LECTURE NOTES on IDEAL MAGNETOHYDRODYNAMICS By
ASSOCIATIE EURATOM-FOM
FOM-INSTITUUT VOOR PLASMAFYSICA
RIJNHUIZEN - NIEUWEGEIN - NEDERLAND
LECTURE NOTES ON IDEAL MAGNETOHYDRODYNAMICS by
J.P. GoedbJoed
Rijnhuizen Report 83-145 ASSCOCIATIE EURATOM-FOM Maart 1983
FOM-INSTITUUT VOOR PLASMAFYSICA
RIJNHUIZEN - NIEUWEGEIN - NEDERLAND
LECTURE NOTES ON IDEAL MAGNETOHYDRODYNAMICS
by
J.P. Goedbloed
Rijnhuizen Report 83-145
Corrected version of the notes of March 1979,
originally printed as internal report at
Instituto de Ffsica, Universidade Estadual de Campinas,
Campinas, Brazil
This work was supported by the "Stichting voor Fundamentaal Onderzoek der Materie" (FOM), the "Nederlandse Organisatie voor Zuiw-WetenschappelijK Onderzoek" (ZWO), EURATOM, the "Fundacio de Amparo i Pesquiw do Ettado de Sao Paulo" (FAPÉSP), and the "Conaelho Nacione) de PesquisM" (CNPQ, Brazil). "Then I saw that all toil and skill in work come from a man's envy of his neighbour. This also is vanity and a striving after wind."
Ecclessiastes 4:4
"Ever since the creation of the world his invisible nature, namely, his eternal power and deity, has been clearly perceived in the things that have been made."
Romans I:20
"Remember then to sing the praises of his work, as men have always sung them."
Job 36:24 PREFACE
These notes were prepared for a course of lectures for staff and students of the Instituto de Flsica, Universidade Estadual de Campinas, Brazil. The course consisted of two-hour lectures twice a week during a period of 9 weeks in the months June-August 1978. It has been my intention to make the subject- matter as much as possible self-contained, so that all needed physical and mathematical techniques and derivations were pre sented in detail. The aim was to bring a physics graduate student with a little previous knowledge of plasma physics to the point where he could sense the possibility of contributing himself to modern developments in the field of n.agnetohydro- dynamics. It has been stated many times during the course that ideal MHD is still full of questions where answers remain to be given, whereas at the same time the framework of the theory is clear-cut enough to provide confidence that eventually a satisfactory picture will emerge. An open field like this should be a fruitful area for academic research. I wish to thank Prof. Paulo H. Sakanaka for the golden opportunity he offered me to visit UNICAMP and to teach this course. His personal help, the interest of Prof. Ricardo M.O. Galvao, and the effort of the students made the visit a very valuable and exciting experience for me. The diligence of Carmen typing the manuscript I have appreciated very much. I am indebted to the foundations CNPQ and FAPESP (Brazil) for the support of this work and to the foundation FOM (The Netherlands) for granting me a leave of absence. I welcome notification of errors, criticism, and suggest ions for improvement of these notes.
Hans Goedbloed Association Euratom-FOM FOM-Instituut voor Plasmafysica Rijnhuizen, Nieuvegein The Netherlands CONTENTS p I. Introduction 1 II. Derivation of macroscopic equations 4 A. Boltzmann equation 4 B. Moments of the Boltzmann equation 6 C. Two-fluid equations T 12 D. One-fluid equations 13 III. The model of ideal MHD 23 A. Introduction 23 B. Differential equations 24 C. Boundary conditions 27 D. Equation of state 29 IV. Characteristics 33 A. Partial differential equations in two independent variables 33 B. Characteristics in ideal MHD 36 V. Conservation laws 49 A. Conservation form of the ideal MHD equations 49 B. Shocks 51 C. Global conservation laws 56 D. Energy conservation for models 2 and 3 59 VI. An example: Dynamics of the screw pinch 63 A. Pinch experiments 63 B. Mixed initial-value boundary-value problem — 66 C. Field-line topology 73 D. Reduction of the plasma equations 77 E. Circuit equations 79 F. Solution of the problem 82 G. Flux and energy conservation 86 VII. Lagrangian and Hamiltonian formulations of ideal MHD 92 A. Summary of some concepts of classical mechanics 92 B. Kinematic considerations ——-—-—-- 96 C. Lagrange and Hamilton equations of motion 101 VIII. Linearized ideal MHD 105 A. Introduction 105 B. Linearized equation of motion 108 C. Boundary conditions — 112 D. Self-adjointness of the force-operator 117 E. Mamilto.i's principle 123 IX. Spectral theory 128 A. Mathematical preliminaries 128 B. Rayleigh-Ritz variational principle 132 C. Initial value problem 135. D. Stability. The energy principle 138 E. o-Stability 145 X. Waves in plane slab geometry 149 A. Waves in infinite homogeneous plasmas 149 B. The continuous spectrum for inhomogeneous media 158 C. Damping of Alfvén waves 170 D. Stability of plane force-free fields. A trap 191 XI. The diffuse linear pinch 204 A. Equilibrium model 204 B. Derivation of the Hain-LÜst equation 208 C. Equivalent system of first order differential equations 216 D. Boundary condition at the plasma-vacuum interface 219 E. Oscillation theorem 222 F. Newcomb's marginal stability analysis. Suydam's criterion 2 33 G. Free-boundary modes 242 H. Fixed-boundary modes - 247 I. o-stable configurations 251 XII. Sharp-boundary high-beta tokamaks 254 A. Introduction B. Equilibrium 260 C. Vacuum field solution for the circle 267 D. Variational principle for stability 273 E. Numerical solution for circular cross-sections 233 .1.
I. INTRODUCTION
In these notes a cross-section through plasma theory is presented which is restricted to ideal magnetohydrodynamics (MHD). This cross-section will again be restricted to my lim ited personal point of view, which is that I wish to deal with a model which
- respects the main physical conservation laws, - has a decent mathematical structure» - permits the analysis of plasma behavior in the complicated geometries considered for the confinement of plasmas for controlled thermonuclear reactions {CTR). Ideal MHD is the only model so far that satisfactorily combines these features. This theory treats the plasma as a perfectly conducting fluid interacting with a magnetic field. If we talk about the model of ideal MHD we mean: " the equations of ideal MHD, - boundary conditions on a prescribed boundary and initial data on and inside that boundary. .2.
In order for the model to be complete both have to be consider ed simultaneously. Nevertheless, different persons put differ ent stress on these two points. The exposition tends to be more physical when the stress is on the first point, whereas consideration of the boundaries tends to lead to more involved mathematics. In the first part of these notes, where we consider simple geometries (homogeneous media, e.g. infinite space or homogeneous slab models), a relatively simple analysis will therefore lead to an abundance of physical phenomena (in par ticular the various kinds of MHD waves), whereas gradually more tedious analysis is needed to correctly treat these phenomena in more complex geometries (inhomogeneous media, e.g. diffuse linear and toroidal pinches). These complicated geom etries also provide interesting new physics, like equilibrium and stability properties, which cannot be analyzed in homo geneous media. Since MHD instabilities are a major threat to CTR confinement, it is essential to have a firm understanding of this subject if one wishes to contribute to this field. It is the aim of these notes to facilitate this understanding.
There are two ways of introducing the equations of ideal MHD: - derive them by appropriate averaging of kinetic equations, - pose them as reasonable postulates for a hypothetical medium called "plasma". Since a satisfactory derivation of the ideal MHD equations does not exist, we basically choose for the second method (starting with chapter III). However, this approach will be supplemented .3.
with a heuristic derivation (chapter II) in order to render some credibility to the equations and also to obtain some understanding of the domain of validity of the ideal MHD de scription. Strict minds may skip this chaotic exposition and start reading at chapter III. The MKSA system of units has been chosen for the next chapter, whereas starting with chapter III u will be put equal to 1 for convenience. The only operation needed to return to the conventional systems of units is then to divide B2 by v (MKSA system of units) or 4TT (Gaussian system of units). .4.
II. DERIVATION OF MACROSCOPIC EQUATIONS
A« BOLTZMANN EQUATION Consider a collection of charged particles in an electromagnetic field- Different species of particles, specifically ions and electrons, will be distinguished by a subscript a. We now define the time-dependent distribution function for particles of species a in six-dimensional phase space: f (r^y^t) . The probable number of particles in the six- dimensional volume element dJr d v centered at r,y will then be f (£,^,t) a3 r d3 v . The variation in time of the distribution function is found from the Boltzmann equation;
3f 3f q 3f 3f 3t £ 3r m v^ * *' 3v k3t 'coll v* l' 'v a *\» Here, E and £ are composed of the contributions of the external fields and the averaged internal fields originating from the long-range interparticle interactions. The PHS of Eq. (2-1) gives the rate of change of the distribution function due to short-range interparticle interactions, which are somewhat arbitrarily called collisions. Neglect of these collisions leads to the Vlasov equation;
3f 3f q 3f 2 2 jr + i-jf * IT
A closed system of equations is obtained by adding Maxwell's equations to determine E and B. In order to determine the charges and currents that occur in Maxwell's equations we take moments of the distri- .5.
bution function. The zeroth moment gives the number of parti cles of species a per unit volume: v*-0 5 K(«-t)d3v' (2_3) whereas the first moment gives the average velocity:
i r 3 ^ua (r.t^ ') * -vu = n— ;(£.t —r)- Ji \v,f a {r,y,t)'t'V ' d v. (2-4)
(The symbol = will always mean: by definition equal to). The charge and current density then follow by summing over species: T<*'fc> - I Va^'0' (2"5) a. a, a
Since all charges and currents in the plasma are supposed to be free, polarization and magnetization effects are negligible so that Maxwell's equations only involve % and £. In the ratio nalized MKSA system of units we then have:
9B *** " " ST • (2"7) 3E VxB - y J + — T7 , (2-8)
V 7«B * 0 , (2-10) -1/2 where c= (e o yo ) . * Average quantities of a function g(r,^,t) are defined as 5(«'fc) 'nTrf-trl9^^'^ **iW'V d3v- (2-4)' .6. In the Viasov theory of plasmas Eqs. (2-2)-(2-10) constitute the complete set of equations for the variables f (r,v,t), E(r,t), and Bfr,t). However, the fact that the distribution function is a function of seven independent variables pre sents us with formidable complications as far as the analysis is concerned. Since we wish to study plasmcs in the conplicated geometries needed in CTR research, we clearly have to get rid of some of the independent variables in order to make progress. The most logical approach is then to remove the velocity as an independent variable by taking moments of the Boltzmann equa tion. This approach will run into the problem of producing an infinite chain of equations which somehow has to be truncated in order to make sense. At that point assumptions need to be made that restrict the validity of the theory. B. MOMENTS OF THE BOLTZMANN EQUATION The different moments of the Boltzmann equation are ob tained by multiplying Eq. (2-1) with powers of v and integrating over velocity space. In the derivations below integration by parts will produce surface integrals over a surface at v = ». It is assumed that the distribution function falls of f rapidly enough at large velocities so that surface integrals do not contribute. Let us abbreviate the RHS of Eq. (2-1) as ° 8t coll 6 ae .7. where the collision term has been decomposed into contribu tions C „ due to collisions of particles of species a with aB particles of species B- Here, we will only consider two kinds of particles, viz. electrons (e) and ions (i) , so that a and 3 run over the two indices e and i. The present derivation will be heuristic enough that we never have to go into the specific form of the collision term. It suffices to list a few general properties following from conservation principles. Since the total number of particles of species a at a certain position is not changed by collisions with par ticles of species 6 (only their velocities change), we have fc d3v = 0 (including 0 = a). (2-12) Also, momentum and energy are conserved for collisions between like particles: f m vC d3v = 0, (2-13) J o^ act ' f ~m v2C d3v = 0, (2-14) J 2 a aa whereas for collisions between unlike particles the following relations hold: Jma*Caed3v + KxCgad3v = °» (2"15) v2C d3v + v2C d3v B (2 16) !K aB 1 K ta °* ~ The separate collision terms in Eqs. (2-15) and (2-16) also would vanish if the distribution function were taken to be a Maxwellian. Taking the zeroth rorrent of Eq. (2-1) then results in the following terms % .8. df 3n _Jid3v , _£ (def. (2.3))i df v.?-SLd3v . V.(noUa) (def. (2-4)), q 3dfi J m % 'Kt *& 3 v ^l*Ko xV "^rt»diy " ° (integrating by parts), J C d3v - 0 (summing Eq. (2-12). Consequently, 3n TT* '^V^ ' °' (2-17) which is the continuity equation for particles of species c. Multiplying Eq. (2-1) by v and integrating over velocity space results in the following terms: 3f 1 nV3v - ^vs.^ r 3 f r V'—— yd3v • V» yyf d3v - 7-(n vv) (where averages are defined in agreement with Eq. (2-4)')/ f q 3f q n J -2-(E+yxB). -SLvd3v - - -SL_2. •> m *v» *v *»# T3v Hence, the first moment of Eq. (2-1) gives r-(n m u ) + V*Utn yy) - n q (E+u xB) - J C .in vd3v, (2-18) which expresses conservation of momentum for particles of species a. .9. The final relevant equation is obtained f*om one of the second moment equations, viz. the scalar one obtained from multiplying Eq- (2-1) by v2. The following terms result: r na 1, r q 3f n q [ _£(E+vxB).a 2d3v = _ 2-2-^E.u , a 'v a 2 3 2 3 f C v d v = fc 0v d v (3i*a). J a Jap Multiplying these terms by y ma gives — (n -z-m v2) + V«(n T-m v2v) - n q E-u = C . rin v2d3v, 3t a2 a sa2 a ^ a a'v ^a J a£> 2 a (2-19) which is the form the energy conservation law takes. These are the only moment equations which will be exploited in the following. In order to turn the Eqs. (2-17) -(2-19) into a closed set a number of assumptions has to be made. Before we do this it is useful to transform the momentum and energy equation into a form that has a more macroscopic appearance. To that end, let us define a random velocity v' of particles with respect to the average velocity ua: v' » v - u . (2-20) The random velocity part of the term yy occurring in the momentum equation (2-18) gives rise to the stress tensor Pa defined as P (r,t) = n m y'y' • p I + ÏÏ , (2-21) 77 where P (r,t) = •=• n m v ti-??