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.     

Experiments on precessing flows in the Earth's liquid core

Experiments simulating flow in the Earth's liquid core induced by luni-solar precession of the solid mantle indicate, to a first approximation, that the core behaves like a rigidized fluid sphere spinning slower than the mantle and with its spin axis lagging the mantle spin axis in precession. Secondary flow patterns are always present. At low precession rates the fluid sphere is subdivided into a set of cylinders coaxial with the fluid spin axis, the cylinders rotating alternately at slightly faster and slower rates relative to the net retrograde motion of the fluid as a whole. Slow non-axisymmetric columnar wave patterns develop between the differentially rotating cylinders. Axial flows between the spheroidal cavity boundary and the interior are observed. Fluid motion becomes turbulent only at precession rates large enough to cause the fluid spin axis to align nearly with the precession axis. There is no evidence that the Earth's liquid spin axis direction departs more than a fraction of a degree from geographic north. Our observations suggest precession induces a complex variety of laminar flows, including slowly varying and/or periodic patterns, in the Earth's liquid core.     

Geomagnetic field analysis—IV. Testing the frozen-flux hypothesis

Summary. We present a model of the magnetic field at the core–mantle boundary, for epoch 1959.5, based on a large set of observatory and survey measurements. Formal error estimates for the radial field at the core are 50 μT, compared with 30 and 40 μT for our previous MAGSAT (1980) and POGO (1970) models.
Current work on the determination of the velocity of the core fluid relies on the assumption that the core behaves as a perfect conductor, so that the field lines remain frozen to the fluid at the core surface. This frozen-flux condition requires that the integrated flux over patches of the core surface bounded by contours of zero radial field remain constant in time. A new method is presented for constructing core fields that satisfy these frozen-flux constraints. The constraints are non-linear when applied to main field data, unlike the case of secular variation which was considered in an earlier paper. The method is applied to datasets from epochs 1969.5 and 1959.5 to produce fields with the same flux integrals as the 1980 model.
The frozen-flux hypothesis is tested by comparing the changes in the flux integrals between 1980/1969.5, 1969.5/1959.5 and 1980/1959.5 with their errors. We find that the hypothesis can be rejected with 95 per cent confidence. The main evidence for flux diffusion is in the South Atlantic region, where a new null flux curve appears between 1960 and 1970, and continues to grow at a rapid rate from 1970 to 1980. However, the statistical result depends critically on our error estimates for the field at the core surface, which are difficult to assess with any certainty; indeed, doubling the error estimates negates the statistical argument. The conclusion is therefore, at this stage, tentative, and requires further evidence, either from older data, if good enough, or from future satellite measurements.     

Effect of the fluid core on changes in the length of day due to long period tides

Summary. The long period luni-solar tidal potential is known to cause periodic changes in the Earth's rotation rate. We find that the effect of a dissipationless fluid outer core is to reduce the amplitudes of these tidal perturbations by ∼ 11 per cent. When the fluid core effect is added to Agnew & Farrell's estimate of the effect of an equilibrium ocean, the result is in accord with observation.
The effects of dissipative processes within the fluid core are also examined. We find out-of-phase perturbations which could be as large as ∼ 10ms at 18.6 yr. We conclude, however, that the poorly understood decade fluctuations in the Earth's rotation rate will prohibit observation of this effect.     

Deciphering some unique paleotemperature indicators in halite-bearing saline lake deposits from Death Valley, California, USA

Saline lake deposits are arguably the best source of mid- to low-latitude terrestrial paleoclimate data. Alternating clastic sediments and evaporites of different chemical composition have long been recognized as sensitive records of changes in inflow and aridity related to a variety of climate parameters. Several sources of paleotemperature information from a halite-bearing saline lake deposit are described here – pseudomorphs of a cold-temperature evaporite mineral, homogenization temperatures of fluid inclusions in halite, and stable-isotope compositions of fluid inclusions in halite. Examples of these paleoclimate data come from analysis of the lower half of a 185-m core drilled in Pleistocene saline lake deposits at Death Valley, California. Daily and seasonal temperature variations in saline lake waters create conditions for the appearance and disappearance of temperature-dependent mineral phases. In the Death Valley core, hexagonal-shaped halite crystals, probable pseudomorphs of the cold-temperature hydrous mineral, hydrohalite (NaCl2H₂O), provide evidence of brine temperatures below about 0 °C. Homogenization temperatures of fluid inclusions in primary halite offer an actual (not proxy) record of surface-brine temperatures. Samples with primary fluid-inclusion textures are carefully selected and handled, and data are collected from single-phase aqueous-brine inclusions chilled to nucleate vapor bubbles. Temperature variations are observable at scales of individual halite crystals (hours to days), single halite beds (weeks to months or years), and multiples of beds to entire facies (hundreds to tens of thousands of years). A ¹⁸O/D stable isotope record from the minute quantities of brines in fluid inclusions in halite is accessible using a method recently developed at the University of Calgary. The stable isotope record from the Death Valley core, a complex response to climate variables including temperature, humidity, storm patterns or seasons, and inflow sources, compliments and expands the interpretation emerging from the stratigraphy and homogenization temperatures.     

