Anisotropic turbulence in weakly stratified rotating magnetoconvection

Numerical simulations of the 3-D MHD-equations that describe rotating magnetoconvection in a Cartesian box have been performed using the code NIRVANA. The characteristics of averaged quantities like the turbulence intensity and the turbulent heat flux that are caused by the combined action of the small-scale fluctuations are computed. The correlation length of the turbulence significantly depends on the strength and orientation of the magnetic field and the anisotropic behavior of the turbulence intensity induced by Coriolis and Lorentz force is considerably more pronounced for faster rotation. The development of isotropic behavior on the small scales—as it is observed in pure rotating convection—vanishes even for a weak magnetic field which results in a turbulent flow that is dominated by the vertical component. In the presence of a horizontal magnetic field the vertical turbulent heat flux slightly increases with increasing field strength, so that cooling of the rotating system is facilitated. Horizontal transport of heat is always directed westwards and towards the poles. The latter might be a source of a large-scale meridional flow whereas the first would be important in global simulations in case of non-axisymmetric boundary conditions for the heat flux.     

The magnetic structure of convection-driven numerical dynamos

The generation of a magnetic field in numerical simulations of the geodynamo is an intrinsically 3-D and time-dependent phenomenon. The concept of magnetic field lines and the frozen-flux approximation can provide insight into such systems, but a suitable visualization method is required. This paper presents results obtained using the Dynamical Magnetic Field line Imaging (DMFI) technique, which is a representation of magnetic field lines accounting for their local magnetic energy, together with an algorithm for the time evolution of their anchor points. The DMFI illustrations are consistent with previously published dynamo mechanisms, and allow further investigation of spatially and temporally complex systems. We highlight three types of magnetic structures: (i) magnetic cyclones and (ii) magnetic anticyclones are expelled by, but corotate with axial flow vortices; (iii) magnetic upwellings are amplified by stretching and advection within flow upwellings, and show structural similarity with helical plumes found in rotating hydrodynamic experiments. While magnetic anticyclones are responsible for the regeneration of a stable axial dipole, here we show that excursions and reversals of the dipole axis are caused by the emergence of magnetic upwellings, which amplify and transport a generally multipolar magnetic field from the inner to the outer boundary of the models. Geomagnetic observations suggest the presence of magnetic structures similar to those found in our models; thus, we discuss how our results may pertain to Earth's core dynamo processes. In order to make DMFI a standard tool for numerical dynamo studies, a public software package is available upon request to the authors (supplementary material is available at: http://www.ipgp.jussieu.fr/~aubert/DMFI.html ).     

Studies of partial rotational remanent magnetization and rotational remanent magnetization at slow speeds of rotation

Summary. Rotational remanent magnetizations and partial rotational remanent magnetizations have been induced in four specimens using alternating magnetic fields of 55 mT maximum peak strength and 128 Hz, and speeds of rotation between 0.0016 and 0.4 rev s⁻¹. Each partial rotational remanent magnetization ( PRRM ), was produced by rotating the specimen only at the maximum setting of the alternating field. The variation of PRRM with (a) speed of rotation, ω, and (b) total angle of rotation, θ, was investigated. In (a), PRRM fell slowly but steadily as ω increased; for (b) it rose sharply as θ increased up to 60° and reached a maximum for θ between 90° and 120°. Alternating field demagnetizations of PRRMs were performed with the specimen (a) at rest, and (b) rotating about an axis perpendicular to the field. Rotation significantly enhanced the demagnetization process. Variation of the time T , taken to remove the inducing alternating field produced no detectable effect in the case of PRRM , but affected the value of ω at which a given feature of the RRM —ω curve appeared, and the product θ_F(=ω T ) appears to be more important than either ω or T separately. Current theories on RRM can be used to explain some of the new experimental data on PRRM .     

