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UBC Theses and Dissertations

Whistler-triggered lower hybrid resonance noise in irregularites [sic] of the ionosphere Michkofsky, Ronald Nick 1974

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WHISTLER-TRIGGERED LOWER HYBRID RESONANCE NOISE IN IRREGULARITES OF THE IONOSPHERE by RONALD NICK MICHKOFSKY B.Sc, Carnegie-Mellon University (formerly, Carnegie Institute of Technology), 1967 M.Sc, University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of GEOPHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1974 In presenting th is thesis in pa r t i a l f u , f M m e n t of the requirements for an advanced deg.ree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l ica t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada - i i -ABSTRACT The mechanism suggested for whistler triggered LHR noise is that of a whistler propagating from a region of the ionosphere where the unperturbed number densities are uniform into one where there is a small spatial irregularity i n number density. To investigate at what frequencies the resulting induced electric f i e l d may be significant compared to the inducing f i e l d (a whistler), steady state solutions were obtained for the elec t r i c and magnetic fields that may exist in a f u l l y ionized plasma that has a small spatial irregularity in number density. The plasma is taken to be i n a constant and uniform background magnetic f i e l d and to have parameters consistent with the upper ionosphere. The irregularity i s taken to be a spatially varying cosine function with wave number K. Assuming the governing equations to be Maxwell's equations and the zeroth and the f i r s t moment equations of the collisionless Boltzmann equation, we obtained solutions with a perturbation scheme. The equations were linearized and terms were only kept to second order. The f i r s t order terms formed a set of equations governing a plasma with unperturbed number densities that were constant in time and space. For f i r s t order variables that are plane waves with wave number k and frequency co, the postulated irregularity gives rise to a second order electric f i e l d with a frequency dependence of CJ. An investigation was made to determine at which frequencies the second order electric f i e l d was significant compared to f i r s t order f i e l d s . For k parallel to K and perpendicular to B ^ , i t was found that the second order f i e l d had a peak value at the LHR (lower hybrid resonance) - i i i --3 -1 frequency. For K of the order of 10 cm , an additional peak occurred for a frequency less than the LHR frequency, when K = 2k. With K -4 -2 -1 increasing from 10 to 10 cm , this frequency increased from 36% to within .3% of the LHR frequency. Neglecting the second order magnetic field, solutions were obtained for k in the x-z plane, B^^ in the positive z-direction, and K in the positive x-direction. For 9, the angle formed by lc and B ^ , not equal to 90°, the second order electric field had a peak that was greater than the LHR frequency. For 6 = 71.57°, the frequency of the peak changes from 1.005 to 31 times the LHR frequency as K varies from 10 to 10 cm . For K = 10 cm , the frequency of the peak changes from 1 to approximately 3.5 times the LHR frequency as 8 varies from 90° to 0°. - iv -TABLE OF CONTENTS ABSTRACT LIST OF FIGURES LIST OF TABLES ACKNOWLEDGEMENTS CHAPTER 1 INTRODUCTION 1.1 General Perspective of Thesis Problem 1.2 Observational Impetus 1.3 Proposed Thesis Problem CHAPTER 2 GOVERNING EQUATIONS CHAPTER 3 COLD PLASMA 3.1 First Order Solutions 3.2 Second Order Solutions 3.2.1 Electrostatic Case a Wave Number Perpendicular to Background Magnetic Field b Wave Number Not Perpendicular to Background Magnetic Field 3.2.2 Electromagnetic Case CHAPTER 4 HOT PLASMA 4.1 First Order Solutions 4.2 Second Order Solutions 4.2.1 Electrostatic Case 4.2.2 Electromagnetic Case - v -TABLE OF CONTENTS (Contd.) Page CHAPTER 5 CONCLUSION 92 5.1 Some Other Theories of Lower Hybrid Noise 92 5.1.1 Polar LHR Noise 92 5.1.2 Mid-Latitude LHR Noise 94 5.2 Brief Summary and Results of Proposed Theory 96 5.2.1 Wave Number ic J _ B ( 0 ) and || K (= K x) 98 a Electrostatic Assumption 98 b Removing the Electrostatic Assumption 99 5.2.2 Wave Number lc in Arbitrary Direction with Respect to B v J and K (= K^x) 100 5.3 Some Additional Comments and Some Possible Future Investigations 101 BIBLIOGRAPHY 103 APPENDIX I FIRST ORDER SOLUTIONS FOR A COLD PLASMA 108 APPENDIX II SECOND ORDER SOLUTIONS FOR A COLD PLASMA-ELECTROSTATIC CASE 110 APPENDIX III SECOND ORDER SOLUTIONS FOR A COLD PLASMA-ELECTROMAGNETIC CASE 115 APPENDIX IV FIRST ORDER SOLUTIONS FOR A HOT PLASMA 117 APPENDIX V SECOND ORDER SOLUTIONS FOR A HOT PLASMA-ELECTROSTATIC CASE 118 APPENDIX VI SECOND ORDER SOLUTIONS FOR A HOT PLASMA-ELECTROMAGNETIC CASE 12° APPENDIX VII INDEX OF REFRACTION OVER FREQUENCY INTERVALS RELATED TO THIS STUDY 122 - v i -LIST OF FIGURES Figure Page 1-1 3-1 Examples of LHR noise recorded on the Alouette 1 VLF receiver for a few invariant latitudes - A (after McEwen and Barrington, 1967). ' - l o g i n | n ^ 0 ) E ( 2 ) / ( n E )\ °101 e x e,s o 1 The value of y i = xo 1 0 in - - is - - / i.n E ; | | as a [ °101 e x e,s o 2 2 (2) function of x ( = w /to ), where E i s given by equation. (3-49) (cold v electrostatic case), k | | K , I -*-(0) k ]_ B v % and X = 0. 31 3-2 3-3 3-4 3-5 The value of = l o g l J n ( 0 ) E ( 2 ) / ( n E )| 610 1 e x e,s o' 1 as a 2 2 (2) function of x ( = to /w^), where E^ ' is given by equation (3-59) (cold, electrostatic case), and X = 0. = 71.57°, K = 10~2 cm"1, x ' The value of y y = iog 1 ( )|n e - JS^ ' f^&tS^0) I I as a 2 2 (2) function of x ( = OJ /to ), where E is given by equation (3-59) (cold, electrostatic case), and X = 0. 9 = 71.57°, K = 10"2 cm"1 x The value of y as a 2 2 (2) function of x ( = co /u T„), where E is given by J-ttl X equation (3-59) (cold, electrostatic case), and X = 0. 6 = 71.57°, K = 5 x 10"3 cm"1 ' x as a The value of y [ = l o g 1 0 | n f } E ^ 2 ) /fr^J I 2 2 (2) function of x ( = to /w ), where E is given by Lri X equation (3-59) (cold electrostatic case), and X = 0. 0 = 71.57°, K = 5 x 10"3 cm"1, x ' 37 38 39 40 - v i i -LIST OF FIGURES (Contd.) Figure 3-6 3-7 3-8 3-9 3-10 3-11 The value of y = log 1 ( )| n^ 0 )E^ 2 ) / ( n , ^ ) | as a 2 2 (2) function of x ( = oo /w^), where E x is given by equation (3-59) (cold, electrostatic case), and X = 0. 9 » 71.57°, K = 10"3 cm"1 x as a The value of y [ = l 6 g 1 0 1 n< 0 )E< 2 ) / ( n , ^ ) | 2 2 (2) function of x ( = GO /u>TU), where E is given by Lai X equation (3-59) (cold, electrostatic case), 6 a 71.57°, K =5 x 10~4 cm"1, x ' and X = 0. The value of y ' = l o g i n | n ( 0 ) E ( 2 ) / ( n E )\ °ini o v v e,s o 1 (2) as a '101 e x is given by equation (3-59) (cold, electrostatic case), and X = 0. 2 2 function of x ( = OJ /a) ), where E Litl X a 71.57°, K = 10"4 cm"1, x The value of y " l ° S i o l n f 0 ) E i 2 ) / ( n , .E,)| e,s o as a 2 2 (2) function of x ( = oo /to ), where E is given by L i l X equation (3-59) (cold, electrostatic case), 9 = 89.94°, K = 10~3 cm"1, and X = 0. x The value of y = l o g l J n ( 0 ) E ( 2 ) / ( n E ) °101 e x e,s o as a 2 2 (2) function of x ( = GO /^JJ)* where E^ is given by equation (3-59) (cold, electrostatic case), and X = 0. 6 = 89.82°, K = 10"3 cm"1, x The value of y = l o g l J n ( 0 ) E ( 2 ) / ( n E )| e,s o as a 2 2 (2) function of x ( = 00 /in ), where E is given by lit! X equation (3-59) (cold, electrostatic case), 9 = 89.43°, K = 10"3 cm"1, and X = 0. x Page 41 42 43 44 45 46 - v i i i -LIST OF FIGURES (Contd.) Figure 3-12 3-13 3-14 3-15 3-16 The value of y = l o g i n | n ( 0 ) E ( 2 ) / ( n E )| &10 1 e x e,s o 1J x as a 2 2 (2) function of x ( = co /to ), where E is given by Lai X equation (3-59) (cold, electrostatic case), and X = 0. 9 = 71.57°, K = 10"3 cm" 1 x The value of y = l O g i n | n ( 0 ) E ( 2 ) / ( n E )| 10 " v ° ° <~> ' e,s o as a 2 2 (2) function of x ( = to /toT U) , where E is given by Jjri X equation (3-59) (cold, electrostatic case), = 50.77°, K = 10 -3 -1 cm , and X = 0. The value of y '=log 1 0|„<°> E( 2>/(„ E ) | ] as a 2 2 (2) function of x ( = to /<oTU), where E is given by J_tTl X equation (3-59) (cold, electrostatic case), and X = 0. 9 = 33.21°, K = 10" 3 cm" 1 x The value of y x as a 2 2 (2) function of x ( = to /<A) t u), where E is given by i-iTl X equation (3-59) (cold, electrostatic case), 18.43°, K = 10 -3 x cm and X = 0. The value of y = l o g i n | n ( 0 ) E ( 2 ) / ( n E )| e.s o as a 2 2 (2) function of x ( = to / ^ J J J ) » where E ^ is given by equation (3-59) (cold, electrostatic case), 9 = 0 ° , K =10 -3 -1 cm , and X = 0 Page 47 48 49 50 51 3-17 The value of y [ - log l f t|nf°>E< 2>/(n e > sE o)f (2) '10' 2, 2 as a function of x ( = <o /to T U), where E ijii x is given by equation (3-80) (cold, electromagnetic case), ^ I I K, , and X = 0. £ l B - ( 0 ) , K ^ l O ^ c m " 1 59 - i x -LIST OF FIGURES (Contd.) Figure 3-18 3-19 The value of y = l o g l J n ( 0 ) E ( 2 ) / ( n E )| TO' p x e,s o ' '10 e2 2 (2) f u n c t i o n of x ( = to / O J , „ ) , where E v ' i s given by Lift X equation (3-80) ( c o l d , electromagnetic case), it | | K , t J _ B " ( 0 ) , K X = 5 x 10~ 3 cm"1, and X = 0. as a The value of y = l o g i n | n ( 0 ) E ( 2 ) / ( n E )l °10 i P x v e , s o 1 J as a 2 2 (2) f u n c t i o n of x ( = OJ /oo^), where E^ i s given by equation (3-80) ( c o l d , electromagnetic c a s e ) , It | | t ] _ $ ( 0 ) , K = 10" 3 cm"1, and X = 0. Page 60 61 3-20 3-21 4-1 4-2 The value of y = l o g i n | n ( 0 ) E ( 2 ) / ( n E )[ 10 1 e x f - e <•» i e,s o as a 2 2 (2) f u n c t i o n of x ( = OJ /fcoT„), where E v ' i s given by J-jit X equation (3-80) ( c o l d , electromagnetic case), ic || K, £ 1 5 ( 0 ), K x = 5 x 10"4 cm" 1, and X = 0. The value of y as a 2 2 (2) f u n c t i o n of x ( = OJ / ^ L H ) , where E^ ' i s given by equation (3-80) ( c o l d , electromagnetic case), k || K, * l B - ( 0 ) ; -4 -1 K = 10 cm , x and X = 0. The value of y = l o g i n | n f 0 ) E ( 2 ) / ( n E )| °i(>' c x e,s o 1 '10'"e2 2 (2) fu n c t i o n of x ( = OJ /u^), where ' i s given by equation (4-39) (hot, e l e c t r o s t a t i c c a s e ) , lc | | K, £ J _ B ( 0 ) , K x = 10" 2 cm"1, and X = 0. as a The value of y as a 2 2 (2) f u n c t i o n of x ( = OJ /OJ ) , where E i s given by L n X equation (4-39) (hot, e l e c t r o s t a t i c case), it | [ K, < ( 0 ), K x - 5 x 10" 3 cm"1, and X = 0. 62 63 74 75 - x -LIST OF FIGURES (Contd.) Figure 4-3 4-4 The value of y = l o g l J n ( 0 ) E ( 2 ) / ( n E )| 10 e "v o G n' I e,s o as a 2 2 (2) function of x ( = OJ / O J,„), where E i s given by equation (4-39) (hot, electrostatic case), k | | K, and X = 0. ti^°\ K x =10" 3 c m " 1 The value of y | = l o g j n f V 2 ^ Q E n ) | '10' e x ' " e,s o' 2 2 (2) function of x ( = OJ / W ^ ) » where E^ ' is given by equation (4-39) (hot, electrostatic case), It | | K, & 1 B"(0), K = 5 x 10 - 4 cm-1, and X = 0. as a Page 76 77 4-5 4-6 4-7 4-8 The value of y as a 2 2 (2) function of x ( = OJ /oo^), where E x y is given by equation (4-39) (hot, electrostatic case), It | | K, £ ] _ S ( 0 ) , K x = 10 - 4 c m 1, a n d X = 0. The value of y • 1 - « u l - i 0 > ^ 2 > « » . . . V l as a e,s o 2 2 (2) function of x ( = OJ /^ L R)» where E v ' is given by in-equation (4-39) (hot, electrostatic case), k || K, r ( 0 ) , K, = 1 0 - 2 cm-1, and X = 0. x The value of y = l o g 1 0 | „ ( 0 > E < 2 > / ( n e ; S E o ) | ' as a 2 2 (2) function of x ( = OJ / " J J J ) » where E^ ' is given by equation (4-39) (hot, electrostatic case), It | | K, ;»• i -K0) -3 -1 k J_ B^ u ;, K x = 10 J cm \ and X = 0. The value of y = l o g i n | n ( 0 ) E ( 2 ) / ( n E )| &m' P x e,s o 1 '101 e2 2 (2) function of x ( = OJ / W ^ ) , where E x 7 is given by equation (4-39) (hot, electrostatic case), It | [ K , £ _ L B ( 0 ) , K X = 10 - 4 cm-1, and X = 0. as a 78 79 80 81 - x i -LIST OF FIGURES (Contd.) Figure 4-9 4-10 4-11 4-12 4-13 The value of y ' - l o 8 l 0 | n f V » / ( n V | ] as a 2 2 (2) function of x ( = to / c o ^ ) , where E w is given by e q u a t i o n (4-69) ( h o t , e l e c t r o m a g n e t i c c a s e ) , k || K, t _ L K = 10"2 c m " 1 , a n d A = 0. The value of y - 1 ° H o l » i ° > E " V , s E o > l ' as a 2 2 (2) function of x ( = to /u^) , where E^ 7 is given by equation (4-69) (hot, electromagnetic case), £ || K, £ 1 B ( 0 ) , K x = 5 x IO - 3 cm"1, and A = 0. The value of y = l o g i n l n ( 0 ) E ( 2 ) / ( n E )| [ 101 e x e,s o 1 as a 2 2 (2) function of x ( = to / ^ L H ) , where E v ' i s given by equation (4-69) (hot, electromagnetic case), k || K, ^ 1 ? ( 0 ) , K = 10"3 cm"1, and A = 0. The value of y = l o g l J n f 0 ) E ( 2 ) / ( n E )| °10' e x e,s o 1 as a 2 2 (2} function of x ( = to / w ^ ) , where E is given by -»• equation (4-69) (hot, electromagnetic case), k || K, -> , ->(0) -4 -1 k 1 B t K = 5 x 10 cm , and A = 0. The value of y = l o g i n | n ( 0 ) E ( 2 ) / ( n E )| "101 e x e,s o 7 1 2 , 2 , , • „(2) as a function of x ( = to / t o T T T ) , where E LH x is given by equation (4-69) (hot, electromagnetic case), k || K, , and A = 0. k l B - ( 0 ) , K x = IO" 4 cm"1 Page 87 88 89 90 91 - x i i -LIST OF FIGURES (Contd.) F igure Page 2 A-VI I -1 The value of n versus x where 6 = 90° 123 2 A-VI I -2 The value of n versus x where 9 = 90° 124 2 A-V1I-3 The value of n versus x where 9 = 90° 125 2 A-VI I -4 The value of n versus x where 9 = 8 9 . 9 4 ° 126 2 A-VI I -5 The value of n versus x where 9 = 8 9 . 8 2 ° 127 2 A-VTI-6 The value of n versus x where 9 = 8 9 . 4 3 ° 128 2 A-VI I -7 The value of n versus x where 8 = 7 1 . 