Time-dependent density anomalies in a stratified,viscoelastic mantle: Implications for the geoid,Earth's rotation and sea-level fluctuations

The effects on the = 2 geoid component and Earth's rotation due to internal mass anomalies are analyzed for a stratified viscoelastic mantle described by a Maxwell rheology. Our approach is appropriate for a simplified modeling of subduction. Sea-level fluctuations induced by long-term rotational instabilities are also considered. The displacement of the Earth's axis of rotation, called true polar wander (TPW) and the induced eustatic sea-level fluctuations, are extremely sensitive to viscosity and density stratification at the 670 km seismic discontinuity. Phase-change models for the transition zone generally allow for huge amount of TPW, except for large viscosity increases; the dominant contribution in Liouville equations comes from a secular term that reflects the viscous behaviour of the mantle. In chemically stratified models, TPW is drastically reduced due to dynamic compensation of the mass anomalies at the upper-lower mantle interface. When the source is embedded in the upper mantle close to the chemical density jump, transient rotational modes are the leading terms in the linear Liouville equations. Long-term rotation instabilities are valuable contributors to the third order cycles in the eustatic sea-level curves. Rates of sea-level fluctuations of the order of 0.05–0.1 mm/yr are induced by displacements of the Earth's axis of rotation compatible with paleomagnetic data.     

天文观测极移运动周期变化的原因解析

宋贯一（1991,2006）相继发现日-地之间存在一种奇特的能量（动量）相耦合的自然现象:当太阳辐射光压作用于地表之后,地球表面物质的特殊物理性质会自然地把太阳辐射光压立刻分解为两部分,即P₁和P₂.其中P₁为全球各纬度带内单位海洋和陆地表面接收到的等值光压（P₁为随时间和纬度而变化的变量）.在一个回归年的时段内,由P₁在北、南两半球上的不平衡分布激发自转轴摆动,引起转动惯量的变化（该变化是目前所了解到的地球转动惯量的最大变化）并产生极移,该项极移运动的周期为12个月左右.由于P₁在北、南半球上的分布相对赤道是基本对称的和规则的,P₁对自转轴摆动的激发可称谓“规则性激发”;P₂为全球各纬度带内单位陆地和海洋表面接收到的光压值之差（P₂为随时间和纬度而变化的变量）.在一个回归年的时段内,由 P₂在北、南两半球上的不平衡分布激发自转轴摆动而产生的极移运动周期为14个月附近.由于P₂在北、南两半球上的分布相对赤道是不对称和不规则的, P₂对自转轴摆动的激发可称谓“不规则性激发”.这种天然存在的力源,恰恰是近百年来世界各国研究地球自转的地球物理学家渴望寻找的那种既能激发自转轴产生自由章动、又使其摆动含有两个不同周期（12个月和14个月）的激发力源.正是这一奇特自然现象的发现,才使长期以来困扰地球物理学领域内的极移、极移所包含两个周期的涨落变化及由此引起的地球自转速度变化等自然之谜得以破解成为可能. 本文作者仅对天文观测的极移运动周期及其极移运动所包含的两个周期分量在一定的范围内变化的成因作出了详细的解析,并得出如下结论： （1）极移运动主要是由太阳光压P₁ 和P₂共同激发引起的.天文观测的极移摆动周期的涨落变化是太阳光压（P₁＋P₂）激发自转轴摆动过程中,在空间上自转轴的摆动中心相对自转轴中心（地心）移动造成的,涨落范围在395～403±2天之间,即天文观测的极移运动的实际计算周期应在13.0～13.3个月之间变化.（2）极移所含的周年期摆动是由太阳光压P₁激发的.天文观测的周年期摆动周期涨落很小,变化于365.24～365.53天之间.在一个回归年内,由于日-地间距离的变化,使地球表面接受到的太阳辐射光能产生微小差异则是造成观测的周年摆动周期稍有拖长的原因.（3）极移所含的钱德勒摆动周期是由太阳光压P₂激发的.天文观测的钱德勒周期涨落较大,变化于426～437±2天之间,即实际计算周期应在14.0～14.4个月之间变化.观测的钱德勒摆动周期的变化是太阳光压P₂激发自转轴摆动过程中,在空间上自转轴的摆动中心相对自转轴中心（地心）移动造成的.上述的定量解析数据均得到实际观测资料的验证,为极移光压成因理论的正确性提供了具体详实的证据.     

