Nonlinear Dynamics, Magnitude-Period Formula and Forecasts on Earthquake

1 Nonlinear Dynamics, Magnitude-Period Formula and Forecasts on Earthquake Yi-Fang Chang Department of Physics, Yunnan University, Kunming 650091, China (E-mail: yifangchang1030@hotmail.com) Abstract Based on the geodynamics, an earthquake does not take place until the momentum-energy excess a faulting threshold value of rock due to the movement of the fluid layer under the rock layer and the transport and accumulation of the momentum. From the nonlinear equations of fluid mechanics, a simplified nonlinear solution of momentum corresponding the accumulation of the energy could be derived. Otherwise, a chaos equation could be obtained, in which chaos corresponds to the earthquake, which shows complexity on seismology, and impossibility of exact prediction of earthquakes. But, combining the Carlson-Langer model and the Gutenberg-Richter relation, the magnitude-period formula of the earthquake may be derived approximately, and some results can be calculated quantitatively. For example, we forecast a series of earthquakes of 2004, 2009 and 2014, especially in 2019 in California. Combining the Lorenz model, we discuss the earthquake migration to and fro. Moreover, many external causes for earthquake are merely the initial conditions of this nonlinear system. Key words: geodynamics, earthquake, prediction, mechanism, fault motion, nonlinear system. PACS number(s): 03.40.Gc, 47.20.Ky, 91.30.Px, 05.45.+b, 91.30.Bi. I. Introduction The earthquakes are very complex nonlinear phenomena, and many theories and some phenomenological formulas about this have been proposed. At present the nonlinear seismology is an exciting direction of the development. Some concepts, a phenomenological description of fractals, and propagation and interaction on the seismic wave in the nonlinear media have been discussed [1-4]. Carlson, Langer, et al. [5-8], presented the Burridge-Knopoff block-and-spring model of an earthquake fault, and discussed basic properties, predictability, etc., of the model. For the forecasts of earthquakes, Kiremidjian, et al. [9] presented a stochastic slip-predictable model based on Markov renewal theory for earthquake occurrences. Borodich [10] described some renormalization schemes for earthquake prediction, which can be reduced to a power-law or the log- periodic approximation of the regional seismic-activity data. Harris [11] summarized more than 20 scientifically based predictions made before the 1989 Loma Prieta earthquake for a large earthquake that might occur in the Loma Prieta region. The predictions geographically closest to the actual earthquakes primarily specified slip on the San Andreas fault northwest of San Juan Bautista. He discussed forecasts of earthquake in 1989 California. Marzocchi, et al. [12] provided insights that might contribute to better formally defining the earthquake-forecasting problem, both in setting up and in testing the validity of the forecasting model, and found that the forecasting capability of these algorithms is very likely significantly overestimated. Helmstetter, et al. [13] developed a time- independent forecast for southern California by smoothing the locations of magnitude 2 and larger earthquakes, and using small m 2 earthquakes gives a reasonably good prediction of m 5 earthquakes. The geodynamics [14] combines the known mantle convection hypothesis, the magma intrusion theory, the phase change theory, and the faulting mechanism, earthquake should be caused by a horizontal fluid layer in a gravitational field that is heated from within and cooled from above, this mantle with very large viscosity moves slowly, and those accompanying momentum-energy transported and accumulated. When these are in excess of a faulting threshold value of rock, a phase transition arises, and an earthquake occurs with the energy releases. Based on a general geodynamics, we present a simplified and fundamental nonlinear dynamical theory on the earthquake, and obtain the magnitude-period formula of the earthquake combining the Carlson-Langer model, so some forecasts can be calculated quantitatively. Further, the theory may be 2 a basis for the development. II. Nonlinear dynamical system of earthquake It is often convenient to think of continents as blocks of wood floating on a sea of mantle rock [14]. The Carlson-Langer model [5-8] consists of a uniform chain of blocks and springs pulled slowly across a rough surface, in which the nonlinear friction force depends only on the velocity of the block. In this paper we extend the fundamental equations to general nonlinear ones of fluid mechanics in which the conservation of momentum-energy exists, so they are more general equations of magnetohydrodynamics. The equations of momentum conservation are DV Dt V t V V F gradp f ( ) . 0 (1) The equations of energy conservation are D Dt v vF vgradp vf pdivV Q ( ) . 1 2 2 0

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