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).     

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．      

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.     

Weak velocity anomaly in the Earth’s outer core from seismic data

Experimental data on the differential travel time t _BC–t _DF of seismic waves PKP_DF and PKP_BC in the Earth’s core under Africa and Australia are analyzed. The differential travel-time residuals beneath Africa in a narrow range of angles from 21° to 25° between the direction of the seismic ray in the core and the Earth’s rotation axis exhibit a scoop-shaped peculiarity not accounted for by cylindrical anisotropy in the inner core. A model with a 0.2–0.8% P-wave velocity anomaly with a radius of 1375 km in the cylindrical region in the outer core is proposed, which closely fits the experimental data. We suggest that the velocity anomaly is generated by the dynamical processes occurring in the outer core, namely, the growth of the inner core and the convection in the outer core, both leading to the formation of a low-density anomaly in the outer core.     

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.     

A reduced model for nonlinear dispersive waves in a rotating environment

Abstract

The simplest model for geophysical flows is one layer of a constant density fluid with a free surface, where the fluid motions occur on a scale in which the Coriolis force is significant. In the linear shallow water limit, there are non-dispersive Kelvin waves, localized near a boundary or near the equator, and a large family of dispersive waves. We study weakly nonlinear and finite depth corrections to these waves, and derive a reduced system of equations governing the flow. For this system we find approximate solitary Kelvin waves, both for waves traveling along a boundary and along the equator. These waves induce jets perpendicular to their direction of propagation, which may have a role in mixing. We also derive an equivalent reduced system for the evolution of perturbations to a mean geostrophic flow.     

Numerical simulations of thermal convection in a rapidly rotating spherical shell cooled inhomogeneously from above

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell with and without inhomogeneous temperature anomalies on the top boundary have been carried out using a three-dimensional, time-dependent, spectral-transform code. The spherical shell of Boussinesq fluid has inner and outer radii the same as those of the Earth's liquid outer core. The Taylor number is 10⁷, the Prandtl number is 1, and the Rayleigh number R is 5R_c (R_c is the critical value of R for the onset of convection when the top boundary is isothermal and R is based on the spherically averaged temperature difference across the shell). The shell is heated from below and cooled from above; there is no internal heating. The lower boundary of the shell is isothermal and both boundaries are rigid and impermeable. Three cases are considered. In one, the upper boundary is isothermal while in the others, temperature anomalies with (l,m) = (3,2) and (6,4) are imposed on the top boundary. The spherically averaged temperature difference across the shell is the same in all three cases. The amplitudes of the imposed temperature anomalies are equal to one-half of the spherically averaged temperature difference across the shell. Convective structures are strongly controlled by both rotation and the imposed temperature anomalies suggesting that thermal inhomogeneities imposed by the mantle on the core have a significant influence on the motions inside the core. The imposed temperature anomaly locks the thermal perturbation structure in the outer part of the spherical shell onto the upper boundary and significantly modifies the velocity structure in the same region. However, the radial velocity structure in the outer part of the shell is different from the temperature perturbation structure. The influence of the imposed temperature anomaly decreases with depth in the shell. Thermal structure and velocity structure are similar and convective rolls are more columnar in the inner part of the shell where the effects of rotation are most dominant.     

The instability of precessing flow

Abstract

An explanation is put forward for the instability observed within a precessing, rotating spheroidal container. The constant vorticity solution for the flow suggested by Poincaré is found to be inertially unstable through the parametric coupling of two inertial waves by the underlying constant strain field. Such resonant couplings are due either to the elliptical or shearing strains present which elliptically distort the circular streamlines and shear their centres respectively. For the precessing Earth's outer core, the shearing of the streamlines and the ensuing shearing instability are the dominant features. The instability of some exact, linear solutions for finite precessional rates is established and used to corroborate the asymptotic analysis. A complementary unbounded analysis of a precessing, rotating fluid is also presented and used to deduce a likely upperbound on the growth rate of a small disturbance. Connection is made with past experimental studies.     

全球流体通道网

新近取得的深反射与全球地震层析成像资料,为地球内部结构和流体通道的研究提供了一些依据．通过研究这些地震资料,可以了解地球内部物质分布状态的几何模式．综合研究表明,地球内存在流体通道网,它连通地球外核、中幔圈、软流圈和岩石圈,是固体地球系统的重要组成部分．这种通道网络象“动脉”那样把地球外核中的流体和热量向外传送,但与热羽说不同的是,不见得有对应的“静脉”存在以保持地幔质量平衡．固体地球作为多层次多要素的巨型复杂系统,其动力学过程要用浑沌理论去解释．研究地球流体活动的轨迹和吸引子,将有助于深入了解地球活动的规律性．     

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.     

