On the effect of oceanic tides on the inertial coupling between the liquid core and the mantle

The amplitudes and phases of forced nutation and diurnal earth tides depend significantly on the moment of forces between the liquid core and mantle of the Earth, resulting from the differential rotation of the core. The solution to the dynamic problem of rotation of an imperfectly elastic mantle with an imperfectly liquid core and an ocean indicates that the predominant role is played by the so-called core-mantle inertial coupling (related to the effect of hydrodynamic pressure in the liquid core on the ellipsoidal core-mantle boundary). The magnitude of this coupling depends significantly not only on the dynamic flattening of the liquid core but also on the elastic and inelastic properties of the mantle, as well as on the amplitudes and phases of oceanic tides. In this paper, the effects of oceanic tides on the magnitude of inertial coupling between the liquid core and the mantle and on the period and damping decrement of free nearly diurnal nutation are estimated.     

核幔耦合对地球自由核章动的激发影响

地球自由核章动（FCN）是地幔与液核相互作用的重要动力学现象,其激发机制涉及地表流体层、地幔和地核等圈层之间的耦合,此前研究多利用地表流体层角动量数据单独研究其对FCN的激发,对核幔耦合的影响考虑不足.本文基于角动量守恒理论分析了核幔耦合对FCN周期及振幅的影响,并结合多个大气及海洋角动量函数时间序列首次估算了核幔耦合在FCN激发过程中的贡献.结果表明核幔耦合对FCN周期产生的固定和时变影响对FCN激发的作用均不可忽视,尤其时变影响可达几十个微角秒,对于进一步解释FCN时变特征非常重要;核幔耦合对FCN振幅的直接影响是地表流体层的激发与实测FCN不相符的主要原因,黏滞、电磁和地形等耗散耦合的存在对地表流体的激发振幅有67%左右的减弱效果.     

超导重力技术在探讨核幔边界黏性特征中的初步应用

旋转椭球型地球的固体地幔与液态地核间相互作用而产生的逆向本征模通常称之为地球自由核章动,自由核章动的品质因子（Q值）能有效反映核幔边界层能量耗散特征,与核幔边界的黏滞度密切相关.本文首次利用全球地球动力学计划网络23个台站27组高密度采样的高精度超导重力仪器观测数据,采用迭积技术,确定了自由核章动参数Q值,进而计算了核幔边界的黏滞系数.数值结果说明获得的核幔边界动力学黏滞系数达到10³ Pa·s量级,与加拿大科学家Smylie等利用VLBI观测资料获得的最新结果一致,这说明重力技术是有效应用于研究地球深内部结构的重要手段之一.     

On the theory of core-mantle coupling

This article commences by surveying the basic dynamics of Earth's core and their impact on various mechanisms of core-mantle coupling. The physics governing core convection and magnetic field production in the Earth is briefly reviewed. Convection is taken to be a small perturbation from a hydrostatic, “adiabatic reference state” of uniform composition and specific entropy, in which thermodynamic variables depend only on the gravitational potential. The four principal processes coupling the rotation of the mantle to the rotations of the inner and outer cores are analyzed: viscosity, topography, gravity and magnetic field. The gravitational potential of density anomalies in the mantle and inner core creates density differences in the fluid core that greatly exceed those associated with convection. The implications of the resulting “adiabatic torques” on topographic and gravitational coupling are considered. A new approach to the gravitational interaction between the inner core and the mantle, and the associated gravitational oscillations, is presented. Magnetic coupling through torsional waves is studied. A fresh analysis of torsional waves identifies new terms previously overlooked. The magnetic boundary layer on the core-mantle boundary is studied and shown to attenuate the waves significantly. It also hosts relatively high speed flows that influence the angular momentum budget. The magnetic coupling of the solid core to fluid in the tangent cylinder is investigated. Four technical appendices derive, and present solutions of, the torsional wave equation, analyze the associated magnetic boundary layers at the top and bottom of the fluid core, and consider gravitational and magnetic coupling from a more general standpoint. A fifth presents a simple model of the adiabatic reference state.     

