D. E. Smylie

The Inner Core Translational Triplet and the Density Near Earth's Center

Four long records from superconducting gravimeters yield evidence of the triplet of translational oscillations of the solid inner core about its central position. Calculations of core oscillation modes allow identification of the three translational resonances at periods of 3.5820 � 0.0008, 3.7677 � 0.0006, and 4.015 � 0.001 hours by their rotational splitting. Each resonance is defined by approximately 20 successive spectral estimates. A new Earth model brings the computed periods into agreement with observation. It has a central density of 12.960 grams per cubic centimeter, inner core radius of 1221.1 kilometers, and a density jump at the inner core boundary of 0.407 grams per cubic centimeter.

Global superconducting gravimeter observations and the search for the translational modes of the inner core

Abstract With the cooperation and collaboration of Courtier (Cantley), Ducarme (Brussels), Goodkind (Pinon Flats), Hinderer (Strasbourg), Imanishi and Seama (Kakioka), and Sun (Wuhan), a very large data set of global superconducting gravimeter observations (294,106 h) has been assembled for analysis. Except for the Brussels record, for which residuals were provided, synthetic tides for each station generated by Merriams G-wave code were used to remove tidal signals, and barometric effects were fitted and corrected off. An examination of the resulting residuals typically shows spikes and offsets (glitches) which cannot be true gravity observations and which raise the background noise level dramatically. In the case of the Cantley record, 1-min samples were available and a spline differentiator was used as an automated glitch detector. Fifteen previously undetected glitches were found and repaired by spline interpolation. Each has been associated with a real physical event by comparison with daily log files from the site and from the data acquisition system. The repaired record is very low noise, comparable to the first long record taken at Pinon Flats. The other stations are in less favourable sites but still provide long continuous records with the exception of Kakioka which has many short gaps. Both the Kakioka and Wuhan instruments have subsequently been moved to better sites. In the absence of detailed station logs as were available for Cantley, a more arbitrary record repair procedure was developed with hour-to-hour changes of more than 1 μgal being replaced by gaps of constant values, and the previous and subsequent record levels adjusted to eliminate discontinuities at the end points of all gaps. This procedure was applied to both the Strasbourg and Kakioka series. Power spectra for each residual series were then estimated using a 12,000-h Parzen window with 75% overlap. Product Spectra for data entirely outside Europe confirm the presence of the three resonances found earlier in data from Europe alone [Smylie, D.E., Hinderer, J., Richter, B., Ducarme, B., 1993. The product spectra of gravity and barometric pressure in Europe. Phys. Earth Planet. Inter., 80, 135–137.], and associated with the three translational motions of the inner core by the strict adherence of their central periods to splitting laws [Smylie, D.E., 1992. The inner core translational triplet and the density near Earths center. Science, 255, 1678–1682.]. The observed periods of the equatorial translational modes provides a very precise measure of viscosity near the ICB [Smylie, D.E., McMillan, D.G., 1998. The Inner Core as a Dynamic Viscometer. Phys. Earth Planet. Inter., this volume.]. The gravity signals from the equatorial translational modes travel in longitude at fixed rates in the opposite sense, while that from the axial mode is axisymmetric and does not move in longitude. The signals also have different dependencies on latitude. We have developed a method of exploiting these differences for simultaneous data from a distributed network of stations and tested it with synthetic data. Initial results from a test on real data using the simultaneous portions of the Brussels, Cantley, Kakioka, Strasbourg and Wuhan records are presented.

