Volume 62, Number 22 P H Y S I C A L R E V I E W L E T T E R S 29 May 1989

Properties of Earthquakes Generated by Fault Dynamics

J. M. Carlson and J. S. LangerInstitute for Theoretical Physics, University of California, Santa Barbara. California 93106

(Received 6 March 1989)

We present a model for fault dynamics consisting of a uniform chain of blocks and springs pulledslowly across a rough surface. The only nonlinear element of our model is a slip-stick friction force between the blocks and the surface. We find that this model gives rise to events of all sizes. Our numericalevaluation of the distribution of earthquake magnitudes results in a power-law spectrum similar to whatis observed in nature. Like certain other dissipative dynamical systems, the observed large fluctuationsin earthquake magnitude persist because the system is in a state of marginal stability.

PACS numbers: 9l.30.Px, 05.40.+J, 05.45.-rb

Systems which have no intrinsic time scales or lengthscales have been the subject of much investigation in recent years. While extensive work has been done tocharacterize certain features of these kinds of systems,until recently there has been little success at providingdetailed mechanisms for their evolution. In a series ofpapers Bak, Tang, and Wiesenfeld1 and, later, Kadanoff,Nagel, Wu, and Zhou2 studied various cellular automata, loosely referred to as "sandpiles." In these models,sand is dropped slowly, one grain at a time, onto randompositions on a lattice. Numerically they found that, formany different rules of evolution of the pile, the distribution of avalanche sizes displays a power-law spectrumand that, at any given time, the configuration of sand onthe pile has a fractal geometry. Bak, Tang, and Wiesenfeld concluded that the observed large fluctuations inavalanche size and the spatial self-similarity result fromthe fact that the attractor of the dynamics is marginallystable; in essence the system is perpetually in a criticalstate. They introduced the term self-organized criticality to describe this behavior.

To the best of our knowledge, the model for earthquakes that we shall describe here is the first demonstrated example of self-organized criticality in a deterministic continuum dynamical system. This model hastwo main advantages compared to the cellular automata.First, because we are working with a system ofdifferential equations, certain analytic results can be obtained. Second, there is no explicit randomness in themodel; instead, the system evolves deterministically fromirregular initial conditions.

The system is illustrated schematically in Fig. 1(a).Several variations of this model were studied previouslyboth numerically and experimentally by Burridge andKnopoff." The model consists of N blocks of equal massm located at positions Xjit) along the x axis, which weimagine to be the axis of a lateral fault. Each block isconnected to its two neighbors by harmonic springs ofstrength kc and, when there are no additional forces onthe system, the equilibrium spacing between blocks is a.The blocks are pulled individually forward through elas

tic couplings (leaf springs or torsion elements) ofstrength kp, which move at constant loading velocity i\This loading mechanism corresponds to slow steady deformation far away from the point of. contact of two tectonic plates whose linear elastic properties are describedby the spring constants kc and kp. Finally, each block issubject to a friction force F(X) which depends only onthe velocity of the block. The important nonlinearities inthe problem are contained in the function F(X) illustrated in Fig. 1 (b).

The equations of motion for this system are

mXj = kc (Xj +, - 2Xj + Xj-1)

- k , , ( X j - v t ) - F ( X j ) . ( 1 )The rest of this Letter will be devoted to describing somesolutions of these equations. Note that, unlike manytraditional earthquake models, Eq. (l) contains no explicit spatial inhomogeneity4-5 or memory effects.6-9 Instead we will show how the dynamics alone is sufficientto generate a wide spectrum of events.

A trivial solution to Eq. (1) is given by uniform

(b)

FIG. I. (a) Block and spring system and (b) friction forceF(X). For the numerical results given in Figs. 3 and 4 the parameters take the values w = l, Af=2500, ftp"40, a = \. andr=0.01. For the friction force we use F(X) =F0/(\ +X) forX>0 with Fo-50.

