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.     

Maximally-efficient-generation approach in the dynamo theory

Abstract

We propose a method of derivation of global asymptotic solutions of the hydromagnetic dynamo problem at large magnetic Reynolds number. The procedure reduces to matching the local asymptotic forms for the magnetic field generated near individual extrema of generation strength. The basis of the proposed method, named here the Maximally-Efficient-Generation Approach (MEGA), is the assertion that properties of global asymptotic solutions of the kinematic dynamo are determined by the distribution of the generation strength near its leading extrema and by the number and distribution of the extrema.

The general method is illustrated by the global asymptotic solution of the α²-dynamo problem in a slab. The nature of oscillatory solutions revealed earlier in numerical simulations and the reasons for the dominance of even magnetic modes in slab geometry are clarified.

Applicability of the asymptotic solutions at moderate values of the asymptotic parameter is also discussed. We confirm this applicability using comparisons with complementary asymptotic expansions and numerical simulations. In particular, this justifies application of the MEGA solutions to estimation of the generation threshold.     

Self-consistent dynamo models driven by hydromagnetic instabilities

The dynamics of the Earth's core are dominated by a balance between Lorentz and Coriolis forces. Previous studies of possible (magnetostrophic) hydromagnetic instabilities in this regime have been confined to geophysically unrealistic flows and fields. In recent papers we have treated rather general fields and flows in a spherical geometry and in a computationally simple plane-layer model. These studies have highlighted the importance of differential rotation in determining the spatial structure of the instability. Here we have proceeded to use these results to construct a self-consistent dynamo model of the geomagnetic field. An iterative procedure is employed in which an α-effect is calculated from the form of the instability and is then used in a mean field dynamo model. The mean zonal field calculated there is then input back into the hydromagnetic stability problem and a new α-effect calculated. The whole procedure is repeated until the input and output zonal fields are the same to some tolerance.     

Reversal sequence in a multiple scale dynamo mechanism

We investigate the temporal evolution of the magnetic dipole field intensity of the Earth through a multiscale dynamo mechanism. On a large range of spatio-temporal scales, the helical motions of the fluid flow are given by a schematic model of a fully developed turbulence. The system construction is symmetric with respect to left-handed and right-handed cyclones. The multiscale cyclonic turbulence coupled with a differential rotation (schematic ω dynamo) or alone (schematic ² dynamo) is the ingredient of the loop through which poloidal and toroidal fields are built from one another. Two kinds of reaction of the magnetic field on the flow are considered: the presence of a magnetic field first favours a larger-scale organization of the flow, and, second, impedes this flow by the effect of growing Lorentz forces. We obtain the general features of the geomagnetic field intensity observed over geological times and describe a general mechanism for reversals, excursions and secular variation. The mechanism happens to keep a memory during the chrons and loose it during the events (excursions and inversions).     

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.     

How strong is the magnetic field in the Earth's liquid core?

A crucial step in the investigation of the energetics of motions in the Earth's core and the generation of the geomagnetic field by the hydromagnetic dynamo process is the estimation of the average strength $\text{B}$ of the magnetic field B = B_p + B_T in the core. Owing to the probability that the toroidal field B_T in the core, which has no radial component, is a good deal stronger than the poloidal field B_p, direct downward extrapolation of the surface field to the core-mantle interface gives no more than an extreme lower limit to $\text{B}$. This paper outlines the indirect methods by which $\text{B}$ can be estimated, arguing that $\text{B}$ is probably about 10^?2 T (100 Γ) but might be as low as 10^?3 T (10 Γ) or as high as 5 × 10^?2 T (500 Γ).     

On the minimal instability time of hydromagnetic flows in the Earth’s core

The dependence of the intensity of geomagnetic field on the intensity of thermal convection in the liquid core of the Earth, which has been empirically derived by a number of the authors from the results of numerical modeling of convective dynamo, is substantiated theoretically. This dependence is used for estimating the characteristic time scale of jerk evolution.     

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.     

古地磁学在地磁场起源研究中的应用

地磁场起源及其倒转是地球科学的难题之一。究其原因一方面是由于无法直接观测地球内部发生的物理过程,另一方面是由于缺乏理论与实验相结合的综合研究。本文以磁流体力学为基础,将古地磁学与αω发电机理论结合在一起进行分析和研究。得出了如下新观点:(1)洛仑兹力在地核发电过程起负反馈作用;(2)较差旋转控制着地磁场西向漂移,(3)α作用使地磁极偏离地球自转轴     

Models for convectively driven hydromagnetic dynamos

Abstract

Models of a convectively driven hydromagnetic dynamo are constructed using a truncated modal expansion. The resulting nonlinear partial differential equations are integrated numerically. The results confirm that rotation is a necessary condition for effective dynamo action, and suggest that equipartition of kinetic and magnetic energies is qualitatively valid, and that toroidal field energies can be much larger than poloidal.     

Numerical simulations of MHD dynamos

The generation of magnetic fields in space plasmas and in astrophysics is usually described within the framework of magnetohydrodynamics. Turbulent helical flows produce magnetic fields very efficiently, with correlation length scales larger than those characterizing the flow. Within the context of the solar magnetic cycle, a turbulent dynamo is responsible for the so-called alpha effect, while the Omega effect is associated to the differential rotation of the Sun.We present direct numerical simulations of turbulent magnetohydrodynamic dynamos including two-fluid effects such as the Hall current. More specifically, we study the evolution of an initially weak and small-scale magnetic field in a system maintained in a stationary regime of hydrodynamic turbulence, and explore the conditions for exponential growth of the magnetic energy. In all the cases considered, we find that the dynamo saturates at the equipartition level between kinetic and magnetic energy, and the total energy reaches a Kolmogorov power spectrum.     

