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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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THE EXACT THEORY OF LINEAR CYCLOTRON INSTABILITIES APPLIED TO HYDROMAGNETIC EMISSIONS IN THE MAGNETOSPHERE by BRUCE RAYMOND JACKS B.Sc, University of 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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s representatives„ I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date S^ j^t. XI- } \ °\LL i ABSTRACT The complex dispersion r e l a t i o n which describes transverse plasma waves propagating i n a cold gyrotropic ambient plasma p a r a l l e l to the back r ground magnetic f i e l d as they interact with charged p a r t i c l e streams i s derived by solving the linearized 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 positive 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 dispersion r e l a t i o n y i e l d two separate equations. Under several assumptions, the equations can be s i m p l i f i e d to y i e l d an expression for the imaginary part of the frequency (the growth rate) 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 for the earth's magnetic f i e l d and a suitable d i s t r i b u t i o n function for 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 interacting with mono-energetic, contra-streaming protons i s considered i n d e t a i l , and the results of t h i s calculation are used i n explaining hydro-magnetic (hm) emissions. In p a r t i c u l a r , i t i s suggested that the high frequency cutoff i s a res u l t of the pitch 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 calculations are done i n order to display the general results 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 rate i s a sharply peaked function of the frequency. TABLE OF CONTENTS Chapter I INTRODUCTION General Discussion Cyclotron Resonance Thesis Outline I I MATHEMATICAL ANALYSIS I I I APPLICATION OF THE GENERAL RESULTS TO THE MAGNETOSPHERE Discussion The Shifted Anisotropic Maxwellian D i s t r i b u t i o n A monoenergetic Pitch Angle D i s t r i b u t i o n IV NUMERICAL CALCULATIONS Normalization of the Equations Parameter Values Results V SUMMARY Discussion Conclusions Appendix I Transformation of Equations and Solution for f t(v,k,(0) II Determination of the Transformed Magnetic Fields I I I Analytic Continuation of the Integrals IV Wave Polarization V Simplifying the General Dispersion Relation VI Coordinate Transformation and Calculation of CJi Bibliography i i i FIGURES Figure Page 1. Variation of sin*iy with y for f i v e values of H . 20 2. Dependence of CJj on the normalized frequency for two p a r t i c l e energies. a)X=0° 1 30 ,b)X-10° 31 c)X=20° 32 3. Dependence of Dx on \ for two p a r t i c l e energies. a) uj'(eq)= 0.3 33 b) 6Jl(eq) = 0.4 33 c) 6J'(eq)=.0.5 34 d) w'(eq)"0.7 35 4. Integration contours for k?0,6Jj;>0. 47 5. Integration contours for k>0,6iJr<0. 47 6. Integration contours for k<0,<* c^>0. 48 7. Integration contours for k<0,CJ 1<0. 48 i v ACKNOWLEDGMENTS I wish to thank sincerely Dr. T. Watanabe for suggesting t h i s problem and for his assistance i n many helpf u l discussions throughout the course of the research. I also wish to thank Professor J. A. Jacobs for providing the opportunity and the f a c i l i t i e s to carry out t h i s work and for his 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 physical study of that region involves the concepts of magnetohydrodynamics and plasma physics. The theory of plasma waves has been used to explain such phenomena as atmospheric whistlers and geomagnetic micropulsations. In such studies, the geomagnetic f i e l d i s fundamental. Atmospheric whistlers are electromagnetic waves which occur i n the frequency range 300 - 30,000 cps and propagate i n the electron resonance mode (fast mode) which has an upper frequency l i m i t at the electron cyclotron frequency - 6Je. They originate i n lightning 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 follow 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 polarized wave exists i n the ion resonance mode (slow mode) for frequencies below the ion gyrofrequency OJc . The dispersion r e l a t i o n for these two types of waves propagating p a r a l l e l to the background magnetic f i e l d B 0 i n a cold, ambient plasma can be written (Astrom, 1950) for a plasma with one singly-charged, ionic component; i s the wave frequency, k i s the wave number and c i s the speed of l i g h t . In t h i s equation, both 6J and k are r e a l quantities. A^ pe and Sk^are the electron and ion plasma frequencies respectively, and are defined by 2 f i r N P 1 x pe. (1-2) where N p i s the electron number density of the plasma, q i s the charge on an electron or a proton and 3. i s negative or positive respectively, and me and m-L are the electron and ion masses respectively. The Gaussian system of units i s used throughout the thesis. In equation 1-1, the upper sign i s used for the ion resonance mode and the lower sign for the electron resonance mode. 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 to present the dynamic spectra (frequency-time plots) of such micropulsations. Those which showed a d i s t i n c t fine structure consisting of re p e t i t i v e r i s i n g tones which often overlapped were called hydromagnetic emissions, or b r i e f l y , hm emissions. They have also presented a theory, which accounted for t h i s fine structure (Wentworth and Tepley, 1962). research done on hm emissions and they emphasize the following points. At hydromagnetic frequencies,, waves of the ion resonance mode tend to be i I guided by the magnetic f i e l d to a much greater extent than waves of the electron resonance mode (Jacobs and Watanabe, 1964). The dispersion of 'hm whistlers' or 'micropulsation whistlers' y i e l d s a th e o r e t i c a l spectrum which agrees approximately with the observed characteristics of the structured hm emissions. The hm whistler signals d i f f e r from those of atmospheric whistlers i n that the signal intensity does not constantly decrease after the f i r s t bounce but often grows before decaying (Tepley and Wentworth, 1964). I t i s possible that waves of the ion resonance mode are d i r e c t l y Jacobs and Watanabe (1965) have described the history of the The idea developed i n t h i s thesis i s that the waves gain energy through a cyclotron i n s t a b i l i t y process involving 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) for electrons interacting with waves of the whistler mode. It i s not a single p a r t i c l e effect (cyclotron radiation) but a plasma i n s t a b i l i t y