The product spectra of gravity and barometric pressure in Europe

Abstract A total of 111 000 hourly values of gravity and barometric pressure from stations in Europe is analysed. The data consist of two sets of records from Brussels, an early set of 36 000 h length and a more recent set of 21 000 h length, a set of records from Bad Homburg of 24 000 h length, and a set of records from Strasbourg of 30 000 h length. All of the gravity measurements were made with similar superconducting instruments and the pressure data were recorded simultaneously at each superconducting gravimeter site. The four sets of records have different time bases, and to bring out common features and suppress individual station systematic errors, the product spectrum is introduced. Spectral density estimates are first computed for a common spectral window for each record, and the product spectrum is formed by multiplying individual spectral estimates across records. The cumulative distribution function is found for the product spectrum and confidence intervals are calculated from it by iteration. The product spectrum in gravity reveals a triplet of resonances in the subtidal band which are shown by an automated computer search to be uniquely associated with the translational modes of the solid inner core. The product spectrum in barometric pressure clearly reveals the first 10 solar heating tides in the atmosphere, but otherwise does not show common features with the gravity product spectrum. In particular, the triplet of resonances in the subtidal band of the gravity product spectrum do not show up in the product spectrum of barometric pressure, climinating the atmosphere as their source.

Coupled motions of the inner core and possible geomagnetic implications

Abstract As the inner core is a good electrical conductor any ambient magnetic field would diffuse into it on a time scale long compared to several thousand years, and conversely be frozen there on shorter time scales. From the observations that the dipole component of the Earths magnetic field has been inclined persistently to the spin axis over hundreds of thousands of years, and that the dipole drifts and decays significantly more slowly than the nondipole field, it is suggested that the external dipole is simply a manifestation of a field frozen in an inclined inner core. It is shown that the much neglected gravitational restoring torque can be significant for an inclined inner core, so much so that its motion is in the main determined by gravity, with electromagnetic and inertial coupling effects being of secondary importance. A regular precession of the inner core is shown to be possible where its spin axis drifts westward relative to the mantle with a period of ∼ 7000 y. Some preliminary calculations of the possible motions of a gravitationally coupled mantle-inner core system are shown.

Viscous and rotational splitting of the translational oscillations of Earth's solid inner core

Abstract The small oscillations of the inner core about its central position present an interesting problem in fluid dynamics with roots in the classical literature. It is a generalization of the problem Stokes [Stokes, G.G., 1851. On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Philos. Soc. 9, 8.] solved of a sphere oscillating in an unbounded, non-rotating, viscous fluid to the case of a sphere oscillating in the contained, rotating fluid outer core. In the inviscid case, the influence of the Coriolis acceleration splits the oscillation into three distinct motions with differing periods, one along the axis of rotation, one prograde in the equatorial plane, and one retrograde in the equatorial plane. In this paper, we consider the effect of the presumed high viscosity at the inner core boundary on the rotational splitting. We obtain novel exterior analytic solutions of the Poincare equation for the flow outside the boundary and solve the Ekman boundary-layer problem to match the no-slip condition at the inner core boundary. Continuity of normal displacement at the inner core boundary, and in far-field approximation at the core–mantle boundary, determine the free constants in the two independent Poincare solutions. Conservation of linear momentum between the inner core, outer core and shell establishes a centre of mass reference frame. Analytic expressions for the pressure and viscous drag are obtained for both the axial and equatorial modes of oscillation. These are then used in the equation of motion for the inner core to deduce viscous and rotational splitting laws. Viscosity is found to reduce the effect of rotation on the periods of all three modes; most noticeably, it reduces the large rotational splitting of the two equatorial modes. We expect the geophysically interesting value of the viscosity just outside the inner core might be recovered from the observed splitting, particularly, that of the two equatorial modes.

Earthquakes and the spectrum of the Brussels Superconducting Gravimeter data for 1982–1986

Abstract Spectral analysis of the Brussels Superconducting Gravimeter data from 1982 to 1986 by earlier workers revealed a decaying spectral peak at 13.89 h following the December 30, 1983, Hindu Kush earthquake. Similar associations were presented for several other large earthquakes. The presence of the observed 13.89-h spectral peak was ascribed to possible excitation of inertial waves in the fluid outer core of the Earth. In this paper, the same data set is analysed with a moving window spectral method that shows the variation of the power in this and other spectral bands. Also, confidence intervals for the spectral estimates are computed. No unambiguous relation with the earthquake sequence used previously is found, and no statistically significant non-tidal spectral peak is found in the band between 12 and 24 h. However, the moving window spectral method applied to the 4.8- and 8-h spectral bands shows a closer association with the earthquake sequence. Again, although confidence interval calculations show some statistically significant spectral features in these bands, there is evidence that at least some of these features are generated by non-linear tidal interactions or possibly over-correction of the raw data for tidal signals. It is speculated that if indeed earthquake-induced, long-period core waves cause a detectable gravity signal, then the spectrum is perhaps richer in the short-period bands.

