Diffusion Coefficients and Thermal Conductivities

The heat flow and ambipolar diffusion equations that contain the higher-order transport effects, such as thermal diffusion and diffusion thermal heat flow, are presented in Section 5.14. The transport coefficients that appear in these equations have been calculated using both the linear (4.129a–g) and semilinear (4.132a,b) collision terms. Here, the more general semilinear transport coefficients are presented, which are valid for arbitrarily large temperature differences between the interacting species. These coefficients reduce to the linear coefficients in the limit of small temperature differences, i.e., when (Ts − Tt)/Tst ≪ l.

The general expressions for the ion and neutral heat flows are summarized as follows

where subscripts s and t refer to either ion or neutral species. The thermal conductivities and diffusion thermal coefficients in equations (I.1) and (I.2) are given by

where

Note that a simple change of subscripts in equations (1.3) to (1.10) yields the other transport coefficients that are needed.

The 13-moment system of transport equations was presented in Chapter 3 and several associated sets of collision terms were derived in Chapter 4. These 13-moment transport equations, in combination with the Maxwell equations for the electric and magnetic fields, are very general and can be applied to describe a wide range of plasma flows in the ionospheres. However, the complete system of equations for a multi-species plasma is difficult to solve under most circumstances, and therefore, simplified sets of transport equations have been used over the years. The simplified sets of equations that are based on the assumption of collision dominance were presented in Chapter 5. In this chapter, certain simplified transport equations are derived in which the plasma is treated as a single conducting fluid, rather than a mixture of individual plasma species. These single-fluid transport equations, along with the Maxwell equations, are known as the single-fluid magnetohydrodynamic (MHD) equations.

The outline of this chapter is as follows. First, the single-fluid transport equations are derived from the 13-moment system of equations. Subsequently, a generalized Ohm's law is derived for a fully ionized plasma. This naturally leads to simplifications that yield the classical set of MHD equations.

The magnetosphere–ionosphere–atmosphere system at high latitudes is strongly coupled via electric fields, particle precipitation, field-aligned currents, heat flows, and frictional interactions, as shown schematically in Figure 12.1. Electric fields of magnetospheric origin induce a large-scale motion of the high-latitude ionosphere, which affects the electron density morphology. As the plasma drifts through the neutrals, the ion temperature is raised owing to ion-neutral frictional heating. The elevated ion temperature then alters the ion chemical reaction rates, topside plasma scale heights, and ion composition. Also, particle precipitation in the auroral oval acts to produce enhanced ionization rates and elevated electron temperatures, which affect the ion and electron densities and temperatures. These ionospheric changes, in turn, have a significant effect on the thermospheric structure, circulation, and composition. At F region altitudes, the neutral atmosphere tends to follow, but lags behind, the convecting ionospheric plasma. The resulting ion-neutral frictional heating induces vertical winds and O/N2 composition changes. These atmospheric changes then affect the ionospheric densities and temperatures.

The ionosphere–thermosphere system also has a significant effect on the magnetosphere. Precipitating auroral electrons produce conductivity enhancements, which can modify the convection electric field, large-scale current systems, and the electrodynamics of the magnetosphere–ionosphere system as a whole. Also, once the thermosphere is set into motion due to convection electric fields, the large inertia of the neutral atmosphere will act to produce dynamo electric fields whenever the magnetosphere tries to change its electrodynamic state.

Throughout the book all equations and formulas are expressed in the MKSA system of units. However, Gaussian-cgs units are still frequently used by many scientists. In this latter system, all electrical quantities are in electrostatic units (esu) except for B, which is in electromagnetic units (emu). Most formulas that are in MKSA units can be converted to Gaussian-cgs units by replacing B with (B/c) and ε0 by 1/4π, where c = (ε0μ0)−1/2 is the speed of light.

For easy reference, the formulas in Table E. 1 are given in both MKSA and Gaussiancgs units. The last four equations are known as the Maxwell equations and, as given here, pertain to a vacuum.

