The Dynamical Effects of Hyperviscosity on Numerical Geodynamo Models

The most fundamental difficulty in the construction of an Earth-like dynamo model is associated with the constraint caused by the rapid rotation of the Earth. To stabilise numerical codes, many workers have introduced hyperviscosity into the governing equations. One of the major effects introduced by hyperviscosity is to offset the rotational constraint, and, consequently, to alter the key dynamics of an Earth-like dynamo. In this paper, an Earth-like convection model with or without the presence of an imposed magnetic field is investigated with or without the effect of hyperviscosity. A nonlinear dynamo model with the mean field approximation is also used to examine the dynamical effect of hyperviscosity. The results suggest that great care should be taken when hyperviscosity is employed in geodynamo models.     

The Karlsruhe Dynamo Experiment. A Mean Field Approach

At the Forschungszentrum Karlsruhe an experiment is in preparation which it is hoped, in view of the geodynamo and other cosmic dynamos, that a homogeneous dynamo will be demonstrated and investigated. This experiment is discussed within the framework of mean-field dynamo theory. Results are presented concerning kinematic cylindrical mean-field dynamo models reflecting some features of the experimental device, as well as results of detailed calculations of the -effect that apply to arbitrarily high magnetic Reynolds numbers. On this basis estimates of the excitation conditions of the dynamo are given and predictions concerning the geometrical structure of the generated magnetic fields are made.     

Inner-core conductivity in numerical dynamo simulations

Speculation about its possible super-rotation has drawn the attention of many geophysical researchers to the Earth’s inner core. An issue of special interest for geodynamo modelling is the influence of the inner-core conductivity. It has been suggested that the finite magnetic diffusivity of the inner core prevents more frequent reversals of the Earth’s magnetic field. We explore the possible influence of the inner-core conductivity by comparing convection-driven 3D dynamo simulations with insulating or conducting inner cores (CIC) at various parameters. The influence on the field structure in the outer core is only marginal. The time behaviour of dipole-dominated non-reversing dynamos is also little affected. Concerning reversing dynamos, the inner-core conductivity reduces the number of short dipole-polarity intervals with a typical length of a few thousand years. Reversals are always correlated with low dipole strength and these short intervals are found in periods where the dipole moment stays low. Polarity intervals longer than about 10,000 years, where the dipole moment has time recover in strength, are equally likely in insulating and CIC models. Since these latter intervals are of more geophysical relevance, we conclude that the influence of the inner-core conductivity on Earth-like reversal sequences is insignificant for the dynamo model employed here.     

A generalized two-disk dynamo model

Abstract

A generalized two-disk dynamo model is considered that includes mechanical friction; this model is intended to simulate in its broad character the behavior of the geodynamo. Fixed points, limit cycles and chaotic attractors are located for different input parameters of the model. The chaotic regimes are of several kinds as are the “routes to chaos”. Several approximate models, helpful for studying the dynamo are discussed. A number of essential differences from the well-known Rikitake dynamo are demonstrated.     

Geomagnetic Reversals and Dynamo Bursts in a Simple Geodynamo Model

The question of what exactly happens with the geodynamo process during the reversal of a geomagnetic field is studied in a simple geodynamo model. The geodynamo action is described by the so called dynamo number characterizing the joint action of the main drivers of the geomagnetic field, i.e., the differential rotation and mirror–asymmetric convection. In mirror-asymmetric convection, for instance, in the northern hemisphere, there are more right vortices than left vortices, whereas in the southern hemisphere, there are more left vortices than right vortices. The effect of the magnetic field on the flow is described by the suppression of the mirror asymmetry: due to this suppression, e.g., in the northern hemisphere, the excess of right vortices over left vortices decreases. It is also assumed that due to this suppression, the mirror asymmetry can change its sign; i.e., the number of left vortices in the northern hemisphere can become larger than the number of right vortices. Correspondingly, the dynamo number can also change its sign. It is shown that the short-term changes of the sign of the dynamo number are responsible for the very short time span accommodating the reversal, when compared to the interval between the reversals.     

Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g., the α-effect, are highly anisotropic in both examples. An α²-mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.     