\ .10. and ir (r,t) is the part due to the anisotropy of the dis tribution function. Likewise, the random velocity part of the scalar v3" occurring in the energy equation (2-19) gives rise to a quantity related to the mean kinetic energy of particles in the frame moving with velocity u .which we define to be the temperature T a m v 2f ( t)d3v 2 23 T (r.,t) = * I « rt M» r < - > a J a a % * 3kna(^,t) * where k is Boltzmann's constant. Notice, P « n kT (2-24) o a a Finally, the random velocity part of the vector v2v occurring in Eq. (2-19) gives rise to a quantity Wfrt} - KVa2^' (2-25) which is the heat flow by random motion of the particles of species a. The collision terms may also be simplified by transforming to the moving frame \ - From Eq. (2-12) it follows that only the random part contributes to the RHS of Eq. (2-18): 3 3 fc 0m yd v - f C .m v 'd v = R , (2-26) which is the mean momentum transfer from particles of species S to particles of species o. By the use of the same relation we find that the RHS of Eq. (2-19) may be written as 2 3 3 ,2 3 J °asim v d v- fc flm u »v'd v + fc _m v d v • u «R + Q , -va ^a o .11. where Q _ fc„ -Ln v,2d3v, (2-27) o = J a3 2 a a which is the generated heat in the system of particles o due to collisions with particles 6. Substituting the definitions (2-21)-(2-27) the equa tions for momentum and energy conservation take the form —(n m u ) + V«(n m u u ) + 7*P - n q (E+u xB) - R , (2-28) n Tr(y „n u*) + — (yn kT ) + V* f7n m u*u + in kT u + u *P + h ) at / a a a dc / a a 2 a a a~a 2 a a'Ca *Ca Jöa "a - n q E'u = u -R + Q . (2-29) The momentum equation (2-28) may be simplified by using the continuity equation (2-17) to remove contributions 3n /3t, whereas the energy equation (2-29) may ba simplified by removing the bulk kinetic energy part by means of both Eq. (2-17) and (2-28). Defining the Lagrangian derivative along flow lines ua, the three moments of the Boltzmann equation then take the form dn (2_31) 7T * V«B * °' n m -~ + 7'? - n q (E+u xB) - R , (2-32) dT n k + ; + V = Q 2 33 7 a dt- Ja ^a 'Sa cc' < " > .12. These constitute the equation of continuity, motion, and heat balance for particles of species a. It will not have escaped the attentive reader that apparent progress has been made by just hiding the problems in simple looking variables. Clear ly, we need additional information concerning the variables Pa, h , R , and Q in order to be able to close the set. C. TWO-FLUID EQUATIONS Let us now specialize to a plasma consisting of elec trons, q = - er and one kind of ions with charge number Z , q± = Ze. From the Eqs. (2-31)-(2-33) one then gets a double set of equations for electrons and ions. From Eq. (2-15) one derives R = R = - R., (2-34) whereas Eq. (2-16) leads to a relation between Q and Q. which by the use of the relation below Eq. (2-26) may be written as Q = Q. - " The two-fluid moment equations then read: dn •— + n V-u » 0, de e ^e ' (2-36) da. dt l T>I n m -r-S- + V*P + en (E+u xB) • R, e e dt 've e 'v *\ nimi IT * 7'£i ' Zeni(^ix^ " " *' dT I nekdT + le'-^e * 7'*e ' ' Q + ^i'^ '«' (2-38) - dT. 4 n.k-r-i + P.SVU. • V-h, - Q' 2 l dt yx 'Vi 'vi .13. whereas the isotropic parts of the pressure tensors read : n kT = n k (2_39) Pe " e e' Pi i V We would have produced a closed set of two-fluid equations if the anisotropic parts ir and £. of the pressure tensor, the heat conduction terms h and h., the momentum transfer R, and the heat production Q were known in terms of the macroscopir c variables n , n., u , u. , T , and T.. A way e l ^e "^i e i to effect this is to simply put all these terms equal to zero. This procedure may be hidden in a long story about large and small parameters, but this is actually what is done to get the two-fluid equations of plasma theory. D. ONE-FLUID EQUATIONS The one-fluid equations of magnetohydrodynamics are produced by combining the pairs of equations (2-36)-(2-38) by means of expressions for the total mass density P , the center of mass velocity vf the charge density t, and the cur rent density j: -v p=nm +n.m.. e e il X s (neme2e + nimiSi)/p' (2-40) T = - en • Zen., e l j = - en u + Zen.u.. i e^e i'vi (Notice the new meaning of the symbol v, which can be used without confusion with the particle velocities since distri bution functions will not be considered anymore). The full information contained in the first two two-fluid equations can be retained if one adds and subtracts each pair of the two-fluid variables in terms of the above defined one-fluid variables by means of the inversion of Eqs. (2 -40): .14. Zp-(m./e)x Zo n = \ e ra.+Zm * m.*Zm ie ie P+(m /e)T p n. =i -\, > 1 ra.+Zm *V m.+Zm ie ie (2-41) Zepv-m.j m. ZeP-m.T Zep ^ e^ -v e . 'ui ep+m t *\« ep ^ where the approximations on the RHS are due to the assunvtion of quasi-neutrality: n - Zn. << n or m.T << ep. (2-42) Quasi-neutrality is a good approximation for the study of plasma phenomena with a scale length L such that L >> *D. (2-43) 1 /2 where the Debye length is defined as X = (e kT/e2n ) = 2 1 /2 = v.th,u e^A >p e~i where u>p e = (n e e /e o me ) ' . For a thermonuclear plasma with n = 10 cm , T « 10 °K, B = 10 Wb/m2 = 10s gauss -4 12 -1 we have XD_ = 7x10 cm and iop e = 6x10 sec . so that this condition is easily satisfied for the global phenomena we want to study. Multiplying the pair of Eqs. (2-36) by the masses and adding them gives the equation of mass conservation: §f + v-(pv) - 0, (2-44) whereas multiplication by the charges and subtraction results in the equation of charge conservation; .15. |f • 7-j » 0. (2-45) Likewise, adding the pair of equations (2-37) while using the approximations on the RHS of Eqs. (2-41) results in the equation of notion: 3v a m. PaT * «>X*VX * -£-L J*7J * v*p - TE - jxB - 0, (2-46) where P = Pe + £i- Notice that this equation transforms to the usual Navier-Stokes equation of hydrodynamics in the case that electric and magnetic effects are absent. Multiplying the pair of Eqs. (2-37) by the quotient charge/mass and subtracting results in an equation for the rate of change of the current density, which is known under the name generalized Ohm's law: -1 + V-f-rvv • iv + vi - ~-(l - z—•> —i 3 3 - —(V»P - Z — V'P.; + _L(1-Z-*)ixB - ^-2.(E+vxB) - - ^-(1+Z-^)R . e i e i ex (2-47) The term with ixg is known as the Hall term. Finally, adding the equations (2-38) results in the heat balance equation: „ „ i e t Zmp.-m.p nt tn. + p.V«(— j) - p„'.#(7T7J> • r :Vu • r.:7u, • V»h = - 7^r(l*Z-S.)j-R (2-48) .16. where p = p + P • » Ï1 = £ + £ • • The equations (2-44)-(2-48) constitute the euolu- lution equations fox the macroscopic one-fluid variables p,T,v,i, and p. Notice that no other approximation has been made than the quasi-neutrality condition (2-42), which is extremely well satisfied. However, a number of two-fluid variables still appear that have to be removed in order to turn the system of equations into a closed set. Therefore, additional assump tions have to be made that are less well satisfied, viz.: - the mass ratio of electrons and ions is small: m < e i - the relative velocity of ions and electrons is small com pared to the bulk velocity: lu. - u I « v, or m.j << epv, (2-50) l e l - the electron and ion viscosity are negligible: le' li * °* (2-51) - heat conduction can be neglected: h - 0, (2-52) - the ion-electron momentum transfer R is proportional to the relative velocity of ions and electrons: R - nen j, (2-53) where the factor of proportionality, the resistivity n, is assumed to be a scalar. These assumptions transform the Eqs. (2-46)-(2-48) into: .17. 3v PTT + py'y + VP - T? - ixB = °. (2-54) r^ + V-(TVV-MV+VJ) - ^£-(E+vxB)=-^£ nj, (2-55) ei ei 2 2 |E + 2 v.7p + 2 pv.v » r)j . (2-56) 2 3t2^K2F,\.J s Together with the Maxwell equations (2-7) and (2-8) and the mass and charge conservation equations (2-44) and (2-45) these equations constitute a closed set of evolution equa tions for the variables p(r,t), T(r,t), y(r,t), j(r,t), p(£,t) , g(£,t), and JJ(£,t). The Maxwell equations (2-9) and (2-10) may then be considered as initial conditions on E and B since they remain satisfied if they are initially satisfied, by virtue of the Eqs. (2-7), (2-8), and (2-45). Although the system (2-7)-(2-10) , (2-44) -(2-45) , (2-54)-(2-56) is mathematically consistent, the corresponding physical problem is quite crazy. Comparing the orders of magnitude of the terms in the generalized Ohm's law, it turns out that the terms 3j/3t and v,(TVV+JV+VJ) are much smaller r\, 'W '\i\ *V\» than the remaining terms. Numerical computation of the evo lution of the current density by means of Eq. (2-55) would be virtually impossible. The terms may be neglected if a condition is met that is slightly more restrictive than the quasi-neutrality condition (2-43), viz.: L >> c/u - K'c/v . . (2-57) pe D ch,e .18. This condition is easily satisfied (for the example given -3 below Eq. (2-43) one finds c/u = 5x10 cm). However, the pe neglect of these terms changes the mathematical nature of the system, since now we no longer have an evolution equation for j. This does not present a real problem because we obtain an algebraic relation between j and E instead by which we may eliminate j from the problem. This relation is properly •v called Ohm's law. Summarizing: Under the conditions (2-43), (2-49)-(2-53) , (2-57) the moment equations of the Boltzmann equation together with Maxwell's equations form the closed set of resistive MHD equations for the macroscopic variables p, T, v, p, Jg, and B: || + V.(pv) = 0, (continuity) <2-58> fl • '•j " °' (charge) (2-59) dv. p— t 7 p • TE - jxB = 0, (momentum) (2-60) 2 ?! * 2 ^"Vp * 2 pV*^ " n^2' JO -r^ + VxE « 0, (Faraday) (2-62) 3t *v. 3E "7 57 + %j ~ "*% " °' ("Ampere") (2-63) where nj - E + vxB, (Ohm) (2-64) 'v *\/ 'Xi yi .19. and initially the following conditions need to be satisfied: 7-E = T/C , (2-65) % o V-B - 0. (2-66) Instead of considering (2-59) as the evolution equation for T, one also could drop it since it is a consequence of (2-63) and (2-65). Then, we do not have an evolution equation for T, but we write T = e 7*E instead so that x may be eliminated o ^ from the equations. Next, we turn to non-resistive MHD. Resistive effects may be considered negligible (e.g., compare the term nj with vxB) if R„ s v vL/n >> 1. (2-67) M 0 Here, IL. is called the Magnetic Reynolds number in analogy with the hydrodynamic Reynolds number R = vL/v, whichmeasures the importance of viscous effects. This turns Eq. (2-64) into Ohm's law of ideal MHD: E + vxB » 0. (2-68) This assumption changes the character of Eq. (2-64) from one that determines } into one that expresses E in terms of v and *\» B. We then need another equation determining j. Let us take Maxwell's equation (2-63) for that purpose. That equation then changes from an evolution equation for Jjj (which is no longer needed since £ is now considered as a known quantity from Eq. (2-68)) into an expression for j. The interpretation we just gave of Maxwell's equation (2-63) is somewhat confusing. It is r.ore consis tent, physically as well as mathematically, to get rid of .20. the displacement current altogether by realizing that we are dealing with flows that satisfy v2/c2 < < 1. (2-69) Thus, we return to the so- -^1 led pre-Maxwell equations, characterized by the fact tha*" Eq. (2-6 3) is replaced by Ampere's law (as Ampere knew it): W j = 7xB. (2-70) But this implies V«j =0 so that Eq. (2-59) now tells us that 3-r/3t = 0. However, this is in conflict with the Eqs. (2-65) and (2-68) , which imply that _L = . ~--e -2- 7-(vxB) * 0 3t e a7 3c o 3t <\. "~ in general. Clearly, for mathematical consistency something more is needed to restore the peace in the system. The best way of finding a consistent set of equations is to apply an ordering in the small parameter v2/c2. One then finds that the term TE in the momentum equation is an order smaller than the other terms so that it may be dropped. After this, all equations are of the same order, except the charge conser vation equation (2-59) which is one order in v2/c2 smaller. In other words, to leading order in the small parameter vVc2 the charge conservation equation may be dropped. Poisson's equation (2-65) may be used to calculate T, but since it does not occur in any of the other equations, it may be dropped as well. The resulting set of equations is a mathematically consistent set, which enjoys the property of being Galilean invariant. In conclusion, in non-resistive ideal MHD resis tivity, displacement current, and space charge effects are .21. neglected, which is expressed by the conditions (2-67) and (2-63). The resulting equations read: |£. • 7-(pv) - 0, (2-71) dy p-^ + Vp - jxB - 0, (2-72) at i\, |E + v-7p + |p 7-v - 0 , (2-73) 3B ~ + 7x| - 0, (2-74) V j - 7xB, (2-75) o* v E + vxB - 0, (2-76) whereas Ï.B - 0 (2-77) need to be satisfied initially. Hence, we now only have evolution equations for the macroscopic variables p, y, p, and £, whereas the determination of j and E is trivi 1. From now on we will put u = 1. 0 REFERENCES 1. H. Grad, Notes on Magnetohydrodvnamies, I, "General Fluid Equations" (New York University, NY0-6486-I, New York, 1956). 2. A.A. Blank, K.O. Friedichs, and H. Grad, Notes on Magneto- hydrodynamics, V, "Theory of Maxwell's Equations without Displacement Current" (New York University, NY0-6486-V, New York, 1957), Sec. 1. .22. REFERENCES (cent.) 3. L. Spitzer, Jr, Physics of Fully Ionized Gases (Interscience, New York, 1962) Chapter 2 and Appendix. 4. S.I. Braginskii, "Transport processes in a plasma" in Reviews of Plasma Physics, Vol. 1, ed. M.A. Leontovich (Consultants Bureau, New York, 1965), p. 205. 5. G. Schmidt, Physics of High Temperature Plasmas (Academic Press, New York, 1966), Chapter 3. 6. T.J.M. Boyd and J.J. Sanderson, Plasma Dynamics (Nelson, London, 1969), Chapter 3. 7. N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics (McGraw-Hill, New York, 1973), Chapter 3. 8. S. Chapman and T.G. Cowling, The Mathematical Theory of Non uniform Gases (University Press, Cambridge, 1958). 9. P.C. Clemmov and J.P. Dougherty, Electrodynamics of Particles and Plasmas (Addison-Wesley, Reading, 1969). 10. B.A. Trubnikov, "Particle interactions in a fully ionized plasma" in Reviews of Plasma Physics, Vol. 1, ed. M.A. Leontovich (Consultants Bureau, New York, 1965), p. 105. 11. A.A. Galeev and R.Z. Sagdeev, "Theory of neo-classical diffusion" in Reviews of Plasma Physics, Vol. 7, ed. M.A. Leontovich (Consultant's Bureau, New York, 1979), p.257. 12. J.D. Jackson, Classical Electrodynamics (John Wiley, New York, 1967). 