Dynamics of the liquid core of the Earth

1972 ◽  
Vol 273 (1233) ◽  
pp. 237-260 ◽  
Keyword(s):  
Boundary Layer ◽  
Inner Core ◽  
Liquid Core ◽  
Stable Stratification ◽  
Earth Model ◽  
Unstable Stratification ◽  
The Core ◽  
Long Period ◽  
Love Numbers ◽  
Neutral Equilibrium

An asymptotic theory is developed for the long-period bodily tides in an Earth model having a liquid core. The yielding inside the core is found to be different in the case of a stable density stratification from the case of an unstable stratification. In the latter case, a boundary layer is formed in which the stress decreases exponentially with depth below the core surface, the scale length of the exponent being proportional to the frequency. In the limit of vanishing frequency the stress tends to zero through most of the liquid core, except near the boundary layer at the surface, where it grows to a finite value. In case of a stable stratification, the stress oscillates with depth below the surface of the core with a wavelength which is proportional to frequency. An infinite number of 'core oscillations' with indefinitely increasing periods exist in a liquid core with stable stratification, but in the case of an unstable stratification, none exist above the fundamental spheroidal oscillation (53.7 min) for n — 2. The assertion made that a liquid core must be in neutral equilibrium is not true. The displacements and stresses within a liquid core in long-period tidal yielding are determinate, even in the static limit, and are not arbitrary. Love numbers are derived for uniformly stable, neutral, and unstable liquid cores, as well as for a model with a rigid inner core.

scholarly journals New measurements of long-period radial modes using large earthquakes

2020 ◽  
Vol 224 (2) ◽  
pp. 1211-1224
Author(s):  
S Talavera-Soza ◽  
A Deuss
Keyword(s):  
Inner Core ◽  
Cross Coupling ◽  
Earth Model ◽  
Cylindrical Anisotropy ◽  
Long Period ◽  
Centre Frequency ◽  
The Earth ◽  
Radial Anisotropy ◽  
Crucial Information ◽  
Quality Factor Q

SUMMARY Radial modes, nS0, are long-period oscillations that describe the radial expansion and contraction of the whole Earth. They are characterized only by their centre frequency and quality factor Q, and provide crucial information about the 1-D structure of the Earth. Radial modes were last measured more than a decade ago using only one or two earthquakes. Here, we measure radial modes using 16 of the strongest and deepest earthquakes of the last two decades. By introducing more earthquake data into our measurements, we improve our knowledge of 1-D attenuation, as we remove potential earthquake bias from our results. For mode 0S0, which is dominated by compressional energy, we measure a Q value of 5982, much higher than previously measured, and requiring less bulk attenuation in the Earth than previously thought. We also show that radial modes cross-couple (resonate) strongly to their nearest spheroidal mode due to ellipticity and inner core cylindrical anisotropy. Cross-coupling improves the fit between data and synthetics, and gives better estimates of the centre frequency and attenuation value of the radial modes. Including cross-coupling in our measurements results in a systematic shift of the centre frequencies of radial modes towards the Preliminary Reference Earth Model. This shift in centre frequencies, has implications for the strength of the radial anisotropy present in the uppermost inner core, with our cross-coupling results agreeing with lower values of anisotropy than the ones inferred from just measuring the modes in self-coupling (isolation). Furthermore, cross-coupling between radial modes and angular-order two modes provides constraints on cylindrical inner core anisotropy, that will help us improve our knowledge of the 3-D structure of the inner core.

Sensitivity studies with different formulations for similarity functions used in surface layer scheme, over the tropical region, in a climate model

2020 ◽  
Author(s):  
Prabhakar Namdev ◽  
Maithili Sharan ◽  
Saroj Kanta Mishra
Keyword(s):  
Boundary Layer ◽  
Surface Layer ◽  
Climate Model ◽  
Stable Stratification ◽  
Turbulent Fluxes ◽  
Surface Fluxes ◽  
Tropical Region ◽  
Similarity Functions ◽  
Unstable Stratification ◽  
Stable Conditions