involving the transfer 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 (Brice, 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 exi s t so that the wave-particle interaction can take place. -The actual 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 discussion, the existence of perturbing hm whistler waves i s assumed. Cyclotron Resonance It 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 velocity. In order to determine whether the wave grows or i s damped, the velocity d i s t r i b u t i o n function for the p a r t i c l e s must be specified. It has often been noted that growing i n s t a b i l i t i e s require an anisotropic d i s t r i b u t i o n ( S t i x , 1962; Montgomery and Tidman, 1964; Cornwall, 1965). It 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 rotation 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 different from the cyclotron frequency because of the Doppler s h i f t -arising from the p a r t i c l e ' s longitudinal velocity u. For the case of protons and a left-hahd polarized'wave, the resonance conditions mentioned above are s a t i s f i e d with a positive r e a l frequency OJ given by CL) - feu = (1-3) • In the magnetosphere, D - (0;. < 0 (Booker, 1962). 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 directions. However, i t must be noted that protons can interact with waves of the whistler mode because of the anomalous Doppler effect (Brice, 1964). When a p a r t i c l e travels faster than the wave and i n the same dir e c t i o n , i t sees a reversal of the wave's polarization. Jacobs and Watanabe (1965) have discussed the different p o s s i b i l i t i e s leading to cyclotron i n s t a b i l i t i e s . Thesis Outline In Chapter I I , a general l i n e a r analysis of the problem i s carried out starting 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 sim i l a r to that outlined by Stix (1962) for longitudinal plasma o s c i l l a t i o n s . This method was suggested by Watanabe (1965a), and the results of these calculations agree with those of Cornwall (1965). Chapter I I I involves the application of the general results to the magnetosphere. The proton streams are assumed to be monoenergetic. The pitch angle d i s t r i b u t i o n function i s chosen to 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 for 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 flu x which has a small normal cross-section (Watanabe, 1964). In Chapter IV, the results of numerical calculations made on a computer are presented. In order to carry out the calculations, several assumptions are made: the earth's f i e l d i s assumed to be a centered dipole f i e l d having a value of 0.3 Gauss on the earth's surface at the geomagnetic equator; the Smith model (Smith, 1961) of the equatorial electron density, which i s v a l i d only for distances up to four earth r a d i i from the earth's surface, i s assumed to hold i n a l l regions of the magnetosphere. These two assumptions l i m i t the exactness of the results. A discussion of the var i a t i o n of electron 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 altitudes 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 li m i t a t i o n s of the thesis. 6 CHAPTER.II MATHEMATICAL ANALYSIS To describe the interaction 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 cold, then the f i e l d s produced by the p a r t i c l e density fluctuations (due to thermal motions) are negligible compared to the f i e l d s of the wave and the equation can be written 3t 3£ *m where t and r are the time and space coordinates respectively, 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 fl u x density, respectively, of the wave. 3/8JC represents the s p a t i a l gradient and 9/9y the gradient i n velocity space. The Maxwell equations used are c u r l B,= * ? j + ^ | f (2-2) c u r l . E = - i H ( 2 " 3 ) where COInAp J defines the current density. The summation i s taken over a l l the components of the plasma. 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 quantities ( B 0 i s zeroth order) and that f can be written f ( v , r , t ) = f 0 (v) + f± ( v , r , t ) (2-5) where f 4 i s a f i r s t - o r d e r perturbation on f 0 . The background f i e l d B G i s taken to be i n the positive z dir e c t i o n . Only transverse waves are con-sidered and the s p a t i a l l y varying quantities 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 I n t r o d u c i n g t h e c y l i n d r i c a l coordinates (u,w,<|>) in vel o c i t y space, the l a s t term i n equation 2-6 can be written where ^ c = ^ r ( 2- 9) i s the p a r t i c l e gyrofrequency which can be positive or negative. I t i s assumed that f t = §i° = $ = ° <2-10> so that equation 2-6 i s s a t i s f i e d . 8 Using a Fourier 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 for ^ (Vjk,^)) (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 function at time t = 0 must be specified. This method was f i r s t used by Landau (1946) i n discussing the longitudinal vibrations of an electronic 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 written (2-11) then fj.(v,k,6j) i s given by -±ii»£[w».*)-ie,c<>.vj) w-ku-wc i x ^ c ^ * /aw 9U B x(w,^)+-c& y(^» CO -feu + ' + •VnCOt (2-12) (2-13) Using equation 2-4, simple algebra gives since f 0(y_) i s constant with respect to Using equation 2-12, and the other two component Maxwell equations after transformation (Appendix I ) , i t can be shown (Appendix II) that Bx(oo)ft)±t&y(oJ^) = (2-14) 9 where x r i 3& . The upper and lower signs correspond to l e f t and right polarized waves respectively (Appendix IV). In p r i n c i p l e , B(t,z) can now be found by applying the inverse transformations to equation 2-14. This means that the response of the plasma system to an i n i t i a l perturbation of p a r t i c l e d i s t r i b u t i o n s by a pa r t i c l e beam can be found. I t i s t h i s result that j u s t i f i e s the use of the transform method, but i n order to make the problem feasible mathematically, i t i s not solved i n general. In the Laplace transformation, the parameter i s allowed to be complex, with the r e s t r i c t i o n that i t s imaginary part be positive. Later, 6J i s i d e n t i f i e d as the wave frequency. The inverse transformation must be carried out along a path which l i e s i n the upper half OJ-plane above the s i n g u l a r i t i e s of B_((0,k). But physically, negative imaginary parts for D should be allowed. The procedure followed i n overcoming t h i s d i f f i c u l t y involves the analytic continuation of a singular i n t e g r a l and has been discussed by St 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 dispersion 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 for both positive and negative imaginary parts of (J . The presence of the sing u l a r i t y i n the inverse transformation results i n the la s t term i n equation 2-16 being evaluated under the condition 60 - + U) t = ° (2-17) This i s how the cyclotron resonance condition enters the problem mathematically. Equation 2-16 can be simp l i f i e d by specifying the cold, back-ground part of f 0 as fa by writing with b C u ) (2-19) where N p i s the number density of the background plasma and S represents the Dirac delta function. The d i s t r i b u t i o n function 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 function. 