, or the time of the
recurrence. The y(t) and dy/dt is the displacement and speed of the sinking sphere,
respectively; x(t) and dx/dt is the displacement and speed of the block, respectively.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
APPLICATION TO A SYSTEM OF THE EARTH-LIKE PROPERTIES
Several assumptions are made in order to apply this mechanical model to an
earth-like subduction zone in which a locked subducted slab has been deformed in a
manner that the shear stress is produced in a fashion that the thrusting is possible along
the ITZ. (Note that forces below are given in Newton per unit length of subduction
zone, or N/m).
First, the viscous forces acting on the sinking sphere must be replaced by a more
realistic geometry. We assume a linear viscous rheology and that the viscous drag
acting over the surface of the descending slab is given by 4η(dy/dt). This value was
derived by Davies (1980) by assuming that there existed a convection cell of size
similar to the length of the subducted plate. With this assumption the viscous forces are
independent of the length of the subducted plate and proportional to the subduction rate
(dy/dt). A constant viscosity (η) of 1021 Pa s is assumed for the drag in the
asthenosphere. With a subduction rate of 100 km/My this viscous force is 1013 N/m.
Secondly, the slab pull and ridge push forces are about 3 x 1013 and 0.3 x 1013
N/m, respectively (Davies, 1980; Lister, 1975; and Turcotte & Schubert, 1982).
Thirdly, we assume that the static frictional forces in the trench region are greater
than the ridge push force, (if they were not the slab would slip continuously), and that
they are less than the slab pull force and than the yield strength of the slab (if they were
not the slab would not slip at all). This assumption is clearly necessary but is also
justified by arguments based on properties of materials in subduction zones (Davies,
1980; Ruff & Kanamori, 1983).
Fourth, we neglect viscous drag on the unsubducted lithosphere (Ranalli, 1987).
In addition, we assume that the pulling force per unit length of the subducted slab
applied on the ITZ is the result of the force balanced by the slab pull, the viscous drag,
and the ridge push. Once the sum of these forces is greater than the static frictional
force in the ITZ, rupture (or slip) occurs. In this study, hydrodynamic pressure induced
by the viscous flow is neglected, primarily because this pressure is not significant in
this simplified model due to use of the spring instead of the elastic plate connected in
between the sinking weight and the resting block.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Fig 4. Diagrams showing thelprocess7of deformation-of subducted lithosphere in the neighborhood of
the ITZ. The initial state of deformation is assumed to be at when the rupture of the lacked
lithosphere along the ITZ just terminated. A displacement of Äu occurs along the lower
boundary of the lithosphere due to the stretching of the deeper subducted lithosphere by the
stab pull force and material in the ITZ is stationary. Fr, is ridge push force, Fp is slab pull force,
FD is drag force, Fs is the static frictional force, and Fshare is the shear force provided by shear
strain εxy, which is assumed to be ∆u/L2 (where u and v represent the displacement components
that are parallel to the x and y coordinates, respectively). L2 is the thickness of the lithosphere at
the trench region and L1 is the length of the ITZ or of the locked lithosphere.
Fifth, plane strain assumption is applied in the calculation of accumulating shear
stresses along the locking area (the ITZ). Normal stresses σx , and σy , on the ITZ are
assumed to be smaller than σxy , such that thrust mechanism is possible there. It is
further assumed that the displacements on the ITZ are zero, then dy/dx is always zero
(parameters are defined in Fig 4), in which the ITZ is assumed to be a straight line and
corresponds to the x axis. As a result, stretching of a slab, which is assumed to be in the
homogeneous deformation in the section just below the ITZ, is able to pull the lower
segment of the slab abuting to the locked ITZ with its motion, resulting in the
underthrusting shear along the ITZ. The spring constant in the mechanical analog will
be a function of Yo ung's modulus E, Poisson's ratio ν, and the dimensions of the slab.
