UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Cnoidal waves generated from a plasma instability Romanin, Benjamin Stephen Joseph 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A6_7 R64.pdf [ 5.34MB ]
Metadata
JSON: 831-1.0052502.json
JSON-LD: 831-1.0052502-ld.json
RDF/XML (Pretty): 831-1.0052502-rdf.xml
RDF/JSON: 831-1.0052502-rdf.json
Turtle: 831-1.0052502-turtle.txt
N-Triples: 831-1.0052502-rdf-ntriples.txt
Original Record: 831-1.0052502-source.json
Full Text
831-1.0052502-fulltext.txt
Citation
831-1.0052502.ris

Full Text

CNOIDAL WAVES GENERATED FROM A PLASMA INSTABILITY by BENJAMIN STEPHEN JOSEPH ROMANIN B.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1984 © Benjamin Stephen Joseph Romanin, 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Crtnpk The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) ABSTRACT ii In this thesis we show that stable cnoidal waves can be generated from a linearly unstable plasma system. We look at a two-stream electromagnetic instability using plasma fluid theory. A reductive perturbation method is used to solve the equations to various order in a smallness parameter, e . To 0(e2) , the set of equations can be reduced to a canonical nonlinear equation: the Korteweg-deVries equation. This equation has stable cnoidal solutions. TABLE OF CONTENTS Title Page Abstract Table of Contents . List of Figures Acknowlegdements Chapter 1 Introduction 1.1 Introduction to Plasma Instabilities 1.2 Linear Theory 1.3 Nonlinear Theory 1.4 Two-Stream Electromagnetic Instability Chapter 2 Plasma Physics 2.1 Plasma Theory 2.2 Derivation of Fluid Equations 2.3 The Electromagnetic Two-Stream System . . . Chapter 3 The Generation of Cnoidal Waves 3.1 Perturbation Method 3.2 Linear Theory 3.3 Second Order Theory 3.4 One Dimensional Problem 3.5 Grouping the Equations 3.6 Reduction to the Korteweg-deVries Equation 3.7 Solution of the Korteweg-deVries Equation ., 3.8 Higher Order Theory Chapter 4 Discussion 4.1 Physical Description of the Solution 4.2 Linear Dispersion Relation 4.3 Nonlinear Dispersion Relation iv Chapter 5 Application to the Magnetosphere 5.1 Magnetospheric Plasmas 50 5.2 Further Study 52 5.3 Conclusion 53 Bibliography 54 Appendix A Plasma Theory A . l Plasma Kinetic Theory 56 A. 2 Plasma Fluid Theory 60 Appendix B Linear Theory B. l Linear Dispersion Relation 64 Appendix C Reformulation of the Problem C l Rearranged Equations 69 C. 2 One Dimensional Problem 77 Appendix D Calculations D. l Eigenvalues and Eigenvectors 79 D. 2 The Coefficients of the KdV Equation 81 Appendix E The Korteweg-deVries Equation E. l Solution of the Korteweg-deVries Equation 83 V LIST OP FIGURES Fig. 1.1 Classification of Instabilities 2 Fig. 1.2 Evolution of an Instability 6 Fig. 4.1 Possibilities 49 Fig. E.l Character of the Solution 85 VI ACKNOWLEDGEMENTS I would like to express my gratitude to Professor Tomiya Watanabe for his continued guidance and support. At no time during our association over the past few years has Professor Watanabe discouraged me, nor has he "managed" me into a course of study which I would not have felt comfortable in. On the contrary, Tomiya has given me the freedom to think and to create — in short, to do research. This requires philosophical maturity on the part of a supervisor, as well as good scientific sense. I would also like to thank Professor Paul LeBlond who took the trouble to read the thesis and to point out errors. The initial motivation to do this thesis was partially inspired by Dr. LeBlonds physics lectures. Next, I would like to thank Peter Whaite for helping me with the printing of this thesis. If he hadn't helped me, I would not have gotten done in time. I would also like to thank Dr. Mat Yedlin for supporting the production of this thesis. Having spent the last five years in the Department of Geophysics and Astronomy (has it been so long?!), I should like to say how pleasant the stay has been. The people in the department are cheerful and stimulating, a combination which makes for a pleasant work environment. More important, however, has been our weekly soccer game. Special mention for Lynda, Joanne, David (he actually didn't show up), Tom, John, Bill, and several others who turned up almost regularly, to be kicked and to kick—ouch! Buy shin pads! Finally, and always, I would like to thank my family for their encouragement, for their friendship, and for their love. C H A P T E R 1 INTRODUCTION 1.1 Introduction to Plasma Instabilities Matter has been divided into four states: solid, gaseous, liquid and plasma. The bulk of matter in the universe is in the plasma state; for example stellar interiors and atmospheres, gaseous nebulae, interstellar hydrogen, planetary magnetospheres and iono-spheres are all in the plasma state. Loosely speaking a plasma is an ionized gas—precisely speaking a plasma is a quasineutral gas of charged and neutral particles which exhibits col-lective behavior. The keywords in the definition are quasineutral and collective behavior. Since the plasma is made up of charged particles some deviation from charge neutral-ity can be expected, however, because the charged particles tend to adjust themselves to counter any charge build up, overall neutrality is maintained.This is what is meant by quasineutrality. In a neutral gas the dynamics of the gas are determined by short range mechanical collisions. In a plasma individual charged particles interact through Coulomb collisions. Also, the motion of charged particles generates currents, electric and magnetic fields, all of which can affect the motion of the particles themselves. It is these long-ranged forces in a plasma which enables one element of a plasma to exert a force on another even at great distances. By collective behaviour we mean motions that depend not only on local conditions, but on the state of the plasma in remote regions as well. The behavior of plasmas in thermodynamic equilibrium is fairly well understood, however this is not the case for plasmas in thermodynamic nonequilibrium. Most plasmas found in nature fall into this category; therefore a better understanding of such plasmas would be of great importance to both space physicists and astrophysicists. The most intriguing aspect of a plasma not in thermodynamic equilibrium is that it may be unstable. A certain amount of free energy is stored in the plasma which, if specific conditions are met, can suddenly be released. A plasma instability is a process in which this free energy is converted into a violent motion of the plasma and/or into radiation of electromagnetic waves in the plasma. Before going on to describe the major types of plasma instabilities we will give a sim-ple mechanical analogue. Let us consider a small ball pushed away from its equilibrium position a distance x (see Fig. 1.1). Now in certain 0) LINEARLY UNSTABLE (EXPLOSIVE) b) LINEARLY UNSTABLE (NONEXPLOSIVE) c) NONLINEARLY UNSTABLE <>) STABLE Fig. 1.1 Classification of Instabilities configurations free energy, in this case in the form of gravitational potential energy, is released so as to promote further movement away from the initial equilibrium position. Mathematically: (ID The solution to Eq. 1.1 implies x(t) oc exp (7*) i.e. the solution grows exponentially with time. This is the basic idea of an instability: namely that the initial, small pertur-bat ion grows larger with time. We see in Fig 1.1 that there are essentially four kinds of instabilities [1]: a) linearly unstable (explosive) b) linearly unstable (nonexplosive) c) nonlinear ly unstable d) absolutely stable In case (a) small movement away from equilibrium causes the ball to roll down the hill. In case (b) only small movement causes unstable motion; large amplitude motion is stable around the other two equilibrium positions. Case (c) is unstable for large amplitude motions, whereas case (d) is always stable regardless of the amplitude of the motion. Now that we have introduced plasma instabilities, let us categorise them. Just as with the ball in the mechanical analogue, free energy in an unstable plasma is converted into fluctuations and/or oscillatory motion. The source of free energy is sought in physical quantities which characterize the plasma in thermodynamic nonequilibrium. Along these lines we classify plasma instabilities into two groups [2]: i) Homogeneous plasma instabilities: The velocity distribution function (see appendix A) of the plasma deviates from the Maxwell-Boltzmann distribution, which is representative of thermodynamic equilibrium. Examples of such instabilities are the two-stream instability and the mirror instability [1]. These kinds of instabilities are also known as microscopic instabilities because changes in the distribution function are only observable microscopically. ii) Inhomogeneous plasma instabilities: Spatial variation of macroscopic physical quantities such as density, velocity, tem-perature etc. implies the plasma is not in thermodynamic equilibrium. The Kelvin-Helmholtz instability and the drift wave instability are examples [1]. These instabilities are also known as macroscopic instabilities. 1.2 Linear Theory As we will see in Chapter 2, the equations which describe a plasma system are nonlinear. The basic assumption of linear theory is that the initial perturbation away from the equilibrium state is small. If we write ri(x, t) for a general perturbation, the assumption is that the solution r(x, t) can be expressed in the form [1] r(x,t) =ro(x,0 + eri(x,*) (1.2) where e is assumed to be a small parameter having the same order as the perturbed amplitude. In the linear theory the perturbed amplitude ri (x, t) can be expressed as r i ( x » 0 = 7 ^ 4 / d<jJ I & n (w,k)exp»(k-x-w<) (1.3) i.e the eigenmodes of the perturbation are independent. For the stability analysis, we usually look for an instability not dependent on any initial conditions of the perturbation. If we set all the initial values to zero, the transformed results have identically the same form as the ones obtainable by substituting a complex amplitude function defined, for example, by: ri (x, t) = Rexpi (k • x — ut) (1.4) In the linear theory an instability is predicted if the dispersion relation D(w,k) =0 (1.5) has a complex root with a positive imaginary part: w = wr + t'7, 7>0 (1.6) The dispersion relation is derived by substituting either Eq. 1.3 or Eq. 1.4 into the equa-tions which describe the plasma system (we will see this done in chapter 3). 