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Instabilities of a Z-pinch discharge Hodgson, Rodney Trevor 1964

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INSTABILITIES OF A Z-PTNCH DISCHARGE by RODNEY TREVOR HODGSON B.Sc, The University o.f B r i t i s h Columbia, I960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COUJMBIA A p r i l , 196U I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t 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 per-m i s s i o n 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 representativesc, I t 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 allowed without my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. The Uni v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of RODNEY TREVOR HODGSON B.'Sc, The Univ e r s i t y of B r i t i s h Columbia MONDAYs FEBRUARY 3, 1964, AT 3:30 P.M. IN ROOM 303, HENNINGS BUILDING (PHYSICS) COMMITTEE IN CHARGE Chairman: F.H. Soward F.L. Curzon . C.P.S. Taylor A.J. Barnard W.A.G. Voss F.W. Dalby T. Watanabe External Examiner: A. F o l k i e r s k i Imperial College U n i v e r s i t y of London INSTABILITIES OF A Z-PINCH DISCHARGE ABSTRACT The c y l i n d r i c a l column of plasma produced i n the f i r s t stage of a z-pinch discharge i s t h e o r e t i c a l l y unstable. For one p a r t i c u l a r type of i n s t a b i l i t y , the amplitude of a surface perturbation increases at a rate dependent on the acc e l e r a t i o n of the surface (Rayleigh-Taylor i n s t a b i l i t i e s ) . An experimental study of these i n s t a b i l i t i e s has been ca r r i e d out by photographing the discharge column with a high-speed framing-camera. Simple r o t a t i o n a l l y symmetric i n s t a b i l i t i e s have been excited i n the normally stable i n i t i a l stage of an argon z-pinch discharge by means of a set of equally spaced glass rings. The framing camera photographs show that the i n s t a b i l i t i e s develope ap-proximately i n accordance with the Rayleigh-Taylor theory. No a x i a l d r i f t of the i n s t a b i l i t i e s i s ob-served, but the new technique of studying i n s t a b i l i -t i e s reveals that the acceleration of the discharge boundary changes appreciably three or four times during the i n i t i a l stage of the discharge. GRADUATE STUDIES F i e l d of Study: Physics Electromagnetic Theory Waves 1... Quantum Theory of Solids Plasma Physics Advanced Plasma Physics Magneto-hydrodynamics Related Studies: D i f f e r e n t i a l Equations G.M. Volkoff R,W. Stewart R, Barrie L.G. de Sobrino F.L. Curzon F. . C.urzon P.R. Smy C A . Swans on PUBLICATIONS - Delayed Implosion of the Z-Pinch i n Nitrogen= F.L, Curzon, R.T. Hodgson and R.J. C h u r c h i l l ; Can.Jour.Phys. 41, 1547, (1963). - E x c i t a t i o n of m=0 I n s t a b i l i t i e s i n a Z-Pinch Discharge-F.L. Curzon, R.T, Hodgson and R.J. C h u r c h i l l ; Accepted for pu b l i c a t i o n i n Jour.Nucl.Energy (Part C). ABSTRACT Simple rotationally symmetric i n s t a b i l i t y modes have been excited i n the normally stable i n i t i a l stage of an argon Z-pinch discharge by means of a set of equally spaced glass rings. High speed framing camera photographs show that the in s t a b i l i t i e s develop approximately i n accordance with the Rayleigh-Taylor theory. No axial d r i f t of the i n s t a b i l i t i e s i s observed but the new technique of studying i n s t a b i l i t i e s reveals that the acceleration of the discharge boundary changes appreciably three or four times during the course of the f i r s t stage of the discharge. F. L. Curzon - v i i i -ACKNOWLEDGEMENTS I would l i k e to thank Dr. F. L. Curzon f o r h i s e x c e l l e n t d i r e c t i o n and advice throughout the course of the experiments reported here. I would a l s o l i k e to thank the other members of the plasma p h y s i c s group and Dr. F. W. Dalby f o r many s t i m u l a t i n g d i s c u s s i o n s . The i n v a l u a b l e a s s i s t a n c e o f Mr. J . H. Turner, Mr. John Lees, and the departmental work shop s t a f f i n the c o n s t r u c t i o n o f apparatus i s g r & t d T u l l y acknowledged. - i i i -TABLE OF CONTENTS Abstract i i L i s t of Illustrations v L i s t of Tables v i L i s t of Plates v i i Acknowledgements v i i i CHAPTER I - Introduction 1 CHAPTER II - Theoretical Background 5 Section A - Unperturbed Discharge 3> Section B - Perturbed Discharge 9 CHAPTER III - Experimental Apparatus and Data Measurement 19 Section A - Apparatus 19 1. Discharge Circuit 22 2. Triggering Operation 25 3. Discharge Perturbation 29 Section B - Measurement Devices and Data Measurement 30 1. Current Measurement 30 2. High Speed Framing Camera 3k CHAPTER IV - Observations and Results . 37 Section A - Unperturbed Discharge 37 Section B - Perturbed Discharge li2 1. Observations li2 2. Treatment of Data 3o Error Analysis U» Results 5. Discussion of Results 6. Suggestions for Further Work CHAPTER V - Conclusions Appendix A Appendix B ... Bibliography LIST OF ILLUSTRATIONS FIG. I . 1 - Gas and E l e c t r o d e s o f Z-Pinch 2 FIG. I I . 1 = Diagram of F i r s t Stage of a Z-Pinch Discharge 6 2 - Diagram of Perturbed Plasma Surface 12 3 - "Long" Wavelength I n s t a b i l i t y 17 k - "Short" Wavelength I n s t a b i l i t y 17 FIG. I I I . l = Block Diagram of Apparatus 20 2 - Schematic Diagram o f Discharge C i r c u i t 22 3 - Condenser Bank 23 k «=• Main Spark Gap Switch 2^ $ - Z-Pinch Discharge Tube 26 6 - C i r c u i t Diagram of I s o l a t e d Discharge C i r c u i t and T r i g g e r Generator 28 7 ~ Diagram of Glass Rings Used to Perturb Discharge 30 8 - Cut-away Drawing o f Rogowski C o i l 31 9 - C i r c u i t Diagram o f Rogowski C o i l and I n t e g r a t o r 31 10,» Current Traces from Rogowski C o i l 33 11 •= Framing Cat^ra and C o n t r o l Equipment 3h 12 - Diagram o f One Frame of a Perturbed Discharge Photograph 36 FIG. IV. 1 - Radius Versus Time and Current Versus Time i n A and N 2 Discharges Under Comparable D e n s i t y C o n d i t i o n s ho _ v i ~ FIG. IV.2 - Pinch Times for Various Discharges Versus ' o 3 - Plots of In A versus t, and "r" versus t I4. ~ LP Versus / a for a) ^ — 2 cm b) ^ ~- 3 cm c) ^ ~= k cm LIST OF TABLES TABLE I I I . l » Essential Characteristics of Apparatus TABLE IV.1 - Dependence of Measuring Errors i n a and uO Number of Frames (n) 2 - Dependence of cO on i/a for Instabilities i n Argon - v i i -LIST OF PLATES PLATE IV. I I I I I I I V PLATE A. I I I I I I -Single Frames o f Unperturbed Discharge i n SOOp/Kg Argon 38 Framing Camera Photograph of 500^vHg Perturbed Argon Discharge hi Single Frames of Perturbed Discharge i n 500^Hg Argon 1*8 Single Frame of Perturbed Discharge with Rings 9 cm Apart 53 F i r s t Stage of Unperturbed Nitrogen Discharge 63 S i n g l e Frames o f Nitrogen Showings a) Unperturbed F i r s t Stage b.) Post-pinch I n s t a b i l i t i e s i n Unperturbed Dis charge c) Perturbed Pre-Pinch Discharge d) Perturbed Post-Pinch Discharge 65 a and b - Perturbed Post-Pinch I n s t a b i l i t i e s i n Argon; X i 2 era and 3 cm c and d - Perturbed Hydrogen Discharge; Pre-Pinch and Post-Pinch 66 CHAPTER I INTRODUCTION I n recent years, there has been a surge of i n t e r e s t i n plasma p h y s i c s , s i n c e s t u d i e s o f the gross p r o p e r t i e s o f plasmas are v i t a l l y necessary t o research i n the f i e l d s of c o n t r o l l e d nuclear f u s i o n (Bishop, 1958), atmospheric physics ( E l l i o t , 1962), and space research (Engel, 1959). S t e l l a r and i n t e r s t e l l a r plasmas are c o n t r o l l e d i n e q u i l i b r i u m by n a t u r a l l y o c c u r r i n g g r a v i t a t i o n a l and magnetic f i e l d s . However, l a b o r a t o r y plasmas can e x i s t o n l y a short time, since contact w i t h any " c o l d " substance r a p i d l y c o o l s and d e - i o n i z e s them. For t h i s reason, magnetic f i e l d s are used to " c o n t a i n " many l a b o r a t o r y plasmas and keep them away from s o l i d o b j e c t s which serve as heat s i n k s . For such l a b o r a t o r y plasmas, r a d i a t i o n i s the most important heat l o s s mechanism ( S p i t z e r , 1956). The r a d i a t i o n rate i n c r e a s e s r a p i d l y w i t h i n c r e a s i n g plasma temperature so t h a t higher temperatures can be achieved o n l y i f the power i n p u t r a t e exceeds the power l o s s r a t e . One way o f a c h i e v i n g a high power i n p u t r a t e i s to discharge the stored e l e c t r i c a l energy of a condenser i n t o the plasma. One of the devices u s i n g t h i s method i s the so c a l l e d Z-pinch. A Z-pinch device r e l e a s e s the energy of the condenser d i r e c t l y i n t o a gas column by di s c h a r g i n g a hi g h c u r r e n t through i t (see F i g . l ) . _2-The ad-vantage o f t h i s device i s t h a t the magnetic f i e l d generated by the high c u r r e n t simultaneously heats the plasma,and keeps i t away from heat s i n k s . F i g . 