A core-mantle interaction in the rotation of the Earth

Effects of an interaction between the mantle and the core of the Earth on its rotational motion are investigated. Assuming that the Earth consists of a rigid mantle and a rigid core with a frictional coupling and a kind of inertial coupling between them, the equations of motion are derived, and they are solved in a close approximation. The solution gives the expressions for the precession, the nutation, the secular changes in the obliquity and the rotational speed, the polar motion and so on as functions of the magnitudes of these forces. A numerical estimation shows that the effect of the friction on the amplitude and phase of the nutation is small for a reasonable intensity of the friction while inertial coupling force has a decisive influence on the amplitude, and an appropriately chosen value of the latter force gives a nutation which closely agrees with observations. It is also indicated that this torque remarkably lessens the rates of the secular changes in the obliquity and the rotational speed. The possibility of a periodical change in the amplitude of the polar motion is suggested as a result of the interaction between the two consituents.     

Forced nutations of a two-layer Earth model

In this paper we present a theory of the Earth rotation for a model composed of an inelastic mantle and a liquid core, including the dissipation in the core–mantle boundary (CMB). The main features of the theory are: (i) to be Hamiltonian, therefore the computation of some complex inner torques can be avoided; (ii) to be self-consistent and non-dependent on a previous rigid Earth theory, so there is no need to use transfer functions; (iii) to be analytical, the solution being derived by perturbation methods. Numerical nutation series deduced from the theory are compared with the IERS 96 empirical series, an accuracy better than 0.8 mas in providing celestial ephemeris pole (CEP) offsets .     

Precession of the lunar core

Goldreich (Goldreich, P. [1967]. J. Geophys. Res. 72, 3135) showed that a lunar core of low viscosity would not precess with the mantle. We show that this is also the case for much of lunar history. But when the Moon was close to the Earth, the Moon’s core was forced to follow closely the precessing mantle, in that the rotation axis of the core remained nearly aligned with the symmetry axis of the mantle. The transition from locked to unlocked core precession occurred between 26.0 and 29.0 Earth radii, thus it is likely that the lunar core did not follow the mantle during the Cassini transition. Dwyer and Stevenson (Dwyer, C.A., Stevenson, D.J. [2005]. An Early Nutation-Driven Lunar Dynamo. AGU Fall Meeting Abstracts GP42A-06) suggested that the lunar dynamo needs mechanical stirring to power it. The stirring is caused by the lack of locked precession of the lunar core. So, we do not expect a lunar dynamo powered by mechanical stirring when the Moon was closer to the Earth than 26.0-29.0 Earth radii. A lunar dynamo powered by mechanical stirring might have been strongest near the Cassini transition.     

Magnetism and thermal evolution of the terrestrial planets

Of the terrestrial planets, Earth and probably Mercury possess substantial intrinsic magnetic fields generated by core dynamos, while Venus and Mars apparently lack such fields. Thermal histories are calculated for these planets and are found to admit several possible present states, including those which suggest simple explanations for the observations; whule the cores of Earth and Mercury are continuing to freeze, the cores of Venus and Mars may still be completely liquid. The models assume whole mantle convection, which is parameterized by a simple Nusselt-Rayleigh number relation and dictates the rate at which heat escapes from the core. It is found that completely fluid cores, devoid of intrinsic heat sources, are not likely to sustain thermal convection for the age of the solar system but cool to a subadiabatic, conductive state that can not maintain a dynamo. Planets which nucleate an inner core continue to sustain a dynamo because of the gravitational energy release and chemically driven convection that accompany inner core growth. The absence of a significant inner core can arise in Venus because of its slightly higher temperature and lower central pressure relative to Earth, while a Martian core avoids the onset of freezing if the abundance of sulfur in the core is ?15% by mass. All of the models presented assume that (I) core dynamos are driven by thermal and/or chemical convection; (ii) radiogenic heat production is confined to the mantle; (iii) mantle and core cool from initially hot states which are at the solidus and superliquidus, respectively; and (iv) any inner core excludes the light alloying material (sulfur or oxygen) which then mixes uniformly upward through the outer core. The models include realistic pressure and composition-dependent freezing curves for the core, and material parameters are chosen so that the correct present-day values of heat outflow, upper mantle temperature and viscosity, and inner core radius are obtained for the earth. It is found that Venus and Mars may have once had dynamos maintained by thermal convection alone. Earth may have had a completely fluid core and a dynamo maintained by thermal convection for the first 2 to 3 by, but an inner core nucleates and the dynamo energetics are subsequently dominated by gravitational energy release. Complete freezing of the Mercurian core is prohibited if it contains even a small amount of sulfur, and a dynamo can be maintained by chemical convection in a thin, fluid shell.     