Then, an effective spring constant of E* applying to this earthlike system is defined,
and it is the ratio between the displacement and the shear force on the ITZ.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
The shear force on the ITZ is L1 σxy , where L1 is the length of the ITZ and σxy is the
shear stress on the surface of the ITZ per unit length of subduction zone. Thus, in the
plane strain, the shear force is equal to L1E/(I+ν) multiplied by åxy, where åxy is the
shear strain. Under the stretch of the slab pull force a locked slab in its ITZ sector
would be deformed as a simple shear and the magnitude of the strain is determined by
the stretching gradient across the subducted lithosphere (Fig 4). Therefore
εxy = (1/2) (du/dy) = (1/2) (∆u/L 2 )
where L2 is the thickness of the subducted lithosphere in the trench region and ∆u is the
displacement at the bottom of the lithosphere in the locked sector. Displacement ∆u
may represent the amount of the stretch of the spring in the mechanical analog.
The effective spring constant, which is defined as the ratio of the shear force and
∆u, then is given as:
E* = ( L1 / L2 )
E
= ( L1 / L2 ) µ
2(1 + υ)
(15)
where ì is the rigidity.
The governing equations for the motions of subducted lithosphere during the time
when the slipping occurs are:
m2 (d2 x/dt2 ) = E* (y + y0 - x) - fd m0 g + Fr
(16a)
(or, m2 (d2 x/dt2 ) + E* x = E* y + (fs-fd ) m2 g)
(16b)
m1 (d2 x/dt2 ) =Fp - 4η(dy/dt) - E*(y+y0 -x)
(16c)
where
m2 = the mass of the subducted lithosphere in the locked sector, which in
accelerated during slipping;
m1 = the mass of the unlocked and subducted lithosphere (ie. this mass is in
sinking);
m0 = the mass of the overlying margin, which contributes to the frictional
resistance;
Fp = slab pull force;
Fr = ridge push force;
η = the viscosity of the upper mantle;
x(t) = the slipping displacement of the subducted lithosphere;
y(t) = the sinking displacement of the subducted lithosphere;
y0 = a constant representing the extensional length at the moment of rupture of
the lithosphere, is (fsm0 g-Fr)/E*; and
t = time
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Following the same procedures we used in deriving equations (8) and (10) we
obtain solutions from equations (16) for y(t), x(t) and their derivatives.
y(t) = -[(fs-fd )m0 g/4η] ( m2 /E*)1/2 sin[t (E*/ m2 ) 1/2]+(Fr+Fp -fd m0 g)t/4η
(17)
dyldt = [(fs-fd )m0 g/4η] cos[t (E*/ m2 ) 1/2]+(Fr+Fp -fd m0 g)t/4η
(18)
x(t) = [(fs-fd )m0 g/E*][1- cos[t (E*/ m2 ) 1/2]
(19)
dx/dt = [(fs-fd )m0 g/ ( m2 /E*)1/2][sin(t (E*/ m2 )1/2)
(20)
Equations (19) and (20) describe the detailed motions of the plate during a
slipping event and are valid at times of 0 toπ (m2 /E*)1/2, when the slipping stops. The
force drop (from equation (19)) is given by 2(fs-fd )m0g, which is two times the force
difference between static friction and dynamic friction. The motion on the thrust zone
during slipping is independent of the subduction mechanisms such as slab pull, ridge
push and viscous forces (ie., the viscosity of the mantle) during the rupture. It is
primarily controlled by the force drop and by the effective spring constant of the
subducted slab. The rate of reaccumulation of strain will be determined by the
subduction rates which are scaled by the viscosity of the upper mantle.
The minimum subduction rate, Umin, is (Fr+Fp -fs m0g) t/4ç when the rupture
begins. The maximum rate, Umax, is [(Fr+Fp -fd m0g)+(fs-fd )m0g] /4ç when the rupture
stops. The total displacement of the unlocked subducted slab per slipping event is
[(Fr+Fp -fd m0 g)+(fs-fd )m0g]/4ç]ð(m2 /E*)1/2 from equation (17). The subduction rate is
increased by the amount of 2(fs-fd )m0g/4ç, which is the ratio of the force drop and the
viscous drag, for each slipping period.
The recurrence time of slip events subduction earthquakes can be derived in the
same way equation (14) was derived, with the following result:
Ät = (4ç/E*) In { l + [2(fs-fd ) m0 g] / [Fr+Fp -fs m0 g]}
(21)
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
L1, L2; THEIR RELATIONSHIP WITH SUBDUCTION PROPERTIES
The thickness of the lithosphere is a function of age and it can be estimated from
plate cooling model (Lister, 1975), or from viscoelastic relaxation method (Turcotte &
Schubert, 1982). Lithosphere acts as a thermal boundary layer of the earth, and
according to thermal model the thickness of the oceanic lithosphere can be
approximated as L2=10(tc ) 1/2 (Molnar et al., 1979; Davies, 1980), where tc is the age, in
the unit of million years, of the lithosphere at trench "region".
