In Chapters 7 and 9, the MHD spectral analysis of an ideal plasma with inhomogeneities in one spatial direction led to singular second order differential equations for the plasma displacement in the direction of inhomogeneity: Eqs. (7.91) and (9.31). The two singularities of these equations give rise to two continuous parts of the MHD spectrum, as demonstrated in Section 7.4 for slab geometry and in Section 9.2.2 for cylindrical geometry. It was shown that the eigenfunctions corresponding to these Alfvén and slow magneto-sonic continua possess non-square integrable tangential components leading to extreme anisotropic behaviour. Clearly, this has a dramatic effect on the dynamical behaviour of inhomogeneous plasmas. In the present chapter, we will discuss the consequences of these continuous spectra for the dynamical response of an inhomogeneous plasma slab or cylinder to periodic, multi-periodic, or random external drivers. This will lead to the concepts of resonant absorption of waves and phase mixing of neighbouring magnetic field lines.

Resonant ‘absorption’ (or ‘dissipation’) and phase mixing are fundamental properties of MHD waves that are studied in many different plasma systems. These phenomena affect the dynamics of plasma perturbations significantly and often dominate the energy conversion and transport in inhomogeneous plasmas. Since they are basic to MHD wave heating and acceleration of plasmas, they deserve special attention. In fact, since all plasmas occurring in nature are – to a higher or lower degree – inhomogeneous and since waves can be excited easily in plasmas, resonant absorption and phase mixing frequently occur.

Two approaches

There are basically two ways of introducing the equations of magnetohydrodynamics:

1. (a) pose them as reasonable postulates for a hypothetical medium called ‘plasma’;

2. (b) derive them by appropriate averaging of kinetic equations.

Our approach, starting with Chapter 4, is mainly along the lines of the first method, pioneered by Grad [98, 32] in a series of lecture notes, using physical arguments and mathematical criteria to justify the results. In this chapter, the main steps of the second method will be discussed and shown to be somewhat unsatisfactory since they involve a number of approximations that are often difficult to justify. The reason for going through this analysis anyway is that it provides understanding of the domain of validity of the MHD description and that it indicates what kind of modifications are in order when this description fails.

Mathematically inclined readers may skip this digression, where most results from kinetic theory are not derived but simply stated, and continue reading with Chapter 4. Also, the serious student of magnetohydrodynamics is advised to turn to a detailed study of the present chapter only after a first reading of Chapters 4–11 on basic MHD since the level of this chapter is essentially that of the advanced theory, but it has been placed here because this is where it logically belongs.

We give a ‘derivation’ of the MHD equations by averaging the kinetic equations for plasmas.

Equilibrium of cylindrical plasmas

We have considered the effects of plasma inhomogeneity on MHD waves and instabilities in Chapter 7 for the model of a plane gravitating plasma slab where inhomogeneity is restricted to the vertical direction. For the description of laboratory and astrophysical plasma dynamics, the concept of magnetic flux tubes is quite central, as we have seen in Chapter 8. This automatically leads to the consideration of cylindrical plasmas where the inhomogeneities are operating in the radial direction. Whereas the model remains one-dimensional, so that most of the analytical techniques developed in Chapter 7 remain valid, the introduction of curvature of the magnetic field brings in qualitatively different physical effects that significantly influence the dynamics of flux tubes. We will now neglect gravity since it plays no role in laboratory plasmas and, for astrophysical plasmas, it is more adequately incorporated in an axi-symmetric model with a central gravitating object. The latter requires a two-dimensional model, which has to be relegated to the more advanced chapters. We will see that curvature of the magnetic field enters the equations in a very similar way to gravity in the plasma slab of Chapter 7.

Diffuse plasmas

For the study of confined plasmas, the diffuse cylindrical plasma column (called ‘diffuse linear pinch’ in the older plasma literature) is one of the most useful models. It is probably the most widely studied model in plasma stability theory. Since we have obtained a basic understanding of the spectrum of inhomogeneous one-dimensional systems, the analysis of the diffuse linear pinch can now be undertaken with more fruit than was possible in the early days of fusion research when this configuration was first investigated.

Hydrodynamics of the solar interior

We have studied the MHD waves for homogeneous plasmas in Chapter 5. This theory was transformed in Chapter 6 to the higher level of spectral theory in order to facilitate the much more complicated analysis of inhomogeneous plasmas, which we want to undertake in the present chapter. Plasma inhomogeneity is not just a complication in the analysis, but also provides qualitatively new physical phenomena like wave damping, wave transformation, and, most important of all, a very wide class of global MHD instabilities of magnetically confined plasmas.

