Comparison Between the Nutations of the Planet Mars and the Nutations of the Earth

The nutations of the planets Mars andEarth are investigated and compared. Alarge number of interior structureparameters are involved in the nutationcomputations. The comparison between the observations and the computationsprovides several constraints on these parmeters andtherefore allows a better understanding of the physics of the interior of theplanet. For the Earth, the high precision of the observations of the nutationshas led to a very good determination of interior properties of the planet. ForMars, observations of nutations are not yet available, and we review how theamplitude of the Martian nutations depends on the hypotheses consideredfor its interior. Although Mars is very similar to the Earth, its interior is not well known;for example, we don't knowif its core is liquid or solid. Only if the core is liquid,the Free Core Nutation (FCN) normal mode exists and can alter the nutationswhich are close to the resonance. From the observed geoids, it is known thatboth planets are not in hydrostatic equilibrium. The departure is larger forMars than for the Earth, and consequently, the implication of considering a convective mantle instead of a mantle in hydrostatic equilibrium described byClairaut's equation for the initial equilibrium state of the planet is largeron the Martian nutations than on the Earth nutations. The consequences of theuncertainty in the core dimensions are also examined and shown to be of a veryhigh influence for Mars if the core is liquid, due to the potential changes inthe FCN resonance. The influence of the presence of an inner core, which isknown to exist for the Earth, could be more important for Mars than for theEarth if the inner core is large. Due to the presence of Tharsis on Mars, thetriaxiality of this planet has, additionally, larger effects than on Earth.     

基于旋转微椭地球模型的内核平动振荡三重谱线理论模拟与实验探测

本文基于旋转微椭地球模型,采用简正模理论计算了地球内核平动振荡三重谱线的本征周期,理论上系统研究了地球内部介质(包括密度、地震波速等)分布异常对三重谱线本征周期的影响,计算了不同的内外核密度差和地核中的不同的P/S波速对应的内核平动振荡理论三重谱线周期;利用全球分布的9个超导台站,迭积每个台站长达54个月的高精度超导重力仪数据,在亚潮汐频段(0.162~0.285cph)检测内核平动振荡三重谱线.结果发现,三重谱线本征周期对内外核边界的密度跳跃非常敏感,随着密度差的增加,以类似于双曲线的特征减小;无论是采用地球质量不变的方法还是采用浮力频率为常数的方法,计算得到的三重谱线本征周期结果相差较小,且随着内外核密度差的增大,差距逐渐减小;内、外核P波波速分布异常对三重谱线周期的影响基本相当,内核S波波速分布异常比P波波速分布异常对三重谱线周期的影响小1个量级;探测到一组信噪比较高且满足谱峰分裂特征的三重谱线的信号(0.19281,0.21456和0.24151cph),有极大的可能是来自于内核平动振荡.基于探测结果可以推断实际的地球模型其内外核密度差应该介于PREM模型和1066A地球模型之间,更接近于1066A模型.     

内核地球自转动力学理论的研究进展(Ⅵ)——章动转换函数和内核的动力学效应

本文是序列文章的第六篇,其主要内容包括：讨论了两种章动转换函数表达式以及它们系数之间的关系,指出了前人给出系数值的缺陷,以及某些公式表述的错误;基于内核地球模型的有关参数,利用Mathematica数学分析软件计算了PREM和1066A地球模型在FULL理论、MTIC和FIC近似下的章动本征模频率和章动转换函数的有关系数;计算了内核动力学对受迫章动的影响,结果表明其影响已在目前VLBI可检测的量级内.本文对地球章动转换函数进行了较完整的论述,期望对进一步研究地球自转动力学起重要作用.     

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.     