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 si m p l i f i e d using integration by parts so that [ d w (Pfju, = ~ ^ U / p L (<*>TW» = - - i k — (2-20) and since [G(f B^ ^  = 0, equation 2-16 becomes - ^ Z ^ j d w l w G ^ i . ^ O (2-21) 11 where * f l p = HirNpl'/'^c. i s the electron plasma frequency of the background plasma and i s taken as the t o t a l plasma frequency since N s i s assumed to be much smaller than N p . I f CO i s written U) «• CO* +lLJx ,(2-22) i t i s also 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 to the period of the wave. Assuming that N 5 and CJj are f i r s t order quantities compared to N P and ( J R , equation 2-21 can be si m p l i f i e d (Appendix V) and setting the r e a l and imaginary components separately equal to zero gives ^ - C ^ - V i t e L = 0 (2-24) and XL) • + J ± ^ (2-25) where Equation 2-24 i s the r e a l dispersion equation which relates and k, and equation 2-25 i s the expression for the growth rate of 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 results of the previous chapter have been derived for 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 to a homogeneous background magnetic f i e l d which extends over a l l space. In applying these results to the magnetosphere, i t i s assumed that the region of interaction 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 well approximated by plane waves. This problem has been mentioned by-Hruska (1966). In order to calculate 6J t using equation 2-25, an e x p l i c i t expression f o r f s must be determined in a meaningful way. Although much has been learned experimentally about p a r t i c l e s contained i n the van Allen 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 altit 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 cyclotron emissions as well as constituting a ring current. Most of the results 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 results w i l l be quoted l a t e r . Two d i s t r i b u t i o n functions are now considered. The shifted, anisotropic 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 several times before (Sudan, 1963; Guthart, 1964; Hultqvist, 1965; Hruska, 1966). The second 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 Shifted, Anisotropic, Maxwellian Di 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 different than the spread p a r a l l e l to i t , and there i s an organized, uniform velocity p a r a l l e l to 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 u a ^ T , , ( 3 . 2 ) f s has been normalized to N s . In t h i s case, Setting u = V R, and integrating over w and ty, o o and (3-4) and so equation 2-25 gives ^ f a ) , ^ ^ l A - t o ^ A ^ , i V ^ t k ^(V*-U.)"\ (3-5) i l l .Ci.'w a* e ^ - r r r & f c " -J 14 I f only one type of p a r t i c l e i s streaming, then the summation can be removed and the condition for positive i s 2± > (3-6) This res u l t i s known ( S t i x , 1962). A Monoenergetic Pitch Angle D i s t r i b u t i o n Using a th 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 for which the scale of s p a t i a l variations i s much larger than the gyroradius of 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 pitch angle of 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 to a tube of flux for which the l i n e a r dimensions of any normal cross-section are much smaller than the scale length of the trapping region. A particular solution i s given by where e< i s an arbitrary constant and C i s constant with respect to 1, *+* , and t and contains the normalization factor. In a 'strong' f i e l d , with no perturbing wave, s i n ^ ^ / B i s a constant of the motion since i t i s proportional to the magnetic moment of a p a r t i c l e , the f i r s t adiabatic invariant (Chandrasekhar, 1960; Alfven and Falthammar, 1963). The result 15 that fg depends on the adiabatic invariant i s to be expected (Cornwall, 1965). The discussion i n the remainder of the thesis concerns only mono-energetic protons. The assumption that the pa 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 to simplify the mathematics. Monoenergetic electrons have been considered previously (Wentworth and Tepley, 1962). In the numerical calculations 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 written W S < v - v . ) a a £ (3-9) where V i s the 'pitch angle d i s t r i b u t i o n parameter'. 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 of the main f i e l d B*, for instance at the equator. The constant 'A' i s determined by normalizing f s to N* at t h i s point, and i t i s necessary that H > -2 so that the in t e g r a l does not diverge. In t h i s case, i . ltfB»* + Q (3-10) * ATTjir V? p ( f | I ) where p = Jf + 1 and P represents the gamma function. The int e g r a l over the pitch 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 positive and the negative z directions, although for a given wave at any point, only one-half the pa r t i c l e s can participate in the cyclotron interaction. Using equations 2-25 and 3-9, GOi i s calculated (Appendix VI) and i s found to be non-zero only when \\\< V,, . In t h i s case, y i N / V ( V o 3 - - v « ) ^ r Y / + i ^ . 1 > 1 ] , M , y ± A » CJT ( 3 _ 1 5 ) 16 The following important qualitative results can be obtained from t h i s , expression. 1. The growth rate i s d i r e c t l y proportional to the density of streaming p a r t i c l e s since A^Ng. 2. The only factor which can be negative i s a) The '-2' term represents a constant damping factor which originates in the expression -2 j ' dw wf s . b) The quantity + G0!e/GJ* - 1 i s positive f o r both the electron-whistler interaction and the proton-hm whistler interaction. Therefore, for wave growth i n either case, must be at least positive. In fact, f must s a t i s f y I f (Oc i s taken as the equatorial value, then t h i s condition allows wave growth at any point on that 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 sp e c i f i c value of If , I t follows that there i s an upper frequency l i m i t for waves that w i l l be amplified. This maximum frequency i s given by TTt <3-17) y Any waves with frequencies higher than t h i s w i l l be damped, and using the value of Uc at the equatorial plane w i l l indicate approximately the maximum frequency 17 of any amplified waves. I f the mechanism for wave amplification suggested here i s correct, 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 q u a l i t a t i v e l y l a t e r . 