Earth's Fluid Core Research Articles

A poincaré model for the earth's fluid core

Abstract The superconducting-gravimeter data of Melchior and Ducarme (1986) has been interpreted as internal motion in the Earth's core by Aldridge and Lumb (1987) using a Poincare model. Several low-order modes with periods of 13–16 hours have been tentatively identified in the core which is taken to be an incompressible, homogeneous fluid within a rigid, rotating container. The identification is based on asymptotic values of the frequencies which change slowly with time while the modes decay with an e-folding time of about 280 days. The slow change in frequency with time implies a small temporal variation in the rotation rate of the core. This mean flow is a nonlinear effect often observed in laboratory experiments designed to excite Poincare modes. Interaction among modes during free ringdown is also observed in those experiments and apparently in the data of Melchior et al. (1988) as well. Laboratory work thus provides the link to extend the Poincare model to include viscous and nonlinear effects in o...

Simulation of the 14‐month chandler wobble in a global climate model

The agent that generates and maintains the 14‐month Chandler wobble of the solid earth about its rotation axis has remained unresolved for a century with first the atmosphere, later earthquakes, and more recently the earth's fluid core proposed as candidates. Here we report that surface air pressure calculated in a coupled ocean‐atmosphere general circulation model (GCM) displays a 14.7 month signal, whose amplitude is similar to that found by Maksimov (1960) in station data; we identify it as the atmospheric Chandler wobble. This result indicates that changes in atmospheric mass distribution excite and maintain the wobble of the solid earth, and that neither earthquakes nor the fluid core are significant contributors. Another result is that in the GCM the amplitude of the wobble at high latitudes is a substantial fraction of the annual cycle, and thus is an important factor in climate formation as Maksimov (1960) suggested.

Asymptotic behaviour of decay rates of internal waves in a rotating stratified spherical shell

Summary. Boundary layer techniques are used to examine the dissipative decay of an internal oscillation that is a member of the inviscid spectrum of normal modes for a rotating fluid shell stratified under a radially directed gravitational field. A formula is derived for the decay factor on the so-called homogeneous spin-down time-scale. Estimates are obtained for the size of the decay factor as a function of wavelength, a function of the frequency and a function of a parameter A which measures the ratio of the stratification strength to the rotation strength. It is shown that all modes decay on the spin-down time-scale. The results are interpreted in the context of a model for the Earth's fluid core. It is observed that the presence of regions of unstable stratification may increase the decay rate for oscillations at frequencies less than twice the rotation frequency.

The palaeogeomagnetic field strength, variations in reversal frequency, and geomagnetic dynamo models

Abstract The geomagnetic field and its frequent polarity reversals are generally attributed to magnetohydrodynamic (MHD) processes in the Earth's metallic and fluid core. But it is difficult to identify convincingly any MHD timescales with that over which the reversals occur. Moreover, the geological record indicates that the intervals between the consecutive reversals have varied widely. In addition, there have been superchrons when the reversals have been frequent, and at least two, and perhaps three, 35-70 Myr long superchrons when they were almost totally absent. The evaluation of these long-term variations in the palaeogeophysical record can provide crucial constraints on theories of geomagnetism, but it has generally been limited to only the directional or polarity data. It is shown here that the correlation of the palaeogeomagnetic field strength with the field's protracted stability during a fixed polarity superchron provides such a constraint. In terms of a strong field dynamo model it leads to the speculation that the magnetic Reynolds number, and the toroidal field, increase substantially during a superchron of frequent reversals.

Hydromagnetic waves in the Earth's fluid core

Abstract Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, electrically conducting, non-dissipative fluid are examined in the geometry of a spherical shell. The basic magnetic field is assumed to be toroidal and is allowed to be an arbitrary axisymmetrical function of space. The basic density distribution is allowed to be an arbitrary function of space, consistent with the gravitational potential and the toroidal magnetic field. Conserved energy integrals are constructed and these constraints are used to determine sufficient conditions for stability. The stable waves are, in general, spatially trapped. Explicit expressions are obtained for the “turning surfaces” that delineate the regions in which waves are trapped. These expressions are used to illustrate trapped hydromagnetic waves in the Earth's fluid core.

Tidal motions within the earth's fluid core: resonance process and possible variations

We intend to show how the amplitude of the tidally induced flow in the Earth's fluid core is perturbed by the elastic yielding of the shell. Some possible variations of the resonance process as a function of the mantle mean rigidity, core geometry and size, and friction at the core-mantle boundary are investigated. The influence of the secular deceleration of the Earth is briefly considered.