<p>The lowest portion of the planetary boundary layer (PBL), where the turbulent fluxes are assumed to be constant, is known as the atmospheric surface layer (ASL). Within the surface layer, the surface exchange processes play an important role in land-atmosphere interaction. Thus, a precise formulation of the surface fluxes is crucial to ensure an adequate atmospheric evolution by numerical models. The Monin–Obukhov Similarity Theory (MOST) is a widely used framework to estimate the surface turbulent fluxes within the surface layer. MOST uses similarity functions of momentum (φ<sub>m</sub>) and heat (φ<sub>h</sub>) for non-dimensional wind and temperature profiles. Over the years, various formulations for these similarity functions have been proposed by the researchers ranging from linear to non-linear forms. These formulations have limitations in the weak wind, stable, and unstable atmospheric conditions. In the surface layer scheme currently available in the Community Atmosphere Model version 5 (CAM5.0), the stable and unstable regimes are divided into four parts, and the corresponding similarity functions are the functions proposed by Kader and Yaglom (1990) for strong unstable stratification, by Businger et al. (1971) for weak unstable stratification, functions by Dyer (1974) for weak stable stratification, and for moderate to strongly stable stratification, the functions from Holtslag et al. (1990) have been utilized. The criteria used for this classification are somewhat ad-hoc, and there is an abrupt transition between different regimes encountered.            </p><p>       In the present study, an effort has been made to implement the similarity functions proposed by Grachev et al. (2007) for stable conditions and Fairall et al. (1996) for unstable conditions in the surface layer scheme of Community Land Model (CLM) for CAM5.0. In the modified version, the similarity functions for unstable conditions are a combination of commonly used Paulson type expressions for near-neutral stratification and an expression proposed by Carl et al. (1973) that takes in to account highly convective conditions. Similarly, the formulation proposed by Grachev et al., for stable conditions, can cover a wider range of stable stratifications. The simulations with CAM5 model using the existing and modified version of surface layer scheme have been performed with 1° resolution for ten years, and the impact of modified functions on the simulation of various important near-surface variables over the tropical region is analyzed. It is found that the scheme with modified functions improving the simulation of surface variables as compared with the existing scheme over the tropical region. In addition, the limitations arbitrarily imposed on particular variables in the existing surface layer scheme can be eliminated or suppressed by using these modified functions.  </p><p>References:</p><p>Fairall CW, Bradley EF, Rogers DP, Edson JB, Young GS (1996) Bulk parameterization of air-sea fluxes for tropical ocean global atmosphere coupled-ocean atmosphere response experiment. J Geophys Res 101(C2):3747–3764</p><p>Grachev, A.A., Andreas, E.L., Fairall, C.W., Guest, P.S. and Persson, P.O.G. (2007a) SHEBA: flux–profile relationships in stable atmospheric boundary layer. Boundary-Layer Meteorology,124, 315–333.</p><p>Keywords:</p><p>Boundary layer, Turbulence, Climate Model, Surface Fluxes</p>

scholarly journals Constraints on Earth’s inner core composition inferred from measurements of the sound velocity of hcp-iron in extreme conditions

2016 ◽  
Vol 2 (2) ◽  
pp. e1500802 ◽  
Author(s):  
Tatsuya Sakamaki ◽  
Eiji Ohtani ◽  
Hiroshi Fukui ◽  
Seiji Kamada ◽  
Suguru Takahashi ◽  
...  
Keyword(s):  
Sound Velocity ◽  
Inner Core ◽  
Earth Model ◽  
Light Elements ◽  
Earth’S Core ◽  
Heated Sample ◽  
Earth's Core ◽  
The Core ◽  
Core Composition ◽  
Earth’S Inner Core

Hexagonal close-packed iron (hcp-Fe) is a main component of Earth’s inner core. The difference in density between hcp-Fe and the inner core in the Preliminary Reference Earth Model (PREM) shows a density deficit, which implies an existence of light elements in the core. Sound velocities then provide an important constraint on the amount and kind of light elements in the core. Although seismological observations provide density–sound velocity data of Earth’s core, there are few measurements in controlled laboratory conditions for comparison. We report the compressional sound velocity (VP) of hcp-Fe up to 163 GPa and 3000 K using inelastic x-ray scattering from a laser-heated sample in a diamond anvil cell. We propose a new high-temperature Birch’s law for hcp-Fe, which gives us the VP of pure hcp-Fe up to core conditions. We find that Earth’s inner core has a 4 to 5% smaller density and a 4 to 10% smaller VP than hcp-Fe. Our results demonstrate that components other than Fe in Earth’s core are required to explain Earth’s core density and velocity deficits compared to hcp-Fe. Assuming that the temperature effects on iron alloys are the same as those on hcp-Fe, we narrow down light elements in the inner core in terms of the velocity deficit. Hydrogen is a good candidate; thus, Earth’s core may be a hidden hydrogen reservoir. Silicon and sulfur are also possible candidates and could show good agreement with PREM if we consider the presence of some melt in the inner core, anelasticity, and/or a premelting effect.

scholarly journals Rotational Velocity of an Earth Model with a Liquid Core

1979 ◽  
Vol 82 ◽  
pp. 313-314
Author(s):  
S. Takagi
Keyword(s):  
Rotational Motion ◽  
Velocity Vector ◽  
Inner Core ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Liquid Core ◽  
Outer Core ◽  
Earth Model ◽  
The Earth ◽  
Rotation Of The Earth

There have been many papers discussing the rotation of the Earth (Jeffreys and Vicente, 1957; Molodenskij, 1961; Rochester, 1973; Smith, 1974; Shen and Mansinha, 1976). This report summarizes the application of the perturbation method of celestial mechanics to calculate the rotation of the Earth (Takagi, 1978). In this solution the Earth is assumed to consist of three components: a mantle, liquid outer core, and a solid inner core, each having a separate rotational velocity vector. Hamiltonian equations of motion were constructed to solve the rotational motion of the Earth.