3. At a given point i n space and for 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 amplified. This fact may be used to explain the observed minimum frequency of hm emissions. 4. For a given p a r t i c l e energy and wave frequency, can vary a. o only i f ^ . v a r i e s . approaches v 0 as k) c gets larger and 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 equatorial region. Past a certain point, V R w i l l always be greater than v e 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 situated near the equatorial plane. The numerical calculations show that t h i s interpretation i s v a l i d . The existence of maximum and minimum frequencies as discussed above can be thought of as roughly defining a band width for the emissions The suggestion that the i n s t a b i l i t y tends to occur i n the equatorial region i s due to Watanabe (1965b) and has been mentioned by Jacobs and Watanabe (1965). The same idea has been put forth by Tepley and Wentworth (1964) for different 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 faster near the equatorial plane, i t i s i n these regions that the proton cyclotron radiation 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 equatorial region, the same process occurs. In order that the wave be reinforced each time, they suggest that the bounce periods of 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 equatorial 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 radiation 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 for 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 interact i n a c o l l e c t i v e manner with the hm waves. Since the particles see anomalously Doppler shifted waves, the sense of the waves' polarization i s opposite to that of the gyration of the p a r t i c l e s (Brice, 1964) and cyclotron resonance cannot occur. Other different 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. It 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 amplification process comes from p a r t i c l e k i n e t i c energy. I t i s possible that although some energy i s transferred 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 velocity 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 pitch angle d i s t r i b u t i o n i s given by sin <y , when V i s positive, more p a r t i c l e s have large pitch 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 longitudinal k i n e t i c energy. When the p a r t i c l e s lose transverse energy to the wave i n a growing i n s t a b i l i t y , there i s a general reduction of pitch angles and some p a r t i c l e s may be l o s t because they have pitch angles which are inside the 'loss-cone' (Cornwall, 1965; Brice, 1964). 20 3-0 15 30 45 60 75 90 M-1 (degrees) 1 Family of curves of sm^H* . Values of the parameter Jf are written beside the corresponding curve i n the diagram. 21 CHAPTER IV NUMERICAL CALCULATIONS Normalization of the Equations It i s assumed that only protons are contained i n the p a r t i c l e stream. The background plasma contains therefore, more electrons than protons by a small amount i n order to preserve o v e r a l l charge neut r a l i t y . Since fg 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 of the i n s t a b i l i t y i s not required. It i s often convenient to write the important equations obtained i n a study i n normalized form involving dimensionless variables so that the general results can be seen without employing numerical values which are v a l i d for a s p e c i f i c case only. In equation 3-15, the term 2 6l)r i n the denominator originates 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 to be put into dimensionless form with the help of the following relationships. <A.i*Ui C J t t = - 6 J e ( 4 _ 1 } where 22 I t i s then convenient to write by defining U = ^ - " W ^ V ~ ( 4 " 3 ) ? U l (4-4) I f the 2CJt term i n equation 3-15 i s neglected, then by dividing both sides of the equation by 60i , i t i s found that _ _ I — t i (4.5) In a si m i l a r manner, the r e a l dispersion r e l a t i o n (equation 2-24) can be put into normal form by neglecting the term for the same reasons as above, and then i t i s written ^ M ^ (4-6) ^ ±1 - UJ* ±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 velocity and pitch angle are known, then the resonant frequency for that p a r t i c l e can be determined from the resonance condition (OR + = (Oi <-4_7) I f the d i s t r i b u t i o n function for the p a r t i c l e s i s anisotropic such that there are more pa r t i c l e s with p a r a l l e l components of velocity s l i g h t l y less 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 transferred to the wave. Transverse Landau damping 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 isotropic (Scarf, 1962; S t i x , 1962). I f 6Jt i s specified, and the emission i s to be of a certain frequency, then the value f o r \u| can be calculated from equation 4-7, and a lower bound for the energies of the p a r t i c l e s involved can be calculated. In order to specify OJi , the earth's main magnetic f i e l d i s assumed to be a centered dipole 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 la 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 dipole 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 \* c o s 4 * L represents the distance, measured i n units of earth r a d i i , that a given f i e l d l i n e i n the equatorial plane l i e s from the centre of the earth. For a dipole f i e l d L = I ^ T . (4-9) where ^ e i s the geomagnetic la t i t u d e at the point where the relevant l i n e of force intersects 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-L i s inversely proportional to L^. At very low frequencies, the wave's phase velocity i s very nearly the Alfven v e l o c i t y . V A , and using equation 4-7, the frequency of emission can be approximated by 24 6J B = ^ - 7 — (4-10) Using the density model of Smith (1961), the l o c a l electron number density i s l i n e a r l y proportional to the gyrofrequency, or N P ~ B ~ - j i ( 4 -n) It i s assumed that t h i s model holds not only i n the equatorial plane below L = 5, but that i t i s v a l i d along f i e l d l i n e s away