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.     

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.     

Towards an Indirect Determination of the Mass-balance Distribution of Glaciers using the Kinematic Boundary Condition

The kinematic boundary condition at the surface is utilized to arrive at an estimate of the mass-balance distribution of the ablation region of Unteraargletscher, Bernese Alps, Switzerland. This is achieved without the use of any ground measurements. The terms of the kinematic boundary condition, involving surface-altitude changes with time, surface slopes, and horizontal surface velocities, are determined using high precision aerial photogrammetry. Estimating the vertical velocity distribution along the surface poses a major problem. Different approaches to solving this problem are discussed, and the potential of one particular approach is evaluated. This approach is, in essence, based on the assumption that the variation of vertical strain rates with depth is simple. The accuracy of the resulting indirect estimate of the mass balance distribution is assessed by a comparison with results from stake measurements made at about 40 different points at the surface. Although calculated values of mass balance are found to be within a reasonable range, they differ, as a function of altitude, in a systematic fashion from stake values. This suggests that the vertical strain-rate variation with depth is too complex to be parameterized in a simple manner.     

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.     

The influence of a density stratification and of an incompressibility modulus on the spectrum of core modes

The concept of a deformation of a simple, non-rotating, spherically symmetric earth model with a fluid outer core, although it is a highly artificial physical situation, provides a useful computational algorithm that allows one lo determine analytically modes of vibration without any Love-number theory. In particular, on these analytically determined modes, we impose regularity conditions at the centre and boundary conditions at the surface, as well as conditions of continuity at the inner-core-outer-core boundary and at the core-mantle boundary. They lead to an eigenvalue equation for the frequency of oscillation. The range of frequencies obtained in this way for different earth models gives an indication of the influence of compressibility and non-homogeneity on the spectrum of eigenfrequencies.     

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.     

Some remarks about the degree-one deformation of the Earth

The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the M_o mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity.
The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale).     

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.     

Analytical approach for the toroidal relaxation of viscoelastic earth

This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.     

Tidal deformation of a viscoelastic body

Summary. The tidal deformation of a homogeneous viscoelastic sphere due to the gravitational attraction of an external body is calculated. The sphere is modelled as an incompressible Kelvin-Voigt solid. An equation for the displacement field is obtained assuming that strains are small and inertia is negligible. This equation has a series solution in terms of Legendre polynomials. The resulting expression for the displacement field reduces to that for an elastic solid and a viscous fluid in the appropriate limits of the material constants. The first term in the viscoelastic solution is used to calculate the moments induced by tidal deformation assuming a circular orbit. In the absence of obliquity and precession, these moments reduced to a torque about the spin axis. This torque is compared to that predicted by a phase lag analysis. These two approaches are formally equivalent if the tidal dissipation function Q ⁻¹ depends in a specific way on the difference of the spin and orbital angular velocities.     

基于空间句法理论的城市群核心区发展边界识别

准确识别当前城市群建设进程中核心区发展边界是研究城市群的一项重要内容。本文提出一种研究思路：采用空间句法分析城市群道路网,将得到的4个分析指标融合成新指标——“城市群集群度”,并提取“城市群集群度”等值线和“城市群集群度”曲线,通过计算找到最佳阈值从而提取出城市群核心区发展边界。以长株潭城市群为例,将基于空间句法的研究结果与基于Densi-Graph方法的研究结果进行对比,在除去数据质量因素后,城市群核心区发展边界识别差异有望控制在10%以内。研究表明：基于空间句法理论的城市群核心区发展边界识别方法容易获取计算数据,适用范围广,可靠性强。     

Fluid territories: Rethinking state territorialisation in Palk Bay,South Asia

The study objective was to gain a better understanding of the transboundary fishing conflict between Indian trawl fishers and Sri Lankan small-scale fishers in Palk Bay using a relational approach to territoriality. The authors employed different ethnographic methods, including open, structured, and semi-structured interviews, and performed a media analysis in order to understand the everyday practices of Indian trawl fishers within the wider geopolitical context of a 30-year war in neighbouring Sri Lanka. The relational approach moves away from seeing cross-border fishing merely as an act of counter-territorialisation. The results revealed that the cross-border fishing underlying the crisis has largely resulted from a complex network of changing relationships between on the one hand Indian trawl fishers and India, and on the other hand Sri Lankan state agencies and Sri Lankan fishers, resulting in a porous international maritime boundary. The authors conclude that this in turn has resulted in a fluid international maritime boundary line.     

