OSCILLATIONS OF THE EARTH'S OUTER ATMOSPHERE AND MICROPULSATIONS by Karl Oskar Westphal Diplom i n Physik, Universitaet Wuerzburg, Germany, 1951 M.A. The University of Toronto, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept th i s thesis as conforming to the required standard The University of B r i t i s h Columbia July, 1961 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. • It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be a l k w e d without my written permission. Department The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of KARL OSKAR WESTPHAL Diplom Physik Wuerzburg, 1951 M.A. Toronto, 1959 Thursday, Aueust 17th, 1961, at 10:00 A.M. in Room 302, Physics Building COMMITTEE IN CHARGE F. H. SOWARD, Chairman J. A. JACOBS R. W. BURLING G. M. VOLKOFF E. LEIMANIS R. D. RUSSELL C. A. SWANSON P. R. SMY G. V. PARKINSON External Examiner: T. WAT AN ABE Tohoku University, Sendai, Japan O S C I L L A T I O N S O F T H E E A R T H ' S O U T E R A T M O S P H E R E A N D M I C R O P U L S A T I O N S A B S T R A C T Micropulsations of the Earth's magnetic field are closely related to the properties of the upper atmosphere and probably interstellar space as well. A model which would explain the observed data in a satisfactory manner would also give information on the properties of the outer atmosphere which at the present time are difficult to obtain otherwise. However, in spite of the vast amount of observa-tional data now available, a satisfactory theoretical analysis of the problem has not yet been given. To a large extent this is due to mathematical and computational difficulties. Geomagnetic micropulsations may be described as magnetohydro-dynamic oscillations of which two different modes, namely poloidal and toroidal oscillations, may exist, and which in general are coupled. In this work only the toroidal mode, which may be under-stood as oscillations of a line of force, is considered. If the phen-nomenon is studied under the simplifying assumption ^ji^ »o where <t> is the longitude it is possible to obtain the eigenvalues of the oscillating lines of force in simple terms by treating the problem in cylindrical rather than in spherical polar coordinates. For a constant and variable charge density distribution, the eigenfrequencies are obtained as functions of the latitude. For a variable charge density distribution, the agreement between theory and observation is good in middle and low latitudes. However, as the latitude increases towards the poles the model gives periods which tend to infinity. As a result of recent studies it appears that the geomagnetic field does not extend as far into outer space as has been assumed, but that to a first approximation it is confined to a cavity. For this reason the equation of toroidal oscillations is applied to a compressed dipole field. Assuming both a constant and a variable charge density distri-bution the eigenperiods of the deformed magnetic lines of force are obtained, using an electronic computer. The results for a variable charge density distribution agree with observational data in the polar regions but not in middle and low latitudes. This may imply that the charge density distribution is in error and it is hoped that further rocket and satellite data will settle this point. PUBLICATIONS 1. O. Blunck and K. O. Westphal. Zum Energieverlust energiereicher Elektronen in duennen Schichten. Zeitschrift fuer Physik, 130, p. 641-649, 1951. 2. J. A . Jacobs and K . O. Westphal. High frequency geomagnetic micropulsations. Physics and Chemistry of the Earth, vol. 5, Pergamon Press. G R A D U A T E STUDIES Field of Study: Geomagnetic Micropulsations Nuclear Physics j . B. Warren Advanced Geophysics j . A . Jacobs Istotope Geology R . D . Russell Related Studies: Applied Electromagnetic Theory G . B. Walker Computational Methods C . Froese Programming of Digital Computers J. R. H . Dempster ABSTRACT Using Maxwell's equations of electrodynamics and the linearized fundamental equation of hydrodynamics neglecting a l l but the ponderomotive force, the two differential equations characterizing toroidal and poloidal modes of oscillations are obtained. Neglecting the coupling between these modes the toroidal mode which appears to be connected with the phenomenon of geomagnetic micropulsa-tions is studied in detail. Substituting for the constant magnetic f i e l d the undeformed dipole f i e l d of the Earth the eigenperiods of the oscillating lines of force are: computed assuming a constant charge density distribution. Using numerical methods the eigenperiods are also obtained in the case of a variable charge density. Since the Earth's dipole f i e l d is presumably deformed by the solar wind a compressed dipole f i e l d is introduced into the equation of toroidal oscillations. The eigen-periods of the oscillating lines of force are obtained in this case, assuming a constant charge density distribution. For the case of a variable charge density a numerical method is described which could yield the eigenperiods. - V -ACKNOWLEDGEMENTS The writer wishes to express his sincerest thanks to Professor J. A. Jacobs and Prof. J. C. Savage for suggest-ing the problem and for many stimulating discussions during the performance of the work. He would also l i k e to thank Mr. H. Dempster, Computation Center, University of B r i t i s h Columbia, for his great i n t e r e s t and assistance i n some of the computational problems. Further thanks are extended to Mr. P. Haas for drawing the diagrams and the p r i n t i n g of the equations and formulae. The f i n a n c i a l assistance of the National Research Council of Canada and the Off i c e of Naval Research i s g r a t e f u l l y acknowledged. - 1 1 1 -TABLE OF CONTENTS A b s t r a c t A c k n o w l e d g e m e n t s CHAPTER I I n t r o d u c t i o n CHAPTER I I M a t h e m a t i c a l F o r m u l a t i o n 1. The B a s i c E q u a t i o n s 2. The E q u a t i o n s o f S m a l l M a g n e t o - H y d r o d y n a m i c O s c i l l a t i o n s 3„ The E q u a t i o n o f t o r o i d a l o s c i l l a t i o n s when t h e c o n s t a n t f i e l d i s a d i p o l e f i e l d 4 0 The E q u a t i o n o f t o r o i d a l o s c i l l a t i o n s i n t h e c a s e o f a c o m p r e s s e d d i p o l e f i e l d CHAPTER I I I S o l u t i o n s 1„ S o l u t i o n f o r t h e n o r m a l d i p o l e f i e l d w i t h c o n s t a n t c h a r g e d e n s i t y 2. S o l u t i o n f o r t h e n o r m a l d i p o l e f i e l d w i t h v a r i a b l e c h a r g e d e n s i t y 3. S o l u t i o n f o r t h e c o m p r e s s e d d i p o l e f i e l d w i t h c o n s t a n t c h a r g e d e n s i t y 4 „ S o l u t i o n f o r t h e c o m p r e s s e d d i p o l e f i e l d w i t h v a r i a b l e c h a r g e d e n s i t y - i v -CHAPTER IV Discussion of the Results 73 APPENDIX I. The compressed dipole f i e l d 83 II. The matrix M 86 III. Solutions of a cubic equation 88 IV. Tables of matrices and their lowest eigenvalues 93 V. Analytical representation of the eigenvalues 98 VI. Solutions of the lines of force equation, compressed dipole 103 VII. Tables of matrices and their lowest eigenvalues 113 References 116 CHAPTER I INTRODUCTION During the last ten years pulsations of the geomagnetic f i e l d have received increasing attention. A great amount of recorded data from stations a l l over the world has accumulated since such pulsations were f i r s t observed by Balfour Stewart in 1861. Amplitudes range from a fraction of a gamma (1 gamma (y) =10 gauss) to as much as a few tens of a gamma, and frequencies from 0.01 to 10 cps. The main reason for studying the phenomenon is that the generation of such pulsations is closely related to the properties of the upper atmosphere and probably interstellar space. A model which would explain the observed data in a satisfactory manner would also give information on the pro-perties of the outer atmosphere which at the present time are d i f f i c u l t to obtain otherwise. However, in spite of the vast amount of observed data now available, a satisfactory theoretical analysis of the problem has not yet been given. To a large extent this i s due to mathematical and computa-tional d i f f i c u l t i e s . -2-In p r i n c i p l e geomagnetic pulsations may be described as magneto-hydrodynamic o s c i l l a t i o n s of which two d i f f e r e n t modes, namely poloidal and t o r o i d a l o s c i l l a t i o n s can exis t and which i n general are coupled. In t h i s thesis we deal only with the t o r o i d a l mode which may be understood as o s c i l l a t i o n s of a l i n e of force. Considering the phenomenon to take place i n a meridian plane i t i s possible to obtain the eigenvalues of the o s c i l l a t i n g l i n e s of force by treating the problem i n c y l i n d r i c a l rather than i n spherical polar coordinates. For a given charge density d i s t r i b u t i o n the eigenfrequencies are obtained as functions of the l a t i t u d e . In the f i r s t attempt to treat t h i s hydromagnetic problem the Earth's main f i e l d i s taken to be that of a geocentric dipole. This has the unfortunate consequence that the eigenfrequencies tend towards i n f i n i t y as the point of i n t e r -section of the l i n e s of force with the Earth's surface approach very high lat i t u d e s - which i s not i n agreement with the observations. By confining the dipole f i e l d to a sphere of f i n i t e radius which may be the case on account of the presence of the solar wind which c a r r i e s the f i e l d along, i t i s possible to remove the discrepancies. From the experimental data obtained at d i f f e r e n t l a t i t u d e s but equal longitudes i t i s possible to determine those eigenperiods which show a maximum frequency occurrence and by comparing th i s frequency-latitutde dependence with - 3 -those calculated i t appears that the model of an o s c i l l a t i n g magnetic l i n e of force i s f e a s i b l e . Before discussing the t h e o r e t i c a l approach to the problem of geomagnetic micropulsations which i s based on the concept of magneto-hydrodynamic waves i t i s i n s t r u c t i v e to outline t h i s phenomenon, which was discovered by Alfve'n i n 1942, for a rather simple case. If an e l e c t r i c a l l y conducting medium moves i n the presence of a magnetic f i e l d , an e l e c t r i c f i e l d i s induced i n the medium, the current being at r i g h t angles to both the d i r e c t i o n of motion of the medium and the magnetic f i e l d . However, an e l e c t r i c current i n a magnetic f i e l d i s acted upon by a mechanical force perpendicular to both the current and the magnetic f i e l d , i . e . either i n the d i r e c t i o n of motion or against i t . It follows from simple energy considera-tions that the mechanical force must be directed so as to impede the o r i g i n a l movement. If the o r i g i n a l motion was at r i g h t angles to the magnetic f i e l d the disturbance w i l l propagate i n the form of a wave along the magnetic l i n e of force. In order that such an in t e r a c t i o n between electromagnetic and hydrodynamic phenomena exist i t was shown by Lundquist (1952) that the inequality »l - 4 -raust be s a t i s f i e d , where L i s the l i n e a r dimension of a l i q u i d conductor, of density and permeability ju0 i n the presence of a magnetic f i e l d B. (mks units are used through-out t h i s thesis. ) It i s e a s i l y seen that under normal laboratory conditions t h i s c r i t e r i o n i s not s a t i s f i e d and magneto-hydrodynamic e f f e c t s are not observed. The s i t u a t i o n , however, i s completely d i f f e r e n t i n problems of cosmic physics. Because of the enormous dimen-sions involved i n such cases Lundquist's c r i t e r i o n i s e a s i l y s a t i s f i e d and interactions between electromagnetic and hydrodynamic phenomena may be considerable. It i s for th i s reason that nearly a l l attempts which have been made i n recent years to explain the phenomenon of geomagnetic micropulsations apply the p r i n c i p l e s of magneto-hydrodynamics to the outer atmosphere. The suggestion that geomagnetic micropulsations might be explained i n terms of magneto-hydrodynamic o s c i l l a t i o n s was f i r s t advanced by Dungey (1954b). Under the assumption that the outer atmosphere i s a very good conductor he derived from Maxwell's equations and from the fundamental equation of f l u i d dynamics two coupled p a r t i a l d i f f e r e n t i a l equations governing the two components E w and V.?, v i z . : - 5 -and (2) From (1) and (2) which are termed the equations of pol o i d a l and to r o i d a l o s c i l l a t i o n s the