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1.
M.R.E. Proctor 《地球物理与天体物理流体动力学》2013,107(3):235-240
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
2.
K. H. Rädler 《地球物理与天体物理流体动力学》2013,107(3-4):191-211
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. 相似文献
3.
Abstract The solution of the full nonlinear hydromagnetic dynamo problem is a major numerical undertaking. While efforts continue, supplementary studies into various aspects of the dynamo process can greatly improve our understanding of the mechanisms involved. In the present study, the linear stability of an electrically conducting fluid in a rigid, electrically insulating spherical container in the presence of a toroidal magnetic field Bo(r,θ)lø and toroidal velocity field Uo(r,θ)lø, [where (r,θ,ø) are polar coordinates] is investigated. The system, a model for the Earth's fluid core, is rapidly rotating, the magnetostrophic approximation is used and thermal effects are excluded. Earlier studies have adopted a cylindrical geometry in order to simplify the numerical analysis. Although the cylindrical geometry retains the fundamental physics, a spherical geometry is a more appropriate model for the Earth. Here, we use the results which have been found for cylindrical systems as guidelines for the more realistic spherical case. This is achieved by restricting attention to basic states depending only on the distance from the rotation axis and by concentrating on the field gradient instability. We then find that our calculations for the sphere are in very good qualitative agreement both with a local analysis and with the predictions from the results of the cylindrical geometry. We have thus established the existence of field gradient modes in a realistic (spherical) model and found a sound basis for the study of various other, more complicated, classes of magnetically driven instabilities which will be comprehensively investigated in future work. 相似文献
4.
Abstract A standard approach to the kinematic dynamo problem is that pioneered by Bullard and Gellman (1954), which utilizes the toroidal-poloidal separation and spherical harmonic expansion of the magnetic and velocity fields. In these studies, the velocity field is given as a combination of small number of toroidal and poloidal harmonics, with their radial dependences prescribed by some physical considerations. Starting from the original paper of Bullard and Gellman (1954), a number of authors repeated such analyses on different combination of velocity fields, including the most recent and comprehensive effort by Dudley and James (1989). In this paper, we re-examine the previous kinematic dynamo models, using the computer algebra approach initiated by Kono (1990). This method is particularly suited to this kind of research since different velocity fields can be treated by a single program. We used the distribution of magnetic energies in various harmonics to infer the convergence of the results. The numerical results obtained in this study for the models of Bullard and Gellman (1954), Lilley (1970), Gubbins (1973), Pekeris et al. (1973), Kumar and Roberts (1975), and Dudley and James (1989) are consistent with the previously reported results, in particular, with the extensive calculation of Dudley and James. In addition, we found that the combination of velocities used by Lilley can support the dynamo action if the radial dependence of the velocity is modified. We also examined the helicity distributions in these dynamo models, to see if there is any correlation between the helicity and the efficiency of dynamo action. A successful dynamo can result both from the cases in which the helicity distributions are symmetric or antisymmetric with respect to the equator. In both cases, it appears that the dynamo action is efficient if the volume integral of helicity over a hemisphere is large. 相似文献
5.
D. J. Ivers 《地球物理与天体物理流体动力学》2014,108(6):716-733
In an electrically conducting fluid, two types of turbulence with a preferred direction are distinguished: planar turbulence, in which every velocity in the turbulent ensemble of flows has no component in the given direction; and two-dimensional turbulence, in which every velocity in the turbulent ensemble is invariant under translation in the preferred direction. Under the additional assumptions of two-scale and homogeneous turbulence with zero mean flow, the associated magnetohydrodynamic alpha- and beta-effects are derived in the second-order correlation approximation (SOCA) when the electrically conducting fluid occupies all space. Limitations of the SOCA are well known, but alpha- and beta-effects of a turbulent flow are useful in interpreting the dynamo effects of the turbulence. Two antidynamo theorems, which establish necessary conditions for dynamo action, are shown to follow from the special structures of these alpha- and beta-effects. The theorems, which are analogues of the laminar planar velocity and two-dimensional antidynamo theorems, apply to all turbulent ensembles with the prescribed alpha- and beta-effects, not just the planar and two-dimensional ensembles. The mean magnetic field is general in the planar theorem but only two-dimensional in the two-dimensional theorem. The two theorems relax the previous restriction to turbulence which is both two-dimensional and planar. The laminar theorems imply decay of the total magnetic field for any velocity of the associated turbulent ensemble. However, the mean-field theorems are not fully consistent with the laminar theorems because further conditions beyond those arising from the turbulence must be imposed on the beta-effect to establish decay of the mean magnetic field. In particular, negative turbulent magnetic diffusivities must be restricted. It is interesting that there is no inconsistency in the alpha-effects. The failure of the SOCA with the two-scale approximation to simply preserve the laminar antidynamo theorems at the beta-effect level is a further demonstration of the restricted validity of the theory and shows that negative diffusivity effects derived by approximation methods must be treated cautiously. 相似文献
6.