Explicit examples of inhomogeneous plasma dynamics abound in the solar system, as we will see in Chapter 8. For the Sun, a number of important phenomena may be described neglecting the magnetic field. Therefore, before we turn to magnetized plasmas in Section 7.3, we will first simplify the model to a purely hydrodynamic one and study the effects of sound and gravity separate from the three MHD waves. Since the hydrodynamic waves are clearly identified in solar observations, we will be able to clarify the potential of observing MHD wave propagation for the investigation of astrophysical objects in general (Section 7.2.4).

We summarize some basic facts of the standard solar model (see Priest [190], Stix [217] or Foukal [69]). The Sun is a sphere of hot material, mainly plasma, of radius R⊙ = 7.0 × 108 m and mass M⊙ = 2.0 × 1030 kg.

In Section 1.4 we stated that astrometry must be developed within an extragalactic reference frame to microarcsecond accuracies. The objective of the present chapter is to provide the theoretical and practical background of this basic concept.

International Celestial Reference System (ICRS)

A reference system is the underlying theoretical concept for the construction of a reference frame. In an ideal kinematic reference system it is assumed that the Universe does not rotate. The theoretical background was presented in Section 5.4.1. The reference system requires the identification of a physical system and its characteristics, or parameters, which are determined from observations and that can be used to define the reference system. In 1991 the International Astronomical Union agreed, in principle, to change to a fundamental reference system based on distant extragalactic radio sources, in place of nearby bright optical stars (IAU, 1992; IAU, 1998; IAU, 2001). The distances of extragalactic radio sources are so large that motions of selected objects, and changes in their source structure, should not contribute to apparent temporal positional changes greater than a few microarcseconds. Thus, positions of these objects should be able to define a quasi-inertial reference frame that is purely kinematic. A Working Group was established to determine a catalog of sources to define this frame that is now called the ICRF.

Astrometry is positional astronomy. It encompasses all that is necessary to provide the positions and motions of celestial bodies. This includes observational techniques, instrumentation, processing and analysis of observational data, positions and motions of the bodies, reference frames, and the resulting astronomical phenomena.

The practical side of astrometry is complemented by a number of theoretical aspects, which relate the observations to laws of physics and to the distribution of matter, or celestial bodies, in space. Among the most important are celestial mechanics, optics, theory of time and space references (particularly with regards to general relativity), astrophysics, and statistical inference theory. These scientific domains all contribute to the reduction procedures which transform the observed raw data acquired by the instruments into quantities that are useable for the physical interpretation of the observed phenomena. The goals of this book are to present the theoretical bases of astrometry and the main features of the reduction procedures, as well as to give examples of their application.

Astrometry is fundamental to, and the basis for, all other fields of astronomy. At minimal accuracy levels the pointing of telescopes depends on astrometry. The cycle of days, the calendar, religious cycles and holidays are based on astrometry. Navigation and guidance systems are based on astrometry, previously for nautical purposes and now primarily for space navigation.

Astronomy and astrophysics are also strongly dependent on astrometry.

The apparent direction in the sky at which a celestial object appears is not the actual direction from which the light was emitted. What is observed is the tangent direction of the light when it reaches the observer. For reasons that will be discussed in this chapter, the light path is not rectilinear and several corrections describing the effects of bending, or shifts in direction, are to be applied to the direction from which the light is observed to determine the actual direction of the emission. We shall not deal here with the various transformations undergone by the light within the observing instrument; they are particular to each case. Some examples are given in Chapter 14. We shall consider only the direction from which the light came when it entered into the instrument. One has to consider the atmospheric refraction, the shift in direction due to the combination of the speed of light with the motion of the observer, called aberration, and the bending of light in the presence of gravitational fields. The latter has been already presented in Section 5.4.2, but will be revisited in Section 6.4. Similarly, the geodesic precession and nutation are to be considered when relating the positions from a moving reference frame of fixed orientation to a fixed reference frame of the same orientation (see Section 7.5).

For many years, the theory of relativity was ignored for astrometry because the effects were much smaller than the accuracies being achieved. For the motions of bodies of the Solar System, Newtonian theory was adequate, provided that some relativistic corrections were introduced. Actually, there was a long period of questioning whether the theory of relativity was correct or not, and thus relativistic corrections were introduced in a manner to determine whether observational data could then be better represented.

However, over the past 30 years, there has been a very significant improvement in the accuracies of observations, all confirming the conclusions of general relativity. So its introduction became not only acceptable, but necessary. In 1976, the International Astronomical Union introduced relativistic concepts of time and the transformations between various time scales and reference systems. In 1991, it extended them to reference frames and to astrometric quantities. Now, with plans for microarcsecond astrometry and with time standards approaching accuraries of one part in 10-16, and better in the future, it is necessary to base all astrometry, reference systems, ephemerides, and observational reduction procedures on consistent relativistic grounds. This means that relativity must be accepted in its entirety, and that concepts, as well as practical problems, must be approached from a relativistic point of view. In 2000, the IAU enforced this approach further by extending the 1991 models for future requirements in such a way that they become valid to accuracies several orders better than those currently achieved in 2000.