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.     

Forced magnetohydrodynamic oscillations of the earth's core

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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.     

On the admissible range for the mass and inertia moments of the liquid core: I. Solving the inverse problem of nutations and free oscillations of the Earth by decomposition of mechanical parameters in an orthogonalized basis

The results of solving the inverse problem of forced nutations and free oscillations of the Earth by decomposing the Q-factor and small depth variations in density in a system of orthogonal functions are considered. These functions are determined by orthogonalization of the functional derivatives of the observed parameters with respect to the depth distributions of the sought parameters (assuming there are no distributions of the velocities of body seismic waves V _p and V _S with depth and unchanged total mass M and inertia moments I of the Earth). The examples are presented to illustrate the numerical solution of the inverse problem on finding the density distributions in the mantle and core of the Earth using orthogonalization of the integral constraints for the probable depth distributions of density describing the conditions of unchanged M and I, as well as the constraints posed by the data on the periods of the free low-order oscillations of the Earth.     

Oscillatory flow in the liquid core

There has been renewed interest lately in the possibility that at least a part of the Earth's liquid core may be stably stratified. A gravitationally stable region would permit the existence of inertia-gravity or gravity-inertia waves in addition to the Rossby and Kelvin waves which exist due to rotational effects and which are well known in oceanography and atmospheric dynamics. These wave motions are of interest because their periods are dependent on the density stratification as specified by the buoyancy frequency N which in turn determines the amplitude of large-scale radial motions in the core.The waves have too high a frequency to be connected dynamically to the magnetic field in the core, but if they do exist they may be detectable by sensitive long-period gravimeters at the Earth's surface. This paper examines the available evidence for the frequency regimes, excitation and damping mechanisms of the core waves. It is concluded that although the waves may exist theoretically, their detection and interpretation as a method for determining N is a difficult proposition.     

Density perturbations in the earth's core due to hydromagnetic oscillations

Summary The equation for density perturbation induced by plane hydromagnetic waves (responsible for geomagnetic secular variation) has been derived based on -plane formulation. It is deduced that for the usual physical properties of the earth's core such a density perturbation would be small and would not be, in the main, responsible for the recently discovered correlation between gravitation and geomagnetic potentials.     

On free oscillations of the earth

This survey concerns mostly the theory of free oscillations, with a section on experimental work included at the end. Developments over the last 15 years are examined. The general theory of free oscillations is reviewed, and the effect on free oscillations of such factors as heterogeneity, the Earth's rotation and non-sphericity, and the source of the oscillations are discussed. Earth models, which have been obtained from oscillation data, are reviewed, and their use in forming theoretical seismograms is described.See note at p. 426.Note     

Magnetohydrodynamic waves in the erath's liquid core

Summary The possibility of generation of magnetohydrodynamic waves in a rotating spherical system under the efect of small disturbances of the medium velocity is investigated. The initial stationary state of the system has been chosen with a view to known geophysical facts, and the effect of the solid inner core is considered. It is proved that two types of waves with distinctly different frequencies can be generated in the liquid region. The behaviour of longperiod magnetic waves is also studied in detail, and it is proved that the direction in which they propagate at a certain point depends on the position of this point in the meridional plane and on the configuration of the initial magnetic field. The phase velocities of the longperiod magnetic waves, propagating towards the west, have been computed within a selected network of points and the region in which they are generated has been defined. It is proved that the phase velocities of these waves are compatible with the observed velocity of the westward drift of the geomagnetic dipole.     

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

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

On global short-periodic synchronous oscillations of the atmosphere

Summary The existence of global synchronous atmospheric oscillations with the period of about 1.4 hour is indicated in terms of a simple theoretical model of the circumpolar circulation.

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Experimental results on the tidal resonance effect of the liquid core

Abstract

Molodenskii's theory on the resonance effect of the liquid core predicts anomalous amplitudes in the diurnal spectrum for earth tide registrations of gravity and tilt. A two years recording of an Askania gravimeter at Kiel University has been analysed using three different numerical methods: harmonic analysis (least squares method), gliding harmonic analysis, and Fourier spectral analysis. Because of difficulties in resolving tidal waves near the resonance frequency, efforts were made towards improving the Fourier spectral analysis by successive subtraction of high amplitudes in the Fourier spectrum. The resonance of the liquid core has been observed with a high degree of confidence thereby supporting Molodenskii's model calculations.     

低频交变压差作用下岩心流动电流和zeta电势实验测量

实验测量岩样zeta电势,对于认识地震诱导的电磁场现象以及探索利用动电效应的测井方法具有重要意义.本文介绍了专门设计的岩样流动电流和流动电势实验测量系统及一种获得岩样zeta电势的测量方法——流动电势和流动电流综合测量法.实验采用交流锁相放大技术,测量了3~100 Hz频率范围内交变压差作用下岩样的流动电势系数和流动电流系数,实验表明,这两个系数均随溶液浓度增大而减小,随pH值增大而增大.实验通过对不同浓度饱和岩样的流动电势和流动电流测量,获得了岩样的有效面积-长度比,进而计算出岩样的zeta电势和表面电导. 实验还论证了忽略表面电导是导致常规流动电势测量法获得zeta电势低于实际值的原因.     