5 7 ° 129 2 A-VIIr-8 The value of n versus x where 9 = 7 1 . 5 7 ° 130 2 A-VII -9 The value of n versus x where 8 = 7 1 . 5 7 ° 131 2 A-VTI-IO The value of n versus x where 8 = 5 0 . 7 7 ° 132 2 A-VII-11 The value of n versus x where 8 = 3 3 . 2 1 ° 133 2 A-VII-12 The value of n versus x where 9 = 1 8 . 4 3 ° 134 2 A-VII-13 The value of n versus x where 9 = 0 ° 135-- x i i i -LIST OF TABLES Numerical Values for Plasma Parameters Consistent with the Ionosphere at 1000 km and 5 5 ° Magnetic Latitude - xiv -ACKNOWLEDGEMENTS I would like to thank Kathy for her moral support and for her patience over the years of this study. For his advice and encouragement and for many long hours of discussion on this and other topics, I would like to thank Dr. T. Watanabe. I am also indebted to him for his careful review of the f i n a l manuscript. Additional thanks are extended to Gerard Nourry for our discussions of my thesis problem. For the speed and the care with which she typed this thesis, I thank Mrs. Yit-Sin Choo. This study has been supported by the people of Canada through the National Research Council. For this support the author i s grateful. CHAPTER 1 INTRODUCTION 1.1 General Perspective of Thesis Problem In studying electromagnetic waves in the ionosphere, one can usually apply the simple model of a uniform density plasma immersed in a uniform, time independent magnetic field. Along with Maxwell's equations, the continuity equations of number density and of momentum have commonly been taken as the governing equations. It would seem that an interesting extension of this would be to allow a small spatial inhomogeneity in the density of the unperturbed plasma (small with respect to the background plasma densities). It would be expected that the solution to the above problem would be quite dependent on various plasma parameters such as plasma frequencies, cyclotron frequencies, etc. and on the frequency range considered. In this study numerical results are obtained for plasma parameters consistent with the ionosphere at 1000 km and for frequencies around the lower hybrid resonance frequency - roughly speaking, for frequencies much greater than the positive ion cyclotron frequencies and much less than the electron cyclo-tron frequency. In the next section, a few remarks are given on some satellite observations which led to this study of electromagnetic waves in an inhomogeneous, anisotropic plasma and with frequencies around the lower hybrid resonance frequency. The last section of this chapter gives a more exact statement of the problem and the various stages of calculations - 2 -pursued. However, let us emphasize that the formalism to be introduced can be applied to a l l frequencies and to plasmas other than the upper ionosphere. 1.2 Observational Impetus In September of 1962 the Alouette 1 s a t e l l i t e was launched into an approximately circular orbit with an altitude of about 1000 km and i n c l i -nation of 80° to the equator. On board was a very low frequency (VLF) receiver that measured electric fields by means of a 150 foot dipole antenna. The frequencies observed were in the range of 6 to 10 kilocycles per second. Frequently within the VLF measurements were noise bands with a sharp lower cutoff frequency (Barrington and Belrose, 1963). Comparison with ground studies show that the new type of noise was not seen by VLF receivers on the ground (Brice and Smith, 1964). As yet no generally accepted theory for the noise generation exists; though i t is f e l t that the noise i s due to a resonant phenomenon and that the lower cutoff frequency occurs at the lower hybrid resonance frequency of the background plasma (Brice and Smith, 1964, 1965). As a consequence, this electric noise has come to be called lower hybrid resonance (LHR) noise. It had also been noted that the lower frequency cutoff varied consistently with latitude. Additional observations were that some noise seemed to be triggered by whistlers and that the triggering waves propagated with wave numbers that were at large angles with respect to the background magnetic f i e l d - nearly perpendicular (Brice et a l . , 1964; Brice and Smith, 1964). - 3 -Examples of LHR noise are given i n F igure (1-1) (McEwen and Bar r ing ton , 1967). Character ized by a l a rge bandwidth, the f i r s t example i s of h igh l a t i t u d e noise - more than 6 5 ° i n v a r i a n t l a t i t u d e 1 . The f a i r l y r a p i d v a r i a t i o n i n the lower cu to f f frequency i s f e l t to be due to v a r i a t i o n s i n the e l e c t r o n dens i ty i n the medium. With t h e i r more narrow bandwidths, the next three examples are of m i d - l a t i t u d e no ise - l e s s than 6 5 ° i n v a r i a n t l a t i t u d e . Examples two and three are burs t types wi th respect ive durat ions of one and f i v e seconds. They are a lso r e f e r r e d to as w h i s t l e r assoc ia ted n o i s e . Comprising three - four th of the m i d - l a t i t u d e no ise i s the continuous type, i l l u s t r a t e d by the l a s t example. However, McEwen and Barr ington (1967) po in t out that even the continuous types have w h i s t l e r s imbedded i n t h e i r records and thus they don ' t d iscount t h e i r be ing w h i s t l e r t r i g g e r e d . From i n v e s t i g a t i o n s of measurements taken from many other s a t e l -l i t e s , LHR noise i s now known to occur at a l t i t u d e s from 400 km to 4 earth r a d i i (Injun 5, Gurnett et a l . , 1969; Ogo 2, Laaspere et a l . , 1969; Ogo 4 , Lasspere and T a y l o r , 1970; Ogo 6, Laaspere and Johnson, 1971, Laaspere et a l . , 1971; A louet te 2, Barr ington et a l . , 1971; Ogo 5, Scar f et a l . , 1972 A , 1972 B; and Ogo 3, B u r t i s , 1973). From these i n v e s t i g a t i o n s i t i s concluded that h igh l a t i t u d e (polar) LHR noise and m i d - l a t i t u d e noise are u s u a l l y found i n l a t i t u d e b e l t s of 6 5 ° to 85° and of 4 0 ° to 6 5 ° , r e s p e c t i v e l y . As to d i u r n a l v a r i a t i o n s i t has been found that po la r LHR noise occurs s l i g h t l y more f requent ly at mid-day whi le m i d - l a t i t u d e noise Invar iant l a t i t u d e s can roughly be taken to be magnetic l a t i t u d e s . - A -Figure (1-1). Examples of LHR noise recorded on the Alouette 1 VLF receiver for a few invariant latitudes - A (after McEwen and Barrington, 1967). - 5 -occurs mostly at night (McEwen and Barrington, 1967; Laaspere et a l . , 1971; and Barrington et a l . , 1971). Besides bandwidth, polar and mid-latitude LHR noise appear to dif f e r i n that the former seems to be electromagnetic (Barrington et a l . , 1971) and the latter i s usually electrostatic (Gurnett et a l . , 1969; and Laaspere et a l . ) . However Laaspere and Taylor (1970) and Burtis (1973) have found some cases of mid-latitude LHR noise that have a magnetic component. Burtis found that his examples of noise always seemed to be electrostatic, except those that were obviously whistler triggered. In 1972 Gross and Larocca pointed out that mid-latitude hiss should not be considered to be caused by one type of source. Gross and Larocca divided LHR hiss into wide and narrow band hiss. Corresponding to McEwen's and Barringtons polar (high latitude) LHR hiss, the wide band hiss has been reported by Laaspere et a l . (1971) and Barrington et a l . (1971) to go from VLF (of the order of ten kHz) to LF (of the order of hundreds of kHz). Gross and Larocca subdivided the narrow band hiss (with a bandwidth of the order of a kHz) into smooth cutoff, whister associated, and erratic cutoff hiss. The c r i t e r i a for the subdivisions were the dura-tion of noise, variation of lower frequency cutoff, and association with other VLF events. The smooth cutoff hiss has a duration of minutes, a smooth slowly Varying lower frequency cutoff, and no apparent association with other VLF events. The whistler associated hiss has a duration of seconds, a smooth cutoff frequency, and an association with whistlers. The erratic cutoff hiss has a duration of seconds, a rapid change of cutoff frequency, and no obvious associations with other VLF events. Similarly - 6 -descr ibed , the wide band h i s s has a dura t ion of minutes, u s u a l l y an e r r a t i c cutof f f requency, and some a s s o c i a t i o n s wi th w h i s t l e r s . With the above s u b d i v i s i o n of m i d - l a t i t u d e h i s s , Gross and Larocca (1972) n a t u r a l l y f e e l that the smooth h i s s needs an o r i g i n other than d i r e c t t r i g g e r i n g by w h i s t l e r s . However they do s ta te that "many of the remaining events seem to be assoc ia ted wi th w h i s t l e r s , and i t appears reasonable to b e l i e v e that wh is t l e rs are t h e i r source" . I ts to these events that the model of a wave (whist ler ) propagating i n a s l i g h t l y inhomogeneous medium would seem most re levan t . 1.3 Proposed Thesis Topic As int imated i n the previous sec t ions the problem to be pursued has been st imulated from the observat ion that w h i s t l e r s seem to t r i g g e r at l e a s t some of the observed LHR n o i s e . I f one thinks of the w h i s t l e r as an inducing f i e l d , then one mxght th ink of the no ise as an induced f i e l d . What i s a simple way of producing an induced f i e l d ? I f there i s a time independent (at l e a s t fo r the time s c a l e under considerat ion) i r r e g u l a r i t y i n the background number densi ty of a neut ra l plasma, then a time vary ing e l e c t r i c f i e l d would tend to produce charge separa t ion , r e s u l t i n g i n an induced (po la r i za t ion ) f i e l d . For a p a r t i c u l a r frequency, say the LHR frequency, the ' induced f i e l d may be r e l a t i v e l y l a r g e , and that frequency would be a type of resonant frequency. Thus the proposed o r i g i n of w h i s t l e r t r iggered LHR noise i s a wave propagating from a region of the ionosphere where the unperturbed ion number d e n s i t i e s are uniform i n t o one where there are s p a t i a l i r r e g u l a r i t i e s . - 7 -With the above preamble i n mind, we have chosen to i n v e s t i g a t e a plasma i n a magnetic f i e l d that i s uniform and time independent and where th each i component of the plasma has a s p a t i a l i r r e g u l a r i t y which i s smal l compared to i t s background densi ty and which v a r i e s as a constant times a cosine f u n c t i o n , wi th wave number K . Thus n. (r) = n cos (K- r + IJJ) (1-1) X) S 1)s where r i s the s p a t i a l v a r i a b l e and i s an a r b i t r a r y phase angle , I t i s assumed that the plasma i s governed by Maxwell 's equations and the equations of c o n t i n u i t y and of momentum. By a standard per turba t ion method, these equations are so lved f o r e q u i l i b r i u m s o l u t i o n s where the f i r s t order terms are assumed to have plane wave s o l u t i o n s : e . g . the x-component of the e l e c t r i c f i e l d i s taken to be E^ l y ( r " , t) = E cos(lc-r - tot - £) (1-2) where E q i s a constant , lc the wave number, and £ a phase angle . The f i r s t order terms form a set of equations that govern waves that may e x i s t i n a plasma where the background number dens i ty i s uni form. Thus the f i r s t order e l e c t r i c f i e l d can be thought of as our inducing f i e l d ( w h i s t l e r ) . Having assumed the existence of plane wave s o l u t i o n s fo r the f i r s t order terms, then the second order equations imply n o n - t r i v i a l second order terms, because of the ex is tence of the postu la ted inhomogeneity i n number dens i ty . I t i s the second order e l e c t r i c f i e l d that we are i d e n t i f y i n g as an induced e l e c t r i c f i e l d (no ise ) . - 8 -The c a l c u l a t i o n s are done i n s e v e r a l cases . Grea t ly s i m p l i f y i n g the problem, the f i r s t s o l u t i o n s are obtained fo r a co ld plasma and wi th an e l e c t r o s t a t i c assumption - assuming no second order magnetic f i e l d . This c a l c u l a t i o n i s done f o r an inducing f i e l d - the f i r s t order e l e c t r i c f i e l d - whose wave normal i s i n an a r b i t r a r y d i r e c t i o n wi th respect to the background magnetic f i e l d . Secondly, the complete e lectromagnet ic problem i s so lved but only f o r an inducing f i e l d whose wave normal i s perpendicu lar to the background magnetic f i e l d . Corresponding cases are done fo r a hot plasma but again only for a wave normal of the inducing f i e l d that i s perpendicular to the background magnetic f i e l d . In these c a l c u l a t i o n s we are t r y i n g to see how the resonant f requencies are changed by tak ing the temperature of the plasma i n t o account. For a l l the above c a l c u l a t i o n s , i n t e r p r e t a t i o n s were only done fo r a postu la ted i r r e g u l a r i t y that could be character i zed by a s p a t i a l wave number that was perpendicu lar to the back-ground magnetic f i e l d - - and was i n the plane formed by the wave number of the induc ing f i e l d and . In numerical c a l c u l a t i o n s , the plasma parameters given i n Table 1-1, were chosen to be cons is ten t wi th ionospher ic condi t ions at 1000 km and at m i d - l a t i t u d e s - fo r example 5 5 ° magnetic l a t i t u d e (Barr ington , 1969; Hoffman, 1969; Hanson et a l . , 1970; Serbu and Maier , 1970; Evans, 1970; B r i n t o n , 1970; Wa ld tenfe l , 1971; Ahmed and Sagalyn, 1972; and F e l d s t e i n and G r a f f , 1972). The background plasma i s assumed to be n e u t r a l and to be composed of e lec t