Introduction

The scattering of electromagnetic radiation by perfect, uniform, spherical particles was described in Chapter 5. However, such particles are rarely found in nature. Most pulverized materials, including planetary regoliths, volcanic ash, laboratory samples, and industrial substances, have particles that almost invariably are irregular in shape, have rough surfaces, and are not uniform in either structure or composition. Even the liquid droplets in clouds are not perfectly spherical, and they contain inclusions of submicroscopic particles around which the liquid has condensed, so that they are not perfectly uniform. At the present state of our computational and analytical capabilities it is not possible to find exact solutions of scattering by such particles, so that it is necessary to rely on approximate models.

The objective of any model of single-particle scattering is to relate the microscopic properties of the particle (its geometry and complex refractive index) to the macroscopic properties (the scattering and extinction efficiencies and the phase function) that, in principle, can be measured by an appropriate scattering experiment. This chapter describes a variety of models that have been proposed to account for the scattering of light by irregular particles. This is not an exhaustive survey; rather, it is a commentary on those models that are most often encountered in remote-sensing applications or that offer some particular insight into the problem.

Introduction

In Chapters 8, 9, and 10, exact expressions for several different types of reflectances and related quantities frequently encountered in remote sensing and diffuse reflectance spectroscopy will be given. Next, approximate solutions to the radiative-transfer equation will be developed in order to obtain analytic evaluations of these quantities. As we discussed in Chapter 1, even though such analytic solutions are approximate, they are useful because there is little point in doing a detailed, exact calculation of the reflectance from a medium when the scattering properties of the particles that make up the medium are unknown and the absolute accuracy of the measurement is not high. In most of the cases encountered in remote sensing an approximate analytic solution is much more convenient and not necessarily less accurate than a numerical computer calculation.

In keeping with this discussion, polarization will be ignored until Chapter 14. This neglect is justified because most of the applications of interest involve the interpretation of remote-sensing or laboratory measurements in which the polarization of the incident irradiance is usually small. Although certain particles, such as Rayleigh scatterers or perfect spheres, may polarize the light strongly at some angles, the particles encountered in most applications are large, rough, and irregular, and the polarization of the light scattered by them is relatively small (Chapter 6) (Liou and Scotland, 1971).

Introduction

In this chapter the specular or mirror-like reflection that occurs when a plane electromagnetic wave encounters a plane surface separating two regions with different refractive indices is discussed quantitatively, along with the accompanying transmission, or refraction, through the interface. Specular reflection is important to the topic of this book for several reasons. First, it is an important tool for investigating properties of materials in the laboratory. Second, it occurs in remote-sensing applications when light is reflected from smooth parts of a planetary surface, such as the ocean. Third, it is one of the mechanisms by which light is scattered from a particle whose size is large compared with the wavelength, so that an understanding of this phenomenon is necessary to an understanding of diffuse reflectance from planetary regoliths.

Boundary conditions in electromagnetic theory

Whenever fields contain a boundary separating regions of differing electric or magnetic constants, certain conditions on the continuity of the fields must be satisfied. It is shown in any textbook on electricity and magnetism that the components of De and Bm perpendicular to the surface and the components of Ee and Hm tangential to the surface must be continuous across the boundary. If the fields constitute an electromagnetic wave propagating through the surface from one medium to another, it is found that these conditions cannot be satisfied unless there is another wave propagating backward from the surface into the first medium, in addition to the wave propagating forward from the surface into the second medium.

In this chapter we continue to study electromagnetic fluctuations in homogeneous, magnetized, collisionless plasmas. The new element here is that we consider the zeroth-order distribution function of each plasma component to be Maxwellian with drift velocity v0j parallel or antiparallel to B0 (Equation (3.1.3)). If two components have a relative drift v0 greater than some threshold, the corresponding free energy can lead to instability growth. Section 8.1 outlines the derivation of the dispersion equation for this case; Section 8.2 discusses electromagnetic ion/ion instabilities; Section 8.3 addresses electromagnetic electron/electron instabilities; Section 8.4 considers electromagnetic electron/ion instabilities; and Section 8.5 examines the consequences of electromagnetic effects on ion/ion instabilities that are electrostatic in the limit of zero β. Section 8.6 is a brief summary.

Space plasma heating and acceleration processes typically act on both species and are likely to give rise to beam/core distributions for both electrons and ions. However, in contrast to the case of T⊥ j > T‖j discussed in the previous chapter, the instabilities driven by beam/core free energies do not clearly separate into low frequency ion-driven and high frequency electron-driven modes. Thus, although we treat relative ion drifts and relative electron drifts separately in this chapter, this separation is due more to our desire to clarify the presentation than to any compelling physical arguments. Thus, in Sections 8.2 through 8.5, we consider a two-species, three-component plasma consisting of a relatively tenuous beam (denoted by subscript b), a relatively dense core (c), and a third component of the other species.

Every plasma is inhomogeneous to some extent, and the associated plasma gradients are sources of free energy that can drive plasma instabilities. In this chapter we consider the linear theory of drift instabilities, modes driven unstable by a plasma gradient perpendicular to B0.