f i e l d vectors E { E r , E^,, E ^ } and •* V { V r, Vj, V j- are determined i n a unique way. However, these two equations are far too complicated to be of much use i n studying geomagnetic micropulsations. For t h i s reason one,assumes a x i a l symmetry, i . e . one supposes that the phenomenon takes place only i n the plane of a meridian - an assumption which seems to be confirmed by the l o c a l time dependence. Under t h i s assumption the coupling term vanishes ( <^ = 0) s p l i t t i n g the coupled system into two separate equations, v i z . : and The f i r s t attempt to compute from equation (4) the eigenperiods of the t o r o i d a l o s c i l l a t i o n s was made by Dungey himself. Taking the magnetic f i e l d of the Earth H Q to be -6-that of a dipole he obtained under the assumption of a constant charge density jo ( = 1 0 - 1 8 kg/m3) for the fundamental period the approximation T ( - ^ X > 3 0 ° ) ( 5 ) where X0 i s the l a t i t u d e at which a p a r t i c u l a r magnetic l i n e of force intersects the surface of the earth. Since this i s a boundary value problem one i n general assumes that the hydro-magnetic wave i s r e f l e c t e d at the point of int e r s e c t i o n . Evaluating equation (5) for p a r t i c u l a r l a t i t u d e s gives the r e s u l t s shown i n table 1. Table 1 *o T l 45° 10 sec 55° 54 sec 65° 11 min 70° 55 min A d i f f e r e n t method of approach to the solution of the boundary value problem which i s i n p a r t i c u l a r permissible for obtaining the periods of the higher modes of o s c i l l a t i o n s has been taken by Kato and Watanabe (1956). Again assuming a constant charge density d i s t r i b u t i o n , they obtained for the period of the n t n order o s c i l l a t i o n -7-X,= F T T ^ G x V>- y f ' - a x ) o/x (6) 0 where y 0 0 the charge density has been taken equal to i t s value at the most distant point and a = sin 2A. Q The r e s u l t s for the fundamental mode as obtained from equation (6) are summarized i n table 2. Table 2 A.0 /> o = 0.8xl0~ 1 8 kg/m3 1.6xlQ-18 kg/m3 r2/4xl'Q- 1 8 kg/m3 45° 6 sec 9 sec 11 sec 55° 33 sec 47 sec 57 sec 65° 381 sec 538 sec 659 sec 70° 2071 sec 2929 sec 3585 sec In equation (4) the only s p a c i a l derivatives which occur are those i n the operator (H 0° V ) operating on the function V y , ; this however can be interpreted as the derivative of v^ i n the d i r e c t i o n of H D. Therefore equation (4) can be i n t e r -preted as a wave.equation where the disturbance propagates along a l i n e of force. Choosing for H 0 a dipole f i e l d , equation (4) represents the wave equation governing the time-space r e l a t i o n s h i p of a disturbance which propagates along -8-the magnetic l i n e s of force of the dipole. It i s then i n th i s sense that one speaks of o s c i l l a t i o n s of the l i n e s of force which, i t must be emphasized, applies i n i t s truest sense only to the case of an i n f i n i t e l y conducting medium. This concept of o s c i l l a t i n g magnetic l i n e s of force has been u t i l i z e d by Obayashi and Jacobs (1958) for computing the charge density d i s t r i b u t i o n of the outer atmosphere making use of the observed periods of micropulsations. To carry out the analysis i t was however necessary to make an assumption of the functional r e l a t i o n s h i p between the height and the charge density. In order to f a c i l i t a t e the solu t i o n of equation (4) Siebert (private communication) chose a charge density d i s -t r i b u t i o n of a mathematical form which reduces equation (4) to the simple d i f f e r e n t i a l equation of a harmonic o s c i l l a t o r . This method of approach however leads to r e s u l t s which are i n disagreement with the observed data i n d i c a t i n g that t h i s approach i s not appropriate. Reviewing the t h e o r e t i c a l work which has been done on the problem so far one arrives at the conclusion that the magneto-hydrodynamical treatment of the problem leads to equations which i n th e i r general form are too complex to be treated mathematically. Even under the reasonable assumption that the phenomenon takes place i n a meridional plane only, the r e s u l t i n g equations are quite intractable - even, i f i n - 9 -addition, a constant charge density d i s t r i b u t i o n and a simple dipole f i e l d of the Earth are assumed. Since coupling e f f e c t s , which manifest themselves through the presence of terms involving i n the case of spherical polar coordinates, are i n i t i a l l y neglected i t seems fe a s i b l e to investigate the problem anew by working i n c y l i n d r i c a l rather than spherical polar coordinates and dropping terms containing ? . The equations then take on a simpler form i n which a mathematical treatment i s possible even for the case of a non-uniform charge density d i s t r i b u t i o n and a deformed dipole f i e l d . -10-CHAPTER I I MATHEMATICAL FORMULATION 1. The B a s i c E q u a t i o n s In o r d e r t o r e l a t e t h e phenomena o f e l e c t r o d y n a m i c s and h y d r o d y n a m i c s , M a x w e l l ' s e q u a t i o n s and t h e b a s i c e q u a t i o n s o f h y d r o d y n a m i c s a r e u s e d . I t f o l l o w s f r o m a c o n s i d e r a t i o n o f t h e o r d e r s o f m a g n i t u d e o f t h e q u a n t i t i e s i n v o l v e d t h a t i n p r o b l e m s o f c o s m i c p h y s i c s , d i s p l a c e m e n t c u r r e n t s a r e n e g l i g i b l e i n c o m p a r i s o n w i t h t h e c o n d u c t i o n c u r r e n t . T h e r e f o r e M a x w e l l ' s f i r s t e q u a t i o n i s s i m p l y where H i s t h e m a g n e t i c f i e l d s t r e n g t h and y t h e c u r r e n t d e n s i t y . I n a d d i t i o n H s a t i s f i e s t h e e q u a t i o n V- H = o (8) M a x w e l l ' s s e c o n d e q u a t i o n w h i c h c o n n e c t s t h e e l e c t r i c f i e l d i n t e n s i t y E w i t h t h e change o f t h e m a g n e t i c f l u x B i s V xE = -I O ) - 1 1 -where B = A H do. Ohm's law f o r a medium moving w i t h t h e v e l o c i t y V i s = <T( £ + V XB ) ( i i ) C o n s i d e r i n g t h e o u t e r a t m o s p h e r e as a p e r f e c t c o n d u c t o r (cr -> oo) t h i s becomes E = ~ V X £ (12) The b a s i c e q u a t i o n o f h y d r o d y n a m i c s c a n be w r i t t e n i n t h e f o r m fir ' C+j*s <"> -> > where p i s t h e d e n s i t y , V t h e v e l o c i t y , and G t h e sum o f a l l e x t e r n a l , n o n - m a g n e t i c f o r c e s . The t e r m 3. x B i s t h e -» m e c h a n i c a l f o r c e e x e r t e d by t h e m a g n e t i c f i e l d B on a volume e l e m e n t c a r r y i n g t h e c u r r e n t d e n s i t y ^ . d / d t r e p r e s e n t s t h e m o b i l e o p e r a t o r -12-It has been shown by Dungey (1954a,b) that for hydro-magnetic waves in the outer atmosphere the e f f e c t of v i s c o s i t y which i s usually the most important cause of attenuation, can < be neglected when considering long period o s c i l l a t i o n s i n the presence of the earth's magnetic f i e l d . Also Plumpton and Ferraro (1953 ) have shown that the periods of o s c i l l a t i o n of a g r a v i t a t i n g l i q u i d star in the presence of a central magnetic pole and are not much reduced i f the g r a v i t a t i o n a l f i e l d i s neglected. Since the magnetic pressure i s much greater than the gas pressure and the pressure gradient due to the disturbance i s small compared with the magnetic force, G = 0 and equation (13) reduces to These are the basic equations which w i l l be applied to the problems of geomagnetic pulsations. 2. The equations of small magneto-hydromagnetic o s c i l l a t i o n s To obtain the equations of small o s c i l l a t i o n s i t i s necessary to eliminate from the system a l l but one dependent vector variable. Because of the nonlinearity of the equations this w i l l be impossible i n general. However i t proves possible in our case to obtain two coupled p a r t i a l d i f f e r e n t i a l equations governing one component of each of the vectors E and V. -13-To t h i s end one has to substitute equations (7) and (10) into equation (15) which yie l d s where H i s the sum of a constant f i e l d H G and a small disturbance h introduced through the motion of the medium. Since "9 -=7 -> |Hj7> I h| and \/,Ho =0 lv ^ "7 equation (16) becomes, under the assumption that \ ~ \ ^ ^ -"7 fTt*y°H°'(7*h) ( l 7 ) Taking the vector product of thi s equation with 3= o^(H6+r)) "y'-'e^ o and taking the time derivative i t follows that { 1 ( 1 1 , \x\ = _ u W x f i r . k W H (is) or by vi r t u e of equation (9) (19) -14-Carrying out the d i f f e r e n t i a t i o n on the left-hand side of equation (19) one obtains However since E = - V x B J t ~ ~ T t t i - v * T T 1 W~ T F * ^ 2 at at v T t ^ Introducing the l a s t expression into the expanded left-hand side of equation (19) ^ t l a t V " ^ V T F + d t at v at2 i s obtained. Since IE I i s proportional )VI the second o term i n the l a s t expression i s of the order V and can be neglected. Hence B) -/»# -15-T h e r e f o r e e q u a t i o n (19) becomes (20) i . e . w h i c h i s a v e c t o r wave e q u a t i o n f o r E. T r a n s f o r m i n g e q u a t i o n (21) i n t o c y l i n d r i c a l c o o r d i n a t e s ( r , < ^ ), s i n c e H Q = -^Hr, H^. , oj , t h e second" t e r m on t h e r i g h t - h a n d s i d e w i l l v a n i s h i n t h e c a s e o f t h e z-component. Thus t h e z-component o f e q u a t i o n (21) becomes a f t e r e x p a n d i n g V * V * E i n jfche" c y l i n d r i c a l c o o r d i n a t e s Mzi_ l I l A o - r » d*Ea. _ I ofefr., d*E^ (22) S i n c e f r o m e q u a t i o n (12) E r =A v 2H^ a n d E^-A^W, e q u a t i o n (22) c a n be f u r t h e r w r i t t e n (23 ) -16-Again, since o (l4 Mr) . ij dU? ^ d r d 2 - _ and the right-hand side of equation (23) becomes d * , r Li _ u ^ " (24) Since the constant f i e l d H Q obeys Vx H Q = 0 which i n c y l i n d r i c a l coordinates i s equivalent to L) -^ Mt =0 (25) equation (23) by virtue of expression (24) and equation (25) f i n a l l y becomes L ? d ' l ^ . d 2 . ! d 2 ) r _ A L u d n b\(*Vi (26 -17-This i s the d i f f e r e n t i a l equation of poloida l o s c i l l a t i o n s and one notices that on the right-hand side the z-component of the v e l o c i t y f i e l d appears. Therefore the term on that side represents the coupling between the two modes E z and V z which i s necessary to describe the magneto-hydrodynamic problem i n a unique way. To obtain a time-space r e l a t i o n s h i p for the z-component of the v e l o c i t y f i e l d i t i s necessary to eliminate the vector -> of the magnetic disturbance h from the system represented by equations (7) to (15). To thi s end one has to consider equation (16) which, neglecting terms of higher order than the f i r s t , becomes ;£!f - - r W v x H ) (27) Using the id e n t i t y 2 H*(V*HH VH -(H-V)'H (28) and writing H = H c + h one obtains for the right-hand side of equation (27) HX(V«H)4V{H^2CH„- IH !} - { C H . 4 ) - V } ( H / M 4vf^va - l04 W?- (H. -V)H i - (l -V)H 0 - fH 0 -V) I -18-> Neglecting higher terms i n h and applying the i d e n t i t y (28) equation (27) becomes -oUn +(l-v)k- w -> -> Since the c y l i n d r i c a l coordinates for any two vectors A and B the z-component of (29) f i n a l l y becomes To eliminate h z in the l a s t equation one has to substitute equation (12) and (10) into equation (9) and one obtains Expanding the right-hand side yields -> jj- - (H-V)V - CV*V)H - HV-V + W'H which since V • H = 0 becomes N -> -» -» -> -> > -ff- = (H-V) V - (V-7)H - UV-V ( 3 2 ) -19-Since H Q i s constant i n time, H z = 0, and )H 0| >> |h| the z-component of equation (32) i s H * = (33) 6 t To eliminate the l a s t terra on the right-hand side of (30 the r - and A} - components of Maxwell's equation are used which after introducing (12) becomes o h . _ _ „ M _ I' O F ; (34) and Multiplying equation (34) by H r and equation (35) by H 2 2 2 and adding y i e l d s , with H r + H^ . = H c , In order to obtain an equation which contains only E z and V z, one d i f f e r e n t i a t e s equation (30) with respect to time and -20-substitutes equation (36), giving - JJ . V l k - L U ^ + _L (Hr X - H (37) Substituting equation (33) f i n a l l y yields which can be rewritten i n the form Equation (38) i s for to r o i d a l o s c i l l a t i o n s . Equations (26) and (38), together with the boundary conditions, describe the behaviour of an i n f i n i t e l y conduct-ing medium of c y l i n d r i c a l symmetry due to a disturbance. Apart from the fact that these two equations are too complicated to be solved i n their general form we are con-cerned here only with phenomena i n the meridional plane. For t h i s reason the si m p l i f y i n g assumption ^/^^ ~ 0 i s made, i . e . the two modes of o s c i l l a t i o n are decoupled, and equations (26) and (38) become -21-and { ^ r - W . - ^ ) ( H L - V ) } ^ =0 (40) It has been stated e a r l i e r that a knowledge of E z and V z i s s u f f i c i e n t for the determination of the other f i e l d components. Assuming that the f i e l d quantities depend on t through the factor exp(iwt), equation (9) yields and - i o ) / ^ * { v * E ^ = " ~ J 7 " ( 4 2 ) while for computing the components of V, equation (17) gives [">f>Vr = ~A^{V4} ? (43) and LCOyOV^ = yfc. H r { V x h } . , (44) However because of equations (9) and (21), -22--» and t h e r - and $ - components o f V r e d u c e t o and I f on t h e o t h e r hand t h e z-component o f t h e v e l o c i t y f i e l d i s known t h e n f r o m e q u a t i o n (12) E,= A K v V z (47) and E,= - A K . V ; (48) w h i l e f r o m e q u a t i o n (33) [ U ^ = H r ^ + "^ Ix (49) I t f o l l o w s f r o m e q u a t i o n s (41) t o (46) t h a t f o r t h e p o l o i d a l mode o f o s c i l l a t i o n t h e f o l l o w i n g s e t o f q u a n t i t i e s a r e i n v o l v e d (0, 0, E z) ; ( h r s h#, o) ; ( V r , V^, 0) -23-while for the toroidal mode of oscillations, equations (47) to (49) show a relationship between (E r, E.j, 0) l (o, o, h z) 5 (0, 0, V z) Hence from a knowledge of E z and V z the two vector fields -> -» E and V may be determined. 