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Magnetohydrodynamics of turbulent accretion discs around black holes Pudritz, Ralph Egon 1979

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HAGNETOHY DRODYNAMICS OF TURBULENT ACCRETION DISCS AROUND BLACK HOLES by RALPH EGON PUDRITZ B. Sc., University of B r i t i s h Columbia, 1973 H.Sc., University of Toronto, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept t h i s t h e s i s as conforming to the required standard The University of B r i t i s h Columbia -7 December, .1979 - • © R a l p h Egon Pudritz, 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e 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 r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbropk Place Vancouver, Canada V6T 1W5 i i Abstract The Cyg X-1 X-ray source i s believed to be comprised of an accretion disc around a central black hole. We apply the methods of Mean F i e l d Electrodynamics to the study of magnetic processes in such an accretion disc. By decomposing the magnetic f i e l d i n the disc into mean and flu c t u a t i n g components, the observed X-ray properties of this system may be accounted f o r . It i s found that intense, short l i v e d magnetic fluctuations may occur which give r i s e to s o l a r - l i k e f l a r e s on the surfaces of the accretion d i s c . The energy releases and time scales of such f l a r e s i s found to provide a physical basis f o r the observed shot^-noise l i k e character of the X-ray emission from the system. It i s demonstrated that a rather strong, large scale magnetic f i e l d can be generated by turbulent dynamo action i n the accretion disc. This r e s u l t i s the reason why magnetic f i e l d s may play a v i t a l role i n these systems. The long time averaged structure of the accretion disc i s determined by the Maxwell-stress due to the mean f i e l d , and i s i n agreement with the "standard" cool accretion disc models. We prove that on intermediate time and length scales, the Maxwell stresses due to the magnetic fluctuations remove the known i n s t a b i l i t y of "standard" accretion disc models to r i n g -i i i l i k e "clumping" and subseguent heating of the gas. This r e s u l t shows that the hard X-ray emission of the Cyg X-1 source must arise from either a hot corona, or intense solar-type f l a r e s above the disc surfaces. If the hard X-ray emission a r i s e s from non-thermal electron populations accelerated i n the f l a r e s , i t i s found that t h i s emission must occur in a rapid "flash-phase" on submillisecond time scales. These f l a r e s occur well away from the inner disc boundaries so that we believe that submillisecond variations of the Cyg X-1 source need not be a test of the rotation of the central black hole. i v Table Of Contents Chapter 1. Introduction .................................... 1 1. Observations Of X-ray Sources Associated With Binary S t e l l a r Systems 1 2. The Basic Physics Of Accretion Discs .............. 10 3. Previous Work On Magnetic Processes In Accretion Discs 22 4. Laboratory "Solar-Flare" Experiments Scaled To Cyg X-1 27 5. Outline Of The Thesis ,...34 Chapter 2. Magnetic Fluctuations In A Turbulent Accretion Disc .........................36 1. Introduction 36 2. Stewart's Analysis Of Energy Balance In Turbulent Accretion Discs ......................... 38 3. Eguations Of Motion Including Magnetic Fields 54 4. Energy Balance For The Fluctuating Fields Including Magnetic Effects 70 Chapter 3. Solution Of The Induction Eguation For The Mean F i e l d B ...88 1. Introduction ....88 2. Analysis Of The Induction Eguation For The Mean Fie l d .. .. 94 3. Solutions To The Eguations In The Stationary Case 107 4. Matching To An External Vacuum Solution ........... 124 5. Small Deviations From Eguilibrium 138 Chapter 4. Implications For Accretion Disc Models 154 1. Introduction 154 2. E q u i l i b r i a t i o n Of The Mean F i e l d And Consequences For Accretion ......155 3. The Long Time Averaged Effects Of The Magnetic Fluctuations ......................................162 4. X-ray Spectra From Solar Type Flares In The Cyg X-1 Source .....178 Cone lu s i ons ......... ............................... ........195 Bibliography ..............199 Appendix A Mean F i e l d Electrodynamics .......203 Appendix B The Calculation Of Correlations Between Velocity And Magnetic F i e l d Fluctuations ......217 Appendix C The Integral Representation For 0 .(z) 225 Appendix D Asymptotic Analysis Of U^CXrZ) 231 Appendix E Expansions Of U„(K,2.) About Z=0 ......244 v i L i s t Of Tables Table 1..Masses MK Associated With X-ray Sources . . . . . . . . . . . 4 . Table 2. Summary Of Shot Noise Aud Millisecond Burst Parameters 10 Table 3. C h a r a c t e r i s t i c s Of Solar Type Flares In Cyg X-1 . . 3 2 Table 4. Asymptotic Analysis: The Functions /UM{K,Z) . . . . . . . 1 1 6 Table 5. Asymptotic Analysis: The Functions/U^(,K,Z) . . . . . . . 1 1 8 Table 6. Asymptotic Analysis: The Functions / ^ ' ( K , Z ) . 1 19 Table 7. Expansions About Z=0: The Functions U/« And Q M . . . . 1 2 2 Table 8. Expansions About Z=0: The Functions u!* And P^  . . . . 1 2 3 Table 9. Saddle Points Of f(r) 234 v i i L i s t Of Figures F i g . 1. X-ray Spectrum Of Cyg X-1: Low State 6 Fig. 2. X-ray Spectrum Of Cyg X-1: High State 7 Fig. 3. Mass Accretion By Roche Lobe Overflow 12 F i g . 4. Geometry Of The F i e l d Line Reconnection Experiments 29 Fig. 5. The <* - E f f e c t In Turbulence With H e l i c i t y 100 Fi g . 6. Contours Defining The Solutions 0 m(z) 111 Fi g . 7. Hard And Soft X-Ray Flux From A Solar Flare 181 Fig. 8. Directions Of Steepest Descent 236 Fi g . 9. Paths Of Steepest Descent ..240 Acknowledgement F i r s t and foremost, I would l i k e to express my most he a r t f e l t gratitude to my supervisor, Prof. G. G. Fahlman. His fine physical i n t u i t i o n , his encouragement, and his emphasis on getting as close to the physical observations as the theory allows have a l l gone to make my work as h i s student a rewarding experience. I wish to thank him for a l l that he has done for me, and to express my deep respect for his s c i e n t i f i c a b i l i t y and his personal g u a l i t i e s . I wish to thank my former supervisor Prof. M.H.L Pryce for so many di f f e r e n t things. Conversation with him led me to work on accretion problems. His f i n a n c i a l support aided me for the f i r s t two years of t h i s work and discussions with him saved me from making some i d i o t i c errors. I thank Dr. John G i l l i l a n d for many stimulating conversations about dynamo theory and for introducing me to so many books and a r t i c l e s on the subject. This in t e r a c t i o n saved me much thrashing about. I thank him f o r valuable comments on the f i r s t draft of t h i s work. I thank the s t a f f and students of the Department of Geophysics and Astronomy, where most of t h i s work was carried out, for the i r h o s p i t a l i t y and friendship. In p a r t i c u l a r , I thank Dr. Serge Pineault and John Davies for many int e r e s t i n g discussions, and Prof,. . J. Auman for valuable considerations about f l a r e models. I am p a r t i c u l a r l y grateful to P a t r i c i a Monger for her help with the actual physical production of t h i s thesis. F i n a l l y , I acknowlege an NRC Postgraduate Scholarship on which two years of the work was carried out and a research assistantship from Prof. Fahlman over the l a s t six month period. X L i s t Of Symbols ffydrodynamics u Boot mean square v e l o c i t y fluctuation c s Speed of sound Mx Mass of central compact object associated kith X-ray source • M Mass transfer rate to accretion disc Mt Turbulent Mach number Vn = (GMx/r)4 Keplerian speed (toroidal) R=VK r/v > 10 ? Reynolds number t|C = r/V K Keplerian time scale t p=r/U r Radial d r i f t time scale T" Correlation time for turbulent eddies 21 Surface density ( v e r t i c a l l y averaged gas density) V e r t i c a l l y averaged component of stress tensor giving r i s e to outward r a d i a l transport of angular momentum in an accretion disc z 0 Half-thickness of accretion disc ftaqnetphydrodynamies b Root mean square magnetic fluctuation £ = u'xb' Mean EMF a r i s i n g from co r r e l a t i o n of flu c t u a t i n g velocity and magnetic f i e l d s Pseudoscalar a r i s i n g i n non-mirror symmetric turbulence and which gives r i s e to regeneration cf the mean magnetic f i e l d *Vr Turbulent d i f f u s i v i t y for mean magnetic f i e l d x i ^ Ambient d i f f u s i v i t y f o r magnetic f i e l d » 1 High conductivity l i m i t E M = V kr/ Ay~10'° Magnetic fieynolds number y = e t/z*\ T Batio of regeneration due to dynamo action to dissipation for mean poloida l magnetic f i e l d %=Vic/r/*vr Hatio of regeneration due to d i f f e r e n t i a l rotation t.o d i s s i p a t i o n for mean toroidal magnetic f i e l d (Chap. 3) P~\XJ)c)/x Measure of dynamo action to d i f f e r e n t i a l rotation (Chap. 3) r= l/X. } r Dimensionless r a d i a l co-ordinate (Chap. 3) z= (fx_ ) H z Dimensionless v e r t i c a l co-ordinate (Chap. 3) Dimensionless r a d i a l wavenumber f c r mean f i e l d modes (Chap. 3) A = |zJ 3 Dimensionless parameter; large when H T « 1 and | Z | ~ Z 0 2T"J Maxwell stress due to mean magnetic f i e l d Maxwell stress due to auto-correlaticn cf fluctuating magnetic f i e l d s 1 Chapter J. In troduction 1=. Observations Of X-ray Sources Associated - With Binary S t e l l a r Systems Since the discovery of the f i r s t g a l a c t i c X-ray source by Giaconi et a l (1962), intensive observational and t h e o r e t i c a l e f f o r t has brought us to the point where strong arguments can be made for the existence of a black hole. Many of these X-ray sources can be accounted for i n terms of a hot gas s p i r a l l i n g into a white dwarf, neutron star, and in one case (the Cyg X-1 source), a black hole. These objects are themselves i n close proximity to a more normal type of star. This thesis analyzes the magnetohydrodynamics of a turbulent disc of hot gas ( the so-called "accretion di s c " ) around a central black hole. I t i s the contribution of t h i s work to apply the methods of mean f i e l d electrodynamics (see Roberts .(1971) and Moffat (1978) for reviews of this theory) to t h i s problem. We intend to show that i f the magnetic f i e l d in a turbulent accretion disc i s regarded as having large-scale mean, and microscale fluctuating components; then the observed rapid v a r i a b i l i t y of the X-ray output of the Cyg X-1 source can be explained i n terms of solar-type f l a r e s a r i s i n g from intense magnetic fluctuations and that the o v e r a l l 2 structure of the accretion disc i s controlled by the large--scale mean magnetic f i e l d . Other authors have concentrated only on the study of chaotic magnetic f i e l d s . Osing our approach, i t i s shown that a large-scale mean magnetic f i e l d can be generated by turbulent dynamo action i n the accretion disc and that an intimate connection exists between the mean and f l u c t u a t i n g magnetic f i e l d s . These types of results are not new to the theory of mean f i e l d electrodynamics, however, to our knowledge, they have never been considered within the physical framework of a turbulent accretion d i s c . I t i s our contention that the p o s s i b i l i t y of dynamo action i n such a system makes the magnetic f i e l d a c r u c i a l element i n the interpretation of the Cyg X - 1 observations. A more detailed outline of the thesis i s presented i n the l a s t section of t h i s chapter. The observations of the Cyg X - 1 source are discussed in the remainder of t h i s section. Section 2 outlines the basic physics of an accretion disc and how the gross observational features can be accounted f o r . Section 3 reviews previous work done on magnetic f i e l d s i n accretion discs while section 4 presents ideas which motivated our own work. The f i r s t important feature of these sources i s t h e i r enormous power output, which for the Cyg X - 1 source i s of order 1 0 erg s which i s ten thousand times the power of the sun. 3 3? -I There appears to be an upper l i m i t of 10 erg s for the known sources. This i s a suggestive observation because t h i s i s of the order of the "Eddington l i m i t " of luminosity L c r for an object of mass M { measuring M in units of solar mass M 0 ) which corresponds to the condition wherein the radiation pressure exerted on a gas eguals the g r a v i t a t i o n a l force of the object. We say more about th i s i n section 2. Of the nine o p t i c a l l y i d e n t i f i e d X-ray sources, seven are known to be spectroscopic binaries. The fac t that X-ray sources are members of binary systems provides a very important handle on the system, that i s , i t s mass. The detailed analysis of mass determination for the observed star M0t,t and of the unseen companion Mx i s reviewed i n Bahcall (1978). The allowed ranges of the masses associated with the X-ray sources are l i s t e d i n Table 1 (adapted from Bahcall (1978)) . 4 Table 1. Masses Mx Associated-With X-ray Sources-SOURCE Mx(SOLAR MASS UNITS) Vela XR-I 1.0* M* f 3.4 SMC X-I 0.5* M„ * 1.8 Cen X-3 0.7* Mx * 4.4 Her X-I O.in Mx4 2.2 3U 1700-37 0.6 * M* Cyg X-I 9 < M«s 15 The Cyg X-1 source stands out because of i t s high mass MX^9M0 . The v i s i b l e i n t h i s system i s an OB supergiant with mass i n the range 15-25 M© , having an op t i c a l magnitude of 9. The binary period of the system i s 5-6 days. The v i s i b l e star varies by 0.07 magnitudes with a double peaked l i g h t curve which i s evidence for t i d a l d i s t o r t i o n since a t i d a l l y distorted star would present a changing area and hence an apparently changing luminosity with a frequency of twice the or b i t a l - r e v o l u t i o n frequency. A comprehensive discussion of the o p t i c a l observations of the Cyg X-1 source may be found i n 5 Bolton ( 1975). We s h a l l henceforth be considering only the Cyg X-1 source, and turn to a summary of the X-ray observations of t h i s source. X-ray Observations Of Cyg. x-1 This source, discovered by Boyer et a l (1965) has a hard X-ray spectrum and i s highly variable at a l l X-ray energies. (1) X-ray Spectrum The Uhuru s a t e l l i t e observations in the 2-10 kev range have been extended into the 15-250 kev range by the OSO 8 s a t e l l i t e ( see Dolan et a l (1979) ). One of the most int r i g u i n g aspects of the spectrum i s that i t undergoes t r a n s i t i o n s between two states: a high luminosity state with LU;5U * S S x io e<r«^ S and a low luminosity state with assuming that the distance to the source i s 2.5 kpc. The high state has an excess of energy in the 2-7 kev band and a lower amount of energy i n the >7 kev domain as compared to the low state. Thus, a high to low t r a n s i t i o n was apparent i n the Uhuru observations during March-April 1971 ( see Sanford et a l (1975) ), while Dolan et a l (1 979) find a low to high t r a n s i t i o n occurring i n Nov 1975. Dolan et a l f i n d that over the 20-150 kev range of the X-6 ray spectrum a power law of the form ol f could be f i t t e d . They state that the spectra may a l l be represented by a single power law expression whose spectral index i s d i f f e r e n t for the two inte n s i t y states. Their high state spectrum i s reproduced i n Fig. 1 while F i g . 2 shows fi v e low state spectra they took. The best f i t single power law parameters , over the 20-150 kev range are 5" a- |.?3 i o.ofo J j 2.z\ i 0.\g C = * 0.5-7 C = 6.«"o i " 0.<?7 Fig.. J[ X-ray Spectrum Of Cyg X-1: H-igh State (From Dplan Et Al--..—Spectrum of Cyg XR-1 observed 1975 November 17, 0200 UT-1530 U T ($ = 0.40-0.50), when the source was in a high state. The values of a, C , and £ 0 of the power law which best represents the data are given in Table 2; the power law is shown in the figure as the solid line. The dashed line is the power law which best represents the low-state spectrum shown in Fig. 2d. 10 20 50 100 300 E (keV) 7 Fig. 2 X-ray Spectrum Of Cyg X-1:_-Low State (From Dolan Et Al (1979)) 10 10 20 50 100 300 E (keV) 10 20 50 100 300 E (keV) 10 20 50 100 300 E (keV) 20 50 100 E (keV) 50 100 E (keV) -Typica l spectra of Cyg XR-1 observed by OSO 8 when the source was in a low state as defined at lower energies. The straight line in each spectrum is* the single power-law expression which gives an ^eptable m . n = ^ observed intensities. The resultant values of a, C , and £ 0 for each spectrum as defined in eq. (1), are given m Table 2 (a) SP^trum observed 1977 November 13,1050 UT-November 14,0015 U T (<X> = 0-30-0.40) (6 Spectrum observed 1977 October 22,q01 1MJT ^ 1445 U T ($ = 0.30-0.40). (c) Spectrum observed 1976, November 11 630 UT-November 12 0600 U T (* - 0 (d) Spectrum observed 1976 November 10,0000 UT-1330 U T (0 = 0.50-0.60). (e) Spectrum observed 1975 November 14,2000 V l November 15, 0930 U T (0 = 0.00-0.10). 8 It i s important to note that Dolan et a l f i n d that about one t h i r d of t h e i r spectra could also be well represented by a double power law i n the 20-150 kev range with an increase i n the spectral index of 0.5 or larger. The break-point between the two power laws occurred between 40 and 125 kev for d i f f e r e n t spectra. F i n a l l y , Dolan et a l considering t h e i r highest energy data points f i n d evidence for an exponential cut-off i n the spectra of both states somewhere between 150 and 200 kev. The system seems to spend most of i t s time in the low state. (2) Short Term X-ray V a r i a b i l i t y For time p r o f i l e s over the 1-50 kev range , the g u a l i t a t i v e appearance i s characterized by a continual aperiodic t r a i n of spiky variations with pulse sizes ranging up to a few times the average i n t e n s i t y on time scales of a fr a c t i o n of a second ( see Oda (1977) f o r a review ). As Oda points out, i t appears that the pulses are a c h a r a c t e r i s t i c of the low state ( greater predominance of hard X-ray component ) and seems to get buried to some degree i n soft X-ray emission during periods when the source i s in the high state. This i s a key point and i s considered again i n Chapter 4. T e r r e l l (1972) successfully simulated the time p r o f i l e over periods of fract i o n s to tens of seconds i n terms of a 9 shot noise model comprised of a superposition of randomly occurring pulses ( instantaneous r i s e and exponential decay ) of constant amplitude and with c h a r a c t e r i s t i c times of fr a c t i o n s of a second-In addition to the variations discussed Oda et a l (1971) and Eotschild et a l (1974) found evidence f o r millisecond bursts, which appear to occur i n bunches- The energy of these bursts appears to be lower than the o v e r a l l emission. The r e a l i t y of these millisecond bursts has been questioned by Weisskopf and Sutherland (1978) who f i n d that "spurious" millisecond bursts may arise as an a r t i f a c t of the data analysis and may have nothing to do with the physical processes associated with the X-ray source. We summarize the shot noise and millisecond burst parameters in Table 2 ( see Rotschild et a l (1977) ). 10 Tabl e 2_. Summary Of Shot Noise - And Millisecond Burst Parameters EVENT TYPE SHOT NOISE MILLISECOND BURST ENERGY/EVENT 10** ergs 10 3 rergs CHARACTERISTIC 10"' s 10 _ i s TIME EVENTS/SECOND 8 s _ l 100 s"' The simplest explanation of the Cyg X-1 source i s that of an accretion disc about a black hole of mass M X=10M© with accretion rates of 10 MGyr'' a r i s i n g from mass outflow ( Roche lobe overflow ) from the v i s i b l e star { see Bolton (1975) ). The reader may consult Kellogg (1975) for a discussion of the v i a b i l i t y of alternate models. Henceforth, we s h a l l only be considering these accretion disc models for the Cyg X-1 source. 2_ The Basic Physics Of Accretion Discs The observations suggest that the v i s i b l e star i s t i d a l l y d i storted. This leads us to consider matter loss from the star 11 by Roche-lobe overflow. We generate the so-called Roche eguipotentials by going into the rotating frame of the binary system and drawing eguipotentials of the g r a v i t a t i o n a l and c e n t r i f u g a l potentials. Consider the case when the lower mass object Mx i s compact and the larger mass star M ^  expands to f i l l i t s Roche-lobe (see Fig. 3). Matter may then leave i t s surface at the point of "zero gravity" ( the inner Lagrangian point L i ) and flow over toward Mx. Such a gas stream w i l l pick up angular momentum vi a the C o r i o l i s forces and go into o r b i t around Mx rather than f a l l i n g d i r e c t l y i n (see numerical calculations of Flannery (1975) f o r a detailed examination of the hydrodynamics). The matter stream and the orbiting gas w i l l stay in the o r b i t a l plane of the binary system. V i s c o s i t y of the gas w i l l lead to the gradual s p i r a l l i n g of the gas i n towards the ce n t r a l object Mx, thereby forming the so-called accretion d i s c . 12 EiSLs. 1 Hiss Accretion By Roche- Lobe Overflow IFrom Novikov And Thorne J197311 The basic physics of an accretion disc around a compact object was worked out by Prendergast (1960), Lynden-Bell (1969), Pringle and Rees (1972), and Shakura and Sunyaev (1973), and comprises the "standard model" of accretion d i s c s . 13 E e l a t i v i s t i c corrections are reviewed by Novikov and Thorne (1972). The p a r t i c l e s of an accretion disc moving i n approximately Keplerian orbits about the central compact object, lose t h e i r angular momentum due to f r i c t i o n between adjacent rings of gas. These p a r t i c l e s therefore gradually s p i r a l towards the central object releasing g r a v i t a t i o n a l pot e n t i a l energy. Part of thi s energy increases the k i n e t i c energy of rotation so that at every radius the velocity i s Keplerian to good approximation, and the other part i s converted into thermal energy by the v i s c o s i t y and subsequently radiated from the disc surface. The angular momentum i s transported out to the outermost portion of the disc with the resu l t that some of the matter a r r i v i n g i n the mass stream i s flung away from the disc and escapes the system. For Cyg X-1, the outer radius of the disc i s of order 5x10^ km. The angular momentum transport i s provided by either the turbulence, the magnetic f i e l d ( present i n the matter that streams to the disc ), or both. Modelling these stresses i s one of the most important aspects of accretion disc theory, although for stationary discs, many of the basic observed properties of the system are independent of such models. The action of the turbulent ( Reynolds ) stress and/or the magnetic ( Maxwell ) stress gives r i s e to a small inward 14 r a d i a l v e l o c i t y by which matter matter slowly d r i f t s into the inner regions of the dis c . In the case where the c e n t r a l compact object i s a black. hole, general r e l a t i v i s t i c considerations show that the innermost stable, Keplerian-like o r b i t that can occur for a p a r t i c l e i n orbit about a non-rotating hole occurs at c* where r^ i s the Schwarzschild radius for a black hole of mass Mx. Matter w i l l drop staight into the hole once i t reaches t h i s inner o r b i t . This r a d i a l i n f a l l from r^ to the event horizon w i l l not l i b e r a t e much energy in the form of heat. The Schwarzschild radius for M=10M© i s 30 km so the inner edge of the accretion disc i s at 90 km. Keplerian o r b i t a l speeds at t h i s innermost radius w i l l be 1.0x10,ocm s "l « 1/3 c. General r e l a t i v i s t i c corrections to the flows are important only at t h i s inner edge, and Newtonian gravity can be assumed i n the disc to good approximation. For a rotating ( Kerr ) black hole, the innermost possible stable o r b i t i s at the event horizon which f o r M=10M© i s at 15 km i n t h i s case. The g r a v i t a t i o n a l potential energy liberated as the p a r t i c l e s traverse ever smaller orbits about the hole w i l l be the binding enery in the l a s t stable o r b i t . Hence, one expects energy releases of 0.057 me1 for the non-rotating case and 0.40 mcl for the rotating one. Thus, energy releases per 15 n u c l e o n i n an a c c r e t i o n d i s c about a c e n t r a l b l a c k h o l e r i v a l o r exceed ( K e r r c a s e ) the e f f i c i e n c i e s f o u n d i n n u c l e a r r e a c t i o n s . The f l u i d p i c t u r e of the gas i s adopted which i s d e s c r i b e d by t h e c o n t i n u i t y e g u a t i o n , the c o n s e r v a t i o n of a n g u l a r momentum ( N a v i e r Stokes e g u a t i o n s ) and c o n s e r v a t i o n of energy, where a x i a l symmetry i s assumed and c y l i n d r i c a l p o l a r c o - o r d i n a t e s a r e employed. I f s t a t i o n a r y c o n d i t i o n s a r e assumed ( <7ot = o ) one f i n d s t h a t most of t h e i m p o r t a n t d i s c p r o p e r t i e s a re indep e n d e n t o f a d e t a i l e d model f o r t h e s t r e s s . The c o n t i n u i t y e g u a t i o n l i n k s t h e mass t r a n s f e r t h r o u g h each annulus of g i v e n r a d i u s t o t h e mass f l o w M ( c o n s t a n t ) a r r i v i n g a t t h e d i s c from the v i s i b l e s t a r . Thus, under s t a t i o n a r y c o n d i t i o n s , N\ = 2T\ Z it' r (,.q) where Z i s t h e " s u r f a c e d e n s i t y " w i t h z 0 the h a l f - t h i c k n e s s of the d i s c , and 0 r the r a d i a l v e l o c i t y . For f u t u r e r e f e r e n c e , the v e r t i c a l average of any g u a n t i t y ¥ i s denoted < f) where <+> « f f Jt, -2, 16 Hydrostatic equilibrium between the z-component of the central object's g r a v i t a t i o n a l force and the gas pressure f> = ^ c * gives a. * C s (i.s) where c s i s the sound speed and the t o r o i d a l v e l o c i t y O9* i s Keplerian to good approximation. U* * V * - (6 M x / r ) *" =" JlK r We note that the s e l f gravitation of the matter i n the disc i s n e g l i g i b l e . Equation 1 .5 shows that a thin disc ( z./r <<1) implies the dominance of the Keplerian v e l o c i t y in the problem. The r a d i a l component of the Navier-Stokes equation ( v e r t i c a l l y averaged) gives, under stationary conditions T ^ D ( [x 4r) - -i 2 ( W r V) M where w""* i s the v e r t i c a l l y averaged stress due to turbulence and/or magnetic f i e l d s . Application of 1.4 to t h i s r e s u l t gives, where the boundary condition has been used. This condition insures that p a r t i c l e s w i l l drop 17 r a d i a l l y into the hole for T<VL . Hence, the stress W has been determined independently of some prescribed ' v i s c o s i t y ' mechanism. From the conservation of energy, we find that the energy flux per unit area i s egual to the energy production by the shear stress Thus afte r the v e r t i c a l average i s taken, we have where Q i s the energy flux per unit surface area. From 1 .8 we have Z " H which shows that a maximal energy flux i s occurring at The t o t a l energy release i s then which gives f o r M^=10M© a luminosity of - 1 Hence an observed luminosity of 4x1037 erg s"' f o r the Cyg X-1 t8 source i s obtained with the deposition of M=10 M^yr"' onto the accretion d i s c . The temperature and radiation spectrum are derived i n a straightforward fashion. For a radiative flux Q, the radiati o n density i s where «" i s the opacity due to electron scattering. The two main types of scattering giving r i s e to X-rays are (1) inverse Compton scattering by which a low energy photon scatters off an energetic electron gaining energy i n the process (2) bremsstrahlung in which X-rays are emitted from the acceleration of electrons in Coulomb f i e l d s . In the f i r s t case, and for very low photon energies compared to electron energies we have the Thompson cross-section ( independent of energy ) giving the opacity <rT = o.'i CIM.* ^v~~' ( / ' i ) while for bremsstrahlung the opacity i s In the standard disc model thermodynamic eguilibrium i s assumed as well as the opticalthickness of the d i s c . Substitution of £ ^ T 4 into 1.12 ( b=Uo-/c=7.65x10"l:rerg cm"* °K W 19 where cr i s the Stefan-Boltzmann constant) , and the use of the opacities 1.13 or 1.14 together with the equation of state for the gas and the radiation gives four equations ( 1-4, 1.5, 1.8, and 1.12 ) for 5 unknowns: z0(r) , 0r(r) , T (r) , and W^(r) . The detailed r a d i a l structure of the disc can be solved i f we specify the stress W""* in terms of the other variables in the problem. This s p e c i f i c a t i o n of the stress i s precisely the most d i f f i c u l t problem i n turbulent hydrodynamics. By-passing these problems, the stress induced by turbulent motions (see Shakura and Sunyaev (1973) as an example) i s modelled with a turbulent viseosit y where equation 1.5 has been used. It i s then assumed that the turbulent Mach number P5 - c A T /-p where u i s the root mean sguare turbulent v e l o c i t y and i s a constant so that Ht P 20 Equation 1.17 now allows the solution of the r a d i a l disc structure. Solving equations 1.4, 1 . 5 , 1 . 8 , and 1.12; and using 1.6 in 1 . 8 , one obtains expressions for \t ~L » 0 r i and T in terms of three parameters: Mx, M, and Mt . The solutions show that t h i s o p t i c a l l y thick disc i s comprised of three zones: A) inner zone; Pr.>>P^ , G^.»