1.1. Gas and E l e c t r o d e s o f Z-pinch. Figure 1 shows the c y l i n d r i c a l l y symmetric system o f gas and electro d e s t h a t forms the e s s e n t i a l p a r t o f a Z-pinch d e v i c e . The a x i a l c u r r e n t ( I ) produces an azimuthal magnetic f i e l d (B*e ). The current c a r r y i n g e l e c t r o n s have an a x i a l d r i f t v e l o c i t y v and experience a Lorentz f o r c e e(v x B) a c t i n g r a d i a l l y inward. The e l e c t r o n stream t h e r e f o r e c o n t r a c t s r a d i a l l y and draws the i o n s i n wi t h i t by means of space charge f i e l d s . T h i s process i s known as the "Pinch E f f e c t " , and, when the plasma column reaches i t s minimum diameter, i t i s s a i d to be "pinched". A pinched plasma column may be maintained i n e q u i l i b r i u m i f the magnetic and k i n e t i c pressures are equal. However, i t can be shown t h a t t h i s e q u i l i b r i u m i s u n s t a b l e , since any sm a l l p e r t u r b a t i o n o f the e q u i l i b r i u m c o n f i g u r a t i o n w i l l i n c r e a s e w i t h time. T h e o r e t i c a l con-s i d e r a t i o n s show t h a t the plasma surface i s a l s o unstable even during the i n i t i a l stage when the discharge column i s c o n t r a c t i n g r a d i a l l y . The time dependence o f the i n s t a b i l i t i e s i s determined by the p h y s i c a l p r o p e r t i e s of the discharge and of the plasma. For t h i s reason, the study of plasma i n s t a b i l i t i e s can provide important i n f o r m a t i o n about the plasma i t s e l f . Plasma surface i n s t a b i l i t i e s have been e x t e n s i v e l y t r e a t e d t h e o r e t i c a l l y (Summer School o f Plasma P h y s i c s , R i s o , I960), but because o f experimental d i f f i c u l t i e s v e r y l i t t l e experimental work has been reported (Curzon et. a l , 1960| Green and N i b l e t t , I960; Curzon and C h u r c h i l l , 1962; H e r t z , 1962). Comparision o f the experimental work with the a v a i l a b l e t h e o r i e s i s a l s o extremely d i f f i c u l t , s i n c e t h e o r i e s p o s t u l a t e s i n u s o i d a l ( s i n g l e mode) surface p e r t u r b a t i o n s , whereas i n t y p i c a l experimental c o n d i t i o n s the plasma surface i s h i g h l y i r r e g u l a r . A F o u r i e r decompo-s i t i o n o f the surface would i n p r i n c i p l e be p o s s i b l e , e n a b l i n g a comparison o f t h e o r e t i c a l and experimental r e s u l t s . However, with present experimental techniques F o u r i e r a n a l y s i s of the plasma surface i s i m p r a c t i c a l . This t h e s i s presents f o r the f i r s t time an a l t e r n a t i v e method o f studying plasma surface i n s t a b i l i t i e s i n t h a t the types o f i n s t a b i l i t i e s are c o n t r o l l e d by the experimental design i n s t e a d of developing randomly. This represents a s i g n i f i c a n t advance over e a r l i e r experimental techniques since measurements can be made more accurately and since the experimen-t a l and t h e o r e t i c a l r e s u l t s can be compared without r e s o r t i n g to Fourier analysis of the plasma surface. The observations on plasma surface i n s t a b i l i t i e s described i n t h i s t h e s i s were made on the i n i t i a l stage of a 5>00p Hg argon Z-pinch discharge. Under these circumstances, the experimental conditions conform most c l o s e l y to those assumed i n simple t h e o r e t i c a l models. The t h e o r e t i c a l and experimental r e s u l t s ( f o r a stable discharge) needed to discuss these i n s t a b i l i t i e s are given i n the f i r s t parts of Chapters I I and IV. The apparatus and measurement methods are out-l i n e d i n Chapter I I I , and the main r e s u l t s of theory and experiment on plasma surface i n s t a b i l i t i e s are reported i n the l a t t e r parts of Chapters II and IV. CHAPTER I I THEORETICAL BACKGROUND This chapter i s div i d e d i n t o two sections corresponding to the d i v i s i o n of the experimental r e s u l t s presented i n Chapter IV. The material i n Section A covers the f i r s t stage of the Z-pinch discharge when the plasma surface of the discharge column i s unperturbed. A simple model (o f the discharge) which gives r e s u l t s agreeing with experimental r e s u l t s i s tre a t e d . I n S e c t i o n B, the development of r o t a t i o n a l l y symmetric per-turbations of the plasma surface i s c a l c u l a t e d . Section A - Unperturbed Discharge The radius of the Z-pinch discharge column as a function of time i s one o f the easiest of i t s p h y s i c a l parameters to measure experimen-t a l l y . Several authors (Roseribluth, 195>1|J Kuwabara, 1963) have formulated models of the i n i t i a l stage of the Z-pinch i n which the physi c a l conditions o f the discharge are s p e c i f i e d . From these models, the radius of the discharge column may be cal c u l a t e d and compared to the experimentally measured radius. The r e s u l t s provided by the various theories do not d i f f e r appreciably. For t h i s reason, only the simplest model, the snow-plow model introduced by Rosenbluth (195k), w i l l be considered i n d e t a i l i n -6. t h i s t h e s i s . The shock wave model t r e a t e d by Kuwabara ( 1 9 6 3 ) i s more complicated, while the f r e e p a r t i c l e model i n t r o d u c e d also by Rosenbluth (l9<h)} i s not a p p l i c a b l e t o hig h d e n s i t y plasmas such as those t r e a t e d i n t h i s t h e s i s . The snow-plow model i s a p p l i c a b l e i f the discharge c u r r e n t flows i n a t h i n c y l i n d r i c a l s h e l l at the boundary o f the plasma column. The model assumes t h a t t h i s s h e l l sweeps up a l l the gas i n t o i t s e l f as i t moves inward. I t i s a l s o assumed t h a t the k i n e t i c gas pressure i s sm a l l compared w i t h the magnetic pressure a r i s i n g from Lorentz f o r c e s . Suppose t h a t the gas column has rad i u s R 0 (meters) at time t - 0 and R(t) (meters) a f t e r a time t (see F i g . 1 ) . The mass of gas swept up i n t o the inward moving s h e l l i s j u s t the mass o f the gas which i n i t i a l l y occupied the c y l i n d r i c a l volume between r a d i i R Q and R ( t ) . The mass per u n i t l e n g t h (M) of the s h e l l i s given by equation ( 1 ) . Return current conductor Glass tube Current c a r r y i n g mass s h e l l Undisturbed gas Vacuum F i g . I I . 1 . Diagram o f F i r s t Stage of a Z-Pinch Discharge. -7-(1) M = 7T p [ R O 2 „ R 2(t)3 M has u n i t s o f kilograms per u n i t l e n g t h of the discharge column, and p i s the i n i t i a l gas d e n s i t y i n KG/rrrl The momentum (P) a s s o c i a t e d w i t h t h i s mass o f gas i s given by (2) P = 7* P Rn2 - R 2(t)j d R ( t ) / d t L 0 The r a t e of change o f t h i s momentum must be equal to the t o t a l Lorentz Force (F) a c t i n g on the surface of the plasma. F i s given by (3) F = - _2KR(t)] p n I 2 ( t ) / 2 \2T\H{t)\ where l ( t ) i s the t o t a l discharge c u r r e n t ( i n amperes) as a f u n c t i o n of time. U s u a l l y , i t i s convenient to work i n terms o f the dimensionless r a d i u s v a r i a b l e (y) def i n e d by (h) y ( t ) = R( t ) / R c By equating the time d e r i v a t i v e of the momentum (equation 2) w i t h the f o r c e (equation 3) and using equation (I4), we may d e r i v e the f o l l o w i n g equation. (5) dt (1 - y2 ) § dt - 8 -Equation {$) i s r e f e r r e d t o as Rosenbluth's snow-plow equation. I f the t o t a l c urrent l ( t ) i s known as a f u n c t i o n o f time, equation (5) can be s o l v e d . A p a r t i c u l a r l y simple s o l u t i o n occurs i f (6) I = IQ s i n ( TT t / t p ) where I 0 and t p are constants. By d i r e c t l y s u b s t i t u t i n g equation (6) i n t o equation (5) and i n t e g r a t i n g , the f o l l o w i n g s o l u t i o n f o r y i s obtained: (7) y = . cos ( TT t / 2 t p ) where Now, when t — t p , y ( t ) = 0 , and t p represents the "pinch time" f o r the snow-plow model w i t h a s i n u s o i d a l l y time dependent c u r r e n t . The p i n c h time i s the time which elapses between the s t a r t of the discharge and the "pinch" ( i . e . minimum discharge r a d i u s ) . I n p r a c t i c e , the discharge c u r r e n t does not have the time , dependence p r e s c r i b e d by equation ( 6 ) . However, when the time depend-ence i s approximately s i n u s o i d a l , one would expect the observed p i n c h times t p to depend on the parameters R 0, /° , and I 0 i n the manner given b y equation ( 8 ) . „9~ The s o l u t i o n f o r y given by equation (7) i s not a p p l i c a b l e to h i g h energy f a s t discharges s i n c e the current v a r i e s l i n e a r l y w i t h time r a t h e r than s i n u s o i d a l l y . However,, i f I i s p r o p o r t i o n a l to t , equation (5) has the power s e r i e s s o l u t i o n * where t p , i n t h i s case, i s p r o p o r t i o n a l to R Q ^ 4 . These v a r i o u s r e s u l t s d e r i v e d above may be checked e x p e r i m e n t a l l y without d i s t u r b i n g the discharge, since the discharge r a d i u s R(t) can be measured p h o t o g r a p h i c a l l y and the discharge current l ( t ) can be measured with a Rogowski c o i l . I n t h i s s e c t i o n , complete a x i a l and r o t a t i o n a l symmetry o f the discharge has been i m p l i e d . I n s e c t i o n B, we s h a l l consider the case where the discharge radius depends a l s o on the a x i a l v a r i a b l e Z. S e c t i o n B - Perturbed Discharge I n s t a b i l i t i e s , o r disturbances which i n c r e a s e with t i m e , are not g e n e r a l l y observed on the plasma surface i n the i n i t i a l stage o f the Z-pinch. Yet, a l l of the proposed models are t h e o r e t i c a l l y unstable to a r b i t r a r y p e r t u r b a t i o n s of the plasma s u r f a c e . As an a n a l y t i c a l l y simple example, consider the snow-plow model of the p i n c h i n which the discharge current (I) i s p r o p o r t i o n a l to the time ( t ) from the s t a r t of the d i s c h a r g e . Each pa r t of the plasma surface obeys equation (£) and moves, independently of any other p a r t , to the a x i s . I f we apply a s m a l l i n i t i a l s i n u s o i d a l p e r t u r b a t i o n t o (9) 1 -10-the plasma surface so that the discharge radius at time t — 0 (RQ) becomes a f u n c t i o n o f Z given by equation ( 1 0 ) , then t h e r a d i i of two d i f f e r e n t p a r t s of the plasma surface (at Z]_ and Z^ say) are given by equations (11) and ( 1 2 ) . (10) R 0(Z) = ~% - k0 s i n kZ (11) R ( Z x , t ) = R 0(Z_) j l - [t/tp(Z_)]?...