Possible relations between the rotational axis of the Earth's inner core and the magnetic dipole axis

The geomagnetic field is maintained by amagnetohydrodynamic dynamo process within the liquid outer core. The distribution of the associated electric currents is modified if the outer core is bounded by electrically conducting material. Then, eddy currents and the related magnetic fields are generated within these regions. In particular, the relative rigid rotation of the inner core produces a secondary magnetic field, which is superimposed on the dynamo field. The angle between the dipole axis of the total field and the rotational axis of the inner core is an important quantity needed for the theory of polar motion of the Earth. This angle is investigated for a broad spectrum of angular velocities of the inner core. To simplify the mathematical procedure, we model the dynamo field using an axisymmetric field generated by a system of electric currents within the outer core. The conductivity of the mantle is neglected. We find that the position of the dipole axis depends on the angular velocity of the inner core as well as on the distribution of the current system within the outer core. Coincidence of both axes can be reached if the angular velocity is high enough and if the current system is concentrated within a thin sheet near the outer core-inner core boundary.     

Venus’ internal structure,temperature and core composition

The mass and radius of our closest neighbour Venus are only slightly smaller than those of the Earth indicating a similarity in composition. However, the lack of self-sustained internal magnetic field in Venus points to a difference in the core structure. The theory of tricritical phenomena has recently been used to study solidification at the high pressures and temperatures of the Earth, revealing how the Earth’s core works. This theoretical approach is here applied to Venus. While keeping Venus’ mantle density similar to the Earth’s, one obtains the gravitational acceleration g inside Venus, its moment of inertia factor, the size, pressure and density of its core, together with the planet’s temperature profile. Mainly due to the temperature difference between the core–mantle boundary and surface being 21% smaller than on the Earth, and the 11.5% smaller gravitational acceleration, Venus’ Rayleigh number Ra parameterizing mantle convection is only 54% of the Earth’s, offering a possible explanation for the present lack of plate tectonics on Venus. The theory as discussed predicts that Venus is molten at the centre, with temperature about 5200 K, and has 8 mol.% impurities there, slightly more impurities than in the Earth’s inner core boundary fluid. These impurities are likely to be a combination of MgO and MgSiO₃.     

The origin of the Moon: Theories involving joint formation with the Earth

A planet the size of the Earth or the Moon is much like a blast furnace; it produces slag-like rock floating on a mass of liquid metal. In the Earth, the mantle and crust are the slag, and the core is the liquid iron.In the Moon, there is clear chemical evidence that liquid iron was separated from the mass, but the Moon has no detectable iron core. This points to some kind of joint origin, which put the metallic iron in the Earth's core. For instance, the Moon might have been a detached part of the rocky matter of the Earth, as suggested by G. H. Darwin in the 1880's. But is is also clear, as Ringwood has pointed out, the there has been an enormous loss of volatiles from both Earth and Moon, but especially from the Moon. It may be that the Moon formed from a sediment-ring of small bodies detached somehow from the outer parts of the Earth, as Öpik has suggested.If tektites come from the Moon, then Darwin's suggestion is probably right; if they come from the Earth, then the Öpik-Ringwood sediment ring may be the origin.Paper presented at the AAAS Symposium on the Early History of the Earth and Moon in Philadelphia on 28 December 1971.     

Chemical composition of the Earth after the giant impact

The giant impact hypothesis for the origin of the Moon has been widely accepted. One of the most important features of this hypothesis is that the impactor's metallic core was incorporated in the Earth after impact. If the mass of the impactor is 0.82 × 10²⁷ g, the mass of the impactor core was estimated to be 0.19 × 10²⁷ g, which is about 1/10 of present Earth's core. Liu (1982) derived the bulk composition of the Earth from CI chondrites, and concluded that the Fe content of his model appears to be low in comparison with the present Earth, which, however, can be rationalized by the addition of impactor core into the proto-Earth developed by Liu (1982). If the impactor's mantle contains 14 wt% FeO as suggested, the mass ratio of impactor/proto-Earth should not exceed 0.22. The same ratio is not likely to exceed 0.30, if a giant blowoff did not occur during impact.     

