Abstract
The philosophy that a single “monolithic” model can “asymptotically” replace and couple in a simple elegant way several specialized models relevant on various Earth layers is presented and, in special situations, also rigorously justified. In particular, global seismicity and tectonics is coupled to capture, e.g., (here by a simplified model) ruptures of lithospheric faults generating seismic waves which then propagate through the solid-like mantle and inner core both as shear (S) or pressure (P) waves, while S-waves are suppressed in the fluidic outer core and also in the oceans. The “monolithic-type” models have the capacity to describe all the mentioned features globally in a unified way together with corresponding interfacial conditions implicitly involved, only when scaling its parameters appropriately in different Earth’s layers. Coupling of seismic waves with seismic sources due to tectonic events is thus an automatic side effect. The global ansatz is here based, rather for an illustration, only on a relatively simple Jeffreys’ viscoelastic damageable material at small strains whose various scaling (limits) can lead to Boger’s viscoelastic fluid or even to purely elastic (inviscid) fluid. Self-induced gravity field, Coriolis, centrifugal, and tidal forces are counted in our global model, as well. The rigorous mathematical analysis as far as the existence of solutions, convergence of the mentioned scalings, and energy conservation is briefly presented.
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References
Ben-Zion, Y.: Dynamic ruptures in recent models of earthquake faults. J. Mech. Phys. Solids 49, 2209–2244 (2001)
Ben-Zion, Y., Ampuero, J.-P.: Seismic radiation from regions sustaining material damage. Geophys. J. Int. 178, 1351–1356 (2009)
Boger, D.: A highly elastic constant-viscosity fluid. J. Non-Newton. Fluid Mech. 3, 87–91 (1977)
Brazda, K.: The elastic-gravitational equations in global seismology with low regularity. Ph.D. thesis, University of Wien (2017)
Brazda, K., de Hoop, M.V., Hoermann, G.: Variational formulation of the earth’s elastic-gravitational deformations under low regularity conditions (2017). arXiv:1702.04741
Dahlen, F.A., Tromp, J.: Theoretical Global Seismology. Princetown University Press, Princetown (1998)
Green, A., Naghdi, P.: A general theory of an elastic–plastic continuum. Arch. Ration. Mech. Anal. 18, 251–281 (1965)
Harris, R.A., et al.: The SCEC/USGS dynamic earthquake rupture code verification exercise. Seismol. Res. Lett. 80, 119–126 (2009)
Huang, Y., Ampuero, J.-P., Helmberger, D.V.: Earthquake ruptures modulated by waves in damaged fault zones. J. Geophys. Res. Solid Earth B9, 3133–3154 (2014)
Kaneko, Y., Lapusta, N., Ampuero, J.-P.: Spectral element modeling of spontaneous earthquake rupture on rate and state faults:effect of velocity-strengthening friction at shallow depths. J. Geophys. Res. 113, B09317 (2008)
Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation- I. validation. Geophys. J. Int. 149, 390–412 (2002)
Komatitsch, D., Tromp, J.: Spectral-element simulations of global seismic wave propagation-II. Three-dimensional models, oceans, rotation and self-gravitation. Geophys. J. Int. 150, 303–318 (2002)
Koot, L., Dumberry, M.: Viscosity of the Earth’s inner core: constraints from nutation observations. Earth Planet. Sci. Lett. 308(3), 343–349 (2011)
Lay, T., Wallace, T.C.: Modern Global Seismology. Academy Press, San Diego (1995)
Lyakhovsky, V., Ben-Zion, Y.: Damage-breakage rheology model and solid-granular transition near brittle instability. J. Mech. Phys. Solids 64, 184–197 (2014)
Lyakhovsky, V., Hamiel, Y., Ampuero, J.-P., Ben-Zion, Y.: Non-linear damage rheology and wave resonance in rocks. Geophys. J. Int. 178, 910–920 (2009)
Lyakhovsky, V., Hamiel, Y., Ben-Zion, Y.: A non-local visco-elastic damage model and dynamic fracturing. J. Mech. Phys. Solids 59, 1752–1776 (2011)
Lyakhovsky, V., Myasnikov, V.P.: On the behavior of elastic cracked solid. Phys. Solid Earth 10, 71–75 (1984)
Maedae, T., Furumura, T.: FDM simulation of seismic waves, ocean acoustic waves, and tsunamis based on tsunami-coupled equations of motion. Pure Appl. Geophys. 170, 109–127 (2013)
Pelties, C., de la Puente, J., Ampuero, J.-P., Brietzke, G.B., Käser, M.: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. 117, B02309 (2012)
Rajagopal, K.R., Roubíček, T.: On the effect of dissipation in shape-memory alloys. Nonlinear Anal. Real World Appl. 4, 581–597 (2003)
Roubíček, T.: Nonlinear Partial Differential Equations with Applications, 2nd edn. Birkhäuser, Basel (2013)
Roubíček, T.: A note about the rate-and-state-dependent friction model in a thermodynamical framework of the Biot-type equation. Geophys. J. Intl. 199, 286–295 (2014)
Roubíček, T.: Geophysical models of heat and fluid flow in damageable poro-elastic continua. Contin. Mech. Thermodyn. 29, 625–646 (2017)
Roubíček, T., Panagiotopoulos, C.G., Mantič, V.: Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity. Z. Angew. Math. Mech. 93, 823–840 (2013)
Roubíček, T., Souček, O., Vodička, R.: A model of rupturing lithospheric faults with re-occurring earthquakes. SIAM J. Appl. Math. 73, 1460–1488 (2013)
Roubíček, T., Vodička, R.: A monolithic model for seismic sources and seismic waves. Geophys. J. Int. (submitted)
Secco, R.A.: Viscosity of the outer core. In: Ahrens, T. (ed.) Mineral Physics and Crystallography: A Handbook of Physical Constants, pp. 218–226. Willey, Hoboken (2013)
Smylie, D.E., Palmer, A.: Viscosity of Earth’s outer core (2007). arXiv:0709.3333
Tosi, N., Čadek, O., Martinec, Z.: Subducted slabs and lateral viscosity variations: effects on the long-wavelength geoid. Geophys. J. Int. 179, 813–826 (2009)
Tsai, V., Ampuero, J.-P., Kanamori, H., Stevenson, D.: Estimating the effect of Earth elasticity and variable water density on tsunami speeds. Geophys. Res. Letters 40, 492–496 (2013)
Wijs, G.A.D., Kresse, G., Vočadlo, L., Dobson, D., Alfé, D., Gillan, M.J., Price, G.D.: The viscosity of liquid iron at the physical conditions of the Earth’s core. Nature 392(6678), 805–807 (1998)
Woodhouse, J.H., Deuss, A.: Theory and observations-Earth’s free oscillations. In: Romanowicz, B., Dziewonski, A. (eds.) Seismology and Structure of the Earth: Treatise on Geophysics, volume 1, chapter 1.02, pp. 31–65. Elsevier, Hoboken (2009)
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Communicated by Andreas Öchsner.
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Roubíček, T. Seismic waves and earthquakes in a global monolithic model. Continuum Mech. Thermodyn. 30, 709–729 (2018). https://doi.org/10.1007/s00161-018-0636-8
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DOI: https://doi.org/10.1007/s00161-018-0636-8