Forward modelling of direct current and low-frequency electromagnetic fields using integral equations

We present a semi-analytical, unifying approach for modelling the electromagnetic response of 3-D bodies excited by low-frequency electric and magnetic sources. We write the electric and magnetic fields in terms of power series of angular frequency, and show that to obey Maxwell's equations, the fields must be real when the exponent is even, and imaginary when it is odd. This leads to the result that the scattering equations for direct current fields and for fields proportional to frequency can both be explicitly formulated using a single, real dyadic Green's function. Although the underground current flow in each case is due to different physical phenomena, the interaction of the scattering currents is of the same type in both cases. This implies that direct current resistivity, magnetometric resistivity and electric and magnetic measurements at low induction numbers can all be modelled in parallel using basically the same algorithm. We make a systematic derivation of the quantities required and show that for these cases they can all be expressed analytically. The problem is finally formulated as the solution of a system of linear equations. The matrix of the system is real and does not depend on the type of source or receiver. We present modelling results for different arrays and apply the algorithm to the interpretation of field data. We assume the standard dipoledipole resistivity array for the direct current case, and vertical and horizontal magnetic dipoles for induction measurements. In the case of magnetometric resistivity we introduce a moving array composed of an electric dipole and a directional magnetometer. The array has multiple separations for depth discrimination and can operate in two modes. The mode where the predominant current flow runs along the profile is called MMR-TM. This mode is more sensitive to lateral variations in resistivity than its counterpart, MMR-TE, where the mode of conduction is predominantly perpendicular to the profile.     

Guided waves in three-dimensional structures

A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal field: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in different directions in the horizontal plane. The influence of 3-D structure is included by allowing coupling between different modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial differential equations for the weight functions for different modal orders.
The representation of the guided wavefield requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as different polarization states of the modal spectrum. The coupling equations describe the interaction between the different polarizations induced by the presence of the 3-D structure.
The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or reflection of the surface wavefield can be included by allowing for transfer between modes travelling in different directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation.     

Directions of anhysteretic and rotational remanent magnetizations in rotating samples

Summary. These experiments support Stephenson's predictions that partial anhysteretic remanent magnetizations produced in rotating samples deviate from the steady field direction, which is collinear with the rotation axis, by amounts depending on the angle, between the alternating magnetic field axis and the rotation axis. A similar effect was observed for partial rotational remanent magnetizations. Possible differences between the two remanence types were noted.     

Magnetic and viscous coupling at the core–mantle boundary: inferences from observations of the Earth's nutations

Dissipative core–mantle coupling is evident in observations of the Earth's nutations, although the source of this coupling is uncertain. Magnetic coupling occurs when conducting materials on either side of the boundary move through a magnetic field. In order to explain the nutation observations with magnetic coupling, we must assume a high (metallic) conductivity on the mantle side of the boundary and a rms radial field of 0.69 mT. Much of this field occurs at short wavelengths, which cannot be observed directly at the surface. High levels of short-wavelength field impose demands on the power needed to regenerate the field through dynamo action in the core. We use a numerical dynamo model from the study of Christensen & Aubert (2006) to assess whether the required short-wavelength field is physically plausible. By scaling the numerical solution to a model with sufficient short-wavelength field, we obtain a total ohmic dissipation of 0.7–1 TW, which is within current uncertainties. Viscous coupling is another possible explanation for the nutation observations, although the effective viscosity required for this is 0.03 m² s⁻¹ or higher. Such high viscosities are commonly interpreted as an eddy viscosity. However, physical considerations and laboratory experiments limit the eddy viscosity to 10⁻⁴ m² s⁻¹, which suggests that viscous coupling can only explain a few percent of the dissipative torque between the core and the mantle.     

The magnetotelluric dispersion relations over 2-D structures

The causal properties of the magnetotelluric response, first derived by Weidelt, and Fischer & Schnegg in the 1-D limit, are confirmed numerically in the 2-D limit for the particular structure consisting of two uniform quarter spaces. This finding is of interest from several points of view: (1) the dispersion relations in their 1-D form seem to retain their validity for all 2-D tensor elements, irrespective of any rotation of the coordinate system chosen at the surface; in particular, (2) they appear equally valid for the E- and B-polarization configurations. (3) Whereas the question recently debated in the literature (whether the Earth can always be considered as a linear, passive and causal system) is perhaps not yet entirely resolved, the present empirical demonstration suggests that it may in general be safe to apply the 1-D dispersion relations to structures with 2-D character.     