rons (e ) and s i n g u l a r l y i o n i z e d hydrogen ( H + ) , + + helium (He ) , and oxygen (0 ) . In a d d i t i o n , the value p icked for the LHR frequency corresponds to a t y p i c a l lower cu to f f frequency observed - 9 -by the Alouette 1 s a t e l l i t e . Picking 5 kH as the LHR frequency and z assuming charge neutrality, one is rather limited in the relative fractional abundances of positive ions that are permitted. In fact with the given magnetic f i e l d , electron number density, and LHR frequency; the relative abundance of hydrogen (H+) can only vary from 0% to approximately 4.4%. Thus for numerical purposes, the respective relative abundances of + + + hydrogen (H ), helium (He ), and oxygen (0 ) were arbi t r a r i l y chosen to be 2%, 12.104%, and 85.896% . - 10 -TABLE 1-1 Numerical Values fo r Plasma Parameters Consistent with the Ionosphere at 1000 km and 5 5 ° Magnetic L a t i t u d e . Magnitude of the e a r t h ' s magnetic induct ion f i e l d = | B ^ | = .35 gauss. Background e l e c t r o n number densi ty = = 10^ cm 3 . Temperature of background plasma = T = 1667°K. T y p i c a l value of LHR frequency = OJ t„/2TT = 5 kH . The respect ive percentages of p o s i t i v e ions - that are hydrogen ( H + ) , hel ium (He + ) , and oxygen (0 + ) - are 2%, 12.104% and 85.896%. i o n type c y c l o t r o n frequency ( rad/sec) plasma frequency ( rad/sec) e l e c t r o n (e ) hydrogen (H ) I to I = 6.16 x 10 1 e 1 3^ = 3.35 x 10-n = 5.64 HT e n H = 1.86 10 hel ium (He ) o)^ = 8.45 x 10 IL, = 2.30 10 He oxygen (0 ) oo 0 = 2.11 x 10 n Q = 3.06 10 ion type e l e c t r o n (e ) thermal v e l o c i t y (cm/sec) a = 1.59 x 10 7 e hydrogen (H ) *H = 3.71 x 10" hel ium (He ) a R e = 1.86 x 10" oxygen (0 ) a Q = 9.31 x 10 - 11 -CHAPTER 2 GOVERNING EQUATIONS The equations, in cgs-Gaussian units, assumed to be governing our plasma are Maxwell's equations : with (2-2) V-t = 0 (2-3) V-E = 4TT q ±n i (2-4) J = V q.n.v. (2-5) h x x x and the equations of mass and momentum conservation (the zeroth and f i r s t moment equations of the Boltzmann-Vlasov equation) for each ion (negative or positive) component : 3n. j± + V- (n.v.) =0 (2-6) J± = ! i (S + I * x f) _ _ M p > (2_7) dt m. c x n.m. rx x x x In the above B and E are the magnetic and electric fields; J the current density; q^, nu, n^, v^, and p_^  the charge, mass, number density, velocity, and pressure of each particle type - designated by subscript i ; t - 12 -the t ime; c the v e l o c i t y of l i g h t ; and £ stands f o r the sum over each i plasma component. The l a s t term of equation (2-7) assumes a s c a l a r p ressure , and we s h a l l now fur ther assume that each ion gas behaves i n an isothermal manner, thus 2 * 1 * a i V n i — v P ± = — — - ( 2 _ 8 ) i 1 l where 2 a^ (commonly c a l l e d the thermal v e l o c i t y ) = k^T^/m^ (2-9) In equation (2-9) k i s the Boltzmann's constant and T. i s the o 1 temperature of the i o n component under c o n s i d e r a t i o n . In order to so lve the equations governing wave phenomenon i n an i r r e g u l a r plasma, we w i l l use a per turbat ion expansion of a l l the v a r i a b l e s , This w i l l done fo r a hot plasma. N a t u r a l l y , the equations governing a co ld plasma emerge i n the l i m i t of zero temperature and thus zero thermal v e l o c i t y . Let -> E = g-U) + f (2) i -• B - ( 0 ) + f C« + J(2) + . + . -> v i = v( 1 > + v ( 2 ) l l + . n. = l nf°> + nf1) + nj2> i i i + . (2-10) where the zeroth order terms are constant average values and the f i r s t and - 13 -second order terms - designated by s u p e r s c r i p t s (1) and (2) - are such that | B ^ | » | B ( 2 ) | , | E ( 1 ) | » | F ^ 2 ) | , e t c . In our problem we have a non-zero J B ^ | and | n £ 0 ^ | that are r e s p e c t i v e l y >> | and | n ^ | , but we have no given , , and thus The problem at hand i s to so lve for p o s s i b l e waves that e x i s t due to a given f i r s t order inhomogeneity i n number d e n s i t y . Thus the f i r s t order dens i ty i s w r i t t e n as a sum of n f"^ , the postu la ted i r r e g u l a r i t y term, and n f 1 ^ , the f i r s t order per turba t ion term - corresponding to the inducing f i e l d : ~(1) _ (1) , (1) . . . n i " n i , s + n i ( 2 _ 1 1 ) Using the expansions given by equations (2-10) and (2-11) i n equations (2-1) to (2 -7) , one f inds that the f i r s t order equations are v x j(D = AijCD + c c 3t (1) v x fCD . _ 1 JL c 9t ^•f ( 1> - 0 v . ^ ^ . l q j n a ) , ^ ) ) = I q.nf°^P V i i i (1) 3n "3t ± - + (n™#») - 0 ^ I. i J HI) N * ( D „ 2+f ( i ) M a)\ 3 v i ^ - i E A / n i / M a . V l n ; ' + n ; M 3t m. m.c1- l ' (0) i i n. 1 l - 14 -The second order equations are -»• -+(2) 4£ +(2) 1 9E c c 3t (2) -»• -K2) V x E v".f<2> -c at V J ( 2 > = 4, I q.n<2> j i i ^ + v . ( n f ) ^ ) + ( n a ) + » a ) ) ? a)l 3n; "3 = 0 8 v < 2 > r - + PP'I)*™ - ^  ? ( 2 ) •+ ^ [ v f 2 ) x J(0) + V ( D x 5 ( D u x in, c l 1 i n (0) Vn (2) .CD n. 1 v" n (0) i i,s .(1) nf°> x Assuming f i r s t order wave terms with a time dependence of the form e-joot^ above two sets of equations have terms with time dependencies which are static, e ± J a ) t , and e ± j 2 u t (j is taken to be v^T). Terms of like time dependence must form independent sets of equations. Only that set of equations whose terms have a time dependence of the form e±'-'ajt w i l l be solved. In doing so one is ignoring the wave coupling terms, (v^ 1^ .V^-^ and m^ c x JCD X , which would exist even i f the postulated irregularity did not. In addition, one has ignored the mechanism for the very existence - 15 -o f the pos tu la ted i r r e g u l a r i t y . However an order of magnitude est imate of the d i f f u s i o n time fo r our postu la ted i r r e g u l a r i t y i s 2 -5 T - 8.5 x 10 sec cm . . AD (1) T r2 U L l ) n K e , s where n g ^ s 1 S t n e magnitude of the number dens i ty of the i r r e g u l a r i t y and K i s the wave number that charac te r i zes the i r r e g u l a r i t y . Equat ion (2-12) i s obtained by d i v i d i n g a c h a r a c t e r i s t i c l e n g t h , the wavelength of our postu la ted i r r e g u l a r i t y , by the c l a s s i c a l d i f f u s i o n v e l o c i t y t ransverse to a s t rong magnetic f i e l d , g iven by S p i t z e r (1962). In h i s c a l c u l a t i o n a s t rong magnetic f i e l d i s one where the gas pressure f o r the plasma i s much l e s s than the magnetic p r e s s u r e ; t h i s i s true i n our study to the extent that the r a t i o of the former to the l a t e r i s about 10 S p i t z e r ' s d e r i v a t i o n was for a two component, f u l l y i o n i z e d plasma. In our case there are three p o s i t i v e ions but a l l are s i n g u l a r l y i o n i z e d and thus f o r an order of magnitude c a l c u l a t i o n S p i t z e r ' s d i f f u s i o n v e l o c i t y should be adequate. In a d d i t i o n , 0 + comprises 86 percent of the p o s i t i v e (1) 3 -3 ions i n the model chosen. In our model n i s l e s s than 10 cm and s , e -2 -4 -1 we s h a l l be cons ider ing K ranging from 10 to 10 cm . Thus 3 7 i s greater than 8.5 x 10 to 8.5 x 10 s e c , r e s p e c t i v e l y . Thus our postu la ted i r r e g u l a r i t y would e s s e n t i a l l y e x i s t i n t a c t f o r phenomenon of the order of a few seconds to a few minutes. In summary, the f i r s t order equations governing wave s o l u t i o n s with a e _ - ' a ) t time dependence are - 16 -v- x JC1) 4TT_ +(1) 1 / 3 E c c at (1) V x E (1) 1 3B (1) c at V\B-(1> = o = 47T I q i n i ( i ) (owi) = 4 q ± n i v ± at + n ± avi^ q = o at i I i * U ) , 4 ± -*(i) -KO) - h. H v. x B -m. m. c I x 4 H n n<°> X (2 -13) (2 -14 ) (2 -15 ) (2 -16 ) ( 2 -17 ) (2 -18 ) (2 -19 ) For second order terms, the equations governing wave s o l u t i o n s with a time dependence of e ± : ' ' o t are V x B V ' v" x E " ( 2 ) -¥ -+(2) +(2) 4TT J ( 2 ) 1 3 E ( 2 ) c c at 1 3B (2) c at = o 3n at J ( 2 ) (2) 1 4, I ^ i M 1 1 i , s x k(0)?(2) • o W l - 0 X X X , S X (2 -20 ) ( 2 -21 ) (2 -22) ( 2 -23 ) (2 -24) (2 -25 ) - 17 --K2) 1 3t m. m. c 1 v ± ( 2 ) x f CO) n (0) -> (2) n i * (1) T O T V n i , s i n ( 1 ) ^ V n ( 1 )  n ( 0 ) x (2-26) - 18 -CHAPTER 3 COLD PLASMA 3.1 First Order Solutions The f i r s t order equations are solved for plane wave solutions assuming a zero temperature plasma. From this set of equations i t is found that plane waves can exist i f a certain dispersion relation i s satisfied. What are these waves? In that the f i r s t order equations are independent of the postulated irregularity, these are waves that can exist i n a palsma that has unperturbed densities which are constant and uniform. We are postulating that an electromagnetic wave exists and are identifying the x-component of the electric f i e l d as the inducing f i e l d in our model. The second order electric f i e l d , as we shall see later, depends not only on the existence of the f i r s t order electric f i e l d but also on our postulated unperturbed irregu-l a r i t y . For a cold plasma the f i r s t order equations are solved i n most standard texts on plasma physics - such as the one by Stix (1962), whose notation w i l l usually be followed. Let us assume that our coordinate system - without loss of generality - i s such that i s zero and that the quan-t i t i e s B ^ , , , and can be written in the form of a constant times exp(j(k*r - u>t)). Then with the zero temperature assumption, equations (2-13) to (2-19) become j ( £ * S ( 1 ) ) - ^ E - ( 1 ) (3-1) jtxg-a) = ^ j ( i ) ( 3_ 2 ) - 19 -= 0 jCk - .E"^ ) - 4 , I q.nP x ( O W l ) - I q ± n ± v . 1 i l "> l J X X (3-3) (3-4) (3-5) (3-6) (3-7) where (0) o)^  ( cyc lo t ron or gyro-frequency) = — m. c x (3-8) and where z i s a u n i t vector i n the p o s i t i v e z - d i r e c t i o n - the d i r e c t i o n of the uniform background magnetic f i e l d . From equation (3-7) v ^ can be w r i t t e n i n terms of F ^ ( 1 ) $P - A T I a<i> < — > x m.w x x (3-9) where <— > .-1 0) JCOIO. 2 2 w - 0). X 2 2 0) - 0)^ 0) 2 2 (0 - to. X 2 2 0) - 0). X (3-10) Using equations (3 -1 ) , (3-2) , (3-5) and (3 -9 ) , we can wr i te an equation only i n the v a r i a b l e E - 20 -t E ( 1 ) = 0 (3-11) where and where n 2 - f - s k - jD - n 2 k k / k 2 x z s - i - Z n 2 X . 2 2 1 00 - 0). X 2 oj .n . x i i t 2 ^ i GJ(OJ - o)^ ) „ 2 - s 2 2 -n k k / k x z 0 2 k ' k (3-12) (3-13) (3-14) X 0) n. (plasma frequency squared) = i m , 2 (0) 4 i rq 1 n i 2 c 2 k 2 n ( index of r e f r a c t i o n squared) = — — ky = 0 (without loss of genera l i ty ) (3-15) (3-16) (3-17) (3-18) To have n o n - t r i v i a l values f o r E - ( 1 ) , the determinant of G must v a n i s h . This y i e l d s a d i s p e r s i o n r e l a t i o n : a cons t ra in t on the values of k , the wave number, allowed f o r a given OJ, the augular frequency. The d i s p e r s i o n r e l a t i o n can be wr i t t en as 4 2 A l n - A 2 n + A 3 = 0 (3-19) - 21 -where ^ ( k x s + k2zP) (3-20) ~ ^ ( S 2 - D2) + PS(k 2 + k 2) (3-21) P(S 2 - D2) (3-22) Thus the index of refraction squared i s given by 2 A 2 ± l \ l N = - 2 T (3-23) where 4 - h 4 k 4 k 4 ( S 2 - D 2 - PS) 2 + 4k 2k 2P 2D 2 x z (3-24) In the case of the wave number being perpendicular to the back-ground magnetic f i e l d - k z being zero - the solution of equation (3-19) becomes 2 2 S Z - D Z (3-25) or n 2 - P (3-26) 2 2 1 For low frequencies - to « co e - P i s negative and thus the only propagating mode is that given by equation (3-25). Our particular interest in the limit of k z being zero i s due to the observation that whistler mode For our plasma to is of the order of II. Thus to2 « to2 implies 2 2 e e OJ « n which implies P, defined by equation (3-15), i s negative. - 22 -waves,whose wave numbers make a large angle with respect to the magnetic line of force, seem to trigger a certain type of LHR noise (Brice and Smith - 1964, 1965). If 8 i s defined as the angle between B ^ and lc, then 2 2 2 2 2 2 k /k and k /k are respectively equal to sin 8 and cos 8 . For a given 8, the right hand sides of equations (3-23), (3-25), and (3-26) are independent of k; thus k is readily computed as a function of co . In this thesis problem we have assumed the existence of a plane wave for the f i r s t order equations, which also govern fluctuations i n a uniform plasma. As we have seen this implies that a certain relationship exists between k and u. In addition, i f one assumes a given value for the x-component of the f i r s t order electric f i e l d , then the other variables - E ( 1 ) , E ( 1 ) , B ( 1 ), B ( 1 ) , B ( 1 ), V j ( 1 ) , v P , v ( 1 ) and y z x 7 y z i»x i>y i,z n P - can be obtained in terms of from equations (3-2), (3-6), (3-9), and (3-11). The variables are li s t e d in Appendix I. 3.2 Second Order Solutions From the second order equations given by equations (2-20) to .CD require at least some can be considered - driving terms. The resulting second order ele c t r i c f i e l d can be thought of as the induced f i e l d in our model. (2-26), i t i s seen that terms involving n. require at least some of the 1, s second order variables not to be zero. In a sense terms involving n* i,s - 23 -3-2.1 E l e c t r o s t a t i c Case The second order equat ions w i l l be so lved by making an e l e c t r o -s t a t i c assumption. By an e l e c t r o s t a t i c assumption, we mean that the second order magnetic f i e l d i s taken as i d e n t i c a l l y zero . Thus to so lve f o r the remaining second order v a r i a b l e s , only equations (2-21) , (2-23) , (2-25) , and (2-26) w i l l be needed. In t h i s s e c t i o n we s h a l l e laborate the a l g e b r a i c steps i n a f a i r amount of d e t a i l . In the succeeding s e c t i o n s the s t e p s , though more compl ica ted , are of ten analogous and w i l l be given i n a l e s s extensive manner. From equat ion (3-25) the term n f ^ v f 1 ^ impl ies the s p a t i a l and 1 y S X time dependence f o r the second order v a r i a b l e s . The v a r i a b l e s nf"^ and i , s vf1^ are r e a l q u a n t i t i e s that can be w r i t t e n as (I),-*. (1) jK~-r ^ (1)* - jK-r" n. (r) = n. e J + n. e J x , s i , s i , s +(1),+ . -»(1) j ( £ . r " - tot) . - K D * - j ( £ - r - tot) v ; ( r , t) = v : e J + v ; e J x ' x i where the terms wi th a s u p e r s c r i p t * stand f o r the complex conjugate of those terms. Again k and to are r e s p e c t i v e l y the s p a t i a l wave number and the angular frequency of the f i r s t order p lane wave s o l u t i o n and K i s the s p a t i a l wave number of the unperturbed i r r e g u l a r i t y . Thus n!