Rotational bulge and one plume convection pattern: Influence on Martian true polar wander

Motion of the entire solid planet with respect to its spin axis have been proposed on Mars. This movement is known as True Polar Wander (TPW). According to the conservation of angular momentum with no external torque, on geological time scales the axis of maximum inertia of a planet is aligned with the rotation axis. Then rearrangement of masses within the mantle disturbs the planet's inertia and induces TPW. The convection pattern on Mars is possibly controlled by a sequence of single plumes originating from the core-mantle boundary. Using a homogeneous model of the martian mantle and modelling the plume as a sphere, we calculate the inertial tensor perturbations caused by the plume mass anomaly. We investigate the stabilizing influence of the remnant rotational bulge due to the lithosphere elasticity on these perturbations. It appears that, during early martian history, the elastic lithosphere was thin enough to allow its fractures under the inertia perturbations induced by a hot plume. Consequently, the lithosphere's behaviour became effectively viscoelastic and the plume could induce large TPW. We conclude that one plume convection pattern should have greatly influenced the rotation pole behaviour during early Mars history: around 4 Gyr ago, Mars already could have experienced two TPW events lasting possibly only a few million years each. We then compare our scenario with others already published in the literature.     

Semi-analytical Solution for Viscoelastic Relaxation in Eccentrically-nested Spheres Induced by Surface Toroidal Traction

A semi-analytical solution to the 2-D forward modelling of viscoelastic relaxation in a heterogeneous sphere induced by a surface toroidal force is derived. The model consists of a concentrically-nested elastic lithosphere, a viscoelastic mantle, and an eccentrically-nested viscoelastic core. Since numerical codes based on finite-element or spectral-finite-difference techniques for modelling viscoelastic relaxation in a spherical geometry in the presence of lateral viscosity variations are becoming more popular, reliable examples for testing and validating such codes are essential. The eccentrically-nested sphere solution has been tested by comparing it with two distinct results: The analytical solution for viscoelastic relaxation in concentrically-nested spheres and the time domain, spectral finite-element numerical solution for viscoelastic relaxation in eccentrically-nested spheres, with excellent agreement being obtained.     

大地测量约束下的阿尔泰山岩石圈流变结构

本文根据大地测量数据得到过去50年左右阿尔泰山富蕴区域形变场,破裂带中段的相对位移最大,平均速率达6mm/a,总体上表现为沿断裂的走滑运动特征．该形变场可以用1931年富蕴8级地震的震后黏弹性松弛模型进行模拟,反演得到下地壳黏滞系数为1.6×10¹⁹～7.9×10¹⁹Pa·s,上地幔黏滞系数为16×1018～63×1019Pa·s,与华北、Nevada等地区利用震后变形资料推算的黏度基本一致．根据该地区最佳黏弹性分层模型,最近五十年由于岩石圈下部应力松弛引起的地震破裂带两侧最大水平速率约为4mm/a． 我们的研究表明：大陆7～8级大震在几十年后仍可能有可观的地表变形,GPS监测得到的现今变形场可能包含震后变形成分．     

Time-space characteristics of viscoelastic post-seismic deformations corresponding to different rheology models

On a long time (> 1 a) scale, the viscoelastic properties of mantle media significantly affect post-seismic deformation. The stress field disturbance in viscoelastic medium caused by fault slip gradually relax, and the relaxation process and its temporal-spatial characteristics are determined by the viscoelastic model. In this paper, we assume that the mantle media are types of common linear rheological models, i.e., the Burgers body, the standard linear solid, and the Maxell body, and we calculate the dislocation Love number and Green function for a spherically symmetric, non-rotating, viscoelastic, and isotropic (SNRVEI) Earth model. The characteristics of the post-seismic relaxation deformations corresponding to the different models are compared. Our results show that for a short time period, the Burgers body and standard linear solid are similar; while for the long time period, the Burgers body and Maxwell body are similar. This suggests that the observations of post-seismic deformation on the surface have a great potential for the inversion of underground viscoelastic structures. However, the potential of using surface displacement to distinguish different rheological models is limited when the observation period is not long enough.     

