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1.
The induction equation of magnetohydrodynamics (MHD) is mathematically equivalent to a system of integral equations for the magnetic field in the bulk of the fluid and for the electric potential at its boundary. We summarize the recent developments concerning the numerical implementation of this scheme and its applications to various forward and inverse problems in dynamo theory and applied MHD.  相似文献   

2.
Paleomagnetic data indicate that there is a north-south asymmetry in the time-averaged magnetic field and that there are small but significant differences between the normal and reverse polarity states. The geographical variation is most likely due to spatial variation in the boundary conditions at the core-mantle interface. The difference in the magnetic fields of the reverse and normal polarity states can be modeled in terms of a “standing field”. The paleomagnetic data are insufficient to determine whether or not this “standing field” is of core origin. However, consideration of mechanisms, including thermoelectric currents, indicates that there probably are important differences in core processes between the two polarity states. At first glance this interpretation is difficult to reconcile with the fact that the magnetic induction equation is antisymmetric with respect to the magnetic field. A way around this problem is the possibility that only certain transitions are allowed between acceptable eigenstates in dynamo models of the Earth's magnetic field.  相似文献   

3.
We address mathematical issues raised by the so-called α?effect of dynamo theory, which is a dynamo mechanism arising in conducting flows with small scale fluctuations. Analytical results on the α?effect concern the linear induction equation, and are usually claimed to hold for the whole magnetohydrodynamics (MHD) system, as long as the amplitude of the perturbations is small. We discuss the justification of that claim, in the case of periodic fluctuations of the fields. We show a nonlinear instability result on the MHD system, that predicts dynamo action for a large class of high frequency periodic flows, up to the fully nonlinear regime.  相似文献   

4.
Abstract

Bayly (1993) introduced and investigated the equation (? t + v·▽-η ▽2)S = RS as a scalar analogue of the magnetic induction equation. Here, S(r, t) is a scalar function and the flow field v(r, t) and “stretching” function R(r, t) are given independently. This equation is much easier to handle than the corresponding vector equation and, although not of much relevance to the (vector) kinematic dynamo problem, it helps to study some features of the fast dynamo problem. In this note the scalar equation is considered for linear flow and a harmonic potential as stretching function. The steady equation separates into one-dimensional equations, which can be completely solved and therefore allow one to monitor the behaviour of the spectrum in the limit of vanishing diffusivity. For more general homogeneous flows a scaling argument is given which ensures fast dynamo action for certain powers of the harmonic potential. Our results stress the singular behaviour of eigenfunctions in the limit of vanishing diffusivity and the importance of stagnation points in the flow for fast dynamo action.  相似文献   

5.
The well-known “toroidal theorem” of Elsasser and Bullard and Gellman rules out dynamo action in a conducting sphere when the velocity field has no poloidal part. It is here shown that for a fixed toroidal velocity field any poloidal velocity must attain a finite size if dynamo action is to be possible. The resulting “anti-dynamo” theorem generalises the earlier result of Childress by giving a bound on the product of the suprema of the toroidal and poloidal velocities.  相似文献   

6.
This article considers magnetic field generation by a fluid flow in a system referred to as the Archontis dynamo: a steady nonlinear MHD state is driven by a prescribed body force. The field and flow become almost equal and dissipation is concentrated in cigar-like structures centred on straight-line separatrices. Numerical scaling laws for energy and dissipation are given that extend previous calculations to smaller diffusivities. The symmetries of the dynamo are set out, together with their implications for the structure of field and flow along the separatrices. The scaling of the cigar-like dissipative regions, as the square root of the diffusivities, is explained by approximations near the separatrices. Rigorous results on the existence and smoothness of solutions to the steady, forced MHD equations are given.  相似文献   

7.
We present a high order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of incompressible fluid dynamics and magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme that has been successfully applied to compressible fluids, with or without magnetic fields. A fractional time-step method is used to enforce the incompressibility condition. Two basic elements of the WENO scheme, upwinding and wave decomposition, are shown to be important in solving the incompressible systems. Numerical results demonstrate that the scheme performs well for one-dimensional Riemann problems, a two-dimensional double-shear flow problem, and the two-dimensional Orszag–Tang MHD vortex system. They establish that the WENO code is numerical stable even when there are no explicit dissipation terms. It can handle discontinuous data and attain converged results with a high order of accuracy.  相似文献   

8.
Abstract

An analysis of small-scale magnetic fields shows that the Ponomarenko dynamo is a fast dynamo; the maximum growth rate remains of order unity in the limit of large magnetic Reynolds number. Magnetic fields are regenerated by a “stretch-diffuse” mechanism. General smooth axisymmetric velocity fields are also analysed; these give slow dynamo action by the same mechanism.  相似文献   

