On the dynamical theory of the rotation of the earth. II. The effect of precession on the motion of the liquid core

1953 ◽  
Vol 49 (3) ◽  
pp. 498-515 ◽  
Author(s):  
H. Bondi ◽  
R. A. Lyttleton
Keyword(s):  
Angular Velocity ◽  
Liquid Core ◽  
Rotation Period ◽  
Steady Motion ◽  
Dynamical Theory ◽  
Rate Of Change ◽  
Body Motion ◽  
Angular Velocity Vector ◽  
The Core ◽  
The Earth

In an earlier paper of the same general title (1) the possibility that the core of the Earth, in view of its supposed liquid nature, does not partake of the rigid-body motion of the outer shell was discussed with particular reference to the secular diminution of the angular velocity. In addition to this small rate of change of the magnitude of the angular velocity vector of the shell there occur changes in its direction consisting of the precession and nutation, but all the rates of change therein involved are small. The secular retardation takes place with extreme slowness, the nutations involve deviations of the axis with small angular amplitudes, while the precession, though of large angular amplitude, is of very long period compared with the rotation period of the Earth. Accordingly, it may be supposed that the effects of these various changes in the angular velocity can be considered separately in their relation to the motion within the core, and it is the object of this paper to give an account of our investigation into what may be termed for brevity the precession problem. It should perhaps be stated at the outset that the work does not constitute a solution of the problem, which our studies have led us to believe is one of the utmost mathematical difficulty presenting features of an exceptional character in hydro-dynamic theory. After first obtaining the equations of steady motion applicable to the interior, and those applicable to the boundary layer, the solution of the latter equations has been obtained; but in respect of the former equations we have been able to carry the question of the interior motion only as far as showing that no motion representable everywhere by analytic functions and consistent with the boundary conditions is possible. The investigation strongly suggests that no steady-state motion of a permanent character is possible for the interior, though the precise nature of the motion that actually occurs poses a problem of special interest from a hydrodynamic standpoint, but it is one to which we are not able to arrive at any definite answer at present. Without making any progress with the problem thus produced, the paper nevertheless makes clear the inherent difficulties of the problem and also serves to emphasize the inadequacy of any simplified mode of attack assuming classical fluid and resembling, for example, Poincaré's method for the nutation problem adopted by Lamb (3). Thus despite its incompleteness it seemed worth while to publish some account of such progress with these highly interesting questions as we have been able to make.

Dynamical theory of the rotation of the Earth

1986 ◽  
Vol 408 (1835) ◽  
pp. 267-275 ◽  
Keyword(s):  
Average Rate ◽  
Liquid Core ◽  
Angular Acceleration ◽  
Dynamical Theory ◽  
Rate Of Change ◽  
Moment Of Inertia ◽  
Iron Core ◽  
The Past ◽  
The Earth ◽  
The Moment

From recent values of improved accuracy of the apparent secular accelerations v and v' of the Moon and Sun, the lunar and solar tidal couples N and N' can be found. The appropriate dynamical theory shows that the moment of inertia of the Earth, C , has been diminishing at an average rate of -1.67 x 10 27 cm 2 g s -1 during the past 3300 years, giving rise to a non-tidal angular acceleration ω ∙ i = 1.52 x 10 -22 s -2 in addition to the retardations of ω resulting from the lunar and solar couples. The intrinsic couple associated with Ċ , the time-rate of change of C , is considerably greater than the solar tidal couple on all values of v and v' so far determined. For an initially all-solid Earth, use of known seismic data shows that the moment of inertia has decreased during the past 3 Ga at an average rate of -1.72 x 10 27 cm 2 g s -1 since a liquid core first began to form, a figure in close agreement with the value based on ancienteclipse data. On the time-honoured hypothesis that the core has resulted from iron separating downwards in an originally homogeneous Earth, the rate of decrease of C is -0.873 x 10 27 cm 2 g s -1 , only about one-half of that based on ancient-eclipse data, while if applied to these data the ratio N / N' = 11.35, which is more than twice the theoretical ratio on any tidal hypothesis. These results show that the iron-core theory is physically unacceptable.