13. A.I. Akhiezer, I.A. Akhiezer, P.V. Polovin, A.G. Sitenko, and K.N. Stepanov, Plasma Electrodynamics, Vol. 1, Linear Theory (Pergamon Press, Oxford, 1975). .23. III. THE MODEL OF IDEAL MHD A. INTRODUCTION Why should a modem theoretical physicist be interested in ideal MHD? Remember: No quantum effects are taken into account, neither are relativistic corrections considered; in the derivation of the preceding section all kinetic effects were removed, whereas finally the neglect of the displacement current even removed electromagnetic waves from the system. In other words, we have moved back ward in time to a period prior to, subsequently 1925 (Schrödinger equation), 1905 (special theory of relativity) , 1872 (Boltzmann equation), and finally 1865 {Maxwell equa tions) . All interesting modern physics seems to have been removed from the system so that we wind up with a completely classical field that could have been studied more than 120 years ago. Nevertheless, three important reasons may be listed that should be sufficient ground for interest in this field: - Ideal MHD is the simplest physical theory that still makes sense in the context of confinement of plasmas for purposes of nuclear fusion. In particular, it is the only theory so far that takes proper account of the global geometry of closed magnetic confinement systems. - The linearized equations of ideal MHD may be cast in a form that is suitable for spectral analysis. In particular, the system can be described by means of self-adjoint linear .24. sidered as the prototype of a theoretical model in physics is to a large extent due to the same fact. Hence, linearized ideal MHD can be endowed with all the mathematical respect ability one wishes to have. - Recent developments in computing, specifically computations in hydrodynamics, make it possible to solve the full non-lin ear initial value problem for realistic geometries. Since non-linearity plays an essential role here, qualitative new physics is to be expected. Mathematically speaking we have effected two major simplifications in the derivations of the preceding chapter: - By integrating over velocity space we have reduced the number of independent variables from seven (r,^,t) to four (r,t), whereas the kind of equations have been changed from integro -differential to differential equations. - The neglect of dissipation has changed the system from a non-conservative to a conservative system. We still have one major mathematical complication in the equa tions viz. non-linearity. This complication will be removed in a later chapter when we linearize the equations. B. DIFFERENTIAL EQUATIONS As stated in the introduction we may just as well postulate the equations of ideal MHD rather than try to give a completely satisfactory derivation from first principles. Therefore, let us no longer worry about the domain of validity and just state the model and start working with it. Consider a perfectly conducting, ideal and compressible fluid inter- .25. acting with a magnetic field. The evolution of the 8-dimen- sional state vector v(£,t) , Jjjtr^t), p(^,t), p(jr,t) is described by the equation of motion for v, Faraday's law for jg, an equation of state for p, and the continuity equation for p. In the Eulerian picture: p3^ = - *rvx. - 7P + — = Vx(vxB) , VB = 0 , (3-2) 3P — = - v-7p - YP7-v , (3-3) H = " V(pv) • (3-4) In the Lagrangian picture: dv p_i = - vp + (VxB) x B , (3-1) ' at 'v *v dT = $*v* " I "'* ' v'l = ° ' (3"2)' ff = " YP?-v , (3-3)' ft - " PV.V . (3-4)' Here, we have substituted Ampere's law (2-75) into the equation of motion and Ohm's law (2-76) into Faraday's law. We have introduced the ratio of specific heats y, which for monoatomic gases has the value 5/3. Although we restrict .26. the analysis to roonoatomic gases (fully ionized plasma!) we will write Y for generality. (See discussion in Sec. Ill D). The equation for incompressible fluids may be found heuristically from the Eqs. (3-1)-(3-4) in the limit y -*• m, V-VH-0, such that dp/dt = - TPV'V remains finite. The latter relation may be dropped from the equations since it merely tells us what the magnitude of the quantity dp/dt is. To make up for the missing relation the constraint V«v = 0 needs then to be added to the equations, so that the equations for incompressible ideal MHD read: dv pdT " ' Vp + (VxS} x I ' <3"5) Jl ' l'*X . »•* - 0 , (3-6) ff - 0 . (3-7) V-y » 0 . (3-8) Usually, the incompressible model leads to a simple analysis, but for some purposes, e.g. spectral analysis, the constraint V-v = 0 spoils the structure of the problem to some extent. The equations above are evolution equations for the macroscopic variables. A different kind of problem is obtained when we set 8/3t = 0 (stationary flow). Making the additional assumption v = 0 leads to the problem of static equilibrium, which is extremely important for confinement of plasmas for CTR purposes: .27. Sp + Bx(7xB) = O , (3-9) V-B = O . (3-10) Here, V«B = 0 may not be considered as an initial condition because it is needed to supply four equations for the four variables p and B. C. BOUNDARY CONDITIONS To complete the model we need to be specific about the kind of problems we wish to consider, in particular we have to specify boundaries and boundary conditions on them. Several models will be considered: (1) PLASMA UP TO THE WALL Let the plasma be surrounded by an infinitely conducting wall screening it away from the outside world. It may be shown (see later sections) that the following boundary conditions are sufficient to determine solutions: %'Z • ° . (3-12) where n is the normal to the wall. The tangential components of ^ and £ and the variables p and p are not subjected to boundary conditions. It is clear from the form of Eqs. (3-1)-(3-4) that initial data v(r,0) , B(r,0), p(r,0), p(r,0) need to be specified on the domain of interest, i.e. the region within the conducting wall. Fixing these and the pr.rtic- .28. uiar shape of the wall then completely determines the problem. At this point one might even resort to the computer to provide us with solutions. (2) PLASMA SURROUNDED BY VACUUM Another possibility is not yet covered, viz. the plasma may be isolated from the wall by a vacuum (a useful model for confined plasmas). The fluid variables are not defined in the vacuum and the magnetic field is determined by vx6 = 0 , V'| = 0 , (3-13) subject to the boundary condition n-Ê = 0 (3-14) at the wall. Here, vacuum variables are distinguished from fluid variables by hats. At the plasma-vacuum interface we now may admit jumps in some of the variables, viz. in p, p, and the tangential components of B: [ + J- B«] - 0, (3-15) «•ia -°. (3-16) ;XW 'J* (3-17) where jumps are indicated by the notation |[fj = f - f. The surface current density j* is obtained in the limit of a surface layer of thickness <5 with current density i, when the limits 6 •*• 0 and 1 •> » are taken in such a way that j* = j<5 remains finite. Notice that j* has the dimension of current density times length. A special case is the static equilibrium problem , which is completely posed by the equations (3-9) , (3-10), (3-13)- (3-17). .29. (3) EXTERNAL COILS Finally, we may also consider a time-dependent boundary-value problem, where the wall is not passive, like in the previous two cases, but carries a surface current J* 11 (r,t) which forces oscillations onto the plasma. This wall may have gaps so that the system is not isolated from the outside world. In that case we have the following boundary conditions at the wall: r [fl - ° . (3-18) In addition, regularity of the vacuum field outside the wall at infinity is required. This case is of course important because all confined plasmas have to be created by means of external coils. Also, external excitation of MHD waves gives rise to this time-dependent problem. We have now provided the complete basis of ideal MHD at the expense of explaining why the above boundary condi tions are sufficient to fix solutions. In a following chapter we intend to make up for this defect. D. THE EQUATION OF STATE In the description of Sec. Ill B we have introduced the parameter EC /C , where C is the specific heat at con- r y' p v p r stant pressure, and C is the specific heat at constant volume, '?he parameters p and p could also be replaced by other .30. thermodynamic variables. In particular, the evolution equa tion for the pressure really arises as a consequence of our having chosen a particular equation of state for the ionized gas, viz. P * f(s)pT, (3-20) where s is the entropy per unit mass. Eq. (3-3) then obtains for adiabatic processes where ff - 0, (3-21) so that #H - »«>"T"1 - ? • •>. <3-22' where c is the velocity of sound. The explicit dependence of the function f is given by f - A exp (s/cv) (3-23) where A is a constant. Another convenient thermodynamic varia ble is the internal energy e: (3-24) (Y-1)P ' The evolution of e is described by 4| . . (y-D e 7-v , (3-25) which is easily derived from Eqs. (3-3) and (3-4). We now have four thermodynamic variables at our disposal, viz. p, p, s, and e, which can be expressed in terms of one another by means of the relations (3-20) , (3-23) and (3-24). Consequently, one can make different choices for the basic equations, depending on which pair of thermodynamic variables one chooses to supplement the basic variables v and B. In the Eqs. (3-1) to (3-4) the variables p and p were .31. chosen. It is instructive to also write down the basic equa tions for some other choice of the variables. The evolution equations for the choice of basic variables y, B, e, and p read: dv = -1 Ve P77 " <* > P ~ CTT-D eVp-Bx(VxB) , (3-26) at *\# *\» dS __ = B*7V - BV-v , V«B = 0 , (3-27) |f = - (Y-l) eV-v , (3-28) || = - pV-v . (3-29) For the choice v, B, p, and s one obtains; pdt " " 7p " £x(Vx£} ' (3-30) —*1 - B-Vy - BV-y , V-B = 0 , (3-31) dt dt = " YpV'X ' (3-32) jf - 0 . (3-33) For the latter choice one should realize that p is a complica ted function of p and s, which one also needs to know explic itly. This relation follows directly from the Eqs. (3-20) and (3-23). For purposes of reference we finally give the evolution equation for the variables v, |, s, and p: .32. 2 pdT = - c Vp - — pVs - Bx(VxB), (3-34) Cp !?- V V v dt S* Ï " § *X • '§ " O . (3-35) dt ~ P7-V » (3-36) d£ dc O • (3-37) where c2 = c2(s,p) is again obtained from the Eqs. (3-20) and (3-23). .33. IV. CHARACTERISTICS A. PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES As a preliminary to the study of the system of par tial differential equations (3-l)-(3-4), recall the method of characteristics for the solution of the second order partial differential equation A * + 2B * + C } - F(p , . Writing £ = *x , n 3 * , Eq. (4-1) is transformed to the system of first order equations A E + B E + B n • C n - F(E,n,x,y). x y x y (4-2) E - n =0. y x The pertinent Cauchy problem is to determine E and n away from the boundary, when they are given on 00 00 (4-3) n **«»x00 A) * n 0(x 0 ) • .34. We wish to investigate tonder which conditions 5(^»x) and n((ji,x) may be otained by means of a power series solution: CU.x) - C(* ,x„) + (•-•„) |f- • (x~x ) |f- oB o o 3 © O3Y + o o (4-4) - + (x x ) •*- nU,x> - n(* ,x ) + (•-•„) IJ " If" 00 0 3 $ O 3 X 3 F 3ni Here, £ *yo'xo*' n ( Transform the partial differential equation (4-2) to $ 1 X coordinates by writing > T7 + X a . etc. X 3$ x ox This gives (4-5) 4 ü _ é ia |i • x *y 3<|> *x 3$ -X y ax *f* »x Consequently/ the derivatives 7-7 and 7-7 may be determined from dm a A* . + Bó B$ + C<* Yx Yy x y - A* 2 - 2B* * - c* 2 - 0, (4-6) A x y y .35. defines two directions at every point in the plane, the characteristic directions, along which Cauchy boundary data do not determine the solution. Curves in the x-y plane that are everywhere tangent to the characteristic direc tions are called characteristics. Along $(x,y) = $ we have d$ = • dx +• 4 dy = 0, so that the characteristic direc- x y tions are given by 2 ÈL = - .!* . B± VB -AC # (4-7) dx 6 A y Three cases can be distinguished: - AC < B2: the characteristics are real, the equation is of a hyperbolic type, example: ii^x = -gr 4>tt, - AC = B2: the characteristics are real and coincide, the equa tion is of a parabolic type, example: ^ •"- *p«C« . llK€ the characteristics are given by — = i c. d t The initial data propagate along the characteristics. In spaces of higher dimension than 2 it is not sufficient for the well-posedness of the Cauchy problem that the boundaries are not coincident with a characteristic. One has to demand .36. in addition that they are space-like*. In physical problems initial data are usually given along space-directions**, so that this does not present a restriction. Finally, it is useful to distinguish two concepts: the domain of influence of I, which is the region in the x-t d6~»..,o« plane where the influence of the initial data I is felt, ie^»;^o< and the domain of dependence of the space-time point P, which is the region which influences the behavior at P. Notice that it does not matter whether the coeffi cients A, B, and C depend on £ and n as well, so that the method of characteristics also works for non-linear equations, specifically quasi-linear partial differential equations. B. CHARACTERISTICS IN IDEAL MHD We generalize the preceding discussion to partial differential equations in more than two independent variables and also more than two dependent variables, in particular the equations of ideal MHD. We wish to show that these equations are symmetric hyperbolic partial differential equations, where the non-linearity is only of a quasi-linear nature. * The reason that we have to demand stronger conditions in spaces of higher dimension than 2 is the fact that the spatial part by itself now contains an elliptic operator, so that Cauchy's problem is ill-posed if we consider time-indepen dent solutions. ** An exception is the excitation of waves by time-dependent forcing terms at the boundary of the plasma. In that case data are given on time-like boundaries. .37. The equations of ideal MHD are partial differential equations with respect to the variables r,t. Consequently, characteristics will be 3-dimensional manifolds C>(rft) = constant = 6 (4-8) in 4-dimensional space-time r,t. These manifolds may be visualized as being generated by the motion of surfaces in ordinary 3-dimensional space r. Let us apply the same tech niques as in the previous section to determine when a(r,t) = 4 is a characteristic. o Assume that boundary data for v(r,t), B(r,t), p(r,t), p(r,t) are given on è . [Notice that the initial value problem will correspond to giving v(£,0), B(r,0) , p(£,0), and p{r,0) on the domain of interest in ordinary 3-space. In order for the initial value problem to be well-posed ordinary 3-space should not be a characteristic. Here, we consider the opposite case that data are given on a characteristic, so that the Cauchy problem is not well-posed]. Like in section IV A we con sider é as a coordinate and introduce additional coordinates X,a, and T, so that 4-space r,t is covered by the coordinates 4>,X,0/ and T. The data may then be written as v(* ,x ,a f") = y 3v 3v ?