In this chapter we generalize the results of Chapters 3 and 4 to the case of quasi-longitudinal whistler-mode propagation. As in Section 2.1 we consider whistler-mode propagation as quasi-longitudinal if either ⃒θ⃒ ≪ 1 (see inequalities (2.14) and (2.17)) in the plasma with arbitrary electron density (inequalities (2.10) and (2.12) are not necessarily valid), or inequalities (2.9) and (2.10) (or (2.12)) are valid simultaneously provided the whistlermode wave normal angle θ is not close to the resonance cone angle θR0 in & cold plasma (see equation (2.13)). First we consider the case ⃒θ⃒ ≪ 1 when whistler-mode waves propagate almost parallel to the magnetic field.

As was shown in Chapter 1, when we impose no restrictions on the electron density we should use the general relativistic expressions for the elements of the plasma dielectric tensor in the form (1.73). Assuming that the waves propagate through plasma with the electron distribution function (1.76) and imposing conditions (1.77) we write these expressions in a much simpler form (1.78). Also, we assume that the electron temperature is so low that it can only slightly perturb the corresponding whistler-mode dispersion equation in a cold plasma, i.e.

where N is the whistler-mode refractive index in a hot plasma, and N0 the whistler-mode refractive index in a cold plasma defined by (2.15).

Approximations

In view of the applications of our theory to the conditions of the Earth's magnetosphere the following assumptions are made:

1. The plasma is assumed to be homogeneous in the sense that its actual inhomogeneity does not influence its dispersion characteristics, instability or damping at any particular point, although in general these characteristics can change from one point to another. For low-amplitude waves this assumption is valid when the wavelength is well below the characteristic scale length of plasma inhomogeneity, a condition which is satisfied for whistlermode waves propagating in most areas of the magnetosphere (except in the lower ionosphere). For finite amplitude waves the condition for plasma homogeneity depends on wave amplitude, but the discussion of these effects is beyond the scope of the book (see e.g. Karpman, 1974).

2. The plasma is assumed to be collisionless in the sense that we neglect the contribution of Coulomb collisions between charged particles as well as collisions between charged and neutral particles leading to charge exchange. More rigorously this assumption can be written as: where qα is the particle's charge (index α indicates the type of particle: α = e for electrons, α = p for protons; 〈r12〉 is the average distance between particles; Tα is the particles' temperature in energy units.

The physical meaning of (1.1) is obvious: the average energy of interaction between charged particles is well below their average kinetic energy.

As was shown in Chapter 5, the quasi-electrostatic approximation becomes invalid when θ approaches the resonance cone angle θR defined by (2.23) or (2.24) (or θR0 defined by (2.13) in the case of a dense plasma, the contribution of ions being neglected). However, at θ equal to or close to θR we can use another approximation based on the following assumptions:

(1) The wave refractive index N is assumed to be so large that only the contribution of the terms proportional to the highest powers of N in the dispersion equation is to be taken into account.

(2) The plasma temperature is assumed to be so low that inequalities (1.77) are valid and the elements of εij can be written in the form (1.78), εijt and εijr being the perturbations of εij0.

The first assumption seems to be a straightforward one in a sufficiently low-temperature plasma, as in a cold plasma limit N20 ∞ and when θ → θR. However, the second assumption does not seem to be an obvious one since εijt ∼ N2 and εijt → ∞ as N2 → ∞. Thus N should be large enough to satisfy the first assumption but small enough to satisfy the second one. The validity of these assumptions can be checked by the actual value of N obtained from the solution.

In contrast to Chapter 2, in this chapter we restrict ourselves to considering the parallel whistler-mode propagation, but impose no restrictions on magnetospheric electron temperature except the condition (1.2) of weakly relativistic approximation. We begin with some general simplifications of the dispersion equation (1.42) with the elements of the plasma dielectric tensor defined by (1.73) in the limiting case θ → 0. In this limit we can assume that ⃒λα⃒ ≡ ⃒k⊥υ⊥/Ωα ⃒ ≪ 1 and expand the Bessel functions in (1.74) using the following formula (Abramovitz & Stegun, 1964).

Remembering (3.1) and keeping only zero-order terms with respect to λα we obtain the following expressions for the non-zero elements of the tensor Πij(n,α) defined by (1.74):

From (1.73) and (3.2) it follows that:

In view of (3.3) and remembering our assumption that θ = 0 we can simplify the dispersion equation (1.42) to

Parallel propagation (weakly relativistic approximation)

This equation can be further simplified if we take into account the contribution of electrons only (which is justified for whistler-mode waves) and present ∈11 = ∈22 as ∈11 = 1 + ∈+ + ∈_ and ∈12 = –∈21 as ∈12 = i(∈+ – ∈_), where ∈+ and ∈_ are the contributions of the electron currents corresponding to n = 1 and n = – 1 respectively in the term ∈11 in (1.73). In this case equation (3.4) can be written as

In order to satisfy (3.5) at least one of three factors in the left-hand side of this equation must be equal to zero. The equation ∈33 = 0 is the dispersion equation for electrostatic Langmuir waves, propagating along the magnetic field (see equation (1.13)).

