Topology of the magnetic field and geomagnetism

We define the topology of an (axisymmetric) magnetic field, a set of qualitative labels characterizing the connectivity of the lines of force. Under the special continuous deformations of theB-field defined by time evolution under the dynamo equation in the convective regime (where the frozen-in behavior dominates the diffusion), this topology is preserved. This theorem should have applications to the study of time-varying magnetic fields in that regime in the case that exact or even approximate solutions are difficult to obtain. A partial generalization to the general case of convection and diffusion is made. As an application, a critique of Hibberd's recent theory of a time-dependent axisymmetric geomagnetic dynamo whose dipole-like field undergoes successive reversals is given.     

Turbulent regeneration of magnetic fields

We calculate an exact form for the regeneration term of the turbulent dynamo equation which is valid for arbitrary values of the magnetic Reynolds number. It is shown that finite conductivity can change the regenerative character of the dynamo, depending on the geometry and character of the fluid motion in the turbulent eddies.On leave from the Departamento de Fisica Teorica e Experimental, UFRN, Natal, RB, Brazil.     

Anelastic convection-driven dynamo benchmarks

Benchmark solutions for fully nonlinear anelastic compressible convection and dynamo action in a rotating spherical shell are proposed. Three benchmarks are specified. The first is a purely hydrodynamic case, which is steady in a uniformly drifting frame. The second is a self-excited saturated dynamo solution, also steady in a drifting frame. The third is again a self-excited dynamo but is unsteady in time, and it has a higher Rayleigh number than the steady dynamo benchmark. Four independent codes have been tested against these benchmarks, and very satisfactory agreement has been found. This provides an accurate reference standard against which new anelastic codes can be tested.     

Mathematical theory of the ionospheric dynamo

A mathematical theory of the three-dimensional ionospheric dynamo is described, which is exact within the context of the assumption that the magnetic field lines are perfectly conducting.Explicit formulae are obtained for the electric fields, and hence for the current systems, which are excited by the dynamo e.m.f. associated with an arbitrary given wind field.They can also be applied to determine the fields arising from an external source of e.m.f., originating for example in the magnetosphere.The convergence properties of the solution are investigated for a simple symmetric wind field; it is found that inclusion of spherical harmonics up to degree 80 is sufficient to yield the features of the electrojet region, now fully coupled to the global current system.     

Spherical Oscillatory alpha2 Dynamo Induced by Magnetic Coupling between a Fluid Shell and an Inner Electrically Conducting Core: Relevance to the Solar Dynamo

A two-layer spherical alpha2 dynamo model consisting of an inner electrically conducting core (magnetic diffusivity lambdai and radius ri) with alpha=0 surrounded by an electrically conducting spherical shell (magnetic diffusivity lambdao and radius ro) with a constant alpha is shown to exhibit oscillatory behavior for values of beta=lambdai&solm0;lambdao and ri&solm0;ro relevant to the solar dynamo. Time-dependent dynamo solutions require ri&solm0;ro>/=0.55 and beta相似文献   

Mean-field approach to spherical dynamo models

The mean-field approach to dynamo theory has proved to be a useful tool in the investigation of cosmical magnetic fields. This paper gives a systematic discussion of this approach for spherical dynamo models as suggested by cosmical bodies. At first some fundamentals of dynamo theory are explained with particular attention to formulations in terms of toroidal and poloidal magnetic fields. Starting from the general ideas of mean-field magnetohydrodynamics the relevant mean-field equations for the models envisaged are derived and discussed. The considerations are not restricted to motions of turbulent nature, motions with more or less regular flow patterns are admitted too. A new representation of the crucial electromotive force caused by the fluctuating motions is given. For an important special case the dependence of this electromotive force on the motions is calculated. The mean-field concept is in particular elaborated under the assumption that the motions show certain symmetry and stationarity properties as to be expected at cosmical bodies. The respective results for the electromotive force caused by the fluctuating motions are discussed in detail. Within this frame the possibilities of dynamo mechanisms are systematically studied. In addition to the well-known α² and αω-mechanisms some others, termed β², βω, and δω-mechanisms, are envisaged, whose relevance for cosmical objects remains to be investigated. In a following paper numerical results are given for dynamo models with mechanisms as envisaged here.     