3. The equation of toroidal oscillations when the constant f i e l d is a dipole f i e l d Considering the motion along a line of force given by 2 r = r0 sin^ (50) where ^ is the co-latitude and r c the value at the furthest point (^ = 90°) the expression (H Q° V ) is given by Since the ^ -component of a dipole f i e l d is given by U " c in f - ^ where M is the magnetic dipole moment, equation (51) becomes M - A ^ r a x (52> -24-and therefore (53 ) Taking the time dependance in equation (40) to be of the form exp(itjt) the equation of toroidal oscillations in the presence of a dipole f i e l d becomes (54) It is frequently convenient to express lengths in units of the Earth's radius, and one thus writes r Q = » 0 a . Remembering that the magnetic dipole moment of the Earth is given by M = H0a* where a is the radius of the Earth and H Q the maximum value of the magnetic f i e l d intensity at the magnetic equator, equation (54) f i n a l l y becomes 7" <L2 2 ,S 2 ft* * 0 (55) This is an ordinary homogeneous differential equation of the second order. Associating the periods of geomagnetic micro-pulsations with the eigenvalues of equation (55) one has to find u such that V z satisfies the boundary conditions -25-at the ends of the i n t e r v a l of integration. It w i l l prove very convenient in comparing re s u l t s to 2 incorporate the charge density d i s t r i b u t i o n f> i n u by 2 - 2 putting p u = u . In this case equation (55) becomes + A * ^ 1 V2 = 0. (56) However th i s i s only legitimate i f the charge density d i s -t r i b u t i o n along the l i n e of force i s constant and not a function of the co-latitude ^ . 4. The equation of t o r o i d a l o s c i l l a t i o n s i n the case of a compressed dipole f i e l d Parker (1958) has considered i n some d e t a i l the i n t e r -action of the "solar wind" with the Earth's geomagnetic f i e l d . The solar wind i s the name give to the outward streaming of gas i n a l l directions from the sun with v e l o c i t i e s i n the range 500-1500 km/sec. The solar wind w i l l compress or sweep away the outer geomagnetic f i e l d down to a l e v e l where the energy density of the f i e l d i s equal to the k i n e t i c energy density of the solar wind. This r e s u l t s i n a deformation of the dipole f i e l d which under these circumstances may as a f i r s t approximation be con-sidered as being confined to a spherical cavity of radius R0. Postulating that the r-component of the magnetic f i e l d S / V J dJ--26-vanishes at r = RQ, i t i s shown in appendix I that ^ - ^ r ( - ^ ^ j r ) ^ (57. and Using the d i f f e r e n t i a l equation of the l i n e s of force, v i z . (59) i t i s further shown i n appendix I that the li n e s of force of a compressed dipole f i e l d are the solutions of the cubic equation r \ L h ± X ^ r , Rl ( 9 0 ) where a i s the radius of the Earth, R D the radius of the confining cavity and the co-latitude of the intersection with the Earth of the l i n e of force which touches the cavity in the equatorial plane. Introducing the radius of the Earth as the unit of length equation (60) becomes -27-where = and o< = A . By l e t t i n g cx tend to large values equation (61) becomes cx^ s / n v which f i n a l l y i n the l i m i t c* -» oo becomes which i s the equation of the l i n e s of force for the ordinary dipole f i e l d expressed i n units of earth r a d i i . Using a d i g i t a l computer \) c has been evaluated from equation (61) as a function of the co-latitude ^ for several values of o< . The r e s u l t s are plotted i n f i g . 2-4. To consider the e f f e c t of a compressed dipole f i e l d with f i e l d components characterized by equations (57) and (58) one has to go back to the operator (H Q" V ) which along a l i n e of force i s given by Introducing for H^ , the expression (58) one obtains field pattern of a moqnetic dir< it - 2 9 -1 2 Fig 2. field pattern of a compressed magnetic d;pole ( size of cavih; 12 Earth radii ) -30-field pattern of a compressed magnetic tiipo ( size of cavity 8 Earth radii ) Fig. 4. field pattern of a compressed magnetic dipole ( size of cavity 4 Earth renin } -32-CH.-V)- N A V r-4 ' r R and hence Sin (62) Introducing this r e s u l t into equation (40) yie l d s co I T (63) Taking the radius of the Earth as the unit of length equation (63 ) can be written \>c4 c^<x3/ <LJ K 2 v 2-u (64) l e t t i n g oC tend to i n f i n i t y and remembering then that equation (64) reduces to (56)„ °-33-CHAPTER III SOLUTIONS 1. Solution for the normal dipole f i e l d with constant charge density As pointed out previously the d i f f e r e n t i a l equation of to r o i d a l o s c i l l a t i o n s i n the case of a constant charge density d i s t r i b u t i o n may be written sin1*} d^ dvz Ho (56) Changing the independent variable by putting (65) equation (56) becomes .2 d V, i , 8 - 2 = o (66) for which a solution i s e a s i l y found. Since one i s p a r t i c u l a r l y interested i n finding the -34-eigenvalues of this equation i t i s necessary to specify the boundary conditions which most l i k e l y apply to the physical s i t u a t i o n . It has been shown by Dungey (1954) that hydromagnetic waves having long periods are almost per f e c t l y r e f l e c t e d when they h i t the ionosphere. Considering the fact that the ionosphere i s very close to the surface of the Earth compared with the distances over which the l i n e s of force extend one can assume without introducing much inaccuracy that the point of r e f l e c t i o n i s situated at the surface of the Earth. Therefore To f i n d the corresponding value of x equation (65) must be integrated, i . e . at 3 = <% From equation (12) i t then follows that at X0 where = 1 8 0 - J ; Since the integrand on the r i g h t -hand side i s symmetrical about the point this -35-becomes which after integration gives A x cos^( Ssin6Jo+ 6sintl + 8 sin2^ + 16) (67) Equation (67) associates the i n t e r v a l from <X to with an i n t e r v a l extending from x Q to Letting the point 3* = ^ correspond to x = o i t follows that the i n t e r v a l tXo to ^ = 180°- 30 corresponds to the i n t e r v a l from -x Q to +xG. Hence X0= CosZ(Ssmit+ 6s'mfy + Zsin^+lb) (68) Using the Alwac III E t h i s expression has been computed for 0 < t % < 90 i n steps of 5° and the r e s u l t i s given i n table 3. F i g . 5 shows the plot of x 0 against J^, . The solution of equation (66) i s then Vt*Acos (69) and V. - B"n ( a*'* A (69b) -36-Table 3 X 0.0 -0.4571 50.0 -0.4366 5.0 -0.4571 55.0 -0.4192 10.0 -0.4571 60.0 -0.3926 15.0 -0.4571 65.0 -0.3549 20.0 -0.4571 70.0 -0.3047 25.0 -0.4570 75.0 -0.2422 30.0 -0.4566 80.0 -0.1685 35.0 -0.4554 85.0 -0.0865 40.0 -0.4527 90.0 0.0000 45.0 -0.4470 Because of the boundary condition i t follows from equations (69a) and (69b) that K l - f ( 7 0 ) fi —6 With a = 6.371 x 10 (m), Q = 1.256 x 10 (voltsec/amp. meter), H Q = 0.312 x 10~ 4 (voltsec/m 2) o Equation (70) can thus be written -38-w= ^ (71) where |X0| i s given by equation (68) and I In equation (71) the odd values of n correspond to a cosine o s c i l l a t i o n while even values of n are associated with an sine o s c i l l a t i o n . Since CD = co V~j5~* (Zrr/r) ^7*~ o n e obtains for the fundamental eigenperiod (n = 1) T, = 3.IS2 x \0Sx.VO4A|X0| X [sec] (72) The eigenperiods of the o s c i l l a t i n g l i n e s of force have been calculated from equation (72) and plotted in f i g . 6 using a — 1 9 "\ constant charge density d i s t r i b u t i o n of 6,5 x 10 (kg/m°). The neutral p a r t i c l e density has not been included i n p because the mean free path i s so long that only charged p a r t i c l e s w i l l contribute to the hydromagnetic wave motion. -39-Table 4 T(sec ) T(sec ) 10° 4.340xl0 5 50° 3.074 15° 1.896xl0 4 55° 1.727 20° 2038 60° 1.036 25° 375.0 65° 0.651 30° 97.61 70° 0.418 35° 31.95 75° 0.267 40° 12.97 80° 0.157 45° 5.972 2. Solution for the normal dipole f i e l d with variable charge density In the preceding section the eigenperiods of the t o r o i d a l o s c i l l a t i o n s of a conducting medium in the presence of a dipole were obtained assuming a constant charge density d i s t r i b u t i o n . Because of t h i s s i m p l i f i c a t i o n i t was possible to obtain the eigenperiods without making any approximations. However, i n general, the assumption of a constant charge density w i l l not be the case. Under these circumstances one has to go back to equation (54) where p i s a function of the distance from the Earth's surface. Since equation (54) governs the o s c i l l a t i o n s along the l i n e s of force p w i l l be a function of both r Q and . For the calculations the -40--41-charge density d i s t r i b u t i o n which was proposed by Dessler (1958) i n his calculations of the propagation v e l o c i t y of sudden commencements was used. This d i s t r i b u t i o n i s shown i n f i g . 7. To obtain the eigenvalues u of equation (54) one has to resort to numerical methods. The d i f f e r e n t i a l equation i s replaced by f i n i t e differences and the problem i s then to f i n d the eigenvalues of a matrix. To this end one writes equation (54) in the form + G ( X , ^ ) V « . 0 (73) where the independent variable has been changed using the transformation (68) and where c ( X , ^ ) « / - y w ^ (74, As in the previous section one takes vanishing values of V z at the boundaries x Q and x^. In order to apply algebraic methods to the problem one divides the i n t e r v a l (x Q, x^) into n parts each of length -42-100 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 3 0 0 1 0 0 0 I IOC 1200 1300 1400 1500 2.5 3.5 4.5 5.5 6.5 7.5 « 5 0 5 IQ.5 H.5 " ' . 5 5 I4.S 15.5 x l O k m Charge Densily Distribution as function cf height above the Parth's surface -43-The l a s t form i s possible because of the symmetry about the point x = 0. The second derivative in equation (73) at points inside (x 0,xj) i s replaced by a difference expression r e s u l t i n g from a truncated power series, v i z . , oix2 * hX The d i f f e r e n t i a l equation can then be approximated by the following l i n e a r and homogeneous recursion r e l a t i o n , V ?Ca-h) - 2 V 2Ca) +Va(»+M + hG C«,w)\4C«)= 0 (76) . 