The linear theory of whistler-mode propagation, growth and damping considered so far has been based on the assumption that waves with different frequencies and wave numbers do not interact with each other (superposition principle) and that the waves do not cause any systematic change in the background particle (electron) distribution function f0. A self-consistent analysis of both these processes creates, in general, a very complicated problem even for modern computers (see e.g. Nunn, 1990). However, in many practically important cases we can develop an approximate analytical theory which takes into account some of these processes and neglects others. This theory, known as the non-linear theory, has been developed during the last 30 years and its results are summarized in numerous monographs and review papers, such as those by Kadomtsev (1965), Vedenov (1968), Sagdeev & Galeev (1969), Tsytovich (1972), Karpman (1974), Akhiezer et al. (1975), Hasegawa (1975), Vedenov & Ryutov (1975), Galeev & Sagdeev (1979), Bespalov & Trakhtengertz (1986), Zaslavsky & Sagdeev (1988), Petviashvili & Pokhotelov (1991) and many others. I have no intention of amending this long list of references by one more contribution. Instead I will restrict myself to illustrating the methods of non-linear theory by two particularly simple examples: the quasi-linear theory of whistler-mode waves (Section 8.1) and the non-linear theory of monochromatic whistler-mode waves (Section 8.2).

Quasi-linear theory

(a) Basic equations

A theory which assumes that the wave amplitude is so small that the superposition principle remains valid, but large enough to provide a non-negligible change of the background electron distribution function ƒo under the influence of the waves, is known as the quasi-linear theory (Akhiezer et al., 1975).

General relativity was for too long the ugly duckling of science. In the 50s and 60s the dominant impression was of the difficulty of the equations, solvable only by arcane techniques inapplicable elsewhere; of the scarcety of significant experimental tests; of the prohibitive cost of computational solutions, compounded by a lack of rigorous approximation techniques; and of the isolation of the subject from the physics of the other fundamental forces. This led to a situation where, even in the 70s, much theoretical work was becoming increasing irrelevant to physics. Exact solutions proliferated but (with the exception of cosmology) attempts at physical interpretation were few and unconvincing. Mathematical investigations in the wake of the singularity theorems became increasingly sophisticated, but few were applied to actual physical models. In the 70s and 80s, however, all this changed, with the growth of experimental relativity, the trend to geometrical methods in high energy physics, and the inception of numerical relativity. The workshop reported in this book marks the complete clearing of this last hurdle, as reliable and practical computational techniques are established.

It brought together numerical and classical relativists, and showed that the cultural gap between them was closing fast. Dramatically increased standards of reliability and accuracy had been set, and were being achieved in many cases, so that numerical work can no longer be seen merely as providing a rough indication for the ‘proper’ work of analysis.

Abstract. The difficulties in solving the characteristic initial value problem in general relativity when matter fields are included are discussed. A scheme is proposed with new dependent variables, coordinates and tetrad which should alleviate some of the problems.

THE CHARACTERISTIC INITIAL VALUE PROBLEM

Almost all calculations in numerical relativity are based on the Arnowitt-Deser-Misner (1962) formalism in which spacetime is foliated by spacelike hypersurfaces. The Einstein equations are decomposed into elliptic constraint equations which are intrinsic to the slices, and hyperbolic evolution equations which govern the evolution from slice to slice. In principle the constraint equations have only to be solved on the initial slice, but even this requires considerable effort and a large computer.

Stewart and collaborators, Friedrich and Stewart (1982), Corkill and Stewart (1983), developed an alternative approach based on a fundamentally different principle. In this spacetime was foliated by null hypersurfaces, and originally two families of null hypersurfaces were used. A helpful analogy is to consider the Schwarzschild solution described by double null coordinates. Examination of that solution will reveal that such coordinates have many theoretical advantages. One can explore the spacetime through the regular event horizon right up to the singularity at r = 0. Further one can proceed out along surfaces u = const, to future null infinity, the area inhabited by distant observers. Indeed one can bring infinity in to a finite point by the technique of conformal compactification, due originally to Penrose.

