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1.
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   

2.
Abstract

In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.  相似文献   

3.
Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.  相似文献   

4.
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   

5.
Abstract

High resolution numerical simulations extending those of Arnold and Korkina (1983) reveal the existence of at least two windows for kinematic dynamo action in a spatially periodic flow with chaotic streamlines. These occur at moderate magnetic Reynolds number R (8–18) and at high R (27 to 200 or more).  相似文献   

6.

We are investigating numerically the nonlinear behaviour of a space-periodic MHD system with ABC forcing. Most computations are performed for magnetic Reynolds numbers increasing from 0 to 60 and a fixed kinematic Reynolds number, small enough for the trivial solution with a zero magnetic field to be stable to velocity perturbations. At the critical magnetic Reynolds number for the onset of instability of the trivial solution the dominant eigenvalue of the kinematic dynamo problem is real. In agreement with the bifurcation theory new steady states with non-vanishing magnetic field appear in this bifurcation. Subsequent bifurcations are investigated. A regime is detected, where chaotic variations of the magnetic field orientation (analogous to magnetic field reversals) are observed in the temporal evolution of the system.  相似文献   

7.
Abstract

The kinematic dynamo problem is considered for certain steady velocity fields with symmetries that are plausible in a rapidly rotating convective system. By generalizing results proved for the mean field dynamo model by Proctor (1977a), it is shown that for a related “comparison problem” with modified boundary conditions, the eigenvalues are degenerate if there is no axisymmetric mean circulation, with modes of dipole and quadrupole parity excited with equal ease. The comparison problem can be shown to be closely similar to the dynamo problem when there is a region unfavourable to dynamo action surrounding the dynamo region. The near-symmetries found by Roberts (1972) for the mean field model are invoked to suggest that a close correspondence is likely even when this region is absent. It is therefore conjectured that such mean motions may be important in explaining the observed preference for solutions of dipole parity by planetary dynamos.  相似文献   

8.
As a step towards a physically realistic model of a fast dynamo, we study numerically a kinematic dynamo driven by convection in a rapidly rotating cylindrical annulus. Convection maintains the quasi-geostrophic balance whilst developing more complicated time-dependence as the Rayleigh number is increased. We incorporate the effects of Ekman suction and investigate dynamo action resulting from a chaotic flow obtained in this manner. We examine the growth rate as a function of magnetic Prandtl number Pm, which is proportional to the magnetic Reynolds number. Even for the largest value of Pm considered, a clearly identifiable asymptotic behaviour is not established. Nevertheless the available evidence strongly suggests a fast dynamo process.  相似文献   

9.
Abstract

The weak-field Benard-type dynamo treated by Soward is considered here at higher levels of the induced magnetic field. Two sources of instability are found to occur in the intermediate field regime M ～ T 1/12, where M and T are the Hartmann and Taylor numbers. On the time scale of magnetic diffusion, solutions may blow up in finite time owing to destabilization of the convection by the magnetic field. On a faster time scale a dynamic instability related to MAC-wave instability can also occur. It is therefore concluded that the asymptotic structure of this dynamo is unstable to virtual increases in the magnetic field energy.

In an attempt to model stabilization of the dynamo in a strong-field regime we consider two approximations. In the first, a truncated expansion in three-dimensional plane waves is studied numerically. A second approach utilizes an ad hoc set of ordinary differential equations which contains many of the features of convection dynamos at all field energies. Both of these models exhibit temporal intermittency of the dynamo effect.  相似文献   

10.
We study the effect of stratification on large-scale dynamo action in convecting fluids in the presence of background rotation. The fluid is confined between two horizontal planes and both boundaries are impermeable, stress-free and perfectly conducting. An asymptotic analysis is performed in the limit of rapid rotation (τ???1 where τ is the Taylor number). We analyse asymptotic magnetic dynamo solutions in rapidly rotating systems generalising the results of Soward [A convection-driven dynamo I. The weak field case. Philos. Trans. R. Soc. Lond. A 1974, 275, 611–651] to include the effects of compressibility. We find that in general the presence of stratification delays the efficiency of large-scale dynamo action in this regime, leading to a reduction of the onset of dynamo action and in the nonlinear regime a diminution of the large-scale magnetic energy for flows with the same kinetic energy.  相似文献   

