One of the brightest and most variable UV emissions in the solar system comes from Jupiter's UV aurora. The auroras have been imaged with each camera on HST, starting with the pre-COSTAR FOC and continuing with increasing sensitivity to the present with STIS. This paper presents a short overview of the scientific results on Jupiter's aurora obtained from HST UV images and spectra, plus a short discussion of Saturn's aurora.

The Earth's aurora: Present understanding

With a long history of ground-based and spacecraft measurements, we now have some understanding of the physics of the Earth's auroral processes. A general picture of the nature of auroral activity on the Earth has evolved, without a complete understanding of the many details. In general, auroral emissions are produced by high energy charged particles precipitating into the Earth's upper atmosphere from the magnetosphere (the region of space where the motions of particles are governed by the Earth's magnetic field). It is well established that the Earth's auroral activity is related to solar activity, and more specifically to conditions in the solar wind reaching the Earth. The precipitating charged particles are accelerated to high energies in the Earth's magnetosphere, with some acceleration occurring in the magnetotail region and some occurring by fieldaligned potentials in the topside ionosphere.

Magnetohydrodynamics, or MHD in short, describes the macroscopic behavior of an electrically conducting fluid – usually an ionized gas called a plasma –, which forms the basis of this book. By macroscopic we mean spatial scales larger than the intrinsic scale lengths of the plasma, such as the Debye length λD and the Larmor radii ρj of the charged particles. In this chapter we first derive, in a heuristic way, the dynamic equations of MHD and discuss the local thermodynamics (Section 2.1). Since most astrophysical systems rotate more or less rapidly, it is useful to write the momentum equation also in a rotating reference frame, where inertial forces appear (Section 2.2). Then some convenient approximations are introduced, in particular incompressiblity and, for a stratified system, the Boussinesq approximation (Section 2.3). In MHD theory the ideal invariants, i.e., integral quantities that are conserved in an ideal (i.e., nondissipative) system, play a crucial role in turbulence theory; these are the energy, the magnetic helicity, and the cross-helicity (Section 2.4). Though this book deals with turbulence, it is useful to obtain an quick overview of magnetostatic equilibrium configurations, which are more important in plasmas than stationary flows are in hydrodynamics (Section 2.5). Also the zoology of linear modes, the small-amplitude oscillations about an equilibrium, is richer than that in hydrodynamics (Section 2.6). Finally, in Section 2.7 we introduce the Elsässer fields, which constitute the basic dynamic quantities in MHD turbulence. In this chapter we write the equations in dimensional form, using Gaussian units, to emphasize the physical meaning of the various terms.

In this chapter we talk about accretion disks, a widespread phenomenon in astrophysics, wherein magnetic turbulence is present not just as a byproduct but rather is essential for its very existence, as is now generally believed. Accretion, the accumulation of mass onto a central object due to gravitational attraction, naturally leads to the formation of disk-like structures, since the infalling matter, due to conservation of angular momentum, tends to rotate about the center of gravity. The system is in approximate equilibrium, in that the radial component of the gravitational force is balanced by the centrifugal force and the axial component by the pressure gradient. Since the disk material moves on Keplerian orbits with angular velocity Ω(r) α r−3/2, the angular momentum α r2Ω decreases with decreasing radius. Hence, when the matter moves inward, conservation of angular momentum requires that the excess is transferred outward. Thus the rate of accretion of mass is determined by the transport of angular momentum, which therefore becomes the crucial issue for understanding the dynamics of these systems. If a transport mechanism is provided, material is spiraling in toward the central object (just as conservation of vorticity leads to a spiraling flow of the water from a bathtub).

One important aspect of turbulence theory is the need to understand how obviously random motions are generated from a smooth flow. There are essentially three approaches to this problem: the dynamic systems approach; the development of singular solutions of the ideal fluid equations, in particular the question of finite-time singularities; and the excitation of instabilities and their effects. The dynamic systems approach, i.e., the transition to a chaotic temporal behavior in some low-order nonlinear dynamic model such as the Lorentz model of thermal convection, had once been considered a very promising way to describe also the transition to turbulence in a fluid. However, these expectations have largely been frustrated, mainly because the low-order approximations of the fluid equations ignore the most important aspect of turbulence, namely the excitation and interactions of a broad range of different spatial scales. We will therefore not discuss dynamic systems theory in this treatise.