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.     

Finite element calculations of very high Rayleigh number thermal convection

Summary. A finite element method with uniform and variable resolution meshes is used to model very high Rayleigh number Ra thermal convection in a square box of infinite Prandtl number, Boussinesq fluid with constant viscosity and thermodynamic properties. Heating is either entirely from below or mostly from within and the boundaries are stress free. The variable mesh is coarse in the interior of the convection cell and it is fine in the very thin boundary layers and plumes surrounding the core. The highest resolution variable mesh has a dimensionless grid spacing of 0.027 in the core and 0.0017 in the boundary layers. The boundary layers contain about 10 mesh points even at the highest values of Ra considered and are thus highly resolved. The variable mesh approach is shown to yield reliable simulations of convection as long as the aspect ratio of the most elongated boundary layer elements is not too large; values of about 4 to 6 work well. This aspect ratio also measures the increase in resolution in the boundary layers as compared with the central core. Steady single-cell rolls are computed for bottom heating and Ra up to 5 × 10⁵ times the marginal instability value of the Rayleigh number Ra_cr. One and two-cell roll solutions are calculated for f = 1, 0.8 and 0.6, where f is the fraction of the heat escaping through the top of the box that is generated internally. The values of Ra_cr for f = 1, 0.8 and 0.6 are 1296, 1024 and 864, respectively. The largest of Ra/Ra_cr at which unicellular convection is stable (steady) are approximately 390, 610 and 970, for f = 1, 0.8 and 0.6.     

On the dynamical effects of a heterogeneous and compressible liquid core in the theory of Chandler wobble

The general 3-D scalar equations of motion of the liquid core (with respect to the radial components of displacements and cubic dilatation) are constructed as a superposition of the solutions of ordinary differential equations describing the dynamics of a stably stratified, heterogeneous, compressible and inviscid rotating fluid inside thin spherical layers ( Molodensky & Sasao 1995 ). The estimation of dynamical effects of a homogeneous and incompressible liquid core on the Chandler period (Groten, Lenhardt & Molodensky 1991) is generalized for the case of a heterogeneous, compressible, inviscid and neutrally stratified liquid core.     

Models of Earth's main magnetic field incorporating flux and radial vorticity constraints

We describe a new technique for implementing the constraints on magnetic fields arising from two hypotheses about the fluid core of the Earth, namely the frozen-flux hypothesis and the hypothesis that the core is in magnetostrophic force balance with negligible leakage of current into the mantle. These hypotheses lead to time-independence of the integrated flux through certain 'null-flux patches' on the core surface, and to time-independence of their radial vorticity. Although the frozen-flux hypothesis has received attention before, constraining the radial vorticity has not previously been attempted. We describe a parametrization and an algorithm for preserving topology of radial magnetic fields at the core surface while allowing morphological changes. The parametrization is a spherical triangle tesselation of the core surface. Topology with respect to a reference model (based on data from the Oersted satellite) is preserved as models at different epochs are perturbed to optimize the fit to the data; the topology preservation is achieved by the imposition of inequality constraints on the model, and the optimization at each iteration is cast as a bounded value least-squares problem. For epochs 2000, 1980, 1945, 1915 and 1882 we are able to produce models of the core field which are consistent with flux and radial vorticity conservation, thus providing no observational evidence for the failure of the underlying assumptions. These models are a step towards the production of models which are optimal for the retrieval of frozen-flux velocity fields at the core surface.     

Electromagnetic core–mantle coupling—I. Explaining decadal changes in the length of day

Measured changes in the Earth's length of day on a decadal timescale are usually attributed to the exchange of angular momentum between the solid mantle and fluid core. One of several possible mechanisms for this exchange is electromagnetic coupling between the core and a weakly conducting mantle. This mechanism is included in recent numerical models of the geodynamo. The 'advective torque', associated with the mantle toroidal field produced by flux rearrangement at the core–mantle boundary (CMB), is likely to be an important part of the torque for matching variations in length of day. This can be calculated from a model of the fluid flow at the top of the outer core; however, results have generally shown little correspondence between the observed and calculated torques. There is a formal non-uniqueness in the determination of the flow from measurements of magnetic secular variation, and unfortunately the part of the flow contributing to the torque is precisely that which is not constrained by the data. Thus, the forward modelling approach is unlikely to be useful. Instead, we solve an inverse problem: assuming that mantle conductivity is concentrated in a thin layer at the CMB (perhaps D"), we seek flows that both explain the observed secular variation and generate the observed changes in length of day. We obtain flows that satisfy both constraints and are also almost steady and almost geostrophic, and therefore assert that electromagnetic coupling is capable of explaining the observed changes in length of day.     

