THE EXACT THEORY OF LINEAR CYCLOTRON INSTABILITIES APPLIED TO HYDROMAGNETIC EMISSIONS IN THE MAGNETOSPHERE by BRUCE RAYMOND JACKS B.Sc, U n i v e r s i t y o f B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of GEOPHYSICS We accept t h i s t h e s i s as conforming t o the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In p r e s e n t i n g the this thesis Columbia, I agree that the Library a v a i l a b l e f o r r e f e r e n c e and s t u d y . mission f o r extensive representatives„ cation of this thesis w i t h o u t my w r i t t e n It i s understood for financial Department o f The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a } i t freely thesis per- for scholarly by t h e Head o f my D e p a r t m e n t o r by permission. S^^jt. XI- s h a l l make I f u r t h e r agree that copying o f t h i s p u r p o s e s may be g r a n t e d Date fulfilment of r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f British his in partial \ Columbia °\LL that gain copying o r p u b l i - shall n o t be a l l o w e d i ABSTRACT The complex d i s p e r s i o n r e l a t i o n which describes transverse plasma waves propagating i n a cold g y r o t r o p i c ambient plasma p a r a l l e l to the back r ground magnetic f i e l d as they i n t e r a c t with charged p a r t i c l e streams i s derived by s o l v i n g the l i n e a r i z e d c o l l i s i o n l e s s Boltzmann equation simultaneously with Maxwell's equations using the Fourier-Laplace transform method. The wave frequency i s allowed to be complex with a p o s i t i v e imaginary part corresponding to a growing i n s t a b i l i t y . The r e a l and imaginary parts of the d i s p e r s i o n r e l a t i o n y i e l d two separate equations. Under s e v e r a l assumptions, the equations can be s i m p l i f i e d t o y i e l d an expression f o r the imaginary part of the frequency (the growth r a t e ) and an equation r e l a t i n g the r e a l wave frequency and the wave number. The theory i s then applied to the magnetosphere by choosing a dipole model f o r the earth's magnetic f i e l d and a s u i t a b l e d i s t r i b u t i o n f u n c t i o n f o r the p a r t i c l e s . The s p e c i f i c case of waves of the ion-resonance mode i n t e r a c t i n g with mono-energetic, contra-streaming protons i s considered i n d e t a i l , and the r e s u l t s of t h i s c a l c u l a t i o n are used i n e x p l a i n i n g hydromagnetic (hm) emissions. In p a r t i c u l a r , i t i s suggested t h a t the high frequency c u t o f f i s a r e s u l t of the p i t c h angle d i s t r i b u t i o n of the p a r t i c l e stream. Computer c a l c u l a t i o n s are done i n order to d i s p l a y the general r e s u l t s of the theory. S p e c i f i c a l l y , when low energy protons (10 - 20 kev), trapped on a f i e l d l i n e with an L value of 5.6 are considered, i t i s found that the region of i n s t a b i l i t y occurs near the geomagnetic equator, and that the growth r a t e i s a sharply peaked f u n c t i o n of the frequency. TABLE OF CONTENTS Chapter I INTRODUCTION General Discussion Cyclotron Resonance Thesis Outline II MATHEMATICAL ANALYSIS III APPLICATION OF THE GENERAL RESULTS TO THE MAGNETOSPHERE Discussion The S h i f t e d A n i s o t r o p i c Maxwellian D i s t r i b u t i o n A monoenergetic P i t c h Angle D i s t r i b u t i o n IV NUMERICAL CALCULATIONS Normalization o f the Equations Parameter Values Results V SUMMARY Discussion Conclusions Appendix I Transformation o f Equations and Solution f o r f (v,k,(0) II Determination o f the Transformed Magnetic F i e l d s III A n a l y t i c Continuation of the Integrals IV Wave P o l a r i z a t i o n V S i m p l i f y i n g the General Dispersion R e l a t i o n VI Coordinate Transformation and C a l c u l a t i o n of CJi t Bibliography iii FIGURES Figure Page 1. V a r i a t i o n of sin*iy with y f o r f i v e values of H . 2. Dependence o f CJj on the normalized frequency two p a r t i c l e energies. a)X=0° 3. 1 20 for 30 ,b)X-10° 31 c)X=20° 32 Dependence of D x on \ f o r two p a r t i c l e energies. a) uj'(eq)= 0.3 33 b) 6J (eq) = 0.4 33 c) 6J'(eq)=.0.5 34 d) w'(eq)"0.7 35 l 4. Integration contours f o r k?0,6Jj;>0. 47 5. Integration contours f o r k>0,6iJ <0. 47 6. Integration contours f o r k<0,<*^ >0. 48 7. Integration contours f o r k<0,CJ <0. 48 r c 1 iv ACKNOWLEDGMENTS I wish t o thank s i n c e r e l y Dr. T. Watanabe f o r suggesting t h i s problem and f o r h i s assistance i n many h e l p f u l discussions throughout the course o f the research. I a l s o wish t o thank Professor J . A. Jacobs f o r providing the opportunity and the f a c i l i t i e s t o carry out t h i s work and f o r h i s patience while i t was being done. 1 CHAPTER I INTRODUCTION General Discussion Since the earth's upper atmosphere contains a s i g n i f i c a n t number of charged p a r t i c l e s , a p h y s i c a l study of that region involves the concepts o f magnetohydrodynamics and plasma physics. The theory of plasma waves has been used t o e x p l a i n such phenomena as atmospheric w h i s t l e r s and geomagnetic micropulsations. In such studies, the geomagnetic f i e l d i s fundamental. Atmospheric w h i s t l e r s are electromagnetic waves which occur i n the frequency range 300 - 30,000 cps and propagate i n the e l e c t r o n resonance mode ( f a s t mode) which has an upper frequency l i m i t a t the e l e c t r o n cyclotron frequency - 6J . e They o r i g i n a t e i n l i g h t n i n g flashes ( H e l l i w e l l and Morgan, 1959) and bounce between the northern and southern hemispheres along paths which approximately f o l l o w the magnetic f i e l d l i n e s ( H e l l i w e l l , 1965). An analogous left-hand c i r c u l a r l y p o l a r i z e d wave e x i s t s i n the i o n resonance mode (slow mode) f o r frequencies below the i o n gyrofrequency OJc . The d i s p e r s i o n r e l a t i o n f o r these two types o f waves propagating p a r a l l e l t o the background magnetic f i e l d B 0 i n a c o l d , ambient plasma can be w r i t t e n (Astrom, 1950) f o r a plasma with one singly-charged, i o n i c component; frequency, k i s the wave number and c i s the speed of l i g h t . equation, both 6J and k are r e a l q u a n t i t i e s . A^pe and Sk^are and ion plasma frequencies r e s p e c t i v e l y , and are defined by i s the wave In t h i s the e l e c t r o n 2 firN 1 x P pe. (1-2) where N p i s the e l e c t r o n number density o f the plasma, q i s the charge on an e l e c t r o n or a proton and 3. i s negative or p o s i t i v e r e s p e c t i v e l y , and m e and m- L are the e l e c t r o n and ion masses r e s p e c t i v e l y . u n i t s i s used throughout the t h e s i s . The Gaussian system o f In equation 1-1, the upper sign i s used f o r the ion resonance mode and the lower sign f o r the e l e c t r o n resonance mode. I t i s possible that waves o f the ion resonance mode are d i r e c t l y involved with the production of micropulsations i n the pc 1 frequency range 0.2-5 cps. Tepley and.Wentworth (1962) were the f i r s t t o present the dynamic spectra (frequency-time p l o t s ) o f such micropulsations. Those which showed a d i s t i n c t f i n e structure c o n s i s t i n g o f r e p e t i t i v e r i s i n g tones which often overlapped were c a l l e d hydromagnetic emissions, or b r i e f l y , hm emissions. They have also presented a theory, which accounted f o r t h i s f i n e s t r u c t u r e (Wentworth and Tepley, 1962). Jacobs and Watanabe (1965) have described the h i s t o r y o f the research done on hm emissions and they emphasize the f o l l o w i n g points. At hydromagnetic frequencies,, waves o f the ion resonance mode tend t o be i I guided by the magnetic f i e l d t o a much greater extent than waves o f the e l e c t r o n resonance mode (Jacobs and Watanabe, 1964). The dispersion o f 'hm w h i s t l e r s ' or 'micropulsation w h i s t l e r s ' y i e l d s a t h e o r e t i c a l spectrum which agrees approximately with the observed c h a r a c t e r i s t i c s o f the structured hm emissions. The hm w h i s t l e r s i g n a l s d i f f e r from those o f atmospheric w h i s t l e r s i n that the s i g n a l i n t e n s i t y does not constantly decrease a f t e r the f i r s t bounce but often grows before decaying (Tepley and Wentworth, 1964). The idea developed i n t h i s t h e s i s i s that the waves gain energy through a c y c l o t r o n i n s t a b i l i t y process i n v o l v i n g low energy protons which are trapped i n the magnetosphere. The process i s exactly analogous to the i n s t a b i l i t y found by B e l l and Buneman (1964) f o r electrons i n t e r a c t i n g with waves of the w h i s t l e r mode. I t i s not a s i n g l e p a r t i c l e e f f e c t ( c y c l o t r o n r a d i a t i o n ) but a plasma i n s t a b i l i t y i n v o l v i n g the t r a n s f e r of some of the transverse k i n e t i c energy of the p a r t i c l e s to electromagnetic energy i n the wave ( B r i c e , 1964;.Neufeld and Wright, 1965a). In order to have an i n s t a b i l i t y at a l l , an i n i t i a l wave disturbance must e x i s t so that the wave-particle i n t e r a c t i o n can take place. The a c t u a l source of t h i s i n i t i a l , small 'seed' wave i s not known at present. The problem has been discussed by Jacobs and Watanabe (1965) and- Obayashi (1965). In the present d i s c u s s i o n , the existence of perturbing hm w h i s t l e r waves i s assumed. Cyclotron Resonance I t i s assumed that the streaming p a r t i c l e s have an i n i t i a l transverse component of v e l o c i t y . In order to determine whether the wave grows or i s damped, the v e l o