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.     

Construction of fast dynamos using unsteady flows and maps in three dimensions

Abstract

Unsteady fast dynamos are constructed using a sequence of rapid movements separated by periods of diffusion of the magnetic field. Motivated by the physical mechanism described by Ze'dovich and by Soward, we examine the effect of single-mode Beltrami waves applied sequentially, and show that they can be approximated by a simple “stretch-fold-shear” (SFS) map of the unit cube onto itself. In the SFS map, the field points in a fixed direction and diffusion is easily computed. The numerical results indicate that fast dynamo action occurs for sufficiently large shear, and that the process is primarily a coherent feature of the larger magnetic scales. Similar results are obtained for smooth flows. Some preliminary analysis of the SFS map using a decomposition method is described.     

Control volume method for hydromagnetic dynamos in rotating spherical shells: Testing the code against the numerical dynamo benchmark

Hydromagnetic dynamos in rotating spherical shells are investigated using the control volume method. We present a validation of our code against the numerical dynamo benchmark. It is successfully benchmarked and we are able to conclude that the control volume method is another numerical method available for numerical modelling of self-consistent dynamos. In addition, the efficiency of our numerical code is tested. Computations provide conclusions that dynamo codes based on the spectral methods are much more efficient than our code based on the control volume method at the study of global fields on small and medium size parallel computers. However, our code could be much more efficient than codes based on the spectral methods on very large parallel computers, especially at the study of turbulence.     

Dynamos on stream surfaces of a highly conducting fluid

Abstract

This paper discusses dynamo action in generalisations of the Ponomarenko dynamo at large magnetic Reynolds number. The original Ponomarenko dynamo consists of a spiralling flow in which the stream surfaces are concentric cylinders of circular cross section, and the flow depends only on distance from the axis in cylindrical polar coordinates.

In this study, the stream surfaces are allowed to be cylinders of arbitrary cross section, and the flow is only required to be independent of the coordinate along the cylinder axes. For smooth flows alpha and eddy diffusion effects are identified, in terms of the geometry of the stream surfaces, and asymptotic formulae for growth rates in the limit of large magnetic Reynolds number are obtained. Numerical support for these results is presented using direct simulation of dynamo action in selected flows at high conductivity. Finally the case is considered when in spherical polar coordinates the flow is independent of the azimuthal coordinate and the stream surfaces, which are tori, have arbitrary cross sections.     

Numerical simulations of MHD dynamos

The generation of magnetic fields in space plasmas and in astrophysics is usually described within the framework of magnetohydrodynamics. Turbulent helical flows produce magnetic fields very efficiently, with correlation length scales larger than those characterizing the flow. Within the context of the solar magnetic cycle, a turbulent dynamo is responsible for the so-called alpha effect, while the Omega effect is associated to the differential rotation of the Sun.We present direct numerical simulations of turbulent magnetohydrodynamic dynamos including two-fluid effects such as the Hall current. More specifically, we study the evolution of an initially weak and small-scale magnetic field in a system maintained in a stationary regime of hydrodynamic turbulence, and explore the conditions for exponential growth of the magnetic energy. In all the cases considered, we find that the dynamo saturates at the equipartition level between kinetic and magnetic energy, and the total energy reaches a Kolmogorov power spectrum.     

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.     

Simple laminar dynamos: from two rolls to one

The study of laminar flows with simple streamlines in conducting fluids that might support kinematic dynamo action has a long history. Early successful examples include the Gailitis (1970 Gailitis, A. 1970. Self-excitation of a magnetic field by a pair of annular vortices. Mag. Gidrod, 6: 19–22. (English translation: Magnetohydrodynamics, 6:14–17) [Google Scholar]) dynamo, which has two meridional rolls. We show here that a single meridional roll, considered as a limit of a modified Gailitis-like dynamo, can also support kinematic dynamo action in a deep spherical shell. This flow has a claim to be the simplest such flow possible.     

Kinematic dynamos associated with large scale fluid motions

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.     

Meteorite paleomagnetism - From magnetic domains to planetary fields and core dynamos

Meteorites represent the earliest records of the evolution of the solar system, providing information on the conditions, processes and chronology for formation of first solids, planetesimals and differentiated bodies. Evidence on the nature of magnetic fields in the early solar system has been derived from chondritic meteorites. Chondrules, which are millimeter sized silicate spherules formed by rapid melting and cooling, have been shown to retain remanent magnetization records dating from the time of chondrule formation and accretion of planetesimals. Studies on different meteorite classes, including ordinary and carbonaceous chondrites, have however provided contrasting results with wide ranges for protoplanetary disk magnetic fields. Developments on instrumentation and techniques for rock magnetic and paleointensity analyses are allowing increased precision. Micromagnetic and an array of geochemical, petrographic and electronic microscopy analyses provide unprecedented resolution, characterizing rock magnetic properties at magnetic domain scales. We review studies on chondrules from the Allende meteorite that reveal relationships among hysteresis parameters and physical properties. Coercivity, remanent and saturation remanence parameters correlate with chondrule size and density; in turn related to internal chondrule structure, mineralogy and morphology. Compound, fragmented and rimmed chondrules show distinct hysteresis properties, related to mineral composition and microstructures. The remanent magnetization record and paleointensity estimates derived from the Allende and other chondrites support remanent acquisition under influence of internal magnetic fields within parent planetesimals. Results support that rapid differentiation following formation of calcium-aluminum inclusions and chondrules gave rise to differentiated planetesimals with iron cores, capable of generating and sustaining dynamo action for million year periods. The Allende chondrite may have derived from a partly differentiated planetesimal which sustained an internal magnetic field.     