1.3 Nonlinear Theory From the linear theory one might conclude that once an instability is triggered it grows indefinitely. Although the plasma may evolve along different paths as the unstable per-turbations grow, it is physically clear that growth must eventually stop. Away from the quiescent equilibrium state increased fluctuations affects the transport processes in the plasma. Ordinarily plasma particles collide Coulombically; however, a strong fluctuating electromagnetic field can also deflect particles. When an instability begins, the rate of deflection, called the effective collision rate, is increased, and in turn transport quantities such as conductivity, diffusivity etc. are increased. Several different approaches have been taken in analysing the nonlinear development of a plasma instability. These will be briefly outlined. i) Weak Turbulence Theory: For weakly nonlinear plasmas the perturbation is expressed as a superposition of linear eigenmodes, but with weak interaction between the modes [1], [3]. In this approach nonlinear saturation is brought about by three fundamental interactions: a) Nonlinear wave-wave interaction A perturbed wave generates other waves through nonlinear coupling. This causes the perturbation spectrum to spread out, and consequently causes the linear instability to saturate (nonlinear saturation). 6 b) Nonlinear wave-particle interaction The perturbed waves scatter plasma particles, and in the process change an unstable distribution function(i.e. one that is not in thermodynamic equilibrium) into a stable one(i.e. one that is in thermodynamic equilibrium). c) Quasilinear wave-particle interaction Energy in unstable linear modes is transferred to stable nonlinear modes thereby saturating the linear instability. This approach works well only for weakly turbulent plasmas(i.e. plasmas with energy in the excited spectrum of modes greater than the thermal noise, but less than the total plasma energy). ii) Strong Turbulence Theory: Another approach is to assume the linear theory is erroneous, and instead postulate the existence of a turbulent stationary state [2]. Such plasma states are frequently ob-served in nature, particularily in geophysical and astrophysical plasmas. In this theory the structure of the turbulent stationary state is determined by solving for the spectral function of the fluctuations in a self-consistent way. Of course this method does not predict linear instability, but rather a turbulent stationary state. In summary an unstable plasma can evolve in the way shown in Fig. 1.2: Fig. 1.2 Evolution of an Instability The unstable plasma can evolve into three states: i) Turbulent Stationary State ii) Saturated State iii) ? The entry is our thesis: " We propose that a linearly unstable solution may evolve into a stable nonlin-ear solution which is quite different from the saturated solution or the turbulent stationary solution." 1.4 Two Stream Electromagetic Instability To present the ideas discussed in the proceeding section, we will look at the two-stream electromagnetic instability. This plasma system consists of ions and electrons streaming relative to one another along a magnetic field. In. chapter 2 we will derive the equations used to describe the plasma system and discuss the approximations being made. In chapter 3 we will introduce a perturbation-theoretical method to analyse the linear and higher order equations of the system. We will see that the linear system gives unstable solutions, which to the next order grow into stable solutions. In chapter 4 we will carefully consider the result, making comparisons with other works related to this one. Finally in chapter 5 we will apply our findings to the magnetosphere where an abundance of two-streaming plasma systems can be found. CHAPTER 2 PLASMA PHYSICS Plasma physics is, of course, a discipline whole unto itself. We shall present two of the major plasma theories, and show how the two theories are related. Finally, we describe the two-streaming electromagnetic plasma system using the two fluid plasma equations and Maxwell's equations. 2.1 Plasma Theory The most complete description of a plasma is given by plasma kinetic theory [2]. In this thesis, however, we will be using plasma fluid theory [4] rather than the more fundamental plasma kinetic theory to study a two stream instability. Often plasma kinetic theory and plasma fluid theory are presented as separate theories, whereas they are actually related. We will outline how the plasma fluid equations can be derived from the kinetic equations. To understand this thesis it is not necessary to follow the derivation, however we feel that by presenting the fluid equations a3 a subset of the more fundamental kinetic equations a richer appreciation of our results may be afforded. 2.2 Derivation of the Flnid Equations One can completely determine a classical plasma system through the Klimontovich equa-tion (Appendix A): « j y + T . | W + * . ^ W _ 0 (2.1) where N(t, V; t) is the Klimontovich distribution function. This equation while complete is extremely impractical because it describes microscopic quantities which are not easily measured. Through a statistical averaging procedure, this fine grained (i.e. microscopic) 9 equation is transformed into a coarse grained (i.e. macroscopic) equation (Appendix A). The basic macroscopic equation for the kinetic theory of plasmas is actually a coupled set of equations known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy: a t 1=1 n i*k = £ , f V{l,s + l)f.+l{l,...,s + l;t)d(s + l) 1=1 J (2-2) where / = (r/, v/) represents the position and velocity of the / -th particle, E«rf(ri,<) + -v/ ABext(Tht) c 3v/ V(l,k) q2n m dri \n -r{| c9v/ and /«(!,..., s; t) = s — particle distribution function If we consider a collisionless plasma (i.e. one without particle correlations) then the BBGKY equation can be reduced to the well known Vlasov equation (Appendix A): ^/y(r,v;0+v ^/y(r,v;*) + 1L 1 E(r,*) + -v AB(r,f) c /i(r,v;0=0 (2.3) By taking the first and the second moments of the Vlasov equation one can derive the fluid equations of continuity and momentum (Appendix A): jnj + V • (nyvy) = 0 (2.4) where j = particle type present in the plasma. 2.3 The Electromagnetic Two-Stream System With the plasma fluid equations we can proceed to set up the problem analysed in this thesis. We assume a cold, collisionless plasma so that the pressure tensor, Py , in Eq. 2.5 equals zero. Now in our work we look at ions and electrons streaming relative to one another along a magnetic field. Following conventional nomenclature we shall write v,-and v e for the ion and electron fluid velocity respectively. (Notice this should not be confused with the particle velocity vj given in the kinetic theory.) The fluid equations for the ion, electron plasma are: ^ny + V • (nyvy) =0 ; = i , e (2.6) l v J + (vy.V)vy = ^ ( E + i v y A B ) j - i , . (2.7) The electromagnetic field is described by Maxwell's equations: 1 d V A E = — — B (2.8) cot V A B = ^ E + 7 ^ 9 W ( 2- 9 ) 11 V • E = 4TT ] P qjtij (2.10) V B = 0 (2.11) Eq. 2.6 through Eq. 2.11 are used to describe the two-streaming electromagnetic plasma system. With the appropriate choice of coordinate system, this coupled set of nonlinear, partial differential equations embodies the physics of the system. The plasma fluid equa-tions are the "equation of motion" of the plasma, with the electromagnetic field playing the role of the accelerating field. CHAPTER 3 THE GENERATION OF CNOIDAL WAVES In this chapter we will show that the electromagnetic two-stream instability can gen-erate cnoidal waves. To do this we must first introduce some "mathematical machinery". Perturbation theory is introduced in the first part of the chapter, and the linear problem is solved using it. The second order problem is handled differently. A mathematical technique known as a reductive perturbation method is applied to the set of equations which describes the two-stream instability. 3.1 Perturbation Method A variety of physical phenomena can be described by differential equations with boundary and/or initial conditions. A given problem may be characterized by a parameter e which is often small. If the problem has the property that the equations which result from setting e equal to zero are easily solved, or are more tractable than the original set of equations, then a solution, say f(x;e) , of such a problem can be constructed by means of a power series in e , i.e. N f(x;e) = Y^enfn(x) + 0(eN+1) (3.1) n=0 fo(x) is the solution of the zero order problem. When the result of this perturbation analysis approximates the solution of the original set of equations, and if it is valid in the range of the independent variables under consideration, such an expansion is called asymptotic [5]. If the solution converges uniformity in x as e —* 0 it is called a regular perturbation problem. If the /(i; e) does not have a uniform limit then it is called a 13 singular perturbation problem [5]. Although questions of convergence and stability are extremely important, in this work little attention is given to these mathematical details. Fortunately, it turns out that the series representations we use are asymptotic. The plasma fluid equations and Maxwell's equations are a coupled set of nonlinear partial differential equations. It is natural to attempt to solve them using a perturbation method, since the equations are difficult to work with as they stand. The dependent variables in these equations are expanded in power series, so that the solution can be expressed as ny = ny0 + enji + e2ny2 • • j = i,e (3.2) vy = vyo + evji + e2vy2 • • j = *', e (3.3) E = E 0 + eEi + e 2E 2 • • • (3.4) B = B 0 + cBx + e 2B 2 • • • (3.5) In the following sections we will solve the two-stream electromagnetic system to succes-sively higher-order in e. 3.2 Linear Theory If we substitute Eq. 3.2 through Eq. 3.5 into the plasma fluid equations and Maxwell's equations, keeping only the 0(e) terms, we have the following equations: — nji + V • (nyoVyi + flyiVyo) = 0 (3.6) { + (Vyo'v)Vj 1} = ^ (El + ~cWjl A B o + C V J'° A B L ) J = *•6 ( 3 - 7 ) d —Bi + cV A Ei = 0 (3.8) d — E x - cV A Bi + 4n £ {nyivyo + nyovyi} = 0 (3.9) V . E i = 4 f ^ W l (3.10) y=«,e V B i = 0 (3.11) We choose a conventional Cartesian coordinate system: x, y, z with a magnetic field in the x direction. Ions and electrons stream along the magnetic field at different velocities, and without colliding. In the initial, unperturbed system vyo = («yo,0,0) j = i,e and Bo = (5x0,0,0) We now imagine that the initial system is "jiggled"— perturbed that is— so that the field quantities all change. Suppose all these perturbed quantities nyi , vyi , Ei , and Bi vary as (see chapter 1) exp [i (kx — u>t)\ Upon substituting this into Eq. 3.6 through Eq. 3.11 we can derive the linear dispersion relation (Appendix B) [6]: U2 - i V - te £ uj ( 7 ^ = 0 (3.12) 15 where ,2. wvi ~ \l — H = P ' o s m o frequency j = i,e rrij qjBXQ Qj — — = cyclotron frequency j = i,e rrijC If we fix our coordinate system to the steady ion stream, then u,o = 0 and the dispersion relation simplifies to 2 .2 2 2 u 2 (u ~ kUeo) u' - k'c* - w*.