Relating stellar cycle periods to dynamo calculations

Stellar magnetic activity in slowly rotating stars is often cyclic, with the period of the magnetic cycle depending critically on the rotation rate and the convective turnover time of the star. Here we show that the interpretation of this law from dynamo models is not a simple task. It is demonstrated that the period is (unsurprisingly) sensitive to the precise type of non-linearity employed. Moreover the calculation of the wave-speed of plane-wave solutions does not (as was previously supposed) give an indication of the magnetic period in a more realistic dynamo model, as the changes in length-scale of solutions are not easily captured by this approach. Progress can be made, however, by considering a realistic two-dimensional model, in which the radial length-scale of waves is included. We show that it is possible in this case to derive a more robust relation between cycle period and dynamo number. For all the non-linearities considered in the most realistic model, the magnetic cycle period is a decreasing function of | D | (the amplitude of the dynamo number). However, discriminating between different non-linearities is difficult in this case and care must therefore be taken before advancing explanations for the magnetic periods of stars.     

On self–killing and self–creating dynamos

Numerical studies with a spherical dynamo model have shown two remarkable phenomena. The model consists of a spherical body of an electrically conducting incompressible uid surrounded by free space. In addition to a rotation of the body an inner motion due to a given forcing is considered which satisfies a no–slip condition at the boundary. The full interaction of magnetic field and motion is taken into account. Starting from a fluid motion capable of dynamo action and a very weak magnetic field it was observed that the growing magnetic field destroys the dynamo property of the motion and then decays, and that the system ends up in a state with another motion incapable of dynamo action and zero magnetic field. In another case with a motion unable to prevent small magnetic fields from decay it proved to be possible that stronger magnetic fields deform it so that a dynamo starts to work which enables the system to approach a steady state with a finite magnetic field.     

Configurations of dynamo waves in a two-layer medium and the 11-year solar cycle

The behavior of dynamo waves in a two-layer medium is investigated in terms of the Parker dynamo model. The solar cycle duration is shown to depend on the ratio of turbulent diffusivities in the layers. Meridional circulation has been incorporated into the Parker system. An increase in the intensity of meridional flows is shown to decelerate the propagation of dynamo waves. The minimum of solar magnetic activity can occur not only in the case of intense meridional circulation in both layers but also when a difference in physical characteristics arises between the layers and the meridional flows are moderate.     

Can turbulent plane‐shear flows support large‐scale dynamos?

Turbulent plane‐shear flow is found to show same basic effects of mean‐fieldMHD as rotating turbulence. In particular, the mean electromotive force (EMF) includes highly anisotropic turbulent diffusion and alpha‐effect. Only magnetic diffusion remains for spatially‐uniform turbulence. The question is addressed whether in this case a self‐excitation of a magnetic field by so‐called sher‐current dynamo is possible and the quasilinear theory provides a negative answer. The streamaligned component of the EMF has the sign opposite to that required for dynamo. If, however, the turbulence is not uniform across the flow direction then a dynamo‐active α ‐effect emerges. The critical magnetic Reynolds number for the alpha‐shear dynamo is estimated to be slightly above ten. Possibilities for cross‐checking theoretical predictions with MHD experiments are discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Size and compositional constraints of Ganymede's metallic core for driving an active dynamo