On stochastic near-critical oscillations in the upper mantle

The problem of the determination of thermomechanical conditions in the upper mantle under a moving lithospheric plate at a given shear stress does not have a unique solution. Given a fixed heat flux from the lower mantle, two types of motion are possible in the mantle: subcritical (slower and colder) and supercritical (faster and hotter). In this work, it is shown that, if these modes are rather close to each other (in the near-critical state of the mantle), transitions from one mode to another and backward are possible. The calculated period of a change in the mantle state amounts to ～0.3 Myr for the Pacific plate. The oscillatory regime of the mantle state can manifest itself in the activity pulsations of hotspots located near fast separating mid-ocean ridges.     

Element partitioning between metallic liquid, silicate liquid, and lower-mantle minerals: implications for core formation of the Earth

We determined the partition coefficients of 19 elements between metallic liquid and silicate liquid at 20 GPa and 2500°C, and between metallic liquid and silicate perovskite at 27 GPa and 2200°C. Remarkable differences were observed in the partitioning behaviors of Si, P, W, Re, and Pb among the silicate liquid, perovskite, and magnesiowüstite coexisting with metallic liquid, reflecting incompatibility of the elements in the silicate or oxide phase. We could not observe any significant difference in the partitioning behaviors of V, Cr, Mn, Co, Ni, and Cu among the phases coexisting with metallic liquid.

Comparison of the present partitioning data with those obtained previously at lower pressure and temperature suggests that the exchange partition coefficients, K_met/sil, of Co, Ni, Mo, and W decrease, whereas those of V, Cr, and Mn increase and tend to approach unity with increasing pressure and temperature. We also made preliminary experiments to clarify the effect of sulfur on the partitioning behaviors. Sulfur lowers the exchange partition coefficients, K_met/sil, of Mo and W between metallic liquid and silicate liquid significantly at 20 GPa and 2300°C.

The mantle abundances of Co, Ni, Cu, Mo, and W calculated for the metal-silicate equilibrium model are lower than those of the real mantle, whereas P, K, and Mn are overabundant in the calculated mantle. The discrepancies in the abundances of Co and Ni could be explained by the chemical equilibrium at higher pressure and temperature. Large discrepancies in Mo and W between the calculated and real mantles could be accounted for by the effect of sulfur combined with the effects of pressure and temperature on the chemical equilibrium. The mantle abundances of P, K, and Cu could be accounted for by volatile loss in the nebula, perhaps before accretion of the Earth, combined with the chemical equilibrium at higher pressure and temperature. Thus the observed mantle abundances of P, K, Co, Ni, Cu, Mo, and W may be consistent with a model of sulfur-bearing metal-silicate equilibrium in lower-mantle conditions.     

Detection of inertial gravity oscillations in the Earth's core with a superconducting gravimeter at Brussels

As stated by Crossley and Rochester the discovery and identification of even a few undertone periods would provide valuable gross constraints on the stability profile and hence, the thermal regime of the core. Three years of continuous measurements of gravity variations with a superconducting gravimeter at Brussels offer a possibility to detect such oscillations in that part of the spectrum where tidal contributions can be eliminated. Spectral peaks of 10–15 ngal amplitude have been detected around a period of 13.9 h which were enhanced immediately after the two deep earthquakes of magnitude 7.2 which occurred during the observation interval.     

On the free oscillations of a laterally heterogeneous earth model

Summary The equations of motion for the free oscillations of a heterogeneous spherical earth model are derived. It is found that the lateral variations of density and elastic moduli couple the odd(even) harmonics of the spheroidal oscillations with themselves as well as with the even (odd) harmonics of the torsional oscillations.List of symbols r, , Spherical coordinates;r is the radial distance from earth's center, is the co-latitude, and is the east longitude - r Space vector denoting a point with coordinatesr, , and - Gradient operator - ² Laplacian operator - _ij Kronecker's delta function - I Identity matrix - i      

用钻孔应变数据探测2011年日本Tohoku大地震激发的低频地球环型自由振荡及其谱峰分裂

地震激发的地球自由振荡中, 环型自由振荡(特别是低频环型自由振荡)的精确探测相比于球型自由振荡更加困难. 环型自由振荡对SH波速度结构更加敏感, 能更好地监测走滑断层运动, 因此精确估算环型自由振荡的谱峰参数可为地球内部大尺度SH波速度结构和大地震的震源机制提供更强约束. 本研究利用2011年日本Tohoku地震后四分量钻孔应变仪的观测资料, 提取了该地震激发的1.3 mHz以下低频环型和球型自由振荡信号, 与STS-1宽频带地震仪的结果相比较, 发现四分量钻孔应变仪对1 mHz以下的环型自由振荡具有更强的探测能力. 本研究还采用频率域自回归方法精确估计了₀T₂—₀T₆的简并频率, 结果与PREM模型理论值相差非常小, 平均相对误差仅为2.52×10^-4.此外还用钻孔应变仪观测资料探测到了₀T₂谱峰分裂现象. 研究结果表明, 钻孔应变仪能有效探测到大地震激发的低频地球环型自由振荡信号, 可为地幔大尺度各向异性结构研究提供高质量的基础数据.