Axially asymmetric velocities in the boundary layer of the nearly symmetric hydromagnetic dynamo

Abstract

An attempt has been made to include the axially asymmetric velocities into the calculation of Braginsky's Z-model of the nearly symmetric hydromagnetic dynamo. In this axisymmetric non-linear model dominated by Lorentz and Coriolis forces and maintained by a specified convection, the α-effect is prescribed. An example is shown of the axially asymmetric Archimedean buoyancy, which can imply an arbitrary alpha effect in the model with viscous core-mantle coupling. The formalisms of Tough and Roberts (1968) is also discussed and a modified α-effect in the Z-model is suggested.     

Magnetic field at the core boundary in the nearly symmetric hydromagnetic dynamo z

Summary The behaviour of the poloidal and toroidal magnetic field at the core-mantle boundary is analysed in more detail, assuming that the conductive layer in the lowest mantle is thin. We can conclude that, in the case of the Z-model of the nearly symmetric hydromagnetic dynamo, the poloidal field may be considered potential everywhere in the mantle and that the azimuthal field depends on the geostrophic azimuthal velocity in the same manner as derived in[1] and[3].

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Time-dependent electromagnetic core-mantle coupling

The magnetic field in the Earth's mantle is computed using a depth-dependent electrical conductivity, of form σ = σ_a(r/a)^?α, and an approximation scheme in which the electromagnetic time constant of the mantle is assumed small compared with the time scales of the secular variation, and in which the induced currents and fields are obtained iteratively. We first associate the toroidal fields in the mantle with motions at the core surface (r = a) which create the observed geomagnetic field by flux rearrangement, and compute the resulting couple, Γ, parallel to the geographical axis. Using only zonal core motions, and values σ_a = 3 × 10³ω^?1m^?1, α = 30 for the conductivity profile, we find that the toroidal induced fields create a couple, Γ_T, that over most of this century has been roughly ten times greater than the poloidal part, Γ_S, of Γ, and has the same sign. The total couple, Γ, has fluctuations of order 10¹⁸ Nm as required for the observed decade fluctuations in the length of the day. Its average is ～ ?1.5 × 10¹⁸ Nm, i.e., it is too large to remain unbalanced. We suppose that an equally important couple in the opposite sense is created by flux leakage from the core, and we estimate the necessary gradient of toroidal field in the core to be of order ?0.5 Gs km^?1 at the core surface. During the course of the data analysis needed for the present work, we found some evidence for a torsional wave in the Earth's core with a period of ～ 60 y.     

Asymmetry in a nearly symmetric dynamo

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On the nearly axially-symmetrical model of the hydromagnetic dynamo of the earth

A short review of the present state of the nearly axially-symmetrical dynamo model is given. A simplified theory for hydromagnetic dynamos taking into account the forces acting in the Earth's core is considered. The role of weak core-mantle friction is discussed and a form of solution is suggested which is characterized by a large geostrophic velocity in the core and by a boundary layer of a new type. The consequences of such a model (called model Z) for the Earth's dynamo are discussed.     

Boundary conditions in the hydromagnetic dynamo problem

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Flow in the earth's core and the hydromagnetic dynamo

Zusammenfassung Den Hauptgegenstand des Artikels bildet die Auswahl von Geschwindigkeiten für ein kinematisches Modell eines hydromagnetischen Dynamos. Die Ergebnisse der theoretischen Analyse betreffend die M?glichkeit der Entwicklung des magnetischen Feldes werden durch die Angaben über die Oberfl?chenstr?mung an der Grenze zwischen dem Erdkern und dem Mantel erg?rzt, die man aus der beobachteten sekul?ren ?nderung des magnetischen Feldes der Erde erwcrben hat. Das dreidimensionale Geschwindigkeitsfeld wurde in der Weise gew?hlt, damit die notwendige Bedingung der Entwicklung, d.i. das Nichtverschwinden des Braginskischen Entwicklungskoeffizienten, erfüllt werde und damit der Charakter von Str?mung auf der Kernoberfl?che den aus der sekul?ren ?nderung gewonnenen Angaben entspreche. Eine m?glichen Form des dreidimensionalen Geschwindigkeitsfeldes wird im Modell des hydromagnetischen Dynamos angewendet, das durch ein System von Integralgleichungen dargestellt wird. Die vorausgesetzte numerische L?sung ist nicht durchgeführt.

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Address: Boční II, Praha 4-Spořilov.     