Cenozoic latitudinal shift of the Hawaiian hotspot and its implications for true polar wander

Pacific plate equatorial sediment facies provide estimates of the northward motion of the Pacific plate that are independent of paleomagnetic data and hotspot tracks. Analyses of equatorial sediment facies consistently indicate less northward motion than analyses of the dated volcanic edifices of the Hawaiian-Emperor chain. The discrepancy is largest 60–70 Ma B.P.; the 60- to 70-Ma equatorial sediment facies data agree with recent paleomagnetic results from deep-sea drilling on Suiko seamount [1] and from a northern Pacific piston core [2]. Equatorial sediment facies data and paleomagnetic data, combined with K-Ar age dates along the Emperor chain [3], indicate a position of the spin axis at 65 Ma B.P. of 82°N, 205°E in the reference frame in which the Pacific Ocean hotspots are fixed. This pole agrees well with the position of the spin axis in the reference frame in which the Atlantic Ocean hotspots and the Indian Ocean hotspots are fixed [4,5], supporting the joint hypotheses that (1) the Pacific Ocean hotspots are fixed with respect to the hotspots in other oceans, (2) the hotspots have shifted coherently with respect to the spin axis, and (3) the time average of the earth's magnetic field 65 Ma B.P. was an axial geocentric dipole. Global Neogene paleomagnetic data suggest that a shift of the mantle relative to the spin axis has been occurring during the Neogene in the same direction as the shift between 65 Ma B.P. and the present. All data are consistent with a model in which the hotspots (and by inference the mantle) have shifted with respect to the spin axis about a fixed Euler pole at a constant rate of rotation for the last 65 Ma.     

The loading problem for a linear viscoelastic earth: I. Compressible,non-gravitating models

Summary A Legendre mode solution is given for deformation of a solid isotropic linear viscoelastic sphere under applied surface stresses. Under the simplifying assumptions that the sphere is elastic in compression and standard linear solid in shear two relaxation times appear; one the creep relaxation time of the material, the other depending on mode. It is shown formally how to reduce the case of a layered viscoelastic sphere to an equivalent unlayered one.     

2011年日本MW9.0地震震后形变机制与震源区总体构造特征

本文利用较为完备的球体位错理论,结合4.5年的震后位移数据,优化了2011年日本M_W9.0地震震源区岩石圈弹性层厚度与地幔黏滞性因子,更新了该强震断层余滑时空演化过程.首先,基于日本列岛215个均匀分布的GPS连续观测站震前2年与震后4.5年的观测数据,提取了2011年日本M_W9.0地震引起的震后位移时空变化;接着,依据断层余滑衰减相对较快的特点,利用黏弹性球体位错理论对震后3~4.5年的GPS观测数据进行反复拟合,确定2011年日本M_W9.0地震震源区地幔黏滞性系数和岩石圈弹性层厚度的最优解分别为6×10¹⁸ Pa·s和30 km;然后,从震后3年内GPS观测数据中剔除地幔黏滞性松弛效应,获取了断层余滑对应的震后位移场;最后,利用基于球体位错理论的反演算法,反演了2011年日本M_W9.0地震断层余滑的时空演化过程.结果表明,2011年日本M_W9.0地震引起的断层余滑在震后半年内变化显著,震后2年主震区域余滑基本停止,断层的两端存在一定的余滑效应,断层余滑的累计矩震级达到8.59;地震后4年,地幔黏滞性松弛效应对震后位移场的贡献在总体上超过断层余滑的贡献.     

Linear Magnetoconvection in Rotating Fluid Spheres Permeated by a Uniform Axial Magnetic Field

A linear analysis of thermally driven magnetoconvection is carried out with emphasis on its application to convection in the Earth's core. We consider a rotating and self-gravitating fluid sphere (or spherical shell) permeated by a uniform magnetic field parallel to the spin axis. In rapidly rotating cases, we find that five different convective modes appear as the uniform field is increased; namely, geostrophic, polar convective, magneto-geostrophic, fast magnetostrophic and slow magnetostrophic modes. The polar convective (P) and magneto-geostrophic (E) modes seem to be of geophysical interest. The P mode is characterized by such an axisymmetric meridional circulation that the fluid penetrates the equatorial plane, suggesting that generation of quadrapole from dipole fields could be explained by a linear process. The E mode is characterized by a few axially aligned columnar rolls which are almost two-dimensional due to a modified Proudman-Taylor theorem.     