«(r, t t fPcr" , t) = nf^Pe j ( ^ ^ X , S X X , S X . Q ) * + ( l ) 1( ( £ - £ ) • r-u>t) + n. v . e + cc x ,s x where cc stands fo r the complex conjugate terms of those e x p l i c i t l y s ta ted - 24 -on the right hand side of the equation. From the above i t was decided to write the second order variables in the form + ( 2 ) ( * t ) = + eJ((fcHC).r-«ot) + r eJ((k-K).r-o,t) + c c ( 3 _ 2 7 ) Thus for a zero temperature plasma equations (2-21), (2-23), (2-25), and (2-26) become -K ->+ (k + K) x E = 0 (3-28) j (£ + K) -E 1 = 4ir J q ±n* (3-29) -am": + (k + K) x — (0)->± , n. v. + x x x,s ( D * n; x,s v<X> x = 0 (3-30) ->+ -jcov. q i *± . ,-»± ~ — E + OJ. (v. x z) m. x x - x (3-31) where nf 1) ^  x,s i,s means that for the + sign nP is chosen and for the x,s - sign n. . Equation (3-31) has the same form as equation (3-7), thus x ,s -v+ "*± ->(1) V7 can be written i n terms of E in an identical form that v. was x i written xn terms of . Thus jq,<-v. x i .-1 £± m. co x x (3-32) < r> -1 where i s defined by equation (3-10). From equation (3-28), which ± ± ± relates E and E to E , and equation (3-29), y z x - 25 -+ FT = x . -J4TT (k + K )T q.ru x — x L. ^1 i (£ + K) • (£ + K) (3-33) If we substitute in equation (3-33) the expression for n^ , which can be found from equation (3-30) , and using the expressions for vT and v f 1 ^ ~*"(D i n terms of E and E , which can be found from equations (3-9) and (3-32), then (k + K ) x — x + E" = x n 2 i to A. E + x 1 , S n f 1 ^ n (0) i (£ + K) • (£ + K) (3-34) i , s We assume that 4 ( D * i,s 'nf^ i s the same for each ion component; thus the . 1 ratio can come out of the summation sign and henceforth may be replaced by e,s n » ( D * e,s n ^ . Using equations (3-10), (3-28), (A-I-l), and (A-I-2) in equation (3-34) and then solving for E x , we find that + (it + KXx* + J Z 1 ) Ex = -r-( n ( 1 ) ] e,s E ( 1 ) na>. e ,s Et X n (0) (3-35) For a two component neutral plasma this assumption is valid. For multiple positive ion components the assumption seems reasonable for lack of a better choice. However, should one have a better choice for individual ion ratios one could easily continue without our assumption, but at the price of a more complicated notation for the solutions. This seems unnessary for our discussion. - 26 -where X ±=(k + K ) 1 - S - - — x — x 2 n - s k k (k + K ) (1 - P) + X z z — z V 2 2 P / 2 k - co P/c x (3-36) Y* = S((k + K ) 2 + K2) + P(k + K ) 2 x — x y z — z z± = + K yD(n 2 - l) s) (3-37) (3-38) Before writing the second order electric field as a real quantity, let's recall that we have written the postulated irregularity and the x-component of the first order electric field as cosine functions -equations (1-1) and (1-2) - and as complex quantities. These are related as follows , \ jK-r , . -iK.r n. cos(K»r + ib) = n. eJ + n. *e J i,s i,s i,s (3-39) thus E cos(k.r - ,ot - O = E ^ 0 ™ ^ + E ( 1 ) V j ( b r - u t ) (3-40) O ^ X X E o E ( D * ^ X J 2 (3-41) i,s N ( D * l . S 1 = o" n. 2 l ,s (3-42) As a real quantity, the second order electric field can then be written in terms of n and E : - 27 -E" ( 2 )(r, t) = ^ - i > f(£ + K)((X +/Y +)cos e - (Z +/Y +)sin e) n(0) 2 e + (£ - K)((X /Y")cos 6 - (Z~/Y )sin fi)J (3-43) where e = (£ + K) .r - cot - £ + $ (3-44) 6 = (ic - K)-r - cot - 5 - (3-45) From equations (3-32) and (3-30) the corresponding formulae were found for v^(r, t) and n^(r, t) and are given in Appendix 11(a). The above second order solution has been found for k and K having arbitrary directions. In our study K i s taken to be i n the x-direction. This greatly simplifies the expressions for the second order variables. a Wave Number Perpendicular to Background Magnetic Field In this section we w i l l look at the very special case of it, the wave number of the f i r s t order variables, being perpendicular to ± ± ± Thus k , K , K are equal to zero and X , Y , and Z become very z z y simple : yr = k + K (3-46) x — x Y* - S(k + K ) 2 (3-47) x — x Z± = 0 (3-48) 2 2 2 where n , the index of refraction, has been taken to be (S - D )/S. With these simplified expressions, the equation for the second order - 28 -electric f i e l d reduces to E ( 2 ) ( r , t) = E x 2 ) ( r , t)x = cos y cos A (3-49) n e where p = it-r - tot - £ (3-50) A = K-r + <Ji (3-51) To derive equation (3-49) and the corresponding formulae for v£2^(r, t) and n£2^(r, t ) , listed in Appendix 11(c), we have used the following trigono-metric identities cos e = cos y cos A - sin y sin A cos 6 = cos y cos A + sin y sin A sin e = sin y cos A + cos y sin A sin 6 = sin y cos A - cos y sin A (3-52) where e = y + A = (it + it) • r - tot - £ + 6 = y - A = (It - it) - tot - £ - i|> (3-53) As shown in equation (3-49) (r, t) is directly proportional to n^1^ /n ^ , which is much smaller than one. In order for the magnitude e, £>/ e of E ^ 2 \ r , t) to be of the order of E q , the x-component of the f i r s t order electric f i e l d , S must be quite small. Thus the angular frequencies for which S has zeros are of interest. For a two component plasma there - 29 -are two zeros and they occur at what are called the lower and the upper hybrid resonance frequency. For a multi-component plasma there are two analogous frequencies and they are approximately given by 2 v _2 a/2 a) ~ to,„ = I — — I (3-54) LH 2 2 to + n e e and CO ~ (0. UH 2 ^ 2 co + n e e 1/2 (3-55) where the subscripts LH and UH stand for lower and upper hybrid and +i til stands for a positively charged i ion component. The above expressions are easily obtained from the definition of S - equation (3_13) - assuming 2 2 2 2 WLH > > ^+1 anc* n > > I n i • These assumptions are consistent with our +i plasma. An additional point contained in equation (3-49) is that the singularity frequency and the magnitude of (r, t) are independent of K , the wave number of our postulated irregularity. Because of our interest in very low frequency waves - particularly (2) around the lower hybrid resonance frequency; E^ , as defined by equation (3-49), is plotted for frequencies near to in Figure (3-1). Specifically, LH. the graph is of the logarithm to the base ten of n<°> |E<2> e 1 x versus x ; .(0) e,s where e.s is the ratio of the background electron density to the postu-lated irregularity in electron density, |E x |/E is the ratio of the magnitudes of the x-components of thq second order electric f i e l d to the 2 2 2 f i r s t order electric f i e l d , and x i s the ratio of to to toT„. For to LH - 30 -2 2 much less than to but greater than toTTJ, there is no propagating wave e LH allowed by the dispersion relation governing the first order fields ; thus (2) we are concerned about E for x less than or equal to one. As x (2) surmised earlier, E^ has a singularity at the lower hybrid resonance frequency. A slight correction to the above statements needs to be made. As defined by equation (3-54) , co^ is only an approximation to the LHR angular frequency ; therefore, to is not quite a zero of S and J-jii (2) hence a singularity of E v . Instead of one, the true LHR angular 2 2 frequency corresponds to an x ( = to /wTU) between 1.0024 and 1.0025. LH (2) It is this x for which the magnitude of E^ is infinite and above 2 2 2 which there is no wave propagation (for to = xto << to ). LH e - 32 -b Wave Number Oblique to Background Magnetic Field In this section we w i l l briefly look at the second order electric f i e l d for the case of It, the wave number of the f i r s t order variables, having an arbitrary direction with respect to B ^ , which is in the positive z-direction. However, the simplifying assumption of K, the wave number of the postulated irregularity, being in the x-direction i s ± ± ± s t i l l made. With the above conditions X , Y , and Z become X" = (k + K ) x — x 1 - S - -, 2-s k k (1 - P) X z ' , 2 2„ . 2 k - OJ P/c x (3-56) Y~ = S(k + K ) 2 + Pk 2 x — X z (3-57) 2T = 0 (3-58) where n , the index of refraction, is given by equation (3-23) We can now write the second order electric f i e l d as fn 1 e,s 0 „ H v. e j 2 (t + £)X +/Y + + ct - it)X~/Y" cos y cos A - |(£ + K")X+/Y+ - (£ - K)X /Y sin y sin A (3-59) where again - 33 -r> -»• y = k-r - tot - £ A = K-r + Corresponding expressions for v P (r, t) and n^2^ (r, t) are given in Appendix II(b). + -K2) -*• Should Y equal zero, equation (3-59) indicates that E (r, t) would be singular. To get some feeling of the frequencies at which singularities can occur, let us consider the following frequency range : 2 2 . 2 ( O j j < < to << to (3-60) The quantities S and P can then be approximated as (a ) 2 + n 2 ) ( c o 2 - . 2 i I ) 2 2 to to e (3-61) and 2/ 2 - n /to e (3-62) Thus, we have f 2 2 2 (k + K ) (to + n ) x — x e e 2 2 2„2 to n e e (k + K ) 2 (to2 + n 2) x — x e e It 2 ^ '(to to ) e (3-63) The approximate zeros of Y~ should be found at 2 , 2 „ 2 to n e e 2 2 2 (k + K ) (to + n ) x — x e e (3-64) 34 -f<°> For k equalling zero - k. perpendicular to B - the z zeros of Y + and Y , for a l l K x > correspond to a common frequency, 2 the lower hybrid resonance frequency. For our plasma, to^ satisfies equation (3-64) and thus corresponds to a singularity i n E^ 2^(r, t ) , as given by equation (3-59). This result i s consistent with what was found in the previous section. For not equalling zero, the expression for 2 co given by equation (3-64) is complicated by the fact that the second 2 term i s i t s e l f a function of to through k and k . Nevertheless i f Z X such zeros exist, their corresponding frequencies are greater than the lower hybrid resonance frequency. Additionally as K x increases, the difference in the two frequencies - corresponding to the zeros of Y + and Y decreases, and both frequencies approach the lower hybrid resonance frequency. This would hold i f K i s greater than k and i f the change X X i n k x i s not too great for the two frequencies. For 6 - the angle between k and — not equal to 90 , i t 2 is not obvious i f there is an to that satisfies equation (3-64) and thus has the characteristics mentioned in the above paragraph. That such 2 an to can exist i s illustrated in Figures (3-2) to (3-6) - for equalling 71.56°. Done for various values of K , the figures are f" ( 0 > E ( 2M 2 7 m e x ,2.2. r.(2) . , versus x (= to /to ), where E is the Liii X graphs of log 10 n E e,s o magnitude1 of the x-component of E ^ - given by equation (3-59) -1 ± ± To calculate the magnitude, we use. the original definitions of X , Y , S, D, P, etc and do not make the approximations used i n the preceeding paragraph. - 35 -for X being zero 1 (or some integral multiple of IT). From the figures one essentially sees the properties predicted earlier : for decreasing K , the singularity frequencies increase even further past the lower hybrid resonance frequency and the s p l i t t i n g of the two singularity frequencies becomes more pronounced. Though consistent with increasing peak frequency for decreasing K^, Figure (3-8) is an exception i n that there is only one peak whose frequency corresponds to a zero of Y +. This -4 qualitative change occurs for some c r i t i c a l between 5 x 10 and 10~ 4 cm"1. (2) In Figures (3-9) to (3-16) the relative plots of versus 2 2 - 3 - 1 x (= co /wTU) i s given for various 0 for K equal to 10 cm i-iH. X Like Figure (3-8), Figures (3-9) and (3-10) have only one peak corresponding to a zero of Y +, the other figures have two. Figure (3-16) is another exception, but with 0 '= 0°, as with 90°, one expects one peak for Y + i s identical to Y . However i t ' s not obvious why a zero for Y should occur for some cases and not others. Empirically i f k +, the wave number corresponding to the zero of Y +, is much less than K^ , then there are two peaks. Such a condition was not satisfied for the cases i l l u s t r a t e d by Figures (3-8) to (3-10). However, the main point to observe in these (2) figures is that for a given K , the peaks in E occur for increasing X X For any X a singularity w i l l occur at a zero of Y~ ; thus choosing a specific X i s not c r i t i c a l in determining the frequency at which a singularity occurs. - 36 -frequency as 6 decreases (as the wave number k becomes more aligned with the background magnetic f i e l d ) . For the angles of 0 considered, other than 90°, wave propagation can occur for frequencies that are not only less than but also equal to and greater than the resonant frequencies of the second order electric f i e l d . See Appendix VII for the index of refraction over the frequency intervals considered in Figures (3-1) to (3-16). o 'in Figure (3-2). The value of y log in|n<°V2)/(n E ) °101 e x e,s o as a function of 2 2 (2) x ( = to /to ), where E is given by equation (3-59) i-ji-l x (cold, electrostatic case), = 71.57°, K = 10 2 cm X, and X x ' = 0. a a a ru 0.2 0.4 0.6 0.8 ~1 1.0 ~ T ~ 1.2 ~1— 1.4 1.6 Figure (3-3). The value of y = l o g l J n < 0 ) E ( 2 ) / ( n E ) °10' e x e,s o as a function of / 2 i 2 \ u w(2) x ( = to /co ), where E J_I1T X is given by equation (3-59) (cold, electrostatic case), 0 = 71.57°, K -2 -1 10 cm , and X = 0. o 03 a to oo a -1F "T 1012.2 x 1012.1 1012.3 ~i 1012.4 n 1012.5 "1 1 1012.6 1X112.7 txio-*) 1 1012.8 ~ l 1012.9 1013.0 Figure (3-5). The value of y as a function of 03 = l o g l J n ( 0 ) E ( 2 ) / ( n E )| 10' e x e,s o 1 2 2 (2) x ( = OJ /OJ ), where E_ is given by equation (3-59) i_iil " x -3 -1 (cold, electrostatic case), 6 = 71.57°, K = 5 x 10 cm , and X o ' r-a ID a ~1 10.41 ~! 