Experiments on precessing flows in the Earth's liquid core

Experiments simulating flow in the Earth's liquid core induced by luni-solar precession of the solid mantle indicate, to a first approximation, that the core behaves like a rigidized fluid sphere spinning slower than the mantle and with its spin axis lagging the mantle spin axis in precession. Secondary flow patterns are always present. At low precession rates the fluid sphere is subdivided into a set of cylinders coaxial with the fluid spin axis, the cylinders rotating alternately at slightly faster and slower rates relative to the net retrograde motion of the fluid as a whole. Slow non-axisymmetric columnar wave patterns develop between the differentially rotating cylinders. Axial flows between the spheroidal cavity boundary and the interior are observed. Fluid motion becomes turbulent only at precession rates large enough to cause the fluid spin axis to align nearly with the precession axis. There is no evidence that the Earth's liquid spin axis direction departs more than a fraction of a degree from geographic north. Our observations suggest precession induces a complex variety of laminar flows, including slowly varying and/or periodic patterns, in the Earth's liquid core.     

A model for energy dissipation at the mantle—core boundary

Summary. An existing experimentally verified model for energy dissipation in a processing spherical cavity filled with liquid assumed to be in a semirigidized state except for a viscous Ekman boundary layer is applied to the Earth's liquid core to assess energy dissipation magnitudes. Application of the model to the best available Earth data occurs at the derived energy dissipation maximum for the model. Other existing research showing that the Earth's atmosphere appears to adjust to a state of maximum dissipation led to generic models for systems of maximum dissipation. The maximum dissipation mantle—core model with core motion driven by Earth precession alone, coupled to the mantle only by viscous shear stresses, and with a spherical mantle—core boundary leads to energy dissipation rates on the order of 10⁴ times those necessary for an Earth dynamo. The maximum dissipation model also leads to excessive magnetic field drift rates and to excessive retardation of the Earth's rotation rate. Effects of the mantle—core ellipticity and of magnetic field coupling are briefly discussed and are used to help develop a less than maximum dissipation model also driven by precession alone but using the additional coupling to yield a model more consistent with observed phenomena.     

Electromagnetic core–mantle coupling—I. Explaining decadal changes in the length of day

Measured changes in the Earth's length of day on a decadal timescale are usually attributed to the exchange of angular momentum between the solid mantle and fluid core. One of several possible mechanisms for this exchange is electromagnetic coupling between the core and a weakly conducting mantle. This mechanism is included in recent numerical models of the geodynamo. The 'advective torque', associated with the mantle toroidal field produced by flux rearrangement at the core–mantle boundary (CMB), is likely to be an important part of the torque for matching variations in length of day. This can be calculated from a model of the fluid flow at the top of the outer core; however, results have generally shown little correspondence between the observed and calculated torques. There is a formal non-uniqueness in the determination of the flow from measurements of magnetic secular variation, and unfortunately the part of the flow contributing to the torque is precisely that which is not constrained by the data. Thus, the forward modelling approach is unlikely to be useful. Instead, we solve an inverse problem: assuming that mantle conductivity is concentrated in a thin layer at the CMB (perhaps D"), we seek flows that both explain the observed secular variation and generate the observed changes in length of day. We obtain flows that satisfy both constraints and are also almost steady and almost geostrophic, and therefore assert that electromagnetic coupling is capable of explaining the observed changes in length of day.     

Effects of dissipation and a liquid core on forced nutations in Hamiltonian theory

In a previous paper, the authors considered the free rotation of an earth model composed of a rigid mantle and a liquid core in the presence of dissipation and under the Hamiltonian formalism, obtaining analytical expressions for the free nutation modes.
In this paper we treat the forced motion. Approximate analytical solutions are worked out by means of Hori's perturbation method, the free solutions obtained in the former paper playing the role of the unperturbed solutions required in the application of the method. These solutions are consistent in the sense that, with the usual terminology, the rigid body solutions and the complex transfer functions are calculated with the same parameters.
Besides in-phase terms, the dissipation at the core–mantle boundary studied in this paper gives rise to out-of-phase terms. From a qualitative perspective, we discuss the issue of the resonance in this context. The presence of dissipation changes dramatically the character of the FCN wobble; that is, it is no longer a regular oscillation but a damped one. A strict resonance phenomenon cannot take place thereby, since the forcing perturbations are oscillations with a real (non-complex) frequency.     