4 i n which terms of the order h are ignored. This recursion formula has to hold at n-1 i n t e r i o r points r e s u l t i n g in n-1 simultaneous li n e a r equations. Starting at the left-hand end of the i n t e r v a l , the following system of equations results \ 4 ( 0 - 2 V,(x 0 4r, ) 4-V z(x 0f 2h) + h 26(x 0 +h }w) Vt(x0+h) - 0-V 2(x 0-rh)-2V zCx 0+2h)4-VHfX < 3f?h)+h 2C(x 0+2h )o5 2) Vi (v-2h) - 0 With the boundary conditions at x Q and x-^ i t follows that for a n o n - t r i v i a l solution of t h i s system of li n e a r equations -44-must have with A l 1 1 0 0 0 0 0 1 A 3 1 0 A 2 1 9 a 6 0 v n - l = 0 (77) Ay = n 2G CXo+y^ co2) - 2. which, because of the r e l a t i o n (74), becomes A 1 (78) This may be written in abbreviated form (79) a. are where because of the dependence on p(d~) , the functions of t¥ along a l i n e of force. Inserting the l a s t expression into equation (77) the problem can be considered to be an eigenvalue problem of the following kind -45-2-2 1 0 0 1 a2W2-2 1 0 • • 0 1 a„<j2-2 1 0 which i s i d e n t i c a l to the matrix problem where A and B are the matrices and -46-Thus i n matrix notation the problem may be written BX = oo2 A X (83) Since t h i s equation i s not i n the usual form suitable for diagonalizing a transformation has to be performed. As shown in appendix III equation (83) may be written My - (84) Because of the symmetry of M the eigenvalues are r e a l as would be expected from the nature of the problem. The next step i s to compute the values of a^ i n equation (79) which according to equation (78) w i l l depend on r Q , i . e . the l i n e of force, and on the charge density d i s t r i -bution which here appears as a function of X j = x Q + j h . However th i s d i s t r i b u t i o n as already mentioned i s given as a function of a l t i t u d e above the surface of the Earth. Since furthermore the a l t i t u d e of a p a r t i c u l a r point above the Earth's surface along a cer t a i n l i n e of force i s given by f-| = r s i n V - a (85) i t follows that for a given X j the corresponding value of s i n 2 d" has to be calculated. To accomplish this one returns -47-to equation (68) which for an a r b i t r a r y x may be written X = ' 35 ( Sf { S $ 3 + 6"S2 + S s + , 6 } < 8 6 ) where S = 5 m' Because of the symmetry about the point x = 0 , i . e . ^ = , equation (86) need be solved for one half of the i n t e r v a l only. Choosing the l e f t -hand half of the i n t e r v a l where x < 0 ( one obtains from equation (86) | x j ~^/,=^L(|-s) (£s+6s2+8s-r/6) (87) where | 5- j, -^ ( > n being an even number of in t e r v a l s . Since for each d i f f e r e n t l i n e of force |xQ| as well as h are d i f f e r e n t i t i s obvious that equation (87) has to be solved a great number of times. For this reason a pro-gramme was written for the Alwac III E using the Newton-Raphson i t e r a t i o n method for solving this equation with respect to s for a r b i t r a r y values of the left-hand s i d e s i . e . for values i n the range 0 < | X 0 J h< |x0|. A short des-c r i p t i o n of the method as applied to thi s p a r t i c u l a r problem i s given i n appendix IV. Table ( I l l - a ) gives the solutions of equation (87) for the pa r t i c u l a r value of |x0| - jh needed in our case. Since s = s i n 2 3 , the al t i t u d e H above the surface of the earth i s r e a d i l y calculated (Table I l l - b ) . -48-With these values the charge density may be determined. Once the r e l a t i o n between p (|x 0|-^)and p (r) has been found for each value of the argument, values of a^ may be i r e a d i l y obtained from the equation 1 . ( 8 8 ) where ju0 - 1.256 x 10" (volt-sec/amp.meter), M 2 = 65.028 x 1 0 3 0 (volt 2-sec 2.meter 2) and r Q = a / s i n 2 ^ In c a l c u l a t i n g the values of a^ one notes the symmetry of H (equation 85) about the center point For this reason p (l*0l w i H also be symmetric about the same point, which implies that the number of points for which p(\*ol~j'h) and therefore the expression (88) have to be calculated, i s approximately halved. Since i n the present case i t was found s u f f i c i e n t to subdivide the entire i n t e r v a l into eight parts, one has to compute values of aj at three points and the center point of the i n t e r v a l only. The r e s u l t i n g values of Aj are compiled in table 5 - from them one may calculate the elements of the matrix M as given in appendix II. These matrices which i n the present case are of order seven have been diagonalized with the help of the Alwac III E using the Jacobi method. The matrices as well as t h e i r eigenvalues are given i n appendix IV. The periods of the fundamental and the next higher mode are given i n table 6. -49-Table 5 & =15° A 1=l4.039xl0 5u 2-2 A 2=l4.039xl05y2_ 2 A3=14.03 9x10^ 2-2 A 4=14.039xl0 5w 2-2 fl> =20° A,=1.6230xl0 4u 2-2 A2=1.6230xl04w2-2 A3=1.6230xl04w2-2 A 4=1.6230xl0 4w 2-2 & =25° A7=6\ 0841X102W 2-2 Ao=5.66l6xl0 2w 2-2 A3=5.4926xlo2w2_2 A4=5.4926xl02u)2_2 & =30° A£=6". 5847x10^2-2 A 2=5.4395x10^ 2-2 A 3=5.1532xl0 1u 2-2 A4=5.0960x10^ 2-2 6^ =3 5 ° ApF.6500u 2-2 A2=6.0166tj2-2 A3=5.7000u2-2 A4=5.5733u2-2 & =40° A7=T.3394w2-2 A2=1.1878w2-2 A3=1.1373W2-2 A4=1.1221u2-2 B„ =45 A =50° x-1.2. Ai=3.7511x10" \t2-2 A 2=3.1081x10"^ 2-2 A 3=2.9473xlO _ 1(j 2-2 A4=2. 8937xl0 _ 1w 2-2 A 1=l. 5053x10 J-wjS-2 A 2=l. 1076x10-^ 2_ 2 A 3=l. 0082x10"^ 2-2 A 4=0.97985xl0" 1u 2-2 &a =55° A 1=12.005xl0 _ 2w 2-2 A2=5.5187xl0"2w2_2 A 3 =4.6 587xl0-2u 2-2 A4=4.4795x10" 2w2-2 d. =60° A"p9".3571xl0-1w2-2 A2=0.70985x10" Hi 2-< A 3=0.34202xl0 _ 1u2-2 A 4=0.30330xl0~ 1w 2-2 8. =65° A 1=5 o0936u 2-2 A2=5. 7303x10-^ 2-2 A 3 =2.1648x10"^ 2-2 A 4=1.6554xl0" 1w 2-2 A =70° A 1=l„ 4209X101CJ 2-2 A 2=0.31575X10 : LCJ 2-2 A3=0. 14472x10^ 2-2 A 4=0.113l4xl0 1w 2-2 & =75° A^T. 2196xl0iw 2-2 A2=0. 83446x10^ 2-2 A3=0. 53491xl0 1w 2-2 A 4=0.44933x10^ 2-2 & =80° AT=X6061u2_2 Aj=3.8719w2-2 A 3=4„2144W 2-2 A 4=4„2442w 2-2 -50-Table 6 <r0 H 1 2 15° 1.907xl0 4 sec. 0.9727xl0 4 sec. 20° 2.052xl0 3 1.046xl0 3 25° 3.804xl0 2 1.965xl0 2 30° 1.174xl0 2 6.184X10 1 35° 3.884X10 1 2.028X10 1 40° 1.735X10 1 9.049 45° 8.871 4.673 50° 5.254 2.837 55° 3.759 2.235 60° 6.513 5.370 65° 1.556X10 1 1.265X10 1 70° 2.908X10 1 2.188X10 1 75° 4.104x10* 2.440X101 80° 3.230X10 1 1.574X10 1 At this point the accuracy of the approximations should be considered - i n other words was the number of interv a l s great enough to y i e l d the lowest eigenvalue even i n the most unfavorable case with s u f f i c i e n t accuracy? With a fix e d - 5 1 -IOOOOQ Fig. 8. Fundamental period, variable charge den:, -y normal dipole field. -52-number of i n t e r v a l s the most unfavorable s i t u a t i o n occurs when the entire i n t e r v a l has i t s greatest value since then \h| has i t s greatest value too which means that the truncation error i s a maximum., To answer th i s question one goes back to the case of constant which permitted a solution of equation (54) i n closed form. In t h i s case the determinant (77) by virt u e of equation (78) can be written i n the form -2 1 0 0 1 f-2 1 0 0 1 ^2_ 2 1 = 0 (89) where l 2 ? X S / i r° P (J (90) It i s shown i n appendix X that y which has to s a t i s f y the determinantal equation (89) i s given by i-z+z^Wi ( 9 1 ) where M i s the order of the determinant and k = 1,2, •°°<' M. z Since i n our case M = 7 the following seven values of ^k -53-are obtained, v i z . , = 3.84776 JH = 2.00000 ^ = 3.41422 = 1.23464 ^ = 2.76536 = 0.58578 y,1 = 0.15224 The most unfavorable s i t u a t i o n occurs when ^ = 15°. In this case j = A r p P co2 = 14.03 9 x I0S co (92) ° M -19 ^ from table 5 where p was taken to be 6.5x10 (kg/m°). Hence for the lowest and next higher eigenvalue one obtains w x 2 = 10.84x10 8 (sec 2 ) and u 2 2 = 41.72x10 8 ( s e c ~ 2 ) On the other hand in the case of constant p , the lowest and next higher eigenvalue can be obtained from equation (71) which, with p = 6.5xlO" 1 9 (kg/m 3), yie l d s w x 2 - 10.98xlO~ 8 ( s e c - 2 ) and u 2 2 = 43.92 ( s e c - 2 ) . Thus the difference between the approximative solution and the exact solu t i o n i s within reasonable l i m i t s . This difference -54-becomes smaller as d'o increases since the number of interv a l s i s kept constant which implies that h becomes smaller. It therefore can be concluded that subdividing the entire i n t e r -val into eight parts yields s u f f i c i e n t accuracy. Using the values of ^ given by equation (91) and diagonalizing the matrices i n the case of constant p one can check the accuracy of the computer calculations by compar-ing the r e s u l t s obtained from combining equations (91) and (92) with those obtained from matrix diagonalization. This was possible only for the cases £f0 = 15° and ^ = 20° since then a. i s e s s e n t i a l l y constant. It was found that the agreement was very good i f the off-diagonal elements did not exceed 0.00005. 3. Solution for the compressed dipole f i e l d with constant charge density In the preceding two sections the eigenperiods of to r o i d a l o s c i l l a t i o n s were obtained under the assumption that the magnetic f i e l d was a dipole f i e l d . As pointed out e a r l i e r this i s not s t r i c t l y true. For this reason i n th i s section the eigenperiods of t o r o i d a l o s c i l l a t i o n s are studied under the assumption that the dipole f i e l d has been deformed, the extent of the deformation being governed by the radius of the cavity i n which the f i e l d i s confined. The d i f f e r e n t i a l equation which governs the motion under -55-these circumstances was derived i n chapter II-4 and i s dd + S,W4& (64) where V t i s obtained from the equation \i 3 . rL? A si^o m , 3 (61) Before computing the eigenvalues of equation (64) we w i l l obtain an approximate solu t i o n . As may be seen from the plots of the magnetic l i n e s of force in figures 2-4 i ) c very soon approaches a constant value for l i n e s of force inter-secting the Earth at very low co-latitutdes (small <^ ). The l i m i t i n g value taken in this case i s approximately the radius of the cavity oC . Therefore with = o( equation (64) becomes id-3 sin& d Vg o(4 4. A Y ^ ' v , = 0 (93) which may be written i n the form _ J _d_ cosec-d' d<& 5 Ho (94) In order to reduce equation (94) to a simpler form one may change, as has been done before, the independent variable by -56-putting cos ecfr d-d' = dx (95) Then equation (94) becomes i2u 2 8 2 d ^ + ^ ^ . r V/2 0 (96 ) d X 2 ^ 2 As before one assumes perfect reflection at the end points as the boundary conditions, viz.: V*=0 a t J , J Q The relation between J^ 0 and x D is found by integrating equation (95), i.e. dx ~ ^ cosect^ dJ' u where, as before, ^ = 180°- i¥0 . Because of symmetry this becomes or after integrating -57-AX = - 2 l o g tan (97 Since &0 < > A X w i l l be a posit i v e quantity as i t was i n equation (67). Therefore equation (97) associates the i n t e r v a l from -do to •J' with an i n t e r v a l extending from x Q to x^. By si m i l a r reasoning to that given i n section 1 i t follows that X0 = to3 tan - ^ 2 - (98) The solution of equation (96) i s V - A c o s ( * « * * y A A (99a) and * V -5 H . X ) (99b 3 H c where GJ) = CO v^o . Using the boundary conditions i t follows from these two equations that a « co vu0 . 7T - X X = n - I • ' 0 I ~ 3 H 0 2 0 — (100) -58-Since a 8 3 H, = 0 . 7 6 2 6 X 1 0 r %. m equation (100) can be written OJ - Y) TT I.S2 S 2 * 10 x. <*4 K |x 0| (101) where a i s the radius of the l i m i t i n g cavity expressed in units of Earth r a d i i and x Q i s given by equation (98). How-ever i n t h i s type of problem one i s more interested in the eigenperiod than i n the angular frequency. From equation (101) the period of the n-th o s c i l l a t i o n i s given by 7 - 3. OFQx 10 x cx x |X0| x \f p j- 5 e c - j (102) However i t should be kept i n mind that this formula applies only to those magnetic l i n e s of force which intersect the Earth at very low co-latitudes ( i¥0 ^ 20°) and i n addition the radius of the cavity should not exceed s i x Earth r a d i i i n order to guarantee that the assumption Vc = o<v = constant i s approximately s a t i s f i e d . Taking -J^ = 10°, yO - 6.5x10 K^jyn equation (102) yi e l d s T = 109 sec for cx-4 and T = 1748 sec fo rcx = 8. -59-In the majority of cases however i t w i l l not be permis-s i b l e to assume that i s constant. For these cases equation (64) has to be solved rigorously using l i n e s of force given by equation (61). In the f i r s t section as well as i n this section we changed the independent variable by putting In the f i r s t section, f C-fJ) =• CoseCi^ . It thus appears l o g i c a l to put i . e . Equation (64) then takes the simple form o (103) The solution of this equation i s (104a) -60-and a co (104b where, as before, co = co Vp . With the same boundary condi-tions as used before, one obtains from these equations -9 6.864- x l O x n r , ~' 4.576 x |0tfx |Xe| IX.I [sec ] (105) where ^ — = 2 . 2 S£x 1 0 From equation (105) the eigenperiods are r e a d i l y obtained, v i z . 7 , 9 . 