Internal waves in a rotating stratified spherical shell: asymptotic solutions

Summary. Small amplitude oscillations of a rotating, density-stratified fluid bounded by a spherical shell are examined. No restrictions are placed on the thickness of the shell. The internal mode spectrum is examined in the complete rotation-stratification parameter range including the regime that is appropriate for a plausible stratification distribution in the Earth's fluid core. A mathematical model is derived in terms of an eigenvalue PDE of mixed type. The existence of oscillatory solutions is exhibited in the limits of no rotation and no stratification. The frequency spectrum is extended asymptotically away from these limiting cases. A reduction in the complexity of the PDE for modes oscillating at the inertial frequency is exploited. A variational formulation is constructed in which the stratification parameter is treated as an eigenvalue of the system for fixed wave frequency. The spectral information is again extended asymptotically away from these ‘accessible’ points. Although the PDE reduces to Laplace's tidal equations (LTE) only under stringent parameter restrictions, it is observed that aspects of the behaviour of low frequency LTE modes are reproduced in the general model.

Possible detection of the Earth's free‐core nutation

Theoretical studies indicate that interactions between the Earth's mantle and fluid core could produce a near 460‐day, retrograde circular component of the nutation, often called the free‐core nutation (FCN). Until now this phenomenon has never been observed. We use the 5.5 years of VLBI observations primarily collected under project IRIS to search for evidence of the FCN. The observations are consistent with an irregular excitation process, and a model which assumes a step excitation in the FCN amplitude to about 2.0 milliseconds of arc in late 1985 fits the data well. Theoretical analysis appears to rule out the strong Mexican earthquake of September 19, 1985, as a cause of the excitation.

Internal oscillations in the Earth's fluid core

Summary. A formulation is given for the eigenvalue problem describing internal oscillations in the Earth’s fluid core. The description is appropriate for either the Boussinesq or the subseismic approximations to the full system of governing equations for a rotating, compressible, inhomogeneous, selfgravitating thick shell of fluid. The depth distribution of the buoyancy frequency N, which measures the strength of the stratification in the core, is highly speculative, although physical observations constrain the maximum value of N. Theories have been proposed in which an outer layer of the fluid core is stably stratified (NZ > 0) whereas the inner regions are allowed to convect. The present mathematical model incorporates arbitrary variation of N with depth. The structure of the ‘turning surfaces’, which delineate the spatially wave-like from the exponentially decaying regimes of an internal mode, are examined for a particular functional choice for N that models a plausible stratification distribution in the core. The shapes of the regions in which the waves are trapped are dependent on the function N and the frequency X of the wave. This fact is used to suggest a mechanism by which hypothetical surface detection of internal waves in the core could be used to estimate both the value of N at the core-mantle boundary and the thickness of the layer of stable stratification.

Stability of the subseismic wave equation for the Earth's fluid core

Abstract The effects of compressibility on the stability of internal oscillations in the Earth's fluid core are examined in the context of the subseismic approximation for the equations of motion describing a rotating, stratified, self-gravitating, compressible fluid in a thick shell. It is shown that in the case of a bounded fluid the results are closely analogous to those derived under the Boussinesq approximation.

Ringdown of inertial waves in a spheroidal shell of rotating fluid

The role of inertial waves in the dynamies of the Earth's fluid core has been investigated through laboratory experiments on a spheroidal shell of rotating fluid. In these experiments inertial waves of azimuthal wavenumber 1, Ekman number 0(10 −5), Rossby number 0(10 −1) were excited by precession of an inner spheroidal body. Proximity to resonance was achieved by adjusting the ratio of the frequency of precession of the inner body to the rotational speed of the container to be near the eigenfrequency of the inertial wave mode being studied. Once the system was near resonance the perturbation was stopped and ringdown records were obtained. Amplitude, eigenfrequency and decay rate were recovered simultaneously for the principal and neighbouring modes excited using an iterative linearized least squares procedure. The recovery of complex eigenfrequencies for non-axisymmetric inertial waves in this shell geometry has given experimental verification of their existence. For those waves of azimuthal wavenumber one, a significant nonlinear interaction among modes is inferred from the simultaneous recovery of neighbouring modes. Other non linear effects include a mean azimuthal flow which appears to be stable for the low spatial order modes studied. These results contrast with highly unstable mean flow found experimentally in similar experiments carried out in cylindrical geometry.

