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Wave-particle interaction around the lower hybrid resonance Horita, Robert Eiji 1968

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WAVE-PARTICLE INTERACTION AROUND THE LOWER HYBRID RESONANCE by ROBERT EIJI B.A.Sc., The University of M.A.Sc, The University of HORITA B r i t i s h Columbia, 1960 B r i t i s h Columbia, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of GEOPHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t he L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department or by hits r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Geophysics The U n i v e r s j t y o f B r i t i s h Columbia Vancouver 8, Canada Date September 30, 1968 i i ABSTRACT Wave-particle in t e r a c t i o n i n the ionosphere is studied t h e o r e t i c a l l y for wave frequencies around the lower hybrid reso-nance (LHR) frequency. Expressions are derived by two methods for the growth rate of whistler-mode waves propagating in a magneto-active plasma penetrated by a tenuous beam of nonthermal p a r t i c l e s . The f i r s t method employs the e l e c t r o s t a t i c disper-sion equation; the second uses the full-wave dispersion equa-tio n which reduces to the e l e c t r o s t a t i c one for large values of r e f r a c t i v e index. The equilibrium d i s t r i b u t i o n function for the plasma is Maxwellian, and that for the diffuse stream-ing p a r t i c l e s i s also Maxwellian, but is shifted by a streaming v e l o c i t y p a r a l l e l to the background magnetic f i e l d . The f i r s t method assumes that the temperatures are i s o t r o p i c , while the second assumes that the d i s t r i b u t i o n s are characterized by the perpendicular and p a r a l l e l temperatures, Tj_ and TJJ . The growth-rate expressions are f a i r l y general, but numerical c a l -culations are performed for the case of a cold plasma consist-ing of electrons, H + , He + , and 0 + ions and a beam of nonthermal electrons. The growth-rate expression obtained using the electro-s t a t i c dispersion equation shows that waves propagating s l i g h t l y of f the d i r e c t i o n perpendicular to the background magnetic f i e l d can grow due to the Landau i n s t a b i l i t y process which is excited by high energy O 10 keV) electrons streaming along the d i r e c t i o n of the magnetic f i e l d of the earth. The growing wave thus triggered i s shown to have a frequency band with a i i i sharp lower cutoff at the LHR frequency and an upper l i m i t at the electron cyclotron frequency or electron plasma frequency, whichever is lower. The previous growth-rate expression is generalized by making use of the full-wave dispersion equation. It is shown that there are two regions i n propagation angle 6 where the Landau i n s t a b i l i t y may occur. The " e l e c t r o s t a t i c " region l i e s just below the resonant angle and, separated by a region of damping, the "low-6" region l i e s above 6 = 0 . The growth-rate values calculated in the " e l e c t r o s t a t i c " region correspond to the values obtained in the previous c a l c u l a t i o n . Generally, the maximum growth rate is larger i n the " e l e c t r o s t a t i c " than in the "low-6" region. It is also seen that with increasing frequency the " e l e c t r o s t a t i c " maximum growth rate increases monotonically and the cyclotron i n s t a b i l i t i e s become important at frequencies above about ten times the LHR frequency. The influence of the following parameters on the growth rate is also examined: temperature r a t i o T|( /T^ , streaming v e l o c i t y of the nonthermal p a r t i c l e s , and the r a t i o of the k i n e t i c energy in the streaming motion to the thermal energy of the streaming electrons. The theory presented is applied to LHR noise bands dis-covered by the Canadian Alouette I s a t e l l i t e . It is shown that many features are in good agreement. Other observations, such as auroral h i s s , also have features which suggest that the theor e t i c a l work may be relevant to these types of ionospheric noise. i v TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES v i LIST OF TABLES v i i ACKNOWLEDGEMENTS v i i i CHAPTER 1 INTRODUCTION 1 1.1 Bri e f Background 1 1.2 Purpose and Scope of Thesis 3 CHAPTER 2 ELECTROSTATIC WAVE ANALYSIS 6 2.1 Landau I n s t a b i l i t i e s i n E l e c t r o s t a t i c Waves Propagating Nearly Perpendicular to the Background Magnetic F i e l d 6 2.2 Basic Equations for Weakly Unstable Elec-t r o s t a t i c Waves in Magneto-Active Plasmas 9 2.3 Electron Plasma Waves Propagating Along the Background Magnetic F i e l d 22 2.4 E l e c t r o s t a t i c Waves Excited Around the LHR Frequency 24 CHAPTER 3 FULL-WAVE ANALYSIS 38 3.1 Power Absorption i n a C o l l i s i o n l e s s Plasma 38 3.2 Wave Energy and E l e c t r i c F i e l d 45 3.3 Growth Rate and Computer Results with Emphasis near the LHR Frequency 51 CHAPTER 4 APPLICATION OF THEORY TO RELEVANT IONOSPHERIC NOISE 69 4.1 LHR Noise 69 4.1.1 Observations and Characteristics of LHR Noise 69 a) General Features 69 b) Polar LHR Noise 70 c) Midlatitude LHR Noise 71 V 4 . 1 . 2 C o m p a r i s o n o f T h e o r y w i t h L H R N o i s e O b s e r v a t i o n s 7 2 4 . 1 . 3 D i s c u s s i o n 7 5 4 . 2 O t h e r R e l a t e d O b s e r v a t i o n s 7 8 C H A P T E R 5 S U M M A R Y A N D C O N C L U S I O N S 8 4 B I B L I O G R A P H Y 8 7 A P P E N D I X I C O M P U T E R P R O G R A M T O C A L C U L A T E G R O W T H R A T E O F W H I S T L E R - M O D E W A V E S 9 1 A P P E N D I X I I P A R T I A L L I S T O F S Y M B O L S U S E D 9 6 v i LIST OF FIGURES Figure Page 1 Reference diagram showing the relationship between the d i r e c t i o n of the streaming motion of the nonthermal p a r t i c l e s and the propaga-tion of the e l e c t r o s t a t i c wave. 8 2 Frequency dependence of the e l e c t r o s t a t i c waves on the angle of propagation 8 near cu-^ , the angular frequency of the lower hybrid resonance. 32 3 Growth rate vs. p a r a l l e l wave number for • » 1. 33 4 Growth rate vs. p a r a l l e l wave number for B y = • 35 5 Growth rate vs. propagation angle 9 for 3 frequencies, 54 6 Maximum growth rate vs. frequency co .^ 56 7 Refractive index vs. propagation angle 9 for 3 frequencies. 58 8 X vs. propagation angle 8 for 3 frequencies. 59 9 Growth rate vs. propagation angle 6 for 3 values of <J>2 . 61 10 Growth rate vs. propagation angle 9 for 3 temperature r a t i o s . 63 11 Growth rate vs. propagation angle 9 for 3 temperature r a t i o s . 64 12 Growth rate vs, propagation angle 8 for 3 values of streaming v e l o c i t y V . 65 13 Growth rate vs. propagation angle 8 for a frequency just below the oxygen ion cyclotron frequency. 68 v i i LIST OF TABLES Table Page I Numerical values of ionospheric parameters at an alt i t u d e of 1000 km i n the auroral zone 14 II Numerical values of the cyclotron frequencies and the B-parameters at an alt i t u d e of 1000 km in the auroral zone 14 III Maximum growth rates and associated values 57 v i i i ACKNOWLEDGEMENTS I wish to thank Dr. T. Watanabe for his patient super-v i s i o n , guidance, encouragement, and for many hours of discus-sion during the progress of this work. I am indebted to him for many valuable suggestions, c r i t i c i s m s , and comments and for c a r e f u l l y reviewing the f i n a l manuscript. I also thank Dr. J. A. Jacobs and more recently Dr. R. D. Russell for encouragement and provision of an atmosphere conducive to study and research, I have benefited greatly by studying i n this environment. Many persons have contributed in one way or another to this work. Dr. N. M. Brice of Cornell University provided stimulating ideas giving impetus to this work. Mr. W. F. Mather of the Defence Research Telecommunications Establishment kindly supplied photographs of LHR noise observed by the Alouette I and II s a t e l l i t e s . Helpful discussions are acknowledged with my colleagues, i n p a r t i c u l a r , Messrs. K. R. Roxburgh, T. P. Ng, M. Berretta, R, W. Harvey, and A. J. Loveless. I also wish to express my appreciation to Miss Judi Kalmakoff for her careful typing of this manuscript. This study would not have been possible without per-sonal awards of a National Research Council of Canada Student-ship and a Defence Research Board of Canada Scholarship which are g r a t e f u l l y acknowledged; the l a t t e r had been accepted. CHAPTER 1 INTRODUCTION 1.1 Br i e f Background Many new types of signals were observed for the f i r s t time by instruments carried by rockets or s a t e l l i t e s into the ionosphere and above. The f i r s t observations of plasma reso-nances i n the ionosphere were made during rocket tests of the topside sounder technique (Knecht et a l . , 1961; Knecht and Russell, 1962). Similar as well as additional (Lockwood, 1963) resonances were detected with the Alouette topside sounder s a t e l l i t e which was launched on September 29, 1962 into a c i r c u l a r , polar orbit at a height of 1000 km. The sounder sweeps from 0.5 to 12.0 MHz in 18 sec. These resonances appear on ionograms as persistent responses at certain c h a r a c t e r i s t i c frequencies often l a s t i n g for several milliseconds. The charac-t e r i s t i c frequencies were found by Calvert and Goe (1963) to be: the electron plasma frequency n , multiples of the electron cyclotron frequency Q , the upper hybrid resonance frequency co^ and i t s harmonic 2o)yH . The resonance at the electron cyclotron frequency is not always observed and, when i t i s , i s much weaker than that at the second or t h i r d harmonic. Calvert and Goe (1963) suggested that the major resonances at the electron plasma frequency and the upper hybrid resonance frequency are caused by e l e c t r o s t a t i c o s c i l -l a t i o n of the ionospheric plasma along and across the earth's magnetic f i e l d respectively. The others he suggested were the r e s u l t of cyclotron resonances. Fejer and Calvert (1964) 2 s u b s e q u e n t l y e x t e n d e d a n d u n i f i e d t h e e x p l a n a t i o n o f r e s o n a n c e p h e n o m e n a , a t t r i b u t i n g a l l t h e o b s e r v e d r e s o n a n c e s t o e l e c -t r o s t a t i c o s c i l l a t i o n s o f t h e i o n o s p h e r i c p l a s m a . T h e y s t a t e d t h a t t h e r e s o n a n t f r e q u e n c i e s n a n d 9, c o r r e s p o n d t o o s c i l l a t i o n s a p p r o x i m a t e l y a l o n g t h e a m b i e n t m a g n e t i c f i e l d ; u) T T U , 2 ° . , 3 f i , e t c . , t o o s c i l l a t i o n s a c r o s s t h e f i e l d . Un e e ' I n o r d e r t o o b t a i n t h i s r e s u l t , t h e y a d a p t e d f o r t h e s p e c i a l c o n d i t i o n s o f t h e i o n o s p h e r e p r e v i o u s w o r k o n e l e c t r o s t a t i c o s c i l l a t i o n s ( L a n d a u , 1 9 4 6 ; B e r n s t e i n , 1 9 5 8 ) . O t h e r r e l e v a n t r e f e r e n c e s a r e : T o n k s a n d L a n g m u i r ( 1 9 2 9 ) , B o h m a n d G r o s s ( 1 9 4 9 ) , G r o s s ( 1 9 5 1 ) , a n d G o r d e y e v ( 1 9 5 2 ) . F e j e r a n d C a l v e r t a l s o s t a t e d t h a t t h e a b s e n c e o f r e s p o n s e s c o r r e s p o n d i n g t o i n t e r m e d i a t e a n g l e s i s a c c o u n t e d f o r b y t h e g r e a t e r g r o u p v e l o c i t y o f t h e e l e c t r o s t a t i c w a v e s , c a r r y i n g t h e e n e r g y a w a y m o r e r a p i d l y . T h e g r o u p v e l o c i t y i s n e a r z e r o f o r f r e q u e n c i e s n e a r t h e r e s o n a n c e s . S u b s e q u e n t l y C r a w f o r d e t a l . ( 1 9 6 7 ) m a d e s t u d i e s o f t h e g r o u p d e l a y o f a w a v e p a c k e t p r o p a g a t i n g b e t w e e n t w o a n t e n n a s . T h e o r e t i c a l p r e d i c t i o n s o f t h e d e l a y w e r e f o u n d t o a g r e e c l o s e l y w i t h l a b o r a t o r y p l a s m a m e a s u r e -m e n t s . T h e y r e p o r t e d e x p e r i m e n t a l r e s u l t s o n a l a b o r a t o r y p l a s m a s i m u l a t i n g t h e i o n o s p h e r i c r e s o n a n c e s . I n a d d i t i o n t o t h e t o p s i d e s o u n d e r , a b r o a d - b a n d v e r y - l o w - f r e q u e n c y ( V L F ) r e c e i v e r w a s i n c l u d e d i n t h e p a y l o a d o f t h e s a t e l l i t e . T h e r e c e i v e r c o v e r s t h e V L F b a n d f r o m 4 0 0 H z t o 1 0 k H z a n d i s c o n n e c t e d t o a 1 5 0 - f t . d i p o l e a n t e n n a . A n u n u s u a l t y p e o f s i g n a l d e t e c t e d b y t h e V L F r e c e i v e r i s b a n d s o f n o i s e c h a r a c t e r i z e d , b y a s h a r p l o w e r f r e q u e n c y c u t o f f w h i c h 3 usually increased with decreasing latitude of the s a t e l l i t e (Barrington and Belrose, 1963). The lower cutoff frequency ranges from about 5-10 kHz and the bandwidth is several kHz or less. Brice and Smith (1964) showed that many features of the noise band lead to the conclusion that i t s lower cut-o f f frequency is the lower hybrid resonance (LHR) frequency WLH ^ o r t^ i e a m ° i e n t plasma. Hence the name LHR noise bands. It w i l l be shown l a t e r that, generally, the ch a r a c t e r i s t i c s of LHR noise bands appear to be i d e n t i c a l to those of elec-t r o s t a t i c waves propagating nearly perpendicular to the earth's magnetic f i e l d . Since the appearance of LHR noise bands i s d i s t i n c t l y d i f f e r e n t at middle and northern l a t i t u d e s , the two types of noise have been c a l l e d midlatitude and polar LHR noise. Certain types of midlatitude LHR noise seem to be simi l a r to the resonance at the upper hybrid resonance fre-quency observed by the topside-sounder technique. Polar LHR noise bands have a diurnal v a r i a t i o n suggesting a r e l a t i o n -ship with electron p r e c i p i t a t i o n (McEwen and Barrington, 1967). Jj&rgensen (1968) has suggested that polar LHR noise is equivalent to auroral h i s s . He states that the hiss i s gener-ated i n the auroral regions of the magnetosphere and that the mechanism of generation is closely connected to very intense fluxes of electrons with energies of the order of several keV. 1.2 Purpose and Scope of Thesis The purpose of this thesis is to present a theory based on wave-particle interaction for the or i g i n of various \ 4 ionospheric noise which have e l e c t r o s t a t i c c h a r a c t e r i s t i c s and are related d i r e c t l y to p a r t i c l e p r e c i p i t a t i o n . The theory may be summarized i n the following two points. The f i r s t is that an e l e c t r o s t a t i c wave is excited in the ionosphere. The second is that a type of Landau i n s t a b i l i t y process due to a stream of higher-energy p a r t i c l e s is responsible for the exci-tation of the e l e c t r o s t a t i c waves. Emphasis is placed on electron p r e c i p i t a t i o n since this is observed frequently with ionospheric noise, especially in the auroral zone. The theory seems quite appropriate to explain the o r i g i n of polar LHR noise, the o r i g i n a l motivation for this work. The growth rates for the e l e c t r o s t a t i c waves are obtained by two methods. The f i r s t employs the dispersion equation for e l e c t r o s t a t i c waves (large r e f r a c t i v e index) while the second uses-the general dispersion equation which reduces to the former for large values of r e f r a c t i v e index. This provides a check on the work and also determines the e f f e c t of r e s t r i c t i n g one method to large values of r e f r a c t i v e index. The growth rate i s obtained d i r e c t l y from the elec-t r o s t a t i c dispersion equation. For the full-wave analysis, the growth rate is obtained by c a l c u l a t i n g the wave energy and the power transfer between the streaming p a r t i c l e s and the propagating wave. The second analysis is much more com-pl i c a t e d than the f i r s t and i t was expedient to use a computer to perform the numerical calculations based on the more com-plex expressions. Most of the growth rate calculations were 5 for frequencies just above . However, some calcula-tions were made at frequencies much greater than In addition, a few calculations were made just below the lowest ion gyrofrequency which i s much less than A short chapter provides more de t a i l s of LHR noise. It deals with the possible application of the theory presented herein to explain the generation of LHR noise as well as other ionospheric noise. The theory is a l i n e a r i z e d one based on the c o l l i s i o n -less Boltzmann equation. The e l e c t r o s t a t i c analysis is limited to those waves with wave, growth rates much smaller than their ' o s c i l l a t i o n frequencies. R e l a t i v i s t i c effects are. excluded. Although the analyses are quite general concerning the stream-ing p a r t i c l e s , only streaming electrons have been studied in d e t a i l . Streaming protons, for example, may also be i n v e s t i -gated but this study i s beyond the scope of this work. Also beyond the scope of this work i s a non-linear analysis. 