from the equatorial plane and at altitudes which correspond to L values greater than about 5 (Brice, 1964; Carpenter and Smith, 1964). Using equation 4-10, i t can be seen that 1 6J R ~ - r ; ;—5TT (4-12) and i t can be seen that for a given emitted frequency, the position i n the magnetosphere at which the interaction takes place strongly determines the energy range of the pa 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), indicate a large f l u x of protons with energies of the order of hundreds of kev at L—3.5, these protons may be very important i n emission processes. The energy range i s the right 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 for 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 flu x of 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) indicates 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 for the guiding l i n e of force below 4.98 for a dipole f i e l d . For the distorted dipole f i e l d which he used, t h i s value becomes 4.75. Taking into account the outline of hm emissions given i n Chapter I, the repetition 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 re 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 interaction 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 calculated 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 calculating an i n t e g r a l numerically. Using t h e i r table, the bounce periods for 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 repetition 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 that these values are not outstanding. This res u l t i s physically reasonable since the dispersion r e l a t i o n indicates 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 c, and so, for a given frequency range, the wave goes slower at higher altitudes since the cyclotron frequency decreases. At the same time, the path which the wave follows i s longer at higher altitudes. Such differences of rep e t i t i o n period between different emissions i s noticeable even by making very rough measurements on different 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 deviation of about 10 sec. In the sonogram given by Jacobs and Watanabe (1965), the period measured between 13^ hr. and 14 hr. i s never less than 130 sec. The difference between these two spectra i s measureable. This type of measurement has been made by Watanabe (1965c). The frequencies of the emission i n the example above in 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 of force given by L = 3.5, then at the lea s t , the protons would have had to have energies of 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 pitch angle parameter must be chosen. Hoffman and Bracken (1965) f i t t e d t h e i r data to a pitch 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 of pitch angles separately, 0° to 30° and 30° to 90°. The parameter values which gave a good f i t ranged from 1 to 4 over the region L = 2 to L = 7. Each calculation here i s done for 3" = 2. The r a t i o Ns/Np 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 re s u l t s , but the calculated growth rates are too large. Jacobs and'Watanabe (1965) assumed the r a t i o to be 10~^. The correct r a t i o probably varies from case to case. One l i m i t i n g factor i n the choice of N 5 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 less than 4.98 i n nine 27 examples considered. Cornwall (1965) suggests that the emissions would not take place at lower altitudes i f the r a t i o NS/NP i s too small and t h i s might occur because the background plasma density i s large. A l l the calculations have been done using an L value of 5.6. For each CJX , the quantity v* - V* i s calculated and i f v^ 4 v£ , 6 J X i s set equal to zero. The wave frequencies have been varied from 6Ji(eq)/20 to 19 6J £(eq)/20 and at each frequency, the growth rate i s calculated for twenty-nine values of the geomagnetic la t i t u d e from 0° to 29° i n half-degree steps. The Smith model of electron density (Smith, 1961) can be written N P= \%,00O ^S* (4-13) This model has been used by Brice (1964) i n the form X l K = (3.T)^. i.OOO CJ e\ (4-14) In the calculations, equation 4-13 i s written N P= v r . 5>5!>0 B e c 7 v f 3 (4-is) and using t h i s value for the density, the Alfven vel o c i t y can be calculated for 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. For L = 5.6, 7 -I the value of V A at the equatorial plane i s 4.95 x 10 cm-sec. A proton energy of 10 kev corresponds to a velocity of 1.38 x 10 cm-sec so i n t h i s case, U 5(eq) = 2.79. Since the vel o c i t y increases as the square root of the energy, for 20 kev, U s(eq) = J~21,(2.79) = 3.95. In the programme, v 0 i s given the following seven values; 2.8Vft(eq), 3.0V A(eq), ... , 4.0V A(eq). 28 I t should be noted that equation 4-13 describes Np empirically and i s an average value. Since Np can vary by as much as a factor of 2 at different times, the calculated results are not exact and differences exist between one s p e c i f i c example of hm emissions and another. The results w i l l serve as an indication of general effects which result from the mechanism which has been considered. The calculations have been done at the University of B r i t i s h Columbia Computing Centre on an I.B.M. 7040 computer using Fortran IV language. Results The largest growth rate at any point was found to occur at the equatorial plane ( X = 0) at a frequency of 0.65 cps. Figure 2-a indicates how the growth rate varies with frequency and p a r t i c l e energy at the equatorial plane. The lower cutoff i s very sharp and the peak, i t s e l f i s narrow. Figures 2-b and 2-c show the same type of plot f o r A = 10° and X = 20° respectively. I t can be seen that as X increases, the frequency band of amplification moves toward higher frequencies for a given v 0 . This i s to be expected and results from the requirement that which means that as 6J; increases with X , must increase i n order to a, r e s t r i c t the size of V^. This effect can be seen i n another way by observing how CJj changes with X for several frequencies. I t i s found that the growth rate i s p r a c t i c a l l y zero for a l l frequencies less than or equal to ( 29 0.2 (^i (eq). Remembering that the largest growth rate occurs near = 0.25 CJi (eq), figure 3-a gives the results for £*JR = 0.3 (eq). At the higher energies, i t i s noticeable that the growth rate maximum occurs near X — 10°. In figures 3-b and 3-C, t h i s effect 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 effect 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 to be large enough, the expression tf(k->t/6JR - 1) - 2 can easily be negative. For energies i n the range 10 - 20 kev, the sharp low frequency cutoff occurs at 0.2 (Ji(eq) and i t i s important to note that the maximum proton energy determines t h i s cutoff point. The proton energy also i s very important i n determining the size of the growth rate. 30 U s = 4.0 o to OS o o X = o° 0.2 0.4 0T6 0.8 Normalized Frequency 1.0 Fig. 2-a Growth rate as a function of frequency at X = 0° for two p a r t i c l e energies. The frequency i s normalized to the equatorial cyclotron frequency, 16.4 sec -!. 