The problem of finite-time singularities has evoked considerable discussion. This is primarily a mathematical problem concerning the nature of the solution of the ideal fluid equations, whose relevance for the generation of turbulence in dissipative systems might be debatable. However, similarly to the theory of absolute equilibrium states of the ideal system considered in Section 5.2, which provides valuable information about the cascade dynamics in dissipative turbulence, the way in which the ideal solution becomes singular gives some indication of the spatial structure of eddies encountered in the dissipative system.

In the derivation of spectral laws presented in Chapter 5 we used only certain general properties of the turbulence, in particular the integral invariants, which lead to the spectral cascades. (Only the Alfvén effect resulting in the IK spectrum is based on a specific dynamic process of the MHD system.) Though phenomenological arguments, especially dimensional analysis, are often very powerful and robust, since they represent basic physical principles, they only predict a few scaling laws but cannot, for instance, specify proportionality factors, such as the Kolmogorov constant and the sign of the residual energy spectrum. Morover, these arguments provide little insight into the turbulence dynamics. Such properties must be treated by a statistical theory derived from the basic fluid equations. Here the most practical approach is two-point closure theory. An alternative method, renormalization-group (RNG) theory, which was originally developed in the context of the theory of critical phenomena, has also been applied to hydrodynamic turbulence (e.g., Yakhot and Orszag, 1986) and MHD turbulence (Fournier et al., 1982; Camargo and Tasso, 1992), but there is still a considerable degree of arbitrariness and even inconsistency. We shall therefore not discuss RNG theory any further but restrict the treatment in this chapter to closure theory.

In Chapter 4 we introduced the one-point-closure approximation consisting of the equations for the average fields and some phenomenological expressions for the correlation functions appearing in these equations, which is appropriate for describing large-scale inhomogeneous-turbulence processes. To study intrinsic small-scale properties, for which correlation functions are of primary interest, one has to go one step further in the hierarchy of moment equations.

Turbulence in electrically conducting fluids is necessarily accompanied by magnetic-field fluctuations, which will, in general, strongly influence the dynamics. It is true that, in our terrestrial world, conducting fluids in turbulent motion are rare. In astrophysics, however, material is mostly ionized and strong turbulence is a widespread phenomenon, for instance in stellar convection zones and stellar winds and in the interstellar medium. Turbulent magnetic fields are therefore expected to play an important role. Despite the fact that, on a microscopic level, astrophysical plasmas exhibit rather diverse properties, a unified macroscopic treatment in the framework of magnetohydrodynamics (MHD) to describe the most important magnetic effects is appropriate. Hence there is much interest in MHD turbulence in the astrophysical community. Considerable interest comes also from the side of pure theory, where MHD turbulence introduces new concepts into turbulence theory, as the large number of articles on this topic in the literature shows. However, to date no monograph on MHD turbulence seems to have been written. I therefore believe that a treatise both introducing the field and reviewing the current state of the art could be welcome.

The solar wind provides an almost ideal laboratory for studying high-Reynolds-number MHD turbulence. Turbulence is free to evolve unconstrained and unperturbed by in situ diagnostics, satellite-mounted magnetometers, probes and particle detectors. We will see that many features of homogeneous MHD turbulence discussed in the previous chapters are discovered in solar-wind turbulence, but there are also unexpected and still unexplained features. Since the turbulence varies significantly in the different regions of interplanetary space depending on the local solar-wind conditions, it is useful to first give at least a rough picture of the mean solar-wind properties, before discussing the properties of the turbulent fluctuations about the mean state.

Mean properties of the solar wind

Stars lose not only energy by radiation but also mass (and angular momentum) by a, more or less, continuous radial flow called the stellar wind, the solar wind in the case of the Sun. The origin of this flow is the high temperature in the corona, which means that the coronal plasma is not gravitationally bound and, if it is not confined by magnetic loops, expands into interplanetary space, giving rise to the supersonic solar wind. The flow extends radially out to a distance beyond the planetary system, before it is slowed down by the termination shock expected at roughly 100 AU (1 AU ≃ 1.5 × 108 km is the Earth's orbital radius). It thus forms a bubble in interstellar space, the heliosphere. The solar-wind plasma consisting of fully ionized hydrogen with a small admixture of helium soon reaches supersonic and super-Alfvénic speeds.