- — - UCT-^-T = 0 (3.13) (Note that uto , ueo etc. now refer to the transformed quantities. For instance, if we designate quantities measured in the ion rest frame with primes we would have u|-0 = 0 , U g 0 = ueo — «,o etc. The primes are dropped for simplicity.) The instability condition is obtained by solving the dispersion relation for complex w : u = ur + t"7, 7 > 0 for then the perturbed quantities grow exponentially with time exp [ t (kx — ujt) ] = exp [ i (^t) ] exp [ t (kx — u>rt) ] If \u\ Qj, j = i,e then Eq. 3.13, which is a quartic equation, can be reduced to a quadratic equation (Appendix B): 1 + i ' w 2 + I * + f ) ^ ( k ^ + f ^ ) = 0 (3.14) It can be shown that to becomes complex when (Appendix B): 16 (ul^i + toliikUeO + Qe))2 + 4 ( f l 2 + w£.) (k2C2 (kueQ + Qe)2 + UJ2pekUeQ (*«<*> + fie)) < 0 (3.15) (Note: fie < 0 .) Therefore, we see linear theory predicts that the 0(e) solution grows indefinitely. 3.3 Second Order Theory Before we expand our equations to O (e2) we will rearrange them into a form which allows us to use a perturbation method developed by Taniuti et al. [7], [8], [9], [10]. The plasma fluid equations and Maxwell's equations together consist of the following dependent variables: rc»> v,-, n e, ve, E, B In rearranging these equations we will elliminate ne and also introduce B = V A A (3.16) E=--5?A (3.17) where A is the electromagnetic vector potential. Details of the rearrangement will be relegated to Appendix C. It should be pointed out that although the reduction is fairly cumbersome, it is conceptually straight forward. The first rearranged equation is: — ni + (ve-V)ni+ni(V-Vi) at 1 d (d \ _ Id. „, „ Id. „ ^ + aiVn,- - —— — v e + Oi Vn , • —— (v e • V) v e + a 2 V n , • —— (v e A V A A) n, <?< \ a / / n, tit at + a, V n , — V • ( l - v , ) + a 3 Vn< • — V • (v e • V) v e + o 4 V n , • — V • v e A (V A A) + a 5 Vrii • — V A (V A A) = 0 where m e 1 fll 4jre2' ° 2 4;rec — m < ! _ 1 4^e 2 ' ivec «5 = .— 4;re The next equation is the u,- component of: + &2 (v, • V) v e + 6 sv,. A (V A A) + 6 4 v , A (V A A) + 6 5 ( ^ ( ^ ) ' V ) V , + 6 S ( ^ { V e V ) V e - V ) V < + , , 6 ( ^ ( V e A V A A ) - V ) V * + 67 ( ^ V - (^ v«) • V ) v ' + 6 7 ( ^ V - ( v e - V ) v e - v ) v < + 6 8 ^ V . v £ A ( V A A ) - v j v , + 69 ( 1 V A (V A A) • V ) + 5 1 0 ( ± &- ( * v . ) • v ) v e + 610 £ (v. • V) v . • v ) v e  + 5„ V • v e A (V A A) • V ) v e + 614 ( I V A (V A A) • v ) v e + 614 ^  ( 1 A ( JL V . ) ) + 6 4 f e v - K - v ) v 0 + 6 4 ( S v - v e A ( V A A ) ) + 6 1 9^ ( ^ V A ( V A A ) ) =0 (3.19) where h - 1 t _ 1 h _ e / m i c Oi = — 7—, f>2 = r~; ; — , Oj = H-me/mi' l + m./m,.' l + me/trii h = -bs, b5 = 6iai, 66 = M2 69 = &ia5, 610 = -6201. bn = -62aa 12 = -&203, 1^3 = -&2<»4» *>14 = ~&2«5 015=610, (»i6 = f»n, h7 = 612 &18 &13, &19 = &14 The reason we take only the u,- component of this equation will become clear later. The Vi and ty,- components are given by the following equation: | r V , + bl(ve- V ) v , at + &2 (v,. • V) ve + b3ve A (V A A) + 64V, A (V A A) + 67 ( ^ V > ( | ( v ' ) V ) v ' + ^ ( ^ V . ( v e V ) v e v ) v , + 6 8 ^ V v E A ( V A A ) - V ^ v , + b, (I V M V * A , . V ) v, + . 4 ( ± £ (£»,)) + , 4 ( I * „ . • V , v , ) + s ( s a A v A A » ) + * s v • + 6 " a ( S ' • <*•' +»'»Ii ( 5 v ' A ' v A A > ) + I i tev A ( v A A ) ) = 0 <" 0 ) The equation for the ue component is: ~ye+h (v e • V ) v e at + 62 (vi • V) v e + b3ve A (V A A) + 64v< A (V A A) + • v ( i l l (Jr- ) ) + M < • v ( £ * ( V e • v ) V e)+ 6 e V< • v ( V e A v A A ) ) + • V V ' ( ^ v « ) ) + b 7 V < ' ^ ( ^ V • (v e • V) v e ) + 6 8 v e - V ^ V - v . A f V A A ) ) + fcv.-v(ivA(VAA))+&5(ii ( « v.) • V ) v, + 64 ( ± « (v. • V) v e • v ) v ( + 6 6 (hh(v<A v A A ) • v ) v - + h ( S v • (£ v«) • v ) v - + 6 7 ( S v •{v< • v ) V e • v ) V t + 6 8 ( V ^ . v t A ( V A A ) . v ) v j + 6 9 ( i v A ( V A A ) . v ) v t + & I 0 ( i i ( ^ ) . v ) v e  + 6 1 0 ( 1 £ (v, - V) v . • V ) v e + 6 U ( 1 Q- (v. A V A A) • v ) v. + b12 (^-V • ( * v.) • v ) v e + &12 ^ V ( v e - V ) v e V ^ v e + 6 „ ( ^ V - v e A ( V A A ) v ) v e + 6 1 4 | ^ V A ( V A A ) - V J V « + ^ M ^ v - ( ^ 0 ) + b 7 M S v - ( v - v ) V e ) + 6 ^ t e V V e A ( V A A ) ) + 6 4 ( ^ V A ( V A A ) ) = 0 ( 3 - 2 1 ) where the constants are defined as before. The equation for the ve and we components is: ^ v e + &!(v e - V ) v e + b2 (v e • V) v, + 63ve A (V A A) + 6 4v, A (V A A) +6-v< • v ( £ £ (Iiv<))+ 6 2 o V e • v {iii <v< •v> v 0 + 6 - v < • y ( ^ <v<A v A A 0 + & 2 2 v e • V ( ^ V • ( ^ v e ) ) + 6 2 2 v e • V ( ^ V • (v. • V) v e ) + 6 2 s v e • V ( ^ V • v e A (V A A ) ) + 6 2 4 v e • V (Iv A (V A A ) ) + 6S ( I I ( £ v . ) • v ) v, + 55 (I * (v. • V) v . • v ) v, + 6 « ( ^ | v ' A V A A | v ) v ' + 6 ' ( S v ( ^ , ' ) v ) y i + t , ( S v ( v " v | v " v ) + ( , . ( X £ v . v , A ( V A A ) . v ) v l + 6 9 ( I v A ( V A A ) . v ) v , + 6 ^ ( i i ( i v , ) ) + 6 ' s ( S v ( v « v | v - ) + 6 , M » 7 v ' v - A ( V A A , ' v ) V i + 6 4 ( S - v a ( v a a O = 0 The equation for the Ax component is: ^ A + c x (\i • V)v, + c 2 (ve • V) V i + cjVe A (V A A ) + c4v, A (V A A ) + 4 ( s v ( W ) + c 7 M s v ( - - v ) v 0 + c ^ t e v - v ' A ( v A A ) ) + c 4 {h v A ( v A A))+ C5V< • v {hoi Uv<))+ c*v< • v {h§i ^ • v^<) + c ^ •v {hi ^ A v A A0 + c* v< • v v • U S ) + C 7 V e • v ( S v •(v< • v ) + c 8v e • V ^  V • v e A (V A A ) ^ + c 9v e • V ^ V A ( V A A ) J = 0 where 1 . Cl = -7 ; 7 , C2 = -Ci, Cj = -bi e/mec + e/m,c ct =-b2, c 5 = c i a i , c 6 = cia 2 c 7 = c i a 3 , c 8 = Ci<i4, c 9 = cia 5 Finally, the equation for the Ay and Az components is: 21 — A + C j v e A (V A A) at + c 6 ( i ^ v e A V A A ) - v ) v « + c 7 (S v 'U v O" v ) v ' + c t (Sv(v-v>v-v)v< + c 8 g v . v J A ( V A A ) . v ) v 1 + , ( l v A ( V A A ) . v ) v i + 4 ( i i ( ^ ) ) + c ^ ( s v - ( v e - v ) v O + c 8 M s v - v e A ( v A A 0 + c ^ ( ^ v A ( v A A ) ) + c 7v e • V ( ^ V • (|^v<)) + c 7v e • V ( ^ V • (ve • V) v e^ + c 8v e • V V • v e A (V A A)) + c 9v e • V ( I V A (V A A)^ = 0 (3.24) 3.4 One Dimensional Problem Next we assume all the dependent variables n,- , vt-, v e , and A are functions of x and t alone. This assumption of one dimensionality is necessary in order to cast the equations in a form prescribed by the reductive perturbation method of Taniuti et al. [5]. We will rewrite Eq. 3.18 to Eq. 3.24 assuming one dimensionality. An outline of how this is done is given in Appendix C. Eq. 3.18 then becomes: + a i i h (§i + ( " a i ) n ' ^ {ii { i u < ) ) + a i ^ {It ( M e ^ u 0 ) + {-ai)nik {ii {u<Txu))+a*oh {i ( 4 A 0 ) +a2o% {i {w'hA')) + ^ n i i { i i O ^ ) ) + ( " a 2 ) n - ^ { i i {w<o%A))+a>T* ( " 4 { i u ) ) Taking the u,- component of Eq. 3.19 gives: d t d of ox d d + &2«t^-«e +hve—Ay OX OX O d d + bswe—Az + 64 tr,- — A y + 64 ur, — A z ox ox ox ("4 (!;"•))+<- t s|S^ ( ! ( " ' ) ) ( u ' l i ("4"-)) + ("•£*)) tef. {"'h teA"))+hifi {"h (teA)) + ( » ^ ) ) + ( £ ( - ^ ) ) ( - £ ( £ - ) ) * (I (h«))+»$h te te-)) + ™>h (I te-)) +*Z£. te teA'))^ihte HA-)h<-<«£ (t teA-)) +<-.>5«£ U (»4 & • ) ) + < - > S £ (f, (fr-)) + i " v £ ("4 ("-^"O)+[-ha)"tti (ft te-'))+b"^i te teA")) -^h te HA)>^ii (f. teA'))^<i (f teA)) +b""ii, te ( f " 4 ) ) + ( f - , (fi*))+h""if, te te"-)) • a tes>))+™>k (h teA>)) - 4 ah ih-)) . d (ut d (d \\ . d (ue d ( d \\ . d (ut d ( d \\ +bi7di U S i + 1 7 ^ v U e ^ S j + 1 8 ^  r ^ S J 6 !L ("l— (w — a X\-Q 18 dt \rii dx \ e dx // (3.26) 23 Taking the v,- and ty,- components of Eq. 3.20 gives: d t d — Vi+biUe—Vi at ox + b2Ued~xVe + (~bs)Uei~xAy + (~b*}U'JxAy + h i l x {Vil (£"•)) + (~MS£ {it { l t u < ) ) + b * i l x (Vidi + {~h)^k (al {Uea3-xUe))+hia3x- {v'Tt {"'Ii**)) + b^tk {"'it { W ' k A ' ) ) + ^ ) U i V ' Y x [Ix (£"•)) + b ^ x (Vilx (U<TxU)) + ^ ) U t V i a % [ix ^ k ( « h H a > ) ) ^ k {«h ( - ^ 0 ) + ^ > k {I H a » ) ) + { - b * ) U i V i k { h ( w < k A * ) ) + b i 4 t ( h h ( i i v < ) ) + b i 4 t { i h ( u < k v ) ) + b i 4 t {^k [ v < k A » ) ) + b i 4 t ( ? i k ( w < k A ) ) + h 4 t H , f x { i h A y ) ) = ° <3-+ hue—We + (-6j)u e— Az + (-b4)ui— Az ox ox dx + h i i k (Wioi (^u0)+ M s ) S^ (£"•))+6»££ («*£ + ( ~ h ) ^ l x {I {"'Ix"')) ^ T x (Wi¥t + h l i d 3 x (WiTt (W<TxA>)) + {-b^w'k {Ix + b ^ k (Wik ( u < k u ) ) + ( - h ) U i W i k ( k (u<ku<)) + (-b')UiWidh (fx (w<fxA*))+hoi ( h i ( i w < ) ) + b i o i ( h i (u<fxw<)) L d ( 1 d ( d . \ \ . d fwt 0 (0 \ \ . d (wtd ( d \ \ + h l d l {-nidl \u<TxA*))+b"dl U"V ) + h 2 d l \ * a i Vedxu<)) , d fwe d ( d t \ \ . d (we d ( d \ \ Taking the u e component of Eq. 3.21 gives: — u e + 6i«e—ue at ox , 9 u d X + o 2u,—« e + b s v e — A y d d d + bitve -r-Az + btVi —Ay + biWi -z-Az ox ox ox + h U c f x (hoi ( I f - - ) ) + b 5 U e f (hoi (u<fu<))+hu<oi (hoi + h u ' a h ( h i (w<ikA)) +b7U*lx (*hfx ( i u < ) ) + h U c f (^lih (u<iLu)) +b*u<oh ( ^ f (v<fA»)) +hu<fx (^f (w<fxA)) + h h f (Uii (iu<)) + ^ u i k (i ( I i - ) ) + h h f ( U i i ( u « £ " 0 ) + U ( u ^ M e ) ) n, ax \ot \ot )) ti{ Ox \ at \ ox / / n, ox \ o i \ ax / / > 6 s - £ {-'ii teA'))ter, ("f, {""iA-))+^"ih ( I { " £ A ' ) ) ( U ' S ( " • £ " • ) ) + M , I S " ' ^ ( " i " 1 ) ) + ! * ^ (-1; teA")) n, ox \ d x \ ox J J n-i ax \ ox \ ox J J ]^fx (fx (V<fxA»)) + ( - h ) ^ U i a h (fx (W<fA)) it d ( d ( d \ \ , 1 d ( d ( d \ \ + t»n,d-x {Uidx { W ^ ) ) + [ - 0 ^ a - x [d  \ V T * * > ) r ™ * * ^ \Tx V + h ° h i (Ueoi ( ^ 0 ) + {-bio)^fx (h ( h u ) ) + b " h f (Uei (Uefu)) + { ~ b ^ f x ( h i (u<fxu<)) +b"hf* (Uei (VefA»)) + b n h f * (Uei (w<fxA*)) , t (u e d (d ( d \ \ . . .