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.     

Large-eddy simulations of convection-driven dynamos using a dynamic scale-similarity model

Dynamo simulations require sub-grid scale (SGS) models for the momentum and heat flux, the Lorentz force, and the magnetic induction. Previous large eddy simulations (LES) using the scale similarity model have represented many aspects of the SGS motion. However, discrepancies are observed due to interchanging the order of filtering operation and spatial differentiation. In this study, we implement a correction term for this commutation error specifically for the scale-similarity model. Furthermore, we implement a dynamic scheme to evaluate time-dependent coefficients for the SGS models. We perform dynamo simulations in a rotating plane layer with different spatial resolutions, and compare results for the time dependence of the large-scale magnetic field. Simulations are performed at two different Rayleigh numbers, using constant values for the other dimensionless numbers (Ekman, Prandtl, and magnetic Prandtl numbers). Both cases show that the dynamic LES can accurately represent the large-scale magnetic field, whereas the dynamo failed in the direct simulations without the SGS terms at the same spatial resolutions. We conclude that the dynamic versions of the SGS and commutation error correction are essential for successful dynamos on coarser grids.     

Convective plan-form of two-scale dynamos in a plane layer

We study generation of magnetic fields, involving large spatial scales, by convective plan-forms in a horizontal layer. Magnetic modes and their growth rates are expanded in power series in the scale ratio, and the magnetic eddy diffusivity (MED) tensor is derived for flows, symmetric about the vertical axis in a layer. For convective rolls we demonstrate that MED is never below molecular magnetic diffusivity. For cell patterns possessing the symmetries of a rectangle, critical values of molecular magnetic diffusivity for the onset of small- and large-scale magnetic field generation are the same. No instances of negative MED in hexagonal cells have been detected. A family of plan-forms has been found numerically, where MED is negative for molecular magnetic diffusivity over the threshold for the onset of small-scale magnetic field generation. However, the region in the parameter space, where large-scale dynamo action is observed, is small.     

Effects of an outer thin stably stratified layer on planetary dynamos

The presence of outer stably stratified layers in planetary cores has been suggested for Earth, Saturn and Mercury. In this study, we use a 3-D numerical dynamo model to investigate the effects of a thin stable layer surrounding a convecting interior on the produced magnetic field. We find that a stable layer can destabilize the field morphology through a thermal wind that produces unfavorable zonal flows throughout the core. The direction of these zonal flows is prograde in equatorial regions, unlike a model with no stable layer that has retrograde equatorial flows. Our models therefore suggest that the Earth does not have a stable layer since we observe a westward drift as opposed to an eastward drift. For Saturn, we find that due to coupling of the flows in the stable and unstable layers, the layer does not act to shear out the non-axisymmetry in the observed magnetic field, and therefore cannot explain Saturn’s axisymmetric magnetic field. For Mercury, we find that if the stable layer is thin, it can actively produce strong or weak surface fields and not necessarily attenuate smaller scale features through the skin effect.     

A unified approach to environmental systems modeling

The paper considers the differences between hypothetico-deductive and inductive modeling: between modelers who put their primary trust in their scientific intuition about the nature of an environmental model and tend to produce quite complex computer simulation models; and those who prefer to rely on the analysis of observational data to identify the simplest form of model that can represent these data. The tension that sometimes arises because of the different philosophical outlooks of these two modeling groups can be harmful because it tends to fractionate the effort that goes into the investigation of important environmental problems, such as global warming. In an attempt to improve this situation, the paper will outline a new Data-Based Mechanistic (DBM) approach to modeling that tries to meld together the best aspects of these two modeling philosophies in order to develop a unified approach that combines the hypothetico-deductive virtues of good scientific intuition and simulation modeling with the pragmatism of inductive data-based modeling, where more objective inference from data is the primary driving force. In particular, it demonstrates the feasibility of a new method for complex simulation model emulation, in which the methodological tools of DBM modeling are used to develop a reduced dynamic order model that represents the ‘dominant modes’ of the complex simulation model. In this form, the ‘dynamic emulation’ model can be compared with the DBM model obtained directly from the analysis of real data and any tensions between the two modeling approaches may be relaxed to produce models that suit multiple modeling objectives.     

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.