31 Fig. 2-b Growth rate as a function of frequency at A = 10° for two p a r t i c l e energies. The frequency i s normalized to the equatorial cyclotron frequency, 16.4 sec--'-. 32 40 ~ o 30 X = 20c TO U o u o 20 . 10 -U s = 4.0 0.2 0.4 , 0.6 0.8 Normalized Frequency Fig. 2-c Growth rate as a function of frequency for two p a r t i c l e energies for X = 20°. The frequency i s normalized to the equatorial cyclotron frequency, 16.4 s e c - x . 40V 33 30h 20L 10 6L)r= 0.3 CJ: (eq) = 4.9 sec" 1 U s = 4.0 5 10 15 20 25 Geomagnetic Latitude (degrees) Fig. 3-a Variation of GOx with X for CO^= 4.9 sec-1. 40 30 20 10 L\ = 0.4 6Ji(eq) = 6.54 s e c - 2 U s = 4.0 10 15 20 25 Geomagnetic Latitude (degrees) F i g . 3-b Variation of 6J xwith X for COR = 6.54 s e c - 1 . 1 5 10 15 20 25 Geomagnetic Latitude (degrees) Fig. 3-d Variation of k\ with X for 60 K = 11.4 s e c - 1 . 35 CHAPTER V SUMMARY Discussion I t i s important to remember that the analysis i s v a l i d only to f i r s t order. The growth of waves can be indicated but soon after i t begins, the l i n e a r theory becomes i n v a l i d and nothing further 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 into the ionosphere so that the wave growth becomes negligible 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 analysis. 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. I t i s conceivable that N s might sometimes be large enough that the r a t i o N s / N p i s of the order of unity and t h i s would invalidate 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 analysis much simpler and general results can s t i l l be obtained. Considering only monoenergetic p a r t i c l e streams i s an over-s i m p l i f i c a t i o n , although the variation of t o r with energy has been calculated numerically. Hoffman and Bracken (1965) indicate that a doubly sloped exponential energy spectrum describes well the d i s t r i b u t i o n of proton fluxes which they observed. I f t h e i r detailed observations were taken at lower energies, then the introduction of an energy d i s t r i b u t i o n of the form e~^/^° where E e i s empirically determined would make the results more quantitative. 36 In order to determine more exactly the actual growth of a wave of some fixed 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 analysis 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 fixed 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 to 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 cyclotron emissions i n the magnetosphere. Both authors f i n a l l y consider only the (L, e) and (R, p) interactions where 'R' and 'L' refer to l e f t and right-hand polarized waves and 'e' and 'p' refer to electrons 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 arise when the transverse velocity 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 that t h i s would never occur except near the mirror points. This assumption does not seem reasonable. He suggests that hm emissions occur when super-luminal protons interact 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 transfer of energy between waves and p a r t i c l e s . He chooses a shifted 37 Maxwellian d i s t r i b u t i o n for the streaming p a r t i c l e s but gives no reason for t h i s choice. Since he takes the temperature d i s t r i b u t i o n to be iso t r o p i c , the contra-streaming (L,p) and (R,e) interactions do not give r i s e to 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 result 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 electron resonance mode of wave and because the dispersion characteristics of the ion resonance mode are the same as the observed spectra of structured hm emissions. Since the signals sometimes increase i n intensity i n time before dying out, i t i s suggested that the wave packets gam energy via a cyclotron i n s t a b i l i t y process as they interact with low-energy protons which are trapped i n the magnetosphere. An expression for the growth of the waves was developed starting 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 factor sin*M'/By/a' results in an upper cutoff frequency of k-V(l + at any given point. I t i s not known whether t h i s effect i s more important than the damping which results from the thermal background plasma. Computer calculations (for Y =2) indicate three important features of 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 well as the frequency of the steep lower cutoff. This suggestion was also made by Qbayashi (1965). F i n a l l y , the largest growth rates for the i n s t a b i l i t y are found tb occur near the equatorial plane, although at some frequencies, there are two regions of largest growth, each s l i g h t l y removed from the equator by ten or twenty degrees. The observed la t i t u d e dependence of hm emissions may be explained by the fact that none of the L waves in the hm packet can have a frequency above the ion cyclotron frequency at the geomagnetic equator and that i t decreases as the latitude of the point where the l i n e of force intersects the earth's surface increases. Another important consideration i s the effect of the ionosphere on the wave as i t travels 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 latitude variation of the duct characteristics. I f measurements on wave propagation above the ionosphere could be made to determine polarizations and i f detailed records of low-energy proton fluxes (1 - 100 kev) could be obtained, many uncertainties i n the theories of hm emissions would be eliminated. 