} (12) a(z 2,t) = R D (z 2 ( [ 1 - [ t / t p ( z 2 ) ] 2 t , | where R = mean i n i t i a l r a d i u s o A Q = constant * p ( Z l ) °< M Z l ) > )(see equation (9) et seq.) y z 2 ) R o ( z 2 ) ) I f we choose Z i H/2k and Z 2 — ~ 7 y 2 k , the maximum d i f f e r e n c e of the r a d i i ( o r the "amplitude" of the disturbance) has the time dependence shown by equation (13)• (13) R(Z_ t ) - R(Z 2,t) - 2 A Q [ l + K ( t / t p ) 2 > ] where K ^ 1 and ) tp - t p ( z x ) ^ t p ( z 2 ) -11-On the b a s i s of t h i s simple model, then, the p a r t i c u l a r disturbance considered above w i l l i n c r e a s e q u a d r a t i c a l l y w i t h time. I n the more general case, the r a d i a l p o s i t i o n o f the plasma surface o f a Z-pinch discharge may be w r i t t e n as a F o u r i e r super-p o s i t i o n o f terms l i k e (lk) A(k,m, to) exp i ( k Z + m.9) e x p e c t I f m "= 0, the discharge i s r o t a t i o n a l l y symmetric and the space v a r i a t i o n of the plasma surface depends o n l y on the a x i a l v a r i a b l e Z. Surfaces o f t h i s type (m = 0) are the e a s i e s t to observe i n the Z-pinch dis c h a r g e , and f o r t h i s reason are considered i n more d e t a i l below. The time dependence of the plasma surface coordinates may be c a l c u l a t e d by s o l v i n g the rel e v a n t equations o f motion and co n s e r v a t i o n . I f an e q u i l i b r i u m s o l u t i o n t o these equations i s found, the s t a b i l i t y o f the s o l u t i o n may be checked by p o s t u l a t i n g an a r b i t r a r y , s m a l l disturbance o f the e q u i l i b r i u m s i t u a t i o n . I f the disturbance increases w i t h time, the e q u i l i b r i u m s o l u t i o n i s u n s t a b l e . The a r b i t r a r y , s m a l l p e r t u r b a t i o n o f the plasma surface i s u s u a l l y p o s t u l a t e d i n the form of the expre s s i o n (lU), s i n c e any p e r t u r b a t i o n of the surface may be made up by combining expressions o f t h i s type. When the equations o f motion are solved, c e r t a i n r e l a t i o n s are found between m, k, and oo. These r e l a t i o n s are c a l l e d d i s p e r s i o n r e l a t i o n s . I f CU has a r e a l , p o s i t i v e p a r t , the disturbance w i l l grow e x p o n e n t i a l l y w i t h time, and the e q u i l i b r i u m s o l u t i o n i s u n s t a b l e . - 1 2 -¥e s h a l l now c a l c u l a t e the d i s p e r s i o n r e l a t i o n between k and U-> f o r r o t a t i o n a l l y symmetric (m -= 0) plasma surfaces i n a Z-pinch discharge. A p h y s i c a l l y r e a l i s t i c model of the discharge w i l l be used, where the assumptions are: 1) The current flows i n a t h i n l a y e r on the surface of the plasma. 2 ) The plasma mass s h e l l i s of f i n i t e t h i c k n e s s . 3) The plasma s h e l l has a constant a x i a l l y d i r e c t e d r a d i a l a c c e l e r a t i o n . F o r a n a l y t i c a l s i m p l i c i t y , we may l e t the rad i u s of the discharge (R) tend t o i n f i n i t y so t h a t the c y l i n d r i c a l surface becomes pllanar (see F i g . 2 ) . 4 Z -» f> a F i g . I I . 2 . Diagram o f Perturbed Plasma Su r f a c e . -13-T h i s problem has been t r e a t e d by K r u s k a l and Schwarzschild (I95h)• A s i m p l i f i e d form o f t h e i r treatment f o l l o w s . Consider an i n f i n i t e l y conducting non-viscous' plasma supported i n e q u i l i b r i u m against g r a v i t y o r an a c c e l e r a t i o n "a" by a magnetic f i e l d B . The i n t e r i o r o f the plasma i s described by the equation of motion ( 1 5 ) , the equation f o r con s e r v a t i o n of mass (16) and the conservation of entropy equation (17). The vacuum e x t e r i o r to the plasma i s described by Maxwell's equations (18) and (19). The supporting magnetic pressure i s provided by a d i s c o n t i n u i t y i n the magnetic f i e l d which a r i s e s from a sheet current on the plasma s u r f a c e . Let "v* = plasma v e l o c i t y i n m/sec p = plasma d e n s i t y i n KG/m^ 2 P — plasma pressure i n newtons/m —» 2 a =. d i r e c t e d a c c e l e r a t i o n i n m/sec Then (15) P ~ = " § R A D P ~ P a ^ (16) d i v ( p v ) a t (17) 1 dP p ^ - p a t (18) d i v B = 0 (19) c u r l l T = 0 The e q u i l i b r i u m s o l u t i o n o f equations (15) - (19) i s given by (20) I vQ = 0 (21) =hy (22) P Q = P y x 0 e-*r (23) B , B Q T X ( 2 W a P r - o - h P y - o (25) P y = 0 = B 0 2 / 2 A where h i s an "atmospheric" constant, T ' r - -w apply a small s i n g l e mode p e r t u r b a t i o n t o the plasma surface so t h a t i t s y coordinate (say ^ ) i s given by equation (26) (26) ^ ( t - 0) - A Q e i k z where A i s a constant and the wavelength ( ) \ ) of the p e r t u r b a t i o n i s o given by (27) ^ = 2 A / k Suppose t h a t ^ stands f o r any of the time dependent v a r i a b l e s . L e t ^ be the e q u i l i b r i u m value o f ^ , and 0 -f i be the perturbed v a l u e . I f we assume th a t (28) \ ^ s Y i t ) = ^ e" k^ e i k z e w t and then s u b s t i t u t e | ^ Q - j - ^ - j . i n t o equations (l£) - (19), we o b t a i n , on n e g l e c t i n g second order terms, the f o l l o w i n g set of equations (29) /? ? 1 - *7 P l k " ®a i k P l " f l a e -(30) - k p 0 v + p0 * \ (-h/>0> = - U J ^ o Equations (29), (30) and (31) represent a set o f l i n e a r homogeneous equations f o r the "amplitude f a c t o r s " % \» A n o n - t r i v i a l s o l u t i o n of t h i s set o f equations e x i s t s i f and o n l y i f the determinant of the c o e f f i c i e n t s v a nishes. I f h << k ( i . e . the pressure and d e n s i t y e q u i l i b r i u m values change l i t t l e over a d i s t a n c e "X ), then the determinant vanishes i f and o n l y i f (32) L J O 2 - ak T h i s r e l a t i o n between the ex p o n e n t i a l growth r a t e <-0 and the wave number k (the d i s p e r s i o n r e l a t i o n ) i s s i m i l a r t o the d i s p e r s i o n r e l a t i o n o f the w e l l known i n s t a b i l i t y o f a l i g h t f l u i d supporting a heavier one against g r a v i t y ( T a y l o r 5 1950). I n s t a b i l i t i e s o f t h i s k i n d are g e n e r a l l y -16-c a l l e d R a y l e i g h - T a y l o r i n s t a b i l i t i e s . The p o s i t i o n of the plasma surface i s given by the equation (33) ? = A 0 e x p ( to t + i k z ) T h i s means t h a t the plasma surface w i l l r e t a i n i t s s i n u s o i d a l shape, but the amplitude (A) o f the disturbances, given i n i t i a l l y by A 0 of equation ( 2 6 ) , w i l l i n c r e a s e e x p o n e n t i a l l y i n time since (3U) A ( t ) = A D exp( w t ) where (35) tjl =/ak S t r i c t l y speaking, the t h e o r i e s given above can o n l y be a p p l i e d to the Z-pinch discharge when c e r t a i n l i m i t i n g c o n d i t i o n s are f u l f i l l e d . These c o n d i t i o n s are discussed below. I n the f i r s t case of the snow-plow i n s t a b i l i t y , we assume i m p l i c i t l y t h a t the wavelength o f the i n s t a b i l i t y i s much longer than the plasma l a y e r t h i c k n e s s (J) as drawn i n F i g . 3 . Since the a n a l y s i s j u s t completed a l s o holds f o r a t h i n mass s h e l l , the disturbance d i e s away i n t o the plasma o n l y as exp(-ky) (equation ( 2 8 ) ) , so t h a t the i n s i d e surface o f the plasma l a y e r w i l l be d i s t o r t e d as w e l l as the outside s u r f a c e . The s h e l l moving i n sweeps up gas i n t o i t s e l f and the mass i n f l u x tends to damp the growth of the i n s t a b i l i t y from exponential -17-F i g . II.3. "Long" Wavelength I n s t a b i l i t y . F i g . I I .U . "Short" Wavelength I n s t a b i l i t y . •18-(Rayleigh-Taylor) time dependence t o a quadratic time dependence. d i e s away r a p i d l y , and the outside surface has v i r t u a l l y no knowledge of the mass i n f l u x o c c u r r i n g at the i n n e r surface o f the s h e l l . On the b a s i s of these arguments^ we would expect an exponential time dependence f o r short wavelength i n s t a b i l i t i e s , and a quadratic time dependence f o r long wavelength ones. The th i c k n e s s o f the plasma s h e l l i s the c r i t i c a l dimension. I n the second case -19-CHAPTER I I I EXPERIMENTAL APPARATUS AND DATA MEASUREMENT This chapter i s d i v i d e d i n t o two s e c t i o n s . The m a t e r i a l i n S e c t i o n A describes the apparatus used to produce a Z-pinch discharge. S e c t i o n B conta i n s a d e s c r i p t i o n o f the measurement devices used to study i t , and an o u t l i n e o f the methods used i n data measurement. S e c t i o n A - Apparatus Of the two p r i n c i p a l d i a g n o s t i c techniques used t o study discharge plasmas, the magnetic probes employed by Burkhardt and Lovberg (1958) d i s t u r b the discharge w h i l e the high speed photographic methods employed by Curzon et, al, ( i 9 6 0 ) do not. The l a t t e r technique i s the more s u i t a b l e f o r the xrork d e s c r i b e d i n t h i s t h e s i s , so the equipment was designed f o r photographic a n a l y s i s o f the Z-pinch. Such a design should have the f o l l o w i n g f e a t u r e s : a) The discharge i s v i s i b l e . b) Evaporation of m a t e r i a l from the w a l l s of the discharge v e s s e l i s minimized to reduce contamination of the plasma. c) The temperature i s high enough to enable comparison o f experimental r e s u l t s w i t h the t h e o r e t i c a l r e s u l t s d e r i v e d from hi g h c o n d u c t i v i t y plasma models. I n order to s a t i s f y these s p e c i f i c a t i o n s , the energy put i n t o the discharge must be kept low, while the r a t e of current r i s e s (dl/dt) (and temperature r i s e dT/dt) i n the gas must be as h i g h as p o s s i b l e . Therefore the inductance of the condenser bank, leads and switches must be as low as p o s s i b l e . The equipment must al s o be s u f f i c i e n t l y rugged t o w ithstand high v o l t a g e s (20,000 v o l t s ) ' a n d high c u r r e n t s (200,000 amperes). The b l o c k diagram below shows the r e l a t i o n of the p r i n c i p a l pieces of equipment needed t o