# Earthquake Processes: A View from Synergetics and the Theory of Catastrophes

- 86 Downloads

## Abstract

The theory of catastrophes and synergetics, developed by mathematicians in the second half of the twentieth century (Haken in Synergetics. Introduction and advances topics. Springer, Berlin, 1978; Arnol’d VI in Catastrophic theory. Springer, Berlin 2004), is used to analyze earthquake processes in active faults. It is shown that the oscillators of the seismic radiation are vibrating blocks at the border of the fault, and the source of the energy supply is the margin of the fault moving with the velocity *V*_{0}. The input or loss of energy is regulated by a nonlinear element, the frictional force *F*(*V*), which depends on the difference *V* between the velocity of the fault margin *V*_{0} and the velocity of the seismic block vibration: *V* = *V*_{0} − d*x/*d*t*. Precise conditions for the occurrence of an earthquake, its attenuation, or the generation of a tectonic tremor in its source are formulated.

## Keywords

Earthquake mechanisms nonlinear friction oscillations of rock blocks in faults## List of symbols

*A*Steepness of linear friction function

*F*(*V*) in Eq. (33)*a*Side of a block

*b*Dimensionless empirical constant in Eq. (3)

*C*Constant in Eq. (34)

- Co
Control parameter in Eq. (22)

*D*Effective coefficient of the frictional damping in Eq. (34)

- Da
Damping of oscillation

- Dis
Dissipative function in Eq. (35)

*E*Total energy (Lagrange’s function)

*e*Dimensionless constant in Eq. (23)

*F**F*^{*}Scale of the strength

*f*Polynomial function in Eq. (21)

*G*Block loading in Eq. (2)

*g*Gravity acceleration

*h*Coefficient of the lateral friction

- In
Integral in Eq. (25)

*k*Coefficient of elastic stiffness in Eq. (1)

*M*Young’s modulus

*m*Mass of a block

*N*Dimensionless load

*N*_{o}Shear stress at the upper boundary of the ocean slab

*p*Empirical dimensionless constant in Eq. (3)

*q*Dimensionless constant in Eq. (42)

*q*_{1}Dimensionless constant in Eq. (42)

*r*Empirical dimensionless constant in Eq. (5)

*S*Actual touch area of the contact zone

*s*Dimensionless velocity of the flank surface of the fault

*T*Temperature of the contact zone

*T*_{i}Tilt of the dropping characteristic of the friction

*t*Time

*t*_{e}Time of the catastrophe

*U*Potential in Eq. (53)

*U*^{/}First derivative of the potential

*U*by amplitude*α**U*^{//}Second derivative of the potential

*U*by amplitude*α**U*^{///}Third derivative of the potential

*U*by amplitude*α**V*Relative velocity

*V*_{o}Velocity of the lower margin of a fault

*V*_{L}Velocity when the friction

*F*is absent*V*^{*}Characteristic relative velocity in Eq. (3)

*W*Dimensionless relative velocity

*x*Displacement of the block

*y*Dimensionless coordinate in Eq. (6)

*Y*Coordinate in Eq. (34)

*z*Dimensionless coordinate in Eq. (16)

*z*_{0}The height of roughness protrusions on the contact surface

## Greek symbols

*α*Dimensionless amplitude of oscillations

*α*_{0}Initial amplitude

*α*_{c}Extreme points for the potential

*β*Coefficient in Eq. (3)

*δ*Dimensionless attenuation of the oscillations

*ε*Dimensionless constant in Eq. (15)

- Θ
_{1,2,3,4} Dimensionless integrative constant in Eqs. (26), (28), (30), (32)

- Θ
_{5} Constant value in Eq. (46)

- Θ
_{6} Constant value in Eq. (50)

*θ*Phase of oscillations in Eq. (18)

*κ**λ*Dimensionless constant in Eq. (23)

*μ*Dimensionless coefficient of the friction in law (2)

*ξ*Dimensionless coordinate in Eq. (5)

*ρ*Density of a basaltic block

*σ*Electrical conductivity of the contact zone

*τ*Dimensionless time

*ω*_{0}Frequency of the block self-oscillations

## Subscripts

*t*Differentiation by time

*τ*Differentiation by dimensionless time

## Notes

### Acknowledgements

The authors would like to thank two anonymous reviewers who thoroughly reviewed the manuscript; their critical comments and valuable suggestions were very helpful in preparing this paper.