Motions of the inner core and mantle coupled via mutual gravitation: regular precessional modes

Abstract According to a model of the Earths interior where the inner core symmetry axis is allowed to deviate from the mantle symmetry axis, motions of the inner core and of the mantle are coupled predominantly via a mutual gravitational torque. Solutions corresponding to regular precession have been studied in detail in this paper in order to produce predictions that may, in principle, be observationally tested. In particular, the Chandler wobble frequency is expected to be split by the presence of the inner core, giving rise to two frequencies on either side. The inner core is found to exhibit a differential rotation at the inner core surface of the order of 1 m s −1 , which corresponds to a period of ∼ 90 days. According to conventional wisdom, which assumes that the inner core co-rotates passively with the rest of the Earth, this value represents an unacceptably fast shear at the inner core-outer core boundary. These two predictions are provisional, however, in the sense that several refinements alluded to in this paper are yet to be undertaken.

A search for free core nutation modes in VLBI nutation observations

In recent decades much effort has been devoted to searching observations for evidence of free core nutations predicted by theoretical models. Polar phototube data, tidal gravity observations and very long baseline nutation measurements have all been used. In this paper we report the first direct detection of the free core nutation in the frequency domain. A data set which consists of nutation series from 1100 very long baseline interferometry observing sessions from January 1984 to May 1992 is used in the spectral analysis. The results of the spectral analysis of the nutation series show that the period of the retrograde free core nutation mode is −431.0 (−425.5 to −436.7) solar days with an estimated amplitude of 0.19 ± 0.07 milliarcseconds. The recovered Q factor is 2000 ± 100. Based on an Ekman boundary layer on the surface of the core, this Q value implies a kinematic viscosity at the core-mantle boundary of 1.2 × 105 cm2 s−1. A spectral peak at the predicted location of the prograde free core nutation is below the 95% confidence interval and cannot be considered significant.

Earth dynamics : deformations and oscillations of the rotating Earth

1. Introduction and theoretical background 2. Time sequence and spectral analysis 3. Earth deformations 4. Earths rotation: observations and theory 5. Earths figure and gravitation 6. Rotating fluids and the outer core 7. The subseismic equation and boundary conditions 8. Variational methods and core modes 9. Static deformations and dislocation theory Appendix A. Elementary results from vector analysis Appendix B. Properties of Legendre functions Appendix C. Numerical Earth models References Fortran index Subject index.

Variational calculation of the free core nutation mode

Abstract Estimates of the period of the free core nutation (FCN) mode from observations of both Earth tides and nutations consistently show a difference of about 25–30 solar days from the period predicted by theoretical models. Reconciliation of this discrepancy has been the focus of some recent research in both analyses of observations and theoretical modelling. In this paper, we present numerical calculations of eigenperiod of the FCN mode based upon a variational principle of the liquid outer core of the Earth. This variational principle is numerically implemented by a finite element approach. This numerical implementation of the variational principle then leads to a nonlinear eigenvalue-eigenvector problem. Solutions to this eigenvalue-eigenvector system will deliver eigenperiods of free core nutation modes and eigenperiods of associated wobble modes. The eigenperiod of the FCN mode obtained in our numerical calculations is approximately 450 solar day which is 10 days shorter than the period predicted by previous theoretical models. One of the possible reasons for this difference may lie in different approaches to the dynamics of the liquid outer core.