produce Z=pinch discharge at a p r e c i s e l y known time. The condenser bank ( (B) o f F i g . l ) discharges i t s energy i n t o the gas i n the discharge t u b e ^ C ) when the main spark gap switch (A) c l o s e s . The t r i g g e r i n g o p e r a t i o n (or sequence of operations to c l o s e the spark gap switch)' s t a r t s w i t h a pulse from the h i g h speed framing camera (D). Each major piece of equipment i s d e s c r i b e d below and a resume'' of the e s s e n t i a l c h a r a c t e r i s t i c s o f the apparatus i s given i n Table I I I . l . I P u l s e Generator Framing Camera D Diagnostic Equipment E Discharge Tube T r i g g e r Generator F \ f Main Spark Gap IK Condenser Bank B 1 Charging U n i t F i g . I I I . l . Block Diagram of Apparatus. TABLE I I I . A . l E s s e n t i a l C h a r a c t e r i s t i c s of Apparatus Capacitoo? Bank p l u s Leads C a p a c i t y C Working v o l t a g e Inductance Maximum discharge c u r r e n t ( I ) Maximum d l / d t R i n g i n g frequency Discharge Tube I n t e r - e l e c t r o d e d i s t ance I n s i d e diameter Outside diameter Return conductor 50 /Jf 10 KV 0.12 pE 130 KA 2 x 1 0 1 0 A/sec 180 K c / s e c 62 cm 15' cm 16.2 cm Brass mesh Vacuum System U M o i l d i f f u s i o n pump w i t h Kinney KS-13 backing pump Base pressure Leak r a t e Framing Camera " 60 frames Average exposure time Average time between exposures Gases and Pressures Used Argon CuHg) 50 100 250 500 - ' v l ^ H g -/vjlj^Hg/hour - 0.25/'sec - 0.25/Jsec N i t r o g e n (/JHg) 37.5 75 15 o 375 750 Hydrogen (mmHg) "675 1.0 2 5 10 Major p a r t of work i s i n 500 jj Hg argon 22 Discharge C i r c u i t The discharge c i r c u i t c o n s i s t s of the condenser bank, main spark gap s w i t c h , high current leads and discharge tube. They are depicted s c h e m a t i c a l l y by F i g . 2 . E l e c t r o d e s Condenser Bank .Discharge Tube C Spark Gap Switch A [ .Triggering Apparatus F i g . I I I . 2 . Schematic Diagram o f Discharge Circuit„ a) Condenser bank and leads The condenser bank (B) o f Figs„ 1 and 2 c o n s i s t s of t e n low inductance N.R.G. type 201 condensers connected i n p a r a l l e l and arranged i n two p i l e s on a sto u t Dexion t r o l l e y ( F i g . 3)» Each c a p a c i t o r i s r a t e d at SJO f a r a d c a p a c i t y and i s operated at a charging v o l t a g e of 10 kV, so t h a t the t o t a l s t o r e d energy of t h i s bank (^ CV^) i s 2500 J o u l e s . The h i g h current leads c o n s i s t of p a r a l l e l p l a t e s o f copper, k inches,wide, l / l 6 of an i n c h t h i c k and k f e e t l o n g . They are clamped c l o s e l y together to minimize inductance, and have 16 sheets of 5 m i l polythene between them to provide adequate i n s u l a t i o n . -23-F i g . I I I . 3 . Condenser Bank -22*. k) Main Spark Gap The design shown i n F i g . k was chosen f o r the main spark gap switch s i n c e t h i s design has been proven v e r s a t i l e , , r e l i a b l e and demountable. The inductance i s not as low as some other types o f switches (Cormack and Barnard, 1962) but i t i s s t i l l s mall compared w i t h the mean inductance of the Z=pinch discharge. To "ground" t e r m i n a l o f condenser bank From top e l e c t r o d e of discharge tube F i g . III.k> Main Spark Gap Switch, (Scale 1 : 1 ) c) Discharge Tube A s e c t i o n through the r o t a t i o n a l l y symmetric, v e r t i c a l l y mounted discharge tube i s shown by the diagram o f F i g . 5>. I t c o n s i s t s of a pyrex glass c y l i n d e r 75 cm l o n g , 15 cm i n t e r n a l diameter, w i t h a w a l l t h i c k n e s s o f 0.6 cm. Plane brass e l e c t r o d e s are l o c a t e d at e i t h e r end w i t h p r o v i s i o n f o r evacuating the tube and i n t r o d u c i n g t he t e s t gas. The atmospheric pressure a c t i n g on the e l e c t r o d e s compresses the O-rings against the tube w i t h s u f f i c i e n t f o r c e to ensure a good vacuum s e a l ( ^10~ 3 mm Hg). The brass gauze which returns the current from the top e l e c t r o d e t o the ground of the condenser bank enables the discharge to be viewed e a s i l y and minimizes i t s inductance. 2. T r i g g e r i n g Operation The discharge tube must be f i r e d i n synchronism with the framing camera i n much the same way as a f l a s h bulb must be f i r e d when an o r d i n a r y camera s h u t t e r opens. The framing camera puts out a pulse when i t i s ready t o record ( i . e . " s h u t t e r open") and the discharge,must be f i r e d from t h i s pulse at a p r e c i s e l y known time ( - 0.1 JU s e c ) . The main spark gap switch has been proven t o break down . 3yt/see t .01 yjsec a f t e r a f a s t t r i g g e r i n g pulse of 10 kV i s a p p l i e d to the t r i g g e r p i n . Therefore, the t r i g g e r i n g system must provide a h i g h v o l t a g e pulse at a known time a f t e r r e c e i v i n g the low v o l t a g e pulse from the framing camera. The sequence of operations to do t h i s i s shown by the block diagram -26^  Exhaust and gas i n l e t Pressure contact Upper e l e c t r o d e Gas column Glass tube Brass gauze Discharge current I Returning current To condenser bank ground From spark gap switch F i g . I l l . 5 . Z-pinch Discharge Tube. ( F i g . 1, page 20). A low v o l t a g e pulse from the framing camera (D) i s put i n t o a standard t h y r a t r o n (l;C33>) pulse generator ( E ) . The s i n g l e output pulse of 9 kV has i t s amplitude doubled by r e f l e c t i o n from the open end of a charged c o a x i a l cable (Theophanis, i960). This high v o l t a g e pulse could be used t o t r i g g e r the main spark gap s w i t c h . However, another stage i n the t r i g g e r i n g o p e r a t i o n (the t r i g g e r generator F of F i g . l ) has been introduced i n order t o : a) i n c r e a s e the energy of the t r i g g e r spark from 0.1 t o 3 j o u l e s , b) i s o l a t e the high energy discharge c i r c u i t - from the t r i g g e r i n g system and measuring instruments. The f i r s t f e a t u r e improves the r e l i a b i l i t y of the main spark gap° The second f e a t u r e makes the discharge equipment much s a f e r and reduces noise s i g n a l s i n the measuring devices by e l i m i n a t i n g spurious e l e c t r i -c a l c o u p l i n g between the h i g h energy discharge c i r c u i t and the measuring devices. F i g u r e 6 shows a c i r c u i t diagram o f the i s o l a t e d discharge c i r c u i t and t r i g g e r generator. The design of the t r i g g e r generator i s adopted from one d e s c r i b e d by Curzon and Smy (I96l) where the h i g h v o l t a g e pulse from the t h y r a t r o n pulse generator i s used to operate an u l t r a v i o l e t l i g h t source. Photons from t h i s source pass through a quartz bulb and produce s u f f i c i e n t e l e c t r o n s i n the spark gap o f the t r i g g e r generator (T-j o f F i g . 6) to cause i t to break down. The r e s u l t a n t pulse across R^ causes the main spark (T-^) to break down. Curzon and Smy (I96l) have shown t h a t the u l t r a v i o l e t t r i g g e r i n g process works w e l l provided the voltage across the a u x i l i a r y spark gap To i s w i t h i n h% o f the o v e r v o l t i n g p o t e n t i a l . I n the design shown i n 2 8 -« a 9-> > > \ F i g . I l l . 6 . C i r c u i t Diagram of I s o l a t e d Discharge C i r c u i t and T r i g g e r Generator. R/ > | R 7 = 100 K J L . ; R 1 = 100 KJl; R 2 - 100 KJ2; R - £0 JL ; R£ = UO M J l j R 6 = 2lj0 M SL C-j_ : C a p a c i t e r bank, SO /J f ,10 kV; C 2 : T r i g g e r c a p a c i t e r .06 /J f; T]_ s main spark gap; T 2 : Discharge tube; T^ : U.V. t r i g g e r e d spark gap Fig. 6 the main condenser bank (C]_) is used to stabilize the potential across the auxiliary spark gap (T^). The potential difference remains within the k% tolerance for a reasonable time after high voltage power supply leads are removed ( A j 1 min.). 3. Discharge Perturbation The equipment described i n the previous parts of this section is necessary to produce a simple Z-pinch discharge. For the results presented in Chapter IV, Section B, however, the gas in the discharge tube must be perturbed so that the radius of the plasma surface varies sinusoidally^with wavelength A along the axis. The discharge is given an i n i t i a l perturbation by a series of glass rings supported at regular intervals inside the discharge tube by four 1/2 inch wide glass tapes. Each ring is made by bending a 3/l6 inch diameter glass rod around a cylindrical form, and fits snugly against the inner wall of the discharge tube with its plane perpendicular to the axis of the tube. When the rings are in position inside the discharge tube, there is nearly complete cylindrical symmetry about the discharge tube axis, except for the four glass tapes and a small gap ( ^ l /U inch) in each ring (see Fig. 7). The rings are threaded through the glass tapes at the proper intervals and introduced into the discharge tube by means of a wooden j ig . After the rings are inserted into the tube and the end rings are firmly wedged, this jig can be collapsed and removed. -30-Fig. III.7. Diagram of Glass Rings Used to Perturb Discharge. Section B - Measurement Devices and Data Measurement 1. Current Measurements - Rogowski Coil The total current (I) through the discharge is measured by a Rogowski coi l "current transformer". A three inch length of RG 65 A/U delay line with i ts outer ground shield removed is placed between the parallel plate high current leads (Fig. 8). v I f the total current (I) is uniformly distributed over the h inch wide leads, the magnetic flux (B) through the coi l L i is proportional to (I). The voltage (V) induced in the coil (Li) i s proportional to dB/dt and therefore to dl/dt. The output signal of the coi l is integrated by the pass-ive integration.circuit (Fig. 9) so as to produce an output signal (V Q u t ) proportional to I . This signal is displayed on a Tektronix 5U5 A - 3 1 -F i g . I I I . 