## References

- Arnol’d, V. I. (2004).
*Catastrophic theory*. Berlin: Springer.Google Scholar - Arsen’yev, S.A. (2016). Sliding friction as mechanism for onset of tectonic earthquakes. In
*Proceed. of the IV All*-*Russian Conf.*“Tectonophysics and actual topics in the Earth’s Sciences”. Russian Academy of Sciences, Moscow, pp. 423–429**(in Russian)**.Google Scholar - Arseny’ev, S. A., & Shelkovnikov, N. K. (1991).
*Dynamics of the long sea waves*. Moscow (in Russian): Moscow University Press.Google Scholar - Bochet, M. H. (1861). Nouvelles recherches experimentales sur le frottemenent de glissement.
*Annales des Mines,**XIX,*27–120.Google Scholar - Castanos, H., & Lomnitz, C. (2012).
*Earthquake disasters in Latin America. A holistic approach*. Berlin: Springer.CrossRefGoogle Scholar - Chen, F. F. (1987).
*Introduction to plasma physics and controlled fusion*. London: Plenum Press.Google Scholar - Conti, P. (1875).
*Sulla resistenza di Attrito*. Rome: Royal Academia dei Lincei. V.II.Google Scholar - Coulomb, C. A. (1821).
*Theorie des machines simples, en ayant egard au frottement de leures parties et a la roideur des cordages*. Paris: Bachelier.Google Scholar - Deiterich, J. H. (1978). Time-dependent friction and mechanics of stick-slip.
*Pure and Applied Geophysics,**116,*790–806.CrossRefGoogle Scholar - Draget, H., Wang, K., & Rogers, G. (2004). Geodetic and seismic signatures of episodic tremor and slip in the northern Cascadia subduction zone.
*Earth, Planets, Space,**56,*1143–1150.CrossRefGoogle Scholar - Eppelbaum, L.V., & Kardashov, V.R. (2001). Analysis of strongly nonlinear processes in geophysics. In L. Moresi & D. Müller (Eds.)
*Proceedings of the Chapman Conference on Exploration Geodynamics*. Dunsborough, Western Australia, pp. 43–44Google Scholar - Gilmore, R. (1981).
*Catastrophe theory for scientists and engineers*. New York: Wiley-Blackwell.Google Scholar - Haken, H. (1978).
*Synergetics. Introduction and advances topics*. Berlin: Springer.Google Scholar - Kardashov, V. R., Eppelbaum, L. V., & Vasilyev, O. V. (2000). The role of nonlinear source terms in geophysics.
*Geophysical Research Letters,**27*(14), 2069–2073.CrossRefGoogle Scholar - Kasahara, K. (1981).
*Earthquake mechanics*. Cambridge: Cambridge University Press.Google Scholar - Khain, V. E., & Lomeeze, M. G. (1995).
*Geotectonics with foundations of geodynamics*. Moscow: Moscow University Press.**(in Russian)**.Google Scholar - Kilgore, B. D., Blanpied, J. H., & Deiterich, J. H. (1993). Velocity dependent friction of granite over a wide range of conditions.
*Geophysical Research Letters,**20*(10), 903–906.CrossRefGoogle Scholar - Kragelskiy, I. V., & Schedrov, V. S. (1956).
*Progress of science on the friction*. Moscow**(in Russian)**: USSR Academy of Sciences.Google Scholar - Landa, P. S. (2015).
*Non-linear oscillations and waves*. Moscow: LIBROCOM.**(in Russian)**.Google Scholar - Landau, L. D. (1944). On the problem of turbulence.
*Doklady of the Academy of Sciences of the SSSR,**44*(8), 339–342.**(in Russian)**.Google Scholar - Lomnitz, C. (1962). Application of the logarithmic creep low to stress wave attenuation in the solid earth.
*Journal of Geophysical Research,**67*(1), 365–368.CrossRefGoogle Scholar - Loskutov, A. Y., & Meehilov, A. C. (2007).
*Theoretical foundation of the complicated systems*. Moscow, Ijevsk: Institute of the Computer Researches.**(in Russian)**.Google Scholar - Marleenatskiy, G. G. (2015).
*Mathematical foundation of synergetic*. Moscow: LIBROKOM (in Russian).Google Scholar - Meeropolskiy, Y. Z. (1981).
*Dynamics of the internal gravitational waves in ocean*. Leningrad: Gidrometeoizdat.**(in Russian)**.Google Scholar - Obara, K. (2002). Nonvolcanic deep tremor associated with subduction in southwest Japan.
*Science,**296*(5573), 1679–1681.CrossRefGoogle Scholar - Petroff, N. P. (1878).
*On the continuous arrester arrangements*. Saint-Petersburg: Bull. of the Saint-Petersburg’s Technological Institute.**(in Russian)**.Google Scholar - Rabinovich, M.I., & Trubetskov, D.I. (2000).
*Introduction to the Theory of Oscillations and Waves*. NITS “Regular and chaotic dynamics”. Moscow-Ijevsk**(in Russian).**Google Scholar - Reutov, V. P. (1988). Explosion non-stability of waves. In A. M. Prokhorov (Ed.),
*Book: “Physical Encyclopedia”*(Vol. I). Moscow: Soviet Encyclopedia.**(in Russian)**.Google Scholar - Scholz, C. H. (2002).
*The mechanics of earthquake and faulting*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Stacey, F. D. (1962). The theory of creep in rocks and the problem of convection in the Earth’s mantle.
*Icarus,**1,*304–313.CrossRefGoogle Scholar - Thompson, J. M. T. (1982).
*Instabilities and catastrophes in science and engineering*. Chichester: Wiley.CrossRefGoogle Scholar - Tohm, R. (1975).
*Structural stability and morphogenesis*. Reading: W.A. Benjamin, Inc.Google Scholar - Tullis, T. E., & Weeks, J. D. (1986). Constitutive behavior and stability of frictional sliding of granite.
*Pure and Applied Geophysics,**124,*383–414.CrossRefGoogle Scholar - Vidal, J. E., & Houston, H. (2008). Slow slip: New kind of earthquake.
*Physics Today,**1,*38–43.Google Scholar