The subseismic approximation in core dynamics—II. Love numbers and surface gravity

This paper extends our earlier examinations of the utility of various approximations for treating the dynamics of the Earth's liquid core on time-scales of the order of 10⁴ to 10⁸ s. We discuss the effects of representing the response of the mantle and inner core by static (versus dynamic) Love numbers, and of invoking the subseismic approximation for treating core flow, used either only in the interior of the liquid core (SSA-1) or also at the boundaries (SSA-2). The success of each approximation (or combinations thereof) is measured by comparing the resulting surface gravity effects (computed for a given earthquake excitation), and (for the Slichter mode) the distribution of translational momentum, with reference calculations in which none of these approximations is made. We conclude that for calculations of the Slichter triplet, none of the approximations is satisfactory, i.e. a full solution (using dynamic Love numbers at elastic boundaries and no core flow approximation) is required in order to avoid spurious eigenfrequencies and to yield correct eigenfunctions (e.g. conserving translational momentum) and surface gravity. For core undertones, the use of static Love numbers at rigid boundaries is acceptable, along with SSA-1 (i.e. provided the subseismic approximation is not invoked at the core boundaries). Although the calculations presented here are for a non-rotating earth model, we argue that the principal conclusions should be applicable to the rotating Earth. Shortcomings of the subseismic approximation appear to arise because both SSA-1 and SSA-2 lower the order of the governing system of differential equations (giving rise to a singular perturbation problem), and because SSA-2 overdetermines the boundary conditions (making it impossible for solutions to satisfy all continuity requirements at core boundaries).     

Geomagnetic evidence for fluid upwelling at the core-mantle boundary

Summary. Previous studies, both geomagnetic and seismic, have been unable to show conclusively whether or not there is fluid upwelling at the core-mantle boundary. Here a new method is developed, in which an attempt is made to invert geomagnetic secular variation data measured at the Earth's surface for a frozen-flux purely toroidal core-mantle boundary (CMB) velocity field, under the assumption that the mantle is electrically insulating and flux is frozen in at the CMB. These data have previously been inverted for the core-mantle boundary radial secular variation, from which the appropriate fit between model and data is known. Two different main field models were used to assess the effect of uncertainty in its radial component at the CMB. The conclusions were the same in both cases: frozen-flux purely toroidal motions provide a poor fit. A statistical test allows very firm rejection of the hypothesis that the residuals are not significantly larger, whereas there is no statistical difference between the residuals of inversions for radial secular variation and frozen-flux velocity fields at the CMB if upwelling and down-welling is included. The inherent non-uniqueness in the velocity field obtained is not of concern, since only their statistical properties are utilized and no physical significance is attached to the flows obtained.     

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.     

Effects of a mushy transition zone at the inner core boundary on the Slichter modes

This article investigates the effects of a mushy inner core boundary on the eigenperiods of the Slichter modes for a simple, but realistic, earth model (rotating, spherical configuration, elastic inner core and mantle, neutrally stratified, inviscid, compressible liquid core). It is found that the influence of the mushy boundary layer is substantial compared with some other effects, such as those from elasticity of the mantle, non-neutral stratification of the liquid outer core and ellipticity of the Earth and centrifugal potential. The results obtained here may set a lower bound on the eigenperiods of the Slichter modes for a realistic earth model. For example, for a PREM model, the lower bound of the central period of the Slichter modes should be about 5.3 hr.     