(*.X.O,T) - vo(xo,Oo,to) * (4-^) ^ • (X-X0) ^" • o o Sv 3v • ((T-0 ) _i_ + (T-T ) JL. + , (4-10) 0 3a ° 3To o likewise for B, p, and p. As in the previous section, we may consider the problem to be solvable if 3v/3$ , 3B/3$ , 3p/3 \ - a* ' \ - n ' p " 3* ' p ' »• ' l ' The different expressions occurring in the MHD equations nay then be written as: 3v 3v 3v V-v • Vifi'vr ' + Vx'T— + VO'T— + Vf— , + • Next, define the following quantities: S = v*» (4-13) u = - Here, jr» is the normal to the space-part of the characteristic ($ can of course be chosen such that |v$| = 1 so that JJ has unit length), and u is the characteristic speed, i.e. the .39. normal velocity of the characteristic $ measured with respect to the fluid which roves with velocity v. For reasons that will soon become clear we will not start from the Eqs. (3-1)-(3-4), but rather from the Eqs. (3-26)-(3-29) in terms of the basic variables v, J3, e, p. Inserting the expressions(4-12) and writing the primed (un known) variables on the left-hand side and the known variables on the right-hand side, we obtain: - -puy* + (y-1) ripe' + (Y D J?e p * n*BB' + nB-B' (4-14) -uB' - n.Bv' + Bn-y' = , (4-15) -ue' + (y-De n«y' = , (4-16) -up' + p n*v' = . (4-17) In order to get equations of the same dimension multiply Eq. (4-15) by \fp, Eq. (4-16) with YP/C, and Eq. (4-17) by c, and introduce the following basic variables that all have the same dimension: pv' , VP B' , (CY-l)p/c)e' , (c/Y)p' , A» 'v where c2 = Y B - (0,0,B) , £ - (nx,0,nzj. (4-19) .40. Furthermore, introduce the Alfvén speed b = B/\TP, (4-20) and the sound speed c S\/YP7P. (4-21) The system of Eqs. (4-14)-(4-17) may then be written as -n b n b n c 0 0 z x x -u 0 0 -nzb 0 0 -u 0 0 0 n c z -nzb 0 0 -u 0 0 0 -n„b 0 0 -u 0 0 n b 0 x 0 0 0 -u 0 (4 nxc 0 n2C 0 0 0 n c 0 x n c 0 0 0 z 2 where n2 = Bn/5, nx = yi-uT/B) . Characteristics are obtained when the determinant of the LHS of Eq. (4-22) vanishes so that solutions cannot be propagated away from the manifold A - zbr «* ("2 " b2) [u1* - (b2 + c2)u2 + b2 c2] - o , Y l n n J * (4-23) where b is the normal Alfvén speed: b = n.B/fó>. Clearly, eight real characteristics are obtained corresponding to the eight variables needed to describe the system. The matrix on the LHS of Eq. (4-22) is real/ symmetric, and has only real eigenvalues. Consequently, the equations of ideal MHD are symmetric hyperbolic equations and the initial value problem, where .41. values are assigned to the variables v, B, e, and P in 3-dimensional space at t = 0, is well-posed. Before we go on to discuss the significance of the solutions obtained above, it is instructive to see what hap pens if we choose as basic variables v, B, p, and s. The Eqs. (3-30) to (3-33) now lead to -puv' + np' - n«B B.' + nB«B* = , (4-24) -uB' - n-By'+ Bn»v' = , (4-25) -up* + Ypn-v' = , 'X, (4-26) -us f __ __ _ (4-27) Multiplying Eq. (4-25) by \fij and Eq. (4-26) by 1/c to get compatible dimensions and inserting the expressions (4-19) for B and JJ we get, -nzb n b n c 0 f-u 0 0 x x 0 -u 0 0 -nzb 0 0 0 0 - u 0 0 0 nzc 0 -n b 0 - u 0 0 (,pH B ' z 0 0 0 x -nzb 0 0 -u 0 0 0 VP By» (4-28) n b o 0 0 -u 0 0 ^ B ; n C 0 0 0 -u 0 n x c z 0 0 0 0 -u Notice that this matrix is again symmetric. Of course, the same characteristics are obtained from this matrix. Thisis also true if we work with the system (3-1)-(3-4) for v, B, p, and p or the system (3-34)-'(3-37) for v, B, p, and s. However, in these two cases the matrix one obtains is not symmetric anymore. Therefore, the representations (4-22) and (4-28) .42. should be considered as more adequate for the present purpose. [Friedrichs' analysis in Notes on MHD VIII makes use of the v, B, P, s representation. His conclusion that this system is symmetric is based on the fact that he considers isentropic processes, where s = constant, so that he omits the term (c"VC )pVs in the momentum equation (3-34)]. Returning now to the discussion of the obtained solutions (4-2 3) for the characteristics, we notice that the characteristic speeds occur in four pairs: u = UQ = ± 0, (4-29) u = uA = + bn (4-30) u u Cb2 + c2) (b2 + c2)2_ 4b c2 1/2 1/2 = s = Ml " K n ] } » (4-31) 2 2 2 2 2 1/ /2 u - uf = ±{|(b + c ) + I[(b2 + c ) - 4b;c ] y . (4-32) The solutions (4-29) correspond to entropy disturbances that just follows the stream-lines of the flow. [Usually, the use of considering degenerate solutions like these is that they remind us of their possible importance when additional physical effects are taken into account that were not included in the model.] The pair of solutions (4-30) correspond to Alfvén waves which move in a backward (-) or forward (+) direction with respect to the flow. The pair of solutions (4-31) are forward and backward slow magneto-acoustic waves, whereas the solu tions (4-32) constitute forward and backward fast magneto-acous tic waves. Notice the following properties of the characteristic speeds: 0 U u U u = 1 J i i J 1 I A( i ( f( < - ' (4-33) .43. In particular, if n // jS: |u | * min (b,c) , |uA| - b, |uf| = max (b,c), (4-34) and if n J. B: =0, |u|= (b2 +c*)l/1 . <4~35> |us| = |uK, ., ,.f It should be noticed that the equations of ideal MHD are general enough that they contain the equations of gas dynamics as a special case. If b = 0 (no magnetic field) the slow and Alfvén waves disappear and the fast magnetoacoustic wave degenerates into an ordinary sound wave. Another limit of interest is the case of incompressible plasma (c •* •) . In that case the speed of the fast magneto-acoustic wave disappears at infinity (instantaneous propagation) , whereas the slow rrtristo- acoustic speed and the Alfvén speed coincide. The waves themselves do not ooincide, of course, because their physical properties (polarization e.g.) are different. Let us now look at the spatial part of a charac teristic at a certain time t = t , the so-called ray surface. This may be considered as a wave front, i.e. a surface across which discontinuities may occur, emitted at time t = 0 from the origin x = y = z = 0. E.g., in the case of vanishingly small magnetic field (b = 0) a characteristic manifold would just be the spherical wave front x2 + y2 + z2 = c2t2. Dropping the z-dependence one may visualize this in x, y, t space as a cone through the point »* x = y = t = 0. The circular (in case C_f*« the z-dependence is kept: spherical) intersection of this cone with the .44. plane t = t then constitutes the ray surface at t = t • For the MHD case we get of course more complicated figures, in particular because the medium is anisotropic and the coef ficients of the partial differential equation are not con stant. To get the ray surface we first of all compute from Eqs. (4-29)-(4-32) the distance ut which a plane wave- front travels along n after having passed the origin at t = 0. The collection of these points gives the following picture for b < c (in terms of the parameter 0 = 2p/B2 this is the extremely high-8 regime: g > 2-y) : This is not the ray surface, but the so-called reciprocal normal surface, [of course, everything is symmetric around the direction of B so that the 3-dimensional pictures are .45. obtained by just rotating the figure around the B-axis.] To get the ray surface we have to take the envelope of the plane wave fronts since the ray surface corresponds to a wave front due to a point disturbance at the origin at t = 0. Taking the envelope of lines indicated by S, A, and F in the figure results in a completely different and, JA particular, more singular picture. For the Alfvén wave, e.g., the reciprocal normal surface consists of two spheres touch ing the origin. Correspondingly, the ray surface consists of two points at x = lb. This shows the extreme degree of anisotropy of the Alfvén waves: point disturbances just travel along the magnetic field. The ray surface for the slow magnetcacoustic wave also exhibits a quite anisotropic character. It consists cf two cusped figures. The fast magneto - acoustic waves exhibit the least degree of anisotropy. In that respect they resemble ordinary sound waves most, [in fact they transform to ordinary sound waves if b -*• 0.| Remember >-o that sound waves in homogeneous media are governed by the equation Ai> = —,- <; for which the reciprocal normal surface and the ray surface coincide and just consist of the ray surdtce sphere x2 + y2 + z2 = c2t2. If A = 0,the equation obtained by putting the LHS of Eq. (4-22) (or Eq. (4-28)) equal to zero has a solution, so that on a characteristic manifold relations between the values .46. v', B', e', and p' exist. The meaning of this is that we may consider these quantities as discontinuities of the flow that are propagated along with the characteristics. In the case that the primed variables represent discontinuities, v' i 3v/3 1) Entropy disturbances (u = 0) : v*x = vy' = v*z = 0, B» = B» . B' = 0, (4-36) x y 2 e'/e = -P'/P = s'/C *0, p'= 0. [Friedrichs has p' = 0 because his momentum equation is differ ent] . Hence, the only perturbations that, occur are in the thermodynamic variables. This explains the name of these distur bances . 2) Alfvén waves (u - u.): These are purely transverse waves where v and E are perpendicular to the plane through n and Bs v^ = v^ = 0, vj * 0, Bx - B' = 0, B^ = Tpv^, j£^ ~ (4-37) e' = p' = p' = s' * 0. y' Here, the thermodynamical variables are not perturbed. 3) Magneto acoustic waves (u = u _ )i s, i These waves are polarized in the plane through n and B: .47. 2 v, = ^ u x n_ u^-b* z Y 2 2 B' n B-..„«! -n.. u b, ^P n_ u2-b2 * ' VP " u^b2^ vz ' By " ° ' z u UP = v _ up 5f','» s^ ;-p'= £*.»• (4-38) The polarizations of the fast and slow magnetoacoustic wave are perpendicular to each other as indicated in the figure. This difference arises through the factor u2 - b2 which is positive for the fast wave and negative for the slow wave. Notice that for all these perturbations the equation V-B = 0 leading to n*B' =0 does not have to be considered separately because it is an automatic result of Eq. (4-15). In case n»B = 0 the root u = 0 is sixfold degenerate. The Eqs. (4-14)-(4-17)then result in the following two condi tions : n.v' = 0, (4-39) 2 p' + |-B/ = 0 (or p' + | 3 ' = 0) , (4-40) whereas now we also have as an independent condition: (4-41) All other components of the variables are arbitrary. These disturbances are called contact discontinuities. An example would be an equilibrium of two adjacent plasmas with differ ent pressure, density, tangential magnetic field, tangential velocity, but satisfying the relations (4-39)-(4-41) . At the contact layer a surface current would produce the disturbance in the tangential field: *' = n x B'. (4-42) .48. Notice that we obtain in a quite unexpected manner all the boundary conditions of Sec. IIIC. REFERENCES 1. K.O. Friedrichs and H. Kranzer, Notes on Magnetohydrodynamics VIII. "Non-linear wave motion". (New York University, NYO-4686- VIII - New York, 1958), Sees. 1-5. 2. R. Courant and D. Hilbert, Methods of Mathematical Physics II (Interscience, New York, 1962), chapter VI. 3. P.R. Garabedian, Partial Differential Equations (John Wiley, New York, 1964), chapters 2, 3, 4, 6, 14. 4. A.I. Akhiezer, I.A. Akhiezer, R.V. Polovin, A.G. Sitenko and K.N. Stepanov, Plasma Electrodynamics I, (Pergamon Press, Oxford, 1975), chapter 3. .49. V. CONSERVATION LAWS A. CONSERVATION FORM OF THE IDEAL MHD EQUATIONS A quasi-linear system of partial differential equations is said to be in conservation form if all the terms can be written as divergences of products of the dependent variables (where in the divergence also the time-derivative is included) : _L(__) + 7.(__) = o. The use of such a form of the equations is the fact that one can easily obtain global conservation laws and shock conditions from them. Starting with the Eulerian form of the equations for v, jg, e, and p: PTT + Pvvy + Vp + Bx(vxB) - 0, p -(rDpe, (5-1) 3B T- - Vx^xjg) =0,7-3 = 0, (5-2) ^- + (y-l)e V-v + v-Ve = 0 , (5-3) j& + V.(pv) = 0 , (5-4) we notice that only the last equation has the required property. In order to bring the other equations in conservation form one makes use of the following vector identities: V'W - *-n * fcv,3 • (5~6) .50. ax(Vx]>) = (Vb)-a - a-7b , (5-7) Vx(axfc) - a7-b + b-7a - bV-a - a.7b = 7.(ba-ab). (5-8) From the Eqs. (5-6) and (5-7) one then has J3X(V X g) -|*B* - V(BB), (5_9) whereas Eq. (5-8) gives 7 x(v x B) = 7-(By - yB), (5-10) so that the magnetic terms then have the required property. From Eq. (5-1) we find by using Eq. (5-4): £(pv) - 7-(pvv + (p + |B2)I - flg] - 0 , (5-11) which is the conservation form of the momentum equation. From the Eqs. (5-2) and (5-9) we have SB — - 7- (Bv - vB) = 0 , (5-12) which is the conservation form of Faraday's law. Finally, the evolution equation (5-3) for the internal energy cannot be brought into conservation form for the obvious reason that it contains only part of the energy which can be converted into other forms of energy. We therefore need a conservation equation for the total energy density. This is obtained by adding the contributions of the kinetic energy, magnetic energy and internal energy: £•(5-11) + §•(5-12) + p(5-3) = 0 . This gives after a considerable amount of manipulations, using Eq. (5-4) and the vector identities (5-5)-(5-8): .51. ^-(jpv2 + pe + I E2) + v [(|pv2 + pe • p + B2)v - v-BB] - O (5-13) (check!) The equations (5-11), (5-12), (5-13) together with Eq. (5-4) constitute the conservation form of the ideal MHD equations. B. SHOCKS Consider the one-dimensional flow of gas where sound waves are excited. The characteristics are straight lines in the t - x plane: dx/dt = +. c. Suppose now that we suddenly increase the pressure, so that the sound speed increases. In the t - x plane this means that the slope of the characteristics decreases. Therefore, we may arrive at the situation where the character- n istics cross, i.e. information ori ginating from different space-time points accumulates. Consequently, gradients in the macroscopic varia bles build up until the point that the idealized model breaks down and dissipative effects due to the large gradients have to be taken into account. Eventually, a steady state will be reached where non-linear and dissipative effects counterbalance each other: a shock-wave has been created. Without specifying the kind of dissipation, one may arrive at the so-called shock relations that relate variables on either side of the propagating shock front. The idea is that the ideal model breaks down inside a layer of infinitesimal thickness ó(i.e., a thickness proportionil to the dissipative coefficient that is assumed to be vanishingly small), but it .52. holds on either side of the layer. In the limit 6 -»• 0 the variables will jump across the layer, and the magnitude of the jumps is determined from the condition that momentum, flux, energy, and mass should be conserved. Thus, one integrates the Eqs. (5-12), (5-13), and (5-4) across the shock front and keeps the leading order contributions arising from the gradients normal to the shock front only, since these gradients are infinitely large in the limit: 3f/3«. •*• ». These contributions then give: lim ( Vf di = n J jf-di = n(f2-f !