1 5 2 x | 0 g x f x . / x f s e c ] ( 1 0 6 ) In p a r t i c u l a r the fundamental eigenperiod (n = 1) i s given by J ~- 9.15-2x10% |XD| *\fjjT [sec] (107) Comparing the equation for the eigenperiods of the o s c i l l a t i n g magnetic l i n e s of force i n the case of an -61-ordinary dipole f i e l d (equ. 72) with that of a compressed dipole f i e l d (equ. 107), one sees that they are of the same that both expressions would y i e l d the same r e s u l t i f the presence of the l i m i t i n g cavity does not influence the shape of the magnetic l i n e s of force too much. This would be the case for those l i n e s of force i n t e r s e c t i n g the Earth at high co-latitudes. In order to evaluate equation (106) i t i s necessary to f i n d the r e l a t i o n between and | XD| • In the present case the change of independent variable was accomplished by the transformation where V c has to be determined from the cubic equation form and would be i d e n t i c a l i f the factor \ ) * could be incorporated into jx | . For this reason we might expect dX = (108) (61) Since i n t e r v a l A X = X,- X 0 i s associated with the i n t e r v a l A3 = - <&Q , integration of equation (108) yields A X ' ft 3 * 0 ^ • » 2 f, » ^ <109' / ( « < + 2 ) sm^T / (oC+2 %3)sm<7 -62-The last expression is due to symmetry considerations. Thus As opposed to the integral which gave x Q in section 1 of this chapter, i t is not possible here to evaluate the integral (110) in terms of known functions. In this case one has to resort to numerical methods in order to obtain X 0 = X 0 (<X;d0 ) The computation of the integral was performed using Simpson's rule with an interval length of one degree. Since under these circumstances the integrand had to be evaluated o at many interior points of the given interval from <j£ to 90°, a computer programme for the ALWAC III E was written to let the machine take care of this laborious step. Appendix VI contains numerical values of the integrand. The evaluation of V c from equation (61) was carried out using the Newton-Raphson method taking as a f i r s t approximation for V c the radius of the limiting cavity. Table 7 gives the numerical values of the integral. These are plotted in Fig. 9 as a function of -cfc for cx - 12 <x = 8 and o< = 4. As pointed out already in the short discussion following the derivation of formula (106) for the eigenperiods, the quantities v/x |X0| in equation (72) and j X j in equation (106) correspond to one another. In order to study -63-t h i s r e l a t i o n s h i p more c l o s e l y values of the expression l * o I * ^ (i?-3o) were computed and the resu l t s tabulated i n table 8 and plotted i n F i g . 10. Table 7 •Jo x0(o<=12) x0(<x =8) x Q(cx=4) 10 57.31xl0 2 I4.53xl0 2 1.247xl02 20 9.827xl02 4.194xl02 0.5869xl02 30 10.07X101 7.656X101 2.398X101 40 15.01 13.92 8.317 50 3.641 3.550 2.864 60 1.235 1.222 1.109 70 0.5001 0.4975 0.4730 80° 0.1902 0.1896 0.1836 Table 8 A5» |Xo|< V01 (<*•'*) 10° 55.293xl0 4 56.873xl0 2 13.410X102 0.99584xl0 2 20° 24.411xl0 2 10.458xl0 2 4.5296xl0 2 0.60472xl0 2 30° 116.89 102.23 79.507 26.004 40° 15.534 15.074 14.102 8.8358 50° 3.6814 3.6479 3.5722 2.9830 60° 1.2408 1.2371 1.2278 1.1464 70° 0.50133 0.50075 0.49953 0.48795 80° 0.19048 0.19040 0.17909 0.17864 -66-We can now evaluate the expression (106) for the eigen-periods of to r o i d a l o s c i l l a t i o n s of a magnetic l i n e of force belonging to a compressed dipole f i e l d . Taking the value f> =6.S'x 10 J_ ^ 3 / m 3 J for the charge density of the conduct-ing medium the following fundamental periods are obtained. Table 9 <(% 10 20 30 40 50 60 70 80 T, (sec ) T, ( c< ^ 8) ( sec ) T, ( c< - 4 ) (sec ) 4228 725.0 74.30 11.07 2.686 0.9112 0.3690 0.1403 1072 309.4 56.48 10.27 2.619 0.9016 0.3670 0.1399 92.00 43.30 17.69 6.136 2.113 0.8182 0.3490 0.1355 The r e s u l t s of table 9 are plotted in F i g . 11 and show c l e a r l y the influence of the f i n i t e cavity. 4. Solution for the compressed dipole f i e l d with variable charge density In the preceding section the fundamental periods of to r o i d a l o s c i l l a t i o n s of a conducting medium i n the presence of a compressed dipole f i e l d were obtained assuming a constant -68-charge density d i s t r i b u t i o n . However, i n general, t h i s assump-ti o n w i l l not be v a l i d . In order to treat the case of a variable charge density i n the presence of a compressed dipole f i e l d one has to go back to equation (103) from section 3, v i z . where the independent variable x has been replaced by y. It was shown i n the l a s t section how the i n t e r v a l ( , ) i s transformed by means of the in t e g r a l (110) into an i n t e r v a l ( y Q,0). Furthermore the evaluation of this i n t e g r a l has shown that the variable y takes a far wider range of values than the corresponding variable x of the normal dipole f i e l d . For th i s reason the matrix method cannot be applied to equation (103) immediately unless one uses a very high order matrix (up to 1,000) which i s p r o h i b i t i v e . Since in general the length of the i n t e r v a l i s ir r e l e v a n t for the c a l c u l a t i o n of the eigenvalue i t seems appropriate to perform a second transformation which compresses the i n t e r v a l (y o,0) into (x o,0) where x G may be chosen a r b i t r a r i l y . However the success of such a transformation depends large l y upon the smooth behaviour of the charge density. For the new i n t e r v a l (x o,0) the matrix method already described can be applied without d i f f i c u l t y . - 6 9 -In order to i l l u s t r a t e the method i t w i l l be assumed that a knowledge of the charge density at 7 intermediate points i s s u f f i c i e n t to determine the fundamental eigenperiod of the o s c i l l a t i n g l i n e of force with reasonable accuracy. Since i t i has been shown that with a 7 order matrix and I h I < 0.\ I the lowest eigenvalue i s approximated with s u f f i c i e n t accuracy, h w i l l be taken to be 0.1 and thus x Q = 0.4. Since i n most cases l y 0 l "7 I X 0 1 i t seems fe a s i b l e to put ( i n ) where p i s a constant factor depending upon C X and ^ . p i s determined from the r e l a t i o n where y G i s found from table 7 (yo= xo)° Confining the ca l c u l a t i o n to the case where cx= 4 only, the following values of p are obtained : ; -70-Table 10 j%_ /?(CX'4) 10° 3.1170X102 20° 1.4672xl02 30° 5.9948X101 40° 20.792 50° 7.1595 60° 2.7733 70° 1.1826 Introducing y = (3x into equation (103 ) yields Jl^+/MfJ VE - 0 (112) where the independent variable x now extends from -x Q to 0 (because of symmetry about the o r i g i n ) . In order to apply the same numerical methods to equation (112) as those used i n section 2 one has to evaluate the expression z (113) Wo at the intermediate points \X0\ ~ subject to the same symmetry properties as mentioned i n section 2. In the -71-present case the intermediate points are X J =0.3; 0.2; 0.1; 0.0. Since p> i s given as a function of a l t i t u d e above the surface of the Earth ( f i g . 7) one has to associate each X j with y- by means of the r e l a t i o n (111) and then, by using the graph i n f i g . 9, f i n d the corresponding co-latitude. Having found the co-latitude ^ , the corresponding \) c of the compressed dipole f i e l d can be obtained from table Vl-a of the appendix. By evaluating [\~~ 6 . 3 7 0 ( V O C M < 1 1 4) the height of a pa r t i c u l a r point corresponding to xj can be found, and hence the expression (113) can be calculated. The r e s u l t i n g values of Aj are compiled i n table 11 - from them one may calculate the elements of the matrix M as given i n appendix II." As before the matrices are diagbnalized with the help of the AlwacIII E using the Jacobi method. The matrices as well as t h e i r eigenvalues are given i n appendix V I I v The periods of the fundamental modes are given i n table 12 and are'plotted in f i g . 11. -72-Table 11 A =io° A 1 =8.39570x10^ 2-2 A 2=5.44448x10^ 2-2 A 3 =4.3 5050xl0lw 2-2 A4=3.56181xl01<j2-2 <% =30° A 1=159.98xl0 1u 2-2 A 2=0.715206xl0 1w 2-2 A 3=0.331253x10^ 2-2 A4=0.188212xl0\/ 2-2 ^ = 5 0 ° A1=622.804w2_2 A2=15.5701w2-2 Ao=0.134225u2-2 A4=0„0566430u 2-2 <>0=20o A-,=34.9494x10^ 2-2 A2=1.94476xK4u2-2 A 3=1.35288xl0 1w 2-2 A4=0.92446 8xl0l<J2-<% =40° A x=151.693x10^ 2-2 A 2=0. 950914x10^ 2-A 3=0„ 0588661x10^2 A4=0.0301123X101<J2 <%=6Q° A1=l04.729w2-2 A2=13.2925u2-2 A3=0.584064u 2-2 A4=0.0604204u2-2 <%-70° Ax=7. 50754u2-2 A2=19.4097u2-2 A 3=3„66222u2_ 2 A4=2.92977w2-2 Table 12 T l (sec) 10° 111 20° 124 30° 252 40° 245 50° 159 60° 69 70° 46 -73-DISCUSSION OF THE RESULTS The aim of t h i s thesis was to calculate the eigenperiods of geomagnetic micropulsations and i n p a r t i c u l a r to i n v e s t i -gate their dependence on l a t i t u d e . The main reason for carrying out the study was the great discrepancy between the observed and calculated values e s p e c i a l l y when the points of observation are i n the polar regions. This discrepancy casts 5 much doubt on the model put forward by Dungey (1954b). For t h i s reason the p r i n c i p l e s of magnetohydrody namics were applied to a somewhat d i f f e r e n t model which, because of the greater s i m p l i c i t y of the r e s u l t i n g equations, promised an easier solution. From Maxwell's equations and the basic equation of hydrodynamics, two p a r t i a l d i f f e r e n t i a l equations were obtained i n c y l i n d r i c a l coordinates. The use of t h i s system of coordinates seems feasi b l e since the phenomenon of geo- j magnetic micropulsations appears to be confined to meridional planes.* Mathematically t h i s can be expressed by disregard-ing - the coupling terms contained i n both equations. Since this i s equivalent to considering the problem i n a plane i t should i n p r i n c i p l e make no difference whether c y l i n d r i c a l * The question of the dependence of micropulsations on G.M.Ti or L.M.T. i s s t i l l rather open. -74-or spherical coordinates are used. On account of c y l i n d r i c a l symmetry solutions are obtained more e a s i l y . The eigenperiods of the t o r o i d a l o s c i l l a t i o n s were obtained as a function of co-latitude f i r s t assuming a medium of constant charge density in a normal dipole f i e l d . As can be seen from f i g . 6 the eigenperiods tend towards i n f i n i t y as zero co-latitude i s approached, a r e s u l t which has not been observed. The r e s u l t i n g high values are e n t i r e l y due to the fact that the l i n e s of force i n the case of a normal dipole extend very far into outer space. The eigenperiods calculated by Dungey using spherical coordinates agree very well with those obtained i n t h i s thesis which j u s t i f i e s the use of c y l i n d r i c a l coordinates i n describing the problem. Figure 6 also shows that the eigenperiods at high co-latitudes tend to zero. Since t o r o i d a l o s c i l l a t i o n s can be understood as o s c i l l a t i n g l i n e s of force i t follows that the period must tend to zero as the length of the l i n e of force decreases as i t does i n t h i s model at higher co-latitudes. This i s the, p r i n c i p a l d i f f i c u l t y of t h i s model whether spherical or t c y l i n d r i c a l coordinates are used. Since the charge density d i s t r i b u t i o n of the medium above the surface of the Earth i s probably not constant the above model was modified using a variable charge density as proposed by Dessler (1958). A comparison of figures 6 and 8 shows the e f f e c t of a variable charge density on the -75-fundamental\ period. The values on the l e f t of f i g . 6 and f i g . 8 agree exactly since the major part of the magnetic l i n e s of force run through regions i n which the charge density can be considered to be constant and equal to that used i n the previous c a l c u l a t i o n . At higher co-latitudes, the magnetic l i n e s of force spend longer i n regions of lower a l t i t u d e i n which the charge density i s higher than that further out. This r e s u l t s i n a somewhat slower drop of the calculated eigen-periods with increasing co-latitude. Above co-latitude 55° the periods increase, the increase i n charge density completely compensating for the shortening of the l i n e s of force. So far a l l the r e s u l t s were obtained on the assumption that the Earth's magnetic f i e l d i s that of a geocentric dipole. Recent investigations (Parker, 1958) however have shown that t h i s i s not so. One of the more important features of the f i e l d i s that i t does not extend as far into outer space as, was o r i g i n a l l y thought. To a f i r s t approximation on the daylight side of the Earth i t appears to be confined to a cavity of variable radius (4-12 Earth r a d i i ) . This may be attributed to the "solar wind" which compresses the f i e l d on the daylight side of the Earth. Since the form of the f i e l d has an important e f f e c t on the eigenperiods of geomagnetic micropulsations a compressed dipole f i e l d was substituted in t o the t o r o i d a l equation and the eigenperiods computed. | The r e s u l t s of this c a l c u l a t i o n have been plotted i n f i g . 