Effects of a mushy transition zone at the inner core boundary on the Slichter modes

This article investigates the effects of a mushy inner core boundary on the eigenperiods of the Slichter modes for a simple, but realistic, earth model (rotating, spherical configuration, elastic inner core and mantle, neutrally stratified, inviscid, compressible liquid core). It is found that the influence of the mushy boundary layer is substantial compared with some other effects, such as those from elasticity of the mantle, non-neutral stratification of the liquid outer core and ellipticity of the Earth and centrifugal potential. The results obtained here may set a lower bound on the eigenperiods of the Slichter modes for a realistic earth model. For example, for a PREM model, the lower bound of the central period of the Slichter modes should be about 5.3 hr.     

Internal waves in the earth's core

This paper investigates possible long-period oscillations of the earth's fluid outer core. Equations describing free oscillations in a stratified, self-gravitating, rotating fluid sphere are developed using a regular perturbation on the equations of hydrodynamics. The resulting system is reduced to a finite set of ordinary differential equations by ignoring the local horizontal component of the earth's angular velocity vector, Ω, and retaining only the vertical component. The angular dependence of the eigensolutions is described by Hough functions, which are solutions to Laplace's tidal equation.
The model considered here consists of a uniform solid elastic mantle and inner core surrounding a stratified, rotating, inviscid fluid outer core. The quantity which describes the core's stratification is the Brunt—Väisälä frequency N , and for particular distributions of this parameter, analytical solutions are presented. The interaction of buoyancy, and rotation results in two types of wave motion, the amplitudes of which are confined predominantly to the outer core: (1) internal gravity waves which exist when N ² > 0, and (2) inertial oscillations which exist when N ²<4Ω². For a model with a stable density stratification similar to that proposed by Higgins & Kennedy (1971), the resulting internal gravity wave eigenperiods are all at least 8 hr, and the fundamental modes have periods of at least 13 hr. A model with an unstable density stratification admits no internal gravity waves but does admit inertial oscillations whose eigenperiods have a lower bound of 12hr.     

A Hamiltonian approach to dissipative phenomena between the Earth's mantle and core, and effects on free nutations

The Hamiltonian formalism was recently applied by Getino (1995a,b) for the study of the rotation of a non-rigid earth with a heterogeneous and stratified liquid core. That earth model is generalized here by including the effect of the dissipation arising from the mantle-core interaction, using a model similar to that of Sasao, Okubo & Saito (1980), which includes both viscous and electromagnetic coupling. First, a solution for the free nutations is obtained following a classical approach, which in our opinion is more familiar to most of the readers than the Hamiltonian treatment. This solution provides a theoretical basis clear enough to study both the qualitative and quantitative effects of the dissipations considered in the hypotheses. The main qualitative features are, besides the delays, that the free core nutation (FCN) suffers an exponential damping, while the chandler wobble (CW) is not damped at first order, by the dissipation considered. The numerical values obtained for the complex compliances agree with the most recent experimental computations.
Next, the problem is studied under a Hamiltonian formalism, and a solution equivalent to the above is obtained. Besides its interest from a theoretical point of view, this formalism is necessary in order to apply canonical perturbation methods in order to obtain analytical nutation series.     

Asymptotic behaviour of decay rates of internal waves in a rotating stratified spherical shell

Summary. Boundary layer techniques are used to examine the dissipative decay of an internal oscillation that is a member of the inviscid spectrum of normal modes for a rotating fluid shell stratified under a radially directed gravitational field. A formula is derived for the decay factor on the so-called homogeneous spin-down time-scale. Estimates are obtained for the size of the decay factor as a function of wavelength, a function of the frequency and a function of a parameter A which measures the ratio of the stratification strength to the rotation strength. It is shown that all modes decay on the spin-down time-scale. The results are interpreted in the context of a model for the Earth's fluid core. It is observed that the presence of regions of unstable stratification may increase the decay rate for oscillations at frequencies less than twice the rotation frequency.     

