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SUMMARY The rotation of the inner core (IC) is influenced by the rest of the Earth through a number of coupling mechanisms. Among four possible coupling mechanisms, gravitational, pressure, viscous and electromagnetic, the first two torques are dominant. Numerous existing IC gravitational torque estimates have been shown to agree very well with one another (Xu & Szeto 1996). It is shown in this paper that diVerent estimates of the IC pressure torque are also in good agreement.

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°.

SUMMARY Gravitational coupling within the Earth had been dismissed until about 10 years ago. Inappropriate application of Newton's theorem to the Earth is cited as a reason for this dismissal. For the theorem a new proof is given, a by-product of which is an exact expression for the gravitational potential within a homogeneous ellipsoidal shell of different outer and inner flattening. The gravitational torque on a spheroid inside such a shell can be analytically calculated. When applied to a simplified model of the earth, the gravitational torque between the mantle and the inner core is found to be enormous*, confirming earlier calculations for realistic earth models based on integration of the Clairaut figure equation. The torque is found to be dominated by a contribution arising from a thin layer surrounding the inner core.

SUMMARY The Earth's core is a contained rotating fluid. In contrast to the classical theory of rotating, incompressible fluids in rigid containers, a realistic treatment of the geophysical problem requires allowance for the compressibility of the fluid core, and for the elasticity of the shell (mantle and crust) and inner core. It is shown in this paper that when these realistic properties are included in the model, the governing sub-seismic wave equation and boundary conditions are fully Hermitean and a general functional governing the problem can be constructed. Compressibility is treated through a decompression factor which multiplies the displacement field to generate a new, solenoidal, displacement vector field, while elasticity of the boundaries is incorporated by means of internal load Love numbers. Both the decompression factor for the fluid core and the internal load Love numbers for the shell and inner core are computed for realistic Earth models.

A brief historical introduction to the study of the Earth's core is followed by a review of recent advances in the dynamics of the liquid outer core and the rotation of the solid inner body. In particular, the scaling of the fluid motion equations is reviewed and a detailed derivation of the 'subseismic equation' which governs small-amplitude, low-frequency oscillations is given following methods first outlined by Rochester. Attention is given to the compressibility of the outer core and its role in dynamo theory as well as its effect on Proudman-Taylor flow, where it produces a fluid motion with helicity. The theorem of J.B. Taylor is also generalised to real core compressibility. Rotational motions of the inner core are examined under the strong gravity torque exerted on it by the rest of the Earth. Both the motion ignoring back reaction on the mantle's rotation and the full coupled problem are solved.

Summary. The widely accepted procedure of separating the contributions to the geomagnetic field arising in the Earth’s core from those arising in the Earth’s crust on the basis of spherical harmonic degree number has been recently challenged by Carle & Harrison. We argue here that their basic objection is valid only for data over line profiles or flat surfaces. We feel that while Carle & Harrison have addressed a number of significant and fundamental aspects of this problem they appear not to have done so clearly. The contribution contained herein is intended as a clarification. In particular we wish to: (1) Acknowledge, as Carle & Harrison asserted, that so-called ‘linear’ data (see text) over line profiles (or flat two-dimensional surfaces) are better represented by Fourier components than by Legendre functions in the following sense: Legendre functions do not have regularly spaced zeros over line profiles (or flat surfaces), and hence are not associated with ‘wavelengths’ in the conventional sense. Thus representation and filtering (such as removing low degree components) using Legendre functions does not satisfactorily separate long- and short-wavelength features (defined in the Fourier sense). (2) Emphasize that the above-mentioned advantage of Fourier analysis over spherical harmonic analysis no longer exists when a (linear) data set over a curved surface is to be represented, since the conventional definition of wavelength breaks down. (3) Demonstrate, by way of squaring a Fourier series and a spherical harmonic series (actually we consider the sample example as given by Carle & Harrison where no +dependence is included), that any non-linear operations on an originally linear data set, irrespective of the ‘basis functions’ in which the data are expanded, invariably create intermodulations. Carle & Harrison appeared to have confused the issue of intermodulations with the issue of wavelength identification when they employed a non-linear operation in showing that ‘the use of spherical harmonicsin describing the core field and in removing the core field from the total field in order to get the field produced by the crustal magnetization is flawed’. (4) Point out that the power spectrum of the residual field after removal of

The inner core rotates under the influence of inertial, frictional, electromagnetic and gravity torques. When its symmetry axis is inclined by an angle to the mantle symmetry axis, calculation shows the gravity restoring torque to be 2.50 x 1024cos01sin01N m, several orders of magnitude larger than the other torques combined. Investigation of its rotational dynamics under this torque shows that it can have two modes of steady precession. One mode involves a slow precession in space and is presumably ruled out because of the nearly diurnal relative motion required with respect to the outer core. The second steady precessional mode involves a rapid, nearly diurnal prograde motion in space and hence is slow in the mantle frame. Because the inner core is a good electrical conductor and because of its large rotational energy it is likely to have a strong interaction with the main magnetic field. In particular, if it is in the second steady precessional mode it may be carrying magnetic fields bodily for periods of several thousand years (its magnetic diffusion time) and the ‘dipole wobble’ of the main field, seen in historical measurements, and apparently extending into the palaeomagnetic record, may be a reflection of the influence of this motion on the outer core dynamo.