Simulation of the spontaneous growth of a dynamic crack without constraints on the crack tip path

The spontaneous growth of a dynamic in-plane shear crack is simulated using a newly developed method of analysis in which no a priori constraint is required for the crack tip path, unlike in other classical studies. We formulate the problem in terms of boundary integral equations; the hypersingularities of the integration kernels are removed by taking the finite parts. Our analysis shows that dynamic crack growth is spontaneously arrested soon after the bending of the crack tips, even in a uniformly stressed medium with homogeneously distributed fracture strengths. This shows that the dynamics of crack growth has a significant effect on forming the non-planar crack shape, and consequently plays an essential role in the arrest of earthquake rupturing.     

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.     

Space Deformation Non-stationary Geostatistical Approach for Prediction of Geological Objects: Case Study at El Teniente Mine (Chile)

This article addresses the problem of the prediction of the breccia pipe elevation named Braden at the El Teniente mine in Chile. This mine is one of the world’s largest known porphyry-copper ore bodies. Knowing the exact location of the pipe surface is important, as it constitutes the internal limit of the deposit. The problem is tackled by applying a non-stationary geostatistical method based on space deformation, which involves transforming the study domain into a new domain where a standard stationary geostatistical approach is more appropriate. Data from the study domain is mapped into the deformed domain, and classical stationary geostatistical techniques for prediction can then be applied. The predicted results are then mapped back into the original domain. According to the results, this non-stationary geostatistical method outperforms the conventional stationary one in terms of prediction accuracy and conveys a more informative uncertainty model of the predictions.     

岩溶管道的迁移-弥散输运模式

污染物在岩溶管道中的输运可以用一个迁移–弥散方程结合初始条件和边界条件来描述。笔者首先分别介绍了无限空间中的初值问题和边值问题,然后推导了半无限空间中的初值问题的格林函数。数值示例表明了格林函数在所生成的穿透曲线上呈现有限的偏态性。最后用初值问题的格林函数模拟了Davies等在Florida西北部的Black Creek和Emerald Sink进行的示踪实验。结果表明迁移–弥散模型能模拟实测穿透曲线中的上升部分,但不适合模拟强偏态的下落部分。     

Scientific attributes and expression methods of geographical boundary

A scientific delineation of geographical boundaries reflects the cognitive level of scientific abstraction and systematic analysis of the spatial variation of geographical objects and is a basic scientific issue of geography. From the perspective of earth system science, this study first explicates the core issues (e.g., basic concepts, scientific contents, and basic properties) of geographical boundaries. Based on the principles of scientificity and systematicness, we then classify geographical boundaries in terms of intrinsic mechanisms, extrinsic appearance and scientific attributes. Furthermore, this paper analyzes the mathematical connotation and representation methods of geographical boundaries, discusses the characteristics of and differences between traditional and modern methods for geographical boundary delineation. Finally, we present a framework for a “geographical boundary model” with an integration of qualitative, quantitative, and positioning methods. Focusing on geographical boundary (a basic theoretical problem in geography), this study engaged in concept definition and method analysis, with the findings enriching the theory and methodology of geographical information science.     

Computing the semantic similarity of geographic terms using volunteered lexical definitions

Volunteered geographic information (VGI) is generated by heterogenous ‘information communities’ that co-operate to produce reusable units of geographic knowledge. A consensual lexicon is a key factor to enable this open production model. Lexical definitions help demarcate the boundaries of terms, forming a thin semantic ground on which knowledge can travel. In VGI, lexical definitions often appear to be inconsistent, circular, noisy and highly idiosyncratic. Computing the semantic similarity of these ‘volunteered lexical definitions’ has a wide range of applications in GIScience, including information retrieval, data mining and information integration. This article describes a knowledge-based approach to quantify the semantic similarity of lexical definitions. Grounded in the recursive intuition that similar terms are described using similar terms, the approach relies on paraphrase-detection techniques and the lexical database WordNet. The cognitive plausibility of the approach is evaluated in the context of the OpenStreetMap (OSM) Semantic Network, obtaining high correlation with human judgements. Guidelines are provided for the practical usage of the approach.