<r f f B) middle zone; P^  >>Pr , <rT » \T4( C) outer zone; P^  » P r , 0^>>crT Other general conclusions that may be drawn are ( see Shakura and Sunyaev for an excellent analysis ) 1) the dependence of T upon r i s of power law form giving r i s e to a "non-thermal" looking power law X-ray spectrum 2) maximal temperatures are of order 10 "K 3) the half-thickness z 0 d i s p l a y s weak dependence on Mtand i s of order 10 cm in the inner region to 10 cm in the outer region 4) Compton processes strongly a f f e c t the shape of the emitted spectrum i n the inner disc region 5) exponential cut-off in spectrum for frequency range ^ v > > k T r n a ) t ( c h a r a c t e r i s t i c for Compton process ) The main short-coming of the standard model i s that i t i s too cool to explain the large X-ray power emitted i n the 10-100 kev band of the spectrum. Lightman and Eardley (1974) demonstrated that t h i s 21 constant Mt model was unstable i n the inner, r a d i a t i o n dominated zone and showed on t h i s basis that t h i s region would probably be extended, hot, and o p t i c a l l y thin ( t h i s i n s t a b i l i t y i s considered again in Chapter 4 ). Shapiro, Lightman and and Eardley (1976) constructed a model, based on 4 thxs observation, which gave electron temperatures T e»10 °K with much higher ion temperatures T-»10"°K ( the so-called "two temperature" model ). Soft photons from the cool ( T t=10 6 °K ) middle region scatter off energetic electrons i n the extented hot inner region undergoing inverse Compton scattering. This accounts for the hard X-ray component. The spectrum has a power law form with an exponential cut-off at 150 kev, which i s consistent with observations. Transitions i n the luminosity of Cyg X-1 ( high and low states ) are thought to ari s e from variations i n the mass transfer rate M (see Alme and Wilson (1976))..A large increase i n mass transfer would increase the low-energy photon f l u x from the outer regions of the disc which would account for a t r a n s i t i o n into the high state. For a s t r i c t l y hydrodynamic explanation of the rapid v a r i a b i l i t y , Shakura and Sunyaev assume that i f Mt*1r strong convective turbulence might occur with the resultant emergence of hot clumps of plasma on the disc surfaces. These authors speculate that solar-type f l a r e s could occur i n the disc i f the f i e l d were b u i l t up enough. 22 3^ Previous Work On Magnetic Processes In Accretion Discs In a short early paper, Blumenthal and Tucker (1972) suggested that the X-ray pulsations from Cyg X-1 could be understood i n terms of giant f l a r e - l i k e events i n a region of high magnetic f i e l d . They proposed that i n a f l a r e , o s c i l l a t i o n s of the f i e l d would be set up leading to plasma heating in the flux tube and thermal emission. The hard X-ray component would arise from synchotron radiat i o n from the high energy p a r t i c l e s known to be emitted i n such f l a r e s . They did not discuss how such f i e l d s could be generated. Later work on magnetic processes focussed on the r o l e of magnetic f i e l d s i n transporting angular momentum i n the disc. One feature of t h i s work i s that only f i e l d s at or below egu i p a r t i t i o n energies with the thermal energy are considered. Another feature i s that the in t e r a c t i o n between fl u c t u a t i n g velocity and fluctuating magnetic f i e l d was not considered. Such interactions however are known to give r i s e to very intense magnetic fluctuations as well as strong large scale "mean" f i e l d s under certain circumstances as much work on the theory of turbulent dynamo action shows. Eardley and Lightman (1975) as an example discuss the angular momentum transport and disc structure a r i s i n g from a chaotic magnetic f i e l d . Assuming that only a chaotic f i e l d can be present i n the turbulent disc, these authors consider the growth of the b p component of the chaotic f i e l d at the expense 23 of the r a d i a l component by the shearing of the r a d i a l f i e l d l i n e s in the Keplerian flow. They assume that d i s s i p a t i o n on micro-scales i s negli g i b l e so that the r a d i a l f i e l d b remains constant. With t h i s picture, there i s seemingly no l i m i t to the growth of the chaotic f i e l d . Consequently f i e l d l i m i t a t i o n by reconnection of the f i e l d l i n e s comprising the "magnetic c e l l s " of flux i s invoked. Since the physics of reconnection events (known to be important i n solar flares) i s poorly understood, Eardley and Lightman write down a phenomenological equation f o r f i e l d l i m i t a t i o n by reconnection, with a c h a r a c t e r i s t i c time for energy loss to the f i e l d as some fraction of the time required for an Alfve"n wave to traverse the distance of the magnetic "eddy". Assuming s t a t i o n a r i t y and magnetohydrostatic balance, they then go on to compute the disc structure with the Maxwell stress due to the flu c t u a t i n g f i e l d s providing the stress W1 .^ Ichiraaru (1977) attacked the problem of angular momentum transport by fluctuating magnetic f i e l d s and theorized that the currents a r i s i n g from reconnection of the "magnetic eddies" would induce an anomalous r e s i s t i v i t y i n the plasma. This anomalous r e s i s t i v i t y a r ises from the scattering of these currents off of the magnetic fl u c t u a t i o n s , and i s very much a plasma theorist's point of view. Ichimaru then solves the induction equation f o r the fluc t u a t i n g f i e l d and finds using his anomalous r e s i s t i v i t y , that under stationary conditions 24 where v i s given by v • ( ? I (=' )'4 / «• and X i s the 0-r component of the Maxwell stress due to the f l u c t u a t i n g magnetic f i e l d s . The disc structure i s now computed and i t i s found that the disc can ex i s t i n two physically d i s t i n c t states for a given value of M. If the r a d i a t i v e loss i s small(large) compared to the rate of viscous heating near the outer disc boundary, the disc w i l l exist i n an o p t i c a l l y thick ( o p t i c a l l y thin) geometrically thin (geometrically thick) configuration. The model has an o p t i c a l l y t h i n , geometrically thick inner radiation dominated zone for both the states mentioned above which i s s i m i l a r to the two temperature regime found by Shapiro, Eardley, and Lightman. These states are i d e n t i f i e d with the high and low luminosity states discussed i n section 1. Transitions between states i s determined by the r a t i o M/T with the low state i d e n t i f i e d with a higher accretion rate and a lower temperature in the outer regions which corresponds to the o p t i c a l l y thick, geometrically thin structure. Production of hard X-rays i s as discussed i n the two temperature model. Numerical ca l c u l a t i o n s show that the outer regions are s i m i l a r to the Shakura and Sunyaev o p t i c a l l y thick solution. Although t h i s is a very i n t e r e s t i n g explanation for the bimodal behaviour of the source, the explanation of the rapid 25 variations i s not discussed. The more recent work on magnetic phenomena i n accretion discs concentrates on the idea of a magnetically confined hot corona as the agency which gives r i s e to the hard X-ray emission. Liang and Price (1977) suggest that the accretion disc i s l i k e l y t o form a hot corona somewhat l i k e the solar corona. They picture a sandwich l i k e disc in which the middle layer i s an o p t i c a l l y thick, geometrically thin disc which generates the energy, surrounded by corona-like layers of much lower density which i s pumped by energy from the inner disc. As i n the sun, the o p t i c a l l y t h i n corona has to reach high temperatures before r a d i a t i v e cooling i s s i g n i f i c a n t . They point to three advantages of such an idea: (1) radiation produced at two temperatures ( cool disc and hot corona ) (2) coronal X-ray emission occurs at much higher temperatures |han for standard models (3) corona expected to be highly dynamical and produce highly variable emission. These authors suggest that a strong "disc wind" could be set up i n these coronae. S p e c i f i c a l l y , a corona of temperature II T^10 °k gives r i s e to X-rays in the ri g h t range. In t h i s picture, energy deposited in the corona i s removed by either bremsstrahlung, Compton scattering, synchotron radiation or 2 6 wind cooling. Whether synchotron or Compton scattering i s the dominant radiation mechanism i s determined by the coronal magnetic f i e l d strength. F i n a l l y Galeev, Bosner, and Viana (1979) work out a more detailed coronal model with the formation of loops of magnetic f i e l d (of scale 10^ cm and e q u i p a r t i t i o n f i e l d strength) protruding through the surface of the disc. They show that the reconnection of magnetic loops within the disc as imagined by the e a r l i e r work i s too slow a process to prevent the amplification of the f i e l d s by shearing. They argue that regions of strong f i e l d are expected, and these emerge from the disc by magnetic buoyancy. They consider only a fluctuating f i e l d . The emergent loops of f l u x are strong enough to confine a coronal plasma which i s heated to high temperatures when the loops undergo reconnections ( f l a r e s ) . The hard X-ray emission then derives from the inverse Compton scattering of soft photons from the cool disc off the hot electrons i n the corona. They show that such a process delivers f l a r e l i k e bursts of hard X-rays on time scales 1s and energies 103Sergs s"1, thereby giving a physical basis of T e r r e l l ' s shot noise model. We believe that t h i s model has time variations that are s l i g h t l y too slow and energy releases that are s l i g h t l y too small to recover the shot noise model but that the idea that the main e f f e c t of the magnetic fluctuations i s to give r i s e 27 to f l a r e s on the disc surface i s correct. It i s important to try to track down the requirements for v a r i a b i l i t y over the rapid time scales i n terms of f l a r i n g mechanisms. In view of the lack of a theory of reconnection and f l a r e a c t i v i t y that can account for both the enormous energy releases on short time scales for solar f l a r e s as an example, we turn to some experimental results f o r help. __ Laboratory " Solar Flare _ Experiments Scaled To Cyg X-1 Suppose we imagine that s o l a r - l i k e f l a r e s are occurring on the surface of the accretion disc model proposed for Cyg X-1. What constraints can be applied on the magnetic f i e l d s and t h e i r length scales i n order that v a r i a b i l i t i e s 10"'s f o r the shot noise and 10~5s for the millisecond bursts, with energy releases of 10 3 f cergs can be accounted f o r . Solar f l a r e s (see Sweet (1969) for a review) t y p i c a l l y involve the release of 10 ergs i n a period of 100s for moderate events and may range as high as 10 ergs ( solar luminosity i s 1033 erg s~' ) . The area of the f l a r i n g regions are 10l7-10*° cm1, the emission region being as high as i t i s broad with a c h a r a c t e r i s t i c scale of 10 cm. The ov e r a l l "mean" f i e l d of the sun i s 1-2 Gauss whereas the active regions may have 300-3,000 Gauss f i e l d s ( exceeding l o c a l e q u ipartition energies by two orders of magnitude in some cases ). Flares originate near regions where the lo n g t i t u d i n a l component of 28 the magnetic f i e l d changes sign ( i.e near a neutral l i n e of the f i e l d ). Succeeding f l a r e s i n an active region may occur with i d e n t i c a l p r o f i l e s and shapes suggesting that the f l a r e configuration i s determined by the l o c a l magnetic configuration. The time dependence and magnitude of X-ray emission are the same for a sequence of f l a r e s at a given s i t e , but are s t r i k i n g l y d i f f e r e n t from s i t e to s i t e . A se r i e s of beautiful experiments on magnetic f i e l d l i n e reconnection were performed ( see Bratenahl and Yeates (1970), and Baum et a l (1973) ) which were scaled to solar f l a r e s and found to be i n good agreement with the observations (see Baum and Brahenahl (1976)). We outline the experiments and r e s u l t s and scale to conditions appropriate in Cyg X-1. The laboratory experiments are done on a double inverse pinch device which creates a neutral l i n e between two magnetic c e l l s . The device consists of two insulation covered, current carrying rods which are p a r a l l e l , 10cm apart , and carry current from discharging capacitor banks ( see Fig. 4 ) . Two c y l i n d r i c a l current sheets are driven r a d i a l l y outward from each rod by JxB forces, and merge upon c o l l i s i o n leaving a neutral l i n e i n the centre of the assembly. 2 9 Ficr^ 4 Geometry Of The F i e l d Line Reconnection Experiments -(From Baum Et Al _Q973JL1 (c) Double inverse pinch device chamber, (a) side, and (b) top view; (c) equipotential lines of the t component of the magnetic vector potential (curl-free magnetic field lines). The dark, line is the separatrix which divides the flux into three regions. Region 3 is accessible from regions 1 and 2 by reconnec-tion at the origin. 30 The main res u l t s of these experiments are: (1) current density i s a r e l a t i v e maximum along the neutral l i n e (2) the e l e c t r i c f i e l d along the neutral l i n e grows as soon as the l i n e i s established with the current density increasing as well (3) a discharge occurs during which the current drops abruptly, the e l e c t r i c f i e l d goes from 100volts/cm —>300volts/cm (in 0.3 sec), and the r e s i s t i v i t y jumps by a factor of 40 (4) X-ray production i s observed which can be interpreted as thick target bremsstrahlung of energetic electrons impinging upon the anode i n the immediate v i c i n i t y of,the neutral l i n e . The power law X-ray spectrum observed can be f i t t e d with a power law electron spectrum. Thermal X-ray production i s hard to estimate. The most important aspect of these experiments for us i s the fact that they scale c o r r e c t l y to moderate energy solar f l a r e s . From the experiments i t i s found: -to (1) f l a r e duration: At[c_\_?= 10 s (2) t o t a l energy release Q u i , =1.6x10 erg under conditions where the length scale and magnetic f i e l d were fixed i n the experiment as : (a) length scale of reconnecting f i e l d : - 1 ^ = 10 cm (b) magnetic f i e l d : By, =10**Gauss 31 Baum and Bratenahl take the observed values L S u w , B s^ for the size of the reconnecting regions and t h e i r magnetic 1 * . f i e l d strengths ( L w = 10 cm and Bs>^ = 10 Gauss ) and derive f l a r e durations of 10 s and energy releases of 10 erg, using the parameters above. The s c a l i n g r e l a t i o n s they use, due to Parker, are a t x = f L x j ^ (,.<„) = Quo. ( B . \ 2 ( U \ 3 (.•*.) where A t x , l < f and Qx are the f l a r e duration, f l a r i n g region s i z e , and energy release. The maximum cosmic ray energy possible from such f l a r e s can be estimated by integrating the e l e c t r i c f i e l d along the entire neutral l i n e length. The scaling i n t h i s case i s 3 where the measured voltage i n the experiment =3x10 v o l t s . For the solar parameters, Vx =30 Gev i n agreement with observations. Assuming that "solar type" f l a r e s are responsible for the rapidly varying X-ray emission i n Cyg X-1, we estimate the time scales and energy releases from the obervational data summarized i n Table 2 for both the shot noise and millisecond 32 burst parameters- Using the s c a l i n g r e l a t i o n s 1.20, 1.21, and 1.22 ; the laboratory parameters; and the time scales and energy releases for shot noise and millisecond bursts: we f i n d the length scales, magnetic f i e l d , and maximal cosmic ray energies associated with f l a r e s on the disc. The r e s u l t s are given in Table 3. Table 3._ Characteristics Of Solar Type Flares In Cyg X-1 EVENT TYPE SHOT NOISE MILLISECOND BURST 6 4 L x 10 cm 10 cm io.r 13 B x 10 Gauss 10 Gauss Vv • 10 l S r ev 1 0 e v For the densities and temperatures i n the standard cool disc model, egu i p a r t i t i o n f i e l d s in the disc are of order 108Gauss- If the r e s u l t s of Table 3 are correct, one i s 33 dealing with f i e l d s above egu i p a r t i t i o n energies. Such strong f i e l d s could not remain in the disc and would emerge from the disc surfaces and there undergo f l a r i n g . The parameters for the shot noise are i n agreement with what one would expect from the disc. The disc thickness varies from 10 cm to about 10 cm. In addition f i e l d strengths two orders of magnitude above egu i p a r t i t i o n strength are predicted which i s s i m i l a r to solar f l a r e s . The bursts represent extreme conditions indeed. The largest astrophysical f i e l d s known are those for pulsars where 10 Gauss f i e l d s have been observed. The r e s u l t s of Chapter 4 w i l l show that the largest f i e l d s present are about 10,i>rGauss, and give an alternative explanation of the millisecond a c t i v i t y i f i t i s indeed r e a l . We note that the maximum cosmic ray energy observed to date i s 10 ev. This analysis shows that i n order to recover the rapid temporal variation of the Cyg X-1 source together with the massive energies associated with these variations, a solar type f l a r e model reguires intense ( greater than e g u i p a r t i t i o n energies i f a standard underlying disc i s assumed ) magnetic flu c t u a t i o n s . A consistent MHD analysis should not therefore impose the condition of eguipartition at the outset. We turn to an outline of the remainder of the t h e s i s . 34 5- Outline Of The Thesis In Chapter 2, the f u l l equations for a magnetized f l u i d are examined. The various f i e l d s are decomposed into mean and fluctuating components and the energy transfer between the various f i e l d s examined. The relations between the fl u c t u a t i n g v e l o c i t y , temperature, and magnetic f i e l d s are developed using the methods of Mean F i e l d Electrodynamics and a regime i s i d e n t i f i e d where large magnetic fluctuations may occur. The magnitude of the mean magnetic f i e l d i s found to be of ce n t r a l importance in assessing the structure of the various f l u c t u a t i n g f i e l d s . In Chapter 3, a detailed analysis of the mean large-scale magnetic f i e l d i s carried out. Since some assumptions about the disc structure and turbulence properties are required, we use the standard disc model as a basis f o r our ca l c u l a t i o n s . Assuming steady conditions, these c a l c u l a t i o n s determine a value f o r the turbulent Mach number independent of any other parameters. In the f i n a l section of the chapter, i t i s found that for appropriately large scales, the mean f i e l d can be generated by turbulent dynamo action. In Chapter 4, the res u l t s of Chapters 2 and 3 are combined to give a coherent view of magnetic processes i n the accretion disc. Arguments are made to show to what strength the mean f i e l d i s expected to grow, which allows the magnitude of the magnetic fluctuations to be fi x e d , and a comparison 35 With the predictions of the scaling arguments cf the previous section of t h i s chapter to be made. & picture i s b u i l t up wherein the mean magnetic f i e l d remains inside the disc at below eguipartition strength and over the long time average i s responsible f o r angular momentum transport in the d i s c . On intermediate tiise scale averages, i t i s shown that the Maxwell stress due to the fluctuations acts to supress the lightoan and Eardley i n s t a b i l i t y so that the underlying accretion disc remains cool and thin. This r e s u l t i s important because i t shows that the hard X-ray emission must derive frcm magnetic f l a r e l i k e processes. The magnetic fluctuations undergo f l a r i n g processes on time scales and with energy releases that explain the shot noise model of the X-ray v a r i a b i l i t y . The hard X-ray emission i s modelled i n terms of a rapid, f l a s h -phase subcomponent of the f l a r e s , in which non-thermal d i s t r i b u t i o n s of electrons account for the power-law X-ray spectrum. wherever possible, the mathematics has teen relegated to a series of appendices. The entire analysis to be presented i s n o n - r e l a t i v i s t i c . Ho major corrections to the r e s u l t s are expected for non-rotating black holes; however, interesting e f f e c t s i n addtion to those presented w i l l arise for rotating black holes i f the region immediately adjacent to the event horizon i s considered. Appendix A reviews the ideas and resu l t s of Mean f i e l d Electrodynamics reguired for t h i s work. 36 Chapter 2-Magnetic Fluctuations In A Turbulent Accretion Disc -Introduction The central aim of t h i s chapter i s to examine the properties of fluc t u a t i n g v e l o c i t y , magnetic and temperature f i e l d s i n a turbulent accretion disc. As mentioned i n the opening chapter, the assumption of s t a t i o n a r i t y allows solution of the disc equations without a detailed knowledge of the d i s s i p a t i o n mechanisms operative in a turbulent disc. However the rapid v a r i a b i l i t y of the disc's X-ray output down to millisecond time scales demands a careful examination of fluctuations i n such a turbulent regime. Let us f i r s t consider energy balance when magnetic f i e l d s are ignored. In a turbulent regime Reynolds stresses act to transfer energy out of the mean flow (in an accretion disc, t h i s i s b a s i c a l l y a Keplerian v e l o c i t y f i e l d ) into the velocity fluctuations. Then pressure fluctuations and v i s c o s i t y act to transfer energy out of the ve l o c i t y fluctuations into i n t e r n a l energy. Stewart (1974) examined thi s process by using various scaling arguments applied to a coupled pair of equations for the fluctuating kinetic and intern a l energies. His techniques and r e s u l t s , b r i e f l y reviewed i n section 2, w i l l prove a useful s t a r t i n g point for 37 our analysis which includes magnetic f i e l d s . When magnetic f i e l d s are included a host of new e f f e c t s may a r i s e . For the moment consider only the flu c t u a t i n g f i e l d s . In addition to the ef f e c t s described i n the preceeding paragraph, energy i s extracted from the vel o c i t y fluctuations by E.T type i n t e r a c t i o n s , and transferred into both the mean and fluctuating magnetic f i e l d s . Ohmic di s s i p a t i o n then transfers energy of the fluctuating magnetic into i n t e r n a l energy fl u c t u a t i o n s . These processes are spec i f i e d and examined i n section 3 . Our object in section 4 i s to solve f o r the k i n e t i c , magnetic and i n t e r n a l energy fluctuations. Now i n order to close t h i s set of three eguations for three unknowns i t w i l l be necessary to give a model for the ultimate value of the mean magnetic f i e l d . We w i l l be using mean f i e l d theory to express the various correlations between the f l u c t u a t i n g velocity and magnetic f i e l d s i n terms of the mean f i e l d . It i s to be expected that the energy in the mean magnetic f i e l d w i l l depend on the k i n e t i c energy fl u c t u a t i o n s . The s p e c i f i c model for t h i s w i l l s pecified i n Chapter 4 and w i l l enable us to close our coupled t r i p l e t of eguations. Comparisons with Stewart's r e s u l t s w i l l also be made in t h i s section As pointed out i n Chapter 1, i f we interpret the shot noise model for the X-ray output as due to solar type f l a r e s occurring on the surface of the accretion d i s c , large magnetic 38 fluctuations are reguired. In section 4 we show under what conditions such large fluctuations are possible, and discuss how they affect the o v e r a l l disc structure. I t w i l l become apparent that i n the regime that allows fluctuating f i e l d s on the order 10 1 0 gauss, i t i s not possible to have a stationary accretion disc on time-scales <10"' sec. We believe that under the conditions mentioned, an accretion disc such as that believed to comprise the Cyg. X-1 X-ray source can be approximated as stationary only on time-scales/v10 sec. 2. Stewart's Analysis Of Energy. Balance In Turbulent Accretion Discs. 2. 1 The Basic Eguations We begin with the eguations of continuity, conservation of momentum and conservation of i n t e r n a l energy f o r a viscous f l u i d i n the presence of an external g r a v i t a t i o n a l f i e l d . Stewart adopts the use of a semicolon notation for covariant d i f f e r e n t i a t i o n which f a c i l i t a t e s the conversion into c y l i n d r i c a l polar co-ordinates (r, 4,The eguations are f * q 1 ^; * = O (*.<) (?o t (r-^ ).p s { r - _ r _ t - p / p (,.„ 39 where f i s the external g r a v i t a t i o n a l potential, t*'* i s the viscous stress tensor, p i s the symmetric s t r a i n tensor ( i . e . = i (m^. p +- u?;«.) ) ; e i s the i n t e r n a l energy, and i s the heat fl u x . The reader may consult Landau and L i f s h i t z (1959) for the detailed derivation of these eguations. In order to discuss turbulent processes, one now decomposes the f i e l d s into mean and fluctuating parts. Proceeding with t h i s decomposition we write f - {o + ^ e = t + e denoting the ensemble average of any guantity 0 by 9 , we have that u« -. If, f - fo e t c - T^e ensemble average of the fluc t u a t i n g quantities vanishes. Let us introduce the decomposition 2.4 into the equations 2.1-2.3 and take t h e i r ensemble averaqe. As a simple example, consider the continuity equation which becomes f. + ( f p U " + f V j. ^  = o Because density fluctuations '^ are important we see that we pick up an extra term '^«V i n addition to the terms present when turbulence i s absent. In , order to make the no e x p r e s s i o n s compact, i t i s u s e f u l to d e f i n e which allows us to w r i t e 2.5 as Stewart d e f i n e s w as the mass flow v e l o c i t y . We i n t r o d u c e the ^  symbol i n order to c l e a r l y d i s t i n g u i s h t h i s from the us u a l mean v e l o c i t y U"--w. i t i s now n a t u r a l to d e f i n e u = u where we n o t i c e t h a t U" + 0'"= U"+lA'* = u* -Using the d e f i n i t i o n s 2.6 and 2.8 as w e l l as the c o n t i n u i t y eguation 2.7 r e s u l t s i n the eguation of motion where and 7*? = . ^ « u 1 i s the Reynolds s t r e s s . We have dropped the v i s c o u s s t r e s s i n w r i t i n g 2.9 because d i s s i p a t i o n on the molecular l e v e l i s n e g l i g i b l e compared with the d i s s i p a t i o n a r i s i n g v i a the t u r b u l e n t s t r e s s e s . The eguation governing the dynamics of the mean flow energy i s found by t a k i n g the dot product of e g a t i o n 2.9 with t o g i v e <^(»') - >a% -f - r e.„ . f„., 41 where we have written and P^p i s the mean symmetric s t r a i n tensor. The equation governing the dynamics of the flu c t u a t i n g flow k i n e t i c energy are found by dot producting the f u l l equations of motion (using decompositions 2.4) by <* and taking the ensemble average r e s u l t i n g i n where u s f w «»/^ i s the mean square velocity fluctuation. The most important feature of these equations i s that the -A. S< Reynolds stress interacting with the mean st r a i n ( TKp- £ ) transfers energy out of the mean flow and into the fluctuations. The f i r s t two terms in the right hand side of 2.11 represent the work done by pressure fluctuations and vi s c o s i t y while the l a s t term represents transport of fluct u a t i n g flow energy by the turbulent stresses. To maintain turbulence at a l l , some external agency (in t h i s case the external g r a v i t a t i o n a l f i e l d due to the central compact 42 object) must provide the s t r a i n . Turning to the inter n a l energy, defining the quantities: setting e-c^T , ( c„ constant) appropriate for a perfect gas so that i - t /c v and the temperature fl u c t u a t i o n 9-e./Ctft the mean square thermal f l u c t u a t i o n P = f # / p6 s a t i s f i e s the equation which may derived in the same manner as equation 2.11 2.2 Consequences Of The Mean Momentum Conservation Equation For Thin Discs This section provides more of the d e t a i l s concerning the r a d i a l structure of accretion discs and provides some background f o r the discussion i n Chapter 1- We apply the following four assumptions to the eguations of motion 2.9 i n c y l i n d r i c a l polar co-ordinates ( see Stewart, p.41 ) A) the disc i s thin with i„"r, where H 0 i s the scale thickness, B) the mean flow i s almost t o r o i d a l with i CiM » l u"| » I w *l 43 i n fact we require [ u*| » r iu*| where C) the mean flow i s axisymmetric and symmetric about i , o D) the size of the largest eddies ^ i s of order *.<> 1. The z-component of equation 2.9 i s f o (. ^  + H 4 ^ * vJ, r U ) = - Vj_Z r 2 so that assumption B) makes the l e f t hand side n e g l i g i b l e compared with the c e n t r i p e t a l force. Using the thin disc approximation gives which i s r e a l l y an eguation of hydrostatic balance. Then applying the estimates ?_ * where c$ i s the speed of sound, gives the resu l t IV)' 1 ~ Z \ 44 N O B i n usual turbulence theory, the duration of a turbulent eddy of size / i s so that eguation 2.17 may be rewritten as (using assumption D) t , ^ t k [ .+ K; Z] 4 ^ ( 1 ) where H^i Cr/ C i i s the turbulent Mach number and "t k s r/V K i s the Keplerian time scale. Eguations 2.17 and 2.19 show that the Keplerian flow V*. i s always supersonic and that the Keplerian time scale i s the shortest time scale i n the turbulent accretion disc problem ( i . e . t t \ tK ). The time scales r e f l e c t i n g fluctuating proceses are t i e d to the Keplerian time scale in thin disc problems. With reference to mean f i e l d theory (see Appendix A), the use of the f i r s t order smoothing approximation to simplify the solution of the flu c t u a t i n g and mean magnetic f i e l d eguations can be s a t i s f i e d i f we imagine the turbulence as a c o l l e c t i o n of random waves where i f \ i s the co r r e l a t i o n time for such a wave of wave-length Jl 2. The r-component of eguation 2.9 i s T + ( r r r - T+0 c \ •r rr T . r- + 45 where by assumption B) the largest term on the l e f t i s - /r and from the re s u l t 2 . 1 7 the dominant term on the rig h t i s - v1* ^ 0/ r so that which gives the mean t o r o i d a l flow as Keplerian to good approxim ation 3. The jg'-component of eguation 2.9 i s and using previous estimates gives — (*.*«0 Taking the v e r t i c a l average (see Chapter 1, section 2) of these eguations and noting that T * 2 must vanish on a free surface the equation becomes which i s equation 1.7. 