Core precession: flow structures and energy

Experiments using a precessing liquid-filled oblate spheroid with ellipticity ( a − b )/ a =1/400 extend and clarify earlier research. They yield flow data useful for estimating flows in the Earth's liquid core. Observed flows illustrate and confirm a nearly rigid liquid sphere with retrograde drift and lagging a cavity (mantle) axis in precession. The similarities of the observed lag angle with that computed for a rigid sphere, and earlier energy dissipation research both support the use of a rigid sphere analytical model to predict energy dissipation and first-order flow within the core–mantle boundary (CMB). Second-order boundary layer and interior cylindrical flow structures also are photographed and measured. Interior flows are never turbulent or unstable at near-Earth parameters, although complex and transient flow patterns are observed within the boundary layer. Other mechanisms proposed to explain net heat loss from the Earth and maintenance of the geodynamo typically require acceptance of some critical but unproven premise. Precession and CMB configuration are known with certainty and precision. Analytical difficulties have been the obstacle. Experiments illustrate the consequences of precession and ellipticity, provide criteria for validating analytical and numerical models, and may yield direct knowledge of the Earth's deep interior with careful scaling.     

Could the energy near the FCN and the FICN be explained by luni-solar or atmospheric forcing?

The observed time-series of precession/nutation show residuals with respect to an empirical model based on the rigid Earth theoretical nutations and a frequency dependent transfer function with resonances to the Earth's normal modes. These residuals display energy mainly in the frequency domain around 430 and 500 days in the inertial frame. In this frequency band, the energy is possibly related to two normalmode frequencies: the free core nutation (FCN) and the free inner core nutation (FICN). In this paper, we examine the possibility of obtaining this energy from the resonance effect induced by a luni-solar (or planetary) forcing, or by an atmospheric forcing at a frequency very close to these Earth free nutations. The amplification factor due to the resonance is computed from an analytical formula expressed in the case of a simplified three-layer ellipsoidal rotating earth (with an elastic inner core, a liquid outer core and an elastic mantle), as well as the empirical formula based on the analysis of VLBI observations. For the tidal forcing, the theoretical results do not show any resonance at the level of precision we have examined but it is still possible to find one frequency near the FCN or FICN frequencies which could be excited. In contrast, for the atmospheric pressure the level of energy needed could be obtained from the diurnal pressure, depending on the noise level of the Earth's global pressure. We also show that the combination of three waves can explain the observed decrease of energy with time. While the tidal potential amplitudes are too small, a pressure noise level of 0.5 Pa would be sufficient to excite these waves.     

A simple model for thermal instability in the asthenosphere

Summary. Motion of the lithosphere over a low viscosity asthenosphere concentrates shear and thus energy dissipation in the asthenosphere. This heat source warms the asthenosphere and, in extreme circumstances, may lead to thermal instabilities. The conditions for thermal stability have been investigated by Melosh who supposed that constant stress acted on the plate, and by Yuen & Schubert who assumed constant velocity boundary conditions. In this paper we investigate a simple analytical model which behaves qualitatively like the more complex systems. This model reproduces the results of Melosh for constant stress and of Yuen & Schubert for constant velocity. The velocity—shear stress characteristic curve for this model shows three branches. The stability of solutions on each branch is a function of the boundary conditions, whether constant stress or constant velocity. The simplicity of the model allows us to investigate stability when neither constant stress nor constant velocity apply and to study the structure of the solutions as these limits are approached. A relation between the velocity of a plate and the driving force is constructed. A loading-line analysis specifies the actual stress and velocity of the plate. Although the solutions are unique for many combinations of the loading-line parameters, there is a region of multiple solutions. These solutions exhibit the characteristics of a 'cusp catastrophe' both a low velocity and a high velocity state are stable, while an intermediate state is unstable. Continental lithosphere may lie in this region, leading to epirogenic movements when the plate changes its velocity with respect to the mantle. Oceanic lithosphere almost certainly moves in the low velocity state.     

Application of A Correspondence Principle to the Free Vibrations of Some Viscoelastic Solids

In this study we apply the correspondence principle for free vibrations of a homogeneous viscoelastic solid derived by Fisher & Leitman to obtain the torsional modes of a homogeneous viscoelastic rod. We also extend the correspondence principle, showing that it may be used to find the frequencies of Love waves in a stratified viscoelastic medium. Finally, we apply the correspondence principle to four viscoelastic materials, the Kelvin-Voigt solid, the Maxwell solid, the standard linear solid, and the Achenbach-Chao solid. We show that in each of these cases some care must be used in applying the correspondence principle because of the presence of multiple solutions. We also examine measures of dissipation of the free vibrations, and we determine the conditions under which the logarithmic decrement may be approximated by the process-independent l/ Q of O'Connell & Budiansky.     