利用地球简正模耦合研究上地幔过渡区方位各向异性

地震方位各向异性广泛存在于地球上地幔中,目前利用地震体波或面波分析研究上地幔各向异性的地球物理方法有很多种,但是由于各自的局限性均难以分析上地幔过渡区中的各向异性特征.方位各向异性可导致球形简正模和环形简正模之间发生耦合.地球长周期自由振荡的简正模可深入到上地幔过渡区.本文利用各向异性地球模型计算各向异性简正模耦合深度敏感核,表明长周期(250~400 s)简正模各向异性耦合(如₀S₂₀-₀T₂₁ 和₀S₂₅-₀T₂₅)的敏感度峰值在400~600 km之间.在不受地球自转影响的台站,如位于南极极点的QSPA站,仍然可以观测到强烈的简正模耦合现象.本文的研究表明:只有在地震观测台站靠近长周期球形振荡的节点时,才能在其观测数据中观测到各向异性耦合现象,许多各向异性耦合在震后18~24 h期间最强,并可导致垂直方向的环形简正模的振幅大于球形耦合简正模的振幅.这些特征是在地震观测数据中寻找各向异性耦合的重要线索.长周期简正模的方位各向异性耦合为我们提供了一个新的认识上地幔过渡区各向异性的窗口.     

A 90° spin on Rodinia: possible causal links between the Neoproterozoic supercontinent, superplume, true polar wander and low-latitude glaciation

We report here new geochronological and paleomagnetic data from the 802±10 Ma Xiaofeng dykes in South China. Together with existing data, these results suggest that Rodinia probably spread from the equator to the polar region at ca. 800 Ma, followed by a rapid ca. 90° rotation around an axis near Greenland that brought the entire supercontinent to a low-latitude position by ca. 750 Ma. We propose that it was the initiation of a mantle superplume under the polar end of Rodinia that triggered an episode of true polar wander (TPW) which brought the entire supercontinent into equatorial latitudes. An unusually extensive emerged land area at the equator increased both atmospheric CO₂ drawdown and global albedo, which, along with waning plume volcanism led directly to the low-latitude Sturtian glaciation at ca. 750–720 Ma.     

1990年青海共和7.0级地震震后垂直形变研究

根据1990年青海共和地震震后地表垂直形变,通过模型拟合得到了支配共和地区震后形变场时空演化的形变源及其力学机制.分析穿过断层的震前1期和震后6期水准数据,结果表明震后垂直形变具有以下特征:①震后震区上盘继续发生继承性的大幅度上升,其中震后头一年上升速率最大;②震后上升区范围显著,范围随时间变化不大,但较同震形变上升区范围增大;③震后相邻测站高差观测值的时间序列明显具有对数衰减特征或指数衰减特征,衰减特征时间分别为0.165年和1.344年.本文还发展了一个利用水准数据与连续介质位错模型研究震后形变机制的新方法.该方法用相邻水准点之间的原始高差观测值而非它们相对参照点的积分值来约束连续介质位错模型,可以有效减少误差累积带来的偏差并充分利用观测数据.利用这一方法的初步分析结果表明,断层震后滑移和介质黏弹性松弛共同导致了共和地震震后形变.前者表现为发生在断层面及其延伸部分的滑移,特别是位于主破裂上方沉积层内的滑移;后者则表现为下地壳与上地幔内的黏弹性松弛,黏滞系数为1020Pa.s量级.     

Dynamic mechanisms of the post-seismic deformation following large events: Case study of the 1999 Chi-Chi earthquake in Taiwan of China

The mechanism of postseismic deformation related to strong earthquakes is important in geodynamics, and presumably afterslip or viscoelastic relaxation is responsible for the postsesimic deformation. The 1999 Chi-Chi, Taiwan of China, earthquake occurred in the region where GPS observation station is most densely deployed in the world. The unprecedented GPS data provides a unique opportunity to study the physical processes of postseismic deformation. Here we assume that the interactions of viscoelastic relaxation, afterslip, fault zone collapse, poroelastic rebound, flow of underground fluids, and all these combined contribute to the surface displacements following the main shock. In order to know the essence of the postseismic deformation after the strong event, fault zone collapse, poroelastic rebound, flow of underground fluids, and so on, are represented equivalently by the variations of the focal medium properties. Therefore, the viscoelastic relaxation, afterslip, and the variations of the equivalent focal medium properties are inverted by applying the GPS temporal series measurement data with viscoelastic finite element method. Both the afterslip rate distribution along the fault and the afterslip evolution with time are obtained by means of inversion. Also, the preliminary result suggests that viscosities of the lower crust and the upper mantle in Taiwan region is 2.7×10¹⁸ and 4.2×10²⁰ Pa·s, respectively. Moreover, the inversion results indicate that the afterslip contributing to postseismic deformation of 44.6% in 450 days after the Chi-Chi earthquake, with 34.7% caused by the viscous relaxation and 20.7% by other factors such as fault zone collapse, poroelastic rebound, and the flow of liquids.     

