The earth's atmosphere

This book is mainly concerned with the effect on radio wave propagation of the ionised regions of the earth's atmosphere. Near the ground the air is almost unionised and its electrical conductivity is negligibly small, because the ionising radiations have all been absorbed at greater heights. When any part of it is in equilibrium, its state is controlled by the earth's gravitational field so that it is a horizontally stratified system. Although it is never in complete equilibrium, gravity has a powerful controlling effect up to about 1000 km from the ground.

The molecules of the neutral atmosphere have an electric polarisability which means that the refractive index for radio waves is very slightly greater than unity, about 1.00026, near the ground. The water vapour also affects the refractive index. Thus the neutral air can very slightly refract radio waves. This can lead to important effects in radio propagation. For example in stable meteorological conditions a duct can form near the surface of the sea, acting as a wave guide in which high frequency radio waves can propagate to great distances (see Booker and Walkinshaw, 1946; Brekhovskikh, 1960; Budden, 1961b; Wait, 1962). Spatial irregularities of the refractive index of the air can cause scintillation of radio signals, and also scattering which can be used to achieve radio propagation beyond the horizon (Booker and Gordon, 1950). These effects are beyond the scope of this book.

Introduction

This and the remaining chapters are mainly concerned with problems in which the approximations of ray theory cannot be made, so that a more detailed study of the solutions of the governing differential equations is needed. The solution of this type of problem is called a ‘full wave solution’. The subject has been studied almost entirely for plane stratified media. When the medium is not plane stratified, for example when the earth's curvature is allowed for, a possible method is to use an ‘earth flattening’ transformation that converts the equations into those for an equivalent plane stratified system; see §§ 10.4, 18.8. See Westcott (1968, 1969) and Sharaf (1969) for examples of solutions that do not use such a transformation.

This chapter is written entirely in terms of a horizontally stratified ionosphere and for frequencies greater than about 1 kHz, so that the effect of positive ions in the plasma may be neglected. The earth's curvature also is neglected.

The need for full wave solutions arises when the wavelength in the medium is large so that the electric permittivity changes appreciably within one wavelength and the medium cannot be treated as slowly varying (§§ 7.6, 7.10). Thus it arises particularly at very low frequencies, say 1 to 500 kHz. But even at higher frequencies the wavelength in the medium is large where the refractive index n is small, and some full wave solutions are therefore often needed for high frequencies too.

Introduction

Before (2.45) can be applied to the theory of wave propagation in a plasma, it is necessary to express the electric displacement D, and therefore the electric polarisation P, in terms of the electric intensity E. The resulting expressions are called the constitutive relations of the plasma and are derived in this chapter. The subject of wave propagation is resumed in ch. 4.

From now on, except in § 3.8, all fields are assumed to vary harmonically in time, and are designated by capital letters representing complex vectors, as explained in § 2.5. Time derivatives of some of the fields appear in the constitutive relations, and since ∂/∂t ≡ iω, it follows that the angular frequency ω appears in the expression for the electric permittivity. When this happens the medium is said to be time dispersive.

In this and the following chapter the plasma is assumed to be homogeneous. In later chapters the results are applied to an inhomogeneous plasma. This is justified provided that the plasma is sufficiently slowly varying. The meaning of ‘slowly varying’ is discussed in § 7.10. It is further assumed, in the present chapter, that the spatial variation of the fields can be ignored. This means that, over a distance large compared with N−⅓, the fields can be treated as uniform. Thus spatial derivatives of the fields do not appear in the expression for the permittivity.

Introduction

Ch. 10 studied the problem of ray tracing in a plane stratified medium. The objective of the present chapter is to extend this theory to deal with media that are not plane stratified. For example the composition of the ionosphere nearly always depends on the horizontal coordinates, and there are cases especially near twilight when it cannot be assumed to be horizontally stratified. The structure of the magnetosphere is controlled largely by the earth's magnetic field and the solar wind, see § 1.9, and it is not plane stratified.

The number of published papers on ray tracing methods is very large and the subject has acquired some mathematical interest that often goes beyond the needs of practical radio engineers. In this book no attempt has been made to give all the references, but a few that have come to the author's attention are given at the relevant points in this chapter. The objective here is to present the basic physical ideas of ray tracing and to describe the methods that have mainly been used for radio waves in the ionosphere and magnetosphere.

In § 10.2 the ray path was expressed by integrals (10.3), and for plane stratified media these can often be evaluated in closed form. For the more general plasma, however, this is rarely possible. The ray path was also expressed by its differential equations (10.5) for dx/dz, dy/dz.