Energy geodynamo parameters compatible with analytical,numerical, paleomagnetic models and observations

A hydromagnetic dynamo is only possible at a sufficiently powerful convection. In the Earth’s core, it is probably the nonthermal convection very much in excess of its critical level with the molecular transporr coefficients. However, in the case of medium- or large-scale fields, the critical energy level caused by the turbulent tranport coefficients is likely to be slightly below the actual level. This probably explains both the 22-year success of this type of simplified geodynamo models and the energy scaling laws for hydromagnetic fields, which generalize these models. Also the review of energy-dependent analytical and observational estimates of vortex fields, hydromagnetic scale sizes, and velocities in the core is presented. These typical parameters are partly in a new way linked to the observed and more ancient magnetic variations. New, albeit, simplified and self-evident, substantiation is given to the paleomagnetic hypothesis about the predominance of the axial dipole under a certain time averaging. In (Pozzo et al., 2012) and more recent works, it is shown that the adiabatic heat flow and electrical conductivity in the Earth’s core are severalfold higher than the generally accepted estimates. Here, the dynamo supporting Braginsky’s convection (Braginsky, 1963) (under the crystallization of the heavy fraction of a liquid onto the solid core) started less than 1 Ga ago, whereas the more ancient geodynamo was supported by the compositional convection of another type. The known mechanisms implementing this convection, which differ by the scenarios of magnetic evolution, are reviewed. This may help identify the sought mechanism through the most ancient paleomagnetic estimates of the field’s intensity and through the numerical models. The probable mechanisms of generation and their absence for the primordial and recent magnetic field of the studied terrestrial planets are discussed.     

Dipole moment scaling for convection-driven planetary dynamos

Scaling laws are derived for the time-average magnetic dipole moment in rotating convection-driven numerical dynamo models. Results from 145 dynamo models with a variety of boundary conditions and heating modes, covering a wide section of parameter space, show that the time-average dipole moment depends on the convective buoyancy flux F. Two distinct regimes are found above the critical magnetic Reynolds number for onset of dynamo action. In the first regime the external magnetic field is dipole-dominant, whereas for larger buoyancy flux or slower rotation the external field is dominated by higher multipoles and the dipole moment is reduced by a factor of 10 or more relative to the dipolar regime. For dynamos driven by basal heating, the dipole moment M increases like M  F^1/3 in the dipolar regime. Reversing dipolar dynamos tend to cluster near the multipolar transition, which is shown to depend on a local Rossby number parameter. The geodynamo lies close to this transition, suggesting an explanation for polarity reversals and the possibility of a weaker dipole earlier in Earth history. Internally heated dynamos generate smaller dipole moments overall and show a gradual transition from dipolar to multipolar states. Our scaling yields order of magnitude agreement with the dipole moments of Earth, Jupiter, Saturn, Uranus, Neptune, and Ganymede, and predicts a multipolar-type dynamo for Mercury.     

The weak taylor state in an αω-dynamo

Abstract

A spherical αω-dynamo is studied for small values of the viscous coupling parameter ε ～ v^1/2, paying attention particularly to large dynamo numbers. The present study is a follow-up of the work by Hollerbach et al. (1992) with their choice of α-effect and Archimedean wind including also the constraint of magnetic field symmetry (or antisymmetry) due to equatorial plane. The magnetic field scaled by ε^1/2 is independent of ε in the solutions for dynamo numbers smaller than a certain value of D _b (the Ekman state) which are represented by dynamo waves running from pole to equator or vice-versa. However, for dynamo numbers larger than D _b the solution bifurcates and subsequently becomes dependent on ε. The bifurcation is a consequence of a crucial role of the meridional convection in the mechanism of magnetic field generation. Calculations suggest that the bifurcation appears near dynamo number about 33500 and the solutions for larger dynamo numbers and ε = 0 become unstable and fail, while the solutions for small but non-zero ε are characterized by cylindrical layers of local maximum of magnetic field and sharp changes of geostrophic velocity. Our theoretical analysis allows us to conclude that our solution does not take the form of the usual Taylor state, where the Taylor constraint should be satisfied due to the special structure of magnetic field. We rather obtained the solution in the form of a “weak” Taylor state, where the Taylor constraint is satisfied partly due to the amplitude of the magnetic field and partly due to its structure. Calculations suggest that the roles of amplitude and structure are roughly fifty-fifty in our “weak” Taylor state solution and thus they can be called a Semi-Taylor state. Simple estimates show that also Ekman state solutions can be applicable in the geodynamo context.     

The dynamo basis of solar cycle precursor schemes

We investigate the dynamo underpinning of solar cycle precursor schemes based on direct or indirect measures of the solar surface magnetic field. We do so for various types of mean-field-like kinematic axisymmetric dynamo models, where amplitude fluctuations are driven by zero-mean stochastic forcing of the dynamo number controlling the strength of the poloidal source term. In all stochastically forced models considered, the surface poloidal magnetic field is found to have precursor value only if it feeds back into the dynamo loop, which suggests that accurate determination of the magnetic flux budget of the solar polar fields may hold the key to dynamo model-based cycle forecasting.     

