This book, together with the preceding Principles of Magnetohydrodynamics (to be referred to as Volume [1]), describes the two main applications of plasma physics, laboratory research on thermonuclear fusion energy and plasma-astrophysics of the solar system, stars, accretion disks, etc., from the single viewpoint of magnetohydrodynamics (MHD). This provides effective methods and insights for the interpretation of plasma phenomena on virtually all scales, ranging from the laboratory to the Universe. The key issue is understanding the complexities of plasma dynamics in extended magnetic structures. In Volume [1], the classical MHD model was developed in great detail without omitting steps in the derivations. This necessitated restriction to ideal dissipationless plasmas, in static equilibrium and with inhomogeneity in one direction. In the present volume on Advanced Magnetohydrodynamics [2], these restrictions are relaxed one by one: introducing stationary background flows, resistivity and reconnection, two-dimensional toroidal geometry, linear and nonlinear computational techniques and transonic flows and shocks. These topics transform the subject into a vital new area with many applications in laboratory, space and astrophysical plasmas.

The two volumes now consist of five parts:

1. I Plasma physics preliminaries (Volume [1], Chapters 1–3),

2. II Basic magnetohydrodynamics (Volume [1], Chapters 4–11),

3. III Flow and dissipation (Volume [2], Chapters 12–15),

4. IV Toroidal plasmas (Volume [2], Chapters 16–18),

5. V Nonlinear dynamics (Volume [2], Chapters 19–21).

Inevitably, with the chosen distinction of topics for Volume [1] (mostly ideal linear phenomena described by self-adjoint linear operators) and topics for Volume [2] (mostly non-ideal, toroidal and nonlinear phenomena), the difference between “basic” and “advanced” levels of magnetohydrodynamics could not be strictly maintained.

We have seen that the MHD description for the macroscopic dynamics of plasmas offers a uniquely powerful, unifying, viewpoint on both laboratory and astrophysical plasmas. The applicability of the MHD viewpoint was discussed previously in Volume [1], along with the various approximations made to arrive at the MHD equations from first principles. For most laboratory plasmas, the single fluid ideal or resistive MHD model eventually needs to be extended towards a multi-fluid model and by including important kinetic effects, since its continuum approach to plasma modeling neglected, e.g., Landau damping as well as many other velocity-space dependent physical phenomena. For many astrophysical plasmas, we face yet another shortcoming of the MHD model used thus far, namely that we restricted all attention to non-relativistic plasma velocities. This is perfectly adequate for most of the plasma found in our own solar system. However, astronomical observations indicate that, e.g., the extragalactic jets associated with Active Galactic Nuclei clearly harbor dynamically important magnetic fields and relativistically flowing plasmas. In order to model these plasmas in a continuum model, the restriction on the plasma velocities must be alleviated, by revisiting the ideal MHD equations in a frame-invariant relativistic formulation within fourdimensional space-time. In this chapter, we present such a formulation, restricting our attention to special relativity where we still have a “flat” geometry. In recent years, modern computational techniques such as those discussed in Chapter 19 have started to be used in this more demanding relativistic MHD regime. Since such activities are necessarily still maturing, we only summarize the numeric algorithmic challenges posed by the ideal MHD model in special relativity.

Transonic MHD flows

Flow in laboratory and astrophysical plasmas

We started the study of the effects of background flow on waves and instabilities of laboratory and astrophysical plasmas in Chapter 12. We also considered the modifications of the equilibrium caused by the flow. These modifications were rather trivial for plane shear flows, but considerable for rotating plasmas due to centrifugal forces. However, except for the forebodings of Chapter 18, the most substantial effects have not been faced yet. The adjective “substantial” on background flows obviously should refer to some standard on what is a sizeable velocity. For transonic gas dynamics, it is clear that the appropriate standard velocity is the sound speed. For the macroscopic description of plasmas, which incorporates the dynamics of ordinary gases, the three MHD speeds (slow, Alfvén and fast) collectively take over the role of the sound speed. This implies that trans“sonic” MHD flows will be characterized by different flow regimes depending on the speed of the background flow relative to those three MHD speeds. In addition, the relative direction of the background velocity, v0, with respect to the direction of the background magnetic field, B0, introduces an anisotropy in plasma dynamics that is not present in ordinary gas dynamics.

Plasmas with dissipation

Conservative versus dissipative dynamical systems

We have already come across the enormous difference between conservative (ideal) MHD and dissipative (resistive, viscous, etc.) MHD in Volume [1], Chapter 4. This difference runs through all of classical dynamics of discrete and continuous media. It involves quite different physical assumptions and corresponding different mathematical solution techniques. An instructive example is spectral theory (Volume [1], Chapter 6) which is classical, consistent and misleadingly beautiful for ideal MHD, but full of unresolved problems in resistive MHD. The classical part concerns self-adjoint linear operators in Hilbert space, analogous to quantum mechanics, and stability analysis by means of an energy principle. When dissipation is important, precisely these two “sledge hammers” are missing in the dynamical systems workshop. Even the definition of what is an important, i.e. physically dominant, contribution to the dynamics deserves extreme care. This is best illustrated by the general description of the dynamics of ordinary fluids which is fundamentally different for ideal fluids, characterised by an infinite Reynolds number, and viscous fluids, characterised by a finite Reynolds number. This is even so for extremely large Reynolds numbers, in a certain sense irrespective of how large this number is. Viscous boundary layers always arise in real fluids. This qualitative difference between ideal and dissipative dynamics, with the occurrence of boundary layers, also applies to MHD when resistivity is introduced.

