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.     

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.     

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.     

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.     

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.     

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.     

A numerical investigation of magnetic field generation in a flow with chaotic streamlines

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).     

Kinematic dynamo induced by helical waves

We investigate numerically the kinematic dynamo induced by the superposition of two helical waves in a periodic box as a simplified model to understand the dynamo action in astronomical bodies. The effects of magnetic Reynolds number, wavenumber and wave frequency on the dynamo action are studied. It is found that this helical-wave dynamo is a slow dynamo. There exists an optimal wavenumber for the dynamo growth rate. A lower wave frequency facilitates the dynamo action and the oscillations of magnetic energy emerge at some particular wave frequencies.     

Magnetic field generation by the motion of a highly conducting fluid

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.     

Kinematic dynamo action in a sphere: Effects of periodic time-dependent flows on solutions with axial dipole symmetry

Choosing a simple class of flows, with characteristics that may be present in the Earth's core, we study the ability to generate a magnetic field when the flow is permitted to oscillate periodically in time. The flow characteristics are parameterised by D, representing a differential rotation, M, a meridional circulation, and C, a component characterising convective rolls. The dynamo action of all solutions with fixed parameters (steady flows) is known from earlier studies. Dynamo action is sensitive to these flow parameters and fails spectacularly for much of the parameter space where magnetic flux is concentrated into small regions, leading to high diffusion. In addition, steady flows generate only steady or regularly reversing oscillatory fields and cannot therefore reproduce irregular geomagnetic-type reversal behaviour. Oscillations of the flow are introduced by varying the flow parameters in time, defining a closed orbit in the space ( D,?M ). When the frequency of the oscillation is small, the net growth rate of the magnetic field over one period approaches the average of the growth rates for steady flows along the orbit. At increased frequency time-dependence appears to smooth out flux concentrations, often enhancing dynamo action. Dynamo action can be impaired, however, when flux concentrations of opposite signs occur close together as smoothing destroys the flux by cancellation. It is possible to produce geomagnetic-type reversals by making the orbit stray into a region where the steady flows generate oscillatory fields. In this case, however, dynamo action was not found to be enhanced by the time-dependence. A novel approach is being taken to solve the time-dependent eigenvalue problem where, by combining Floquet theory with a matrix-free Krylov-subspace method, we can avoid large memory requirements for storing the matrix required by the standard approach.     

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.     

Kinematic dynamo action in spherical Couette flow

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.     

On a Physically Realistic Fast Dynamo

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.     

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.     

Subharmonic Dynamo Action in the Roberts Flow

The paper deals with the dynamo action of the Roberts flow, that is, a flow depending periodically on two cartesian coordinates, X and Y , but being independent of the third one, Z . In particular the case is considered in which the magnetic fields, which are periodic in X, Y and Z , have period lengths in the XY -plane being integer multiples of that of the flow. Two approaches are used. Firstly, the equations governing the magnetic field are reduced to a matrix eigenvalue problem, which is solved numerically. Secondly, a mean magnetic field is defined by averaging over proper areas in the XY -plane, corresponding equations are derived, in which the induction effect of the flow occurs as an anisotropic f -effect, and analytic solutions are given. The results are of particular interest for the Karlsruhe dynamo experiment, which works with a Roberts type flow consisting of 52 cells inside a cylindrical volume. In order to check the reliability of predictions concerning self-excitation based on the mean-field approach, analogous predictions are derived for a rectangular box containing 50 cells, and are compared with results obtained with the help of direct solutions of the eigenvalue problem mentioned. It turns out that the simple mean-field approach in general underestimates the requirements for self-excitation. The corresponding results agree with those obtained in the subharmonic approach only if the side length L of the box, its height H and the edge length l of a spin generator satisfy $ L \gg H \gg l $ . In Appendix B, some comments on previous results concerning $\cal {ABC}$ dynamos are made in the light of the subharmonic formalism used in the paper.     

地震波在不规则界面上的散射效应及其某些应用

本文运用数值解法,求解了两类散射问题:(1)在声学近似下,平面P波在半无限介质空间表面上任意形状的三维空腔上的散射;(2)平面SH波在半无限弹性空间中埋藏着任意形状截面的无限长、且平行于地面的弹性柱体上的散射,得到了几种几何形状的物体所引起的散射数值结果。把某些特殊情况下的散射结果与已知的精确解作对比,两者能很好地吻合。     

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.     

Large-scale convective dynamos in a stratified rotating plane layer

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.     

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.