VLBI observations of free core nutations and viscosity at the top of the core

Very long baseline interferometry (VLBI) nutation measurement series, both in excess of 23 years length, from Goddard Space Flight Center (GSFC) and the United States Naval Observatory (USNO), have been analyzed for free core nutation resonances.VLBI nutation observations can only be made when the radio sources being used are visible, rendering the data sequence inherently non-equispaced. This poses the problem of the approach to be taken with unevenly spaced sampling. Both the conventional Discrete Fourier Transform (DFT) and the Fast Fourier Transform Algorithm (FFT) for its computation strictly require a fixed sampling interval.Our approach is to find the Discrete Fourier Transform of the non-equispaced record by minimizing an objective function which weights the error between the DFT representation and the measured values in inverse proportion to the square of their standard errors. The resulting conditional equations have a coefficient matrix of Toeplitz form but the recursive Levinson algorithm has been found inadequate for their solution, even when implemented in double quad precision. Instead, we employ the Singular Value Decomposition technique to solve the least squares problem of fitting the Discrete Fourier Transform to the non-equispaced VLBI nutation observations. A novel feature of our procedure is to use the Parseval relation to determine the number of singular values of the coefficient matrix to be eliminated.We report the observation for the first time of the prograde mode predicted by Jiang [Jiang, X., 1993. Wobble–nutation modes of the earth, Ph.D. thesis, York University, Toronto, Canada]. The long series of observations allow the determination of the time evolution of the free core nutations. We observe both the prograde and the retrograde modes to be in free decay. In addition to providing measures of the viscosity just below the core–mantle boundary (CMB), the free decays suggest impulsive excitations rather than continuous excitation by electromagnetic core–mantle coupling or the atmosphere. The average recovered viscosity at the top of the core is of the order 615 Pa s in contrast to the value of $8\times 10^{-3}$ Pa s found by Gans [Gans, R., 1972. Viscosity of the Earth’s core, J. Geophys. Res. 77, 360–366] from the extrapolation of laboratory measurements, and commonly used by dynamo theorists.     

On the Mechanism of the Secular Tidal Acceleration of the Solid Inner Core and the Viscosity of the Liquid Core

As is known, the secular deceleration of the Earth's diurnal rotation is explained mainly by the tidal friction in the ocean. Below we consider this mechanism in some detail, taking into account also elastic deformations of the mantle under the action of ocean loading and the interaction between the tide-generating body, ocean tidal wave, liquid outer core, and solid inner core. It is shown that elastic displacements of the core-mantle boundary under the action of ocean loading are of about the same amplitude and phase as the elastic loading displacements of the Earth's outer surface. As a result, side by side with the mechanism of secular deceleration of diurnal rotation of the mantle, there are also (1) the opposite mechanism of secular acceleration of diurnal rotation of the outer liquid core and of the solid inner core and (2) the mechanism of excitation of differential rotation in the liquid core. Taking these effects into account, we compare theoretical and modern observed data on the eastward drift of the solid inner core. It is shown that the best agreement may be obtained if the turbulent viscosity of the liquid core is about 2 × 10 ³ Poise     

Detecting thermal boundary control in surface flows from numerical dynamos

The geomagnetic field and secular variation exhibit asymmetrical spatial features which are possibly originating from an heterogeneous thermal control of the Earth's lower mantle on the core. The identification of this control in magnetic data is subject to several difficulties, some of which can be alleviated by the use of core surface flow models. Using numerical dynamos driven by heterogeneous boundary heat flux, we confirm that within the parameter space accessible to simulations, time average surface flows obey a simple thermal wind equilibrium between the Coriolis and buoyancy forces, the Lorentz, inertial and viscous forces playing only a secondary role, even for Elsasser numbers significantly larger than 1. Furthermore, we average the models over the duration of three vortex turnovers, and correlate them with a longer time average which fully reveals the signature of boundary heterogeneity. This allows us to quantify the possibility of observing mantle control in core surface flows averaged over a short time period. A scaling analysis is performed in order to apply the results to the Earth's core. We find that three vortex turnovers could represent between 100 and 360 years of Earth time, and that the heat flux heterogeneity at the core-mantle boundary could be large enough to yield an observable signature of thermal mantle control in a time average core surface flow within reach of the available geomagnetic data.     