If the maximum depth of intraplate earthquakes in the oceanic lithosphere is used
to infer the thickness of elastic lithosphere, L2e , which will be decreased by a factor of
two according to the estimate of 10(tc ) 1/2 (Stein, 1987). Nevertheless, Liboutry (1987)
estimated the elastic thickness L2e to be about 4(tc ) 1/2.
The length of the ITZ occupies the region of subduction zone where stick-slip
motion of subducted slab is dominant. Byerlee & Brace (1968) and Stesky et al. (1974)
suggested that stick-slip behavior will occur at low temperatures (<250°C) and high
pressures (>3kbar) for gabbro and dunite. At higher temperature stable-sliding occurs.
The temperature distribution on the ITZ would suggest that at depths shallower than 50
km one would expect the existence of stick-slip motion of the plates (Bird, 1978).
However, it is also noted that stable-sliding behavior, which can occur at low
temperatures for carbonate and serpentine rich rocks, may be a factor on the thrust
interface. In addition, asperities, which exist due to irregularities of the contact
thrusting surfaces and rock type difference in the ITZ, can also complicate processes of
stick-slip motion on the thrust zone, such that the variety of the dimensions of the ITZ
is among subduction zones, or even in a subduction zone.
The depth to the lower end of the ITZ generally varies among island arcs but does
not extend below 70km (Kelleher et al. 1974). In the Aleutians, where great thrust
earthquakes have been studied in detail, Davies & House (1979) and Reyners & Coles
(1982) mentioned that the ITZ ends at a depth of 40 km where the subducted slab starts
to bend. Accordingly, Spence (1987, 1989) concluded that great thrust earthquakes
trigger at 20-40 km depth, just updip of the major slab bend.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
In the paper L1 of an ITZ is referred to be slant distance between the axis of the
trench and the place where the major slab begins to bend. This value is strongly
dependent on coupling strength between the subducted slab and overlying lithosphere,
and thus relates to the age of the subducted slab, the widening of the accretionary prism
and other environmental and mechanical factors.
The effective spring constant of a subducted slab, E*, is a function of L1/L2 and it
will vary among subduction zones such that the rate at which force is accumulated on
the locked ITZ is controlled at least by geometric properties of subduction zones (under
the consideration that the rigidity is site-independent).
Since E* is proportional to L1/L2 then it is inversely proportional to the age of the
oceanic lithosphere at the trench. In general, a young slab with a long and shallow
dipping ITZ (large L1) will construct a stiffer spring with larger value of E*, and the
rate of accumulation of shear force onto the ITZ will be higher than older steeply
dipping slabs with the same stretching rate.
PREDICTION OF RECURRENCE RATES
In the largest earthquakes the magnitude Ms (measured with the 20 second surface
wave magnitude) is saturated above Ms∼8. Thus Kanamori (1977a) suggested a new
magnitude scale for these large events, denoted by Mw , which is determined by the
seismic moment. The seismic moment W0 , is the product of ì, A and d; where ì is the
rigidity, A the fault area (estimated from the geometry of the fault plane and the rupture
propagation), and d the average seismic slip displacement.
The validity of the simplified model needs to be tested before prediction of
recurrence rates of great thrust earthquakes is made from equation (21). This will be
done by comparing estimated values of the rupture displacements (or distance of
coseismic slip) derived from the seismic moment of great earthquakes with estimates,
using the relation "rupture displacement = force drop/effective spring constant", from
this model.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
1. Coseismic slip & force drop
From equation (19) the rupture displacement, d, can be estimated from
2(fs-fd )m0g/E* for a slip event of duration Ät = ð(m2 /E*)1/2. Thus to induce a higher
seismic displacement we need a larger force drop (2(fs-fd )m0g) together with a smaller
effective spring constant (E*).