Tides and nutation of the Earth: II. Realistic models of the liquid core

In the first part of this study (Molodensky, 2004; hereinafter, paper I), a survey of the theory of tides and nutation of the Earth was given with the inclusion of the dynamical effects of the elastic mantle and liquid core in the framework of a very simple model of a homogeneous, incompressible core. Here, effects of the inhomogeneity, compressibility, and viscosity of the liquid core are considered, along with those of electromagnetic coupling of the liquid core with the mantle and solid inner core. Errors in the present-day measurements of the amplitudes of forced nutation (of the order of 20 arc s) are well below not only the dynamical effects of the Earths liquid core but also the effects of the inelasticity of the mantle, the dynamical effects of the solid inner core, and the possible effects of electromagnetic coupling between the liquid core, solid inner core, and mantle. This opens up new avenues for astrometric studies of the mechanical properties and electrical conductivity of the mantle and solid inner core at very low frequencies. The modern data on the amplitudes and phases of the Earths forced nutation cannot be interpreted entirely unambiguously, since the following factors remain unknown: (i) the role of the core-mantle electromagnetic coupling compared to the coupling due to core viscosity near the core-mantle boundary and (ii) the frequency dependence of the creep function of the mantle at low frequencies. In large measure, the effects of electromagnetic and viscous coupling can be separated if high-precision data on the tidal variations in the gravitational force at periods of about a day are invoked and allowances are made for the differences between the effects of viscous and electromagnetic coupling on the amplitudes and phases of forced nutation and on the tidal variations in the gravitational force. Here, ranges of possible values of the creep function are constructed for periods from one hour to one day; for these calculations, values consistent with the entire set of data on the forced nutation of the Earth are assumed for the effective dynamical flattening of the liquid core-mantle and liquid core-solid inner core boundaries (these values describe the ellipticity of the boundaries of the inner core, outer core, and mantle, as well as the electromagnetic coupling between the liquid core, mantle, and solid inner core).Translated from Astronomicheskii Vestnik, Vol. 39, No. 1, 2005, pp. 61–80.Original Russian Text Copyright © 2005 by Molodensky.     

The large-number hypothesis and the Earth's expansion-II

Careful study by Owen of the displacements of the continents renders it likely that the Earth radius increased by almost 20% during the past 200 million years. Ramseys proposal that the Earth liquid core is a high-pressure phase of the mantle material is adopted to calculate the Earth radius under variableG. It is shown that ifG varies according to the large-numbers hypothesis of Dirac, the primordial Earth would have had radius 700 km less than the modern value. If it is assumed that the Earth began to respond and, hence, expand only 2 × 10⁸ yr B.P., the expansion as large as 700 km is shown to be energetically plausible.     

On diffusion of the tangential fluctuations of the geomagnetic field through the mantle

In this paper we develop a statistical approach to resolve the transport problem for the tangential fluctuations of the geomagnetic field in the mantle. For the sake of simplicity we treat the mantle as a thick layer of vacuum and assume in addition that only a radial component of the magnetic field of the core penetrates through the core-mantle boundary. These assumptions allow us to find exact expressions for the tangential field components throughout the mantle. By using such expressions we construct a correlation tensor of tangential components and then, since the mantle is thick enough, study its asymptotic properties on the Earth surface. Incidentally, the correlation tensor trace happens to be equal to the correlation function of the radial component that was obtained by Pilipenko and Sokoloff (1992). Indeed, we provide a simple boundary problem which initially describes the diffusion functions. We also pay a special attention to transformation properties of the correlation tensor and find here some interesting analogies with secular variation data of the geomagnetic field     

Non-linear equations for the rotation of a viscoelastic planet taking into account the influence of a liquid core

Non-linear equations governing the temporal evolution of the vector of instantaneous rotation are developed for an Earth with a homogeneous mantle having a viscoelastic Maxwell rheology and with a homogeneous inviscid fluid core.This general theory is investigated using the angular momentum theorem applied to the coupled core-mantle system. It allows to study the influence upon the planetary rotation of a quasi-rigid rotational motion in the liquid core. It also enables to investigate the consequences of excitation sources (e.g. pressure), located at the core-mantle interface. Especially, the influence of viscoelastic variations in the inertia tensors resulting from the rotation itself or from various excitation sources are detailed with the help of a Love number formalism. The equations of the linear theory for an elastic Earth with a liquid core, and the non-linear theory for a viscous planet with a quasi-fluid behavior are shown to be particular cases of our generalized system of equations. Some planetological applications may be derived from the quasi-fluid approximation.     

