Modified theory of disturbances for stationary problems of a charged spherical body in a rarefied plasma

The modified theory of disturbances, which differs from the existent similar theory by the consideration of an additional term in disturbances of concentration and flux density of infinite plasma particles, has been developed for stationary problems of a charged spherical body in a rarefied plasma. It has been indicated that, in the simplest constant centrally symmetric field, the neglect of the above term results in a substantial error in the stationary disturbance of the concentration of infinite particles attracted by this field.     

Magnetic instabilities in rapidly rotating spherical geometries. III. The effect of differential rotation

Abstract

We investigate the influence of differential rotation on magnetic instabilities for an electrically conducting fluid in the presence of a toroidal basic state of magnetic field B ₀ = B_MB₀(r, θ)1 _φ and flow U₀ = U_MU₀ (r, θ)1_φ, [(r, θ, φ) are spherical polar coordinates]. The fluid is confined in a rapidly rotating, electrically insulating, rigid spherical container. In the first instance the influence of differential rotation on established magnetic instabilities is studied. These can belong to either the ideal or the resistive class, both of which have been the subject of extensive research in parts I and II of this series. It was found there, that in the absence of differential rotation, ideal modes (driven by gradients of B ₀) become unstable for A_c ? 200 whereas resistive instabilities (generated by magnetic reconnection processes near critical levels, i.e. zeros of B₀) require A_c ? 50. Here, Λ is the Elsasser number, a measure of the magnetic field strength and Λ_c is its critical value at marginal stability. Both types of instability can be stabilised by adding differential rotation into the system. For the resistive modes the exact form of the differential rotation is not important whereas for the ideal modes only a rotation rate which increases outward from the rotation axis has a stabilising effect. We found that in all cases which we investigated Λ_c increased rapidly and the modes disappeared when R_m ≈ O(Λ_C), where the magnetic Reynolds number R_m is a measure of the strength of differential rotation. The main emphasis, however, is on instabilities which are driven by unstable gradients of the differential rotation itself, i.e. an otherwise stable fluid system is destabilised by a suitable differential rotation once the magnetic Reynolds number exceeds a certain critical value (R_m )_c. Earlier work in the cylindrical geometry has shown that the differential rotation can generate an instability if R_m ) ?O(Λ). Those results, obtained for a fixed value of Λ = 100 are extended in two ways: to a spherical geometry and to an analysis of the effect of the magnetic field strength Λ on these modes of instability. Calculations confirm that modes driven by unstable gradients of the differential rotation can exist in a sphere and they are in good agreement with the local analysis and the predictions inferred from the cylindrical geometry. For Λ = O(100), the critical value of the magnetic Reynolds number (R_m )_c Λ 100, depending on the choice of flow U₀ . Modes corresponding to azimuthal wavenumber m = 1 are the most unstable ones. Although the magnetic field B ₀ is itself a stable one, the field strength plays an important role for this instability. For all modes investigated, both for cylindrical and spherical geometries, (R_m )_c reaches a minimum value for 50 ≈ Λ ≈ 100. If Λ is increased, (R_m )_c ∝ Λ, whereas a decrease of Λ leads to a rapid increase of (R_m )_c, i.e. a stabilisation of the system. No instability was found for Λ ≈ 10 — 30. Optimum conditions for instability driven by unstable gradients of the differential rotation are therefore achieved for ≈ Λ 50 — 100, R_m ? 100. These values lead to the conclusion that the instabilities can play an important role in the dynamics of the Earth's core.     

球坐标系中图形单元应变与旋转张量及其误差解算

利用泰勒级数展开和弹性力学的几何方程,推导了球坐标系中由GPS位移数据解算图形单元应变和旋转张量的解析公式. 通过线性化处理,并利用误差传播定律,详细推导了图形单元应变和旋转张量的误差公式. 利用川滇地区最新的GPS测站位移速率数据,采用图形单元法解算了该区的面应变率及最大剪切应变率分布,并对其进行了初步的分析. 同时阐述了图形单元应变数学模型的局限性,讨论了图形单元法计算应变的意义以及图形单元的选择问题,进一步分析了应变计算的定权方法,还讨论了GPS观测网图形单元的地心半径差异与应变的关系问题.      

Dirichlet's problem for a body limited by two intersecting spherical surfaces

Summary The solution of the internal and external Dirichlet problem is given without assuming rotational symmetry for a body, limited by two intersecting spherical surfaces. The problem is treated in a toroidal system of co-ordinates, using the generalized Mehler integral transform. The results are compared with the solution, presented by Lebedev [1] for the case of rotational symmetry.     

On free nutation of the rotation vector of a rigid body

aam, m ¶rt;-mau ¶rt;uu a ma au am m¶rt; ma m unmuu m m. ¶rt;a, u a u unmuu annuuam, m am n una u 0,1 u a u ammuuu ¶rt;mau aa m m.     

