4:45 PM - 5:00 PM
[SCG52-24] Reconsideration of the energy balance in earthquake faulting – II: Evaluation of radiated seismic energy
Keywords:Earthquake faulting, elastic strain energy, radiated seismic energy, slip-dependent friction law, slip acceleration
What we discussed in Part I (SCG50-18, JpGU 2021 Meeting) is summarized as follows. The earth is a self-gravitating body, and so we should include a change in gravitational potential energy together with a change in elastic potential energy in the calculation of energy balance in earthquake faulting (Kostrov, 1974; Dahlen, 1977). However, considering the rheological property of the earth's mantle and the steady seafloor spreading and oceanic plate subduction in long time scale, the stress field caused by self-gravitation must be nearly isotropic, and so it does not affect energetics in earthquake faulting substantially. Then, in quasi-static problems, the decrease of elastic potential energy balances with the work done for shear faulting.
In Part II, we consider the mechanical energy balance of a non-gravitating earth in dynamic shear faulting. In this case, from basic equations in continuum mechanics, we obtain the following energy balance equation after enough time has passed since faulting: radiated seismic energy (K) = released elastic potential energy (ΔE ) – work done for shear faulting (ΔW). In a point approximation of rupture area S, assuming the shear stress τ acting on a fault to be a single-valued function of fault slip D, we can rewrite the above equation as K/S = ½(τi+τf )Df – ∫0Dfτ(D)dD. This energy balance equation, which is often depicted as a τ–D diagram (e.g., Kanamori & Rivera, 2006), has been widely used for the evaluation of radiated seismic energy, but something is wrong. When the rupture process is quasi-static, the dynamic energy balance equation is reduced to the quasi-static energy balance equation because the radiated seismic energy becomes zero. Curious to say, the work done for shear faulting (the second term on the right-hand side) appears to be independent of the rate of rupture growth even in dynamic cases.
In dynamic problems, not only the fault slip D but also the rupture area S increases with time t, and so the point source representation using a seismic moment tensor M0(t)Npq is more convenient for the present discussion. The disturbances generated by a moment tensor source are formally categorized into the near-, intermediate-, and far-field terms. The first two terms, which decay in amplitude with the square of the source-receiver distance r, remain as permanent deformation after the disturbance died down. On the other hand, the far-field term, which decays with r, radiates from the source as traveling P and S waves. The total energy radiated as traveling waves is theoretically evaluated as K = c∫0T [∂t2M0(t)]2dt with c = [2/VP5+3/VS5]/60πρ. The important thing is that the amount of radiated seismic energy depends on the time history of rupture growth accompanied by fault-slip acceleration and deceleration, which is controlled by the inflow rate of the elastic strain energy released in the region surrounding the source.
In Part II, we consider the mechanical energy balance of a non-gravitating earth in dynamic shear faulting. In this case, from basic equations in continuum mechanics, we obtain the following energy balance equation after enough time has passed since faulting: radiated seismic energy (K) = released elastic potential energy (ΔE ) – work done for shear faulting (ΔW). In a point approximation of rupture area S, assuming the shear stress τ acting on a fault to be a single-valued function of fault slip D, we can rewrite the above equation as K/S = ½(τi+τf )Df – ∫0Dfτ(D)dD. This energy balance equation, which is often depicted as a τ–D diagram (e.g., Kanamori & Rivera, 2006), has been widely used for the evaluation of radiated seismic energy, but something is wrong. When the rupture process is quasi-static, the dynamic energy balance equation is reduced to the quasi-static energy balance equation because the radiated seismic energy becomes zero. Curious to say, the work done for shear faulting (the second term on the right-hand side) appears to be independent of the rate of rupture growth even in dynamic cases.
In dynamic problems, not only the fault slip D but also the rupture area S increases with time t, and so the point source representation using a seismic moment tensor M0(t)Npq is more convenient for the present discussion. The disturbances generated by a moment tensor source are formally categorized into the near-, intermediate-, and far-field terms. The first two terms, which decay in amplitude with the square of the source-receiver distance r, remain as permanent deformation after the disturbance died down. On the other hand, the far-field term, which decays with r, radiates from the source as traveling P and S waves. The total energy radiated as traveling waves is theoretically evaluated as K = c∫0T [∂t2M0(t)]2dt with c = [2/VP5+3/VS5]/60πρ. The important thing is that the amount of radiated seismic energy depends on the time history of rupture growth accompanied by fault-slip acceleration and deceleration, which is controlled by the inflow rate of the elastic strain energy released in the region surrounding the source.