dynamical models of earthquakes

PHYSlCA ELSEVIER Physica A 229 (1996) 530-539

Dynamical models of earthquakes Hiizu Nakanishi 2

Department of Physics, Faculty of Science and Technology, Keio University, Yokohama 223, Japan

Abstract

Recent results on the uniform one-dimensional chain of spring-block model for earthquakes are reviewed. A two-dimensional extension of the model is also discussed and some important parameters are estimated.

1. Introduction

Earthquakes are, as we all know, one o f the most drastic phenomena that are hap-

pening on the earth. Due to the development o f plate tectonics, basic mechanism of

earthquakes is now understood as the fracture phenomenon in the earth crust. Most im-

portant questions for ordinary people are, o f course, when and where big earthquakes

will occur, but it is fair to say that answering to such questions is still not possible in practical sense.

On the other hand, earthquakes have some interesting features if they are regarded

as a physical phenomenon, and there arise more academic questions to ask for better

understanding of the phenomenon.

(1) Why are there many power laws in the earthquakes? - One of the most well-

known properties is the Gutenberg-Richter law [1], which says that the smaller the earthquakes are, the more frequent they occur, and the frequency distribution of the

earthquake size follows a power law. Another power law holds for the frequency decay

of aftershocks after big earthquakes; It has been observed that aftershock rate decays as 1/t in most cases (Omori law)[2]. It is also claimed that epicenters of earthquakes form

a fractal structure which leads to a power law for the spatial correlation o f epicenter.

Invited paper presented at the Second IUPAP Topical Conference and Third Taipei International Symposium on Statistical Physics: Nonlinear and Random Processes, 18-24 July 1995, Academia Sinica, Taipei, Taiwan. The Proceedings has already been published in Physica A 221 (1995) Nos. 1-3. 2 Present address: Department of Physics, Kyushu University 33, Fukuoka 812-81, Japan.

0378-4371/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved Pll S0378-4371 (96)00040-4

H. Nakan ish i / Physica A 229 (1996) 5 3 ~ 5 3 9 531

Since, the power-law distributions imply lack of characteristic scale, there may be some mechanisms like Self-Organized Criticality which brings the system into a "cri- tical state" as has been claimed by Bak et al. [3].

(2) Why do earthquakes occur irregularly? It is obvious that real faults have com- plicated structure and that makes earthquakes very difficult to predict, but there seems to be intrinsic dynamical instability in the system and this may also cause unpredictability. The earthquakes system may be a deterministic spatio-temporal chaotic system.

(3) How does the earthquake rupture propagate? - Earthquakes usually occur along pre-existing faults, thus they may give a simple example of fracture problems. In the simplest case, namely, earthquakes in isolated straight fault, we can study basic

problems of fracture propagation. In this report, we will present the simplest possible model of earthquakes to address

these questions more specifically and review briefly what we know about it. Then we will present an extension of the model to improve some of its short comings.

2. Simple picture and minimal model

Let us start by discussing the simplest picture for earthquakes we can have. The earth consists of several layers of shells which have different physical and chemical

properties. The outer most shell of the earth is elastic and called the lithosphere. The lithosphere

is being driven by convective motion in the viscous layer called asthenosphere under that. Since different parts of the lithosphere are driven in different directions, it is broken up into a number of tectonic plates which are moving typically a few centimeters a year, and stick-slip motion is being taken place occasionally along the plates boundaries. Thus, the simplest picture of earthquakes is this stick-slip motion along the preexisting

fault. Now we try to construct "a minimal model" of earthquakes (Fig. 1 ). Consider a one-

dimensional chain of blocks (mass m) and springs (spring constant k,), and suppose it represents the elastic stripe of the crust along the fault which undergoes the stick-slip motion. The system is being driven very slowly (at speed vp) through leaf springs (spring cons t an t k p ) as a plate moves due to the convective flow in the asthenosphere. For simplicity, the other plate on the opposite side of the fault is assumed to be fixed. Friction force operates between the plate and the blocks.