9.
An inverse dynamo problem is presented in which we search for either kinematic dynamos which produce the same external magnetic fields or an invisible dynamo. The existence of flows which produce the same external magnetic fields is proved. However, we have not found general conditions necessary for such kind of dynamos. An “invisible dynamo” operates in an electrically conducting region surrounded by vacuum and generates a magnetic field trapped in the electrically conducting region so that no magnetic field exists in the vacuum. Invisible magnetic decay modes exist in cylinders, but no invisible growing field supported by the dynamo mechanism has been found.  相似文献   

10.
We consider an unforced, incompressible, turbulent magnetofluid constrained by concentric inner and outer spherical surfaces. We define a model system in which normal components of the velocity, magnetic field, vorticity, and electric current are zero on the boundaries. This choice allows us to find a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity and current. The model dynamical system represents magnetohydrodynamic (MHD) turbulence in a spherical domain and is analyzed by the methods similar to those applied to homogeneous MHD turbulence. We find a statistical theory of ideal (i.e. no dissipation) MHD turbulence analogous to that found in the homogeneous case, including the prediction of coherent structure in the form of a large-scale quasistationary magnetic field. This MHD dynamo depends on broken ergodicity, an effect that is enhanced when total magnetic helicity is increased relative to total energy. When dissipation is added and large scales are only weakly damped, quasiequilibrium may occur for long periods of time, so that the ideal theory is still pertinent on a global scale. Over longer periods of time, the selective decay of energy over magnetic helicity further enhances the effects of broken ergodicity. Thus, broken ergodicity is an essential mechanism and relative magnetic helicity is a critical parameter in this model MHD dynamo theory.  相似文献   

11.
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.  相似文献   

12.
ABSTRACT

The Archontis dynamo is a rare example of an MHD dynamo within which forcing drives a dynamo where the flow and magnetic fields are almost perfectly aligned and the energies are approximately equal. In this paper, I expand upon our knowledge of the dynamo by showing that the intermediate steady states of the kinetic and magnetic energies observed by Cameron and Galloway are not a necessary feature of aligned dynamos. Furthermore, I show that the steady state into which the flow and magnetic fields eventually evolve is remarkably robust to the addition of time dependence and asymmetry to the forcing.  相似文献   

13.
Turbulent magnetofluids appear in various geophysical and astrophysical contexts, in phenomena associated with planets, stars, galaxies and the universe itself. In many cases, large-scale magnetic fields are observed, though a better knowledge of magnetofluid turbulence is needed to more fully understand the dynamo processes that produce them. One approach is to develop the statistical mechanics of ideal (i.e. non-dissipative), incompressible, homogeneous magnetohydrodynamic (MHD) turbulence, known as “absolute equilibrium ensemble” theory, as far as possible by studying model systems with the goal of finding those aspects that survive the introduction of viscosity and resistivity. Here, we review the progress that has been made in this direction. We examine both three-dimensional (3-D) and two-dimensional (2-D) model systems based on discrete Fourier representations. The basic equations are those of incompressible MHD and may include the effects of rotation and/or a mean magnetic field B o. Statistical predictions are that Fourier coefficients of the velocity and magnetic field are zero-mean random variables. However, this is not the case, in general, for we observe non-ergodic behavior in very long time computer simulations of ideal turbulence: low wavenumber Fourier modes that have relatively large means and small standard deviations, i.e. coherent structure. In particular, ergodicity appears strongly broken when B o?=?0 and weakly broken when B o?≠?0. Broken ergodicity in MHD turbulence is explained by an eigenanalysis of modal covariance matrices. This produces a set of modal eigenvalues inversely proportional to the expected energy of their associated eigenvariables. A large disparity in eigenvalues within the same mode (identified by wavevector k ) can occur at low values of wavenumber k?=?| k |, especially when B o?=?0. This disparity breaks the ergodicity of eigenvariables with smallest eigenvalues (largest energies). This leads to coherent structure in models of ideal homogeneous MHD turbulence, which can occur at lowest values of wavenumber k for 3-D cases, and at either lowest or highest k for ideal 2-D magnetofluids. These ideal results appear relevant for unforced, decaying MHD turbulence, so that broken ergodicity effects in MHD turbulence survive dissipation. In comparison, we will also examine ideal hydrodynamic (HD) turbulence, which, in the 3-D case, will be seen to differ fundamentally from ideal MHD turbulence in that coherent structure due to broken ergodicity can only occur at maximum k in numerical simulations. However, a nonzero viscosity eliminates this ideal 3-D HD structure, so that unforced, decaying 3-D HD turbulence is expected to be ergodic. In summary, broken ergodicity in MHD turbulence leads to energetic, large-scale, quasistationary magnetic fields (coherent structures) in numerical models of bounded, turbulent magnetofluids. Thus, broken ergodicity provides a large-scale dynamo mechanism within computer models of homogeneous MHD turbulence. These results may help us to better understand the origin of global magnetic fields in astrophysical and geophysical objects.  相似文献   