Core-mantle interactions

Several aspects of core-mantle interactions were considered during a Royal Astronomical Society Discussion Meeting on 12th May 1989, including modelling the geomagnetic field at the core surface, the morphology of the field between 1600 and 1820 AD, dynamo theory, Taylor's constraint, fluid motions at the top of the core that reproduce the observed secular variation, pressure coupling between the core and mantle and its geophysical consequences, topographic core-mantle coupling, angular momentum transfer at the core-mantle interface, the detection and implications of core oscillations, particularly those with associated fluctuations in the Earth's rotation rate, and the seismological determination of the core-mantle boundary topography from lateral inhomogeneities in the mantle.     

Torque balance and energy budget for the precessionally driven dynamo

The feasibility of a precessionally driven dynamo is investigated. The relative orientation of the angular-velocity vectors of the mantle and core and the precession vector of the earth are determined from a torque balance. The core and mantle are acted upon by separate gravitational torques and mutual interaction torques resulting from pressure, viscous and magnetic stresses at the core-mantle interface. The viscous and magnetic torques are determined using the results of a detailed analysis of the Ekman-Hartmann and magnetic diffusion layers generated at the core-mantle interface by the misalignment of the mantle and core angular-velocity vectors. The dissipative torques are found to be weaker by a factor of 10^?4 than those estimated by Malkus (1968) and Stacey (1973), resulting in only 3.5 · 10⁷ W being extracted from the rotational kinetic energy of the earth by these mechanisms. Furthermore, it is found that all of this energy is dissipated in the boundary layers at the core-mantle interface and none is available to drive the geodynamo.     

On the Models of the Lower Mantle Viscosity Consistent with the Modern Data of Core-Mantle Boundary Flattening

In Forte and Claire Perry (2000) models of mantle viscosity (by using the data on tectonic plate velocities, of global free-air gravity anomalies, of surface topography corrected for crustal isostasy, and the excess of dynamic ellipticity of the core-mantle boundary in accordance with Herring et al., 1986 and Mathews et al., 1999) have been constructed. In the following investigation we reconsider the results which are obtained from the presently available data on core-mantle boundary flattening (abbreviated by CMBF). In contrast with the aforementioned work, we use below the value of CMBF (Molodensky and Groten, 1998) which is based on a new approach to the theory of diurnal Earth tides and nutation which takes into account the second-order terms of expansions of a small parameter (for a detailed discussion of this subject see (Molodensky and Groten, 1998)). Below we find the area of admissible values of mantle viscosity which does not contradict the following data sets: (a) the numerical value on CMBF; (b) the value of the whole Earth's dynamical flattening, and (c) the data on the secular deceleration of the Earth's rotation. Our estimations show, that the maximal viscosity at depth 2000 km may be of the order of 10²⁷ Poise. This value is consistent with the distribution obtained by Trubitsin (2000) who adopted the viscosity dependence on temperature and pressure by an exponential function with olivine parameters under the assumption that the activation energy varies only weakly with pressure, and the activation volume varies in inverse proportion to temperature. Under these assumptions, his solution of convection equations gave the depth dependence of temperature and thereby the viscosity distribution.     

Topographic heterogeneity of the core-mantle boundary in subduction zones and its effect on kinematics of main geomagnetic field sources

This work considers the kinematics of the source of the main geomagnetic field (MGF) near the core-mantle boundary under the Caribbean region. This source was selected because (i) its trajectory for the reviewed 110 years crosses this boundary, (ii) the region belongs to the so-called cemeteries of the tectonic plates, and (iii) numerous works have studied the structural heterogeneities of the lower mantle in this region with seismic tomography. Our study of the structural heterogeneities of the lower mantle and the trajectory of the MGF source indicates that the relics of the ancient tectonic plates in this region not only reach the coremantle boundary but could penetrate to the liquid core as well to the depth of 300 km. The “cemeteries” of the tectonic plates span significant areas in size. If the topographic heterogeneities of the core-mantle boundary, which are formed by the relics of the ancient tectonic plates, reach several hundreds of kilometers, then they can significantly affect the kinematics of individual structured flows in the liquid core and, consequently, change the spatial structure of secular MGF variations on the Earth’s surface.     