On oscillations of the Earth's fluid core

Summary. In the conventional spheroidal and toroidal representation, the differential equations governing long-period free oscillations of the Earth’s fluid core form an infinitely coupled system. Because of the mathematical complexity, solutions of this system are customarily attempted by numerical integration. The approach, however, necessitates a severe truncation of the coupling chain and this, in turn, renders it difficult to interpret the results. An alternative approach is sought in this work. By invoking the ‘solenoidal flow’ approximation which neglects the dilatation, and the ‘subseismic’ approximation which neglects the flow pressure respectively, we have succeeded in developing analytic solutions for the inertial and gravitational oscillations. This results in a significant simplification of the eigenvalue problem because the infinitely coupled system of differential equations is now reduced to an algebraic one. More importantly, the analytic solutions reveal immediately the roles played by the rotation and density stratification in core dynamics. We find that the inertial oscillations, which arise from the Earth’s rotation, are essentially independent of the core’s density stratification. Thus the fluid core can be approximated by a homogeneous, incompressible and non-gravitating inviscid Newtonian fluid with only minor modifications. The gravitational oscillations, on the other hand, are governed by both the rotation and density stratification. In a non-rotating earth, gravitational oscillations are not possible for neutrally or unstably stratified cores. Because of rotation, however, gravitational oscillations become possible for any density stratification in the core, and the eigenfrequencies are divided into alternating allowed and forbidden zones. We show that in each allowed zone, there exist an infinite number of gravitational oscillations with their eigenfrequencies approaching asymptotically the eigenfrequency of a corresponding inertial oscillation. Both the solenoidal flow and subseismic approximations ignore the fluid flow arising from the variation in gravitational potential. To correct for this neglect, as well as taking into account the deformation in the solid part of the Earth, an asymptotic formulation is developed. It is shown that the eigenfrequencies can be determined without explicitly solving for the ‘correction’

A search for Chandler and nearly diurnal free wobbles using Liouville equations

Summary. The basic equations describing the dynamical effects of the Earth's fluid core (Liouville, Navier-Stokes and elasticity equations) are derived for an ellipsoidal earth model without axial symmetry but with an homogeneous and deformable fluid core and elastic mantle. We develop the balance of moment of momentum up to the second order and use Love numbers to describe the inertia tensor's variations. The inertial torque takes into account the ellipticity and the volume change of the liquid core. On the core—mantle boundary we locate dissipative, magnetic and viscous torques. In this way we obtain quite a complete formulation for the Liouville equations. These equations are restricted in order to obtain the usual Chandler and nearly diurnal eigenfrequencies. Then we propose a method for calculating the perturbations of these eigenfrequencies when considering additional terms in the Liouville equations.

The effects of the atmosphere and oceans on the Earth's wobble -- I. Theory

Summary The theory of wobble excitation for a non-rigid earth is extended to include the effects of the earth's fluid core and of the rotationally induced pole tide in the ocean. The response of the solid earth and oceans to atmospheric loading is also considered. The oceans are shown to be affected by changes in the gravitational potential which accompany atmospheric pressure disturbances and by the load-induced deformation of the solid earth. These various improvements affect the excitation equations by about 10 per cent. Atmospheric and oceanic excitation can be computed using either an angular momentum or a torque approach. We use the dynamical equations for a thin fluid to relate these two methods and to develop a more general, combined approach. Finally, geostrophic winds and currents are shown to be potentially important sources of wobble excitation, in contrast to what is generally believed.

Earth's core, climate, and magnetic field

H.H. Lamb (Climate, Present, Past and Future, vol. 2., Methuen Co., N.Y., 1977) postulated convincingly that there is a tight correlation between variations in the earth's climate and solar exposure (as represented by mean surface temperature 8). Given constant radiation and surface, solar exposure is a function of the length of day, and thus it has been tempting to correlate, as well, mechanisms that could cause or be the result of changes in the earth's rotation rate. Such mechanisms have been reviewed by Lambeck (The Earth's Variable Rotation, Geophysical Causes and Consequences, Cambridge University Press, London, 1980).Lambeck examined the variations in terms of fluid mechanical coupling between the earth's fluid core and the mantle but deduced that the electromagnetic coupling process between core and mantle probably plays the most dominant role in affecting the spin rate. The factors, climate (9), length of day, variation in the earth's rotation rate, and variations in the geomagnetic field, have been recently correlated by a group of French geophysicists (V. Courtillot et al., Nature, 297, 386, 1982) to form the basis of what is described as an ‘attractive causal chain’ to forecast future climatic variations. The predictive correlations are not subject to circular reasoning as to which—climate variations or variations within the core—was the ultimate cause. Courtillot et al. note a lag in time between core changes and climate changes—in terms of 0—of 15–25 years to predict the climate.