Computation of strain and rotation tensor as well as their uncertainties for small arrays in spherical coordinate system

Based on Taylor series expansion and strain components expressions of elastic mechanics, we derive formulae of strain and rotation tensor for small arrays in spherical coordinates system. By linearization process of the formulae, we also derive expressions of strain components and Euler vector uncertainties respectively for subnets using the law of error propagation. Taking GPS velocity field in Sichuan-Yunnan area as an example, we compute dilation rate and maximum shear strain rate field using the above procedure, and their characteristics are preliminarily car- ried on. Limits of the strain model for small array are also discussed. We make detailed explanations on small array method and the choice of small arrays. How to set weights of GPS observations are further discussed. Moreover relationship between strain and radius of GPS subnets is also analyzed.     

Dynamics of escape of high-energy electrons from a spherical plasma volume in the geomagnetic field

Using results of numerical modeling, the dynamics of escape of electrons, produced as a result beta-decay, to the external magnetic field from a spherical plasma volume with an expelled magnetic field is studied. The dependence of the fraction of escaped electrons on time and radius of the plasma volume has been obtained for two kinds of electron sources: a point isotropic source, located at the center of the sphere, and a volume isotropic source. It is shown that for a point source some part of electrons remains in the sphere, whereas for a volume source all electrons, at different values of the magnetic cavern radius, leave it at an identical relative escape rate.     

磁偶极子构造法球形磁性矿体磁异常正演

球形磁性地质体是地质勘探中广泛遇到的基本地质体之一,选择合适的磁异常分析方法对其磁场进行准确正演计算具有重要的意义.为了准确快速地求解球形磁性地质体磁场特征,本文提出了一种基于磁偶极子构造原理求解球形磁性地质体磁场的数值计算方法,对球形磁性地质体按半径由小增大进行多层单元分割,再对每层单元进行块单元分割,将每个块单元视为"磁偶极子",利用基本磁偶极子公式计算了所有块单元在探测平面(线)的磁场大小,同时分析了球体磁场的剖面特征,并将数值解与解析解结果进行了对比和误差分析,最后建立双球组合模型结合Comsol多物理场仿真工具对其仿真对比,验证了磁偶极子构造法求解球体磁场的准确性.研究结果表明:数值解法结果与解析解结果磁异常波动趋势完全一致;不同测点处绝对误差有所差别,但磁异常值均在1 nT以下.本文提出的数值解法无需复杂的数学推导,计算结果稳定可靠.     

球谐旋转变换结合非全次Legendre方法的局部六边形网格重力场球谐综合

本文首次提出基于六边形网格剖分的全球重力场结构,并解决了局部六边形网格点模型重力异常快速计算问题.首先,采用全新的方法给出缔合Legendre函数值从稳定振荡区到快速衰减区分界线的理论表达式,并基于该公式提出一种基于跨阶次递推的非全次Legendre方法,实现了高纬度地区点的快速球谐综合.其次,引入球谐旋转（Spherical Harmonic Rotation）理论,实现了2160阶次的球谐系数在坐标系旋转下的变换,结合非全次Legendre方法,解决了中低纬度地区点的快速球谐综合.通过计算南极洲（高纬）低分辨率和加里曼丹岛（低纬）高分辨率六边形网格重力异常表明,非全次Legendre方法以10^-19m·s^-2精度水平与传统全阶次方法计算结果吻合,且计算效率提升1倍多,旋转变换结合非全次Legendre方法的计算精度在10^-16m·s^-2,效率提升近5倍.本文提出的方法不仅提升了球谐综合的计算效率,凡是有高纬度的缔合Legendre函数计算的问题,都可利用该方法提升效率,同时,超高阶次球谐旋转变量变换的实现将在地磁场模型构建、计算机视觉、量子物理等领域发挥重要作用.

     

Three-dimensional convection in spherical shells

Abstract

In this study, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number. An iterative method is used to find stationary solutions. The spherical parts of the operators are treated using a Galerkin collocation method while the radial and time dependences are expressed using finite difference methods. A systematic search for stationary solutions has led to eight different stream patterns for a low Rayleigh number (1.28 times the critical number). They can be classified as:

I) Axisymmetrical solutions, analogous to rolls in plane geometry.

II) Solutions which have several ascending plumes within a large area of ascending current, and also several descending plumes within an area of descending current. This type of flow is analogous to bimodal circulation in plane geometry.

III) Solutions characterized by isolated ascending (or descending) plumes separated from each other by a closed polyhedral network of descending (or ascending) currents. This type of circulation is called ‘polygonal’ in analogy with hexagonal circulation in plane geometry.