The subseismic approximation in core dynamics—II. Love numbers and surface gravity

This paper extends our earlier examinations of the utility of various approximations for treating the dynamics of the Earth's liquid core on time-scales of the order of 10⁴ to 10⁸ s. We discuss the effects of representing the response of the mantle and inner core by static (versus dynamic) Love numbers, and of invoking the subseismic approximation for treating core flow, used either only in the interior of the liquid core (SSA-1) or also at the boundaries (SSA-2). The success of each approximation (or combinations thereof) is measured by comparing the resulting surface gravity effects (computed for a given earthquake excitation), and (for the Slichter mode) the distribution of translational momentum, with reference calculations in which none of these approximations is made. We conclude that for calculations of the Slichter triplet, none of the approximations is satisfactory, i.e. a full solution (using dynamic Love numbers at elastic boundaries and no core flow approximation) is required in order to avoid spurious eigenfrequencies and to yield correct eigenfunctions (e.g. conserving translational momentum) and surface gravity. For core undertones, the use of static Love numbers at rigid boundaries is acceptable, along with SSA-1 (i.e. provided the subseismic approximation is not invoked at the core boundaries). Although the calculations presented here are for a non-rotating earth model, we argue that the principal conclusions should be applicable to the rotating Earth. Shortcomings of the subseismic approximation appear to arise because both SSA-1 and SSA-2 lower the order of the governing system of differential equations (giving rise to a singular perturbation problem), and because SSA-2 overdetermines the boundary conditions (making it impossible for solutions to satisfy all continuity requirements at core boundaries).     

On the Stability of a Spherical Gravitating Compressible Liquid Planet without Spin

During the past ten years or so there has been considerable discussion in the literature regarding the author's 1963 contention that (neglecting temperature effects and spin) the Earth's liquid core cannot be stable unless the Adams-Williamson condition relating density distribution and compressibility holds there.
The present paper throws light on this question by showing mathematically that a sphere of gravitating compressible liquid cannot be internally stable unless this condition is fulfilled. Physical reasons for the necessity of this condition, which implies that particles of the liquid are in neutral equilibrium, are also discussed. By internal stability is meant stability of the density distribution while the spherical shape is maintained.
The question of shape stability is not treated here, since it may be assumed that the Earth's mantle is sufficiently rigid to keep the core essentially spherical.
The liquid is assumed to be a perfect fluid, elastic, and in the discussion only small strains are considered from an equilibrium configuration of initial hydrostatic stress. Furthermore thermodynamic effects are neglected and there is no spin.     

A variational principle for the subseismic wave equation

Summary. A variational principle is developed for the subseismic wave equation governing the normal modes of the outer liquid core with frequencies below seismic frequencies (>300 μHz). The calculation of these modes is important both in determining the core contribution to the Earth's dynamical response to tidal and other forces and because their detection at the surface could provide valuable new insight into the density structure of the core, critical to theories of the geomagnetic dynamo. Included as a special case is a variational principle for the Poincaré equation governing inertial oscillations studied in the laboratory by Aldridge and others. This opens up the possibility of 'tracking' laboratory results over to the real Earth.     

Seismic Modelling of Lateral Heterogeneity At the Base of the Mantle

Summary. Results from several recent studies suggest that there are lateral heterogeneities of up to a few per cent in the lowermost 150–200 km of the mantle (Bullen's D " region). Inferred anomaly sizes span the range from less than 50 km to greater than 1000 km.
In this study differences in the velocity structure among regions at the base of the mantle were inferred from an analysis of amplitude ratios of PKPAB and PKPDF for given earthquake-station pairs at distances greater than 155° (Sacks, Snoke & Beach). We distinguish two kinds of regions: A (anomalous) regions in which the mean, median and spread in AB/DF amplitude ratios are significantly higher (> 50 per cent) than for a reference radial earth model and N (normal) regions in which the distribution of the amplitude ratios is as expected.
The AB branch has near-grazing incidence to the core and therefore maximum sensitivity to velocity structure compared to the near-normal incident DF phases. Using an iterative, forward-modelling approach, we have determined general characteristics of the velocity structure for regions at the base of the mantle which can produce amplitude-ratio distributions similar to those for an A region. Agreement between model and data is obtained over the period range from 0.5 s to greater than 10 s using a laterally heterogeneous model for the D " region. the model consists of cells which are 200 km in lateral extent with velocity variations of up to ±1 per cent. This structure is modulated by a region-wide (1000km) perturbation which increases smoothly from zero at the edges of the region to a negative 1 per cent at the centre. Small cells (∼40 km) cannot produce anomalously large amplitude, long-period AB arrivals, and larger cells (∼1000km) cannot match the observed scatter. the ∼200 km scale anomalies could be small-scale convection cells confined to the D " region.     