c i t y d i s t r i b u t i o n function f o r the p a r t i c l e s must be s p e c i f i e d . I t has often been noted that growing i n s t a b i l i t i e s require an a n i s o t r o p i c d i s t r i b u t i o n ( S t i x , 1962; Montgomery and Tidman, 1964; Cornwall, 1965). I t can be seen i n t u i t i v e l y that a 'resonance' might occur i f a p a r t i c l e i s gyrating with the same sense of r o t a t i o n as the wave's p o l a r i z a t i o n , and i f the p a r t i c l e sees a wave frequency equal to i t s own cyclotron frequency. In a laboratory reference frame, the resonant-frequency i s 4 d i f f e r e n t from the cyclotron frequency because o f the Doppler s h i f t -arising from the p a r t i c l e ' s l o n g i t u d i n a l v e l o c i t y u. For the case o f protons and a left-hahd polarized'wave, the resonance conditions mentioned above are s a t i s f i e d with a p o s i t i v e real frequency OJ given by CL) - feu = • In the magnetosphere, D - (0;. < 0 (Booker, 1962). (1-3) I t can then be seen that the product ku must be negative, v i z . , the wave and p a r t i c l e s must t r a v e l i n opposite d i r e c t i o n s . However, i t must be noted that protons can i n t e r a c t with waves o f the w h i s t l e r mode because o f the anomalous Doppler e f f e c t ( B r i c e , 1964). When a p a r t i c l e t r a v e l s f a s t e r than the wave and i n the same d i r e c t i o n , i t sees a r e v e r s a l o f the wave's p o l a r i z a t i o n . Jacobs and Watanabe (1965) have discussed the d i f f e r e n t p o s s i b i l i t i e s leading t o cyclotron instabilities. Thesis Outline In Chapter I I , a general l i n e a r analysis o f the problem i s c a r r i e d out s t a r t i n g from Maxwell's equations and the c o l l i s i o n l e s s Boltzmann equation. The Fourier-Laplace transform method i s used, the general procedure being s i m i l a r t o that outlined by S t i x (1962) f o r l o n g i t u d i n a l plasma o s c i l l a t i o n s . This method was suggested by Watanabe (1965a), and the r e s u l t s o f these c a l c u l a t i o n s agree with those o f Cornwall (1965). Chapter I I I involves the a p p l i c a t i o n o f the general r e s u l t s t o the magnetosphere. The proton streams are assumed t o be monoenergetic. The p i t c h angle d i s t r i b u t i o n function i s chosen t o s a t i s f y a d i f f e r e n t i a l equation which i s v a l i d f o r p a r t i c l e s trapped i n a strong, steady, magnetic 5 f i e l d i n a tube of f l u x which has a small normal cross-section (Watanabe, 1964). In Chapter IV, the r e s u l t s o f numerical c a l c u l a t i o n s made on a computer are presented. In order t o carry out the c a l c u l a t i o n s , several assumptions are made: the earth's f i e l d i s assumed t o be a centered dipole f i e l d having a value o f 0.3 Gauss on the earth's surface at the geomagnetic equator; the Smith model (Smith, 1961) o f the e q u a t o r i a l e l e c t r o n density, which i s v a l i d only f o r distances up t o four earth r a d i i from the earth's surface, i s assumed t o hold i n a l l regions o f the magnetosphere. two assumptions l i m i t the exactness o f the r e s u l t s . These A discussion o f the v a r i a t i o n o f e l e c t r o n density along f i e l d l i n e s has been given by Carpenter and Smith (1964). Watanabe (1965c) has indicated,how information about the d i s t r i b u t i o n of electrons at a l t i t u d e s greater than about four earth r a d i i may-be obtained. The f i n a l chapter summarizes several relevant papers which deal with hm emissions and cyclotron i n s t a b i l i t i e s i n the magnetosphere and discusses the l i m i t a t i o n s of the t h e s i s . 6 CHAPTER.II MATHEMATICAL ANALYSIS To describe the i n t e r a c t i o n between the waves and the p a r t i c l e stream, one must determine the evolution i n time of the p a r t i c l e d i s t r i b u t i o n function. Knowing the i n i t i a l conditions, the electromagnetic f i e l d s i n the plasma can then be determined. I t i s assumed that the d i s t r i b u t i o n function f ( v , r , t ) s a t i s f i e s the c o l l i s i o n l e s s Boltzmann equation, and i f the plasma i s c o l d , then the f i e l d s produced by the p a r t i c l e density f l u c t u a t i o n s (due t o thermal motions) are n e g l i g i b l e compared to the f i e l d s of the wave and the equation can be w r i t t e n 3t 3£ *m where t and r are the time and space coordinates r e s p e c t i v e l y , v i s the p a r t i c l e v e l o c i t y , m i s the p a r t i c l e mass and E and B are the e l e c t r i c f i e l d strength and the magnetic f l u x density, r e s p e c t i v e l y , of the wave. 3/8JC represents the s p a t i a l gradient and 9/9y the gradient i n v e l o c i t y space. The Maxwell equations used are c u r l B,= * ? j + ^ | f curl.E=-iH (2-2) ( 2 " 3 ) where COInAp J defines the current density. of the plasma. The summation i s taken over a l l the components 7 Equation 2-1 i s expanded by assuming that the f i e l d s B and E are f i r s t order q u a n t i t i e s ( B 0 i s zeroth order) and that f can be w r i t t e n f(v,r,t) = f where f 4 0 (v) + f ± (v,r,t) i s a f i r s t - o r d e r perturbation on f . 0 taken to be i n the p o s i t i v e z d i r e c t i o n . (2-5) The background f i e l d B is G Only transverse waves are con- sidered and the s p a t i a l l y varying q u a n t i t i e s are assumed to depend only on the coordinate z, and not on x and y. Neglecting terms of second order i n equation 2-1, the zeroth and f i r s t - o r d e r equations are found to be and Introducingthe c y l i n d r i c a l coordinates (u,w,<|>) i n v e l o c i t y space, the l a s t term i n equation 2-6 can be w r i t t e n where ^ c = ^ r (-) 2 i s the p a r t i c l e gyrofrequency which can be p o s i t i v e or negative. 9 It i s assumed that f t = §i° = $ so that equation 2-6 i s s a t i s f i e d . = ° <- > 2 10 8 Using a F o u r i e r transform i n space and a Laplace transform i n time and using two component equations obtained>from equations 2-2 and 2-3, the transformed equation 2-7 can be solved f o r ^ ( V j k , ^ ) ) (Appendix I ) . In t h i s way, equation 2-7 i s handled as an i n i t i a l value problem, where the p a r t i c l e d i s t r i b u t i o n f u n c t i o n at time t = 0 must be s p e c i f i e d . This method was f i r s t used by Landau (1946) i n d i s c u s s i n g the l o n g i t u d i n a l v i b r a t i o n s of an e l e c t r o n i c plasma. I f the i n i t i a l d i s t r i b u t i o n can be w r i t t e n (2-11) then fj.(v,k,6j) i s given by -±ii»£[w».*)-ie,c<>.vj) w-ku-w i x ^ c ^ * c + /aw 9U B (w,^)+-c& (^» x y (2-12) CO -feu + • 'VnCOt Using equation 2-4, simple algebra gives (2-13) since f (y_) i s constant with respect t o 0 Using equation 2-12, and the other two component Maxwell equations a f t e r transformation (Appendix I ) , i t can be shown (Appendix I I ) that B (oo ft)±t& (oJ^) = x ) y (2-14) 9 where x r . The upper and lower signs correspond respectively i 3& t o l e f t and r i g h t polarized waves (Appendix IV). In p r i n c i p l e , B(t,z) can now be found by applying the inverse transformations t o equation 2-14. This means that the response o f the plasma system t o an i n i t i a l perturbation o f p a r t i c l e d i s t r i b u t i o n s by a p a r t i c l e beam can be found. I t i s t h i s r e s u l t that j u s t i f i e s the use o f the transform method, but i n order t o make the problem f e a s i b l e i t i s not solved i n general. mathematically, In the Laplace transformation, the parameter i s allowed t o be complex, with the r e s t r i c t i o n that i t s imaginary part be positive. L a t e r , 6J i s i d e n t i f i e d as the wave frequency. The inverse transformation must be c a r r i e d out along a path which l i e s i n the upper h a l f OJ-plane above the s i n g u l a r i t i e s of B_((0,k). negative imaginary parts f o r D should be allowed. But p h y s i c a l l y , The procedure followed i n overcoming t h i s d i f f i c u l t y involves the a n a l y t i c continuation o f a s i n g u l a r i n t e g r a l and has been discussed by S t i x (1962). Using the Cauchy P r i n c i p a l Value ( P ) , i t i s found (Appendix V) that the d i s p e r s i o n r e l a t i o n i s given by where 10 and V ~ * Equation 2-16 i s v a l i d f o r both p o s i t i v e and negative imaginary parts o f (J . The presence o f the s i n g u l a r i t y i n the inverse transformation r e s u l t s i n the l a s t term i n equation 2-16 being evaluated under the c o n d i t i o n 60 - + U) = ° (2-17) t This i s how the c y c l o t r o n resonance c o n d i t i o n enters the problem mathematically. Equation 2-16 can be s i m p l i f i e d by s p e c i f y i n g the c o l d , background part of f 0 as f a by w r i t i n g with b where N p C u (2-19) ) i s the number density o f the background plasma and S represents the Dirac d e l t a f u n c t i o n . The d i s t r i b u t i o n f u n c t i o n f ^ i s normalized t o N p and f s represents the streaming p a r t i c l e d i s t r i b u t i o n f u n c t i o n . The 3$e/3U term i n the p r i n c i p a l value i n t e g r a l vanishes under the ,integral over w. The 2§i/dw term can be s i m p l i f i e d using i n t e g r a t i o n by parts so t h a t [ and since d w (Pfju, [G(f ^ B ^= = ~ ^ U / p L (<*>TW» = - - i k — (2-20) 0, equation 2-16 becomes -^Z^jdwlwG^i.