Dynamo theory of the earth's varying magnetic field

Various forms of the dynamo theory are presented in a graphic manner. Each of them depends on a flow pattern of presumably thermal convection in the earth's fluid core. The conducting fluid moves in magnetic fields generated by currents induced by the motion. Each of the flow patterns includes vortices, with helicity induced by the Coriolis force, that twist the magnetic fields in such a way as to regenerate an initial field. Some of them involve also differential rotation, with the parts of the core near the axis rotating more rapidly than the outer parts. Some forms of the theory are more successful than others in accounting qualitatively for the various observed aspects of the field, particularly the westward drifts and the occasional polarity reversals. The thermal energy source may be radioactivity of heat or crystallization at the inner-core surface.

Magnetic waves in the core of the earth. II

Abstract Small oscillations of electrically conducting fluid under the influence of magnetic, Archimedean (buoyancy) and Coriolis forces, taking into account Ohmic diffusion, are considered for conditions corresponding to the Earth's fluid core. The significant rôle played by the nonuniform rotation of the core is stressed. Its effect is investigated by means of greatly simplified planar models. A qualitative picture of the magnetic wave regime probably established in the Earth's core is proposed. On the basis of this picture a simple model of torsional oscillations of the core is constructed.

Wobble and nutation of the Earth

ummary Elastic-gravitational normal mode theory is used to study theoretically the free wobble and free nutation of a suite of geophysically plausible rotating slightly elliptical earth models. We principally find that observations of wobble eigenfrequencies are not likely to constrain strongly the structure of the Earth's fluid core, that the results of our calculations, together with Dahlen's estimate of the ocean correction, yield the observed Chandler wobble period within its observational uncertainty and that estimates of wobble excitation by earthquakes based on quasi-static calculations of the Earth's response are correctly computed. Also we summarize several analytical treatments of both wobble and internal core modes of rotating Earth models and use that information to assess analytically and numerically the approximate theory exploited here. We show that the approximation we use is valid for the Earth's major free wobbles but not for studying internal modes in the fluid core. summary By exploiting the existence of analytic solutions to certain members of the general class of problems of interest to us here, we have been able to explore the range over which the numerical theory used here is valid. In particular, the theory is adequate to describe the major mantle wobbles of the Earth – namely the Chandler wobble and the nearly diurnal free wobble. It is also adequate to describe the response of the Earth to tidal potentials, at least to the extent that ‘internal’ modes in the fluid core do not participate in the Earth's response. The theory has been demonstrated to be sorely inadequate for two other classes of normal modes: internal core modes and the Chandler Wobble of the solid inner-core. In both cases the failure of the theory is associated with the inability of our finite representation to adequately represent motion in the fluid. These failures are not evident from inspection of the numerical results alone; their detection was only made possible by the availability of analytic solutions. A corollary of these results is that recent published calculations of internal core modes for rotating geophysically plausible earth models by Crossley (1975) and Shen & Mansinha (1976) are almost surely invalid. The theoretical basis for those studies, like the one exploited here, is wholly inadequate to describe internal core modes.

Spectral line similarity in the geomagnetic dipole field variations and length of day fluctuations

Power spectral density analysis using Burg's maximum entropy method (MEM) was applied to the geomagnetic dipole field and its rate of change for the years 1901-1969. Both spectra indicate relative maxima at 0.015 cycle/yr and its harmonics. These maxima correspond approximately to 66-, 33-, 22-, 17-, 13-, 11-, and 9-year spectral lines. The application of the same analysis techniques to the length of day (l.o.d.) fluctuations for the period 1865-1961 reveals similar spectral characteristics. The existence of the common spectral peaks with periods of 66 and 33 years in the l.o.d. fluctuations and the geomagnetic dipole field is clearly established. The existence of the higher harmonics is somewhat uncertain because of the line-splitting problem in the MEM spectral analysis. It is suggested that the spectral line similarity in the l.o.d. fluctuations and the dipole field variations is related to the motion within the earth's fluid core during the past 100 years.

Convection at the melting point: A thermal history of the earth's core

Abstract Higgins and Kennedy (1971) concluded that the Earth's fluid core has a stable stratification if it is at its melting point. Busse (1972) and Elsasser suggested as an alternative that a hydrostatic-isentropic distribution of particulate solid can produce neutral stability in a partially molten core. Here this suggestion is quantified and a determination is made of the efficiency of the production of fluid motion from the heat flux. This is used to establish that macroscopic convection can exist only if the particulate solid is of sufficiently small size. A thermal history of the core compatible with upper mantle heat flux is advanced in which it is suggested that the inner core is a fairly recent feature. The implication of these results for convection-driven and precession-driven dynamos is that both can function for a small enough suspended particulate, that the convection-dynamo will fail for particles greater than one micron in diameter, and that the precession-driven dynamo probably cannot su...