Introduction

We now have to consider how results such as those of ch. 12 for an isotropic plasma must be modified when the earth's magnetic field is allowed for. The dispersion relation (4.47) or (4.51) is now much more complicated so that, even for simple electron height distribution functions N(z), it is not possible to derive algebraic expressions for such quantities as the equivalent height of reflection h′(f) or the horizontal range D(θ). These and other quantities must now be evaluated numerically. The ray direction is not now in general the same as the wave normal. Breit and Tuve's theorem and Martyn's theorems do not hold. These properties therefore cannot be used in ray tracing. Some general results for ray tracing in a stratified anisotropic plasma have already been given in ch. 10.

The first part of this chapter §§ 13.2–13.6 is concerned with vertically incident pulses of radio waves on a stratified ionosphere, since this is the basis of the ionosonde technique, § 1.7, which is widely used for ionospheric sounding. The transmitted pulses have a radio frequency f which is here called the ‘probing frequency’. It is also sometimes called the ‘carrier frequency’. The study of this subject involves numerical methods that are important because of their use for analysing ionospheric data and for the prediction of maximum usable frequencies. It has already been explained, § 10.2, that the incident pulse splits into two separate pulses, ordinary and extraordinary, that travel independently.

Radio waves in the ionosphere was first published in 1961. Since then there have been many major advances that affect the study of radio propagation, and three in particular. The first is the use of space vehicles and rockets, which have enabled the top side of the ionosphere to be studied and have revealed the earth's magnetosphere and magnetotail. The second is the advance of the subject ‘plasma physics’ which has transformed our knowledge of the physical processes in ion plasmas. The third is the use of computers, large and small, which, by removing the need for laborious calculations, have changed our attitude to theoretical results, even though they are sometimes wrongly used as a substitute for clear physical thinking. Moreover some important textbooks have appeared in the intervening 20 years. They are too numerous all to be mentioned, but four have a special bearing on the subjects of this book. These are: T.H. Stix (1962), Theory of plasma waves; K. Rawer and K. Suchy (1967), Radio observations of the ionosphere; K. Davies (1969), Ionospheric radio waves; and V.L. Ginzburg (1970), Propagation of electromagnetic waves in plasma.

Some of the topics mentioned in Radio waves in the ionosphere, here abbreviated to RWI, are of importance in other branches of physics. Two in particular have formed the subject of numerous mathematical papers. These are (a) ‘W.K.B. solutions’, and (b) the study of ordinary linear homogeneous differential equations of the second order.

Introduction

This chapter is concerned with various applications of the full wave methods discussed in ch. 18. The object is to illustrate general principles but not to give details of the results. The number of possible applications is very large and only a selection can be given here. The topics can be divided into two groups: (a) problems where the solutions can be expressed in terms of known functions, and (b) problems where computer integration of the differential equations, or an equivalent method as in §§ 18.2–18.11, is used. In nearly all applications of group (a) it is necessary to make substantial approximations. The group (a) can be further subdivided into those cases where the fourth order governing differential equations are separated into two independent equations each of the second order, and those where the full fourth order system must be used. For the separated second order equations the theory is an extension of ch. 15 which applied for an isotropic medium, and many of its results can be used here.

In all the examples of this chapter the ionosphere is assumed to be horizontally stratified, with the z axis vertical. The incident wave is taken to be a plane wave with its wave normal in the x–z plane. It is assumed that the only effective charges in the plasma are the electrons.

Introduction

The first order coupled wave equations (16.22) were derived and studied in § 16.3 for the neighbourhood of a coupling point z = zp where two roots q1, q2 of the Booker quartic equation are equal. If this coupling point is sufficiently isolated, a uniform approximation solution can be used in its neighbourhood and this was given by (16.29), (16.30). It was used in § 16.7 to study the phase integral formula for coupling. The approximations may fail, however, if there is another coupling point near to zp, and then a more elaborate treatment is needed. When two coupling points coincide this is called ‘coalescence’. This chapter is concerned with coupled wave equations when conditions are at or near coalescence.

A coupling point that does not coincide with any other will be called a ‘single coupling point’. It is isolated only if it is far enough away from other coupling points and singularities for the uniform approximation solution (16.30) to apply with small error for values of |ζ| up to about unity. Thus an isolated coupling point is single, but the reverse is not necessarily true.

Various types of coalescence are possible. The two coupling points that coalesce may be associated with different pairs q1, q2 and q3, q4 of roots of the Booker quartic. This is not a true coalescence because the two pairs of waves are propagated independently of each other.