Extinct isotope heterogeneities in the mantles of Earth and Mars: Implications for mantle stirring rates

Heterogeneities in terrestrial samples for ¹⁸²W/¹⁸³W and ¹⁴²Nd/¹⁴⁴Nd are only preserved in Hadean and Archean rocks while heterogeneities in ¹²⁹Xe/¹³⁰Xe and ¹³⁶Xe/¹³⁰Xe persist to very young mantle‐derived rocks. In contrast, meteorites from Mars show that the Martian mantle preserves heterogeneities in ¹⁸²W/¹⁸³W and ¹⁴²Nd/¹⁴⁴Nd up to the present. As a consequence of the probable “deep magma ocean” core formation process, we assume that the Earth and Mars both had a very early two‐mantle‐reservoir structure with different initial extinct nuclide isotopic compositions (different ¹⁸²W/¹⁸³W, ¹⁴²Nd/¹⁴⁴Nd, ¹²⁹Xe/¹³⁰Xe, ¹³⁶Xe/¹³⁰Xe ratios). Based on this assumption, we developed a simple stochastic model to trace the evolution of a mantle with two initially distinct layers for the extinct isotopes and its development into a heterogeneous mantle by convective mixing and stretching of these two layers. Using the extinct isotope system ¹⁸²Hf‐¹⁸²W, we find that the mantles of Earth and Mars exhibit substantially different mixing or stirring rates. This is consistent with Mars having cooled faster than the Earth due to its smaller size, resulting in less efficient mantle mixing for Mars. Moreover, the mantle stirring rate obtained for Earth using ¹⁸²Hf‐¹⁸²W is consistent with the mantle stirring rate of ~500 Myr constrained by the long‐lived isotope system, ⁸⁷Rb‐⁸⁷Sr and ¹⁴⁷Sm‐¹⁴³Nd. The apparent absence of ¹⁸²W/¹⁸³W isotopic heterogeneity in modern terrestrial rocks is attributed to very active mantle stirring which reduced the ¹⁸²W/¹⁸³W isotopic heterogeneity to a relatively small scale (~83 m for a mantle stirring rate of 500 Myr) compared to the common sampling scale of terrestrial basalts (~30 or 100 km). Our results also support the “deep magma ocean” core formation model as being applicable to both Mars and Earth.     

Determining the possible building blocks of the Earth and Mars

Abstract— To determine the possible building blocks of the Earth and Mars, 225,792,840 possible combinations of the bulk oxygen isotopic and chemical compositions of 13 chondritic groups at 5% mass increments were examined. Only a very small percentage of the combinations match the oxygen isotopic composition, the assumed bulk FeO concentration, and the assumed Fe/Al weight ratio for the Earth. Since chondrites are enriched in silicon relative to estimates of the bulk Earth, none of the combinations fall near the terrestrial magmatic fractionation trend line in Mg/Si‐Al/Si space. More combinations match the oxygen isotopic composition and the assumed bulk FeO concentration for Mars. These combinations fall near the trend for shergottite meteorites in Mg/Si‐Al/Si space. One explanation for the difficulty in forming Earth out of known chondrites is that the Earth may be composed predominately of material that did not survive to the present day as meteorites. Another explanation could be that significant amounts of silicon are sequestered in the core and/or lower mantle of the Earth.     