A multi-layered kinematic dynamo model: implications of a stratified upper layer in the Earth's core

It has been suggested that there exists a stably stratified electrically conducting layer at the top of the Earth's outer fluid core and that lateral temperature gradients in the lower mantle is capable of a driving thermal-wind-type flow near the core–mantle boundary. We investigate how such a flow in a stable layer could influence the geomagnetic field and the geodynamo using a very simple two-dimensional kinematic dynamo model in Cartesian geometry. The dynamo has four layers representing the inner core, convecting lower outer core, stable upper core, and insulating mantle. An α² dynamo operates in the convecting outer core and a horizontal shear flow is imposed in the stable layer. Exact dynamo solutions are obtained for a range of parameters, including different conductivities for the stable layer and inner core. This allows us to connect our solutions with known, simpler solutions of a single-layer α² dynamo, and thereby assess the effects of the extra layers. We confirm earlier results that a stable, static layer can enhance dynamo action. We find that shear flows produce dynamo wave solutions with a different spatial structure from the steady α² dynamos solutions. The stable layer controls the behavior of the dynamo system through the interface conditions, providing a new means whereby lateral variations on the boundary can influence the geomagnetic field.     

Numerical simulation of the electric field and zonal current in the Earth’s ionosphere: The dynamo field and equatorial electrojet

The work is devoted to the numerical simulation of the dynamo electric field and its effects in the Earth’s ionosphere within the scope of the thermosphere-ionosphere-protonosphere global self-consistent model developed at WD IZMIRAN. The new electric field calculation block, which was used to obtain results of the self-consistent calculations of the electric field potential generated by the dynamo effect of the thermospheric winds (the dynamo field) and the equatorial electrojet for March 22, 1987, is briefly described in this work. A comparison of the obtained results with the experimental data showed a satisfactory agreement. Moreover, the proposed model was used to calculate the diurnal variations in the ionospheric parameters for Jicamarca equatorial station under the same conditions with the help of the new block of the electric field. The results of these calculations are also presented and discussed in this work. It has been indicated that the new model satisfactorily describes the specific features of electric field distribution at the geomagnetic equator and the well-known phenomenon of equatorial electrojet.     

Analytic scaling laws in planetary dynamo models

ABSTRACT

Numerical models of planetary MHD dynamos have led to many advances in the last twenty years. However, the accessible parameter space of these models is somewhat limited due to computational constraints. A number of numerically based scaling laws have been proposed, but it has been difficult to reliably confirm them from the simulation data alone. Since these scaling laws have to be used to extend the numerical models into the planetary parameter regime, getting correct scalings is an important issue. Here some possible balances between the relevant terms in the dynamo equations are considered, and a number of different known and new scaling laws are proposed and justified from analytical point of view. Some of these laws are compatible with the widely known Christensen and Aubert (Geophys. J. Int. 2006, 166) scaling for the magnetic field strength. Plausible estimates of the typical values of the important quantities in the Earth's core and other Terrestrial planets are made, and the extent to which the various scaling laws are consistent with these estimates is 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.     

高压直流输电对地磁场观测的影响

2009年12月,甘肃天水、陕西乾陵等地磁台Z分量观测差值出现台阶式异常变化,同期陕西关中地区宝鸡、乾陵、周至等台站地电场观测也出现同步异常变化.经调查落实,确认此变化是由宝鸡—德阳±500kV直流输电工程试运行引起的.高压直流输电是近年来新出现的地磁观测中的一类干扰异常.根据对宝鸡—德阳高压直流输电线路邻近区域各地磁台站实际观测资料的分析,并结合安培定理和毕奥-萨伐尔定律计算了理想状态下高压直流输电对地磁观测的影响.结果表明,理论值与实际观测值基本相符,从而从机理上解释了高压直流输电对地磁观测影响的原因.由于计算时没有考虑不同地磁观测系统的差异和地下电性结构等因素的影响,因此计算方法有待进一步完善.     

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.     

The palaeogeomagnetic field strength,variations in reversal frequency,and geomagnetic dynamo models

Abstract

The geomagnetic field and its frequent polarity reversals are generally attributed to magnetohydrodynamic (MHD) processes in the Earth's metallic and fluid core. But it is difficult to identify convincingly any MHD timescales with that over which the reversals occur. Moreover, the geological record indicates that the intervals between the consecutive reversals have varied widely. In addition, there have been superchrons when the reversals have been frequent, and at least two, and perhaps three, 35-70 Myr long superchrons when they were almost totally absent. The evaluation of these long-term variations in the palaeogeophysical record can provide crucial constraints on theories of geomagnetism, but it has generally been limited to only the directional or polarity data. It is shown here that the correlation of the palaeogeomagnetic field strength with the field's protracted stability during a fixed polarity superchron provides such a constraint. In terms of a strong field dynamo model it leads to the speculation that the magnetic Reynolds number, and the toroidal field, increase substantially during a superchron of frequent reversals.     

A search for axisymmetric dynamos

Some general theorems on the necessary level of symmetry in the velocity field of a kinematic dynamo have been proposed: the analytic difficulties of direct proofs of these appear insurmountable at present. Explicit axisymmetric counter-examples are here sought in an attempt to disprove one such general hypothesis numerically. The usual Bullard-Gellman technique, and a new time-dependent extension of it, are used. Despite extensive computation on a number of different models, no axisymmetric dynamo is found: possible reasons for this are discussed.     

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.