11.
The turbulent cross helicity is directly related to the coupling coefficients for the mean vorticity in the electromotive force and for the mean magnetic-field strain in the Reynolds stress tensor. This suggests that the cross-helicity effects are important in the cases where global inhomogeneous flow and magnetic-field structures are present. Since such large-scale structures are ubiquitous in geo/astrophysical phenomena, the cross-helicity effect is expected to play an important role in geo/astrophysical flows. In the presence of turbulent cross helicity, the mean vortical motion contributes to the turbulent electromotive force. Magnetic-field generation due to this effect is called the cross-helicity dynamo. Several features of the cross-helicity dynamo are introduced. Alignment of the mean electric-current density J with the mean vorticity Ω , as well as the alignment between the mean magnetic field B and velocity U , is supposed to be one of the characteristic features of the dynamo. Unlike the case in the helicity or α effect, where J is aligned with B in the turbulent electromotive force, we in general have a finite mean-field Lorentz force J ?×? B in the cross-helicity dynamo. This gives a distinguished feature of the cross-helicity effect. By considering the effects of cross helicity in the momentum equation, we see several interesting consequences of the effect. Turbulent cross helicity coupled with the mean magnetic shear reduces the effect of turbulent or eddy viscosity. Flow induction is an important consequence of this effect. One key issue in the cross-helicity dynamo is to examine how and how much cross helicity can be present in turbulence. On the basis of the cross-helicity transport equation, its production mechanisms are discussed. Some recent developments in numerical validation of the basic notion of the cross-helicity dynamo are also presented.  相似文献   

12.
ABSTRACT

The magnetic fields in the inner parts of some spiral galaxies are understood quite well. Their generation is connected with the dynamo mechanism that is based on the joint action of turbulent diffusion and the α-effect. Usually the galactic dynamo is described with the so-called no-z approximation which takes into account that the galaxy disc is quite thin, with the implication that some spatial derivatives may be replaced by algebraic expressions. Some galaxies have outer rings that are situated at some distance from the galactic centre. The magnetic field can be described there also using the no-z model. As the thickness of such objects is comparable with their width, it is necessary to take into account the z-dependence of the field. We have studied the magnetic field evolution using the no-z approximation and torus dynamo model for the torus with rectangular cross-section in the axisymmetric case.  相似文献   

13.
Abstract

A spherical mean-field dynamo model is considered in which both the mean motion and the mean electro-motive force due to fluctuating motions show some spherical symmetry. It is shown that under some reasonable assumptions the magnetic field is bound to decay to zero.  相似文献   

14.
Abstract

A simple nonlinear model is developed for the solar dynamo, in which the real convective spherical shell is approximated by a thin flat slab, and only the back-reaction of the field B on the helicity is taken into account by choosing the simple law α = α(1-ζB 2), where α and ζ are constants, to represent the decrease in generation coefficient ζ with increasing field strength. Analytic expressions are obtained for the amplitude of the field oscillation and its period, T, as functions of the deviation d - dCT of a dynamo number d from its critical value dcr for regeneration. A symmetry is found for the case of oscillations of small constant amplitude: B(t+½T)= -B(t). A Landau equation is obtained that describes the transition to such oscillations.  相似文献   

15.
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   

16.
Abstract

In this paper a method for solving the equation for the mean magnetic energy <BB> of a solar type dynamo with an axisymmetric convection zone geometry is developed and the main features of the method are described. This method is referred to as the finite magnetic energy method since it is based on the idea that the real magnetic field B of the dynamo remains finite only if <BB> remains finite. Ensemble averaging is used, which implies that fields of all spatial scales are included, small-scale as well as large-scale fields. The method yields an energy balance for the mean energy density ε ≡ B 2/8π of the dynamo, from which the relative energy production rates by the different dynamo processes can be inferred. An estimate for the r.m.s. field strength at the surface and at the base of the convection zone can be found by comparing the magnetic energy density and the outgoing flux at the surface with the observed values. We neglect resistive effects and present arguments indicating that this is a fair assumption for the solar convection zone. The model considerations and examples presented indicate that (1) the energy loss at the solar surface is almost instantaneous; (2) the convection in the convection zone takes place in the form of giant cells; (3) the r.m.s. field strength at the base of the solar convection zone is no more than a few hundred gauss; (4) the turbulent diffusion coefficient within the bulk of the convection zone is about 1014cm2s?1, which is an order of magnitude larger than usually adopted in solar mean field models.  相似文献   

17.
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   

18.
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   

19.
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   

20.
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   

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