6 CHAPTER 2 ELECTROSTATIC WAVE ANALYSIS 2.1 Landau I n s t a b i l i t i e s in E l e c t r o s t a t i c Waves Propagating  Nearly Perpendicular to the Background Magnetic F i e l d Lower hybrid resonance is known to take place in plasma waves propagating perpendicular to the background magnetic f i e l d . In a cold plasma which consists of elec-trons and only one po s i t i v e ion species, the lower hybrid resonance frequency i s given approximately by the f o l -lowing formula (see Eq. (2-17), p. 32 of the text by Stix (1962) which w i l l be referred to as Ref. I ) : W 2„ n 2 n °. L H i i e where rr. i s the ion plasma frequency and ft. and Q X X G • are the ion and electron cyclotron frequencies respectively. Of the two wave modes propagating perpendicular to the back-ground magnetic f i e l d , that which i s c a l l e d the extraordinary (X) wave (p. 18, Ref. I) shows resonance as the wave frequency CO approaches the lower hybrid resonance frequency from below; i . e . , the square of the re f r a c t i v e index, V 2 , tends to positi v e i n f i n i t y . Just above the lower hybrid resonance frequency, the X wave is an imaginary wave; V 2 is negative and tends to negative i n f i n i t y as the wave frequency TO approaches U^H from above. (See Eq. (2-15), p. 32, Ref. I. Note that 0 < TO2 < W2 < U>2 < I D 2 in that equation.) In 7 Ref. I (p.p. 223-224), i t i s shown that i f the r e f r a c t i v e index v takes a large value, the wave is e l e c t r o s t a t i c , i . e . , the e l e c t r i c f i e l d of the wave i s almost p a r a l l e l to the wave number vector k and the wave magnetic f i e l d is very small. Accordingly, the X wave is an e l e c t r o s t a t i c wave around the lower hybrid resonance frequency. As shown by the theory of Landau i n s t a b i l i t i e s (see e.g., p.p. 132-136, Refo I ) , an e l e c t r o s t a t i c wave grows i f there is a stream of charged p a r t i c l e s which flows i n the dir e c t i o n of wave propagation with a mean v e l o c i t y almost equal to but s l i g h t l y greater than the wave phase v e l o c i t y . In the case of an e l e c t r o s t a t i c wave propagating exactly perpendicular to the background magnetic f i e l d , i t cannot grow i f the streaming p a r t i c l e s flow i n the di r e c t i o n of the back-ground magnetic f i e l d as assumed in this thesis. However, the Landau i n s t a b i l i t y process can take place for an elec-t r o s t a t i c wave propagating o f f the perpendicular d i r e c t i o n ( i f such a wave exists at a l l ) . The condition for the Landau i n s t a b i l i t y process, viz that the mean ve l o c i t y of the streaming p a r t i c l e s i n the direc-tion of wave propagation i s almost equal to the wave phase v e l o c i t y , i s given as follows (see Fig. 1), - * V cos e (2-2) k s The wave vector k with magnitude k is taken i n the x-z 8 Fig. 1 . Reference diagram showing the relationship between the d i r e c t i o n of the streaming motion of the non-thermal p a r t i c l e s and the propagation of the elec-t r o s t a t i c wave. plane with the z axis taken p a r a l l e l to the background mag-netic f i e l d B . The angle between the wave number vector o k and the background magnetic f i e l d i s denoted by e and the mean vel o c i t y of the streaming p a r t i c l e s by V g . The phase v e l o c i t y of an e l e c t r o s t a t i c wave is r e l a t i v e l y small as the r e f r a c t i v e index is required to take a large value. The streaming p a r t i c l e s , however, might be of r e l a t i v e l y higher energy and therefore the above condition for the Landau i n -s t a b i l i t y process i s s a t i s f i e d only for smaller values of cos 6, i . e . , only for 6 near (but not equal to) 90°. Since the relevant mode of the plasma wave at 0 = 90° is electro-s t a t i c around the lower hybrid resonance frequency, waves with 6 near 90° are expected to be e l e c t r o s t a t i c for fre-9 quencies which are not too much d i f f e r e n t from the lower hybrid resonance frequency. Note in the CMA diagram (Fig. 2-1 of Ref. I) that the X mode suffers a destructive t r a n s i t i o n between regions 10 and 9 and between regions 3 and 2 but under-goes a reshaping t r a n s i t i o n between regions 8b and 11 when the lower hybrid resonance i s crossed (p. 37, Ref. I ) . It w i l l be shown i n this thesis that an e l e c t r o s t a t i c wave in regions 7 and 8 can grow due to the Landau i n s t a b i l i t y process. Since the X mode is also e l e c t r o s t a t i c for frequencies near the upper hybrid resonance frequency, i t is expected that the Landau i n s t a b i l i t y process might work for those waves as well and one might also observe upper hybrid resonance noise using an appropriate receiver. This noise band is expected to have a sharp upper frequency cutoff at the upper hybrid reso-nance frequency (in contrast to LHR noise) because resonance occurs in region 3 of the CMA diagram. 2.2 Basic Equations for Weakly Unstable E l e c t r o s t a t i c Waves  in Magneto-Active Plasmas In this section, the dispersion equation for elec-t r o s t a t i c waves in the upper ionosphere w i l l be derived. The background plasma i s assumed to consist of electrons and three species of posi t i v e ions, v i z , protons and singly charged helium and oxygen ions. Each of the four species of gases, the electron gas and the three ion gases, is assumed to have a Maxwellian v e l o c i t y d i s t r i b u t i o n and the same temperature, 10 denoted by T . The temperature of each gas i s assumed to be i s o t r o p i c ; the temperature in the d i r e c t i o n along the back-ground magnetic f i e l d is thus equal to that i n the d i r e c t i o n perpendicular to i t . In addition to these background gases, i t is assumed that there i s a gas of nonthermal charged par-t i c l e s of one kind which streams along the d i r e c t i o n of the background magnetic f i e l d . The mean streaming v e l o c i t y is denoted by V . The d i s t r i b u t i o n function for the stream-s ing p a r t i c l e s is also assumed to be Maxwellian with an iso-tropic temperature T g as viewed from a reference system moving with the streaming v e l o c i t y V g . The number density of each of the five species of gases, the four background gases plus the streaming p a r t i c l e s , is assumed to be uniform in space in the unperturbed state. C o l l i s i o n s between par-t i c l e s are ignored. The background magnetic f i e l d i s assumed to be uniform i n space, and i t s d i r e c t i o n i s taken along the z axis while the wave vector k is taken i n the x-z plane. The dispersion equation for e l e c t r o s t a t i c waves given in Ref. I (Eq. (9-103), p. 225) holds under conditions more general than in our case. From the conditions mentioned above and Eq. (8-34) of Ref. I (p. 178), the dispersion equation may be written as + Ttf ^ ZL I„ %> { ' - * 5 ("n )} 1 1 oO 0 0 /e) v i . • o 1 7T*- -\ T 4 T A \ ~/ e ) ' «6 <7> A °° (S) 1 TT>- A T 4 4 T /> \ ' / S > } (2-3) This i s the dispersion equation for plane e l e c t r o s t a t i c waves where the wave quantities are assumed to vary as exp i ( k x * + k zz - mt) with respect to time and space. The same notation as that i n Ref. I is used except for g , g. , and g , r- e -j s which are defined as follows, m m. m »J - — . B? - . 6 2 = - 2 - ( 2 - 4 ) E 2<T J 2KT S 2KT where K is Boltzmann's constant, and m , m. , and m ' e ' j ' s are the masses of the electrons, ions, and streaming p a r t i c l e s respectively. The symbols , , and a^s^ (n=0, ± 1 , ±2, ...) are defined as follows: • e ) - S- » . . °ie) S- S E ( 2 - 5 . 1 ) o k e n k e 12 r • >  f . s TO + nfi . o k J n k j z z (s) _ z s (s) _ z s s - o c o a Q Bs , a n es (2-5.3) z z The notation k z is for the z component of the wave vector k . The electron, ion, and streaming p a r t i c l e cyclotron f r e -quencies are taken posi t i v e and denoted by Q , ft. , and E J Qs ; the corresponding plasma frequencies are represented by II , n . , and n . The symbols A , X • , and A e 3 s e j s are defined as follows, A = H k 2 n - 2 B " 2 , A . = H k 2 n : 2 e : 2 , A = H k 2 f r 2 f r 2 (2-6) e a x e e ' j x j j ' s x s s v J where k i s the x component of the wave vector. The t h i r d , fourth, and f i f t h terms i n the dispersion equation represent the reactive terms due to the background electron gas, the background ion gases, and the streaming par-t i c l e s respectively. The sixth and seventh terms represent Landau damping (n = 0) and cyclotron damping (n f 0) by the background electron gas and the background ion gases. The l a s t term gives the Landau and cyclotron i n s t a b i l i t i e s or damping due to the streaming charged p a r t i c l e s . In order to simplify Eq. (2-3), the case i s con-sidered where the value of the parameter A is much smaller than 1 for every kind of p a r t i c l e . If A is much smaller 13 t h a n 1 f o r t h e r m a l oxygen i o n s ( 0 + ) , i t i s so f o r the o t h e r t h e r m a l p a r t i c l e s because X i s p r o p o r t i o n a l to the p a r t i c l e mass. N u m e r i c a l v a l u e s o f s e v e r a l i o n o s p h e r i c parameters i n the a u r o r a l zone are g i v e n i n T a b l e I . I f i t i s assumed t h a t the t e m p e r a t u r e i n the i o n o s p h e r i c r e g i o n o f i n t e r e s t i s 1000° K, v a l u e s o f the 3 p a r a m e t e r s f o r t h e r m a l p a r t i c l e s are as g i v e n i n T a b l e I I . The n u m e r i c a l v a l u e s o f the c y c l o t r o n f r e q u e n c i e s a r e a l s o g i v e n i n the t a b l e . In the c a l c u l a t i o n o f these c y c l o t r o n f r e q u e n c i e s , the s t r e n g t h o f the l o c a l m a g n e t i c f i e l d a t an a l t i t u d e o f 1000 km has been assumed to be 0.38 g a u s s . The c o r r e s p o n d i n g ground magnetic f i e l d s t r e n g t h i s 0.58 g a u s s , a t y p i c a l v a l u e i n the a u r o r a l zone. 9 From the n u m e r i c a l v a l u e s f o r and , i t can be shown t h a t the c o n d i t i o n A . < < I i s s a t i s f i e d i f the f o l -0 + l o w i n g c o n d i t i o n i s met, |k | << 3.18 x l O " 3 c m " 1 (2-7) For waves p r o p a g a t i n g n e a r l y p e r p e n d i c u l a r to the b a c k g r o u n d m a g n e t i c f i e l d , 2ir/|k | i s a lmost e q u a l to the w a v e l e n g t h , and the above c o n d i t i o n can be r e w r i t t e n as j t >> 19.7 m (2-8) E q . (9-101) o f R e f . I ( p . 224) can be used w i t h the c o l d plasma d i e l e c t r i c t e n s o r to g i v e an a p p r o x i m a t e v a l u e f o r the o p p o s i t e l i m i t f o r k . For f r e q u e n c i e s c l o s e to the LHR f r e q u e n c y 14 TABLE I N u m e r i c a l v a l u e s o f i o n o s p h e r i c p a r a m e t e r s a t an a l t i t u d e o f 1000 km i n the a u r o r a l zone. S t r e n g t h o f the e a r t h ' s m a g n e t i c f i e l d = B Q = 0.38 gauss Number d e n s i t y o f t h e r m a l e l e c t r o n s = N g = 10 1 1 c m " 3 Number d e n s i t y o f n o n t h e r m a l e l e c t r o n s = N = 1 c m " 3 Mean s t r e a m i n g v e l o c i t y o f n o n t h e r m a l e l e c t r o n s = V = 6 x 1 0 9 cm/sec E l e c t r o n p l a s m a f r e q u e n c y = n = 5.64 x 1 0 6 r a d / s e c T y p i c a l v a l u e o f LHR f r e q u e n c y = o>L^/2ir = 5 kHz T y p i c a l v a l u e o f the mean i o n plasma f r e q u e n c y = J^ns^j 4.11 x 1 0 4 r a d / s e c TABLE II N u m e r i c a l v a l u e s o f the c y c l o t r o n f r e q u e n c i e s and the B - p a r a m e t e r s a t an a l t i t u d e o f 1000 km i n the a u r o r a l zone. C y c l o t r o n f r e q u e n c y I n v e r s e o f the ( r a d / s e c ) mean t h e r m a l v e l o c i t y ( c m ~ 1 - s e c ) E l e c t r o n "e -• 6. 69 X 106 *e = 5 .74 X i o -B P r o t o n ( H + ) V = = 3. 64 X 1 0 3 B H + = 2 .46 X 10" 6 H e l i u m ( H e + ) fiHe+ " = 9. 15 X 1 0 2 6 H e + = 4 .92 X 10" e Oxygen ( 0 + ) V : = 2. 29 X 1 0 2 V = 9 .84 X 10" 6 15 (where the propagation angle i s near ir/2 ) the Stix cold plasma d i e l e c t r i c tensor component P dominates the others, S and D, and the r e f r a c t i v e index is given by v « k c/co , where c i s the speed of l i g h t . It can then be seen from Eq. (9-101) that v >> 180 fc>LH/w . This leads to |k | >> 1.88 x 10"4* cm - 1 (2-9) and Jt << 334 m (2-10) From the above figures i t can also be shown that the approxi-mate range for the r e f r a c t i v e index is given by 180 << vii)/coLH << 3030 (2-11) It is assumed that the streaming p a r t i c l e s are elec-trons with an average streaming v e l o c i t y V g = 6 x 10 9 cm/sec (corresponding to a k i n e t i c energy of 10 keV). This assump-tion i s made simply because electrons with energy of that order are known to be related to various geophysical phenomena. It i s not implied that the 10-keV energy is optimum for t r i g -gering the i n s t a b i l i t y process under consideration. Although this energy i s mildly r e l a t i v i s t i c , r e l a t i v i s t i c effects have been neglected i n this analysis. The r e l a t i v i s t i c results can be obtained to a f i r s t approximation by making approxi-mate changes i n the e f f e c t i v e mass of the streaming electrons. In our case we are dealing with the factor [1 - ( V g / c ) 2 ] ^ = 98%, 16 so t h a t the r e l a t i v i s t i c c o r r e c t i o n i s s m a l l . A l s o , o t h e r k i n d s o f s t r e a m i n g c h a r g e d p a r t i c l e s such as p r o t o n s are n o t e x c l u d e d . I f each s t r e a m i n g e l e c t r o n i s assumed to have a t h e r m a l energy comparable t o the k i n e t i c energy o f the s t r e a m -i n g m o t i o n (i.e. 10 keV) , the v a l u e o f $ g i s 2.04 x 1 0 " 1 0 c m " 1 - s e c . Then the p a r a m e t e r X g i s much s m a l l e r than 1 i f |k | << 1.93 x 1 0 " 3 c m - 1 (2-12) T h i s i s a p p r o x i m a t e l y s a t i s f i e d i f the c o n d i t i o n i n e q u a t i o n (2-7) i s s a t i s f i e d . The c o n t r i b u t i o n from the s t r e a m i n g e l e c t r o n s to the r e a c t i v e term i n the d i s p e r s i o n e q u a t i o n can be i g n o r e d s i n c e t h e i r number d e n s i t y i s much s m a l l e r t h a n t h a t o f the t h e r m a l e l e c t r o n s . In t h i s t h e s i s , the number d e n s i t y o f the t h e r m a l e l e c t r o n s i s assumed to be lO 1* c m " 3 , and t h a t o f the s t r e a m i n g e l e c t r o n s 1 c m " 3 , b e i n g comparable to t h a t i n the case o f a t y p i c a l a u r o r a l d i s p l a y . W i t h t h i s c h o i c e o f p a r t i c l e d e n s i t i e s , the f i f t h term i n e q u a t i o n (2-3) can be shown to be n e g l i g i b l e compared to the t h i r d t e r m . The f u n c t i o n I (X) i s a m o d i f i e d B e s s e l f u n c t i o n n (see p . 