We need to know first the magnitude of the difference between the static and
dynamic frictional coefficients on the ITZ before we can estimate the rupture
displacement. In this model, the mathematical expression of the stress drop is
represented as 2(fs-fd )m0g/L1. If the overlying mass m0 and L1 can be well defined by the
rupture area of great thrust earthquakes, then the value of " fs-fd " can be obtained by the
observed stress drop.
For the 1964 Alaska earthquake, the estimated stress drop is 28 bar (Kanamori,
1970a), and mo is 1.38xl013kg (based on 180km long thrust fault), giving (fs-fd ) = 0.002.
For the 1963 Kuril earthquake, the estimated stress drop is 23 bar (Kanamori ' 1970b)
and the estimated mo is 4.5x1012 kg (Based on 110 km fault length), giving (fs-fd )=0.003.
Both regions suggest small values of " fs-fd ", which are of the same order and may
indicate that smaller effective stresses are induced by high pore pressures on the thrust
zone. This could be caused by the dewatering rates being smaller than, the subduction
rate of the sediment.
Using d=2(fs-fd )m0 g/E* and assuming (fs-fd )=0.003: the coseismic slip length of
the Alaska earthquake is ranged from 2.5m (for L1/L2 being 3) to 5.0m (for L1/L2c being
6). The Kuril earthquake of 1963 gives d value 2.0 to 4.0m for L1/L2 (or L1/L2c ) being
1.1 to 2.2. All estimates are in reasonable agreement with the slip displacements, d,
estimated from the seismic moments (Table 1).
2. Recurrence rates
To use equation (21) to calculate the recurrence time of large subduction thrust
earthquakes involves many "guesses" of the parameters. We can bypass some of the
guess-work by using data from actual subduction zones. For example, we can use the
seismic moment or the stress drop of large thrust earthquakes to estimate some of the
parameters. This can be done as follows:
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Table 1. The predicted minimum recurrences for the selected subduction zones, where the greatest
interplate earthquakes occurred. The values of Mw, which is derived from M w = 1.5 x
log(M0 )-10.7, and Ut are Ruff & Kanamori, 1980; Except for the Taiwan, 1966 earthquake
from Pezzopzne & Wesnousky, 1989. Rupture areas, A, are estimated according to McCann
et al., 1979. The values of L1 and L2 are estimated from the age of the subducted zones and
the geometry of the fault planes. Displacements, d, are derived from the seismic moments. Ät
(obs) is the observed recurrence time, and the values are complied from Mogi, 1968;
Kelleher, 1972; Kanamori, 1977b; McCann, et al., 1979; and Pezzopane & Wesnousky, 1989.
Ät (min) is the minimum recurrence time, and the values are calculated according to
equation (22) in the text. Total slip displacement, dt, is derived from Ut x Ät (obs).
Event
Mw
Ut
A
L1 /L2
4
2
(km/My) (10 km )
d
(m)
Ät (obs) Ät (min) d/Ut
(years) (years) (year)
d/dt
(%)
Alaska, 1964
9.2
59
18.0
3.1
.6
130
89
95
72
E. Aleutians, 1957
9.1
75
14.3
1.7
5.0
100
65
67
67
W. Aleutians, 1965
8.7
75
5.0
1.4
3.6
100
47
48
48
Kamchatka, 1952
9.0
93
10.2
1.9
5.0
100
52
54
53
Kuril, 1963
8.5
93
3.8
1.5
2.4
45
26
26
57
Colombia, 1906
8.8
77
4.8
2.2
5.3
>60
66
69
<115
Chile, 1922
8.5
110
3.3
1.4
27.0
40
24
25
61
S. Chile, 1960
9.5
111
18.0
4.5
15.8 85-130
123
142
>100
Taiwan, 1966
7.8
68
0.7
2.0
1.1
16
16
95
17
(a) The hydrostatic stress and the frictional resistance on the rupture zone are
largely unknown. However, if the slip displacement, d, during a large earthquake is
known, it can be used to replace 2(fs-fd ) m0 g/E*. Displacement d can be obtained from
the seismic moment (M0 =ìAd) if we can independently estimate A, or the observed
stress drop derived as in the previous section.