^O (2-21) 11 where * f l = HirNpl'/'^c. i s the e l e c t r o n plasma frequency o f the background p plasma and i s taken as the t o t a l plasma frequency since N much smaller than N p i s assumed t o be s . I f CO i s w r i t t e n U) «• CO* +lLJx ,(2-22) i t i s a l s o assumed that M « 6 0 R (2-23) This condition means that the i n s t a b i l i t y grows or decays v e r y , l i t t l e during a time i n t e r v a l corresponding t o the period of the wave. and CJj are f i r s t order q u a n t i t i e s compared t o N P Assuming that N 5 and ( J , equation 2-21 R can be s i m p l i f i e d (Appendix V) and s e t t i n g the r e a l and imaginary components separately equal t o zero gives ^ - C ^ - V i t e L =0 (2-24) and (2-25) XL) • + J ± ^ where Equation 2-24 i s the r e a l d i s p e r s i o n equation which r e l a t e s and k, and equation 2-25 i s the expression f o r the growth rate o f the i n s t a b i l i t y . 12 CHAPTER I I I APPLICATION OF THE GENERAL RESULTS TO THE MAGNETOSPHERE Discussion The r e s u l t s of the previous chapter have been derived f o r the case of plane waves i n f i n i t e i n extent propagating p a r a l l e l t o a homogeneous background magnetic f i e l d which extends over a l l space. In applying these r e s u l t s to the magnetosphere, i t i s assumed t h a t the region of i n t e r a c t i o n i s small enough that the geomagnetic f i e l d can be considered homogeneous there, but large enough that the hm waves are w e l l approximated by plane waves. This problem has been mentioned by-Hruska (1966). In order to c a l c u l a t e 6J using equation 2-25, an e x p l i c i t t expression f o r f s must be determined i n a meaningful way. Although much has been learned experimentally about p a r t i c l e s contained i n the van A l l e n b e l t s , almost nothing i s known about the d i s t r i b u t i o n of low energy protons at higher a l t i t u d e s . Davis and Williamson (1962) have reported data obtained from the s a t e l l i t e 'Explorer 12' and Cornwall (1965) suggested these protons might be important i n c y c l o t r o n emissions as w e l l as constituting a r i n g current. Most of the r e s u l t s concerned protons i n the energy range 50 kev - 5 mev. Hoffman and Bracken (1965) have given a more complete report of the same data. Some of these r e s u l t s w i l l be quoted later. Two d i s t r i b u t i o n functions are now considered. The shifted, a n i s o t r o p i c Maxwellian d i s t r i b u t i o n i s used as an example since i t has been used s e v e r a l times before (Sudan, 1963; Guthart, 1964; H u l t q v i s t , Hruska, 1966). The second 1965; d i s t r i b u t i o n chosen i s discussed i n d e t a i l below. The S h i f t e d , A n i s o t r o p i c , Maxwellian D i s t r i b u t i o n This type of d i s t r i b u t i o n represents a p a r t i c l e stream whose spread of random thermal v e l o c i t i e s perpendicular to the background f i e l d i s d i f f e r e n t than the spread p a r a l l e l t o i t , and there i s an organized, uniform v e l o c i t y p a r a l l e l t o the f i e l d . In t h i s case, the d i s t r i b u t i o n function i s written i - M i A \ - ^ - < - ^ w l (3-1) where a^T,, u f s has been normalized to N s . 2 ) . In t h i s case, Setting u = V , and i n t e g r a t i n g R ( 3 over w and ty, (3-4) oo and so equation 2-25 gives ^fa), ^ ^ l A - t o ^ A ^ , ill .Ci.'w a* ^ e and i V ^ k ^(V*-U.)"\ -rrr&fc" -J t (3-5) 14 I f only one type o f p a r t i c l e i s streaming, then the summation can be removed and the condition f o r p o s i t i v e is 2± > (3-6) This r e s u l t i s known ( S t i x , 1962). A Monoenergetic P i t c h Angle D i s t r i b u t i o n Using a t h e o r e t i c a l approach, Watanabe (1964) has obtained a d i f f e r e n t i a l equation which governs the d i s t r i b u t i o n function of p a r t i c l e s trapped i n a 'strong' magnetic f i e l d , v i z . , one f o r which the scale o f s p a t i a l v a r i a t i o n s i s much l a r g e r than the gyroradius o f the p a r t i c l e . I f the f i e l d i s steady in,time, where *+• i s the l o c a l p i t c h angle o f a p a r t i c l e , 1 i s distance measured along a f i e l d l i n e , and B i s the l o c a l magnetic f i e l d strength. This equation i s v a l i d only i n the one-dimensional case, when the p a r t i c l e s are confined t o a tube of f l u x f o r which the l i n e a r dimensions o f any normal cross-section are much smaller than the scale length of the trapping region. A particular s o l u t i o n i s given by where e< i s an a r b i t r a r y constant and C i s constant with respect t o 1, *+* , and t and contains the normalization f a c t o r . perturbing wave, s i n ^ ^ / B In a 'strong' f i e l d , with no i s a constant of the motion since i t i s p r o p o r t i o n a l t o the magnetic moment of a p a r t i c l e , the f i r s t adiabatic i n v a r i a n t (Chandrasekhar, 1960; A l f v e n and Falthammar, 1963). The r e s u l t 15 that f g depends on the adiabatic i n v a r i a n t i s t o be expected (Cornwall, 1965). The discussion i n the remainder of the t h e s i s concerns only mono- energetic protons. The assumption that the p a r t i c l e s are monoenergetic i s not too r e s t r i c t i v e and helps t o s i m p l i f y the mathematics. electrons have been considered Monoenergetic previously (Wentworth and Tepley, 1962). In the numerical c a l c u l a t i o n s which are done l a t e r , the p a r t i c l e energy i s varied as a parameter. The d i s t r i b u t i o n function i s w r i t t e n W S < v - v . ) a a £ ( where V i s the 'pitch angle d i s t r i b u t i o n parameter'. 3-9) I t i s assumed that the number density of the streaming p a r t i c l e s i s known at some point i n the magnetosphere, that i s f o r some value o f the main f i e l d B*, f o r instance at the equator. f s t o N* The constant 'A' i s determined by normalizing at t h i s p o i n t , and i t i s necessary that H > -2 so that the i n t e g r a l does not diverge. In t h i s case, i . ltfB»* * ATTjir V? where p = Jf + 1 +Q p(f|I) (3-10) and P represents the gamma function. The i n t e g r a l over the p i t c h angle 4* , i s taken from 0 to TT because the p a r t i c l e s are supposed to stream i n the p o s i t i v e and the negative z d i r e c t i o n s , although f o r a given wave at any point, only one-half the p a r t i c l e s can p a r t i c i p a t e i n the cyclotron interaction. Using equations 2-25 and 3-9, GOi i s c a l c u l a t e d (Appendix VI) and i s found t o be non-zero only when y iN/V , M \\\< V,, . In t h i s case, (Vo --v«)^r / i^. > 3 Y + , y ± A » CJT 1 1 ] ( 3 _ 1 5 ) 16 The f o l l o w i n g important q u a l i t a t i v e r e s u l t s can be obtained from t h i s , expression. 1. The growth rate i s d i r e c t l y p r o p o r t i o n a l t o the density of streaming p a r t i c l e s since A^Ng. 2. The only f a c t o r which can be negative i s a) The '-2' term represents a constant damping f a c t o r which o r i g i n a t e s i n the expression -2 j ' dw wf . s b) The quantity + G0!e/GJ* - 1 i s p o s i t i v e f o r both the e l e c t r o n - w h i s t l e r i n t e r a c t i o n and the proton-hm w h i s t l e r interaction. Therefore, f o r wave growth i n e i t h e r case, must be at l e a s t p o s i t i v e . In f a c t , f must s a t i s f y I f (O i s taken as the e q u a t o r i a l value, then t h i s c condition allows wave growth at any point on t h a t f i e l d l i n e provided v^" > . c) Suppose a p a r t i c l e stream trapped i n the magnetosphere can be described by a s p e c i f i c value of If , I t f o l l o w s that there i s an upper frequency l i m i t f o r waves that w i l l be a m p l i f i e d . This maximum frequency TTt i s given by <3 y Any waves with frequencies higher than t h i s w i l l be damped, and using the value of U c at the e q u a t o r i a l plane w i l l i n d i c a t e approximately the maximum frequency 17) 17 of any a m p l i f i e d waves. I f the mechanism f o r wave a m p l i f i c a t i o n suggested here i s c o r r e c t , then the existence of a maximum frequency gives a method of determining Y , provided the guiding magnetic f i e l d l i n e can be determined. This problem i s discussed qualitatively later. 3. At a given point i n space and f o r a given p a r t i c l e energy, as waves of lower frequencies are considered, V -* R v„ , and the waves are not a m p l i f i e d . This f a c t may be used t o e x p l a i n the observed minimum frequency of hm emissions. 