2.3 Energy Balance For The Fluctuating Fields Since the turbulence time scale tt«rtK, the turbulence 4 6 w i l l be well mixed i n both the z and $ directions. However, long r a d i a l scale variation of mean guantities such as the mean s t r a i n w i l l r e s u l t i n the r a d i a l variation of the turbulence. For t h i n discs then, the turbulence i s s t a t i s t i c a l l y homogeneous in each annulus r=constant. We turn to the study of the fluctuation equations 2.11 and 2.13.. On the l e f t hand side of these equations we have the appearance of the operator D = 2 + u • ^ where 3 / a - t measures the rate of change at a point and U-2 represents the advective rate of change following the mean flow. The P/Dt operator then measures the rate of change for a hypothetical point moving with the mean flow. For a steady flow .7 =o and i f we have homogeneity of turbulent quantities, i . e . they are independent of r, 4 , and z, then U.V= o as well. A stationary disc requires D ^ / D t = o ; where = £*or 6>1 . Another approximation which i s employed i s to note that the diverqence terms on the r i g h t hand side of the equations 2.22 and 2.23, which represent the s p a t i a l r e d i s t r i b u t i o n of energy vanish i n the case that our turbulence i s independent of z and <j>. This can be seen by imagining that we integrate these terms over an annular volume with cross sec t i o n a l dimension z„ and then applying Gauss's theorem. Equation 2. 11 then reduces to 47 In this equation, the l e f t hand side represents the rate at which enerqy i s being supplied to the turbulence via the reynolds stress interaction with the mean s t r a i n whereas the right hand side comprises the dissipation of energy by buoyancy and vi s c o s i t y respectively. This type of r e l a t i o n i s common to many analyses of turbulent processes (see Tennek.es and Lumley (1972)). Equation 2.26 concentrates on those features of turbulence not d i r e c t l y related to s p a t i a l energy transport. The turbulence interacts with the mean flow and in j e c t s energy into the turbulence on the largest scale . The dis s i p a t i o n by v i s c o s i t y ( second term on ri g h t hand side ) occurs at microscales however- As Tennekes and Lumley point out, a l l evidence suggests that the viscous d i s s i p a t i o n at these microscales occurs at a rate dictated by the energy i n the largest eddies. With an energy density in the largest scales of ^B £*" and a c h a r a c t e r i s t i c t i m e Z / G r , the viscous di s s i p a t i o n should occur at a rate (.u*//, i . e . Denote the components of the flu c t u a t i n g velocity by in c y l i n d r i c a l polar co-ordinates. The main contribution to i s the 4-r component denoting the s t r a i n associated 48 with the Keplerian flow \/K • The l e f t hand side i s then ^ " P E „ p * ^>0cu ul C-rJi') where the c o r r e l a t i o n c o e f f i c i e n t i s In the case of small pressure fluctuations the buoyancy term i s approximately P,« * * . Now the g r a v i t a t i o n a l force in the r a d i a l direction i s balanced by the c e n t r i p e t a l force (the gas i s d r i f t i n g r a d i a l l y at a much slower rate than i t s o r b i t a l period) so that the l o c a l gravity i s mainly in the z d i r e c t i o n . If we define V = 8*. as the l o c a l gravity then the buoyancy term i s » h e r e ^ U o ; - j ) , F i n a l l y , assuming a perfect gas allows us to relate the density fluctuation f' with (5 4- * -A <• f so that we have with the velocity-temperature co r r e l a t i o n c o e f i c i e n t •21 ^ i A — 49 Combining equations 2.27 , 2.28 and 2.3o with 2.26 then g i v e s (^rri) c u u2- ^ - iL, c e u 6 + ( i - 3 i > which shows that the buoyancy i s c o u p l i n g the v e l o c i t y f l u c t u a t i o n s t o the thermal f l u c t u a t i o n s , which we study next. Osing the same c o n d i t i o n s t h a t l e a d to eguation 2.26, eguation 2.13 becomes .A where h e a t i n g due to kinematic v i s c o s i t y has been n e g l e c t e d . The l e f t hand s i d e i s approximately «. while the heat f l u x \ i s d i v i d e d up i n t o i t s conductive and r a d i a t i v e p a r t s with r « P o c , [ - l e v a ' * r where X i s the thermal d i f f u s i v i t y and t> i s the time s c a l e f o r r a d i a t i v e c o o l i n g . I n t r o d u c i n g a thermal l e n g t h s c a l e a llows us to w r i t e 2.32 as ( i t + yc„) <0 8"a: = - ©jT . O-ssO 50 or Equations 2-31 and 2.33 re p r e s e n t two equations i n the unknowns 9 and u . Usinq equation 2.33 (b), the equation f o r the v e l o c i t y f l u c t u a t i o n s becomes Stewart makes t h i s equation more manageable by m u l t i p l y i n g throughout by and by d e f i n i n g the parameters and the dimensionless t u r b u l e n t i n t e n s i t y i n terms of which equation 2.34 may be w r i t t e n as ' = ______ + X (a-ss) ( x + y ) 51 The parameter k\ i s the Richardson number but more ins i g h t into i t s role may be obtained i f we note that using the equation of state f o r a perfect gas where f\ i s the gas constant and ^ the molecular weight and where K/u = j ^  for a monatomic gas, we have f>= Taking the ' 3 z c</ T derivative with respect to z of t h i s r e l a t i o n gives - — = 1 { T_ t l J- 1£ ) so that the Richardson number i n t h i s approximation takes the form In a stably s t r a t i f i e d disc s o that R; > o and consequently the buoyancy extracts enerqy from the turbulent k i n e t i c energy. Conversely, an unstably s t r a t i f i e d disc with >o and Ri<o shows that energy i s pumped into the turbulence via r i s i n g f l u i d elements. We s h a l l assume a stably s t r a t i f i e d disc where R i> 0 . The freguency 52 (the so-called Brunt-Vaisala frequency) represents the freguency of o s c i l l a t i o n of a neutrally buoyant element i n a stable density gradient and the Richardson number ft; given by 2.37 compares this frequency with the Keplerian freguency; the basic frequency i n the problem. Returning to equation 2.35 we define (*•»-*) where f(x) i s the r a t i o of the rate of energy d i s s i p a t i o n to energy production. The condition f(x)=1 corresponds to stationary turbulence. I f f(x)>1 as an example (not stationary), then we have greater d i s s i p a t i o n than production so that the turbulence ultimately damps out. Solving f(x)=1 e n t a i l s nothing more than solving a guadratic eguation i n x. Two cases are in t e r e s t i n g . Case J V= o : Here i - 4 P Rc z so that stationary turbulence i s possible only f o r ^Ri * ^ with damping ( i . e . ) occurring when ^ ^ ; > ^ . Experimentally i t i s known that turbulence dies out when & ^ o-2 ( Tennekes and Lumley p. 99 ) . Stewart goes on to argue that of the two roots, ( assuming ) the larger one i s stable while the smaller root i s unstable in that a flu c t u a t i o n about th i s root may either lead to ultimate damping or increasing of the 53 turbulent i n t e n s i t y to the larger root. Since B R : - > o as i-> o ( i . e . ^ = o at ?=o ) so that one always has a stationary a* turbulence about the mid-plane of the disc with x=1. Using x=1 at z=0, together with the d e f i n i t i o n of x and u = V t K , shows that the co r r e l a t i o n c u i s c u = t K / t t . Case 2 }f ^ o •• Again solving the quadratic equation for x shows that for y < pRc < i C 1 + Y*) two stationary turbulent flows are possible only the flow corresponding to the larger root being stable. For f> Ri < y , one stationary flow i s possible (i.e only one root has x>0) and i t i s stable. Again we find that near z=0 (with pR; -> o ) stationary turbulence i s possible but near the surface regions the r e l a t i o n of p>Ri to Y i s c r u c i a l . When p>R; i s large r e l a t i v e to if these regions must have a laminar flow because the turbulence damps out. Conversely, for y large r e l a t i v e to f>Ki , the disc i s turbulent r i g h t to the surface. As Stewart points out, t h i s i s explained by noting that large Y corresponds to rapid radiative cooling which destroys the temperature fluctuations and hence decreases the buoyancy e f f e c t . In conclusion, i t should be noted that i n Stewart's analysis t t ^ t K with tt*f«: only when the turbulence i s sonic or supersonic. With Keplerian time scales of 10~ 3 sec. for the 54 inner region of an accretion disc with a rotating black hole and 10"z sec. for a non-rotating hole ( of mass M** loKa) t i n order for the turbulence to be responsible for the rapid X-ray variation (hot blobs r i s i n g to surface as an example) Stewart's analysis would suggest a sonic or perhaps supersonic turbulence to be present. Our own work w i l l show ( Chapters 3 and 4 ) that energetic solar type f l a r e s may account for these rapid v a r i a t i o n s even i n the case of subsonic turbulence i n the disc. We now turn to generalize the study of fluctuations i n a turbulent accretion disc to the case that magnetic f i e l d s are included. 3-. Equations Of Motion Including Magnetic Fields 3__ The Basic Eguations For A F l u i d In A Magnetic F i e l d We now wish to study the dynamics of a turbulent, t h i n accretion disc when magnetic f i e l d s are included. To t h i s end the complete set of eguations describing a magnetic f l u i d are l i s t e d a f t e r which the decompostion of these f i e l d s (including the magnetic f i e l d ) into mean and fluctuating parts w i l l lead to the same type of analysis for the fluctuations as found i n section 2.3. For a magnetic f l u i d the continuity eguation 2.1 i s s t i l l v a l i d . The Navier-Stokes eguations 2.2 must now contain a 55 c o n t r i b u t i o n on the r i g h t hand s i d e from the Lo r e n t z f o r c e . 1 * -^ * W c = ( V x b ) x t / 4 T T T h i s f o r c e may be w r i t t e n as the divergence of the Maxwell s t r e s s t e n s o r , i . e where <r°<P - ( V t P " ^  ^  / l f i r and hence eguation 2 . 2 becomes The e l e c t r o m a g n e t i c f i e l d s i n moving conductors are given by the equations c St rjx _ - 4 » j- / £ - __[__" [ e + U x where v" t> = o and where cr i s the c o n d u c t i v i t y . S o l v i n g f o r the e l e c t r i c f i e l d e i n terms of k bY the second eq u a t i o n , and s u b s t i t u t i n g i n t o the f i r s t , assuming th a t the c o n d u c t i v i t y i s uniform (or n e a r l y so ) g i v e s r i s e t o the s o - c a l l e d i n d u c t i o n equation o f magnetohydrodynamics, 5 t 56 where the magnetic d i f f u s i v i t y ^ i s defined by = cz/W<r- The use of t h i s eguation implies that the conductivity i s independent of the magnetic f i e l d which requires that the mean free path of the electrons be small compared with the radius of curvature of the i r o r b i t s i n a magnetic f i e l d . This condition may breax down i n regions of s u f f i c i e n t l y low density or high magnetic f i e l d strength. More d e t a i l s of the derivation may be found i n Moffat's book (1978, Chapter 2). F i n a l l y , the conservation of energy equation must be amended to include the magnetic f i e l d . The f u l l eguation i s ( i u 2 t £ + - I/O + u . t . + + c J where the energy density Sir contributed by the magnetic f i e l d has been included on the l e f t hand side, and the energy flux density £ ex (the Poynting vector) has been included on the r i g h t hand side. Writing e i n terms of k again, the Poynting vector takes the form which i s then substituted into equation 2.42. Equation 2.42 may be si m p l i f i e d by the same procedure 57 that leads from the conservation of t o t a l energy to equation 2.3. S p e c i f i c a l l y , we take the dot product of ^ with equation 2. 4o ( using the continuity equation for additional s i m p l i f i c a t i o n ) , the dot product of b with equation 2.41, and subtract the resulting equations from 2.42 ( with the substitution 2.43 ). After some algebra one finds where we r e c a l l $ ~ fjir ^ *" - . Comparison of 2.44 with 2.3 shows that the equation for the i n t e r n a l energy i s modified by the addition of the Joule heating term j 2/<r giving the rate of evolution of heat due to Ohmic dissip a t i o n . Eguations 2.3, 2.40, 2.41 and 2-44 along with the equation of state [" form the basic framework of our analysis. 3.-2 Eguations Governing The Mean And Fluctuating Fields The decomposition of the various f i e l d s i n t o mean and fluct u a t i n g components i s now introduced where i n addition to equation 2.4 we introduce b = B •»-k with ^ = B ( and hence _ - o ~) with 2* ^  s — [ B- B p - x ^sK?~] 4r z J 58 (a) The Navier-Stokes Eguation-The equation f o r the mean velocity f i e l d i s where the l a s t two terms are the ef f e c t s due to the Maxwell stress. To f i n d the equation for the energy i n the mean v e l o c i t y f i e l d we take the dot product of with equation 2.45 and find f + ( T « P - P5"P + Z " ) % < r - f where we see the Maxwell stresses interactinq with the s t r a i n F Kp of the mean ve l o c i t y f i e l d to transfer energy out of the mean flow. The energy in the fluctuating v e l o c i t y f i e l d i s found by taking the dot product of u' with the f u l l y decomposed equation 2.10 and then averaging. The new term that arises as compared with the equation i n the absence of maqnetic f i e l d s ( 2. 11 ) i s 59 We manipulate t h i s factor as follows: We note that A 1 I i f the d e f i n i t i o n of u * in terms of i s used (equation 2.8). Osinq the d e f i n i t i o n of cr" "P a rearrangement of t h i s term gives It i s most i n s t r u c t i v e to introduce the e l e c t r i c f i e l d s induced by the presence of fluc t u a t i n g v e l o c i t i e s by the de f i n i t i o n s If we remember J = c v * B and ;1 . £ ^ x (,' , then The fluctuating kinetic-energy equation can therefore be 60 written as - [ ( t - N . c»- a'')u . j J , * with the new magnetic terms added on the la s t l i n e . The term it**", a +- <r"*l\ O * has t^ i e same e f f e c t as -f'" Z\ in that i t adds the pressure (k z + >» ^/ffT to the f l u i d pressure I 5 and hence contributes to the further damping of the turbulence by buoyancy processes as already discussed. In the absence of density flu c t u a t i o n s , the most important e f f e c t s of the magnetic f i e l d are the terms - _> T - r'. f . Here the e l e c t r i c f i e l d s , induced by the inte r a c t i o n of the flu c t u a t i n g velocity f i e l d and the magnetic f i e l d , do mechanical work on the system i n the presence of currents. We w i l l l a t e r confirm that these terms extract energy out of the vel o c i t y fluctuations and pump i t into the mean and flu c t u a t i n g magnetic f i e l d s respectively. (b) The Induction Equation Assuming the decomposition )z & + k as already discussed, and using the same decomposition f o r the ve l o c i t y . 61 equation 2.41 becomes afte r averaging Dt We notice that i n turbulent f l u i d s , the co r r e l a t i o n of the flu c t u a t i n g magnetic and velocity f i e l d s gives r i s e to an electromotive force _'x w' not present when the flows are laminar. The counterpart of t h i s term i n the mean Navier-Stokes equation i s the Reynolds stress. Here the s i m i l a r i t y ends however because as Moffat ( p. 248 ) points out the Navier-Stokes equations, being non-linear i n u do not permit the ready c a l c u l a t i o n of u u J i n terms of mean guantities such as U . Because the induction equation 2.41 i s li n e a r i n the magnetic f i e l d however, i t i s possible to calculate (At in terms of mean f i e l d quantities such as 6 i n a sat i s f a c t o r y manner. The detailed discussion of t h i s theory i s presented i n Appendix A. It i s important to note that the length and time scales over which the mean magnetic f i e l d s vary i s assumed to be much larger than the scales involved for the flu c t u a t i n g f i e l d k . This idea of separation of scales has been used in the development of the theory of equation 2.50 and has received some support from detailed computer simulations by Pouquet et a l (1976). Equation 2.50 w i l l be studied i n d e t a i l i n the next chapter. The equation governing the energy i n the mean 62 magnetic f i e l d may be found by t a k i n g the dot product of eguation 2.50 with B/?ir- Making repeated use of the v e c t o r i d e n t i t y . C A K B ) « B . VK _ - /\ . V K B g i v e s B W 4-lT where r e c a l l V- _• - ^ (i7x&)<§ . F u r t h e r use of v e c t o r i d e n t i t i e s i n c l u d i n g allows the above eguation to be w r i t t e n as B sir ^7. where | i s the symmetric s t r a i n tensor a r i s i n g from U . i t i s easily shown that Now 417 (U* § ) x B " U . | so t h a t with a l i t t l e manipulation with t h i s f a c t o r and use of the d e f i n i t i o n I - U'K b /c g i v e s Bl] 7.U + Z : E + §. J - jVo srr L tiTT 4 7T J 0 *<J 63 The r i g h t hand s i d e of t h i s eguation c o n t a i n s the work done by the magnetic pressure 8* htr, the i n t e r a c t i o n of the Maxwell s t r e s s with mean s t r e s s , our I•T term which here e n t e r s with a p o s i t i v e s i g n while i n equation 2-49 i t e n t e r s with a negative s i g n , the d i s s i p a t i o n due to Ohmic h e a t i n g , and the divergence of the electromagnetic energy f l u x which takes the form of the Poynting v e c t o r f o r a moving conductor. The appearance of i n equation 2.51 and -1-2 i n equation 2.49 f o r the f l u c t u a t i o n s i n d i c a t e s t h a t energy t r a n s f e r from the f l u c t u a t i n g v e l o c i t y f i e l d i n t o the mean magnetic f i e l d i s o c c u r r i n g . In order t o have an a d v e c t i v e term on the l e f t hand s i d e of W- • C Msn) we use U u + u with p - f + _ (using equations 2.6 and 2.8) to o b t a i n , with a l i t t l e rearrangement, $ \ + u"fe>* \ . J B M % V-P ,t_ ~\ \ L V itr V HIT ' *TT i ; The i n d u c t i o n equation f o r the f l u c t u a t i n g magnetic f i e l d i s found by decomposing the f u l l i n d u c t i o n equation 2.42 and s u b t r a c t i n g the mean i n d u c t i o n equation 2.50 to g i v e i 64 where c i I i ~ r Neglect of the term £ (the " f i r s t order smoothing approximation ", see Appendix A.1) i s possible when the turbulence i s imagined to be a c o l l e c t i o n of random waves with « lu. / ^  . We s h a l l be adopting t h i s assumption i n our calculations, and more about i t s role may be found i n Appendix A.2. To find the eguation for the energy i n the magnetic fluctuations we take the dot product of eguation 2.53 with {p'/fir . Proceeding by exactly the manipulations we used to f i n d eguation 2.51, the result i s : analagous to eguation 2.51. In t h i s eguation the f l u c t u a t i n g magnetic pressure b *" / S"n~ i s doing work, the f l u c t u a t i n g magnetic stress which transferred energy out of the mean flow i s here acting as an energy source, the £'. j-' term i s transferring energy into b'1 from the fluctuating v e l o c i t y f i e l d , Ohmic d i s s i p a t i o n due to the fluctuating currents i s di s s i p a t i n g energy as heat, and f i n a l l y the divergence of the energy f l u x vector ( which takes the form again of a Poynting vector i n a moving conductor ) i s the l a s t term i n 2.53L. Again, i n order to have the same type of advective term on the l e f t hand side of 2.53bas Stewart has, we follow the same 65 procedure used i n going from 2.51 to 2.52 to f i n d (c) The Internal Energy Equation The correction due to the magnetic f i e l d s i s simply the addition of the Ohmic heating term appears in equation 2.44. I f . t h i s equation i s averaged (using the standard decomposition), we find that the mean int e r n a l energy eguation for E has the terms ( l * + j'*) / c as source terms on the rig h t hand side of the equation i n d i c a t i n q that the Ohmic dis s i p a t i o n TIV of the mean f i e l d B and the Ohmic dis s i p a t i o n j' z / cr from the fluc t u a t i n q f i e l d k are being transferred into the mean inter n a l energy. The eguation for the mean-square thermal fl u c t u a t i o n i s just 66 It turns out that because the d i f f u s i v i t y ^ and not a turbulent magnetic d i f f u s i t i v i t y appears in t h i s l a s t term, that i t i s negli g i b l e as far as the 0 fluctuations are concerned. (d) Summary The energy flow a r i s i n g only from the new magnetic terms i s summarized b r i e f l y . These ef f e c t s are i n addition to those discussed i n section 2. Here we ignore the terms appearing as divergences ( v7. [ ] ) as discussed i n section 2. (1) Mean-flow Kinetic Energy: ( eguation 2.46 ) 1. Energy loss - ( 4 a-""4!*) e ^  due to inte r a c t i o n of Maxwell stresses with the "mean" s t r a i n . (2) Fluctuating Flow Kinetic Energy:. ( eguation 2.49 ) 1. Energy loss - [ Z " r + <r"*p ) due to interaction of Maxwell stresses wxth s t r a i n e.^ 2. Energy loss ~ ( . + £ * " j U ) d u e t o t n e presence of f l u c t u a t i n g e l e c t r i c f i e l d s in a turbulent medium. (3) Mean Magnetic Energy: (eguation 2.53) 1. Energy gain + I * ! 5 ( E^p t ) which arises from the mean and fluctuating velocity f i e l d s respectively 2. Energy gain -t-f^Xi which aris e from the fl u c t u a t i n g 67 velocity f i e l d 3. Energy loss - 1* /cr which goes into the mean inte r n a l energy. (4) Fluctuating Magnetic Energy:-(eguation 2.54 ) 1- Energy gain +• cr"K|i (E*p. + *KJ») a r i s i n g from the mean and fl u c t u a t i n g velocity f i e l d s respectively 2. Energy gain £ ' " j 1 * a r i s i n g from the fl u c t u a t i n g velocity f i e l d 3. Energy loss j 1 /°~ which goes into the mean i n t e r n a l energy The next section w i l l concern i t s e l f with the eff e c t the magnetic f i e l d s have on the structure of the mean flow. 3.. 3 Consequences Of The Mean Momentum Conservation Eguation (Including Magnetic Effeetsj_ For Thin Discs This section i s entirel y analagous to the analysis i n section 2.2 except eguation 2.45 ( which includes the Maxwell stresses ) i s used instead of equation 2.9. We use exactly the same approximations ( thin disc, a x i a l symmetry, etc. ) as given i n section 2.2. In addition, and i n conformity with the usual MHD assumptions,we add (E) the length scale l t ' over which the magnetic fluctuations b occur are of order of those over which the v e l o c i t y f luctuates; X_' a «£u . 68 In the low Mach number regime; Mt<<1, we have l ^ « z a (see Shakura et a l (1978) and Chapter 3) and i t i s consistent from the "separation of scales" idea to imagine that the mean f i e l d varies on length scales l>.z0. In the high Mach number regime; M^.~ 1, we have l u * z 0 and the idea of a mean f i e l d i s r e a l l y only v a l i d on scales L>>ze. Chapter 3 w i l l compute the structure of the mean magnetic f i e l d i n the low Mach number regime r and i n Chapter 4 i t i s shown that t h i s regime produces magnetic fluctuations of a magnitude s u f f i c i e n t to explain the shot-noise model as has been discussed i n Chapter 1. 1. The z-component of equation 2.45 gives with the assumptions given (compare with 2.15) and using the estimates l " 4 e B z It**. shows that Cs + u + J ^ ( H + * l ) 69 where the Alfven v e l o c i t i e s have been used. Thin discs then require that i n addition to the requirements i n section 2 that v / / l / K l « l (M t * i) The previous i n e q u a l i t i e s are derived under the assumption that we are averaging the fluctuations over the large-scales and long-times c h a r a c t e r i s t i c for the mean flow. These scales may be i d e n t i f i e d as r for s p a t i a l variation and the d r i f t time scale t 0=r/i/>>t K. It i s important to note that the p o s s i b i l i t y of intense fluctuations on short length scales l u and on very short time scales %. <t K i s not ruled out. 2. The r-component of eguation 2.45 gives to lowest order provided that the r e s u l t s for 2-58 with *»/<- «I hold. 3. The ci-component of equation 2.45 qives 70 More w i l l be said about these Maxwell stresses i n t h i s chapter and i n Chapter 4. ii*. Energy Balance For The- Fluctuating F i e l d s l l n c l u d i n q Magnetic Effects) This section proceeds in s i m i l a r fashion to section 2.3 except we have now three coupled eguations for u z j ez and ID2" instead of just two for and 0 1 . We assume s t a t i o n a r i t y ( i . e . D/Dt = o ) and discard the terms in these eguations involving divergences of guantites. With these stated approximations and assumptions, and by s p l i t t i n g up the Maxwell stresses into diagonal ( the magnetic pressure ) and off-diagonal parts, the equations governing the fluctuations are from equation 2.49 p from equation 2.54 . 1 2 - \ mi . 'j. > a .A - O H I T ' pir and from equation 2.55 where i n writing 2.60 the remaining term a r i s i n g from the off 71 diagonal Maxwell stresses may be written as a divergence and hence ignored. A l l of the purely hydrodynamic effects w i l l remain unchanged from the estimates used in section 2.3 i f we assume that the magnetic f i e l d s do not a l t e r the underlying turbulence too much. This w i l l be an assumption we s h a l l employ throughout t h i s thesis and j u s t i f i c a t i o n for t h i s procedure w i l l be found in Chapter 4. (1) Analysis Of Equation 2.60 We note that the f i r s t term i s approximately and from equation 2.56 which expresses maqnetohydrostatic equilibrium we have The next terms are the viscous d i s s i p a t i o n and the enerqy source which are qiven by equations 2.27 and 2.28 respectively. In comparison to the f i r s t three terms, the term so that P 72 w i l l be treated as ne g l i g i b l e since only the small s t r a i n e l c p i s involved. F i n a l l y , we consider the terms £.T and . These terms are computed i n Appendix B where we f i n d ( equations B.31 and B.34 ) where the length scales for the mean density L * and magnetic f i e l d L i are defined and where the turbulent d i f f u s i v i t y ^ T i s As already discussed, j 1 and i - T are the rates at which energy from the vel o c i t y fluctuations i s being pumped into the fluc t u a t i n g and mean magnetic f i e l d s respectively. From equation 2.65, we see that £.T i s positive only i f the v e r t i c a l scale of the mean f i e l d L z i s larqer than the scale L? for the v e r t i c a l density p r o f i l e . I t i s precisely under these conditions that the magnetic f i e l d i s stable with respect to the maqnetic buoyancy process (see discussion l a t e r in t h i s section). When L j / L , <1, the maqnetic f i e l d cannot exis t i n a stable configuration and such a region of gas becomes buoyant. 73 The other important feature about equations 2 . 6 5 i s that the r a t i o holds i n the low Mach number regime M^«1. We see that energy i s being transferred into the magnetic fluctuations at a rate (L|/i« ) faster than the rate into the large-scale mean magnetic f i e l d - If we approximate Z 0 as an example, and denote A T 0 as the growth time scale for the mean f i e l d , and ATj as the growth time scale for the fluctuating magnetic f i e l d , then We w i l l assume that we are i n the low Mach number regime for which iTjT» f.T . Col l e c t i n g a l l these approximations and substituting into equation 2 . 6 0 gives In t h i s eguation we find the energy source of the ve l o c i t y fluctuations given by the Reynolds stress on the l e f t hand side balanced by losses to the turbulent k i n e t i c energy by buoyancy, v i s c o s i t y , and energy transfer into the magnetic 74 f i e l d s . Equation 2.67 shows that we must provide an analysis of This i s a very d i f f i c u l t problem i n general. I f the mean f i e l d grows to equi p a r t i t i o n strengths < via dynamo action ) i t must very strongly e f f e c t the turbulence i n such a way that no further growth i s possible. However, Malkus and Proctor (1975) have analyzed a mechanism by which the mean f i e l d growth i s arrested at below equipartition strengths, a mechanism which involves the generation of large scale velocity f i e l d s instead of the suppression and a l t e r a t i o n of the underlying turbulence. As already mentioned, t h i s i s discussed in Chapter 4 where i t i s shown that i n l i n e with Malkus and Proctor we estimate the ultimate mean f i e l d energy tf/w i n terms of the turbulent k i n e t i c energy i f our equations are to be closed. 0 ( J Z . ^ T ) (2) Analysis Of Eguation 2__6__: The use of equation 2.65 to approximate yiel d s = 0 where we have noted that i s axisymmetric and U r » U z and where 75 i n which I- i s the relevant scale length for the f l u c t u a t i n g current. We have grouped the f i r s t two terms together because they represent magnetic interactions with the mean v e l o c i t y f i e l d , whereas the l a s t two terms represent processes on the microscales. We f i r s t note that the Ohmic d i f f u s i o n time scale i s Jy ( see Moffat ) so that we set J£J = lu and hence U'X 1 I trU') 4 L ' ^ UJl') In the absence of a mean flow f i e l d , equation 2.70 reduces to \ which i s exactly the res u l t found by Krause and Roberts (1976) and t h e i r analysis i s summarized i n Appendix A.2. This important r e s u l t shows that i n astrophysical settinqs where A| ^ r the fluctuating f i e l d s can be much more powerful than the mean magnetic f i e l d . It i s very important to note that t h i s r e s u l t ( eguation 2.71 ) does not v i o l a t e the f i r s t order smoothing approximation (see Appendix A.2). As Krause and Roberts point out, the part of that i s correlated with u ( t> , ) i s of order * This r e s u l t i s v a l i d provided that T^u/./^an. approximation v a l i d f o r the sun and assumed v a l i d for the accretion disc as w e l l . 76 where the l a s t inequality i s a consequence of the f i r s t order smoothing approximation in which the turbulence i s id e a l i z e d as a c o l l e c t i o n of random waves with Tu <.< \ . We now consider the effect of a mean flow on the r e s u l t s of Krause and Roberts. These authors obtained the result 2.71 in the absence of a mean flow by solving the induction equation for the fluctuations 1% - ^ V'b' = Vx C u' x B ) H as an inhomogeneous equation usinq the Green's function for the d i f f u s i o n operator 1 ' \ Krause and Roberts (1973) consider how these results are affected by the presence of mean flows. When we have a non-zero mean flow the induction equation for the fluctuations may be written ( taking V-U =0 ) Regarding U as a constant over the scales that b' varies then i f we Fourier transform the above equation for b the frequencies w over which b vary are Doppler shifted by the mean vel o c i t y 0 to 77 so that the mean flow has a n e g l i g i b l e e f f e c t on the r e s u l t 2.71 provided oo or in other words, i f T i s the time scale and r the length scale for 0, T r This same r e s u l t i s found using the Green's function approach i f we note that the Green's function becomes roughly QLX,%) * (w^T ) ^ « „ r ( - ( S - W r ) z / i r ) Hence, in the l i m i t 2.72, we find that eguation 2.71 holds so that from eguation 2.70 In the small correlation time l i m i t , and for T^. < < T , the time scales over which the.last two terms i n equation 2.70 are in balance are much shorter that the time scales over which enerqy i s beinq transferred out of the mean flow ( the f i r s t two terms i n 2.70). The l i m i t 2.72 may be regarded as the smallest time T« that gives the r e s u l t 2.71 (and conseguently 2.73). Eguation 2-73 shows that on Keplerian time scales, the fluctuations 78 b^are much larger than b r . This i s understood by observing that on such time scales, the strong shear of the Keplerian flow i s stretching the ra d i a l f i e l d l i n e s of b'r into t o r o i d a l f i e l d l i n e s of b'* , Rearranging 2.73 s l i g h t l y U IT "X. where b ^ » b r implies that b*« h"*"1 . Noting that we may write eguation 2.74 in an int e r e s t i n g way: ( 2 -TO r"4>v U' 0 U UL «• 4 Now £ C U U i s the magnitude of the stationary stress as discussed i n Chapter 1 . Qn the.