Earthquake Energy Balance Raúl Madariaga Laboratoire de Géologie Ecole Normale Supérieure de Paris Spice training network EU & ANR cattel

Modern test of earthquake scaling law collapsed Circular crack model fp Universal Spectral Shape fs Test by Prieto et al JGR, 2004

Consequence of the scaling law (Savage and Wood,1971, Madariaga, 1976, Boatwright, 1980-84) Ide et al 2004 Apparent stress is a fraction of stress drop App sa Where C2 = 0.23 for Brune’s spectrum C2= 0.33 for Boatwright’s Ds Ds and sa are almost scale independent

es = Dw - Gc A simple shear crack (earthquake) moving at fast speed vr Local energy balance es = Dw - Gc friction peak external ini Gc Dw residual slip Dc

More than 1/3 of the stress drop is used to move the rupture!! Energy balance Seismic energy sa Fracture energy Ds Self-energy change slip More than 1/3 of the stress drop is used to move the rupture!! (used by Abercombrie and Rice, 2005) Gc scales with slip !

What controls rupture propagation? = Dw/Gc -1 = es/Gc Kunlun EQ vr > vs 0.2 1.4 sa=0.33Ds vr= 0 1 Tsunami EQ How to reduce speed?  increase Gc

Far field radiation from circular crack Displacement pulse Spectrum 0.25 Hz 4s w-2 decay is controled by the stopping phases not by the duration

slip weakening friction Slip rate Slip Stress change Elliptical crack dynamics Fully spontaneous rupture propagation under slip weakening friction Stopping phase (S wave)

Far field radiation from an elliptical fault Radiated signals are very variable Spectra are not

Isochrones and stopping phases 1st Stopping phase 1 2 2nd Stopping phase

Stopping phases in the near field In plane (P-SV) section Antiplane (SH) section

Conclusions Current spectral models contain information about fracture and friction Naive interpretation is that rupture speed is related to apparent stress Gc scales with slip Rough energy release rate estimates Gc can be done directly from apparent stress (Abercombrie et Rice, 2005) Actually spectral model only sees energy release rate changes! Modern numerical methods simulate stopping phases very well

Rupture process for a circular crack Slip rate Slip P st.phase Rayleigh S The rupture process is controlled by wave propagation!

Global Energy Balance L L = length scale S = surface Es = radiated energy Ws = strain energy change Gc = surface energy Kt = Kostrov's term Ws Seismologists mesure Es directly (still difficult) Estimate Wc from seismic moment (easily measured)

Energy flow and rupture speed Radiated energy density es v Husseini et al, 1978 Static energy density g(v) v b From Kostrov, Eshelby, Husseini, Freund

Radiation from an antiplane crack with a kink  Velocity z Stress zy Stresszx S    kink S wave (-2 ) Rupture front vr < vs Corner stresses Starting asperity

Scaling of energy Landers Gc ~ Es ~ DW ~ 106 J/m2 W~15 km Sumatra Gc ~ Es ~ DW ~ 107 J/m2 W~150 km