In the previous chapters we assumed, explicitly or tacitly, that turbulence is isotropic, in particular that there is no mean magnetic field, so that spectra depend only on the modulus of the wave vector k and structure functions only on the distance l between two points. We also assumed that the kinetic and magnetic energies have similar magnitudes, which implies that the magnetic field is distributed in a space-filling way. However, even if the turbulence is globally isotropic, the local dynamics is not, differing strongly between the directions parallel and perpendicular to the local magnetic field. As discussed in Section 5.3.3, the small-scale fluctuations are dominated by perpendicular modes, and Alfvén waves propagating parallelly are only weakly excited, which gives rise to the observed Kolmogorov energy spectrum k−5/3 instead of the IK spectrum k−3/2.

In nature magnetic turbulence often occurs about a mean magnetic field, just as hydrodynamic turbulence occurs about a mean flow. However, whereas the latter can be eliminated by transforming to a moving coordinate system, the presence of a mean magnetic field has a strong effect on the turbulent dynamics. If this field is much larger than the fluctuation amplitude, turbulence becomes essentially 2D in the plane perpendicular to the field, since the stiffness of field lines suppresses magnetic fluctuations, Alfvén waves, with short wavelengths along the field. Hence turbulent motions tend to simply displace field lines without bending them. To deal with this situation quantitatively we derive a set of equations for a plasma embedded in a strong magnetic field B0 = B0ez.

Turbulence is a ubiquitous phenomenon. Wherever fluids are set into motion turbulence tends to develop, as everyday experience shows us. When the fluid is electrically conducting, the turbulent motions are accompanied by magnetic-field fluctuations. However, conducting fluids are rare in our terrestrial world, where electrical conductors are usually solid. One of the rare examples of a fast-moving conducting fluid, which has been of some practical importance and concern and to which authors of theoretical studies sometimes referred, is, or better, was the flow of liquid sodium in the cooling ducts of a fast-breeder reactor. It is therefore not surprising that, in contrast to the broad scientific and technical literature on ordinary, i.e., hydrodynamic, turbulence, magnetic turbulence has not received much attention.

The most natural conducting fluid is an ionized gas, called a plasma. It is true that laboratory plasmas, which are confined by strong magnetic fields, notably in nuclear-fusion research, exhibit little dynamics, except in short disruptive pulses. Only the reversed-field pinch, a toroidal plasma discharge of relatively high plasma pressure, exhibits continuous magnetic activity, such that it is sometimes considered more as a convenient device for studying magnetic turbulence rather than as a particularly promising approach to controlled nuclear fusion.

The interstellar medium (ISM), which had formerly been better known for the allegoric shapes of its nebulae and dark clouds than for its physical properties, has developed into a fascinating area of astrophyscical research during the past few decades. An important aspect is the turbulent flows observed in many regions of the ISM, with velocities that, at least in the cooler parts, far exceed the speed of sound. These flows seem to play a decisive role in the cloud dynamics, slowing down gravitational contraction and star formation. In contrast to the objects studied in the two previous chapters, which had well-defined physical and geometrical properties and thus allowed a detailed analysis, the ISM is a rather diffuse system, whose modeling is far more uncertain and arbitrary. While the various atomic processes, such as transition probabilities and excitation rates, are well known and also the thermodynamic properties are fairly well understood, we have only a coarse picture of their hydrodynamics, including the effect of magnetic fields. Thus it is, for instance, difficult to apply hydrodynamic-stability theory without very special assumptions regarding geometry and flows. Hence the results we discuss in this chapter are mainly of qualitative nature, in which general arguments, such as equipartition and virialization, play an important role. The precise numerical factors, which are often found in the astrophysical literature, imply special choices of geometry and profiles and should not be taken too literally. In Section 12.1 we give a brief overview of the characteristic properties of the ISM.

Turbulence is usually associated with the idea of self-similarity, which means that the spatial distribution of the turbulent eddies looks the same on any scale level in the inertial range. This is a basic assumption in the Kolmogorov phenomenology K41 and, on the same lines, the IK phenomenology introduced in Section 5.3.2. It is, however, well known that this picture is not exactly true, since it ignores the existence of small-scale structures, which cannot be distributed in a uniform space-filling way. In fact, in a real turbulence field experiments as well as numerical simulations show that smaller eddies, or higher frequencies, become increasingly sparse, or intermittent, which apparently violates self-similarity. This chapter deals with the various aspects of intermittency.