Ganymede has an intrinsic magnetic field which is generally considered to originate from a self-excited dynamo in the metallic core. Driving of the dynamo depends critically on the satellite's thermal state and internal structure. However, the inferred structure based on gravity data alone has a large uncertainty, and this makes the possibility of dynamo activity unclear; variations in core size and composition significantly change the heat capacity and alter the cooling history of the core. The main objectives of this study is to explore the structural conditions for a currently active dynamo in Ganymede using numerical simulations of the thermal history, and to evaluate under which conditions Ganymede can maintain the dynamo activity at present. We have investigated the satellite's thermal history using various core sizes and compositions satisfying the mean density and moment of inertia of Ganymede, and evaluate the temperature and heat flux at the core-mantle boundary (CMB). Based on the following two conditions, we evaluate the possibility of dynamo activity, thereby reducing the uncertainty of the previously inferred interior structure. The first condition is that the temperature at the CMB must exceed the melting point of a metallic core, and the second is that the heat flux through the CMB must exceed the adiabatic temperature gradient. The mantle temperature starts to increase because of the decay of long-lived radiogenic elements in the rocky mantle. After a few Gyr, radiogenic elements are exhausted and temperature starts to decrease. As the rocky mantle cools, the heat flux at the CMB steadily increases. If the temperature and heat flux at the CMB satisfy these conditions simultaneously, we consider the case as capable of driving a dynamo. Finally, we identify the Dynamo Regime, which is the specific range of internal structures capable of driving the dynamo, based on the results of simulations with various structures. If Ganymede's self-sustained magnetic field were maintained by thermal convection, the satellite's metallic core would be relatively large and, in comparison to other terrestrial-type planetary cores, strongly enriched in sulfur. The dynamo activity and the generation of the magnetic field of Ganymede should start from a much later stage, possibly close to the present.     

On the Saturation of Astrophysical Dynamos: Numerical Experiments with the No-Cosines Flow

In the context of astrophysical dynamos we illustrate that the no-cosines flow, with zero mean helicity, can drive fast dynamo action and we study the dynamo’s mode of operation during both the linear and non-linear saturation regimes. It turns out that in addition to a high growth rate in the linear regime, the dynamo saturates at a level significantly higher than normal turbulent dynamos, namely at exact equipartition when the magnetic Prandtl number Pr_m∼ 1. Visualization of the magnetic and velocity fields at saturation will help us to understand some of the aspects of the non-linear dynamo problem.     

A theoretically derived energy coupling function for the magnetosphere

The energy coupling function between the solar wind and the magnetosphere can be obtained for two extreme situations, in which the magnetospheric geometry is determined primarily by either (i) the interplanetary magnetic field, or (ii) the solar wind pressure. In this paper, we obtained an expression for the energy coupling function by assuming a simple interpermeation of the interplanetary and geomagnetic fields. Two important quantities in this case are the potential difference between the two neutral points and the amount of open flux. From these two overall quantities, the voltage and the current of the magnetospheric dynamo are calculated. The dynamo power output represents the rate at which energy is transferred from the solar wind to the magnetosphere. The derived functional dependence on the interplanetary conditions provides a theoretical basis for the energy coupling function previously deduced from observations.     

Steady dynamos in finite domains: an integral equation approach

The paper deals with the integral equation approach to steady kinematic dynamo models in finite domains based on Biot‐Savart's law. The role of the electric potential at the boundary is worked out explicitly. As an example, a modified version of the simple spherical α‐effect dynamo model proposed by Krause and Steenbeck is considered in which the α‐coeffcient is no longer constant but may vary with the radial coordinate. In particular, the results for the original model are re‐derived. Possible applications of this integral equation approach for numerical simulations of dynamos in arbitrary geometry and for an “inverse dynamo theory” are sketched.     

The nonlinear dynamo problem: Small oscillatory solutions in a strongly simplified model

For a dynamo model obtained by idealizing features of the Sun, an analytic proof of the existence and a recursion formula for the determination of small time periodic solutions in a finite vicinity of the critical dynamo numbers are given. The nonlinear problem is solved by transforming the boundary value problem of the induction equation into a fixed point problem in an infinite dimensional sequence space and then applying the LYAPUNOV -SCHMIDT method to reduce it to a relationship between dynamo number, amplitude and frequency.     