On the dynamical theory of the rotation of the earth

1948 ◽  
Vol 44 (3) ◽  
pp. 345-359 ◽  
Author(s):  
H. Bondi ◽  
R. A. Lyttleton
Keyword(s):  
Rigid Body ◽  
Inner Core ◽  
Deformable Body ◽  
A Priori ◽  
Dynamical Theory ◽  
Outer Shell ◽  
Angular Velocity Vector ◽  
The Core ◽  
The Earth ◽  
Rotation Of The Earth

In the dynamical theory of the motion of the Earth relative to its centre of mass, the planet is usually regarded as a rigid or at most only slightly deformable body, and moments of inertia are adopted that are taken to refer to the Earth as a whole, while the motion itself at any instant is assumed capable of representation by a single angular velocity vector. This procedure, however, appears to involve unwarranted assumptions the recognition and removal of which may lead to conclusions of considerable importance. For it is well known from the theory of earthquake waves that the material of the central core of the Earth behaves like a liquid in that it transmits only longitudinal wave vibrations, while there is also other evidence suggesting that the material of the core is a true liquid (1). There is accordingly no a priori reason for supposing that the core will behave like a rigid body firmly attached to the surrounding shell if more or less permanent shearing forces are applied to it. In particular, in respect of any couple known to act on the outer shell, it is not permissible to assume, without examination of the assumption, that its effect will be transferred immediately to the inner core in a way preserving rigid-body rotation of the whole. If the material of the core behaves like a liquid where wave-motion is concerned, this suggests that it will probably also behave like a liquid whatever shearing forces act on it, and the extent to which changes in the rotatory motion of the outer shell can be communicated to the core, and what effects direct gravitational forces acting on the core may have, must in the first instance be questions of hydrodynamics and not rigid dynamics.

Constraints on core composition from shock-waves data

1982 ◽  
Vol 306 (1492) ◽  
pp. 37-47 ◽  
Keyword(s):  
Equations Of State ◽  
Liquid Core ◽  
Light Element ◽  
Outer Core ◽  
Atomic Ratio ◽  
Core Material ◽  
Solar Photosphere ◽  
The Core ◽  
The Earth ◽  
Core Composition

Seismic data demonstrate that the density of the liquid core is some 8-10 % less than pure iron. Equations of state of Fe-Si, C, FeS 2 , FeS, KFeS 2 and FeO, over the pressure interval 133-364 GPa and a range of possible core temperatures (3500- 5000 K), can be used to place constraints on the cosmochemically plausible light element constituents of the core (Si, C, S, K and O ). The seismically derived density profile allows from 14 to 20 % Si (by mass) in the outer core. The inclusion of Si, or possibly G (up to 11 %), in the core is possible if the Earth accreted inhomogeneously within a region of the solar nebulae in which a C :0 (atomic) ratio of about 1 existed, compared with a G : O ratio of 0.6 for the present solar photosphere. In contrast, homogeneous accretion permits Si, but not C, to enter the core by means of reduction of silicates to metallic Fe-Si core material during the late stages of the accumulation of the Earth. The data from the equation of state for the iron sulphides allow up to 9-13 % S in the core. This composition would provide the entire Earth with a S:Si ratio in the range 0.14-0.3, comparable with meteoritic and cosmic abundances. Shock-wave data for KFeS 2 give little evidence for an electronic phase change from 4s to 3d orbitals, which has been suggested to occur in K, and allow the Earth to store a cosmic abundance of K in the metallic core.

