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.     

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.     

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.     

SNREI地球对表面负荷和引潮力的形变响应

基于PREM模型，利用非自转、球型分层、各向同性、理想弹性（SNREI）地球的形变理论，讨论了地球在不同驱动力作用下的形变特征．采用地球位移场方程的4阶Runge Kutta数值积分方法，解算了在表面负荷和日月引潮力作用下地球表面和内部形变和扰动位，并给出了地球表面的负荷Love数和体潮Love数．结果表明在固体内核中的形变很小，液核中低阶(n＜10)负荷位移随半径的变化非常复杂．当负荷阶数超过10时，地核中的形变和扰动位都很小，地球的响应主要表现为弹性地幔中的径向位移，且随深度增加急剧减弱，负荷阶数越高这种衰减的速度越快．SNREI地球的地表负荷Love数和体潮Love数与信号频率的依赖关系很弱．在计算体潮Love数的过程中，采用了SNREI地球的运动方程，同时考虑了由于地球自转和椭率引起的核幔边界附加压力，这一近似处理方法获得的结果能很好地符合地球表面重力潮汐实际观测结果．     

Core mantle boundary topography from short period PcP, PKP, and PKKP data

We use a total of 839,369 PcP, PKPab, PKPbc, PKPdf, PKKPab, and PKKPbc residual travel times from [Bull. Seism. Soc. Am. 88 (1998) 722] grouped in 29,837 summary rays to constrain lateral variation in the depth to the core-mantle boundary (CMB). We assumed a homogeneous outer core, and the data were corrected for mantle structure and inner-core anisotropy. Inversions of separate data sets yield amplitude variations of up to 5 km for PcP, PKPab, PKPbc, and PKKP and 13 km for PKPdf. This is larger than the CMB undulations inferred in geodetic studies and, moreover, the PcP results are not readily consistent with the inferences from PKP and PKKP. Although the source-receiver ambiguity for the core-refracted phases can explain some of it, this discrepancy suggest that the travel-time residuals cannot be explained by topography alone. The wavespeed perturbations in the tomographic model used for the mantle corrections might be too small to fully account for the trade off between volumetric heterogeneity and CMB topography. In a second experiment we therefore re-applied corrections for mantle structure outside a basal 290 km-thick layer and inverted all data jointly for both CMB topography and volumetric heterogeneity within this layer. The resultant CMB model can explain PcP, PKP, and PKKP residuals and has approximately 0.2 km excess core ellipticity, which is in good agreement with inferences from free core nutation observations. Joint inversion yields a peak-to-peak amplitude of CMB topography of about 3 km, and the inversion yields velocity variations of ±5% in the basal layer. The latter suggests a strong trade-off between topography and volumetric heterogeneity, but uncertainty analyses suggest that the variation in core radius can be resolved. The spherical averages of all inverted topographic models suggest that the data are best fit if the actual CMB radius is 1.5 km less than in the Earth reference model used (i.e. the average outer core radius would be 3478 km).     

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.     

Earth tides and polar motion

We establish a general theory that describes the rotational motion of a layered, oblate, elastic Earth under the influence of tidal forces when account is taken of the liquid outer core. We obtain a linearized version of the Navier-Stokes equation; within it not only have we retained the Coriolis and centrifugal acceleration terms, but also have included the nutational terms. We also make use of the Euler equation for angular momentum to analytically relate the nutational motion of the rotational axis with the oscillations of the liquid core and obtain a constraint for the nutational amplitude. Consideration of the Poisson equation for density variation completes our analytical model.We primarily discuss the equations of motion for the liquid core and present the solution as the sum of two terms: one being a component of the spheroidal displacement field, the other of the toroidal field. We also formulate the equations valid for the solid mantle when rotational effects are included, and establish the boundary conditions that must hold at the various interfaces in order that a complete integration of the differential system of equations be accomplished.We assume that the outer core consists of an inviscid fluid and ignore the existence of any boundary layer. We do not impose, however, any restriction on the stratification of the fluid. The dynamical coupling between liquid core and solid mantle is represented by a torque which is generated by the forced oscillations within the liquid core; these oscillations are in turn triggered by the diurnal tides.The expected influence of the liquid core/solid mantle boundary on the nutational motion is discussed in view of Poincare's results concerning a liquid core surrounded by a rigid shell. Comparison is finally made of our model with Molodenskii's 1961 theory for a neutral core and the 1976 Shen-Mansinha nutational theory for an unrestricted core.     

内核地球的自转运动和地球固定参考系的研究

本文研究了内核地球模型下的地球表面的旋转运动和地球形变场的复数矢量球函数表示,以及外壳固定参考架、地球参考系的理论定义和它们之间等价性的理论证明.同时给出了液体外核（FOC）、固体内核（SIC）和整体地球的转动惯量张量和角动量的具体表达式.在考虑到引潮力位对地球形变场的影响下,研究了地幔相对角动量的具体表述.本文的工作是对前人有关理论的扩展和改进,对进一步研究内核地球自转的动力学理论是非常重要的.     