14.
In Kim et al. (Kim, E., Hughes, D.W. and Soward, A.M., “An investigation into high conductivity dynamo action driven by rotating convection”, Geophys. Astrophys. Fluid Dynam. 91, 303–332 ().) we investigated kinematic dynamo action driven by rapidly rotating convection in a cylindrical annulus. Here we extend this work to consider self-consistent nonlinear dynamo action in which the back-reaction of the Lorentz force on the flow is taken into account. In particular, we investigate, as a function of magnetic Prandtl number, the evolution of an initially weak magnetic field in two different types of convective flow – one chaotic and the other integrable. On saturation, the latter shows a systematic dependence on the magnetic Prandtl number whereas the former appears not to. In addition, we show how, in keeping with the findings of Cattaneo et al. (Cattaneo, F., Hughes, D.W. and Kim, E., “Suppression of chaos in a simplified nonlinear dynamo model”, Phys. Rev. Lett. 76, 2057–2060 ().), saturation of the growth of the magnetic field is brought about, for the originally chaotic flow, by a strong suppression of chaos.  相似文献   

15.
The Navier–Stokes-α equation is a regularised form of the Euler equation that has been employed in representing the sub-grid scales in large-eddy simulations. Determined efforts have been made to place it on a secure deductive foundation. This requires two steps to be completed. The first is fundamental and consists of establishing from the equations governing the fluid flow, a relationship between two velocities called by Holm (Chaos, 2002a, 12, 518) the “filtered” and “unfiltered” velocities. The second consists of the relation between these two velocities. Until now, the preferred route to the first objective has been variational, by varying the action using Hamilton's principle. Soward and Roberts (J. Fluid Mech., 2008, 604, 297) followed that variational route and established the existence of an important but unwelcome term omitted by Holm in his derivation. It is shown here that the Soward and Roberts result may be derived from Euler's equation by a direct approach with considerably greater efficiency. Holm achieved the second objective by making a “Taylor hypothesis”, which we use here to evaluate the unwelcome term missing from his analysis of the first step. The resulting model equations differ from those of Holm's α model, and the attractive mean Kelvin's circulation theorem that follows from his α equations is no longer valid. For that reason, we call the term omitted by Holm unwelcome.  相似文献   

16.
The first attempt at numerical MHD simulations of the appearance of several current sheets above an active region before a series of elementary flares is described. Energy accumulates in the field of each sheet that can be released during one of the flares. The computations started three days before the appearance of a series of flares, i.e., before the emergence of a new magnetic flux in the active region. The initial (potential) magnetic field was calculated by solving the Laplace equation with an oblique derivative. The boundary conditions on the photosphere were specified from maps of the measured magnetic field in the active region for various instants of time. The Peresvet program solving the full system of MHD equations with dissipative terms was used in the computations. An absolutely implicit scheme conservative relative to the magnetic flux was used. The problem of properly choosing the size of the computational domain and finding the positions of singular magnetic field lines is discussed.  相似文献   

17.
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.  相似文献   

18.
Abstract

A spherical αω-dynamo is studied for small values of the viscous coupling parameter ε ～ v1/2, 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 ε1/2 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.  相似文献   

19.
2017年8月8日四川九寨沟发生了MS7.0地震,距离汶川地震震中约为140km.地震的发生会调整区域应力分布和影响区域地震活动性变化.那么,汶川地震的发生对此次九寨沟地震有何影响?区域应力如何演化?为回答这些问题,本研究基于高性能并行有限元方法和网格自适应加密技术,采用含地形起伏的黏弹非均匀椭球型地球模型,分别计算了汶川地震引起的同震-震后应力时空动态演化及其对此次九寨沟地震的影响.同时,考虑同震静态应力和震后黏弹性应力调整,计算了汶川、芦山和九寨沟地震的发生对周边断层应力积累的影响.计算结果显示汶川地震对九寨沟发震断层的同震库仑应力加载约为0.008 MPa,震后9年的黏弹性应力加载约为0.012~0.016 MPa,可能会使九寨沟地震提前发生.此外,汶川、芦山和九寨沟地震的发生引起龙门山断裂映秀以南段、东昆仑断裂、龙日坝断裂东段和岷江断裂北段及鲜水河断裂康定段应力加载大于0.01 MPa,增加了这些断裂带发生地震的危险性.  相似文献   

20.
Non-linear - dynamo waves existing in an incompressible medium with the turbulence dissipative coefficients depending on temperature are studied in this paper. We investigate of - solar non-linear dynamo waves when only the first harmonics of magnetic induction components are included. If we ignore the second harmonics in the non-linear equation, the turbulent magnetic diffusion coefficient increases together with the temperature, the coefficient of turbulent viscosity decreases, and for an interval of time the value of dynamo number is greater than 1. In these conditions a stationary solution of the non-linear equation for the dynamo waves amplitude exists; meaning that the magnetic field is sufficiently excited. The amplitude of the dynamo waves oscillates and becomes stationary. Using these results we can explain the existence of Maunders minimum.  相似文献   

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