Computational magnetohydrodynamics is a very active research field due to the increasing demand for quantitative results for realistic magnetic configurations on the one hand and the availability of ever more computer power on the other [373]. Many MHD phenomena can not be described by analytical methods in all of their complexity although simplified analytical models have led to indispensable insight into the fundamental physics of various magnetohydrodynamic processes. The intricate geometry of present tokamaks, for instance, forces theory to resort to computer simulations as the mathematics is not fully tractable anymore. The fast increase of computer speed and memory allows simulations with ever more “physics” in the equations and taking into account the full 3D geometrical effects.

While the governing ideal MHD equations form a set of nonlinear, hyperbolic, partial differential equations, we already encountered many magneto-fluid phenomena which are adequately modeled by means of the linearized MHD equations. In this chapter, we concentrate mostly on computational approaches for linear MHD problems, and introduce several basic numerical concepts and techniques along the way. We give a brief overview of the most frequently encountered spatial discretizations to translate any problem expressed as a (set of) differential equation(s) into a discrete linear algebraic problem, and discuss commonly used strategies for solving the resulting linear systems and generalized eigenvalue problems. Representative applications cover MHD spectroscopic computations for diagnosing eigenoscillations and stability of given, possibly pre-computed, MHD equilibria, as well as steady-state and time dependent solutions to externally driven MHD configurations. Both ideal and non-ideal linear MHD problems are encountered.

Laboratory and astrophysical plasmas

Grand vision: magnetized plasma on all scales

In Chapter 1 of the preceding Volume [1] we pointed out that, since more than 90% of visible matter in the Universe is plasma, the dynamics of plasmas and the associated magnetic fields are an important constituent of the description of nature. In Chapter 4 [1], we then showed that the equations of magnetohydrodynamics (MHD) are scale-independent: the scales of length, density and magnetic field strength of a magnetically confined plasma may be divided out. This simple fact has the amazing consequence that the macroscopic dynamics of plasmas in both laboratory fusion devices (tokamaks, stellarators, etc.) and astrophysical objects (stellar coronae, accretion disks, spiral arms of galaxies, etc.) may be described by the same equations, viz. the equations of MHD. We encountered several examples of this before, in Volume [1]. In the present Volume [2], we will continue the investigation of this common field of research by means of the new “wide-angle MHD telescope”.

Figure 12.1 shows two representative, but very different, examples from science and technology, viz. the design drawing of the international tokamak experimental reactor ITER, presently under construction, and an image made by the Hubble Space Telescope of the Pinwheel Galaxy M101. The consequence of scaleindependence is that the most obvious difference of the two configurations, their length scale indicated next to the figure, is actually irrelevant for the description of macroscopic plasma dynamics!

Human beings are curious by nature and have marveled at the night sky ever since our Homo sapiens ancestors first gazed up into the heavens. What is “up there”? Why do stars shine? How did the Universe begin? Does life exist elsewhere? What is on the other side of the Moon?

Astronomy is one of the oldest sciences, but modern physics and technology, coupled with observations from space, have recently generated a stupendous wave of new knowledge. Most of our earliest questions about the nature of the Universe have now been answered, and many unexpected, intriguing new findings have been made, findings that invite us to be both humble and bold. And one needs not be a professional astronomer or physicist to understand them.

Our intent in writing this book has been to offer to the general reader a summary of current astronomical knowledge, generously illustrated and provided with rigorous but simple explanations, while avoiding mystifying professional jargon.

The 250 “windows” on astronomy in this book do not exhaust the topic, but we hope that they will pique the curiosity of our readers and stimulate them to explore further, by navigating on the World Wide Web or by consulting some of the many fine publications on astronomy, such as those suggested at the end of this book. Most important of all, we hope that they will find renewed wonder in the night sky!

Introduction

As we pointed out in §1.6, most of our knowledge about the astrophysical Universe is based on the electromagnetic radiation that reaches us from the sky. By analysing this radiation, we infer various characteristics of the astrophysical systems from which the radiation was emitted or through which the radiation passed. Hence an understanding of how radiation interacts with matter is very vital in the study of astrophysics. Such an interaction between matter and radiation can be studied at two levels: macroscopic and microscopic. At the macroscopic level, we introduce suitably defined emission and absorption coefficients, and then try to solve our basic equations assuming these coefficients to be given. This subject is known as radiative transfer. At the microscopic level, on the other hand, we try to calculate the emission and absorption coefficients from the fundamental physics of the atom. Much of this chapter is devoted to the macroscopic theory of radiative transfer. Only in §2.6, do we discuss how the absorption coefficient of matter can be calculated from microscopic physics. The emission coefficient directly follows from the absorption coefficient if the matter is in thermodynamic equilibrium, as we shall see in §2.2.4.

Theory of radiative transfer

Radiation field

Let us first consider how we can provide the mathematical description of radiation at a given point in space. It is particularly easy to give a mathematical description of blackbody radiation, which is homogeneous and isotropic inside a container.