10.42 ~I 10.43 10.4 n 1 1 — i — 10.44 10.45 ID.46 10.47 (X10-f) n — 10.48 10 Figure (3-6). The value of y a 'to 2 2 x ( = co /co T „) , where E LU X f = log i n|n (°>E ( 2 )/(n E ) | ] (2) as a function of x e,s o is given by equation (3-59) -3 -1 (cold, electrostatic case), 8 = 71.57°, K x = 10 cm , and A = 0. a in a Figure (3-10). The value of y y o "to = l o g i n | n ( 0 ) E ( 2 ) / ( n E )j 10' e x e,s o 1 as a function 2 2 (2) of x ( = to /to T t I), where E ' is given by equation (3-59) X LH' (cold, electrostatic case), = 89.82°, K = 10 - 3 cm 1, and A X = o. a in a • a a rv a 1 10.12 n — 10.13 X 10.1 10.11 1 10.14 T T 10.15 ID.16 (X1(H) 1 10.17 1 10.18 10 a "ID Figure (3-13). The value of y = l o g l J n ( 0 ) E ( 2 ) / ( n E ) J10 1 e x e,s o as a function of 2 2 (2) x ( = co /co ), where is given by equation (3-59) (cold, electrostatic case), = 50.77°, K = 10 3 cm \ and A x ' = 0, a if! a .0 "1 2.0 3.0 4.0 5.0 6.0 H 7.0 8.0 9.0 "1 10 lO Figure (3-14). The value of y = log 10 „n ( ( V 2 >/( n E ) x e,s o as a function of 2 2 (2) x ( = OJ /<i>L T 1), where E ^ is given by equation (3-59) (cold, electrostatic case), 33.21°, K = 10 3 cm \ and X = o. a "in a • p > a 2.0 "~1— 4.0 6.0 8.0 10.0 12.0 14.0 1 16.0 "I 18.0 n x 20.0 Figure (3-16). The value of 10 = l o 3 i n | n ( 0 ) E ( 2 > / ( n E °101 e x e.s o ) as a function of 2 2 (2) x ( = to /co ), where E is given by equation (3-59) Liti X (cold, electrostatic case), = 0°, K = 10 3 cm 1, x 1 and X = 0. o "in a a a 'ni 2.0 4.0 "~1 6.0 ~1 8.0 ~ l 10.0 12.0 14.0 1 16.0 18.0 ~1 20 - 52 -3 .2 .2 . Electromagnetic Case S t i l l assuming a cold plasma, we w i l l now solve the second order order equations without making the electrostatic assumption. For the reasons given i n section 3 . 2 .1 , a l l variables w i l l be assumed to be of the form * ( 2 ) ( ? , t) = / e J ( ( » . ^ t ) + -J<(M).r^,t) + cc (3-65) Thus the second order equations become -K -H (k + K) x B~ J4TT -tt c J c (3-66) ii* -»\ -*± (k + K) x E CO yfc c (3-67) (k + K).B" = 0 (3-68) (k + K)-E = 4TT } q.nT — ? x x (3-69) J = I q i ( 0 ) + ± , n. v. + l x CD ^ i,s L ( D * i,s n n. l (3-70) ± ,i>- -K -con. + (k + K) i — ( 0 ) + ± n; v. 4 1 x ' n ( 1> ^  x,s i.s v. .(1) = 0 (3-71) 1*+ + to. (vt x z) -jcov. = •— x m. l " xx (3-72) where z is a unit vector in the positive z-direction and co^  i s the cyqlotron frequency defined by equation (3-8) . Combining equations (3-65) and (3-66) to eliminate B* yields - 53 -\ - K 2 - (k + K ) 2 2 y z — z c + E~ (k + K ) (+ K ) y x — x — y + E^k +K )(k + K ) = - j ^  r z x — x z — z J 2 x c (3-73) E x ( k x ± K x ) ( ± K y ) + E±( ^ " ( \ ± V 2 - ( k z ± K z ) 2 ] + E ± ( + K )(k + K ) = - j • 4-~ J * z — y. z — z J c2 y (3-74) E* (k + K ) (k + K ) + E* (+ K ) (k + K ) x x — x z — z y — y z — z + E" ^ - (k + K ) 2 - K 2 2 x — x y c J . 4irco + ~ 2 r z (3-75) Equation (3-72) can be solved for v_^  to give v. I i ,-1 g± m. co i l (3-76) where A i is defined by equation (3-10). Using this result and the :t(D expressions for v^ given by equations (A-I-14) to (A-I-16) (found in Appendix I), we can find expressions for J * . With these expressions for -M-J and after a b i t of algebra, equations (3-73) to (3-75) can be rewritten as i ^ S - K 2 - (k - K ) 2 2 y z z + E" . 2 (k + K ) (+K ) - JJJL. D x — x — y 2 3 c n<X> ' e,s , n ( D * + E:(k„ + K„) (k„ + K J =4 — c n e z x — x z — z 1 - S n 2 - S ,(D (3-77) - 54 -x 1 (k + K ) ( + K ) + ^ r D x — x — y 2 J c + E~ ^ S - ( k + K ) 2 - (k + K ) 2 2 x — x z — z c e,s + • 2 + E;(+K )(k + K ) = - J \ z — y z — z 2 c » e,s .(0) D(n 2 - D E ^ n 2 - S X (3-78) E* (k + K ) (k + K ) + E* (+K ) (k + K ) x x — x z — z y — y z — z e,s + E" — P - (k + K ) 2 - K 2 2 x — x y c • n e ,s n (0) k k (1 - P ) E ( 1 )  X z x _ 0 2 k 2 - ^ p x 2 c (3-79) The above equations are valid for ic in the x-z plane and K having an arbitrary direction. For this general orientation, the electrostatic pro-blem solved in the previous section was algebraically complex. Consequently, for the more general electromagnetic problem, we w i l l only choose the simple case of £ and K lying in the x-direction and thus both being perpendicu-lar to the background magnetic f i e l d . + + After setting k , K , and K equal to zero, we find E , E , 6 z' y z H x y and E* from equations (3-77) to (3-79). In turn we use these expressions in equations (3-76), (3-71), and (3-67) to find v 7 , nJ7, and . In terms of real quantities x , ( 2 ) , x ne,s ^o [, + , - N , , + - x . . , E x (x, t) = ~(o) "2~ K a a-) cos u cos A - (a - a )sm u sm A n e (3-80) - 55 -where B<2>(X, t) e, s o n (0) 2 (3 + + 3 )sin y cos X + (3 + - 3 )cos y sin X (3-81) (3-82) ( k (3 + + 3_) + K (3 + - 3 _))sin y cos X I X X + ( k (3 + - B~) + K (3 + + 3~))cos y sin X X X (3-83) B x 2 ) ( x , t) = B< 2 )(x, t) - 0 n cE e, s _o (0) 2OJ n a = -2 2 ? 9 9 co (S - D - 2S) + c (k + K ) x — X 2 2 2 9 9 u> (S* - D Z) - c (k + K ) S x — X o o 2 f ( S 2 - D 2 ) S - ( S 2 - D 2 ) l D (OJ 2(S 2 - D 2 ) - c 2(k + K )2S) k x - tot - E X ^ X = K x + A x y (3-84) (3-85) (3-86) (3-87) (2) (2) The variables v± y(x, t) and n? y(x, t) are listed in Appendix III. To make some of the physical properties of equations (3-80) to (3-84) more apparent, the following combinations of a* and can be deduced : a + + a = 2 2 £ - 3kZ) x x + •24] S X 2 c ; -(K 2 - 4k2)S x x (3-88) To get these expressions the dispersion relation given by equation (3-25) was used : w/^^k 2) = s / (S 2 - D 2 ) . - 56 -+ a - a = 4k kx + x 9 2 1 k 2 - 2 — x 1 2 c v 9 9 x (K - 4k )S x x (3-89) i + + 3" = 2Qo 2/c 2)fs(S 2 - D2) - S 2 - D2^ (K 2 - 4k2)DS x x (3-90) 4 k x ( . 2 / c 2 ) f s 2 ( S 2 - D 2 ) - S 2 - D 2 ^ Kx (K 2 - 4k2)DS x x (3-91) + _ -2uA (S(S 2 - D2) - S 2 - D2) k (3' + B ) + K (3" - 3 ) = x x 2 2 9 c (IT - 4kZ)DS x x (3-92) + _ + _ 2a,2(K2 - 2k2) (S(S 2 - D2) - S 2 k ( S + - B ) + K (3 + 3 ) = x x . , - A c K (K - 4k )DS X X X (3-93) As with the electric fields in the electrostatic case, the above electromagnetic fields are inversely proportional to "S and consequently also have a singularity at L H R f r e c l u e n c y - Unlike the electro-static case, there i s a dependence on K , the wave number of the postulated 2 2 t irregularity. However for K >> 4k , x x 2 2 2 a), . << a) << oj , and X not +1 e near 90 , the electric f i e l d magnitude essentially reduces to (0)1 n. /n. i,s/ 1 E /S , the electrostatic result given by equation (3-49). If X is also not near 0°, a simple approximation"'" can be written for the (2) (T) ratio of the magnitudes of ~& to E : f 2 As long as this condition is satisfied K i s also much greater than 2 2 2 2 2 X 2 (k — ai /c )S, at least when O J , . << to << to x +1 e We use the fact that, for . << to • << to , |D| » |s - 57 -| B ( 2 ) | / | E ( 2 ) | z COD(S + l ) / ( c K x ) (3-94) (2) From the above i t is seen that the relative importance of B with (2) 1 respect to E decreases for increasing From equations (3-88) to (3-93) an additional singularity seems 2 2 possible, when 4k^ equals K - that i s the wavelength of the f i r s t 2 order wave equals twice the wavelength of the postulated irregularity This singularity has no correspondence in our electrostatic solution; thus i t arises from the electromagnetic nature of the second order fields. (2) To get a feeling for the behavior of the magnitude of E , as f n ( 0 > E ( 2 > l Q. X given by equation (3-80), we plot the log n — versus lu n E ^ e,s o . 2 2 3 x (= OJ /co T „) in Figures (3-17) to (3-21). In these figures the plots are done for various values of the parameter The figures a l l show the singularity at x equal to one, corresponding to the LHR frequency. For the figures with the additional singularity, the singularity occurs for an x such that K does equal to 2k . For Figures (3-17) and X X -2 -3 -1 (3-18), where K x equals 1 x 10 and 5 x 10 cm respectively, a similar singularity does exist, but i t occurs for an x so close to one 1 For A near 0° the ratio decreases, but for A near 90° it increases. 2 A similar result holds for the backscattering of waves from an irregu-l a r i t y (Bowles et a l . , 1963; and Wheelon, 1959). 3 For these plots A was again taken to be zero, TT, 2TT, - 58 -as not to be resolvable from the singularity at the LHR frequency. Thus for the values of K x considered, the frequency of the additional singula-rit y decreases for decreasing K^. An additional point discussed earlier seems to be born out. The plots in Figures (3-17) and (3-18), for which 2 2 K is >> 4k i f .01 < x < .99, are essentially identical to a similar X X plot done for the electrostatic case - Figure (3-1). - 64 -CHAPTER 4 HOT PLASMA The non-zero temperature of the plasma w i l l now be considered. The main aim of the following calculations i s to see the effect, i f any, of temperature on the singularities of the second order solutions of our cold plasma calculations. The outline of this chapter essentially parallels chapter 3. The main exception i s that only wave numbers perpendicular to the background magnetic f i e l d are considered. 4.1 F i r s t Order Solutions Before looking at the temperature effects on the second order solution, the f i r s t order variables and the dispersion relation governing to and k w i l l have to be recalculated. Again we w i l l assume that our coordinate system is such that k is zero and that the f i r s t order y variables can each be written as a constant times exp(j(k«r - t o t ) ) . Then equations (2-13) to (2-19) become jGtxf ( 1 )) - ^ L j W - Jj=f<l> (4-1) c c £ x E " ( 1 ) - £ iT ( 1 ) (4-2) c - 0 (4-3) jkM* ( 1 ) = 4ir I q . n ^ (4-4) i J ( 1 ) = Iq.n<°>v ( 1 ) (4-5) - 65 -i i i (4-6) -jcov. m. + co.v. i l (1) . 2r> (1) / (0) (4-7) where UK, a^, and z are the cyclotron frequency, the thermal velocity and a unit vector in the positive z-direction - the direction of the back-ground magnetic f i e l d . The above equations differ from those of a cold plasma only in the last term of equation (4-7) , but the complications of this term soon become apparent. Solving for n£"^ in equation (4-6) and substituting in equation (4-7) yields v f X ) - -1 c feci) V . X z 1 * f HH > ] - ^ f ( i ) i m. co CO V J 1 (4-8) This equation can be written as a matrix times vector v ^ equals vector E^ 1^. Taking the inverse of this matrix allows one to write v f 1 ^ in terms of where l m.co T,i l ' (4-9) 1_ ? 4 r 2, 2 . 2, co (co -k a.) z 1 2 2 2 -jcoto (^co ~ ^ z a ^ ) 2 2 co k k a. x z 1 ,2.2 2. jcoco. (co -k a. ) 1 z 1 2, 2 2 2. co (co -k a^) icoco.k k a. J 1 x z 1 2, , 2 > co k k a. X Z 1 -icoco.k k a. J i x z l 2, 2 2 . 2 2, co (.co -co.-k a.) 1 x i (4-10) - 66 -where r - ( 2 2 i 2w 2 ,2 2. ,2,2 4 C. = (to - to. - k a )(to - k a.) - k k a? 1 1 X 1 z i x z i (4-11) Of course A ^ reduces to A ± , the corresponding cold plasma matrix, when a. is zero, l Using equations (4-1), (4-2), (4-5), and (4-9); an equation only containing the variable can be written : where % S<« - o (4-12) where <—> GT g l l 812 '21 '31 B32 '13 822 g23 '33 (4-13) 2 k '  z 811 = n 72 1 + 1 k i L 2 . 2 . 2 2 , ' II. (to - k a. ) . l z i y ^1 '22 i n 2 / 2 . 2 2, , '33 ,«2 2 x L 2 , 2 ^ A 1 _ L V T T ^ / Z Z ,22., n —=- - 1 + } n.(to - to. - k a . ) / r . 1 ^ V I 1 X 1 S 1 '2 2 2 2 812 = 821 = d ? n i oj i ( to - k z a i ) / ( t o ? 1 ) (4-14) 813 831 , k k 2 x z , r -n —5- + 1 2 2 nfk k af/c. 1 x z 1 1 '23 &32 2 2 n.to.k k a. / ( t o ? . ) 1 1 x z 1 ^1 - 67 -To have non-trivial values of E ^ , the determinant of "G <—> T must vanish. In general, the resulting dispersion relation would be quite complicated; we greatly simplify the situation by taking £ to be perpendicular to B (0) Thus having taken k to be zero, z where " S l JD 1 -JD 1 n - s u n2-P L2.f 2 2 . 2 2,1 n. / (to - to. - k a.) 1 1 X X s l = 1 - I i c _ •, V fn 2c 2 i 2 2 N / 2 , 2 2 , 2 2S1 S H ~ 1 - I | n ± ( to - k xa ±yto (to - to ± - k x a p °1 = I ^ ^ ( t o 2 - to 2 - k 2a 2) i 1 -V x x x P (as defined previously) = 1 - £ n / ,2, 2 to (4-15) (4-16) (4-17) (4-18) (4-19) The corresponding dispersion relations are and n 2 (= c 2 k 2 / t o 2 ) = ( S ^ - D 2)/S 1 n 2 = P (4-20) (4-21) In that P is negative for low frequencies - to « o / , the Remember that k = k xx where x is a unit vector in the positive x-direction. - 68 -governing dispersion relations is taken to be equation (4-20) . Unlike the cold plasma dispersion relation, the right-hand side of equation (4-20) 2 2 is e x p l i c i t l y dependent on k ; thus computing k as a function of UJ is not as easy as for a cold plasma. When actual values of k are needed, they are computed numerically"''. Of course equation (4-20) reduces to the 2 cold plasma dispersion relation when the temperature and thus a^ are zero. For OJ and k satisfying the dispersion relation and for a given value of , the other f i r s t order variables - through the use of equations (4 -2) , (4-6) , (4-9) , and (4-12) - can be written in terms of E^"^ . These results appear in Appendix IV. 4.2 Second Order Solutions 4.2.1 Electrostatic Case Making our electrostatic assumption of no second order magnetic f i e l d , we w i l l now solve the second order equations governing our hot plasma. As in sections 3.2.1 and 3.2.2 we w i l l assume the variables to have the form * ( 2 ) ( ? , t) - ^jaM).