> = n | f 1 . (5-14) 6 •*• 0 i • The time-derivative 3f/3t also contributes as may be seen by transforming to a frame moving with the normal speed u of the shock-front: ) - — 3t Idt Jshock U 3£ ' d where ( f/<3t)snoc]c denotes the rate of change in a frame moving with the shock. Since this quantity remains finite and 3f/3J. -*• », we must have 3f/3t •+ » as well. Hence: t i lin i|fd£.-u\|fdt--uïfTl. (5-15) The shock relations then simply follow from the conservation equations by replacing Vf by n ft f 31 and 3f/3t by -u I f"fl . One may wonder what it means to integrate equations across a region where the equations do not hold. The answer is that the additional terms due to dissipation do not con tribute. E.g., take the Navier-Stokes equation of ordinary hydrodynamics. In this equation terms like 3v/9t, v3v/3x, .53. u32v/3x2 appear, where v is the viscosity coefficient. If v displays a jump, both 3v/3x and 32v/3x2 will be infinite. However, since 3v/3x is an even function and 32v/3x2 an odd function, the latter term will not contribute upon integration across the -\T7 layer. In general, a more sophisticated boundary layer analysis may be required -^ to show that the net result is just integrating the ideal equations across a layer of infinitesimal thickness. Of course, one then exploits the fact that the ad ditional terms in the equation have a small coefficient in front of them. Making the above substitutions in the conservation equations (5-11), (5-12), (5-13), and (5-4) then results in the following jump conditions: - u fpvj +n*iTP^ • (P + \ B2) I - ££] = 0 , (5-16) - uÏBl - r[ Bv. - vBl= 0 , rlB] = 0 , (5-17) - u i j pv2 + pe + j B2! + n- I (| pv2 + pe + p + B2)v - v'BBÜ - 0 , (5-18) -ul pB + n- I py 1 - 0 . (5-19) Projecting transverse and normal components of the Eqs. (5-16) and (5-17) results in a set of equations that are very similar to tlvj equations (4-14)-(4-17) determining the relation between .54. the perturbations j/', £', e', and p' of characteristics. In fact, one may obtain one-one correspondence between the characteristics and the shocks by making certain substitutions. Thus, one obtains Alfven, slow magnetoacoustic, and fast magneto- acoustic shocks. The same kind of relationships between the different components of the shock variables may be produced. If we now demand that u and n*v remain continuous across the shock to guarantee coherence of the fluid, the following set of equations is obtained: ln-vl = 0 , ( 5-20) Jn-B] = 0 , (5-21) n-v(n-v - u) I Q\ + ftp + ^ B21 = 0 , (5-22) (n«v - u) I pnxv]] + n.B ïnxBl = O , (5-23) 2 (%'Z - U)Ï|PV -Pe +|B2l+p.v Up +1 B2]-n-Btv-Bll = 0 , (5-25) (Jj-v - u)üpl = 0 . (5_26) We will not continue the discussion of these shocks further. Our interest here is the case n/v. - u = 0, which was not so interesting in the context cf characteristics, but which is quite important in the context of shocks. Usually, when n-v - u = 0 one does not speak about a shock since the front is just carried with the fluid velocity, but one calls this a contact discontinuity. From the Eqs.(5-20)-(5-26) it is clear that in this case none of the .55. variables except p could jump unless n*B = 0. In other words: contact discontinuities require that B be paralel to the surface of discontinuity (i.e., excluding the case I vl = ÏBÜ = Ï pi = 0, I ol f 0). The reason for this is clear. E.g., suppose that B would intersect a surface of discontinuity of the pressure. Then, the pressure on both sides of the surface v/ould immedi ately equalize by flow along the field line. If n*B = 0, the only jump conditions left over from the set (5-20) -(5-26) are: In.vl = 0 , (5-27) tp + |B21 = 0 , C 5-28) whereas all other variables, i.e. v , B , and p may display arbitrary jumps. The jump of the tangential field component B , in particular, would give -ise to a surface current density of arbitrary magnitude: 4* = nx{[Bj. (5-29) Thus, we have obtained the boundary conditions to be posed on a surface separating two moving plasmas of different densities, tangential velocities, tangential magnetic field components, and pressures. If one replaces one of the fluids by a vacuum one ob tains precisely model (2) of Sec. III-C, which proves the cor rectness of the boundary conditions that were posed there with out proof. .56. C. GLOBAL CONSERVATION LAWS Let us continue with the discussion of the conserva tion laws. In order to understand the physical meaning of the different terms, define the following quantities: - momentum density it = py , (5-30) - stress tensor T = pvv + (p + - B2) I - BB , (5-31) tfi = vB - Bv , (5-32) - total energy density 'K = •=• pv2 + pe + y B2 , (5-33) - energy flow U = (^ pv2 + pe + p)v + B2v - vBB . (5-34) The equations (5-11)-(5-13), and (5-4) may then be written — + V«T - 0 , (conservation of momentum) (5-35) — + y.jj, = o , (conservation of flux) (5-36) — + 7-U * 0 , (conservation of energy) (5-37) vr + V*ir =0 , (conservation of mass) . (5-38; These are the evolution equations for *, £,*>*•, and p in conser vation form, where it should be noticed that the quantities appearing in the divergence terms can all be expressed in terms of TT, jg/K , and p, so that these four variables constitute another basic set of variables to describe ideal MHD. The stress tensor J£ is composed of the Reynolds stress tensor p y y, the isotropic pressure pi, and the magnetic part =• B2I - B | of Maxwell's stress tensor. In a projection based on y, the only non vanishing contribution to the Reynolds stress is a positive stress (pressure) pv2 along jr. In a projection based on B the remaining part of the stress tensor may be written as .57. p • ^ B2 O O \ 2 l O p • y B2 O O O -W «so that the £-field gives a positive stress (pressure) in directions perpendicular to £ and a negative stress (tension) in directions parallel to B. On purely formal grounds we have introduced the tensor P = vB - Bv in the evolution equation for B. We have not given a name to this symbol because it appears to have no direct physical meaning (at least we do not know of any) - The only reason we wrote the flux equation in this way is the fact that one obtains the jump conditions of Sec. V B roost easily. To get global conservation laws for the momentum, the energy, and the mass one should apply Gauss' theorem on the equations for n,1*, and p, but to get a global conservation law for the flux one should apply Stokes* theorem on the equa tion for B. For that reason the previously exploited fonn of Fara day's law with the term vx(v x £) appearing is to be preferred over that of Eq. (5-36) . The different terms appearing in the total magnetic energy density o*. may be grouped in two parts: »» = >*+ *r, (5-39) where OC is the kinetic energy density; rK 3 y pv2 , (5-40) and «^ is the potential energy density: *T = pe • i B2 - ^y + I B2 . ( 5-41) .58. The energy flow vector n is composed of a hydrodynamic part and a magnetic part. The latter part may be transformed to the usual Poyntinq vector S: S = ExB = - (vxB)xB = B2v - v-BB . (5-42) Consider now a plasma surrounded by a perfectly conducting wall (model (1) of Sec. Ill C), so that both vn = 0 and n*B = 0 at the wall. Define the following quan tities: - total momentum: " = { jdt , (5-43) ^ - total flux through a surface O bounded by a closed curve I on the wall: " totel energy: H = J* dT ^ (5-45) - total mass: M = \ pdx. (5-46) By applying Gauss' theorem to Eq. (5-35) we find: I - I - " S V-Tdr = - $ (p + I B2) ndo . (5-47 This is the total force excerted by the wall, which has to vanish if che configuration is to remain in place. By the application of Stokes' theorem to Eq. (5-36), or rather Eq. (5-2), we obtain * = \ Vx(vxB)-ndO = 4vxB-d«, - 0 , (5-48) since v_» £ and d£ are tangential to the wall. Hence, flux can not leave or enter the vessel. Applying Gauss' theorem again to Eq. (5-37) gives H = - U«Udt = - iu-ndi - 0 , (5-49) which states that total energy is conserved. .59. Similarly, Eq. (5-38) gives M » - W*irdT » iw-ndo » 0 , (5-50) so that the total mass is constant. Finally, one of the most inqportant conservation laws of a perfectly conducting fluid is obtained by integrating Faraday's law over a surface moving with the fluid. The result is that the flux through a contour moving with the fluid is constant: • = J B-jjda = constant. (5-51) c This theorem is proved in Sec. VII B, Eq. (7-24). The intu itive picture associated with this conservation law is to say that the field lines are frozen into the fluid (Alfvén) . Indeed, in ideal MHD the concept of magnetic field lines obtains more physical reality than it even had in the old days of Faraday. D. ENERGY CONSERVATION FOR MODELS 2 AND 3 In the previous section we proved energy conser vation of the non-linear system of ideal MHD equations for model 1 (plasma enclosed by a wall). In a later chapter (Sec. VIII E) we will need the law of conservation of total energy for a plasma-vacuum system (model 2). The generalization to model 2 is straightforward. The total energy for plasma and vacuum is H .[** dTP +\>7? dtV , (5-52) where *P = i pv2 • pe • \ B2 ,*V.l S2 . (5-53) In the time dependence of these energies one needs to take into account the rate of change of the volume elements. In Sec. VII B (Eq. (7-23)) we will derive that .60. ^ (dT) - 7-vdT , (5-54) so that JM*--!^ «•(*£<«> (2^L + v.yijt + rj(v.v)d \ St ^ -v. dx 1-TT- • fa v-nda (5-55) Although v is only defined in the plasma so that Eq. (5-54) is only valid there, the latter result obviously also applies to the vacuum as it merely tells us that the rate of change of the energy is due to the rate of change of the energy density and to the rate of change of the total volume. According to Eq. (5-37) we may integrate the plasma contribution by parts to get: "It" = ~ l*ï py2 + pe + p + B2)?*£do - -fap rtdo - 5(p+ \ B2>redo» so that *— " - [(P • J B2>v-n.do. (5-56) For the vacuum contribution we find B# v *ir " ^7 - - S* *f - - v-(fxB) + e-7xS - - 7-(ËXS) , so that dHv faaiv . v fmv .61. = \ ÊxE-nda -\— B2vnda To remove the electric field from this expression we need to apply Faraday's law just outside the surface of the ideally conducting plasma. This gives at the plasma-vacuum inter face: nxl = n-vB . (5-57) Hence, ~r = \B2y-ndo - \ \ B2vnda = \ | B2vnda . (5-58) Combining the Eqs. (5-56) and (5-58) and using the jump condition (5-28) we obtain |f - J Ip + \ B21 v-ndo = 0 , (5-59) q.e.d. For model 3, where the vacuum is enclosed by coils with surface currents j* , there is no conservation of /^waii energy for the interior region because the surface currents pump energy into the system. If we assume that these currents are arranged in such a way that no energy is lost external to the wall, i.e. Bex t =0, the rate of change of the energy is given by .int f c -j-wa11 f r. • * J wall ,. ,_. t = - Jl "S•v n'v .d o = - J xE*i rwal* ,l, do ,* \(5-60 " /) where we have used the jump condition (3-19). Hence, the rate of change of the energy internal to the coils is given by the Poynting flux across the wall. .62. REFERENCES 1. K.O. Friedrichs and H. Kranzer, Notes en Ma<^etohydrodynanics, VIII. "Non-linear wave motion". (New York university, NYO - 6486 - VIII, New York, 1958), Sees. 6-9. 2. W.A. Newcomb, Notes on Magnetohydrodynamics (unpublished). 3. T.J.M. Boyd and J.J. Sanderson, Plasma Dynamics (Nelson, London, 1969), Chapter 4. .63. VI. AN EXAMPLE: DYNAMICS OF THE SCREW PINCH A. PINCH EXPERIMENTS We will consider an explicit example of flux and energy conservation in a non-trivial geometry. In pulsed plasma confinement systems the magnetic fields are usually created by discharging a capacitor bank over a coil which induces currents in the plasma column. These currents create a magnetic field which pinches the plasma through the result ing jxB force. The electrostatic energy of a capacitor bank is 1 -> j CV , where C is the capacity and V the voltage over the capacitor. The question then arises how this electrostatic energy can be optimally converted into magnetic field energy needed for the confinement of plasma. The simplest scheme is undoubtedly the linear 6-pinch where the primary current I is created by dischargina the capacitor over a single turn coil surrounding a straidit cylin drical plasma column of circular cross-section. The induced plasma current I will mainly flow on the plasma surface if the plasma is well conducting. This surface current will create a drop in the longitudinal Bz-field, which produces an inward pinching force on the plasma column. [Vfe assume that a bias- linear 9-pinch .64. field B has been created prior to the induction of the plasma current I , so that 3 = 2y p/B2 < l]. The plasma is squeezed until the magnetic pressure -=—(B )•"• balances the o total pressure p + -—(B11^)2 of the plasma. The resulting hot 2UQ z plasma would be well-confined if a trivial effect did not occur, viz. rapid flow out of the ends of the 8-pinch. This results in loss of the plasma. These end losses operate on the usee time-scale and, therefore, block the road to con trolled fusion. To avoid the endloss problem one could close the plasma onto itself by exploiting a toroidal vessel. How ever, the toroidal 8-pinch rapidly expands due to the curva ture of the B -field (which is then toroidal). This lack of equilibrium also leads to plasma losses on the visec time- scale. A configuration curing both defects is the toroidal z-pinch, where the toroidal current I on a reel-shaped pri- mary z-coil induces a longitudinal current I in the plasma. This toroidal plasma current produces an external poloidal magnetic field B which again pinches the plasma. However, this configuration is violently unstable toroidal z-pinch *^s .65. with respect to external kink modes driven by the toroidal plasma current or, equivalently, the poloidal curvature of B - Again, plasma is lost on the ijsec time-scale. Also notice that in the case of a z-pinch the energy is already not opti mally used because the flux sticking through the inner hole of the torus is to be considered as lost, i.e. not used for plasma confinement. A combination of the two, avoiding endloss due to open-ended systems and instabilities due to the absence of a stabilizing toroidal fie^i, is the toroidal screw