«e d (d ( d . \ \ u ue d ( d i d ni0- [n,dl \U'dxUe))+bllnidx- {"'Ft {V'dx^)J +b"nidx \^Tt \W'di ' , d (ue d ( d \\ , d (ue d { d \\ Taking the v e and twe compinents of Eq. 3.22 gives: d u d d< dx + b2ue~Vi + (-bs)ue^-Ay + (-bi)ui^Ay +b™u<i {ii +b*oU<i [hit iu<tv<)) +b*iU<l {-hi ( u <l^)) +b»u<lx ( S £ (iu)) + 6 " U e £ ( S £ ( " < £ " < ) ) + 6 2 3 U < ^ fe£ ( ^ 0 ) - (1 ( 4 ( • . £ - ) ) - < - *>2£ U ( ^ ) ) + f c 6 ^ ( 4 {^hAy))+hii {Vii +w%h {i (v<lA»)) +»H ( 4 W W ) + ™ > k {h ( ^ ) ) + ( h HA>)) +hi (hrt (iv')) +hoi [hi ( " • £ " • ) ) + b 7 d i {iu<)) +b4tH,l{lAy)) = ° <3-3C 26 + b2ue — Wi + {-bi)ue—Az + {-bi)ui—Az +b™u<l [ h i {ltWe))+hoUefx [ h i {UeiWt))+hiUgi {-hi iu<iA>)) + 6 2 s M 4 [w<iLA)) +b**u<lx { j i A ' ) ) + 6 » £ £ {Wio\ U S ) + (~b^l {1 ( « U + { - b ^ i U U ^ S ) + u u ^ s ) ^ u u s ) + a u s ) K K S ) + < - * W U ( * ^ 0 ) + ™ 2 - £ ( £ + U * 7 U S ) + b 4 t ( i a l { ^ w ' ) ) + b * T t (-JZTt { U < o h A ) ) a (we d ( d \\ t d (w< d ( d \\ , d (we d ( d \\ + b7dt U a i U S J + 6 r * V ^ S j + &8*7 U a i V e ^ S j d fwe d ( d \\ , d ( 1 d ( d \\ , Taking the A x component of Eq. 3.23 gives: 9 A 9 — A x + ci«, — «, ot ox 9 9 A ox ox 9 A 9 A 9 A + c s w e — A z + c4 vt -r-Ay + c4 Wi dx ox ox + c 4 U * 7 U S ) + c 4 U * 7 ( U e £ U e ) ) + C 6 o 7 Q U ( V 4 S ) U l * ( " 4 S ) + c 4 ( S £ U S ) + c 4 ( 2 £ ( u 4 S ) 5 + C6o7 27 + c4t {%k (v^))+e'Tt (%k (w'kA)) + c>u<k (£"«)) +Ci"e^ (hi ("•£"•)) + C 6 u e^ ( ^ 0 ) + C e u e ^ ( t t , e ^ ) ) d (ue d (d \\ d fue d ( d \\ d (ut d ( d \\ Taking the J4.J, and Az components of Eq. 3.24 gives: +c*hk (Vii (iu<))+ ( _ C s ) S ^ (hi (iu<))+c*hk (Vii (u<ku<)) + {-c^ik (i (u<ku<)) + c«hk (Vii (v<kA»)) +C6hk (Vii {w*kA')) ( i ( ^ ) ) + i - < i a ( ^ ) h - u k a - ) ) + (" C 7 )S 1'*^ (£"0) + C 7 S ^ ( M e ^ M e ) ) + ( _ C 7 )n7 t' i^ (<£ ("'I"')) + C^k (^*)) + (Vio% (W^XA)) + ^UiVid% (TX + i-e^Vik (k (w<kA*))+c*i (hi (iv<)) + c 4 (hi (u<kv<)) +c«i (~hi (u<kA»))+C7i (*hk (iu<)) +C7i {%k (Utku)) +c*i {%k {v'k^))+etk (^k (w<kA*)) +c»i (~hk (kA»)) + c*u<k (hi (iv<))+C5U<k (hi (u<kVe))+C6Uek (~hi (UekA»)) 8 (vt d (d \\ d fve d ( d \\ d (ve d ( d \\ + C7U<d-x U " v J + C 7 U e ^ V<rxu<)) +c*u<d-x {nid-x \v<c-xA»)) - A z + ( - c 3 ) u e - > U + (- C 4K->l z + ( " C s ) S ^ ( U e ^ " e ) ) + C 6 ^ W + {WiFt {w<kA)) +(-c°^k {k ( ^ 0 ) + ( _ C 6 ) S ^ {k ( w ' k A ' ) ) + e i * k {Wik (o\U)) + {-C7)^Wik {k {ku'))+C7wik {Wik (u<ku<)) + { - c ^ W i k {k {u<ku)) +c*utk {Wik { v - k A y ) ) + c ^ k {Wik { w ^ ) ) + { - c ^ W i k {k {v<kA»)) +(-c*)tr7iWik {k {w<kA*))+c4t {iii {lw<))+c*ii {iii {u<kw<)) 9 ( 1 9 ( 9 a \\ d (we d (d \\ d (we d ( d \\ +C6di {-nidi {u<d-xA*))+c-'oi \diu<))+cidi\*ai r ^ U e J J +c*di {v<rxA»))+c>di \w'diA'))+C9oi {-nidi W*)) +CiU<k {ik {iiw))+CiU<k {ik {u<kw))+c«u<k {-iii {u<kA)) d (wt d (d \\ d fwe d ( d \\ d (wt d ( d , \\ +c*u<k {^k {w<kA*))+c*u<k {~ik {kA))=° ^ 29 3.5 Grouping the Equations Now we are in the position to express Eq. 3.25 to Eq. 3.34 in the form presented by Taniuti et. al. We shall show how the set of equations can be reduced to a single nonlinear equation by using a reductive perturbation method. First we group Eq. 3.25 to Eq. 3.34 into a a single vector equation, as prescribed by Taniuti et al. [7]: »=13p=3 3=1a=l v 7 where U Ui Vi Wi Ue Ve We Ax i 4' i \AJ / U e Tli 0 0 0 0 0 0 0 0 0 M e 0 0 M i 0 0 0 -429 -^ 2 10 0 0 M e 0 0 M e 0 0 -4j9 0 0 0 0 bl «e 0 0 M e 0 0 -44io 0 0 0 0 -45 5 0 0 0 -4s9 -4s 10 0 0 M e 0 0 M e 0 0 -46 9 0 0 0 0 M e 0 0 M e 0 0 -4710 0 -48 2 0 0 0 0 0 0 -4 8 9 -4g 10 0 0 0 0 0 0 0 0 -4.9 9 0 v o 0 0 0 0 0 0 0 0 -4-1010 J with -4 2 9 = M e + M t , -^ 2 10 = M e + M i , -4 J 9 = ~ M e ~ M . -44 10=-4s9, -455 = M « + M e , -459 =-4 2 9 , -45 10 = -42 1 0 J469 = A S 9 , A 7 1 O = - 4 S 9, 4^g 9 = C3t)e +ctv{ A 9 9 = C3We + C4IU,, A 9 9 =-csue - C4Ui, A 1 0 10=^99 The matrices H f and K f are: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bio 0 0 0 0 0 0 0 0 &10 0 0 0 0 0 0 0 0 h 0 0 0 0 0 0 0 0 65 0 0 0 0 0 0 0 0 h 0 0 0 0 0 0 0 0 C5 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 j_ »l 0 0 0 0 0 0 0 0 1_ n,-0 0 0 0 0 0 0 0 J_ 0 0 0 0 0 0 0 0 _l_ 0 0 0 0 0 0 0 0 x 0 0 0 0 0 0 0 0 J_ ni 0 0 0 0 0 0 0 0 x n i 0 0 0 0 0 0 0 0 U 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 Vo 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 bn 0 0 0 0 0 0 0 0 612 0 0 0 0 0 0 0 0 6 1 2 0 0 0 0 0 0 0 0 b7 0 0 0 0 0 0 0 0 67 0 0 0 0 0 0 0 0 b7 0 0 0 0 0 0 0 0 c 7 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 0 A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 cs 0 0 cj 0 0 "\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J_ m 0 0 0 0 0 0 0 0 0 0 o) 0 °\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C7 0 0 c7J /O 0 0 0 0 0 ^ 0 0 0 o d ^ o o 0 0 0 ^ 0 0 0 o o ' ^ o o o o d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0^ 0 0 0 0 0 0 0 0 0 > 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -^  0 0 0 0 d ^ o o o o d ^ o o 0 0 0 ^ 0 0 0 0 0 ^ , fii J 0 0 0 0 0 0 0 0 o\ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 Vo 0 0 0 1 0 0 0 0 07 f -axn, 0 0 0 0 0 0 0 K2l 0 65 — 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K$ 1 65 «e 0 0 0 0 0 0 0 0 0 b ^ 0 620We 0 0 0 0 0 0 0 b ^ 0 &20 We 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C5We 0 0 0 0 0 0 0 0 CsW« 0 0) where K21 = -610 We n, 05 — n. 1 KS1 = K2 1 / I 0 0 0 0 0 0 0 0 0\ 0 J_ n, 0 0 0 0 0 0 0 0 0 0 w, 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 0 Vi 0 0 0 0 0 0 0 0 0 0 Wi 0 0 0 0 0 0 0 0 0 0 J_ n l 0 0 0 0 0 0 0 0 0 0 _L n j 0 0 0 0 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 07 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 OJ ( 0 0 0 0 -a sn, 0 0 as 0 °\ K21 M e £ 0 0 0 0 &12«e^T 0 0 0 Kix 0 M e £ 0 0 0 0 0 0 0 0 0 M e £ 0 0 0 0 0 0 K51 0 0 M e 0 0 6l2«e^T 0 0 K61 0 M e ^ 0 0 62  «e 0 0 0 0 Kn 0 0 M e ^ 0 0 622 «e 0 0 0 0 0 0 0 C7«e 0 0 0 0 0 IUi 0 0 0 C7«e 0 0 0 0 \Kio 1 0 0 0 0 C7"e 0 0 0) where #21 = "Me — «i:- &12«e — «e, Ks 1 = -b7Ue — Vi, K41 = -b7Ue Wi, Ksl = K21 Tli >*i tli Tie Kei=Jin, K7i = Ku, Kgi = -c7ue — w,-, KiQi = -c7ue — «>, H f1 0 0 0 0 0 0 0 0 °\ 0 u, 0 0 0 0 0 0 0 0 0 0 Vi 0 0 0 0 0 0 0 0 0 0 w, 0 0 0 0 0 0 0 0 0 0 u9 ft i 0 0 0 0 0 0 0 0 0 0 ^ 0 0 0 0 0 0 0 0 0 0 fl j 0 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 07 0 0 0 1 0 0 0 0 o\ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 — 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 0) (0 0 0 0 0 0 0 0 0 o\ bis 0 0 0 0 &16 bi6 0 0 0 0 bio 0 &u 0 0 0 0 0 0 0 0 bio 0 611 0 0 0 0 0 bs 0 0 0 0 be b6 0 0 0 0 bs 0 0 0 0 0 0 0 0 0 bs 0 h 0 0 0 0 0 C5 0 0 0 0 C6 C6 0 0 0 0 CS 0 C6 0 0 0 0 0 0 V 0 0 Cs 0 c 6 0 0 0 0 f l 0 0 0 0 0 0 0 0 o\ 0 _L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 _ J_ ii,- 0 0 0 0 0 0 0 0 0 0 _ J_ ni 0 0 0 0 0 0 0 0 0 0 J_ nt- 0 0 0 0 0 0 0 0 0 0 _L «i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V o 0 0 0 0 0 0 0 0 oJ 0 0 0 «e 0 0 0 0 o \ 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 u 0 0 0 0 0 0 0 0 0 J (0 0 0 0 0 0 0 0 0 0 bn 0 0 bis 0 0 bi8 0 0 0 0 612 0 0 bu 0 0 6is 0 bn 0 0 bi2 0 0 bu 0 0 6l3 0 b7 0 0 b* 0 0 h 0 0 0 0 b7 0 0 0 0 bs 0 0 0 0 b7 0 0 h 0 0 bs 0 Cl 0 0 c 8 0 0 C8 0 0 0 0 C7 0 0 C8 0 0 C8 0 eg V o 0 C7 0 0 C8 0 0 c 8 0 / tlj 0 0 0 0 0 0 0 0 0 \ 0 Is. 0 0 0 0 0 0 0 0 0 0 wf n j 0 0 0 0 0 0 0 0 0 0 «1 0 0 0 0 0 0 0 0 0 0 tl i 0 0 0 0 0 0 0 0 0 0 w. 0 0 0 0 0 0 0 0 0 0 »A fl j 0 0 0 0 0 0 0 0 0 0 ««. ft j 0 0 0 0 0 0 0 0 0 0 We 0 V o 0 0 0 0 0 0 0 0 _ J _ «l / 0 0 0 "e 0 0 0 0 o \ 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 w« 0 0 0 0 0 0 0 0 0 0 0 0 0 tfe 0 0 0 0 0 0 0 0 0 ve 0 0 0 0 0 0 0 0 0 fe 0 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 u 0 0 0 0 0 0 0 1 0 J ( 0 0 0 0 0 0 0 0 0 o\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bu 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 69 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V Cg 0 0 0 0 0 0 0 0 0 / ( __!_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \ 0 0 0 0 0 0 0 0 0 07 /o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 1\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 07 / 0 0 0 0 a 2 0 0 0 \ Ki i 0 0 0 •^ 2 6 0 0 - M , £ 0 0 0 0 0 0 0 0 0 0 - M i ^ 0 0 Me 0 0 Ks7 0 0 - M < £ 0 0 0 0 0 6 6 -0 n ; 0 0 0 0 0 -M i j f : 0 0 0 0 Cs«e 0 0 0 0 0 0 0 -wh 0 0 0 0 0 0 V 0 0 0 0 0 0 / where = - M t 1 - 6io«« 1 " i i ^26 = _ M i 1 flj -6ll« e -i 1 — Ki i •K57 = •^ 2 6 / I 0 0 0 0 0 0 0 0 0 "e 0 0 0 0 0 0 0 0 0 0 _L n . 0 0 0 0 0 0 0 0 0 0 «t 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V, 0 0 Wi 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 w, 0 0 0 0 0 0 0 0 0 0 V, 0 \0 0 0 0 0 0 0 0 0 W{ J /O 0 0 0 We 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 •-0 0 0 0 0 0 0 0 We W« 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 fe 0 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 t'e 0 \0 0 0 0 0 0 0 0 Ve o) ( ° —a2r», a 2 0 0 0 0 0 0 i X 71; 0 ^2 3 bn 1_ 0 1 0 0 0 0 0 0 -M,£ 0 0 0 0 0 0 0 0 0 0 0 0 b 6 n , 0 0 M e M e b ^ 0 0 0 0 0 0 -M.^T 0 0 0 0 &2lWe 0 0 0 0 0 0 0 b 0 &21 We 0 C6«e 0 0 CeWe 0 0 0 0 0 0 0 0 0 0 0 C6We 0 V o 0 0 0 0 0 < 0 CeWe J ^6 s = K? 0 0 0 0 0 0 0 0 0 \ 0 J_ m 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J_ 0 0 0 0 0 0 0 0 0 0 u. 0 0 0 0 0 0 0 0 0 0 f. 0 0 0 0 0 0 0 0 0 0 Wi 0 0 0 0 0 0 0 0 0 0 _ J_ m 0 0 0 0 0 0 0 0 0 _ J_ J 0 0 0 0 0 0 0 ve 0 "\ 0 0 0 0 0 0 0 0 vt 0 0 0 0 0 0 0 0 0 0 w e 0 0 0 0 0 0 0 0 0 wc 0 0 0 0 0 0 0 0 0 we 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 « e 0 Vo 0 0 0 0 0 0 0 0 ( 0 0 0 0 0 0 0 0 0 o\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 2 0«« 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C s « e 0 0 0 0 0 0 0 0 0 V o C 5 « e 0 0 0 0 0 0 0 OJ f 1_ n,' 0 0 0 0 0 0 0 0 0 \ 0 J_ fl 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V 0 0 0 0 0 0 0 0 0 0 0 0 0 «e 0 0 0 o \ 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 vo 0 0 0 0 0 0 0 0 07 ( 0 as - a 3 f i j 0 0 0 0 0 0 0\ #2 i 0 0 0 M e £ 0 0 0 0 #JI 0 0 0 0 0 M < £ 0 0 0 #4i 0 0 0 0 0 0 0 0 #5 1 b 7 u e 0 0 0 0 0 0 #6 1 0 0 6 2 2u e 0 0 M e ~ 0 0 0 #7 1 0 0 0 6 2 2u e 0 0 0 0 0 0 0 c 7 u e 0 0 0 0 0 0 #9 1 0 0 C7«« 0 0 0 0 0 \#io i 0 0 0 c 7 u e 0 0 0 QJ where #2 1 = ~MeM, b l 2 u \ — , KS1 = - b 7 U e V i — , K n = - b 7 U e W i — , #5 1 =#2 1 ft I ttj fig ftf #7 1 = #4 1) #9 1 = -C7UeVi 1 m' #10 1 = —c / I 0 0 0 0 0 0 0 0 °\ 0 «e 0 0 0 0 0 0 0 0 0 0 U e tlj 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w. fl 0 0 0 0 0 0 0 0 0 0 «i 0 0 0 0 0 0 0 0 0 0 f. 0 0 0 0 0 0 0 0 0 0 Wt 0 0 0 0 0 0 0 0 0 0 0 0 vo 0 0 0 0 0 0 0 0 o ) 0 0 0 «e 0 0 0 0 0\ 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Vo 0 0 0 0 0 0 0 0 o J 38 K}2 = ( 0 04 -04". 0 0 0 0 0 0 #2 i 6l3«e^T 0 0 0 0 0 0 0 #3 1 0 0 0 0 0 0 0 0 #4 1 0 0 0 0 0 0 0 0 #5 1 &13«e^7 0 0 0 0 0 0 #6 1 0 0 0 0 bstie± 0 &24«e 0 #7 1 0 0 0 623 «« 0 0 0 0 0 0 c 8 « e 0 0 0 0 0 0 0 #9 1 0 0 c 8w e 0 0 0 Cc,Ue 0 V #10 1 0 0 0 C 8«e 0 0 0 0; where #21 = -6 8«e«.