-long time T » t r , long scale L>>z o magnetohydrostatic balance i s maintained. Then from eguation 2.58 /- T where we use L, T to denote these long scale averages. Thus from eguation 2.7 3 I cr"*- ) so that the Maxwell stress due to the magnetic fluctuations i s neg l i g i b l e and cannot determine the long scale structure of the accretion disc. However, on short time and length scales -, i t i s possible 79 to have magnetic fluctuations up to a maximum amplitude of For these scales, eguation 2.73 shows that so that the Maxwell stress from the fluctuations, in small regions and for short times, are of order of the mean, long scale averaged stationary stress (of magnitude f. WW* ). We return to t h i s point in Chapter 4-We note that using the r e s u l t 2.73, and examining the induction eguation for b using the assumptions of a x i a l symmetry and disc thinness, that the l i m i t 2.7 2 may be relaxed somewhat. The point here i s that since the fluctuations b/,r are small compared to b'^  , the eguation for b'1* shows that the terms involving the mean velocity are assuming that the fluctuations are independent of z and 0. I f the fluctuations b are small enough compared to b T (as eguation 2.73 shows) then the mean f i e l d terras for the predominant fluctuations b^ nearly cancel out. This implies that we may take the correlation time l i m i t up to values T M £ t K without too seriously a f f e c t i n g the v a l i d i t y of 2.71 and 2.73. 80 F i n a l l y , i t i s important to note that when bVwfo > c\ , the region wherein we f i n d such a high f i e l d strength becomes buoyant and r i s e s i n the di s c , a phenomenon f i r s t noted by Parker (1955a) and discussed more generally by Gilman (1970). B a s i c a l l y when the magnetic presure becomes large enough i n a region, and assuming at least i n i t i a l l y that we have pressure egualization, the gas pressure must drop and i f f><< ^  , we see that the density i n t h i s region decreases. Conseguently, t h i s region of high magnetic f i e l d strength f l o a t s upward i n the gas ( assuming <• o ) . This phenomenon i s c a l l e d "magnetic buoyancy" (coined by Parker ), and must of course transfer energy out of the turbulence and cool the region of gas i n which t h i s high magnetic f l u c t u a t i o n i s prevalent. I t i s thought to occur i n sunspot regions on the 3" sun where very intense fields~10 Gauss ( much higher than egu i p a r t i t i o n strength ) emerge from the solar surface. We believe the same mechanism i s operative here. The presence of intense magnetic fluctuations k ~ l/* does not v i o l a t e the thin disc assumption because instead of bulging the disc so that z 0-^ , these fluctuations are associated with a buoyant region that r i s e s up and eventually results in the emergence of these f i e l d s from the surface of the thin d i s c . We note that the r i s e time of these regions i s roughly the Keplerian time scale. The condition that t h i s magnetic buoyancy mechanism be operative i s that the length scale of 81 the magnetic fluctuation be less that the density scale height, i.e that i t < * 0 . The analysis leading to eguation 2.73 leads to the r e s u l t _}•>_* /Hr & < />. C As we have seen for maximal magnetic fluctuations ' W H T T f D * , the stresses induced are cr"*"' « ^ .U-'u^. since magnetohydrostatic equilibrium on time scales T « i u / V K i s being violated we see that we have large fluctuations 0 C{o _n the stresses responsible f o r angular momentum transport outward ( r a d i a l l y ) and net r a d i a l inflow. Only i f averaging i s done on length scales t0 o r time scales T A ^ / ^ K i s i t possible to discuss magnetohydrostatic equilibrium qiven by equation 2.59 or equivalently, equation 2.63. These larqe magnetic fluctuations then are to be considered as deviations from the mean magnetic fluct u a t i o n s , which over time scales T» **/\J__ w i l l be of order tW^rrfo £ cl . (3) Analysis Of Equation 2.62 Using a l l the approximations that lead to equation 2.33 (a) , equation 2.62 i s ( X * + ____ \ c e e u - J. t e ' (v4) 1 -- - _y - T Estimating the Ohmic dissipation term gives 82 The important factor here, the turbulent magnetic Reynolds number (_R^  ) = X I l i m i t s the role of Ohmic heating even in the case of maximal magnetic fluctuations..Eguation 2.78 shows that because of Ohmic heating, the temperature fluctuations take a higher value than i n the absence of magnetic f i e l d s . Consider f i r s t the case where time scales are of order *o/\]K with maximal fluctuations. With a mean f i e l d below eguipartition we have ( ( T , £ • M - i A' ) C6u9 - -d Z\_t + x 1 V ' L c * 4rr f„ / ( le tr J where we have used eguation 2.77 and /cr * l • If the mean f i e l d i s at egui p a r t i t i o n strength BV^ TT f« * cl , then the heating due to those large magnetic fluctuations i s s i g n i f i c a n t and acts to make regions containig the fluctuations hotter than i n t h e i r absence. However, since we deal with mean f i e l d s below e g u i p a r t i t i o n , t h i s magnetic effect w i l l be taken as ignorable. 83 For longer time scales where we have magnetohydrostatic equilibrium the magnetic terms are < Cs/fu - Now since (RM ) ' " 1 ' the magnetic e f f e c t s are e n t i r e l y n e g l i g i b l e . We conclude that even for the largest magnetic fluctuations, as long as the mean f i e l d i s below equipartion strenqth, the effe c t s of Ohmic heating are n e g l i g i b l e and that equation 2.33 (a) i s s t i l l a qood approximation to use for the maqnitude of the thermal fluctuations. Hence (>/^& + i / t f . ) where we notice that g, given by equation 2.63, contains the maqnetic pressure as well and that only the f l u i d pressure appears i n equation 2.78. We now combine equations 2.78 and 2.68 into equation 2.67 usinq the estimate 2.66 and where q i s given by equation 2.63. We then obtain I f we note that u*-2 u/t t and that the l a s t term i n equation 2.79 may be written i*.* Jl CTu./tt)2' , the magnetic term adds an •* dependence which i s the same dependence as u SI 84 found for the Reynolds stress. As in the analysis leading to eguation 2.53, we multiply by (J cu J l u ( - f j i ) z ~ ] , and by using the d e f i n i t i o n s of [2 and f , as well as equation 2.37 for the Richardson number ( where g i s given by equation 2.63 ) we have - I M + * + i = i u.to) where -fc t and as before The last term i n equation 2.80 represents the energy extracted from the turbulent kinetic energy by the term £•j * $ • 7 . As pointed out e a r l i e r , we f e e l the assignment Tu & t K i s appropriate i n the disc so that we have C u V As i n section 2 we need only solve equation 2.80 i n x. Again i t i s easier to s p l i t the analysis into two cases for convenience. Case 1. (T= o : Here x = ( i - s) t 1/ ( * -o i - f ^ T 1 85 so that stationary turbulence i s only possible provided with of course the requirement that £<| . As $-» [ ; pRi-* o * so that turbulence in the whole disc i s shut o f f . If S ~ ("t* \ Z - ' w e s e e t n a t t h i s corresponds to the l i m i t of supersonic turbulence i f Cu i s estimated as t«./tt • This begins to defy the v a l i d i t y of the theory we have used to derive these r e s u l t s , however, and the situation for S - * ' i s probably considerably more complicated. Case 2 Y j= 0 : Here we f i n d that for two stationary turbulent flows are possible, only the flow corresponding to the larger root being stable. For only one stationary flow i s possible and i t i s stable. Comparing these results with those found by Stewart i n the absence of magnetic f i e l d s , shows that for small Mach numbers M t <•<• I , the extraction of energy from the turbulent 86 k i n e t i c energy by the magnetic fluctuations makes buoyancy a more e f f e c t i v e agent i n damping the turbulence ( i . e . i s r e s t r i c t e d to small values i f we are to have stationary turbulence ). Furthermore, l o c a l extreme magnetic fluctuations 1~V ^ ( o V>cl cause large l o c a l values for pRi which as we see above res u l t s in the shutdown of the turbulence i n the region. Our analysis may be put into perspective i f we r e s t r i c t ourselves to low Mach numbers ft_« i , wherein " f c i < / t t « ( . The f i r s t order smoothing assumption i s v a l i d provided we are considering time scales for turbulent disturbances TM « t t A_ ' -a Since the Keplerian time scale seems to be the fa s t e s t one i n the problem, by focussing on the time scale TU < -fcK r the f i r s t order smoothing assumption i s being s a t i s f i e d for M t « I . On t h i s time scale, equation 2 . 7 1 holds even in the presence of mean flows and hence large magnetic disturbances can occur. These are to be thought of as l o c a l strong perturbations of the o v e r a l l f l u i d . Our scenario suggests that as the energy i s being transferred into the l o c a l fluctuating magnetic f i e l d , the turbulence i s damped out, the f l u i d cools and the magnetic f i e l d r i s e s to the surface of the disc on times for the largest fluctuation IYMT ^ 0 =• V*. by some magnetic buoyancy type process. These loops of intense f i e l d escape from the disc by undergoing reconnection with a neighbouring loop i n the manner described i n Chapter 1 . We leave to Chapter 4 the calc u l a t i o n of the strength of these l o c a l intense f i e l d s . 87 We t u r n now t o i n v e s t i g a t e under what c o n d i t i o n s a mean f i e l d can be generated by t u r b u l e n t dynamo a c t i o n i n the a c c r e t i o n d i s c . 88 Chapter-3 Solution Of The Induction Equation For - The Mean F i e l d B i i Introduction As has been pointed out i n Chapter 2 and Appendix A, additional terms appear in the induction equation for the magnetic f i e l d i n a turbulent conductor as contrasted with one in which only laminar flow i s occurring. When only a mirror-symmetric turbulence i s present, a turbulent d i f f u s i v i t y * T i s added to the usual molecular d i f f u s i v i t y <f of the mean magnetic f i e l d , and i n conditions of high magnetic Reynolds numbers (as usually found in astrophysical flows), MT » ^  This indicates that the idea of "frozen- i n " f i e l d l i n e s of mean magnetic f i e l d i s incorrect for strongly turbulent flows. When the turbulence possesses h e l i c i t y , a mean current J p a r a l l e l or a n t i - p a r a l l e l to B arises and has the e f f e c t of regenerating the mean f i e l d . Steenbeck, Krause, and Radler (1966) were able to show that the presence of l o c a l rotation and a density gradient induces h e l i c i t y i n the turbulence, thereby providing a mechanism by which dynamo action ( s e l f e x citation of the mean f i e l d at the expense of turbulent k i n e t i c energy ) could sustain mean f i e l d s ( the so-called I I* - e f f e c t " ) . When the mean f i e l d remains weak (i.e much below equi-89 p a r t i t i o n with the energy i n the turbulence ) the Lorentz-force a r i s i n g from these f i e l d s appearing i n the Navier-Stokes equation may be regarded as n e g l i g i b l e . Conseguently, i n t h i s s i t u a t i o n , prescription of the flow and the turbulence c h a r a c t e r i s t i c s allows the ca l c u l a t i o n of £ - _' x b' i n terms of B and various quantities a r i s i n q from averaginq over turbulent velocity fluctuations. The solution of the induction equation for the mean f i e l d ( equation 2.50 ) i s then possible i f an appropriate set of boundary conditions for the problem on hand i s provided. For s u f f i c i e n t l y vigorous h e l i c a l turbulence, an i n i t i a l l y weak mean f i e l d of s u f f i c i e n t l y large scale w i l l be amplified by dynamo action as has been shown i n work on t e r r e s t i a l , solar , and g a l a c t i c magnetic f i e l d s (see Moffat (1978)). Ultimately the f i e l d becomes strong enough so that a back-reaction on the flows occurs thereby preventing further growth. The magnetic f i e l d can act to suppress or a l t e r the turbulence or induce large scale " mean " flows, both of which arrest further growth of the f i e l d . The importance of the large scale mean magnetic f i e l d for accretion problems i s three f o l d : 1. Generation Of Magnetic Fluctuations. The previous chapter has shown that the fl u c t u a t i o n magnetic f i e l d energy density i s related to the mean magnetic f i e l d energy density by L** _• 6 Z • The reason for t h i s i s understood i f the induction \ 90 eguation for the fluctuating magnetic f i e l d i s considered ( eguation 2.53 ). A ve l o c i t y f l u c t u a t i o n u'interacts with the mean f i e l d B and over a length scale 1* . and time scale twists up the f i e l d l i n e and creates a fluctuating magnetic f i e l d b on these same length and time scales. Consequently, information about the amplitude and orientation of B allows us to estimate what type of maqnetic fluctuations are to be expected. 2- Transport Of Angular Momentum. I t was shown in Chapter 2 that the fluctuating Maxwell stress <r"4 ir was s i g n i f i c a n t only i n those l o c a l i z e d regions where fluctuations t> * occur. Over s u f f i c i e n t l y long time scales, these l o c a l intense fluctuations are unimportant so far as angular momentum transport in the disc i s concerned. However, the mean Maxwell stress contributes to the o v e r a l l stresses ( again on s u f f i c i e n t l y long time scales ) and hence i s important i n determining the disc structure. 3. The Presence Of A Magneto-sphere. The intense magnetic fluctuations emerge through the upper and lower surfaces of the accretion disc , and i n our picture, engage i n subsequent reconnections giving r i s e to solar type f l a r e s . The region exterior to the disc i s expected to be of low density. If i t i s imagined to be a vacuum as an example ( an i d e a l i z a t i o n of course ) , then the reguirement that the exterior vacuum f i e l d s match continuously to the i n t e r i o r disc f i e l d at the upper and 91 lower surfaces of of the disc implies the presence of a large-scale, current-free magneto- sphere when large-scale mean f i e l d s are present i n the disc. The strength and structure of such a large scale vacuum f i e l d i s important for determining the t r a j e c t o r i e s that energetic p a r t i c l e s leaving the disc region would take and the radiation that they would emit as they s p i r a l along the f i e l d l i n e s . More s p e c i f i c a l l y , Blandford (1976) assuming a force-free magneto-sphere and Lovelace (1976), assuming a current free magneto-sphere have t r i e d to construct models of double radio- sources r e s u l t i n g from the presence of magnetized accretion discs around a central compact object. Their work however does not discuss i n d e t a i l how a large-scale magnetic f i e l d may be generated and maintained i n the disc. I t i s f e l t that the work to be presented here can act as a f i r s t step towards a more comprehensive treatment of such theories. With the previous arguments as a motivation, i t w i l l be the object of t h i s chapter to solve the mean f i e l d induction equation under conditions appropriate to an accretion disc. It w i l l be the assumption of t h i s chapter that the mean f i e l d i s i n i t i a l l y weak. The mean flow w i l l be taken to be Keplerian and the assumptions made about the turbulence as discussed i n chapters 1 and 2 w i l l be employed. Hence, we s h a l l determine on what lenqth and time scales we may expect the mean maqnetic f i e l d to grow i n a prescribed hydrodynamic se t t i n g . 92 In section 2, we begin with equation 2.50 and simplify i t as much as possible using the approximations under which the disc structure was solved in the absence of magnetic f i e l d s . In p a r t i c u l a r , i t w i l l be assumed that the disc i s t h i n , the mean flow i s Keplerian, and that the mean f i e l d i s axisymmetric. As discussed i n Appendix A, i t w i l l be assumed that the underlying turbulence i s mildly anisotropic so that h e l i c i t y i s present. It w i l l be necessary to specify the v e r t i c a l density p r o f i l e and the disc half-thickness z<> at a l l r a d i i . The density i s approximately Gaussian i n the gas-pressure dominated zone and t h i s permits solution by a n a l y t i c a l methods. The parameter z 0 however, depends on r and t h i s i s very d i f f i c u l t to deal with when matching to an exterior solution for the magnetic f i e l d . We w i l l assume that z 0=const for the purpose of the analysis. As long as the radius of curvature B 0 i s large, t h i s defect can be corrected by a perturbation procedure involving a power series expansion i n *./R. -Section 3 i s devoted to the solution of the induction equation af t e r the s i m p l i f i c a t i o n s discussed i n section 2 have been applied. In p a r t i c u l a r , the v e r t i c a l structure of the mean f i e l d w i l l be analyzed extensively. The point of the analysis w i l l be to determine accurately the behaviour of the f i e l d near | z h ? 0 so that matchinq with an external vacuum 93 f i e l d can be accomplished. It w i l l be assumed in t h i s section that dynamo action and dissipation exactly balance one another so that stationary conditions p r e v a i l . When M t « ( , the problem can be solved a n a l y t i c a l l y . In section 4 we match the disc solutions to an external vacuum f i e l d assuming s t a t i o n a r i t y (on long time-scales). This procedure w i l l r e s u l t i n a r e l a t i o n between the turbulent Mach number M* and the r a d i a l wavenumber of the f i e l d . The f i n a l section attacks the same problem again assuming non-stationary conditions. Small deviations from eguilibrium are assumed so that the dynamo action and dissipation are very nearly i n balance. The procedure results i n a dispersion r e l a t i o n l i n k i n g the complex growth time scale to the turbulent Mach number and the r a d i a l wave-number. It i s important to point out that the entire theory being discussed i s n o n - r e l a t i v i s t i c so that the study of the mean f i e l d s at the innermost edge of the disc i s not considered. In particular,the electromagnetic boundary conditions at the event horizon are not considered. Recently Znajek (1978) has shown that the boundary conditions s a t i s f i e d by the electromagnetic f i e l d s at the horizon of a Kerr hole may be interpreted i n terms of equal e l e c t r i c and magnetic conductivities of such an object. In addition, Blandford and Znajek (1977) showed that electromagnetic f i e l d s could extract energy from a rotating hole ( Kerr ). Various i d e a l i z a t i o n s 94 about t h e f i e l d s were made i n t h i s work, and we b e l i e v e t h a t t h e p r e s ence o f a t u r b u l e n t d i s c ( not a p e r f e c t c o n d u c t o r ) c o u l d c o n s i d e r a b l y c o m p l i c a t e the p h y s i c s . 2 . A n a l y s i s Of The I n d u c t i o n E q u a t i o n For The Mean F i e l d 2 . 1 S i m p l i f i c a t i o n s A r i s i n g From The Assumptions Of A T h i n D i s c And A x i s y mmetric F i e l d , . -We r e c a l l the i n d u c t i o n e q u a t i o n f o r the mean f i e l d i s 2| - V K ( U x V + J" - ^ S ) where £ = U'K__ and where we s h a l l be workinq i n c y l i n d r i c a l c o - o r d i n a t e s (r,0,z) . The mean f l o w U i s assumed t o be ^ * C o , \)Ktr) to) which i s v a l i d p r o v i d e d t h a t the d i s c i s t h i n . T h i s d i f f e r e n t i a l r o t a t i o n o f t h e qas c o n t a i n s s h e a r i n q motions on the l e n q t h s c a l e r , which i s i m p o r t a n t i n t h e a n a l y s i s of £ S p e c i a l i z i n g t o c y l i n d r i c a l c o - o r d i n a t e s , d e f i n i n g i i = £ - ^ tfx B Cx.O and assuming a x i a l symmetry g i v e s 95 Since V.$-0lthe assumption of a x i a l symmetry for the f i e l d B allows us to use the representation B - ( - ^  , T * 2 ti-n) (vo in which . P and T are a r b i t r a r y functions of r, z, and t , and which w i l l be fixed by substitution of equation 3.6 i n t o equations 3.3-3.5. Equations 3.3 and 3.5 are then e n t i r e l y equivalent so that only the coupled set 3.3 and 3.4 need be considered. Hence, these equations become, respectively, D _ r = si ( 3 . - 7 ) at (V«) E q u a t i o n 3.8 may be s i m p l i f i e d by noting = «o"s"f so that To proceed, i t i s necessary to calculate £ , which i s done i n Appendix B using the following approximations: 1. Contributions to the h e l i c i t y are taken to arise from the i n t e r a c t i o n of the antisymmetric component of the mean st r a i n tensor ( R„<p = J-l^-p - Up-*.)) w i f h the density gradient 2,. Gradients of the turbulent intensity are ignored. As discussed i n Chapter 2 , we expect the turbulent 96 c o e f f i c i e n t s t o depend only on r (the t u r b u l e n c e i s homogeneous i n z and a t each radius) . Appendix B shows t h a t 4 ir L s 3 6, 8 2 2 r J ^ -- ^ - /n T (X7< B ) ( 3 . . 0 B 2 _ Br- - •£ 7 i 8 a* J * ' where • . ( . v ( *1r * and where 2fr i s the r a t i o of the d e n s i t y s c a l e s i n 2>«- ' "it. the r and z d i r e c t i o n s . We emphasize t h a t f i r s t order smoothing i s assumed i n these c a l c u l a t i o n s ( <2 & £?u/iu. « i ) as w e l l as the high c o n d u c t i v i t y l i m i t " IT/^ » 1 • For a t h i n d i s c , <? << | . I n a d d i t i o n e quation 3.15 shows th a t 97 so even i f we r e l a x the small c o r r e l a t i o n time l i m i t ( i . e say Q * \ ) then x l u / i a c , so t h a t ^-A/* «i f o r t h i n d i s c s . We s h a l l have more t o say about t h i s r e s u l t a f t e r we s u b s t i t u t e the equations 3.10 - 3.12 i n t o 3.7 and 3.9. Before we do t h i s we note t h a t f o r t h i n , axisymmetric d i s c s . Equation 3.12 shows t h a t i f the maqnetic f i e l d l e n q t h s c a l e s are of the same order as the d e n s i t y l e n g t h s c a l e s , then a l l the terms m u l t i p l i e d by <=< should be of the same order of magnitude. However, s i n c e and £^ e n t e r s i n t o eguation 3.9 i n the combination 2>£,- 2£i ' w e estimate that (i.n) a r e s u l t v a l i d only f o r t h i n d i s c s . Noting t h a t the r e p r e s e n t a t i o n 3.6 i m p l i e s where »~ ^ r- 3 * s u b s t i t u t i o n of equations 3.10 - 3.12 i n t o 3.7 and 3.9 g i v e s , using the approximation 3.17 H L T -. 1 (/» J i - « . + ( * r * - » 0 A T ( 5 . Z i ) U 2 r " a* 1 98 In the l i m i t of high turbulent magnetic Reynolds number, we have - * i r » - ^ so that *t r 1 - ^ ^ A<t - Let us rearrange equations 3.20 and 3.21 as ( using A\t y> ) - /M A P r tx. I h.zz) at The f i r s t thing to note about these equations i s the 3 appearance of the d i f f u s i o n operator ft ~ & on the l e f t hand sides. I t i s untenable to assume that the the mean f i e l d remains frozen-in to the gas since ^ _ » ^  . In a s u f f i c i e n t l y vigorous turbulence, i s so large that the mean f i e l d s quickly damp out i n the absence of sources. The source terms for the poloid a l and tor o i d a l f i e l d s have been written on the riq h t hand sides of the equations. The source for the reqeneration of the poloidal f i e l d P i s the t o r o i d a l f i e l d T. Dynamo action i s qenerating the poloidal f i e l d at the expense of to r o i d a l f i e l d . If <x = o , ( i. e absence of h e l i c i t y ) the poloidal f i e l d has no source and hence decays exponentially with a time constant ? 0 Z/^ T which i s not much longer that the Keplerian time scale . With a damped out poloidal f i e l d , i t i s not possible to sustain the t o r o i d a l f i e l d by d i f f e r e n t i a l r o t a t i o n , and so, very guickly, the entire mean f i e l d i s dissipated. 99 There are two sources available for the regeneration of the t o r o i d a l f i e l d T. The f i r s t term on the right hand side of equation 3.23 i s a term involving the int e r a c t i o n of the shearing of the t o r o i d a l flow ( — K ) with the r a d i a l f i e l d B r { » - ) . This term i s absent for pure rotation ( - ( = o ) . Physically what i s happening can be seen i n terms of the r a d i a l f i e l d l i n e s being stretched out i n the o7 direction by the d i f f e r e n t i a l r o t a t i o n , thereby creating t o r o i d a l f i e l d T. The second source term again represents the e f f e c t of h e l i c a l turbulence, t h i s time resulting in the creation of t o r o i d a l f i e l d from poloidal f i e l d . The source terms involving oc are best understood by Parker's (1955b) arguments. Consider' an almost uniform long scale f i e l d B in the r - 0 plane, in the presence of a v e r t i c a l density gradient (see Fig. 5). Imagining that small-scale upwellings of f l u i d occur the f i e l d l i n e s w i l l be bent i n t o horseshoe shaped loops. I f , i n addition, a l o c a l rotation i s present, these horseshoes get twisted out of the i r i n i t i a l planes. Averaging over a l l of these small-scale twisted horseshoe shaped loops, we see that an i n i t i a l l y t o r o i d a l mean f i e l d should give r i s e to a r a d i a l component of the f i e l d ( source term i n equation 3.22 ) and an i n i t i a l l y r a d i a l mean f i e l d should give r i s e to a t o r o i d a l component of the f i e l d ( second source term i n equation 3.23 ). These arguments explain why oc i s dependent on both VPD and tfx U and why the 100 to r o i d a l and poloid a l f i e l d dependences arise i n the source terms as they do. Fig,. 5 The °t-Effect In Turbulence With H e l i c i t y When the term proportional to CK. i n equation 3.23 dominates the term a r i s i n g from d i f f e r e n t i a l r otation, we see that the <K - e f f e c t i s responsible for regenerating both components of the f i e l d , a s i t u a t i o n c a l l e d the " <*x -dynamo ". When the d i f f e r e n t i a l r o t a t i o n dominates the « term i n 3.23, we have a " -dynamo 11. We have already noted that •x/V*. « ( i n the case of thi n discs. Writing the right hand side of equation 3.23 as and estimatinq : — , & if/a*  L* we see that the d i f f e r e n t i a l rotation dominates the < - e f f e c t source provided -7 f « i . Using the estimate 3.15 for we see that t h i s condition becomes Q ^ ^ J ^ i - Since consistency 101 of our whole analysis requires use of the small c o r r e l a t i o n time l i m i t ( ( ) the d i f f e r e n t i a l rotation w i l l act as the major mechanism for convertinq B r to B^  . If the l i m i t Q2<< I i s relaxed, our condition i s that the turbulent eddies should be of a scale iC L_ xr i e . The previous discussion indicates that our set of equations for the f i e l d s becomes these equations show that t o r o i d a l f i e l d qives r i s e to poloidal f i e l d by the - e f f e c t while poloidal f i e l d gives r i s e to t o r o i d a l f i e l d by d i f f e r e n t i a l rotation, i . e . P ^ T If the strength of these sources i s s u f f i c i e n t to overcome the d i s s i p a t i o n due to "£T, the f i e l d s w i l l be amplified. Energy i s being extracted from the turbulent k i n e t i c energy i n order to rlin the - e f f e c t source for P while the Keplerian flow i s the source of energy necessary to regenerate T. Further progress requires that we specify the v e r t i c a l density d i s t r i b u t i o n and the expression of turbulent c o e f f i c i e n t s i n terms of mean f i e l d quantities. 102 2.2 V e r t i c a l Density P r o f i l e And The Calculation Of <K And In the standard model of accretion d i s c s , Shakura and Sunyaev (1973) showed that in the regions where the gas pressure dominates the radiation pressure, the gas density Po may be written, For \zt>z„ , the density f a l l s o ff exponentially at a faster rate. As we s h a l l imagine the disc to have a discrete boundary at \z\=za , we use eguation 3.26 as the density p r o f i l e i n the gas pressure dominated zones. Hence Their paper showed that i f e f f e c t s of turbulence are ignored, the inner, radiation-pressure dominated zone has a density p r o f i l e independent of z. This result would imply that • regeneration of the f i e l d by dynamo action would occur here^ although i n our case, a d d i t i o n a l d i f f i c u l t matching problems a r i s e . More recent work by Shakura et a l (1978) shows that convective turbulence should occur i n t h i s radiation dominated zone which a l t e r s the v e r t i c a l energy transport i n such a way that the v e r t i c a l density p r o f i l e i s no longer independent of z. For ^hi<o and assuming a polytropic r e l a t i o n between pressure and density they show that 103 where n ranges from n=0-85 f o r Mt=1 t o n=1..17 as ^t-»0. The v e r t i c a l d e n s i t y p r o f i l e i s then F o r | z K< Z o t h e d e n s i t y p r o f i l e s 3.29 and 3.27 a r e t h e same whereas i n t h e s u r f a c e r e g i o n s t h e d e n s i t y i n the r a d i a t i o n dominated zone f a l l s o f f more q u i c k l y than i n t h e gas dominated zone. We s h a l l adopt t h e Gaussian p r o f i l e 3.27 th r o u g h o u t t h e e n t i r e d i s c . With t h e v e r t i c a l d e n s i t y p r o f i l e s p e c i f i e d , we t u r n t o c a l c u l a t e t h e t u r b u l e n t c o e f f i c i e n t s <*. and/v\T . . Shakura e t a l show t h a t t h e v e l o c i t y a m p l i t u d e o f t h e c o n v e c t i v e t u r b u l e n c e ( dominant i n the r a d i a t i o n zone ) and t h e s h e a r t u r b u l e n c e ( dominant i n t h e gas dominated zone ) a r e of t h e same o r d e r o f magnitude. S p e c i f i c a l l y u / c & -- K t * lu I iB (3.30) so t h a t M<C<1 r e q u i r e s i u / l o « : | . So f o r low Mach numbers, we see t h a t t h e d i s c u s s i o n l e a d i n g t o e q u a t i o n 3.25 means t h a t t h e s h e a r i n q dominates t h e oc - e f f e c t r e q e n e r a t i o n of the t o r o i d a l f i e l d T even when the assumption o f f i r s t o r d e r smoothing i s r e l a x e d . T u r n i n g t o t h e e x p r e s s i o n s 3.14 and 3.15 f o r ^ and , we see t h a t we would l i k e t o r e p r e s e n t ZT and f M i n terms o f mean f l o w q u a n t i t i e s . We adopt t h e assumption used i n t h e s t a n d a r d d i s c models t h a t M 4 =const. I t has a l r e a d y been 1 0 4 s h o w n t h a t h y d r o s t a t i c e g u i l i b r i u m i n t h e d i s c i m p l i e s c s / l / k *" * o / r f o r M t <<1. C o n s e q u e n t l y we w r i t e w h e r e f o r t h e t h e o r y we w a n t t o p u r s u e M<<1 a n d z = c o n s t . F i r s t o r d e r s m o o t h i n g t h e o r y i s v a l i d i f we r e s t r i c t o u r s e l v e s t o s h o r t c o r r e l a t i o n t i m e s , a - ruu/iu - ru/tt «\ Now i n t h e l i m i t o f s m a l l M a c h n u m b e r s e q u a t i o n 2 . 