pinch. Here, both I. and I are switched simultaneously, so that the plasma 6 Z experiences an inward force consisting of the two components j„6 B z and j z B„6 . The current and field lines are helices wound on nested toroidal surfaces, the so-called magnetic surfaces. The magnetic configuration is similar to that of a tokamak, the difference being tha the toroidal field is created by a pulsed poloidal current in the case of a screw pinch, whereas it is quasi-stationary in the case of a tokamak. Both confi gurations may be in stable equilibrium due to the large toroi- toroidal screw pinch. dal magnetic field and also through the vicinity of a conduc ting wall surrounding the plasma. We will consider the screw .66. pinch case here because this provides the best illustration of flux and energy conservation in a dynamical system con sisting of two circuits and a plasma with two time-dependent magnetic field components. We wish to answer the following question: at t = 0 two charged capacitors C and C„ of voltage V and Vn are Z o Z 6 switched to the 9- and z-coil surrounding the plasma. What is the resulting plasma motion and how are the available electro static energies -s- C VJ; and ~- CQ v| converted into magnetic i. Z Z too field energies | ~— B* di and \ ~— B~ di ? [in this chapter \i is written again to facilitate comparison with experimental data] . B. MIXED INITIAL-VALUE BOUNDARY-VALUE PROBLEM In the present problem energy and flux conservation play the important role. To strip the problem of unnecessary complications, we therefore neglect the effect of the pressure and the density on the plasma dynamics. Comparing terms in the momentum equation (5-1) we see that the neglect of the pressure implies that we consider a low B plasma, B = 2n p/B2 << 1, which is certainly a valid approximation. On the other hand, the neglect of the plasma density implies that we consider veloc ities much smaller than the Alfvin velocity, v « b, which is a rather poor assumption for typical pinch implosions. Formally, we may justify this assumption by considering a slow compression experiment where the density is small enough for the Alfvén transit time to exceed the time-scale of the compression. Under these assumptions the ideal MHO equations to be used reduce to .67. (V x B) x B = O , (6-1) 5B 3T = V X ^ X V • (6-2) The first equation tells us that the current is everywhere parallel to the magnetic field so that no magnetic forces occur in the interior of the plasma. The second equation im plies that we consider the idealized Ohm's law E + v x B - 0 (6-3) "\i ^ K to be valid throughout the entire plasma. We idealize the two coils to be one copper shell closely fitting the plasma vessel with a poloidal cut (e.g. by the plane $ = 0) over which the voltage V is applied and a toroidal cut (e.g. by the plane 9=0, the curves labelled 3 and 3' in the picture) over which the voltage VQ is applied. The geometry of the toroidal vessel is fixed by prescribing the major radius R and the minor radius a of the torus. Tc simplify things we consider the inverse aspect ratio E = a/R of the torus to be a small L T*. parameter e << 1, and we onlv keep leading order effects. For the present analysis this implies that the only genuine toroidal effect r* considered is the ;oupling of the toroidal current I to the induced plasma current I by the z zp changing flux *T through the hole of the torus. For all other purposes the configuration is considered as a straight circular cylinder of length 2TTR. ( Hence, the use of z rather than 41 as the longitudinal coordinate). .68. In the approximation of a straight cylinder the plasma is described by the three variables v(r,t), B (r,t) , 6 and B (r,t) : z v = (v.0,0) , B = (0,B9,B2) , which from Eqs. (6-1) and (6-2) satisfy B9 T (r BQ)' + Bz Bz' = 0 , (6-4) n (v V' - <6-5> _£= _ _ (r v Bz). f (6_6) where primes denote differentiation with respect to r. The external circuits impose conditions on the values of the three variables v, BQf and B at the wall. Let us indi- 9 Z cate the value of a variable f(r,t) at the wall r = a by a bar: I(t) = f(a,t) . (6-7) Since we assume Ohm's law (6-3) to be valid for the entire plasma, the most exterior layers of the plasma experience an electric field given by Vz(t) - J E-dA % 2*R Ë, (t) - - 2*R v(t) B (t) (6-8) tor v (where the toroidal contour is taken along any one of the curves 1, 2, 3, or 4) , V (t) = d 2lTa 6 £ £' £ * ËQ(t) » 2ira v(t) T (t) (6-9) pol z (where the poloidal contour is taken through the points 3-2-1-4-3»). At this point one may wonder how there could be a radial plasma velocity at the wall. It was experimentally .69. observed that the pinching of the plasma column in a screw pinch is not perfect, i.e. the dense plasma is not swept up completely by the inward motion of the field, but a low den sity plasma is left behind. This tenuous plasma is hot enough to permit currents to flow there which create a magnetic field that strongly deviates from a vacuum configuration. Since the pressure and the density of the tenuous plasma may be ne glected, the magnetic field has to be force-free. This force- free magnetic field strongly influences the equilibrium and stability properties of the screw pinch, whereas the dynamics is also strongly influenced as we will see. Without going into detail about the origin of the tenuous force-free plasma we may simulate the creation of such a field by endowing the quartz wall surrounding the configuration with the property to be able to emit tenuous plasma. Thus, the plasma velocity v(t) at the wall is thought to originate from the creation of hot plasma that instantaneously sticks to the inward moving field lines. The equations (6-8) and (6-9) still have another defect that has to be removed before we can start solving the problem. If we assume that the dense plasma initially fills up the whole tube and that the bias field B (t=0) is purely toroidal, Eq^ . (6-8) tells us that the electric field Ëz (t=0) is not balanced. This leads to a contact discontinuity as described in Sec. V B where the ideal MHD model breaks down locally in a layer of thickness 6 which is considered tD be infinitesimal. To balance the electric field a surface current j*(0) is created that produces a jump in the poloidal field Li .70. given by Eq. (5-29): 3*(0) = B (0). Likewise, we may assume that the toroidal field displays a discontinuity produced by a possible initial imbalance between the poloidal electric field Ea(0) and the term v(0) B, (0) so that we have j*<0> = Bz (0) - Bz p (0). As the plasma moves inward the surface of discontinuity also moves inward. At this surface pressure balance expressed by the jump condition (5-28) has to be satisfied. Since we also neglect the pressure of the "dense" interior plasma this gives: B* (t) = B2(r (t),t) + B2( (t),'t) at r - r (t) . (6-10) zp z o 6 r o o The function r (t) will only be known after we have solved the problem so that v(r,t) is known. To recapitulate: The equations (6-4)-(6-6) are valid in both the inner region 0 £ r < r (t) , where Bfl = 0, and the O 8 exterior region r (t) < i± a, where the force-free field has both B and B components. At the surface of discontinuity r = r (t) Eq. (6-10) relates the fields on both sides. Notice that the inner plasma is considered to be a "dense" plasma with a pressure and a density much larger than that of the low den sity exterior plasma, but still small enough to be neglected in the momentum equation. P P Schematically: » i a •u cnn nwOu) 777777,/ -*• r Here, the value 1 indicates the order of p and p where their effect would have to be ireluded in the momentum equation. .71. We may now state the problem as follows: 1 - V* *- r rjLo\a r„{\) Given the initxal data 3 (0), v(0), B (0) , B (0), and the Zp u Z boundary data v(t), B~„ (t), B (t) , what is the magnitude of the plasma variables B (t), r (t), v(r,t), B_(r,t), B,(rft) zp o Ö z at a later tine? From these quantities one may then calculate the fluxes and energies and investigate how they are distri buted. Of course, we will only be interested in the dynamics during the compression phase when v(t) < 0. This phase is terminated at t = t when the two circuits are clamped, i.e. when the toroidal and poloidal gaps in the copper shell are closed, so that after t = t the configuration would become static if no dissipation or instabilities were to occur. The clamping time t is chosen such that the plasma motion would just reverse, i.e. v(t ) = 0, so that from the Eqs. (6-8) and c (6-9): Vz (t c ) = V„6 (t c ) = 0. At this moment the electrostatic energy of the capacitor banks is fully converted into magnetic field energy. The boundary data v(t), B (t) , B (t) will not be forced onto the system, but they are determined by the circuit equations, which in turn are coupled to the plasma equations by virtue of Eqs. (6-8) and (6-9). Therefore, the complete prob lem consists in simultaneously solving the plasma and circuit equations. Of course, the circuits are described by Maxwell's equations as well, but one would not readily give up the sim- .72. plicity of network analysis for that part of the problem. On the other hand, a circuit-like description of the plasma is not adequate. E.g., we will see that a concept like self-in ductance, which is very useful in circuit theory, loses most of its sense for a plasma in motion. This is due to the non-linearity of the plasma equations. The circuit equations will be derived in Sec. VI E. The problem we have stated above is a mixed initial- value boundary-value problem. It has the special peculiarity that the initial data are contained in the boundary data, because Bz p (0) is the only variable that has to be known over the cross-section of the cylinder at t =0. Its value follows from the boundary data by applying Eq. (6-10) at t = 0. If we — boundary A a\ a. consider for this problem what the •*• y characteristics are, we see from \'\eS A contact disturbance with the flow of the fluid (like the jump at r -- r (t)). o C. FIELD LI;.'S TOPOLOGY The plasm?, equations (6-4)-(6-6) provide us with three equations for the unknowns v(r,t), B (r,t), and B (r,t). However, we only have two circuits and, accordingly, only two equations to determine V (t) and V (t) from which the boundary Z 6 values v BQ and v B follow by the application of Eqs. (6-3) and (6-9). Therefore, we will have to reduce the number of plasma equations. This can be done by introducing a new vari able describing the field line topology. At the same time we will use the opportunity to dwell on some of the consequences of flux conservation since these properties have a much wider validity than the problem we are presently treating. Let us cut the torus (a cylinder of radius r actual ly) in two ways: a transverse and a longitudinal cut as indi cated in the figure. With these two cross-sections two fluxes may be associated: the flux the long way r * (r,t) = 2-n f B (r,t)r dr , (6-11) z Jo z $8 the flux the short way r 4>e(rft) = 2TTR J BQ(r,t) dr. (6-12) o Calculating the rate of change of these fluxes from Faraday's law as expressed by the Eqs. (6-5) and (6-6) we find: .74. 3B a* r f 2itR | ~ dr * - 2irR J(v B )'dr = at Q 2irR v Be - - v *6 ,(6-13) o o 3$ r 3B r dr = - 2w J (r v B )'dr 3t J at 2irr v B * - v • , z z (6-14) so that fluxes through contours moving with the fluid remain constant: d Another consequence of the Eqs. (6-13) and (6-14) is that one may define a local (i.e. depending on r) variable dif>, rB q(r,t) = dT (6-16) which is extremely useful for the description of field line topology in toroidal systems. The geometrical meaning of the parameter q is indicated in the figure where we have unfolded a cylinder of radius r. In this projection the field lines become anK straight lines and q measures the 1 pitch of the field lines; Ö|.WR q - d*/d9 1 Cq<0 lUr along a field line (where $ is the toroidal angle) . Traditionally, the name safety factor has been in use for the parameter q because stability with respect to internal kink modes in tokamaks requires q(axis) > 1, whereas stability with respect to external kink modes requires q(bound ary) > 1. In a straight circular cylinder q is related to the toroidal current I. (r): .75. 2ar2B , . i . (6-17, tor This is the reason for the use of q in connection with external kink mode stability criteria, because these modes are driven I a by the toroidal current. For stability, tort ) should not sur pass a critical value, called the Kruskal-Shaf ranov limit viiich corresponds precisely with q(a) = 1. Th2 fact that q = 1 also corresponds to a topology where the fluid lines close on them selves after one revolution the short way and one revolution the long way around the torus has been the source of much con fusion in the literature. The point is that the latter fact has nothing to do with external kink mode stability in a gen uine torus because the linear relationship between q and I expressed in Eq. (6-17) does not hold there, whereas the interpreta tion of q as a topological property of the field lines reir.ains valid in toroidal geometry. The reader should be warned in ad vance that statements to the opposite ara encountered in the literature. Some more concepts that are frequently encountered: - Rational surface: This is a surface (in this case a cylinder of a certain radius) where the field lines close upon them selves after M revolutions the short way and N revolutions the long way around the torus: q = N/M , (6-18) If q is irrational the field lines GO not close ax themselves and just cover a magnetic surface ergodically. - Rotational transform: This quantity has been used traditionally in connection with stellarators. It just measures the angle i .76. (iota) over which a field line tui proceeds after one revolution ® the long way around the torus: 0. l = 2ir/q (6-19) Hence, the use of i is fully equivalent to that of q to de scribe field line topology. - Normalized inverse pitch of the field lines; \i = 1/q R • (6-20) This synbol expresses the sane property as q. It has been, used exten sively in screw pinch literature for straight cylinders. A disadvantage of this variable is that it has nc geometrical meaning in genuine toroidal geometry. For this reason the use of q is to be preferred. Returning now to our discussion of the dynamics of the plasma, the most important property of q still remains to be exploited. The rate of change of q may be calculated by using Faraday's law (6-5), (6-6) : »„ „ 3B rB 3Ba , rB at RB„ at at RB V ^2 v V q 0 RB2 0 e e e Hence, moving with the fluid we find dq - 19. ^ • « (6-21) Td*t = 73Tt + v q -0 , so that the pitch of the field lines is conserved. This property also generalizes to toroidal geometry. .77. D. REDUCTION OF THE PLASMA EQUATIONS It is clear that