39 APPENDIX I Transformation of Equations and Solution for f t (y_,k,6j). Let G(t,z) represent any of the quantities which are to be transformed. I f G(t,z) i s well-behaved, then the Fourier-Laplace transform of G exists and i s defined by G(OJ,M= ( d t J j i i G t t . i ) t " t ( * * " C o t ) ( A i - i ) o -eo For the Fourier transform to exi s t i t i s s u f f i c i e n t that G(t,z) be of bounded variation and absolutely integrable, i . e . , +00 ( | & ( t , ^ a * < co ( A 1-2) and i t i s implied that G(t,z)-^0 as z-* + oo. In order to assure existence of the Laplace transform, i t i s convenient to assume ( S t i x , 1962; Sokolnikoff and Redheffer, 1958) that for some choice of the constants M and , |Gct»u n^1 ( A I - 3 ) and ImGj>ja. Applying the transformations defined above to the following component Maxwell equations 2* c a t ( A 1 _ 6 ) ^ c 9 t ( A 1 - 7 ) 40 and using integration by parts and the conditions outlined above, i t i s found that -c-fe & yCoJ,&)= TTfcto.fe) -£[«.uEx(u>,fe) + E x ( o , ^ ( A I - 8 ) t * B l c C u ) > ) = M y y C w > ) - i [ t ^ E y ( W j W + E Y C o J M ] (Ai-9) t * E y ( c o , I O = . - i [ t ( 0 ^ ( w » + & X < © A ) ] (Ai-io) L^E^iuiM^ {[iufyCuM + B y ( o ^ ) ] ( 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 written + c M - V y W l = ° (Al-12) Applying the combined transformation defined by equation Al-1 to t h i s equation and eliminating Ex(k>,k) and Ey(60,k) from the resulting 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 function f x(y,k,(0). 41 I t i s now convenient to transform coordinates from ( v x , v y) to (w,4>) using the following relations 3V X 9w'^vK 3 * ' ? ^ (Al-14) ^ . 2 - + ( A 1 _ 1 5 ) and i t was assumed above that 9f 0/94> vanishes. Since w2 = v£ + v 2 5 i t i s easy to show that _ - w - c o s * - ^ ( A 1 _ 1 6 ) 2* = * = si* • - < ^ J L £ 1 _ ( A 1 . 1 7 ) By writing ^ ( Y j k j O ) as a Fourier series in<p, i ^ . t . o i^ Z r ' e ^ * <Ai-i8) equation Al-13 can f i n a l l y be written " 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 co 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 function on an i n t e r v a l I. A l l solutions must have the form = § C X ) = e " •ax X r J e a * b C t ) d t + c e - 4 X ( M _ 2 1 ) where x 0 i s i n I and c i s any constant. Note that i f the anti-derivative of the integrand i s evaluated at x„, then t h i s 'constant 1 can be grouped with 'c' i n multiplying e~ a x and the i n t e g r a l evaluated at the upper l i m i t of 'x' i s just the i n d e f i n i t e i n t e g r a l . The new constant i s then chosen to be zero, and the res u l t i s the same. In t h i s way, the term involving 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 solution of the inhomogeneous equation and the term containing the redefined constant i s the general solution 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 solution to the homogeneous equation must be dropped from the general solution of the inhomogeneous equation. The resulting solution for ^ ( v , k,(o) i s given by T Z _ j 60--feu•'w^t 43 APPENDIX I I Determination of the Transformed Magnetic Fields Applying the Fourier-Laplace transform defined i n Appendix I to equation 2-13 gives ^A)±c>y(ojA)=Z S.UveTt*w51(yA>w) (A2-1) and using the expression for f ^ ( ^ , 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 for the current density can also be found from the transformed Maxwell equations. By eliminating EK(w,k) and Ey(o;,k) from the four transformed equations i n appendix I, i t i s found that i(CO*-Ca 6 X {tOj (?) = -HTVC- - CJ &„( 0 » + c& EyCoA) (A2-2) and c(^-^\t)ZyCLoM= H i r c f e y ^ w . ^ - w a y C o ^ ) - c ^ E x ( o ^ ) ( A 2 _ 3 ) and hence + c f e [ e x ( o y f e ) ± t E y c o ^ ( A 2 _ 4 ) Noting that when the solution for f 1(y,k,6j) i s substituted into equation A2-1, the in t e g r a l involving f ^ e i l 1 n ^ i s non-zero only when m = + 1, i t i s found that 44 ">*p J 60 - -feu ^ 60c L f 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 Analytic Continuation of the Integrals The outline of the procedure as given by Stix (1962) w i l l not be copied,in d e t a i l . There, the case of longitudinal plasma o s c i l l a t i o n s i s considered as an example. The differences for the case of transverse waves interacting with a p a r t i c l e stream are noted. The problem i s simp l i f i e d by calculating the asymptotic value of B(t,k) as t-9°o . The expression for the magnetic f i e l d s given i n equation 2-14 i s v a l i d for 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. The analytic continuation of equation 2-14 must be determined. I t i s assumed that B^(0,k), B y ( 0 , k ) , Ex.(0,k), E v(o,k), 9f„/aw, r\ (-1) 2 f o / 9 u , and f j . are a l l entire functions of u. One must then consider integrals of the form + 00 -oo with \, 6J +• 60c V = ^ (A3-2) where the i n t e g r a l i s to be taken along the r e a l axis. F(u) i s assumed to be an entire function of u. I f one i s considering u as a complex variable, then the path of integration i n equation A3-1 can be changed in accordance with complex variable 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 of integration can be raised ( f i g . 10) from the r e a l axis above the singularity 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 analytic continuation i s obtained by deforming the path of integration down from the r e a l axis ( f i g . 11) so that K v K p a U i ) = I c v ) C p a t k a . ) - ^ i F ( v ) ( A 3 _ 4 ) and path 2 can easily be chosen to be the r e a l axis. 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 to write ( f i g . 12) I ( V H p * t k l ) =• K V K p . t k i ) (A3-5) Case 4. k<0, 0. Im(V)>0 and the continuation by contour deformation i s analogous to case 2. From figure 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) - KvXpatK 2) « - ^  F^V) ( A 3 _ 7 ) 47 path 2 X W = V path 1 ->Re( V a) Fig. 4 Contours for integration when k> O and 6 J X "> ° 48 path 2 V 4 = V path 1 Fig. 7 Contours for integration when k < O and OJj< O 49 and so I ( V H p * ^ 1 ) = I ( V ) < W «xiS) - i ^ F ( V ) ( A 3 - 8 ) since k<0 i n t h i s case. I t can be seen from the above discussion that the integrals over 'u' for kJjX) are taken s t r i c t l y as they appear, along the r e a l u,', axis (cases 1 and 3). F o r ^ i < 0, (cases 2 and 4), again the integrals are taken along the r e a l axis but there i s an additional residue term i n each of the two cases. Following Stix (1962), i t i s seen that both the numerator and denominator of Bx(W,k) + iBy(lO,k) are analytic functions of u i n the whole plane (entire functions) and so the poles of B(k>,k) come only from the zeroes of the denominator. There are two equations, one for each of positive and n e g a t i v e ^ . 60 for lx)x> 0, and A ™ If* 6J -fcu * CUc + W - T O w - ^ ^ ^ + i ^ U J = ° ( A 3 - 1 0 ) •V-for GJi< 0. Using the Cauchy p r i n c i p a l value ((P) notation, these two equations can be combined into one equation which i s v a l i d for both positive and negative values of the imaginary part of 6J . ( A 3 - 1 1 ) 50 where the functional form of G i s given by frttiJ-jLJCl 7 ^ ) ^ + "7J (A3-12) 51 APPENDIX IV Wave Polarization Consider the case of 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. Left hand polarization i s defined by writing t V t , " * ) - K ^ y t t , ^ = O ( A 4 _ 2 ) I t then follows from Maxwell's equations that E * ( t , % ) + £ E y C t ^ ) - 0 (A4-3) The Fourier-Laplace transformation of these two equations i s straight forward so that at t = 0, the condition B * ( 0 , k ) ± l B y ( 0 > ) =r O (A4-4) implies that E x C 0 , f c ) t (o,fe) = O (A4-5) The t o t a l f i e l d i s written as the sum of the separate f i e l d s of two waves of opposite polarization. 