The GRACEFUL project : probing the Earth’s deep interior with satellite observations of the gravity field, magnetic field and earth’s rotation

2020 ◽  
Author(s):  
Julia Pfeffer ◽  
Anny Cazenave ◽  
Mioara Mandea ◽  
Véronique Dehant ◽  
Anne Barnoud
Keyword(s):  
Magnetic Field ◽  
Gravity Field ◽  
Liquid Core ◽  
Temporal Variations ◽  
Champ Satellite ◽  
The Core ◽  
Temporal Correlations ◽  
The Earth ◽  
Deep Earth ◽  
Order Of Magnitude

<p><span id="divtagdefaultwrapper" dir="ltr"><span lang="en-US">Convective motions in the Earth’s liquid core are known to  generate temporal variations of the magnetic field and of the length of day. Mass redistribution associated with these motions and exchange of matter with the lower mantle at the core mantle boundary (CMB) may eventually also contribute to the temporal variations of the gravity field, possibly detectable in the data of the GRACE and GRACE Follow On missions. In a pioneering work, Mandea et al., 2012 detected compelling spatio-temporal correlations at interannual time scale between the gravity and magnetic fields measured respectively by the GRACE and CHAMP satellite missions. These correlations were later interpreted by these authors as the results of physico-chemical interactions between the core and the mantle at the CMB. While such mechanisms are plausible, their mere existence, order of magnitude and  time scales remain an open question. Here we present the </span><span lang="en-US"> GRACEFUL project, recently selected by the  "Synergy" programme of the </span><span lang="en-US">European Research Council</span><span lang="en-US">, which objective is to  explore in more detail the previously reported observations described above, in particular the interannual co-variations of the magnetic and gravity fields, as well as their link with deep Earth processes.  This presentation is focussed on the  gravity field component, in particular on the search for the deep Earth signal that we hope to be able to detect i</span><span lang="en-US">n the  GRACE/GRACE FO data,  </span><span lang="en-US">after removing all other contributions due to water mass redistributions  occuring in the surface fluid evelopes, as well as  unrelated solid Earth signals associated with the Glacial Isostatic Adjustment and large earthquakes.</span></span></p>

scholarly journals Can the Core-Mantle Boundary Topography Influence the Earth's Nutation?

1990 ◽  
Vol 141 ◽  
pp. 161-162
Author(s):  
V. V. Bykova
Keyword(s):  
Perturbation Theory ◽  
Liquid Core ◽  
Theory Method ◽  
Mantle Boundary ◽  
First Order ◽  
Core Mantle Boundary ◽  
The Core ◽  
The Earth ◽  
Annual Term ◽  
Boundary Form

The nutation of the Earth with slightly nonelliptical liquid core is investigated by the perturbation theory method. It is shown that first-order terms affect the core ellipticity and its triaxiality. The most sensitive nutation terms in the second approximation were found to be retrograde 18.6-year term and retrograde annual term. The observed nutation amplitude values can be satisfied by special core-mantle boundary form.

Towards a theory of irregular variations in the length of the day and core-mantle coupling

1977 ◽  
Vol 284 (1326) ◽  
pp. 547-554 ◽  
Keyword(s):  
Angular Momentum ◽  
Liquid Core ◽  
Theoretical Research ◽  
Artificial Satellites ◽  
Momentum Exchange ◽  
Planetary Scale ◽  
The Core ◽  
Long Baseline ◽  
The Earth ◽  
Length Of The Day

Determinations of fluctuations in the length of the day reveal changes due to the transfer of angular momentum between the Earth’s ‘solid’ mantle and the overlying atmosphere on time scales upwards of a few weeks, as well as the slower but more pronounced ‘decade variations’ due largely (according to current ideas) to angular momentum transfer between the mantle and the Earth’s liquid core. Improvements in techniques for monitoring the Earth’s rotation, such as those afforded by recent advances in methods of ranging to artificial satellites and the Moon and of very long baseline interferometry, should therefore lead to results of interest to meteorologists concerned with planetary-scale motions in the atmosphere and to geophysicists con­cerned with the magnetohydrodynamics of the core and the origin of the main geo­magnetic field. The consideration of the stresses at the Earth’s surface and at the core­mantle interface that bring about angular momentum exchange between the solid and fluid parts of the Earth raises a number of basic hydrodynamical questions requiring further experimental and theoretical research. In the case of the core, quantitative difficulties encountered by the suggestion that the stresses are electromagnetic in origin led to the idea of topographic coupling associated with hypothetical undulations of the core-mantle interface.