Rapid afterslip and short-term viscoelastic relaxation following the 2008 M_W7.9 Wenchuan earthquake

Significant postseismic deformation of the 2008 M W 7.9 Wenchuan earthquake has been observed from GPS data of the first 14 days after the earthquake. The possible mechanisms for the rapid postseismic deformation are assumed to be afterslip on the earthquake rupture plane and viscoelastic relaxation of coseismiclly stress change in the lower crust or upper mantle. We firstly use the constrained least squares method to find an afterslip model which can fit the GPS data best. The afterslip model can explain n...     

Viscoelastic representation of surface waves in patchy saturated poroelastic media

Wave-induced flow is observed as the dominated factor for P wave propagation at seismic frequencies. This mechanism has a mesoscopic scale nature. The inhomogeneous unsaturated patches are regarded larger than the pore size, but smaller than the wavelength. Surface wave, e.g., Rayleigh wave, which propagates along the free surface, generated by the interfering of body waves is also affected by the mesoscopic loss mechanisms. Recent studies have reported that the effect of the wave-induced flow in wave propagation shows a relaxation behavior. Viscoelastic equivalent relaxation function associated with the wave mode can describe the kinetic nature of the attenuation. In this paper, the equivalent viscoelastic relaxation functions are extended to take into account the free surface for the Rayleigh surface wave propagation in patchy saturated poroelastic media. Numerical results for the frequency-dependent velocity and attenuation and the time-dependent dynamical responses for the equivalent Rayleigh surface wave propagation along an interface between vacuum and patchy saturated porous media are reported in the low-frequency range (0.1–1,000 Hz). The results show that the dispersion and attenuation and kinetic characteristics of the mesoscopic loss effect for the surface wave can be effectively represented in the equivalent viscoelastic media. The simulation of surface wave propagation within mesoscopic patches requires solving Biot’s differential equations in very small grid spaces, involving the conversion of the fast P wave energy diffusion into the Biot slow wave. This procedure requires a very large amount of computer consumption. An efficient equivalent approach for this patchy saturated poroelastic media shows a more convenient way to solve the single phase viscoelastic differential equations.     

Postseismic Deformation Following the 2010 $$M = 7.2$$ El Mayor-Cucapah Earthquake: Observations,Kinematic Inversions,and Dynamic Models

Due to its location on a transtensional section of the Pacific-North American plate boundary, the Salton Trough is a region featuring large strike-slip earthquakes within a regime of shallow asthenosphere, high heat flow, and complex faulting, and so postseismic deformation there may feature enhanced viscoelastic relaxation and afterslip that is particularly detectable at the surface. The 2010 \(M = 7.2\) El Mayor-Cucapah earthquake was the largest shock in the Salton Trough since 1892 and occurred close to the US-Mexico border, and so the postseismic deformation recorded by the continuous GPS network of southern California provides an opportunity to study the rheology of this region. Three-year postseismic transients extracted from GPS displacement time-series show four key features: (1) 1–2 cm of cumulative uplift in the Imperial Valley and \(\sim\)1 cm of subsidence in the Peninsular Ranges, (2) relatively large cumulative horizontal displacements \(>\)150 km from the rupture in the Peninsular Ranges, (3) rapidly decaying horizontal displacement rates in the first few months after the earthquake in the Imperial Valley, and (4) sustained horizontal velocities, following the rapid early motions, that were still visibly ongoing 3 years after the earthquake. Kinematic inversions show that the cumulative 3-year postseismic displacement field can be well fit by afterslip on and below the coseismic rupture, though these solutions require afterslip with a total moment equivalent to at least a \(M = 7.2\) earthquake and higher slip magnitudes than those predicted by coseismic stress changes. Forward modeling shows that stress-driven afterslip and viscoelastic relaxation in various configurations within the lithosphere can reproduce the early and later horizontal velocities in the Imperial Valley, while Newtonian viscoelastic relaxation in the asthenosphere can reproduce the uplift in the Imperial Valley and the subsidence and large westward displacements in the Peninsular Ranges. We present two forward models of dynamically coupled deformation mechanisms that fit the postseismic transient well: a model combining afterslip in the lower crust, Newtonian viscoelastic relaxation in a localized zone in the lower crust beneath areas of high heat flow and geothermal activity, and Newtonian viscoelastic relaxation in the asthenosphere; and a second model that replaces the afterslip in the first model with viscoelastic relaxation with a stress-dependent viscosity in the mantle. The rheology of this high-heat-flow, high-strain-rate region may incorporate elements of both these models and may well be more complex than either of them.     