A search for Chandler and nearly diurnal free wobbles using Liouville equations

Summary. The basic equations describing the dynamical effects of the Earth's fluid core (Liouville, Navier-Stokes and elasticity equations) are derived for an ellipsoidal earth model without axial symmetry but with an homogeneous and deformable fluid core and elastic mantle.
We develop the balance of moment of momentum up to the second order and use Love numbers to describe the inertia tensor's variations. The inertial torque takes into account the ellipticity and the volume change of the liquid core. On the core—mantle boundary we locate dissipative, magnetic and viscous torques. In this way we obtain quite a complete formulation for the Liouville equations.
These equations are restricted in order to obtain the usual Chandler and nearly diurnal eigenfrequencies.
Then we propose a method for calculating the perturbations of these eigenfrequencies when considering additional terms in the Liouville equations.     

Core precession: flow structures and energy

Experiments using a precessing liquid-filled oblate spheroid with ellipticity ( a − b )/ a =1/400 extend and clarify earlier research. They yield flow data useful for estimating flows in the Earth's liquid core. Observed flows illustrate and confirm a nearly rigid liquid sphere with retrograde drift and lagging a cavity (mantle) axis in precession. The similarities of the observed lag angle with that computed for a rigid sphere, and earlier energy dissipation research both support the use of a rigid sphere analytical model to predict energy dissipation and first-order flow within the core–mantle boundary (CMB). Second-order boundary layer and interior cylindrical flow structures also are photographed and measured. Interior flows are never turbulent or unstable at near-Earth parameters, although complex and transient flow patterns are observed within the boundary layer. Other mechanisms proposed to explain net heat loss from the Earth and maintenance of the geodynamo typically require acceptance of some critical but unproven premise. Precession and CMB configuration are known with certainty and precision. Analytical difficulties have been the obstacle. Experiments illustrate the consequences of precession and ellipticity, provide criteria for validating analytical and numerical models, and may yield direct knowledge of the Earth's deep interior with careful scaling.     

Holocene paleolimnological changes in Lake Skallen Oike in the Syowa Station area of Antarctica inferred from organic components in a sediment core (Sk4C-02)

Antarctic climate changes influence environmental changes at both regional and local scales. Here we report Holocene paleolimnological changes in lake sediment core Sk4C-02 (length 378.0 cm) from Lake Skallen Oike in the Soya Kaigan region of East Antarctica inferred from analyses of sedimentary facies, a range of organic components, isotope ratios of organic carbon and nitrogen, and carbon-14 dating by Tandetron accelerator mass spectrometry. The sediment core was composed of clayish mud (378.0–152.5 cm) overlain by organic sediments (152.5 cm-surface). The age of the surface and the core bottom were 150 (AD1950-1640) and ca. 7,030 ± 73 calibrated years before present (cal BP), respectively, and the mean sedimentation rate was estimated to be 0.55 mm/year. Multi-proxy analyses revealed that the principal environmental change in the core is a transition from marine to lacustrine environments which occurred at a depth of 152.5 cm (ca. 3,590 cal BP). This was caused by relative sea level change brought about by ongoing retreat of glaciers during the mid-Holocene warming of Antarctica, and ongoing isostatic uplift which outpaced changes in global (eustatic) sea level. The mean isostatic uplift rate was calculated to be 2.8 mm/year. The coastal marine period (378.0–152.5 cm, ca. 7,030–3,590 cal BP) was characterized by low biological production with the predominance of diatoms. During the transition period from marine to freshwater conditions (152.5-approximately 135 cm, ca. 3,590–3,290 cal BP) the lake was stratified with marine water overlain by freshwater, with a chemocline and an anoxic (sulfidic) layer in the bottom of the photic zone. Green sulfur bacteria and Cryptophyta were the major photosynthetic organisms. The Cryptophyta appeared to be tolerant of the moderate salinity and stratified water conditions. The lacustrine period (approximately 135 cm-surface, ca. 3,290 cal BP-present) was characterized by high biological production by green algae (e.g. Comarium clepsydra and Oedegonium spp.) with some contributions from cyanobacteria and diatoms. Biological production during this period was 8.7 times higher than during the coastal marine period.     