深内部地球结构对内核平动振荡本征周期的影响

地球固态内核的平动振荡是地球的基本简正模之一,又称Slichter模,其本征周期大约为几个小时,与地球内部结构密切相关.为了研究影响内核平动振荡的本征周期与内部结构的依赖关系,本文利用球对称、非自转、弹性和各向同性地球模型（SNREI）,通过自由振荡运动方程的数值积分,以地球模型PREM为基础,理论上系统研究了地球内部介质（包括密度、地震波速等）分布异常对Slichter模本征周期的影响.数值结果表明,Slichter模周期随着内外核边界（ICB）密度差的增加以类似于双曲线的特征显著减小,当ICB密度差从597 kg·m^-3减小到200 kg·m^-3时,周期增大66.44%,当ICB密度差从597 kg·m^-3增大到1000 kg·m^-3时,周期减小21.48%;Slichter模周期随着核幔边界（CMB）密度差的增大而缓慢增大;相对于PREM,地球模型1066A在ICB和CMB的密度差分别相差45.321%和1.132%,内部地震波速度和密度梯度也存在差异,但是,当密度差减小到1066A模型提供的数值时,得到的Slichter模周期与基于1066A获得的结果（4.599 h）非常接近,差异分别只有3.762%和0.037%;表明Slichter模本征周期与地球内部介质的精细结构关系不大,而对ICB的密度差非常敏感.内、外核P波波速分布异常对Slichter模周期的影响基本相当,当内核和外核P波波速均增加5%时,Slichter周期分别减小1.02%和1.69%,P波波速分别减小5%时,Slichter模周期分别增加1.27%和1.847%,内核S波波速分布异常比P波波速分布异常对Slichter模周期的影响小1个量级;与地核相比,地幔中的地震波速异常对Slichter模本征周期的影响小1~2个量级;表明地核中地震波速异常对Slichter模周期的影响很小,目前有关Slichter模周期理论计算的差异主要来自于所采用的地球模型中内核边界的密度差的差异,本文结果可以为Slichter模的研究、探测及其对地球深内部结构的约束提供理论依据.     

内核地球自转动力学理论的研究进展(Ⅰ)

地球固体内核（SIC）和地球其余部分之间的引力和压力的耦合作用引起了一个力矩，从而对地球的章动运动产生影响.由于SIC的转动惯量和整体地球转动惯量相比是非常小的，因此可以认为SIC的动力学效应只是导致一个新的章动本征模，其频率与自由核章动（FCN）相差不太远，且对地球章动产生了一个微弱的共振影响.本文在文献〔1〕理论的基础上，对内核地球自转动力学理论进行了更加深入和详细的研究，顾及到高阶引潮力位的影响，介绍了研究内核地球自转的基本假设和定义，引潮力位的复数球函数表示，复数矢量球函数的基本理论等.     

Notes on the variability of reflected inner core phases

Recent events beneath Central America have produced excellent sets of inner core reflection (PKiKP phase) at high frequency recorded by USArray ranging from 18° to 30°. However, the amplitude of this phase displays considerable scatter with a factor of six or more. Such scatter has been attributed to upper-mantle scattering and the Inner Core Boundary (ICB) in combination. Here, we show that neighboring events share upper-mantle scatterers beneath the receivers, and their ratio allows a clearer image of deep earth structure. After confirming some of the measured variation is indeed due to deep structure, we stacked nearby traces to reduce fine scale variations which are mostly due to shallow structure. Then, the remaining relatively large scale variation pattern of PKiKP phase is caused by the inner core boundary, as demonstrated by numerical experiments. After migration of data to the ICB, we observe a consistent image. We find such a pattern can be explained by a patch of mushy material of a few kilometers high where the material changes gradually from that of the outer core to that of the inner core.     