The equation of motion for the system can be written down easily as

mii i k c ( u i - i - 2u i ÷ ui+l ) - k p ( u i - v p l ) - F ( t J i ) , (1)

where ui is the position of the ith block and the dots denote the time derivative. Note that we have assumed the system is completely uniform, therefore the mass and the spring constants do not depend on i. This is, of course, not true for a real faults, where inhomogeneity should play important roles, but we will see even this "oversimplified"

532 H. Nakanishi/Physica A 229 (1996) 530-539

[ t

Fig, 1. One-dimensional spring-block system.

model is capable to show a complex behavior similar to real earthquakes and, we believe, capture some essence of the phenomena.

The friction term is important for the stick-slip motion because Eq. (1) would be linear without it. The friction consists of two parts, namely, static friction and dynamic friction. The static friction can take any value up to the maximum static friction F0 and the dynamic friction is smaller than the maximum static friction. We assume the dynamic friction depends only on the velocity and is velocity weakening, i.e. as the speed of block becomes large, the friction decreases. We know that the velocity- dependent friction is not true for many cases, but it should be noted that the friction function used here is a phenomenological one. We will not discuss it any further.

This type of spring-block model was originally invented by Burridge and Knopoff about 30 years ago [4] and is called Burridge-Knopoff (BK) model, but the simplest version of it that we consider here was studied very carefully by Carlson and Langer, and others recently [5, 6].

3. Properties of BK m o d e l

3.1. Parameters in the model

There are some parameters that characterize the system behavior. The time scale

~d - ( 2 )

is the duration time of a big event at a certain point along the fault. This is order of 1 second in a real earthquake. Typical displacement of a big event is given by

F 0 D =_ kp (3)

which is order of 1 m. Thus, the slipping speed vs during a big event is order of

D vs - (4)

H. Nakanishi/Physica A 229 (1996) 530 539 533

An important dimensionless parameter is the ratio v of the loading speed Vp to the slipping speed vs,

Vp v = (5)

Vs

and this is as small as 10 -8 in the real earth. Due to the smallness of this parameter, the duration time of earthquakes is virtually zero and the whole system is stuck for almost all the time and earthquakes finish instantaneously.

As for the velocity weakening friction function F(v), we use the simple form:

( - o c , F0) , (v = 0 ) , F(v) = Yo (v > 0). (6)

1 + 2~(v/vs) '

The parameter ~ in F(v) characterizes the initial negative slope of F(v); Large represents strong dynamical instability and small :~ represents weak instability. We assume c~ to be order of 1.

There are three length scales in the model. The block spacing a sets the smallest length scale in the model. One may think this does not play any role in the "macro- scopic" scale, but it turns out that the existence of the smallest length scale is important in this model [6]. This length scale may be of the order of meters in the real crust.

The length of the whole system L corresponds to the fault length, which is order of 100 kin. If L is large enough, the system behavior does not depend on L although many real faults in the earth may not be in this regime.

The last length scale ~ defined by

= V~pp ( 7 )

is the distance that the sound wave travels during the slipping time zj. This is order of l0 kin.

3.2. System behavior

Due to the dynamical instability caused by the velocity weakening friction, the uniform stable creeping motion is unstable and any disturbance will be enhanced to realize the stick-slip motion. It can be seen also that the uniform stick-slip motion where all the blocks slip synchronously is unstable.

For almost any initial configurations the system falls into the statistically stationary state after a transient period. A typical behavior of the system is shown in Fig. 2, where the time sequence of the configuration at every sticking time is plotted. The lowest line represents an arbitrarily chosen initial configuration and the system is being pulled upwards. Every time any part of the system slips, a new line is added. It is obvious that the time sequence of the system behavior is spatio-temporally chaotic. Remember that the model does not have any randomness, thus these irregularities are the results of the dynamical instability caused by the velocity weakening dynamical friction.


5

4

"~3

(a)

(b)

E

- I ~ ) i i i ii 0 100 200 300 400 500

B l o c k N u m b e r

Fig. 2. System configurations at every sticking time. (a) (c) show enlarged plots in the dotted circles in the previous ones.

