The concept of self-organized criticality (SOC) has recently been suggested to be relevant for understanding the processes underlying earthquakes. In this work, we attempt to clarify the basic physical and geological mechanisms responsible for the self-organization of the crust within a continental plate. Toward this goal, a field theory is proposed which is not deduced from constitutive or microscopic rheological and mechanical laws but rather from symmetry and conservation laws, in a way similar to Landau theory of second-order critical phase transitions. In order to construct the theory, one has first to identify the relevant field or order parameter, which we propose to be the coarse-grained fluctuating strain tensor. We then discuss the existence of global conservation law. It stems from two facts: (1) a continental plate, which is in mechanical equilibrium and which is faulted over its whole surface, corresponds to a state of the crust nearly obeying the rupture criterion almost everywhere, a condition intimately related to the observation of active faults almost everywhere within plates. We thus argue that stress fields obeying the rupture criterion almost everywhere are attracting states of the plate dynamics. (2) The different components (dilation/compression, shear, and rotation) of the plate deformation must be kinematically compatible with each other, Both conditions (1) and (2) can be casted under the form of a diffusion-like conservation equation. Using this equation and the symmetries imposed by the tensorial character of the order parameter, we propose a generalized tensorial nonlinear diffusion equation controlling the spatiotemporal evolution of the strain tensor.

The nonlinear part of the equation is devised to take into account the threshold nature of rupture in the upper crust. From the diffusion-like structure of the governing equation, we deduce that the dynamical mechanical equilibrium of the crust is characterized by long-range and long-time correlations in the strain fluctuations. These long-range correlations are the signature of SOC and are intimately related to the existence of a global conservation equation. In order words, this means that local fluctuations are relaxed via a generalized diffusion process progressively invading adjacent domains. The spatiotemporal correlation functions of the strain tensor define a set of critical exponents: the dynamic exponent z governs the spatial diffusion with time of the strain fluctuations; the field exponent describes the renormalization of strain with changes of scales; the spatial anisotropy exponents control the spatial anisotropy of the strain diffusion. A solution of the generalized tensorial nonlinear diffusion equation would involve the determination of these exponents. We have not succeeded in this goal for the full problem but present a scalar analogy which has been examined recently and solved exactly (Hwa and Kardar, 1989). Finally, we discuss the relation between our field theory and earthquakes and especially the connection between the critical exponents introduced in our approach and the b value entering the Richter-Gutemberg law, linking the number of earthquakes to their magnitude. The connection relies on the idea that the average energy released in an earthquake gives the order of magnitude of typical fluctuations of the elastic energy in the crust. ¿1990 American Geophysical Union