Tomographic inversion for three-dimensional anisotropy of Earth’s inner core

Seismological studies generally suggest that the Earth’s inner core is anisotropic and the anisotropic structure changes significantly both laterally and with depth. Previous body-wave studies of the inner core have relied on ray tracing or waveform modeling using one-dimensional (1D) models. Here we present non-linear tomographic inversions of the inner core anisotropy using three-dimensional (3D) ray tracing, spline parameterization, and a large collection of PKP differential travel times. We adapt a pseudo-bending ray tracing (PBR) method in spherical coordinates for seismic rays that traverse the inner core (PKP(DF) phase). The method iteratively perturbs each discontinuity point and continuous segment of the ray through 3D earth structure so that its travel time is minimum. The 3D anisotropic structure of the inner core is approximated to the first order as 3D heterogeneous (but isotropic) structure for a given ray. The data are corrected using a scaled mantle tomographic model. The inner core anisotropy model obtained has the following major features. (1) The model has strong hemispherical and depth variation. The isotropic velocity in the topmost inner core is greater in quasi-eastern hemisphere (QEH) (40–160°E) than in quasi-western hemisphere (QWH) (other longitudes). The anisotropy is weak in QEH to the depth of 600–700 km below the inner core boundary (ICB), while in QWH, the anisotropy increases at much shallower depth (about 100–200 km below the ICB) to about 3–4%, then remains at about 2–4% throughout the rest of the inner core. (2) The anisotropy form changes abruptly (over a depth range of about 150 km) at the radius of about 600 km, slightly less than half of the inner core radius, forming a distinct inner inner core (IIC). The velocity in the IIC has maximums at equatorial and polar directions and minimum at an angle of about 40° from the equatorial plane. The velocity in the outer inner core (OIC), however, changes little for ray directions 0–40° from the equatorial plane. (3) Despite large variation of the anisotropy, the isotropic velocity (Voigt average) throughout the inner core is nearly uniform. The results suggest that the OIC is likely composed of the same type of iron crystals with uniform chemistry, but the IIC may be composed of a different type of crystal alignment, a different iron phase, or a different chemical composition. Our tests on model parameterization, mantle correction, and linear and non-linear inversion suggest the main features of our model are very robust. However, fine scale structures are likely to differ, particularly in the major transition zones, e.g., in the topmost QWH (isotropy to anisotropy), between OIC and IIC (change in the form of anisotropy), and between QEH and QWH in OIC (difference in anisotropy strength). Searches for possible waveform complications from these boundaries need to be aware of the directional dependence and geographical variation to be successful.     

Earth''''s deformation due to the dynamical perturbations of the fluid outer core

Introduction The fluid outer core separates the solid inner core from the solid elastic mantle, and as a result, makes the free and forced movement of this mechanical system more complicated and profuse. As the elastic mantle, the free oscillations may occur within the Earths fluid outer core (FOC) due to excitation of a strong and deep earthquake (Crossley, 1975b; Friedlander, Siegmann, 1982; Shen, 1983; Friedlander, 1985). However, compared with the oscillations of the elastic mantle, i…     

液核动力学扰动引起的地球形变

讨论了地球固体部分对液核动力学效应引发的核幔边界和内核边界上压力和引力扰动的形变响应．采用弹性-引力形变理论描述地幔和内核的形变，给出了内部负荷Love数的一般表达式．以初始参考地球模型为例，分别计算了在地球表面、核幔边界和内核边界上的内部负荷Love数．探讨了液核边界上压力和引力扰动导致的地球形变场的空间和频率分布特征．本文的结果可以为中短周期液核动力学理论模拟提供必要的边界条件．      

Velocity constraints for the inner core inferred from long-period PKP amplitudes

Long-period PKP amplitudes from 16 earthquakes in the distance range 110– 170° are compared with theoretical amplitudes which are derived from synthetic seismograms calculated for 56 systematic modifications of Earth model 1066B in the inner core. A suitable normalization procedure allows for the common representation of all observed amplitudes as a function of epicentral distance. Using the theoretical amplitude distributions it can be shown that the parameters of a regression line through the logarithmic and normalized amplitudes between 110 and 134° are related to the velocity and density jump at the inner-core boundary (ICB). The analysis shows that the dominant influence on the PKP amplitudes is the P-velocity jump which can be restricted to 0.64 ± 0.05 km s^?1. There exists a trade-off between the S-velocity jump and the density jump. Restricting the latter to the reasonable range 0–1.2 g cm^?3 the S-velocity jump at the ICB can be inferred to be 2.5–3.0 km s^?1. A rather strong S-velocity gradient below the ICB follows from the condition that the S-wave travel-time through the inner core agrees with that implied by free oscillation observations. This leads to central S-velocities between 3.81 and 4.15 km s^?1, assuming a parabolic velocity law.     