地球发电机过程的实验室模拟研究

关于地磁场起源的研究依赖于理论研究、实验室实验、数值模拟和实地观测等四个方面的工作：理论研究给出物理框架,提出物理思想;实验室实验检验理论预言,发现新现象;数值模拟可以对很大参数空间的复杂过程和对象给出多方面的过程描述;而实地观测不仅仅是上述三方面研究工作的出发点和基础,而且也是对研究结果的最终检验.本文介绍地球发电机过程实验研究的主要结果,特别是2000年后的突破性进展.目前,大多数实验研究还停留在运动学发电机水平,相对于数值模拟的巨大成就来说,实验研究需要大力发展,这就是正在发展中的第二代实验发电机.     

Small Scale Kinematic Dynamos: Beyond the α-Effect

In geodynamo simulations which simulate the generation of an axial dipolar magnetic field, the generation mechanism appears to be adequately described as an α²-dynamo with an anisotropic α-effect. The anisotropy in the α-effect favors an equatorial dipole field, however, which calls into question the interpretation in terms of an α²-dynamo. It is shown in this article with kinematic dynamo calculations and exemplary velocity fields with an anisotropic α-effect that both types of dipolar magnetic field can be generated. Two examples of working dynamos in a sphere with flows with zero α-effect are also provided.     

Mantle-driven geodynamo features – Effects of compositional and narrow D″ anomalies

Lower mantle heterogeneity could cause deviations from axial symmetry in geodynamo properties. Global tomography models are commonly used to infer the pattern of core–mantle boundary heat flux via a linear relation that corresponds to a purely thermal interpretation of lower mantle seismic anomalies, ignoring both non-thermal origins and non-resolved small scales. Here we study the possible impact on the geodynamo of narrow thermal anomalies in the base of the mantle, originating from either compositional heterogeneity or sharp margins of large-scale features. A heat flux boundary condition composed of a large-scale pattern and narrow ridges separating the large-scale positive and negative features is imposed on numerical dynamos. We find that hot ridges located to the west of a positive large-scale core–mantle boundary heat flux anomaly produce a time-average narrow elongated upwelling, a flow barrier at the top of the core and intensified low-latitudes magnetic flux patches. When the ridge is located to the east of a positive core–mantle boundary heat flux anomaly, the associated upwelling is weaker and the homogeneous dynamo westward drift leaks, precluding persistent intense low-latitudes magnetic flux patches. These signatures of the core–mantle boundary heat flux ridge are evident in the north–south component of the thermal wind balance. Based on the pattern of lower mantle seismic tomography (Masters et al., 2000), we hypothesize that hot narrow thermal ridges below central Asia and the Indian Ocean and below the American Pacific coast produce time-average fluid upwelling and a barrier for azimuthal flow at the top of the core. East of these ridges, below east Asia and Oceania and below the Americas, time-average intense geomagnetic flux patches are expected.     

Anisotropic turbulent thermal diffusion and thermal convection in a rapidly rotating fluid sphere

Estimates of the molecular values of magnetic, viscous and thermal diffusion suggest that the state of the Earth’s core is turbulent and that complete numerical simulation of the geodynamo is not realizable at present. Large eddy simulation of the geodynamo with modelling of the sub-grid scale turbulence must be used. Current geodynamo models effectively model the sub-grid scale turbulence with isotropic diffusivities larger than the molecular values appropriate for the core. In the Braginsky and Meytlis (1990) picture of core turbulence the thermal and viscous diffusivities are enhanced up to the molecular magnetic diffusivity in the directions of the rotation axis and mean magnetic field. We neglect the mean magnetic field herein to isolate the effects of anisotropic thermal diffusion, enhanced or diminished along the rotation axis, and explore the instability of a steady conductive basic state with zero mean flow in the Boussinesq approximation. This state is found to be more stable (less stable) as the thermal diffusion parallel to the rotation axis is increased (decreased), if the transverse thermal diffusion is fixed. To examine the effect of simultaneously varying the diffusion along and transverse to the rotation axis, the Frobenius norm is used to control for the total thermal diffusion. When the Frobenius norm of the thermal diffusion tensor is fixed, it is found that increasing the thermal diffusion parallel to the rotation axis is destabilising. This result suggests that, for a fixed total thermal diffusion, geodynamo codes with anisotropic thermal diffusion may operate at lower modified Rayleigh numbers.     

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.     