12. -76-One of the important features i s that the large values of the eigenperiods have vanished, the eigenperiods assuming f i n i t e values as the co-latitude approaches zero. To obtain the l i m i t one cannot use equation (98). For t h i s purpose however X o i n f i g . 9 as computed from i n t e g r a l (110) i s nearly equal to / X Q | x "i)o c 4 i n f i g . 10 where | X 0 | i s to be taken from equation (68) on putting 0 . Using the l i m i t i n g value for | X 0 | at 0 and the fact that \ ) o c tends to o< , the radius of the cavity., the eigenperiod of the fundamental mode takes the l i m i t i n g value 8 4 I— - 4 J 8 3 4 x |0 xcrf x Vyo [>ecj which, with o( - and p ** &.5x/0 gives the value 86 seconds. The small disagreement between the above value and T, for c*=4 i n table 9 i s due to the use of the approximation rather than the i n t e g r a l . As one would expect the radius of the cavity influences the size of the eigenperiods considerably, i n p a r t i c u l a r those which are associated with l i n e s of force in t e r s e c t i n g the Earth's surface at low co-latitudes. As higher co-latitudes are approached the eigenperiods corres-ponding to d i f f e r e n t values of CX almost coincide with those obtained i n the case of a normal dipole f i e l d . i Phy s i c a l l y this can be understood from figures 2, 3 and 4 . The l i n e s of force which intersect the Earth's surface at -77-high co-latitudes ( > + 5 " ° - 50°) hardly suffer any deforma-tion even for o< = 4- and hence have the same shape as those of a normal dipole with the r e s u l t that the eigenperiods are equal to those of a normal dipole. As a consequence the eigenperiods tend to zero as the equator i s approached, a fact which i s i n disagreement with the observed data. F i n a l l y i t seemed promising to calculate the eigenperiods for a compressed f i e l d assuming a variable charge density d i s -t r i b u t i o n . In that way one could expect to incorporate both the advantages of model 2 and model 3 (sections 2 and 3 of chapter III) into one model which would remove the trend to i n f i n i t y at low co-latitudes and at the same time r a i s e the small values of the eigenvalues at high co-latitudes. The ca l c u l a t i o n was carr i e d out for the case cx = 4 and the re s u l t indicated i n f i g . 11. The r e s u l t as i t stands i s not too s a t i s f a c t o r y . One would have expected from simple con-siderations that at low co-latitudes the curve would approxi-mately follow that obtained for the compressed dipole f i e l d with constant charge density while at high co-latitudes the behaviour of the curve would be s i m i l a r to that of a normal dipole with variable charge density d i s t r i b u t i o n . Instead the curve r i s e s s l i g h t l y with increasing co-latitude (up to about 30°) and there decreases continuously. Before drawing any conclusions from t h i s r e s u l t one must consider the following point. As already mentioned the i n t e r v a l over -78-which the d i f f e r e n t i a l equation had to be integrated was quite large a d i f f i c u l t y which was overcome by introducing a second variable. Subdividing the i n t e r v a l into 8 parts and associat-ing the equally spaced points with points along the li n e s of force, those points near the ends of the l i n e s of force are favored. Since these points are situated i n that part of the medium having the higher charge density an incorrect picture may e a s i l y be obtained. In order to avoid t h i s the i n t e r v a l should be subdivided into more parts i n order to obtain a better representation with respect to the charge density d i s -t r i b u t i o n along a l i n e of force. Since for an i n t e r v a l of length 0.1 a seven order matrix gives the lowest eigenvalue with s u f f i c i e n t accuracy i n the case of constant charge density one can be sure that any change i n the eigenperiod with increase i n the number of interv a l s i s e n t i r e l y due to the increased information on the charge density along the ( l i n e s of force. The number of points which would be s u f f i c i e n t could only be found by r e f l e c t i n g the c a l c u l a t i o n using 16 interva l s instead of 8 as i n the present case. Before actually performing the c a l c u l a t i o n i t would be advisable to supplement table 7 by c a l c u l a t i n g the values of x Q for <t- 15° 2 5° 7 5 ° That would allow higher accuracy i n the inverse interpolation for determining the value of d' which belongs to a certa i n value of x. In view of the d i f f i c u l t i e s involved i n carrying out the -79-improvement described above i t seems fe a s i b l e to consider a d i f f e r e n t method of approach. Assuming that the disturbance travels along a l i n e of force the time required to t r a v e l from one end point to the other i s given by where V i s the Alfven v e l o c i t y given by v - H° fa \ mU This i s the formulation which has been used by Jacobs and Obayashi (1958) to f i n d the charge density from a knowledge of the period of micropulsations. The integration has to be c a r r i e d out along the l i n e of force using the compressed dipole f i e l d . The d i f f i c u l t y l i e s i n the evaluation of the i n t e g r a l and arises mainly from the fact that the equation of a l i n e of force i s a cubic equation which makes the l i n e element ds more complicated. Using modern computing devices however the problem should be tractable. So far the discussion has been concerned only with the r e s u l t s of the c a l c u l a t i o n s . Since i t was our aim to determine a model which would explain the phenomenon of micropulsations i t i s necessary to consider the experimental data which have been obtained. The usual procedure to obtain -80-/ l Va: 4 'iable chaj -ge density © t Exper im< ;ntal valu< !S > ©® \. oc - 4-\ Cons' ;ant charge i density Referent :es • Kato & £ laita Q95< 1) Campbell . (1959) Maple (] Ber thole C 1M 1 -3- i-k .909 ) I (1960) ocnoiic Duffus 8 & vexuKamj c Shand (1£ > v.J.yoo / >58) 0.1-F i g . 12 Comparison of experimental and th e o r e t i c a l values, -81-the dependence of the eigenperiods on the co-latitude i s to take those eigenperiods which exhibit a maximum frequency of occurrence at a ce r t a i n l a t i t u d e . Only data which have been observed simultaneously at stations l y i n g on the same meridian should be used. In spi t e of the greatly increased e f f o r t s which have been put into research on micropulsations, data s a t i s f y i n g these requirements are very d i f f i c u l t to obtain. For thi s reason i t i s necessary to resort to data obtained under less rigorous conditions. In f i g . 12 experimental values of periods are plotted against geomagnetic co- l a t i t u d e . The tendency of the eigenperiods to decrease with increasing co-latitude i s c l e a r l y apparent. For comparison some of the calculated eigenperiods are plotted i n the same diagram. The agreement between the observed and calculated values i s not too s a t i s f a c t o r y either for the compressed dipole f i e l d with constant charge density, or for the compressed f i e l d with variable charge density. In the f i r s t case the magnitude as well as the slope do not agree; the disagreement i n the second case i s only with respect to magnitude. Apart from the fact that the model may not be appropriate the discrepancy might be due to an incorrect charge density d i s t r i b u t i o n or to the wrong choice of oc. Since the value of CX i s not c r i t i c a l i n the range of co-lat i t u d e s under discussion (see f i g . 10) the major cause for the discrepancy has to be sought i n the charge density -82-d i s t r i b u t i o n . From single measurements i n low co-latitudes which y i e l d eigenperiods of the order of 100 seconds, i t appears almost cert a i n from our model that during disturbed conditions must be around 4 i n agreement with other c a l c u l a t i o n s . The th e o r e t i c a l explanation of geomagnetic micropulsations i n terms of t o r o i d a l o s c i l l a t i o n s of a compressed dipole f i e l d i s promising and mathematically t-ractabie i f c y l i n d r i c a l coordinates are used. Further studies on the subject using the methods described i n th i s thesis are strongly recommended. -83--.X-r APPENDIX I "'" The f i e l d i n t e n s i t y of a magnetic dipole can be derived from the potential V = - : — — — 2 ; — coiJ ( i - i ) where i s the co-latitude and M the magnetic moment of the dipole. In order to allow for the influence of the solar wind, one modifies equation (1-1) according to Obayashi (I960) superimposing a potential associated with a constant f i e l d . Hence one writes (1-2) where the constant A i s chosen so that the r-component of the magnetic f i e l d vanishes at r = R G (compressed f i e l d ) . Therefore, since one takes (1-3) -84-With th i s value of A, the r - and -components of the com-pressed dipole f i e l d are and The l i n e s of force are obtained by integrating the d i f f e r e n t i a l o equation dr _ rdd' which upon introducing the equations (1-4) and (1-5) becomes l .e (1-6) Integrating this equation yi e l d s = 2Ln sir>& f C -85-l . e . <-7R = C sin d- (1-7) To determine the constant C i n equation (1-7) we postulate that the l i n e of force intersects the surface of the Earth (r = a) at & - J^, . r CL Thus L and equation (1-7) becomes -86-APPENDIX II -1/2 -1/2 Letting X = A Y, where A i s defined l a t e r , equation (83) becomes BA" 1 / 2Y = U 2 A A " 1 / 2 Y Multiplying this from the left-hand side one obtains -1/2 -1/2 2 - 1 / 2 - 1 / 2 2 A BA Y = u*A AA Y = w Y or MY = u Y where one has abbreviated -1/2 -1/2 M = A BA With /foi" 0 0 0 0 - '/^ 0 0 - A A = -87-the r e a l symmetric matrix M becomes -88-APPENDIX III In order to apply the Newton-Raphson method, equation (87) has to be written i n the form ij f CS j = - L . ( | - s ) 1 ( 5 s 3 4 - 6 s 2 + S s +16) -(\Xo\-jh)= 0 ( I I I - D Taking the derivative of f ( s ) with respect to s yiel d s ^ - f o i ( I I I - 2 ) If therefore an approximative solution s^ of equation ( I I I - l ) i s known an improved solution s i + - ^ i s obtained from s'«a v ffS (iii"3) As may be seen from equation (III-2) f'(s) w i l l never vanish in the i n t e r v a l 0 < S < I and hence the method may be used quite s a f e l y to obtain a solution of any desired accuracy. Leaving out a l l d e t a i l s concerned with sca l i n g , etc. the "flow chart" looks as follows -89-~ l 5Sj+6 I ~r.... (5S,+<5)S,-(5S/+0 + S Ccmp. L ^ (*V-1 <*"» [(5S(-46)S/+8]Sj+l6 i/ Ccmp. ^ /35ci-s)A{r(ss,+^ s,-+as,-+i6} (/ Coma fe5gi-S/r/K5st^)s,-»g3Si.-n^} C C T . p . Accuracy Test J -90-The necessary s t a r t i n g values s^ were obtained from the graph i n f i g . 5. For speeding up the input both l * 0 l " ^ n and S i were punched out on tape. The following table contains the solutions of equation (87) which were required for the computations of the matrix elements. The pairs of values are arranged i n groups each corresponding to a ce r t a i n co-latitude of inte r s e c t i o n . -91-Table I l l - a =0.342855 =0.228570 =0.114285 X =20c k i - * i . =' =25c 0.342839 0.220559 0.114280 =0.342754 =0. 228503 =0.114251 <% =30c 0.342447 0.228298 0.114149 ^ = 3 5 ° 0.341575 0.227717 0.113858 J 0 =40° IXj-jji, : =0.339506 =0.226337 =0.113169 =0.335244 =0.223496 =0.111748 s=0.83872 =0.94129 =0.98658 s=0.83874 =0.94130 =0.98659 s=0.83886 =0.94133 =0.98659 s=0.83927 ; =0.94145; =0.98662 ; s=0.84o45 ; =0.94178 ; =0.98669 ; s=0.84321 ; =0.94258 ; =0.98685 ; s=0.84873 J =0.94419 ; =0.98719 ; <% =50c 0.327425 0.218283 0.109142 s=0.85836 =0.94706 =0.98779 ki- 3Jl =0.314405 =0.209603 =0.104802 8=0.87306 =0.95163 =0.98877 =60° k i =0.294476 =0.196317 =0.098158 s=0.89284 =0.95811 =0.99017 <% | X 0 | -=65° =0.266161 =0.177441 =0.088720 =0.91639 =0.96633 =0.99200 =70° =0.228556 =0.152371 =0.076185 s=0.94130 =0.97563 =0.99413 A, =75° k l =0.181626 =0.121084 =0.060542 s=0.96460 =0.98489 =0.99631 =80° lx0| =0.126380 =0.084254 =0.042127 s=0.98349 =0.99280 =0.99822 -92-Table I l l - b A =15< <£=50 c s=0.83872 =0.94129 =0.98658 =1.0000 <^=20c s=0.83874 =0.94130 =0.98659 =1.0000 <% =25c s=0.83886 =0.94133 =0.98659 =1.0000 <% =30c s=0.83927 =0.94145 =0.98662 =1.0000 -^-35° s=0.84045 =0.94178 =0.98669 =1.0000 ^ = 4 0 ° s=0.84321 =0.94258 =0.98685 =1.0000 t%=45 c s=0.84873 =0.94419 =0.98719 =1.0000 H=82259km =92319 =96761 =98077 H =40736km =46496 =49040 =49793 H=24486km =28255 =29921 =30414 H=15686km =18371 =19558 =19910 H=104l4km =12437 =13334 =13600 H=7039km =8618 =9322 =9531 H=4782km =6037 =6602 =6770 s=0.85836 =0.94706 =0.98779 =1.0000 & =55° s=0. =0. =0. =1. & =60° s=0. =0. =0. =1. 