(b) The term Fr+Fp -fs m0g is the minimum viscous drag, which is determined from
the viscosity of the mantle and the sinking rate of the subducted slab. This value is
equal to 4çUmin where Umin is the minimum sinking rate of the subducted slab and it
occurs as soon as the ruptue is triggered; or is equal to 4çUmax minus E*d, where Umax
is the maximum sinking rate occurring as the rupture is ended.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Equation (21) can therefore be given by:
∆t = [4η/E*] ln[1+E*d/4ηUmin ]
(22)
∆t=[4η/µ[L2 /L1 ] ln [ l + (Umax - Umin )/Umin ]
(23)
or
The problem in applying (22) or (23) to real subduction zones is that Umin or Umax
is not measurable. What is well determined is the average convergence rate (Ut ) from
magnetic anomalies. If Umin is replaced by the observed convergence rate (Ut) in
equation (22) the estimate of ∆t will be a lower limit of recurrence time, and great
earthquakes will occur at the time interval greater than the estimated value.
Using seismic moments and thrust distances for the largest thrust earthquakes
(McCann, et al., 1979), we have computed minimum recurrence times predicted by
equation (22) assuming E=160 Gpa, ν=0.25 (or µ=0.64x1012 dyne/ cm2), η = 1021 pa s,
and Umin =Ut . These were compared with the observations, and are shown in Table 1.
The relationship between the calculated and the inferred recurrence times suggests that
the model has a sound physical basis.
The 1966 Taiwan earthquake is located north of the presumed westward extension
of the Ryukyu trench (Fig 5), along which the Philippine Sea plate has subducted
beneath the Eurasian plate. Its mainshock occurred at the depth of 22 kilometers, and
focal mechanisms shows P axes oriented near horizontal trending normal to the
presumed axis of the trench (Pezzopane & Wesnousky, 1989), indicating the thrust
mechanisms. The aftershocks occurred and propagated updip along the thrust plane to
form the rupture area of the event that is indicated in Fig 5. Using the seismic moment
M0 of 48.6x 1026 dyne cm and the estimated area A of 7000 km2 (Rowlett & Kelleher,
1976; Fezzopane & Wesnousky, 1989), a slip displacement of 1.1 meters and
recurrence time of about 16 years are calculated (Table 1).
It is interesting to know that another thrust earthquake occurred in the year of
1983 at almost the same place with the 1966 earthquake, but with smaller magnitude
(M0 = 6xl025 dyne cm) and rupture area (Fig 5). Recurrence time of 17 years was
indicated in this region, as compared to the time interval of 16 years predicted by the
model. However, a smaller 1983 earthquake may imply that the rate of seismic
recurrence should be nonuniform on various time scales along a subduction zone.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Fig 5. Rupture regions of 1966 (dashed line) and 1983 (thin solid line) earthquakes near Taiwan. The
regions were determined by Pezzopane and Wesnousky (1989) who mapped all aftershocks (the
pluses) that occurred 48 hours after the main shocks (the stars). The Ryukyu trench is indicated
by solid lines with large open bars on the overriding plates. Arrow and the near-by number
indicate the relative plate motion between the Philippine and Eurasian Plates, the motion that
provides the converging mechanisms triggered the 1966 and 1983 earthquakes (Pezzopane and
Wesnousky (1989)).
CONTROLLING PARAMETERS FOR
THE OCCURRENCE AND RECURRENCE
1. Implications of the model
If all the convergence takes place by stick-slip motion (as compared to aseismic
subduction) then the recurrence rate of thrust earthquakes should be d/Ut . In other
words, the motion at a subduction zone is totally taken place by a series of ruptures
(with the distance of d) separated in time. Defining Us the seismic slip rate which is
given by the estimated slip displacement for a thrusting event divided by its observed
average recurrence time, and there should be equality between Us and Ut, according to
the model.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
Displacement, d, estimated from seismic moment may be only a fraction of the
real plate motion and plate motions are not so easy to assess in subduction zones. Thus,
the estimates of d/Ut in the Table 1 may not reflect the true recurrence times or the
equality between Us and Ut has some uncertainties:
(a) Not all of the plate motion occurs seismically (Ruff & Kanamori, 1983; Stein,
1987), resulting in Us being less than Ut and
(b) Not all of large shallow subduction earthquakes were thrust events reflecting
plate motion. These earthquakes may include the normal and strike-slip faulting events,
such as 1977 Sumba normal-faulting earthquake (Spence, 1986). In this case, Us is less
than Ut .