9 . C i r c u i t Diagram of E o g p w s k i C o i l and I n t e g r a t o r . L i ; Rogowski c o i l , 11 h; R i ~ 180 j R 2 - 100 K j R3 — R^ — 50-0- 1 C-L - .005 JOf. •32-o s c i l l o s c o p e screen and photographed w i t h a P o l a r o i d camera. The c o i l i s p r o t e c t e d against e l e c t r o s t a t i c pickup by a c a r e f u l l y grounded s p l i t r i n g s h i e l d . S e v e r a l techniques suggested by Daughney (1963) are i n c o r p o r a t e d i n t o the c u r r e n t measuring device i n order to i n c r e a s e the s i g n a l t o noise r a t i o ( i . e . grounding and s h i e l d i n g procedures). The frequency response of the c o i l and i n t e g r a t o r network are o p i t i m i z e d by f o l l o w i n g the procedure described by Segre and A l l e n (i960). F i g u r e 10.a i s a t y p i c a l example of a current t r a c e obtained w i t h t h i s system. Fi g u r e 10.b shows another current t r a c e on a longer time s c a l e . T h i s t r a c e was measured a c c u r a t e l y u s i n g a Z e i s s X - Y t r a c k i n g micro-scope. The measurements^ taken every l / 3 yUsec, were used t o i n t e g r a t e the t r a c e and c a l i b r a t e the c o i l and i n t e g r a t o r c i r c u i t . The c a l i b r a t i o n was done by c a l c u l a t i n g the t o t a l charge of the condenser bank (Q) from the r e l a t i o n (1) Q = CV and equating i t t o the i n t e g r a l o f the discharge c u r r e n t , i . e . (2) Q '0 o where C = c a p a c i t y of condenser bank V - charging voltage Since the output v o l t a g e of the i n t e g r a t o r network of F i g . 10 (V o u^.) i s p r o p o r t i o n a l t o the t o t a l c u r r e n t ( I ) -33-F l g . I I I . 1 0 . Current Traces a. 1 Aj sec/cm h o r i z o n t a l 5 volts/cm v e r t i c a l b. 10jjsec/cm h o r i z o n t a l 5> volts/cm v e r t i c a l (3) V t = * 1 cx -— constant then (h) V o u t d t = W The c a l i b r a t i o n constant c< obtained i n t h i s way was o< = 1.07 x 10™^ volt/amp Three independent c a l i b r a t i o n s of.the c o i l - i n t e g r a t o r network agreed t o w i t h i n 1 $ . 2* High Speed Framing Camera The B a r r and Stroud Model CP 5 high speed r o t a t i n g m i r r o r framing camera can photograph 60 frames at a maximum framing rate o f 8 x 10^ frames/sec. I t c o n s i s t s e s s e n t i a l l y o f an o b j e c t i v e l e n s L-^  (see F i g . 11} and a ma g n i f i e r , l ^ , which focuses the image on the surface of the r o t a t i n g m i r r o r . A l e n s quadrant, L3, focuses the m i r r o r surface on the curved f i l m t r a c k . The p o l i s h e d s t a i n l e s s s t e e l m i r r o r runs i n f l e x i b l y mounted white metal bearings. I t i s d r i v e n at a maximum r a t e of 330,000 r.p.m. by an a i r t u r b i n e operated at 180 P„S.I„G. PM2 £ F i g . I l l . 1 1 . Framing Camera and C o n t r o l Equipment. The event to be photographed must be synchronized w i t h the p o s i t i o n ' o f the r o t a t i n g m i r r o r , since r e c o r d i n g can o n l y t a k e p l a c e during 15>° of each h a l f r e v o l u t i o n of the m i r r o r . A pulse t o i n i t i a t e the high c u r r e n t discharge i s provided by one o f the two p h o t o m u l t i p l i e r s , PM 1 or PM 2. L i g h t from a 6 v o l t lamp i s r e f l e c t e d from a p a r t of the m i r r o r surface and d i r e c t e d by a p r i s m to the two p h o t o m u l t i p l i e r s , so they give pulses synchronized w i t h the m i r r o r p o s i t i o n . The m i r r o r speed i s monitored d i r e c t l y by a frequency meter which r e c e i v e s a s i g n a l from a search c o i l adjacent t o a magnetized c o l l a r on the m i r r o r s h a f t . I n order to determine the exact frame exposure time, the pulses from one p h o t o m u l t i p l i e r are counted f o r l / l O sec. j u s t when the discharge i s f i r e d . The discharge was photographed from the s i d e . U s u a l l y , about t e n discharges were photographed and then the f i l m ( I l f o r d HP 3) was developed (10 minutes i n I l f o r d developer I.D. 11). A f t e r development, the framing camera f i l m o f each experimental run was numbered and f i l e d . The images of the discharge on the developed f i l m are about £ mm by 8 mm. They were magnified 10 times by the o p t i c a l system of a J a r r e l l -Ash microdensitometer, which p r o j e c t e d each image onto a ground glass screen. The discharge diameter was then measured from these enlarged images w i t h an o r d i n a r y r u l e . The photographs of the perturbed discharge appear as d e p i c t e d i n F i g . 12. The maximum and minimum diameter of the discharge ( D m a x and D .„ of F i g . 12) were measured f o r each frame. The frames were measured min ° i n a random order to prevent systematic e r r o r s . -36-Fig. TII.13. Diagram of one frame of perturbed discharge photograph. The mean discharge radius r and the amplitude of the instability A were calculated from the relations r = (Dmax + Dmin)A A = (Dmax " Dmin)A The errors i n r and A estimated from the standard deviation of a number of independent measurements of the same film are (?) 1.5 ™ l A — 0.8 mm The results obtained i n this way are treated in detail i n the next chapter. -37-CHAPTER IV OBSERVATIONS AND RESULTS Thi s chapter i s d i v i d e d i n t o two s e c t i o n s . I n s e c t i o n A some r e s u l t s of observations made on the unperturbed Z-pinch discharge i n argon and n i t r o g e n are presented. The purpose o f t h i s s e c t i o n i s to e s t a b l i s h those p r o p e r t i e s of the unperturbed discharge which are r e q u i r e d i n the d i s c u s s i o n o f the c h a r a c t e r i s t i c s o f the perturbed d i s c h a r g e . S e c t i o n B contains a f u l l d i s c u s s i o n o f the r e s u l t s obtained by-p e r t u r b i n g the discharge. This work forms the main body o f the t h e s i s . S e c t i o n A. Unperturbed Discharge A d e t a i l e d study o f the Z-pinch i n argon and n i t r o g e n was c a r r i e d out i n order t o determine the best c o n d i t i o n s f o r the experiments des-c r i b e d i n S e c t i o n B o f t h i s chapter. Photographs of the n i t r o g e n discharge column (see Appendix, P l a t e A.I, page 64 ) r e v e a l many p e c u l i a r f e a t u r e s i n the e a r l y stages o f the discharge (see also Curzon and C h u r c h i l l , 1962, and Curzon, Hodgson and C h u r c h i l l , 1963)• The a x i a l streamers shown i n the Appendix ( P l a t e A.IKx/page65) demonstrate t h a t the discharge column i s by no means r o t a t i o n a l l y symmetric, and there i s t h e r e f o r e l i t t l e p o i n t i n u s i n g such discharges t o i n v e s t i g a t e r o t a t i o n a l l y symmetric discharges, i n s t a b i l i t i e s . - 3 8 -On the other hand, the discharge i n argon appears q u i t e r e g u l a r (see P l a t e I . a ) . There i s no marked i r r e g u l a r i t y i n the curve of discharge radius as a f u n c t i o n of time, and the a x i a l streamers do not appear. However, end on photographs r e v e a l t h a t the discharge gas ( i n both argon and n i t r o g e n discharges) forms a t h i c k luminous s h e l l (see P l a t e I , b ) . - . . • - — - —*— . - -- j P l a t e IV . 1 . S i n g l e frames o f unperturbed discharge i n 500/jHg argon. Time: 3 psec a f t e r s t a r t o f dischargej Exposure time: 0.25 p sec. a) side-on b) end-on (through cathode). Since the snow-plow model i s p a r t i a l l y based on the assumption t h a t the c u r r e n t and mass s h e l l i s i n f i n i t e s i m a l l y t h i c k , i t i s not s t r i c t l y a p p l i c a b l e . However, the theo r y (Chapter I I , S e c t i o n A, pages 6-9) was t e s t e d by measuring the dimensionless r a d i u s y = R ( t ) / R Q -39-photographically for discharges in the entire range of pressures in nitrogen and argon. The discharge current I(t) was measured as a function of time, and the measured values bf I were used i n integrating the Rosenbluth snow-plow equation (equation ( *> )> page 7 ) numerically with an I.B.M. 1620 computor. 'The (smoothed) measured radius, the cal-culated radius,, and the current were plotted on the same graphs as functions of time (see Fig . l ) . In order to check the scaling laws for a Z-pinch with a sinusoidally time dependent current (Chapter II , equation ( 8 ) , page ( 8 ) ) the pinch times were taken from these graphs and plotted versus R0 pz/l0 (see Fig. 2 ) . In view of the differences between the theoretical model and the experimental system,, the agreement between the theoretical and experi-mental y versus t curves is satisfactory, while the agreement of the measured pinch times with the theory is surprisingly good. The observations of the unperturbed discharge column in argon therefore show that i t is rotationally symmetric and that its dynamic characteristics agree well with the predictions of the snow-plow theory. Because of these features, argon was chosen as the discharge gas for the experiments on perturbed plasma surfaces described in the next section. F i g . IV. 1. Radius versus time and Current versus time i n A and N 2 discharges under comparable d e n s i t y c o n d i t i o n s . P l o t s o f dimensionless r a d i u s y aga i n s t time ; broken curves c a l c u l a t e d from "snow-plow" theory; f u l l curves measured from framing camera photographs; discharge current waveform p l o t t e d as measured by a Rogowski c o i l , (a) 25>Hg, (b) 50-//Hg, (c) 100//Hg, (d) 2$Cy/Hg, .