Does the Moon Have a Metallic Core?: Constraints from Giant Impact Modeling and Siderophile Elements

The issue of whether the Moon has a small metallic core is reexamined in light of new information: improved dynamical modeling, new constraints on core size, and high temperature and pressure metal-silicate partition coefficients. Addressed specifically is the question of whether the Moon's siderophile element budget can be explained by derivation of the Moon from a differentiated impactor or proto-Earth (stage 1), followed by formation of a small metallic core within the Moon (stage 2). If the Moon is made of mantle material from either a “hot” impactor or a “warm” impactor or proto-Earth, a small metallic core (0.7 to 2 mass%) is predicted. If the Moon is made from mantle material from a “hot” proto-Earth, the lunar mantle would be more depleted in W or Re than is observed. Scenarios in which the Moon is made from impactor or proto-Earth mantle material that has equilibrated with metal at low pressures and temperatures (“cold” scenarios) would yield a much larger metallic core than observed. Finally, the greater depletions of Ni, Mo, and Re in the Moon (relative to the Earth) can be explained by low PT and reduced metal-silicate equilibrium in an impactor without later core formation in the Moon (i.e., no stage 2), but depletions of Co, Ga, and W cannot. Altogether, geochemically unlikely or geophysically inadequate non-metallic core alternatives, substantial geophysical evidence for a metallic core, and the successful models presented here for siderophile element depletions all favor the presence of a small lunar metallic core. Previous geochemical objections to an impactor origin of the Moon are eliminated because siderophile element concentrations in the lunar mantle are consistent with separation of a small core from a bulk Moon derived from impactor mantle material.     

The Rotation of a Non-Rigid, Non-Symmetrical Earth I: Free Nutations

This work is the first of a series of papers in which the canonicaltheory of the rotation of the non-rigid Earth of Getino and Ferrandiz isextended to the case of the non-symmetrical Earth. Here, the freeHamiltonian for an Earth composed of a rigid mantle and a liquid corewith A B (A and B principal moments ofinertia) is developed and integrated, obtaining the complete analyticalsolution which includes the free frequencies, Chandler Wobble and Fluid CoreNutation, corresponding to the non-symmetrical Earth. We have evaluatednumerically the effect of the Earths equatorial non-symmetry on thefree nutations. Although the effects of the second order are negligeable,the changes in the normal mode periods (about 1 day) may be important.     

Planetary core size: A seismological approach

A key parameter for understanding the geodynamics of a terrestrial planet is the size of its core. Numerical evaluation of 28 different interior structure models of Mercury, Venus, Earth, the Moon, and Mars suggests that there is an almost linear relationship between the core radius and the extent of the seismic P-wave core shadow. A scaling law is derived from a simple mantle density and velocity model that permits the interpretation of respective seismic measurements on terrestrial planetary bodies.     

用Hamilton方法计算弹性地球的章动序列

本文利用Hamilton方法研究弹性地球自转运动，采用地球模型PREM参数，给出了形状轴的章动序列．结果表明我们的方法是可行的，计算是可靠的．弹性地幔对地球章动的影响仅在毫角秒量级上，它相对液核对地球竟动的影响要小得多．     

An interpretation of the variations of the magnetic core-mantle coupling torques and the core drift rate

The influence of recently computed axial magnetic core-mantle coupling torques on the Earth's rotation was investigated. These torques derived from poloidal geomagnetic field within the mantle and at the core-mantle boundary are retarding torques. An accelerating torque due to the action of unknown parts of the core field was estimated by inverse solution of the equation of the mantle rotation for the periodic variations of the quantities of the magnetic field and the length of day. The variations of the drift rate of the Earth's core were compared with those of the mantle rotation velocity for a force-free Earth. The time constants of the coupling process were estimated and discussed in connection with the magnetic coupling of the mantle with an upper core layer. Der Einfluß kürzlich berechneter axialer Kern-Mantel-Kopplungsmomente auf die Erdrotation wurde untersucht. Diese Lorentz-Drehmomente, abgeleitet vom poloidalen geomagnetischen Feld im Mantel und an der Kern-Mantel-Grenze, sind retardierende Momente. Ein beschleunigendes Drehmoment, das der Wirkung unbekannter Feldanteile zugeordnet wird, wurde durch inverse Lösung der Mantelrotationsgleichung für die periodischen Variationen der Magnetfeldgrößen und der Tageslänge abgeschätzt. Die Variationen der Kerndriftgeschwindigkeit wurden mit denen der Mantelrotationsgeschwindigkeit für eine kräftefreie Erde verglichen. Die Zeitkonstanten des Kopplungsprozesses wurden ermittelt und im Zusammenhang mit der magnetischen Kopplung des Mantels mit einer oberen Kernschicht diskutiert.