Periodicity of Huge Earthquakes by Numerical Simulation
Atsushi Tanaka1 and Yuuki Sasaki111

Graduate School of Science and Engineering, Yamagata University, Japan

Introduction

The massive earthquake on March 11, 2011 in Japanbrought us down enormous damage. Though it is wellknown that earthquakes happen almost periodically, it has
not been explained exactly. The well-known empirical ruleis that small scale ones often occur, however large ones
rarely occur. Gutenberg and Richter formulated it as exponential rule, that is the number of occurrences decrease
exponentially with the scale of earthquakes[1]. It is called
Gutenberg-Richter rule and described as follows,
log N (m > M ) = a − bM(2)

so the power-law in N (m > M ) and E as follows
2N (m > M ) ∝ E − 3 b .

(3)The main aim of this paper is to show the periodicityof the occurrence of earthquakes in numerical simulation.
The most famous model of the earthquake is the stickslip model[2]. Using that model, we can obtain size, spacial and temporal distribution of the magnitude of earthquakes. In most researches, they only confirm GutenbergRichter rule, and researches of the periodicity are rare.
Thus we carry out numerical simulation using stick-slip
model and confirm Gutenberg-Richter rule and the periodicity of earthquakes.
2Model and Simulation

2.1

The Stick-Slip Model

Springs(1)

where M is the magnitude and N (m > M ) is the numberof occurrences of earthquakes which is larger than M . The
coefficient b is called b value and is an important parameter
which shows the property of earthquakes. It is estimatedbetween 0.7 to 1.1 in real earthquakes all over the world.
There are many researches of the relation between themagnitude of an earthquake and its released energy. In
most of them, they suppose the formula log Es = α+βMs .
Gutenberg and Richter estimated as follows,
log E = 11.8 + 1.5M

Ceiling
Plate

The stick-slip model expresses one fault using several
blocks and springs between them. As shown in Fig.1, they
have a fixed ceiling plate and an unidirectionally moving
floor plate with some friction, and some blocks contact
the latter. Each block connects each other with springs
in four directions, and also connects the ceiling plate in asimilar way. If we make a modeling of the fault around theboundary of the continental plate and the ocean plate, thefloor and blocks stand for the ocean and continental plates
respectively. The subduction movement of the ocean plateis expressed by the movement of the floor.

FloorPlateFig. 1. The schematic diagram of stick-slip model.
As the floor moves, the elastic force of springs connectedto the ceiling and other blocks becomes larger than thestatic friction, then some block starts to slip. The movement of that block causes stretch or shrink of other springs
and increase of their elastic force. Then some surrounding
blocks also start to slip in a similar way. The propagation of these slips occur successively until no other blocks
slip. We consider all these processes as one earthquakes
and all released spring energy means the energy of thatearthquake.
2.2

Simulation

In simulation first we set placements of all blocks 0, andfix the static friction of them as FN = 5.0 × 105 . We carryout simulation as follows.
1. Set static frictions for 100 × 100 blocks.
2. Set spring constants of springs between the ceiling
plate and blocks and between blocks.
3. For each block, calculate the sum of elastic force ofsurrounding springs, and ingenerate the slip if it islarger than the static friction.
(a) For slipping blocks, calculate the displacementof the cell after the slip.
(b) Check whether all other blocks would slip again.
(c) Count the number of new slipped blocks, andrepeat (a) and (b) until it becomes 0.
4. Record the sum of released energy.
5. Increase the displacement of all blocks by 1.
6. Repeat from 3 to 5.
We consider the procedures from 3 to 5 as 1 step andcarry out simulation until 5 × 105 steps. Here spring constants are given by random number according to normal

PeriodEnergyk

Step

Fig. 4. The relation between periodicity and k.

bFrequency

Fig. 2. The change of released energy.

kFig. 5. The relation between b and k.

EnergyFig. 3. The relation between energy and frequency.
distribution with the mean k and the variance σk2 , and wevary k and σk and measure the change of the periodicity
and b value.
33.1

Results and Discussion
Verification of Gutenberg-Richter Rule

The numerical results of released energy with k = 10and σk = 0.5 are shown in Fig.2. We can easily find earthquakes occur almost periodically, however those scales areslightly different. From this data, we can obtain the relation between energy and the number of occurrences ofearthquakes as shown in Fig.3. We estimated the power
exponent b′ = −0.703 and thus from Eq.(3) b = 1.0546.
3.2

Periodicity and b value

We observed the periodicity and b value for several
spring constants with k = 5, 10, 20, 30, 40, 50, 60, 70, 80, 90
kand σk = 20
. Results for the periodicity are shown in
Fig.4. Simulation data are well fitted by f (x) = xa , a =1.127 × 105 , thus periods are inversely proportional tospring constants. On the other hand, the relation between
b value and spring constants is almost logarithmic(Fig.5).

3.3

Discussion

In stick-slip phenomena, if we think of only one block,the slip distance is proportional to the displacement, andthe released energy is proportional to the square of thedisplacement. Therefore we can easily speculate that thelarger spring constant becomes, the more frequently small
earthquakes will occur, and the smaller it becomes, theless large slips occur.
In our 2-D stick-slip model, springs between blocks affect the condition of the occurrence of slip. If spring constant is strong, the propagation of slips is easy to occur.
Therefore since some complicated elements affect it, theperiodicity and b value are unpredictable by spring constants.
4Summary

In summary we carry out numerical simulations ofearthquake using 2-D stick-slip model. we obtained that if
the spring constant is high, the period and b value of earthquake become short and high respectively. In real earthquakes the relation between the periodicity and b value is
unknown though, it is known that there exist specific bvalue and the period of large earthquakes in each place.
Since in our simulation we focus attention only on thechange of spring constants showing the hardness of plates,
it cannot transpose to an actual earthquake related to complicated factors as it is. However it might be contributory
to the analysis and prediction of earthquakes near future.
References[1] B. Gutenberg and C.F. Richter, Seismicity of the Earthand Associated Phenomena, 2nd ed. Princeton, N.J.:
Princeton University Press, 1954.
[2] R. Burridge, and L. Knopoff, Model and theoretical seismisity, Bull. Seism. Soc. Am., 57, pp.341–371, 1967.