On the Stability of a Spherical Gravitating Compressible Liquid Planet without Spin

During the past ten years or so there has been considerable discussion in the literature regarding the author's 1963 contention that (neglecting temperature effects and spin) the Earth's liquid core cannot be stable unless the Adams-Williamson condition relating density distribution and compressibility holds there.
The present paper throws light on this question by showing mathematically that a sphere of gravitating compressible liquid cannot be internally stable unless this condition is fulfilled. Physical reasons for the necessity of this condition, which implies that particles of the liquid are in neutral equilibrium, are also discussed. By internal stability is meant stability of the density distribution while the spherical shape is maintained.
The question of shape stability is not treated here, since it may be assumed that the Earth's mantle is sufficiently rigid to keep the core essentially spherical.
The liquid is assumed to be a perfect fluid, elastic, and in the discussion only small strains are considered from an equilibrium configuration of initial hydrostatic stress. Furthermore thermodynamic effects are neglected and there is no spin.     

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.     

Numerical models of mantle convection with secular cooling

A 2-D time-dependent finite-difference numerical model is used to investigate the thermal character and evolution of a convecting layer which is cooling as it convects. Two basic cooling modes are considered: in the first, both upper and lower boundaries are cooled at the same rate, while maintaining the same temperature difference across the layer; in the second, the lower boundary temperature decreases with time while the upper boundary temperature is fixed at 0°C. The first cooling mode simulates the effects of internal heating while the second simulates planetary cooling as mantle convection extracts heat from, and thereby cools, the Earth's core. The mathematical analogue between the effects of cooling and internal heating is verified for finite-amplitude convection. It is found that after an initial transient period the central core of a steady but vigorous convection cell cools at a constant rate which is governed by the rate of cooling of the boundaries and the viscosity structure of the layer. For upper-mantle models the transient stage lasts for about 30 per cent of the age of the Earth, while for the whole mantle it lasts for longer than the age of the Earth. Consequently, in our models the bulk cooling of the mantle lags behind the cooling of the core-mantle boundary. Models with temperature-dependent viscosity are found to cool in the same manner as models with depth-dependent viscosity; the rate of cooling is controlled primarily by the horizontally averaged variation of viscosity with depth. If the Earth's mantle cools in a similar fashion, secular cooling of the planet may be insensitive to lateral variations of viscosity.     

On seismic-wave scattering by a rough core—mantle boundary

summary . A formulation is given for the seismic-wave scattering by a rough solid—liquid interface, in analogy to results derived for a solid—solid interface and a heterogeneous volume. Using Kennett's approach and the reciprocity theorem, the scattering is formulated as the excitation by an equivalent dislocation. Using interface parameters relevant to the core—mantle boundary (CMB), computational results for several types of body-wave scattering are given and compared to scattering by a heterogeneous volume. In an application to the generation of PKP precursors it is concluded that, whereas some data groups point to heterogeneity (which may not be small) in the lower mantle above CMB, in other cases a rough CMB may be considered equally feasible. Scattering at the source or receiver side of the core by both a slightly rough CMB (radial variations up to a few hundred metres) and a slightly heterogeneous lower mantle (relative variations in physical parameters up to a few per cent) produces the energy level that is observed in most of the PKP precursors; also the relevant scale lengths of variation are about the same ie both mechanisms (10–20 km with possibly somewhat higher values at relatively long epicentral distances).     

Mantle convection under simulated plates: effects of heating modes and ridge and trench migration, and implications for the core—mantle boundary, bathymetry, the geoid and Benioff zones

Summary. Numerical convection models are presented in which plates are simulated by imposing piecewise constant horizontal velocities on the upper boundary. A 4 × 1 box of constant viscosity fluid and two-dimensional (2-D) flow is assumed. Four heating modes are compared: the four combinations of internal or bottom heating and prescribed bottom temperature or heat flux. The case with internal heating and an isothermal base is relevant to lower mantle or whole mantle convection, and it yields a lower thermal boundary layer which is laterally variable and can be locally reversed, corresponding to heat flowing back into the core locally. When scaled to the whole mantle, the surface deflections and gravity and geoid perturbations calculated from the models are comparable to those observed at the Earth's surface. For models with migrating ridges and trenches, the flow structure lags well behind the changing surface 'plate'configurations. This may help to explain the poor correlation between the main geoid features and plate boundaries. Trench migration substantially affects the dip of the cool descending fluid because of induced horizontal shear in the vicinity of the trench. Such shear is small for whole mantle convection, but is large for upper mantle convection, and would probably result in the Tonga Benioff zone dipping to the SE, opposite to the observed dip, for the case of upper mantle convection.     