Dating of the Dome Fuji shallow ice core based on a record of volcanic eruptions from AD 1260 to AD 2001

We measured the concentration of non-sea-salt sulfate () in the Dome Fuji shallow ice core (Antarctica) from the surface to 40 m depth with the aim of dating the core with reference to the record of volcanic eruptions. Three huge spikes related to large-scale volcanic eruptions were detected at depths of 12.5, 29.9, and 38.8 m, correlated to the eruptions of Tambora (AD 1815), Kuwae (AD 1452) and an unknown event (AD 1259), respectively. We identified another nine spikes related to accurately dated eruption events. The shallow ice core was dated from AD 1260 to AD 2001 based on these 12 eruption events and the assumption of constant annual snow accumulation in the periods between eruption events. The results yield a maximum correction of ∼20 years compared with the dating proposed in a previous study. The annual accumulation varied within ±∼15% of the average water equivalent value over the study period (25.5 mm).     

Quantitative assessment of the reliability of chironomid remains in paleoecology: effects of count density and sample size

Random distributions for a wide range (1–100,000) of chironomid head capsules (HC) were simulated on a 1-m² surface. The number of HC found in circular surfaces equivalent to standard core diameters (90 and 63 mm) was estimated 1000 times, over the range of tested densities. For each number of HC found in the samples, the range of simulated densities was estimated using a threshold probability (p > 0.95). This enabled us to develop equations to infer HC density from sample counts. Because of the threshold probability for comparable sample counts, the equations yield higher estimated densities under a random distribution than for a regular distribution. The probability of sampling at least one HC was >0.95 for densities of 900 HC m^?2 for the 90-mm core and 1400 HC m^?2 for the 63-mm core. For a specific sample count, the range of actual densities was ~10 times higher for the 63-mm core than the 90-mm core. Comparison with field larval densities revealed that most densities were too low to be suitable for annually resolved reconstruction of a quantitative signal, using current corer sizes, although a large number of populations can support sub-decadal analyses. Nonetheless, some lakes exhibit population sizes large enough to reconstruct robust quantitative estimates of past chironomid abundances. This work provides guidance to reconstruct species dynamics and fine-scale time series analyses in paleoecology.     

Variationsl solution of long-period oscillations of the Earth

Summary The linearized equation of motion for the slightly elliptical rotating earth is obtained and using Phinney & Burridge's generalized spherical harmonics, the variational principle is derived for the normal mode oscillations of the Earth. The numerical solutions of two earth models 1066B and B1S6 are searched by minimizing the energy functional for the terrestrial spectral range longer than the lowest order free oscillation. The periods of core modes computed for the earth model B1S6, with stably stratified outer core, ranges from about 4 to 13hr and the periods for the 1066B are much more spread without clustering around the periods of 6 and 12 hr as in B1S6. The results for the earth model 1066B indicate that an outer core can support long-period oscillations even when it is not stably stratified. The Chandler wobble periods obtained are 402.3 day for B1S6 and 402.7 day for 1066B.     

Partitioning of the grain-size components of Dali Lake core sediments: evidence for lake-level changes during the Holocene

We recovered a sediment core (DL04) from the depocenter of Dali Lake in central-eastern Inner Mongolia. The upper 8.5 m were analyzed at 1-cm intervals for grain-size distribution to partition the grain-size components and provide a high-resolution proxy record of Holocene lake level changes. Partitioning of three to six components, C1, C2, C3 through C6 from fine to coarse modes within the individual polymodal distributions, into overlapping lognormal distributions, was accomplished utilizing the method of lognormal distribution function fitting. Genetic analyses of the grain-size components suggest that two major components, C2 and C3, interpreted as offshore-suspension fine and medium-to-coarse silt, can serve as sediment proxies for past changes in the level of Dali Lake. Lower modal sizes of both C2 and C3 and greater C3 and lower C2 percentages reflect higher lake stands. The proxy data from DL04 core sediments span the last 12,000 years and indicate that Dali Lake experienced five stages during the Holocene. During the interval ca. 11,500–9,800 cal year BP, lake level was unstable, with drastic rises and falls. Following that interval, the lake level was marked by high stands between ca. 9,800 and 7,100 cal year BP. During the period from ca. 7,100 to 3,650 cal year BP, lake level maintained generally low stands, but displayed a slight tendency to rise. Subsequently, the lake level continued rising, but exhibited high-frequency, high-amplitude fluctuations until ca. 1,800 cal years ago. Since ca. 1,800 cal year BP, the lake has displayed a gradual lowering trend with frequent fluctuations.     