4. For a given p a r t i c l e energy and wave frequency, only i f ^ . v a r i e s . a. can vary o approaches v 0 as k ) gets l a r g e r and c t h i s occurs as the region under consideration moves down the f i e l d l i n e away from the e q u a t o r i a l region. point, V R w i l l always be greater than v e Past a c e r t a i n and i n s t a b i l i t y can no longer take place so i n t u i t i v e l y i t seems that the unstable region tends to be s i t u a t e d near the e q u a t o r i a l plane. The numerical c a l c u l a t i o n s show that t h i s i n t e r p r e t a t i o n i s v a l i d . The existence of maximum and minimum frequencies as discussed above can be thought of as roughly d e f i n i n g a band width f o r the emissions The suggestion that the i n s t a b i l i t y tends to occur i n the e q u a t o r i a l region i s due to Watanabe (1965b) and has been mentioned by Jacobs and Watanabe (1965). The same idea has been put f o r t h by Tepley and Wentworth (1964) f o r d i f f e r e n t reasons. They suggest that streaming protons i n the magnetosphere can sometimes be superluminous with respect 18 to hm waves and that since the p a r t i c l e s move f a s t e r near the e q u a t o r i a l plane, i t i s i n these regions that the proton c y c l o t r o n r a d i a t i o n i s subject to the anomalous Doppler s h i f t and i s l i k e l y to be most intense. They suggest that on each pass through the e q u a t o r i a l region, the same process occurs. In order that the wave be reinforced each time, they suggest that the bounce periods o f the wave packet and p a r t i c l e s be approximately equal so that the p a r t i c l e s pass through the wave packet at the equator each time. I t i s not obvious that by the time the p a r t i c l e s return to the e q u a t o r i a l plane, they w i l l s t i l l be i n phase with the wave which they i n i t i a l l y generated. Tepley and Wentworth also had to assume that the p a r t i c l e stream was coherent to begin with i n order to obtain a s i g n i f i c a n t amount of r a d i a t i o n i n the f i r s t emission. Obayashi (1965) has discussed t h i s point. Besides t h i s , there i s no 'a p r i o r i ' reason f o r the two t r a v e l time periods to be the same. But the most important f a u l t i n the theory i s the suggestion that such a superluminous p a r t i c l e stream can i n t e r a c t i n a c o l l e c t i v e manner with the hm waves. Since the p a r t i c l e s see anomalously Doppler s h i f t e d waves, the sense of the waves' p o l a r i z a t i o n i s opposite t o that of the gyration of the p a r t i c l e s ( B r i c e , 1964) and cyclotron resonance cannot occur. Other d i f f e r e n t attempts have been made to determine l i k e l y regions of wave growth i n the magnetosphere and several of these are discussed i n Chapter V. I t was suggested above that X must exceed a minimum value before wave growth can occur. The energy which i s gained by the wave i n the a m p l i f i c a t i o n process comes from p a r t i c l e k i n e t i c energy. It is possible that although some energy i s t r a n s f e r r e d from the p a r t i c l e s , i t i s not enough to balance the constant damping which i s present and the wave decays. I t i s seen then, that the transverse component of v e l o c i t y 19 of the p a r t i c l e s cannot be a r b i t r a r i l y small (Neufeld and Wright, 1965b); Obayashi, 1965). , Since the p i t c h angle d i s t r i b u t i o n i s given by s i n <y , when V i s p o s i t i v e , more p a r t i c l e s have l a r g e p i t c h angles than small ( f i g . 1 ) . Increasing from zero e f f e c t i v e l y increases the average transverse k i n e t i c energy of the p a r t i c l e s while decreasing the average l o n g i t u d i n a l k i n e t i c energy. When the p a r t i c l e s lose transverse energy t o the wave i n a growing i n s t a b i l i t y , there i s a general reduction of p i t c h angles and some p a r t i c l e s may be l o s t because they have p i t c h angles which are inside the 'loss-cone' (Cornwall, 1965; B r i c e , 1964). 20 3- 0 15 30 45 M- 1 1 60 75 (degrees) Family of curves of sm^H* . Values of the parameter Jf are w r i t t e n beside the corresponding curve i n the diagram. 90 21 CHAPTER IV NUMERICAL CALCULATIONS Normalization o f the Equations I t i s assumed that only protons are contained i n the p a r t i c l e stream. The background plasma contains t h e r e f o r e , more e l e c t r o n s than protons by a small amount i n order t o preserve o v e r a l l charge n e u t r a l i t y . Since f g i s non-zero only f o r protons, the summation over components m the numerator of the expression f o r the growth rate o f the i n s t a b i l i t y i s not required. I t i s often convenient t o w r i t e the important equations obtained i n a study i n normalized form i n v o l v i n g dimensionless v a r i a b l e s so that the general r e s u l t s can be seen without employing numerical values which are v a l i d f o r a s p e c i f i c case only. In equation 3-15, the term 2 6l) i n the denominator o r i g i n a t e s r i n the displacement current term i n Maxwell's equations and at hm frequencies i t can be neglected (Jacobs and Watanabe, 1965). Eliminating t h i s term allows the equation t o be put i n t o dimensionless form with the help of the f o l l o w i n g r e l a t i o n s h i p s . <A.i*Ui where CJ =-6J t t e ( 4 _ 1 } 22 I t i s then convenient to write U = ^-"W^ V~ ( 4 " 3 ) by d e f i n i n g Ul ? (4-4) I f the 2CJt term i n equation 3-15 i s neglected, then by d i v i d i n g both sides of the equation by 60i , i t i s found that _ _ I — t (4.5) i In a s i m i l a r manner, the r e a l d i s p e r s i o n r e l a t i o n (equation 2-24) can be put i n t o normal form by neglecting the term f o r the same reasons as above, and then i t i s w r i t t e n ^ ^ ±1 - UJ* M ^ (4-6) ±h +60' Parameter Values The f i r s t requirement of any theory of hm emissions i s that the emitted frequency be i n the Pc 1 range from 0.2 to 5 cps. I f a proton's v e l o c i t y and p i t c h angle are known, then the resonant frequency f o r that p a r t i c l e can be determined from the resonance condition (OR + = (Oi <- _ 4 I f the d i s t r i b u t i o n function f o r the p a r t i c l e s i s a n i s o t r o p i c such that there are more p a r t i c l e s with p a r a l l e l components of v e l o c i t y s l i g h t l y l e s s than |ul than there are p a r t i c l e s with components s l i g h t l y greater, then energy w i l l be t r a n s f e r r e d to the wave. Transverse Landau damping 7) 23 of the wave occurs i f the p a r t i c l e d i s t r i b u t i o n i s i s o t r o p i c (Scarf, 1962; S t i x , 1962). I f 6J i s s p e c i f i e d , and the emission i s t o be of a t c e r t a i n frequency, then the value f o r \u| can be c a l c u l a t e d from equation 4-7, and a lower bound f o r the energies of the p a r t i c l e s involved can be c a l c u l a t e d . In order t o s p e c i f y OJi , the earth's main magnetic f i e l d i s assumed to be a centered d i p o l e f i e l d with a value B«J = 0-3 on the surface of the earth at the geomagnetic equator. I f X i s the geomagnetic l a t i t u d e and L i s the Mcllwain coordinate (Mcllwain, 1961) i n t h i s case applied to a d i p o l e f i e l d , then the t o t a l f i e l d strength at a point with coordinates ( L , X ) i s given by \* U o cos * 4 L represents the distance, measured i n u n i t s of earth r a d i i , that a given f i e l d l i n e i n the e q u a t o r i a l plane l i e s from the centre of the earth. For a dipole f i e l d L where ^ e = I^T. (4-9) i s the geomagnetic l a t i t u d e at the point where the relevant l i n e of force i n t e r s e c t s the earth's surface. I t i s recognized that a dipole representation of the earth's main f i e l d i s not perfect because of the compression on the daytime side but i t i s a good approximation and very easy to describe mathematically. Under the dipole model, iO- i s i n v e r s e l y p r o p o r t i o n a l to L^. L At very low frequencies, the wave's phase v e l o c i t y i s very nearly the A l f v e n v e l o c i t y . V , and using equation 4-7, the frequency of emission can be A approximated by 24 6J B = ^-7— (4-10) Using the density model of Smith (1961), the l o c a l e l e c t r o n number density i s l i n e a r l y p r o p o r t i o n a l t o the gyrofrequency, or N P ~ B ~ - j i (4-n) I t i s assumed that t h i s model holds not only i n the e q u a t o r i a l plane below L = 5, but t h a t i t i s v a l i d along f i e l d l i n e s away from the e q u a t o r i a l plane and at a l t i t u d e s which correspond to L values greater than about 5 ( B r i c e , 1964; Carpenter and Smith, 1964). Using equation 4-10, i t can be seen that 6J R ~ -r; 1 ;—5TT (4-12) and i t can be seen t h a t f o r a given emitted frequency, the p o s i t i o n i n the magnetosphere at which the i n t e r a c t i o n takes place strongly determines the energy range of the p a r t i c l e s involved. Cornwall (1965) has suggested that since the data from the Explorer 12 s a t e l l i t e , f i r s t reported by Davis and Williamson, (1963) and l a t e r i n more d e t a i l by Hoffman and Bracken (1965), i n d i c a t e a large f l u x of protons with energies of the order of hundreds o f kev at L — 3 . 5 , these protons may be very important i n emission processes. The energy range i s the r i g h t order f o r resonance i n the Pc 1 range i n a dipole f i e l d . Cornwall (1965) also mentions that 10 - 20 kev protons at L— 5.6 have been suggested as the energy source f o r the emissions. Hoffman and Bracken (1965) have reported the presence of protons i n the region of the magnetosphere between these two extremes, with the f l u x o f low energy p a r t i c l e s increasing with increasing a l t i t u d e . I f these 25 energies are i n the correct range, then resonance could occur on any l i n e of force having an L value between about 3.5 and 5.6. Qbayashi (1965) i n d i c a t e s that a l l hm emissions should occur i n the region between L = 4.0 and L = 5.6. However, out of nine examples, Watanabe (1965c) found no L values f o r the guiding l i n e of force below 4.98 f o r a dipole f i e l d . For the d i s t o r t e d dipole f i e l d which he used, t h i s value becomes 4.75. Taking i n t o account the o u t l i n e of hm emissions given i n Chapter I , the r e p e t i t i o n of r i s i n g tones separated by a constant time i n t e r v a l i s interpreted as an hm wave packet bouncing between ionospheric r e f l e c t i o n s i n the northern and southern hemispheres, being guided by the geomagnetic f i e l d l i n e s . I t i s suggested that the wave i s strengthened by the cyclotron i n t e r a c t i o n with the proton stream each time i t traverses the f i e l d l i n e . The bounce period of hm waves has been c a l c u l a t e d t h e o r e t i c a l l y by Jacobs and Watanabe (1965) as a function of the frequency and the L value, and i t involves c a l c u l a t i n g an i n t e g r a l numerically. Using t h e i r t a b l e , the bounce periods f o