Using a magnetic dynamo model, suggested by Kazantsev (J. Exp. Theor. Phys. 1968, vol. 26, p. 1031), we study the small-scale helicity generation in a turbulent electrically conducting fluid. We obtain the asymptotic dependencies of dynamo growth rate and magnetic correlation functions on magnetic Reynolds numbers. Special attention is devoted to the comparison of a longitudinal correlation function and a function of magnetic helicity for various conditions of asymmetric turbulent flows. We compare the analytical solutions on small scales with numerical results, calculated by an iterative algorithm on non-uniform grids. We show that the exponential growth of current helicity is simultaneous with the magnetic energy for Reynolds numbers larger than some critical value and estimate this value for various types of asymmetry. 相似文献
7.
John T. Donald & Paul H. Roberts 《地球物理与天体物理流体动力学》2013,107(5):367-384
Intermediate dynamos are axisymmetric, spherical models that evade Cowling's theorem by invoking an α-effect to create the meridional magnetic field from the zonal. Usually the energy source maintaining the motions is a specified thermal wind, but here the dynamo is driven by the buoyancy created by a uniform distribution of heat sources. It has been argued by Braginsky and Meytlis (this journal, vol. 55, 1990) that, in a rapidly rotating, strongly magnetic system such as the Earth's core, heat is transported principally by a small-scale turbulence that is highly anisotropic. They conclude that the diffusion of heat parallel to the rotation axis is then significantly greater than it is in directions away from that axis. A preliminary study of the consequences of this idea is reported here. Solutions are derived numerically using both isotropic and non-isotropic thermal diffusivity tensors, and the results are compared. It is shown that even a small degree of anisotropy can materially alter the character of the dynamo. 相似文献
8.
M. M. Vishik 《地球物理与天体物理流体动力学》2013,107(1-3):151-167
Abstract Using an asymptotic expansion of Green's function for the problem of magnetic field generation by 3D steady flow of highly conducting fluid a general antidynamo theorem is proved in the case of no exponential stretching of liquid particles. Explicit formulae connecting the spectrum of the magnetic modes with the geometry of the Lagrangian trajectories are obtained. The existence of the fast dynamo action for special flows with exponential stretching is proved under the condition of smoothness of the fields of stretching and non-stretching directions. 相似文献
9.
The magnetohydrodynamic dynamo problem is solved for an electrically conducting spherical fluid shell with spherically symmetric distributions of gravity and heat sources. The dynamics of motions generated by thermal buoyancy are dominated by the effects of rotation of the fluid shell. Dynamos are found for low and intermediate values of the Taylor number, T ? 105, if the scale of the nonaxisymmetric component of the velocity field is sufficiently small. The generation of magnetic fields of quadrupolar symmetry is preferred at Rayleigh numbers close to the critical value Rc for onset of convection. As the Rayleigh number increases, the generation of dipolar magnetic fields becomes preferred. 相似文献
10.
We investigate numerically kinematic dynamos driven by flow of electrically conducting fluid in the shell between two concentric differentially rotating spheres, a configuration normally referred to as spherical Couette flow. We compare between axisymmetric (2D) and fully 3D flows, between low and high global rotation rates, between prograde and retrograde differential rotations, between weak and strong nonlinear inertial forces, between insulating and conducting boundaries and between two aspect ratios. The main results are as follows. Azimuthally drifting Rossby waves arising from the destabilisation of the Stewartson shear layer are crucial to dynamo action. Differential rotation and helical Rossby waves combine to contribute to the spherical Couette dynamo. At a slow global rotation rate, the direction of differential rotation plays an important role in the dynamo because of different patterns of Rossby waves in prograde and retrograde flows. At a rapid global rotation rate, stronger flow supercriticality (namely the difference between the differential rotation rate of the flow and its critical value for the onset of nonaxisymmetric instability) facilitates the onset of dynamo action. A conducting magnetic boundary condition and a larger aspect ratio both favour dynamo action. 相似文献
11.
M. R. E. Proctor 《地球物理与天体物理流体动力学》2013,107(1):127-145
Abstract A magnetohydrodynamic, dynamo driven by convection in a rotating spherical shell is supposed to have averages that are independent of time. Two cases are considered, one driven by a fixed temperature difference R and the other by a given internal heating rate Q. It is found that when q, the ratio of thermal conductivity to magnetic diffusivity, is small, R must be of order q ?4/3 and Q of order q ?2 for dynamo action to be possible; q is small in the Earth's core, so it is hoped that the criteria will prove useful in practical as well as theoretical studies of dynamic dynamos. The criteria can be further strengthened when the ohmic dissipation of the field is significant in the energy balance. The development includes the derivation of two necessary conditions for dynamo action, both based on the viscous dissipation rate of the velocity field that drives the dynamo. 相似文献
12.