6(to, fc)- B % , k ) +aa )cu) i«,> ( A 4 _ 6 ) (A4-7) 52 where the superscript ( 1 ) indicates the left-hand mode and ( 2 ) the right-hand mode. Using equations A4-6 and A4-7, i t i s found that £ > ( 0 , lO = &\oM + &\o,to ( A 4 - 8 ) and EioM = E 0 ) ( o » +• E a )(o» (A4-9) Substitution of the above values of B^(6J,k), By(6J,k), B x(0,k), B^Ojk), Ex(O j^) an& Ey(0,k) into equation 2-14 and choosing, say, the upper sign everywhere, y i e l d s an equation involving only the superscript ( 1 ) because those quantities involving the superscript ( 2 ) a l l vanish because of the d e f i n i t i o n of the right and l e f t modes of polarization. When the lower sign i s chosen, the resu l t i n g equation contains only the superscript ( 2 ) . Therefore, i n equation 2-14, the upper sign corresponds to left-hand polarization and the lower sign corresponds to right-hand polarization. 53 APPENDIX V Simplifying the General Dispersion Relation 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 fact 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 of G ( f s ) and UJ and the product CO£Hs i s second order. At the- same time, two of the three terms i n G ( f s ) contain a multiplying factor 1/co, and when t h i s factor i s written i n the normal form of 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 carried out for the denominator of the integrand of the fourth term, with the result that V|Gc(5Q ^ W £(^s) (A5-5) In the l a s t term, the multiplying 60 can be replaced by6J*. One further 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. Expanding the numerator of 54 the integrand i n a Taylor series about u = V R, i t i s found that « [ w 6 ( W ] u « V ) t (A5W5) since V - V ^ ^ a n d G ( f s ) contains the f i r s t order quantity N s. Using a l l the above approximations, equation A5-1 can be written + i H ^ + S f f e ^ i - ^ £ i > ^ 4 - ° ( « - ) Setting the imaginary part of t h i s equation equal to zero gives equation 2-25. A second equation results when the r e a l part i s set equal to zero. The fourth term can be neglected since i t i s of f i r s t order while the t h i r d term i s zeroth order. This approximation yi e l d s the dispersion r e l a t i o n given i n Chapter I. 55 APPENDIX VI Coordinate Transformation and Calculation of 6Jx In c y l i n d r i c a l coordinates (u,w) Equation 3-9 gives f s i n spherical coordinates (V,HO as i,*AS<v-v.) S i l t l g . (A6-2) I t i s convenient to transform G ( f s ) using the chain rules 9w ~ 0V 3W 0q> SW (A6-3a) 0 u 9 V * u 9 u and the transformation V = +• (Wl4- ) ^  (A6-3b) (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 si m p l i f i e d because the in t e g r a l i n the numerator of equation 2-25 must be evaluated at u = V R. It w i l l be 56 written (A6-6) In equations Al-5a and Al-5b, u and w are eliminated by putting u = V R, and w = + ( v 2 - V^)3. This result 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 follows by algebra that R 6_>R L V** 1 V** 1 V- 1] * w ; i C V V ' ) v ^ ^ J ( A 6-9) with 60* W * y has been r e s t r i c t e d to values greater than -2. I f IVr\ > v a then the i n t e g r a l i s zero. I f I v J = v 0, the int e g r a l may be different from zero because of the presence of the delta functions. In t h i s case, )S must 57 be further r e s t r i c t e d i n order to avoid d i v i s i o n by zero i n factors containing the expression v^ - V*. From the t h i r d term i n the integrand, i t can be seen that Y^ -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 delta-function derivative means that the function which multiplies i t i n the integrand must be f i r s t d i fferentiated and then evaluated at v = v 0. This calculation leads to an expression containing terms with the factors (v* - V*)k and (v** - v j ^ 1 * 1 . I t i s seen, then, that for \\«\ = v., V must be at least zero, and i n t h i s case, whether V i s zero or positive, the int e g r a l i s zero. This result i s not unexpected since | v R l = v 0 represents the resonance condition for the case of protons interacting with contra-streaming left-hand polarized waves when the proton beam i s 'linear', that i s , when the beam has no transverse k i n e t i c energy. Neufeld and Wright (1963) have shown that for t h i s case, a contra-streaming i n s t a b i l i t y does not e x i s t . It i s found then, that for W*l < v„ and V> -2, Substituting t h i s result 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 Alfven, H., and C.-G. Falthammar, (1963), Cosmical Electrodynamics, Oxford University Press. Astrom, E., (1950), On waves i n an ionized gas, Ark. Fys., 2, 443-457. B e l l , T.F., and 0. 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Mcllwain, C.E., (1961), Coordinates for 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., and D.A. Tidman, (1964), Plasma Kinet i c Theory, McGraw-Hill Book Co. 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 " electron beam, Phys. Rev., 137A, 1076-1083. Obayashi, T., (1965), Hydromagnetic whistlers, J. Geophys. Res., 70, 1069-1078. Scarf, F.L., (1962), Landau damping and the attenuation of whistlers, Phys. F l u i d s , 5, 6-13. Smith, R.L., (1961), Properties of the outer ionosphere deduced from nose whistlers, J. Geophys. Res., 66, 3709-3716. Sokolnikoff, 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. Fluids, 6, 57-61. Tepley, L.R., and R.C. Wentworth, (1962), Hydromagnetic emissions, X-ray bursts and electron 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 excitation of hydro-magnetic emissions, Rep. Contr. NAS5-3656, Lockheed Miss 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 applications to trapped radiation, Can. J. Phys., 42, 1185-1194. Watanabe, T., (1965a), Private communication (May). Watanabe, T., (1965b), Private communication (October). Watanabe, T., (1965c), Determination of the electron d i s t r i b u t i o n i n the magnetosphere using hydromagnetic whistlers, J. Geophys. Res., 70, 5839-5848. Watanabe, T., (1966), Quasi-linear theory of transverse plasma i n s t a b i l i t i e s with applications to 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 electron bunches, part 2: t h e o r e t i c a l interpretation, J. Geophys. Res., 67, 3335-3343. 

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