In the direction parallel to a magnetic field, pressure gradients give rise to electric fields that lead to currents and bulk plasma motion; that is, such gradients do not correspond to a steady-state description under a macroscopic description of the plasma. However, pressure gradients perpendicular to a magnetic field can correspond to a steady-state situation; that is, ∇P in the momentum equation of a one-fluid description of the plasma can be balanced by the J × B0/c term. Nevertheless, such gradients do not correspond to an equilibrium plasma configuration; the zeroth-order distribution functions are non-Maxwellian and lead to the growth of plasma instabilities which act to dissipate the gradients. In this chapter we consider, as before, collisionless plasmas with a uniform zeroth-order magnetic field B0 = ẑB0. In Section 4.1 we discuss a model distribution function for density gradients perpendicular to a uniform magnetic field, examine the associated linear dispersion equation and discuss the two most popular density drift instabilities. Section 4.2 describes the instability properties that result when a plasma with a density gradient is subject to a uniform acceleration, and Section 4.3 briefly summarizes some properties of temperature drift instabilities.

In this chapter we examine, as before, electromagnetic fluctuations in a homogeneous, magnetized, collisionless plasma. In contrast to the previous chapter, however, we admit anisotropies in the distribution functions. In particular we consider a two-temperature bi-Maxwellian zeroth-order distribution function; this permits the growth of temperature anisotropy instabilities. Section 7.1 outlines the derivation of the dispersion equation; Section 7.2 discusses the properties of modes driven unstable by a proton temperature anisotropy, whereas Section 7.3 discusses the properties of electron temperature anisotropy instabilities. Section 7.4 is a brief summary.

Our emphasis in this chapter is on instabilities driven by T⊥j > T‖ a condition that is observed more often in space plasmas than the converse T‖ > T⊥. The reason for this discrepancy is simple: although space plasmas do not necessarily exhibit a bias toward perpendicular heating processes, perpendicular heating does not much change the mobility of the heated particles, whereas parallel heating enables the particles to move more rapidly along B0. Thus parallel-heated particles may leave the region of energization more quickly, implying that T‖ > T⊥ should be a less frequently observed condition. Of course, parallel-heated particles may appear elsewhere as a magnetic-field aligned beam streaming against a cooler background plasma; the electromagnetic instabilities driven by such configurations have quite different properties from temperature anisotropy instabilities, and are studied in detail in the next chapter.

Plasma instabilities are normal modes of a system that grow in space or time. Thus the word “instability” implies a well-defined relationship between wavevector k and frequency ω; this in turn implies that the associated plasma fluctuations are relatively weak so that linear theory is appropriate to describe the physics.

This book uses linear Vlasov theory to describe the propagation, damping and growth of plasma modes. Linear theory cannot describe the ultimate fate of a plasma instability, nor its interactions with other modes. Of course the questions of how an instability reaches maximum amplitude, whether and how it contributes to plasma transport and whether such transport affects the overall flow of mass, momentum and energy at large scales are crucial for establishing the relevance of microphysics to large scale modelling of space plasmas. But these questions must be addressed by nonlinear theory and computer simulation, which are beyond the purview of this book. Our relatively modest goal is to use computer solutions of the unapproximated Vlasov dispersion equation to firmly establish the properties of plasma normal modes; our hope is that this information will provide a useful foundation for the interpretation of computer simulations and spacecraft observations under conditions of relatively weak fluctuation amplitudes.

Micro- vs macro-

The most general classification of growing modes in a plasma divides them into two broad categories: macroinstabilities at relatively long wavelengths and microinstabilities at shorter wavelengths.

This chapter begins our consideration of instabilities, plasma modes that grow in time or space. The source of growth of a plasma microinstability is what is imprecisely called “free energy:” an anisotropy or inhomogeneity in the zeroth-order velocity distribution function. In this chapter we consider free energy sources associated with the relative drifts of plasma components, and find that different types of relative drifts each can give rise to several different unstable modes.

As in Chapter 2, we consider uniform collisionless plasmas in which the evolution of the distribution function of the jth component is described by the Vlasov equation (1.3.2). Again we restrict ourselves to electrostatic fluctuations; that is, we assume there are no fluctuating magnetic fields and the fluctuating electric fields are derived from Poisson's equation (1.2.4). In Section 3.1 we state zeroth-order distribution functions representing several different free energy sources. Section 3.2 considers electrostatic instabilities driven by component/component relative drifts in unmagnetized plasmas and Section 3.3 considers electrostatic instabilities driven by the same free energy sources in magnetized plasmas. Throughout this chapter, we assume the plasma to be charge neutral and to bear no steady-state electric field.

Although virtually all space plasmas bear ambient magnetic fields, some waves and instabilities have properties that are essentially independent of B0. In Section 3.3 we demonstrate this by showing that the magnetized electrostatic dispersion equation at k × B0 = 0 reduces to the unmagnetized form.