Gravitational coupling in a triaxial ellipsoidal Earth

Gravitational coupling within an axially symmetric, ellipsoidal Earth gives rise to torques among the mantle, outer core, and inner core. Such torques act along “lines of nodes”, i.e., orthogonal to axes of symmetry. In a triaxial Earth model, small deviations from axial symmetry modify the torques in direction as well as magnitude. The perturbative torques depend on three Euler angles specifying the long axis of the inner core relative to that of the mantle shell. Their magnitudes, excluding angular dependences, are ∼400 times (i.e., of the order of the inner core boundary flattening) smaller than their biaxial analogs. While triaxiality has a negligible impact on coupling considerations along the line of nodes, it creates gravitational coupling along orthogonal directions that have not been previously investigated. The gravitational coupling torques on the inner core and the mantle that take these triaxial perturbations into account are summarized. To the extent that fluid pressure and electromagnetic effects have not been explicitly modeled here, the present calculations are intended to provide only an order of magnitude estimate of coupling within the Earth along directions not previously examined. It is found that the axial torque on the inner core produces a differential rotation of less than 0.02°/yr and thus fails by at least one order of magnitude in supplying a mechanism for inner core differential rotation suggested by seismological evidence [Song and Richards, 1996]. This conclusion breaks down only if one is prepared to accept an inner core obliquity surpassing 10°.