A boundary layer model for mantle convection with surface plates

Summary. A simple, analytical model for mantle convection with mobile surface plates is presented. Our aim is to determine under what conditions free convection can account for the observed plate motions, and to evaluate the thermal structure of the mantle existing under these conditions. Boundary layer methods are used to represent two-dimensional cellular convection at large Rayleigh and infinite Prandtl numbers. The steady-state structure consists of cells with isentropic interiors enclosed by thermal boundary layers. Lithospheric plates are represented as upper surfaces on each cell free to move at a uniform speed. Buoyancy forces are concentrated in narrow rising and decending thermal plumes; torques imparted by these plumes drive both the deformable mantle and overlying plate. Solutions are found for a comprehensive range of cell sizes. We derive an expression for the plate speed as a function of its length, the mantle viscosity and surface heat flux. Using mean values for these parameters, we find that thermal convection extending to 700 km depth can move plates at 1 cm yr^-1, while convection through the whole mantle can move plates at 4–5 cm yr^-1. Analysis of the steady-state temperature field, for the case of heating from below, shows that the upper thermal boundary layer develops a complex structure, including an 'asthenosphere' defined by a local maximum in the geotherm occurring at depths of 50–150 km.     

Viscosity profile of the lower mantle

Summary. We determine the variation of effective viscosity η across the lower mantle from models of the Gibb's free energy of activation G * and the adiabatic temperature profile. The variation of G * with depth is calculated using both an elastic strain energy model, in which G * is related to the seismic velocities, and a model which assumes G * is proportional to the melting temperature. The melting temperature is assumed to follow Lindemann's equation. The adiabatic temperature profile is calculated from a model for the density dependence of the Grüneisen parameter. Estimates of η depend on whether the lower mantle is a Newtonian or power law fluid. In the latter case separate estimates of η are obtained for flow with constant stress, constant strain rate, and constant strain energy dissipation rate. For G * based on the melting temperature, increases in η with depth range from a factor of about 100 for Newtonian deformation or power-law flow with constant stress to about 5 for non-Newtonian deformation with constant strain rate. For G * based on elastic defect energy, increases in η with depth range from a factor of about 1500 for Newtonian deformation or power-law flow with constant stress to about 10 for non-Newtonian deformation with constant strain rate. Among these models, only a non-Newtonian lower mantle convecting with constant strain rate or constant strain energy dissipation rate is consistent with recent estimates of mantle viscosity from post-glacial rebound and true polar wander data.     

B-polarization induction in two thin half-sheets coupled to the mantle by a conducting crust—II. Solution by the mode-matching method, the fields above the conductors, and example results

The mode-matching method is used to obtain an exact analytical solution to the problem of B -polarization induction in two adjacent thin half-sheets, lying on a conducting layer that is terminated by a perfect conductor at finite depth. These components of the model represent, respectively, the Earth's conducting surface layers, crust, and mantle. In dimensionless variables, the model has three independent parameters, these being the two thin-sheet conductances and the layer thickness. The mode-matching solution obtained in this paper is shown to be identical lo that derived via the Wiener-Hopf method in a companion paper (Dawson 1996), and so provides additional verification of that solution. As was shown in the companion paper, the solution for the present model contains, as special limiting cases, those for three models considered earlier by various authors. The second part of the present paper addresses the solutions for the electric fields in the non-conducting half-space above the conductors, which represents the atmosphere. In the final part, sample numerical calculations are presented to illustrate the solution.     

Internal waves in the earth's core

This paper investigates possible long-period oscillations of the earth's fluid outer core. Equations describing free oscillations in a stratified, self-gravitating, rotating fluid sphere are developed using a regular perturbation on the equations of hydrodynamics. The resulting system is reduced to a finite set of ordinary differential equations by ignoring the local horizontal component of the earth's angular velocity vector, Ω, and retaining only the vertical component. The angular dependence of the eigensolutions is described by Hough functions, which are solutions to Laplace's tidal equation.
The model considered here consists of a uniform solid elastic mantle and inner core surrounding a stratified, rotating, inviscid fluid outer core. The quantity which describes the core's stratification is the Brunt—Väisälä frequency N , and for particular distributions of this parameter, analytical solutions are presented. The interaction of buoyancy, and rotation results in two types of wave motion, the amplitudes of which are confined predominantly to the outer core: (1) internal gravity waves which exist when N ² > 0, and (2) inertial oscillations which exist when N ²<4Ω². For a model with a stable density stratification similar to that proposed by Higgins & Kennedy (1971), the resulting internal gravity wave eigenperiods are all at least 8 hr, and the fundamental modes have periods of at least 13 hr. A model with an unstable density stratification admits no internal gravity waves but does admit inertial oscillations whose eigenperiods have a lower bound of 12hr.     