2632 1989 The American Physical Society

FC93

•jpnjs sq 01 usddEq qsiqM apis jsqjis uo sjjooiq ussMjsqs|oqM b sb sdijs sjjoojq jo dnojg siqj jBqj siunssE pue 'Xi-|booi uoijEjngijuoo snosusitouioq X|jBijEds b ui s>po|q up dnoj§ b jspisuoo saa 'japoiu aqj jo uoisjsa psyi|dunsb guuspisuoo Xq—siqdojjsBjss puB dssjs—sjusasjo ssdXj omj aqi szusjsBJEqs ubo _m 'Xi|BsijX|Euy

paads punos aqj XisjBiuixojddB jb XjBpunoq aqi 01 Xbmaqi ijb saiEi.Bdo.id qojqM juojj .poqs b sjsSSuj jusas pazi-[bdo[ b ajduiExa siqj u\ (q)£ -3ij ui UMoqs iBqi sb qonssajjBnbqyBa jsaj3 Xjnjj aas 3m '.ja japjo jo X|q3nojXouanbaij sSejsab ub qjiM 'X||buo;sb330 'IUsas 3uid-dqs uuojiun b joj z '8ij ui UMoqs ssoqj oj sfqBJEduiooSSIJIOOJSA UIBJJB S1U3A3 Dll{dOJJSDWD SSSqj Ut SJBdlSlJJEdIBqi s-pojq sqj puB 'sipousd iou sjb (b)£ -gij u; UMoqsos[B 'siusas js3jb-] -ssjbss js3jb| uo uisjsXs aqi qjoouisoj idb puB Xjioojsa Sutjind aqi uibjjb jaAau siusas ddduosssqj, '5|00[q suo Xjuo 3uia|oaui suoijoui oipousd ijelus3JB (E)e '8lJ Ul UMOqS SJU3A3 1S3JIBU1S SqjL 7 SUIIJ pUBf uoijisod jo uoijounj b sb ty Xjioojsa sqj jo sqdEj3 joujjoj sqj ui e -3ij ui psjBJjsnj'ii sjb qoiqM jo aiuos 'sjuaAajo s3ubj apiM b paAJasqo 3m 'ajBjs XpBsjs Xijboijsijbjsb aq oj pajBaddB JBqM psqsssj uiajsXs aqj Jajjv uoij-ouj oijbjs uinuiixBiu aqj paaaxa oj qSnoua jbj paqajajjsaq oj s8uujs 3uijjnd aqj joj ss^ej ji suiij uinuiixBiuaqj si qoiqM ldya/0jzmm7i spousd SuipEOj U91 *PWHH-IXOlddB JOJ 3AJ0A3 OJ UISJSXS 3qj p3M0||B 3M 'SOIJSIJBJS3ui|iduioo us3sq sm sjojsg "XjiauaSouioqui JBijBds ||buisb qjiM uoijBJn3guoa .pnjs Xjjnj b ui luajsXs aqj pajJBjs3m 'X[|Bjau39 Q|-sjins3J jbjiiuis pjsiX qjog -jsejuoojb sojoj sqj duiBp qsiqM suoijipuoo Xjspunoq puB '(spu3Xjdiuis uisjsXs sqj) XjBpunoq sqj jb ssjoj ojaz si ajaqjqoiqM ui suoijipuoo XjBpunoq psjspisuos a^ s^oojq002 oj dn sszis luajsXs pus 'sjajaiuBJBd aqj jo sssioqojusjsjjip snouBA joj XuBsusuinu (i) psAjos saa