— -&13«e—, #31 = - M e f i — , #4 1 = ~ & 7 " « — , #5 1 = #2 1 n,- fl,- fl; fl, K 2 2 = 0 0 0 0 0 0 0 VO K s 2 — #91 = 1 -C7UeVi—, ni # 101 -0 0 0 0 0 0 0 0 «e 0 0 0 0 0 0 0 0 HA n,' 0 0 0 0 0 0 0 0 n; 0 0 0 0 0 0 0 0 n; 0 0 0 0 0 0 0 0 «i 0 0 0 0 0 0 0 0 Vi 0 0 0 0 0 0 0 0 Wi 0 0 0 0 0 0 0 0 _J_ n; 0 0 0 0 0 0 0 0 f° 0 0 0 0 0 0 0 Ve o\ 0 0 0 0 0 0 0 0 ve 0 0 0 0 0 0 0 0 0 ve 0 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 Ve 0 0 0 0 0 0 0 0 0 1 0 Vo 0 0 0 0 0 0 0 0 07 °\ 0 0 0 0 0 0 0 0 o7 39 I 0 « 4 - a 4 r » i 0 0 0 0 0 0 o\ #21 &13«e£ 0 0 0 M e £ 0 0 0 0 # J 1 0 0 0 0 0 M e i 0 0 0 # 4 i 0 0 0 0 0 0 M . ^ T 0 0 M e 0 0 M e i 0 0 0 0 #6 1 0 0 &2S«e 0 0 0 0 0 #7 1 0 0 0 &2J«e 0 0 M e ^7 &24«e 0 0 0 C8"e 0 0 0 0 0 0 0 #9 1 0 0 C8«e 0 0 C8»e 0 0 0 V #10 i 0 0 0 C8«e 0 0 C*U<1 C9«e 0) where #21 = " M e " . &1S"2; —, #31 = -& 8 Uet>« — , Kil = - b 6 U e W i — , « i n , rij n , #5 l = #211 #61 = #s 11 #71 = #41 #91 = - C g U e V i — , #10 1 = - C 8 « e U 7 i — f 0 0 0 0 0 0 0 0 0 we > 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 We 0 0 0 0 0 0 0 0 0 1 vo 0 0 0 0 0 0 0 0 0 J The remaining H £ = O and K£ = 0 3.6 Reduction to the Korteweg-deVries Equation We shall now reduce the set of equations, written in the form Eq. 3.35, into a single nonlinear equation. Discussions will be based on the following assumptions [7]: A.1) There exists a "steady state" solution Uo such that U , A , 's and 's are developed as power series in the smallness parameter e , U = Uo + eUi + e 2 U 2 + • • • A = A 0 + e A i + e A 2 + etc. where Ui , U 2 • • • are 3-times differentiable with respect to x and t. In our case the "steady solution" is the unperturbed system. For ions and electrons streaming along an ambient magnetic field, with the reference frame fixed to the steady ion stream U 0 = "•id 0 0 0 0 0 Ayo{y,z {Az0{y,z)J Notice the vector potential Ao must be a function of y and z in the steady state because we have a magnetic field in the x direction i.e. B 0 = V A A 0 A.2) At least one of the eigen values of Ao(= A(Uo)) is real, nondegenerate and finite, and the eigen space does not comprise any invariant subspace. The eigenvalues and eigenvectors of Ao are calculated in Appendix D. We now introduce the coordinates £ , n , through the transformations Z = ea (x - AoO (3.36) _ _ ,a+l_ t] = £ X a = l / ( p - l ) (3.37) (3.38) Substituting the perturbation series into Eq. 3.35 and performing the transformations Eq. 3.36 through Eq. 3.38 yields: 0 0 / a \ OO CO « e ( - ^ + £ ^ / a - - - £ u ' -t=i v s ' i=oj=i s «=oy=i ' + £ £ I I (-AoHf + K f ) ^ + eKf 3=la=l d ..a d d£ adr}_ Vj = 0 (3.39) Then equating the various powers of e to zero we obtain the equations 0 = 0, O(0) (3.40) {A0/ + A 0 } - | r U i = 0 > O(e) (3.41) {-Ao/ + Ao} • ^ U 2 + A 0 | - U 1 + { U 1 . ( V n A ) 0 } ~ U l «=13p=3 3 + E II ( - A o H « o + Kjo) • = 0, 0(e2) (3.42) /?=la=l S in which V u is the gradient operator with respect to U , U - V u stands for Yli=i^i^ and A i is written as U i • (VuA) 0 (this follows by expanding each component of A in a Taylor series about Uo ). From Eq. 3.41 it follows that U ^ r o u W + Vifa) (3.43) where is one of the components of U i , ro is the right eigenvector of Ao (see Appendix D), and V i (r)) is an arbitrary vector valued function of r\. Multiplying Eq. 3.42 by the left eigenvector 1Q for Xc we obtain Aolo • ^ U i +1 0 • (Ui • (V nA )o) • ^ U i 8=13p=3 3 + lo EH ( - A o H « o + K J , ) • ^ 3 U i = 0 (3.44) 0=1a=l ^ Using Eq. 3.43 this becomes 3 A u(i) + p 2 U(i)A u(i) + p 3l_ u(i) where 9i = A 0 (lo • V x (r,)) fi = lo • (V x (V) • ( V n A ) 0 ) • r 0 Pi = Ao (lo • ro) P2 = lo • (ro • ( V U A ) 0 • r 0) /pi »=13p=3 P3 = l o - E n ( - A 0 H « 0 + ^ o ) - ' o / p i 3=\a=\ In our problem we have the initial condition U —• UQ for t —> —oo We then have V i (n) — 0 and Eq. 3.45 reduces to + p2 uW-^uW + P 3 -^UW = 0 (3.46) This is a well known nonlinear equation called the Korteweg-deVries equation. We see that an initial 0(e) perturbation evolves into a perturbation , which when 0(e2) quan-tities are included, satisfies the Korteweg-deVries equation. In the next section we will present the solution to the equation. 3.7 Solution of the Korteweg-deVries Equation The solution to Eq. 3.46 is (Appendix E): (x,t) = T2 + (f3 - r2)cn2 ]/^{*-2f(n + r1 + n)tlk}\ (3.47) where k = y*3 - r 2 - n and cn = Jacobian Elliptic Function The period of cn is Tp = 4K rz - n where K = Complete Elliptic Integral of the First Kind We will now look at two special cases: i) ?2 —> ri(=^ k —* 1) In this case the elliptic function cn degenerates into a hyperbolic function sech and Tp —> oo [11]. (x, t) = r\ + (r3 — ri)sech2 ra - r i 12 {x - (2ri + r3)<> (3.48) If we set u^1) = ue , r i = Ueo (i.e.the upstream value of ue or the"steady state" value), and a = r$ — ri ue (x, t) = ueo + asech2 12 {x - ( u e 0 + a/Z)t) (3.49) ii) r 2 —> r3 In this case the cnoidal solution degenerates into a sinusoidal solution. We will not discuss this case in detail, but refer the reader to an outside source [12]. It should be noted that the roots ri , r 2 and r$ are functions of the "steady state" variables Ueo , n,o , etc.(Appendix D and E). The roots determine the character of the solution, and in this way the initial conditions of the plasma system ultimately determine what type of solution emerges from the unstable growth. A complete analysis would lead us too far astray, so we shall simply reiterate that cnoidal, solitary or sinusoidal solutions may evolve from a linear instability. The reader might wonder why such a fuss has been made of the results presented in the previous sections. The main reason is that the 0(e2) solutions are stable. One can show that solutions to the KdV equation are stable with respect to small pertur-bations [13], [14]. We must conclude that once such a solution is reached the linear unstable growth is arrested. This result also follows from the formalism of the reductive perturbation method, as will be discussed next. 3.8 Higher Order Theory According to the reductive perturbation method once the 0(e) solution is determined all subsequent higher order solutions are also determined. This result can easily be seen in Eq. 3.40 through Eq. 3.42. Once is determined, Ui is also known. Solving Eq. 3.42 determines the components of ^ U 2 in terms of arbitrary one of them, say j|u(2) and Ui . Thus one can in a like manner proceed to higher order [7]. Since Ui is a stable solution, all higher order solutions must also be stable — reaffirming our assertion that the linear unstable solution gives way to a higher order stable solution. This is a physically satisfying result for in nature unstable solutions do not grow indefinitely, but eventually stabilise. CHAPTER 4 DISCUSSION In this chapter we give a physical explanation of the results obtained in chapter 3. 4.1 Physical Description of the Solution We recall that the solution was expressed as a perturbation series: where e was some unspecified "small parameter". In this section we will show that this parameter can be taken to be characteristic of the amplitude. For example we can write where a is the characteristic amplitude, and L is the characteristic length in the vertical direction. The following discussion will be based on this definition of e . 0(e) Theory: Initially the unstable system is perturbed only slightly, and therefore the character-istic amplitude o is small. At this stage higher order terms in the expansion Eq. 4.1 i.e. terms 0(e2) , 0(e3) , etc. may be neglected. We saw in chapter 3 that the 0(e) equations are satisfied by U = U 0 + eUi + e 2U 2 • • • (4.1) Ui oc exp [7<] exp [i (kx — ujrt)] (4.3) where Since 7 > 0 , the linear solution grows with time. The question naturally arises: does the growth predicted by the linear theory continue? This leads us to the 0(e2) theory. 0(e2) Theory: As the perturbation grows the higher order terms can no longer be neglected. One can proceed to the higher order problem in various ways, but in this work we adopted a reductive perturbation method. In this theory the 0(e) problem is solved by including the 0(e2) system of equations. It should be pointed out that in some perturbation methods the 0(e) solution is obtained first and then used to solve the 0(e2) problem. What we are doing is really quite different. In summary this approach gives 0(e) => Ui = r0ttW 0(e2) => -i t t(i) + P 2 ttW + P3 £suM = 0 drj d£ dtl6 which implies that the first order solution is stable and non-growing. Higher Order Theory: Once the 0(e) problem is solved all higher order terms in Eq. 4.1 can be solved. 4.2 Linear Dispersion Relation The linear dispersion relation predicts growth through the the presence of a positive imaginary part: u(k) = ur(k) + i~f(k) Notice that the linear dispersion relation is a function of the wavenumber k alone. We will see that when the higher order terms are included the dispersion relation becomes a function of both the wavenumber and the amplitude. 4.2 Nonlinear Dispersion Relation: In appendix E we find that the speed of the nonlinear wave is V . It follows that the nonlinear dispersion relation is given by [12]: u = Vk Using Eq. E.5 for V gives w = ~ - { n + r2 + r3}k (4.3) Remembering (see Appendix E) that r\ , r2 ,and r$ represent the amplitude of the cnoidal wave or solitary wave, this can be simply expressed as [12]: u} — u)(k, a) (4.4) where a is a characteristic amplitude. Wave breaking is a common phenomenom in nonlinear propagating waves. In certain instances it can be mathematically shown that when a wave's amplitude is sufficiently large, phase dispersion may cause a shock to form, which then eventually breaks [12]. Interestingly, in certain circumstances the phase dispersion which causes the breaking can be balanced by amplitude dispersion, preventing the onset of such discontinuities. These waves are not thought of as growing from an instability, but rather as existing in the medium to start with, hence the generation of the waves is not dealt with. What we have found is really quite similar, except in our case the amplitude dispersion balances 49 the phase dispersion preventing further growth in the wave instead of the formation of shock discontinuities. We can diagramatically represent this as The main point is that somewhere in the linear growth the imaginary part of the linear dispersion relation goes to zero, being balanced by an amplitude term. The details of this requires further analysis. A similar result has been derived for the Kelvin-Helmholtz instability [15]. This macroscopic instability is shown to reduce to the nonlinear Schroedinger equation— an equation which also has cnoidal and solitary solutions. A multi-scale perturbation technique is used to obtain these results, whereas we have used a reductive perturbation method. Another fundamental difference in their work is that the authors assume the system is linearly stable, which, in our opinion, obscures the wave generation mechanism. Admittedly, we have not looked at many instabilities, either microscopic or macro-scopic, but we feel bold enough to speculate. In the first chapter of the thesis we indicated that a linear instability might evolve along different paths (see Fig. 1.2). We now reassert this statement: u(k) => (j(k, a) (4.5) Fig. 4.1 Possibilities CHAPTER 5 APPLICATION TO THE MAGNETOSPHERE 5.1 Magnetospheric Plasmas A magnetosphere arises from the interaction of a continuously streaming, hot collisionless plasma with a magnetized body. In this interaction a cavity-the magnetosphere proper— is carved out in the flow by the magnetic field of the central body. This magnetic