Nonlinearities in dynamo

By the example of the dynamo model in the rotating plane layer heated from below, the effects are examined that lead to the stabilization of an exponentially growing magnetic field in the magnetostrophic convection in passing from the kinematic dynamo mode to the nonlinear mode. The estimates of the energy redistribution in the spectrum are given, and the mechanisms of suppression of helicity are presented. Equalization of the field of velocity and the magnetic field is analyzed. The modes examined are close to those utilized in the up-to-date models of the planetary dynamo in the cores of planets.     

Torque balance and energy budget for the precessionally driven dynamo

The feasibility of a precessionally driven dynamo is investigated. The relative orientation of the angular-velocity vectors of the mantle and core and the precession vector of the earth are determined from a torque balance. The core and mantle are acted upon by separate gravitational torques and mutual interaction torques resulting from pressure, viscous and magnetic stresses at the core-mantle interface. The viscous and magnetic torques are determined using the results of a detailed analysis of the Ekman-Hartmann and magnetic diffusion layers generated at the core-mantle interface by the misalignment of the mantle and core angular-velocity vectors. The dissipative torques are found to be weaker by a factor of 10^?4 than those estimated by Malkus (1968) and Stacey (1973), resulting in only 3.5 · 10⁷ W being extracted from the rotational kinetic energy of the earth by these mechanisms. Furthermore, it is found that all of this energy is dissipated in the boundary layers at the core-mantle interface and none is available to drive the geodynamo.     

A shell model for a large-scale turbulent dynamo

This article addresses the interesting and important problem of large-scale magnetic field generation in turbulent flows, using a self-consistent dynamo model recently developed. The main idea of this model is to consider the induction equation for the large-scale magnetic field, integrated consistently with the turbulent dynamics at smaller scales described by a magnetohydrodynamic shell model. The questions of dynamo action threshold, magnetic field saturation, magnetic field reversals, nature of the dynamo transition and the changes of small-scale turbulence as a consequence of the dynamo onset are discussed. In particular, the stability curve obtained by the model integration is shown in a very wide range of values of the magnetic Prandtl number not yet accessible by direct numerical simulation but more realistic for natural dynamos. Moreover, from our analysis it is shown that the large-scale dynamo transition displays a hysteretic behaviour and therefore a subcritical nature. The model successfully reproduces magnetic polarity reversals, showing the capability to generate persistence times which are increasing for decreasing magnetic diffusivity. Moreover, when the system reaches a statistically stationary dynamo state, where the large-scale magnetic field can abruptly reverse its polarity (magnetic reversal state) or not, keeping the same polarity (steady state), it shows an unmistakable tendency towards the energy equipartition for the turbulence at small scale.     

On dynamo action in the high-conductivity limit

Abstract

This paper builds on a speculation by Moffatt (1979) on an apparent conflict between two results of dynamo theory in the high conductivity limit. Firstly, the finding by Bondi and Gold (1950) on the boundedness of the magnetic dipole moment of a perfectly conducting fluid body is, for a sphere, extended to all magnetic multipole moments. Secondly, a refined version is considered of the simple spherical mean-field dynamo model proposed by Krause and Steenbeck (1967). Some constraints on the mean electromotive force near the boundary of the conducting body are taken into account, which have not been recognized up to now. In the framework of the second order correlation approximation it is shown that it is just these constraints that ensure the boundedness of the magnetic multipole moments in the high conductivity limit. Thus the apparent conflict is resolved. In this context another possible source of error in mean-field dynamo models is pointed out. The present theory also adds insight into dynamo process in cosmical objects, in a way that is briefly discussed.     

二维电离层发电机理论模式及其初步应用

在地理坐标系下推导出二维电离层发电机理论方程,采用逐线迭代法求解得到全球二维电离层发电机电流函数,进而得到电离层发电机电流和电场.模式中使用的电导率是根据外部经验模式给出的背景大气和电离层参数,采用理论公式计算得出;输入的中性风场和磁场分别由HWM93和IGRF2000模型给出,该电离层发电机理论模式很好地给出了全球Sq电流形态及电离层E层发电机电场的基本特征.利用该模式研究了外部模式风场以及地磁场随高度的变化对模拟结果的影响,发现在90~180 km高度上,风场随高度变化对电流影响较大,而地磁场影响较小;重点模拟研究了地磁平静时期,Sq电流涡旋中心位置和总电流强度的变化规律,初步研究发现,电流中心位置在地理纬度±30°附近,不同的地方时电流随地磁纬度线平行移动,且南北半球两个电流涡中心电流强度之和变化不大.分析发现这种规律与发电机高度上的磁场总强度及地磁倾角的全球分布有很好的相关性.