1 9 s h o w s t h a t P u t t i n q 3 . 3 3 a n d 3 . 3 2 t o q e t h e r g i v e s Q = H t f « /-6K « ' s o t h a t i f T u i t f e , t h e a s s u m p t i o n M^C<1 d e l i v e r s Q < < 1 . T h e r e f o r e a s l o n g a s M^C<1, r e s t r i c t i o n o f t h e c o r r e l a t i o n t i m e t o Tu$tK s h o w s t h a t Q « M t « 1 a n d h e n c e t h e mean f i e l d a n a l y s i s i s i n t e r n a l l y c o n s i s t e n t . C o m b i n i n g r e s u l t s , i n t h e l i m i t M<<1 a n d f o r T^tK, e x p r e s s i o n s 3 . 1 4 a n d 3 . 1 5 f o r <nT a n d b e c o m e A\TCy) * H t 2 * C3-3 0 w h e r e r e c a l l r / l / K - H e r e h a s t h e d i m e n s i o n s c m * s " ' a n d t h e d i m e n s i o n s o f a v e l o c i t y cm s _ l . We n o t e f o r f u t u r e 105 reference the property The s i m p l i f i c a t i o n s a r i s i n g from these assumptions are considered i n the next subsection. 2.3 The Equations In Dimensionless Form Dividing 3.24 and 3.25 by /v\r, and using the results 3.35 and 3.36 gives £ L p - \ 1 + x - - J. +}L P = r^T (5.3-0 where 3 / z We note that i s the r a t i o <*/*tT and X. i s the r a t i o W/rvtj r and are measures of the strength of dynamo and shear processes which amplify the f i e l d with respect to the dis s i p a t i o n by ^ T - Since both Y and X are constants with dimension cm"* we see that the guantity ( X Y) has dimension cm"' and we use t h i s to make the equations dimensionless. Thus 106 i s a measure of the product of the two source terms ( <* VK/r) divided by the sguare of the d i s s i p a t i o n and i s therefore a measure of o v e r a l l source strength r e l a t i v e to the d i s s i p a t i o n . This parameter i s e n t i r e l y analagous to the parameter -A defined by Parker (1971) i n his study of the ga l a c t i c dynamo. It w i l l prove convenient to solve the coupled eguations 3.37 and 3.38 i f we use B> and B^ , instead of P and T. Thus dividing both 3.41 and 3.42 by ( t o . ) and defining the dimensionless variables A we have where and = _J ^  - The guantity yB i s the r a t i o of source 107 strength due to dynamo action to the source strength due to d i f f e r e n t i a l rotation- In the l i m i t M<<1, Js <<1 which i s showing the dominance of d i f f e r e n t i a l rotation over dynamo action i n the small Mach number regime. Eguations 3.44 and 3.45 indicate that a natural choice for the dimensionless time co-ordinate would be "ts J t / t k which suggests that the temporal v a r i a t i o n of the mean f i e l d should be roughly on scales t k v > t t , assuming M<<1. The d i f f i c u l t y here i s that t K * r V z so that eguations 3.44 and 3.45 have the problem of inhomogeneity i n the r a d i a l d i r e c t i o n due to the r a d i a l dependence of *\r «. This i s i n general a d i f f i c u l t problem but section 5 provides an analysis of a solvable regime. In the next section, we assume that the d i s s i p a t i o n and " i cx.ui » dynamo action are i n balance so that the f i e l d i s time independent. 3- Solutions To The Eguations-In The Stationary Case We consider here the si t u a t i o n when =. o ; i . e the di s s i p a t i o n and dynamo action exactly compensate one-another. Eguations 3.44 and 3.45 then become 108 These equations admit separable solutions. Writing and introducing the separation constant - K. we f i n d that R obeys the eguation while the z dependent factors are governed by the coupled equations J 1 Equation 3.52 gives which is substituted i n t o eguation 3.51 to give a fourth order equation for 0 - AKJJU . % l\X + (V-i)U - 0 (3^5) «U* us 1 Xi With K r e a l , K >0 results i n solutions R beinq a l i n e a r combination of the Bessel functions J, C ) and whereas K <0 leads to a l i n e a r combination of the modified Bessel 109 functions K, ((-K)'4-?) and I/Hc/^rj. Matching to a vacuum with proper behaviour in the l i m i t z -* <o and r - * 0 , give k >0 and Equation 3.53 may most elegantly be solved by using an int e g r a l representation of the solution. S p e c i f i c a l l y , we seek a solution to eguation 3.53 of the form U(z) = J K<*.*) *U) Jit Since the d i f f e r e n t i a l operator acting on U contains polynomials of only order one i n z, the choice of the Laplace kernel w i l l require only the solution of a f i r s t order d i f f e r e n t i a l equation in t for the as yet unknown function v (t). This i s carried out i n Appendix C.1 where i t i s shown that 0 i s qiven by J • -TI ( 3-rt) where C i s any contour for which the inteqrand vanishes at the end points. Of course there w i l l be a number of d i f f e r e n t contours in the complex-t plane which accomplish t h i s , and these w i l l correspond to the d i f f e r e n t independent solutions of ( w | r o . The appearance of the factor t"K in equation 3.56 implies the presence of a branch cut i n the complex-t plane. 110 Let us f i r s t consider the type of contours which are admissible. Consideration of the integrand shows that as It I -> , the factor e i s the most important. I f Re(t )<0, the i n t e g r a l diverges as we integrate out to ltl-^-*o . Convergence of the in t e g r a l occurs only in regions where Re (t ) >0. Writing t--[t\ e' , t h i s requirement becomes Solving the r e l a t i o n shows that there are four zones of convergence as diagrammed in F i g . 6 . Therefore, our contours are most simply chosen to have as asymptotes the co-ordinate axes i n the complex t plane. These contours are drawn in F i g . 6 where we also show the branch cut extending up the p o s i t i v e imaginary a x i s . The end points of these contours are at Itl = i n the directions indicated, so that the requirement that the inteqrand vanish at the end points of the contour i s s a t i s f i e d . 111 112 The presence of the branch cut implies that the four solution s with the contours shown i n Fig. 6 are l i n e a r l y independent. I t may readily be shown, that the sum 2T U,^ ( ? ) ^ O « A phase s h i f t i s picked up due to the presence of the branch cut which insures that the sum i s non-zero ( see Appendix C .2 ) . The requirement that our disc solutions be matched continuously to an exterior vacuum solution at the upper and lower disc surfaces z=± z„ demands the analysis of K^c?) as z—-*±z 0. I t w i l l fee remembered that for M<C<1, as z — > ± z o f so that we may use the asymptotic form f o r U^ii) . The contintuity of the f i e l d throughout the disc w i l l reguire matching conditions at z=0. Conseguently we must examine the behavoiur of U«ti) i n the v i c i n i t y of z=0. These two tasks occupy the rest of t h i s section. It i s important to point out that the assumption of separation of scales for the f l u c t u a t i n g and mean f i e l d s allows us to r e s t r i c t attention to the l i m i t i<<=<| . This i s readily seen by noting that IAP = (V1/^^) r so that the r a d i a l wavelength i s \ r ~ i ^ t l ^ ' ' ) Since we reguire our mean f i e l d to vary on r a d i a l scales larger than z o t we see that the l i m i t K << | focusses on the correct regime for analysis of the 113 mean f i e l d s - This observation w i l l prove very h e l p f u l i n simplifying the analysis to come. We now digress to a discussion of the asymptotic analysis before going on with the problem of matching to the external vacuum f i e l d . Asymptotic form of the solutions a s z - » ? _ In the l i m i t z -» z0, the l i m i t M<X1 implies that z » 1 . It i s convenient to define the variable f as t \ % \ * r and the r e a l positive guantity X by [kit*)'*1 i n terms of which the solutions 3.58 may be written C T where the positive sign i s adopted for z>0 and the negative sign for z< 0 . Introducing the d e f i n i t i o n s 4 cr) s -Z H t r allows us to write equation 3.61 i n the form 0-•**)/. f Aftr) + * _cr) 114 We evaluate the i n t e g r a l in 3.62 in the l i m i t A-*** (i.e z -> oo ) by the method of steepest descents. This method i s usually applied to in t e g r a l s of the form c where i f T0 i s a saddle point ( i.e where -fcr) ), and the contour may be deformed to pass through the saddle point onto the path of steepest descent (this must be j u s t i f i e d by Cauchy's theorem), one finds that in the l i m i t A -* ^ , fu) becomes where ot i s such that -f"crB) e 2 " * i s real and negative. This method requires extension i n order to handle an in t e g r a l such as that appearinq i n equation 3.61, where we have the appearance of an addit i o n a l parameter - Physically we want to focuss on modes such that K« I , so that as Appendix D. 1 shows, for f ( f ) and g ( r ) defined by 3.61, we have i n the l i m i t A -> , !<<./ ; where we more s p e c i f i c a l l y reguire that KlX''1- <:< 1 • The necessary mathematical d e t a i l s for the asymptotic analysis are found i n Appendix D. The c r i t i c a l points of the 115 integrand are found i n D.2 whereupon the paths of steepest descent from each saddle point i s determined ( section D.3 ). Our contours Cw are then deformed onto the paths of steepest descent , the contributions due to each saddle point picked up assessed, and the res u l t s summed to give the f i n a l asymptotic representation of W„ u ^ ) ( section D. 4 ) . We find that the solutions may be written, i n the asymptotic l i m i t * -* where the functions yU„ are l i s t e d i n Table 4. 116 Table 4._ Asymptotic Analysis: The - Functions\MM (K > z) -2 > o 2 <: o e -TT/}CK: +.)." X \ *- e - e - e e e e ^ ) - e a - e. e and where we r e c a l l that lAI = l i l 4 ' 3 . The numerical factors p, q, and g are ci - - 5 - i 3 JJ These re s u l t s show that two solutions (0, and ) grow exponentially for z>0 and exhibit exponentially damped o s c i l l a t i o n s in z<0. The solutions U 3 and 0^ behave conversely. The r e l a t i o n between U„{K,Z) and 0^(K,-Z) found i n the preceeding table are t r i v i a l applications of r e l a t i o n s C.26-C.29. We r e c a l l that the z dependence of B r ( Q(£) ) i s given 1 1 7 by equation 3.52 so that we must calculate <i zU/'J.¥~ . We s h a l l also need to find P (z) i n order to evaluate B z ( r , z ) . With the i n t e g r a l representation 3.58, t h i s i s e a s i l y done. S p e c i f i c a l l y /» .-t\ + K i % ? t Jr.. j . * 1 Using the d e f i n i t i o n s 3.59 and 3.60 these i n t e g r a l s take the form In the asymptotic regime , making the same assumptions leading to the expression for gives where the functions ^ (.*.,%) are l i s t e d i n Table 5, 118 0) T a b l e A s y m p t o t i c - A n a l y s i s : The - F u n c t i o n s y U ^ l K,Z) 2 > ° 2 < o J(,), "\ e € « — e e e z. , e. z 77"*:'<.' f"( e e yMH CK,1 ) e e -/77<2c' p/| - e e 119 And A ^ o i l ( v i s ) w i t h t h e f u c t i o n s l i s t e d i n T a b l e 6. (z) ^ T a b l e 6.. A s y m p t o t i c A n a l y s i s : The- Functionsv f l U (K,z)^ 9 > o S <^  O e e 5 e r A - e e e 7rcKx_/)c ^1 e e - c e - e. e These r e s u l t s show t h a t i n t h e l i m i t z o i l .1* hence I, S i n c e K/A * ^  | , we have 1 J? ( S . 7 f ) 1 2 0 We t u r n f i n a l l y t o the behaviour of U (z) at z = 0 . Behaviour of U/h.(z)_ as z - * 0 . The i n t e g r a l r e p r e s e n t a t i o n 3 . 5 8 a l l o w s the ready expansion of OV, (z) about z= 0 s i n c e we only need t o expand e i n a power s e r i e s i n z, e v a l u a t i n g the i n t e g r a l s i n t (the c o e f f i c i e n t s of the expansion) as we go. In order to f a c i l i t a t e the i n t e g r a t i o n s , i t i s convenient t o deform the contours t o run along the a p p r o p r i a t e p o r t i o n s of the imaginary and r e a l t axes. As an example, the contour C, i s r e p l a c e d by an i n t e g r a t i o n running from to the o r i g i n , and form the o r i g i n out to + ^> . The c o e f f i c i e n t s are then p r o p o r t i o n a l t o i n t e g r a l s of the form which, with the v a r i a b l e T-t z can be w r i t t e n i n g e n e r a l as f - ^ - ^ v - , where f - if a n < a ~X- {< with s (*•«-M-* *,)/*. T h i s l a t t e r i n t e g r a l can be represented i n terms of p a r a b o l i c c y l i n d e r f u n c t i o n s as ( see Gradshteyn and Ryzhik ( 1 9 6 5 ) p. 3 3 7 , formula 3 - 4 6 2 . 1 ) (3.1C) 121 where fte j? >^ o and ^ e * > 0 and ° y 1 S the parabolic cylinder function. The condition Re. i s ea s i l y met with our constraint that K « | The d e t a i l s are given i n Appendix E. We only need the values of the expansions at z=o i n order to do the matching problem of section 4. Defining and taking the l i m i t of as z —» o + , we have 1 ^ \A„u,%) ifrr c with and Q-w given i n Table 7 1 2 2 Table 7. Expansions About Z_\0z The Functions _TM And m 1 £y. Z 3 e f e £y. where X H -I fl K (Wo) 6VC*) S DyCx) - X P 123 Writing z e ifrr e (3.^3) ? -— ' -J we tabulate U„ and in Table 8. Table 8.,Expansions About Z=0: The Functions And / 2 e Dy,., f-x) -77V,; 3 -27?)/, ^ e 0 ; . , " 0 -277 1/, t 6 fry., (X) -3/f >/, i -37 / l/, 1' The l i m i t s as z —> 0" of the functions , , may be found using the above results plus the 4 ^ * P-124 important symmetry relations Tic ti-fc.' ) U 5 U - * ) - e U, £ K S) -TTi ' C l-fc*) derived in Appendix C.2. 4. Matching To An External Vacuum-Solution. We w i l l suppose that outside the disc (i.e lzl>z e) the gas i s so tenuous that the region may be regarded as a vacuum. VAC A vacuum cannot support currents so that we demand (7* B = o , whxch together wxth v-_ = o , requires that B be derivable from a potential *t , with , VAC If we assume a x i a l symmetry of the vacuum f i e l d , and require that Q — * O *S * -* t h * as r -> O 125 t h e n f may be w r i t t e n i n s e p a r a b l e f o r m a s 2 >+••?. (3 S><0 w h i c h , u s i n g e g u a t i o n 3 . 8 6 g i v e s i n t h e r e g i o n z > z 0 a n d i n t h e r e g i o n z< - £ „ w h e r e e v e r y w h e r e i n t h e vacuum a n d w h e r e E(4.) and F (4.) r e p r e s e n t a m p l i t u d e s t o be f i x e d by t h e b o u n d a r y c o n d i t i o n s . A t t h e u p p e r a n d l o w e r s u r f a c e s o f t h e d i s c | z | = z „ , we r e q u i r e t h a t B r , , and B s b e c o n t i n u o u s , i . e . T h e s e r e q u i r e m e n t s i m p o s e a s e t o f s i x c o n s t r a i n t s . 126 At the plane z=0, we require continuity of the f i e l d s and thei r qradients, i . e (s.<?l) which are four more conditions t o t a l l i n q ten altoqether. Noting that U i s a superposition of the four independent solutions ^U^K,?) , and writinq the disc mean f i e l d components as 1 where the other components follow form equations 3.49, 3.52, and 3.6 the boundary conditions 3.93 and 3.94 qive a set of ten equations for the ten unknowns E, F, C^ ,, and ( n=1,2,3,4 ). We use the res u l t s of the asymptotic analysis to evaluate at t z c ( s p e c i f i c a l l y equations 3.66, 3.72, and 3.73 toqether with the results 3.74 and 3.75 ) i n order that the conditions 3.93 can be met with the vacuum solutions ( 3.90 and 3.91 ) specified at ± z„. The constraints at z=0 require equations 3.78 - 3.84 toqether with the r e l a t i o n s 3.85 to establish the form of the solutions i n the l i m i t z —> o" . The r a d i a l behaviour of these solutions may be r e a d i l y matched i f we pick a vacuum mode with dimensionless wavenumber 127 i ; such that -4 = K . with i< > o we see that R e * ; ? ) " -As i t stands, a set of ten simultaneous equations i n ten unknowns must be solved. This set may be reduced to a set of fi v e i f we l i m i t the discussion to modes i n thich the dominant disc f i e l d B^is either an even or an odd funcion of z (see Parker (1971)). Turninq to equations 3.24 and 3.25, and rememberinq that ac*) - -<x.c-£) , we have that f o r even modes U (z) =0 (-z) ,so that P(z) =-P(-z). S i m i l a r l y , for odd modes U(z)=-U(-z) so that equations 3.24 and 3.25 are s a t i s f i e d i f P(z)=P(-z). For even modes then, the t o r o i d a l f i e l d B ^  w i l l be an even function of z, the r a d i a l f i e l d B r w i l l be an even function of z, and the v e r t i c a l f i e l d B_ w i l l be an odd function of z. Exactly the reverse i s true for odd modes. Consequently f o r even modes P cL_ z o (Jit = o ) (s-u) J l i and for odd modes U = <L P -- a C J * =<> ) 0-17) AS I f the f u l l matchinq conditions are written out, i t i s found that the choice -H i. Li2-i e + e ') o, --TTlCu1-e 0 ^ 3 t>n --t -n; t i c 2 -e F -. + E ( 3 . U) 128 s a t i s f i e s eguation 3.96 for even modes whereas the choice -!UC*2-.) P. = " * C, -it i U 1-*) D , - - - e C i - I T : Cu2-t) s a t i s f i e s eguation 3.97 for odd modes. The analysis has therefore been reduced to the study of the f i v e eguations: together with 3.96 f o r even modes; and 3.100 together with 3.97 for odd modes. The l i m i t z —* 0 + i s to be used i n a l l res u l t s for z=0. Restriction to either even or odd modes has allowed us to consider only the z^ -0 region. We have f i v e equations for the f i v e unknowns E ( K . ), C ( < ) where the other c o e f f i c i e n t s are qiven by either 3.98 or 3.99. These two sets of equations 3.98 and 3.99 are a di r e c t consequence of the symmetry re l a t i o n s 3.85. Written out i n d e t a i l , equations 3.100 are for the r a d i a l component P *LKJ») I C, e - Qze t 0, e £ - e e B t 129 t h e $ c o m p o n e n t a n d t h e z c o m p o n e n t w h e r e f r o m e q u a t i o n 3 . 6 0 = £ (3.(01) (>- io«/) F o r e v e n modes t h e s e e q u a t i o n s a r e s u p p l e m e n t e d by t h e r e l a t i o n s 3 . 9 6 w h i c h w r i t t e n o u t a r e Even Modes -TV.; -ist^.i -iirv.i C, D^/") + C 2 e ^.(-x) * C3e 0y_/x) + C, e • D Y_,C-x).0 C 6 V I 6 0 t C t e <-«> + e ^ , C , e ^ , . 0 = q ( V i o l ) a n d f o r o d d modes a r e s u p p l e m e n t e d by t h e r e l a t i o n s 3 . 9 7 w h i c h 130 written out are Odd Modes (5 -10*) where a l l these r e s u l t s follow from Tables 7 and 8. These eguations may be put i n matrix form fj * = o where A ' i s the 5x5 matrix of c o e f f i c i e n t s of the above sets of eguations and x i s the column vector of the unknowns ^ j E . S p e c i f i c a l l y 4 i s the matrix of c o e f f i c i e n t s of eguation 3.101 - 3.103 and 3.106 and 3.107; while for odd modes A i s the matrix of c o e f f i c i e n t s of eguations 3.101 -3.103 and 3.108 and 3.109. We reguire that for n o n - t r i v i a l solutions. The point of t h i s analysis w i l l be to f i n d a r e l a t i o n between /I, ( i . e . M^. ) and k , which are the only parameters appearing in the matrices. Manipulation of the determinants shows that in both cases, the problem reduces to setting a 3x3 determinant egual 131 to zero. T h i s a r i s e s because i n both cases, o(«.t f\ ' takes the form ft : o JU* Be-° . jut c e ' where E>e° are 2x2 matrices and C-6'" are 3x3 matrices. T h i s being the case, the s o l u t i o n can be found g u i t e e a s i l y . The c o e f f i c i e n t s of B e" are dependent only on K v i a the v a r i o u s p a r a b o l i c c y l i n d e r f u n c t i o n s we have d e f i n e d , and XtX p/'° vanishes only f o r <- o i n g e n e r a l . 132 e, o Hence o u r c o n d i t i o n s f o r e v e n and o d d modes become oU-t C* _ o w h e r e li SUM. Hiu 2 I S fc 2 l l / , e e e / L - e. +- e e 7 e / i -i- e Aa A 2; n 127T/3 I - e -2nv;i in ,k z , : Cpi*)-i< e + e. e. e 1 +• e (3...I) a n d w h e r e we n o t e t h a t V, = 1 - K and t h a t t h e o n l y d i f f e r e n c e 2. b e t w e e n ^ a n d <- i s a s i g n d i f f e r e n c e i n t h e s e c o n d r o w -A f t e r some a l g e b r a i c m a n i p u l a t i o n , f o r e v e n modes t h e r e q u i r e m e n t J e t C = 0 q i q i v e s [ cos *f + e_J = o 133 and for odd modes the requirement <>U-t C° = o gives ( J.M3) where We note that equation 3.113 takes exactly the same form as 3.112 i f for odd modes we take ^ < f = <P - IT . In each of equations 3.112 and 3.113, we demand that the real and imaqinary parts vanish separately. We s h a l l also take e _*0 since Aa —* co . Noting that A_ y> i , we have from equation 3.112 for even modes Cos with the eguation for odd modes qiven by 3.115 where ^ i s replaced by ^ , and From the vanishing of the r e a l parts, we have to f i r s t approximation for even modes cos 1*3 X„) - o C'-"4^ and for odd modes , ( ^  J Co i — • Ao V t 134 where f o r both cases we have from the v a n i s h i n g of the imaginary p a r t s if Uv. ( 3 H i0\ * V Own) IC We note t h a t the s e t s of equations 3.116 (a) and 3.117 (for even modes) and 3.116 (b) and 3.117 ( f o r odd modes) are compatible s i n c e we have with / L ^ l , ct><, * o , and hence (_3y "* . This i s compatible with 3.116 s i n c e /U /V'* -* • Equation 3.117 should be reqarded as a f i r s t order c o r r e c t i o n i n «.*- t o equations 3.116. To z e r o t h order i n •<-'*- then, y f o r even modes and f o r odd modes . S u b s t i t u t i o n of these r e s u l t s i n t o equation 3.116, and subsequent expansion i n K l q i v e s the f i r s t order c o r r e c t i o n s . Even Modes Odd Modes ( 3 . no) 135 I f we r e c a l l that \. Q/rrr)V* , and express J» i n terms of Mf using 3.118 and 3.119, we find that and M° should f a l l i n narrow ranges ( governed by ) about the values e M where the f i r s t values are H° - o. o z The reader i s cautioned that these r e s u l t s may change s l i g h t l y i f more precise information about the spectral energy density of the turbulence i s available ( see Appendix B ). Solving for the amplitudes C^(K) , and E ( K ) we f i n d c * c l ° » e c r Cc<) E t K ) = -I p / I . c ' e / j CCk) 136 where C (K) i s some complex amplitude to be determined by conditions on the r a d i a l behaviour of the solutions. The r e l a t i v e strength of the vacuum f i e l d i s from 3 .124 Eoc) p *4 - H t t e where p~ Hi , Ao^-^t''/ and M£<1- Hence the vacuum f i e l d i s very weak compared to the f i e l d i n the disc for the long-range " mean-field " modes. Powerful large scale vacuum f i e l d s d e f i n i t e l y require conditions i n the disc such that MFC «-i . We now wish to write out the components of B for each mode tc i n the regime z —* z c . To do t h i s we use the r e s u l t s 3. 124 together with the asymptotic forms of the f i e l d components. For each mode K , we s h a l l have to sum a l l the contributions a r i s i n q f o r each n, from ^ and M {°, given by 3 . 1 2 2 . Since ' L , ^ ^H^ , the dominant contributions w i l l a r i s e from M|(I for even modes and for odd modes. The results 3 . 122 then show that the dominant f i e l d B^ in a stationary s e t t i n g w i l l be even ( i . e . B^ w i l l have "dipole symmetry" whereas Bt w i l l be an odd function of z and so i s of "quadrttpole symmetry" ) since M^, = 0 . 1 9 >> ( = 0 . 0 4 . The r e l a t i v e amplitudes of even to odd modes i s approximately 1 F ° > k . r * 137 away from the c e n t r e plane of the d i s c . Hence, we expect a t o r o i d a l f i e l d of " d i p o l e symmetry" with Mt ~ M^ 6 = 0.19 under s t a t i o n a r y c o n d i t i o n s . For even modes then, we f i n d - 0 + In X ~. \ (3.nr) where we have absorbed common f a c t o r s i n t o B(K) and where 1 2 , with p = (tltwV M t e , We a l s o note the vacuum amplitude i s 138 F i n a l l y , the r a d i a l behaviour of our modes goes as ~ f v ( -K l r ) where y = o or i . Now noting that A0 = z a we have 2 . so that the scale L of the modes i s Focussing on long scale modes L>>z 0 requires from equation 3.129 Equation 3.130 shows that to have scales L » z „ for the mean f i e l d i s equivalent to considering K2- i n the range Havinq worked out the consequences of s t a t i o n a r i t y , we now investigate the s i t u a t i o n when dynamo action and d i s s i p a t i o n do not exactly compensate one another. 5. Small Deviations From Equilibrium In t h i s concluding section we s h a l l assume _\\ ^  o and it 139 therefore investigate the p o s s i b i l i t y of having dynamo action dominating the di s s i p a t i o n . Returning to our f u l l equations 3.44 and 3.45, we seek to define an appropriate dimensionless time co-ordinate t. Noting that t K = r l/A where A= (GMX ) 1 (M^is the mass of the central object). Making r dimensionless ( r e c a l l r={J>C)^r) allows t K to be written t k - r /# (yy.) '* so that the choice t = }t / (s..*0 results in the eguations 3* r »i 38f _ J 1*" •* J- £ Again, these eguations are separable assuming a time dependence where ^ i s a complex constant. Me should remember that the r factor arises because our turbulent d i f f u s i v i t y *_r * c~y''L . Introducing 3.134 into equations 3.132 and 3.133, we may aqain seek separable solutions i n r and z with a 140 separation constant - k to f i n d that R(r) now obeys and with Q (z) and 0 (z) obeying eguations 3.51 and 3.52 as before. Since we w i l l be matching to external vacuum solutions, we take < as real and positive. We may regard P(£) 3 cV r / z as an e f f e c t i v e dimensionless, sguare, r a d i a l wave-number (complex i n general). Introducing RcS) = I C KW) which re s u l t s i n the This equation may be solved exactly. In the case where J f = - Yo ( ^ 0 and real ) we fin d decaying f i e l d s and from Abramowitz and Stegun ( (1968) formula 9.1.53 ) the solution i s we investigate the two extremes: (1) IK|^ I Pi , Then K.'~ tc and (2) I K! *<• I r I f Then K'~ Vc? ) eguation + i :! -141 which using d e f i n i t i o n 3.136 i s which supports the claim about '""'c?) made above. Res t r i c t i o n to the case \K[ « I Pi amounts to studying slowly varying temporal variations of the mean f i e l d , and as we w i l l see, the analysis w i l l follow along the l i n e s as given in sections 3 and 4. The second l i m i t IK|<< i r | , corresponding to rapid variations of the mean f i e l d , greatly complicates the mathematics. In pa r t i c u l a r , we have modes of r a d i a l dependence Rf, (• ^ t r ) in the disc ( R symbolizing the appropriate Bessel function ) whereas the vacuum modes have a r a d i a l dependence going as 3, ( r) . The matching of disc and vacuum solutions then i s complicated. On short enough time scales, the mean disc f i e l d i s strongly influenced by the inhomogeneity introduced by a r a d i a l l y varying turbulent d i f f u s i v i t y - * i r . The mathematical problems introduced by rapid time variations are probably best handled by a boundary-layer type analysis, where we note with Braginskii (1965) that i n general, when matching to an exterior vacuum solution, a boundary layer of thickness a, i s expected about z=tz e. The previous chapter showed that the disc i s not stationary on time scales ^ iot_ so that a rapidly varying mean f i e l d on these time scales could not develop- i n ( 3 . I 3 J - ) 142 s t a t i o n a r y c o n d i t i o n s . We s h a l l t h e r e f o r e only concern o u r s e l v e s with slow temporal v a r i a t i o n s of the mean f i e l d i n the l i m i t IK1'» IP| . In t h i s l i m i t , we may t h i n k of the background hydrodynamic s e t t i n g as s t a t i o n a r y . In a d d i t i o n , no new mathematical procedures need t o be i n t r o d u c e d . C o n c e n t r a t i n g on the IM IP I l i m i t then, we seek s o l u t i o n s t o our d i s c equations and match t o an e x t e r n a l vacuum s o l u t i o n as we d i d before. For i't o ; t h i s w i l l r e s u l t i n a d i s p e r s i o n r e l a t i o n which r e l a t e s X to K and other c o n s t a n t s . To deal with our r a d i a l inhomogeneity, we note with Whitham (1974) t h a t i n working with non-uniform media, the d i s p e r s i o n r e l a t i o n t o f i r s t approximation i s the d i s p e r s i o n r e l a t i o n f o r a uniform medium pro v i d e d t h a t the t y p i c a l p e r i o d s and wavelengths over which the medium v a r i e s are long compared to the waves being c o n s i d e r e d . In these cases, the procedure i s t o compute the d i s p e r s i o n r e l a t i o n i n the case where the v a r i o u s parameters i n eguation are taken to be c o n s t a n t , and then t o r e i n s e r t t h e i r s p a t i a l and temporal dependences when the r e l a t i o n has been e s t a b l i s h e d . T h e r e f o r e , we w i l l regard 9 = P / K. t o be a s m a l l parameter, and t o regard the v a r i a t i o n s of P with r as n e g l i g i b l e i n order t o determine the d i s p e r s i o n r e l a t i o n t o lowest o r d e r . Beginning with the «• time dependence i n the manner alr e a d y d i s c u s s e d , we write { l 3 r 3r 1 p 143 with the intent now of introducing the separation constant - K , with P regarded as constant and with \«I ^ | P ] . We then f i n d , using 3.49 that R (r) s a t i s f i e s equation 3.50 while equations 3.51 and 3.52 become .A1 where K H K + P with the understandinq that 1*1 » I T I . Equations 3.139 and 3.140 are exactly those we dealt with before except that With our vacuum f i e l d assumed to have & time dependence, the matchinq procedure i n section 4 may again be used, along with the various forms of the solutions ^.X) etc where in a l l cases we replace < with K = K <• P . Since we assume I M » . IP I , our solutions are n e g l i g i b l y affected by any complex component introduced by P . Our strategy i s to expand the r e l a t i o n s C e ' ° = 0 to f i r s t order i n 8 = P / K , and then to solve for & . We 144 s t r e s s t h a t P w i l l be c o m p l e x i n g e n e r a l . T h e d i s p e r s i o n r e l a t i o n JUt C e ' °= o d e r i v e d f r o m t h e m a t r i c e s 3.110 a n d 3.111 i s i n g e n e r a l f o r m (T, TZ - u r „ ) *' ° = o (3./</<j w h e r e f o r e v e n modes a n d f o r o d d modes I, - - e. - « e T - — " = — ° — C w h e r e r 3 R 5 k t P 145 Noting that we expand 3.141 to f i r s t order in 8 using 3.142 for even modes and 3. 