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In this final chapter we focus on the interactions between convection, magnetic fields and rotation in stars that, like our Sun, possess deep outer convection zones, with the aim of relating theory to observations. Following on from the treatment of planetary dynamos in Chapter 7, we begin by considering the large-scale fields that are responsible for the solar cycle and survey attempts to model solar and stellar dynamos, ranging from mean-field dynamo theory to the results of the latest massive computations (Charbonneau 2010).
Then we turn to small-scale behaviour at the solar surface. Over the past two decades detailed observations – from the ground, from the stratosphere and from space – have revealed a wealth of detailed information about the structure and properties of magnetic features on the Sun and on other magnetically active stars. Although the idealized theoretical models that we have described in previous chapters do explain the general behaviour of magnetic fields at the surface of a vigorously convecting star, any more detailed confrontation of theory with observations demands a more precise description of the stellar plasma. Two properties are particularly important. The first is the role of ionization: in the Sun, hydrogen is ionized just below the visible photosphere, with resulting changes to the equation of state and the value of γ that affect the superadiabatic gradient and lead to the presence of a deep convection zone (Stix 2002).
The last thirty years have seen great leaps forward in the subject of magnetoconvection. Computational techniques can now explain exotic nonlinear behaviour, transition to chaos and the formation of structures that can be observed on the surface of the Sun. Here, two leading experts present the current state of knowledge of the subject. They provide a mathematical and numerical treatment of the interactions between electrically conducting fluids and magnetic fields that lead to the complex structures and rich behaviour observed on the Sun and other stars, as well as in the interiors of planets like the Earth. The authors' combined analytical and computational approach provides a model for the study of a wide range of related problems. The discussion includes bifurcation theory, chaotic behaviour, pattern formation in two and three dimensions, and applications to geomagnetism and to the properties of sunspots and other features at the solar surface.
The original motivation for studying magnetoconvection came from the interplay between magnetic fields and convection that is observed in sunspots. Since then this subject has developed into a fascinating and important topic in its own right. We therefore decided to write a comprehensive monograph that would cover all aspects of magnetoconvection from the viewpoint of applied mathematics, and as a branch of astrophysical (or geophysical) fluid dynamics. Thus we shall emphasize the role of nonlinear dynamics, and focus on idealized model problems rather than on ambitious realistic simulations.
The properties of convection in an electrically conducting fluid with an imposed magnetic field are interesting not only in themselves but also as the richest example of double-diffusive behaviour. Linear theory allows both steady and oscillatory solutions, while theoretical descriptions of nonlinear behaviour demonstrate the power of bifurcation theory, with examples of bifurcation sequences that lead to chaos, as well as of group-theoretic applications to pattern selection. These mathematical results can all be related to carefully constructed numerical experiments.
Although we shall adopt an applied mathematical approach, our discussion is particularly relevant to the behaviour of magnetic fields at the surface of the Sun, which are now being observed in unprecedented detail, both from the ground and from space. Convection also interacts with magnetic fields in the solar interior, as it does in other stars, and is a key component of solar and stellar dynamos.
We have seen that convection may set in at either a Hopf or a pitchfork bifurcation, giving rise to branches of nonlinear oscillatory or steady motion. In this chapter we consider weakly and mildly nonlinear behaviour, in regimes that are accessible to an analytical approach, without having to rely on large-scale computation. Our treatment relies on mathematical developments in nonlinear dynamics – a subject that has its roots in the work of Poincaré more than a century ago but has grown explosively during the past few decades. In what follows we shall adopt a straightforward approach that is aimed at traditional applied mathematicians rather than at experts in nonlinear mathematics. Magnetoconvection provides a rich and fascinating demonstration of the power of bifurcation theory, and of its ability to explain a wide range of interactions between branches of solutions that may be stable or unstable, steady, oscillatory or chaotic.