Seismic scattering in the upper crystalline crust based on evidence from sonic logs

Telluric distortion occurs when electric charges accumulate along near-surface inhomogeneities. At low frequencies, the electric currents associated with these charges can be neglected compared to currents induced deeper in the Earth. At higher frequencies, the magnetic fields associated with these currents may be significant. Some parameters describing the distortion magnetic fields can be estimated from measured magneto-telluric impedance matrices. For regional magnetic fields aligned with regional strike directions, parameters associated with the distortion magnetic field component parallel to the regional magnetic field are undeterminable, whereas parameters associated with the distortion magnetic field component perpendicular to the regional magnetic field can be estimated. Optimal estimates are straightforward even for the realistic case of measurement errors that are correlated between elements of a measured impedance matrix. In a simple example of a 1-D anisotropic model with anisotropy direction varying with depth, the modelling of distortion magnetic fields results in regional impedance estimates corresponding more closely to the responses of uncoupled isotropic models, allowing sensible interpretation of an additional one and a half decades of data.     

A normal mode expansion for the forced response of a rotating earth

Summary. It is well known that the forced motion of a conservative physical system can be represented as a sum of the normal modes of that system. Although normal mode expansions have been successfully developed and applied to examine the excitation of a non-rotating earth, these techniques have been less successful when the Earth's rotation is included. We develop here an expansion formalism for the rotating Earth which completely decouples all non-trivial normal modes. We consider the response of the Earth both to applied body forces and surface tractions. We consider forces which are purely sinusoidal in time and forces which have power distribution over a continuous frequency interval.     

Symmetry of the wave equation and excitation of body waves

Summary. The symmetry of the differential system for elastic waves, previously noted for plane geometry, is extended to any linear differential system and, in particular, the elastic-gravitational vibrations in a spherical earth. The result remains valid in a linearly viscoelastic medium. The symmetry allows the inverse of the propagator matrix to be obtained by simply 'transposing' the elements of the propagator. With this result, it is shown how the source excitation using a particular integral can be put in a more instructive form, comparable with the result for the excitation of normal modes.     

Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media

An algorithm for the numerical modelling of magnetotelluric fields in 2-D generally anisotropic block structures is presented. Electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor. The problem leads to a coupled system of partial differential equations for the strike-parallel components of the electromagnetic field. E _x, and H _x These equations are numerically approximated by the finite-difference (FD) method, making use of the integro-interpolation approach. As the magnetic component H _x, is constant in the non-conductive air, only equations for the electric mode are approximated within the air layer. The system of linear difference equations, resulting from the FD approximation, can be arranged in such a way that its matrix is symmetric and band-limited, and can be solved, for not too large models, by Gaussian elimination. The algorithm is applied to model situations which demonstrate some non-trivial phenomena caused by electrical anisotropy. In particular, the effect of 2-D anisotropy on the relation between magnetotelluric impedances and induction arrows is studied in detail.     

Separation of local and regional information in distorted GDS response functions by hypothetical event analysis

Electromagnetic investigations are usually intended to examine regional structures where induction takes place at a given period range. However, the regional information is often distorted by galvanic effects at local conductivity boundaries. Bahr (1985) and Groom & Bailey (1989) developed a physical distortion model for decomposing the MT impedance tensor, based upon local galvanic distortion of a regional 2-D electromagnetic field. We have extended their method to predict the magnetic variation fields created at an array of sites. The magnetic response functions at periods around 1000 s may be distorted by large-scale inhomogeneities in the upper or middle crust. In this period range, the data measured by a magnetometer array contain common information that can be extracted if the data set is treated as a unit, for example by using hypothetical event analysis. With this technique it is always possible to recover the regional strike direction from distorted data, even if a strong, spatially varying regional vertical field component is present in the data set. The determination of the regional impedance phases, on the other hand, is far more sensitive to deviations from the physical distortion model.
The approach has been used to investigate the Iapetus data set. For the array, which covers an area of 200  km × 300  km in northern England/southern Scotland, the technique revealed a common regional strike azimuth of ca . N125° E in the period range 500–2000  s. This direction differs from the strike indicated by the induction arrows, which seem influenced mainly by local current concentrations along the east–west-striking Northumberland Trough and a NE–SW-striking mid-crustal conductor. Both impedance phases are positive and differ by ca . 10°, which supports the assumptions of distortion fields in the data set and that the regional structure is 2-D.     