Seismic evidence for a sharp lithospheric base persisting to the lowermost mantle beneath the Caribbean

Broad-band data from South American earthquakes recorded by Californian seismic networks are analysed using a newly developed seismic wave migration method—the slowness backazimuth weighted migration (SBWM). Using the SBWM, out-of-plane seismic P -wave reflections have been observed. The reflection locations extend throughout the Earth's lower mantle, down to the core–mantle boundary (CMB) and coincide with the edges of tomographically mapped high seismic velocities. Modelling using synthetic seismograms suggests that a narrow (10–15 km) low- or high-velocity lamella with about 2 per cent velocity contrast can reproduce the observed reflected waveforms, but other explanations may exist. Considering the reflection locations and synthetic modelling, the observed out-of-plane energy is well explained by underside reflections off a sharp reflector at the base of the subducted lithosphere. We also detect weaker reflections corresponding to the tomographically mapped top of the slab, which may arise from the boundary between the Nazca plate and the overlying former basaltic oceanic crust. The joint interpretation of the waveform modelling and geodynamic considerations indicate mass flux of the former oceanic lithosphere and basaltic crust across the 660 km discontinuity, linking processes and structure at the top and bottom of the Earth's mantle, supporting the idea of whole mantle convection.     

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.     

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.     

Body tides on an elliptical, rotating, elastic and oceanless earth

Summary. The Earth's deformation caused by the luni-solar tidal force is defined as the 'body tide'. We compute the effects of the Earth's rotation and elliptical stratification on the body tide for a number of modern elastic structural models. Rotation and ellipticity within the mantle are found to affect tidal observations by about 1 per cent. A consequence is an improved estimate for the fluid core resonance in the diurnal tidal band. Agreement between results for the different structural models is very good. As a result, the results computed here can be used to model the tidal effects of a globally averaged, oceanless, rotating, elliptical and elastic earth to an accuracy of at least one part in 300.     

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.     

Revised value of the dynamical ellipticity of the Earth

Summary. A new value of the Earth's dynamical ellipticity H , defined as the ratio of the difference between the Earth's polar and mean equatorial moments of inertia to its polar moment of inertia, is derived from the most recent, and accurate, values of the Earth's equinoxial precession, the Earth—Moon mass ratio μ and other appropriate data, and a re-evaluation of the numerical procedures involved. This value is an order of magnitude more accurate than its previous values and yields an equivalent improvement in accuracy in other geodynamical quantities derived from H . The new value is consistent with the new System of Astronomical Constants and the new Geodetic Reference System 1980 and is suitable for use in the many astronomical, geophysical and geodetic applications of H .     

Speculations on the Thermal and Tectonic History of the Earth

Summary. The connection between the Earth's thermal history and convection in the mantle is exploited to elucidate the early evolution of the Earth. It appears probable that convection extending over almost all of the mantle has dominated vertical heat transport throughout the whole of the Earth's history. Only in boundary layers at the surface and at a depth of 650–700 km is conduction likely to be important. The resulting evolution appears to be consistent with geological observations on early Precambrian rocks.
Various arguments are put forward in favour of two horizontal scales of convective flow in the mantle at depths less than 650 km. The large scale flow is related to the motion of major plates, and must be ordered over distances of more than 5000 km. Its evolution and energetics are discussed and there are no obvious problems in maintaining the proposed convective motions. Small scale flow with an extent of the order of 500 km appears necessary both to explain the heat flow through older parts of the Earth's surface and to reconcile the geophysical observations with the results of numerical experiments. Though the existence of the small scale flow is at present speculative, various tests of its presence are proposed.