The configuration inside the circles ( a ) - (c ) shows the enlarged view of the parts enclosed by the dotted circles. It can be seen that the events down to the smallest length scales are occurring.

The size of a event can be measured by the area enclosed by the lines that represent configurations before and after the event (ui and u~, respectively), namely, the seismic moment ms in this model is

m , - - a ~ _ ~ ( u ~ - u i ) (8) i

and the magnitude #s is defined by its logarithm:

#s = in ms • (9)

The distribution of magnitude is shown in logarithmic scale in Fig. 3. The distribution consists of two parts: a straight line in the small magnitude side and a peak in the large magnitude side. The straight line represents power-law distribution of the moment, which implies a scaling behavior. The slope of the scaling region is close to 1 and does not change for certain range of the model parameters. This scaling feature in the magnitude distribution is seen in real earthquakes and known as the Gutenberg-Richter law [1]. It can be shown that the scaling region ends at the magnitude fi,

f i = l n ( ~ D - ) . (10)

For the events whose magnitude is larger than this, inertia effects dominate the dynamics and the scaling behavior is not seen. There is no analytic theory of the scaling behavior itself.

Fig. 4 shows how a big event develop in time. After the initial shock breaks at the epicenter, the configurations are shown at equal time intervals. The series of vertical

H. Nakanishi/Physica A 229 (1996) 530 539 535

:zL

~4 C

0

-5 L

OD

- 1 0 ~ ' ' ' J - 1 0 - 5 0 5

ta

Fig. 3. Magnitude distribution in the semi-logarithmic scale.

0.5

E o.o

-o.5

X~

-1.0

I I

, ,' i' ',

I

0 50 100 150 Block Number

Fig. 4. Time evolution of a big event. Configuration at equal interval of time are plotted. The arrow indicates the epicenter.

lines shows that the two slipping pulse propagate with very narrow fronts in the oppo-

site directions and the propagation speed has been analyzed by [7] using the marginal stability condition.

4. Two-dimensional extension of BK model

As we have seen the simple uniform BK model shows many aspects of real earthquakes, but there are also limitations.

Since the model is one-dimensional, there is no stress concentration at the propagation front which should have important effects on the dynamics of front propagation. There is also no long-range interaction between different parts of fault through stress field in the plate. Energy radiation into the plate does not exist in this model.


4I~i ! ~ '

I ll , , ~ .

0 : i'ii~i'~!i' i' '. i , , 1 0 100 200 0 100 200

B l o c k P o s i t i o n

Fig. 5. Time sequences of events for (a) 1-D BK model and (b) 2-D visco-elastic coupling model.

The most obvious one, however, can be found if you look at the time sequence of

events. In Fig. 5(a), you can see prominent accumulation o f foreshocks before large

events and virtually no afiershock after them, which situation is just opposite: in the real

earth there are very few foreshocks, if any, and a lot o f aftershocks whose frequency

decays as 1/t [2]. This is because BK model does not have any relaxation mechanism

during the sticking periods.

In order to improve these aspects, we will construct a two-dimensional extension

of the BK model. Regard a plate as a two-dimensional elastic sheet that occupies the

upper half plane and suppose it is driven by the visco-elastic coupling with the viscous

flow under it, then the equation o f motion in the continuous limit is

t

/ (y > 0), (11) P

ii(t,x, y ) = c2V2u(t ,x, y ) -

o ~

where F(t) is a memory function with the normalization

O<3

/ d t F ( t ) = (12) 1

0

and 7 is a viscous coupling constant. We assume F(t) is a decreasing function o f t

with the height F(0) = 1/Zve and the width o f Zve. The viscous fluid is assumed to behave elastic for the time scale shorter than Zve. The boundary condition along the fault at the plate boundary at y = 0 is g]ven by

•u = f ( f i ( y = 0 ) ) , (13) '//' ~YY y=O

H. NakanishiIPhysica A 229 (1996) 530-539 537

where p is the elastic modulus and f0~) is the velocity dependent friction stress

function. The sound speed c is given by

= ~pp, (14) c

with p being the mass density of the elastic sheet. During the sticking time, fi would not change very much over the time scale of ~ .....

thus Eq. (11) can be approximated as

y) - 2v2,(t,x, y ) , (ls

where we have ignored the inertia term. This is a diffusion equation for the plate deformation.