Viscoelastic Love numbers and long period geophysical effects

2021 ◽  
Author(s):  
A Michel ◽  
J-P Boy
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Earth Tides ◽  
Rheological Parameters ◽  
Earth Model ◽  
Ice Melting ◽  
Long Period ◽  
Love Numbers ◽  
The Laplace Transform ◽  
The Fourier Transform

Summary Long term deformations strongly depend on the Earth model and its rheological parameters, and in particular its viscosity. We give the general theory and the numerical scheme to compute them for any spherically non rotating isotropic Earth model with linear rheology, either elastic or viscoelastic. Although the Laplace transform is classically used to compute viscoelastic deformation, we choose here instead, to implement the integration with the Fourier transform in order to take advantage of the Fast Fourier Transform algorithm and avoid some of the Laplace transform mathematical difficulties. We describe the methodology to calculate deformations induced by several geophysical signals regardless of whether they are periodic or not, especially by choosing an adapted time sampling for the Fourier transform. As examples, we investigate the sensitivity of the displacements due to long period solid Earth tides, Glacial Isostatic Adjustment (GIA), and present-day ice melting, to anelastic parameters of the mantle. We find that the effects of anelasticity are important for long period deformation and relatively low values of viscosities for both Maxwell and Burgers models. We show that slight modifications in the rheological models could significantly change the amplitude of deformation but also affect the spatial and temporal pattern of the signal to a lesser extent. Especially, we highlight the importance of the mantle anelasticity in the low degrees deformation due to present-day ice melting and encourage its inclusion in future models.

scholarly journals Thermal Equation of State of Fe3C to 327 GPa and Carbon in the Core

Minerals ◽  
2019 ◽  
Vol 9 (12) ◽  
pp. 744 ◽  
Author(s):  
Suguru Takahashi ◽  
Eiji Ohtani ◽  
Daijo Ikuta ◽  
Seiji Kamada ◽  
Tatsuya Sakamaki ◽  
...  
Keyword(s):  
High Temperature ◽  
Equation Of State ◽  
Inner Core ◽  
Velocity Structure ◽  
Light Element ◽  
Scale Model ◽  
Earth Model ◽  
Thermal Equation ◽  
The Core ◽  
Pressure Scale

The density and sound velocity structure of the Earth’s interior is modeled on seismological observations and is known as the preliminary reference Earth model (PREM). The density of the core is lower than that of pure Fe, which suggests that the Earth’s core contains light elements. Carbon is one plausible light element that may exist in the core. We determined the equation of state (EOS) of Fe3C based on in situ high-pressure and high-temperature X-ray diffraction experiments using a diamond anvil cell. We obtained the P–V data of Fe3C up to 327 GPa at 300 K and 70–180 GPa up to around 2300 K. The EOS of nonmagnetic (NM) Fe3C was expressed by two models using two different pressure scales and the third-order Birch–Murnaghan EOS at 300 K with the Mie–Grüneisen–Debye EOS under high-temperature conditions. The EOS can be expressed with parameters of V0 = 148.8(±1.0) Å3, K0 = 311.1(±17.1) GPa, K0′ = 3.40(±0.1), γ0 = 1.06(±0.42), and q = 1.92(±1.73), with a fixed value of θ0 = 314 K using the KBr pressure scale (Model 1), and V0 = 147.3(±1.0) Å3, K0 = 323.0(±16.6) GPa, K0′ = 3.43(±0.09), γ0 = 1.37(±0.33), and q = 0.98(±1.01), with a fixed value of θ0 = 314 K using the MgO pressure scale (Model 2). The density of Fe3C under inner core conditions (assuming P = 329 GPa and T = 5000 K) calculated from the EOS is compatible with the PREM inner core.