175, R e f . I ) . F o r X << 1 , the v a l u e s o f * n ( x ) d e c r e a s e v e r y r a p i d l y f o r i n c r e a s i n g ( i n a b s o l u t e v a l u e ) n , as can be seen from the f o l l o w i n g f o r m u l a (Watson, 1922; W h i t t a k e r and Watson, 1940; T e a r a z a w a , 1960): 17 v -T — — ^TTj m! (n+m) t .-vn+2m A (2-13) In equation (2-3), only those terms are retained which are second order or less i n A . Accordingly, the summations with re-spect to n should be taken up to |n| = 2 f e~) The parameter ) i s assumed to be much larger than 1 i n absolute value. Since we are interested i n frequencies higher than the proton cyclotron frequency and lower than the f el electron cyclotron frequency, |o.^  J\ for n ^ 0 is also much greater than 1 i f | ot^e-^ | >> 1 . Since the frequency range of interest i s about 5 - 1 0 kHz, the condition |a^e^|>> 1 is s a t i s f i e d i f |k | << 1.80 x 10" 3 cm"1 (2-14) The absolute value of the parameter for each ion species is also much greater than 1 i f inequality (2-14) is s a t i s f i e d . This i s because the value of aQ is proportional to the square root of the p a r t i c l e mass. Since |a | >> 1 , we expand the function s C a n ) into the following asymptotic series (equation (8-45), p. 180, Ref. I) S(« ) = — — + — + — — + ... (2-15) n 2 a 2-2 a 3 2 • 2 • 2 a•> n n n 1 8 Since to = to r + ico^ has an imaginary part which is much smaller (in absolute value) than i t s real part, so does Therefore, the function U in the above formula i n Ref. I is taken equal to zero. For thermal electrons, l 0 1 ^ 6 ^ ! > > * for any n as already mentioned, and the expansion i n Eq. (2-15) can always be used no matter what value n may take. For the (e) fe) frequency range of i n t e r e s t , the function (1 - 2a^ 1 S(a^ J ) ) is almost equal to 1 except when n = 0 , and accordingly i t is safe to ignore terms with [n| greater than 2 i n Eq. (2-3). On the other hand, for thermal ions, a r \ ^ with negative n decreases at f i r s t as |n| increases. For such an n s a t i s -fying to + nn. « 0 (2-16) r j a^J^ could be smaller than 1. Therefore, i t must be i n v e s t i -n ' gated c a r e f u l l y whether the ion terms with |n| greater than 2 i n Eq. (2-3) can be neglected or not. This point has been studied numerically, and i t has been found that those terms of higher order are n e g l i g i b l e . Since slowly growing or decaying waves are of i n -terest here (| to^ | << u>r) , we f i r s t consider only the re-active terms in Eq. (2-3), ignoring the reactive term due to the streaming p a r t i c l e s (the f i f t h term in Eq. (2-3)). Regard-ing 1/°^ a s a f i r s t - o r d e r term and retaining only those terms which are second-order or less in X and/or 1/a 2 , and then 19 regarding the damping or i n s t a b i l i t y terms (the la s t three terms in Eq. (2-3)) as a small perturbation and applying an i t e r a t i v e method, we obtain: 3KT TT _ .ij 3 K T ¥ and O ") r A(io r) u = - — 1 2 H(cor) (2-17) (2-18) where the function H(cor) is given by * 7" sjcT T T V f t . ^ - * f l > ) + 3KJ lU1 L (2-19) m + x a t 20 The summation n o t a t i o n i n E q s . (2-17) and (2-19) i n d i c a t e s t h a t c o n t r i b u t i o n s from a l l s p e c i e s o f t h e r m a l p a r t i c l e s be t a k e n . In our c a s e , t h e r e are f o u r terms - one due to t h e r m a l e l e c t r o n s and t h r e e to the t h r e e k i n d s o f t h e r m a l i o n s . The l a s t t h r e e terms i n E q . (2-3) are r e p r e s e n t e d by i A ( a i r ) where the argument u>r i n d i c a t e s t h a t the r e a l p a r t o f ID be t a k e n . F o r f r e q u e n c i e s which a r e much h i g h e r than the i o n c y c l o t r o n f r e q u e n c i e s but much lower than the e l e c t r o n c y c l o t r o n f r e q u e n c y , E q s . (2-17) and (2-19) may be a p p r o x i -mated as f o l l o w s , AL IOM J r tons " (2-20) and -e 2 1 fe e r i o n s J ( 2 - 2 1 ) l * ! 1 J - 7 " T T 1 I t can be seen t h a t H ( u > r ) i s always p o s i t i v e . The o n l y nega-t i v e term on the r i g h t - h a n d s i d e o f the above e q u a t i o n i s much s m a l l e r than the f i r s t term because o f the c o n d i t i o n A << 1 In E q , ( 2 - 2 0 ) , each o f the l a s t t h r e e terms i s p r o p o r t i o n a l to the t e m p e r a t u r e T and i s much s m a l l e r than the o t h e r t e r m s . I f t h e s e s m a l l terms are i g n o r e d , the d i s -p e r s i o n e q u a t i o n ( 2 - 2 0 ) can be s i m p l i f i e d as f o l l o w s , X 1^ 21 TT; + i\(X+Ltf) The terms w h i c h are p r o p o r t i o n a l to the t e m p e r a t u r e T are a l s o much s m a l l e r compared to the o t h e r terms i n E q . ( 2 - 2 1 ) and t h e r e f o r e the e x p r e s s i o n f o r the f u n c t i o n H ( k > r ) can be 22 s i m p l i f i e d to g i v e , 2„3 E l e c t r o n Plasma Waves P r o p a g a t i n g a l o n g the Background  M a g n e t i c F i e l d In the case o f l o n g i t u d i n a l plasma waves p r o p a g a t -i n g p a r a l l e l t o the b a c k g r o u n d m a g n e t i c f i e l d , the m a g n e t i c f i e l d has no e f f e c t on the m o t i o n o f c h a r g e d p a r t i c l e s , and t h e r e f o r e the s o l u t i o n o b t a i n e d i n the p r e v i o u s s e c t i o n s h o u l d reduce to t h a t f o r e l e c t r o s t a t i c waves i n the absence o f the b a c k g r o u n d m a g n e t i c f i e l d . L e t us assume t h a t the i o n s a r e i n f i n i t e l y heavy i n o r d e r to c o n s i d e r e l e c t r o s t a t i c waves i n an e l e c t r o n p l a s m a . Then the f o l l o w i n g r e s u l t from E q . (2-17) i s o b t a i n e d , co2 n 2 . _ r _ _e 3jc k 2 - k 2 in z z e S i n c e the s e c o n d term on the r i g h t - h a n d s i d e o f t h i s e q u a -t i o n i s much s m a l l e r than the f i r s t (|aQ 6^| > > 1) » <»T i n the s e c o n d term can be r e p l a c e d by n 2 , r e s u l t i n g i n the w e l l - k n o w n f o r m u l a f o r e l e c t r o n plasma o s c i l l a t i o n s (2-23) _e_ .2 (2-24) 23 ( s e e , f o r example, E q . (3-17) i n the t e x t by S p i t z e r (1962)) The f u n c t i o n d e f i n e d by E q . (2-19) becomes as f o l -lows i n the p r e s e n t c a s e , H ( O = k 2 + S £ i (2-25) m n ID": r e r The s e c o n d term on the r i g h t - h a n d s i d e i s much s m a l l e r than the f i r s t . The f r e q u e n c y u i s a l m o s t e q u a l to n as can be s e e n i n E q . ( 2 - 2 4 ) . The f u n c t i o n H (u>r) i s , t h e r e f o r e , a l m o s t e q u a l to k 2 . In the c a l c u l a t i o n o f A(ui ) , the c o n t r i b u t i o n from the i o n s (the s e v e n t h term i n E q . ( 2 - 3 ) ) i s i g n o r e d . In the s i x t h and l a s t t e r m s , i t can be s e e n t h a t o n l y the terms w i t h n = 0 are n o n z e r o . Then f i n a l l y the growth r a t e i s o b t a i n e d as f o l l o w s , (2-26) T h i s e x p r e s s i o n f o r the growth r a t e can a l s o be d e r i v e d from the f o r m u l a gj*ven i n R e f . I ( e q u a t i o n ( 1 4 ) , p . 136). In our c a s e , the e l e c t r o n d i s t r i b u t i o n f u n c t i o n f (v) i n R e f . I i s g i v e n by (2-27) 24 The plasma f r e q u e n c y n 2 i n the f o r m u l a i n R e f . I s h o u l d be r e p l a c e d by n 2 + n 2 i n our n o t a t i o n . W i t h t h e s e s u b -e s s t i t u t i o n s , i t can be seen t h a t the f o r m u l a g i v e n i n R e f . I l e a d s to the e x p r e s s i o n g i v e n i n E q . ( 2 - 2 6 ) . 2.4 E l e c t r o s t a t i c Waves E x c i t e d around the LHR F r e q u e n c y In the case o f p r o p a g a t i o n p e r p e n d i c u l a r to the b a c k g r o u n d m a g n e t i c f i e l d (k = 0) , the d i s p e r s i o n e q u a -t i o n f o r f r e q u e n c i e s which are much h i g h e r than the i o n c y c l o t r o n f r e q u e n c i e s b u t much lower than the e l e c t r o n c y -c l o t r o n f r e q u e n c y becomes, (2-28) The l a s t term i n t h i s e q u a t i o n i s much s m a l l e r t h a n the s e c o n d and t h i r d t e r m s . I f the l a s t term i s i g n o r e d , i t can be seen t h a t the f r e q u e n c y o)r d e t e r m i n e d from the above e q u a t i o n (or a l t e r n a t i v e l y from E q . ( 2 - 2 2 ) ) i s e q u a l to the p a r t i c u l a r f r e q u e n c y U J t h d e f i n e d as f o l l o w s , (2-29) T h i s e q u a t i o n g i v e s an a p p r o x i m a t e e x p r e s s i o n f o r the lower h y b r i d r e s o n a n c e f r e q u e n c y and can be r e w r i t t e n i n the form g i v e n by E q . (6) i n the p a p e r by B r i c e and Smith (1965) by 25 u s i n g the wave f r e q u e n c y f i n Hz i n s t e a d o f the a n g u l a r f r e q u e n c y u i r i n r a d / s e c . I f the l a s t term i n E q . (2-28) i s t a k e n i n t o a c c o u n t , the f r e q u e n c y u>r i s g i v e n a p p r o x i m a t e l y by u„ ^ w T I l + w* ( 2 - 3 0 ) ' r ~ LH where the c o r r e c t i o n term a' i s g i v e n by Zn. 1 U * t " ife; mj ^ (2-31) In the case when k z i s n o t z e r o b u t v e r y s m a l l , i . e . , the p r o p a g a t i o n i s s l i g h t l y o f f the p e r p e n d i c u l a r d i r e c -t i o n , the f r e q u e n c y u i s d e t e r m i n e d from E q . (2-20) and i s g i v e n a p p r o x i m a t e l y by u, flfc w + u>" (2-32) where the f r e q u e n c y w i s the s o l u t i o n o f E q . (2-28) (the case o f p e r p e n d i c u l a r p r o p a g a t i o n ) and i s a p p r o x i m a t e l y e q u a l to the r i g h t - h a n d s i d e o f E q . ( 2 - 3 0 ) . The c o r r e c t i o n term o i " i s g i v e n by w - .... • i t n S J (2-33) 26 The right-hand side of Eq„ (2-33) is seen to be always p o s i t i v e . Inside the bracket of the second term in the numerator, the l a s t term i s much smaller than the f i r s t term, n 2 , because of the condition A„ << 1 . The second ' e ' e term is also much smaller than the f i r s t term. Since n e , the electron plasma frequency, i s much higher than the lower hybrid resonance frequency u > ^ , the second term in the numerator which is approximately equal to ne/'aiLH * S m u c ^ greater than the f i r s t term, -1. Equation (2-33) can be considerably s i m p l i f i e d by ignoring the smaller terms which are proportional to the temperature T , giving the follow-ing r e s u l t , (2-34) This equation can also be derived from Eq. (2-22) on the following condition, c o t 2 e << ( £ _ n?)/n 2 (2-35) ions In our case, the values i n Table I lead to 89° 35• < 6 < 90° 25'. Since w" is positive for k_ + 0 , the frequency w r is greater than u Q ( * o> L H ) for off-perpendicular propa-gation (e f j) , and tends to u>0 as the d i r e c t i o n of propagation approaches the perpendicular d i r e c t i o n (e -»• j) The frequency u increases very rapidly as the propagation 27 angle 0 d e v i a t e s from the p e r p e n d i c u l a r d i r e c t i o n , s i n c e n 2 i s much l a r g e r than ions -1 In Eq. (2-18) which gives the growth r a t e , the f u n c t i o n H (io r) i s always p o s i t i v e , and t h e r e f o r e A(cur) should be ne g a t i v e i n order that the growth r a t e OK be p o s i t i v e . As mentioned a l r e a d y , the f u n c t i o n A(o) r) con^ s i s t s of three p a r t s , A 1(to r) , A 2(o> r) , and A 3(o) r) which correspond to the seventh, e i g h t h and n i n t h terms i n Eq. (2-3) r e s p e c t i v e l y . The f u n c t i o n s A 1(co r) and A 2(u» r) are always p o s i t i v e and only the f u n c t i o n A 3(u> r) , the c o n t r i b u t i o n from the streaming p a r t i c l e s , can be n e g a t i v e . The f u n c t i o n A 3(o> r) i s given by oO (2-36) As can be seen from the e x p o n e n t i a l f a c t o r ( i n s i d e the summa-t i o n s i g n ) , a r e p r e s e n t a t i v e term i s very s m a l l unless the f o l l o w i n g c o n d i t i o n i s s a t i s f i e d , u - k V + nft « 0 (2.-37). r z s s For n = 0 , t h i s becomes o) - k V « 0 r z s (2-38) 28 which gives the same condition as Eq. (2-2). For frequencies near the lower hybrid resonance frequency TO^ , the value of k z which s a t i s f i e s this condition is approximately given by k « 5,24 x l O " 6 cm - 1 (2-39) i f we take co /2TT «ioT„/2Tr = 5 kHz and V = 6 x 10 9 cm/sec. X* Li r i S The value of k z i s within the l i m i t imposed by inequality (2-14). For n = 1 , the value of k z to s a t i s f y the con-d i t i o n in Eq. (2-37) is about 1.12 x 10" 3 cm - 1 which is barely within the l i m i t given by inequality (2-14). For |n| = 2 , there are no values of k within the l i m i t set by condition z (2-14). In order that the e l e c t r o s t a t i c wave take a frequency near the lower hybrid resonance frequency, k 2 must be much larger than k 2 as seen from Eq. (2-34). For a given value of k 2 , k 2 is smaller for a higher wave frequency. For a frequency about twice as large as the lower hybrid resonance frequency ( v i z , tor » 2to L H ; such a frequency may even be larger than the upper frequency l i m i t of an LHR noise band), the value of Ik I i s about 6.44 x 10" k cm - 1 for k •» ' x 1 z 1.05 x 10" 5 cm"1 (the case for n=0) and 7.00 x 10" 2 cm"1 for k « 1.14 x 10' 3 cm"1 (the case for n=l). The value of k x in the l a t t e r case is far outside the l i m i t imposed by inequality (2-7). Therefore, only the term with n=0 is important in the function £3(10^) 29 U s i n g the c o n d i t i o n t h a t A g << 1 and r e t a i n i n g the n=0 term o n l y , we o b t a i n (2-40) owth r a t e be p t ive) , the f o l l o w i n g c o n d i t i o n must be met, In o r d e r t h a t A 3 ( u ^ ) be n e g a t i v e (the gro t o s i « r " k z V s < 0 (2-41) E q u a t i o n (2-38) and the above i n e q u a l i t y show t h a t to cause the i n s t a b i l i t y p r o c e s s , the p r o j e c t i o n o f the s t r e a m i n g v e l o c i t y o f the n o n t h e r m a l p a r t i c l e s i n the d i r e c t i o n o f wave p r o p a g a t i o n s h o u l d be i n the same sense as the wave phase v e l o c i t y ( i . e . , V and w /k s h o u l d have the same s i g n ) s r z and the p r o j e c t e d s t r e a m i n g v e l o c i t y V cos 6 s h o u l d be a l m o s t e q u a l to but s l i g h t l y g r e a t e r than the wave phase v e l o c i t y . From e x p r e s s i o n (2-23) f o r the f u n c t i o n H ( c o r ) and E q . ( 2 - 4 0 ) , the growth r a t e i s a p p r o x i m a t e l y g i v e n by •*S * (2-42) 30 The inequality sign is used since the contributions from AiCoij.) and A 2(o) r) to o)^ are negative. As far as fr e -quencies near the lower hybrid resonance frequency are con-cerned, A 1(o) r) , the contribution from the thermal elec-trons, may be unimportant since the lower hybrid resonance frequency i s much smaller than the electron cyclotron fre-quency. The quantity A^w ) is s i g n i f i c a n t only for a frequency oi r that is near the electron cyclotron frequency or i t s harmonics. However, the quantity A 2 ( w r ) , the con-t r i b u t i o n from the thermal ions, could be s i g n i f i c a n t since the frequency u could take a value at a certain harmonic of an ion cyclotron frequency. Generally, the contribution is riot s i g n i f i c a n t . If the temperature of each thermal back-ground gas i s assumed to be zero, both Aj (oO and A 2 ( w r ) are zero, and the inequality sign can be dropped in expression For the purpose of numerical c a l c u l a t i o n , i t is convenient to rewrite expression (2-42) i n a dimensionless hybrid resonance frequency. The unit for wave numbers i s defined as follows, (2-42). form. The unit for frequencies is taken as OJ LH the lower (2-43) 31 Expression (2-42) can then be rewritten as o + | : ) ^ 4 i KI K;I < (2-44) where o>r = <«)r/w^pj tu^ = ^ / ^ L H ( 2 - 4 5 . 