Why have the single force and torque been excluded from seismic source models?

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SUMMARY A theoretical justification is made for the physical existence of single-force and torque (asymmetrical part of moment tensor) components from indigenous earthquake sources occurring even deep within the solid earth. The elimination of the total-force and total-torque components of the equivalent volume force in Backus & Mulcahy's theory does not result from an explicit requirement of conservation of the total linear and angular momenta of the earth, but simply results from their formulation of equivalent volume force as a spatial gradient of stress glut. The effect of mass advection is not described by the stress glut introduced by Backus & Mulcahy to describe a failure of the linear elastic stress-strain relation, whereas mass advection plays a significant role in the generation of single force and torque components. We have extended the definition of the equivalent volume force in Backus & Mulcahy's formulation to cover additional non-linear terms specifying mass advection. A finite magnitude of the total-force or the total-torque component of the equivalent volume force is shown to be caused by mass advection leading to the density redistribution within the source region. Rigorous definition of the terms ‘single force’ and ‘torque’ is made by ‘linear and angular momenta exchange’ between the source region spanning a finite volume containing a finite mass and the rest of the earth. The single force and torque defined by momentum exchange include not only the total force and total torque components but also other higher order polynomial moments of the equivalent volume force. Our results show that single force, similar in principle to that from the fall of an apple, can occur even in the deep interior of the earth as a continuum with neither a completely detached mass nor the effect of the earth's free surface. A seismic event releasing gravitational energy beneath a volcanic area is expected to be a possible source which would generate single force and torque in the earth's interior.