(e) 500/;Hg, ( f ) 37.5/VHg, (g).75^Hg, (h) l ^ H g , ( i ) 375/jHg, (j) 750yUHg. .1 -U2-Section B. Perturbed Discharge 1. Observations Side-on photographs of the discharge in argon (SOOjjKg pressure) with the perturbing rings in place show that each ring makes a "dent" in the plasma surface. The disturbance on each side of the "dent" spreads out in the axial direction until i t meets the disturbance from the next ring. Then the surface appears sinusoidal. Plate II (page Ifi) shows a typical sequence of frames of a Z-pinch discharge (taken side-on) with rings 2 cm apart. For ring spacings of 2 , 3 and h cm the perturbed surface of the plasma appears sinusoidal with wavelength "X equal to the ring spacing (see Plate I l i a and I l lb , page I i 8 ) . For larger ring spacing (6 and 9 cm) the plasma surface does not become sinusoidal (see Plate IV, page 53 ) since the disturbances from each ring do not spread rapidly enough to meet before the first pinch. 2» Treatment of Data Attempts to relate the measured values of the discharge radius (r~) and instability amplitude (A(t)) to each other were made on the basis of the following considerations. In section A of this chapter, i t was shown that the snow-plow theory is a satisfactory model for the dynamic properties of the Z-pinch in argon. This theory predicts that the acceleration "a" of the discharge boundary should be directed radially inward, and should be approximately constant (equation (9) , page 9 )• Therefore the conditions favour growth of Rayleigh-Taylor instabilities in the f irst stage of an argon Z-pinch •U3-discharge. From Chapter H , Section B, then, a radial perturbation of the discharge surface of the form A(t) sin 2J\Z/ X is expected to be unstable, and the amplitude A(t) should increase according to equation (3U) derived on page 1 6 . For convenience, the equation is repeated here Hence, i f "a" is constant, then plots of ln A(t) versus t should be straight lines of slope , where the "growth rate" oJ is given by equation ( 2 ) . Logarithmic plots of A(t) versus t were therefore made, and in fact consist of several linear segments of different slopes, (see F ig . 3&> page 50 ). The time intervals over which co remains constant, ^ T( ui) say, vary in length from about 1 to 2 /jsec. Hence, the number of frames (n) for which remains constant, or the number of points (t, ln A) which define the line segment, varies from n = 3 to n •= 1 0 . In order to verify equation (2) the acceleration "a" was assumed to be constant during the time intervals A T ( iO) . This assumption is qualitatively justified by the appearance of the ~r versus t curves (see, for example, Fig . 3b, page £o). The acceleration for each time range L\ T( cO) was calculated by (1) A(t) - A 0 exp(cot) where (2) ud - Jl-Ka/ X -kh' fitting parabolic arcs to the-r versus t curves in each time range. The least squares method used is described i n detail in Worthing and Geffner (19U3). •In order to check for the linear correlation of OJ with v T predicted by equation (2), uOwas plotted versus /"a for ring spacings /\ , of 2, 3, and k cm. However, before discussing the qualitative and quantitative results the errors involved in calculations of "a" and must be mentioned, since a l l "least squares" fits use the points weighted according to thei accuracy. 3 o Error Analysis The errors i n >/a and uJ may be calculated from the measured errors in A(t) and r(t) by using the standard formula for propagation of errors According to this formula, the expected error ( <£ f) in an arbitrary function f of M variables (XJ_, say) is given by the expression (3) ( where <5 x^  is the expected error i n the variable Xj_. The growth rate ( ) depends on the n values of In A i n the range A T( cJ ), and is given by equation (k) 2 Aj 2 A J In Aj ( j . At) - 1 Aj ( i At) 2 Aj In Aj ( } ^ 2 A ± 2 A i ( i A t ) 2 - [1 A i ( i * t ) J se7^~ where i A t is the l^1 point in the range AT( uj ) on the In A versus • curve. A t is the time interval between points on this curve and the interval between frames ( ^ 0.2$^f sec). The estimated error in uJ ( i .e . & u) ) c a n be derived by applying the error propagation formula (equation (3)) to the expression for ^ (equation (U)). The following approximate result is obtained,, (?) Su3T_ ^ / I 2 -- 1 A (At ) 7 (n-l)(n)(n l) sec where o A is the estimated error in A and A is the mean value of A over the range A T ( oJ ). as a function of n is proportional to <^  (n) of Table I , and is shown as the error flags of Fig . )j (page E>1). The acceleration (a) of the discharge-boundary may be calculated from the points (i & t 5 r±) in a time range CiT( <^ ) by using the fol-lowing expression ( ' " 7 7 ^ ( „ -2 ) (n - : i ) (n ) („ f l )<„*2 ) ^ The estimated error c^a derived from equations (3) and (6) is given by equation (7) m £ a = _AjL / — 360 _ V s e c " 2 U ; ( L\t)2 V (n-2)(n-l)(n)(iHl)(n +2) where ^ r is the estimated error in ~rt ^ a is proportional to >vP(n) of Table 1. The corresponding error in Va"is given by the error flags of Ja of Fig. k' - U 6 -TABLE IV.1 Dependence of measuring e r r o r s i n a and L O on number of frames (n) n 2 3 U 5 6 7 8 9 00 a < £ ( n ) U 0 2 0 13 9 7 5 . 5 1+.5 3 - 7 v ^ ( n ) « ° 8 0 33 18 11 7 5 l i I t i s evident from Table 1 t h a t values of "a" f o r s m a l l values of n are not v e r y r e l i a b l e . For t h i s reason, "a" was not c a l c u l a t e d f o r n l e s s than 6 ( i . e . f o r AT( ) l e s s than 1 . 5 y f s e c ) . This means i n e f f e c t , t h a t the a c c e l e r a t i o n o f the discharge boundary cannot be measured a c c u r a t e l y unless i t i s constant f o r at l e a s t 1 . 5 /) sec ( 6 frames). U. R e s u l t s S i n g l e frames ( P l a t e s I I I . a and I l l . b ) taken from a sequence of framing camera photographs (such as P l a t e II) show t h a t the perturbed surface of the plasma i s o f the form A(t) s i n ( 2/TZ/^.), where A i s equal t o the r i n g spacing. S i n g l e mode m — 0 plasma surface i n s t a b i l i t i e s are thus a v a i l a b l e f o r study. Next page; P l a t e I V . I I . Framing Camera Photograph of argon discharge ( 5 0 0 / J Hg pressure) w i t h p e r t u r b i n g r i n g s i n p l a c e . Time incr e a s e s l e f t t o r i g h t ; frame p o s i t i o n s are staggered; exposure time P l a t e I V . I I I . S i n g l e frames of perturbed discharge i n argon (exposure time 0.25/Osec) a) Ring spacing 2 cm, l+Jsec a f t e r s t a r t of discharge b) Ring spacing U cm, £^sec a f t e r s t a r t o f discharge. -U9-Graphs of the logarithmic amplitude of the instability (In A) as a function of time show clearly the exponential time dependence of the amplitude A (see Fig. 3a)• The linear correlation between uJ and ~Ja is well demonstrated by Fig. Ua for A - 2 cm. The "least squares" straight line fitted to these points gives the relation (8) <jJ — (5.0 t l ) / a + (O.l; t 1) x IO 5 sec"1 where </& ~ 10 to I O 5 om's sec _ J - (see page 51). The Rayleigh-Taylor relations ^equations (l) and (2;) predict that uJ versus Ja should be a straight line through the origin. Since the relation (8) agrees with the theory within experimental error, i t i s concluded that the observed instabilities for "X - 2 cm are Rayleigh-Taylor instabilities. The evidence is not conclusive that the 3 and k cm wavelength instabilities are Rayleigh-Taylor instabilities. However, on the basis of the theory and the results for X - 2 cm, i t is reasonable to assume that the ")\ = 3 and "X = k cm perturbations are also Rayleigh-Taylor instabilities. Therefore, the straight lines on Fig . iia, Ub and kc were fitted using "least squares" procedures in which the lines are con-strained to pass through the origin of the ^ and /a" - axes,as required by the theory. The numerical values of cO / / a so obtained are given in Table u.2. -50-1 I I I I ° TIME FROM START OF DISCHARGE - MICROSEC. ( a ) 500 p Hg., ARGON RING SPACING 3 Cffl. J I I I I 0 1 2 3 4 5 6 TIME FROM START OF DISCHARGE - MICROSEC. ( b ) Fig . 17.3. Plots of: a) In A versus t b.) T versus t. -51-Fig. IV.a. °^ versus /IT for: a) "X - 2 cm b) >> - 3 cm c) > - h cm . C I -TABLE IV. 2 Dependence o f UJ on f o r i n s t a b i l i t i e s i n argon ( i n i t i a l pressure 500/fHg) >\(cm) bJ/fa (cm"2) l i J / </ 2 Tfa/> 2 5-7 t 1 0.32 t 0 . 0 6 3 5 . 1 t 1 0.35 t 0.07 ii.3 - 0 . 5 0.3k t 0.0k The experimental e r r o r s are too great to make d e f i n i t e conclusions about the wavelength dependence of the growth,rate Uj . The theory (equation (2)) p r e d i c t s t h a t UJ/ J2 IT a/A should be a constant and Table 2 v e r i f i e s t h i s r e l a t i o n , w i t h i n experimental e r r o r , f o r )\ - 2 , 3 and k cm. One i n t e r e s t i n g f e a t u r e of the r e s u l t s i s t h a t the a c c e l e r a t i o n (a) of the plasma boundary i s not constant as p r e d i c t e d by the simple theory o f Rosenbluth (195U) ( i . e . equation ( 9 ) , page 9)» The a c c e l e r a t i o n , i n f e r r e d from the p l o t s o f I n A versus t , appears t o change three o r fo u r times during the disc h a r g e . F i n a l l y , i t should be noted t h a t the p e r t u r b a t i o n s do not d r i f t along the discharge tube a x i s to any measureable extent. - 5 3 -P l a t e IV.IV. S i n g l e frame o f perturbed discharge i n 500 Hg argon k/J sec a f t e r s t a r t o f discharge. Ring spacing: 9 cm, exposure time: 0 .25 J) sec. 5o D i s c u s s i o n of R e s u l t s The experimental c o n d i t i o n s described e a r l i e r i n t h i s t h e s i s d i f f e r c o n s i d e r a b l y from those s p e c i f i e d by the' simple R a y l e i g h - T a y l o r i n s t a b i l i t y theory. However, a comparison of the measured r e s u l t s w i t h theory can be p a r t i a l l y j u s t i f i e d by the f o l l o w i n g d i s c u s s i o n . For the simple Rayleigh-Taylor theory of plasma i n s t a b i l i t i e s , K r u s k a l and Schwarzschild (1951;) assume t h a t a s e m i - i n f i n i t e , p e r f e c t l y conducting, non-viscous plasma i s supported by a magnetic f i e l d against a - g r a v i t a t i o n a l f i e l d o r a c c e l e r a t i o n ( a ) . -5U-The plasiria surface i n a Z-pinch i s c y l i n d r i c a l , . n o t plane as assumed by K r u s k a l and SchwarzschLld. However, H a r r i s (1962) p o i n t s out t h a t the c y l i n d r i c a l case approximates