深部盐矿勘探钻井液研究与应用

ZK001井钻遇地层为大段泥岩,部分地层为页岩、含盐泥岩、盐膏层。在泥页岩地层进行大口径全面钻进,钻速较快,固控设备能力有限,引起钻屑中粘土颗粒重复性膨胀、软化及裂解分散,最终可能造成粘土侵,对井内安全造成威胁。深部岩膏层易溶于水,从而导致盐层蠕变、井径扩大、井壁失稳等问题。以ZK001井为例,通过优化钻井液体系及现场施工工艺,解决了地层强造浆、盐层失稳等问题。通过岩石力学测试、阳离子交换容量测试、钻井液体系优选及性能测试,确定了上部地层采用聚合物钻井液体系,下部含盐层取芯钻进转换为饱和盐水钻井液体系。通过选用合适的钻井液密度、粘度、切力量、滤失量、含盐量等指标,可以有效控制上部泥岩井壁稳定、含盐层蠕变、井径扩大的问题。钻进过程中钻井液性能稳定且易于维护,从而保证了钻进工作安全、高效、顺利的进行。     

A physical explanation of the static core paradox

Summary. The introduction of Rayleigh friction and Newtonian cooling into the dynamical problem of determining the excitation of the normal modes of oscillation of an earth model with a fluid core by a transient earthquake source is shown to provide a fully satisfactory resolution and a clear physical explanation of the difficulties and paradoxes which have arisen in previous treatments of the corresponding static deformation problem. The source of the previous difficulties is that the dissipation-free limit is associated with an essential singularity in the static response, unless the stratification in the core is neutral. This singularity, in turn, exists because the eigenfrequency spectrum of any earth model with a non-neutrally stratified core has an accumulation point at zero frequency.     

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.     

A diurnal resonance in the ocean tide and in the Earth's load response due to the resonant free 'core nutation'

Summary. The luni-solar forced nutations and body tide are believed to be resonant at frequencies near (1 + 1/460) cycle sidereal day⁻¹ as seen from the rotating Earth. This resonance is due to the Earth's rotating, elliptical fluid core. We show here that tides in the open ocean and the Earth's response to those tides must also be resonant at (1 + 1/460) cycle day⁻¹. We examine these resonant oceanic effects on the Earth's nutational motion and on the body tide. Effects on the forced nutations might be as large as 0.002 arcsec at 18.6 yr. The effects on the observed resonance in the body tide are more important. For tidal gravity, for example, the difference between K ₁ and 0 ¹ which is usually used to determine the resonance, can be perturbed by 30 per cent or more due to the oceanic resonance effects.     

Geomagnetic field analysis - I. Stochastic inversion

Summary. Most of the Earth's magnetic field and its secular change originate in the core. Provided the mantle can be treated as an electrical insulator, stochastic inversion enables surface observations to be analysed for the core field. A priori information about the variation of the field at the core boundary leads to very stringent conditions at the Earth's surface. The field models are identical with those derived from the method of harmonic splines (Shure, Parker & Backus) provided the a priori information is specified appropriately.
The method is applied to secular variation data from 106 magnetic observatories. Model predictions for fields at the Earth's surface have error estimates associated with them that appear realistic. For plausible choices of a priori information the error of the field at the core is unbounded, but integrals over patches of the core surface can have finite errors. The hypothesis that magnetic fields are frozen to the core fluid implies that certain integrals of the secular variation vanish. This idea is tested by computing the integrals and their standard and maximum errors. Most of the integrals are within one standard deviation of zero, but those over the large patches to the north and south of the magnetic equator are many times their standard error, because of the dominating influence of the decaying dipole. All integrals are well within their maximum error, indicating that it will be possible to construct core fields, consistent with frozen flux, that satisfy the observations.     

An excitation mechanism for the free 'core nutation'

Summary. The Earth is believed to possess a free nutational mode due to its rotating, elliptical, fluid core, with an eigenfrequency of approximately (1 + 1/460) cycle sidereal day⁻¹ as seen from the sidereally rotating Earth. This free 'core nutation' has not yet been undisputably observed. Furthermore, there has been considerable doubt that any known mechanism could excite this mode to an observable level. We show here that diurnal atmospheric and oceanic loading of the Earth's surface provide an efficient excitation mechanism which depends critically on the physical damping of the mode. Possible effects of the mode on geodetic measurements are discussed. We also consider the effects of 'wobble' and 'nutation' on astrometric observations.     

The effects of the atmosphere and oceans on the Earth's wobble — I. Theory

Summary The theory of wobble excitation for a non-rigid earth is extended to include the effects of the earth's fluid core and of the rotationally induced pole tide in the ocean. The response of the solid earth and oceans to atmospheric loading is also considered. The oceans are shown to be affected by changes in the gravitational potential which accompany atmospheric pressure disturbances and by the load-induced deformation of the solid earth. These various improvements affect the excitation equations by about 10 per cent. Atmospheric and oceanic excitation can be computed using either an angular momentum or a torque approach. We use the dynamical equations for a thin fluid to relate these two methods and to develop a more general, combined approach. Finally, geostrophic winds and currents are shown to be potentially important sources of wobble excitation, in contrast to what is generally believed.