r a wave with a frequency of 1.3 cps for L = 5.6 and L = 3.5 are found to be approximately 280 sec. and 60 sec. respectively. Tepley and Wentworth (1964) mention that the r e p e t i t i o n period of the r i s i n g tones i n hm emissions can vary from one to f i v e minutes so t h a t these values are not outstanding. This r e s u l t i s p h y s i c a l l y reasonable since the d i s p e r s i o n r e l a t i o n i n d i c a t e s that the phase v e l o c i t i e s of ion resonance mode waves tend toward zero as 6J« approaches 6o , and so, f o r a given frequency range, the wave goes slower c at higher a l t i t u d e s since the cyclotron frequency decreases. At the same time, the path which the wave follows i s longer at higher a l t i t u d e s . Such d i f f e r e n c e s of r e p e t i t i o n period between d i f f e r e n t emissions i s noticeable even by making very rough measurements on d i f f e r e n t 26 dynamic spectra. In the example presented by Cornwall (1965), the period measured over the i n t e r v a l between 7 min. and 14 min. i s approximately 84 sec. with a mean d e v i a t i o n o f about 10 sec. In the sonogram given by Jacobs and Watanabe (1965), the period measured between 13^ hr. and 14 hr. i s never l e s s than 130 sec. these two spectra i s measureable. The d i f f e r e n c e between This type of measurement has been made by Watanabe (1965c). The frequencies o f the emission i n the example above i n which the bounce period i s 130 sec. are low, around 0.3 cps. I f t h i s event i s to have occurred on the l i n e o f force given by L = 3.5, then at the l e a s t , the protons would have had t o have energies o f about 30 mev. At t h i s energy, the protons are r e l a t i v i s t i c and such p a r t i c l e s are not mentioned i n Cornwall's presentation (Cornwall, 1965). Besides the p a r t i c l e energy, the streaming p a r t i c l e density and the p i t c h angle parameter must be chosen. Hoffman and Bracken (1965) f i t t e d t h e i r data t o a p i t c h angle d i s t r i b u t i o n and found that the best f i t was made when they considered two ranges o f p i t c h angles separately, 0° to 30° and 30° t o 90°. The parameter values which gave a good f i t ranged from 1 t o 4 over the region L = 2 t o L = 7. Each c a l c u l a t i o n here i s done f o r 3" = 2. The r a t i o Ns/N p i s taken to-be 1 at the equator so that r e l a t i v e sizes can be seen from the r e s u l t s , but the calculated rates are too large. 10~^. growth Jacobs and'Watanabe (1965) assumed the r a t i o t o be The correct r a t i o probably v a r i e s from case t o case. f a c t o r i n the choice of N 5 One l i m i t i n g i s that the growth rate must be very much smaller than the r e a l frequency. I t was mentioned above that Watanabe (1965c).found no emissions taking place on a l i n e of force with an L value l e s s than 4.98 i n nine 27 examples considered. Cornwall (1965) suggests that the emissions would not take place at lower a l t i t u d e s i f the r a t i o N /N S i s too small and P t h i s might occur because the background plasma density i s large. A l l the c a l c u l a t i o n s have been done using an L value of 5.6. For each CJ , the quantity v* - V* X i s set equal t o zero. i s c a l c u l a t e d and i f v^ 4 v£ , 6 J X The wave frequencies have been varied from 6Ji(eq)/20 to 19 6J (eq)/20 and at each frequency, the growth rate i s £ c a l c u l a t e d f o r twenty-nine values of the geomagnetic l a t i t u d e from 0° to 29° i n half-degree steps. The Smith model of e l e c t r o n density (Smith, 1961) can be w r i t t e n N= P \%,00O ^S* (4-13) This model has been used by B r i c e (1964) i n the form Xl = K (3.T)^. i.OOO C J \ (4-14) e In the c a l c u l a t i o n s , equation 4-13 i s w r i t t e n N = P v r . 5>5!>0 B c 7 v f (4-is) 3 e and using t h i s value f o r the density, the A l f v e n v e l o c i t y can be c a l c u l a t e d f o r the magnetosphere. Values corresponding to p a r t i c l e energies i n the range approximately 10 - 20 kev can then be assigned to Uj. 7 the value of V A at the e q u a t o r i a l plane i s 4.95 x 10 energy of 10 kev corresponds t o a v e l o c i t y of 1.38 x 10 case, U (eq) = 2.79. 5 For L = 5.6, -I cm-sec. A proton cm-sec so i n t h i s Since the v e l o c i t y increases as the square root of the energy, f o r 20 kev, U (eq) = J~2 (2.79) = 3.95. 1, s In the programme, v 0 i s given the f o l l o w i n g seven values; 2.8V (eq), 3.0V (eq), ... , 4.0V (eq). ft A A 28 I t should be noted that equation 4-13 describes Np e m p i r i c a l l y and i s an average value. Since Np can vary by as much as a f a c t o r of 2 at d i f f e r e n t times, the c a l c u l a t e d r e s u l t s are not exact and d i f f e r e n c e s e x i s t between one s p e c i f i c example o f hm emissions and another. The r e s u l t s w i l l serve as an i n d i c a t i o n o f general e f f e c t s which r e s u l t from the mechanism which has been considered. The c a l c u l a t i o n s have been done at the U n i v e r s i t y of B r i t i s h Columbia Computing Centre on an I.B.M. 7040 computer using Fortran IV language. Results The l a r g e s t growth rate at any point was found t o occur at the e q u a t o r i a l plane ( X = 0) at a frequency o f 0.65 cps. Figure 2-a i n d i c a t e s how the growth rate v a r i e s with frequency and p a r t i c l e energy at the e q u a t o r i a l plane. i s narrow. The lower c u t o f f i s very sharp and the peak, i t s e l f Figures 2-b and 2-c show the same type o f p l o t f o r A = 10° and X = 20° r e s p e c t i v e l y . I t can be seen that as X increases, the frequency band o f a m p l i f i c a t i o n moves toward higher frequencies f o r a given v . 0 This i s t o be expected and r e s u l t s from the requirement which means that as 6J; increases with X , that must increase i n order t o a, r e s t r i c t the s i z e of V^. This e f f e c t can be seen i n another way by observing how CJj changes with X f o r s e v e r a l frequencies. I t i s found that the growth rate i s p r a c t i c a l l y zero f o r a l l frequencies l e s s than or equal t o ( 29 0.2 (^i (eq). Remembering that the l a r g e s t growth rate occurs near = 0.25 CJi (eq), f i g u r e 3-a gives the r e s u l t s f o r £*JR = 0.3 (eq). At the higher energies, i t i s noticeable that the growth rate maximum occurs near X — 10°. In f i g u r e s 3-b and 3-C, t h i s e f f e c t i s much more noticeable and as the wave frequency i s increased s t i l l further(figures 3-d and 3 - e ) , the wave i s damped s l i g h t l y near the equator and amplified i n the region around 25° geomagnetic l a t i t u d e . This damping e f f e c t occurs near the equator because at that point on the f i e l d l i n e , the r a t i o i s smallest and i f X does not happen t o be large enough, the expression tf(k->t/6JR - 1) - 2 can e a s i l y be negative. For energies i n the range 10 - 20 kev, the sharp low frequency c u t o f f occurs at 0.2 (Ji(eq) and i t i s important t o note that the maximum proton energy determines t h i s c u t o f f point. The proton energy also i s very important i n determining the s i z e o f the growth rate. 30 U s = 4.0 X = o° o to OS o o 0.2 0.4 0T6 0.8 Normalized Frequency F i g . 2-a Growth rate as a function of frequency at X = 0° f o r two p a r t i c l e energies. The frequency i s normalized to the e q u a t o r i a l cyclotron frequency, 16.4 s e c ! . - 1.0 31 F i g . 2-b Growth rate as a function of frequency at A = 10° f o r two p a r t i c l e energies. The frequency i s normalized to the e q u a t o r i a l cyclotron frequency, 16.4 sec -'-. - 32 40 ~ X= 30 20 c o TO 20 . U U s = 4.0 o u o 10 - 0.2 0.4 , 0.6 0.8 Normalized Frequency F i g . 2-c Growth rate as a function of frequency f o r two p a r t i c l e energies f o r X = 20°. The frequency i s normalized to the e q u a t o r i a l cyclotron frequency, 16.4 s e c . - x 33 40V 6L) = 0.3 CJ: (eq) = 4.9 s e c " r 30h U 20L s = 4.0 10 5 F i g . 3-a 10 15 20 Geomagnetic L a t i t u d e (degrees) 25 V a r i a t i o n of GO with X f o r CO^= 4.9 sec-1. x 40 L\ = 0.4 6 J i ( e q ) = 6.54 sec - 2 30 20 U s = 4.0 10 10 15 20 25 Geomagnetic L a t i t u d e (degrees) F i g . 3-b V a r i a t i o n of 6 J w i t h X f o r CO = 6.54 x R sec . - 1 1 1 5 10 15 20 25 Geomagnetic L a t i t u d e (degrees) F i g . 