87306 95163 98877 0000 89284 95811 99017 0000 ^ 0=65° s=0. =0. =0. =1. 91639 96633 99200 0000 <% =70° s=0. =0. =0. =1. 94130 97563 99413 0000 s=0. =0, =0, =1, <% =80° s=0. =0. =0. =1. ,96460 ,98489 ,99631 ,0000 98349 99280 99822 0000 H=3240km =4233 =4689 =4826 H=2178km =2948 =3311 =3421 H=l451km =2023 =2304 =2390 H=959km =1359 =1564 =1628 H=633km =889 =1026 =1070 H=4 22km =565 =646 =672 H=292km =355 =392 =404 -93-APPENDIX IV As can be seen from table 5 a common factor i n the aj which i s of course d i f f e r e n t for the d i f f e r e n t l i n e s of force appears quite frequently. For th i s reason i t seems fe a s i b l e o to incorporate t h i s factor into the value of u . One then has the following matrices and the i r eigenvalues: J;=i5° 1.42460 -0.71230 0 0 0 0 0 -0.71230 1.42460 -0.71230 0 0 0 0 0 -0.71230 1.42460 -0.71230 0 0 0 0 0 -0.71230 1.42460 -0.71230 0 0 0 0 0 -0.71230 1.42460 -0.71230 0 0 0 0 0 -0.71230 1.42460 -0.71230 0 0 0 0 0 -0.71230 1.42460 Eigenvalues Scaling: 106w 2=X2 A.^2=0.10844 ; X22=0.41725 5 \32=0.87944 <£=20c 1.232286 -0.616143 0 0 0 0 0 •0.616143 1232286 -0.616143 0 0 0 0 0 -0.616143 1232286 -0.616143 0 0 0 0 0 -0.616143 1232286 -0.616143 0 0 0 0 0 -0.616143 1232286 -0.616143 0 0 0 0 0 -0.616143 1232286 -0.616143 0 0 0 0 0 -0.616143 1232286 Eigenvalues Scaling: 104u2=X2 2 2 2 X! =0.093802 ; X2 =0.360924 j X 3 =0.760715 -94-<X=25° 0.657451 -0.340771 0 0 0 0 0 -0.340771 0.706514 -0.358650 0 0 0 0 -0.358650 0.728252 -0.364128 0 0 0 0 0 -0.364128 0.728252 -0.364128 0 0 0 0 0 -0.364128 0.728252 -0.358650 0 0 0 0 0 -0.358650 0.706514 -0.340771 0 0 0 0 0 -0.340771 0.657451 Eigenvalues Scaling; 2xl0 2w 2=X 2 A.!2=0.054579 ; A22=0.204527 ; A32=0.426525 <#=30° 0 -o ,607469 ,334182 O 0 0 O O -0.334182 0.735362 -0.377756 0 O 0 O 0 0.377756 0.776217 390280 0 0 0 -0 O O -0.390280 0.784929 -0.390280 0 0 0 0 O -0.390280 0.776217 -0.377756 O 0 0 O 0 -0.377756 0,735362 -0 -0.334182 O 0 0 O 0 0 ,334182 ,607469 Scaling: 2xl0 1u 2=A. 2 Eigenvalues: A-x^O. 057250 j A.2^ =0. 206443 J A.3*=0.422851 A =35° • 0.601504 -0.316186 0 O 0 0 0 -0.316186 0.664827 -0.341521 O 0 0 O O -0.341521 0.701754 -0.354844 0 0 0 O 0 -0.354844 0.717708 -0.354844 0 O 0 0 0 -0.354844 0.701754 -0.341521 0 0 0 0 O -0.341521 0.664827 -0.316186 0 0 O O 0 -0.316186 0.601504 Eigenvalues Scaling: 2w2=A.2 1^2=0.052352 j A22=0.191944 » 132=0.401217 -95-<%=40° 1.493206 -0.792820 -0.792820 0 0 0 0 0 1.683785 -0.860385 0 0 0 0 0 0 -0.860385 0 1.758551 -0.885214 -0.885214 1.782372 0 -0.885214 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.885214 0 0 1.758551 -0.860385 0 -0.860385 1.683785 -0.792820 0 -0.792820 1.493206 Scaling: w2=A2 Eigenvalues: A^^O. 131089 j A.22=0.482142 ; A.3"=l. 00078 <%=45° 0.533177 -0.292869 0 -0.292869 0.643480 -0.330400 0 -0.330400 0.678587 0 0 -0.342421 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.342421 0 0 O 0.691157 -0.342421 O 0 -0.342421 0.678587 -0.330400 0 0 -0.330400 0.643480 -0.292869 0 0 -0.292869 0.533177 Scaling: l O " 1 ^ 2 ^ 2 Eigenvalues: A.^0.050164 ; A.22=0.180788 j X3"=0.371086 £,=50' 1.3281639 -0.774455 -0.774455 O 0 0 0 0 1.805706 -0.946316 0 O 0 O 0 0 0 -0.946316 0 O 1.983733 -1.006113 0 -1.006113 2.041129 -1.006113 0 0 0 0 0 0 O 0 0 0 -1.006113 1.983733 -0.946316 0 O -0.946316 1.805706 -0.774455 O 0 -0.774455 1.328639 Eigenvalues: Scaling: lO _ 1w 2=X 2 X^O. 143013 I \ 2 2=0.490445 .} A. 2=0.990559 -96-X =55c Oo333194 -0.245714 0 0 0 0 0 -0.245714 0.724808 -0.394438 0 0 0 0 0 -0.394438 0.858609 -0.437807 0 0 0 0 0 -0.437807 0.892957 -0.437807 0 0 O 0 0 -0.437807 0.858609 -0.394438 O 0 0 0 0 -0.394438 0.724808 -0.245714 0 0 0 0 0 -0.245714 0.333194 Scaling; 2xlO~2w2=A.2 Eigenvalues: Xl =0-055870 A22=0.158036 5 X 3 =0.303705 ^=60 c 0.427483 -0.776024 -0.776024 5.63499 0 -4.059018 0 0 O O 0 O 0 0 0 -4.059018 11.69522 -6.209676 0 0 O -6 13 -6 0 0 209676 18826 209676 0 0 -6 11 -4 0 0 0 209676 69522 059018 0 0 0 0 0 -4.059018 5.63499 -0.776024 0 0 0 0 0 -0.776024 0.427483 Scaling: 2xlO - 1w 2=A 2 Eigenvalues: A.^2=0.18612 ; Xg2=0.27378 § A.32=0.153292 t£=65 c 0.0392650 -0.0585326 O 0 -0.0585326 0.349022 -0.283924 0 O -0.283924 0.923873 -0.528251 0 0 -0.528251 1 6 O O 0 0 0 O 0 0 0 0 0 0 208170 -0.528251 -0.528251 0.923873 -0.283924 O -0.283924 0.349022 0 O 0 O O -0.0585326 0 O -0.0585326 0.0392650 Scaling: 10~\*2=A.2 Eigenvalues: Xx2=0.016305 ; A.22=0.024666 ; X32=0.118095 -97-^ = 7 0 ° 1,40756 -1,49296 0 0 0 0 0 ,49296 ,33412 ,67806 0 0 0 0 0 -4,67806 13,8198 -7,81500 0 0 0 0 0 -7.81500 17.6772 -7.81500 0 0 0 0 0 -7.81500 13.8198 -4.67806 0 0 0 0 0 -4.67806 6.33412 -1.49296 0 0 0 0 0 •1.49296 1.40756 Scaling: 101y2=A.2 Eigenvalues: A.12=0.4669 ; A.22=0.8250 I A.32=2.3658 <&=75° 1.63988 -0.991257 0 -0,991257 2.39676 -1.49678 0 -1.49678 3.73895 0 0 -2.03975 0 0 0 0 0 0 0 0 0 -2 4 -2 O o ,03975 ,45107 ,03975 0 0 0 O 0 ,03975 ,73895 -1.49678 0 -2, 3 t 0 0 0 0 -1.49678 2.39676 -0.991257 O O 0 0 0 -0.991257 1.63988 1 2 2 Scaling: 10 y =X Eigenvalues: A-i2=0. 234426 ; A.22=0.662810 J X 2 = l . 45850 ^=80 ( 0.767430 -0.314805 -0.314805 0 O O 0 O 0 -O, 516542 -0.247554 247554 0.474563 0 O O 0 -0 236447 O 0 0 0 O -0.236447 0.471231 -0.236447 0 0 0 0 0 -0.236447 0.474563 -0.247554 0 O 0 0 0 -0.247554 0.516542 -0.314805 0 0 0 O 0 -0.314805 0.767430 Scaling: w 2=X 2 Eigenvalues : A.^0.037834 j \2*=0« 159243 j A.32=0.352488 -98-APPENDIX V Putting Y - Z= X the determinant i n (89) becomes x 1 0 0 1 x 1 0 O l x l = O This determinant i s of the n* n order and can be expanded to obtain a recursion formula, viz„ D n = x D n - l - Dn-2 (V-l) where D_n= 0, D Q = 1, = x 0 In order to f i n d a solut i o n of equation ( V - l ) , one writes (V-2) Substitution then y i e l d s , after c a n c e l l i n g the terms on both sides, -99-n n-i n-2 U - X U. - U (V-3) Therefore for n = 2 u 2 = xu - 1 or x z. D X = u + 1/u = u 1 ( l + u 2 ) (V-4) From th i s i t follows from equation ( V - l ) , that D 2 = x D x - l = u~ 2(l+u 2+u 4) Therefore by induction -n ,, 2 4 2n D n = u (l+u^+u*+ •••• u ) (V-5) As required t h i s expression w i l l vanish i f the numerator i s -100-zero without the denominator being zero at the same time. Hence the solu t i o n i s NX. = e I 2 ; — M , (v-7) Substituting (V-7) into equation (V-4) one obtains or since X =- J - 2 1 2. jrK (V-8) As. an i l l u s t r a t i v e example l e t us take the simple case given by i r 2 0 Expanding y i e l d s the equation fjr - 0 = 0 -101-l 1 which has the solutions fy - 3 and =' • ^ n other hand from equation (V-8) with M = 2 for the order of the determinant one obtains z ft = 2 + 2 cooz"/3 I -102-vAPPENDIX VI To evaluate the i n t e g r a l (110) using Simpson's rule i t i s necessary to compute values of the integrand at equally spaced points depending on the length of the i n t e r v a l A & . Since the integrand contains powers of being the solutions of the cubic equation (61) i t was e s s e n t i a l to incorporate a subroutine into the program which by means of the Newton-Raphson method solved equation (61) whenever necessary. The program was written i n such a way that after inputting <J^ , A and C< the values of ~S)C and those of the integrand (I) were computed and the i r numerical values printed out. The advancing from was done automatically. The r e s u l t s of the computation for d i f f e r e n t l i n e s of force are given i n table Vl-a. -103-Table Vl-a 0 I r cx = V I 10 1. 0000 5. 7521 1. 0000 5. 7364 1. 0000 5. 5843 n 1. 2069 11. 0967 1. 2056 10. 9978 1. 1940 10. 1121 12 1. 4320 20. 1544 1. 4282 19. 7855 1. 3946 16. 7721 13 1. 6746 34. 7677 1. 6663 33. 6590 1. 5964 25. 6174 14 1. 9339 57. 3412 1. 9179 54. 4304 1. 7939 36. 2638 15 2. 2090 90. 8584 2. 1808 83. 9858 1. 9822 47. 9687 16 2. 4983 138. 8311 2. 4519 123. 9768 2. 1578 59. 8586 17 2. 8004 205. 1456 2. 7278 175. 4592 2. 3188 71. 1593 18 3. 1133 293. 7638 3. 0049 238. 5493 2. 4646 81. 3427 19 3. 4347 408. 3312 3. 2794 312. 2418 2. 5954 90. 1328 20 3. 7620 551. 6578 3. 5479 394. 4669 2. 7123 97. 4616 21 4. 0925 725. 2407 3. 8074 482. 3869 2. 8164 103. 3907 22 4. 4233 928. 8699 4. 0555 572.8415 2. 9091 108. 0535 23 4. 7515 1160. 4488 4. 2905 662. 7687 2. 9917 111. 6149 24 5, 0744 1416. 0477 4. 5113 749. 5470 3. 0653 114. 2377 25 5. 3896 1690. 3194 4. 7177 831. 1975 3. 1311 116. 0797 26 5. 6951 1977. 0020 4. 9096 906. 3749 3. 1901 117. 2750 27 5. 9890 2269. 5168 5. 0874 974. 3327 3. 2431 117. 9420 28 6. 2702 2561. 5905 5. 2517 1034. 8203 3. 2908 118. 1819 29 6. 5379 2847. 6279 5. 4034 1087. 9201 3. 3339 118. 0798 30 6. 7917 3123. 0187 5. 5434 1133. 9743 3. 3729 117. 7023 31 7. 0314 3384. 2850 5. 6723 1173. 4612 3. 4083 117. 1082 32 7, 2571 3628. 9160 5. 7912 1206. 9340 3. 4405 116. 3423 33 7. 4693 3855. 4458 5. 9009 1234. 9973 3.4699 115. 4458 34 7. 6684 4063. 1582 6. 0022 1258.2190 3. 4968 114. 4475 35 7. 8549 4252.0059 6. 0957 1277. 1676 3. 5214 113. 3745 36 8. 0296 4422. 4043 6. 1822 1292. 3617 3. 5440 112. 2483 37 8.1931 4575. 082 6. 2623 1304. 2547 3. 5648 111. 0844 38 8. 3460 4711. 0649 6. 3365 1313. 2942 3. 5840 109. 8966 39 8. 4891 4831.4375 6. 4053 1319. 8595 3. 6017 108. 6974 40 8. 6230 4937. 3962 6. 4692 1324. 2966 3. 6181 107. 4964 41 8. 7484 5030. 1391 6. 5287 1326. 8861 3. 6333 106. 2999 42 8. 8657 5110.8491 6. 5840 1327. 9041 3. 6474 105. 1135 43 8. 9756 5180. 6476 6. 6356 1327. 5862 3. 6605 103. 9431 44 9. 0786 5240. 5982 6. 6837 1326. 1190 3. 6728 102. 7932 45 9. 1752 5291. 7328 6. 7287 1323. 6923 3. 6841 101. 6644 46 9. 2658 5334. 8917 6. 7707 1320. 4662 3. 6947 100. 5600 47 9. 3508 5371. 0117 6. 8100 1316. 5588 3. 7046 99. 4820 48 9. 4307 5400.8423 6. 8468 1312. 0941 3. 7139 98. 4321 49 9. 5057 5425. 1042 6. 8813 1307. 1784 3. 7226 97. 4111 50 9. 5763 5444. 3859 6. 9137 1301. 9001 3. 7307 96. 4195 -104-I t I I \ T 51 9.6426 5459. 3161 6. 9440 1296. 3296 3 .7383 95. 4581 52 9.7051 5470.4115 6. 9726 1290. 5468 3 .7454 94. 5266 53 9.7639 5478. 1292 6. 9994 1284. 6012 3 .7522 93. 6262 54 9.8192 5482. 8620 7. 0246 1278. 5458 3 .7585 92. 7557 55 9.8714 5485. 0732 7. 0483 1272. 4262 3 .7644 91. 9152 56 9.9205 5485. 0166 7. 0706 1266. 2790 3 .7700 91. 1055 57 9.9668 5483. 0422 7. 0916 1260.1454 3 .7752 90.3244 58 10.0105 5479. 4430 7. 1114 1254. 0516 3 .7801 89. 5726 59 10.0517 5474. 4204 7. 1300 1248. 0180 3 .7848 88. 8499 60 10.0905 5468. 2313 7. 1476 1242. 0700 3 .7891 88.1546 61 10.1270 5461. 0382 7. 1641 1236. 2260 3 .7932 87. 4896 62 10.1615 5453. 0566 7. 1797 1230. 5043 3 .7971 86. 8504 63 10.1940 5444. 4419 7. 1944 1224. 9273 3 .8008 86. 2390 64 10.2246 5435.3191 7. 2081 1219. 4890 3 .8042 85. 6543 65 10.2534 5425. 8129 7. 2211 1214. 2064 3 . 8074 85. 0962 66 10.2805 5416. 0598 7. 2333 1209. 0962 3 .8104 84. 5626 67 10.3059 5406. 1360 7. 2448 1204. 1642 3 .8133 84. 0552 68 10.3299 5396. 1925 7. 2556 1199. 4082 3 .8160 83. 5731 69 10.3523 5386. 2273 7. 2657 1194. 8457 3 .8185 83. 1147 70 10.3734 5376. 3632 7. 2751 1190.4759 3 . 8208 82. 6811 71 10.3931 5366. 6736 7. 2840 1186. 3048 3 .8230 82. 2706 72 10.4115 5357. 1850 7. 2922 1182. 3376 3 .8251 81. 8831 73 10.4286 5347. 9847 7. 2999 1178.5742 3 .8270 81. 5194 74 10.4445 5339.0997 7. 3071 1175. 0145 3 .8288 81. 1787 75 10.4593 5330. 5594 7. 3137 1171. 6661 3 .8305 80. 8594 76 10.4730 5322. 4454 7. 3199 1168.5232 3 .8320 80. 5616 77 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9046 5. 5131 570. 8075 3 .3645 59. 8121 79 6. 7652 1570. 9725 5. 5289 573. 4011 3 .3689 59. 7853 80 6. 7917 1585. 5905 5. 5434 575. 7346 3 .3729 59. 7598 81 6. 8155 1598. 7578 5. 5563 577. 8140 3 .3765 59. 7353 82 6. 8367 1610.5143 5. 5678 579. 6540 3 .3796 59. 7113 83 6, 8554 1620.8639 5. 5779 581. 2599 3 .3824 59. 6904 84 6. 8716 1629. 8008 5. 5867 582. 6381 3 .3848 59. 6713 85 6, 8852 1637. 3445 5. 5940 583. 7936 3 .3868 59. 6554 86 6. 8963 1643. 5011 5. 6000 584. 7314 3 .3885 59. 6418 87 6. 9050 1648. 2942 5. 6046 585. 4532 3 .3898 59. 6305 88 6. 9111 1651. 7061 5. 6080 585. 9724 3 .3907 59. 6231 89 6.. 9148 1653. 7504 5. 6100 586. 2844 3 .3912 59. 6192 90 6. 9160 1654. 4251 5. 6106 586. 3856 3 .3914 59. 6165 o I \ I I 30 1. 0000 1. 9977 1. 0000 1. 9922 1 .0000 1. 9394 31 1. 0609 2. 4565 1. 0607 2. 4460 1 .0580 2. 3455 32 1. 1230 2. 9964 1. 1224 2. 9780 1 .1163 2. 8081 33 1. 1861 3. 6267 1. 1850 3. 5970 1 .1748 3. 3289 34 1. 2501 4. 3575 1. 2485 4. 3119 1 .2334 3. 9091 35 1. 3150 5. 1995 1. 3127 5. 1317 1 .2918 4. 5487 36 1. 3807 6. 1632 1. 3776 6. 0655 1 .3499 5. 2463 37 1. 4470 7. 2599 1. 4431 7. 1219 1 .4076 6. 0000 38 1. 5140 8. 4998 1. 5089 8. 3091 1 .4646 6. 8058 39 1. 5815 9. 8942 1. 5752 9. 6350 1 .5209 7. 6592 -107-I v c . 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8992 80.1584 2. 8162 67. 4952 2. 3670 25.8894 60 2. 9568 85.6966 2. 8674 71. 4807 2. 3941 26.5508 61 3. 0131 91.3521 2. 9172 75.4805 2. 4200 27.1784 62 3. 0680 97.1022 2. 