1 ) ne = ne / u )LH ' "e = ne / u )LH ' ns = n s / w L H ( 2 - 4 5 . 2 ) and k' = k /k k = k /k (2-46) X X o z z' o and the dimensionless quantity ^ s ^ s ftas been expressed in terms of a parameter <f>2 which is the ra t i o of the ki n e t i c energy i n the streaming motion to the thermal energy of the streaming electrons, &2sV2s = 3/2 * 2 (2 -47) and t a n 2 9 = k 2/k 2 i s obtained from Eq. ( 2 - 2 2 ) . From this equation, the relationship between the frequency u/ and the angle 9 has been graphed in Fig. 2 for our case. D i f f e r e n t i a t i o n of Eq. (2-44) with respect to shows that for given values of frequency it/ and <|> , the maxi-mum and minimum growth rates occur when k' s a t i s f i e s 15 10 32 85 86 87 88 89 90 6 (degrees) Fig. 2. Frequency dependence of the e l e c t r o s t a t i c waves on the angle of propagation 6 near <»>,„, the angular frequency of the lower hybrid resonance. The dimen-sionless frequency i s defined by <D* = to /ID,„ , where OO is the re a l part of the Complex angular frequency" OJ Calculations were made for where the values of the constants used are those given in Section 2-2. Figure 3 shows the growth rate U J | plotted as a function of the p a r a l l e l component of the wave number vector, k' , for three values of frequency a/ . The value of the parameter <t> is unity, that i s , the k i n e t i c energy in the streaming motion and the thermal energy of the streaming p a r t i c l e s both equal 10 keV. Note the change i n scale of u>! for posi t i v e and negative k' . It 1 z is clear from the curves that damping occurs (o^ < 0) when k < oi . I n this damping region, O J . has two minima and z r x t t i passes through zero at k = 0 . For k > u> , the growth z z r rate increases to a maximum and then decreases rapidly. A F i g . 3. Growth rate vs. p a r a l l e l wave number for <J> = 1 Note the change i n scale for k'z < 0 . See Eqs. (2-43), (2-45), and (2-46) for d e f i n i t i o n s of the dimensionless co-ordinates. The parameter <f>2 i s the r a t i o of the k i n e t i c energy i n the streaming motion to the thermal energy of the streaming electrons. 34 general feature is the increasing magnitude of the maximum and the minima values of 0 / for increasing frequency. Figure 4 i s a similar plot of the growth rate versus except that the parameter <|> i s varied with a/ = 1.1 . For negative k z , the magnitude of u>^  diminishes rapidly with increasing $ . In the region of posit i v e k , i t i s clear that the larger the value of , the larger is the maximum value of w! and the narrower the wave number 1 bandwidth. When <j> = 1 , the maximum value of is 8.04 x 10" 6 at k' = 1.523 . Then from Eq. (2-22) i s obtained k' = 568 . This gives k = 2.98 x IO" 3 cm - 1 which i s just within the l i m i t imposed by inequality (2-7) but just outside that given by inequality (2-12). The upper bound of the maxi-mum growth rate is 0.253 sec" 1 and the growth time (the e-folding time of the wave amplitude) i s about 3.96 sec. When di2 = 10 , the maximum value of to! i s 1.53 x 10 - l t at k' = 1.290. 1 z Also, k = 2.53 x 10" 3 cm - 1 and the maximum growth rate is 4.81 sec" 1, about twenty times greater than i n the previous case with $ = 1 . F i n a l l y , when <j>2 = 100 , the following values are obtained: the maximum value of I«K =1.95 x 10" 3 , k z = 1.163 , k x = 2.28 x 10" 3 em - 1 , and a maximum growth rate of 61.3 sec" 1. For <j> f 1 , equation (2-12) must be modified to |k I << 1.93 x IO" 3 <j> . Thus, when .<j>2 i s 10 or 100, the value of k^ is within the l i m i t imposed by both in e q u a l i t i e s (2-7) and (2-12), where the l a t t e r has been appro-p r i a t e l y modified. -4 -3 -2 t o l C x l O " 6 ) TO - 2 4 5 0 F i g . 4. Growth r a t e v s . p a r a l l e l wave number f o r U>R = 1.1 Note change i n s c a l e f o r n e g a t i v e v a l u e s o f k . Curves are not drawn f o r <j>2= 10 and 100 when k z < 0 s i n c e v a l u e s o f TO!^  are n e g l i g i b l e h e r e , 04 tn 36 The above c a l c u l a t i o n s show t h a t the growth t imes f o r f r e q u e n c i e s n e a r u>L^  are i n the o r d e r o f 1 - 1 0 " 2 sec f o r v a l u e s between 1 and 100 o f <l>2 , the r a t i o o f the stream-i n g k i n e t i c energy to the t h e r m a l energy o f the s t r e a m i n g e l e c t r o n s . I f the v a l u e o f < r 2 i s much g r e a t e r than u n i t y , t h e n the growth time i s v e r y s m a l l . From e q u a t i o n s (2-34) and (2-42) and F i g . 3, i t can be s e e n t h a t as the wave f r e q u e n c y ccr tends to u i ^ , the lower h y b r i d r e s o n a n c e f r e q u e n c y , the growth r a t e co^  tends to z e r o . A l s o , from E q . (2-22) can be o b t a i n e d the upper l i m i t i n g v a l u e o f it/ f o r e l e c t r o s t a t i c waves. The f r e q u e n c y u>r i n c r e a s e s m o n o t o n i c a l l y w i t h d e c r e a s i n g 6 , g i v i n g a t 9 = 0 , ai 2 = n 2 + / n? so t h a t we o b t a i n ' r e iSrTs J u>2 = a)2 = n| + n? (2-49.) i o n s f o r the f r e q u e n c y range o f the e l e c t r o s t a t i c waves. The m a x i -mum growth r a t e g i v e n by E q s . (2-44) and (2-48) i n c r e a s e s w i t h f r e q u e n c y u>r and n e a r the upper l i m i t a t 8 = 0 t a k e s a v a l u e o f the o r d e r o f a* 1 0 " 2 f o r V * - 1 . I t s h o u l d be m e n t i o n e d t h a t i n the d e r i v a t i o n o f t h e s e e q u a t i o n s i t had been assumed t h a t U>2 << n 2 , so t h a t c a r e must be t a k e n r e i f n i s o f the same o r d e r as n (as i t i s i n our c a s e ) . 6 6 A b a s i c l i m i t a t i o n to the v a l i d i t y o f the growth r a t e e x p r e s s i o n g i v e n by E q . (2-44) i s t h a t the r e f r a c t i v e i n d e x must be l a r g e . But E q s . (2-22) (or ( 2 - 3 4 ) ) and F i g . 2 37 show that the wave frequency U> becomes smaller as the propa-gation angle e i s larger. To meet the Landau i n s t a b i l i t y condition (Eq. (2-2)), therefore, the wave phase vel o c i t y should be larger (the re f r a c t i v e index V smaller) for a higher wave frequency U>R (holding the streaming ve l o c i t y V"s constant). If the r e f r a c t i v e index V becomes small enough, the wave may no longer be considered e l e c t r o s t a t i c . 38 CHAPTER 3 FULL-WAVE ANALYSIS 3.1 Power A b s o r p t i o n i n a C o l l i s i o n l e s s Plasma The p r e v i o u s c h a p t e r has i n d i c a t e d t h a t an e x t e n s i o n i n the a n a l y s i s would be u s e f u l where no r e s t r i c t i o n s are p l a c e d on the r e f r a c t i v e i n d e x v . T h i s e x t e n s i o n may be made by e m p l o y i n g the g e n e r a l i z e d d i s p e r s i o n e q u a t i o n i n s t e a d o f the e l e c t r o s t a t i c one. T h i s g e n e r a l i z e d e q u a t i o n i s t h e n used to o b t a i n the wave e l e c t r i c f i e l d w h i c h i s then i n s e r t e d i n t o the growth r a t e e x p r e s s i o n ; the growth r a t e i s a f u n c t i o n o f the*-power t r a n s f e r and the wave e n e r g y . The r e s t r i c t i o n s i n the e l e c t r o s t a t i c a n a l y s i s t h a t X and l/<* n are much l e s s t h a n u n i t y are removed i n the f u l l - w a v e a n a l y s i s . The a s s u m p t i o n s a r e s i m i l a r to those made i n the p r e v i o u s c h a p t e r . A tenuous s t r e a m o f p a r t i c l e s impinges upon a b a c k g r o u n d p l a s m a . A l l s p e c i e s o f p a r t i c l e s have a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n , a l b e i t w i t h v a r i o u s p a r a l l e l and p e r p e n d i c u l a r t e m p e r a t u r e s , T„ and Tj_ , and a p a r a l l e l d r i f t v e l o c i t y V g i n the case o f the s t r e a m i n g p a r t i c l e s . As b e f o r e , the number d e n s i t i e s o f the b a c k g r o u n d e l e c t r o n s and p o s i t i v e i o n s p l u s the s t r e a m i n g p a r t i c l e s a r e assumed t o be u n i f o r m i n space i n the u n p e r t u r b e d s t a t e ; the b a c k g r o u n d m a g n e t i c f i e l d B Q i s assumed u n i f o r m i n space a l s o . The power a b s o r p t i o n by the p a r t i c l e s p e r u n i t volume P i s found by an e x p r e s s i o n i n R e f , I ( E q . ( 9 - 5 4 ) , p . 2 0 5 ) , 39 where c i s the v e l o c i t y o f l i g h t , i s the number d e n s i t y o f p a r t i c l e s w i t h c h a r g e o f magnitude Z^e , and i s the s i g n o f the c h a r g e , ±1 . M i s the m o b i l i t y t e n s o r and E i s the wave e l e c t r i c f i e l d ; p a r a m e t e r s a p p r o p r i a t e to p a r t i c l e s o f the kth type are to be used i n e v a l u a t i n g M . The summa-t i o n i s over a l l p l a s m a components and the s t r e a m i n g p a r t i c l e s . E q u a t i o n (3-1) may be expanded and e x p r e s s e d i n terms o f the components o f the wave e l e c t r i c f i e l d and m o b i l i t y t e n s o r a s , 00 00 n= -°° n= - 0 0 + | E y|2 ( M ^ + cc) + | E Z | 2 (Unzz * cc) + (E E - E E * ) ( M n - c c ) y x x y X Y + (E E * + E * E ) ( M n + cc) v X Z X ZJ v xz J + (E E * - E * E ) ( M n - cc)' I (3-2) where the a s t e r i s k denotes the complex c o n j u g a t e and c c denotes the complex c o n j u g a t e o f the companion term w i t h i n the b r a c k e t s . The s u b s c r i p t k has been o m i t t e d f o r s i m -p l i c i t y o f n o t a t i o n . The components o f the t e n s o r M n are 40 given by (see Eq. (9-8), p. 188, Ref. I ) , M xx Mn yy fi e KTI £ A x x <©>„ 2mk z fieicTi .n 2mk z _ yy ^W-^ 2 k A z z ^ <vz *>n - <vz V ^ z i n < T , *y 2mk *y ^< e<Ti k . _ , M x z — Axz < ^ ® > xz 2 m k XZ M ' n i KTI k M n = . 1-x A n s Q\ yz 2mk ^ z M n = - M yx n •xy Mn = Mn ZX xz M n 'zy M n 'yz where fi is the cyclotron frequency, m the mass, k and 41 k z the x - and z-components o f the wave v e c t o r k , K the Boltzmann c o n s t a n t , and v the p a r t i c l e v e l o c i t y component a l o n g the z - d i r e c t i o n „ We have d e f i n e d A A y y - 2 e " X [ ( ^ i + 2 x ) I n - 2 A i ; ] (3-4) A* = -2 e - A I zz n A n - -2 e " A n [ I - I»] xy 1 n n J An = -2 e " X 2. I xz x n A n „ - 2 e " X [ I - I»] yz 1 n n J where I and I 1 a r e the m o d i f i e d B e s s e l f u n c t i o n o f the n n f i r s t k i n d and i t s d e r i v a t i v e . The f o r m u l a f o r I n U ) n a s a l r e a d y been g i v e n i n E q . (2-13). S i n c e , i n t h i s s e c t i o n , we a l l o w a r b i t r a r y x , i t i s u s e f u l to have the a s y m p t o t i c e x p a n s i o n f o r i n ( * ) • When -3tr/2 < a r g X < TT/2 we have ( p . 373, W h i t t a k e r and Watson, 1940) e I M . , 7 / ^ [ 4 n2 - l 2 ] [ 4 n 2 - 3 T - > [ 4 n a - ( 2 r - l f ] 42 + - (n+i)lfi - A e _ e I + r = L > n 2 - 1&] [4n 2 -3 z ]"- frn*-r ! r a * ) r (3-5) where the second series i s neg l i g i b l e when |arg x| < ir/2 XH. the computer calculations of Sec. 3,3 a subroutine (BESIK) provided by the UBC Computing Centre was used to evaluate I n U ) • For values of X greater than 88.029692 this sub-routine cannot be used but another subroutine can be written based on the asymptotic formula of Eq. (3-5). It was not found necessary to write this second subroutine at this time. Some recurrence relations for the modified Bessel function are: : n - l - ln+l - F 1 (3-6) X I„(X) + n I„(X) = X I ^ . U ) n n (3-7) The general expression for the bracketed factors i n the mobility tensor components i n Eq. (3-3) are given i n Ref. I (Eqs. (9-12) and (9-13), p.p. 190-191). We are interested here i n some of the relations which may be reduced after some algebraic manipulation: <&> + cc = - ±. (JUL. \2 « \K \ 2 K V (3-8) 43 l_ J. J (3-10) where V"s i s the m a c r o s c o p i c s t r e a m i n g v e l o c i t y i n the z -d i r e c t i o n . I t can be e a s i l y shown t h a t E q s . ( 3 - 3 ) , ( 3 - 8 ) , ( 3 - 9 ) , and (3-10) l e a d t o : Mn + cc = C A n XX XX Mn + cc yy = C A n, "yy Mn + cc zz M n - cc xy = - i e C A n xy (3-11) Mn + cc xz C A n x to + hp, xz k fi z M n - cc = i E C A yz n _x to '+' rifi yz k 0 z 44 where we have d e f i n e d f o r c o n v e n i e n c e , (3-12) * I n s e r t i n g the e x p r e s s i o n s (3-11) and (3-12) i n t o E q . (3-2) l e a d s to a s i m p l e e x p r e s s i o n f o r the power a b s o r p t i o n p e r u n i t volume by p a r t i c l e s o f the Yth s p e c i e s . I' P k n - M n e x p ( - a * ) (3-13) where (3-14) and z^-z.tj - A " A - w + u l l ( 3 . 1 5 ) 45 Thus, summing P^ n over n and k y i e l d s the t o t a l power a b s o r p t i o n per u n i t volume by a l l charged p a r t i c l e s , P & 3.2 Wave Energy and E l e c t r i c F i e l d The energy W of the wave may be obtained from Eq. (3-7) of Ref. I (p. 48) W = - I - [ B * . "B + E* • — [wK, ]•"£] (3-16) 16ir au where B and E are the complex F o u r i e r amplitudes of the wave magnetic and e l e c t r i c f i e l d s and i s the Hermitian pa r t of the d i e l e c t r i c t e n s o r . The magnetic f i e l d components may be expressed i n terms of the e l e c t r i c f i e l d components by the Maxwell r e l a t i o n V x E = - — — (3-17) c 3t which becomes a f t e r F o u r i e r a n a l y s i s i n space and time i k x f = i - B (3-18) c Thus we have ^ B = - k E n x z y 46 SL B c -\ k E z x k E X z (3-19) ^ B c z k E x y since we have chosen a co-ordinate system such that k y = 0 , without loss of generality. Therefore the f i r s t term i n the brackets of Eq. (3-16) becomes B*-B = v2|E x|2 + v 2|E | 2 + v 2 | E z | 2 - v x v z ( E * E x + E ^ J ) (3-20) where v and v are the x- and z-components of the refrac-x z ^ _^  tive index v = kc/u The expressions for the power absorption P & and the wave energy W may be evaluated once the wave e l e c t r i c f i e l d components are determined. These components may be obtained from Eq. (9-18) of Ref. I (p. 193), v 2 + K Z XX K xy x z ^x v 2 - v 2 + K x z yy v v + K „ x z xz K K zy yz v 2 + K X zz = 0 (3-which i s the matrix form, with k = 0 , of the r e l a t i o n y 47 v x(v x E) + K-E = 0 (3-22) where K i s the d i e l e c t r i c tensor which, i n t u r n , i s given by K = 1 + i L M ( k ) (3-23) where 1 i s the u n i t dyad and the m o b i l i t y tensor has alre a d y been given i n Eqs. (3-3). From Eq. (3-21) we may o b t a i n f o r the e l e c t r i c f i e l d component r a t i o s , E fv v + K )K - K (-v 2 + K . ) . . . . y x z xzJ yx yzK z xxJ (3-24) E K K - (v v + K ) (-v 2 - v 2 + K ) x yz xy x z x z ' v x z yy' E K K - (-v2 - v 2 + K ) (-v2 .