On the theory of core-mantle coupling

This article commences by surveying the basic dynamics of Earth's core and their impact on various mechanisms of core-mantle coupling. The physics governing core convection and magnetic field production in the Earth is briefly reviewed. Convection is taken to be a small perturbation from a hydrostatic, “adiabatic reference state” of uniform composition and specific entropy, in which thermodynamic variables depend only on the gravitational potential. The four principal processes coupling the rotation of the mantle to the rotations of the inner and outer cores are analyzed: viscosity, topography, gravity and magnetic field. The gravitational potential of density anomalies in the mantle and inner core creates density differences in the fluid core that greatly exceed those associated with convection. The implications of the resulting “adiabatic torques” on topographic and gravitational coupling are considered. A new approach to the gravitational interaction between the inner core and the mantle, and the associated gravitational oscillations, is presented. Magnetic coupling through torsional waves is studied. A fresh analysis of torsional waves identifies new terms previously overlooked. The magnetic boundary layer on the core-mantle boundary is studied and shown to attenuate the waves significantly. It also hosts relatively high speed flows that influence the angular momentum budget. The magnetic coupling of the solid core to fluid in the tangent cylinder is investigated. Four technical appendices derive, and present solutions of, the torsional wave equation, analyze the associated magnetic boundary layers at the top and bottom of the fluid core, and consider gravitational and magnetic coupling from a more general standpoint. A fifth presents a simple model of the adiabatic reference state.     

On the topographic coupling at the core-mantle interface

The mean tangential stresses at a corrugated interface between a solid, electrically insulating mantle and a liquid core of magnetic diffusivity λ are calculated for uniform rotation of both mantle and core at an angular velocity Ω in the presence of a corotating magnetic field B. The core and mantle are assumed to extend indefinitely in the horizontal plane. The interface has the form z = η(x, y), where z is the upward vertical distance and x, y are the zonal and latitudinal distances respectively. The function η(x, y) has a planetary horizontal length scale (i.e. of the order of the radius of the Earth) and small amplitude and vertical gradient. The liquid core flows with uniform mean zonal velocity U₀ relative to the mantle. Ω and B possess vertical and horizontal components.The vertical (poloidal) component B_p is uniform and has a value of 5 G while the horizontal (toroidal) field B_T = B_pαz, where α is a constant. When |α| ? 1, the mean horizontal stresses are found to have the same order of magnitude (10^?2 N m^?2) as those inferred from variations in the decade fluctuations in the length of the day, although the exact numerical values depend on the orientation of Ω as well as on the wavenumbers in the zonal and latitudinal directions.The influence of the steepness (as measured by α) of the toroidal field on the stresses is investigated to examine whether the constraint that the mean horizontal stresses at the core-mantle interface be of the order of 10^?2 N m^?2 might provide a selection mechanism for the behaviour of the toroidal field in the upper reaches of the outer core of the Earth. The results indicate that the restriction imposed on α is related to the value assigned to the toroidal field deep into the core. For example, if |α| ? 1 then the tangential stresses are of the right order of magnitude only if the toroidal field is comparable with the poloidal field deep in the core.     

Axial poloidal electromagnetic core-mantle coupling torque: A re-examination for different conductivity and satellite supported geomagnetic field models

We investigate the temporal behaviour of the axial component of the electromagnetic core-mantle coupling torque that is associated with the poloidal part of the geomagnetic field observable at the Earth surface. For its computation, we use different models of the geomagnetic field, expanded into spherical harmonics (Wardinski and Holme, 2006; Sabaka et al., 2004), and the mantle conductivity. The geomagnetic field, which we have to know at the core-mantle boundary for the associated computations, will be inferred from the field at the Earth surface by the non-harmonic field continuation through a conducting mantle shell. The aims of this investigation are (i) to check how sensitive is the computation of the torque with respect to the different geomagnetic field models, (ii) to check its dependence on the spherical harmonic degree n, and (iii) to determine the difference between the mechanical torque derived from the observed length-of-day variations (atmospheric influence subtracted) and the poloidal electromagnetic torque in dependence on the assumed conductivity. To use the non-harmonic field continuation for the torque calculation and to obtain an insight into the influence of the different geomagnetic field models on the EM torques are the major aspects of this paper. grm@gfz-potsdam.de     

Numerical treatment of a model of the hydromagnetic dynamo with a selected system of convection in the earth's core

Zusammenfassung In der Arbeit[3] wurde das Problem der Auswahl des Geschwindigkeitsfeldes für kinematisches Modell eines hydromagnetischen Dynamos im Erdkern gelöst. Den Inhalt des vorliegenden Artikels bildet die Beendigung dieser Problematik in numerischer Beziehung und die Diskussion der Ergebnisse. Man hat die numerischen Methoden beantragt und daran zu arbeiten angefangen, die Kennzahl und daher auch die charakteristische Geschwindigkeit im Modell gefunden. Die numerische Stabilität der Lösung wurde nicht untersucht. Man beglaubigte die zur Existenz eines stationären Mechanismus der Generierung des geomagnetischen Feldes nötigen Voraussetzungen gleichzeitig mit den Bedingungen, die zur Einführung des Generierungs-KoeffizientenP notwendig sind. Der Koeffizient ist in sphärischen Koordinaten angeführt und es wird gezeigt, dass die Kennzahl seine Grösse nicht beeinflusst.     