Introduction

Empirical models of the Venus and terrestrial upper atmospheres have been developed. Tables K. 1 and K.2 provide representative values of the Venus neutral temperature and densities for noon and midnight conditions, respectively. The values are from the Venus International Reference Atmosphere (VIRA) model.1 Representative neutral temperatures and densities for the Earth's thermosphere are given in Tables K.3 to K.6. The tables provide typical values at noon and midnight for both solar maximum and minimum conditions, and for quiet geomagnetic activity. The neutral parameters are from the Mass Spectrometer and Incoherent Scatter (MSIS) empirical model.

Neutral atmospheres play a crucial role with regard to the formation, dynamics, and energetics of ionospheres, and therefore, an understanding of ionospheric behavior requires a knowledge of atmospheric behavior. A general description of the atmospheres that give rise to the ionospheres was given in Chapter 2. In this chapter, the processes that operate in upper atmospheres are described, and the equations presented have general applicability. However, the discussion of specifics is directed toward the terrestrial upper atmosphere because our knowledge of this atmosphere is much more extensive than that for all of the other atmospheres (i.e., other planets, moons, and comets).

Typically, the lower domain of an upper atmosphere is turbulent, and the various atomic and molecular species are thoroughly mixed. However, as altitude increases, molecular diffusion rapidly becomes important and a diffusive separation of the various neutral species occurs. For Earth, this diffusive separation region extends from about 110 to 500 km, and most of the ionosphere and atmosphere interactions occur in this region. At higher altitudes the collisional mean-free-path becomes very long and the neutral particles basically follow ballistic trajectories. For the case of light neutrals, such as hydrogen and helium, and more energetic heavier gases, some of the ballistic trajectories can lead to the escape of particles from the atmosphere.

The topics in this chapter progress from the main processes that operate in the diffusive separation region of an upper atmosphere to the thermal escape of atoms from the top of the atmosphere.

A wide variety of plasma flows can be found in the various planetary ionospheres. For example, gentle near-equilibrium flows occur in the terrestrial ionosphere at midlatitudes, while highly nonequilibrium flow conditions exist in the terrestrial polar wind and in the Venus ionosphere near the solar terminator. The highly nonequilibrium flows are generally characterized by large temperature differences between the interacting species, by flow speeds approaching and exceeding thermal speeds, and by flow conditions changing from collision-dominated to collisionless regimes. In an effort to model the various ionospheric flow conditions, several different mathematical approaches have been used, including collision-dominated and collisionless transport equations, kinetic and semikinetic models, and macroscopic particle-in-cell techniques. However, the transport equation approach has received the most attention, because it can handle most of the flow conditions encountered in planetary ionospheres. Typically, numerous assumptions are made to simplify the transport equations before they are applied, and therefore, it is instructive to trace the derivation of the various sets of transport equations in order to establish their intrinsic strengths and limitations.

Boltzmann Equation

The Boltzmann equation not only is the starting point for the derivation of the different sets of transport equations but also forms the basis for the kinetic and semikinetic theories. With Boltzmann's approach, one is not interested in the motion of individual particles in the gas, but instead with the distribution of particles.

Introduction

The determination of physical constants and the definition of the units with which they are measured is a specialised and, to many, hidden branch of science.

A quantity with dimensions is one whose value must be expressed relative to one or more standard units. In the spirit of the rest of the book, this section is based around the International System of units (SI). This system uses seven base units (the number is somewhat arbitrary), such as the kilogram and the second, and defines their magnitudes in terms of physical laws or, in the case of the kilogram, an object called the “international prototype of the kilogram” kept in Paris. For convenience there are also a number of derived standards, such as the volt, which are defined as set combinations of the basic seven. Most of the physical observables we regard as being in some sense fundamental, such as the charge on an electron, are now known to a relative standard uncertainty, ur, of less than 10–7. The least well determined is the Newtonian constant of gravitation, presently standing at a rather lamentable ur of 1.5 – 10–3, and the best is the Rydberg constant (ur = 7.6 – 10–12). The dimensionless electron g-factor, representing twice the magnetic moment of an electron measured in Bohr magnetons, is now known to a relative uncertainty of only 4.1 – 10–12.

No matter which base units are used, physical quantities are expressed as the product of a numerical value and a unit. These two components have more-or-less equal standing and can be manipulated by following the usual rules of algebra.