Section 7.1 gives a brief introduction. We illustrate the concept of self-similarity by some simple examples and clarify the notion of intermittency, distinguishing between dissipation-range and inertial-range intermittency. Section 7.2 deals with structure functions, in particular the set of inertial-range scaling exponents, which are convenient parameters for a quantitative description of the statistical distribution of the turbulence scales. We discuss the important constraints on these exponents imposed by basic probabilistic requirements. Since experiments and, even more so, numerical simulations deal with turbulence of finite, often rather low, Reynolds number, the scaling range may be quite short, or even hardly discernable, especially for higher-order structure functions, which makes determination of the scaling exponents difficult. The scaling properties can, however, be substantially improved by making use of the extended self-similarity (ESS), which often provides surprisingly accurate values of the relative scaling exponents.

Until now we have viewed the turbulent motions and fields mainly in space, or configuration space. Though spatial structures are important in MHD turbulence, for instance as final states of selective decay processes, the most characteristic property of fully developed turbulence is the presence of a wide spectrum of different scales. Turbulence scales have already been used in a very loose way in Section 4.2.2 in discussing the mechanism of dynamic alignment. In this chapter these ideas will be given a more precise meaning. In Section 5.1 we introduce the concept of homogeneous turbulence, a very useful idealization of a turbulence field far away from boundary layers. Here the Fourier components of the field play the role of the amplitudes at a certain scale l ∼ k−1. Section 5.2 considers an approximation that, at first sight, has little resemblance to real turbulence, namely a nondissipative system of Fourier modes truncated at a finite wavenumber and its relaxed states, which are called absolute equilibrium distributions. In spite of their artificial character such states can provide valuable information about the tendencies of the spectral evolution in dissipative systems, in particular the direction of the spectral fluxes called cascades. In Section 5.3 we then switch on dissipation in order to study the spectral properties of MHD turbulence.

In the previous chapters turbulence was assumed incompressible. As discussed in Section 2.3, this assumption is valid if either the sonic Mach number of the flow is small, Ms = υ/cs « 1, or the Alfvén Mach number is small, MA = υ/υA « 1. The former condition applies to a weakly magnetized plasma, in which υA « cs, or to motions along the magnetic field, while the latter applies to motions perpendicular to the field. If the flow is turbulent, there is some arbitrariness in the definition of the Mach numbers, since one may choose (a) the mean flow velocity, (b) the r.m.s. velocity fluctuation υ = 〈ῦ2〉1/2 = (Ek)1/2, or (c) the local velocity. Following convention in turbulence theory, we refer to the Mach number in terms of the r.m.s. velocity, noting that local Mach numbers may be considerably higher.

Since laboratory plasmas are usually confined by a strong magnetic field, they can be considered incompressible, the dynamics consisting mainly of cross-field motions. Also the motions in the liquid core of the Earth, which drive the Earth's dynamo, are incompressible, since Ms « 1 (here inertial effects are often neglected altogether, which is called the magnetostrophic approximation). By constrast, most astrophysical plasmas are compressible, for instance the interstellar medium, which is rather cold, such that, in the turbulent motions observed, Ms, and possibly also MA, tend to be large (see Chapter 12), or the turbulence in the interplanetary plasma, which is riding on the supersonic and super-Alfvénic solar wind (Chapter 10).

In the preceding chapters we considered the dynamics of an individual system. Starting from a smooth state, fine structures develop, which, in general, become unstable at some point. After the onset of instability the structure of the flow is very complex and irregular and, most importantly, the further behavior is unpredictable in the sense that minimal changes would soon lead to a completely different state. Such a behavior is commonly called turbulent. Though a direct view of the continuously changing patterns is certainly most eyecatching and fascinating, a pictorial description of these structures is not very suitable for a quantitative analysis. On the other hand, it is just this chaotic behavior which makes turbulence accessible to a theoretical treatment involving statistical methods. While individual shapes and motions are intricate and volatile, the average properties of the turbulence described by the various correlation functions are, in general, smooth and follow rather simple laws. A well-known paradigm is the turbulent behavior in our atmosphere. We try to predict the short-term changes, called weather, in a deterministic way for as long as is feasible, which, as daily experience shows, is not very long, while predictions of the long-term behavior, called climate, can be made only on a statistical basis.

Dividing the fields into mean and fluctuating parts, we derive equations for the average quantities, the generalized Reynolds equations, which contain second-order moments of the fluctuating parts, the turbulent stresses.