It appears that only a fraction of the plate motion is released seismically and
shown as the thrust events, and
dt = d + da + dns,
(24)
where dt is the total slip displacement reflecting a real plate motion, d is the slip
displacement for the thrust seismic events (the estimates given in the Table 1), da is the
aseismic slip displacement, and dns is the slip displacement for the normal and
strike-slip seismic events.
As a result, there is a possibility that the predictions made by d/Ut or equation (22)
were underestimated (as compared to ∆t(obs) given in Table 1). Alternatively, Ut should
be an upper bound on Us.
However, southern Chilean subduction zone, for example, is the other extreme.
The predicted recurrence time estimated from the slip, d, in the great 1960 earthquake
(Table 1) may exceed the recurrence time (130 years) of major earthquake occurred in
this area in the last 400 years (Kanamori, 1977b; McCann el al., 1979). One possibility
is that the slip may vary as the function of time and position along a subduction zone,
such that the seismic slip is overestimated.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
If the indentity (24) holds, the observed recurrence times should be equal to the
value of dt/Ut (or d/Us). Unfortunately, neither dt nor d (even Us) can be measured
directly. However, if dt is approximated by the product of Ut and ∆t(obs), as high as
50% of the total slip that released in the form other than the thrust event in the
subduction zones is indicated by the values of d/dt in Table 1. In addition, young
subduction zones, in the Colombia and the Chile, for instance, they seen to favor the
pure stick-slip motion (d/dt =100%).
In the identity (23) if ∆t is set to be equal to d/U (for stick-slip process only, then
U= Us= Ut ), then the model indicates that Ut is related to Umin and Umax in the following
form:
U =( Umax - Umin ) / ln(Umax / Umin )
(25)
Strongly locked slabs are characterized by a large difference between Umin and
Umax. In a totally decoupled slab without the stickslip process, U can be equal to Umax
and Umin (approximated by Umax ≅ Umin in the identity (25), then U ≅ Umax ≅ Umin ), and
U is the so-called terminal velocity (or steady-state sinking velocity), and ∆t is equal to
zero.
In the identity (22), ∆t can be approximated as d/Umin only if E*d (force drop in
earthquake)<<4η Umin (minimum viscous drag from the mantle). It is noted that
recurrence rates ∆tmin calculated from equation (22) using the average convergence rate
U are almost near the values estimated by d/Ut , except for the subduction zones of
higher L1 /L2 value (at the Alaska and southern Chile). It thus indicates that the
possibility that the force drop is much less than the viscous drag (or that most of the
available driving force is absorbed by viscous drag and not by frictional resistance on
the ITZ).
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
2. Controlling parameters
The ratio L1 /L2 plays an important role in the subduction earthquake process. It
basically determines the effective spring constant of the subducted slab. Larger ratios
produce stiffer springs and smaller ratios correspond to weaker springs. Larger ratios,
which cause a smaller recurrence interval of earthquakes, usually correlate with young
subducted slabs which generally have a large mass overlying their broad interface
zones. It is expected that young and fast convergence subduction zones are the
boundaries with highest potential for impending great thrust earthquakes, resulting
from the large force drop being achieved quickly and easily in the strong slab. In older
subduction zones, even though they may have a large overlying mass, such as
Kamchatka subduction zone, their potential is offset by their longer recurrence intervals
(the stretching of a weak spring) unless they are fast in subduction rates.
If stick-slip motion is the only process in the ITZ (so U=Us=Ut , and Davies and
Brune (1971) showed that this equality generally holds in the subduction zones Ut
greater than 5 cm/year), to review the primary physical parameters controlling the
recurrence rate we note that
∆t=d/U=2(fs-fd )m0 g/(E*U) ∝ L2 h/U
(26)
The overlying mass m0 is estimated by the product of ρ, h and Ll , where h is the
depth to the lower end of the ITZ from the surface of the earth and ρ is the density of
the mass m0.
Following the relationship (26), it is apparent that recurrence time of the thrust
earthquakes is inversely proportional to the convergence rate, and proportional to the
age of the lithosphere at the trench. It predicts that ∆t, although, is a weak function of
the length of the ITZ (L1 ), but depends upon the depth to the lower end of ITZ, h, a
parameter that is determined by the coupling behavior between a subducted slab and its
overlying plate.