the plane case i f 2 7TR/ X >> 1* where R i s the rad i u s of curvature of the c y l i n d r i c a l plasma s u r f a c e . T h i s c o n d i t i o n i s f u l f i l l e d i n a l l experiments reported i n t h i s 1 t h e s i s . The f i n i t e t h i c k n e s s o f the plasma l a y e r i n the Z-pinch does not • p r o h i b i t comparison o f the experimental w i t h the t h e o r e t i c a l r e s u l t s , s ince T a y l o r (195>0) has shown t h a t the s e m i - i n f i n i t e f l u i d theory i s v a l i d provided the thic k n e s s o f the plasma l a y e r i s g r e a t e r than A/3 S where N i s the wavelength of the p e r t u r b a t i o n a p p l i e d to the su r f a c e . P l a t e l b (page 38) shows t h a t * f o r the times of i n t e r e s t i n our e x p e r i -ment, t h i s c o n d i t i o n h o l d s . Another r e s t r i c t i o n which the experimental c o n d i t i o n s must s a t i s f y i n order to conform to the f i r s t order K r u s k a l - S c h w a r z s c h i l d theory has been e s t a b l i s h e d b y Lewis (195>0) f o r a s i m i l a r problem i n hydrodynamics. He demonstrates t h a t the f i r s t order i n s t a b i l i t y theory i s v a l i d u n t i l A ~ /N/2. Th i s r e s t r i c t i o n i s s a t i s f i e d i n a l l experiments reported here. However, the remaining assumptions o f i n f i n i t e e l e c t r i c a l conduct-i v i t y and zero v i s c o s i t y made i n the Kruskal-Schwarzschild theory are not v a l i d , and may a f f e c t the comparison between experimental and t h e o r e t i c a l r e s u l t s . I n a d d i t i o n to those assumptions made i n the Kruskal-Schwarzschild theory, one other assumption has al s o been made i n e v a l u a t i n g the data. I n applying " l e a s t squares" curve f i t t i n g techniques, one i m p l i e s t h a t e r r o r s i n the measurements of F, A, uJ , and are normally d i s t r i b u t e d . Since the number of measurements i s i n s u f f i c i e n t to v e r i f y t h i s assumption, the use of " l e a s t squares" procedures must be regarded as a convenient non-subjective method of t r e a t i n g the data. The experimental r e s u l t s derived i n t h i s way may, however, be compared with those reported by other observers. The experimental work o f Curzon et a l ( l ° 6 o ) ; Green and N i b l e t t ( 1 9 6 0 ) 5 and Curzon and C h u r c h i l l ( 1 9 6 2 ) shows that the fastest growing i n s t a b i l i t y modes, observed a f t e r the pinch, have growth rates roughly one h a l f the value predicted by simple theory. The r e s u l t s presented here are also of t h i s order of magnitude. However, the experimental techniques adopted f o r the work described i n t h i s t h e s i s represent a considerable improvement on those used by e a r l i e r workers. For example, Curzon et a l ( I 9 6 0 ) evaluated UJ , "X , and a by taking s t a t i s t i c a l averages of measurements made on a large number of sin g l e Kerr c e l l photographs. Only one photograph was obtained from each discharge, so that the s t a t i s t i c a l averages of ^ 3 "X and / a can not be r e a d i l y r e l a t e d to each other. I n a s i m i l a r study with a framing camera Green and N i b l e t t ( i 9 6 0 ) have estimated the growth rate of the m ~ ^ 5 i n s t a b i l i t i e s of a 9-pinch. They f i n d that cJ//2T1a />s i s "of the order" of unity. From t h i s estimate, and from the s i m i l a r i t y between photographs of the l a t e r stages of t h e i r i n s t a b i l i t i e s and those photo-graphed by Lewis (195>0), they have concluded that these 9-pinch i n s t a b i l i t i e s are of the Rayleigh-Taylor type. Further comparison of t h e i r work with the r e s u l t s presented here i s not possible, however,.because of the lack of d e t a i l i n t h e i r paper. -56. They do not state, for example, whether they established that the in s t a b i l i t i e s grow exponentially with time or not. In addition, i t i s d i f f i c u l t to compare their measured results with the Kruskal Sehwarzs-child theory because of the considerable differences between the plasma geometry assumed i n the theory and that observed i n practice. Curzon and Churchill (1962) have also used a framing camera to study post-pinch i n s t a b i l i t i e s (possibly Rayleigh-Taylor in s t a b i l i t i e s ) i n argon and nitrogen Z-pinches. In an argon discharge they observed a single train of m = 0 sinusoidal i n s t a b i l i t i e s , three wavelengths long. However, the acceleration of the discharge boundary was calculated from only five frames so that i t may be very inaccurate. Furthermore, since the occurrence of such trains i s fortuitous ( i . e . one discharge i n f i f t y ) , the measuring accuracy can not be greatly improved. The only report on i n s t a b i l i t i e s i n the f i r s t stage of the Z-pinch has been given by. Hertz (1963). He uses the naturally occurring m - 0 i n s t a b i l i t i e s found i n a discharge i n hydrogen, and photographs them with an image converter. Hertz reports li n e a r l y time dependent amplitudes and exponentially increasing wavelengths for these i n s t a b i l i t i e s . He then compares his data with some theories of i n s t a b i l i t i e s i n the f i r s t stage of the Z-pinch put forward by Wyld (1958). Hertz' published photographs show that the discharge surface i s not at a l l sinusoidal and i t i s d i f f i c u l t to see how he can (with his equipment) measure the i n s t a b i l i t i e s to within the accuracy he claims. Although the observations referred to above demonstrate a correlation between the growth rate oo and the acceleration, a, none of the observers has made sufficiently accurate measurements to comment •57-on the wavelength dependence of (AJ , The growth rate, as a function of the wavelength X has been calculated for fluids of f i n i t e viscosity (Chandresekhar, 1955) and f i n i t e conductivity (Furth, Killeen, and Roseribluth, 1963)• Sufficiently accurate ^ versus A curves could be compared to the results of these theories i n order to estimate the plasma conductivity and viscosity. Whereas the i n s t a b i l i t i e s on the discharge column have been investigated by other workers, no one has reported fluctuations i n the acceleration of the discharge boundary. Perhaps this phenomenon i s caused by magneto-acoustic waves propagating in the layer of shock heated gas trapped between the shock front and the driving magnetic f i e l d . Further discussion of this phenomenon must be deferred, how-ever, u n t i l further experimental data i s available. 6 . Suggestions for Further Work The work described i n this thesis can be extended i n two ways. F i r s t l y , the techniques of exciting and studying i n s t a b i l i t i e s during the i n i t i a l stage of the Z-pinch can be improved and, secondly, i t i s possible to use the techniques developed to study i n s t a b i l i t i e s produced during the later stages of the discharge (e.g. post-pinch). The possible developments along these lines are considered below. I n i t i a l Stage of the Discharge The three main defects of the work described i n this thesis are: a) The technique employed f a i l s to excite longer wavelength ( X > k cm) i n s t a b i l i t i e s , b) L i t t l e i s known at present about the physical properties of ' - 5 8 -the discharge plasma. c) The measuring accuracy i s low. The wavelength l i m i t a t i o n on the i n s t a b i l i t i e s which can be s t u d i e d by the techniques d e s c r i b e d above i s s e r i o u s because i n s t a b i -l i t i e s cannot be produced i n c o n d i t i o n s where the snow-plow model i s a p p l i c a b l e . Since no experimental v e r i f i c a t i o n o f the snow-plow theory of i n s t a b i l i t i e s i s at present a v a i l a b l e , the problem of extending the range of i n s t a b i l i t y - w a v e l e n g t h s i s o b v i o u s l y of considerable importance. One way o f improving the present s i t u a t i o n i s to use b e t t e r methods of c o r r u g a t i n g the i n i t i a l surface of the discharge column. Instead of u s i n g glass r i n g s one could, f o r example, use a discharge v e s s e l i n which the r a d i u s of i t s c i r c u l a r c r o s s - s e c t i o n v a r i e s s i n u s o i d a l l y as a f u n c t i o n of the a x i a l p o s i t i o n . U n f o r t u n a t e l y , f o r l a r g e r tubes, the cost of production would be too h i g h . However, an experiment u s i n g a smaller system i s now being completed i n the Plasma P h y s i c s Labora-t o r i e s at t h i s u n i v e r s i t y . More i n f o r m a t i o n about the p h y s i c a l p r o p e r t i e s of the discharge plasma i t s e l f may be obtained by measuring the c u r r e n t d i s t r i b u t i o n w i t h magnetic probes. The e l e c t r o n and i o n d e n s i t i e s and temperatures can be measured u s i n g Langmuir probes and time r e s o l v e d spectroscopy. The accuracy of the measurements, as shown by Table I , page k6 , depends c r i t i c a l l y on the number of framing camera photographs (n) which can be obtained while the a c c e l e r a t i o n of the discharge boundary i s constanto I f the discharge c o n d i t i o n s are not changed, n can be i n c r e a s e d o n l y by decreasing the frame exposure time. U n f o r t u n a t e l y , t h i s -5> procedure would considerably reduce the l i f e of the bearings i n the framing camera turbine, and therefore a compromise between maximum accuracy and maximum turbine l i f e must be sought. However, n can be readily increased by 2%% with a corresponding reduction i n errors i n and CO . For values of /a" calculated from n = 8 instead of n - 6 points, for example, the estimated absolute error i n /a would drop from t 5 x 10 ^ sec" 1 to t 3,5 x 10^ afi sec" 1 . Similarly the absolute error i n (JJ would be changed from i " 1.3 x 10^ sec" 1 to 1 1 x 1 0 ^ sec" 1. The s t a t i s t i c a l errors could be reduced by taking more photographs. Here again, the f i n i t e l i f e of the camera turbine must be considered. Alternatively, n may be increased by changing the discharge conditions and maintaining the frame exposure time constant. The collapse of the discharge column could be slowed down either by increasing the i n i t i a l argon gas density or reducing the peak discharge current. Both methods would slow down the dynamic characteristics and hence increase n and the measuring accuracy. Unfortunately, such procedures result i n an undesirable reduction i n the plasma temperature. To slow down the collapse stage of the discharge column and maintain high temperatures, gases of high atomic number such as cesium or xenon could be used. These two gases seem to offer the best prospects of improving the experiments described i n this thesis. It must of course f i r s t be established that the i n i t i a l stages of the Z-pinches i n these gases do not possess the undesirable characteristics observed i n the Z-pinch i n nitrogen (see Plate A . l ) . 