利用前临界PcP-PKiKP资料研究中国东部内核边界性质

前临界内核边界反射震相PKiKP与核幔边界反射震相PcP构成组合,能有效压制浅部结构及震源因素的干扰,提供了对内核边界精细结构的直接约束. 本研究从华北克拉通西北部密集流动地震台阵一年观测资料中筛选出8个地震事件,得到共计 73对PcP-PKiKP组合,覆盖了从朝鲜半岛到我国东北及华中地区下方的内核边界. 本文系统分析了走时残差和振幅比数据,结果显示:(1)密集台阵资料有助于前临界PKiKP震相拾取,而浅源地震亦可提供高质量的PcP-PKiKP观测资料.(2)走时残差呈现了自西北向东南从正常到负异常的迅速变化(沿内核边界70 km范围内>0.5 s), 限制了研究区域内核界面不超过3 km的起伏. (3)相对振幅比变化表明了研究区内核边界密度差北西-南东向的系统增加, 揭示了内核结晶环境的小尺度扰动.     

Structure of the inner core boundary

Abstract

A model of the inner-core boundary (ICB) is constructed which is consistent with current ideas of the dynamic and thermodynamic state of the core and which is capable of reflecting seismic waves with period of one second. This requires the mass fraction of solid below the ICB to grow to an appreciable fraction in roughly one kilometer. This rapid growth of solid with depth is a result of downward fluid flow from the outer core which is a part of the convective motions which sustain the geodynamo. The solid which crystallizes from this descending fluid after it crosses the ICB continually coats the dendrites which occur there. The gradual cooling of the outer core causes the ICB to advance by growth of dendrites at their tips. The balance of these two effects gives an equilibrium profile for the mass fraction of solid with depth below the ICB which is capable of yielding sharp reflection of seismic waves.     

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.     

Earth’s deformation due to the dynamical perturbations of the fluid outer core

The elasto-gravitational deformation response of the Earth’s solid parts to the perturbations of the pressure and gravity on the core-mantle boundary (CMB) and the solid inner core boundary (ICB), due to the dynamical behaviors of the fluid outer core (FOC), is discussed. The internal load Love numbers, which are formulized in a general form in this study, are employed to describe the Earth’s deformation. The preliminary reference Earth model (PREM) is used as an example to calculate the internal load Love numbers on the Earth’s surface, CMB and ICB, respectively. The characteristics of the Earth’s deformation variation with the depth and the perturbation periods on the boundaries of the FOC are also investigated. The numerical results indicate that the internal load Love numbers decrease quickly with the increasing degree of the spherical harmonics of the displacement and depend strongly on the perturbation frequencies, especially on the high frequencies. The results, obtained in this work, can be used to construct the boundary conditions for the core dynamics of the long-period oscillations of the Earth’s fluid outer core. Foundation item: State Natural Science Foundation of China (40174022 and 49925411) and the Projects from Chinese Academy of Sciences (KZCX2-106 and KZ952-J1-411).     

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.     

The monosulfid solid solution in the FeNiS system: relationship to the earth's core on the basis of experimental high-pressure investigations

Experimental high-pressure results on phase stability, electrical conductivity and compression behavior up to 5 and 21 GPa respectively are used to calculate an isothermal equation of state for a monosulfid solid solution (MSS-composition) in the FeNiS system. The high-pressure relations in the range 1–8 GPa are very complex. A continuous electrical transition, from semiconducting to metallic, takes place at high pressures and temperatures and results in anomalous compression behavior at pressures in this region. No polymorphic transition from the NiAs-structure to another type could be observed; however, density increases by as much as 8.8%. Using compression values for pressure greater than 10 GPa, the bulk modulus, a zero-pressure density and a core density were calculated. Extrapolation for the conditions of the outer core yields a difference in the density of up to 20%, relative to seismological models.In a composition model with (Fe, Ni)+MSS, a MSS-content must be assumed to be in the range of 30–35 wt% at the core-mantle boundary (CMB) and 13–17 wt% at the inner-core boundary (ICB). That corresponds to a sulfur content of 10.8–13.3 wt% (CMB) and 4.9–6.5 wt% (ICB), respectively, the values increasing with increasing Ni content of the MSS-phase.     