9654 79.4786 2. 4447 27.7729 63 3. 1215 102.9280 3. 0120 83. 4606 2.4681 28.3345 64 3. 1734 108.8072 3. 0570 87. 4116 2. 4904 28.8644 65 3. 2237 114.7186 3. 1004 91. 3192 2. 5115 29.3637 66 3. 2724 120.6401 3. 1421 95. 1683 2. 5316 29.8335 67 3. 3195 126.5468 3. 1822 98. 9453 2. 5505 30.2741 68 3. 3648 132.4143 3. 2206 102. 6409 2. 5684 30.6875 69 3. 4084 138.2215 3. 2573 106. 2417 2. 5854 31.0744 70 3.4501 143.9427 3. 2923 109. 7360 2. 6013 31.4352 71 3. 4901 149.5540 3. 3256 113. 1146 2. 6163 31.7716 72 3. 5281 155.0343 3. 3572 116. 3692 2. 6303 32.0849 73 3. 5643 160.3621 3. 3871 119. 4890 2. 6435 32.3749 74 3. 5985 165.5119 3. 4152 122. 4669 2. 6558 32.6440 75 3. 6308 170.4646 3. 4416 125. 2953 2. 6672 32.8917 76 3. 6611 175.1987 3. 4663 127. 9699 2. 6778 33.1198 77 3. 6893 179.6961 3. 4893 130. 4802 2. 6876 33.3289 78 3. 7156 183.9359 3. 5105 132. 8249 2. 6966 33.5191 79 3. 7398 187.9045 3. 5301 134. 9987 2. 7048 33.6921 80 3. 7620 191.5831 3. 5479 136. 9951 2. 7123 33.8481 81 3. 7821 194.9566 3. 5640 138. 8117 2. 7190 33.9872 82 3. 8001 198.0139 3. 5784 140. 4472 2. 7250 34.1102 83 3. 8160 200.7384 3. 5911 141. 8940 2. 7302 34.2177 84 3. 8299 203.1251 3. 6021 143. 1531 2. 7347 34.3099 85 3. 8416 205.1584 3. 6115 144. 2227 2. 7385 34.3876 86 3. 8512 206.8364 3. 6191 145. 1003 2. 7416 34.4509 87 3. 8586 208.1467 3. 6250 145. 7821 2. 7440 34.4997 88 3. 8640 209.0845 3. 6292 146. 2713 2. 7458 34.5345 89 3. 8672 209.6508 3. 6317 146. 5650 2. 7468 34.5556 90 3. 8682 209.8398 3. 6326 146. 6644 2.7471 34.5617 -108-0 Sf I I I 40 1. 0000 1.5539 1. 0000 1. 5497 1. 0000 1. 5086 41 1. 0416 1.7920 1. 0415 1. 7853 1. 0397 1. 7205 42 1. 0835 2.0565 1. 0831 2. 0463 1. 0792 1. 9507 43 1. 1255 2.3486 1. 1248 2. 3340 1. 1186 2. 1993 44 1. 1675 2.6697 1. 1666 2. 6496 1. 1577 2. 4660 45 1. 2096 3.0214 1. 2083 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2947 2. 0811 14.7443 84 2. 3766 31.5875 2.3386 28. 6439 2. 0866 14.8462 85 2. 3844 31.9463 2.3459 28. 9419 2. 0913 14.9326 86 2. 3908 32.2429 2.3519 29. 1873 2. 0951 15.0032 87 2. 3958 32.4751 2.3566 29. 3799 2. 0981 15.0582 88 2. 3994 32.6419 2.3599 29. 5176 2. 1002 15.0974 89 2. 4015 32.7420 2.3619 29. 6006 2. 1015 15.1208 90 2. 4022 32.7748 2.3625 29. 6277 2. 1019 15.1287 V, r I o\=4 I 50 1 .0000 1. 3039 1 .0000 1. 3003 1.0000 1. 2659 51 1 .0291 1. 4416 1 .0290 1. 4366 1.0278 1. 3888 52 1 .0581 1. 5882 1 .0578 1. 5815 1.0552 1. 5177 53 1 .0867 1. 7437 1 .0863 1. 7349 1.0823 1. 6524 54 1 .1151 1. 9080 1 .1145 1. 8968 1.1089 1. 7926 55 1 .1431 2. 0810 1 .1424 2. 0670 1.1351 1. 9377 56 1 .1708 2. 2624 1 .1699 2. 2452 1.1607 2. 0875 57 1 .1981 2. 4520 1 .1969 2. 4310 1.1859 2. 2413 58 1 .2250 2. 6495 1 . 2236 2. 6242 1.2105 2. 3988 59 1 .2514 2. 8542 1 .2497 2. 8242 1.2345 2. 5593 60 1 .2773 3. 0658 1 .2754 3. 0306 1.2580 2. 7223 61 1 .3027 3. 2839 1 .3005 3. 2427 1.2808 2. 8870 62 1 .3275 3. 5075 1 .3251 3. 4599 1.3029 3. 0530 63 1 .3517 3. 7361 1 .3490 3. 6816 1.3244 3. 2196 , 64 1 .3754 3. 9690 1 .3723 3. 9068 1.3453 3. 3862 65 1 .3983 4. 2052 1 .3950 4. 1350 1.3654 3. 5522 66 1 .4206 4. 4438 1 .4170 4. 3651 1.3848 3. 7169 67 1 .4423 4. 6842 1 .4383 4. 5962 1.4035 3. 8797 68 1 .4632 4. 9250 1 .4589 4. 8275 1.4214 4. 0401 69 1 .4833 5. 1655 1 .4787 5. 0578 1.4386 4. 1974 70 1 .5027 5. 4046 1 .4978 5. 2865 1.4550 4. 3511 71 1 .5212 5. 6409 1 .5161 5. 5122 1.4707 4. 5007 72 1 .5390 5. 8738 1 .5335 5. 7340 1.4856 4. 6455 73 1 .5559 6. 1019 1 .5501 5. 9508 1.4997 4. 7853 74 1 .5720 6. 3242 1 .5659 6. 1619 1.5130 4. 9194 75 1 .5872 6. 5395 1 . 5808 6. 3661 1.5256 5. 0474 76 1 .6015 6. 7470 1 .5948 6. 5621 1.5373 5. 1691 77 1 .6149 6. 9453 1 .6079 6. 7495 1.5482 5. 2838 78 1 .6273 7. 1336 1 .6201 6. 9272 1.5583 5. 3915 79 1 .6388 7. 3106 1 .6313 7. 0942 1.5677 5. 4916 -110-0 <*=IZ I I v f . arty 80 1 .6494 7. 4760 1.6416 7. 2495 1 .5762 5.5841 81 1 .6590 7. 6282 1.6510 7. 3926 1 .5839 5.6684 82 1 .6676 7. 7668 1.6594 7. 5227 1 .5908 5.7445 83 1 .6752 7. 8912 1.6669 7. 6393 1 .5969 5.8120 84 1 .6818 8. 0000 1.6733 7. 7413 1 .6022 5.8710 85 1 .6874 8. 0933 1.6788 7. 8285 1 .6066 5.9212 86 1 .6920 8. 1705 1.6833 7. 9008 1 .6103 5.9625 87 1 .6956 8. 2309 1.6868 7. 9573 1 .6131 5.9945 88 1 .6982 8. 2740 1.6893 7. 9979 1 .6152 6.0175 89 1 .6997 8. 3001 1.6908 8. 0222 1 .6164 6.0313 90 1 .7002 8. 3086 1.6913 8. 0301 1 .6168 6.0361 I 60 1. 0000 1. 1534 1. 0000 1.1502 1. 0000 1.1197 61 1. 0199 1. 2356 1. 0198 1.2316 1. 0190 1.1933 62 1. 0394 1. 3201 1. 0392 1.3151 1. 0375 1.2681 63 1. 0584 1. 4065 1. 0582 1.4005 1. 0556 1.3440 64 1. 0770 1. 4946 1. 0766 1.4874 1. 0731 1.4203 65 1. 0950 1. 5839 1. 0945 1.5754 1. 0901 1.4972 66 1. 1125 1. 6742 1. 1119 1.6644 1. 1065 1.5742 67 1. 1295 1. 7651 1. 1288 1.7540 1. 1223 1.6508 68 1. 1459 1. 8564 1. 1451 1.8437 1. 1376 1.7270 69 1. 1617 1. 9475 1. 1608 1.9332 1. 1523 1.8023 70 1. 1769 2. 0380 1. 1759 2.0220 1. c-664 1.8766 71 1. 1915 2. 1276 1. 1904 2.1099 1. 1799 1.9494 72 1. 2055 2. 2160 1. 2043 2.1964 1. 1927 2.0204 73 1. 2188 2. 3026 1. 2174 2.2810 1. 2049 2.0895 74 1. 2314 2. 3869 1. 2300 2.3636 1. 2164 2.1561 75 1. 2434 2. 4687 1. 2418 2.4435 1. 2273 2.2203 76 1. 2546 2. 5474 1. 2529 2.5204 1. 2375 2.2816 77 1. 2651 2. 6228 1. 2634 2.5939 1. 2470 2.3398 78 1. 2749 2. 6944 1. 2730 2.6636 1. 2558 2.3947 79 1. 2840 2. 7617 1. 2820 2.7294 1. 2640 2.4460 80 1. 2923 2. 8247 1. 2902 2.7904 1. 2715 2.4935 81 1. 2998 2. 8826 1. 2977 2.8468 1. 2782 2.5372 82 1. 3066 2. 9352 1. 3044 2.8981 1. 2843 2.5768 83 1. 3126 2. 9825 1. 3103 2.9441 1. 2897 2.6120 84 1. 3178 3. 0242 1. 3155 2.9845 1, 2943 2.6428 85 1. 3222 3. 0596 1. 3198 3.0189 1. 2982 2.6690 86 1. 3258 3. 0889 1. 3234 3.0475 1. 3015 2.6906 -111-0 OUIZ <x = g CX--4 jd_ }± E v± I v± 1 87 1.3286 3 .1119 1.3262 3 .0697 1.3040 2.7076 88 1.3306 3 .1284 1.3282 3 .0857 1.3058 2.7198 89 1.3319 3 .1384 1.3294 3 .0953 1.3069 2.7270 90 1.3323 3 .1417 1.3298 3 .0985 1.3072 2.7295 CA=IZ & I r L 70 1.0000 , 1 .0629 1.0000 1 .0600 1.0000 1.0319 71 1.0124 1 .1098 1.0124 1 .1065 1.0119 1.0740 72 1.0243 1 .1559 1.0242 1 .1521 1.0232 1.1150 73 1.0356 1 .2012 1.0354 1 .1968 1.0339 1.1551 74 1.0463 1 .2452 1.0461 1 .2404 1.0441 1.1940 75 1.0565 1 .2881 1.0562 1 .2827 1.0538 1.2314 76 1.0661 1 .3292 1.0658 1 .3234 1.0628 1.2674 77 1.0750 1 .3686 1.0747 1 .3622 1.0713 1.3016 78 1.0833 1 .4061 1.0829 1 .3991 1.0791 1.3339 79 1.0910 1 .4414 1.0906 1 .4339 1.0864 1.3642 80 1.0981 1 .4742 1.0976 1 .4664 1.0930 1.3924 81 1.1045 1 .5046 1.1040 1 .4963 1.0990 1.4182 82 1.1103 1 . 5322 1.1097 1 .5234 1.1044 1.4417 83 1.1154 1 .5569 1.1148 1 .5478 1.1092 1.4626 84 1.1198 1 .5786 1.1192 1 .5691 1.1133 1.4810 85 1.1236 1 .5973 1.1230 1 .5875 1.1169 1.4967 86 1.1267 1 .6126 1.1260 1 .6026 1.1197 1.5096 87 1.1291 1 .6246 1.1284 1 .6144 1.1220 1.5198 88 1.1308 1 .6332 1.1301 1 .6228 1.1236 1.5270 89 1.1318 1 .6385 1.1311 1 .6279 1.1245 1.5314 90 1.1322 1 .6402 1.1315 1 .6297 1.1249 1.5328 -112-a I T I 80 1 .0000 1.0143 1 .0000 1 .0115 1 .0000 .9847 81 1 .0058 1.0351 1 .0058 1 .0322 1 .0056 1 .0034 82 1 .0111 1.0541 1 .0111 1 .0510 1 .0106 1 .0204 83 1 .0157 1.0712 1 .0157 1 .0679 1 .0150 1 .0357 84 1 .0198 1.0861 1 .0197 1 .0826 1 .0189 1 .0490 85 1 .0232 1.0989 1 .0231 1 .0953 1 .0222 1 .0604 86 1 .0260 1.1095 1 .0259 1 .1057 1 .0248 1 .0698 87 1 .0282 1.1178 1 .0281 1 .1140 1 .0269 1 .0771 88 1 .0298 1.1238 1 .0296 1 .1199 1 .0284 1 .0824 89 1 .0307 1.1273 1 .0306 1 .1234 1 .0293 1 .0856 90 1 .0310 1.1285 1 .0309 1 .1245 1 .0296 1 .0866 -113-APPENDIX VII h =10° 0.238217 -0.147909 0 0 -0.147909 0.367344 -0.205472 0 0 -0.205472 0.459717 -0.254037 0 O -0.254037 0.561512 0 0 0 -0.254037 0 0 0 0 0 0 0 0 0 0 O 254037 0 0 O O -0, 0.459717 -0.205472 -0.205472 0.367344 0 -0.147909 -0 0 0 0 0 0 0 ,147909 ,238217 Scaling. lo \ i 2=X 2 Eigenvalues; A.^0.031931 ) A.22=0.096597 ; \g2=0.201148 \ = 2 o° 0.057226 -0.121296 0 0 O 0 O -0.121296 0 0 0 1.028404 -0.616504 O 0 -0.616504 1.478328 -0.894182 O 0 -0.894182 2.163406 -0.894182 O 0 -6.894182 1.478328 0 0 0 -0.616504 0 0 0 0 O 0 0 O -0.616504 1.028404 -0 -0.121296 0 O 0 0 0 O 121296 057226 Scaling. lo\i2^\2 Eigenvalues; A-i2=0.025495 I X22=0.037008 § Xo2=0.225847 -114-^ =30c 0.01250 -0.09349 0 0 0 0 0 -0.09349 2.79640 -2.05449 0 O 0 0 0 -2.05449 6.03768 -4.00495 0 0 0 0 0 -4.00495 10.62632 -4.00495 0 0 0 0 0 -4.00495 6.03768 -2.05449 0 0 0 0 0 -2.05449 2.79640 -0.09349 0 0 0 0 0 -0.09349 0.01250 Scaling: l0 1u 2=A. 2 Eigenvalues: 1^=0.00620 J A. ^ =0.00830 ; A,3*=0.74523 2=, =40° 0.01318 -0.08326 0 0 0 0 0 -0.08326 2.10324 -4.22665 0 O O 0 0 -4.22665 33.97541 -23.75178 0 0 0 0 O -23.75178 66.41804 -23.75178 O O 0 0 0 -23.75178 33.97541 -4.22665 0 0 0 0 0 -4.22665 2.10324 -0.08326 0 0 0 O O -0.08326 0.01318 Scaling: l o h i 2 * * \ 2 Eigenvalues: A.^0.00655 ; X22=0.00877 ; X32=0.97673 & =50° 0.00321 -0.01016 0 0 0 0 0 -0.01016 0.12845 -0.69173 0 0 0 0 0 -0.69173 14.90035 -11.46859 0 0 0 0 O -11.46859 35.30886 -11.46859 O O 0 0 0 -11.46859 14.90035 -0.69173 0 0 0 0 0 -0.69173 0.12845 -0.01016 O 0 0 O 0 -0.01016 0.00321 Scaling: wz=A. Eigenvalues: X12=0.00157 j 122=0.00210 j A.32=0.06522 -115-=60° 0.01909 -0.02680 0 0 0 0 0 -0.02680 0.15046 -0.35889 0 0 0 0 0 -0.35889 3.42428 -5.32325 0 0 0 0 0 -5.32325 33.10140 5.32325 0 0 0 0 0 -5.32325 3.42428 -0.35889 0 0 0 0 0 -0.35889 0.15046 -0.02680 0 0 0 0 0 -0.02680 0.01909 Scaling: u2=A.2 Eigenvalues: X± =0.00829 ; \ 2 =0.01197 ; A.3^ =0.08253 £ 0 =70° 0.266399 -0.082840 -0.082840 0.103041 -0.118609 0 -0.118609 0.546117 0 0 0 0 0 O O -0.305288 0 O O O -0.305288 0.682647 -0.305288 O O 0 0 -0.305288 O 0 O 0 0.546117 -0.118609 -0.118609 0.103041 0 0 0 0 O -0.082840 -0.082840 0.266399 Eigenvalues: Scaling: <J2=\2 Xx2=0.018424 ; \^=0.044137 \ A32=0.211214 ( -116-REFERENCES Dessler, A.J. (1958) The propagation v e l o c i t y of world-wide sudden commencements of magnetic storms. J. Geophys. Res. 63, 405-408. Dungey, J.W. (1954a) The propagation of Alfven waves through the ionosphere. Penn. State Univ. Ionos. Res. Lab. S c i . Rep. No. 57. Dungey, J.W. (1954b) Electrodynamics of the outer atmosphere. Penn. State Univ. Ionos. Res. Lab. S c i . Rep. No. 69. Kato, Y. and Watanabe, T. (1956) Further study on the cause of giant pulsations. S c i . Rep. Tohoku Univ. Ser. 5, Geophys. 8, 19-23. Lundquist, S. (1952) Studies i n magneto-hydrodynamics, Arkiv for Fysik, Bd. 5, nr 15, p.297. Obayashi, T. and Hakura, Y. (I960) Enhanced i o n i s a t i o n i n the polar ionosphere caused by solar corpuscular emission. Rep. of Ionos. and Space Res. i n Japan, Vol. XIV, No. 1. Obayashi, T. and Jacobs, J.A. (1958) Geomagn. pulsations and the Earth's outer atmosphere. Geophys. Journal of the Royal Astron. Soc. Vol. 1, No. 1 (1958). Parker, E.N. (1958) Interaction of the solar wind with the geomagn. f i e l d , Physics of Fluids 1, 171-187. Plumpton, C. and Ferraro, V.C.A. (1953) On the magn. o s c i l l a t i o n s of a gra v i t a t i n g l i q u i d s t a r . Mon. Not. Roy. Astr. Soc. 113, 647-652.
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Oscillations of the earth's outer atmosphere and micropulsations Westphal, Karl Oskar 1961
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Title | Oscillations of the earth's outer atmosphere and micropulsations |
Creator |
Westphal, Karl Oskar |
Publisher | University of British Columbia |
Date Issued | 1961 |
Description | Using Maxwell's equations of electrodynamics and the linearized fundamental equation of hydrodynamics neglecting all but the ponderomotive force, the two differential equations characterizing toroidal and poloidal modes of oscillations are obtained. Neglecting the coupling between these modes the toroidal mode which appears to be connected with the phenomenon of geomagnetic micropulsations is studied in detail. Substituting for the constant magnetic field the undeformed dipole field of the Earth the eigenperiods of the oscillating lines of force are computed assuming a constant charge density distribution. Using numerical methods the eigenperiods are also obtained in the case of a variable charge density. Since the Earth's dipole field is presumably deformed by the solar wind a compressed dipole field is introduced into the equation of toroidal oscillations. The eigenperiods of the oscillating lines of force are obtained in this case, assuming a constant charge density distribution. For the case of a variable charge density a numerical method is described which could yield the eigenperiods. |
Subject |
Astronomy -- Spherical and practical Earth -- Rotation |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-11-28 |
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. |
IsShownAt | 10.14288/1.0085477 |
URI | http://hdl.handle.net/2429/39305 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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