内核地震波速各向异性的成因

地球内核是轴对称各向异性的,其对称轴与地球的极轴之间有１１°左右的夹角,本 文根据地球内核相对于外部地球有差异转动这一观测结果,利用晶体生长理论,对内核地震波 速度各向异性的成因进行了探讨．当从熔融状态结晶时,晶体的生长速度与晶体和熔融态之 间相对运动的线速度成正比涸此当固态内核在液态外核中生长时,沿赤道方向的生长速度比 两极方向快．在万有引力场的作用下内核始终保持近似球形,生长速度较快的赤道附近的物 质会向两极区域流动,形成轴对称的流变场。这一轴对称的流变场伴随着轴对称的应力场,使 得构成地球内核的ｈｃｐ型铁晶体的ｃ轴沿着内核自转轴的方向排列,导致观测到的地球内核地 震波速度各向异性。作为推论,内核相对于外部地球可能同时存在着进动和章动。     

Short-period PKP phases and the anelastic mechanism of the inner core

Short-period seismograms are synthesized for PKP phases in anelastic Earth models. The synthetics were constructed using a synthetic technique valid at grazing incidence, a source-time function appropriate for deep-focus earthquakes, and an instrument response for either a short-period WWSSN or SRO seismograph. The agreement between predicted and observed amplitudes and spectral ratios requires neither a low-Q_α zone at 0.2–2 Hz nor a low or negative P-velocity gradient at the bottom of the outer core. Thin low-Q_α zones beneath the inner core boundary fit spectral ratio data that sample the upper 200 km of the inner core but fail to fit data that sample the lower inner core. Only a model having Q_α^?1?[0.003, 0.004] at 0.2–2 Hz, nearly constant with depth in the inner core, satisfies all of the spectral ratio and amplitude data. The assumption of a bulk viscosity of 10-10³ Pa s for the liquid phase of a partially molten inner core combined with the observation of low shear attenuation in the inner core at frequencies less than 0.005 Hz limit the physical parameters associated with two possible attenuation mechanisms: (1) fluid flow and viscous relaxation due to ellipsoidally shaped inclusions of melt, and (2) the solid-liquid phase transformation induced by the stress change during the passage of a seismic wave. Both mechanisms require an order of 0.1% partial melt to reproduce the observed Q_α^?1. In the outer core, the time constant of the mechanism of phase transformation is predicted to be 10⁴–10⁶ s. Confirmation of small shear attenuation in the inner core in the frequency band of seismic body waves would favor the mechanism of phase transformation.     

On the modified Taylor constraint

A dynamo driven by motions unaffected by viscous forces is termed magnetostrophic. Although such a model might describe magnetic field generation in Earth’s core well, a magnetostrophic dynamo has not yet been found even though Taylor [Proc. R. Soc. Lond. A 1963, 274, 274–283] devised an apparently viable method of finding one. His method for determining the fluid velocity from the magnetic field and the energy source involved only the evaluation of integrals along lines parallel to the Earth’s axis of rotation and the solution of a second-order ordinary differential equation. It is demonstrated below that an approximate solution of this equation for a broad family of magnetic fields is immediate. Furthermore inertia, which was neglected in Taylor’s theory, is restored here, so that the modified theory includes torsional waves, whose existence in the Earth’s core has been inferred from observations of the length of day. Their theory is reconsidered.     

A real <Emphasis Type="Italic">P</Emphasis>-wave and its dependence on the presence of gas

Attenuation of seismic compression waves leads to the real existence of a fast P₁ wave in rocks which are fully saturated with dropping fluid and a slow P₂ wave in the rocks containing gas in their pores. This accounts for the seismic blanking zones below the gas horizons for the P₁ waves. Oscillations of gaseous inclusions ensure the energy transfer to the dominant frequencies which are different for the cases of passive seismic (few Hz) and active source seismic (10–20 Hz). The intervals of dominant frequencies are determined from the negative attenuation of these low-frequency waves. According to the observations and the suggested equation, random noise amplifies the signal at these frequencies. Thus, the P₂ waves at the dominant frequency of the active source seismics are applicable for elaborating on the details of the saturation of the production layer by hydrocarbons. The relation to the AVO method (Amplitude Variation with Offset) and dilatancy effect during the preparation of an earthquake is noted.     