143 f o r odd modes. Multiplying the r e s u l t i n g equations by & / z then gives the equations for even and odd modes where T r e t| and Tf* are the r e a l and imaginary parts of equation 3.112 and T° , and T ° „ are the r e a l and imaginary parts of equation 3.113. Note that 6'- o qives back our o r i g i n a l dispersion r e l a t i o n s . The factors F and G are the same for both even and odd modes and are f s - 27? 3 77 * 7° Before we proceed, l e t us analyze the r e l a t i v e magnitudes 146 of the terms i n F and G. The f i r s t term i n G i s G, Xr \o CoS t" while the second term i s of order £ J v ^ ^ Ac Taking the r a t i o 6, K 1 1 - t ^ f l Near equilibrium, we may use eguation 3.117 in both even and odd cases to give 6* Since we w i l l want to focuss on long r a d i a l scale mean f i e l d s , eguation 3.130 shows that for L » z 0 together with our usual l i m i t gives [ ^ I . Hence, for near eguilibrium conditions and focussing on long r a d i a l scale modes gives G •» F 147 Using a l l these approximations i t may be shown from eguation 3.144 that '° - - - I - [ ( T e ' T F + T e'° 6). i( T<-° F _ T'\ t,\ F + 6 L 1 ' 6 I * ~S J (3..fo) Assuming e terms are n e g l i g i b l e we f i n d from 3.150 wh ere 0 ~ - 0 The r e s u l t 3 .152 shows that for the same \„, frequencies i n the odd modes are exactly the negative of those for the even modes. We comment on t h i s l a t e r . Noting that $ 5 P/K , equation 3.151 qives f o r even modes V ' \ - — — 1 ( C + t-«P ) (»-T3) P e - i K f \ n e i ° For qrowinq modes to occur one must have 1 r<<_( >o , which for even modes qives Z ^ 5 /L > K 148 We have already seen that under equilibrium conditions tcK*^ < o for both even and odd modes. Hence, we see that a c r i t i c a l dimensionless wavenumber K c enters the problem so that when '< i s small enouqh so that K'!7- S ZB / ( 3 . l « ) then exponential qrowth of the f i e l d takes place. Those modes K'1 such that > K? w i l l decay. Using the r e s u l t 3.129 for the scale of the mean f i e l d components, we see that even modes of scale zd Xo w i l l grow exponentially while scales of L<L C w i l l decay. We r e c a l l that with \ a a function of the turbulent Mach number Mt ( see 3.104 ) and that with L » z c ;|t*»rl The mode for which ^ ' / c / ^ " ' 4 ) = o i s found to be and this mode i s the most rapidly amplified component in the spectrum. The r e l a t i o n 3.153 may be written i n more suggestive form as 149 For consistency, we note that when [ r I * « the analysis can no longer be applied- From 3.159 the range of dimensionless wavenumbers considered here i s therefore ± k£ < K z < 1 «C 1 2 2 Within t h i s range, the fastest decaying mode occurs at K_a = 1 , which gives a fas t e r decay rate than the 7. rate at which the mode K+. = | Ke. i s growing- S p e c i f i c a l l y , we find that when P e ( > o , and when I i <• o then These growth rates correspond to the scales U 2 Lc z for the growing mode «"+ and to for the decaying mode K- where we demand \ < - < 1 • 3 Lc The r e s u l t 3.154 shows that one necessarily has an (J K-- «•+) ( 3- Uo) 150 o s c i l l a t o r y dynamo at work here, whose frequency increases as we qo to smaller and smaller scales. Usinq the r e s u l t 3.152, we see that for odd modes - rc - - r' so that odd modes of scale L<L care qrowing exponentially while those of longer scales L>LC are.exponentially damped with r i =2 « c and K+= ^ 2^K*. The dynamo action i s again o s c i l l a t o r y . The r e s u l t 3.152 may be traced to the general symmetry of our underlying eguations 3.41 and 3.42. The product <Tx which i s a measure of the o v e r a l l dynamo strength to di s s i p a t i o n strength i s for our analysis found to be positive. I f < o then U (z) =U (-z) implies P(z)=P(-z); and U(z)=-U(-z) implies P (z) =-P (-z) . In our system, i t may then be shown that what were even mode dispersion relations for /X>o become odd mode dispersion r e l a t i o n s when YX < ° . This kind of behaviour has been noted by Moffat (1978) p. 230 for the study of A W ' 151 dynamos. In his discussion. X plays the role of YX. and i n a disc system containing two thin regions of dynamo action we quote: "so that i f a dipole o s c i l l a t o r y mode e x i s t s for X= Xo say, then a quadropole o s c i l l a t o r y mode with the same (complex) growth rate p exists for X - ' Xa . ". The r e s u l t 3.152 i s an expression of t h i s basic symmetry property. The fact that our solutions necessarily have an o s c i l l a t o r y character may be traced to our approximation that our mean vel o c i t y f i e l d was taken to be t o r o i d a l . As Moffat points out (p. 213), t h i s i s a property of «w 1 dynamos and when the poloidal velocity f i e l d s are non-zero we may expect a freguency s h i f t of !4|> • _ to occur for a mode of the mean f i e l d of wavenumber K . I f Y\~«<^ i s the frequency when = ° , the new frequency i s K ;vwi^ - Y:^ - Uj, . K Since our main poloidal flow i s r a d i a l , we expect that the e f f e c t s of such a flow w i l l be negl i g i b l e provided that the d r i f t time scale U i s much longer than the ch a r a c t e r i s t i c tirwe over which the mean f i e l d changes. With t B » f i t i n our analysis, t h i s requires f - * . «• t 6 <•< t 0 Puttinq a l l our re s u l t s together we find that the time dependence for our solutions i s qiven by r f 1 ! f - " K * 4 , ["(-CO 4- i 1 K 152 where + i s for even modes and - for odd modes and where t K = r 3 / l /(GH^is the Keplerian time. Focussing on k - <•*. - i n c for even modes (i.e the mode whose growth rate i s the quickest) corresponding to the scale L+ we have This shows that the o s c i l l a t i o n freguency i s a factor <c I\B "I smaller than the inverse growth time scale of the f i e l d . The growth time of the f i e l d s depends on Mt and Kc , the entire process being scaled by the Keplerian time scale. Now the exact value of Kc depends very much on the exact value of Et . However, we may introduce the parameter S such that L = tojS with 1 . Since <• sV-U'*' with t h i s parameter, we have K.c-h/Aoz' where Using the dependence of \„ on Mt we find that the even mode with scale L+. = grows exponentially as Conseguently, the smaller the value of the turbulent Mach number Mt, the longer i s the growth time of the f i e l d , the relevant time constant being t j - /(o.z) [Ht SC) L (%./c<t) 153 Thus, the most rapidly growing even mode K+ w i l l occur with M _ = 0.19, with *SC being a measure of (Mtc - | t and B — - t x For the odd modes, the most rapidly growing mode i s K = K° = i K<: corresponding to the scale 1 i , • d / ' . I n thi s case however, the largest value of H_ i s M t°, = 0.04 so that the exponential growth occurs with a t y p i c a l time of These r e s u l t s c l e a r l y show that the most rapidly growing ''z ''x mode i s the even mode of wavenumber «•+ - £ Kc . We note that 3 the exact amplification rate i s sensitive to the exact value of MT . The build up of the f i e l d occurs on time scales -£5 « io 0 - b K which i s of the order of seconds to tens of seconds i n the inner regions of the disc where "f. ' * <D * s and approaches the d r i f t time scale. The results show that the longer the length scale of the mode, the longer the time required to bu i l d up the amplitude by dynamo action. We leave to chapter 4 the analysis of how such growing f i e l d s ultimately e g u i l i b r i a t e to some steady value. 154 Chapter 4 Implications For - Accretion Disc Models Introduction In t h i s chapter, we bring together the various ideas investigated i n chapters 2 and 3 and examine the e f f e c t of our analysis on accretion disc structure. We f i r s t turn our attention to the eguilibrium value of the mean large-scale magnetic f i e l d , and investigate what type of stresses are set up by such a f i e l d . Section 2 i s devoted to this analysis and i t i s shown that the mean, large-scale, (long time average) Maxwell stresses give r i s e to the same type of accretion disc as studied by Shakura and Sunyaev (1973) . In section 3, we t r y to assess the long time averaged ef f e c t the l o c a l i z e d intense magnetic fluctuations w i l l have on angular momentum transport and disc structure. Arguments are introduced which, although not completely rigorous because of a lack of detailed information about the spectrum of the the magnetic turbulence, nonetheless show that the Lightman and Eardly i n s t a b i l i t y mentioned i n the opening chapter may be suppressed. A consistent cool thin accretion disc (averaged over long enough time scales) can therefore be imagined. The source of the hard X^-ray spectrum would then appear to be 155 associated with a c o l l e c t i o n of intense loops of magnetic f i e l d , emerging from the disc surfaces, and undergoing solar-type f l a r e s . Section 4 i s a rather crude analysis of the type of spectrum one could expect from the f l a r i n g regions discussed. Again, we make the analogy with solar f l a r e s and model the hard X-ray emission as a r i s i n g from a rapid flash-phase of the f l a r e , wherein bremsstrahlung emission a r i s e s from a non-thermal electron population (accelerated i n the f l a r e region) interacting with the denser gas towards the disc surface.. Zi. E g u i l i b r i a t i o n Of The Mean Magnetic F i e l d And Consequences For Accretion Disc Structure.-In t h i s section, we study the angular momentum transport (over long time scales) generated by the mean f i e l d B. In order to do t h i s , some estimate of the ultimate eguilibrium value of B must be made. We have shown in Chapter 3 that i n i t i a l l y weak magnetic disturbances of long enough scale w i l l grow exponentially on time scales loo iK . i t was assumed in t h i s case that the flow was Keplerian. We now ask, what does this f i e l d do to the turbulence and/or the mean flow to l i m i t i t s own growth. Two p o s s i b i l i t i e s come to mind. The mean f i e l d may act to a l t e r the turbulence ( reaction on micro-scale ) when the f i e l d approaches eguipartition energies (see Moffat (1972)). 156 Another point of view i s that the large-scale f i e l d may be determined by r o t a t i o n a l constraints acting d i r e c t l y on the large-scale flows, and may be i n s e n s i t i v e to the detailed structure of the underlying turbulence responsible for magnetic regeneration (see Malkus and Proctor (1975), Proctor (1977), and a b r i e f review by Moffat (1978) p. 303-307 ). Which mechanism predominates i s a question which has no general answer yet, however, the l a t t e r point of view i s the one of immediate i n t e r e s t in accretion processes. The idea i s that growing large-scale magnetic f i e l d s w i l l give r i s e to large-scale Lorentz forces. These forces in turn generate a large scale velocity f i e l d . The magnitude of the induced v e l o c i t y can be determined from the induction eguation and i t s estimation i s independent of the magnitude of B. The ultimate l e v e l of mean magnetic f i e l d energy i s then determined by the magnetostrophic balance i n which Lorentz forces and C o r i o l i s forces are of the same order of magnitude (provided certain conditions are met). In t h i s picture, we imagine the induced velocity f i e l d as a r i s i n g as a re s u l t of angular momentum transport by the mean f i e l d , which i s how we connect with the accretion problem. Let us b r i e f l y discuss the Malkus and Proctor (1975) analysis for " <*• " dynamos. The idea i s to assume that CK i n the mean induction i s unaffected by the large-scale magnetic f i e l d . Defining the quantity 157 where <x_ i s the value of <x for which excitation of the large scale mean f i e l d can occur. When > i ^ r the f i e l d grows exponentially u n t i l the Lorentz force back-reaction on the possible for the growth of the f i e l d to be arrested by the appearance of a mean vel o c i t y d i s t r i b u t i o n driven by the Lorentz force; a l l t h i s occurring before modification of <* by the mean f i e l d B i s important. The mean velocity f i e l d w i l l continue to grow u n t i l i t can compensate for the Ohmic losses of the growing magnetic f i e l d . For the problem of rotating f l u i d s i n a sphere, the mean magnetic f i e l d l e v e l should then be roughly determined by the balance of C o r i o l i s and Lorentz forces. The exact l e v e l of the mean magnetic f i e l d depends on The eguations studied for •V2"" dynamos i n the rotating frame of reference are (neglecting Reynolds stresses; we return to t h i s point later) flows i s s i g n i f i c a n t . I f R* = R* t ( l 4 _ e ) where 0 i t i s (R* -R<xt) ( M . I ) where <^ cx) i s prescribed and with i n i t i a l conditions _ L_,o) = O _>Cx,o) -- B„ C x ) 158 where BJx) i s the eigenfunction in the problem when U=0 and =R« . With ft*. = ^ ( 1 + * ) , the mean f i e l d B i n i t i a l l y grows exponentially and generates a velocity f i e d given by 4.1. The velocity f i e l d grows u n t i l i t has s i g n i f i c a n t e f f e c t in 4 . 2 . The relevant magnitude of U when t h i s stage i s reached i s found by comparing tf*£u*B) with ^ so that we expect 0 i s of order In the s i t u a t i o n where the C o r i o l i s forces i n 4.1 are more important than the i n e r t i a l forces ( at least away from the boundary ) so that the relevant magnitude of B i s deduced from the balance of the Lorentz and C o r i o l i s forces, and i s of order 2. \ = Si U o <r - ^ J2 ^ <4 rr where both the estimates for B Dand U 0should by multiplied by a function of ^ . We wish to make similar estimates for the 11 " dynamo 4 studied i n Chapter 3 . For t h i s problem, both and ^ * are prescribed. Previous arguments have shown that for thin discs the t o r o i d a l v e l o c i t y i s always Keplerian to good 159 approximation and that t h i s comes about because the p a r t i c l e s o r b i t i n the powerful external g r a v i t a t i o n a l f i e l d of the central black hole. The growing mean f i e l d B i s imagined to give r i s e to a r a d i a l v e l o c i t y U r. The e f f e c t i v e magnetic d i f f u s i v i t y i n the problem i s "L. . He then estimate the order of magnitude of the induced v e l o c i t y f i e l d as ar i s i n g from the Lorentz force reaction on the disc. Neglecting the Reynolds stress for the moment, we then expect that the v e r t i c a l l y averaged Maxwell stress ^ S*"^/mr i s (see eguation 1 .7) over long time, steady state conditions iAj* r = ^ BV> - - r u ' u * Equation 4 . 6 i s nothing new; however, the r a d i a l v e l o c i t y 0 r has been set by the mean induction eguation (relation 4 . 5 ) . This means that we now have a s u f f i c i e n t number of eguations to compute the disc structure, assuming that the Maxwell stress from the mean f i e l d B dominate the Reynolds stress. Before we analyze t h i s l a s t assumption, we note that from the d e f i n i t i o n 3 . 3 5 for <*T t that U r i s just 160 so that the Maxwell stress 4.6 becomes ( assuming 0<0 for r a d i a l inflow ) irr We note that 4.8 expresses the Maxwell stress i n terms of the pressure i n the same fashion as the standard model outlined i n Chapter 1, that i s , except for our factor of M^  . Again, the eguilibrium values for l ^ r ' and / B* JB") found above should be multiplied by functions of e i n order to arrive at exact values. Assuming that the eguilibrium f i e l d B has a si m i l a r structure as computed i n Chapter 3 (the ultimate f i e l d has roughly the same structure as the li n e a r i z e d analysis derives, provided that i t i s below equ i p a r t i t i o n strength) we estimate from equations 3.126 and 3.127 as the r e l a t i o n between the r a d i a l and azimuthal f i e l d components i n the reqion z zo. With M « 1 , and using the d e f i n i t i o n 3.46 for p and 3.60 for \ we have where numerical factors from the z integration have been dropped and only the scaling with Mt retained. With r e l a t i o n 161 4.9, we fin d from 4. 8 which shows that the mean magnetic f i e l d energy i n the l i m i t Mj. -*• 0 i s below equipartition with the thermal energy of the gas by a factor Considering the Reynolds stress contribution to Wr^ r we have already noted that the eddy v i s c o s i t y model (see Chapter 1) gives * K t Zc* . I t i s however also reasonable to estimate the Reynolds stress ( v e r t i c a l l y averaged ) as The model 4.10 shows that a competition between the Reynolds stress and the Maxwell stress due to the mean f i e l d may be expected. The dominant stress i s l i k e l y to be decided by the detailed v e r t i c a l structure and the magnitude of £ . Here, we assume the Maxwell stress dominates. This simple order of magnitude analysis shows that the mean f i e l d Maxwell stress gives r i s e to the same steady-state disc structure eguations as given i n Chapter 1- This a r i s e s because we have assumed that <* = K t cs i n our calcu l a t i o n s for *1 T. The discussion so far has ignored the contribution to the stress Wr<^ made by the intense f l u c t u a t i n g f i e l d s we discussed 162 in Chapter 2. We regarded these intense, s h o r t - l i v e d , s p a t i a l l y l o c a l i z e d fluctuations as ne g l i g i b l e as far as angular momentum transport averaged over long time scales was concerned. We examine t h i s assumption i n the following section. 3^ The Long Time Averaged Effects-Of The Magnetic Fluctuations We examine the r e s t r i c t i o n s that can be put on M ^  i n the / V . 1 .2. case where magnetic fluctuations as large as t> / rrf° ~ ^n. occur. Using the re s u l t k - ( ^T/ n ) &l and the eguilibrium value for B discussed i n the previous section, we obtain -—• — —- — * Z. c, v and for maximal fluctuations we have Using the expression for /MT , t h i s i s ^ * Mt z c* or with the magnetic Reynolds number - r^ l t /-^  we have 163 where we have noted t h a t S i n c e we a r e d e a l i n g w i t h v e r y l a r g e magnetic Reynolds numbers, the r e s u l t 4.13 shows t h a t maximal f l u c t u a t i o n s may occur i n regimes where M t i s s m a l l . As an example, w i t h K M ~ 10 and w i t h ranges f o r t o / r a p p r o p r i a t e f o r t h e d i s c of * » / , r ^ i o _ z , we f i n d t h a t -2-4 _ . - 1 . 2 T h i s i s i n t e r e s t i n g i n t h a t our model f o r l a r g e f l u c t u a t i o n s i s c o n s i s t e n t w i t h an a n a l y s i s where M<C<1. Let us more c l o s e l y examine how l a r g e the l o c a l magnetic f l u c t u a t i o n s can become. The d e n s i t i e s ^ i n the t h r e e r e g i o n s of t h e s t a n d a r d a c c r e t i o n d i s c model a r e i n terms o f t h e d i m e n s i o n l e s s v a r i a b l e s (see Shakura and Sunyaev) ^ 5 ^ /Ha r» s r / r • (a) i n n e r r e g i o n : ( \ ± <> * ) 0^ -c 1.1 K\0~1 M t m " 2 -v^"1 ^ 3 / l (b) m i d d l e r e g i o n : ( ^ rA < ^5<>o) 164 (c) o u t e r r e g i o n : (l,*00 £ i~* ) w h e r e t h e c r i t i c a l a c c r e t i o n r a t e M C ( - i s d e t e r m i n e d f r o m t h e E d d i n g t o n l i m i t . We n o t e t h a t t h e o n l y d i f f e r e n c e b e t w e e n S h a k u r a a n d S u n y a e v , a n d t h e f o r m u l a s we g i v e i s t h a t t h e y u s e M-t when we u s e M* ( t h i s i s b e c a u s e W r < < - M t P f o r t h e i r e d d y v i s c o s i t y m o d e l a n d \S)r* * t\\ P f r o m t h e mean f i e l d M a x w e l l s t r e s s ) . A g l a n c e a t t h e r a d i a l d e p e n d e n c e s o f t h e s e d e n s i t i e s s h o w s s h o w s t h a t t h e maximum d e n s i t y o c c u r s a t Q - i^a ; a t t h e b o u n d a r y o f t h e i n n e r a n d m i d d l e r e g i o n s w i t h r„ -*• ifo . F o r t h e C y g X-1 s o u r c e , t h e b l a c k h o l e m a s s i s 10 M 0 s o m -10 a n d s i n c e t h e l u m i n o s i t y i s 10 e r g s ; m -10 . 1 z W r i t i n g ^ r^ , ' w e f i n d t h a t f o r m a x i m a l f l u c t u a t i o n s L> /V = V K - , t h a t i n t h e i n n e r z o n e o f t h e d i s c = s-A x 10 r i M t I f we t a k e r i = i r ° ; we t h e n f i n d t h a t ; o.r We u s e t h e e s t i m a t e o f 10 G a u s s , b e c a u s e a s s h o w n i n 165 Chapter 1, we found that such a f i e l d strength can explain the shot noise model i n terms of solar-type f l a r e s . The value for the turbulent Mach number of M=0.26 derived by the above arguments i s i n agreement with the eigenvalue of M =0.19 found in Chapter 3. We have used two e n t i r e l y d i f f e r e n t approaches and come down to similar estimates for the value of M^ . This leads us to the view that at equilibrium, the o v e r a l l structure of the disc i s determined by the mean maqnetic f i e l d ( i t provides the angular momentum transport ) which sets the value for M * 0.19. This disc structure, i n turn, w i l l e f f e c t the magnitude of the magnetic fluctuations that can be expected. The shot-noise model can be accounted for by randomly ( i n time ) occurring f l a r e s , which have maximum energies i f originating i n the region rs l o o - t r o * ^ . Flares occur everywhere on the disc surfaces, but the energies emitted by f l a r e s in regions other than r * * w o - i r o cA w i l l be to low to stand out above the o v e r a l l background emission. We note that with Mt=0.19 say, the interpretation of millisecond bursts as f l a r e events becomes d i f f i c u l t . We return to t h i s point in section 4. Before leaving t h i s discussion, we note that i f we take '— Id S" b*10 Gauss, then the t o t a l energy contained i n the f i e l d i s Now t h i s considerably overestimates the energy release of 166 10 J Oergs per event. The explanation i s that only about 3? of the t o t a l magnetic f i e l d energy i s being converted into ether ferns of energy during every f l a r e event. This compares favourably with the results of the experiments discussed in Chapter 1. A l l of the discussion to t h i s point has concentrated on a thin disc as the underlying model for our c a l c u l a t i o n s . As mentioned i n the opening chapter however, the assumption that vJ r*-Mt p where H_ i s a constant was shown ty Lightman and Eardley (1974) to lead to a secular i n s t a b i l i t y cf the inner (radiation dominated) zone of the standard thin disc model. We b r i e f l y discuss t h i s i n s t a b i l i t y and l a t e r show that the ccntribution made to the o v e r a l l stress by an intermediate t i B € average of c"r,» can s t a b i l i z e the inner zene, so that i t i s consistent to think of a cool thin accreticn d i s c . Raking no assumptions about s t a t i o n a r i t y , the disc structure equations deliver the following eguation f o r the evolution of the surface density a result which follows from the continuity eguation 31 • i 3 M U r ) S D 0 / Z o ) 167 and the conservation of angular momentum Now i n the inner radiation pressure dominated zone of the d i s c , one can show that i n the constant Mt model for the turbulence a result which follows from the independence of P,_ ( and the temperature T ) on the density Z~ i n t h i s inner region. I f eguation 4.22 i s substituted into 4.19, there results a non-li n e a r d i f f u s i o n type eguation for the surface density L that turns out to have a negative e f f e c t i v e d i f f u s i o n c o e f f i c i e n t . Lightman (1974(a) and (b)) studied t h i s eguation- both a n a l y t i c a l l y and numerically and confirmed that the r e s u l t of t h i s negative d i f f u s i o n c o e f f i c i e n t i s f o r material to "clump" into rings, with higher density zones getting higher i n density and lower density zones getting lower. This clumping occurs on a l l wavelengths (secular) and on a time scale W i f ^ t p * 1 t " Z t K ( 4 . 2 * 0 where the l a s t equality i s a consequence of 168 where t 0 i s the d r i f t time scale t D = r / . Physically, with IAT^ «. I " 1 (for constant K_) one has low stress i n high- Z regions and high stress i n low- Z regions so that matter i s pushed in t o regions of low stress resulting i n increasing density contrast and the formation of dense rings of gas. The wavelengths X of these regions must be /( > X*. because of turbulent mixing on smaller scales. This r i n g structure i s not thermally stable and should heat rapidly, resulting' i n the swelling of the o p t i c a l l y thick, radiation pressure dominated cool regime into a much hotter, gas pressure dominated, o p t i c a l l y thin one. I t i s t h i s observation which lead to the two temperature model dicussed i n Chapter 1-Lightman and Eardley ( 1 9 7 4 ) point out that i f M± i s not a constant however, but f a l l s at lea s t as fast as Z , then a stationary, stable, thin cool disc i s possible. This may be seen by substituting * Z'M (n}.1) into 4.21 and then into equation 4 . 1 8 , where one finds then a positive e f f e c t i v e d i f f u s i o n c o e f i c i e n t . Physically, what i s happening i s that the e f f i c i e n c y of angular momentum transport i s decreased so that the i n s t a b i l i t y no longer occurs. Now in the magnetically dominated disc we have been discussing, the Maxwell stress due to the mean f i e l d takes the form P so that t h i s long time averaged stress cannot a l t e r the i n s t a b i l i t y discussed in the previous paragraph. Let us however examine the Maxwell stress a r i s i n g from the 169 f l u c t u a t i n g f i e l d s . He r e c a l l t h a t from Chapter 2 we had the s t r e s s a r i s i n g from the magnetic f l u c t u a t i o n s corresponding t o the s i t u a t i o n where l» ' _y B . Now over the n hydrodynamic time s c a l e rl^ we found t h a t s t a t i o n a r i t y was not p o s s i b l e and so the above ex p r e s s i o n f o r <r"*v' denotes f l u c t u a t i o n s i n the o v e r a l l Maxwell s t r e s s o c c u r r i n g on s h o r t time s c a l e s and s m a l l l e n g t h s c a l e s . Talcing the v e r t i c a l average and using the r e s u l t s 4.11 and 4.5 we f i n d Now from the long time averaged s t r u c t u r e of the d i s c , assuming steady s t a t e g i v e s which, i n view of eguation 4.5 shows t h a t Mr - t\/zWZ and hence, 4.23 may be w r i t t e n where have def i n e d an e f f e c t i v e t u r b u l e n t Mach number M. as 170 In other words, the l o c a l i z e d Maxwell stress fluctuations corresponding to our intense magnetic fluctuations can be characterized by a v i s c o s i t y parameter or e f f e c t i v e Mach number which has a density dependence of Z ~ z . This i s just the type of density dependence that would s t a b i l i z e the inner disc region- However, t h i s analysis must be taken one more step. We must average these fluctuations 4.25 i n the Maxwell stress over longer length and time scales i n order to determine what t h e i r average ef f e c t w i l l be. The relevant scales for averaging are the length and time scales over which the i n s t a b i l i t y discussed could a r i s e , which are scales intermediate between the hydrodynamic time scale and turbulent eddy s i z e on the one hand, and the very long time and length scales assumed fo r the stationary disc models we have disc ussed. The most obvious ef f e c t of magnetic fluctuations i s to cause density f l u c t u a t i o n s , since we have noted that regions of intense f i e l d should drive down the density i n that region making i t buoyant. Now our dynamo parameter depends on P o " ' so that we expect random fluctuations i n the magnetic f i e l d to be associated with random fluctuations i n cx . The averaging problem then i s to regard « as having a randomly varying component, which we average over a time scale t*. 1 7 1 and over a length scale of The analysis to follow, f i r s t investigated by Kraichnan ( 1 9 7 6 ) shows that these intermediate time and length scale averages of the induction eguation with random fluctuations of r e s u l t s i n a modification of ^ to a new value . Hence, our fluctuations over the intermediate scales l i s t e d above allow us to assess the e f f e c t of cr"*"* fluctuations over these scales. We follow Kraichnan's analysis s t a r t i n g from the mean induction eguation 36. -. VK (A B + U * ft ) + «, !7lB 2>t and consider the e f f e c t of s p a t i a l and temporal fluctuations of over the scales t ^ and i * . To do t h i s , a double averaging procedure over scales a, and a z s a t i s f y i n g i w « a , «• JL*. c i t « r- (Hit?) i s introduced. Preliminary averaging over a gives r i s e to the induction eguation 4 . 2 7 . We treat *~ as having a randomly varying component which w i l l be averaged over the scale ax. The double overbar " ~ w i l l denote averaging over a t guantities that have already been averaged over a,. 172 S p l i t t i n g B and oc i n t o mean and f l u c t u a t i n g p a r t s where B = Bo ; \, = o one then f i n d s where The term Vic 6, i n eguation 4.32 may be n e g l e c t e d ( f i r s t order smoothing) provided t h a t 173 where o^,^*., J i s the root mean sguare of the fluctuation <*, . We notice that the induction eguation 4.27 has been modified by the appearance of a new term <x• k> . We estimate t h i s c o r r e l a t i o n i n the same manner as done i n Appendices A and B. S p e c i f i c a l l y , we Fourier transform eguation 4.32 to solve for b, (the Fourier transform of _ ) , compute jo, ' , and then inverse transform. This procedure i s complicated by the presence of terms depending on the mean ve l o c i t y U and ua ( not considered by Kraichnan ). Moffat (1978, p. 177) sketches out the case for (x.0 - U - o where one finds where B c and t7'x are treated as uniform over the length scale 1«. One then obtains oc where and t^/i^) i s the spectrum function of the f i e l d * , . Now l e t us compare the terms I7x(rf. and P x xt,) with MTVZ_, i n eguation 4.32. The dominant contribution from (7x f«i>b,) i s of order , so that the AS- term dominates i f 174 where the l a s t inequality i n 4.38 follows from the estimates 3.35 and 3.36. Now for small Mach numbers Mt we noted that A / ? 