核幔耦合对地球自由核章动的激发影响

地球自由核章动（FCN）是地幔与液核相互作用的重要动力学现象,其激发机制涉及地表流体层、地幔和地核等圈层之间的耦合,此前研究多利用地表流体层角动量数据单独研究其对FCN的激发,对核幔耦合的影响考虑不足.本文基于角动量守恒理论分析了核幔耦合对FCN周期及振幅的影响,并结合多个大气及海洋角动量函数时间序列首次估算了核幔耦合在FCN激发过程中的贡献.结果表明核幔耦合对FCN周期产生的固定和时变影响对FCN激发的作用均不可忽视,尤其时变影响可达几十个微角秒,对于进一步解释FCN时变特征非常重要;核幔耦合对FCN振幅的直接影响是地表流体层的激发与实测FCN不相符的主要原因,黏滞、电磁和地形等耗散耦合的存在对地表流体的激发振幅有67%左右的减弱效果.

     

Core motions,electromagnetic core-mantle coupling and variations in the earth's rotation: New constraints from geomagnetic secular variation impulses

Recently observed secular acceleration impulses (SAI) of the geomagnetic field are interpreted in terms of organized motions of the outer core layers. Such motions have planetary dimensions (5000 km) and a large amplitude (3 × 10^?4 m s^?1) and are established in very short times (less than one year). The correlation of SAI observed in the Northern Hemisphere with minima in the Earth's rotation rate (around 1840, 1905 and 1970) is shown to be consistent with a simple model involving electromagnetic coupling of the weakly conducting (of the order of 100 ω^?1 m^?1) mantle, of a coherent outer core layer (thickness 100 to a few hundred kilometres) and of the rest of the core. The magnitude of the torque which acts suddenly on both parts of the core at the time of the impulses is estimated.     

Control volume method for hydromagnetic dynamos in rotating spherical shells: Testing the code against the numerical dynamo benchmark

Hydromagnetic dynamos in rotating spherical shells are investigated using the control volume method. We present a validation of our code against the numerical dynamo benchmark. It is successfully benchmarked and we are able to conclude that the control volume method is another numerical method available for numerical modelling of self-consistent dynamos. In addition, the efficiency of our numerical code is tested. Computations provide conclusions that dynamo codes based on the spectral methods are much more efficient than our code based on the control volume method at the study of global fields on small and medium size parallel computers. However, our code could be much more efficient than codes based on the spectral methods on very large parallel computers, especially at the study of turbulence.     

On asymptotics of low-frequency oscillations of the liquid core: 2. Frequency dependence of the inertial core-mantle coupling

New, unique information on the inertial and dissipative coupling of the liquid core and the mantle has been retrieved from modern high-precision (radiointerferometer and GPS) data on tidal variations in the rotation velocity and nutation of the Earth. Comparison of theoretical and observed data provided new estimates for the dynamic flattening of the outer liquid and the inner solid cores, mantle quality factor, viscosity of the liquid core, and electromagnetic coupling of the liquid core and the mantle [Molodensky, 2004, 2006]. As was shown in the first part of the paper [Molodensky, 2008] (further referred to as [I]), generation of eddy flows in Proudman-Taylor columns, whose orientation is controlled by the topography of the liquid core-mantle boundary, should be taken into account for correct estimation of the inertial coupling (see formulas (8) and (34) in [I]). The range of periods within which this effect plays a significant role is determined by the decay time of these flows. This time is estimated in the paper for the case where dissipation is related to viscous friction at the core-mantle boundary or with the electromagnetic coupling of the liquid core and the mantle. Because of significant uncertainties in modern data on the viscosity of the liquid core, the magnetic field intensity at the core-mantle boundary, and the electrical conductivity of the lower mantle, the dissipative coupling of the liquid core and the mantle cannot be calculated as yet. However, as shown in the paper, the decay time of eddy flows is connected with the attenuation time of subdiurnal free nutation and with the liquid core viscosity. This enables the estimation of the frequency dependence of the dissipative coupling in a fairly wide range. It is shown that the range of periods for which relations (8) and (34) in [I] are valid encompasses the best-studied length-of-day variations and, therefore, these relations are applicable to analysis of the majority of modern data.     