We shall confine our attention here to idealized models of Boussinesq magnetoconvection, and focus on two-dimensional behaviour. In subsequent chapters these restrictions will be progressively relaxed. We shall mainly be concerned with imposed magnetic fields that are vertical, but horizontal fields will be considered briefly in the final subsection. As in Section 3.1.4, we assume that the velocity u and the magnetic field B are confined to the xz-plane and independent of y.
In this chapter we penetrate further into the nonlinear domain, relying principally on the results of careful numerical experiments, and confining our attention to the simplest and most thoroughly studied configurations. Our primary aim is to extract qualitative understanding from the computations. Once interpreted, they provide a basis for investigating the more complicated structures and patterns that will be treated later in the book.
We begin by extending the mildly nonlinear results in Chapter 4 to cover convection in a rectangular box when the magnetic Reynolds number is large and the magnetic field becomes dynamically important. Then we study the analogous problem in a cylindrical domain with axial symmetry imposed. Next we return to Cartesian models and to the chaotic behaviour that was introduced in Section 4.3, in order to confirm that the Shilnikov effect is present in the full system; in addition, we find a regime with Lorenz-like chaos. Thereafter we consider the effects of relaxing the lateral constraints and thereby allowing travelling waves, together with steady convection in tilted cells and vigorous pulsating waves. That leads us to consider patterns of convection in extended regions, where rolls are modulated at longer wavelengths and localized (or isolated) states can appear. Then we proceed to the strong field limit, and consider behaviour when cells are vertically elongated and very slender. Finally, we discuss the effects of inclined magnetic fields on nonlinear convection.
In this chapter we introduce the effects of rotation into the study of magnetoconvection. While these effects can safely be neglected when discussing the dynamics of the solar photosphere, since typical timescales are much less than a solar day, the large-scale motions occurring deeper in the solar convection zone and in the Earth's liquid core are strongly affected by rotation. Indeed, rotation would appear to be a crucial ingredient in the dynamo mechanisms that are responsible for the geomagnetic field and the solar magnetic cycle. A full discussion of dynamo theory is outside the scope of this book (though see, for example, Dormy and Soward 2007) but we shall discuss dynamo models in which convection plays a prominent role. As such, we shall depart later in this chapter from consideration of convective flows in simple planar models and in addition discuss what happens in spherical geometries.
A necessary preliminary to understanding the complex interaction of magnetic fields with rotating convection is a discussion of the rotating, nonmagnetic case. This is first done in a Cartesian geometry. Then the effect of a vertical magnetic field is introduced. We restrict ourselves to the problem of convection in a layer rotating about a vertical axis. Then we can discuss the effects of a vertical magnetic field (this makes comparison with previous chapters easier, but such a configuration is not one that can readily be recognized in nature).
Interest in magnetoconvection arose initially from astrophysics, following the discovery of strong magnetic fields in sunspots, and the realization that their relative coolness (and hence their darkness) was a consequence of magnetic interference with convection. As theoretical studies progressed from linear to nonlinear investigations, and ultimately to massive numerical experiments, it became clear not only that magnetoconvection poses in itself a fascinating challenge to applied mathematicians but also that it serves as a prototype of double-diffusive behaviour in fluid dynamics, oceanography and laboratory experiments.
In this opening chapter we first summarize the development of our subject and then provide a brief survey of the chapters that follow in the book. Although we shall focus our attention on idealized configurations that are mathematically tractable, we also discuss more complex behaviour in the real world.
Background and motivation
The original motivation for our subject came from astrophysics. Stars like the Sun, with deep outer convection zones, are magnetically active. Their magnetic fields are maintained by hydromagnetic dynamo action, resulting from interactions between convection, rotation and magnetic fields in their interiors – just as the geomagnetic field is maintained by a dynamo in the Earth's liquid core. The most prominent magnetic features on the Sun are sunspots, like that shown in Figure 1.1. Although such a spot covers less than of the solar disc, there are other more active stars with huge spots that spread over significant fractions of their surfaces (Thomas and Weiss 2008).