Abstract. Criteria are presented for choosing a matter model in analytical or numerical investigations of the Einstein equations. Two types of matter, the perfect fluid and the collisionless gas, are treated in some detail. It is discussed how the former has a tendency to develop singularities which have little to do with gravitation (matter-generated singularities) whereas the latter does not seem to suffer from this problem. The question of how the concept of a matter-generated singularity could be defined rigorously is considered briefly.

INTRODUCTION

In any investigation of the Einstein equations it is necessary to make some assumptions about the energy-momentum tensor. One possibility is simply to require that some energy conditions be satisfied. (In that case it might be more appropriate to say that the object of study is the ‘Einstein inequalities’.) Despite the fact that this is sufficient to obtain important results including the singularity theorems and the positive mass theorem, it is very likely that there are significant results concerning the qualitative behaviour of solutions of the Einstein equations which require more specific assumptions. In any case the choice of a definite matter model is indispensable for numerical calculations and for analytic work based on the use of a well-posed initial value problem. The particular kind of matter chosen will of course depend on the problem being studied.

Abstract. We review the classical and modern work on the stability of rotating fluid configurations with particular interest in astrophysical scenarios likely to produce gravitational radiation. We describe a hybrid method for numerically generating axisymmetric equilibrium models in rapid differential rotation based on the self consistent field approach. We include a description of the 3-D hydrodynamics code that we have developed to model the production of gravitational radiation, and present the results of a 3-D test case simulating the growth of the dynamical bar mode instability in a rapidly rotating polytrope.

INTRODUCTION

Overview

The study of the effects of rotation on equilibrium fluid bodies was begun by Newton in Book III of the Principia where he investigated the consequences of rotation on the figure of the earth, and concluded that the result would be a flattening at the poles to give the earth a slightly oblate shape. Much of the classical work accomplished since that time has been concerned with equilibrium configurations for fluids with uniform density and/or rigid rotation. Recent advances in computing technology, however, have allowed more detailed investigations involving differential rotation, various equations of state, and dynamical evolution of self gravitating systems. This work has fostered a variety of astrophysical applications, notably in the study of the formation of single and binary stars from collapsing gas clouds, and of the structure of compact objects, such as white dwarfs and neutron stars.

Abstract. The characteristic initial value problem is reviewed and a number of possible schemes for implementing it are discussed. Particular attention is given to choosing variables and choosing a minimal set of equations in the Newman–Penrose formalism. A particular scheme which is based on null cones and involves giving the free gravitational data in terms of Ψ0 is presented. The question of regularity at the vertex is briefly discussed and asymptotic expansions for the spin coefficients near the vertex are given.

INTRODUCTION

In studying problems in which gravitational radiation plays an important rôle, a description of the geometry which is adapted to the wavefronts of the radiation is obviously useful. Thus both the Bondi formalism and Newman–Penrose formalism have proved very helpful in understanding gravitational radiation at null infinity J+. From the point of view of an initial value problem this suggests that rather than specifying data on a spacelike surface one should specify data on a null surface and look instead at the characteristic initial value problem (CIVP).

There are a number of technical advantages that one gets from looking at the CIVP. The first of these is that the variables one uses are precisely those one needs to calculate the physically important quantities such as the amount of gravitational radiation, the Bondi momentum and so on. The second advantage is that the elliptic constraints which play such an important rôle in the spacelike case are effectively eliminated and one can freely specify the appropriate null data.

Abstract. We use the multiquadric approximation scheme for the solution of a three-dimensional elliptic partial differential equation occurring in 3 + 1 numerical relativity. This equation describes two-black-hole initial data, which will be a starting point for time-evolution computations of interacting black holes and gravitational wave production.

INTRODUCTION

Adopting the Arnowitt-Deser-Misner (ADM) 3 + 1 description of general relativity (1962) has, over the years, proved to be a fruitful approach for numerical relativity calculations. Using this description spacetime is constructed as a foliation of spacelike hyper surfaces. This split into space plus time leads to a constrained system of equations so that initial data must be specified on a spatial hypersurface and evolved into the future. The specification of initial data necessarily involves the solution of elliptic partial differential equations; these being the Hamiltonian and momentum constraints. When combined with York's conformal approach (1979) the system of elliptic equations is well-posed for solution by numerical techniques.

Until a few years ago the standard approach adopted by numerical relativists for the construction of initial data consisted of finite-differencing the constraint equations and applying iterative techniques, such as simultaneous-over-relaxation, to the resulting matrix of algebraic equations. More recently direct matrix solvers such as conjugate gradient and its variations have been employed (Evans 1986; Oohara and Nakamura 1989; Laguna et al 1991). A sophisticated multilevel iterative scheme developed by Brandt (1977) has also been used in numerical relativity calculations, mainly by Choptuik (1982, 1986), Lanza (1986, 1987) and Cook (1990, 1991).