The period of the free core nutation: towards a dynamical basis for an ‘extra-flattening’ of the core-mantle boundary

Indirect observations and theoretical predictions for the period of the free core nutation (FCN) differ by anywhere from 15 to 30 days, and various effects have been invoked in attempts to explain this difference. The favored explanation remains as much as 5% departure in the flattening of the core-mantle boundary (CMB) from that of its hydrostatic reference figure. This 5% ‘extra-flattening’ of the CMB is not seen at the Earth's surface, where the difference is only about 0.5%. In contrast to the a posteriori model adjustments used to determine this up to 5% value, and the kinematic results available from viscous flow modeling using the seismically determined lateral heterogeneity in density data, we consider this problem from the perspective of a forward-modeling dynamical study. More specifically, we investigate the related problem of flow-induced surface and CMB topography, arising from convection in the mantle. As such, we have completed a comparative and systematic study of relative surface and CMB topography resulting from numerical models of mantle convection. When effects resulting from boundary curvature are isolated, it appears that the magnitude of CMB topography produced is insufficient in producing a significant extra-flattening of the CMB. However, results concerning effects solely resulting from a depth-dependent mantle viscosity profile, indicate that this factor may indeed lead to enhanced topography at the CMB of the magnitude required to produce the extra-flattening there.     

Magnetic probing of planetary interiors

The following general question is addressed: what can be learned about a planetary interior from measurements of the global planetary magnetic field at (or near) its surface? The discussion is placed in the context of Earth, for clarity, but the considerations apply to terrestrial planets in general (so long as the observed magnetism is either predominantly of internal origin, or else external source effects can be successfully filtered out of the observations). Attention is given to the idealized but typical situation of a rotating but spherically symmetric planet containing a highly conducting uniform fluid core surrounded by a nearly insulating rigid mantle, whose conductivity, a function of at most radius only, falls monotonically from its largest value at the base of the planetary mantle to zero at the planetary surface; the largest value of mantle conductivity as well as the mean value for the whole mantle and the mantle conductance are assumed small compared to the corresponding values of the core. Exterior to the planet is vacuum in the sense of an electrically uncharged insulator. The core fluid is inviscid, Boussinesq and gravitationally driven.Complete and perfect observations of either the instantaneous internal vector magnetic field together with its secular variation at a single epoch, or more realistically, the instantaneous internal vector magnetic field alone at two separated epochs are presumed available; the time separation between measurement epochs is long compared the Ohmic diffusion time of the planetary mantle, but small compared to that of the liquid core.Under such circumstances we describe how information about each of the following planetary properties can, in principle (though not without practical difficulty) be retrieved from the observations: (1) depth of the core-mantle boundary (a result of Hide); (2) depth to the current and motion sources responsible for the planetary dynamo; (3) presence or absence of small-scale turbulence in the upper reaches of the core; (4) large-scale horizontal fluid motion at the top of the core; (5) strength of horizontal currents, zonal magnetic fields, Coriolis and Lorentz forces at the top of the core; and (6) current system in the mantle and strength of electromagnetic core-mantle coupling.     

Models of density and mechanical <Emphasis Type="Italic">Q</Emphasis>-factor distributions according to new data on the nutations and free oscillations of the Earth: 2. Comparison with astrometric data

In the first part of the paper [Molodenskii, 2011], we considered the problem of ambiguity in the solution of the inverse problem of retrieval of density distribution in the Earth’s core and mantle and determination of the Q factors in the mantle from the entire set of modern data on seismic velocities (V _p and V _S ), the frequencies f _i and quality factors Q _i of free oscillations of the Earth, and the amplitudes and phases of its forced nutations. We have constructed the model distributions of these parameters, in which the root-meansquared (rms) deviations of all observed values from the predicted ones are much smaller than in the PREM model. Below, we compare the observed amplitudes of the forced nutation with the values predicted by our model. In order to understand how rigid are the constraints imposed by the amplitudes of forced nutation, we not only calculate the deviations of the observed amplitudes of nutation from the predictions by our model but also estimate the changes in these deviations caused by small variations in several parameters of the model. To the parameters to be varied we refer those which have no or barely any effect on the periods and damping constants of free oscillations but have a pronounced effect on the amplitudes of forced nutation. These parameters include (1) the rheological properties of the mantle in the interval of periods from an hour to a day; (2) the dynamical flattening of the liquid core; (3) the dynamic flattening of the solid inner core; (4) the viscosity of the liquid core; and (5) the moment of inertia of the solid inner core. In addition, we estimate the effects of variations in the moment of inertia of the liquid core to be small (±0.2%) and not to affect, within the observation error, the periods of free oscillations. We show that the uncertainty in the model depth distributions of density considerably decreases when the new data on the amplitudes and phases of the forced nutation of the Earth are taken into account. With these data, it is possible to estimate the creep function for the lower mantle in a wide range of periods from a second to a day.     