Nonlinear analysis on the coupling process of electromagnetic vibrator and earth

Vibrator is an excitation equipment of the vibratory source in the seismic exploration[1―3]. In order to ap-proach the δ function, dynamite can be used to release high energy in an instantaneous explosion and form the seismic wavelet. On the other hand, a portable vi-brator sweeps a low energy signal for a long time to simulate the function of an explosion source. The re-flection signal of the portable vibrator is estimated for time-delay by digital cross correlation technology, and the disp…     

Review of electromagnetic coupling between the Earth's atmosphere and the space environment

We review our understanding of the electrical properties of the lower and upper atmosphere along with various possible sources of the electromagnetic energy near and far above the Earth's surface. The transport of electromagnetic energy from the atmosphere to the ionosphere and then to the magnetosphere and back to the Earth's surface via ionosphere and lower atmosphere is discussed. The electromagnetic coupling of various regions is also discussed.     

Amplitudes of long-period PcP and the core-mantle boundary

The amplitudes of the core reflection PcP are sensitive to the wave velocities and densities in the neighborhood of the core-mantle boundary (CMB). We study the amplitude ratio of the long-period phases PcP and P from two South American deep-focus earthquakes with favorable fault-plane solution, depth and magnitude, as recorded by WWNSS and CSN stations in North America.Comparison is made with long-period PcP/P amplitude ratios, derived from theoretical seismograms for a variety of CMB models. Models from previous studies, which were mainly derived from short-period PcP observations and which are characterized by discrete layers above the CMB, are almost all inconsistent with the long-period data. The data also discriminate against low nonzero S velocities below the CMB. Simple first-order-discontinuity models of the CMB, for instance according to the Jeffreys-Bullen earth model or according to recent models based mainly on free oscillations, explain the data reasonably well.Model improvements are attempted by varying the P-velocity gradient above the CMB. The best amplitude fit is obtained for a rather strong decrease in P velocity with depth in this zone which, however, gives no acceptable traveltime fit for PcP. The scatter in body-wave amplitudes is considerable even for long-period waves and may prevent the correct assessment of that part of the amplitude variation of a phase with distance that is due to the variation of velocities and densities with depth alone.     

On the effects of a bumpy core-mantle interface

Motivated by the high degree of correlation between the variable parts of the magnetic and gravitational potentials of the Earth discovered by Hide and Malin (using a harmonic analysis approach and utilizing the geomagnetic data) when one field is suitably displaced relative to the other, Moffatt and Dillon (1976) studied a simple planar model in an attempt to find a quantitative explanation for the suggestion that this high degree of correlation may be due to the influences produced by bumps on the core-mantle interface. Moffatt and Dillon assumed that the core-mantle interface was z = η(x) where |?η/?χ| ? 1 and such that in the core [z < η(x)] a uniform flow (U₀, 0, 0) prevails in the presence of a uniform ‘toroidal’ field (B₀, 0, 0); (here z is the vertical coordinate and x is the eastward distance). The whole system rotates uniformly about the vertical with angular velocity Ω. The present work extends the model discussed by Moffatt and Dillon to include a horizontal component of angular velocity Ω_H and a uniform small poloidal field B_p. In addition, the uniform toroidal field is here replaced by one which vanishes everywhere in the mantle and increases linearly, from zero on the interface, with z. It is shown that the presence of Ω_H and B_p, together with the present choice of toroidal magnetic field, has a profound effect both on the correlation between the variable parts of the magnetic and gravitational fields of the Earth, and on how far the disturbances caused by the topography of the interface [which is necessarily three-dimensional i.e. z = η(x, y) here] can penetrate into the liquid core. In particular it is found that the highest value of the correlation function is +0.79 which corresponds to a situation in which the magnetic potential is displaced both latitudinally and longitudinally relative to the gravitational potential.     

Numerical simulation of Martian historical dynamo: Impact of the Rayleigh number on the dynamo state

The observed Mars remnant magnetism suggests that there was an active dynamo in the Martian core. We use the MoSST core dynamics model to simulate the Martian historical dynamo, focusing on the variation of the dynamo states with the Rayleigh number Ra (a non-dimensional parameter describing the buoyancy force in the core). Our numerical results show that the mean field length scale does not vary monotonically with the Rayleigh number, and the field morphology at the core mantle boundary changes with Rayleigh number. In particular, it drifts westward with a speed decreasing with Rayleigh number. Supported by National Natural Science Foundation of China (Grant No. 40328006)