The behaviour of each of the eight solutions has been studied by increasing the Rayleigh number up to 15 times the critical number. A trend towards transitions from type (I) and type (II) solutions to type (III) solutions is observed. It is inferred that only the “polygonal” solutions are stable for a Rayleigh number greater than 15 times the critical number.     

Poles of rotation

One of the concepts introduced into the theory of plate tectonics is that of poles of rotation. It is supposed that the relative motion between two rigid plates can be described by the rotation of one of the plates about a pole of rotation, which shows the direction of the rotation vector. It has been pointed out that in general poles of rotation will not remain fixed. The consequences of poles of rotation which move slowly with time are examined. In particular, it is shown that fracture zones may lie very close to small circles, but that the centers of these small circles may lie as much as 20° away from the average pole of rotation. The establishment of poles of rotation from individual fracture zones is thus an inaccurate method unless it can also be shown that the pole of rotation has remained fixed.     

Geomagnetic spherical harmonic analysis

Summary An attempt is made to determine the optimum representation of the geomagnetic field in terms of spherical harmonics with regard to the number of parameters required for an adequate accuracy of fit, and considering the effect of allowing for the oblateness of the earth. A representative set of geomagnetic data was subjected to a set of analyses with from 8 to 120 coefficients for both spherical and oblate models of the earth. It is concluded that a sixth order analysis (48 coefficients) with no allowance for oblateness is the most compact representation of the field compatible with tolerable accuracy.     

Finite prandtl number convection in spherical shells

Abstract

Finite amplitude convection in spherical shells with spherically symmetric gravity and heat source distribution is considered. The nonlinear problem of three-dimensional convection in shells with stress-free and isothermal boundaries is solved by expanding the dependent variables in terms of powers of the amplitude of convection. The preferred mode of convection is determined by a stability analysis in which arbitrary infinitesimal disturbances are superimposed on the steady solutions. The shell is assumed to be thick and only shells for which the ratio ζ of outer radius to inner radius is 2 or 3 are considered. Three cases, two of which lead to a self adjoint problem, are treated in this paper. The stable solutions are found to be l=2 modes for ζ=3 where l is the degree of the spherical harmonics and an l=3 non-axisymmetric mode which exhibits the symmetry of a tetrahedron for ζ=2. These stable solutions transport the maximum amount of heat. The Prandtl number dependence of the heat transport is computed for the various solutions analyzed in the paper.     

Seismic decoupling for explosions in spherical cavities

Summary A series of paired explosions in a salt mine near Winnfield, Louisiana, has been conducted to test a theory by Dr.A. L. Latter concerning seismic decoupling by underground cavities. The theory predicted a decoupling of about 100. Free-field and surface measurements from an explosion in either a 6-ft-or a 15-ft-radius spherical cavity were compared with similar measurements from a completely tamped explosion. Shot sizes were from 20 Ib up to a few tons. Surface measurements were made out to 100 km and covered the frequency range from 0.5 to 100 cps. The experiment confirmed that decoupling does occur. The actual decoupling factor as a function of frequency is presented and compared with theLatter theory.Presented at the international meeting of the International Union of Geodesy and Geophysics at Helsinki, July 27, 1960.     

Kinematic dynamo action in spherical Couette flow

We investigate numerically kinematic dynamos driven by flow of electrically conducting fluid in the shell between two concentric differentially rotating spheres, a configuration normally referred to as spherical Couette flow. We compare between axisymmetric (2D) and fully 3D flows, between low and high global rotation rates, between prograde and retrograde differential rotations, between weak and strong nonlinear inertial forces, between insulating and conducting boundaries and between two aspect ratios. The main results are as follows. Azimuthally drifting Rossby waves arising from the destabilisation of the Stewartson shear layer are crucial to dynamo action. Differential rotation and helical Rossby waves combine to contribute to the spherical Couette dynamo. At a slow global rotation rate, the direction of differential rotation plays an important role in the dynamo because of different patterns of Rossby waves in prograde and retrograde flows. At a rapid global rotation rate, stronger flow supercriticality (namely the difference between the differential rotation rate of the flow and its critical value for the onset of nonaxisymmetric instability) facilitates the onset of dynamo action. A conducting magnetic boundary condition and a larger aspect ratio both favour dynamo action.     

