Advances in Nonlinear Dynamos

Antonio Ferriz-Mas, Manuel Nunez

• Publisher: CRC Press
• Year: 2019
• Print ISBN: 9780415287883
• eBook ISBN: 9781134443901
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Nonlinear dynamo theory is central to understanding the magnetic structures of planets, stars and galaxies. In chapters contributed by some of the leading scientists in the field, this text explores some of the recent advances in the field. Both kinetic and dynamic approaches to the subject are considered, including fast dynamos, topological methods in dynamo theory, physics of the solar cycle and the fundamentals of mean field dynamo. Advances in Nonlinear Dynamos is ideal for graduate students and researchers in theoretical astrophysics and applied mathematics, particularly those interested in cosmic magnetism and related topics, such as turbulence, convection, and more general nonlinear physics.

1 The field, the mean and the meaning

Peter Hoyng

SRON Laboratory for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands, E-mail: [email protected]

This chapter is about the fundamentals of mean field dynamo theory, with an emphasis on the statistical aspects of the theory. The equations for passive vectorial transport (dynamo equation) and passive scalar transport are derived on a par, as applications of the theory of stochastic equations with multiplicative noise. Only approximate transport equations for mean quantities exist, and the approximations are scrutinized in relation to the ensemble average and the azimuthal average. A summary of the elementary physics of the dynamo equation is followed by an analysis of the influence of mean shear flows and resistive effects on the transport coefficients α and β. It is argued that the dynamo equation contains in practical situations no higher than second-order spatial derivatives. The physics of turbulent diffusion and the information content of the dynamo equation is analysed, and it is concluded that transport equations for the mean can only make probabilistic statements about the physical systems to which they apply. They cannot predict the evolution for all times. This is elucidated for ensemble and azimuthal averages at the hand of examples, referring to the phase memory and magnetic energy losses of the solar dynamo, and to the variability and reversals of the geodynamo. The accent is on clarity of presentation, and on linear theory. A few remarks on nonlinear theory are made.

1.1. Introduction

Mean field dynamo theory seems to have a controversial status. To some it is their natural habitat, while others shy away from it because they maintain that the basic tenets are not understood or do not apply to real dynamos. In spite of this, much work in the literature on global magnetic fields is based on mean field theory. There are two main reasons for that. The first is that there is often no viable alternative. It is true that the magnetic Reynolds number R_m of the geodynamo is sufficiently small that several groups have succesfully performed fully resolved three-dimensional simulations (Glatzmaier and Roberts, 1995; Kageyama and Sato, 1997; Kuang and Bloxham, 1997; Christensen et al., 1998). While these computations have demonstrated that self-consistent dynamo action is able to overcome resistive decay of the currents, the demands on computing resources are extreme. A routine study of many different cases, and runs extending over a long time and in the correct parameter regime are impossible. And stellar dynamos have such high Reynolds numbers that fully resolved MHD simulations of the whole dynamo will be impossible for the forseeable future. Only a small section of the dynamo can be addressed, and that is helpful for an understanding of the physical mechanisms at work (Nordlund et al., 1992, 1994; Brandenburg et al., 1996).

A second important reason for the popularity of mean field theory is that it works rather ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• List of contributors
• Preface
• 1 The field, the mean and the meaning
• 2 Fast dynamos
• 3 On the theory of convection in the Earth’s core
• 4 Dynamo action of magnetostrophic waves
• 5 Magnetic flux tubes and the dynamo problem
• 6 Physics of the solar cycle
• 7 Highly supercritical convection in strong magnetic fields
• 8 Thin aspect ratio αΩ-dynamos in galactic discs and stellar shells
• 9 Computational aspects of astrophysical MHD and turbulence
• 10 Topological quantities in magnetohydrodynamics
• Name index
• Subject index