On the admissible range for the mass and inertia moments of the liquid core: II. Results of numerical calculations

We analyze the present-day data on the periods of free oscillations and amplitudes of the forced nutations of the Earth for evaluating the admissible range of the mass and moment of inertia for the liquid core. The initial model for this study is taken in the form of the model distribution of density and mechanical Q parameters of the mantle suggested in (Molodenskii, 2010; 2011a; 2011b). This model was constructed by the steepest descent method in the space of 64 parameters, which determine the distribution of density and parameters of mechanical Q in the mantle, liquid outer core, and solid inner core of the Earth. We assumed the Q parameter of the mantle and inner solid core to be constant and sought for the density variations for the simplest two-parameter model of the piecewise-linear functions with the jumps on the boundary between the liquid core and the mantle and at the olivine-spinel phase transition at a depth of 670 km in the mantle. After this, the computations were repeated for the other distributions of Q (which were also assumed to be unchanged) that correspond to their limiting admissible values. Using this approach, we managed to find the most probable values of the mass and moment of inertia of the liquid core and determine the admissible range of their values. According to our estimates, the ratios of the mass and moments of inertia of the liquid core to the mass and moment of inertia of the whole Earth fall in the intervals 0.317996 ± 0.00065 and 0.110319 ± 0.00022, respectively. These values are lower than the corresponding values for the PREM model (0.322757 and 0.112297) by (1.48 ± 0.30)% and (1.76 ± 0.35)%, respectively. The interpretation of these results requires the revision and thorough analysis of the data on the admissible temperature range of the liquid core and (or) its chemical composition.     

利用VLBI和超导重力资料研究自由核章动周期时变特征

本文根据自由核章动在章动观测和时变重力观测中引起的与其频率相近的受迫章动项或固体潮潮波的共振特性,选取6个不同机构解算的VLBI天极偏差序列和全球超导重力仪网络提供的7个台站高精度时变重力观测资料,根据加权最小二乘方法拟合地球自由核章动参数,研究其本征周期的时间变化特征.VLBI资料获得的本征周期变化幅度在1天之内,存在十年尺度的时间变化特征,迭积多台站同时段重力资料获得的自由核章动本征周期时变特征与VLBI获得的结果相比变化幅度较大,但趋势大致符合.在此基础上,通过自由核章动的理论模型探讨了影响其本征周期的几个主要因素,结果表明FCN周期变化与电磁耦合存在相关性,核幔边界磁感应强度径向分量变化导致的电磁耦合的变化可能是造成FCN周期时变性的原因.     

On asymptotics of low-frequency oscillations of the liquid core: 2. Frequency dependence of the inertial core-mantle coupling

New, unique information on the inertial and dissipative coupling of the liquid core and the mantle has been retrieved from modern high-precision (radiointerferometer and GPS) data on tidal variations in the rotation velocity and nutation of the Earth. Comparison of theoretical and observed data provided new estimates for the dynamic flattening of the outer liquid and the inner solid cores, mantle quality factor, viscosity of the liquid core, and electromagnetic coupling of the liquid core and the mantle [Molodensky, 2004, 2006]. As was shown in the first part of the paper [Molodensky, 2008] (further referred to as [I]), generation of eddy flows in Proudman-Taylor columns, whose orientation is controlled by the topography of the liquid core-mantle boundary, should be taken into account for correct estimation of the inertial coupling (see formulas (8) and (34) in [I]). The range of periods within which this effect plays a significant role is determined by the decay time of these flows. This time is estimated in the paper for the case where dissipation is related to viscous friction at the core-mantle boundary or with the electromagnetic coupling of the liquid core and the mantle. Because of significant uncertainties in modern data on the viscosity of the liquid core, the magnetic field intensity at the core-mantle boundary, and the electrical conductivity of the lower mantle, the dissipative coupling of the liquid core and the mantle cannot be calculated as yet. However, as shown in the paper, the decay time of eddy flows is connected with the attenuation time of subdiurnal free nutation and with the liquid core viscosity. This enables the estimation of the frequency dependence of the dissipative coupling in a fairly wide range. It is shown that the range of periods for which relations (8) and (34) in [I] are valid encompasses the best-studied length-of-day variations and, therefore, these relations are applicable to analysis of the majority of modern data.