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 model for energy dissipation at the mantle—core boundary

Summary. An existing experimentally verified model for energy dissipation in a processing spherical cavity filled with liquid assumed to be in a semirigidized state except for a viscous Ekman boundary layer is applied to the Earth's liquid core to assess energy dissipation magnitudes. Application of the model to the best available Earth data occurs at the derived energy dissipation maximum for the model. Other existing research showing that the Earth's atmosphere appears to adjust to a state of maximum dissipation led to generic models for systems of maximum dissipation. The maximum dissipation mantle—core model with core motion driven by Earth precession alone, coupled to the mantle only by viscous shear stresses, and with a spherical mantle—core boundary leads to energy dissipation rates on the order of 10⁴ times those necessary for an Earth dynamo. The maximum dissipation model also leads to excessive magnetic field drift rates and to excessive retardation of the Earth's rotation rate. Effects of the mantle—core ellipticity and of magnetic field coupling are briefly discussed and are used to help develop a less than maximum dissipation model also driven by precession alone but using the additional coupling to yield a model more consistent with observed phenomena.     

风沙运动中MAGNUS效应的数值研究

在风沙运动中,跃移沙粒一般都会伴随高速旋转,同时引起一种升力效应,即Magnus效应。采用数值模拟的方法计算了0.1Re400,转速ω为100~1000 rev.s-1范围时均匀来流中单个球形沙粒受到的Magnus力,并与Rubinow和Keller推导的Magnus力公式进行了比较。计算结果表明,在0.1Re200范围内,升力和沙粒的旋转速度成正比,随雷诺数的增大而减小。同时,Magnus效应和固定沙粒在均匀来流中的流动结构也有关系,即计算结果和Magnus力公式的升力系数比K在沙粒流动是附体的时随Re数增加逐渐减小,在Re=20~30时达到最小值,然后在沙粒流动是对称分离的时K随着Re数增加逐渐增大。当雷诺数继续增大到超过200时,沙粒本身流动状态非对称所引起的升力超过Magnus效应产生的升力,但Magnus效应的作用也不可以忽略。根据计算结果,对Magnus力公式进行了修正。另外,还发现流场速度梯度与Magnus效应并没有耦合作用,速度梯度对升力的影响可以和Magnus效应线性叠加。     

Effect of 3-D viscoelastic structure on post-seismic relaxation from the 2004 M= 9.2 Sumatra earthquake

The 2004 M = 9.2 Sumatra–Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress–strain relation; an aspherical perturbation in viscoelastic structure; a 'static' mode basis set consisting of Earth's spheroidal and toroidal free oscillations; a "viscoelastic" mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra–Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth's surface. Comparison with post-seismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.     

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.     

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.     

中国国内旅游流空间场效应分析

从空间场效应的角度采用因子分析法,利用1999-2003年中国国内旅游抽样调查资料,分析国内旅游流的产生、分配、集聚、扩散的基本特征以及旅游流空间网络的相关性。结果表明:(1)经济是旅游流产生的主要驱动力,旅游资源是旅游流空间分配的主要驱动力; 目前国内已形成4大旅游流产生地和5大旅游流集聚地;(2)旅游集聚场与扩散场的等级结构受市场原则作用,表现为K=3序列的空间模式;(3)旅游集聚场地域结构呈"U"型特征,而旅游扩散场呈"三级阶梯"状格局;(4)β、γ指数揭示旅游流空间网络相关性较差。今后加快长江、黄河流域旅游带以及南部沿海、北部沿海和东北旅游区等"二带三区"的旅游网络化建设,是我国推进区域旅游合作的主要方向和重点区域,是实现旅游流空间流向合理调整与流量结构优化的关键。