3-d V a r i a t i o n of k\ with X f o r 60 = 11.4 s e c . - 1 K 35 CHAPTER V SUMMARY Discussion I t i s important to remember that the a n a l y s i s i s v a l i d only to f i r s t order. The growth of waves can be i n d i c a t e d but soon a f t e r i t begins, the l i n e a r theory becomes i n v a l i d and nothing f u r t h e r can be said about the behaviour of the system. I t might happen that when the p a r t i c l e s have l o s t a s u f f i c i e n t amount of t h e i r energy, many are dumped i n t o the ionosphere so that the wave growth becomes n e g l i g i b l e compared to i t s attenuation. In t h i s case, the thermal background plasma may be an important damping agent, but i t was assumed to have zero temperature i n the above a n a l y s i s . Kennel and Petschek (1966) have considered the s t a b i l i t y of trapped p a r t i c l e s i n d e t a i l and Cornwall (1966) and Watanabe (1966) have discussed some non-linear aspects of the problem. conceivable that N s It is might sometimes be large enough that the r a t i o N /N s p i s of the order of unity and t h i s would i n v a l i d a t e the l i n e a r theory. I t i s also known that the earth's f i e l d i s not accurately represented by a dipole but t h i s representation makes the a n a l y s i s much simpler and general r e s u l t s can s t i l l be obtained. Considering only monoenergetic p a r t i c l e streams i s an overs i m p l i f i c a t i o n , although the v a r i a t i o n of t o with energy has been r c a l c u l a t e d numerically. Hoffman and Bracken (1965) i n d i c a t e that a doubly sloped exponential energy spectrum describes w e l l the d i s t r i b u t i o n of proton f l u x e s which they observed. I f t h e i r d e t a i l e d observations were taken at lower energies, then the i n t r o d u c t i o n of an energy d i s t r i b u t i o n of the form e~^/^° where E r e s u l t s more q u a n t i t a t i v e . e i s e m p i r i c a l l y determined would make the 36 In order to determine more exactly the a c t u a l growth of a wave of some f i x e d frequency, the wave amplitude must be integrated over the region i n which i s non-zero. Jacobs and Watanabe (1965) have used Sturrock's a n a l y s i s of growing waves (Sturrock, 1961) and shown that the mechanism which has been considered here gives r i s e to a non-convective i n s t a b i l i t y (the point where the i n s t a b i l i t y occurs i n i t i a l l y remains f i x e d i n space, although the disturbance can spread out around i t ) and they have discussed b r i e f l y the problem of how such a disturbance might come t o be observed on the earth's surface. In Chapter I I I , the theory presented by Tepley and'Wentworth (1964) has been discussed and references to the papers by Cornwall (1965) and Obayashi (1965) have been made i n several places. Gendrin (1965) and Hruska (1966) have considered the problem of c y c l o t r o n emissions i n the magnetosphere. Both authors f i n a l l y consider only the (L, e) and (R, p) i n t e r a c t i o n s where 'R' and 'L' r e f e r to l e f t and right-hand p o l a r i z e d waves and 'e' and 'p' r e f e r to e l e c t r o n s and protons respectively. Gendrin comments b r i e f l y on i n s t a b i l i t i e s which a r i s e when the transverse v e l o c i t y components of the p a r t i c l e s are important but suggests i m p l i c i t l y t h a t t h i s would never occur except near the mirror points. This assumption does not seem reasonable. He suggests t h a t hm emissions occur when super-luminal protons i n t e r a c t with R waves and describes the process of repeated emissions as Tepley and Wentworth (1964) do. This idea has been c r i t i c i z e d above. Hruska (1966) considers a plasma i n s t a b i l i t y by considering the net t r a n s f e r of energy between waves and p a r t i c l e s . He chooses a s h i f t e d 37 Maxwellian d i s t r i b u t i o n f o r the streaming p a r t i c l e s but gives no reason f o r t h i s choice. Since he takes the temperature d i s t r i b u t i o n t o be i s o t r o p i c , the contra-streaming (L,p) and (R,e) i n t e r a c t i o n s do not give r i s e t o a growing i n s t a b i l i t y and he does not discuss them any further. Conclusions I t i s suggested that hm emissions r e s u l t from hm wave packets propagating along the earth's magnetic f i e l d l i n e s guided between ionospheric r e f l e c t i o n s i n the northern and southern hemispheres. The ion resonance mode of wave i s considered because i t i s guided by the earth's f i e l d at hydromagnetic frequencies much more than the e l e c t r o n resonance mode of wave and because the dispersion c h a r a c t e r i s t i c s o f the ion resonance mode are the same as the observed spectra o f structured hm emissions. Since the s i g n a l s sometimes increase i n i n t e n s i t y i n time before dying out, i t i s suggested that the wave packets gam energy v i a a c y c l o t r o n i n s t a b i l i t y process as they i n t e r a c t with low-energy protons which are trapped i n the magnetosphere. An expression f o r the growth o f the waves was developed s t a r t i n g from Maxwell's equations and the c o l l i s i o n l e s s Boltzmann equation. Choosing a pitch-angle d i s t r i b u t i o n function containing the f a c t o r sin*M'/B ' r e s u l t s i n an upper c u t o f f y/a frequency o f k - V ( l + at any given point. I t i s not known whether t h i s e f f e c t i s more important than the damping which r e s u l t s from the thermal background plasma. Computer c a l c u l a t i o n s ( f o r Y = 2 ) i n d i c a t e three important features o f the theory. F i r s t , the growth rate i s a sharply peaked 38 function of the frequency. Second, changes i n the proton energy greatly influence the magnitude of the growth rate as w e l l as the frequency of the steep lower c u t o f f . This suggestion was also made by Qbayashi (1965). F i n a l l y , the l a r g e s t growth rates f o r the i n s t a b i l i t y are found tb occur near the e q u a t o r i a l plane, although at some frequencies, there are two regions o f l a r g e s t growth, each s l i g h t l y removed from the equator by ten or twenty degrees. The observed l a t i t u d e dependence of hm emissions may be explained by the f a c t that none of the L waves i n the hm packet can have a frequency above the i o n cyclotron frequency at the geomagnetic equator and that i t decreases as the l a t i t u d e of the point where the l i n e of force i n t e r s e c t s the earth's surface increases. Another important consideration i s the e f f e c t of the ionosphere on the wave as i t t r a v e l s from the lower regions of the magnetosphere to the observation point on the earth's surface. Ionospheric wave guiding may r e s t r i c t the wave packet frequencies because of a l a t i t u d e v a r i a t i o n o f the duct c h a r a c t e r i s t i c s . I f measurements on wave propagation above the ionosphere could be made to determine p o l a r i z a t i o n s and i f d e t a i l e d records of low-energy proton f l u x e s (1 - 100 kev) could be obtained, many u n c e r t a i n t i e s i n the theories of hm emissions would be eliminated. 39 APPENDIX I Transformation o f Equations and S o l u t i o n f o r f (y_,k,6j). t Let G(t,z) represent any of the q u a n t i t i e s which are t o be transformed. I f G(t,z) i s well-behaved, then the Fourier-Laplace transform of G e x i s t s and i s defined by G(OJ,M= ( d t J j i i G t t . i ) t " o -eo t ( **" (Ai-i) C o t ) For the F o u r i e r transform t o e x i s t i t i s s u f f i c i e n t that G(t,z) be of bounded v a r i a t i o n and absolutely integrable, i . e . , +00 (|&(t,^a* < co ( -2) A1 and i t i s implied that G(t,z)-^0 as z-* + oo. In order t o assure existence of the Laplace transform, i t i s convenient t o assume ( S t i x , 1962; S o k o l n i k o f f and Redheffer, 1958) that f o r some choice of the constants M and , |Gct»u n^ (AI-3) 1 and ImGj>ja. Applying the transformations defined above t o the f o l l o w i n g component Maxwell equations 2* ^ c a t c 9t ( A 1 _ 6 ) ( A 1 - 7 ) 40 and using i n t e g r a t i o n by parts and the conditions o u t l i n e d above, i t i s found that (AI-8) -c-fe & CoJ,&)= T T f c t o . f e ) -£[«.uE (u>,fe) + E x ( o , ^ y t*B l c x Cu)>)=My Cw>)-i[t^E ( y y t*E (co,IO=. - i [ t ( 0 ^ ( w » L^E^iuiM^ {[iufyCuM y W j W+E Co M] Y (Ai-9) J + & <©A)] (Ai-io) X + B (o^)] y ( A 1 _ n ) The f i r s t - o r d e r Boltzmann equation which was derived i n Chapter I I (equation 2-7) can be w r i t t e n + c M - V y W l = ° (Al-12) Applying the combined transformation defined by equation A l - 1 t o t h i s equation and e l i m i n a t i n g E (k>,k) and Ey(60,k) from the r e s u l t i n g x equation using equations Al-10 and Al-11, i t i s found that i s the d i f f e r e n t i a l equation governing the transformed d i s t r i b u t i o n f u n c t i o n f (y,k,(0). x 41 I t i s now convenient t o transform coordinates from ( v , v ) t o x y (w,4>) using the f o l l o w i n g r e l a t i o n s 3V 9w'^v X ^.2- 3*'?^ K (Al-14) + _ ( A 1 and i t was assumed above that 9f /94> vanishes. 0 Since w = v£ + v 2 2 5 1 5 ) i t is easy t o show that _ - w - cos* - ^ ( A 1 2 * = * = si* • - < ^ J L £ 1 _ _ ( A 1 1 6 ) . 1 7 ) By w r i t i n g ^ ( Y j k j O ) as a F o u r i e r s e r i e s in<p, i^.t.oi^Zr'e^* <Ai-i8) equation Al-13 can f i n a l l y be w r i t t e n " i ^ M f - u)|fc • *2i°] .-«[ 6 x C ( u ,i) + 1 This equation i s l i n e a r , non-homogeneous, f i r s t - o r d e r with constant c o e f f i c i e n t s and can be solved using standard methods. Coddington (1961) 42 uses the notation y 1 + ay = b(x) (Al-20) where a i s a constant and b i s a continuous f u n c t i o n on an i n t e r v a l I. A l l s o l u t i o n s must have the form = §CX) where x 0 = e" •ax X re J * b C t ) d t + c e - a i s i n I and c i s any constant. ax X ( M _ Note that i f the a n t i - d e r i v a t i v e the integrand i s evaluated at x„, then t h i s 'constant 'c' i n m u l t i p l y i n g e~ 4 1 2 1 ) of can be grouped with and the i n t e g r a l evaluated at the upper l i m i t of 'x' i s j u s t the i n d e f i n i t e i n t e g r a l . zero, and the r e s u l t i s the same. The new constant i s then chosen to be In t h i s way, the term i n v o l v i n g the i n d e f i n i t e i n t e g r a l represents the p a r t i c u l a r s o l u t i o n of the inhomogeneous equation and the term containing the redefined constant i s the general s o l u t i o n of the homogeneous equation. The redefined constant i s chosen to be zero because £j_(v, z,t) must be zero when the wave and p a r t i c l e stream perturbations are removed, so the s o l u t i o n to the homogeneous equation must be dropped from the general s o l u t i o n of the inhomogeneous equation. s o l u t i o n f o r ^ ( v , k,(o) i s given by T Z_j 60--feu•'w^t The resulting 43 APPENDIX I I Determination of the Transformed Magnetic F i e l d s Applying the Fourier-Laplace transform defined i n Appendix I t o equation 2-13 gives ^A)±c> (ojA)=Z S.UveT *w5 (yA w) t y 1 and using the expression f o r f ^ ( ^ , (A2-1) > k,lo) found i n the f i r s t appendix, equation A2-1 gives the current density (transformed) i n terms of the transformed magnetic f i e l d s . An expression f o r the current density can a l s o be found from the transformed Maxwell equations. By e l i m i n a t i n g E (w,k) and E (o;,k) K y from the four transformed equations i n appendix I , i t i s found that i(CO*-C 6 {tOj (?) = -HTVC- a X - CJ &„( » 0 + c& EyCoA) (A2-2) and c(^-^\t)Z CLoM= y H i r c f e y ^ w . ^ - w a y C o ^ ) - c ^ E x ( o ^ ) ( A 2 and hence +cfe[e (o fe)±tE co^ x y y ( A 2 _ Noting that when the s o l u t i o n f o r f (y,k,6j) i s s u b s t i t u t e d i n t o 1 equation A2-1, the i n t e g r a l i n v o l v i n g f ^ e m = + 1, i t i s found that i l 1 n ^ i s non-zero only when 4 ) _ 3 ) 44 ">*p J f 60 - -feu ^ 60 L c r C : F l ) F i n a l l y , using equations A2-4 and. A2-5, i t i s found that ^ - . M - (A2-6) where 1 " J - CO-fea^COc (A2-7) 45 APPENDIX I I I A n a l y t i c Continuation of the I n t e g r a l s The o u t l i n e of the procedure as given by S t i x (1962) w i l l not be copied,in d e t a i l . There, the case of l o n g i t u d i n a l plasma o s c i l l a t i o n s i s considered as an example. The d i f f e r e n c e s f o r the case of transverse waves i n t e r a c t i n g with a p a r t i c l e stream are noted. The problem i s s i m p l i f i e d by c a l c u l a t i n g the asymptotic value of B(t,k) as t-9°o . The expression f o r the magnetic f i e l d s given i n equation 2-14 i s v a l i d f o r Imk) >jX , where /* was defined i n Chapter I i n connection with the d e f i n i t i o n of the Laplace transform. equation 2-14 must be The a n a l y t i c continuation of determined. I t i s assumed t h a t B ^ ( 0 , k ) , B ( 0 , k ) , Ex.(0,k), E ( o , k ) , y r\ v 9f„/aw, (-1) 2 f o / 9 u , and f j . are a l l e n t i r e functions o f u. One must then consider i n t e g r a l s o f the form + 00 -oo with \, 6J +• 60c V= ^ where the i n t e g r a l i s t o be taken along the r e a l a x i s . be an e n t i r e f u n c t i o n of u. (A3-2) F(u) i s assumed t o I f one i s considering u as a complex v a r i a b l e , then the path of i n t e g r a t i o n i n equation A3-1 can be changed i n accordance with complex v a r i a b l e theory. There are four cases, depending on the signs of k and 46 Case 1. k>Q, Ux> 0. In t h i s case, Im(V)>0. The path o f i n t e g r a t i o n can be raised ( f i g . 10) from the r e a l axis above the s i n g u l a r i t y using the residue theorem. I(v)(p*tki) =• K V K P * ^ ) - F(V) (A3-3) Case 2. k>Q,0Ji< Q. In t h i s case, Im(V)<0 and the a n a l y t i c continuation i s obtained by deforming the path of i n t e g r a t i o n down from the r e a l axis ( f i g . 11) so that KvKpaUi) = Icv)Cp tka.)-^iF(v) a ( A 3 _ 4 ) and path 2 can e a s i l y be chosen t o be the r e a l a x i s . Case 3. k<0, 0->JL> 0. In t h i s case, Im(V)<0. The path i s taken along the r e a l axis and i t i s t r i v i a l t o write ( f i g . 12) I ( V H p * t k l ) =• K V K p . t k i ) Case 4. k<0, (A3-5) 0. Im(V)>0 and the continuation by contour deformation i s analogous to case 2. From f i g u r e 13, i t can be seen that I C v K p o c t k l ) = K v H p a t K 2.) (A3-6) Using the residue theorem I ( V ) ( W a.cs) - K v X p a t K 2 ) « - ^ F^V) ( A 3 _ 7 ) 47 path 2 X W = V ->Re( ) path 1 Fig. 4 Contours f o r i n t e g r a t i o n when Va k> O and 6 J "> ° X 48 path 2 V 4 = V path 1 Fig. 7 Contours f o r i n t e g r a t i o n when k < O and OJj< O 49 and so I(VHp*^1) = I ( V ) < W «xi ) - i ^ F ( V ) S (A3-8) since k<0 i n t h i s case. I t can be seen from the above d i s c u s s i o n that the i n t e g r a l s over 'u' f o r k J j X ) are taken s t r i c t l y as they appear, along the r e a l u,', a x i s (cases 1 and 3). F o r ^ i < 0, (cases 2 and 4 ) , again the i n t e g r a l s are taken along the r e a l a x i s but there i s an a d d i t i o n a l residue term i n each o f the two cases. Following S t i x (1962), i t i s seen that both the numerator and denominator of B (W,k) + iBy(lO,k) are a n a l y t i c functions o f u i n the whole x plane ( e n t i r e f u n c t i o n s ) and so the poles of B(k>,k) come only from the zeroes of the denominator. There are two equations, one f o r each of p o s i t i v e and n e g a t i v e ^ . 60 f o r lx) > 0, and x A + 6J - f c u * CUc ™ If* W - T O w - ^ ^ ^ + i ^ U •VJ = ° ( A 3 - 1 0 ) f o r GJi< 0. Using the Cauchy p r i n c i p a l value ((P) n o t a t i o n , these two equations can be combined i n t o one equation which i s v a l i d f o r both p o s i t i v e and negative values o f the imaginary part o f 6J . (A3-11) 50 where the f u n c t i o n a l form of G i s given by frttiJ-jLJCl 7 ^ ) ^ + "7J (A3-12) 51 APPENDIX IV Wave P o l a r i z a t i o n Consider the case o f pure wave propagation when no p a r t i c l e stream perturbation i s present. In t h i s case, D = 0, since f A = 0* " Consider waves of the form with only x and y components present. L e f t hand p o l a r i z a t i o n i s defined by writing tVt,"*) - K ^ y t t , ^ = O ( A 4 _ 2 ) I t then f o l l o w s from Maxwell's equations that E * ( t , % ) + £ E y C t ^ ) - 0 (A4-3) The Fourier-Laplace transformation o f these two equations i s s t r a i g h t forward so that at t = 0, the condition B * ( 0 , k ) ± l B ( 0 > ) =r O (A4-4) E C0,fc) t (A4-5) y implies that x (o,fe) = O The t o t a l f i e l d i s w r i t t e n as the sum o f the separate f i e l d s of two waves of opposite p o l a r i z a t i o n . 6(to, fc)- B % , k ) +a cu) «,> a) i ( A 4 _ 6 ) (A4-7) 52 where the superscript ( 1 ) i n d i c a t e s the left-hand mode and ( 2 ) the right-hand mode. Using equations A4-6 and A 4 - 7 , i t i s found that £>(0, l O = &\oM + &\o,to (A4-8) = E ( o » +• E ( o » (A4-9) and EioM 0 ) a ) S u b s t i t u t i o n of the above values o f B^(6J,k), By(6J,k), B ( 0 , k ) , B ^ O j k ) , x Ex(Oj^) & an Ey(0,k) i n t o equation 2-14 and choosing, say, the upper sign everywhere, y i e l d s an equation i n v o l v i n g only the superscript ( 1 ) because those q u a n t i t i e s i n v o l v i n g the superscript ( 2 ) a l l vanish because of the d e f i n i t i o n of the r i g h t and l e f t modes o f p o l a r i z a t i o n . When the lower sign i s chosen, the r e s u l t i n g equation contains only the superscript ( 2 ) . Therefore, i n equation 2-14, the upper sign corresponds t o left-hand p o l a r i z a t i o n and the lower sign corresponds t o right-hand p o l a r i z a t i o n . 53 APPENDIX V S i m p l i f y i n g the General Dispersion R e l a t i o n Equation 2-21 i s (A5-1) The f i r s t and t h i r d terms can be approximated.using the f a c t that 6 0 * ^ 6J« *a<J*<Jx (A5-2) and CO ^ t\J« ±CJc (A5-3) The fourth term i n equation A5-1 contains the product o f G ( f ) and UJ s and the product CO£H i s second order. At the- same time, two of the s three terms i n G ( f ) contain a m u l t i p l y i n g f a c t o r 1/co, and when t h i s s f a c t o r i s w r i t t e n i n the normal form o f a complex number with a r e a l denominator, 6J tJfi + i W i £o£+-6j£ UK £0* (A5-4) Exactly the same procedure can be c a r r i e d out f o r the denominator of the integrand of the fourth term, with the r e s u l t that V|Gc(5Q ^ W £(^s) In the l a s t term, the m u l t i p l y i n g 60 can be replaced by6J*. s i m p l i f i c a t i o n can be made i n the l a s t term. (A5-5) One f u r t h e r Expanding the numerator of 54 the integrand i n a Taylor s e r i e s about u = V , i t i s found that R «[w6(W] « u (A5W5) V ) t since V - V ^ ^ a n d G ( f ) contains the f i r s t order quantity N . s s Using a l l the above approximations, equation A5-1 can be w r i t t e n + i H ^ + S f f e ^ i - ^ £ i > ^ 4 -° («-) S e t t i n g the imaginary part o f t h i s equation equal t o zero gives equation 2-25. A second equation r e s u l t s when the r e a l part i s set equal t o zero. The fourth term can be neglected since i t i s o f f i r s t order while the t h i r d term i s zeroth order. r e l a t i o n given i n Chapter I. This approximation y i e l d s the d i s p e r s i o n 55 APPENDIX VI Coordinate Transformation and C a l c u l a t i o n of 6Jx In c y l i n d r i c a l coordinates (u,w) Equation 3-9 gives f i n s p h e r i c a l coordinates (V,HO as s i,*AS<v-v.) Siltlg. (A6-2) I t i s convenient t o transform G ( f ) using the chain r u l e s s 9w ~ 0V 3W 0 u 9 V * u 0q> SW 9 (A6-3a) (A6-3b) u and the transformation V = +• (Wl4- ) ^ (A6-4a) »+> = tav>-' M! (A6-4b) Equations Al-3a and Al-3b then become and u and w can be eliminated using the inverse transformation. However, the problem can be s i m p l i f i e d because the i n t e g r a l i n the numerator o f equation 2-25 must be evaluated at u = V . R I t w i l l be 56 written (A6-6) In equations Al-5a and Al-5b, u and w are eliminated by p u t t i n g u = V , and R w = + ( v - V^)3. 2 This r e s u l t f o r w implies that W<Jw » V<W (A6-7) and s W l * . * . J v ^ ( A 6 . 8 ) Using these r e s u l t s , i t then f o l l o w s by algebra that R 6_> V** L R V** 1 * w ; i C V V ' ) V- ] 1 1 v ^ ^ J ( -9) A6 with 60* W * y has been r e s t r i c t e d t o values greater than -2. I f IVr\ > then the i n t e g r a l i s zero. v a I f I v J = v , the i n t e g r a l may be d i f f e r e n t from 0 zero because of the presence of the d e l t a functions. In t h i s case, )S must 57 be f u r t h e r r e s t r i c t e d i n order t o avoid d i v i s i o n by zero i n f a c t o r s containing the expression v^ - V*. i t can be seen that Y^ From the t h i r d term i n the integrand, -2, and t h i s i s weaker than the o r i g i n a l r e s t r i c t i o n . The second term requires that ^ ^ 0. In the f i r s t term, the presence of the d e l t a - f u n c t i o n d e r i v a t i v e means that the function which m u l t i p l i e s i t i n the integrand must be f i r s t d i f f e r e n t i a t e d and then evaluated at v = v . 