+. K ) JL = x y yx K x z yyiK z xxJ ( 3 - 2 5 0 E (-v 2 - v 2 + K ) ( v v + K ) - K K x K x z y y n x z xz' xy yz These r a t i o s may be used to express the power a b s o r p t i o n P and the wave energy W i n terms of |E | 2 . Then i s obtained the r a t i o PQ/W which i s p r o p o r t i o n a l to the growth r a t e TO. Equation (3-1) i n d i c a t e s t h a t c o n t r i b u t i o n s are made to P^ n from a l l the plasma components as w e l l as the tenuous stream of p a r t i c l e s . We may s i m p l i f y c o n s i d e r a b l y the a n a l y s i s by assuming a c o l d plasma so that c o n t r i b u t i o n s are made to 48 P k n from only the streaming p a r t i c l e s . In a d d i t i o n , simple expressions may be obtained f o r the e l e c t r i c f i e l d components ^from the c o l d plasma form of Eq. (3-21) s - «i iD v v X z - iD S - v 2 X z p - V 2 = 0 (3-26) where 5 = R = L-j (R -»-L) , D=^(R-L) Ui ( *» ) P- I -ZL TT: (3-27) (3-28) (3-29) (3-30) These exp r e s s i o n s are E iD v v X z (3-31) (3-32) 49 P u t t i n g t h e s e v a l u e s i n t o G n i n Eq. (3-15) g i v e s the power a b s o r p t i o n P^ as a f u n c t i o n o f wave f r e q u e n c y u , propa-g a t i o n angle 6 , and the p h y s i c a l parameters o f the plasma and the s t r e a m i n g p a r t i c l e s . Some a l g e b r a i c m a n i p u l a t i o n y i e l d s n where (3-33) H n vz-5 A + v^e -PR L n = 2 e~* In (3-35) (3-36) I t s h o u l d be borne i n mind t h a t X , e , 9. are parameters o f the s t r e a m i n g p a r t i c l e s which alone c o n t r i b u t e t o the power a b s o r p t i o n . The v a l u e s o f S , D , R , L , and P are f u n c t i o n s o f the wave f r e q u e n c y and the c o l d plasma con-s t i t u e n t s o n l y . The wave energy W a l s o t a k e s a s i m p l e form when a c o l d plasma i s assumed. The second term o f Eq. (3-16) can be ex p r e s s e d as: E * . ^ _ ( a ) K h ) -E = E*-T-E 8 co = IE | 2T + IE | 2T + IE | 2T 1 x 1 xx 1 y 1 yy 1 z 1 zz 50 + (E E* - E E*)T + (E E* + E*E )T y x x y ' x y K x z x zJ xz + (E E* - E*E )T z y z y y yz (3-37) In the case o f a c o l d plasma, the e x p r e s s i o n f o r the tensor T may be found to be, T . = fcf«pj = i + Z 1 XX *7 (3-38) where, again, the summation over k excludes the tenuous beam of p a r t i c l e s . From Eqs. (3-16), (3-20), (3-37) and (3-38) can be obtained the e x p r e s s i o n f o r the wave energy W , 16 K W - V X + (4^fi -|)eos*<? + 1 7 ^ - 5 (3-39) 51 3.3 Growth Rate and Computer R e s u l t s w i t h Emphasis n e a r the  LHR F r e q u e n c y From the r e s u l t s o f S e c t i o n s 3.1 and 3.2 may be o b t a i n e d the wave growth r a t e by computing the energy b a l a n c e . T h i s i s a c c o m p l i s h e d by e q u a t i n g the r a t e o f i n c r e a s e i n energy f o r the p a r t i c l e s to the r a t e o f d e c r e a s e i n energy i n the plasma wave, ^ = 2 u. W = - P a (3-40) dt 1 a T h e r e f o r e the growth r a t e i s g i v e n by P co. = -h (3-41) 1 W T h i s e x p r e s s i o n was programmed f o r computer c a l c u l a t i o n (see A p p e n d i x I) where the r e l e v a n t i o n o s p h e r i c p a r a m e t e r s g i v e n i n T a b l e I were u s e d . S i n c e t e m p e r a t u r e a n i s o t r o p y has been i n t r o d u c e d , the p a r a m e t e r s f o r the s t r e a m i n g p a r t i c l e s X g f s i and aK J become n to - k_V_ + nfi. O J - . a (3-43) (s) z s s_ K z where B 2 = — ( 3 - 4 4 ) 2KT„ 52 and (3„ , T | ( and Tj^  r e f e r to the s t r e a m i n g p a r t i c l e s . F i n a l l y , the s o l u t i o n f o r the d i s p e r s i o n e q u a t i o n f o r the wave p r o p a g a t i n g t h r o u g h the background c o l d plasma i s g i v e n i n R e f . I ( E q . ( 1 - 2 6 ) , p . 1 2 ) ; we are i n t e r e s t e d i n the w h i s t l e r -mode s o l u t i o n whose wave-normal s u r f a c e i s a dumbbel l l e m n i s c o i d . T h i s s o l u t i o n i s v 2 „ E ' - F (3-45) 2A where A = S s i n 2 e + P c o s 2 e (3-46) B» = RL s i n 2 9 + PS (1 + c o s 2 e ) (3-47) F 2 = (RL - P S ) 2 s i n 4 6 + 4 P 2 D 2 c o s 2 e (3-48) These e q u a t i o n s complete the s e t r e q u i r e d to p e r f o r m the computer c a l c u l a t i o n . The b a c k g r o u n d plasma i s assumed t o c o n s i s t o f e l e c t r o n s and s i n g l y - c h a r g e d p o s i t i v e i o n s o f h y d r o g e n , h e l i u m , and oxygen. The f r a c t i o n a l abundance o f oxygen i o n s i s assumed to be 90%, a r e a s o n a b l e v a l u e i n the p o l a r i o n o s p h e r e ( B a r r i n g t o n et a l . , 1965; B a r r i n g t o n and McEwen, 1967). T h i s v a l u e and the e l e c t r o n d e n s i t y g i v e n i n T a b l e I l e a d t o number d e n s i t i e s o f 2.05 x 1 0 2 and 7.95 x 1 0 2 c m " 3 f o r the H + and H + i o n s because o f c h a r g e n e u t r a l i t y and S = 0 at £ L H = 5 kHz , The s t r e a m i n g p a r t i c l e s , as b e f o r e , are t a k e n to be e l e c t r o n s w i t h a number d e n s i t y much l e s s t h a n 5 3 t h a t o f t h e b a c k g r o u n d e l e c t r o n s ( T a b l e I ) . W i t h t h e s e n u m e r i -c a l v a l u e s , n o r m a l i z e d g r o w t h r a t e OK = " ^ / " L H v e r s u s p r o p a -g a t i o n a n g l e e p l o t s w e r e o b t a i n e d w i t h t h e p a r a m e t e r s : n o r m a l i z e d f r e q u e n c y u / = 0^/10^ , t h e r a t i o o f t h e p a r a l l e l a n d p e r p e n d i c u l a r t e m p e r a t u r e s o f t h e s t r e a m i n g e l e c t r o n s T | | / T j ^ , a n d t h e r a t i o o f t h e k i n e t i c e n e r g y i n t h e s t r e a m i n g m o t i o n t o t h e t h e r m a l e n e r g y o f t h e s t r e a m i n g e l e c t r o n s , <|>2 ( E q . ( 2 - 4 7 ) i s m o d i f i e d b y : $ g -»• B | ( a n d 3 / 2 -»- 1 / 2 + T j ^ / T , , . ) F i g u r e 5 s h o w s t h e n o r m a l i z e d g r o w t h r a t e v e r s u s e w i t h a' = 1 . 1 , 2 , a n d 5 w h e r e T ( | / T J ^ = 1 a n d <j>2 = 1 . N o t e t h a t t h e s c a l e s a r e d i f f e r e n t f o r n e g a t i v e a n d p o s i t i v e u ) | . I t s h o w s t h a t w i t h i n c r e a s i n g 6 , toj i n c r e a s e s f r o m a s m a l l p o s i t i v e v a l u e a b o u t z e r o t o a m a x i m u m , d e c r e a s e s t h r o u g h z e r o t o e x h i b i t o n e o r t w o n e g a t i v e m i n i m a b e f o r e b e c o m i n g p o s i -t i v e a g a i n , r e a c h e s a p o s i t i v e m a x i m u m , a n d d r o p s t o z e r o a t 6 = 6 r e s w h e r e e r e s i s S i v e n  b y t a n 2 e T . o c = - - ( 3 - 4 9 ) r e s s T h i s " e l e c t r o s t a t i c " r e g i o n f o r p o s i t i v e co! n e a r 9 o c c u r s o v e r a n e x t r e m e l y s m a l l i n t e r v a l i n 6 a n d c o n s e q u e n t l y t h e c u r v e p l o t s o n t h e f i g u r e a s a s t r a i g h t l i n e h e r e . ( T h e r e a s o n f o r t h e " e l e c t r o s t a t i c " d e s i g n a t i o n w i l l b e c o m e c l e a r s h o r t l y . ) F o r t h e s e f r e q u e n c i e s u> r , o n l y t h e L a n d a u t e r m w h e r e n = 0 i s s i g n i f i c a n t a n d c o n t r i b u t i o n s f r o m t h e c y c l o t r o n t e r m s ( n / 0 ) t o P = P = / P a r e n e g l i g i b l e . T h e c y c l o t r o n a s f s n n=-=o ( t e r m s d o n o t b e c o m e s i g n i f i c a n t u n t i l u> r i s a b o u t 1 0 . 5. Growth rate vs. propagation angle e for 3 frequenci 55 T h e r e f o r e F i g . 5 shows t h a t Landau i n s t a b i l i t i e s o c c u r f o r low v a l u e s o f 8 as w e l l as f o r v a l u e s n e a r 9 , and t h a t t h e s e two r e g i o n s are s e p a r a t e d by a r e g i o n where Landau damp-i n g o c c u r s . Note a l s o t h a t the i n s t a b i l i t y r e g i o n f o r 6 above 6 = 0 ( " l o w - e " r e g i o n ) d i m i n i s h e s w i t h i n c r e a s i n g co^ . I t has been found to have d i s a p p e a r e d at about co^ . = 10 The r e s o n a n c e a n g l e e r e s a l s o d e c r e a s e s w i t h i n c r e a s i n g co^ . becoming z e r o when P = 0 , t h a t i s , when the f r e q u e n c y i s g i v e n by (3-50) i o n s F i g u r e 5 a l s o i n d i c a t e s t h a t the maximum growth r a t e n e a r 6 i n c r e a s e s w i t h f r e q u e n c y . F i g u r e 6 i s a p l o t o f t h i s maximum growth r a t e as a f u n c t i o n o f f r e q u e n c y showing t h i s b e h a v i o u r as w e l l as the lower c u t o f f at the lower h y b r i d r e s o n a n c e f r e q u e n c y (co^ . = 1) . The p o s i t i v e v a l u e s of. O K n e a r e = 90° are r e l a t e d to the p o s i t i v e v a l u e s o f <D! p l o t t e d i n F i g s . 2 and 3 which are b a s e d on the e l e c t r o s t a t i c d i s p e r s i o n e q u a t i o n . A c o m p a r i s o n can t h e r e f o r e be made b e -tween the maximum growth r a t e s ( w ^ ) m a x o b t a i n e d from the e l e c -t r o s t a t i c and the g e n e r a l d i s p e r s i o n e q u a t i o n . These v a l u e s were t a k e n from F i g s , 2 and 6 or c a l c u l a t e d i f n e c e s s a r y and are l i s t e d i n T a b l e I I I . V a l u e s f o r ( u i ) m a x computed u s i n g the g e n e r a l d i s -p e r s i o n e q u a t i o n were t a k e n from F i g . 6 and a r e l i s t e d i n the f o u r t h column o f T a b l e I I I . C o r r e s p o n d i n g v a l u e s f o r the I Fig. 6. Maximum growth rate vs. frequency u/ e l e c t r o s t a t i c case were taken from Fig. 2 or calculated and are l i s t e d i n the f i f t h column of this table. The last column gives the ratios of the corresponding (wi^max • T h e departure from unity of these ratios r e f l e c t s the eff e c t of the assumptions made in the e l e c t r o s t a t i c case, such as X << 1 and large r e f r a c t i v e index v . In the table, the r a t i o of the maximum growth rates approaches 1 as X becomes much less than 1. Figures 7 and 8 show the var i a t i o n of re f r a c t i v e index v and X on the propagation angle 6 for three values of u/ . In the ca l c u l a t i o n for X , the parameters T|| /"I^ and <(>2 both are taken to be 1 for Fig. 8; the r e f r a c t i v e index v , however, is here independent of these two para-meters since they describe the streaming p a r t i c l e s assumed TABLE I I I Maximum growth rates and associated values 1 V A v l'max x 10 " 6e l e c t r o s t a t i c ^i^max' e l e c t x 10" 6 ^ i ^ m a x ' e l e c t ^ P max 1.1 1851 1.22 3.13 8.04 2.57 2 523 0.323 44.8 63.0 1.40 5 186 0.256 151 202 1.34 10 91 0.243 312 413 1.32 60 t o be tenuous i n the a n a l y s i s . I t can be seen i n F i g . 7 t h a t the r e f r a c t i v e i n d e x i s i n g e n e r a l lower f o r h i g h e r f r e q u e n c i e s e x c e p t n e a r 6 , where v -»• « , s i n c e the h i g h e r f r e q u e n c i e s r © s have lower v a l u e s o f 8 as n o t e d b e f o r e i n F i g . 5. The r e s ° p a r a m e t e r X can be seen i n F i g . 8 to i n c r e a s e m o n o t o n i c a l l y w i t h 9 , a p p r o a c h i n g i n f i n i t y as 6 approaches 8 At a g i v e n v a l u e o f 8 , X i s always g r e a t e r f o r h i g h e r f r e q u e n c i e s . I t has a l s o been o b s e r v e d t h a t i f e i t h e r $ z o r T|| i s i n c r e a s e d , t h e n the v a l u e o f X d e c r e a s e s . F i g u r e s 5, 6, and 8 i n t h i s c h a p t e r showed the v a r i a -t i o n o f growth r a t e and X w i t h p r o p a g a t i o n a n g l e 8 f o r f r e -q u e n c i e s = 1.1, 2, 5 ( F i g s . 5 and 8) and the maximum growth r a t e as a f u n c t i o n o f co r where the o t h e r two p a r a -meters had v a l u e s <j>2 = 1 and T|( /Tj_ = 1 . L e t us now c o n -s i d e r the e f f e c t o f t h e s e two p a r a m e t e r s , k e e p i n g to = 1.1 The dependence o f to! on <|>2 i s shown i n F i g . 9 . T h r e e c u r v e s f o r <t>2 = 1, 2, and 3 are p l o t t e d where ' w » 1.1 and T|| /Tj_ - 1 • Note t h a t , as b e f o r e , the s c a l e s are d i f -f e r e n t f o r p o s i t i v e and n e g a t i v e to^ . One c u r v e o n l y was drawn n e a r 8 , where a l l the c u r v e s r e a c h z e r o growth r 6 s r a t e , to a v o i d a t a n g l e o f l i n e s h e r e . G e n e r a l l y , an i n c r e a s e i n <|>2 l e a d s to an i n c r e a s e i n the magnitude o f the e x c u r s i o n s from the h o r i z o n t a l a x i s . F o r example, the v a l u e s o f ( to ! ) r ' l max are 3.13 x I O - 6 , 11.9 x 1 0 " 6 , and 24.0 x 1 0 " 6 f o r <J>2 = 1, 2, and 3. i N e x t , the dependence o f to^ on the t e m p e r a t u r e 3 + 2 + 1 + -10 + -20 + u ! ( x 1 0 " 6 ) JO -30 T F i g ' 9 . Growth r a t e p r o p a g a t i o n a n g l e v a l u e s o f r a t i o o f the k i n e t i c energy i n the s t r e a m i n g m o t i o n to the t h e r m a l energy o f the s t r e a m i n g e l e c t r o n s . 62 anisotropy is depicted on Figs. 10 and 11. Curves are drawn for three values of T | | • 0.5, 1, and 2 . The other para-meters are w = 1.1 , <(>2 = 1 . In Fig. 10, only one curve is drawn near 6 for the same reason as in the previous figure. But Fig. 11 shows the three curves in the immediate v i c i n i t y of e r e s a n c* demonstrates the very narrow i n t e r v a l of e i n which the Landau i n s t a b i l i t y occurs. Figs. 10 and 11 show that the two posi t i v e maximum growth rates are greater for smaller T||/T_]_ > i . e . , for streaming p a r t i c l e s with more thermal energy in the p a r a l l e l than in the perpendicular direc-t i o n . Although not shown here, i t was noted that this behaviour with T | | / T j _ i s reversed for the " e l e c t r o s t a t i c " maximum growth rate i f the p a r a l l e l thermal energy i s kept constant and the perpendicular thermal energy is varied. It had been assumed in Table I that the streaming electrons have a streaming v e l o c i t y of 6 x 10 9 cm/sec cor-responding to an energy of 10 keV. It is well-known that streaming electrons of 6 keV energy are prevalent too (Mcllwain, 1960). Therefore calculations were made for with co^ , = 1.1 , <|>2 = 1 , T | | / T j _ = 1 f ° r a n u m ° e r of values of V s . Three curves are plotted i n Fig. 12 for values of V s = 4.6 x 10 9, 6 x 10 9, and 7.3 x 10 9 cm/sec . Again, only one curve for V = 6 x 10 9 cm/sec was drawn near where s res a l l the curves meet at zero growth rate. The values for (co!) near e are nearly the same for the three curves. v l'max res 1 However of the two minima in each curve, the one with 6 F i g . 10. Growth r a t e v s . p r o p a g a t i o n a n g l e 6 f o r 3 t e m p e r a t u r e r a t i o s . Fig. 11. Growth rate vs. propagation angle 6 for 3 temperature r a t i o s . 3 66 closer to 6 is greater i n absolute value for smaller V res & s For the next minima (which are drawn in Fig. 12) the reverse is true and the absolute value of the minimum growth rate is smaller for smaller V g . In the "low-6" region, the maxi-mum growth rate decreases with decreasing V g . Another feature to be noted is the decrease in the region for Landau i n s t a b i l i t y for low e values as V g decreases. In fact, as V g decreases below 4.6 x 10 9 cm/sec (6 keV) to about 2 x 10 9 cm/sec (1 keV) , this region vanishes altogether leav-ing only the region near e r e s (the " e l e c t r o s t a t i c " region) in which the Landau i n s t a b i l i t y occurs. As V g is decreased s t i l l further this " e l e c t r o s t a t i c " region with p o s i t i v e growth rate also vanishes, leaving only Landau damping of the wave to take place. Thus there is a minimum energy for the stream-ing p a r t i c l e s below which the Landau i n s t a b i l i t y process cannot be triggered. The results of the computer c a l c u l a t i o n show that p o s i t i v e growth rates occur for frequencies above the lower hybrid resonance and that there exists a sharp lower cutoff at u>LH . Although emphasis has been placed on •frequencies near the LHR frequency, p o s i t i v e growth rates were obtained i n the " e l e c t r o s t a t i c " region for frequencies well above the LHR frequency where cyclotron damping or i n s t a b i l i t y becomes s i g n i f i -cant. The upper l i m i t i s given either by the electron cyclotron frequency or the electron plasma frequency, whichever is lower. To demonstrate that the equations are f a i r l y general 67 and to determine whether i n s t a b i l i t i e s can be excited at other frequencies, computer calculations were made at frequencies much less than the lower hybrid resonance frequency. For example, Fig. 13 shows the results of such a ca l c u l a t i o n when the parameters are the same as those used for Fig. 5 except that now the frequency is u>r = 7 x 10 3 . This frequency is just below the lowest ion gyrofrequency, the oxygen ion gyrofrequency, which i s much less than co^^ . In this c a l -culation, only the Landau i n s t a b i l i t y term (n = 0) was found to be s i g n i f i c a n t . The figure shows that Landau i n s t a b i l i t y may take place at a l l propagation angles below S . Such results may be relevant to ion cyclotron whistlers (Gurnett et a l . , 1965) and noise observed around the proton cyclotron frequency (Harvey, 1967). 