MS8.0汶川地震对MS7.0芦山地震成核失稳的影响

依据速率-状态依赖性摩擦本构关系,并结合中下地壳和上地幔的黏弹性松弛效应,以震后库仑破裂应力变化和同震动态库仑破裂应力变化的计算为基础,模拟了两种应力变化对芦山地震断层的失稳发震时间的影响,研究了2008年M_S8.0汶川地震与2013年M_S7.0芦山地震之间的触发关系.计算得到,汶川地震在芦山地震震源断层面上产生的动态应力变化的峰值为0.127 MPa;此外,经过近5年中下地壳和上地幔的黏弹性松弛效应,芦山地震震源断层面上受到的震后应力变化值为0.025 MPa.结果表明,芦山地震的震源断层在应力积累逐渐接近临界状态的某一特定时期内,受到了汶川地震产生的动态应力变化、静态应力变化以及黏弹性松弛效应造成应力变化的共同触发作用,且动态应力的延迟触发作用可能更为显著.最后对芦山地震之后研究区域的应力变化场进行了初步探讨.     

Estimation of inner-core rotation rate by using the Earth’s free oscillation

We estimate a rate of inner-core differential rotation from time variations of splitting functions of seven core modes of the Earth’s free oscillations excited by eight large earthquakes in a period of 1994–2003. The splitting functions and moment tensor elements are simultaneously determined for each core mode by a spectral fitting technique. The estimated moment tensor well agrees with Harvard CMT solution. The splitting functions are corrected for the effect of mantle heterogeneity using a 3D mantle velocity model. Inner-core rotation angle about the Earth’s spin axis is determined for each core mode as a function of event year by comparison of the corrected and reference splitting functions. Mean rotation rate of six core modes is estimated at 0.03±0.18° per year westward, and this value is insignificantly different from zero. Therefore, the inner core is not rotating at a significant rate relatively to the crust and mantle.     

Mechanisms of Transient Postseismic Deformation Following the 2001 Mw 7.8 Kunlun (China) Earthquake

Using global positioning system (GPS) technology, significant postseismic surface displacements were observed within the first 4 months after the 2001 Mw 7.8 Kunlun earthquake which occurred in China. In this study, we investigated the mechanisms that may have possibly contributed to the postseismic deformations that have been observed. Based on the modeling results, we find that an afterslip model can interpret postseismic displacements in the near field even when the fault plane is extended to the bottom of the crust (~70 km). Models based on the viscoelastic relaxation theory showed a large discrepancy in the spatial pattern of the deformation compared with what has been observed. Thus, we infer that both mechanisms cannot interpret the observed postseismic deformation independently. A combination of afterslip and viscoelastic relaxation can further improve the data fit, especially at sites far from the fault. With maximum afterslip of ~0.4 m occurring at a depth of 10 km in the central section, the combined model shows that the estimated afterslip occurred mostly on and below the coseismic rupture plane, as well as on its eastern extension. The estimated moment released by the afterslip in the first 4 months is almost 40% of that released by the coseismic slip. The best-fitting viscoelastic relaxation model shows a “weak” upper mantle with a viscosity of ~1.0 × 10¹⁸ Pa s. The combined model also suggests the existence of a lower crust with viscosity larger than 1.0 × 10¹⁸ Pa s, although it cannot be constrained accurately.