Takesi Yukutake 《Physics of the Earth and Planetary Interiors》1981,24(4):253-258
An examination of the westward drift of the geomagnetic field indicates that the drift velocity is almost independent of latitude, suggesting a uniform rigid rotation of spherical shape. When the geomagnetic field is separated into standing and drifting components and expressed in a spherical harmonic series, a lack of sectorial terms is noted in the standing field. It is shown that these features are well explained by a stratified core model.The core is supposed to be stratified near the surface where toroidal fluid motions are predominant. In the deeper part, the fluid motion is two-dimensional, forming Taylor columns. A simplified core model is assumed to represent these features, in which the core is divided into two parts, an outer spherical shell that rotates westwards at a uniform rate of 0.3° y?1 and a central sphere in which the two-dimensional columnar motions reside. The toroidal motions in the outer spherical interact with the dipole field to induce the drifting field, whereas the columnar motions generate the standing field through interaction with a toroidal field. It follows that a small velocity as 5 × 10?3 cm s?1 for the stratified motion is sufficient to create the observed drifting field. 相似文献
13.
It is known that a sharp decrease in the angular velocity of the accretion disk around a black hole could in principle produce a kinematic axisymmetric dynamo, in contrast to the classical situation described by Cowling's antidynamo theorem. Here the effect of a nontrivial poloidal velocity of the disk is studied, showing that a strong gradient of this velocity enhances the possibilities of a working dynamo. 相似文献
14.
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. 相似文献
15.
A. M. Soward 《地球物理与天体物理流体动力学》2013,107(1-2):81-107
Abstract Dynamo action in a highly conducting fluid with small magnetic diffusivity η is particularly sensitive to the topology of the flow. The sites of rapid magnetic field regeneration, when they occur, appear to be located at the stagnation points or in regions where the particle paths are chaotic. Elsewhere only slow dynamo action is to be expected. Two such examples are the nearly axially symmetric dynamo of Braginsky and the generalisation to smooth velocity fields of the Ponomarenko dynamo. Here a method of solution is developed, which applies to both these examples and is applicable to other situations, where magnetic field lines are close to either closed or spatially periodic contours. Particular attention is given to field generation in the neighbourhood of resonant surfaces where growth rates may be intermediate between the slow diffusive and fast convective time scales. The method is applied to the case of the two-dimensional ABC-flows, where it is shown that such intermediate dynamo action can occur on resonant surfaces. 相似文献
16.
We study semi-analytical time-dependent solutions of the relativistic MHD equations for the fields and the fluid emerging from a spherical source. We assume uniform expansion of the field and the fluid and a polytropic relation between the density and the pressure of the fluid. The expansion velocity is small near the base but approaches the speed of light at the light sphere where the flux terminates. We find self-consistent solutions for the density and the magnetic flux. The details of the solution depend on the ratio of the toroidal and the poloidal magnetic field, the ratio of the energy carried by the fluid and the electromagnetic field and the maximum velocity it reaches. 相似文献
17.
The effects of variable viscosity on flow dynamics within spherical shells are investigated using a finite-element thermal convection model, and preliminary result for cases with relatively low Rayleigh numbers and small viscosity contrasts are reported. These results demonstrate some general effects of viscosity variation on mantle dynamics, and, in particular, the generation of toroidal energy. Since lateral viscosity variations are necessary in the generation of toroidal motion in a thermally driven convective system, it is not surprising our results show that flows with greater viscosity contrasts produce greater amounts of toroidal energy. Our preliminary study further shows that solutions become more time-dependent as viscosity contrasts increase. Increasing the Rayleigh number is also found to increase the magnitude of toroidal energy. Internal heating, on the other hand, appears to lead to less toroidal energy compared wth bottom heating because it tends to produce a thermally more uniform interior and thus smaller viscosity variations. 相似文献
18.
Siegfried Franck 《Physics of the Earth and Planetary Interiors》1982,27(4):249-254
This paper is concerned with some new problems of the dynamics and energetics of the Earth's core. The model of the so-called gravitationally-powered dynamo is investigated under the assumption of liquid immiscibility in the FeS system as a possible core material. In this way the growing inner core causes nucleation of small FeS-droplets that ascend under the release of gravitational potential energy. This energy is enough to drive a dynamo with a toroidal magnetic field of mean size. 相似文献
19.
The shear viscosity of a suspension of deformable bubbles dispersed within a Newtonian fluid is calculated as a function of the shear rate and strain. The relative importance of bubble deformation in the suspension is characterized by the capillary number (Ca), which represents the ratio of viscous and surface tension stresses. For small Ca, bubbles remain nearly spherical, and for sufficiently large strains the viscosity of suspension is greater than that of the suspending fluid, i.e. the relative viscosity is greater than 1. If Ca>O(1) the relative viscosity is less than one. In the limit that Ca→∞ (surface tension is dynamically negligible), numerical calculations for a suspension of spherical bubbles agree well with the experimental measurements of Lejeune et al. (1999, Rheology of bubble-bearing magmas. Earth Planet. Sci. Lett., vol. 166, pp. 71–84). In general, bubbles have a modest effect on the relative viscosity, with viscosity changing by less than a factor of about 3 for volume fractions up to 50%. 相似文献
20.
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. 相似文献