Dynamo action and the period gap in cataclysmic variables

The conjecture is presented that the gap in the distribution of the orbital periods of cataclysmic variables is related to a particular kind of hydromagnetic dynamo, called an interface dynamo, operating near the base of the convective envelope of their secondary components. Such a dynamo is characterized by the spatial separation of the regions where differential rotation and the α effect operate. Unlike conventional dynamos, the linear growth rate of an interface dynamo becomes negative for highly supercritical dynamo numbers, leading to the disappearance of the dynamo action. If such a result, from linear theory, is confirmed by non-linear calculations, it may provide a physical basis for the so-called disrupted magnetic braking hypothesis, invoked to explain the existence of the period gap by several evolutionary models of cataclysmic variables.     

Dynamo action in an ambient field

The origin of global magnetic fields in celestial bodies is generally ascribed to dynamo action by fluid motions in their electrically conducting interiors. Some objects – e.g. close‐in extra‐solar planets or the moons of some giant planets – are embedded in ambient magnetic fields which modify the generation of the internal field in these bodies. Recently, the feedback of the magnetospheric field by Chapman‐Ferraro currents in the magnetopause onto the interior dynamo has been proposed to explain the observed weakness of the intrinsic magnetic field of planet Mercury. We study a simplified mean‐field dynamo model which allows us to analytically address various issues like positive and negative feedback situations, stationary versus time‐dependent solutions, and the stability of weak and strong field branches. We discuss the influence of the response function on the solutions when the external field depends on the strength of the intrinsic field like in the situation of the feedback dynamo of Mercury. We find that the feedback mechanism works only for a narrow range of dynamo numbers in the case of Mercury which makes him unique in our solar system. We conclude with some implications for extra‐solar planets (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the possibility of a bimodal solar dynamo

A simple way to couple an interface dynamo model to a fast tachocline model is presented, under the assumption that the dynamo saturation is due to a quadratic process and that the effect of finite shear layer thickness on the dynamo wave frequency is analogous to the effect of finite water depth on surface gravity waves. The model contains one free parameter which is fixed by the requirement that a solution should reproduce the helioseismically determined thickness of the tachocline. In this case it is found that, in addition to this solution, another steady solution exists, characterized by a four times thicker tachocline and 4–5 times weaker magnetic fields. It is tempting to relate the existence of this second solution to the occurrence of grand minima in solar activity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A two-layer αω-dynamo model with dynamic feedback on the ω-effect

In an attempt to produce a simple representation of an interface dynamo, I examine a dynamo model composed of two one-dimensional (radially averaged) pseudo-spherical layers, one in the convection zone and possessing an α-effect, and the other in the tachocline and possessing an ω-effect. The two layers communicate by means of an analogue of Newton's law of cooling, and a dynamical back-reaction of the magnetic field on ω is provided. Extensive bifurcation diagrams are calculated for three separate values of η, the ratio of magnetic diffusivities of the two layers. I find recognizable similarities to, but also dramatic differences from, the comparable one-layer model examined by Roald &38; Thomas. In particular, the solar-like dynamo mode found previously is no longer stable in the two-layer version; in its place there is a sequence of periodic, quasi-periodic and chaotic modes probably created in a homoclinic bifurcation. These differences are important enough to provide support for the view that the solar dynamo cannot be meaningfully modelled in one dimension.     

Magnetic activity and evolution of Algol-type stars – II

We examine the possibility of probing dynamo action in mass-losing stars, components of Algol-type binaries. Our analysis is based on the calculation of non-conservative evolution of these systems. We model the systems U Sge and β Per where the more massive companion fills its Roche lobe at the main sequence (case AB) and where it has a small helium core (early case B) respectively. We show that to maintain evolution of these systems at the late stages which are presumably driven by stellar 'magnetic braking', an efficient mechanism for producing large-scale surface magnetic fields in the donor star is needed. We discuss the relevance of dynamo operation in the donor star to the accelerated mass transfer during the late stages of evolution of Algol-type binaries. We suggest that the observed X-ray activity in Algol-type systems may be a good indicator of their evolutionary status and internal structure of the mass-losing stellar components.