Observational constraints on the chemical and thermal structure of the Earth's deep interior

Summary. The observations of the periods of free oscillation of the Earth provide direct constraints on the density distribution in the Earth. These in turn allow constraints to be placed on the size of departures from a state of adiabaticity and chemical homogeneity. These departures are quantified in terms of a stratification parameter '8' first introduced as an index of chemical homogeneity. The resolving power theory of Backus & Gilbert is used to determine the ability of the observed free oscillations to constrain η in the lower mantle and outer core. The results suggest that the outer core is not strongly chemically stratified although a significantly thermally stable core cannot be excluded. The free oscillations also apparently require a compositional difference between the inner and outer cores.     

A stratified optimization method for a multivariate marine environmental monitoring network in the Yangtze River estuary and its adjacent sea

An efficient monitoring network is very important in accessing the marine environmental quality and its protection and management. In an estuary, there are fronts that separate distinctly different water masses and affect material transport, nutrient distribution, pollutant aggregation, and diffusion. This stratified heterogeneous surface neither satisfies the stationary requirements of kriging, nor can be handled adequately by removing a spatially continuous trend. This article presents a stratified optimization method for a multivariate monitoring network. In this method, principal component analysis (PCA) was used to reduce the dimensionality of the correlated targets, and the mean of surface with nonhomogeneity (MSN) method was adopted to produce the best linear unbiased estimator for a spatially stratified heterogeneous surface that failed to satisfy the requirements for a kriging estimate. The existing monitoring network in the Yangtze River estuary and its adjacent sea, which was designed by purposive sampling year ago was optimized as an illustration. The optimization consisted of two steps: reduce the redundant monitoring sites and then optimally add new sites to the remaining sites. After optimization, the inclusion of 51 sites in the monitoring network was found to produce a smaller total estimated error than that of the current network, which has 70 sites; moreover, the use of 55 sites can produce a higher precision of estimation for all three principal components (PCs) than that of the current 70 sites. The results demonstrated that the proposed method is suitable for optimizing environmental monitoring sites that have dominant stratified nonhomogeneity and that involve multiple factors.     

Dislocation-mediated melting of iron and the temperature of the Earth's core

Summary . Dislocation theories of melting provide a possibility of calculating the melting temperature, from first principles, as the temperature at which the free energy of a crystal saturated with dislocations becomes equal to that of the dislocation-free crystal. After a brief review of the physical bases of the dislocation melting theories, Ninomiya's theory is used to determine the melting temperature as well as the volume and entropy of melting and the slope of the melting curve for iron at atmospheric pressure and under conditions prevailing at the Earth's inner core boundary. The necessary parameters (elastic moduli, Grüneisen parameter) are drawn from seismological earth models. We find a melting temperature of the material of the inner core of about 6150 K, independent of shock-wave experiments but in good agreement with them and with extrapolations using Lindemann's law. With usually accepted values of the melting point depression due to light elements in solution, the temperature at the inner core boundary is found to be T _ICB≅ 5000 K. This temperature is compatible with a temperature of the outer core at the core-mantle boundary T _CMB≅ 3800 K. Dislocation melting theories can thus help constrain the temperature profile in the Earth's core.     

On Boussinesq's problem for Maxwell continua subject to an external gravity field

Summary. Boussinesq's problem is solved for a uniform and incompressible Maxwell half-space subject to an external gravity field. The solution is based on momentum equations which account for stress advection in the hydrostatically pre-stressed continuum during its deformation. The analysis shows that disregarding the pre-stress term renders the theoretical stress distribution incorrect and the deformation singular in the inviscid limit of the Maxwell continuum. Our solution is contrasted with a recently published alternative solution of the same problem, where regularity in the inviscid limit was forced by modified boundary conditions.