During the slipping events, fi is large only for the time scale much less than ~,,e, then Eq. (l l) is reduced to

ii(t,x, y) = c2V2u(t,x, y) -- 7- (u(t,x, y) _ u(t = O,x, y) ) , (16) ~ve

where we have assumed a slipping event starts at t = 0 and ignored the vp term. Eq.

(16) is a simple 2-D extension of BK model. A discretized version of (15) and (16) was studied by the author [8] and it was

shown that the time sequence of events is completely different from that of the one- dimensional BK model and capable to show the Omori law due to the relaxation

processes inside the plate (Fig. 5(b)) although the magnitude distribution remains the

same (Fig. 6). The parameters in the model can be estimated as follows. The system width W over

which the plate deformation extends can be estimated by the stress balance equation between the maximum friction and the viscous driving:

F0 W ~ - - (17)

pTvp

From Eq. (16), it can be shown that a big event is characterized as follows; The displacement D and the duration time at a certain point ~j are

p D ~ F o ~ c , ~ , (18t

z- d ~ ( 1 9 )

and the width of the plate w where the stress is relaxed during the big events is given

by

w ~ c . (20)

538 H. Nakanishi/Physica A 229 (1996) 530 539

£ _

2

0

-2

-4

-6

- 8 -"

-10 -10

+ "~'+~'++ C~c~ +~ o

++ + +

+4+~+t,I. -I+ + + +

__d I I I I I _I ~- +1 4. i , r , , J

+ + +

+

, , , , I , , ~ J I ~ , , , I

-s g 0 s

Fig. 6. Magnitude distributions for 2-D visco-elastic coupling model for c2/?, =0(top), 0.1, and 0.5 (bottom).

It should be noted that the width w is the distance that the sound wave travels during

the duration time ra:

w ,-- Tac, (21 )

which corresponds to ~ in the BK model (7).

From Eqs. (17) - (20) , the viseo-elastic relaxation time q,e can be expressed as

"eve " - ' 7J d , (22)

with T - D / v p being the interval between the big earthquakes.

The diffusion length Waiff over which the displacement deformation diffuses during

the sticking time T is given by

W~iff :- ~ x / W ~ . (23)

These parameters, r~e and Wjffj should be important to understand the earthquake phe-

nomena and it is worth noting that if we assume

W ,--5000 km, D--- 1 m, T ,-~ 100 yrs, za "~ 1 s, c ,-~ 10 km/s ,

Tu and Wj~. can be estimated from Eqs. (22) and (23) as

r~,e "" 2.5 months, Wji /7 "~ 200 km.

Such estimate may not be very accurate, but gives us some feeling on the phenomenon.

H. Nakanishil Physica A 229 (1996) 530 539 539

Acknowledgements

A part o f the r ev iew is results o f c lose col laborat ion with J.S. Langer , J. Carlson,

C. Tang, B. Shaw, and C. Myers.

References

[1] B. Gutenberg and C.F. Richter, Ann. di Geofis. 9 (1956) 1. [2] F. Omori, J. Coll. Sci. Imp. Univ. Tokyo 7 (1894) 111; T. Utsu, Geophys. Mag. 30 (1961) 521. [3] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381; Phys. Rev. A 38 (1988) 36. [4] R. Burridge and L. Knopoff, Bull. Seismol. Soc. Am. 57 (1967) 341. [5] J.M. Carlson and J.S. Langer, Phys. Rev. Lett. 62 (1989) 2632; Phys. Rev. A 40 (1989) 6470. [6] J.M. Carlson, J.S. Langer, B.E. Shaw and C. Tang, Phys. Rev. A 44 (1991) 884. [7] J.S. Langer and C. Tang, Phys. Rev. Lett. 67 (1991) 1(/43. [8] H. Nakanishi, Phys. Rev. A 46 (1992) 4689.

dynamical models of earthquakes

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