The cowling anti-dynamo theorem

Abstract

The Cowling anti-dynamo theorem is proved using the Bullard and Gellman spectral formulation of kinematic dynamo theory.     

On the robustness of the toroidal velocity theorem

In dynamo theory the toroidal velocity theorem in its classical version (Elsasser, Phys. Rev. 1946, vol. 69, pp. 106–116, Bullard and Gellman, Phil. Trans. R. Soc. Lond. 1954, vol. A247, pp. 213–278) rules out dynamo action in a spherical conducting volume provided that the fluid is incompressible, the conductivity is uniform, and the velocity field is purely toroidal. We prove in this note that this result is robust in the sense that slight compressibility of the fluid, small non-radial variations and even large radial variations in conductivity, and the presence of a small non-toroidal velocity component do not invalidate the theorem. Moreover, by proper choice of the conductivity distribution modelling the conducting volume, small deviations from spherical symmetry of the conductor can also be taken into account.     

An extension of the Toroidal Theorem

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.     

Spherical harmonic analysis of the Navier-Stokes equation in magnetofluid dynamics

The equation of motion (Navier-Stokes equation) for a uniformly rotating, compressible, magnetic, viscous fluid is analyzed in terms of infinite series of spherical surface harmonics. Differential equations are obtained for the radial functions of the poloidal and toroidal harmonics of the velocity, corresponding to those obtained by Bullard and Gellman for the magnetic field from the electromagnetic induction equation. This new analysis opens the way for the dynamical problem of electromagnetic induction in the earth's core to be considered by the spherical harmonic method.     

The role of mean circulation in parity selection by planetary magnetic fields

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.     

Fast dynamo action in the Ponomarenko dynamo

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.     

Dynamo in Asymmetric Square Convection

Linear and nonlinear dynamo action is investigated for square patterns in nonrotating and weakly rotating Boussinesq Rayleigh-Bénard convection in a plane horizontal layer. The square-pattern solutions may or may not be symmetric to up-down reflections. Vertically symmetric solutions correspond to checkerboard patterns. They do not possess a net kinetic helicity and are found to be incapable of kinematic dynamo action at least up to magnetic Reynolds numbers of , 12 000. There also exist vertically asymmetric squares, characterized by rising (descending) motion in the centers and descending (rising) motion near the boundaries, among them such that possess full horizontal square symmetry and others lacking also this symmetry. The flows lacking both the vertical and horizontal symmetries possess kinetic helicity and show kinematic dynamo action even without rotation. The generated magnetic fields are concentrated in vertically oriented filamentary structures. Without rotation these dynamos are, however, always only kinematic, not nonlinear dynamos since the back-reaction of the magnetic field then forces the solution into the basin of attraction of a roll pattern incapable of dynamo action. But with rotation added parameter regions are found where stationary asymmetric squares are also nonlinear dynamos. These nonlinear dynamos are characterized by a subtle balance between the Coriolis and Lorentz forces. In some parameter regions also nonlinear dynamos with flows in the form of oscillating squares or stationary modulated rolls are found.     

Magnetic Helicity in fast dynamos

The behaviour of magnetic helicity in kinematic dynamos at large magnetic Reynolds number is considered. Hughes, et al . [ Phys. Lett. A 223 , 167-172 (1996)] observe that the relative helicity tends to zero in the limit of large magnetic Reynolds number. This paper gives upper bounds on the helicity, by relating the helicity spectrum to the energy spectrum. These bounds are confirmed by numerical simulation and the distribution of helicity over scales is considered. Although it is found that the total helicity becomes small in the limit of high conductivity, there can remain significant, but cancelling, helicity at large and small scales of the field. This is illustrated by considering the evolution of helicity in the stretch-twist-fold dynamo picture.     

A unified approach to a class of slow dynamos

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.     

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.     

Geomagnetic polarity reversals in a turbulent core

The behavior of the main magnetic field components during a polarity transition is investigated using the α²-dynamo model for magnetic field generation in a turbulent core. It is shown that rapid reversals of the dipole field occur when the helicity, a measure of correlation between turbulent velocity and vorticity, changes sign. Two classes of polarity transitions are possible. Within the first class, termed component reversals, the dipole field reverses but the toroidal field does not. Within the second class, termed full reversals, both dipole and toroidal fields reverse. Component reversals result from long term fluctuations in core helicity; full reversals result from short term fluctuations. A set of time-evolution equations are derived which govern the dipole field behavior during an idealized transition. Solutions to these equations exhibit transitions in which the dipole remains axial while its intensity decays rapidly toward zero, and is regenerated with reversed polarity. Assuming an electrical conductivity of 3 × 10⁵ mho m^?1 for the fluid core, the time interval required to complete the reversal process can be as short as 7500 years. This time scale is consistent with paleomagnetic observations of the duration of reversals. A possible explanation of the cause of reversals is proposed, in which the core's net helicity fluctuates in response to fluctuations in the level of turbulence produced by two competing energy sources—thermal convection and segregation of the inner core. Symmetry considerations indicate that, in each hemisphere, helicity generated by heat loss at the core-mantle boundary may have the opposite sign of helicity generated by energy release at the inner core boundary. Random variations in rates of energy release can cause the net helicity and the α-effect to change sign occasionally, provoking a field reversal. In this model, energy release by inner core formation tends to destabilize stationary dynamo action, causing polarity reversals.     

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.     

Kinematic Dynamos in the Kerr Metric

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.     

Mechanisms for magnetic field generation in precessing cubes

ABSTRACT

It is shown that flows in precessing cubes develop at certain parameters large axisymmetric components in the velocity field which are large enough to either generate magnetic fields by themselves, or to contribute to the dynamo effect if inertial modes are already excited and acting as a dynamo. This effect disappears at small Ekman numbers. The critical magnetic Reynolds number also increases at low Ekman numbers because of turbulence and small-scale structures.     

Generation and maintenance of bisymmetric spiral magnetic fields and interaction with spiral density waves

Abstract

The paper consists of two parts. The first introduces the dynamo equation into a rotating gaseous disk of finite thickness and then searches for its solution for the generation and maintenance of large-scale bisymmetric spiral (BSS) magnetic fields. We determine numerically the dynamo strength and vertical thickness of the gaseous disk which are necessary for the BSS magnetic fields to rotate as a wave over large area of the disk.

Next we present linearized equations of motion for the self-gravitating disk gas under the Lorentz force due to the BSS magnetic fields. Since the angular velocity of the BSS field is very close to that of the spiral density wave, a nearly-resonant interaction is caused between these two waves to produce large-amplitude condensation of gas in a double-spiral way. The BSS magnetic field is considered as a promising agency to trigger and maintain the spiral density wave.     

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.     

Magnetic helicity generation in the frame of Kazantsev model

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.     

Higher helicity invariants and solar dynamo

Modern models of nonlinear dynamo saturation in celestial bodies (specifically, on the Sun) are largely based on the consideration of the balance of magnetic helicity. This physical variable has also a topological meaning: it is associated with the linking coefficient of magnetic tubes. In addition to magnetic helicity, magnetohydrodynamics has a number of topological integrals of motion (the so-called higher helicity moments). We have compared these invariants with magnetic helicity properties and concluded that they can hardly serve as nonlinear constraints on dynamo action.     

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.