field also physically ties the points of the magnetosphere together, guiding charged particles, plasma waves and electric currents; trapping thermal plasma and energetic particles; and transmitting hydromagnetic stresses between the exterior flow and the resistive central body [16]. We will not discuss the structure of the magnetosphere in this work, but will only focus on the areas where our results might find an application. Magnetospheric plasmas can often be considered cold and collisionless, and therefore can be described by the plasma two-fluid equations and Maxwell's equations. Since the nonlinear evolution of an electromagnetic two-stream instability has already been studied, we would now like to apply our theory to magnetospheric plasma flows. There are a few regions in the magnetosphere where a two-streaming, field- aligned plasma flow might become unstable and generate the nonlinear waveforms we have been discussing. i) Auroral Field Lines: Plasma from the outer magnetosphere streaming down auroral field lines into the ionosphere is known to carry a current (which implies two- streaming) [16], [17]. Electric fields parallel to the magnetic field are thought to be responsible for the acceleration of particles into the ionosphere. Also, field-aligned electromagnetic turbulence along auroral field lines has been measured by satellites [17]. Nishida has proposed that a two-stream instability might be operating when a beam of auroral electrons penetrates into the ionospheric plasma [6]. In all these examples the field-aligned currents are considered small so that d.c. magnetic field can be ignored. ii) Cusp Field Lines: In this case the field-aligned current is probably more irregular to start with, and parallel electric fields are not as prominent as they are along auroral field lines. Gradients in the flow velocity might also support the macroscopic Kelvin-Helmholtz instability [17]. Further out in the magnetosphere neutral plasma is thought to be forced down along cusp field lines. Near the ionosphere parallel electric fields and currents set the stage for a two-stream instability. iii) Tailward Reconnection: According to one theory of magnetospheric substorms, plasma in the tail region of the magnetosphere is accelerated toward the earth upon reconnection of the tail field [18]. Electrons and ions separate as they move towards the auroral zones because of differing mirror points, forming field aligned electric fields in the process. These electric fields may be responsible for generating the auroral field-aligned currents often observed during substorm activity. A two-stream instability might be operating during these times, although the parallel electric fields are certainly not constant. iv) Plasma Mantle: Along the flanks of the magnetosphere solar wind plasma flows parallel to the mag-netic field towards the tail region. This plasma probably remains neutral, however if the impinging solar wind makes an angle with the tail flanks of the magnetosphere, a field-aligned current might develop [16], [18]. v) Plasma Reflected from the Bow Shock: As solar wind plasma is forced against the bow of the magnetosphere some of it is reflected upstream [17], [19]. The reflected plasma streams through the ambient plasma, forming a two-streaming system. It is currently unclear whether the reflected beam is neutral or current carrying. Interesting waves of cnoidal character and turbulence are often observed by satellites passing through this region. 5.2 Further Study Each one of the plasma systems listed in the previous section merits careful study on its own, both in the linear and nonlinear regimes; unfortunately we are not fully equipped to do so with the theory presented in the previous chapters. We will outline some of the main difficulties in applying our findings to the magnetosphere. In most theoretical studies at least some simplifying assumptions are made and this one is no exception. We list and discuss the main assumptions made in this work. i) Cold, collisionless, plasma: As was mentioned above, magnetospheric plasmas can often be considered cold and collisionless. The mean free path for a typical solar wind proton, for instance, is roughly 10 A.U. (Astronomical Units) [1], Also, in space the plasma flow velocity often exceeds the thermal velocity of the plasma; therefore thermal motion of particles can often be ignored. (Of course there are times when thermal motion can not be ignored, and one must include the pressure tensor in the fluid equations, or use plasma kinetic theory.) We feel this assumption is not completely unrealistic for the kinds of plasma systems described in the previous section. ii) One Dimensional Geometry: Obviously, any realistic model of a magnetospheric plasma system must be multi-dimensional. The magnetic fields, the electric fields, the plasma flows, the currents, etc. all vary in space and in time. More seriously, the one dimensional assumption disallows the presence of parallel electric fields (see Eqs. 3.6 through Eq. 3.11). For instance, if 53 B 0 = (Bx0,0,0) and vyo = («yo,0,0) , j = i,e then in equilibrium Eq. 3.9 implies Q — E 0 - cV A B 0 + 47T (en l 0v t 0 - en e 0v e 0) = 0 With Eo = constant (i.e. a "steady state"), Bo = constant and nto = neo we have v,o = v e 0 i.e. no two-streaming. Currents parallel to magnetic fields are difficult to explain using the plasma fluid equations as are parallel electric fields. To understand them properly one must use plasma kinetic theory. It should be pointed out that the 0(e) theory is not affected by these considerations, only the 0(0) theory. 5.3 Conclusion In conclusion, a more realistic treatment would require the development of a reductive perturbation method in several variables, which could then be applied to magnetospheric plasmas. We have only shown, that in the most idealised case, stable nonlinear waveforms can grow from a linear instability. 54 BIBLIOGRAPHY [1] A. Hasegawa, "Plasma Instabilities and Nonlinear Effects", Physics and Chemistry in Space, 8, (1975), Copyright ©1975 by Springer-Verlag, Berlin • Heidelberg. [2] S. Ichimaru, "Basic Principles of Plamsa Physics: A Statistical Approach", Frontiers in Physics, Copyright ©1973 by W. A. Benjamin Inc. [3] R. Z. Sagdeev and A. A. Galeev, "Nonlinear Plasma Theory", Copyright ©1969 by W. A. Benjamin Inc. [4] F. F. Chen, "Introduction to Plasma Physics", Copyright ©1974 Plenum Press, New York. [5] A. Jefferey and T. Kawahara, "Asymptotic Methods in Nonlinear Wave Theory", Copyright ©1982 A. Jefferey and T. Kawahara, Pitman Publishing Inc. [6] A. Nishida, "Theory of Irregular Magnetic Micropulsations Associated with a Mag-netic Bay", Journal of Geophysical Research, 69, (1964), 947-954. [7] T. Taniuti and Chau-Chin Wei, "Reductive Perturbation Method in Nonlinear Wave Propagation", Journal of the Physical Society of Japan, 24, No.4, (1968), 941-946. [8] T. Taniuti, "Part I. General Theory: Reductive Perturbation Method and Far Fields of Wave Equations", Supplement of the Progress of Theoretical Physics, 55, (1974), 1-35. [9] T. Kakutani, "Part HI. Application to Collisionless Plasma in Fluid Model: A. Plasma Waves in the Long Wave Approximation", Supplement of the Progress of Theoretical Physics, 55, (1974), 97-119. [10] T. Kakutani, H. Ono, T. Taniuti, and C. C. Wei, "Reductive Perturbation Method in Nonlinear Wave Propagation II. Application to Hydromagnetic Waves in Cold Plasma", Journal of the Physical Society of Japan, 24, No.5, (1968), 1159-1166. [11] A. Jeffery and T. Kakutani, "Weak Nonlinear Dispersive Wave: A Discussion Cen-tered around the Korteweg-deVries Equation", SIAMRev., 14, (1972), 582-643. [12] G. B. Whitham, "Linear and Nonlinear Waves", Copyright ©1974 by John Wiley & Sons, Inc. [13] A. Jeffery and T. Kakutani, "Stability of the Burgers Shock Wave and the Korteweg-deVries Soliton", Ind. U. Math J., 20, (1970), 463-468. [14] A. Scott, F. Y. F. Chu, and D. Mclaughlin, "The Soliton: A New Concept in Applied Science", Proceedings of the IEEE, 61, No.10, (1973), 1443-1483. [15] R. Kant and S. K. Malik, Untitled, Astrophysics and Space Science, 86, (1982), 345-360. [16] C. F. Kennel, L. Z. Lanzerotti, E. N. Parker, and Editors, "Magnetospheres", So-lar System Plasma Physics, U, (1979), Copyright ©1979 North-Holland Publishing Company. [17] C. F. Kennel, L. Z. Lanzerotti, E. N. Parker, and Editors, "Solar System Plasma Pro-cesses", Solar System Plasma Physics, III, (1979), Copyright ©1979 North-Holland Publishing Company. [18] A. Galeev, T. Sato, A. Nishida, G. Haerendel, G. Paschmann, M. Ashour-Abdalla and C. F. Kennel, "Magnetospheric Plasma Physics", Developments in Earth and Plan-etary Sciences, 04, Copyright ©1982 by Center for Academic Publications, Japan. [19] A. Hasegawa, "Excitation and Propagation of an Upstreaming Electromagnetic Wave in the Solar Wind", Journal of Geophysical Research, 77, (1972), 84. APPENDIX A P L A S M A THEORY In this appendix we introduce plasma kinetic theory [2] and derive the plasma fluid equations from the basic plasma kinetic equations. A . l Plasma Kinetic Theory In plasma kinetic theory one considers a classical system containing N identical particles in a box of volume V ; n = N/V denotes the average number density. Each of the particles has a charge q and mass m , and a smeared background of opposite charge is assumed so that the average space-charge field is zero [2]. The * -th particle has position and velocity given in a six- dimensional phase space as Since we are dealing with point particles, the microscopic density of the particles in phase space may be expressed by where X = (r,v) . N(X;t) is called the Klimontovich distribution function. In phase space the distribution function satisfies the continuity equation: Xi(t) = [r,-(0,v,-(01 (ii.1) m (A.2) dX d •N = 0 (A.3) which can be written in terms of (r,v) —N + w- — N + v- — N = 0 dt OT ov (AA) where v is the acceleration at the point (r,v) . In plasma physics the electromagnetic acceleration is the most important one, therefore v = i -m E(r, t) H—v A B(r, t) c (A.5) It must be realized that E(r, t) and B(r, t) consist of contributions from local and external sources: E(r, t) = Eexl (r, t) + e(r, t) B(r, *) = Bext (r, *) + b(r, t) (A.6) Both the microscopic and macroscopic fields can be determined from Maxwell's equations. Ignoring the microscopic magnetic field one can rewrite the Klimontovich equation in the following form [2]: d_ di + L(X) - j V(X, X') N(X'; t) dX' N(X;t)=:0 (A.7) where or m Eext(r,0 + -vABe^(r,t) c d_ and o2n f d 1 The Klimontovich equation describes the microscopic state of the system, whereas we can only observe the macroscopic state. To establish a connection between the mi-croscopic and macroscopic descriptions, we introduce an averaging process based on the Liouville distribution over the six-dimensional phase space (the T space) [2]. The mi-croscopic state of system is expressed in the V space by a point which we call a system point. We imagine N macroscopically identical systems, each with a different microscopic configuration. Because they are microscopically different, these systems form a scatter of points in the Y space. Then one defines the Liouville distribution function D({Xi};t) in the T space by With the aid of the Liouvile distribution, we may carry out a statistical averaging of a microscopic quantity such as N(X; t) . For example Using this definition we can calculate the average products of the Klimontovich functions N(X; t) . For example we can calculate {Xi} = (Xi,X2,...,XN) (A.8) (A.9) (N(X;t)N(X';t)) or (N{X-1t)N{X,-1t)N{Xnit)) etc. Using these and higher order average products we can arrive at a Liouville average of the Klimontovich equation, which is actually a hierarchy of equations known as the Bogoliubov-Born Green-Kirkwood- Yvon (BBGKY) hierarchy: d_ dt » i 8 /«(!