5 0 i w r = . 0 0 7 a[ ( x 10" 1 2) 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 8 (degrees) Fig. 13. Growth rate vs. propagation angle 6 for a frequency just below the oxygen ion cyclotron frequency. 69 CHAPTER 4 APPLICATION OF THEORY TO RELEVANT IONOSPHERIC NOISE 4.1 LHR N o i s e 4.1 .1 O b s e r v a t i o n s and C h a r a c t e r i s t i c s o f LHR N o i s e (a) G e n e r a l F e a t u r e s I t has been m e n t i o n e d a l r e a d y i n C h a p t e r 1 t h a t the l o w e r - f r e q u e n c y c u t o f f o f the LHR band has been i d e n t i f i e d as the f r e q u e n c y o f the lower h y b r i d r e s o n a n c e at the l o c a t i o n o f the s a t e l l i t e and t h a t the c h a r a c t e r i s t i c s o f LHR n o i s e bands appear to be r e l a t e d to e l e c t r o s t a t i c waves p r o p a g a t i n g n e a r l y p e r p e n d i c u l a r to the e a r t h ' s m a g n e t i c f i e l d . What then are • ! (. i: > • t h e s e c h a r a c t e r i s t i c s ? They have been r e p o r t e d i n p a p e r s by a number o f a u t h o r s . Some o f t h e s e are papers b y : B a r r i n g t o n and B e l r o s e ( 1 9 6 3 ) , B r i c e and Smith ( 1 9 6 4 ) , B r i c e et a l . ( 1 9 6 4 ) , B r i c e and Smith ( 1 9 6 5 ) , and McEwen and B a r r i n g t o n (1967). The l a s t two p a p e r s i n t h i s l i s t are p a r t i c u l a r l y i m p o r t a n t c o n t r i -b u t i o n s , LHR n o i s e has been seen i n the upper i o n o s p h e r e by the VLF r e c e i v e r o f A l o u e t t e I b u t , i t i s c l a i m e d , n e v e r by a VLF r e c e i v e r on the ground ( B r i c e and S m i t h , 1964). The n o i s e i s o b s e r v e d i n the f r e q u e n c y range from about 5 to 10 kHz w i t h a bandwidth which v a r i e s from one to s e v e r a l k H z . The lower f r e q u e n c y c u t o f f o f the band v a r i e s c o n s i s t e n t l y w i t h l a t i t u d e , i n c r e a s i n g i n f r e q u e n c y w i t h d e c r e a s i n g l a t i t u d e o f the s a t e l l i t e . These LHR n o i s e bands have a l s o been found i n I n j u n 3 VLF r e c o r d -i n g s , a l t h o u g h not so o f t e n as i n A l o u e t t e r e c o r d i n g s . A l s o 70 they have been observed on OGO II records; the LHR noise bands triggered by fractional-hop whistlers are found to be among the most intense signals observed on these records (Laaspere et a l . , 1967). Both the Alouette I and the OGO II s a t e l l i t e s use elec-t r i c dipole antennae whereas the Injun 3 s a t e l l i t e employs a magnetic loop antenna. Research is s t i l l in progress on the phenomenon since the LHR frequency is a function of the p o s i t i v e -ion composition of the plasma and is a p o t e n t i a l l y valuable diagnostic tool for investigation of the ionosphere. (b) Polar LHR Noise The appearance of LHR noise bands i s d i s t i n c t l y d i f -ferent at middle and northern latitudes and accordingly the two types of noise have been c a l l e d midlatitude and polar LHR noise (McEwen and Barrington, 1967). Polar LHR noise bands have r e l a -t i v e l y large bandwidths and a cutoff frequency which i s e r r a t i c and exhibits rapid fluctuations (for invariant latitudes above 70°), although on the average i t is lower at higher l a t i t u d e s . The fluctuations are believed to be due to variations in the electron density of the medium. The noise is usually continuous, l a s t i n g for several minutes. Polar LHR noise has a maximum occurrence i n the region 70-80° invariant l a t i t u d e , where the noise bands are observed as much as one hal f of the time. Although i t has no seasonal v a r i a t i o n , i t has a diurnal v a r i a t i o n and shows two maxima in frequency of occurrence, one s l i g h t l y before midmight and another secondary peak about 0600-0800 hours l o c a l time. These diurnal 71 maxima are s i m i l a r to those for auroral a c t i v i t y and p a r t i c l e p r e c i p i t a t i o n and lead one to postulate a possible r e l a t i o n between polar LHR noise bands and p a r t i c l e p r e c i p i t a t i o n . (c) Midlatitude LHR Noise Midlatitude LHR noise bands exhibit smooth lower-frequency cutoffs consistently, unlike polar LHR noise bands. Also, they show a wide va r i a t i o n in appearance. They range from short wedge-shaped signals d i r e c t l y related to whistlers (referred to as "whistler-triggered" emissions) to continuous emissions of unknown o r i g i n . The l a t t e r emissions account for about three quarters of the observed midlatitude LHR noise. Whistlers are observed on most of the records with continuous emissions so that i t is not possible to resolve whether the steady midlatitude noise bands are triggered by whistlers or not. The whistler-triggered emissions have durations from a f r a c t i o n of a second to several seconds, and indicate that frequencies near the lower cutoff frequency are enhanced for longer periods than the higher frequencies. Short as well as long f r a c t i o n a l -hop whistlers are observed to trigger LHR emissions with occur-rences of almost equal frequency. The triggering long f r a c t i o n a l -hop whistlers are rarely observed on corresponding ground-based recordings. On the other hand, none of the whistlers with echoes are seen to trigger LHR emissions. Consequently i t has been suggested that whistlers with echoes are ducted, whereas the whistlers which trigger LHR emissions are non-ducted. Also, recently Brice (private communication, 1967) has suggested 72 that these whistler-triggered emissions are not emissions but are non-ducted whistlers propagating with very large wave-normal angles where the group ve l o c i t y becomes quite small and the time delay large. Midlatitude LHR noise has a maximum in occurrence at 50-60° invariant l a t i t u d e . It is observed mainly during the months of June-October and primarily at night, with maxima at approximately the same times as polar LHR noise. Generally, there exists good c o r r e l a t i o n between occurrence of whistlers recorded on the ground and midlatitude LHR noise. 4.1.2 Comparison of Theory-with LHR Noise Observations In an e l e c t r o s t a t i c wave, the e l e c t r i c f i e l d of the wave is almost p a r a l l e l to the wave vector and accordingly, the magnetic f i e l d of the wave is very small. This is one of the c h a r a c t e r i s t i c s of LHR noise for i t has been reported that LHR noise i s observed more often by Alouette I using an e l e c t r i c dipole antenna with the VLF receiver than by the Injun 3 s a t e l l i t e which employs a loop antenna. Also, LHR noise bands triggered by fractional-hop whistlers have been found to be among the most intense signals observed by the OGO II s a t e l l i t e which also uses an e l e c t r i c dipole antenna. In addition, from the conclusion that whistlers which trigger LHR noise are non-ducted while those that do are ducted, i t was suggested that triggering is more l i k e l y for signals propa-gating with large wave-normal angles (Brice and Smith, 1965). 73 (The wave-normal angle referred to by these authors i s the same as the propagation angle 6 defined in this thesis.) We have already seen i n Chapter 3 that the e l e c t r o s t a t i c region is in the v i c i n i t y of e = 90° for frequencies near the lower hybrid resonance frequency. It had been mentioned e a r l i e r that the quantity A. (to ) , the contribution from the thermal ions, could be t r s i g n i f i c a n t for <or at a harmonic of an ion cyclotron fre-quency. At these values of tor the maximum contribution from the thermal ions i s proportional to (x^/2) n/(n!) . Since the smallest values of n are 9, 35, and 138 for the hydrogen, helium, and oxygen ions respectively, only the contribution from the protons are s i g n i f i c a n t . Generally, this contribution leads to a value for A 2 ( t o r ) which is less than that for A (to ) . However, under certa i n conditions (for example, i f <t>2 i s small) the s i t u a t i o n may be reversed and damping results at the proton cyclotron frequency harmonics just above the lower hybrid resonance frequency. This damping might result in absorp-tion bands within the LHR noise band. It would be intere s t i n g to look for this e f f e c t in spectrograms. It had been noted that as the wave frequency a T tends to to^^ , the lower hybrid resonance frequency, the growth rate to^ tends to zero. This explains the observational fact that an LHR noise band has a sharp lower frequency cutoff at the lower hybrid resonance frequency. 74 We had seen i n Sec. 2.4 that the growth times are in the order of 1 - 10" 2 sec for values between 1 and 100 of <|>2 , the r a t i o of the streaming k i n e t i c energy to the thermal energy of the streaming electrons. If the value of <f>2 is much greater than unity, then the growth time is very small and the t o t a l duration of the signal may also be very small for the following reason. According to the quasi-linear theory of Landau insta-b i l i t i e s for an e l e c t r o s t a t i c wave propagating in an electron plasma with no external magnetic f i e l d , the wave amplitude does not grow i n d e f i n i t e l y as predicted by the l i n e a r theory but decays f i n a l l y with time, with a decay time comparable to the maximum i n i t i a l growth time (Bernstein and Engleman, 1966). Therefore, the t o t a l duration is comparable to the growth time in order of magnitude. This mechanism i s one of a number of possible generation mechanisms for whistler-triggered LHR noise whose duration ranges from a f r a c t i o n of a second to several seconds. The appearance of LHR noise of long duration seems to indicate that conditions are such that the quasi-linear theory no longer applies and a wave introduced into a growing i n s t a b i l i t y no longer decays with time but develops into random noise (Kadomtsev, 1965). It is not known how an unstable wave chooses between these two p o s s i b i l i t i e s in the time-evolution of the s i g n a l . This i s a problem which needs investigating in the non-linear theory of LHR noise. Since polar LHR noise is continuous and i s related to p a r t i c l e p r e c i p i t a t i o n , our th e o r e t i c a l work appears most relevant to this type of LHR noise. 75 4 , 1 . 3 D i s c u s s i o n The t h e o r y d e v e l o p e d i n C h a p t e r 2 p r e d i c t s the s h a r p lower f r e q u e n c y c u t o f f i n the LHR n o i s e b a n d , b u t f a i l s to e x p l a i n the e x i s t e n c e o f the upper f r e q u e n c y l i m i t (though i t i s o b s e r v e d l e s s c l e a r l y than the lower f r e q u e n c y c u t o f f ) . As m e n t i o n e d b e f o r e , the o b s e r v e d LHR f r e q u e n c y ranges from about 5 - 10 k H z , and the bandwidth i s about s e v e r a l kHz o r l e s s , i . e . , the upper f r e q u e n c y l i m i t i s u s u a l l y l e s s t h a n t w i c e the LHR f r e q u e n c y . The growth r a t e c a l c u l a t e d from e q u a t i o n (2-44) becomes n e i t h e r z e r o nor n e g a t i v e as the wave f r e q u e n c y u becomes s e v e r a l t imes as l a r g e as the lower h y b r i d r e s o n a n c e f r e q u e n c y to^^ The g e n e r a l i z e d a n a l y s i s was p e r f o r m e d s i n c e i t was c o n j e c t u r e d the upper f r e q u e n c y l i m i t might be o b t a i n e d by p l a c i n g no r e s t r i c t i o n s on the r e f r a c t i v e i n d e x . A l s o , Kennel (1966) had shown t h a t a w h i s t l e r can be u n s t a b l e over a s i g -n i f i c a n t cone o f p r o p a g a t i o n a n g l e s 6 when e n e r g e t i c r e s o n a n t e l e c t r o n s have a s u f f i c i e n t l y h a r d energy s p e c t r u m , and a p i t c h a n g l e a n i s o t r o p y c o r r e s p o n d i n g to more e l e c t r o n energy p e r p e n -d i c u l a r than p a r a l l e l to the m a g n e t i c f i e l d . Growth r a t e c u r v e s were not drawn beyond 9 = 80° i n F i g . 3 o f h i s p a p e r because an a p p r o x i m a t i o n i n the a n a l y s i s b r o k e down beyond t h i s p o i n t . However, i t appeared t h a t some o f the c u r v e s may r e - e n t e r the growth r e g i o n at some v a l u e o f 8 between 80° and 90° b e f o r e t e r m i n a t i n g at z e r o growth r a t e at 8 = 90° . T h i s s u g g e s t e d t h a t the upper f r e q u e n c y l i m i t o f the LHR band might be 76 d e t e r m i n e d by e x t e n d i n g the a n a l y s i s g i v e n i n C h a p t e r 2 or i n the p a p e r by K e n n e l (1966) w i t h a p p r o p r i a t e m o d i f i c a t i o n s . However, the upper f r e q u e n c y l i m i t was found to o c c u r at e i t h e r the e l e c t r o n plasma f r e q u e n c y o r the e l e c t r o n g y r o f r e q u e n c y , w h i c h e v e r was s m a l l e r . These f r e q u e n c i e s are w e l l above the LHR f r e q u e n c y and c o u l d not e x p l a i n the upper f r e q u e n c y l i m i t o f LHR n o i s e . One p o s s i b l e e x p l a n a t i o n f o r the upper f r e q u e n c y l i m i t may l i e i n the method o f measurement or the i n s t r u m e n t a -t i o n i n v o l v e d . F o r example, the s i g n a l s may be so i n t e n s e c l o s e to the LHR f r e q u e n c y so t h a t the AGC l e v e l s are such t h a t h i g h e r f r e q u e n c i e s i n the band are n o t o b s e r v e d . Or t h e r e may be some p h y s i c a l p r o c e s s i n v o l v e d w h i c h may n o t always y i e l d an upper f r e q u e n c y l i m i t . F o r example, the p r e c i p i t a t e d f l u x o f e l e c t r o n s may have an energy s p e c t r u m q u i t e d i f f e r e n t from t h a t assumed i n the t h e o r e t i c a l work i n C h a p t e r s 2 and 3. A s t r e a m o f e l e c t r o n s , each o f w h i c h has an energy comparable to t h a t o f a u r o r a l p a r t i c l e s p r e c i p i t a t i n g i n t o the upper i o n o s p h e r e a l o n g the d i r e c t i o n o f the e a r t h ' s m a g n e t i c f i e l d , i s assumed to be the cause o f an i n s t a b i l i t y p r o c e s s . At h i g h l a t i t u d e s , i t appears r e a s o n a b l e to assume t h a t such an e l e c t r o n s t r e a m e x i s t s . In f a c t , n e a r l y m o n o e n e r g e t i c e l e c -t r o n s w h i c h appear to be s u i t a b l e f o r e x c i t i n g the Landau i n -s t a b i l i t y p r o c e s s p r o p o s e d i n t h i s p a p e r had been d e t e c t e d by M c l l w a i n ( 1 9 6 0 ) . V e r y r e c e n t l y ( A l b e r t , 1967a;. A l b e r t , 1967b), n e a r l y m o n o e n e r g e t i c e l e c t r o n s w i t h peak energy between 10 -77 14 keV have been o b s e r v e d d u r i n g a v i s i b l e a u r o r a . I t has a l s o been p r o p o s e d ( E v a n s , 1967) t h a t t h e s e e l e c t r o n beams may e x c i t e v a r i o u s p l a s m a o