0 * M-t • so i t i s consistent to estimate L^z,, i n order for 4.38 to hold. So near the surface reqions z-z<,the two terms become comparable. The term V x C u x lo, ) i s more d i f f i c u l t because U*can be large. However, usinq the same arguments as used in Chapter 2, terms involving U w i l l not be important provided that b,*" i s small compared to bf . The important contribution from U i s then the r a d i a l v e l o c i t y 0 rso that ^ 7 dominates i f which i s well s a t i s f i e d . Conseguently, for scales l^*zo>>l(it (only i f Mt « 1 ) , and regions z<z D with axisymmetric fluctuations <*, and b. , the r e s u l t s 4.36 and 4.37 are s t i l l applicable and eguation 4.31 becomes , X7K (« A B + [ U + Y ) x B 0 ) + [^T- X) T7* g o H Eguation 4.40, under the approximations l i s t e d i n the previous paragraph, shows that the e f f e c t of the fluctuations <*, give r i s e to a correction Y to the v e l o c i t y f i e l d and modify the d i f f u s i v i t y of the f i e l d B 0to where the l a s t ineguality arises from the fact that X>0. For 175 f l u c t u a t i o n s * i independent of p'; eguation 4 . 37 shows t h a t Y =0 ( independent of k ). Hence we expect c o r r e c t i o n s t o the r a d i a l i n f l o w v e l o c i t y 0 r to be the main e f f e c t of Y. The c r u c i a l p o i n t i s t o examine the magnitude of X. Moffat shows t h a t when then X may be estimated as Now, we have a l r e a d y c o n s t r a i n e d l K t o be 1 ^ -z 0, so t h a t with * T » Mt**a/t*. , the i n e q u a l i t y 4.4 2 g i v e s t IC so t h a t the time s c a l e t ^ i s c o n s t r a i n e d which by 4.22k i s where t was the time s c a l e over which the Lightman and Ea r d l e y i n s t a b i l i t y occurred. Let us estimate i n terms of the de n s i t y f l u c t u a t i o n s we imagine a r i s i n g from i n t e n s e magnetic f l u c t u a t i o n s b (on a s c a l e a , ) . Since ot depends on the d e n s i t y as (>„"' 2p> , we 176 estimate where ^ i s the root mean squared d e n s i t y f l u c t u a t i o n averaged out over s c a l e s L^and t*. Then S t r i c t observance of 4.44 guarantees t h a t X<<*v, however, pushing the time s c a l e up to t l w V+ ci, i m p l i e s from 4.47 t h a t the magnitude of the d e n s i t y f l u c t u a t i o n s a s s o c i a t e d with our l o c a l magnetic f l u c t u a t i o n s i s a l l important. From our d i s c u s s i o n and using 4.41 together with 4.47 we f i n d I f we were to average over l o n g e r times t , we cou l d expect t h a t (\ / approaches some constant value so that always A\- > o T h i s i s only s p e c u l a t i o n however. The g e n e r a l problem remains t h a t i f X >/nT _, a negative d i f f u s i v i t y o f the mean f i e l d would r e s u l t , and t h e r f o r e , the concept of the f i e l d s b' and B e x i s t i n g on two widely separated s c a l e s i s i n doubt. In g e n e r a l , the magnitude of X probably depends s e n s i t i v e l y on the d e t a i l e d spectrum of the tu r b u l e n c e (see 177 K r a i c h n a n {1976) ) . We note from r e l a t i o n s 4.5 and 4.24 t h a t a decrease of A\_ i s a s s o c i a t e d with a decreased r a d i a l i n f l o w 0 r and an i n c r e a s e d v a l u e o f the s u r f a c e d e n s i t y Z . The c o r r e c t i o n t o o"" i s the term Y? We may now r e t u r n t o the d i s c u s s i o n of ths Lightman and E a r d l e y i n s t a b i l i t y . In Chapter 2 we saw t h a t the H a x w e l l s t r e s s a r i s i n g from the f l u c t u a t i o n s can be r e l a t e d t o the magnitude of t h e long time averaged s t r e s s as ^L^lO- k I*" T° I IM""^  . On s h o r t t i m e s c a l e s , t h e magnetic energy f l u c t u a t i o n s can reach e q u i p a r t i t i o n w i t h the r o t a t i o n a l energy d e n s i t y and thus t h e i r a s s o c i a t e d s t r e s s i s o f the same o r d e r of magnitude as w""t Over much l o n g e r l e n g t h and t i m e s c a l e s m a g n e t o - h y d r o s t a t i c b a l a n c e must be m a i n t a i n e d so t h a t <!% Ic* and c o n s e q u e n t l y (lW> « on these s c a l e s . We now note that V\)r*= Zc$ i w i t h M t c o n s t a n t and N °_ b ' -  /(</o 2. Cs. where /n/0 depends on (iAT /. Dn the l o n g e s t s c a l e s , we e x p e c t -YI7 and hence ini0 t o t a k e i t s s m a l l e s t v a l u e so t h a t m a g n e t o - h y d r o s t a t i c b a l a n c e can be s a t i s f i e d . However, on s h o r t e r s c a l e s t « and 1«., i t i s ap p a r e n t t h a t W0* can be o f t h e same o r d e r as . S i n c e W*^>^ Z f the Lightman and E a r d l e y i n s t a b i l i t y w i l l be d e f e a t e d by t h e s t r e s s due t o the f l u c t u a t i n g f i e l d . These arguments show t h a t a t h i n , c o o l a c c r e t i o n d i s c i s c o n s i s t e n t when t h e e f f e c t s of magnetic f l u c t u a t i o n s a r e c o n s i d e r e d . We e x p e c t t o f i n d a corona o f i n t e n s e magnetic f i e l d f l u c t u a t i o n s of maximum s t r e n g t h b ~ 10 Gauss o v e r l y i n g t h i s d i s c . These f i e l d s w i l l undergo 178 f l a r i n g processes, and we turn f i n a l l y to a b r i e f analysis of the X-ray emission that might be expected. 4. X-ray Spectra From Solar-Type Flares In The-Cyg X-1 Source The observed power law X-ray spectrum of the Cyg X-1 source i s usually interpreted as a r i s i n g from a r a d i a l l y dependent temperature integrated over the disc surface. The model of Galeev et a l (1979) imagines that a magnetically confined, hot corona of material i s heated by reconnection of the looped coronal f i e l d s giving r i s e to thermal bremsstrahlung emission. Their mechanism explains the hard component of the X-ray spectrum with the soft photon flux (<10 kev) a r i s i n g from the cool underlying accretion disc. We note that for an energy release of 10 ergs in a "thermal" f l a r e then where T i s the temperature, n the plasma p a r t i c l e density and V the volume of the magnetically confined plasma. To get the lowest estimate for nV, we adopt a temperature of 1.8x10 K corresponding to the maximal X-ray energies of 150 kev and fi n d the number of p a r t i c l e s ^"V Sr I.I X X tO which for a maximum loop radius of 1-10 cm, V»10 cm and so we 179 are t a l k i n g about plasma densities which i s less than the p a r t i c l e density * I O " c>~~3 i n the outer portion of the inner radiation dominated zone of the accretion disc. One c r i t i c i s m we have of the Galeev et a l model i s that with loops of dimension 1-3x104 cm and f i e l d s b~10 Gauss, i f 3x10 ergs are to released i n each f l a r e event then v i r t u a l l y the entire magnetic f i e l d energy must be converted into thermal energy of the plasma, which i s contrary to the observation that only about 5% of the magnetic energy i s so converted. We f e e l therefore that t h e i r time scales are overestimated by 5 and their f i e l d s underestimated by a factor 20. However, their basic physical p r i n c i p l e s provide a consistent model for thermal heating of the coronal plasma due to f l a r e s -There i s a large amount of uncertainty i n the analysis of the hard X-ray component (>10 kev) of the solar f l a r e X-ray emission, as to whether thermal or non-thermal ( power-law ) populations of electrons are responsible f o r generating the observed power-law X-ray spectra (see Kane (1975) for a seri e s of a r t i c l e s dealing with t h i s question). It does seem clear that for solar f l a r e s bremsstrahlung i s the dominant radiation mechanis m. Observations of solar f l a r e s of duration 100 s shows that 180 there i s a flash phase l a s t i n g 1 s during which much of the hard X-ray emission i s occurring. The t h e o r e t i c a l work suggests that a non-thermal electron population may be responsible f o r hard X-ray emission during t h i s i n i t i a l short flash-phase of the ov e r a l l f l a r e . Datlowe et a l (1974) have studied a sample of 123 hard solar X-ray bursts using the solar X-ray experiment on the OSO-7 s a t e l l i t e . During a t y p i c a l event, the hard X-ray flux peaked e a r l i e r and decayed rapidly compared to the soft X-ray f l u x . This i s c l e a r l y shown i n Fig. 7 taken from their paper. The hard and soft X-ray components of a f l a r e exhibit very d i f f e r e n t behaviour. In addition, the OSO-7 data most commonly show a steady softening of the spectrum throughout the burst. These authors f i n d that there i s a detectable time difference between the time i n t e r v a l in which the f l a r e energy grows and the time i n t e r v a l over which hard X-ray producing non-thermal energy input takes place, suggesting that the soft (thermal) X-ray emission does not arise from energy input of the non-thermal electrons within the hot f l a r e plasma i t s e l f . F i n a l l y , for the solar bursts studied, the spectral indices ranged from 3.5 to 5.5. 181 F i c j i 7 Hard And S o f t X-Ray Fl u x From A S o l a r F l a r e IFrom Datlowe E t A l (1974)) 10.24 Second Average 03=30 03=38 03=46 03=54 04=02 04=10 04 = 18 04=26 . H a r d and soft X - r a y flux from an S N flare at 03:46 U T on January 26,1972. The upper trace gives the 5.1-6.6 keV channel o f the soft X- ray detector, and is characteristic o f the thermal X-ray flux. The lower trace gives the 20-30 keV channel of the hard X - r a y detector, representative of the hard X - r a y flux. Each point represents 10.24 s of data. H a r d X - r a y analysis was carried out from 3:47:31 to 3:49:24. The background was taken to be the flux from 04:00 to 04:08. We s h a l l t h e r e f o r e i n v e s t i g a t e what c o n d i t i o n s are necessary i n order t h a t the hard X-rays from the Cyg X-1 source be i n t e r p r e t e d as a r i s i n g from non-thermal e l e c t r o n p o p u l a t i o n s t h a t are maintained i n f l a r e s over some time ^ftuti<*" A T+W W e a s s u m e t h a t : (1) Each f l a r e of t o t a l d u r a t i o n 10"'s has an i n i t i a l f l a s h phase of d u r a t i o n ^t^L <10 _ 1 s during which the hard 182 component (>20 kev) of the X-ray emission occurs, l i b e r a t i n g hie 10 ergs of energy in t h i s band. (2) The spectral index for the X-ray photons of energy >20 kev i s constant from f l a r e to f l a r e . Both of these assumptions are crude because conditions at diffe r e n t f l a r e s i t e s on the accretion disc are apt to be di f f e r e n t . We follow the same procedures as used for solar f l a r e work as outlined by Korchak (1976). Suppose that at each moment i n time, the X-ray spectrum from an emitting region of volume V can be described by a power law form for photon energies in some range €t s i it*. and where Z i s the "spect r a l index". Then assuming a d i f f e r e n t i a l cross-section appropriate for bremsstrahlung by Coulomb c o l l i s i o n s , the instantaneous spectrum for the electrons may be written as where Kc may be determined i n terms of KK and the average gas density a. in the emitting region, and which includes a factor Ra where R i s the distance from the source to the observer ( R= 1A.0. For solar f l a r e s ). 183 One must now go one more step and co n s i d e r the r e l a t i o n between the e l e c t r o n spectrum i n the e m i t t i n g r e g i o n ( J |U « / O(E ) and the spectrum of the e l e c t r o n s i n the source r e g i o n . These two r e g i o n s need not be the same, and i n s o l a r f l a r e s , the a n a l y s i s suggests the source e l e c t r o n s are a c c e l e r a t e d i n the lower d e n s i t y r e g i o n s h i g h e r i n the atmosphere and move downward i n t o denser r e g i o n s c l o s e r to the s t e l l a r s u r f a c e where the emission of X-rays o c c u r s . I f J^eldt i s the in s t a n t a n e o u s average spectrum a l r e a d y d i s c u s s e d , and j JLE i s the power of the source, then the r e l a t i o n between these two i s given by a c o n t i n u i t y equation which under q u a s i -s t a t i c c o n d i t i o n s may be s o l v e d t o give where T C i s the c h a r a c t e r i s t i c time f o r the l o s s of e l e c t r o n s due to Coulomb c o l l i s i o n s of the e l e c t r o n s by the ambient gas of d e n s i t y n and Te i s the c h a r a c t e r i s t i c time of escape from the e m i t t i n g r e g i o n . The l i f e t i m e i n the Coulomb c o l l i s i o n s i s gi v e n by f S A r c = z.x 10 _B s (4.J-0 where E i s given i n u n i t s of kev while the minimum estimate f o r the escape time i s given by f r e e escape, 184 where 1 i s the l i n e a r dimension of the emitting region and v i s the electron speed. When % « % , , then the escape time i s much smaller than the c o l l i s i o n time and nest of the electrons escape without c o l l i d i n g . This case i s the so-called "thin target" approximation. Conversely , for Tc <<Te, the c o l l i s i o n term dominates and most of the electrons lose t h e i r energy through c o l l i s i o n s . This i s the "thick-target" apprcxiaation and i s obviously more e f f i c i e n t at producing X-rays. It has been noted by Brown (1975) that the thick target case may over-estimate the t o t a l number of electrons reguired for the X-ray emission by an order of magnitude. Koxchak (1976) notes in his analysis that either of these cases are U n i t i n g approximations useful only f o r a q u a l i t a t i v e analysis of a f l a r e problem. Let us f i r s t consider what conditions are required for the thick-target approximation. Here Taking a t y p i c a l electron energy of 50 kev say, with a dimension 1-10 cm for the emitting region, and o a 2 say, we find that for the inequality 4.54 to be s a t i s f i e d , the gas density i n the thick-target oust be n>>10 cm . With coicnal atmospheres of about 10 -10 cm , t h i s could be well s a t i s f i e d . We r e s t r i c t ourselves to the analysis of the thi c k -185 target case for the moment. As Korchak points out, a lower electron energy cut-off E,kev must be introduced i n order to estimate the t o t a l number of electrons and the i r t o t a l energy. It i s possible to make and order of magnitude error i n the theory because i t i s impossible to evaluate E, within the context of the theory i t s e l f . Nevertheless, introducing the low energy cut-off for the electrons E,in the electron power-law spectrum, at a distance of 1A.U the t o t a l flux F c of electrons with an energy E^E (and t h e i r t o t a l power P for thick-target emission are given by ( see Korchak ) r a - 0 P c E - s-x (o 1" $ rcs-\) r c s - i ) i-5 K, where E, i s given in terms of kev and Kx i s such that K* £' has the units cm^s^kev"1 with £ given i n kev. To sp e c i a l i z e to Cyg X-1 conditions we f i r s t correct the resu l t s 4.55 and 4.56 by factors of (Rc^x-i/B S u« ) where g R c ^ x - i =2. 5kpc=5. 15x10 A. 0. From the data of Dolan et a l i t seems two p o s s i b i l i t i e s for the power law X-ray spectrum are possible. 186 (1) There i s no break i n the electron spectrum i n the energy band 20-150 kev- In t h i s case, from equations 1.2 and 1.3 we estimate S ~ 2.0 which gives the value f o r Kx as K x * Then for thick target bremsstrahlung, we f i n d for Cyg X-1 f l a r e s F p C E>, E.) « ' / . z r x i / ' E,"' (V"> p ( E V E, ) - ' ^ 2 * E l ' *" Eguation 4.59 gives the power contained i n the non-r e l a t i v i s t i c electrons. If we want the entire hard X-ray emission per f l a r e (>20kev) of* 10 ergs to arise from these non-thermal electrons, we reguire (note that the t o t a l id emission per f l a r e i s 4x10 ergs) so that with a cut-off of E*20 kev; which in turn implies that the t o t a l number of electrons required i s (4.42) 187 There i s one d i f f i c u l t y with these r e s u l t s . We have i n Chapter 1 made a case for taking 1=10 cm as the size of the magnetic fluctuations. We are imagining i n our f l a r e model that the f l a r i n g region i s a neutral tube or sheet between two adjoining magnetic loops. Now the time for a l i g h t s i g n a l to propagate over 10 cm i s Hence, case (1) v i o l a t e s causality considerations by an order of magnitude. I f eguation 4.56 i s examined, i t i s seen that a higher spectral index (^>2) i s favourable for lengthening the flash time. This leads us to our second case. (2) There i s a break i n the hard part of the X-ray spectrum at - 50 kev say. Dolan et a l mention that a t h i r d of t h e i r spectra demonstrate the break with spectral indices of 2.5 or more for the higher energy domain. We assume that the hard X-ray emission (E >50 kev say) i s the f i r s t r adiation produced i n the source, and that t h i s emission has a spectral index of & =3.0 say. As the f l a r e continues, the lower energy ( s t i l l hard) X-rays are produced but that these have a lower sp e c t r a l index of & =2.0 because the power law electron population i s being degraded by inverse Compton scattering of cool disc photons. I t i s known that a power law electron population of spectral index ^ w i l l give r i s e to an X-ray spectral index of ( r 1+1)/2 when Compton scattering occurs. 188 Thus, t a k i n g <T =3.0 as the photon index f o r £ >50 kev say i n s p e c t r a with an observed break i n the 20-150 kev band, and a d j u s t i n g k* by a f a c t o r of 1.3 due to the change i n slope of the spectrum (estimated from F i g . 3), we f i n d t h a t using E=20 kev g i v e s Fc » /.o6 * i o * * s-< f«uO ? = r . 2 r x to € ^ s (M.6*0 from which using eguation 4.60 we d e r i v e a f l a s h time and hence a t o t a l number of e l e c t r o n s per event of This f l a s h time s a t i s f i e s c a u s a l i t y c o n s t r a i n t s . Now with 5 =3.0, f o r £ >50 kev we assume t h a t we have a s i n g l e e l e c t r o n power law i n the range 20-150 kev with s p e c t r a l index Assuming t h a t X-ray photons i n the 20-50 kev range are emitted i n the l a t t e r phases of the f l a s h , one expects the e l e c t r o n s i n t h i s energy range to be degraded by i n v e r s e Compton s c a t t e r i n g (we support t h i s c l a i m l a t e r ) . Thus, the 189 electrons with spectral index Pe = 2.5 give r i s e to X-ray spectra for the 20-50 kev range with a spectral index of which i s i n agreement with the low state spectral index of 1.83i0.06 given by Dolan et a l . This model, involving a break i n the X-ray spectrum due to the degradation of electrons in the l a t t e r phases, of the f l a s h by inverse Compton scattering, delivers a plausible picture of the f l a r i n g region. The idea that the spectrum of each f l a r e s teadily "softens" during the course of the f l a r e seems to correspond with the solar f l a r e r e s u l t s . We note that the f l a s h time of 4x10 rs i s much smaller than the escape time of 10 s so that a thick-target process must be assumed. Hence, the requirement < ^'Tpt.l, gives r i s e to (equation 4.52) a lower l i m i t for the gas density of n>1.8x10'rcm"3. To make further progress we consider two possible extreme cases. We have noted that an upper l i m i t on the coronal gas %\ -i density was n*10 cm p while the lower l i m i t i s established by ir h*1.8x10 cm"3 so that £ &%Ul\- Hence, we may consider IS" _•} i | ^ 10 cm <n£<10 cm". We consider the conseguences of either extreme. C) Rex10. £fi" 1 = ^ TA^L 1" T n e n from 4.66 we reguire a zy -j i volume of 10 cm or 1*-10 cm dimension of electrons to be 190 swept up in just one f l a r e . This i s c l e a r l y u n r e a l i s t i c since i t approaches the r a d i a l extent of the disc i t s e l f . (2) -St^iO cm 1 "i" c ~ «°^ T' L Then i f no reacceleration of electrons occurs, one requires a volume of 10 cm or 1-3 1-10 cm. This i s s t i l l larqe by an order of maqnitude. However, with nc~10 cm" as the electron density, we see that the electrons are being accelerated and halted i n the same s p a t i a l region. Onder these conditions. Brown (1975) has 43 suggested that the t o t a l number of electrons (Nt*10 ) could be reduced i f electrons were reaccelerated. The number of electrons required would then be reduced by a factor depending on how freguently electrons , already having undergone c o l l i s i o n , could be reaccelerated. I f t h i s mechanism worked at high e f f i c i e n c y , then since fe./^f ^ * 10~r, we would suppose r that each electron would be reaccelerated a maximum of 10 times. The t o t a l number of electrons now required i s reduced 18 -y n - 3 to N =10 and with n,=10 cm , a volume of 10 cm i s involved in each f l a s h . An emission region consisting of a neutral tube of length 10 cm then would have a radius of ~10 cm while a neutral sheet of area 10 cm would have a thickness of 10 cm,. We conclude that the electrons are most l i k e l y being accelerated i n the the same region where emission occurs, and that with the electron density the same as the gas density of n=10 cm , an adequate f i t to the date can be entertained. 191 We notice that the 10 cm" 3coronal density i s to occur i n the region 100-150 r^ where we have already shown that the most powerful f l a r e s should originate. This model fo r the spectrum should be regarded as crude because no physics of the acceleration mechanism has been produced. However, no comprehensive treatment as yet exi s t s for solar f l a r e mechanisms. I t i s interesting to note however that a c o l l i s i o n time 10 s for n ~ 10 cm gives r i s e to a mean free path for electrons of 100kev energies say of A ~ 10cm. Now, as noted i n Chapter 1, scaling arguments suggest that one may have e l e c t r i c f i e l d s equivalent to 10 volts/cm so that over a mean free path of 10 cm, a p a r t i c l e 3 could aquire ** 10 kev energies- With a 100kev electron as our starting point, t h i s suggests that i f the e l e c t r i c f i e l d s are d i r e c t l y involved i n some manner, then the acceleration mechanism operates at about a 10% e f f i c i e n c y f o r converting magnetic energy into p a r t i c l e acceleration. These considerations suggest that the combination of a high ambient gas density i n the f l a r i n g region ( compare to solar f l a r e s where n~10 cm i n the f l a r i n g region) together with an o v e r a l l e f f i c i e n c y of <10% for pumping magnetic energy into p a r t i c l e acceleration , i s responsible for l i m i t i n g the bulk of the electrons to <150 kev energies. This explains the observed high energy cut-off at 150 kev. Our model for the spectrum i s then that randomly 192 occurring f l a r e s of 10/s l a s t i n g for 0.1s each emit t o t a l 3t> energies of order 10 ergs. The dominant f l a r e s occur i n a region 100-150 r^. Each such f l a r e i s characterized by a double power law X-ray spectrum with 2.0 for the 20-50 kev range, and %" 3.0 for the 50-150 kev range. The hard X-ray emission (>20kev) of t o t a l energy 10 ergs per f l a r e i s modelled as a r i s i n g in a very rapid f l a s h phase of duration 4x10 rs during which the electron spectrum i s taken to be a power law dependence, so that non-thermal bremsstrahlung i s the dominant emission mechanism. The explanation of the double power law X-ray spectrum i s that the electron population i s progressively degraded by inverse Compton scattering of cool disc photons. In the f i n a l stages, thermal emission of soft photons i s the predominant process. Such f l a r e s must have electron densities n t-10 electrons cm for the thick target case. The soft X-ray f l u x ( E<20 kev ) arises both from soft photons emitted by the f l a r e after the flash phase, as well as the soft photon flux from the accretion disc i t s e l f . Let us investigate the the e f f i c i e n c y of the inverse Compton process by which soft photons from the accretion disc scatter o f f the non-thermal electrons i n the f l a s h phase. Now the c h a r a c t e r i s t i c time f o r Compton cooling i s of order ( Tucker ( 1975) ) 193 where V?i£ i s the energy d e n s i t y of the photon f i e l d and If i s the Lorentz f a c t o r f o r the e l e c t r o n s . From Shakura and Sunyaev, the photon energy d e n s i t y i n the r a d i a t i o n dominated i n n e r zone of the standard d i s c model i s given by which, when s u b s t i t u t e d i n t o eguation 4.67, using M-0.19 and m=10 g i v e s which s p e c i a l i z i n g t o the r e g i o n r^~100 g i v e s I , 3. O-if > to % y For m i l d l y r e l a t i v i s t i c e l e c t r o n s ( X - 3 ) , the Compton c o o l i n g time i s of order of the f l a s h d u r a t i o n T ^ * 4 / - 10 s. We note t h a t the r a d i a t i o n zone photon energy d e n s i t y i s independent of m so t h a t v a r i a t i o n s i n m w i l l not e f f e c t t h i s r e s u l t . We see t h a t Compton s c a t t e r i n g becomes dominant towards the end of the f l a s h phase. The photon energy d e n s i t y from the middle zone of the d i s c depends on m however. I t i s s m a l l e r than the energy d e n s i t y of the r a d i a t i o n dominated zone so t h a t c o o l photons from t h i s p a r t of the d i s c would not be expected to a l t e r the Compton time s i g n i f i c a n t l y . However, v a r i a t i o n s i n m w i l l e f f e c t t h e o v e r a l l s o f t X-ray output of the source. T h i s 194 exp l a i n s the o b s e r v a t i o n t h a t the hard X-ray c h a r a c t e r i s t i c s are v i r t u a l l y u n a l t e r e d i n e i t h e r high or low s t a t e s , whereas the s o f t photon f l u x c e r t a i n l y does change between these two s t a t e s . F i n a l l y , we c o n s i d e r the m i l l i s e c o n d b u r s t s . I f these b u r s t s are a t a l l r e a l , we suggest t h a t they may be some s o r t of "naked" f l a s h phase of a f l a r e not accompanied by s o f t photon emission (see Canizares (1976)). We emphasize t h a t r a p i d f l a s h times are r e q u i r e d f o r the model we have d i s c u s s e d . These f l a r e s occur w e l l away from the event h o r i z o n . Rapid f l a s h phenomena may very w e l l e x p l a i n any v a r i a t i o n s at m i l l i s e c o n d and s u b m i l l i s e c o n d time s c a l e s so th a t o b s e r v a t i o n s of such v a r i a t i o n s would not c o n s t i t u t e a t e s t of whether a r o t a t i n q or n o n - r o t a t i n q b l a c k - hole e x i s t s at the c e n t r e of an a c c r e t i o n d i s c . 195 Conclusions The analysis of magnetohydrodynamics in a turbulent accretion disc using the methods of Mean F i e l d Electrodynamics shows that magnetic fluctuations and mean magnetic f i e l d s are important on di f f e r e n t time and length scales. In Chapter 2, the analysis shows that i f small co r r e l a t i o n time scales < t K are considered, then intense fluctuations of the magnetic f i e l d are possible on short l * « z 0 length scales. On such time scales, the accretion disc cannot be stationary. I f the mean properties of the fluctuations are considered over longer time and length scales, we fi n d that because energy i s being drained out of the turbulent fluctuations to support the fluc t u a t i n g and mean magnetic f i e l d s , that buoyancy forces become more prominant a factor i n damping out the turbulence, especially i n the surface regions of the disc. The analysis of the v e l o c i t y , temperature, and magnetic fluctuations shows that the mean magnetic f i e l d can determine how large such fluctuations w i l l be. This guestion was furthur studied in Chapters 3 and 4. In Chapter 3, we show that assuming a steady mean f i e l d , that matching a disc mean f i e l d to an external vacuum f i e l d reulted i n an estimate for the turbulent Mach number of Mt~ 0.19. I f an underlying standard disc model i s assumed, this would reduce to two the t o t a l number of parameters 196 required to f i t the observations (assuming steady state). The t o r o i d a l f i e l d dominates i n the disc and the favoured configuration i s a mode of "dipole symmetry" for B^ , (and conseguently "quadropole symmetry" for B 2 ) . The vacuum f i e l d i s expected to be very weak i n comparison to the disc f i e l d i n the l i m i t of small Mach numbers. These results are a l l obtained by analytic means in the low Mach number l i m i t . The central r e s u l t of t h i s Chapter 3 i s the demonstration that a turbulent dynamo i s possible i n a "standard" cool accretion disc model. The "dipole symmetry" mode for B^ i s aqain the favoured configuration since for wavelengths larger than a c r i t i c a l wavelength (dependent on Mt) t h i s mode has the fastest growth rate. The time scales for t h i s growth are of order 100tk. in the l i m i t that small deviations from equilibrium are considered. The roles of the mean and fluctuatinq maqnetic f i e l d s as f a r as accretion d i s c structure and observational consequences are found to be quite d i f f e r e n t , even thouqh the two f i e l d s are i n t e r r e l a t e d as examined i n Chapter 2. In Chapter 4, i t was shown that on lonq length and time scales, that Maxwell stresses due to the mean f i e l d dominate those a r i s i n g from the f l u c t u a t i n g f i e l d s , and that they provide a stress Wr* of the same form as assumed for the analysis of the "standard " accretion disc model. On intermediate time scales however, the magnetic fluctuations contribute a stress which acts to 197 s t a b i l i z e the accretion disc to "clumping" i n the inner radiation dominated zone. Conseguently, such standard models are consistant and the hard X-ray emission from the source must aris e from either a hot corona or intense solar-type f l a r e s . Chapters 2 and 4 show that magnetic fluctuations are s u f f i c i e n t l y strong to account for these phenomena and provide a physical basis for the shot noise model. The interpretation of the hard X-ray component as a r i s i n g from the "flash-phases" of solar type f l a r e s on the accretion disc shows that rapid flash times />t u^*L4x.10~r s are expected. This means that sub-millisecond bursting of the Cyg X-1 source need have nothing to do with processes occurring near a black hole event horizon. Such rapid variations, i f found, cannot r e l i a b l y be used to discriminate between either a rotating or non-rotating black hole. Many features of our analysis may be extended to other astrophysical phenomena. Immediate application to the g a l a c t i c dynamo problem i s possible. The idea that active galaxies may be powered by accretion discs around central massive black holes can be further tested by applying these methods to the system and determining the role of magnetic f i e l d s i n such energy releasing processes. Double radio sources seem to reguire twin opposed beams of r e l a t i v i s t i c electrons to power them. Magnetic f i e l d s generated by accretion discs may have long range structure capable of collimating such beams. 