地球的差异旋转

对于具有流体对流层的旋转星球 ,由于星球自转对对流的影响 ,必然会在星球对流层内部不同部分间以及星球的不同圈层间产生差异旋转 (differentialrotation) .所谓差异旋转是指旋转角速度随着深度 (星球不同圈层的旋转角速度不同 )以及纬度 (同一圈层内部不同部分间的角速度不同 )具有差异的现象 .地球是一个多圈层的旋转系统 ,主要由大气圈、水圈、岩石圈、地幔、外核以及内核组成 .大气圈和水圈具有明显的流体性质 ,并且在漫长的地质年代中 ,地幔、岩石圈和地幔之间的软流层以及外核均具有流体性质 ,而且在大气压力、热、重力和电磁力等的作用下发生了对流 .这些对流运动一旦受到地球自转的影响 ,就必然会致使地球各圈层间以及对流层内部不同部分间产生差异旋转 .几个重要现象 :基本地磁场的长期西漂、岩石圈的长期西漂、地球自转速率变化 (周日长度波动 )和固体内核各向异性对称轴的移动表明在固体地球内部各圈层—岩石圈、地幔、外核和内核间存在差异旋转 .来自地震学上的数据证明了固体内核与地幔之间存在较明显的差异旋转 ,速率可达 1.1°～ 3.0°/a.这跟其它数据如自由振荡数据以及地磁场长期西漂数据获得的结果具有很大的差异 .导致固体地球内核相对于地幔差异旋转的主要宏观机制为电磁力矩、引     

Atmosphere and Earth's rotation

The theoretical aspects of the transfer of angular momentum between atmosphere and Earth are treated with particular emphasis on analytical solutions. This is made possible by the consequent usage of spherical harmonics of low degree and by the development of large-scale atmospheric dynamics in terms of orthogonal wave modes as solutions of Laplace's tidal equations.An outline of the theory of atmospheric ultralong planetary waves is given leading to analytical expressions for the meridional and height structure of such waves. The properties of the atmospheric boundary layer, where the exchange of atmospheric angular momentum with the solid Earth takes place, are briefly reviewed. The characteristic coupling time is the Ekman spin-down time of about one week.The axial component of the atmospheric angular momentum (AAM), consisting of a pressure loading component and a zonal wind component, can be described by only two spherical functions of latitude : the zonal harmonicP ₂ ⁰ (), responsible for pressure loading, and the spherical functionP ₁ ¹ () simulating supperrotation of the zonal wind. All other wind and pressure components merely redistributeAAM internally such that their contributions toAAM disappear if averaged over the globe. It is shown that both spherical harmonics belong to the meridional structure functions of the gravest symmetric Rossby-Haurwitz wave (0, –1)^*. This wave describes retrograde rotation of the atmosphere within the tropics (the tropical easterlies), while the gravest symmetric external wave mode (0, –2) is responsible for the westerlies at midlatitudes. Applying appropriate lower boundary conditions and assuming that secular angular momentum exchange between solid Earth and atmosphere disappears, the sum of both waves leads to an analytical solution of the zonal mean flow which roughly simulates the observed zonal wind structure as a function of latitude and height. This formalism is used as a basis for a quantitative discussion of the seasonal variations of theAAM within the troposphere and middle atmosphere.Atmospheric excitation of polar motion is due to pressure loading configurations, which contain the antisymmetric functionP ₂ ¹ () exp(i) of zonal wavenumberm=1, while the winds must have a superrotation component in a coordinate system with the polar axis within the equator. The Rossby-Haurwitz wave (1, –3)^* can simulate well the atmospheric excitation of the observed polar motion of all periods from the Chandler wobble down to normal modes with periods of about 10 days. Its superrotation component disappears so that only pressure loading contributes to polar motion.The solar gravitational semidiurnal tidal force acting on the thermally driven atmospheric solar semidiurnal tidal wave can accelerate the rotation rat of the Earth by about 0.2 ms per century. It is speculated that the viscous-like friction of the geomagnetic field at the boundary between magnetosphere and solar wind may be responsible for the westward drift of the dipole component of the internal geomagnetic field. Electromagnetic or mechanical coupling between outer core and mantle may then contribute to a decrease of the Earth's rotation rate.     

Investigation of array-derived rotation in TAIPEI 101

We investigate rotational motions derived from measurements by arrays of translational seismometers??hereafter called array-derived rotations (ADRs)??and compare these to measurements made by a commercially available point rotation sensor (eentec? R-1?). We focus on two aspects of the array problem: (1) the requisite conditions for calculating an ADR well and (2) the effect of array configuration on the result. Our data set consists of translational accelerations and rotation rates recorded by an array of Kinemetrics? EpiSensor? accelerometers and two R-1? rotational sensors in the TAIPEI 101 building in Taipei, Taiwan. Our results indicate that (1) array configuration affects the accuracy of ADRs about orthogonal components in horizontal plane, (2) coherence between two point rotation measurements (two R-1?) can determine the maximum frequency of translations viably used for the calculating ADRs, and (3) the performance of the R-1? is adequate, at least above a frequency of 0.12?Hz (periods shorter than 8?s). We also discuss deriving strain from the same array.