Axial and equatorial rotations of the Earth's cores associated with the Quaternary ice age

The differential axial and equatorial rotations of both cores associated with the Quaternary glacial cycles were evaluated based on a realistic earth model in density and elastic structures. The rheological model is composed of compressible Maxwell viscoelastic mantle, inviscid outer core and incompressible Maxwell viscoelastic inner core. The present study is, however, preliminary because I assume a rigid rotation for the fluid outer core. In models with no frictional torques at the boundaries of the outer core, the maximum magnitude of the predicted axial rotations of the outer and inner cores amounts to ∼2° year⁻¹ and ∼1° year⁻¹, respectively, but that for the secular equatorial rotations of both cores is ∼0.0001° at most. However, oscillating parts with a period of ∼225 years are predicted in the equatorial rotations for both cores. Then, I evaluated the differential rotations by adopting a time-dependent electromagnetic (EM) torque as a possible coupling mechanism at the core-mantle boundary (CMB) and inner core boundary (ICB). In a realistic radial magnetic field at the CMB estimated from surface magnetic field, the axial and equatorial rotations couple through frictional torques at the CMB, although these rotations decouple for dipole magnetic field model. The differential rotations were evaluated for conductivity models with a conductance of 10⁸ S of the lowermost mantle inferred from studies of nutation and precession of the Earth and decadal variations of length of day (LOD). The secular parts of equatorial rotations are less sensitive to these parameters, but the magnitude for the axial rotations is much smaller than for frictionless model. These models, however, produce oscillating parts in the equatorial rotations of both cores and also in the axial rotations of the whole Earth and outer and inner cores. These oscillations are sensitive to both the magnitude of radial magnetic field at the CMB and the conductivity structure. No sharp isolated spectral peaks are predicted for models with a thin conductive layer (∼200 m) at the bottom of the mantle. In models with a conductive layer of ∼100 km thickness, however, sharp spectral peaks are predicted at periods of ∼225 and ∼25 years for equatorial and axial rotations, respectively, although these depend on the strength of radial magnetic field at the CMB. While the present study is preliminary in modelling the fluid outer core and coupling mechanism at the CMB, the predicted axial rotations of the whole Earth may be important in explaining the observed LOD through interaction between the equatorial and axial rotations.     

Research on the scatter features of the PKiKP/PcP amplitude ratio and the inner core boundary density contrasts beneath Northeast Asia

Quantifying the density contrasts of the Earth's inner core boundary(ICB) is crucial to understand core-mantle coupling and the generation of the geodynamo. The PKiKP/PcP amplitude ratio is commonly used to obtain the density contrast at the ICB, but its applications are limited by scattered observed data. In this study, we selected the PKiKP and PcP phases reflected at the same region of inner-core and core-mantle boundaries beneath Northeast Asia from different earthquakes for the first time, and the observations suggested that the PKiKP/PcP amplitude ratio is widely scattered. We also compared the PKiKP and PcP amplitudes, which demonstrated that the scatter cannot be attributed only to ICB anomalies but might also arise from raypath differences and heterogeneities throughout the crust and mantle. By fitting the observed PKiKP/PcP amplitude ratio, we obtained a density contrast of approximately 0.65 g cm~(-3) and a compressional velocity contrast of approximately 0.87 km s~(-1) at the ICB beneath Northeast Asia. The larger contrast values indicate the possible occurrence of local crystallization occurring at the inner core surface.     

Core cooling by subsolidus mantle convection

Although vigorous mantle convection early in the thermal history of the Earth is shown to be capable of removing several times the latent heat content of the core, we are able to construct a thermal evolution model of the Earth in which the core does not solidify. The large amount of energy removed from the model Earth's core by mantle convection is supplied by the internal energy of the core which is assumed to cool from an initial high temperature given by the silicate melting temperature at the core-mantle boundary. For the smaller terrestrial planets, the iron and silicate melting temperatures at the core-mantle boundaries are more comparable than for the Earth, and the cores of these planets may not possess enough internal energy to prevent core solidification by mantle convection. Our models incorporate temperature-dependent mantle viscosity and radiogenic heat sources in the mantle. The Earth models are constrained by the present surface heat flux and mantle viscosity. Internal heat sources produce only about 55% of the Earth model's present surface heat flow.     