On the phase-change hypothesis of the structure of the Earth

1965 ◽  
Vol 287 (1411) ◽  
pp. 471-493 ◽  
Keyword(s):  
Phase Change ◽  
Average Rate ◽  
Liquid Core ◽  
Rotation Period ◽  
Crystal Form ◽  
Outer Radius ◽  
Gravitational Energy ◽  
The Earth ◽  
Slow Expansion ◽  
Central Regions

The hypothesis that the liquid core of the Earth represents a phase-change at high pressure (and suitable temperature) of the mantle material is further investigated. A more accurate series of two-zone models have been computed, and also a new series of three-zone models. The change of overall radius as between an original all-solid Earth and the present size is shown to be at least 370 km. In the outer regions, greater pressure may be needed with rising temperature to effect the transition to denser crystal form (associated with the 20°-discontinuity), and from this cause acting alone slight expansion of the Earth would result but to an extent less than one-tenth the overall contraction. Epochs of rapid contraction (mountain-building eras) could thus be separated by longer intervals of very slow expansion. The initial liquefaction of the central regions brings about pressure increase at the boundary of the core that renders the Earth unstable in that about 6 per cent of the entire mass liquefies extremely rapidly to cause a sudden collapse of the planet as a whole. The accompanying decrease of outer radius is about 70 km. Thereafter the planet remains thoroughly stable and contracts only slowly. The total contraction to date would have reduced the moment of inertia by a factor about 4/5, and the corresponding reduction in rotation period (through conservation of angular momentum) would be an effect comparable with tidal friction. The contraction also leads to release of gravitational energy at an average rate comparable with that from radioactive sources. An important consequence of the phase-change hypothesis is that the melting-point gradient changes sign after sufficient depth, thereby permitting melting of the central regions to occur at moderate temperatures explicable by a reasonable content of radioactive elements.

scholarly journals SOME PROBLEMS ABOUT THE MOTION OF A RIGID BODY IN A RESISTIVE MEDIUM

2021 ◽  
Vol 3 (2) ◽  
pp. 6-17
Author(s):  
D. Leshchenko ◽  
◽  
T. Kozachenko ◽  
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Modern Technology ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
The Body ◽  
Body Motion ◽  
Body Rotation ◽  
Angular Velocity Vector ◽  
Resistive Medium

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

Ups and Downs

2018 ◽  
Author(s):  
Roy Livermore
Keyword(s):  
Mantle Convection ◽  
Laboratory Experiments ◽  
Inner Core ◽  
Computer Modelling ◽  
Great Depth ◽  
New Horizons ◽  
The Core ◽  
The Earth ◽  
The World ◽  
The 1960S

Despite the dumbing-down of education in recent years, it would be unusual to find a ten-year-old who could not name the major continents on a map of the world. Yet how many adults have the faintest idea of the structures that exist within the Earth? Understandably, knowledge is limited by the fact that the Earth’s interior is less accessible than the surface of Pluto, mapped in 2016 by the NASA New Horizons spacecraft. Indeed, Pluto, 7.5 billion kilometres from Earth, was discovered six years earlier than the similar-sized inner core of our planet. Fortunately, modern seismic techniques enable us to image the mantle right down to the core, while laboratory experiments simulating the pressures and temperatures at great depth, combined with computer modelling of mantle convection, help identify its mineral and chemical composition. The results are providing the most rapid advances in our understanding of how this planet works since the great revolution of the 1960s.

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