^t) + r e j ( ( « ) . ? - w t ) + c c ( 4 _ 2 2 ) Thus the second order equations to be solved are These computations were done with a double precision version of Janet Streat's ZFUN, a library program of the computing center of the University of British Columbia. - 69 -"K -H-(k + K) x E~ = 0 (4-23) (k + K) -E- = -4irj £ q n~ (4-24) -con* + (k + K) ( 0 )^± , n. v. + l l i,s I J D * ^ n i , s ' V (1) = 0 (4-25) . ->± i 2± , ,-*± -v J i -jtov = — E + co. (v. x z) r~ 1 m i i (0) 1 n. (£ + K) l + n. - r (1) ( n. ^ i,s „(!)* (0) (4-26)' Solving for n^1) and n* from equations (4-6) and (4-25), and then substituting in equation (4-26), one can write an equation in terms of v. and E : I V i " " V ( v i x z ) " - | ( k ± K> $ ± K) -v* CO m.co l x (4-27) where - > + _ J V i R i = E " + — W 9 (1) ,_2 r n : ' \ (D* coq.n; - ( n ; I i i ^ i,s J i,s ( £ ± ^ ) ( ^ 1 ) ) (4-28) Equation (4-27) has an identical form to equation (4-8) ; thus the solutions of these two equations should also have the same form. Therefore from equation (4-9) , we can immediately write v. = A. , R. i m.co x,± x (4-29) w h e r e A±^± i s g i v e n by e q u a t i o n (4-10) , e x c e p t t h a t £ i s r e p l a c e d by - 70 -(k + K) , and where is zero. To make the problem more tractable, k z and K z are also assumed to be zero. Under these conditions equation (4-3) implies that + + E~ =.. E~ = 0 y z (4-30) Thus the equations to solve become -J4TT x (k + K ) x — x i I q i n i (4-31) 1 CO n n ( 0 ) ( k + K ) v ± + 1 x — X 1 , X (1) i,s (D* ^ n i , s (k + K ) v ( 1 ) x — X 1 , X (4-32) v. l J H i A - l # t A. R. m.co x,± l l (4-33) -1 -*± where A. and R. are simplified : x,± x t o 2 - c o 2-(k +K ) 2 a 2 x x— x x - j c o o ^ 2 2 fir J . V \ 2 2 co - c o . — (k +K ) a, x x— x i JC0C0, 2 2 /L. oxr ^  2 co —co.—(k +K ) a. x x— x x co 2-(k +K ) 2 a 2  x— X X 2 2 ,, ± v .2 2 co -co . -(k +K ) a. x x— x x (4-34) t - x R. = x i.x ja.m. + —• i i X „ r,(0) coq.n. (1) x,s 1 , 8 ' n. (k + K )K v ( 1 ) x — X X x , x (4-35) Combining equations (4-31) to (4-33) to form an equation whose - 71 -only variable i s E and using the expression for v; given by x r i,x e J equation (A-IV-4), we find a solution for E* : x + E~ = x (1) f ne,s ^  n ( D * v e,s J n (0) E ( 1 ) ( l + \r)/s< x — 1 (4-36) where + (k + K )K x — x x a i n i r 0 toto.S.. 2 _i i 1 (0 + — — r 2 2 2 2 to - to. - (k + K ) a . x x — x 1 ( 2 2 (to - to .2 2, k a.) x x (4-37) = 1-1 n 2 / to 2 - to 2 - (k + K ) 2 a 2 x / x x — X X (4-38) -H-From E we can find v. and n. from equations (4-33) and (4-32). In terms of real quantities E ( 2 ) f x M = x e > s -SL U, t; - x ( n e x x cos y cos X X X sin y sin X (4-39) where = (1 + W-) / x (4-40) y = k x - tot - £ x (4-41) X = K X + ij; X r (4-42) ->(2) (2) The variables v ^ ^ x , t) and n^ 7 (x, t) are found in Appendix V. (2) From equation (4-39), E y appears to have a singularity where X - 72 -either S* or S 1 has a zero. If 2 2 K a x e 2 k 2 a 2 x e << co e and ( 4 _ 4 3 ) „2 2 V H 2 . 2 2 , « co (for co » to ), 2 2 "x then 1 and are approximately equal to S ; therefore and would each appear to have a zero or at least a very small value around f n<°> lE<2>l 1 the LHR frequency. Plots of i°8^o n E e,s o 2 2 versus x (= co /coTU) Lrl for various values of K x - Figures (4-1) to (4-5) - appear to show that (2) E x is relatively large at x equals one, corresponding to the LHR frequency. For the values of x considered, Figures (4-1) to (4-5) also seem to indicate that |E | has l i t t l e dependence on K - for x x -4 -1 -2 -1 + 10 cm < K < 10 cm . In fact these figures are almost identical — x — e to Figure (3-1), which concerns the cold electrostatic case. In the above paragraph we were careful to say that I** I 1 S relatively large for x near one rather than an actual singularity exists there. Such caution was exercised because S^ i s a function of both to These inequalities are satisfied for our plasma i f x < .995 and - 2 - 1 ~ K < 10 cm . x — t ± This observation could be expected since equation (4-43) as well as W « 1 hold for .2 < x < .995 . - 73 -and k. Because to and k are also related by a dispersion relation, i t i s not obvious that there is an to for which vanishes. This caution was enhanced by the fact that the dispersion relation t e l l s us , 4 that no waves exist for x > 1.0025 . In fact, from Figures (4-6) to (4-8) we see that there is no singularity in the frequency range where the waves may exist . Nevertheless at x equals 1.0024, |E^ 2)| is of the order of five hundred times x ^ / " e j V I f n e , s / n e 0 ) i s 8 r e a t e r t h a n -4 i (2) i 5 x 10 , then |E • | is already within an order of magnitude of the f i r s t order f i e l d . Thus around the LHR frequency the second order elec-t r i c f i e l d i s s t i l l significant even though no resonance in the rigorous sense of the word exists. From equation (4-37) one might conclude that W* and thus | E X ^ | - as given by equation (4-39) - might become large near the cyclo-tron frequencies of the ions. This interesting possibility might seem likely to occur i f 2 2 2 2 to = to. + k a. (4-44) 1 X 1 or to 2 = to 2 + (k + K ) 2 a 2 (4-45) 1 x — x 1 2 2 2 2 2 u c o i S i However in the limit of w to. + k a. , to H — -* 0 in such a way l x i ' 3 ± that W does not have a singularity when equation (4-44) is satisfied. 2 2 2 2 ± Now in the limit of to to. + (k + K ) a. , not only W but also l x — x l 3 S? has a singularity; consequently (1 + W~)/S? and therefore |E^2) | do not have singularities when equation (4-45) is satisfied. Thus what f i r s t seemed an interesting result does not emerge as a prediction of our model. cn o Figure (4-2). The value of y 2 2 of x ( = co /to ), where as a function ,(2) x is given by equation (4-39) -> rr i - M (hot, electrostatic case), k || K, k ]_ B ^ ° \ K x = 5 x 10 3 cm 1, and 0. 0.1 I 0.2 " T -0.3 0.4 ~1 0.5 ~I— 0.6 0.7 o.e 0.9 1.0 - 82 -4.2.2 Electromagnetic Case Without making the electrostatic assumption, the second order equations governing our hot plasma w i l l be solved. Even though we w i l l assume that k , K , K are zero 1, the solutions are s t i l l quite z y z ^ complicated and w i l l only be used to check i f the apparent singularities mentioned in preceding sections s t i l l exist. Again a l l variables are assumed to have the form cc (4-46) Thus the second order equations become jot+ ib x 3* . *i 3* - -^ 2* — c c (4-47) (it + it) x 2 * = £ B 1  — c (it ± K) -5 = 0 (k~ + £) -E* = 4TT 7 q.n* — h l l (4-48) (4-49) (4-49') -H- n. v. + l l i,s L(D* i,s n. -KD V. 1 (4-50) ton* + (it + K) (0)+± . n; 'v. + I I 1 , 8 n. (D* i , s 1 = 0 (4-51) Without loss of generality, we had already chosen k to be zero in our handling of the f i r s t order equations. ^ - 83 -. -*± q i -*± -*± A - j c o v . = — E + to. ( v . x z) l a 2 ( S + K) X .CD n n. -i,s ( D * i.s • nf°> l (4-52) In that the last two equations are the same as Equations (4-25) and (4-26) and with the assumption of k y, k z, K , YL^ equalling zero, is given in terms of E*1 by equation (4-33) : jq A. , R. m.co x, x ' where A is given by equation (4-34) and 1 , x R i " + x ~ ~ ( o T coq.n. nx x r n ( D . "i.s x,s •> (k + K )K x — x x (4-53) (4-54) - H -From equations (4-47) and (4-48), B~ can be eliminated to give = —4TTj J /co 1 -1 -2 2 c (k + K ) x — X CO c 2(k + K ) 2) x — X 1 -4iri J /c y -4iri J /c J z (4-55) (4-56) (4-57) With the results giving v* and as functions of ET and , (D 'x E v"' respectively, J can be written in terms of ET and E U ' . Using , ( D 'x - H -these expressions for J in equations (4-55) to (4-57) one finds - 84 -+ + + + 1 x 1 y ', n ( 1 ) e,s (D* ^ ne,s > (1 - W*)E(1> x (4-58) + + jD7E~ + J 1 x c (k + K n x — X 1 '11 = - J t (1) 1 e,s 1 e,s (0) D 2 + (1 - S )S +' D, - WD E ( 1 ) (4-59) x E~ = 0 z (4-60) where S^, D.^  W , and S 1 are given by equations (4-16) to (4-18), (4-37), and (4-38) respectively; and S L " 1 " I l f n 2 ( u 2 - (k + K )V) i X — x' 1 J 2 2^  I co2(co2 - to? - (k + K ) Za Z) V 1 x — X x} 2 2^  (4-61) D: = I 2 co.n. i 1/ 2 2^  to (to - coT - (k + K ) afl v x x — X XJ (4-62) (k + K ) K x — x x n 2 2 o). n. a. X I X r 2 w o ^ S ^ CO + (co -co -(k +K ) a.) (to -co.-k a.) ^ x x— X xJ v x x x - 1 2 2> r 2 2 , 2 2> (4-63) ± ± Expressions for E x and E y are obtained from equations (4-58) and (4-59) + E". = x ( n 1 e, s 1 n ( D * k e,s J i T x (4-64) - 85 -E~ =? r n n (1) . ~e,s (D* e,s •* PJE ( 1 ) T x (4-65) where + + + +. ( D - / D 1 ) ( D l - S 1 S 1 1 + S 1 ) + S - 1 ± ( S I l W - + D ^ ) 2 2 c (k + K r ^ - W ) DENa (4-66) ( D i " s n s i ) s i + s i s i + D i D i i d I ( S I W D + D i w ± ) + 5T D 1(DENa) , + 2 c 2 ( k x ± K x ) 2 + DENa - S - S ^ - (D~) Z 2 S l (4-67) (4-68) 2 + + Of course f o r the zero temperature case ( a i = 0 ) , and $~ reduce to the cold value c o e f f i c i e n t s , a* and - equations (3-84) and (3-85) . From equations (4-48), (4-53), and (4-51) , expressions f o r B , vT , and nT can be obta ined. In terms of r e a l quant i t i es we can wr i te E f (X, t) n E e , (0) ~2~ [^(aT+aT)cos y cos X - ( a T - a T ) s i n y s i n X (4-69) (2) n e s E o ( + + ^ E y ( x ' C ) = ~ f o f T~ (S T +B~)sin y cos X + (fB*-g~)cos y s i n X (4-70) E ^ 2 ) ( x , t) = B < 2 ) ( x , t) = B < 2 ) ( x , t) = 0 (4-71) B«»(K, t) n cE e , s o (0) 2io n s i n y cos X cos y s i n X (4-72) - 86 -where V •- k xx - tat - K (4-73) A = K xx + i|> (4-75) The corresponding expressions for v^ 2^(x, t) iand (x, t) are given in Appendix VI. fn |E<» Figures (4-9) to (4-13) contain plots of l°g 1 0 e ,s 1 x (0) E n o e 2 2 versus x (= co / c o T r r ) for various values of K . Comparing these figures to the ones for a cold plasma - Figures (3-17) to (3-21) - we see there are practically no differences i n either ordinate magnitudes or frequencies of the peaks. As for the cold plasma, the peak at x = 1 is at the LHR 2 2 frequency and the other peaks occur at a frequency where K x = 4 k x . . y O Figure (4-12). .The value of y 2 2 x ( = co / > where = l o g i n | n ( 0 ) E ( 2 ) / ( n E ) 101 e x e,s o as a function of ,(2) is given by equation (4-69) (hot, electromagnetic case), It || K, It J_ B and X = 0. ( 0 ) , K =5 x lO" 4 cm"1, x ' y o o Pi a ni Figure (4-13) The value of y 9 9 x ( = l o g i n | n ( 0 ) E ( 2 ) / ( n E ) ° 1 0 1 e x e,s o as a function of 2 2 (2) u /io T T J), where E is given by equation (4-69) LH II i -K0) -4 -1 (hot, electromagnetic case) , k | | K, k J_ B , K = 10 cm , and A = 0. 0.1 a • "~l 0.2 0.3 - r — 0.4 0.5 0.6 0.7 0.8 - r — 0.9 - 92 -CHAPTER 5 CONCLUSION 5.1 Some Other Theories of Lower Hybrid Resonance Noise 5.1.1 Polar LHR Noise As a source for atmospheric radio noise at frequencies of several hundred kHz , E l l i s (1957) proposed that auroral particles might emit this v noise by a process known as Cerenkov (often spelled Cerenkov or Cherenkov) radiation. This radiation i s emitted by a charged particle moving at a velocity that is greater than the local phase velocity of light (Jackson, 1962; Panofsky and P h i l l i p s , 1962). An analogy often made i s of a projec-t i l e travelling faster than the speed of sound (Mansfield, 1967) . Using Mansfield's (1967) equations of power emitted by an electron spi r a l l i n g along the background magnetic f i e l d of a homogeneous plasma, Jorgensen (1968) - contrary to Liemohn's (1965) estimates - concluded that the power levels of observed polar LHR noise could be explained by incoherent Cerenkov radiation produced by electrons precipitating in the auroral zone. From later s a t e l l i t e observations, Jorgensen's power estimates appear to be three order of magnitudes too small (Gurnett and Frank, 1972; and Rao et a l . , 1973). However recent studies (Laaspere et a l . , 1971; Mosier and Gurnett, 1972; Hoffman and Laaspere, 1972; and Gurnett and Frank, 1972) have shown that polar LHR noise i s associated with precipitating electrons with energies from .1 to 1 kev, whereas Jorgensen had assumed energies from 1 to 10 kev. With the softer energy spectrum, Lim and Laaspere (1972) have - 93 -redone Jorgensen's calculations and have obtained intensities of incoherent Cerenkov radiation that was larger by two orders of magnitude. For frequencies above 70 kHz, Lim and Laaspere f e l t that the agreement between the observed and calculated intensities was satisfactory; though below 70 kHz, they conceded up to an order of magnitude difference. Lim and Laaspere f e l t that a possible reason for the difference was the seve-r i t y of one of their constraints : that emitted waves have a phase velocity greater than twice the thermal velocity of background electrons. The reason for the constraint i s that collisionless damping occurs i f the thermal velocity of background electrons i s of the order of the phase velocity (Stix, 1962). However an additional observational challenge to the incoherent Cerenkov radiation theory comes from some indications (Gurnett and Frank, 1972) that, before polar LHR noise i s observed, there must be a threshold flux for low energy electrons and above the threshold the noise may not be proportional to flux. Should these points be upheld some coherence of the precipitating electrons w i l l have to be taken into account. The problem of finding growth rates for whistler mode waves in a magneto-active plasma, due to a collective interaction with a flux of electrons streaming along the background magnetic f i e l d , has been inves-tigated by Horita and Watanabe (1969A, 1969B; and Horita, 