198 Intense e l e c t r i c f i e l d s generated i n f l a r e events on an a c c r e t i o n d i s c may provide b u r s t s of extremely r e l a t i v i s t i c e l e c t r o n s. 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G. 1978, Ap.J., 221 , 228 Whitham, G.B. 1974, L i n e a r and Non-Linear Waves (John W i l e y and Sons) Znajek, R.L. .1978, Mon. Not. R. A s t r . S o c , 185 , 833 203 Appendix A Mean f i e l d Electrodynamics; A. 1 Basic Ideas We refer the reader to the excellent reviews by Moffat (1978, Chapter 7) and Roberts (1971). Introducing the decomposition of the magnetic and velocity f i e l d s u ; U + u ; u = o the mean and fluctuating parts of the induction eguation are H 3f [ where a mean electromotive force i s seen to arise due to the co r r e l a t i o n of the flu c t u a t i n g velocity and magnetic f i e l d s and It i s the objective of mean f i e l d electrodynamics to express £ as a l i n e a r functional of B. Then A.2 i s a closed i eguation for B which may be studied in i s o l a t i o n from b. 204 We f i r s t note that the computation of £ i s simplest when known as the f i r s t order smoothing approximation. I f 1«. i s the cor r e l a t i o n length of the turbulent fluctuations and'fc the corre l a t i o n time, t h i s term i s small when as an example, * z • s r . . fu « i O - o We consider the conseguences of A.7 more in section A.2. i With G~0 , and writing equation A.3 as + (u.T7t ' - t ' . VU ) - ^ \7*t' = T 7 X ( U ' * B ) we see that the flu c t u a t i n g f i e l d b i s being created from B by u. Hence the correlation between u and b reduces to determining the co r r e l a t i o n tensor of u. Solving A.8 i n terms of a Green's function G- (x,t;x,t) ( the boundaries should have n e g l i g i b l e e f f e c t i n evaluating £ since correlations over l«are the only important considerations ) i * 4 f f Using A.9, the d e f i n i t i o n A.4 then gives 205 where the c o r r e l a t i o n tensor (x,t;x,t) i s CS.*; = «ltx^) < * ' , 0 (ft.,,) When the ensemble average i n eguation A. 11 depends on x but not t , the t u r b u l e n c e i s " s t a t i s t i c a l l y steady " and i n t h i s case Q^x c*,*; « Q^i t-t' j I f the ensemble average does not depend on x, the t u r b u l e n c e i s " homogeneous " so t h a t f o r a steady, homogeneous tu r b u l e n c e £«i 6.*.') « Q„x (*-*'; +-4') In a steady t u r b u l e n c e , the ensemble average may be r e p l a c e d by a time average over any one member of the ensemble, and f o r a homogeneous t u r b u l e n c e , the ensemble average may be r e p l a c e d by a s p a t i a l average. I f the s t a t i s t i c a l p r o p e r t i e s of the t u r b u l e n c e are independent of the o r i e n t a t i o n of the c o - o r d i n a t e frame (at a point) the t u r b u l e n c e i s 11 i s o t r o p i c " and i f the p r o p e r t i e s are independent of whether the frame i s r i g h t or l e f t handed i t i s " m i r r o r symmetric " Returning t o eguation A.10 we see t h a t i f we expand B(x',t) i n a power s e r i e s about x, t h a t i s & ~ B e * ) + (x -* ' J .V§u) then because Q ^ { vanishes with I x - V | the dominant terms should a r i s e from the lowest d e r i v a t i v e s . Hence, the g e n e r a l form f o r <E; t o f i r s t order i n the s p a t i a l d e r i v a t i v e s i s - * : J B j • k]k >3i (*•«*) 206 where the tensors ^CJ and k'j-i. depend on U and on the s t a t i s t i c a l properties of u but not on B. The exact s p e c i f i c a t i o n of acj and b : j - 4 for a given case involves i n t r i c a t e c a l c u l a t i o n but very general conclusions can be drawn from the form of eguation A. 13. Since £. i s a polar vector and B i s an a x i a l vector, we require that both a ; j and bcjiL be a x i a l . As an example, i f U=0 and the turbulence i s steady, homoqeneous, and i s o t r o p i c ; then the only i s o t r o p i c skew tensors of deqree two and three are where i s a pseudoscalar ( dot product of a polar and an ax i a l vector ) and n T i s a scalar. In t h i s case A.13 becomes i - oc B ^B, (A',S^ . > 0 L ' S L a ^ : I - a ;« ( I +• •< B) fr-'O G~<« 5 ('Vnr) [ i l . In a mirror symmetric turbulence a l l associated pseudoscalars must vanish. Hence, i f we have a non-mirror symmetric turbulence, an electromotive force proportional to B arises ( known as the 1 * - e f f e c t ' ) which i s a type of term capable of the reqeneration of the mean f i e l d . 207 The term - "\T 7 x § makes the t o t a l d i f f u s i v i t y appearing i n the mean i n d u c t i o n equation egual t o ^ =M tt ^ . Mean F i e l d E l ectrodynamics t h e r e f o r e d e l i v e r s a t u r b u l e n t d i f f u s i v i t y f o r the mean f i e l d B which i n the high c o n d u c t i v i t y l i m i t 1* Tv \M 5 > I dominates the ambient d i f f u s i v i t y . In t h i s l i m i t t h e r e f o r e , the mean f i e l d i n a t u r b u l e n t conductor cannot be imagined to be " f r o z e n - i n " to the plasma. The important q u e s t i o n i s what causes a l a c k of m i r r o r symmetry. The s m a l l e r the s i z e of some t u r b u l e n t eddy, the q r e a t e r the tendency towards i s o t r o p y . Hence, as f a r as the s m a l l eddies are concerned, we can imaqine t h a t as a f i r s t approximation, the t u r b u l e n c e i s homoqeneous, i s o t r o p i c , and mirror-symmetric. D e v i a t i o n s from t h i s s t a t e w i l l be s m a l l and w i l l depend on X ; the d i r e c t i o n of a n i s o t r o p y . A n i s o t r o p y e x i s t s i f we have a l o c a l r o t a t i o n ~[l or the presence of a d e n s i t y q r a d i e n t g i n the problem. Summing up, we imagine t h a t our t u r b u l e n t f l u c t u a t i o n s can be w r i t t e n y - u 0 + y, r ' where U o i s an i s o t r o p i c , homogeneous, and mirror-symmetric t u r b u l e n c e with s m a l l d e v i a t i o n s u, depending on the presence of some a n i s o t r o p y i n the problem. For the presence of both l o c a l r o t a t i o n and a d e n s i t y g r a d i e n t , one may show t h a t o< =pc ) (which i s a pseudoscalar) . The s m a l l , non-mirror symmetric c o n t r i b u t i o n u, i s r e s p o n s i b l e f o r g e n e r a t i n g the <* -e f f e c t . 208 A.2 The F i r s t Order Smoothing Approximation Krause and Roberts (1976) showed t h a t i n the f i r s t order smoothing approximation (eguation A.6, s a t i s f i e d by the i n e g u a l i t y A.7 as an example), i t i s s t i l l p o s s i b l e to have l a r g e magnetic f l u c t u a t i o n s compared t o the mean magnetic f i e l d amplitude i f one i s i n the high c o n d u c t i v i t y l i m i t where u l i s the mean squared v e l o c i t y f l u c t u a t i o n and X* i s a time s c a l e t y p i c a l of the f l u c t u a t i o n b. The reason f o r t h i s r e s u l t i s t h a t i n computinq 2 , i t i s o n l y the pa r t hcav.r , the part o f b t h a t i s c o r r e l a t e d with u' t h a t i s important and t h i s p a r t i s of order I g n o r i n g the e f f e c t s of the mean flow; they s o l v e equation A.3 by a Green's f u n c t i o n technique o b t a i n i n q (see equations 6 and 10 i n t h e i r paper ) r (A -2°) 209 where the velocity c o r r e l a t i o n Q(r' ) i s defined by A. 11 and the Green's function G i s G(r) = ( w ^ r ) ^ « r(-i7<vr) r"-2'J appropriate f o r the d i f f u s i o n operator. For equation A.19; i n the high conductivity l i m i t &<* /ru, * *i so that for Q < \ < 4 u . / ^ , the Green's function i s b a s i c a l l y a ^ function. Equation A.19 can then be approximated as which shows that the f i r s t order smoothing assumption i s giving the r e s u l t A.18. In t h i s high conductivity l i m i t however, eguation A.20 for the magnitude of the mean squared magnetic fluctuations shows that because u^c-i) t.' c-t + r) i s correlated for a time J I ~ $ » T'U , then the estimate of A. 20 i s T7 *• air- & * i r » ^ n. \ I t i s t h i s r e s u l t which i s c r u c i a l to the theory we investigate i n the text. A_. 3 For malism For Computing Various Correlations Arising. In Mean F i e l d Electrodynamics The approach we employ to calculate * , <*iT etc was 210 developed by Roberts and Soward (1975). Here we summarize some of the r e s u l t s t h a t they worked out and which we s h a l l need f o r our own c a l c u l a t i o n s . I n t r o d u c i n g the concept of a l a r g e s c a l e on which the mean v e l o c i t y , magnetic e t c f i e l d s vary and a m i c r o s c a l e f o r a d e s c r i p t i o n of the tu r b u l e n c e , i t i s convenient t o i n t r o d u c e mean and r e l a t i v e c o - o r d i n a t e s f o r two po i n t s x, and x^as X a i ( < , . * , ) . « i ( j , . x x ) ; T « J f t . * * . ) . f . - ^ - t , [A-**) i n terms of which the two-point, two-time c o r r e l a t i o n f u n c t i o n s such as $ ; j C*> . t, • ^ ) -- u!(K, ) U- U t , t j may be expressed as ^ j ^ . T ; ,,-t) , « l ( X ^ , T . ^ J uj- (K-J* , T-i-tr) C*-IS) The t u r b u l e n c e i s steady i f and l o c a l l y mirror symmetric i f (-1.1 x.*) - c*< T- + ) The method used by these authors r e l y s on a double F o u r i e r t r a n s f o r m a t i o n and expansion method whereby the F o u r i e r t r a n s f o r m with r e s p e c t to x of A.26 i s 211 and the Fourier transform for the mean variables X, T The idea i s to regard variations of mean quantities as ne q l i q i b l e over scales l^and times which characterize the turbulence- Hence, the induction equation for the flu c t u a t i o n i s f i r s t Fourier transformed, the various correlations computed i n K ~JI ; i_ - w space, and then an expansion of the re s u l t s carried out i n powers of K ( that i s a power series i n larqe scale derivative ^1M ) . The r e s u l t s are then transformed back so that the various c o e f f i c i e n t s i n the problem w i l l be in t e g r a l s over the microscale spectrum i , and u> . The idea of separation of scales i s obviously central to the whole process. As an example, the c a l c u l a t i o n of L proceeds by taking the Fourier transform of equation A.3 to find ^ (4-zl) where i s defined as so that i n the hiqh conductivity l i m i t which, as t —* ^ > goes to 212 We may now compute &i using equation A.26 as This r e s u l t i s expanded i n power of K and converted into X-space. In the case of homogeneous turbulence so that where A = 4 +j. K . inverting t h i s r e s u l t with respect to K and integrating over a l l and ^ then gives where ^ , u > ) &(-u.) JLJLJL^ ^-^) We d i g r e s s s l i g h t l y t o p o i n t o u t t h a t f o r i s o t r o p i c 213 turbulence, the c o r r e l a t i o n tensor w i l l contain a mirror symmetric part and and a non^mirror symmetric part. To lowest order in the form of i s (4 •%(.) J 2C J where $ i s the Fourier transform of «'.u' and i s the spectrum of the turbulent i n t e n s i t y while H i s the h e l i c i t y spectrum, which i s the Fourier transform of tfx <*' . m addition, the tensor P- (k) i s defined as ft: (4 ) . Sii ~ *±*i {AZn) Substitution of A.36 i n t o the expressions A.35 and A.34 results i n the reduction of A.33 to where the positive d e f i n i t e turbulent d i f f u s i v i t y «fT i s 3 J and the parameter <* ( dimensions of velocity ) i s related to the h e l i c i t y (equations A.38 - A.40 are re s u l t s 3.51 - 3.53 i n the Roberts 214 and Soward paper). We s h a l l require equations A.26 and A.33 for the analysis in Appendix B. We must specify how the h e l i c i t y i s to arise i n our shear flow, and include the e f f e c t s of density gradients in the problem. We f i r s t write down the effects of l o c a l rotation i n generating h e l i c i t y . For us the l o c a l rotation comes about through the antisymmetric part of the l o c a l s t r a i n . Consider f i r s t the e f f e c t of rotation. In the rotating frame, the velocity fluctuation obeys H Suppose that the turbulence i s imagined to be predominantly i s o t r o p i c and mirror symmetric ( uj ) with the rotation 2 JL» introducing a small deviation u| to t h i s state. In the l i m i t u0 f u / " t the i n e r t i a l term i n A. 41 i s ignorable and v j I i s well s a t i s f i e d so that £ ^ , - ry f' _ zJL* x uj ("^ Fourier transforming A.42, and finding the c o r r e l a t i o n * {>) ^ * i« J ~ z J * 215 one may show th a t expansion t o 0(K) and i n v e r s e t r a n s f o r m i n g t h a t where the h e l i c i t y i s • 00 We n o t i c e t h a t the v o r t i c i t y of the mean flow i s 2 JL* = SL . Roberts and Soward show th a t i f i n s t e a d of pure r o t a t i o n , we have a mean flow with a non-zero s t r a i n on the long s c a l e s , then the antisymmetric p a r t o f t h i s s t r a i n t e n s o r Jl* -- 1 Jl = j. Vx U — 2. - Z w i l l generate h e l i c i t y H a s given by e x a c t l y 1/2 the r e s u l t found i n A.44. T h i s i s p h y s i c a l l y s e n s i b l e because the v o r t i c i t y "P* ^  of the mean flow i s a c t i n g as a l o c a l r o t a t i o n . Thus F i n a l l y , these authors note that the e f f e c t s of c o m p r e s s i b i l i t y may be taken i n t o account by r e p l a c i n g a l l g r a d i e n t s 3 $ / D Xs of $ W by 216 I f we do not include derivatives of * as important, then the mirror symmetric part of the i s o t r o p i c turbulence i s to f i r s t order i n the density gradient ( see eguation 3.38 of Roberts and Soward ) while the h e l i c i t y A. 44 we w i l l use may be corrected by using A.46 and ignoring the gradient of v-We close by noting with these authors that neglect of the effects of l o c a l rotation ( to zeroth order ) on the turbulence requires Also in the analysis of the v e l o c i t y spectrum, the requirement i s used; a behaviour which has usually been accepted i n the dynamical theory of turbulence. 217 Appendix B The Calculation Of Correlations Between Velocity And Magnetic F i e l d Fluctuations B. 1 The Calculation Of £ = w'xU1 We assume i n t h i s c alculation that the h e l i c i t y i s generated by the antisymmetric part of the mean s t r a i n i (7xU - We s h a l l include the e f f e c t of density z — gradients ^ \J (= and w i l l assume that for our purposes the turbulence i s homogenous over the scales that the density varies. Hence, we ignore gradients of the turbulent i n t e n s i t y vi . With = tfxU and with the assumptions above, we have to f i r s t order i n ( • " ' ( s e e eguations A. 36 and A. 47 ) /A with the h e l i c i t y ( see equation A.44) v. ^Xi J Substituting B.2 into B.1, we use eguation A.33 for the 218 c a l c u l a t i o n of Ei-- u'*|,'. assuming that we may approximate the tu r b u l e n c e as homogeneous to z e r o t h o r d e r . We s h a l l keep terms up t o f i r s t order i n . Noting t h a t only when an even number o f d i r e c t i o n c o s i n e s 4^ appear i n the i n t e g r a t i o n over ^ do we get a non-zero r e s u l t , we f i n d Performing the i n t e g r a t i o n over X using «H 4 with the i d e n t i t i e s J 5 J e q u a t i o n B .3 r e d u c e s t o r i i ^xi 3B,~ + z. w h e r e t h e i n t e q r a l s l m a r e I , s So r r 3 f A*AC-^) 6*c~>)~] $®ciu) 220 a n d w h e r e we h a v e e m p l o y e d t h e i d e n t i t y Summing o v e r t h e i n d i c e s , t h e t h i r d t e r m i n B.7 v a n i s h e s l e a v i n g I = . r, VxB + TV (VBS) J2t + We now t u r n t o t h e s i m p l i f i c a t i o n o f t h e i n t e g r a l s l f , I, , and Iq. O s i n g t h e r e l a t i o n A.27 f o r A(<*>) , a n d t a k i n g t h e h i g h c o n d u c t i v i t y l i m i t s one o b t a i n s •t -# °° ^ -£,*c„) \ -- 1 - 1 i P i Sc^o) - z $U^) - Sc-*>)] T a k i n g t h e r e a l p a r t s we f i n d o If J I T~ 1 , is- ± 1 * £ j* (s.n) 221 Assuming t h a t ? ^ I ^ ) - f '(£.») <^zas UJ-»0, we have roughly For the t o r o i d a l v e l o c i t y f i e l d we have 1 -. ( ^ o j j ; jB e , l/ K {13.20) so t h a t eguation B. 13 becomes | - - * l T x & + if f (Vs 2 ) JZj + Eguations B.21 show t h a t a term of the form 5 appears i n the e x p r e s s i o n f o r 2 . Hence, d e f i n i n g (8-22) g i v e s I  H JL* 4 it J E x p r e s s i n g B.23 i n c y l i n d r i c a l p o l a r c o - o r d i n a t e s , with 222 the assumption of axisymmetry and defining the r a t i o gives r i s e to eguations 3.10 to 3.12 i n Chapter 3. 33,2 The Calculation Of u'x(Uxt') = f Beginning with formula A.26 for b and noting that ^j2-»1;^x^ in the Fourier domain we f i n d which, with the assumed homogeneity ( to zeroth order ), of the turbulence leads to ( see steps leading to A.33 ) Integrating over 4 and u> and inverting with respect to , we have i n analogy with eguation B .7 J J ( 8 - " ) 223 I f we now substitute eguations B.1 and 2 for <£ % into B.27 and keep terms to f i r s t order in <=V<2XS , we find that the l a s t two terms i n B.27 make no contribution. The f i r s t term i n B.27 contains the product f so integration over ot-t' only gives non-zero results for those parts of <£ «.i containing an even number of factors . Hence, using i d e n t i t y B. 12 twice, we f i n d AJL <*- A' * ; J L # i + U i = - EK cU ^ i ~ X; J L *? ( °U w) where we note that Integrating B.28 over ° U and using B.5 then gives 7*7^*77 ) - - f j i ^ d ^ e i i B We calculate the quantity £. j i d e n t i t y A . CA. from Chapter 2 usinq the so that 4M 224 where we have used the r e s u l t B.29. E v i d e n t l y £_'. j i s p o s i t i v e which shows t h a t the i n t e r p r e t a t i o n of t h i s term as a l o s s term f o r u* and a source term f o r b* i s v a l i d . Turning t o the e s t i m a t i o n of 7 we have using B.23 U n l i k e the C j term, the magnitude of £• T depends on the comparison of the <x term with r e p e c t to e f f e c t s due t o the d i s s i p a t i o n *\x . Using equation B.22 we have oc * so t h a t Now, f o r our sma l l c o r r e l a t i o n time l i m i t , we set T u < t K , so t h a t the crude order of maqnitude estimate B. 33 becomes 225 Appendix C The I n t e g r a l Representation For 0(z) -C.I The S o l u t i o n Of Equation 3^53 For 0(z) Given the equation ^Ct)--o , the r e p r e s e n t a t i o n f -- f K(e.t) ATti) di (c.i) r e s u l t s i n Choice of an a p p r o p r i a t e k e r n e l l<cz,t) such that where ^-t i s some new d i f f e r e n t i a l o p erator i n t , q i v e s * Jtit(Kc*.i)) <r(i) dt - o I n t e q r a t i o n by par t s t r a n s f e r s the d i f f e r e n t i a l o p e r a t i o n s from to nit) r e s u l t i n g i n (cs) 226 w h e r e i s t h e a d j o i n t o p e r a t o r t o a n d '0 i s t h e s o - c a l l e d b i l i n e a r c o n c o m i t a n t . O b v i o u s l y i f s a t i s f i e s M 4 ( \r) = o a n d t h e p a t h o f i n t e g r a t i o n i s c h o s e n s u c h t h a t P^^) v a n i s h e s a t t h e e n d p o i n t s , t h e n t h e d i f f e r e n t i a l e g u a t i o n i s s a t i s f i e d . I t may r e a d i l y b e s h o w n t h a t i f M t h e n Mf(l-) ^ - ^  (ocu-) + p V ( C l ) a n d PCtr l<) cx ir fC T h e r e a d e r may r e f e r t o M o r s e a n d F e s h b a c h ( 1 9 5 3 ) f o r m o r e d e t a i l . We now a p p l y t h i s t o t h e s o l u t i o n o f e q u a t i o n 3 . 5 3 . T h e c h o i c e o f t h e L a p l a c e k e r n e l i s m o t i v a t e d b y t h e o b s e r v a t i o n t h a t J-fr o*fc (cf) 227 so that use of t h i s kernel w i l l require only a f i r s t order d i f f e r e n t i a l equation i n t to be solved to f i n d AT-U) . Thus with 2 K / - t A + ( K Z - l ) we fi n d It (c . , 0 Employing the r e l a t i o n C. 9 then gives where Comparing C.12 with C.6 we see that with ecCi) s -t we f i n d the equation for AX as D i f f e r e n t i a t i n q we have *y - o Jy f f f 3 _ 2 K t + i £ z 1 W - O 228 whose s o l u t i o n i s AT = Hence, w i t h t h e L a p l a c e k e r n e l and w- g i v e n by C.16, 0 (z) t a k e s t h e form where t h e c o n t o u r s C a r e chosen such t h a t v a n i s h e s a t t h e e n d - p o i n t s o f i n t e g r a t i o n . C^2 R e l a t i o n s Between S o l u t i o n s U(z) We b e g i n w i t h t h e i n t e g r a l r e l a t i o n s d e f i n e d by e g u a t i o n 3 . 5 8 f o r •< and z r e a l w i t h •< <1, and f o r t h e c o n t o u r s s k e t c h e d i n F i g . G We deform the v a r i o u s c o n t o u r s so t h a t they run a l o n g t h e c o - o r d i n a t e axes of t h e r e l e v a n t quadrant of t h e complex-t p l a n e . As an example, 0,(z) i s e x p e*. j> ( - t % 4- let 1 *-* t ) Let us d e f i n e t h e i n t e q r a l c i t 229 Using t h i s d e f i n i t i o n , and changing v a r i a b l e s ; i n the f i r s t i n t e g r a l of C. 19 as t-»e lTwe f i n d t h a t U(z) can be w r i t t e n as U,CK,S) 1 ( M ) - e z Tt-*, it) (cn) S i m i l a r deformations of the other contours d e f i n i n g the s o l u t i o n s Uz , 0 3 , ty, lead s to Summing the f o u r s o l t u i o n s given by C.21 t o C.24 g i v e s 2. «,iK,J) , (e - . ) e I K ^ ? ) (c.zr) T h i s sum i s non-zero f o r rm where m i s any i n t e g e r . In t h i s case, our f o u r s o l u t i o n s are l i n e a r l y independent. The f a c t o r L e - » j i s expected to a r i s e i n the case where we eval u a t e a f u n c t i o n on a contour t h a t runs around a branch c u t i n the complex plane. 230 Comparing the representation C.21 f o r U, with C. 23 for 0 3 we see that Similar comparison of C.27 with C.24 shows that Then from C.26, rearrangement gives and from C.27 U A < / i ) = e U , ^ , fez*) U 4 Cic, *) = e ( c .d) The set of relations C.26 to C.29 gives the r e l a t i o n between solutions f o r z>0 and z<0, and comprise the set of symmetry relations mentioned i n the text. I t may also be shown that (P.H. Roberts, pr i v a t e communication) a r e s u l t which follows from C.21 and C.22. We also have as seen from C.23 and C.24. The four symmetry r e l a t i o n s C.26, C.27, C.30 and C.31 allow the l i n e a r independence of the solutions corresponding to the four contours (n=l,...,4) to be established. 231 Appendix D Asymptotic Analysis O f a (z) D .J Asym ptotic Form For The Solutions jJ (z) Taking eguation 3.62 as our s t a r t i n g point, we expand the function H 1 Cr) = AUr) + K <j<r) as a Taylor series about a saddle point giving where, since To i s a saddle point; -f'«".)= 0. With q(f) = T Z there i s a term O fc 3 V ) ) ( r - r , ) z which we have ignored with respect to which i s v a l i d i n the l i m i t I* " 1 <»« Hence, i n the l i m i t A - » , the representation 3.62 becomes where 232 and i s the contour deformed to run through the saddle point t ; . Defining the new r e a l variable where '-re ( r re a l ) and chosen such that /I i i s r e a l and negative, one may write the i n t e g r a l X as (in the l i m i t X-* o* ) -1 Z , \ \ oil 1 { <rt) I * which on completing the sguare for the i n t e g r a l so i t takes the form gives ''•L (.-*')/,/ - "I 233 Noting that 5T equation D .7 then reduces to the r e s u l t given in eguation 3.65 where the term X1 * ^^"^ i s negligible with respect to At<-r<>) i n the l i m i t D. 2. The angle « i s the d i r e c t i o n of steepest descent from the saddle point. D.2 Saddle Points, C r i t i c a l Points^-And Directions Of Steepest Descent To evaluate the saddle points of the problem, we set where $cr) - - ^  and where the + sign i s for z>0 and the -sign for z<0. Hence the saddle points s a t i s f y ^ 3 so that f o r z>0; the three saddle points are at 1 2 % L f and for z<0; the saddle points are at (D.M) With these values for f 0 , fc fo) and + are then computed. The angle ot (double valued) i s then computed as mentioned i n section D.1. These r e s u l t s are gathered into Table 9. 234 Table 9 Saddle P o i n t s Of F.(r 1 • f. + 1 - 3 0 , 77 Z > o e ) +B>3 -27i >'/3 e - 3 e - / - 3 o , TT i < o e * T - 3 e ) ~ % e . 3 +•*!?.• - 3 e The f u n c t i o n %^TJ = T" has a c r i t i c a l p o i n t a t Tc =0. We must then assess the path of s t e e p e s t descent from t h i s p o i n t as w e l l . Now $  cTc) =0 and "^"0 = 1. C o n s u l t i n g Table 7.1 of B l e i s t e i n and Handelsman ( 1975 ) , the s t e e p e s t path i n t h i s case i s a > o : ot - TT - A 2 < o cX. = O 235 This r e f e r e n c e provides an e x c e l l e n t d i s c u s s i o n of the method of s t e e p e s t descents. We diagram the saddle p o i n t s "To , c r i t i c a l p o i n t 7 C , and the d i r e c t i o n s of steepest descent (arrows) f o r both z>0 and z<0 i n F i g . 8. We have placed the branch cut on the p o s i t i v e imaginary a x i s f o r convenience so as not t o i n t e r f e r e with the saddle p o i n t s at ± 1 . 236 237 D.3 Paths Of Steepest Descent I f we w r i t e T=x+:^ } -H <r) -. UC*;<A) + ivix,}) (D. iz) i t may be shown ( see B l e i s t e i n and Handelsman ) t h a t curves of s t e e p e s t descent and ascent from any p o i n t ?0 = * + ^  are those curves d e f i n e d by To p i c k out t h e descent paths ( two from each saddle p o i n t , i n opposite d i r e c t i o n s ) , we use the d i r e c t i o n s computed i n Table 9 f o r each saddle p o i n t . From the d e f i n i t i o n of l*r) and from Table 9, equation D. 13 then q i v e s , f o r f o = ± 1 (+ s i q n f o r z>0, - s i g n f o r z<0) ; f o r T.= e (both f o r z>0) : „ + 77 ;A and f o r i0-~ e (both f o r z<0) : I t i s not necessary to have a d e t a i l e d knowledge of y(x) at every p o i n t , however the g e n e r a l p r o p e r t i e s of the paths of steepest descent are r e g u i r e d f o r the a n a l y s i s i n s e c t i o n D.4 238 From eguation D.14 i t follows that y s a t i s f i e s either of Now from the r e s u l t s of Table 9, we know that the directions * = O,TT correspond to steepest descent paths from these saddle points. Hence, the curve y=0 corresponds to the steepest descent paths from these saddle points. The curve j = * + - then corresponds to steepest ascent paths from these points. In the l i m i t x -*»o ; y=±x so that these are the asymptotes f o r the ascent paths. In addition, for 7o=+1, x>1 and To = -1, x^ -1-Writing eguation D.15 in the form we see that i n the l i m i t x - * 0 , D.18 may be s a t i s f i e d by f ^ ^ z • x < o Rearranging equation D.14 in the form shows that i n the l i m i t x-*-<*>; aqain Now the directions y — x are out of the zones C o - * ) of convergence 239 f o r our i n t e g r a l r e p r e s e n t a t i o n s , so t h a t the l i m i t y= 0 f o r x - » m u s t be the behaviour of the st e e p e s t descent curve. This same kind of reasoning may be a p p l i e d t o equation D . 1 6 where the r e s t r i c t i o n x> 0 must be made f o r s o l u t i o n s t o e x i s t . The r e s u l t s of t h i s a n a l y s i s are i l l u s t r a t e d i n F i g . 9 f o r the case z> 0 and z< 0 , . The descent paths are l a b e l l e d i n s o l i d l i n e and ascent paths i n dotted l i n e s . 240 F i g . 9_. P a t h s Of S t e e p e s t D e s c e n t -2 4 1 Us.H F i n a l Results For Asymptotic Expansion - Of The Solutions In t h i s section, the asymptotic form for each solution BVjUr'z) i s determined by deforming the contour onto one or more paths of steepest descent, and assessing the contribution from each of the saddle points or c r i t i c a l points that are picked up. Only the dominant contributions w i l l be kept. We s h a l l do the analysis f o r z >0 since similar considerations apply for the z<0 case. ( 1 ) U M f K , Z ) The contour C 4 i s deformable into the contour ( see Figures 9(a) and 6 ) Hence, only the contribution from the saddle point atT i s picked up. Dsing formula 3.65 we then have e e < x iff - 211/ i ^ic1 - '3 e f(*l-V>?])/* fl + K1)/^ - 2 e ( 2 ) O i J ^ z l The contour Cj may be deformed onto 2 4 2 ^ ( l + I C l ) / f so t h a t the saddle p o i n t a t To = e i s picked up. He then have x j e - e J (3) 0* (K, z) The contour C* i s deformable onto CL = - 0, + D z - Or + D ' (D-Z«0 so that c o n t r i b u t i o n s from the saddle p o i n t a t f o =+1 , the c r i t i c a l p o i n t U = 0 , and the saddle p o i n t To = e are p i c k e d up. However, the saddle p o i n t at To =+1 c o n t r i b u t e s the f a c t o r ''1 ^  ^ € which i s e x p o n e n t i a l l y growing, whereas the U =0 has no e x p o n e n t i a l f a c t o r and To = e c o n t r i b u t e s an e x p o n e n t i a l l y damped f a c t o r e . Hence, to very good accuracy the c o n t r i b u t i o n from 70 =+1 i s the o n l y f a c t o r we need c o n s i d e r and we have (<0 O.l^xZl Here we note t h a t the contour C, l i e s e n t i r e l y t o the 243 r i g h t of our branch cut whereas the contour D , - C ^ - D j + Dw runs to the l e f t of the branch cut- The dominant c o n t r i b u t i o n i s s t i l l from Tc= + 1 f o r the same reasons as d i s c u s s e d f o r U a. Hence we f i n d These r e s u l t s are summarized i n Chapter 3 by eguations 3.66 and 3.67 and Table 4. 2 4 4 Appendix E Expansions Of 0>j[f< f z) About Z=0 Taking the f u n c t i o n U,(K,Z) as an example we had 0, as (z>0) from eguation C.21 where the i n t e g r a l I ( K , Z ) was given by C . 2 0 . Expanding I as a power s e r i e s i n z about z=0 we have where t - 1 Changing v a r i a b l e s i n E. 2 to T = d e f i n i n g = " - ^ J / * one gets Using the i d e n t i t y 3.76 given i n Chapter 3 , Q « C K ) then becomes 2-245 where i s the parabolic cylinder function. Doing the same expansion for I (-*, <-z) and substituting everything back into the expression for 0, ( K , Z ) we find 2 The coefficents i n the expansion E.5 may be s i m p l i f i e d using the i d e n t i t y which may be found i n Gradshteyn and Ryzhik (1965), p.1066, formula 9.248.2. Using formula E.6, the the expansion E.5 becomes (z>0) Similar expansions are arrived at by st a r t i n g with formulas C.22, C.23 and C. 24 for Uz , U3 , and UH respectively. Derivatives of E.7 are e a s i l y found, as well as the l i m i t z -* 0 . As an example which i s shown by equation 3.78. 246 The expansions f o r z<0 are found by u s i n g the r e s u l t s above ( f o r z>0) and a p p l y i n g the symmetry r e l a t i o n s C.26 -C.29 i n order t o get the z<0 expansions. 

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