"XjJUSJSISJSd p3JBJ3U33 3JB SJU3A3jsjibuis 'snqx '3u.PHs 3JB X"aqj sjiqM pstjqduiB sq jjimsjfoojq Suiddqs jo dnoj3 sqj ui ssuisusSoiuoqui |BijBdsXub 'uoijounj uoijsuj sqj qjiM psjEisosss XjqiqBjsui aqjoj anp 'puooag -sjuaAa js3jbj juanbasqns joj ji Suued-ajd sjbss js3jbj b uo uiajsXs aqj sqjooius juaAa s8jb[ aqj'JU3A3 5f30|q-3j3uiS Sqj 35{l| 'jsjij "5{onjs uibuisj qoiqMs)poiq omj ussMjsq Jsqj33oj 3uiddns S-pqq jo dnoj3 bsajoaui qoiqM sju3as ssoqj jo sssusnbssuos omj sjb sjsqj'puBq jsqjo sqj uq uoijeooj jBqj jb jsjbj /Cpsfij sjoiujusas Suiddqs 5jooiq-s3jqj b 3ui)jblu (>|onjs uibuisj jBqjs>|30jq sqj puoXsq pssBsjsj sq jouubo uibjjs ssnBosq)S-pojq ssjqj jo sjbos sqj uo uisjsXs sqj sqjooius jusassqj '5fonjs uibuisj qoiqM sJoqq3isu omj ussMjsq sdijs5joojq suo Xiuo usqM 'jusas 3uidd;js [|BUis b 3uunQ -suiijsiubs sqj XjsjBUiixojddB jb Suiddqs joj pjoqssjqj sqjqoBSj snqj pus '5jonjjs sjb Xsqj sjiqM uoijBjnStjuoo jboojsnosusSoiuoq Xjsaijbjsj b ui sq oj usddsq Xsqj ssnBssqos op Jsqjs3oj difs qoiqM sjjooiq jo sjsjsnp -sjusas jos3ubj spiM b ssjbjsusS soiujBuXp sqj Moq SSJBJJSnjIJ UISJ-sXs sqj jo uoijnjOAS sqj jo uoijduossp oijBiusqas 3uimoji°J 9M1 'S9Z!S IIB jo sjusas 3u;dd;is sjiqiqxs uisjsXs sqj

•iJ>t/0JZ=->iSI 31UU 3UIPE0| 3qj pun '-/|(^//i/)_.3=Si S, guj;, gU|d■d!Is 3qi snoauaSoiuoq X|]BijHds s; uisjsXs aqj uaqM / auii] saX uoijisod (jssui) pun y XjpopA :suonn|os oipouaj -z 0lj

1 3WI1

qoiqM ui suoijBnjis xsjduioo sjoui jspisuoo sm jxs{yi•JOiABqsq

3u;jssjsjui sqj oj spBSj qsiqM sjnjBSj siqj si ji -psijiidiUBsi Xjisus3oiuoqui JBijBds Xub 'Jsqjs3oj sdijs S5jso|qjo jsjsnp b jsASusqM jBqj si sisXjbub siqj jo ssusnbssuoojuBjJodui; uy -3u;ddiis sjiqM o > {X)J ssnBosq sssubXjqiqBjsui siqj uiBSy "SUOIjbuba JBijBds oj sjqBjsunsjb suoijnjos oipousd jBqj sjbsasj sisXjbub XjqiqBjs jbsuijpjBMJojjqSiEjjs b 'uoijouj uuojiun jo ssbs sqj ui sy(A/Df+ t)X— Oyty :a poods suios jb 3uijB3Bdojd s)jui>|

jo uuoj sqj sifBj ueo suoijnjos oipousd 'Xubiiuiis (^•Sij sss) -poiq sjSuis b sjsm ji ji sb 'uosiun uj (uoijouisqj dojs oj pssssjduioo Xijusioujns sjb s3uuds 3uinndsqj lijun) sdqs pus (uoijouj oijbjs umuiixBui sqj suibj-jb ji jijun) s_pns Xjsjbujsjib uisjsXs sqj pus 'psAJSssjdsi XjjsuiuiXs ibuoijeisubjj 'ssbs jsqj uj -snosusSouioqXijEijBds sje suoij;puoo ibij;ui sqj usqM 'sjduiExs JOJ'ssub qsiqM suoijn[os oipousd ssq osjb (i) uoijEnbg