60-P o s t - P i n c h I n s t a b i l i t i e s The e x i s t i n g photographs o f the discharges i n argon f o r 2, 3, h and 6 cm wavelengths show reasonably s i n u s o i d a l post-pinch i n s t a b i l i t i e s (see P l a t e A.Ilib,page 66). These i n s t a b i l i t i e s have not been evaluated e x t e n s i v e l y , since they do not conform to the t h e o r e t i c a l models. For example, the r a t i o of wavelength t o discharge r a d i u s ( X /R) i s of the order of u n i t y and plane geometry can not be assumed. Furthermore, post-pinch i n s t a b i l i t i e s are l i k e l y to be complicated by unknown f a c t o r s (such as plasma contamination by w a l l m a t e r i a l s ) . However, f o r a c o n t i n u i n g c o n s i s t e n t research program these i n s t a b i l i t i e s should be s t u d i e d i n more d e t a i l . -61-CHAPTER V CONCLUSIONS The original results presented in this thesis ares 1. Single mode m = 0 instabilities can be produced on the normally-stable plasma surface of an argon Z-pinch. The experimental conditions are such that i t is possible to compare the development of the instabi-l i t ies with the predictions of the linearized magneto hydrodynamic theory. 2. Within experimental error the amplitude of these instabilities grows exponentially with time. 3. For the wavelength X - 2 cm, the growth rate ( ^ ) of these instabilities is correlated with the acceleration of the plasma boundary in accordance with the simple Rayleigh-Taylor theory, i . e . ^ = (5.7 t 1) sTa" sec"1 k» The acceleration of the plasma boundary does not appear to be constant during the f irst stage of the Z-pinch i n argon. -62 APPENDIX A T h i s s e c t i o n presents i n t e r e s t i n g r e s u l t s which are not necessary to the main body of the t h e s i s . These i n c l u d e photographs o f "post-pinch" i n s t a b i l i t i e s i n argon discharges, as w e l l as photographs of n i t r o g e n and hydrogen discharges. The s i n g l e frames shown have an average exposure time of 0 . 2 5 / Jsec* The time (T) at which the photograph i s taken, the gas, the i n i t i a l p r essure, and the r i n g spacing are g i v e n i n the p l a t e c a p t i o n s . The time T i s measured from the s t a r t of the discharge. P l a t e I shows a sequence of frames of an unperturbed discharge i n n i t r o g e n . The enlarged frame ( P l a t e I I . a ) taken from t h i s sequence shows the streamers and l a c k of u n i f o r m i t y q u i t e c l e a r l y (compare w i t h argon discharge P l a t e I V . I . a , page 3 8 ) . Another s i n g l e frame taken from t h i s same set of photographs shows a t y p i c a l example of post-pinch i n s t a b i l i t i e s i n discharges of most gases (see P l a t e I I . b , T = 11 s e c ) . The best example of a perturbed pre-pinch i n s t a b i l i t y i n n i t r o g e n i s given i n P l a t e I I . c . I n no case are these i n s t a b i l i t i e s as uniform as those i n the argon discharges. P l a t e I I . d may be compared with P l a t e I I , b . The e f f e c t of the r i n g s on the post-pinch i n s t a b i l i t i e s i s q u i t e c l e a r . The s i n g l e frames of the perturbed SOpEg argon discharge shown i n P l a t e I I I . a and I l l . b i n d i c a t e t h a t the technique of studying •63-i n s t a b i l i t i e s can be e a s i l y extended For completeness, examples of a are included (see Plate I I I . c and d) to the region a f t e r the pinch, perturbed discharge i n hydrogen Next Pages PLATE A.I. F i r s t stage of a nitrogen discharge ( i n i t i a l pressure 375// Hg). -6U-c d PIATE A . I I . N i t r o g e n discharge ( i n i t i a l pressure 375 Hg) a) Unperturbed pre-pinch streamers (T - U/Jsec) b) P o s t - p i n c h n a t u r a l l y o c c u r r i n g i n s t a b i l i t i e s (T r 10yusec ) c) Perturbed pre-pinch i n s t a b i l i t i e s (T - U/Jsec, r i n g spacing u cm) d) Perturbed post-pinch i n s t a b i l i t i e s (T - lOpsec, r i n g spacing k cm) c d P l a t e A . I l l <> a) Perturbed p o s t - p i n c h i n s t a b i l i t i e s i n argon (T - 6/^  sec, 50^Hg, r i n g spacing 3 cn) b,) Perturbed post-pinch i n s t a b i l i t i e s i n argon (T "^y^sec, £ 0 / j H g , ring spacing 2 cm) c) Perturbed post-pinch i n s t a b i l i t i e s i n hydrogen (T - hy sec, 0.5" mm Hg, r i n g spacing 2 cm) d) Perturbed pre-pinch i n s t a b i l i t i e s i n hydrogen (T - 5 ^ s e c 3 2 mm Hg, r i n g spacing 2 cm). APPENDIX B The values of mean rad i u s (F) and i n s t a b i l i t y amplitude (A.) ( f o r each frame) are t a b u l a t e d below f o r wavelengths ( ^ ) o f .2, 3 and k cm. r". and A are i n m i l l i m e t e r s , w h i l e the time i n t e r v a l 6 t i s given below each column i n microseconds. a) > - 2 cm 66.6 63.0 6U.3 63.0 61.9 59.5 57.2 56.0 56.6 51.8 U7.6 U6.ii kh.5 1*2.1 38.5 36.7 1.68 1.68 2.52 2.52 2.52 2.52 3.36 3.36 2.9k 2.9k 2.52 2.52 3.78 3.78 2 .9U 3.36 61.9 6 2 . 5 6 2 . 5 60.7 57.2 5U.8 52.1, 51.2 5 0 . 0 U5.7 k2.l 1,2.1 36.7 3 5 . 1 32.7 30.3 1.68 1.68 2.52 2.52 2.52 3 .36 U.20 3 .36 3.36 3.78 -3.36 30-78 3.36 3.78 3.78 3.78 6U.3 6U.3 6U.3 63.1 61.9 58.9 56.0 5U.2 52. k 1+8.8 U7.0 U3.9 kl.k 38.5 36.7 1.68 1.68 2.52 2.52 2.52 2 .9U 3.36 2.9k 2.52 3.36 2.9U 3.36 2 .9U U.62 U.20 65.5 65.5 6U.3 59.6 57.8 57.2 5U.2 51.2 U7.6 U6.U U2.1 38.5 A 1.68 1.68 1.68 2.52 2.10 2.52 2.9k 3.36 3.36 3.78 3.78 3.78 A t —0.25 4 t - 0.25 A t - 0 . 2 5 A t = 0.30 b) )\ 3 cm 72.6 72.6 71.5 71.5 69.3 66.0, 6L*.9 61.6 59.U 56.1 55.0 50.6 1*8.2 U5.1 1*2.9 39.6 35.2 l l . o 2.2 2.2 3.3 3.3 h.h 5.5 h •5 5 5 .6 6 7 6 7 8.8 At-0.33 72.6 72.6 71.5 70.1* 69.3 67.1 63.8 62.7 59.1* 58.3 55.0 50.6 1*8.1* 1*7.3 1*5.1 1*2.9 39.6 3.3 h.h h.h h.h .h .3 .1* .1* .5 5.5 6 6 3  h  h. 5  6. 6 6.6 7.7 A t =0.31 72.6 72.6 72.6 70.1* 70.1* 68.2 61*. 9 61.6 59.1* 57.2 53.9 52.8 1*7.3 1*5.1 U5.1 2.2 2.2 3.3 3.3 3.8 h.h 6.0 3.8 3.3 h.h h.h 5.5 6.0 5.5 6.0 A t - 0.33 70.1* 68.2 62.7 6U.9 62.7 58.3 57.2 52.8 50.6 1*8.1* U6.2 1*2.9 1*1.8 At - 0.31 69. c) X = U cm 70.3 70.3 69.1 69.1 66.7 66.7 6U.3 63.1 58.3 57.2 53.6 52.ii U8.8 U8.8 U3.9 Uo.3 39.1 35.7 37.9 1.68 1.6.8 2.52 2.52 2.52 2.52 2.52 3.36 3.36 2.52 3.36 U.20 3.36 3.36 5.0U 5.88 5.0U 5.88 7.56 A t = o.2U5 71 .5 70.3 69.1 69ol 69.1 67.9 65.5 65.5 63.1 58.3 56.0 53.6 51.2 U8.8 U8.8 Ul.U Ul.U Uo.3 39.1 35.7 3U.5 A t 0.8U 1.68 2.52 2.52 2.52 3.36 3.36 3.36 5.0U 3.36 3.36 3.36 3.36 3.36 3.36 5.0U 5.0U U.20 3.78 U.20 3.36 ^0.2U8 73.9 69.1 69.1 70.3 70.3 6 7 . 9 67.9 6U.3 6U.3 59.6 57.2 5U.8 53.6 51.2 U5.1 U2.7 Ul.U A 1.26 :2.52 2.52 , 1.68 1.68 1.68 3.36 U.20 U.20 2.52 2.52 2.52 3.36 3.36 5.88 5.88 5.0U At =o.2U5 71.5 72.7 71.5 70.3 67.9 66.1 63.7 61.9 58.9 57.8 56.0 53.0 51.8 U8.2 U7.6 U5.7 U3.9 U2.1 39.1 35.1 31.5 30.9 1.68 2.52 2.52 3.36 3.36 3.78 U.62 U.20 5.U6 5.U6 5.0U U.62 6.30 5.U6 5.88 7.1U 7.56 7.1U 7.56 7.1U 7. Hi 7.56 At = o.2U5 70.9 70.9 69.1 66.1 6U.9 63.1 61.9 59.6 56.6 5U.2 53.0 50.0 U6.U 1*2.1 39.7 37.3 33.9 32.7 32.1 2.9U 2.9U 3.36 U.62 U.62 U.20 U.20 U.20 •U.62 U.62 5.U6 5.88 5.88 7.56 6.30 6.30 7.1U 7.1U 7.98 6t = 0o262 71.5 69.1 68.5 67.9 67.3 6U.3 62.5 59.6 57.8 56.6 56.0 5U.2 50.6 U9.U 1.68 3.36 2.9U 3.36 2.9U 3.36 3.78 U.20 U.62 3.78 U.20 3.78 U.62 U.62 A t = 0.218 BIBLIOGRAPHY Bishop, A., ,(195"8) P r o j e c t Sherwood, Addison Wesleyo Burkhardt, L. C. and Lovberg, R. H. (1958) P r o c . Geneva Co rife re nc Chandresekhar, So (1955) Proc. Camb. P h i l . Soc. 5 l 5 162. Cormaek, G„ D . and Barnard, A. J. (1962) Rev. S c i . I n s t . _335 6o6. Curzon, F. L. and C h u r c h i l l , R. J . (1962) Can. J . Phys. iuO, 1191. Curzon, F. L„, F c l k i e r s k i , A., Latham, R. and Nation, J.A. (i960) Proc, Roy, Soc. A 257, 386. Curzon, F. L., Hodgson, R. T. and C h u r c h i l l , R. J . (1963) Can. J . Phys. iO., I 5 u 7 . Daughney, C. C . (1963) Master's Thesis, U.B.C. E l l i o t , H. (1962) Rep. Prog. Phys. 26, 11*5. E n g e i , A, von (1959) Nature 183, 573. F u r t h , H. P., K i l l e e n , J . and Rosenbluth, M. N. (1963) Phys. F l u . 6, Ii59. Green, T. S, and Nib L e t t , G. B. (i960) Nuc. Fusi o n 1, u2. H a r r i s , E, G» (1962) Phys. F l u , 5 5 1057. H e r t z , W. (1962) Z. Naturf. ITa^ 681 . K r u s k a l , M, and Sehwarzschild, M. (195U) Proc. Roy. Soc. Lond, A 2 3 3 , 3)48, Kuwabara, S, (1963) J. Phys. Soc. Jap. 18, 713« Lewis, D. J . (1950) Proc. Roy, Soc. Lond. A 2 0 2 , 81 . -71-Rosenblufch, M. N. # Garwin, R... and Roseribluth, A. (195U) Los Alamos Report 18?0. Segre, 3. E, and A l l e n , J , E. ( i 9 6 0 ) J , S c i . I n s t . 37, 369. S p i t z e r , L. (1956) P h y s i c s of F u l l y I o n i z e d Gases, I n t e r s c i e n c e Summer School of Plasma P h y s i c s Ris8 ( i 9 6 0 ) . T a y l o r , G. (1950) Proc, Roy. S o c Lond. 201_A, 193. Theophanis, C A. ( i960) Rev, S c i * I n s t c 31 , U27. Worthing, A. G. and Geffner, J . (19U3) Treatment of Experimental Data, John W i l e y and Sons 0 

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