Ups and Downs

2018 ◽  
Author(s):  
Roy Livermore
Keyword(s):  
Mantle Convection ◽  
Laboratory Experiments ◽  
Inner Core ◽  
Computer Modelling ◽  
Great Depth ◽  
New Horizons ◽  
The Core ◽  
The Earth ◽  
The World ◽  
The 1960S

Despite the dumbing-down of education in recent years, it would be unusual to find a ten-year-old who could not name the major continents on a map of the world. Yet how many adults have the faintest idea of the structures that exist within the Earth? Understandably, knowledge is limited by the fact that the Earth’s interior is less accessible than the surface of Pluto, mapped in 2016 by the NASA New Horizons spacecraft. Indeed, Pluto, 7.5 billion kilometres from Earth, was discovered six years earlier than the similar-sized inner core of our planet. Fortunately, modern seismic techniques enable us to image the mantle right down to the core, while laboratory experiments simulating the pressures and temperatures at great depth, combined with computer modelling of mantle convection, help identify its mineral and chemical composition. The results are providing the most rapid advances in our understanding of how this planet works since the great revolution of the 1960s.

Serial Homology as a Challenge to Evolutionary Theory

2017 ◽  
Author(s):  
Stéphane Schmitt
Keyword(s):  
Biological Sciences ◽  
Evolutionary Theory ◽  
Serial Homology ◽  
The Core ◽  
Long Period ◽  
Colonial Theory ◽  
Experimental Approaches ◽  
Evo Devo ◽  
The 19Th Century ◽  
Development And Evolution

The problem of the repeated parts of organisms was at the center of the biological sciences as early as the first decades of the 19th century. Some concepts and theories (e.g., serial homology, unity of plan, or colonial theory) introduced in order to explain the similarity as well as the differences between the repeated structures of an organism were reused throughout the 19th and the 20th century, in spite of the fundamental changes during this long period that saw the diffusion of the evolutionary theory, the rise of experimental approaches, and the emergence of new fields and disciplines. Interestingly, this conceptual heritage was at the core of any attempt to unify the problems of inheritance, development, and evolution, in particular in the last decades, with the rise of “evo-devo.” This chapter examines the conditions of this theoretical continuity and the challenges it brings out for the current evolutionary sciences.

scholarly journals Development of Depth-Limited Wave Boundary Layers over a Smooth Bottom

2020 ◽  
Vol 9 (1) ◽  
pp. 27
Author(s):  
Hitoshi Tanaka ◽  
Nguyen Xuan Tinh ◽  
Xiping Yu ◽  
Guangwei Liu
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Boundary Layers ◽  
Wave Motion ◽  
Numerical Study ◽  
Experiment Data ◽  
Tidal Motion ◽  
Long Period ◽  
Simulation Results ◽  
Wave Boundary

A theoretical and numerical study is carried out to investigate the transformation of the wave boundary layer from non-depth-limited (wave-like boundary layer) to depth-limited one (current-like boundary layer) over a smooth bottom. A long period of wave motion is not sufficient to induce depth-limited properties, although it has simply been assumed in various situations under long waves, such as tsunami and tidal currents. Four criteria are obtained theoretically for recognizing the inception of the depth-limited condition under waves. To validate the theoretical criteria, numerical simulation results using a turbulence model as well as laboratory experiment data are employed. In addition, typical field situations induced by tidal motion and tsunami are discussed to show the usefulness of the proposed criteria.

scholarly journals The appearance of boundary layers and drift flows due to high-frequency surface waves

2012 ◽  
Vol 707 ◽  
pp. 482-495 ◽  
Author(s):  
Ofer Manor ◽  
Leslie Y. Yeo ◽  
James R. Friend
Keyword(s):  
Boundary Layer ◽  
Surface Waves ◽  
High Frequency ◽  
Slip Boundary Condition ◽  
Inertial Effects ◽  
Velocity Amplitude ◽  
Slip Boundary ◽  
Long Period ◽  
Bulk Waves ◽  
Schlichting Streaming

AbstractThe classical Schlichting boundary layer theory is extended to account for the excitation of generalized surface waves in the frequency and velocity amplitude range commonly used in microfluidic applications, including Rayleigh and Sezawa surface waves and Lamb, flexural and surface-skimming bulk waves. These waves possess longitudinal and transverse displacements of similar magnitude along the boundary, often spatiotemporally out of phase, giving rise to a periodic flow shown to consist of a superposition of classical Schlichting streaming and uniaxial flow that have no net influence on the flow over a long period of time. Correcting the velocity field for weak but significant inertial effects results in a non-vanishing steady component, a drift flow, itself sensitive to both the amplitude and phase (prograde or retrograde) of the surface acoustic wave propagating along the boundary. We validate the proposed theory with experimental observations of colloidal pattern assembly in microchannels filled with dilute particle suspensions to show the complexity of the boundary layer, and suggest an asymptotic slip boundary condition for bulk flow in microfluidic applications that are actuated by surface waves.

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