Viscous gravitational relaxation

Summary This paper is concerned with a detailed examination of the response of Maxwell models of the planet to surface mass loads. Particular attention is devoted to an examination of the factors which determine the isostatic response since the understanding of this response is crucial in a number of different geodynamic problems. One particular example which we discuss in detail is concerned with the prediction of free air gravity anomalies produced by large-scale deglaciation events. Using the methods developed here we are able to provide the first direct assessment of the importance of initial isostatic disequilibrium on the observed relative sea-level variations and free air gravity anomalies forced by the melting of the Laurentide ice sheet. We are therefore able to estimate the extent to which such initial disequilibrium might influence the inference of mantle viscosity from isostatic adjustment data. Our calculations establish that free air gravity data, although they are sensitive to the degree of initial disequilibrium, provide an extremely high quality constraint upon the viscosity of the lower mantle.     

A new nutation series for a more realistic model earth

The frequency-dependent correction coefficients with respect to the forced nutations of a rigid earth are computed using the complex scalar gravitational-motion equations for an earth model with an anelastic mantle. Oceanic loads and tidal currents enter the model via outer boundary conditions. The ellipticity of the core-mantle boundary and the dynamical ellipticity are adjusted to observations. This requires the behaviour inside the model earth to be regarded as non-hydrostatic. Some relevant equations for the evaluation of boundary conditions and some terms in the equations of motion are expanded to second order in ellipticity. The computation of the equipotential-surface ellipticity profile is carried to second order as well. These second-order expansions lead to increased accuracy of the results in general. Moreover, one achieves a better reliability for the integration at frequencies close to a resonance. This allows the integration of the equations of motion at any relevant nutation period without the need for a normal-mode expansion. A complete new nutation series for a realistic model earth is presented.     

Analytical approach for the toroidal relaxation of viscoelastic earth

This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.     

Pleistocene deglaciation and the Earth's rotation: implications for mantle viscosity

Summary. Recent results from the analysis of postglacial rebound data suggest that the viscosity of the Earth's mantle increases through the transition region. Models which fit both relative sea-level and free air gravity data have viscosities which increase from a value near 10²² poise in the upper mantle beneath the lithosphere to a value of about 10²³ poise in the lower mantle. In this paper we analyse the effect of deglaciation upon the Earth's rotation and thereby show that the observed secular trend (polar wander) evident in the ILS—IPMS pole path, and measurements of the non-tidal acceleration of the length of day, are both consistent with the viscosity profile deduced from postglacial rebound. The two analyses are therefore mutually reinforcing.     

Effect of the overconsolidation ratio of soils in surface settlements due to tunneling

Construction of urban tunnels requires the control of surface subsidence to minimize any disturbance to nearby buildings and services. Past study of surface subsidence has been limited to mainly empiri...     

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.     

Magnetic and viscous coupling at the core–mantle boundary: inferences from observations of the Earth's nutations

Dissipative core–mantle coupling is evident in observations of the Earth's nutations, although the source of this coupling is uncertain. Magnetic coupling occurs when conducting materials on either side of the boundary move through a magnetic field. In order to explain the nutation observations with magnetic coupling, we must assume a high (metallic) conductivity on the mantle side of the boundary and a rms radial field of 0.69 mT. Much of this field occurs at short wavelengths, which cannot be observed directly at the surface. High levels of short-wavelength field impose demands on the power needed to regenerate the field through dynamo action in the core. We use a numerical dynamo model from the study of Christensen & Aubert (2006) to assess whether the required short-wavelength field is physically plausible. By scaling the numerical solution to a model with sufficient short-wavelength field, we obtain a total ohmic dissipation of 0.7–1 TW, which is within current uncertainties. Viscous coupling is another possible explanation for the nutation observations, although the effective viscosity required for this is 0.03 m² s⁻¹ or higher. Such high viscosities are commonly interpreted as an eddy viscosity. However, physical considerations and laboratory experiments limit the eddy viscosity to 10⁻⁴ m² s⁻¹, which suggests that viscous coupling can only explain a few percent of the dissipative torque between the core and the mantle.     