ssssbui sqj jo suoijisod sqj ui suoijBqjnjjod3uiXjea XjiBijEds puB uuojiun qjoq oj sjqBjsun si uisjsXssqj 'sssesjsui Xjioojsa sqj sb sssBSJosp uoijouj sqj ssnss-sq 'spjOM jsqjo u\ o > (a)J si XjinqBjsui sqj joj uoij-ipuos jusioyjns y ('SJsqumu sabm jjb jb sjiuy suibuisjsjbj qjMOjg sqx) -XiiBijusuodxs moj3 sspoui jsunoj jjejBqj sfBSAaj uonnjos siqj jo sisXjbub XjqiqBjs jEauq pjBM-JOjjqSiBJjs y Q?±a jjb joj ajqBjsun si uoijnjos siqj '(q) i•3;j ui pajBJjsnn! mj0J 3M1 J° si mbj uoijouj sqj usqAA

(I)

6861 avjaj 62 S"a3JLXHl M3IA3^[ MVOISAHd

( " ) J — - W - t y

:Xjisojsa Suiijnd sqj jb uoijoui

22*Hewr_N '29 3^moA

Volume 62, Number 22 P H Y S I C A L R E V I E W L E T T E R S 29 May 1989

(a)

0.03

•x 0.02

0.01

TIME t

(b)

TIME t

FIG. 3. Complex solutions: velocity X, vs position j andtime t. The smallest features in (a) are periodic creep eventswhich involve only one block. The larger features are catastrophic events which involve a larger number of blocks. Thegreat event illustrated in (b) involves all of the blocks andessentially resets the system.

The equation of motion for the slipping blocks is givenby

nmX=-2kcX-nkp(X-vt)-nF(X) , (3)

where X describes the coherent motion of the n blocks,and we have absorbed an unimportant constant on theright-hand side (which is determined by the positions ofthe two blocks which remain fixed) into our definition ofX. Observe that all forces, except those due to the twosprings which connect the slipping blocks to the rest ofthe system, act equally on all the slipping blocks and arethus preceded by a factor of n. In the limit of large n,(3) reduces to the equation for periodic events involvingthe whole system discussed previously. The importantfeature of the model which has been left out of Eq. (3) isthe amplification of spatial irregularities while blocksslip. Our assumption in (3) is that the system is locally

sufficiently smooth for an event of size n to occur. Equation (3) allows us to determine analytically the approximate amplitude and shape of an event of size n. However, in the original system [Eq. (1)], the blocks are left ina scrambled state after a slipping event, and smaller orlarger events will follow.

During the small creep events, the slipping blocks attain only very small velocities, and thus to a good approximation we can linearize F(X): FCY)«F0— F'(0)X. In this approximation (3) becomes the equation of motion for an antidamped harmonic oscillatorwith a slowly increasing bias force. Letting X(t)-~ea',we find that the solution oscillates (a is complex) when

Am kp-r2kc >[F'(0)]2 (4)

This condition can be satisfied, for example, when kc islarge and n is sufficiently small. In the limit thatF'(0)=0 and kp<Kkc,u the velocity X simply oscillatessinusoidally between zero and nkpv/kc. WhenF'(0) < 0, the antidamping results in a residual force onthe blocks when they restick. Because the maximum velocity of these small-amplitude events is less than theloading speed v, these events never release much of thestrain accumulated in the pulling spring.

When the condition given in Eq. (4) is violated, thesolution of (3) in the linearized regime grows exponentially with time (a is real). Eventually the group ofblocks attains a velocity for which F(X) is small. Thelarge catastrophic events exhibit this type of behavior.We can obtain an approximate solution for X(t) duringa large event by letting F(X) ~ 0 while the blocks areslipping. In this approximation and in the limit of slowpulling velocities, while the blocks slip (3) has a solutionof the form

X(t) = - Focos(cot), (5)

to mwhere co-=kp/m + 2kc/nm, and the coefficient Fo/co2mis determined by the initial conditions X(0) =0 and— oj~X(0)=Fo/m, which specifies that the spring forcesbalance the threshold static friction. A more carefulanalysis including the linear approximation to F(X) forsmall velocities produces similar results. In either case,unlike the small-amplitude creep events, during a largeevent the blocks always attain a velocity greater than theloading speed. When the blocks eventually stop slippingthere is a residual force which compresses the pullingsprings, resulting in a relatively long delay before theseblocks are involved in another slip event.