s c i l l a t i o n s and t h a t c o n d i t i o n s may be f a v o u r a b l e f o r the growth and subsequent decay o f some o f these waves i n a time o f about 0.1 s e c . The s p e c t r a l form o f n e a r l y monoener-g e t i c e l e c t r o n streams can be r e p r e s e n t e d by a M a x w e l l - B o l t z m a n n energy d i s t r i b u t i o n , but c h a r a c t e r i z e d by a p a u c i t y o f h i g h -energy e l e c t r o n s . The p a r a m e t e r <j>2 , the r a t i o o f the k i n e t i c energy i n the s t r e a m i n g m o t i o n t o the t h e r m a l energy o f the s t r e a m i n g p a r t i c l e s , i s then much g r e a t e r than u n i t y and t h e r e -f o r e t h e s e e l e c t r o n s may f u l f i l l the r e q u i r e m e n t s o f the t h e o -r e t i c a l a n a l y s i s p r e s e n t e d i n t h i s p a p e r , f o r example, the e l e c t r o n s have l a r g e enough v a l u e s f o r V s and <j>2 to g i v e i n s t a b i l i t i e s . However, a l t h o u g h d i f f e r e n t i n appearance from t h a t a t n o r t h e r n l a t i t u d e s , LHR n o i s e w i t h c o n t i n u o u s e m i s s i o n s i s o b s e r v e d at m l d l a t i t u d e s a l s o . At t h e s e l a t i t u d e s , a p o s s i b l e s o u r c e o f e l e c t r o n s might be from the p r e c i p i t a t i o n o f h i g h e r -energy p a r t i c l e s from the t r a p p e d r a d i a t i o n z o n e s . I t may be w o r t h - w h i l e c o n s i d e r i n g o t h e r p o s s i b i l i t i e s as the cause o f the i n s t a b i l i t y p r o c e s s . One p o s s i b i l i t y i s t h a t the i n s t a b i l i t y might be caused by a n i s o t r o p y i n the p a r -t i c l e v e l o c i t y d i s t r i b u t i o n i n one or a l l o f the component gases (such as a n i s o t r o p y i n the p i t c h a n g l e d i s t r i b u t i o n o f the c h a r g e d p a r t i c l e s or d i f f e r e n c e s i n the t e m p e r a t u r e s p a r a l l e l and p e r p e n d i c u l a r to the b a c k g r o u n d m a g n e t i c f i e l d ) . E l e c t r o -s t a t i c wave i n s t a b i l i t i e s due to a n i s o t r o p i c t e m p e r a t u r e have been i n v e s t i g a t e d by H a r r i s (1961), 78 F i n a l l y , a n o t h e r p o s s i b l e g e n e r a t i o n mechanism f o r the s h o r t wedge-shaped s i g n a l s known as " w h i s t l e r - t r i g g e r e d " e m i s s i o n s i s s u g g e s t e d . The p r o p o s e d mechanism i s a r e s o n a n c e s i m i l a r to t h a t p r o d u c i n g the s t r o n g upper h y b r i d r e s o n a n c e ( a b b r e v i a t e d UHR) s p i k e s i n A l o u e t t e t o p s i d e sounder i o n o g r a m s . S i n c e the t o p s i d e sounder sweeps the f r e q u e n c y range from 0.5 MHz to 12.0 MHz which i s w e l l above the LHR f r e q u e n c y , LHR s p i k e s a r e n o t o b s e r v e d on t h e s e i o n o g r a m s . However, a w h i s t l e r which sweeps the f r e q u e n c y range about the LHR f r e q u e n c y may p l a y the same r o l e as the t o p s i d e sounder and y i e l d LHR s p i k e s which appear as the wedge-shaped s i g n a l s o b s e r v e d on the VLF sonograms; T a k i n g t y p i c a l v a l u e s f o r the UHR s p i k e o f 5 x 10" 3 s e c and 1.5 MHz f o r the d u r a t i o n and f r e q u e n c y r e s p e c t i v e l y , the r a t i o o f the d u r a t i o n t o the wave p e r i o d (2Tr / u i y ^ ) i s o f the o r d e r o f 1000 c y c l e s . I t i s i n t e r e s t i n g to n o t i c e t h a t t h i s r a t i o i s comparable to t h a t f o r the LHR n o i s e , f o r t a k i n g a t y p i c a l v a l u e o f f L H = 5 kHz y i e l d s an LHR d u r a t i o n o f the o r d e r o f 1 s e c . F o r c o m p a r i s o n , the o b s e r v e d d u r a t i o n o f " w h i s t l e r - t r i g g e r e d " LHR n o i s e ranges from a f r a c t i o n o f a s e c o n d to s e v e r a l s e c o n d s . 4.2 O t h e r R e l a t e d O b s e r v a t i o n s O t h e r o b s e r v a t i o n s have been made, b e s i d e s the LHR n o i s e b a n d s , o f i o n o s p h e r i c n o i s e which may a l s o be r e l a t e d to the t h e o r y p r e s e n t e d i n t h i s t h e s i s . These o b s e r v a t i o n s may be r e l a t e d to the i n s t a b i l i t i e s f o r s m a l l wave-normal a n g l e s as w e l l as t h o s e i n the " e l e c t r o s t a t i c " r e g i o n . Some o f them, 79 unlike the LHR noise bands, do not exhibit an upper frequency l i m i t so close to the LHR frequency. Gurnett (1966) made a study of VLF hiss observed by the Injun 3 s a t e l l i t e during i t s 10-month l i f e t i m e . He described the frequency spectra of the VLF hiss events chosen, the occurrence i n latitude and l o c a l time, and the r e l a t i o n -ship between the VLF hiss and energetic charged p a r t i c l e fluxes. T y p i c a l l y the frequency spectrum of the VLF hiss extended from a lower frequency l i m i t of about 2 - 4 kHz to above the upper frequency l i m i t of the VLF receiver ( 8 . 8 kHz). One form of VLF hiss has a lower frequency cutoff which changes systemati-c a l l y with l a t i t u d e . The cutoff frequency f i r s t decreases with increasing l a t i t u d e , reaches a minimum of about 2 - 4 kHz, then increases with increasing l a t i t u d e thereby giving a V-shaped appearance on the spectrogram. The VLF hiss was found to occur i n a zone about 7° wide i n latitude centred on 77° invariant latitude at 1400 magnetic l o c a l time (MLT). A diurnal varia-t i o n was found with occurrence predominant during l o c a l after-noon and evening, from 1200 to 2400 MLT. This d i f f e r s from the diurnal occurrence of ground-based VLF observations (Jtfrgensen, 1966) with a maximum during the evening (about 1900 to 2100 MLT). This difference led Gurnett to suggest that the s a t e l l i t e VLF hiss occurring during the l o c a l after-noon (1200 to 1800 hours MLT) may not be observed on the ground. The occurrence of VLF hiss was also found to be associated with the intense fluxes of soft electrons with energy of about 10 keV 80 often found during the early evening. J^rgensen (1966) refers to two kinds of h i s s , namely hiss below about 2 kHz, which is a morning or day phenomenon, and hiss above about 2 kHz which is almost exclusively seen at night, except at high l a t i t u d e s . The wide-band hiss above 2 kHz is sometimes c a l l e d auroral hiss because of i t s associa-tion with auroral arcs and bands. Jjrirgensen (1968) stated that the upper frequency l i m i t for auroral hiss is not known and that hiss at 500 kHz i s often observed at Byrd Station. This statement appears to be inconsistent with another statement made in his paper, v i z that the emission named polar LHR by some workers is equivalent to the emission c a l l e d auroral hiss by others, since polar LHR noise has an upper frequency l i m i t close to the LHR frequency. J^rgensen postulates that auroral hiss may be generated by incoherent Cerenkov radiation from electrons with energies of the order of 1 keV. It i s interest-ing to note that Cerenkov radiation i s produced by a charged p a r t i c l e moving i n a plasma i f the component in the wave normal d i r e c t i o n of the p a r t i c l e ' s v e l o c i t y along the background mag-netic f i e l d equals the phase v e l o c i t y of the wave. This is equivalent to Eq. (2-2) , the condition for the Landau insta-b i l i t y process. However, the Landau process i s a c o l l e c t i v e i n t e r a c t i o n between a beam of p a r t i c l e s and the propagating wave whereas the Cerenkov emission i s from i n d i v i d u a l p a r t i c l e s traversing a plasma. J^rgensen's analysis assumes a wave normal angle of zero i n section 2 of his paper and may be inappropriate in i t s present form for polar LHR noise which appears to be 81 related to large wave normal angles, as noted previously. The Cerenkov radiation theory proposed by J^rgensen and the theory proposed i n this thesis both predict an upper frequency l i m i t equal to the electron gyrofrequency at the bottom i n the iono-sphere. However, the Landau i n s t a b i l i t y process defines a lower cutoff at the LHR frequency whereas the Cerenkov radia-tion i s also generated at frequencies below the LHR frequency, although the power is several orders of magnitude below the power generated above The c h a r a c t e r i s t i c s of auroral hiss suggest that perhaps this type of VLF emission may originate from electro-s t a t i c waves excited around the lower hybrid resonance fre-quency by the Landau i n s t a b i l i t y process analyzed i n this thesis. As an e l e c t r o s t a t i c wave generated in the ionosphere presumably cannot propagate out of the ionosphere, i t must be converted into an electromagnetic wave somewhere in the ionosphere in order to be observed as a VLF emission on the ground. Such a conversion might take place i f the plasma density d i s t r i b u t i o n i s nonuniform i n space. A process whereby an e l e c t r o s t a t i c wave is converted into an electromagnetic wave when propagating i n a nonuniform plasma or when penetrating a surface of dis-continuity i n the plasma density d i s t r i b u t i o n has been i n v e s t i -gated for a type of radio wave emission from the sun (Tidman, 1960; Tidman and Weiss, 1961; Tidman and Boyd, 1962). Even i f the plasma density d i s t r i b u t i o n is uniform in the unperturbed state, conversion of an e l e c t r o s t a t i c wave to an electromagnetic 82 wave can take p l a c e as a r e s u l t o f a n o n l i n e a r wave-wave i n t e r a c t i o n p r o c e s s i n the e l e c t r o s t a t i c wave (Tidman and D u p r e e , 1965; T i d m a n , 1965). In the case o f a u r o r a l h i s s type VLF e m i s s i o n s , t h e r e i s e v i d e n c e t h a t such a s u r f a c e o f d i s -c o n t i n u i t y o r a s t e e p g r a d i e n t i n the plasma d e n s i t y d i s t r i b u -t i o n , does e x i s t i n the i o n o s p h e r e . A u r o r a - a s s o c i a t e d VLF h i s s e m i s s i o n s are r e c e i v e d a t a p o l a r ground s t a t i o n o n l y i n the p r e - b r e a k u p p e r i o d (the N-1 phase (Morozumi and H e l l i w e l l , 1966)) d u r i n g the c o u r s e o f a u r o r a l a c t i v i t y (Morozumi, 1962; Harang and L a r s e n , 1964). The a u r o r a l a c t i v i t y d u r i n g t h i s p e r i o d i s c h a r a c t e r i z e d by q u i e t a r c s or b a n d s . I t has been o b s e r v e d t h a t h i s s type VLF e m i s s i o n s are always r e c e i v e d s i m u l t a n e o u s l y w i t h the o c c u r -r e n c e o f a r c o r band type o f a u r o r a e (Morozumi, 1962). An a r c extends a v e r y l o n g way (1000 km o r more) n e a r l y i n the m a g n e t i c e a s t - w e s t d i r e c t i o n ( S t o r m e r , 1955), b u t i s v e r y t h i n i n the n o r t h - s o u t h d i r e c t i o n . Q u i e t a r c s are u s u a l l y c h a r a c -t e r i z e d by t h i s t h i n - b l a d e s t r u c t u r e . The t h i c k n e s s (or r a t h e r t h i n n e s s ) o f the s t r u c t u r e i s about a few h u n d r e d meters o r even l e s s ( D a v i e s and McCormac, 1967). The d i s t r i b u t i o n o f l u m i n o s i t y a g a i n s t h e i g h t has a v e r y s h a r p maximum i n the h e i g h t r a n g e , n o r m a l l y from about 100 - 110 km ( S t o r m e r , 1955). I t i s b e l i e v e d t h a t an a r c i s c a u s e d by c h a r g e d p a r t i c l e s ( p r e -sumably e l e c t r o n s ) s t r e a m i n g a l o n g the d i r e c t i o n of the e a r t h ' s m a g n e t i c f i e l d . As s u g g e s t e d by the l o c a t i o n of the s h a r p maximum i n the l u m i n o s i t y - h e i g h t d i s t r i b u t i o n , the i o n i z a t i o n 83 w i t h i n an a r c i s most enhanced p r e s u m a b l y i n the lower i o n o -s p h e r i c r e g i o n (100 - 110 km). Because o f the t h i n - b l a d e s t r u c t u r e , the i o n i z a t i o n i n the lower i o n o s p h e r e i s s u r m i s e d to change v e r y a b r u p t l y a c r o s s the b o u n d a r i e s o f an a r c . I t would be i n t e r e s t i n g to make s i m u l t a n e o u s o b s e r v a t i o n s o f LHR n o i s e i n the i o n o s p h e r e , p a r t i c u l a r l y i n the lower r e g i o n by a space v e h i c l e and VLF e m i s s i o n s o f a u r o r a l h i s s type at a ground s t a t i o n . A n o t h e r i n t e r e s t i n g o b s e r v a t i o n has been made by B a r r i n g t o n and McEwen (1966) which may be the same as the V -shaped VLF h i s s s t u d i e d by G u r n e t t (1966). They r e p o r t e d an i n t e r e s t i n g n o i s e b u r s t phenomenon a t h i g h l a t i t u d e s , t h a t appears to be r e l a t e d i n some way to the LHR n o i s e b a n d s . Because o f t h e i r a p p e a r a n c e , they have been termed VLF s p l a s h e s G e n e r a l l y , t h e y are V - s h a p e d and the w i d t h o f the V v a r i e s from a c o u p l e o f seconds to a few m i n u t e s . O f t e n , the V - s h a p e d n o i s e b u r s t s seem to grow out o f o r r e p l a c e a c o n t i n u o u s LHR n o i s e b a n d . In a d d i t i o n , the o c c u r r e n c e p a t t e r n f o r the s p l a s h i s not i n c o n s i s t e n t w i t h an a s s o c i a t i o n between s p l a s h e s and a u r o r a l p a r t i c l e p r e c i p i t a t i o n . 84 CHAPTER 5 SUMMARY AND CONCLUSIONS W a v e - p a r t i c l e i n t e r a c t i o n around the lower h y b r i d r e s o n a n c e f r e q u e n c y has been i n v e s t i g a t e d i n t h i s t h e s i s . E x p r e s s i o n s f o r the growth r a t e o f w h i s t l e r - m o d e waves p r o p a -g a t i n g i n a plasma p e n e t r a t e d by a tenuous beam o f n o n t h e r m a l p a r t i c l e s were o b t a i n e d u s i n g the e l e c t r o s t a t i c d i s p e r s i o n e q u a t i o n and the f u l l - w a v e d i s p e r s i o n e q u a t i o n g i v e n i n the t e x t by S t i x (1962). A l l the s p e c i e s o f p a r t i c l e s i n v o l v e d were assumed t o have a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n . In the case o f the s t r e a m i n g p a r t i c l e s , the d i s t r i b u t i o n was s h i f t e d by a s t r e a m i n g v e l o c i t y p a r a l l e l to the b a c k g r o u n d m a g n e t i c f i e l d . The e l e c t r o s t a t i c a n a l y s i s assumed i s o t r o p i c t e m p e r a t u r e s and the f u l l - w a v e a n a l y s i s assumed t h a t the d i s -t r i b u t i o n s are c h a r a c t e r i z e d by two t e m p e r a t u r e s , p a r a l l e l and p e r p e n d i c u l a r to the b a c k g r o u n d magnetic f i e l d . The e x p r e s s i o n s d e r i v e d f o r the growth r a t e a r e somewhat g e n e r a l , but i n the n u m e r i c a l c a l c u l a t i o n s we c o n s i d e r e d the case o f s t r e a m i n g e l e c t r o n s i n t e r a c t i n g w i t h w h i s t l e r - m o d e waves p r o p a g a t i n g i n a c o l d m a g n e t o - a c t i v e - p l a s m a c o n s i s t i n g o f e l e c t r o n s and H + , H + , and 0 + i o n s . ' e ' The g r o w t h - r a t e e x p r e s s i o n o b t a i n e d by u s i n g the e l e c t r o s t a t i c d i s p e r s i o n e q u a t i o n showed t h a t the p r o p a g a t i n g waves can grow due to the Landau i n s t a b i l i t y p r o c e s s f o r f r e -q u e n c i e s above the lower h y b r i d r e s o n a n c e f r e q u e n c y . The i n -s t a b i l i t y r e g i o n has a s h a r p l o w e r - f r e q u e n c y c u t o f f a t the 85 LHR frequency and an upper frequency l i m i t at the smaller of the electron cyclotron or electron plasma frequency. With increasing frequency within this i n s t a b i l i t y region, the maxi-mum growth rate increases monotonically. The full-wave analysis gave an i n s t a b i l i t y region in the "low-e" region above 9 = 0 in addition to that in the " e l e c t r o s t a t i c " region near resonant angle 8 which cor-responds to the i n s t a b i l i t y found from the electrostatic-wave analysis. The growth rates in the "low-8" region are not as large as those in the " e l e c t r o s t a t i c " region and become nega-tive (damping) before the l a t t e r with decreasing streaming ve l o c i t y V g . The cyclotron i n s t a b i l i t i e s or damping (n = ± 1, ± 2,...) become important at frequencies greater than about ten times the LHR frequency. It was also observed that the growth rates decrease with increasing Tj| /T^ i n both the "low-8" and the " e l e c t r o s t a t i c " region. An increase in $ 2 (the r a t i o of k i n e t i c energy in the streaming motion to the thermal energy of the streaming electrons) generally leads to an increase in the magnitude of Landau i n s t a b i l i t y or damping. The theory developed in this d i s s e r t a t i o n was then applied to various relevant ionospheric noise. Many charac-t e r i s t i c s of the lower hybrid resonance (LHR) noise bands dis-covered by the Alouette I s a t e l l i t e suggest that the theory can explain t h e i r generation. This is true p a r t i c u l a r l y in the case of polar LHR noise. However, the polar LHR noise 86 has an upper f r e q u e n c y l i m i t v e r y c l o s e to the LHR f r e q u e n c y whereas the t h e o r y g i v e s one many o r d e r s o f magnitude h i g h e r . The method o f measurement o r some p h y s i c a l p r o c e s s may be the cause o f the o b s e r v e d upper f r e q u e n c y l i m i t . O t h e r types o f n o i s e , such as a u r o r a l h i s s w i t h an upper f r e q u e n c y l i m i t g r e a t e r than 500 k H z , have a l s o been d i s c u s s e d . B r i e f l y , t h i s t h e s i s has p r e s e n t e d a t h e o r y w h i c h has f e a t u r e s w h i c h agree w i t h many i o n o s p h e r i c n o i s e o b s e r v a t i o n s . 