Variability of the topographic core-mantle torque calculated from core surface flow models

With the prospect of studying the relevance of the topographic core-mantle coupling to the variations of the Earth’s rotation and also its applicability to constraining the core surface flow, we investigate the variability of the topographic torque estimated by using core surface flow models accompanied by (a) uncertainty due to the non-uniqueness problem in the flow inversion, and (b) variance originating in that of geomagnetic secular variation models employed in the inversion. Various flow models and their variances are estimated by inverting prescribed geomagnetic models at the epoch 1980. The subsequent topographic torque is then calculated by using a core-mantle boundary topography model obtained by seismic tomography. The calculated axial and equatorial torques are found subject to the variability of order 10¹⁹ and 10²⁰  Nm, respectively, on which (b) is more effective than (a). The variability of the torque is attributed even to (a) and (b) of the large-scale flows (degrees 2 and 3). Yet, it still seems unlikely for the decadal polar motion with the observed amplitude to be excited exclusively by the equatorial topographic torque associated with any of reasonable core surface flow models. It is also confirmed that, with the topography model adopted here, the axial topographic torque on a rigid annulus in the core (coaxial with the Earth’s rotation axis) associated with any of reasonable flow models is larger by two orders of magnitude than the plausible inertial torque on such cylinders. This implies that any core surface flow model consistent with the topographic coupling does not exist, unless the topography model is appropriately modified. Nevertheless, the topographic coupling might provide not only a weak constraint for explaining the decadal LOD variations, but also the possibility to probe the core surface flow and the core dynamics.     

Dependence of the Love numbers upon the inner structure of the Earth and comparison of theoretical models with results of measurements

Summary The question this paper is examining is the following: to what extent are the Love numbers dependent on certain characteristics of the inner structure of the Earth? It has been proven — on the basis of calculations carried out by the author-that these quantities are only in a small degree dependent on the density values measured on the surface of the Earth and on the selection of the density function in the mantle of the Earth. On the other hand the value of Love numbersh, k andl is considerably influenced by the assumptions made about the core of the Earth, namely by the position of the boundary between the core and the mantle and by the magnitude of the rigidity coefficient presumed in the core in the vicinity of the core-mantle boundary.The results of the calculations are compared with those mean values of Love numbers obtained from the data of stations operating at different places of the Earth. By reason of this it can be assumed that the core of the Earth has, in the vicinity of the core-mantle boundary, a coefficient of effective rigidity of the order of 10¹⁰ dyn/cm², if the core-mantle boundary is placed at the relative Earth radius of 0.545 from the centre of the Earth.     

Effect of lateral viscosity variations in the core-mantle boundary region on predictions of the long-wavelength geoid

Seismic studies of the lowermost mantle suggest that the core-mantle boundary (CMB) region is strongly laterally heterogeneous over both local and global scales. These heterogeneities are likely to be associated with significant lateral viscosity variations that may influence the shape of the long-wavelength non-hydrostatic geoid. In the present paper we investigate the effect of these lateral viscosity variations on the solution of the inverse problem known as the inferences of viscosity from the geoid. We find that the presence of lateral viscosity variations in the CMB region can significantly improve the percentage fit of the predicted data with observations (from 42 to 70% in case of free-air gravity) while the basic characterisics of the mantle viscosity model, namely the viscosity increase with depth and the rate of layering, remain more or less the same as in the case of the best-fitting radially symmetric viscosity models. Assuming that viscosity is laterally dependent in the CMB region, and radially dependent elsewhere, we determine the largescale features of the viscosity structure in the lowermost mantle. The viscosity pattern found for the CMB region shows a high density of hotspots above the regions of higher-than-average viscosity. This result suggests an important role for petrological heterogeneities in the lowermost mantle, potentially associated with a post-perovskite phase transition. Another potential interpretation is that the lateral viscosity variations derived for the CMB region correspond in reality to lateral variations in the mechanical conditions at the CMB boundary or to large-scale undulations of a chemically distinct layer at the lowermost mantle.