Introduction

Although the earth's magnetic field has a very important influence on the propagation of radio waves in the ionospheric plasma, it is nevertheless of interest to study propagation when its effect is neglected. This was done in the early days of research in the probing of the ionosphere by radio waves, and it led to an understanding of some of the underlying physical principles; see, for example, Appleton (1928, 1930). In this chapter, therefore, the earth's magnetic field is ignored. For most of the chapter the effect of electron collisions is also ignored, but they are discussed in §§ 12.3, 12.11. The effect both of collisions and of the earth's magnetic field is small at sufficiently high frequencies, so that some of the results are then useful, for example with frequencies of order 40 MHz or more, as used in radio astronomy. For long distance radio communication the frequencies used are often comparable with the maximum usable frequency, §§ 12.8–12.10, which may be three to six times the penetration frequency of the F-layer. They are therefore in the range 10 to 40 MHz. This is large compared with the electron gyro-frequency which is of order 1 MHz, so that here again results for an isotropic ionosphere are useful, although effects of the earth's magnetic field have to be considered for some purposes.

This chapter is largely concerned with the use of pulses of radio waves and the propagation of wave packets, and uses results from ch. 10, especially §§ 10.2–10.6.

Introduction

In ch. 7 it was shown that at most levels in a slowly varying stratified ionosphere, and for radio waves of frequency greater than about 100 kHz, the propagation can be described by approximate solutions of the differential equations, known as W.K.B. solutions. The approximations fail, however, near levels where two roots q of the Booker quartic equation approach equality. In this chapter we begin the study of how to solve the differential equations when this failure occurs. For an isotropic ionosphere there are only two values of q and they are equal and opposite, and given by (7.1), (7.2). The present chapter examines this case. It leads on to a detailed study of the Airy integral function, which is needed also for the solution of other problems. Its use for studying propagation in an anisotropic ionosphere where two qs approach equality is described in § 16.3.

For an isotropic ionosphere, a level where q = 0 is a level of reflection. In § 7.19 it was implied that the W.K.B. solution for an upgoing wave is somehow converted, at the reflection level, into the W.K.B. solution for a downgoing wave with the same amplitude factor, and this led to the expression (7.151) for the reflection coefficient R. The justification for this assertion is examined in this chapter and it is shown in § 8.20 to require only a small modification, as in (7.152).

Introduction

This chapter continues the discussion of radio waves in a stratified ionosphere and uses a coordinate system x, y, z as defined in §6.1. Thus the electric permittivity ε(z) of the plasma is a function only of the height z. It is assumed that the incident wave below the ionosphere is a plane wave (6.1) with S2 = 0, S1 = S so that wave normals, at all heights z, are parallel to the plane y = constant, and for all field components (6.48) is satisfied. It was shown in § 6.10 that if the four components Ex, Ey, ℋx, ℋy of the total field are known at any height z, they can be expressed as the sum of the fields of the four characteristic waves, with factors f1, f2, f3, f4. In a homogeneous medium the four waves would be progressive waves, and these factors would be exp (− ikqiz), i = 1, 2, 3, 4. We now enquire how they depend on z in a variable medium. This question is equivalent to asking whether there is, for a variable medium, any analogue of the progressive characteristic waves in a homogeneous medium. The answer is that there is no exact analogue. There are, however, approximate solutions, the W.K.B. solutions, which have many of the properties of progressive waves.

This problem has proved to be of the greatest importance in all those branches of physics concerned with wave propagation.

Introduction

The earlier chapters have discussed the propagation of a plane progressive radio wave in a homogeneous plasma. It is now necessary to study propagation in a medium that varies in space, and this is the main subject of the rest of this book. The most important case is a medium that is plane stratified, and the later chapters are largely concerned with the earth's ionosphere that is assumed to be horizontally stratified. Because of the earth's curvature the stratification is not exactly plane, but for many purposes this curvature can be neglected. In cases of curved stratification it is often possible to neglect the curvature and treat the medium as locally plane stratified. Some cases where the earth's curvature is allowed for are discussed in §§ 10.4, 18.8.

The theory of this chapter is given in terms of the earth's ionosphere. A Cartesian coordinate system x, y, z is used with x, y horizontal and z vertically upwards. These coordinates are not in general the same as the x, y, z of chs. 2–4. The composition of the plasma, that is the electron and ion concentrations Ne, Ni are assumed to be functions of z only. For the study of radio propagation the frequency is usually great enough for the effect of the ions to be ignored. When radio waves are reflected in the ionosphere the important effects occur in a range of height that is small enough for the spatial variation of the earth's magnetic induction B to be ignored.