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     

Tidal components of geomagnetic variations

The natural geomagnetic field is constantly disturbed. The total registered effect of geomagnetic variations depends on both planetary and local processes. Planetary sources and sources in the Earth’s core respond to tidal effects. In the accepted model, the complex MHD processes in the Earth’s outer core are approximated by the assumed ring current in the equatorial plain of the liquid core. The geomagnetic variation originating as a result of tidal deformations of ring currents are ～10^?4 and 0.10–1 nT in the liquid core and magnetosphere, respectively. The calculated values coincide in order of magnitude with the processed geomagnetic measurements at Paratunka observatory (Kamchatka region).     

Excitation of the Slichter mode by collision with a meteoroid or pressure variations at the surface and core boundaries

We use a normal-mode formalism to compute the response of a spherical, self-gravitating anelastic PREM-like Earth model to various excitation sources at the Slichter mode period. The formalism makes use of the theory of the Earth’s free oscillations based upon an eigenfunction expansion methodology. We determine the complete response in the form of Green’s function obtained from a generalization of Betti’s reciprocity theorem. Surficial (surface load, fluid core pressure), internal (earthquakes, explosions) and external (object impact) sources of excitation are investigated to show that the translational motion of the inner-core would be best excited by a pressure acting at the core boundaries at time-scales shorter than the Slichter eigenperiods.     

Tidal effects in the controlled-source ULF electromagnetic field on the Baltic crystalline shield

The results of the experiment on studying the dynamics of the electromagnetic field (EM) generated by the stationary controlled ULF-band source during 30 days on the Baltic crystalline shield are presented. Diurnal variations in the EM fields and slow variations in the surface impedance with a period of about 14 days are revealed. The diurnal variations in the fields are mainly due to the fluctuations in the ionospheric parameters caused by the changes in the daytime ionization of the ionosphere by solar radiation. By comparing the harmonic component with a period of about 14 days, which was established in the time series of surface impedance, with the slow tidal deformations of the Earth’s crust, we revealed the correlation between the EM variations and tidal processes in the Earth. The estimates for the probable changes induced by tidal deformations in the structure and conductivity of the underlying medium are obtained by modeling.     

液核自由运动的变分方法

从流体静力平衡地球的弹性-引力运动动量方程出发,以角动 量方程控制弹性地幔、液态外核和弹性内核之间的相对转动,在球对称近似下考虑了地幔和 内核对液核边界扰动的形变响应,并以此作为液核边界运动的约束条件．根据地球简正模对 称性的一般特征,建立了自转、非黏性、椭球分层流体外核自由振荡运动的变分原理并给出 了相应的泛函．     

Basic Relations for Studying the Influence of Geophysical Processes on the Earth’s Rotation: The Angular Momentum Approach

Relations to study the influence of geophysical processes on the temporally varying rotation of the Earth are considered. Liouville’s equations of rotational motion are derived for a two-component Earth model (consisting of a solid mantle and a fluid core) and suitably simplified for calculations of the influence of mass redistributions on the Earth’s rotational behaviour. Excitation functions, or effective angular momentum functions, describing the influence of mass redistributions on the equations of rotational motion are derived, and their calculation is elucidated by some examples. Relations between temporally varying second degree Stokes coefficients of the gravity field and excitation functions are discussed. Different solutions of the equations of rotational motion are described. The identification of exciting geophysical processes by the kinematics of the inverse calculated excitation function is portrayed.     

内核地球自转动力学理论的研究进展--Ⅲ液核自转动力学方程

本文是序列文章的第三篇,其内容包括：基于连续介质力学的基本理论,给出了液体外核（FOC）两种形式的角动量方程,对作用在FOC上外力矩进行了详细研究,同时对液核作用在固体内核（SIC）上的压力产生的压力矩进行了讨论,在O(ms)的量级上给出了它们的表达式.本文改正了文献〔1〕在推导过程中的某些错误（例如：（B18）、（B28）、（B29）、（B30）、（B35a）和（B35b）式）.本文是对文献〔1〕有关理论的扩展和改进,对进一步研究内核地球自转的动力学理论是非常重要的.     

内核地球自转动力学理论的研究进展(Ⅴ)——地球自转耦合运动方程组

本文是序列文章的第五篇,其内容包括：基于连续介质力学的基本理论,在考虑到地球的自引力、液核对核幔边界的压力和外部引潮力的作用下,严格地给出了地幔的角动量方程.利用前文的有关结论,进而给出了整体地球自转的动力学方程和内核地球模型的地球自转耦合运动学方程组.本文顾及了高阶岁差章动力矩对地球自转的影响,因而在理论上扩展了文献〔1〕给出的理论模型.本文的理论对进一步研究在高阶岁差章动力矩作用下的内核地球章动是非常有意义的.     