> • • • >a! 0 = i2f V(l,s + l)ft(l,...,s;t)d(s + l) 1=1 J (All) where I = X(,k = Xk This coupled set of equations form a basis for the kinetic theory of plasmas [2]. By adopting a power series representation /. = /.(0)+/.(1)+/.(2) and assuming /.(1,...>«;*) = IIF(') 1=1 we find the single particle distribution function F(l; t) satisfies the following equation: d_ dt + 1(1 ) - j V{\,2)F{2;t) d2 F(1;0 = 0 Extending this analysis to a multicomponent plasma consisting of j components, we write /y(r,v;i) = F(l;t) and arrive at the convential form of the Vlasov equation: = 0 (A.12) where 3 dv1 | r - r ' | B(r,t) =Bext{T,t) It is perhaps appropriate at this point to discuss the meaning of the single particle distribution functions, /y(r,v;<) [4]. The the number of j -particles per cm3 at position r and time t with velocity components between vx and vx + dvx , vy and vy + dvy , vz and vz + dvz is Finally, it should be pointed out that the Vlasov equation is only valid for collisionless plasmas, since it is an equation for the single particle distribution function /y(r, v; t) con-taining no information on particle correlations (i.e. 2-particle 3-particle,..., s-particle distribution functions). Collisionless plasmas occur rather frequently in geophysical and astrophysical settings; for instance, in the solar wind the mean free path between collisions of charged particle is « 10 A.U. which can be considered collisionless [1], A . 2 Plasma Fluid Theory From the Vlasov equation we can derive the collisionless plasma fluid equations [4]. To accomplish this we simply take various moments of the Vlasov equation. The lowest fj{*> y. z> vx, ^  vz,t) dvx dvy dvz moment is obtained by integrating Eq. A.12 over v : Integrating the first term gives JTt!'dy = Ttjf'dv = h n ' The second term gives /d d P d v ' a r " 7 ^ = d r y V / J ' * * = ai' = v • ( w ) where The third term can be shown to equal zero. For instance, /*E.A/ yrfv= f ^•(/yB(r,t))*r= / /,-E • «fS = 0 The last integral goes to zero because /y —» 0 faster than v - 2 as v —• ±oo for a distribution with finite energy. A similar result holds for the | v A B term. Collecting the terms then gives the lowest moment of the Vlasov equation: jnj + V • (nyvy) = 0 (A.13) which is the fluid equation of continuity for the j -th component of the plasma. fj dv = 0 The next moment of the Vlasov equation is obtained by multiplying Eq. A. 11 by v and integrating. We have my j v^/yrfv + my j v (v • V) fj dv + qj j v j^E+^vAB 9 r , fjdv = 0 <9v The first term gives m y / W o \ f j d V = m j a \ IV/j d V = mjdl The second integral is J V ( V - V ) f j d v J V- (/yw) rfv = V • J /ywdv Separating v into the average fluid velocity vy and a thermal velocity w v = vy + w Since vy is already an average, we have V • (nyvv) = V • (nyvyvy) + V • (nyww) + 2V • (nyvyw) The average w is obviously zero. The quantity mynyWW is precisely what is meant by the stress tensor Py : 63 The remaining term can be written V • (flyVyVy) = Vy V • (flyVy) + fly (vy • V ) Vy The third integral is AH v A B d_ f i d v = j £ - { / > v [ B + r r A B ] } * r - y j / y v r f v - ^E .+ ^ VABJJ dv - / { / y [ E + I v A B ] } . A v ^ The first two integrals vanish for the same reasons given previously. Therefore, we have VAB d_ 3v fjdv = - , y / [ E + i v A B -tyfiy j E + -Vy A B fjdw Finally, collecting our results from above we can write mJQl (n>V>) + »»yVyV • (n>Vy) + my fly (vy • V) Vy + V • Py - fjyfly E + -Vy A B c J = 0 which can be rearranged into a recognisable fluid equation: dt U + ^  • V) v y = ^  ( E + iv, A B) - • P,- M.14) APPENDIX B LINEAR THEORY In this appendix we derive the linear dispersion relation from the linearised plasma fluid equations and Maxwell's equations. B . l Linear Dispersion Relation We assume a one dimensional geometry with ions and electrons streaming in the x -direction parallel to a constant magnetic field, Bo . The field quantities are written in the following notation: Vy = (tiy, Vy, Wj) B = (Bx,By,Bz) E = (Ex,Ey,Ez) We assume each field quantity can be written as perturbation series (see chapter 3): Vy = Vy 0 + €Vy! + € 2Vy 2 H B = B 0 + eBi + e 2B 2 + • • • E = Eo + eEi -f- e 2E 2 + • • • where subscripts are added to the various components to designate 0(e) , 0(e2) etc, terms. We can express the plasma fluid equations and Maxwell's equations to any order in e . The linearised plasma fluid equations are presented in chapter 3 (Eq. 3.6 to Eq. 3.11). We assume all the 0(e) quantities vary as exp [t (A;i — ut)] Substituting this into Eq. 3.6 through Eq. 3.11 we derive the linear dispersion relation [6]. The steps of the derivation are outlined in the remainder of this appendix. Substituting the first order quantities into Eq. 3.9, we obtain the following compo-nents. B x l = 0 (BA) B y l = - — E z l (B.2) UJ ck Bzi = — E y \ (B.3) Similarily for Eq. 3.8 IUJ j=t,e J ' I r V j yl = 7 ^ Wo«yi (B.5) j=i,e . f u) k2c 1 4T  <r-^  % \ ~c UJ~ j ~c~ qin&w3i (B-6) j=i,e From the fluid momentum equations Eq. 3.7: -t'my {(J - kujo} Uji = qjExi (B.7) -irrij {UJ - kujo} vji = qj 11 - ^ | Eyi + ^wjxBXQ (B.8) -irrij {u - fcuyo } VJji = qj j 1 - J Ezi - y V y i ^ o (5.9) The reader should notice Eq. B.5, Eq. B.6, Eq. B.8 and Eq. B.9 are uncoupled from Eq. B.10 and Eq. B.13. This uncoupling occurs for propagation parallel to the ambient magnetic field Bo . We will derive the dispersion relation for the transverse mode (i.e Vji , Wji , Byi , Bzi , Eyi ,and Ez\ ). The longitudinal mode (i.e uji , Bxi , and Exi) has a separate dispersion relation which will not be discussed in this thesis. Instead, we assume that the perturbed longitudinal quantities are zero. It turns out that all higher order perturbed longitudinal quantities are then also equal to zero. Solving Eq. B.9 for Vji and substituting into Eq. B.8 we get after minor rearrang-ments: q)B*«l., ...... I * . , ^ {u - kujQ} Eyi - i-^- {w - kuj0}2 Ezi ,2 Wjx (B.10) Similarity, we can solve Eq. B.8 for tuyi and substitute into Eq. B.9 resulting in: i Q2B o ^ {u - kuj0}2 Eyi + - i - J - {UJ - kuj0} Ezl urn,- ujm-c J 3 Vjl (5.11) Finally, we obtain the linear dispersion relation. Solving Eq. B.10 for tuyi , substi-tuting into Eq. B.6, and assuming we are studying circularily polarised waves Ey\ = ±iEzi we find upon some minor reduction [6]: w 2 _ k2c2 _ W ^r^L = 0 (B.12) Alternately, we could have solved Eq. B.ll for wyi and substituted into Eq. B.5. The results are identical. Choosing the lower sign (i.e — fly) gives a dispersion relation which has complex roots with positive imaginary parts. Eq. B.12 is the linear dispersion relation presented in chapter 3. Eq. B.12 is a quartic equation in u , which can be shown to have complex roots with positive imaginary parts (a quartic equation can be solved analytically). Although the computations are straightforward they are long. Instead we will reduce the quartic equation to a quadratic equation by restricting the frequency range. We assume: \u\ < fly j = i,e Also, we fix our coordinate system to the steady ions stream so that u,o = 0 • (Notice, we have made a transformation to another reference frame. To be completely accurate, the velocities should be written as u'iQ = 0 , u'e0 = — u,o . For ease of notation we drop the primes, but remember that we are in the moving coordinate system). Eq. B.12 then reads: 2 1.2 2 2 u 2 (OJ - kue0) / D I O \ u - k c - - " ^ - f c u e 0 - n e = 0 ( R 1 3 ) Expanding the third term: OJ UJ u2 w-ft,- a n2 Also, UJ - kllef) - Q e « - k u e 0 - fte Substituting these approximations into Eq. B.13 gives a quadratic in w 1 + This equation has complex roots with positive imaginary parts provided that the dis-criminant of the solution is less than zero: (ul^i + UjliikUeO + Qe))2 + 4 (n2 + w£) (k2c2 (fcu c 0 + nef + w2>ekueO (kua + Oe)) < 0 (B.15) Note: Qe < 0 . APPENDIX C REFORMULATION OF THE PROBLEM In this appendix we rearrange the plasma fluid equations and Maxwell's equations into a form prescribed by the reductive perturbation method of Taniuti et al. [7]. We also outline how these equations are reduced to the one dimensional geometry studied in this thesis. C l Rearranged Equations In this appendix we rearrange the plasma fluid equations and Maxwell's equations into a form prescribed by the reductive perturbation method. The basic equations are the plasma fluid equations and Maxwell's equations: — t i i + V • (n(v.) = 0 (Cl) —ne + V • (neve) = 0 (C.2) (C.3) {^vc + (ve.V)ve} e _ 1 _ E + -ve A B me [ c (C.4) V A B = -— E + (n,v,- - neve) cat c (C.5) 1 d „ V A E = — — B cdt (C.6) V • E = Aire (n,- - ne) (C.7) V • B = 0 70 (C.8) We proceed to rearrange Eq. C l to Eq. C.8 to obtain Eq. 3.18 to Eq. 3.24. First we derive an equation for vt- and v e which has third order derivatives in it. From Eq. C.5 we have I d c nr U = —A ^ T E + - V A B + A C (C.9) 47ren,- at 47ren,- n,-From Eq. C.7 we have n e . V • E 1 - (CIO) Substituting Eq. CIO into Eq. C.9 gives * 47ren,- dt 47rent-Taking V- and §-t of Eq. C.4 gives V A B + v e - v e (C.ll) V . E = - ^ V • | ^ v e + (ve • V) v e | - V • |^v e A B J (C.12) Substituting Eq. C.12 and Eq. C.13 into Eq. C.l l for V • E and JjE , and also using the electromagnetic vector potential B = V A A gives an equation for v; or v e in terms of third order derivatives alone: | i s / a \ 1 3 , „ . 1 I d , _ . . + 0 3 ' (!