It also shows that slip displacement is proportional to the product of L2 and h.
Therefore, the enhancement of the slip displacement is favored in subduction zones
with a thicker elastic lithosphere and a deeper slab bend. However, a thicker lithosphere
is easily characterized by a steeper dip (jarrard, 1986) such that the value of d is the
result of a counterbalance of the dip and the thickness (or the age) of the slab in the
"trench" region.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
The importance of overlying mass can be stated: Occurrence of great thrust
earthquakes is primarily governed by the mass loading on the ITZ. A larger weight is
able to generate a greater earthquake with possibly longer recurrence times, and the
length of the ITZ will be a determinant parameter for the occurrence.
Great earthquakes favor to occur in a subduction zone with a long interface zone
such that A (rupture area) is large enough to produce a high force drop or M0. The
heavier load has the tendency to prolong the time of recurrence. To shorten the time of
the recurrence due to the loading effect, two factors then are incorporated such that
highest potential of greatest thrust earthquakes can be achieved. They are by decreasing
L2 and increasing U, a feature of young, fast subducted slabs.
CONCLUSION
A simplified model of the subduction process that includes stick-slip behavior in
the thrust zone where major subduction earthquakes occur is given. This model
generates recurrence intervals for the slip events that are in reasonable agreement with
the observations, indicating that the model has a sound physical basis.
In this model the subduction rate is considered to be primarily controlled by the
slab pull force and the viscosity of the asthenosphere. The stick-slip behavior arises
because the time taken by the slab pull force to accumulate sufficient stress on the
thrust zone to overcome frictional resistance is very long compared to the time taken to
release the stress in the form of an earthquake.
The model predicts that slip recurrence intervals will be smaller with larger
subduction rates and younger slabs, and that the mass overlying the slip zone will
increase both the force drop and recurrence interval. Subduction zones with larger
amount of sediments, which are lithified and have substantial shear strength, may be
expected to have larger recurrence intervals and larger earthquakes.
Nevertheless, the model indicates the ridge push has no effect on the magnitude or
details of the slipping motion. This body force has the tendency to decrease the amount
of extensional strain in the subducted slab needed to trigger the slipping and in
increasing the subduction rate. However, it can not determine these magnitudes. For
instance, the slab being pushed by a large ridge push force is not going to be
characterized by a faster subduction rate.
Acta Oceanorgraphica Taiwanica
No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
ACKNOWLEDGEMENTS
This study has benefited from the useful discussions with T. Keffer, R. Sternberg
and L. Ruff. During the course of the research G.S. Song had been partially supported
by the National Science Council of the Republic of China for his graduate study in the
University of Washington, Seattle, Washington, USA.
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No.25, PP. 1-18, 1 Table, 5 Figs., March, 1990
一個簡化的下衝板塊拉力模式：
對板塊間隱沒性大地震發生及其發生循環之檢視
宋國士
摘
路易士
要
現今隱沒作用範本的主要營力是由於較冷且重的板槐，沉入較熱且輕的地
函。這個力將在地函中受到黏性力及在大地震發生之「海溝」地域內受到摩擦力
的阻擋。本報告的簡化模式，對於這種隱沒過程的表現，是一個靜置於平板上的
質體(m2 )，其一端由彈簧(彈性常數為 k)連接至一個沉入黏性係數η流體的質體(m1 )
所產生的拉力運動。置於平板上的質體 m2 ：和平板間的靜摩擦係數為 fs ，動摩擦
係數為 fd 。於物理和數學的推演上，這簡化模式提供了質體 m2 週期性的瞬間滑動
行為，此行為可代表隱沒作用中「黏附一滑勳」的運動過程。
黏性係數η為控制隱沒速率的最主要因素，楊氏模數 E (即彈簧之彈性常數為
k)決定瞬間滑動之位移量。另外η/E 比值可初步估計瞬間滑動行為的再發生率。
當這簡化模式代入地球參數系統中，它能合理地預估隱沒帶大地震發生的週
期時間，並發現其和觀測的地震週期相吻合。這模式並可進一步對控制發生循環
的臨界因素給予一有效的探察。