Finally, from our numerical results we have calculatedthe distribution of earthquake magnitudes. The magnitude M is defined to be the natural logarithm of theearthquake moment Mo, which for our model can be taken to be

i * /cv t X jd t , (6)

2634

Volume 62, Number 22 P H Y S I C A L R E V I E W L E T T E R S 29 May 1989

'

0 5 1 0 1 5 2 0MAGNITUDE yft

FIG. 4. Distribution of earthquake magnitudes N(M). Forthis result the number of blocks is ^ = 200 and the constant Cin Eq. (6) is IO6*.

where C is a constant. The distribution N(M) producedby our numerical results is illustrated in Fig. 4, and is instrikingly good agreement with what has been observedfor real earthquakes:12 \nN(M) ~A —BM, where A is aconstant and _?«!. Even the obvious discrepancy— the frequency of great events is too large—may beconsistent with observations. We tentatively attributethe latter phenomenon to the fact that both our computer model and real earthquake faults necessarily havefinite lengths.

The observed large fluctuations in the magnitudes ofearthquakes in nature suggest that certain dynamicalmodels of faults may be ideal candidates for the study ofgeneral features of self-organized criticality. Many faultmodels incorporate an intrinsic spatial variation which isimportant for certain features of earthquakes. For example, Burridge and Knopoff3 showed that local high-

viscosity regions incorporated into this model can generate sequences of aftershocks in qualitatively goodagreement with what is observed. In this paper we haveshown that the dynamics alone is sufficient to generate awide range of events. Somewhat alarmingly, our resultssuggest that small events lead to smoothing on a largerscale, resulting in larger events later on. Currently weare analyzing our results in more detail to determine thecorrelations between large events.

We have benefitted from many useful discussions withJ. R. Rice, Chao Tang, Franco Nori, Hiizu Nakanishi,and Glen Swindle. This research was supported by U.S.Department of Energy Grant No. DE-FG03-84ER45108and also in part by the National Science Foundation under Grant No. PHY82-17853, supplemented by fundsfrom the National Aeronautics and Space Administration, at the University of California at Santa Barbara.

'P. Bak. C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59,381 (1987); Phys. Rev. A 38, 36 (1988); C. Tang and P. Bak,Phys. Rev. Lett. 60, 2347 (1988); J. Stat. Phys. 51, 7977(1988).

2L. Kadanoff, S. Nagel, L. Wu, and S. Zhou, Phys. Rev. A(to be published).

3R. Burridge and L. Knopoff, Bull. Seismol. Soc. Amer. 57,341 (1967).

4S. T. Tse and J. R. Rice, J. Geophys. Res. 91, 9452 (1986).5For related work on the theory of random pinning forces in

charge-density waves, see S. N. Coppersmith and P. B. Little-wood. Phys. Rev. B 36, 31 I (1987).

6A. L. Ruina, J. Geophys. Res. 88, 359 (1983).7J. R. Rice, Pure Appl. Geophys. 121, 443 (1983).8F. Horowitz, Ph.D. thesis, Cornell University, Ithaca, 1989

(unpublished).9J. H. Dieterich, Pure Appl. Geophys. 116, 790 (1978).'"Periodic boundary conditions, however, always seem to lead

to smooth propagating modes."If the system is to have sensible elastic properties in the

continuum limit, a—0, then ma, kca, and kp/a must remainfinite. Thus we choose kc^kp.

12B. Gutenberg and C. F. Richter, Ann. Geofis. 9, 1 (1956).

2635