0 This c a l c u l a t i o n leads t o an expression containing terms with the f a c t o r s (v* - V * ) k and (v** - v j ^ * 1 1 . I t i s seen, then, that f o r \\«\ = v., V must be at l e a s t zero, and i n t h i s case, whether V i s zero or p o s i t i v e , the i n t e g r a l i s zero. This r e s u l t i s not unexpected since | v l R = v 0 represents the resonance condition f o r the case of protons i n t e r a c t i n g with contrastreaming left-hand p o l a r i z e d waves when the proton beam i s ' l i n e a r ' , that i s , when the beam has no transverse k i n e t i c energy. Neufeld and Wright (1963) have shown that f o r t h i s case, a contra-streaming instability does not e x i s t . I t i s found then, that f o r W*l < v„ and V> -2, S u b s t i t u t i n g t h i s r e s u l t i n equation 2-25, i t i s immediately found that ( A 6 - U ) 58 where N s i s defined by Ns - N s ( f ^ (A6-12 and represents the streaming p a r t i c l e density at a point where the magnetic f i e l d has the value B. 59 BIBLIOGRAPHY A l f v e n , H., and C.-G. Falthammar, (1963), Cosmical Electrodynamics, Oxford U n i v e r s i t y Press. Astrom, E., (1950), On waves i n an ionized gas, Ark. Fys., 2, 443-457. B e l l , T.F., and 0. Buneman, (1964), Plasma i n s t a b i l i t y i n the w h i s t l e r mode caused by a gyrating e l e c t r o n stream, Phys. Rev., 133A, 13001302. Booker, H.G., (1962), Guidance of r a d i o and hydromagnetic waves i n the magnetosphere, J . Geophys. Res., 67, 4135-4162. B r i c e , N., (1963), An explanation of triggered VLF emissions, J . Geophys. Res. 68., 4626-4628. B r i c e , N., (1964), Fundamentals of very low frequency emission generation mechanisms, J . Geophys. Res., 69, 4515-4522. B r i c e , N., (1965), Generation of very low frequency and emissions, Nature, 206, 283-284. hydromagnetic Carpenter, D.L., and R.L. Smith, (1964), W h i s t l e r measurements of e l e c t r o n density i n the magnetosphere, Rev. Geophys., 2, 415-441. Chandrasekhar, S., (1960), Plasma Physics, compiled from notes by Trehan, U n i v e r s i t y , of Chicago Press. S.K. Coddington, E.A., (1961), An Introduction to Ordinary D i f f e r e n t i a l Equations, P r e n t i c e - H a l l , Inc. Cornwall, J.M., (1965), Cyclotron i n s t a b i l i t i e s and electromagnetic emission i n the u l t r a low frequency and very low frequency ranges, J. Geophys. Res., 70, 61-69. Cornwall, J.M., (1966), Micropulsations and the outer r a d i a t i o n zone, J . Geophys. Res., 71, 2185-2199. Davis, L.R., and J.M. Williamson, (1962), Low energy trapped protons, Space Res., 3, 365-375. Gendrin, R., (1965), Gyroresonance r a d i a t i o n produced by proton and e l e c t r o n beams i n d i f f e r e n t regions of the magnetosphere, J . Geophys. Res., 70, 5369-5383. Guthart, H., (1964), W h i s t l e r s i n a thermal magnetosphere, Stanford Research I n s t i t u t e , Menlo Park, C a l i f o r n i a . H e l l i w e l l , R.A., (1965), W h i s t l e r s and Related Ionospheric Phenomena, Stanford U n i v e r s i t y Press, Stanford, C a l i f o r n i a . 60 Hoffman, R.A., and P.A. Bracken, (1965), Magnetic e f f e c t s of the quiet time proton b e l t , J . Geophys. Res., 70, 3541-3556. Hrfiska, A., (1966), Cyclotron i n s t a b i l i t i e s i n the magnetosphere, J. Geophys. Res., 71, 1377-1384. H u l t q v i s t , B., (1965), On the a m p l i f i c a t i o n of ELF emissions by charged p a r t i c l e s i n the exosphere with s p e c i a l reference t o the frequency band around the proton cyclotron frequency, Plan. Sp. Sc. 13, 761-772. Jacobs J.A. and T. Watanabe, (1964), M i c r o p u l s a t i o n w h i s t l e r s , J . Atmos. Terr. Phys., 26, 825-829. Jacobs, J.A., and T. Watanabe, (1965), A m p l i f i c a t i o n of hydromagnetic waves i n the magnetosphere by a cyclotron i n s t a b i l i t y process with a p p l i c a t i o n s to the theory of hydromagnetic w h i s t l e r s , Rep't. Boeing S c i . Res. Lab., Dl-82-0398, ( l a t e r published i n J . Atmos. Terr. Phys., (1966), 28, 235-253). Kennel, C.F., and H.E. Petschek, (1966), L i m i t on stably trapped p a r t i c l e f l u x e s , J . Geophys. Res., 71, 1-28. Landau, L., (1946), On the v i b r a t i o n s of the e l e c t r o n i c plasma J . Phys. (U.S.S.R.), 10, 25-34. Mcllwain, C.E., (1961), Coordinates f o r mapping the d i s t r i b u t i o n of magnetically trapped p a r t i c l e s , J . Geophys. Res., 66, 3681-3691. Montgomery, D.C., Book Co. and D.A. Tidman, (1964), Plasma K i n e t i c Theory, McGraw-Hill Neufeld, J . , and H. Wright, (1963), I n s t a b i l i t i e s i n a Plasma-Beam System Immersed i n a Magnetic F i e l d , Phys. Rev., 129, 1489-1507. Neufeld, J . , and H. Wright, (1965a), Hydromagnetic i n s t a b i l i t i e s caused by a gyrating proton stream, Nature, 206, 499-500. Neufeld, J . , and H. Wright, (1965b), I n s t a b i l i t i e s produced i n a stationary plasma by an "almost c i r c u l a r " e l e c t r o n beam, Phys. Rev., 137A, 1076-1083. Obayashi, T., (1965), Hydromagnetic w h i s t l e r s , J . Geophys. Res., 70, 1069-1078. Scarf, F.L., (1962), Landau damping and the attenuation of w h i s t l e r s , Phys. F l u i d s , 5, 6-13. Smith, R.L., (1961), P r o p e r t i e s of the outer ionosphere deduced from nose w h i s t l e r s , J . Geophys. Res., 66, 3709-3716. S o k o l n i k o f f , I.S. and R.M. Redheffer, (1958), Mathematics of Physics and Modern Engineering, McGraw-Hill Book Co. S t i x , T.H., (1962), The Theory of Plasma Waves, McGraw-Hill Book Co. 61 Sturrock, P.A., (1961), Amplifying and evanescent waves, convective and nonconvective i n s t a b i l i t i e s , Chap. 4, i n Plasma Physics, Ed. by J.E. Drummond, McGraw-Hill Book Co. Sudan, R.N., (1962), Plasma electromagnetic i n s t a b i l i t i e s , Phys. F l u i d s , 6, 57-61. Tepley, L.R., and R.C. Wentworth, (1962), Hydromagnetic emissions, X-ray bursts and e l e c t r o n bunches, part 1: experimental r e s u l t s , J. Geophys. Res., 67, 3317-3333. Tepley, L.R., and R.C. Wentworth, (1964), Cyclotron e x c i t a t i o n of hydromagnetic emissions, Rep. Contr. NAS5-3656, Lockheed M i s s i l e s and Space Co. Watanabe, T., (1964), D i s t r i b u t i o n of charged p a r t i c l e s trapped i n a varying strong magnetic f i e l d (one-dimensional case).with a p p l i c a t i o n s to trapped r a d i a t i o n , Can. J . Phys., 42, 1185-1194. Watanabe, T., (1965a), Private communication (May). Watanabe, T., (1965b), Private communication (October). Watanabe, T., (1965c), Determination of the e l e c t r o n d i s t r i b u t i o n i n the magnetosphere using hydromagnetic w h i s t l e r s , J . Geophys. Res., 70, 5839-5848. Watanabe, T., (1966), Q u a s i - l i n e a r theory of transverse plasma i n s t a b i l i t i e s with a p p l i c a t i o n s t o hydromagnetic emissions from the magnetosphere, Can. J . Phys., 44, 815-835. Wentworth, R.C, and L.R. Tepley, (1962), Hydromagnetic emissions, X-ray bursts and e l e c t r o n bunches, part 2: t h e o r e t i c a l i n t e r p r e t a t i o n , J. Geophys. Res., 67, 3335-3343.
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The exact theory of linear cyclotron instabilities applied to hydromagnetic emissions in the magnetosphere Jacks, Bruce Raymond 1966
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Title | The exact theory of linear cyclotron instabilities applied to hydromagnetic emissions in the magnetosphere |
Creator |
Jacks, Bruce Raymond |
Publisher | University of British Columbia |
Date Issued | 1966 |
Description | The complex dispersion relation which describes transverse plasma waves propagating in a cold gyrotropic ambient plasma parallel to the background magnetic field as they interact with charged particle streams is derived by solving the linearized collisionless Boltzmann equation simultaneously with Maxwell's equations using the Fourier-Laplace transform method. The wave frequency is allowed to be complex with a positive imaginary part corresponding to a growing instability. The real and imaginary parts of the dispersion relation yield two separate equations. Under several assumptions, the equations can be simplified to yield an expression for the imaginary part of the frequency (the growth rate) and an equation relating the real wave frequency and the wave number. The theory is then applied to the magnetosphere by choosing a dipole model for the earth's magnetic field and a suitable distribution function for the particles. The specific case of waves of the ion-resonance mode interacting with mono-energetic, contra-streaming protons is considered in detail, and the results of this calculation are used in explaining hydro-magnetic (hm) emissions. In particular, it is suggested that the high frequency cutoff is a result of the pitch angle distribution of the particle stream. Computer calculations are done in order to display the general results of the theory. Specifically, when low energy protons (10 - 20 kev), trapped on a field line with an L value of 5.6 are considered, it is found that the region of instability occurs near the geomagnetic equator, and that the growth rate is a sharply peaked function of the frequency. |
Subject |
Cyclotrons Magnetohydronamics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0302270 |
URI | http://hdl.handle.net/2429/36845 |
Degree |
Master of Science - MSc |
Program |
Geophysics |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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