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Res. 65, 2727-2747 (1960). Morozumi, H. M. A Study of Aurora A u s t r a l i s i n Connection with an Association between VLF Hiss and Auroral Arcs and Bands Observed at South Geographic Pole 1960, M.S. Thesis SUI-62-14, State Univ. Iowa, 1962. Morozumi, H. M. and R. A. H e l l i w e l l . A Correlation Study of the Diurnal Variation of Upper Atmospheric Phenomena in the Southern Auroral Zone, Tech. Rep. SU-SEL-66-124, Radioscience Lab., Stanford Electronics Lab. (1966). Spitzer, L., J r . Physics of Fully Ionized Gases (Interscience Publishers, New York, 196Z) , Znd rev. ed. Stix, T. H. The Theory of Plasma Waves (McGraw-Hill Book Company" Inc. ,~ITew York, 196Z) . Stormer, C. The Polar Aurora (Oxford Univ. Press, London, 1 9 5 3 7 7 90 Tearazawa, K. Mathematics for Natural S c i e n t i s t s (Iwanami Book Company, Tokyo, 1960) „ Tidman, D. A. Radio Emission by Plasma O s c i l l a t i o n s i n Non-uniform Plasmas. Phys. Rev. 117 , 366-374 (1960). Tidman, D. A. Radio Emission from Shock Waves and Type II Solar Outbursts. Planet. Space Sci. 13, 781-788 (1965). ~~ Tidman, D. A. and T. J. M. Boyd. Radiation by Plasma O s c i l -lations Incident on a Density Discontinuity. Phys. Fluids 5 , 213-218 (1962) . Tidman, D. A. and T. H. Dupree. Enhanced Bremsstrahlung from Plasmas Containing Nonthermal Electrons. Univ. Maryland Tech. Note BN-384 (1965). Tidman, D. A. and G. H. Weiss. Radio Emission by Plasma O s c i l -lations in Nonuniform Plasmas. Phys. Fluids 4_, 703-710 (1961). Tonks, L. and I. Langmuir. O s c i l l a t i o n s in Ionized Gases. Phys. Rev. 33_, 195-210 (1929). Watson, G. N. Theory of Bessel Functions (Cambridge Univ. Press, London, 1922). Whittaker, E. T. and G. N. Watson. A Course of Modern Analysis (Cambridge Univ. Press, London, 1940). A D D I T I O N A L B I B L I O G R A P H Y Publications based on the early stages of this work (esse n t i a l l y Chap. 2 of the thesis) are given below. Horita, R. E. and T, Watanabe. Some Remarks on the Origin of Lower Hybrid Resonance Noise in the Ionosphere. Space Res. (1968) In press. Horita, R. E. and T. Watanabe. E l e c t r o s t a t i c Waves i n the Iono-sphere Excited around the Lower Hybrid Resonance Fre-quency. Planet. Space Sci. In press. 91 APPENDIX I COMPUTER PROGRAM TO CALCULATE GROWTH RATE OF WHISTLER-MODE WAVES Fortran Source L i s t C GROWTH RATE OF WHISTLER-MODE WAVES C C INPUT DATA FOR PLASMA COMPONENTS AND STREAMING PARTICLES DIMENSION XM(10),SGN(10),DENS(10),ZP(10),ZC(10),ZC2(10) DATA ECG,CVEL,BOLTZ/4. 802 8 6E-10, 2 .9979 3E10, 1.38044E-16/ PI = SNGLC3.14159265359D+0) SCALEF = 10 . s: 2 . 302585 C INPUT IN CGS UNITS C A. NUMBER OF SPECIES INVOLVED C B. MASS (GM), SIGN OF CHARGE (+1 OR - 1 ) , AND NUMBER DENSITY C (1/CC) OF EACH SPECIES C C. BACKGROUND MAGNETIC FIELD (GAUSS), STREAMING VELOCITY C (CM/SEC), ENERGY RATIO OF STREAM, PARALLEL/PERPENDICULAR C TEMPERATURE OF STREAMING PARTICLES, LOWER HYBRID C RESONANCE FREQUENCY (HZ), AND HIGHEST ORDER OF C CYCLOTRON INSTABILITY CONSIDERED C D. FREQUENCIES—INITIAL, INCREMENTAL, AND FINAL (NORMALIZED C BY THE LHR FREQUENCY)(INCREAS ING) C E. WAVE NORMAL ANGLES--INITIAL, INCREMENTAL, AND FINAL C (DEGREES) (DECREASING FROM RESONANT ANGLE) 66 READ (5,1) NSPEC 1 FORMAT (13) DO 2 1=1,NSPEC 2 READ (5,3) XM(I),SGN(I),DENS(I) 3 FORMAT (E15.5, F4.0, E10.3) READ (5,4) B0,SVEL,PHI2,TRATIO,FREQLH,N C N IS THE HIGHEST ORDER OF CYCLOTRON INSTABILITY CONSIDERED 4 FORMAT (5E10.3,I3) READ (5,5) OMEGAS, OMEGAI, OMEGAF 5 FORMAT (3E10.3) READ (5,6) THETAS, THETAI, THETAF 6 FORMAT (3E10.3) C THETA VALUES INPUT ARE IN DEGREES C CALCULATE CONSTANTS FROM INPUT DATA WLH = 2.:s PI s: FREQLH WLH2 = WLH :J WLH ECG4P = k. ECG ECG « PI BETA = SQRT ( ( 0 . 5 + l./TRATIO) :c PHI2)/SVEL DO 7 1=1,NSPEC Z C ( I ) = ECG :J B0/(XM(I)5«CVEL) ZC 2 ( I ) = Z C ( I ) :s Z C ( I ) 7 Z P ( I ) = ECG4P :c DENS (I )/XM( I ) C FIND THE TWO UNKNOWN DENSITIES (NEG) AND COMPUTE THEIR VALUES C THEY ARE FOUND FROM CHARGE NEUTRALITY AND S = 0 AT THE LOWER 92 C HYBRID ( L H ) RESONANCE FREQUENCY. YI = 0. Y2 = 0. INSP = NSPEC - 1 DO 9 I = 1,INSP J = I I F ( D E N S ( I ) ) 1 0 , 9 , 9 9 CONTINUE 10 N J = J NIX = NJ + 1 DO 12 I = N I X , I N S P K = I I F ( D E N S O ) ) 1 3 , 12, 12 12 CONTINUE 13 NI = K DO 16 I = 1,INSP I F ( I - N J ) 1 4 , 1 6 , 1 4 14 I F ( I - N I ) 1 5 , 1 6 , 1 5 15 YI = YI + S G N ( I ) " D E N S ( I ) Y2 = Y2 + D E N S ( I ) / ( X M ( I ) : c ( W L H 2 - Z C 2 ( I ) ) ) 16 CONTINUE AI = 1 . / ( X M ( N I ) : J ( W L H 2 - Z C 2 ( N I ) ) ) A J = 1 . / ( X M ( N J ) : s ( W L H 2 - Z C 2 ( N J ) ) ) DENS(NJ)= ( S G N ( N I ) : e ( l . /ECG4P-Y2 )+AI :cYI ) /(SGN(NI ):sAJ - -SGN(NJ)::AI) D ENS(NI) = ( - l . / S G N ( N I ) ) : c ( Y l + S G N ( N J ) " D E N S ( N J ) ) C PRINT DATA AT BEGINNING OF OUTPUT WRITE ( 6 , 7 0 ) N S P E C 70 FORMAT (7H0NSPEC = , I 3) DO 500 1=1,NSPEC 500 WRITE ( 6 , 7 1 ) X M ( I ) , S G N ( I ) , D E N S ( I ) 71 FORMAT (6H MASS = ,1PE12.5,5X,6H SIGN=,0PF4.0,5X,9H DENS ITY=, - 1 P E 1 0 . 3 ) WRITE ( 6 , 7 2 ) B0,SVEL,PHI2,TRATIO,FREQLH,N 72 FORMAT (4H B0 =, E10 . 3, 5X, 6H SVEL=, 1PE 10.. 3, 5X, 8H PHI:c::2 = , -1PE10. 3,5X,11H TPAR/PERP = ,1PE10.3,5X,8H FREQLH=, 1P.E10'.. 3, -5X,3H N=,I3) WRITE ( 6 , 7 3 ) OMEGAS,OMEGAI,OMEGAF 73 FORMAT(12H OMEGA INIT=,1PE12.3,5X,6H INCR=,1PE12.3,5X, -7H FINAL=,1PE12.3) WRITE ( 6 , 7 4 ) THETAS, THETAI, THETAF 74 FORMAT(12H THETA INIT=,2PE12.3,5X,6H INCR=,1PE12.3,5X, -7H FINAL=,2PE12.3) I F ( D E N S ( N J ) ) 67,64,64 67 WRITE ( 6 , 6 8 ) 68 FORMAT (74H THE VALUES FOR THE INPUT DENSITIES AND THE -LH FREQUENCY ARE INCOMPATIBLE./) GO TO 66 64 I F ( D E N S ( N I ) ) 67,65,65 65 Z P ( N J ) = ECG4P :c DENS ( N J )/XM(N J ) Z P ( N I ) = ECG4P :e DENS (NI )/XM(N I ) WRITE ( 6 , 6 9 ) 93 69 FORMAT ( 7 0 H 0 T H E S C A L E IS TO BE ADDED TO THE EXPONENTS FOR -POWER AND GROWTH R A T E S . ) WRITE ( 6 , 9 0 ) 90 F O R M A T ( / 6 H T H E T A , 1 0 X , 1 7 H R E F R A C T I V E I N D E X , 2 X , 8 H LAMBDA , - 8 X , 6 H POWER, 9 X , 7 H E N E R G Y , <+X, 14H NORM GROWTH R , 3 X , - 1 2 H G R O W T H R A T E , 4 X , 6 H S C A L E / / ) C C A L C U L A T E F R E Q U E N C Y - D E P E N D E N T PARAMETERS W = OMEGAS WLH 34 W2 = W W OMEGAP = W/WLH WRITE ( 6 , 8 0 ) OMEGAP 80 FORMAT ( 22H NORMALIZED F R E Q U E N C Y = , 1 P E 1 0 . 3 / ) R=l . XL =1. P = 1. HXX=1. HXY = 0 . D017 1= 1 , I N S P R = R - Z P ( I ) / ( W :c (W + SGN(I):c Z C ( I ) ) ) XL =XL - Z P ( I ) / ( W a (W - SGN(I):c Z C ( I ) ) ) P = P - Z P ( I ) / W 2 HXX = HXX + Z P ( I ) : e ( W 2 + Z C 2 ( I ) ) / ( ( W 2 - Z C 2 ( I ) ) : : ! < 2 ) 17 HXY = HXY + W : i S G N ( I ) 5 c Z C ( l ) : ! Z P ( l ) / ( ( W 2 - Z C 2 ( I ) ) ! ? « 2 ) HZZ = 2 . - P S = (R + X L ) / 2 . IF ( P ) 7 7 , 7 7 , 7 5 75 WRITE ( 6 , 7 6 ) 76 FORMAT (15H P IS P O S I T I V E . ) GO TO 33 77 I F ( S ) 8 1 , 8 1 , 8 3 81 WRITE ( 6 , 8 2 ) 82 FORMAT (15H S IS N E G A T I V E . ) GO TO 33 83 D = (R - X L ) / 2 . C C A L C U L A T E A N G L E - D E P E N D E N T PARAMETERS THETAR = A T A N 2 ( S Q R T ( - P ) , S Q R T ( S ) ) THETAR = THETAR 180 . / P I I F ( T H E T A R - T H E T A S ) 1 9 , 1 9 , 18 18 THETA = THETAS GO TO 20 19 T H E T A = THETAR - T H E T A I 20 THETA = THETA P I / 1 8 0 . XCOS = C O S ( T H E T A ) XSIN = S I N ( T H E T A ) XCOS2= XCOS:c:s2 XSIN2= XSIN5s : :2 A = S !! XSIN2 + P :cXCOS2 B = R 5: XL5SXSIN2 + P"Sss(1. +XCOS2) F = S Q R T ( ( ( R : c X L - P « S)55XSIN2) : 5 ! s 2 + h . !cp:«p::D5«D"XCOS2) C R E F R A C T I V E INDEX AND PARAMETERS RIN2 = ( B - F ) / ( 2 . « A ) RIN = SQRT ( R I N 2 ) XK = RIN J : W / C V E L 94 XKZ =XK XCOS XKX =XK sc XSIN XLAMBD = ( X K X / ( Z C ( N S P E C ) : : B E T A ) ) : { : c 2 / ( 2 . : :TRATI 0) IF (XLAMBD - 8 8 . 0 2 9 6 9 2 ) 5 6 , 5 4 , 5 4 54 T H E T A = T H E T A a 1 8 0 . / P I WRITE ( 6 , 5 5 ) XLAMBD , THETA 55 FORMAT (34H0LAMBDA EXCEEDS 8 8 . 0 2 9 6 9 2 , LAMBDA=, - E 1 3 . 4 , 5 X , 7 H T H E T A = , 2 P E 1 5 . 6 ) GO TO 38 56 V = S G N ( N S P E C ) D / ( R I N 2 - S ) Y = RIN2 X S I N 2 / ( R I N 2 « XSIN2 - P) C C A L C U L A T E MODIFIED B E S S E L FUNCTIONS I WITH SUBROUTINE B E S I K C COULD ADD ASYMPTOTIC FORMULA FOR MODIFIED B E S S E L FUNCTION FOR C LAMBDA GREATER THAN 8 8 . 0 2 9 6 9 2 , I F REQUIRED LATER DIMENSION B I N ( I O O ) C A L L B E S S E L (BI N O , B I N , N , X L A M B D ) DIMENSION B I N P ( I O O ) C A L L BESSP ( B I N P O , B I N P , N , X L A M B D , B I N O , B I N ) C C A L C U L A T E POWER TRANSFER DIMENSION P K N ( I O O ) , A L P H A ( I O O ) , A L P H A 2 ( 1 0 0 ) , ZMN(IOO) J J = 1 NN = 0 ZMNC = Z P ( N S P E C ) : c B E T A / ( T R A T I O > : l 6 . » S Q R T ( P I ) : : A B S ( X K Z):JWLH) 21 IF ( N N ) 2 3 , 2 2 , 2 3 22 XLN = 2.:: E X P ( - X L A M B D >!CBIN0 XLNP = 2.:: E X P ( - X L A M B D > « B I N P 0 GO TO 24 23 NNA = I A B S ( N N ) XLN = 2 . :s EXP(-XLAMBD ):cBIN(NNA) XLNP= 2 . EXP(-XLAMBD ):JBINP(NNA) 24 ZNN = NN RN = V a XLAMBD + Y " ( W + Z N N « Z C ( N S P E C ) ) / Z C ( N S P E C ) - Z N N GN = ( X L N / X L A M B D ) : : (V5 :V5 : (ZNN5CZNN+XLAMBD : J : :2 ) + RNssRN)--2 . : :XLNP:5V«RN Z M N ( J J ) = ZMNC (W-XKZ:cSVEL + ZNN:sZC(NSPEC)s: - ( 1 . - T R A T I O ) ) ! ! G N / W ALPHA ( J J ) = ( W - X K Z : « S V E L + Z N N : : Z C ( N S P E C ) ) " B E T A / X K Z I F ( J J - 1 ) 5 8 , 5 7 , 5 8 57 U K = A L P H A ( J J)J : : s 2 / S C A L E F I S C A L E = -10 JS U K Z U K = U K 58 A L P H A 2 ( J J ) = A L P H A ( J J ) : s » 2 - Z U K s« S C A L E F PKN ( J J ) = Z M N ( J J ) x EXP ( - A L P H A 2 C J J ) ) J J = J J + 1 IF ( N N ) 2 6 , 2 5 , 2 7 2 5 NN = NN + 1 GO TO 28 26 NN = - ( N N - 1 ) GO TO 2 8 27 NN = -NN 28 IF ( N + 1 - N N ) 2 1 , 3 0 , 2 1 30 PSP = 0 . NTOT = 25JN+1 95 DO 31 I = l , N T O T 31 PSP = PSP + P K N ( I ) C C A L C U L A T E ENERGY ENERGY = ( R I N 2 : cV5 : V + R I N 2 « ( 1 . - Y ) » " 2 "XCOS2 + HXX::( 1. + V : : V ) -- 4 . : s H X Y x S G N ( N S P E C ) x V + HZZ::Y:sY5{XC0S 2 / X S I N2 ) / C 1 6 . :cp I ) C C A L C U L A T E GROWTH RATE (NORMALIZED AND R E A L ) GROWTH = - P S P / ( 2 . : : E N E R G Y ) REGROW = GROWTH :s WLH WRITE ( 6 , 3 5 ) ( P K N U ) , I = l , N T O T ) 35 FORMAT (32H PKN ( I = 0 , 1 , - 1 , 2 , - 2 , . . „ ) = , I P 1 0 E 1 2 . 3 ) THETA = THETA « 1 8 0 . / P I C OUTPUT WAVE NORMAL ANGLE ( D E G R E E S ) , R E F R A C T I V E I N D E X , LAMBDA, C POWER TRANSFER ( D I M E N S I O N L E S S ) , WAVE ENERGY ( D I M E N S I O N L E S S ) , C "GROWTH RATE (NORMALIZED BY LH F R E Q U E N C Y ) , GROWTH RATE ( 1 / S E C ) , C AND THE S C A L E FACTOR WRITE ( 6 , 3 7 ) T H E T A , R I N , X L A M B D , P S P , E N E R G Y , G R O W T H , R E G R O W , I S C A L E 37 FORMAT ( 2 P E 1 5 . 6 , 1 P 6 E 1 5 . 4 , 115 ) 38 THETA = THETA - T H E T A I IF ( T H E T A - T H E T A F ) 3 3 , 2 0 , 2 0 33 W = W + O M E G A l J t W L H I F (W - O M E G A F x W L H ) 3 4 , 3 4 , 6 6 39 STOP END SUBROUTINE B E S S E L ( B I N O , B I N , N , X L A M B D ) DIMENSION B I N ( N ) M = N-1 BINO = BESS I 0 ( X L A M B D ) I F (XLAMBD - . 0 1 ) 102 , 102 , 101 102 B I N ( l ) = X L A M B D / 2 . B I N ( 2 ) = XLAMBD :c : :2 /8 . G O TO 103 101 B I N ( 1 ) = B E S S I 1 ( X L A M B D ) B I N ( 2 ) = BIN0 - 2 . ! 5 B I N ( 1 ) / X L A M B D 103 IF ( N . L T . 3 ) RETURN DO 100 1= 2 , M A I X =1 100 B I N ( I + l ) = B I N ( I - l ) - 2 . " A I X B I N ( I ) / X L A M B D RETURN END SUBROUTINE B E S S P ( B I N P 0 , B I N P , N , X L A M B D , B I N O , B I N ) DIMENSION B I N P ( N ) DIMENSION BIN ( N ) BINP0 = B I N ( l ) B I N P ( 1 ) = B I N 0 - B I N ( 1 ) / X L A M B D I F ( N . L T . 2 ) RETURN DO 200 1=2,N AI Y =1 200 B I N P ( I ) = B I N ( I - 1 ) - A I Y : « B I N ( I ) / X L A M B D RETURN END 96 APPENDIX II PARTIAL LIST OF SYMBOLS USED A c o l d plasma p a r a m e t e r A ^ c o n s t a n t B' c o l d plasma p a r a m e t e r B wave magnetic f i e l d B Q b a c k g r o u n d m a g n e t i c f i e l d C c o n s t a n t c v e l o c i t y o f l i g h t D c o l d plasma p a r a m e t e r E wave e l e c t r i c f i e l d e c h a r g e o f e l e c t r o n e as s u b s c r i p t o r s u p e r s c r i p t , r e f e r s to b a c k g r o u n d e l e c t r o n s F c o l d plasma p a r a m e t e r f f r e q u e n c y I m o d i f i e d B e s s e l f u n c t i o n n i i m a g i n a r y number j as s u b s c r i p t or s u p e r s c r i p t , r e f e r s to b a c k g r o u n d i o n s K d i e l e c t r i c t e n s o r H e r m i t i a n p a r t o f d i e l e c t r i c t e n s o r k wave (number) v e c t o r L c o l d p l a s m a p a r a m e t e r Ji w a v e l e n g t h M m o b i l i t y t e n s o r M c o n s t a n t n 97 mass number d e n s i t y o r d e r o f Landau and c y c l o t r o n i n s t a b i l i t y or damping c o l d plasma parameter power a b s o r p t i o n c o l d plasma parameter c o n s t a n t c o l d plasma parameter f u n c t i o n o f complex e r r o r f u n c t i o n as s u b s c r i p t o r s u p e r s c r i p t , r e f e r s t o s t r e a m i n g p a r t i c l e s t e mperature c o n s t a n t time s t r e a m i n g v e l o c i t y p a r t i c l e v e l o c i t y i n z - d i r e c t i o n wave energy c o o r d i n a t e s a t o m i c number S t i x parameter i n v e r s e o f the mean t h e r m a l v e l o c i t y s i g n o f charge (±1) p r o p a g a t i o n a n g l e ( o r wave-normal a n g l e ) Boltzmann c o n s t a n t 98 Stix parameter re f r a c t i v e index plasma frequency summation square root of r a t i o of the k i n e t i c energy in the streaming motion to the thermal energy of the stream-ing electrons cyclotron frequency angular frequency 

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