*•) + S V ' ( v < ' V ) v<} + ° % V T ' A < V A A » + a5—V A(V A A ) + v e (C.H) where me 1 me oi = - — a 2 = , a3 = - — T : , a4 = a2 05 47re2' 47rec' 47re2' Aire We will write Eq. 0.14 compactly as v,- = f(n,,v e ,A)+v e (C.15) where f represents the third order derivative terms in Eq. 0.14. We are now ready to generate the equations presented in chapter 3. i) Equation 3.18: From Eq. C l Q —Hi + Vn$- • v,- + m (V • v$) = 0 (C.16) Substituting v» = f + v e from Eq. C.15 gives g —ni + v e • n, + n,- (V • v,-) + Vn,- • f = 0 (C.17) Substituting the expression for f from Eq. 0.14 then gives Eq. 3.18 in chapter 3. ii) Equation 3.19: Solving Eq. C.3 for E and substituting into Eq. C.4 gives £v,.+ (v...V)v,. me , _. e l „ + — ( V e - V)v e + V e A B m,- m,c c e 1 _ me d , _ v , A B + -^—v e = 0 (C.18) Substituting J j v e = JjV,- — Jjf and rearranging + 62 (ve • V)v e + 6 3-v e A B c 1 d + t 4 - V i A B + ( 4 2 ) - f = 0 (C.19) where the constants 61 , 62 etc. are defined in chapter 3. We make the following substitutions to generate Eq. 3.19 Q ^v,-+ 61 {(ve + f) • V}v,-+ b2 {(v,- - f) • V} v e + 63ve A B + 6 4 v,AB + (-62)— f=0 (C.20) at Finally, substituting Eq. C.15 for f and using the electromagnetic vector potentials gives Eq. 3.19. iii) Equation 3.20: Making the following substitutions into Eq. C.19 generates Eq. 3.20 73 ^V,+fc!{(ve + f)- V}V,-+ 62 (v e -V)v e + 6 3 v e AB a ]ai + 6 4 v t AB + (-62)—f=0 (C.21) Substituting for f and using the electromagnetic potential gives Eq. 3.20. iv) Equation 3.21: Rearranging Eq. C.18 9 < ™ ^ V e + (V e • V) V e 171; , e „ + — (vt- • V) v.- + v e A B e ^ m. d -vt ABH = 0 mec me dt Substituting J^ v,- = J^ve + Jjf and grouping the constants ^ v e + 6!(vt-V)v,-+ h (ve • V) v e + 63ve A B + 64v,-AB + 61—f =0 (C.22) To generate Eq. 3.21 we make the following substitutions into Eq. C.22: ^ v e + 6i{(ve+f)V}v,-+ b2 {(v,- - f) • V} v e + 63ve A B + 64v,- A B + —f = 0 (C.23) Next expand the curly bracket terms and substitute v,- = v e 4- f Q-^C + hi (ve • V) {ve + f} + 6 1(f-V)v t + 6 2(v,-V)v e + (-62)(f-V)ve + 63ve A B + 64v,- A B + 61 —f = 0 (C.24) Finally, substituting for f and using the electromagnetic vector potential gives Eq. 3.21 v) Equation 3.22: Expanding Eq. C.24 gives J^ve + 61 (ve • V) v e + 62 (v,-V)v e + 6 3 v e AB + 64vt- A B + 61 (ve • V) f + h (f.V)v, + (-6 2)(f.V)v e + 6i^f=0 (C.25) Writing 62 {v,- • V} v e = b2 {(ve + f) • V} v e into Eq. C.25 and expanding ^ v e + 6i(v e-V)v e + b2 (v e -V)v e + 63ve A B + 64v,- A B + 61 (ve • V) f + h (f- V)v,- + 6i—f=0 75 (C.26) Substituting 6 2 (veV)v e = 6 2 ( v e V ) {v,-f} into Eq. C.26 and expanding gives ^ v e + 6i (v e -V)v e + 62 (v e-V)v, + 6 3 v e AB + 6 4 v e AB + (-/32)(v e-V)f + 61 (v e-V)f+f+6i (f- V)v,-(C.27) Finally, substituting Eq. 0.14 for f and using the electromagnetic vector potential gives Eq. 3.22. vi) Equation 3.23: Start with Eq. C.4, rearranging and introducing the electromagnetic vector poten-tials: e d mec dt A e v e A V A A (C.28) Substituting v e = vt- — f gives e_d_ i e c di v e A V A A {^v. + tv .VJv,} }.. 76 (C.29) Adding and subtracting a term (vt- • V)v,-, using | v , + (v...V)v, = A E + -vt- A B c and using the electromagnetic vector potential A gives upon rearranging: — A + ci(v t-V)v,-+ c 2 (ve • V) v, + c 3 v e A V A A -I- c4v,- A V A A + c s — f at + c6 (ve- V)f =0 (C.30) where the constants c\ , c2 etc. are defined in chapter 3. To generate Eq. 3.23 we substitute Eq. 3.14 for f. vii) Equation 3.24: To generate Eq. 3.24 we make the following substitution into Eq. C.30 — A + C l(v, • V)v, + c 2 {(vt- - f) • V} v,- + c 3 v e A V A A + c4v,- A V A A + ci + (ve • V)f j = 0 (C.31) Substituting Eq. 3.14 for f gives, upon expanding, Eq. 3.24. 77 C.2 One Dimensional Problem In analysing the electromagnetic two-stream instability we make the simplifying assump-tion that all dependent variables vary only as function of x and t (see chapter 3). Eq. 3.18 to Eq. 3.24 can easily be rewritten in component form as presented in chapter 3. For example the equation for n,- is given by Eq. 3.18. The various terms can be expressed as a function of x and t straightforwardly. For example: d d ve • Vn,- d_ d~x ,,-(V vt) = rii—Ui ox axVrii — — (ve • V)ve - a i n * o l ( ^ ( U e a | U e ) ) a z V n ' nidi { V e A V A A ) = °2 l^nV n7 + J = a2Tx{dl \v<TxAv + Wed-xA>)) = a 2 l k { § i {VelA»)) +a2ah{h(WehA)) + ( - ° 2 ) n ^ ( n 7 £ ( V e £ ^ ) ) etc. Likewise, the remaining terms of Eq. 3.18, and the remaining equations, Eq. 3.19 to Eq. 3.24 can be simplified. We will not write out these rather tedious computations, but instead, leave it to the reader to verify, if they desire. 79 APPENDIX D CALCULATIONS In this appendix we compute the left and right eigenvectors of the matrix Ao , and the constants pi , P2 > P3 °f * n e Korteweg-deVries equation presented in chapter 3 . D.l Eignevalues and Eigenvectors First we determine the eigenvalues of Ao = A (Uo) . i) Eigenvalues: The equation to be solved is A 0 r o = A 0 r 0 (D.l) The eigenvalues are determined in the standard way by evaluating Using det (A 0 - XQI) - 0 (D.2) / " e O 0 0 0 0 0 0 0 0 0 h u e 0 0 0 0 0 0 0 0 0 0 0 0 0 b2ue0 0 0 -6 3« e 0 0 0 0 0 biuc0 0 0 & 2 « e 0 0 0 -6s«e0 0 0 0 0 0 0 0 0 0 0 0 0 0 M«o 0 0 -bzue0 0 0 0 0 & 2 « e 0 0 0 bi»eO 0 0 -bs ue0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 V 0 0 0 0 0 0 0 0 0 -c 3« eo J 80 we find that AQ = &iueo (D.3) is an eigenvalue(notice that C3 = -61). This implies that the perturbed waveform travels at near the speed of the streaming electron fluid since &i ~ 1. ii) Eigenvectors: Next we calculate the left and right eigenvectors according to the following equations: (A 0 - A07) • r 0 = 0 (DA) 10 • (A 0 - A0J) = 0 (A 0 - A 0 I) r • l j = 0 (D.5) where (Ao - A 07) / u e 0 ( l -61) 0 0 0 0 0 0 0 0 0 neo 0 0 0 0 0 0 0 0 \ 0 0 0 0 0 0 0 0 0 0 0 0 0 &2«e0 0 0 -6s«e0 0 0 0 0 0 0 &2u«o 0 0 -&S«eO 0 0 0 0 0 0 0 0 0 0 h "«o 0 0 0 0 0 - & 3 « e 0 0 0 0 6 2U e 0 0 0 0 0 0 C2«<:0 0 0 0 0 0 - M e O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / Solving these equations vis a vis row reduction we obtain /-DrAx \ -BrAy -BrAz -BrAy -BrAz TAX r 0 rAy V rAz J (D.e) where 81 A = (i-&iKo' B 63 62 C2 D = -AC and lo / 0 \ 0 0 0 0 0 lT l A y (D.7) The components rAx , r^,, , rAz , r „ , ljx , ljx , l\ , and lrAz are arbitrary. D.2 The coefficients of the KdV Equation In this section we calculate the coefficients of the KdV equation: £.<. ) + B .< .>*.< .> + f t »> Pi y P2 and p3 . i) pi = A0lo • ro Substituting Eq. D.6 and Eq. D.7 into the expression for pi gives: Pi = &l«eO {UxTAx (c/cme) + lexrex + lAy^Ay + ^Az^Az) ii) P2 = lo • (('o • VA 0 ) • r 0) /pi Evaluating the various terms: 82 Pi = tixbirex I ] rAx + lixb% ( ) rAxrex + lixbs -\cmeJ \cmeJ i c(me+mi) A y € 2 i L ^ 2 J L ^ ; r f/iu + hxbs—j ; \rAz + 'ixb* —, ; r c(mi+me) v c(mi+me) c(mi + me) + ' « i&2 TAxrtx + l e x b i r l x +lexbs— rrAv cme c(m, + me) y r 2 rAz __ C 2 i i ^ 2 i t ^ 2 + lexbi—, ; rr A „ + IcxDs—. TTa + l e x b t - j — — ; :^ c(m, + m e) y c(mi+me) A Z c(mi+me) i e i e i e i e ~ <Ay c S f A i f A y — lAyCt TAxTAy — lAzCs TAxTAz — lAzCi TAxTAl cmt cm e cm e cm e / & l " e O I lix^Ax + hx^ex + tAvrAy + l A z r A z \ {cme ) (D.9) iii) P3 = lo • E8fiiUP==\ (-AoHjo + Kfo) • r 0 Inserting Eq. D.6, Eq. D.7 and the matrices 0 , K£ 0 : •0 . 1 Ps = — Pi 1 + — Pi 1 + — Pi 1 + — Pi 1 + — Pi 1 + — Pi 1 + — Pi 1 + — Pi 1 + — Pi -Ao) 5 Ao)2 Ao)2 -Ao) Ao)2 1 . . 1 , 1 , 1 Oilix^ex r- bilexrex h CslAyrey 1- c^lAzrcz n,o n,o n.o n i o 1 . . 1 bnl%xrex h OTiexrex — « i O n,0. b^Ueolexfex H Ci^e^Ayrey H C 5U e 0 l A z r e z  n,o n,o n,o b7Ue0lexrex 1 n.oj 1 n.o 1 »io ^15U«0Ux^ex- H 67« e o ' « z r < ! i - C o ^ y f / t y Cgt^.ryi . 1 " i O 1 n,o - A 0 ) 6 i i « 2 0 ' . a » - e j — + — ( - A o ) f t i ( 6 5 " ? o r e i - — - 6 i i " « o r « a - — ) n.o Pi \ "io n.o / - A 0 ) ' / l y ( - C 6 W 2 o ^ y — + C s W ^ r c j , — ) , \, ( 2 1 2 1 ,\ - A 0 ) » / U I - C 6 « e 0 r ^ z tc5ue0rez J &7W*o'« rei C9U e o'Ay' 'Ay Cc>UeolAzrAz  n.o (D.10) APPENDIX E THE KORTEWEG-DeVRIES EQUATION E.1 Solution of the Korteweg-deVries Equation The Korteweg-deVries (KdV) equation is d tu tu d ^ u ( i ) + p 2 U ( i ) ^ u ( i ) + p 3 ^ _ u ( i ) = 0 (E.1) For simplicity we henceforth write Introduce u ( i / , f l = « W , X=i-Vrt (E.2) Substituting this into Eq. E.1 gives - W + P2UU' + P3U"' = 0 or - W + ^ ( u 2 ) ' + p 3u'" = 0 ml Integrating -Vu + p2u2 + p 3u" + A = 0 Multiplying through by u' and integrating T u 2 + ¥ u 3 + f («') + * » + * = <> Finally, rearranging this gives 2 3 , „ ..3 S3U* + S2U6 + SiU + s0 = P (u) (E.3) where _ -2p 2 _ 27 3p3 ' 2p3 si = —A(constant of integration), SQ = —fJ(cons£ara£ o/ tnfegrraiion) It is obvious that for real solutions ( £ - ) ' = i , M - 0 Because there are no bounded real solutions if P (u) has only one real root, we conclude that it must have three real zeros [11]: P (u) = (u - ri) (u - r2) (u - r3) (EA) where r\ < r2 < r 3 which implies [11] V = ^ { t i + n + r3} (E.5) A = — {rir2 + nr 3 + r2r3} 3p 3 81 B = -p- { r i r 2 r 3 } In order that the roots be real the following conditions must hold: Q 3 + R2 < 0 where Q S3 £1 3 1 s3 R=-s{sls2-3so}-—2-S3 S3Z7 The character of the solution can be gleaned graphically (Fig. D.l). f(u) Fig. E.1 Character of the Solution Real solutions lie between r2 and r$ . These solutions are nonlinear and oscillatory. We will look at three limiting cases: i) r2 —* n (curve B): This is a solitary wave since j^u =0 at only one value of u. ii) r 2 —* rz (curve C): This is the constant solution since j^u = 0 for all real values of u. iii) r\ = r2 = rz This is a constant solution as well. The explicit solution of Eq. E.3 is given in terms of the Jacobian Elliptic function. u (x, t) = r2 + {rz - r2) cn7 rz - r i 12 {x-2p2/3{ri + r2 + rz)t;k} (E.6) where rz - r2 rz - ri Special cases of this general solution are examined in chapter 3, section 3.4. It is perhaps prudent to point out that the general solution applies to the components of the vector / ni \ Ui Vi U = ne We A x Ay \AZJ The electromagnetic field components are calculated from the vector potential A = (Ax, Ay, Az) according to This implies that the electromagnetic field components are proportional to the x and t derivatives of the Jacobian Elliptic function. For example dx -cn2 = -2cnsn y/l — K2sn2w —w dt where w = argument of Jacobian Elliptic Integral, and 0 < K < 1 . We see the elec-tromagnetic perturbations are different from the plasma oscillations, although they are still represented in terms of Jacobian Elliptic integrals. As a further exposition on this point let us consider the solitary solution (see chapter 3). The solitary(soliton) solution is proportional to seek2 The electromagnetic field quantities are proportional to the x and t derivatives. For example a a ——u = —2a sechw2 tanhw—w dx dx While still a solitary wave, this solution is not a single humped solitary wave, but rather a double humped solitary wave. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0052502/manifest

Comment

Related Items