Magnetic instability in a rapidly rotating cylindrical annulus with a finitely conducting inner core

Abstract

We consider the stability of a toroidal magnetic field B = B(s*) (where (s*,φ,z*) are cylindrical polar coordinates) in a cylindrical annulus of conducting fluid with inner and outer radii s_i and s _o rotating rapidly about its axis. The outer boundary is taken to be either insulating or perfectly conducting, and the effect of a finite magnetic diffusivity in the inner core is investigated. The ratio of magnetic diffusivity in the inner core to that of the outer core is denoted by ηη→0 corresponding to a perfectly conducting inner core and η→∞ to an insulating one. Comparisons with the results of Fearn (1983b, 1988) were made and a good match found in the limits η→0 and η→∞ with his perfectly conducting and insulating results, respectively. In addition a new mode of instability was found in the eta;→0 regime. Features of this new mode are low frequency (both the frequency and growth rate →0 as η→0) and penetration deep into the inner core. Typically it is unstable at lower magnetic field strengths than the previously known instabilities.     

Restoring forces on vertical circular cylinders forced by earthquakes

An eigenfunction solution is presented for the dynamic response of vertical circular cylinders to earthquake excitation in a compressible fluid of finite depth. This single eigenseries expansion eliminates the need for a double summation over both the eigenfunctions and the trial functions as required by Rayleigh-Ritz methods. Revised definitions for the added mass and hydrodynamic radiation damping coefficients per unit length are derived from the hydrodynamic fluid pressures. Based on comparisons between these newly defined coefficients, the compressibility of the fluid is found to be relatively more important at dimensionless frequencies greater than unity (ω > 1.0) when analysing both rigid and flexible cylinders having relatively large diameter to water depth ratios,r₀/h > 0.25 (squatty type). This conclusion regarding the relative importance of the fluid compressibility is derived from a comparison between the relative magnitudes and the vertical distributions over depth of both the added mass and radiation damping coefficients per unit length for both rigid and flexible squatty cylinders. From additional comparisons with Rayleigh-Ritz solutions that require trial functions, the results for totally immersed flexible slender cylinders (r₀/h< 0.10) are shown to be equivalent; but the results for totally immersed flexible squatty cylinders (r₀/h > 0.25) are not. The reason for this difference appears to be in the truncation of the trial function series in the Rayleigh-Ritz methods, which excludes the higher mode shapes, and in the definitions of the added mass coefficients. Comparisons with laboratory data for both rigid and flexible cylinders confirm the accuracy of the solutions obtained by the eigenseries in the limited frequency interval above the highest frequency for surface gravity waves (f > 1.0 Hz) and below the first dimensionless cut-off frequency for acoustic waves (ω< 1.0).     

An attenuating structure beneath the Aso Caldera determined from the propagation of seismic waves

The attenuation of amplitude is seen in seismic waves which pass through the central region of the Aso caldera, in Kyushu, Japan. It is also recognized from spectral analysis of seismic waves that the higher frequencies of the P-wave are reduced in the waves which pass through the central region of the caldera. It is shown that the relative attenuation increases remarkably for the frequency range of 5 to 10 Hz. The specific attenuation factor Q of the P-wave train is about 100. From the surface projection of the ray paths with low Q values through the Aso caldera to each station, the attenuating region is located beneath the center of the caldera, extending to the north of the central cones. In conjunction with the low Q value of the P-wave and the decreases of S-wave amplitudes, the relative P-wave residual times have comparatively large values for seismic waves passing through the central region beneath the caldera. In order to attempt to provide additional information on the depth configuration of the attenuating material, the ray paths of P-wave's first arrivals are located in three-dimensional space. It indicates that the low-velocity material is located beneath the center of the caldera at depths of about 6 to 9 km. However, lowvelocity anomalies above the depth of 6 km and below the depth of 15 km were not able to be detected, because most of the available seismic ray paths had crossed the caldera at depths of about 6 to 15 km. Furthermore, the relative residual times have numerous errors resulting from incorrect hypocenter locations, origin times, inhomogeneities in the structure and uncertainty of the velocity structure. At shallow depths in the Aso caldera, refraction or reflection studies are required for an accurate estimate of the structure and more detailed properties of the attenuating material.     

Convection at the melting point: A thermal history of the earth's core

Abstract

Higgins and Kennedy (1971) concluded that the Earth's fluid core has a stable stratification if it is at its melting point. Busse (1972) and Elsasser suggested as an alternative that a hydrostatic-isentropic distribution of particulate solid can produce neutral stability in a partially molten core. Here this suggestion is quantified and a determination is made of the efficiency of the production of fluid motion from the heat flux. This is used to establish that macroscopic convection can exist only if the particulate solid is of sufficiently small size. A thermal history of the core compatible with upper mantle heat flux is advanced in which it is suggested that the inner core is a fairly recent feature. The implication of these results for convection-driven and precession-driven dynamos is that both can function for a small enough suspended particulate, that the convection-dynamo will fail for particles greater than one micron in diameter, and that the precession-driven dynamo probably cannot survive particles greater than ten microns in diameter.     

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.