1973) and by Singh and Singh (1971; and Singh, 1973). For a given frequency, Horita and Watanabe found a positive growth rate i f the phase velocity was approximately equal to but slightly greater than the wave normal component of the velocity of the streaming electrons. This i s identical to the - 94 -condition that a single charged particle needs to satisfy to emit Cerenkov radiation. As to frequencies for which this condition i s satisfied, i t was found in both formulations to range from the LHR frequency to the smaller of either the electron cyclotron or the electron plasma frequency. This frequency range is consistent with the observed frequency band of the polar LHR noise - from a few kHz to a few MHz. 5.1.2 Mid-Latitude LHR Noise After the discovery of LHR noise at about 1000 km (Barrington and Belrose, 1963), Smith et a l . (1966) proposed that mid-latitude LHR noise results from a horizontal ionospheric duct that traps electromag-netic waves which propagate nearly transverse to the earth's magnetic f i e l d . The duct i s a consequence of the upper ionosphere having more than one kind of positive ion, with different height profiles. As a function of altitude, the value of the LHR frequency has a relative minimum at approximately 1000 km (Kimura, 1966). For whistler mode waves with fre-quencies above but near the minimum LHR frequency and with wave normals that are nearly perpendicular to the background magnetic f i e l d , there i s a duct (approximately a horizontal layer) containing the minimum LHR frequency. The duct i s a region of high index of refraction sandwiched between regions of low index of refraction. Once the above waves are in the duct, they are trapped by refraction. The duct was estimated to be a few hundred km thick and to trap waves having frequencies in a band of a few kHz. Travelling within the duct, a satellite's instruments would measure waves having frequencies with a lower bound at the local LHR frequency (Gross, 19 70). Finding LHR noise up to 3000 km, McEwen and - 95 -Barrington (1967) discarded the duct theory. Later Gross (1970, 1972) investigated the duct theory more completely and showed that the duct could be 2000 km thick. Consequently, the duct theory might s t i l l explain LHR noise observed from 1000 to 3000 km. However observations of mid-latitude LHR noise at several earth radii (Scarf et a l . , 1972a, 1972b; Burtis et a l . , 1973) require some additional mechanism (s). In 1968 Storey and Cerisier proposed that narrow banded LHR noise could be obtained from VLF noise approximately propagating down the earth's magnetic f i e l d lines, from some source i n the distant magnetosphere. If the noise is in the non-ducted whistler mode, the wave normal angle with respect to the earth's magnetic f i e l d becomes large (Yarboff, 1961; Smith and Angerami 1968). When a wave's frequency is equal to the local LHR frequency, i t is reflected (until approximately 1000 to 1500 km, the local LHR frequency would be increasing as the noise propagates toward the earth). At a given altitude, one would observe noise with frequencies at approximately the local LHR frequency, reflected at that altitude, and at higher frequencies, composed of wave components s t i l l traveling downward and those already reflected back from lower altitudes. The intensity of the noise f a l l s with increasing frequency. In a model calculation, Storey and Ceisier obtained a narrow band frequency spectrum similar to mid-latitude noise with a maximum intensity ocurring near the local LHR frequency and about .6 kHz above the lower cut off frequency. Like the duct theory, this mechanism claims mid-latitude LHR noise to be a propagation as opposed to an emmission effect. - 96 -Because of the very motion of a s a t e l l i t e with respect to the ionosphere, Budko (1969) investigated the possibility that a whistler, in the vicinity, would necessarily excite LHR noise. The basic idea is that a body moving in a plasma would produce inhomogeneities in the plasma that would then t r a i l the s a t e l l i t e . This inhomogeneity would then interact with a whistler to produce a secondary electric f i e l d . For a s a t e l l i t e moving perpendicular to B ^ - the background magnetic f i e l d , the interaction was investigated for a cold plasma and for a whistler whose wave normal was parallel to B^^ . The calculations, for large time t - for the model picked, greater than .01 sec with respect to the i n i t i a l presence of the whistler - indicated that a secondary electric f i e l d would exist at approximately the LHR frequency and that the bandwidth would be proportional to 1/t. According to Budko's mechanism, LHR noise should be observed whenever a whistler reaches the s a t e l l i t e ; however LHR noise is observed much less often than whistlers (Laaspere et a l . , 1969). 5.2 Brief Summary and Results of Proposed Theory Whistler triggered LHR noise in the upper ionosphere was the physical motivation of this study. The mechanism put forth is of a wave (whistler) propagating from a region of the ionosphere where the unperturbed ion number densities are uniform into one where there are spatial - 97 -irregularities in the number densities'1". Caused by the propagating wave, charge separation would inturn produce an induced ele c t r i c f i e l d , which is being identified as the source of LHR noise. The question then asked i s at what frequencies, i f any, is the induced f i e l d at resonance or at least significant compared to the f i r s t order ele c t r i c f i e l d . To attempt to answer the above question, an equilibrium solution was obtained for the ele c t r i c and magnetic fields that may exist in a slig h t l y inhomogeneous, ful l y ionized plasma. The plasma was assumed to be in a constant and uniform background magnetic f i e l d , and the irregularity i n number density was taken to be a spatially varying cosine function with wave number K. Assuming the plasma to be governed by Maxwell's Equations and the zeroth and f i r s t moment equations of the collisionless Boltzmann Equation, we obtained solutions by a perturbation scheme. The equations were linearized and only terms to second order were kept. The f i r s t order terms formed a set of equations governing a plasma whose unperturbed number densities were constant and uniform. With f i r s t order wave solutions our postulated irregularity required that The idea was put forth publicly by Dr. T. Watanabe and the author at the URSI-IUPAP Symposium on Waves and Resonances in Plasmas held at St. John's, Newfoundland in July of 1971. Along with topics put forth by several others our idea appeared in the last section of a summary report of R.W. Lorenz that was published in Radio Science (vol. 7, pp. 885-886, 1972). Many of the results of this thesis were delivered by the author in an oral paper at the 1973 F a l l Annual Meeting of the American Geophysical Union in San Francisco, California (abstract is i n EOS, Vol 54, 1973). - 98 -the second order solutions be nonzero. The main part of the thesis has been to see at what frequencies (around the LHR frequency) the second order electric f i e l d (induced field) i s significant compared to the f i r s t order f i e l d - the inducing f i e l d . 5.2.1 Wave Number £ 1 B Q and | | K ( = K^x) A review w i l l now be made of the results when both it, the wave number of the f i r s t order propagating plane wave, and K, the postulated irregularity wave number, are parallel and both are perpendicular to B^^, the background magnetic f i e l d . a Electrostatic Assumption Let us f i r s t review the electrostatic solutions, which ignores the second order magnetic f i e l d . Solving the problem for a cold plasma (temperature taken to be 0° K), we found that the second order ele c t r i c f i e l d , independent of K, had a resonant - a singularity - frequency at the LHR frequency 1. When the temperature of the plasma was taken into 2 2 2 account - for every co between .1 to and .99 co and for a K LH LH X -2 -4 -1 between 10 and 10 cm , change, i n the second order e l e c t r i c f i e l d was negligible. (compare Figure (3-1) with Figures (4-1) to (4-5)). Though there no longer i s an actual singularity, there i s s t i l l a peak 2 4 2 in the second order electric f i e l d . When to = 1.002^ co , the second j LH 1 The true LHR angular frequency squared i s between 1.0024 and 2 2 1.0025 times coTTJ, where toTTJ i s defined by equation (3-54). LH LH - 99 -order f i e l d i s of the order of five hundred times the product of n fvS^ , e, s/ e the ratio of the electron irregularity density to the background electron density, and the f i r s t order electric f i e l d . For n IvS^ greater 6 ) S/ 6 -4 than 5 x 10 , the second order electric f i e l d w i l l be within an order of magnitude of the f i r s t order electric f i e l d . b Removing the Electrostatic Assumption For a cold plasma and without making our electrostatic assumption, the second order electric f i e l d i s s t i l l found to have a singularity at the LHR frequency ( - coTTJ). However an additional singularity occurs LH 2 2 at the frequency where K = 4k (that i s , where the wavelength of the X X f i r s t order f i e l d i s twice that of the postulated irregularity). With -4 -1 K = 10 cm , the singularity is located at a frequency approximately X equal to .36 c o ^ . As K x increases the singularity's frequency -3 -1 increases up to t o ^ . With > 5 x 10 cm , the singularity occurs at 2 2 a frequency that i s within .3% of t o T „ . For co between .1 to and LH Ln 2 -2 -4 -1 .99 toTTT and for K between 10 and 10 cm , there is l i t t l e LH x change, taking temperature into account, in the magnitude of the second order electric f i e l d (compare Figures (3-17) to (3-21) with Figures (4-9) to (4-13)). Unlike the peak at toTtl - as mentioned in section a - the LH 2 2 peak at the frequency where K x = 4k^ remains a singularity. For an irregularity that i s characterized by a band of wave numbers from 10 to 10 cm or to 10 cm , one would expect peaks in the second order electric f i e l d to cover a band of frequencies - 100 -from co to .97 c o T I I or .36 c o T „ , respectively. If one identifies LH LH LH the second order electric field with whistler triggered LHR noise, one would have a noise band with a lower cutoff frequency that would depend on the minimum K characterizing the irregularity and would be below the LHR frequency. 5.2.2 Wave Number it in Arbitrary Direction with Respect to and K ( = Kx) For a cold plasma and with our electrostatic assumption, we found a solution for the second order electric field in a coordinate system with B ^ in the z-direction and for it arbitrarily oriented in the x - z plane. Though the solution was found for K in an arbitrary direction, only the most tractable case was investigated : K in the x-direction. For 0 (the angle between k and ) equal to 7 1 . 5 6 ° , the second order electric field was found to have two singularities for Kx equal to 10 , 5 x 10 , 1 0 , and 5 x 10 cm and one for -4 -1 10 cm (see Figures (3-2) to (3 -8) ) . In a l l cases the frequencies of the singularities were greater than the LHR frequency. As Kx decreased the frequencies of the singularities increased from 1.001 c o T T J to 30 c o T „ , LH LH -2 -4 -1 for Kx respectively equal to 10 and 10 cm . On the other hand, -3 -1 for Kx equal to 10 cm , the second order electric field's singularities had frequencies that increased for decreasing 0 (see Figures (3-9) to (3-16)): from approximately co to 3.4 c o T U for 0 LH LH respectively equal to 90° and 0 ° . - 101 -If we again identify the second order electric f i e l d with LHR. noise and i f we have a distribution of f i r s t order ele c t r i c fields with wave normals corresponding to 9's from 90° to 71.5° and with an -2 -3 -1 irregularity characterized by K for 10 to 10 cm , then the above would imply a noise band with frequencies from 1.5 co to wTU-5.3 Some Additional Comments and Some Possible Future Investigations In the present study, i t was assumed that the unperturbed inhomogeneity in density was small compared to the background plasma density. This assumption is consistent with observations that density irregularities, in the upper ionosphere and at mid-latitudes, are usually less than one percent of the background plasma density. This i s even true at night, when irregularities are most often measured (Herman, 1966; Dyson, 1969; Huang Chun-ming, 1970; and McClure and Hanson, 1973). In terms of our mechanism, the diurnal occurrence in irregularities i s also consistent with the diurnal occurrence of mid-latitude LHR noise - maximum at night (McEwen and Barrington, 1967; Laaspere et a l . , 1971; and Barring-ton et a l . , 1971) . With respect to irregularity scale size, small scale irregularities are usually of the order of 1 km or less, which is compatible with our choosing irregularities with wave lengths that vary from .006 km to .6 km (Herman, 1966; Dyson, 1969; Rufenach, 1972; and Cunnold, 1972). In that upper ionospheric density irregularities are f i e l d aligned, a future study could assume irregularities with a cylindrical geometry. - 102 -Using Ogo 4 data, Laaspere and Taylor (1970) made a comparison of the frequency of the maximum intensity of mid-latitude LHR noise (close to but always greater than the lower frequency cutoff) and the LHR frequency, as calculated from simultaneous spectrometer measurements. From this comparison i t appeared that the frequency of maximum intensity, and thus the lower cutoff frequency, was almost always less than the LHR frequency, as calculated from the spectrometer data. The maximum intensity frequency usually varied for 0 to 20% below the calculated LHR frequency. As indicated in the summary, our mechanism can be consistent with the lower cutoff frequency being lower than the LHR frequency. The reflection theory of Storey and Cerisier predicts the maximum intensity frequency to be at the LHR frequency, while the duct theory predicts the lower cutoff frequency to be at the LHR frequency. Thus given Laaspere's result, one would infer that at least some of the observed mid-latitude LHR noise can not be explained by the reflection or the duct theory. For 6 - the angle between the f i r s t order electric f i e l d (whistler) and the background magnetic f i e l d - other than 90°, second order electric fields (noise) were only found assuming no second order magnetic f i e l d . Without this assumption for 0 = 90°, singularities could be found at frequencies below the LHR frequency. Thus i t might be interesting to drop this assumption for 0 other than 90°. However, a more useful study would be to correlate observed whistler triggered LHR noise with observed density irregularities. - 103 -BIBLIOGRAPHY Ahmed, M. and R.C. 