The forced nutations of an elliptical, rotating, elastic and oceanless earth

Summary. We compute the luni-solar forced nutations of an elliptical, rotating, self-gravitating, elastic, hydrostatically prestressed and oceanless earth. Several recent structural models are considered, each possessing a fluid outer core and solid inner core. Complete results are given for the nutation of the 'axis of figure for the Tisserand mean surface' which best represents the observational effects of the Earth's nutational motion. Differences between results for different structural models are observationally insignificant. Differences between our results and Molodensky's are as large as ∼ 0.002 arcsec at six month and at 18.6 yr.     

Excitation of normal modes on non-rotating and rotating earth models

Summary. The method of spectral decomposition for linear operators, formulated in Dirac's bra-ket notation, gives the excitation formulae for the normal modes of infinitesimal oscillations of a non-rotating earth. The formalism is then extended, in parallel with Lancaster's λ-matrix treatment, to obtain the corresponding formulae for a rotating earth. The algebraic structure of these formulae is carefully examined based in Chao's group-theoretical results, and some particular cases of earth models and sources are discussed.     

Excitation of the normal modes of a rotating earth model by an earthquake fault

Summary. The motion excited in a rotating earth model by a kinematically prescribed earthquake fault is solved for in closed form. In addition, expressions for the total energy released and the energy dissipated by bodily friction subsequent to faulting are obtained in terms of the normal-mode excitation amplitudes.     

Exact and asymptotic synthetic multiplet spectra on an ellipsoidal earth

Summary. Exact spectra of a normal mode multiplet _nS_l or _n T_l on a non-rotating ellipsoidal earth are composed of l + 1 lines. Asymptotically, in the limit of large l , such spectra appear, however, as a single line or peak broadened by attenuation alone. The location within the multiplet of this single peak depends only upon the angle of inclination of the source-receiver great circle path to the axis of ellipsoidal symmetry. The appearance of a single peak is produced by the cancellation of nearby singlets. To assess the utility of the single peak approximation, exact and asymptotic synthetic spectra have been compared directly for a number of multiplets and for a variety of source-receiver configurations. Except in the immediate vicinity of the source and its antipode, the approximation appears to be satisfactory on an ellipsoidal earth down to about l ∼ 10. Additional studies will be required to determine the limits of validity on a laterally heterogeneous earth.     

Evaluation of stress and displacement fields due to an elliptical plane shear crack

Summary. Following the classic work of Eshelby, the slip and stress distributions due to an elliptical plane shear crack are evaluated. The relation between average (or maximum) slip on the crack and the (constant) static stress drop, for faults of different aspect ratios, is found. The slip vector is not parallel to the applied stress but makes a small angle to it, except when the stress is applied along the major or minor axis of the ellipse. The stress -distribution around the crack shows that in addition to the expected stress concentration along the crack edge, there are broad regions of stress increase off the crack plane for circular and elliptical cracks, similar to those known to exist for in-plane but not for antiplane shear cracks. Whether the stress- intensity factor at the end of one axis is greater or less than that at the end of the other axis ( k_a ≶ k_b ), depends on the condition: √ b/a ≶ (1 − v ) where a and b are the semi-axes of the ellipse, k_a and k_b are the stress-intensity factors at the end of the a- and b -axes and v is Poisson's ratio. The total stress-intensity factor varies smoothly along the edge of the ellipse from one axis to the other and it is found that this variation is rather small.     

Rotational remanent magnetization and the torque exerted on a rotating rock in an alternating magnetic field

Summary. Using an air turbine at rotation frequencies of between 1.5 and 275 revolutions per second (rps), the dependence of rotational remanent magnetization (RRM) on rotation frequency has been investigated for two igneous samples in and alternating field of 51 mT peak at 50 Hz. The same experimental arrangement has also been used to measure the dependence on rotation frequency of the torque exerted by the alternating field on the rock samples. The dependence of torque and RRM on peak field has also been measured at a rotation frequency of 112 rps and a linear relationship between RRM and torque has been demonstrated.
In an attempt to elucidate the way in which RRM arises, analytical and numerical models of the rock have been developed in order to calculate the torque curves and these agree quite closely with those observed experimentally. While the precise factor responsible for RRM has not yet been identified from the numerical model it is suggested that RRM may arise as a result of particle moments suddenly flipping into the field direction, and thus by virtue of their intrinsic angular momentum acquiring a transient component of magnetic moment antiparallel to the rotation vector describing the flip. This component, due to the hysteresis of the assembly of particles, will not then entirely disappear when the alternating field is removed. An estimate of the transient axial field which can be considered to deflect each moment towards the rotation axis during the flip yields a value of the order of 1 mT.