A new analytical solution estimating the flexural rigidity in the Central Andes

We present a new 2-D analytical solution of the fourth-order differential equation, which describes the flexure of a thin elastic plate.
The new analytical solution allows the differential equation for an elastic plate to be solved for any irregular shaped topography with a high spatial resolution. We apply the new method to the Central Andes. The flexural rigidity distribution calculated by this technique correlates well with tectonic units and the location of fault zones, for example, the Central Andean Gravity High correlates with the presence of a rigid, high-density body.     

Comments on the Chandler wobble Q

Summary. The Chandler wobble Q _w, as obtained from the astronomical data cannot be equated with the Q _m of the source of damping, as an examination of Chandler wobble energetics reveals. We find that if dissipation occurs in the mantle then Q _w≃ 9 Q _m, implying that either the mantle Q is frequency dependent or the wobble Q is much larger than 100. If the dissipation is in the oceans then Q _w≃ 20 Q _o, and the pole tide must be far from equilibrium.     

Secular variation of the poloidal magnetic field at the core–mantle boundary

A region of enhanced conductivity at the base of the mantle is modelled by an infinitesimally thin sheet of uniform effective conductance adjacent to the core–mantle boundary. Currents induced in this sheet by the temporally varying magnetic field produced by the geodynamo give rise to a discontinuity in the horizontal components of the poloidal magnetic field on crossing the sheet, while the radial component is continuous across the sheet. Treating the rest of the mantle as an insulator, the horizontal components of the poloidal magnetic field and their secular variation at the top of the core are determined from geomagnetic field, secular variation and secular acceleration models. It is seen that for an assumed effective conductance of the sheet of 10⁸  S, which may be not unrealistic, the changes produced in the horizontal components of the poloidal field at the top of the core are usually ≤10 per cent, but corrections to the secular variation in these components at the top of the core are typically 40 per cent, which is greater than the differences that exist between different secular variation models for the same epoch. Given the assumption that all the conductivity of the mantle is concentrated into a thin shell, the present method is not restricted to a weakly conducting mantle. Results obtained are compared with perturbation solutions.     

Finite element analysis of flow patterns near geological lenses in hydrodynamic and hydrothermal systems

We use theoretical and numerical methods to investigate the general pore-fluid flow patterns near geological lenses in hydrodynamic and hydrothermal systems respectively. Analytical solutions have been rigorously derived for the pore-fluid velocity, stream function and excess pore-fluid pressure near a circular lens in a hydrodynamic system. These analytical solutions provide not only a better understanding of the physics behind the problem, but also a valuable benchmark solution for validating any numerical method.
  Since a geological lens is surrounded by a medium of large extent in nature and the finite element method is efficient at modelling only media of finite size, the determination of the size of the computational domain of a finite element model, which is often overlooked by numerical analysts, is very important in order to ensure both the efficiency of the method and the accuracy of the numerical solution obtained. To highlight this issue, we use the derived analytical solutions to deduce a rigorous mathematical formula for designing the computational domain size of a finite element model. The proposed mathematical formula has indicated that, no matter how fine the mesh or how high the order of elements, the desired accuracy of a finite element solution for pore-fluid flow near a geological lens cannot be achieved unless the size of the finite element model is determined appropriately.
  Once the finite element computational model has been appropriately designed and validated in a hydrodynamic system, it is used to examine general pore-fluid flow patterns near geological lenses in hydrothermal systems. Some interesting conclusions on the behaviour of geological lenses in hydrodynamic and hydrothermal systems have been reached through the analytical and numerical analyses carried out in this paper.     

Postglacial rebound and lateral viscosity variations: a semi-analytical approach based on a spherical model with Maxwell rheology

A spectral method is employed to study the response to surface loads of a Maxwell earth including lateral viscosity variations. In particular, we focus on the effects of lithospheric cratons on the long-wavelength time-dependent displacement field for simple earth models. The viscosity contrast of the craton with respect to the surrounding mantle is kept fixed, whereas its thickness is allowed to vary. We show that the long-wavelength vertical displacement is not greatly affected by the presence of a lithospheric craton, while the tangential displacement is severely modified for the case of a homogeneous mantle. With increasing harmonic degree and thickness of the craton, the load-deformation coefficients deviate from those pertaining to a homogeneous mantle with a viscosity of 10²¹ Pa s. These deviations are particularly enhanced on timescales larger than a few hundred years. These findings indicate that the interpretation of the viscosity structure of the mantle inferred from postglacial rebound signatures based on radially stratified models is affected by the presence of lateral viscosity variations.