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Mansfield, N.N., Radiation from a Charged Particle Spiraling i n a Cold Magnetoplasma, Astrophys. J., 147, 672-681, 1967. McClure, J.P., and W.B. Hanson, A Catatog of Ionospheric F Region Irregu-l a r i t y Behavior Based on Ogo 6 Retarding Potential Analyzer Data, J. Geophys. Res., 78., 7431-7440, 1973. - 106 -McEwen, D.J., and R.E. Barrington, Some Characteristics of the Lower Hybrid Resonance Noise Bands Observed by the Alouette I Satellite, Can. J. Phys., 45, 13-19, 1967. Michkofsky, R.N., and T. Watanabe, EOS (Transactions, American Geophysical Union) 54, 1187, 1973. Mosier, S. R. and D.A. Gurnett, Observed Correlations between Auroral and VLF Emissions, J. Geophys. Res., 77_, 1137-1145, 1972. Panofsky, W.G. and M. Phi l l i p s , Classical El e c t r i c i t y and Magnetism, Addison-Wiley Publishing Co., Inc., Reading, Massachusetts, 1962. Rao, M., S.K. Dikshit, and B.A.P. Tantry, Incoherent Cerenkov Radiation in the Magnetosphere and Ground Observations of VLF Hiss, J. Geophys. 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Katsufrakis, Lower Hybrid Resonance Noise and a New Ionospheric Duct, J. Geophys. Res., 71, 1925-1927, 1966. Smith, R.L. and J.J. Angerami, Magnetospheric Properties Deduced from OGO 1 Observations of Ducted and Unducted Whistlers, J. Geophys. Res., 73, 1-20, 1968. Spitzer, T.H., Physics of Fully Ionized Gases, Interscience Publishers, New York, 1962. - 107 -Stix, T.H., The Theory of Plasma Waves, McGraw-Hill Book Cos., Inc., New York, 1962. Storey, 0. and J.C. Cerisier, Une Interpretation des Bandes de Bruit au Voisinage de l a Frequency Hybride Bases Observers au Moyen de Satellites A r t i f i c i e l s , Compt. Rend. (C. R. H. Acad. Sci.), Ser. B, 266, 525-528, 1968. Waldtenfel, P., Exospheric Temperature from Rockets and Incoherent Scatter Measurements, J. Geophys. Res., 7_6, 6990-6994, 1971. Wheelon, A.D., Radio Wave Scattering by Tropospheric Irregularities, J. Res. N. B. S., 63D, 205-234, 1959. Yabroff, I., Computation of Whistler Ray Paths, J. Res. NBSD, Radio Propagation, 65D, 485-505, 1961. - 108 -APPENDIX I FIRST ORDER SOLUTIONS FOR A COLD PLASMA Given a plane wave form of the x-component of the f i r s t order electric f i e l d , then the rest of the variables can be written as ,(D jDE( 1> x n 2 - s (A-I-l) .(1) n2k k E<X> X Z X 2 2 2 n k - k^P x (A-I-2) (1) -jck DE( 1 )  J z x w (n 2 - s) (A-I-3) ,(D ck E( 1> ' Z X 1 - X 2 i 2 , 2„ n k - k P x (A-I-4) ,(D jck DE( 1 )  J x x w (n 2 - s) (A-I-5) v. (1) Jq. E ( 1 ) ' J^X X i , x nu co coco. D l l 2 2 / 2 2w 2 [ co - coi (co - OK) (n - S) (A-I-6) v. (1) X, z n f x,y nuco 2^ co D I 2 2 r 2 2, , 2 _, I [ co - coi (co - co i)(n - S) J J q . E ( 1 ) ( J^x X m. co x n 2k k X z 2 1 2 , K n k - k P x (A-I-7) (A-I-8) - 109 -(0)„(1) , 2 (1) = 1 1 X X i k coco.D x 1 + k k 2 X z m.co l 2 2 2 2 W 2 _ ' 2. 2 ' 2_ co -co. (co -co.)(n -S) n k - k P 1 1 X X (A-I-9) 2 In the limit of = 0 and where n is given by equation (3-25), E ^ = y E^> = 0 J D XJ X - j S E ^ J x D B^> = 0 (A-1-10) (A-I-ll) (A-I-12) (1) -jck S E( 1 )  J x x coD (A-I-13) v (1) J q .E ( 1 )  J l x 2 2 m.co(co - co.) 1 X „ coco. S 2 , l co + — (A-I-14) i.y v. q . E ( 1 ) 1 X 2 2 m.co(co - co.) 1 X . to2S coco. H — l D (A-I-15) i,z = 0 (A-1-16) - 110 -APPENDIX II SECOND ORDER SOLUTIONS FOR A COLD PLASMA-ELECTROSTATIC CASE a. These solutions are for the case of £ i n the x-z plane, K having an arbitrary direction, and the background magnetic f i e l d in the positive z-direction. S < 2 > n E (r, t) = (0) 2 at + K ) 'X + , cos E - —- sin e Z_ Y-v f 2 ) ( ? , t) 1,X ' + (k - K ) X z — cos 6 - -.— sin <S (A-II-1) % „ / n (0) 2to(co2 - co2)m. l l R cos e + R cos 6 -v. v. i,x i,x q i n / n ( « E f K e,s/ e J o 2o)(a)2 - to2)nu I sm e - I sin 6 v. v. i,x i,x R cos e + R cos 6 -v. v. !»y i»y (A-II-2) v f 2 ) ( ? , t) i,z q. n / n (0) ik e,s/ e I + sin e - I sin 6 v. v. i.y i»y (A-II-3) 2com. l R cos E + R cos 6 v. v. i , z l , z I sin e - I sin 6 v. v. i»z i,z (A-II-4) 42\t, t) -q.n. E l i,s o 2a)2m. l R cos e + R cos 6 - 1 sin e - I sin 6 n j n n. n. i i i i (A-II-5) where - I l l -e = (£ + K) - r - cot - E + 6 = (t - t) - r - cot - E ~ ip = ( l / Y * ) l , x ^ co 2(k + K ) Z~ + coco.K x — x — i y (A-II-6) (A-II-7) (A-II-8) i ; = (-I/Y1) i , x oj2(k + K ) X ± + coco.K TT x — x l y (A-II-9) = ( 1 / Y 1 ) coco, (k + K )Xt + co2K Z± x x — x y (A-H-10) i ; = ( l / Y * ) coco, (k + K ) Z ± + co2K X* l x — x — y (A-II-11) IT = ( l / T T X k + K ) Z ± v. z — z x,z (A-II-12) V . x.z (-1/Y ±)(k + K )X^ z — z (A-II-13) n. x ( ( 2 2 W±1 Ceo - U K ) Y - i f 2 + + (k + K ) (co (k + K )Z~ + coco.K X~l x — x ^ x — x , — x y ' + K coco i ( k x ± K X ) X ± + u K Z ± + 0^ - - U K D / ^ - S ^ Y 1 ] (A-II-14) + ( k, ± K ) 2(co 2 - oAz 1 z z X + / 2 2 + I " = - (oT - co7)Y~ n i I 1 (k + K ) x — x co2(k + K ) X ± + coco.K 77 x — x x y + (co2 - coco iD/( n 2-S))Y ± 4- K - y coco, (k + K ) Z + + co2K X + > l x x — x — y + (k + K ) (co2 - co2) (k + K ) X f + n 2 k k Y V C n ^ - k 2 ? ) ) z — z x z / ' x J (A-II-15) - 112 -b. These are the solutions for the case of 1c in the x-z plane, and the background f i e l d in the positive z-direction. E"(2> (r, t) n E e,s o „ ( 0 ) 2 (k + K)X . (k - K)X ^  cos y sin A v ( 2 ) ( r , t) x,x (k + K)X + jt - K)X~ sin y sin A (A-II-16) q. (n / n ( 0 ) ) E f - 1 e,s/ e J o | 2 2 2to (co - to .) m. I + 1 sxn y cos A v. v. ' 1,X X,XJ v ( 2 ) ( r , t) + n + - r V . V . X,X x,x-cos y sin A (A-II-17) q. (n / n ( 0 ) ) E [ ^x1- e,s/ e •> o I + 2 2 2to (to - co )^nu R + R , I v. v. , [ 1 i.y i,y J cos y cos A v f 2 ) ( r , t) X , z - |R+ - R" V . V . sin y sin A (A-II-18) q.(n /n (°))E f x*- e,s/ e J o 2com. i + +r V , V . X , Z 1 , z' sin y cos A + i + - 1 ~ V . V . X , Z X , ZJ cos y sin A (A-II-19) (2) * ± (r, t) q-n- E „ i x,s_o 2co m. " + - 1 I + 1 n. n x v sin y cos A + i + - r n. n. X X' cos y sin A (A-II-20) y = .'ic«'r - tot - £ (A-II-21) - 113 -X = K-r + ip (A-II-22) v. x,x -(l/Y ±)X ±co 2(k + K ) x — X (A-II-23) v. (l/Y±)X ±o)u.(k + K ) i x — X (A-II-24) v. x,z -(l/Y^JTk (A-II-25) n. - i f 9 + r ~ cow.D' ( kx± v h ( k x ± K x ) x _ + K " " T " n -s 2 ± 9 9 f + n k k Y > \ n k -k P x (A-II-26) c. These are the solutions for the case of It = k .and £ = K x x Thus k and K are parallel and perpendicular to the background magnetic f i e l d which i s in the positive z-direction. E< 2 )(x. t) n E ex l o y s~ c o s y c o s x n e (A-II-27) where e is a unit vector in the positive x-direction. vf 2>(x, t) x,x ' q. co X (0) n / n e.s/ e ( 2 2 ^ c (.0) - to.Jin.S X X sin u cos X (A-II-28) v< 2 )(x, t) i , y q. to. n / n (0M x x e.s/ e 2 2 (to - to^)nuS cos u cos X (A-II-29) v ( 2 ) ( x , t) = 0 x,z ' (A-II-30) - 114 -2 coco . S •n I CO , I , 1 q.n. E — + co + — - — n i i,s o[ S D 2 2 2 co (co - co.)m. l l K cos y sin A + k sin y cos A X X (A-II-31) where in this case y = k x - c o t - E x ^ A = K x + ill x (A-II-32) (A-II-33) - 115 -APPENDIX III SECOND ORDER SOLUTIONS FOR A COLD PLASMA-ELECTROMAGNETIC CASE and in the x-direction, the solutions are n E r . ^ G S O I T — _ 1 fc) = y - ^(a +u )cos y cos A - (a -a )sin y sin A J t) = n E e,s o (0) 2 (A-III-1) 1 ( B + 4 £ )sin y cos A + (8 +-8~) cos y sin A t) = B x 2 ) ( x , t) = B^ 2 )(x, t) = 0 x y (A-III-2) (A-III-3) t) = n cE e,s o_ (0) 2to + k (B +4^ _) + K (B+-8 )^ A . X sin y cos A k (B -B ) + K (6 +8 ) cos y sin A t) = H e,s/ e ) o 2 2 2u)(io - to.)m. i i 2 + - + -co (a +a ) + coco. (8 +8 0 (A-III-4) sin y cos A 2 + - + -co (a -a ) + cocojB -8 ) j cos y sin A (A-III-5) t) = n /n<°> E I e,s/ e J o 2 2 2co (to - UK)IIU + - 2 + -COOK (a +a ) + to (3 +3 ) cos y cos A + — 2 + - 1 totoja -a ) + to (8 -8 ) sin y sin A (A-III-6) - 116 -where v f 2 ) ( x , t) = 0 1,Z (A-III-7) n < 2 ) ( X , t) - q . n . E i i , s o 2co (co -10 x 2 + - + -co (a +a ) + (00^(3 +3 ) „ coco.S ^ _ -) + K (to (a -a ) + coco. (8 -B~)) s i n y cos X + K 2 + - + -co (a +a ) + toco J B +8 ) „ coco S + 2(co + ) + k x ( a i Z ( a -a") + coco.(B -8")) cos y s i n X (A-III-8) a = 2 2 2 2 2 -co (S - D - 2S) + c (k + K ) x — x 2 2 2 2 2 a) (S Z - D Z) - c (k + K ) Z S x — x (A-III-9) - CO 2 f ( S 2 - D 2 )S - ( S Z + D Z) 2 . „ 2 S 2 2 2 2 2 w (S Z - - IT) - c (k + K T S x — x (A-III-10) y = k x - c o t - r x X = K x + il) x r (A-III-11) (A-III-12) - 117 -APPENDIX IV FIRST ORDER SOLUTIONS FOR A HOT PLASMA If the x-component of f i r s t order electric f i e l d is given by E ^ e x p ( j ( k x x - cot ) ) , where k and co are related by 2 2 2 2 c k /co = (sIS-L1 ~ 1\)/S^, then the other f i r s t order variables can be written as -iS F ( 1 ) - B ( 1 > - B ( 1 ) - 0 x -ick S..E J x 1 x toD, y (1) (A-IV-1) (A-IV-2) (A-IV-3) v ( 1 ) 1,X J q . E ( 1 ) X ,2 2 .2 2. m. to (to - co. - k a. ) i i x I f 2 ^ V l 1 co + — (A-IV-4) v ( 1> i»y q.E ( 1 ) l x / 2 2 ,2 2. m.to(co - co. - k a.) i I x i y coco. + 1 2 2 2 (to - k a.)S, x i 1 D, (A-IV-5) v ( 1 ) 1, z = 0 (A-IV-6) x Jt q.n<0)E(1> J x n i i x 2. 2 2.2 2. m.co (co - to. - k a.) 1 1 x x' \ 2 ^  ^ i S l ) to + - — (A-IV-7) - 118 -APPENDIX V SECOND ORDER SOLUTIONS FOR A HOT PLASMA-ELECTROSTATIC CASE For k and K in the x-direction and in the positive z-direction, the solutions are t) -= X v ( 2 ) ( x , t) 1,X v P ( K , t) i»y n E ( e.s o n " q i (0) 2 x x cos y cos A 4-h X X sin y sin X (A-V-l) L /n<°>l n / n 2com. I + + 1 -II V . V . ^ i,x i,x + I - I I cos y sin X i,x V i , x J sin y cos X (A-V-2) q.fn /n<°>l i j e,s/ e 2tom. + R + R ' x,y ity-> cos y cos X R + - R" v. v. i»y i,y sin y sin A (A-V-3) (A-V-4) -q.n. E l i,s c 2to m. l I +1 sin y cos A n. n. l x} + i + - r v n ± n ± cos y sin A (A-V-5) where - 119 -x = (i + ir)/s* V. 2 2 co - CO . 1 _co ^ _ (k + K ) 2 a 2 x — X 1 l + \r (A-V-6) + — a.(k + K )K l x — x x r 0 coco.S.,-2 , I 1 co + 2.2 2 , 2 , co (co - co. - k a.) 1 X I (A-V-7) R" v. CO v. 1,X (A-V-8) I * = (k + K ) n. x — x l 2 , " V l co + — _ + 2 2 2 ' i , x co - co. - k a. l x x (A-V-9) k x - cot - E x A = K x + f x (A-V-10) ( A - V - l l ) - 120 -APPENDIX VI SECOND ORDER SOLUTIONS FOR A HOT PLASMA-ELECTROMAGNETIC CASE For % and S in the x-direction and in the positive z-direction, the solutions are E< 2 )(K, t, 1 1 E I 1_ e,s o - + n (0) IT ^aT + a T ^ c o s u c o s * - (ax ~ a T ) s i n u sin X (A-VI-1) e.s o |, + t o f Y~ (®T + ^T^ S i n y C 0 S X + (®T ~ 3 )^ cos y sin X (A-VI-2) B< 2 >(x, t) - B ' 2 ) ( X , t) - 0 x y (A-VI-3) n cE e,s o n ( 0 ) 2 U k x ( B T + B ? + K x ( 3 T " ^ T ^ S i n y C 0 S A + cos y sin X (A-VI-4) v< 2!(*. t> ..[» /„»>] e,s/ e E r. 2iom. l i+ + r v 1 , X 1 , X sin y cos X + u + - r V . V . 1,X 1 , X cos y sin X (A-VI-5) - 121 -v ! 2 ) ( X > « f / (0)1 n / n 2tom. R+ + R_ , . v. v. ^ i,y i,y cos y cos A R + -R-v. v. sin y sin A v f 2 ) ( x , t) = 0 1,2 ' (A-VI-6) (A-VI-7) -q.n. E l i,s o i  2 /to m. i + +r n. n. l l sm y cos A + I -I n. n. l l cos y sin X (A-VI-8) where a T and 6 T are given by equations (4-66) and (4-67), v. l . X 2 ± ± to a T + toa^g 2 2 .2 2 to -to. - (k +K ) a. l x— x 1 a. (k + K )K l x — x x f 2^ ^ i ^ to + — - — + 2 2 , 2 2 to -to.-k a. l x i 2 2 , x f f ,2 2 to -to.-(k +K ) a. l x— x 1 (A-VI-9) + ± 2 ± toto ia T + to 8 T j r ~ coto.S ' to.af(k + K )K L, + — l l x — x x D, + X- ,r 2 2 fir -i-v N 2 2 -— f 2 2 . 2 2 i,y to -to.-(.k +K ; a. to to -to.-k a. l x— x I i x i to2-to 2-(k +K ) 2 a 2 l x— x 1 (A-VI-10) n. l (k + K ) x — x 2 ± ± to a T + toto^ ^Brp 2 2 , i p . 2 2 to - t o . - ( k + K ) a. l x — x 1 + 2 A W a , i S l to + — - — D l 2 2 , 2 2 to - to. - k a. 1 x i 1 + a. (k + K )K l x — x x ~ 2 2 n ^ v \2 2 to - t o . - ( k + K ) a. l x — x 1 (A-VI-11) k x - tot - E x (A-VI-12) A = K x + x (A-VI-13) - 122 -APPENDIX VII INDEX OF REFRACTION OVER FREQUENCY INTERVALS RELATED TO THIS STUDY 2 The following pages contain graphs of n , index of refraction 2 2 squared (defined by equation (3-23) versus x ( = to /to T t r) . As in the text of the thesis, the negative sign is taken in equation (3-23) (with the intervals of frequency considered, the plus sign always corresponds to a negative index of refraction squared). The graphs are done for each value of 6 that was chosen in section 3 -2-1 , where 0 is the angle between the wave normal of the first order plane wave solutions and the background magnetic field. For a given 0, the graph(s) are done for intervals of x that include the intervals used in plotting the x-component of the second order electric field (see Figures (3-1) to (3-16)) . O Figure (A-VII-8). The value of n versus x where 0 = 71.57°. r in 

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