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Earthquake Energy Balance

Raúl MadariagaLaboratoire de Géologie

Ecole Normale Supérieure de Paris

Modern test of earthquake scaling law

Test by Prieto et al JGR, 2004

collapsed

fs

fp

6.1/ sp ff

Circular crack model

7.1/ sp ff

)/1()(

20

20

M

Universal Spectral Shape

Consequence of the scaling law

2300

0

CCMM

Esa

App

Apparent stress is a fraction of stress drop

Where C2 = 0.23 for Brune’s spectrum

C2= 0.33 for Boatwright’s

a

Ide et al 2004

(Savage and Wood,1971, Madariaga, 1976, Boatwright, 1980-84)

and a are almost scale independent

vr

A simple shear crack (earthquake) moving at fast speed

Local energy balance

Gc w

Dc

slip

friction

ini

es = w - Gc

peak

external

residual

Energy balance

sc EWSG

Self-energy change

Seismic energy

Fracture energy

S

MG ac

022

1

More than 1/3 of the stress drop is used to move the rupture!!

(used by Abercombrie and Rice, 2005)

slip

Gc scales with slip !

= w/Gc

1

1.4

-1 = es/G

c

0

0.2

Tsunami EQ

Kunlun EQ

How to reduce speed? increase Gc

vr= 0

vr > v

s

What controls rupture propagation?

acG

w

2

a=0.33

Far field radiation from circular crack

SpectrumDisplacement pulse

4s

0.25 Hz

decay is controled by the stopping phases not by the duration

Elliptical crack dynamics

Stopping phase (S wave)

Slip rate Slip Stress change

Fully spontaneousrupture propagation

underslip weakening friction

Far field radiation from an elliptical fault

Radiated signals are very variable

Spectra are not

Isochrones and stopping phases

1st Stopping phase

2nd Stopping phase

1

2

Stopping phases in the near field

Antiplane (SH) sectionIn plane (P-SV) section

Conclusions

1. Current spectral models contain information about fracture and friction

2. Naive interpretation is that rupture speed is related to apparent stress

3. Gc scales with slip

4. Rough energy release rate estimates Gc can be done directly fromapparent stress (Abercombrie et Rice, 2005)

5. Actually spectral model only sees energy release rate changes!

6. Modern numerical methods simulate stopping phases very well

P st.phase Rayleigh

S

Slip rate Slip

Rupture process for a circular crack

The rupture process is controlled by wave propagation!

Global Energy Balance

Seismologists mesure Es directly (still difficult)

Estimate Wc from seismic moment (easily measured)

L = length scaleS = surfaceE

s = radiated energy

W

s = strain energy change

Gc

= surface energy

Kt

= Kostrov's term

L

W

s

KostrovSGWE css

Energy flow and rupture speed

Radiated energy density es

Static energy density

From Kostrov, Eshelby, Husseini, Freund

v)(1 v

w

es

(v)

v0

Husseini et al, 1978

ass w

e 2

Radiation from an antiplane crack with a kink

S

kink S wave(-2 )

Starting asperity

Velocity z

Stress zy

Stresszx

Rupture front

vr < v

s

Corner stresses

Scaling of energy

Landers Gc ~ Es ~ DW ~ 106 J/m2

Sumatra Gc ~ Es ~ DW ~ 107 J/m2 W~150 km

W~15 km