Torus dynamo in the outer rings of galaxies

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.     

The DRESDYN project: liquid metal experiments on dynamo action and magnetorotational instability

ABSTRACT

Magnetic fields of planets, stars and galaxies are generated by self-excitation in moving electrically conducting fluids. Once produced, magnetic fields can play an active role in cosmic structure formation by destabilising rotational flows that would be otherwise hydrodynamically stable. For a long time, both hydromagnetic dynamo action as well as magnetically triggered flow instabilities had been the subject of purely theoretical research. Meanwhile, however, the dynamo effect has been observed in large-scale liquid sodium experiments in Riga, Karlsruhe and Cadarache. In this paper, we summarise the results of liquid metal experiments devoted to the dynamo effect and various magnetic instabilities such as the helical and the azimuthal magnetorotational instability and the Tayler instability. We discuss in detail our plans for a precession-driven dynamo experiment and a large-scale Tayler–Couette experiment using liquid sodium, and on the prospects to observe magnetically triggered instabilities of flows with positive shear.     

Cross helicity and related dynamo

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.     

Periodic and aperiodic dynamo waves

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.     

Inverse dynamo problem in a cylinder

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.     

A nonlinear dynamo driven by rapidly rotating convection

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.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

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.     

The galactic dynamo: Axisymmetric and non-axisymmetric modes

Abstract

We discuss recent developments in the theory of large-scale magnetic structures in spiral galaxies. In addition to a review of galactic dynamo models developed for axisymmetric disks of variable thickness, we consider the possibility of dominance of non-axisymmetric magnetic modes in disks with weak deviations from axial symmetry. Difficulties of straightforward numerical simulation of galactic dynamos are discussed and asymptotic solutions of the dynamo equations relevant for galactic conditions are considered. Theoretical results are compared with observational data.     

Properties of nonlinear dynamo waves

Abstract

Dynamo theory offers the most promising explanation of the generation of the sun's magnetic cycle. Mean field electrodynamics has provided the platform for linear and nonlinear models of solar dynamos. However the nonlinearities included arc (necessarily) arbitrarily imposed in these models. This paper conducts a systematic survey of the role of nonlinearities in the dynamo process, by cousidering the behaviour of dynamo waves in the nonlinear regime. It is demonstrated that only by considering realistic nonlinearities that are non-local in space and time can modulation of the basic dynamo wave be achieved. Moreover this modulation is greatest when there is a large separation of timescales provided by including a low magnetic Prandtl number in the equation for the velocity perturbations.     

On dynamo action in the high-conductivity limit

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.     

Necessary conditions for the magnetohydrodynamic dynamo

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.     

Kinematic turbulent dynamo in a shear flow

Abstract

We consider the turbulent dynamo action in a differentially rotating flow by making use of a kinematic approach when the effect of a generated magnetic field on turbulent motions is neglected. The mean electromotive force is calculated in a quasilinear approximation. Differential rotation can stretch turbulent magnetic field lines and break the symmetry of turbulence in such a way that turbulent motions become suitable for the generation of a large scale magnetic field. The presence of shear changes the type of an equation governing the mean magnetic field. Due to shear stresses the mean magnetic field can be generated by a turbulent dynamo action even in a uniform turbulence. The growth rate depends on the length scale of the mean field being faster for the field with a smaller length scale.     

Note on the scalar dynamo model

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.     

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.     

Hydromagnetic dynamo model with spiral convection

Summary The present paper deals with a hydromagnetic dynamo model of the generation mechanism of the Earth's magnetic field. An attempt has been made at selecting a flow-velocity field in the Earth's core which would satisfy the condition 0 for regenerating the field according to [2], and which would yield a velocity field pattern on the core surface as given in the papers by Kahle et al. [9]. These conditions are satisfied by the velocityv=V ₁+U ₂ ^c –V ₂ ^c and, geometrically, this velocity field is represented in space by a spiral convective motion. On the core surface two downflows and two upflows with the corresponding rotating cells may then be found. Only the axisymmetric harmonic component regeneration of the magnetic field has been considered. Adequate regeneration equations have been obtained by means of Braginski's method of quantity estimates in order of magnitude.     

The flux ejection dynamo effect

Abstract

The mean-field effects of cyclonic convection become increasingly complex when the cyclonic rotation exceeds ½-π. Net helicity is not required, with negative turbulent diffusion, for instance, appearing in mirror symmetric turbulence. This paper points out a new dynamo effect arising in convective cells with strong asymmetry in the rotation of updrafts as against downdrafts. The creation of new magnetic flux arises from the ejection of reserve flux through the open boundary of the dynamo region. It is unlike the familiar α-effect in that individual components of the field may be amplified independently. Several formal examples are provided to illustrate the effect. Occurrence in nature depends upon the existence of fluid rotations of the order of π in the convective updrafts. The flux ejection dynamo may possibly contribute to the generation of field in the convective core of Earth and in the convective zone of the sun and other stars.     

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).     

Evolution of aligned states within nonlinear dynamos

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.     

Kinematic dynamo action with helical symmetry in an unbounded fluid conductor: Part 2. Further development of an explicit solution for the prototype case of lortz

Abstract

An explicit example of a steady prototype Lortz dynamo is elaborated in terms of a previously derived illustrative, exact, closed form solution to the nonlinear dynamo equations. The eigenvalue character of the dynamo problem is now introduced which simplifies the solution. The magnetic field lines, which lie on circular cylinders, and velocity streamline pattern are then displayed and discussed. Analysis of the magnetic energy balance by way of the Poynting flux reveals the existence of a finite critical cylinder across which zero net magnetic energy flows, thereby proving that the material inside is a self-excited dynamo, despite the fact that the total magnetic energy is unbounded.     

A steady state of the disc dynamo

Abstract

We discuss the steady states of the αω-dynamo in a thin disc which arise due to α-quenching. Two asymptotic regimes are considered, one for the dynamo numberD near the generation thresholdD ₀, and the other for |D| ? 1. Asymptotic solutions for |D—D ₀| ? |D ₀| have a rather universal character provided only that the bifurcation is supercritical. For |D| ? 1 the asymptotic solution crucially depends on whether or not the mean helicity α, as a function ofB, has a positive root (hereB is the mean magnetic field). When such a root exists, the field value in the major portion of the disc is O(l), while near the disc surface thin boundary layers appear where the field rapidly decreases to zero (if the disc is surrounded by vacuum). Otherwise, when α = O(|B|^?s) for |B| → ∞, we demonstrate that |B| = O(|D|^1/s ) and the solution is free of boundary layers. The results obtained here admit direct comparison with observations of magnetic fields in spiral galaxies, so that an appropriate model of nonlinear galactic dynamos hopefully could be specified.