Post-glacial rebound and present-day three-dimensional deformations

The WEGENER^† activities related to the study of post-glacial rebound are presented together with a review of the present state-of-the-art in this study field. Post-glacial rebound research is an unique tool for studying the viscoelastic behaviour of the Earth's mantle on time scales of thousands of years. The viscosity structure of the Earth's mantle determined from an inversion of observations of glacially induced deformations is a basic requirement for modelling long-term phenomena such as the convection in the Earth's mantle, and for better understanding unsolved questions such as the nature of the mantle discontinuities or the vertical scale of convection.First, an introduction to the scientific background is given, and the three principal ingredients for post-glacial rebound studies, namely the ice model, the Earth model, and the observations are briefly considered. For the ice model, the uncertainties due to a trade-off between ice model and Earth rheology are outlined. The different approaches used to model the Earth and its deformations in post-glacial rebound studies are discussed emphasising the preliminary nature of the derived rheologies and depth dependencies. The observations, in particular the relative sea-level changes and three-dimensional surface deformations, are described with special emphasis on observational gaps. Based on the discussion of the ingredients, an outline of the future developments in post-glacial rebound research is attempted with particular emphasis on the Earth model and the theory of deformations.For several decades extreme efforts have been made to precisely monitor the land uplift in Scandinavia. However, for the height component the existing data still are associated with large uncertainties while reliable data on the horizontal component are practically nil. The ongoing long-term (longer than ten years) spacegeodetic measurements are likely to provide the three-dimensional deformations with the spatial resolution and accuracy required in order to substantially contribute to post-glacial rebound studies. Thus, present-day three-dimensional deformations of the Earth's surface beneath and around the former ice sheets as a constraint for the mantle rheology and viscosity structure will increasingly become important as they become known from space-geodetic measurements with high spatial resolution and an accuracy approaching the mm/a-level.     

Axial poloidal electromagnetic core-mantle coupling torque: A re-examination for different conductivity and satellite supported geomagnetic field models

We investigate the temporal behaviour of the axial component of the electromagnetic core-mantle coupling torque that is associated with the poloidal part of the geomagnetic field observable at the Earth surface. For its computation, we use different models of the geomagnetic field, expanded into spherical harmonics (Wardinski and Holme, 2006; Sabaka et al., 2004), and the mantle conductivity. The geomagnetic field, which we have to know at the core-mantle boundary for the associated computations, will be inferred from the field at the Earth surface by the non-harmonic field continuation through a conducting mantle shell. The aims of this investigation are (i) to check how sensitive is the computation of the torque with respect to the different geomagnetic field models, (ii) to check its dependence on the spherical harmonic degree n, and (iii) to determine the difference between the mechanical torque derived from the observed length-of-day variations (atmospheric influence subtracted) and the poloidal electromagnetic torque in dependence on the assumed conductivity. To use the non-harmonic field continuation for the torque calculation and to obtain an insight into the influence of the different geomagnetic field models on the EM torques are the major aspects of this paper. grm@gfz-potsdam.de     

Plume heat flow is much lower than CMB heat flow

Plumes rising from the core–mantle boundary (CMB) are often assumed to transport most, or all, of the heat conducted across the CMB. Here this assumption is explored using numerical convection models in idealized geometries that lead to a single plume under steady-state or near steady state conditions. Plume heat transport is calculated for different internal heating rates using two methods and compared to the CMB heat flux. For these conditions, it is found that the heat flux transported by plumes in the upper mantle is only a fraction of the core heat flux and, thus, core heat flow estimates derived from observed hotspots could be multiplied by a factor of several.     

Study of the Earth's free core nutation by tidal gravity data recorded with international superconducting gravimeters

By stacking high-precision tidal gravity observations obtained with superconducting gravimeters at six stations in China, Japan, Belgium, France, Germany and Finland, the local systematical discrepancies in the parameter fitting, caused by atmospheric, oceanic tidal loading and the other local environmental perturbations, are eliminated effectively. As a result, the resonance parameters of the Earth's free core nutation are accurately determined. In this study, the eigenperiod of free core nutation is given as 429.0 sidereal days, which is in agreement with those published in the previous studies. It is about 30 sidereal days less than those calculated in theoretical models (about 460 sidereal days), which confirms the real ellipticity of the fluid core of the Earth to be about 5% larger than the one expected in assumption of hydrostatic equilibrium. The quality factor (Q value) of free core nutation is given as about 9543, which, compared with those determined before based on the body tide observations, is much larger, but more close to those obtained using the VLBI observations. The complex resonance strength is also determined as (?6.10(10?4, ?0.01(10?4)(/h, which can principally describe the deformation characteristics of an anelastic mantle.