Eq. (6-21) constitutes an enormous simplification for the present problem because it enables us to calculate q(r,t) at a certain position r and a certain time t if we know the time t' at which the plasma element was emit ted from the plasma: q(r,t) = ^(t') . (6-22) The time t' in turn may be calculated if we know v{r,t). For force-free fields in a cylinder the reduction of the number of plasma equations is obtained by the fact Jiat the Eqs. (6-4) and (6-16) imply that both field components B (r,t) 9 and B (r.t) can be derived from the sinqle quantity q(r,t): z \ / q r r r/R B (r,t) = A(t)\/ exp I - dr L \/q2+r2/R2 J q2+r2/R2 % A(t) [l - \ r2/R2q2 - J(r/R2q2) dr \ B (r.t) - A(t) \LJ^ZSL exp [- [-L&L- dr 1 9 Vq2+r2/R2 L J q2+r2/R2 (6-23) 2 2 2 2 2 A(t)(r/Rq) [l - \ r /R q - j"(r/R q ) dr ] % as may easily be verified by substituting these expressions into Eq. (6-4). Here, A is an inteqration constant and the approximation»on the RHS result from the ordering q ^ 1, e << 1. Clearly, wo have succeeded in reducing the number of variables needed to describe the force-free field to two: q(r,t) and v(r,t). .78. The equation for v(rft) to be used in combination with Eq. (6-21) is obtained by differentiating Eq. (6-4) with respect to time, inserting the expressions (6-5) and 3BB 3Bz (6-6) for -r-^- and -r— and finally substituting the (unap- 31 9 C proximated) expressions (6-23) for Bfl and B : o z . q2-r2/R2 v« » + _i vt r qi+rZ/R1 (6-24) 2 2 2 2 2 - JL [x + 2 -Jllï .. - r /R (rg '+4r /R n y . Q . r2 q2+r2/' (q2+r2/R2)2 This would be a horrible non-linear equation in v(r,t) and q(r/t) if we did not have our small parameter e available. Just dropping small terms in e we get: v" +iv' - -L v - 0 . (6-25) r r2 Since q no longer appears, this equation is valid throughout the interval 0 <_ r <^ a, so that the solution is simply v(r,c) - (r/a) v"(t) . (6-26) The expressions ( 6-22) and (6-26) virtually solve the problem for the plasma. The field B (t) in the "dense" plasma is found from zp F flux conservation, Eq. (6-15): na2 B (0) - ir r2(t) B (t) , (6-27) zp o zp where r (t) follows from dr (t)/dt = v(r ) ~ (r/a) v(t), so o o o o that t rQ(t) - a exp [ j(v(t)/a> dt] . (6-28) .79. The fields B (r (t)) and B (r (t)) are found from pitch conser vation, Eq. (6-22): q(r (t)) = ^"(0), (6-29) and pressure balance: B2(r (t)) + B2(r (t)) = [l + r2/q2(r )R2] B2(r(t)) = B2 (t) . (6-30) 6 o z o v o OJE° zp These expressions determine the value of A(t) to be used in Eq. (6-2 3) if the integral is taken from r (t) : 2 2 A(t) = B (t) = (a /r (t)) B (0) . (6-31) zp o zp Hence, if v(t) and q(t) are known, everything inside the plasma is known. These two quantities have to be determined by the two circuit equations. E. • CIRCUIT EQUATIONS Let us now derive the circuit equations which describe the time evolution of V (t) and V\ (t) . Since we have assumed z e that the coupling of the coils to the plasma is perfect, and since we will neglect stray inductances and resistive losses in the supply cables, the equivalent z-and 9-circuit look like: I.p 1 1—> 1 * f. i 1 ' 1 =T f. E,* vBe=o pi A* •» a .80. Here, the self-inductance LT couples the primary toroidal current Iz to the induced plasma current I zp , whereas the primary poloidal current Ia directly determines the plasma magnetic field B . If *T is the flux through the hole in the torus the toroidal ring voltage V is given by z d* cl dl 6 2) V*--IT--LT <ÏT*T!*> • < -' where LT is the self-inductance of the torus, which is just a geometrical factor which does not depend on the distribution of the plasma current to leading order in e: L_ J. p RUn 8/e - 2) . (6-33) 1 Ti O The rate of change of the voltage V is related to the circuit I by the usual relation Z dV - ir-rS*' (6~34' so that the equation for the z-circuit becomes: 2 C d V £ + _L, v = . _^2d I . . (6_35) z dci LT z dt Since the self-inductance L_ has served its purpose in provid ing us with a picture of how the toroidal current is induced into the plasma, we will now push the ordering in the inverse aspect ratio to its very limit and neglect the term V /L altogether. This is allowed if I»T is much larger than typical internal self-inductances of the plasma. We will see that these are given by l, ^ y u R, so that we assume L_>> I . Formally, Z £ 0 * Z this assumption is justified by taking e small enough. However, it is clear that this assumption is a rather pcor one for prac tical purposes due to the weak'logarithmic growth of LT with e. The value of V is related to the boundary values of .81. the plasma variables by means of Eq. (6-8) and the value of I may be expressed in terns of B by I = [j-n do = — fVxB-n da = — £ B-dZ = — B„ • (6-36) v o o pol ^° Inserting these relations intq Eq. (6-35) and neglecting the term with LT leads to the equation for the z-circuit: A A = (6 37) dT ^ V - z *e • 2 - TT • " 0 Z This equation has the required form of providing the evolution of the boundary data to be posed on the solution of the plasma equations. Iri the e-circuit I. directly determines the field in- side the c^il, so that (as fas as the outer boundary of the plasmi -_s concerned) we need not know I . Thus, analogous to Eq. (6-35) we have dVe l dt C9 -If9l • (6-38) Here, the relation of VQ to the boundary values of the plasma variables is given by Eq. (6-9), whereas I is related to the 6 toroidal field Bz means of l« " — Y P*d* s — B > (6-39) o tor o where we have t?ken a contour on the in side of the torus as indicated. Inserting these relations into JJq. (6-30; provides the equation for the e-circuit in the required form of an evolution equation for the boundary data: dT <* V - Ae *z • Ae E VTT ' C6-40) o Ö .82. Clearly, the two circuit equations are non-linear and, more over, they are coupled through the occurrence of v in both equations. F. SOLUTION OF THE PROBLEM We have solved the plasma equations already, i.e. we have expressed v(r,t) and q(r,t) in terms of the boundary data v(t) and q(t') by means of Eqs. (6-22) and (6-26). To determine v(t) and q(t') we have the two circuit equations (6-37? and (6-40). From these two equations one easily finds an evolution equation for q(t): *£ = (A9 - V< • (6"41) whereas the evolutior equation for v(t) is found by substitut ing the expressions 6-23) and (6-31) into Eq. (6-40) , while using Eq. (6-26) and neglecting terms of order e2: £ -1 *2 - *,- («-«) The solution of the latter equation is easily found by observing that it ia just the Ricatti equation corresponding to a linear homogeneous second order differential equation with constant coefficients which permits harmonically oscillating solutions. This gives: m aü) cot v(t) ~ 2 8 («t + a) f (6-^3) where the phase angle a is determined by the initial velocity: a = - arccotg (2v(0)/aw), (6-44) f and the frequency u is determined by the constant kQ o 'rhe e-circuit: .83. 1/? 1/2 u> = (2AQ/a) = UqCg) - (6-45) Here, we have introduced a self-inductance like quantity for the 9-circuit: a2 R *0 = 7 ^. / • (6-46) 6 2 o Since Eq. (6-41) is a linear equation in q the solution is easily found after substituting v(t) of Eq. (6-43): q"(t) = q"(0) [cos (lot + a)/cos a] X , (6-47) where the parameter A is determined by the relative difference of the constants A„ and A of the circuits: 6 Z A A (£ C ) 1 C )_1 \ = -fl L" „5- = —Lfl fli " " <^5-,5 . (6-48) £ = \ u R . (6-49) z 2 o Clearly, a natural frequency u> appears in these solutions which is entirely determined by the G-circuit. It is instructive to consider the time-dependence of the components corresponding to Eq. (6-47): B" (t) = B (0) sin (ut + a)/sin a, (6-50) z z Sin (a)t + a /sirt B (t) - B (0) > °| . (6-51) [cos(ut+a)/cosaj From these solutions it is clear that the e-circuit oscillates harmonically with a frequency u> = u> = U C.) ' . The reason 8 0 9 is clear: the z-field is very large and hardly affected by the small plasma current I , so that to leading order the 9p .84. 8-circuit merely sees a vacuum field and, consequently, oscil lates with a frequency determined by l., which is just the vacuum self-inductance of the 6-coil. On the other hand, the 6-field is entirely determined by the plasma current I , so zp that the z-circuit is strongly affected by the i.on-linear plasma dynamics. Consequently, the z-circuit displays anhar- nonic time-dependence. There is one case in which the z-circuit also displays a harmonic time-dependence. This is the case X = 0, which may be written as a sort of resonance condition between the two circuits: «e = wz , (6-52) 172 s 1 where „e , U^)" - «", "«V* '* • If the condition (6-52) is satisfied the two circuits are strongly coupled and produce a constant pitch q(t) =q(0) at the wall. By virtue of Eq. (6-22) a constant-pitch force-free field with q(r,t) =q(0) is then created in the tube. In this case and only in this case the solution of the problem nay be represented by means of .i simple equivalent circuit diagram: I j \ equivalent circuits for _ _!_ eg J the resonant case (X = 0) . This picture, intuitively appealing as it may be, should be considered with considerable reservation. First of all, it is simply a representation a posteriori of the solution of the complicated non-linear differential equations in a special case, viz. when the condition for the creation of a constant-pitch .85. force-free field is satisfied. An equivalent circuit repre sentation for the general case X f 0 does not exist. There is no way around solving the full set of equations. Furthermore/ even in the resonant case the interpretation of the quantity l z as the internal self-inductance associated with the plasma current I is extremely doubtful as we will see in the follow- zp J ing section. If the resonant condition (6-52) is not satisfied the z-circuit does not follow the oscillation of the e-circuit and, consequently, shear of the field lines is produced: W*=»- Wtr.l q0(-0 qlo") creation of shear in the non-resonant case (X f 0) -*• r q(r,t) = q(0) [{l - (rVa") sin2 (u,t + a))/cos\]Xf] (6-53) This solution develops a pathology at the time ut + a = TT/2 , when v •+ 0. Depending on whether A is positive or negative q(t) either goes to zero or blows up. The reason for the singu larity appears to be the mismatching of what would be the natural frequencies of the two circuits/ which jauses trouble at the end of the inward motion. Since the z-circuit cannot follow the oscillation of the e-circuit, the voltage V (t) given by Eq. (6-8) either lags behind or runs ahead of the evolution of .86. V (t) . In order to balance the electric field at the moment of reversal of the plasma notion (v -+ 0) , infinite current densities then arise which cause q(t) to 9° to zer° or to blow up. (This is called self-crowbarring of the plasma). Hence, if X ^ 0 the ideal MHD model breaks down at the end of the compression. G. FLUX AND ENERGY CONSERVATION In this section we wish to study the consequences of flux and energy conservation for the two circuits and the two magnetic field components separately. We have already seen in Eq. (6-15) that the fluxes trapped in a contour moving with the fluid remain constant. However, it is of more interest here to investigate the rate of change of the total flux in the tube by the influx of magnetic field from the boundaries. From Sqs. (6-13) and (6-14) it is clear that, as far as fluxes are concerned, it makes sense to associate B_ with the 6-circuit, z which ''sees 3* /3t, and B with the z-circuit, which sees 34> /3t. (Here $(t) = *(a,t)). o Let us now introduce the apparent self-inductances of the plasma as seen by the circuits: 1* /3t • IT II = 0 (6-54) z " 31 /3t zp 3* /at z ut ii - _ (6-55) where the reason for the use of quotation marks will become clear below. .87. The rate of change of the flux $_ can be calculated • by applying Faraday's law to a contour along the magnetic axis; so that 3 4 /at = - a u V v B" & "L •• = 2_5 = - M Ra - = i- p R z 2 2 2Tra 3Be/3t ° 3Be/8t ° 1 + Atg (u)t + a) (6_56s where we have subsequently substituted the Eqs. (6-36), (6-8), (6-43) , and (6-51) . Clearly, the z-circuit sees a self-inductance "L " of the plasma that changes in time. This change in time is only known after the problem has been solved. Also notice that "L " ^ l , as given by Eq. (6-49) except for the resonant case X = 0. For the self-inductance of the plasma as seen by the e-circuit we have from Eq. (6-55) : u V v" B , 2 9 2*R 3B /at ° R 9B /3t 2 ° R z z where we have substituted the expressions (6-39), (6-9), (6-43), and (6-50) , respectively. Here, the expected result is obtained, viz. "L" = la, as defined in Eq. (6-46) , because the 6-circuit 9 8 mainly sees a vacuum magnetic field configuration. The reason that we have put quotation marks on the self- inductances above is that, although the circuits see the plasma as having these self-inductances, they cannot be interpreted as the internal self-inductances of the plasma. It is well-known that the self-inductance of a current-carrying conductor is properly defined in terms of the total current flowing through .88. the conductor and the magnetic energy of the field created by that current according to the definition o Accordingly, the internal self-inductances of the plasma should be defined as follows: \ Lz l2zp • we - I at Be dT ' (6"59) r plasma ™ plasma o For W and I we have the following expressions: B (t) X.P • T7 e • so that the internal self-inductance of the plasma associated with the B -component is given by: ö L w R(1 r /alt) = L l (6 61) z " \ o " o " z" * z • " Hence, the apparent self-inductance "L " as seen by the z-circuit is at least twice as large as L as calculated from energy z considerations. For VJ and Ia + I. we have the following expressions: Z o üp „ . iili rB2 rdr „ilMlj2it) z y J Z ! + j ,ll«B (t) * lli B (t) , 6 6p ]iQ zp * M& z which gives the following expressions for L : Again, the expected result is obtained due to the fact that the z-field approximately Is a vacuum field. (This result is only true to leading order in e!). .89. The reason for the discrepancy between L and "L " is the fact that fluxes are conserved for the separated fields components but energies are not. The rate of change of the magnetic field energy density W = -L_ ( 2 + B2) (6-63) 2u B8 z o is found from the Eqs. (6-5) and (6-6): ^ + I (r vUT)' = 0. (6-64) From this expression we obtain for the total magnetic field energy: a SW 3 L' _ 2 \l . = - TT2R a V/ÜT. <6_65) 3t = 3t J dT = 47T R J r (r vvJ) r dr 8 o This rate of change of energy is due to the flow of energy from the circuits into the plasma as represented by the Poynting vector. From Eqs. (5-37) and (5-42)we have ^f + V-S = 0 , (6-66) so that |« = - f .n do = - -L f ExB-n do = ^ (Ê B - f B > - 3t J ^S ^ yj'b^^ u Q0z z9a o o v v J = ( c v ) + ( C v 2) (6 67) • e h ~ z Zp £ T e e £ l z 2 • - where we have applied subsequently the Eqs. (6-8), (6-9), (6-36), (6-39), (6-35), and (6-38). This expression shows the contribu tions of the two circuits separately. From the Eqs. (6-8), (6-9), (6-36) and (6-39) these contributions turn out to be 2 2 V. 1 = (4IT R/M ) a v B , (6-68) u V O Z - V I = (4*2R/ ) a v B2 z zp M o 9 ' .90. so that this seems to indicate that the z-component of the magnetic field is associated with the 0-circuit and the 8-component with the z-circuit. Let us examine whether the latter statement makes sense with respect to the change in time of the separate magnetic energy components: 3W 2 2 2 z 2ir R 3 r„2 ., 4TT R - ^ 2TT R f o' —— • — IB* r dr = a v B" + v B* r dr , 3t y 3t J z vn z u J z O 0 0 au <6_69) 2 2 2 2 0 2ir R 3 r„2 , An R - =? * R f n2i z -rr » — \ B* r dr » a v B* v B ' r dr. 3t y 3t J 0 y 0 y J z Wi 0 ° ° (6-70) Consequently, we find that the rate of change of magnetic energy of the B -component is not uniquely determined by the z energy influx from the 0-circuit, and vice versa for the B -component and the z-circuit, but there is a flow of energy F 0 from the z-circuit to the B -component: 3W 2 -~ » - T- (T Cfl V ) + F , (6-71) 3t 3t 2 6 0 3W 2 _Jt 3 - i (i C V ) - F , (6-72) 3t 3t v2 z z' * where 2 a F = ^-^ fv Bz' r dr . (6-73) p J z 0 O For the resonant case (A = 0) we find from the solutions ob tained in Sees. VI D and F: F ."2^- a v B? , (6-74) yo 6 so that 3WQ/3t = 2F - F = F. Consequently, the energy influx 6 from the z-circuit in to the plasma is equally devided between the increase of magnetic energy of the B -component and flow of energy to the B -component. The conversion of electrostatic energy of the Z capacitor banks to magnetic energy of the plasma turns out to be .91. much more to the advantage of the large z-component of the magnetic field than to the small e-component. As a result of this effect the self-inductances Lfl and L as defined in Eqs. (6-59) and (6-60) lose their mean- ing, whereas the apparent self-inductances "LQ" and "I^" seen by the circuits can only be calculated after the solution to the complete problem has already been obtained. We repeat: The basic reason is that flux conservation holds for the two components B and B separately, whereas energy conservation does not. 9 2 * 3W Diagram of the rich and the ( C v ) z Tt 2 e e poor countries. During compression you lose your energy, during expansion y 3W <* i If v2\-* e you gain it back. Morale: Don't ~ 7t(2C*V 9 at let yourself be squeezed. Fight back! REFERENCES 1. P.C.T, van der Laan, W. Schuurman, J.W.A. Zwart, and J.P. Goedbloed, Proc. Fourth Intern. Conf. on riasma Physics and Controlled Nuclear Fusion Research, Madison (1971) I, 217. "On the decay of the longitudinal current in toroidal screw pinches". 2. J.P. Goedbloed and J.W.A. Zwart, Plasma Physics V7 (1975) 45 "On the dynamics of the screw pinch". .92. VII. LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF IDEAL MHD A. SUMMARY OF SOME CONCEPTS OF CLASSICAL MECHANICS One of the roost powerful and beautiful parts of classical physics are the Lagrangian and Hamiltonian for mulations of classical mechanics. In particular, the formu lation of a Lagrangian from which the equations of motion can be derived by means of Hamilton's principle is one of the most concise descriptions of dynamical systems. One may consider a branch of physics to have become part of the classical curriculum if one succeeds in constructing the ap propriate Lagrangian. For ideal MHD this was accomplished by Newcomb in a paper of 1962 (Nuclear Fusion, Suppl. 2_, 1962, 451). Let us first collect a few pertinent concepts and formulas from classical mechanics. For a classical dynamical system the Lagrangian L may be defined as the difference of the kinetic and potential energy: L - T - V , (7-1) which is a function of the generalized coordinates q. and the generalized velocities q. : L - L(qk* q^. f t) • Hamilton's principle then states that the notion of the system from time t. to time t- is such that the line integral x z *k JLdt is an extremum: « j" L(qk, qk> t)dt - 0 . (7-2) .93. Here, 6 indicates the variation of the line-integral while keeping the end points fixed. The differential equations corresponding to this variational problem are Lagrange's equations: d / 3L ^ 3L ,. ,. re UcJ - -sq - ° • <7"3) From the Lagrangian one may construct generalized momenta conjugate to q. : pv - -^- (7-4) k »«* Conservations laws in classical mechanics are connected with the fact that one or more of the generalized coordinates may be ignorable, i.e. L does not depend on it (L ^ L(q, )). From Eq. (7-3) we then have p. = 0, so that p. = constant: The gen eralized momentum corresponding to an ignorable coordinate is a conserved quantity. One may change from a description in terms of gener alized coordinates and generalized velocities to one in terms of generalized coordinates and generalized momenta. In such a description of classical mechanics the role of L is taken by the Hamiltonian: H (Pk, qk, t) - £ pk qk - L = T + V . (7-5) The corresponding Hamiltonian equations of motion are easily found from Bq. (7-3) : (7 6 Conservation laws in this description are connected with situations where H does not depend on one of the generalized .94. coordinates. The same conclusion as above follows, viz. that the corresponding generalized momentum is a constant of the motion. For continuous systems and fields the motion is not described by a discrete set of generalized coordinates q, (t) , but by a continuous set n(x,t) where the discrete label k is replaced by the continuous label x. In three-dimensional space the continuous label becomes x and the generalized coordinate also may become a vector field ^j(x,t). (For a general field n could have more than three components. However, for our purpose a vector field in 3-space suffices as we will see). In this case the Lagrangian is an integral over all avail able space of the Lagrangian density Ad : L = J* dx , (7-7) where H now becomes a function not only of the continuous set of generalized coordinates n*t but also of the partial deriv atives of TK with respect to x. and t: * - *(n.., *i> ni' xi» fc) • (7-8) Here, we have adopted the notation j Hamilton's principle now takes the form 6 j dtj&dt - 0 , (7-10) S where the variation is to vanish at the endpoints t. and t~ and at the spatial boundaries over which the volume integration is taken. The differential equations corresponding to this varia tional problem now become partial differential equations with x. and t as independent variables: .95. dt 8n. • 3x. 3n-• 3n. Hence, instead of the n ordinary differential equations we had in point mechanics (k =1, 2, n) , we have fewer but partial differential equations in cmtinuum mechanics (i = 1,2,3). Again, one may construct generalized momentum den sities: ir. = ~- • (7-12) l dr\. The Hamiltonian formulation exploits the Hamiltoniar. density; ^-TtCn... n£, *£, x£ t) = £*. *i -*- > (7-13) whereas the total Hamiltonian becomes H = JT. dt . (7-14) The Hamiltonian equations corresponding to Eq. (7-11) are: 11; = ^„ » 1 d TT . • _ y _L_ a* _ JO. • (7_lj) 11 i " T Sx. 3D- • an. 3 j 1J 1 If ^ does not explicitly depend on time the total Hamiltonian is conserved: ~ » 0 . (7-16) a t This is the form energy conservation takes in the Hamiltonian formulation. A particular example of a continuous system is presented by the propagation of sound in a gas. In that case, the generalized coordinates n(>:,t) can be taken to be the displacement of the gas. In the case of ideal MHD our first question thus becomes what to use as generalized coordinates. .96. B. KINEMATIC CONSIDERATIONS Recall the ideal MHD equations in the Lagrangian form (not to be confused with the Lagrangian formulation of the dynamics which we discuss presently): d£ 1 pd7 + 7(P + I B2) " %'™ = ° » (7_17> d * dt B*7v - BV«v , V-B - 0 , (7-18) d£ = - YP V-v , (7-19) dt ff - " P *'% • (7-20) Our program consists in formulating a Lagrangian density such that the equation of motion (7-17) is obtained as Lagrange's equation corresponding to the variational problem expressed by Hamilton's principle. Also, a Hamiltonian density is to be constructed such that Eq. (7-17) is just obtained as the Hamil ton i an equation of motion. We have to address two questions first: What to take as generalized coordinates? Which role are the three additional equations (7-18)-(7-20) to play in this formulation? Let us start with the latter problem. To understand the meaning of the three evolution equations for B, p and p recall the discussion in Sec. V C, where we derived global conservation laws for Jg, ^(yes, what in this chapter will become the Hamiltonian density), and p. Here, we wish to discuss the local meaning of these equations. To that end we need ex pressions for the Lagrangian rates of change of elements of length, surface, and volume moving with the fluid. Without .97. proof we state the required equations: •h ± (d4) - - C?v)'d£ • (V-v)do , (7-22) -*- (dr) « (V-y)dT . (7-23) at "v These equations form the kinematic basis for fluid mechanics. From the second relation and Eq. (7-18) we may cal culate the rate of change of the local flux B*d -rat- (pdx) » -T—at dt + pa—t (dt) » - pV'vd^ T + pV«vd«v t * 0 . (7-25) This is the local counter part of Eq. (5-43): the mass in a vol ume element moving with the fluid is conserved. Finally, Eq. (7-19) for the evolution of the pressure may be usefully combined with Eq. (7-20) to prove that £ (PP"Y) » 0 , (7-26) which is of course nothing else than a restatement of the equa tion for entropy conservation (cf. Eqs. (3-20) and (3-21)). In .98. other words: we have succeeded in integrating the Eqs. (7-18)- (7-20), so that these equations are to be considered as holonomic constraints in the Lagrangian and Hamiltonian for mulation. Concerning the question of generalized coordinates: In analogy to the example of sound waves mentioned at the end of the previous section one could take for the general ized coordinates the displacement vector field £ of the plas ma elements from their initial position x ; x(x , t) = x + £(x , t) . (7-27) In fact, such a description will be employed extensively in the follow ing chapters. However, for the present purpose, there is no need to use £. We may just as well exploit x(% , t) itself as the continuous set of generalized coordinates. (Remember: x is to be considered as the continuous label of the generalized coordinate x). This is precisely what is called the Lagrangian description of fluid mechanics. The generalized velocities corresponding to £ are then denoted as *(*o't} -ïïl -7Ï • (7"28) where the derivative is taken with x held fixed. In general, we may now expect that the Lagrangian density )L wil be a function of the generalized coordinates and velocities as in Eq. (7-8): * -K (x. ., x. , x. , x . , t) , (7-29) .99. where X (7-30) ij 9x oj The latter matrix connects the positions x of the fluid ele ments at time t to their initial positions x . The Jacobian of the tranformation from x to x is then just the determinant of x. .: J=Det(x..)=re.,.e. x. . x. x „ , ij 2 ik£ jmn tj km £n (7-31) where c . ., is the Levi-Civita pseudo tensor: 1 if ijk is an even permutation of 123 - 1 if ijk is an odd permutation of 123 (7-32) ijk 0 if i = j, or j =k, or i = k (We adopted the summation convention to sum over equal indices). Defining the cofactor A. . as the determinant obtained from (x. .) by taking out the i-th row and the j-th column, e.g. Xll X12 x , we have the following identities: 23 X31 X32 3 J ij 2 e ik. £ jmn km In 3x. . (7-33) ij J 5. . = A, . x, . , (7-34) ij ki kj 3A. . —U. - 0 . (7-35) 3x0j The latter relations are sufficient to provide the in tegrated form of the kinematic relations (7-21)-(7-2 3) : dx. = x.. dx . , (7-36) do. • A. . do . (7-37) 1 1J Oj di = J dT o . (7-38) .100. From the conservation laws (7-24)-(7-26) we ha^'e: _Y Y PP - PoP0" » (7-40) pdt » p dT , (7-41) o o which by the application of Eqs. ( 7-36)-(7-38) gives the vari ables B, p, and p in terms of their initial values: B. = x.. B ./J , (7-42) P - P /J\ (7-43) o p - Po/J . (7-44) The equation v»B = 0 finally gives the only relation that the initial values have to satisfy: 3 B . •r—— - 0 . (7-45) 3 xoj For future reference we also give the expressions in terms of the displacement |: J * Det (£ + V £) , (7-46) * " lomil* Vo$)/J • (7'47) I» To recapitulate: We have shown that the Eqs. (7-18)-(7-20) may be considered as holonomic constraints by explicitly integra ting them to obtain the Eqs. (7-42)-(7-44). In this form B, p, and p are given as functions of x , t. It remains to construct a Lagrangian density from the generalized coordinates x. . (x ,t), x. (x ,t) , x. (x ,t) , which provides the equation of motion (7-17) 10 1 O as Lagrange's equation. .101. C. LAGRANGE AND HAMILTON EQUATIONS OF MOTION Hamilton's principle states that the evolution of the system is such that 6 f dt[g.(x. ., x. , x. , x , t)dx - 0 , (7-48) JJ IJllO O where the variation is to vanish at the endpoints t. and t_ and at the spatial boundaries of the system. The usual pro cedure for constructing a Lagrangian density is to try to find kinetic and potential energy densities and to postulate it as the difference between the two quantities. The sole jus tification of this procedure is the result in which the correct equation of motion is obtained. Fortunately, we have already constructed the kinetic and potential energy densities in chap ter V (Eqs. (5-33) and (5-34)): 2 2 ^= \ P v , UT= p/(Y-l) + | B . (7-49) A minor modification is needed to account for the fact that the Lagrangian is defined as the integral over the initial volume T0 : L - fJ*d T - S*'dT -5cX-*T)dT - J(*-*T )(pQ/p)dTo . (7-50) Hence, we postulate po L2 x (T-DP 2pl' (7 51) or, in terms of the proper generalized coordinates u 1 Po 1 a, - a- p x - - — x. . x., B . B . . (7-52) (Y-1)JY"1 J °J ° Lagrange's equation' corresponding to the variational problem (7-48) reads: JL p. + £_!_ -IX. . M. . o . (7-53) dt «x. Y x . ïx.. 3x, .102. Inserting the expression (7-52) gives: P X. + AT (TT ) 1 Oj 1] + ( B x B B } ? 3TT lk "TTT "k£ \m V om " 7 ik oj J ok = ° > i oj IJ which by the use of Eqs. (7-33), (7-35), and (7-45) becomes: p V. + Z [A. - rr— (-^ + -V x x. B B ) *o l . L ij 3x . jY 2J2 K- D* K™ oa ora - B . T-2— d x-, B . )] =0 . (7-54) oj 3x . J ik ok J J oj To get the equation of motion in more transparent form we transform back to the Eulerian picture, i.e. the inde pendent variable is changed from x to £. The Eulerian velocity will be expressed as v(x,t) - x(x ,t) . (7-55) *v 'u */ MO Furthermore, we need to convert derivatives with respect to x ' *• ^o to derivatives with respect to x. This is done as follows: 3x . 3x, 3x • . oi k _ oi IJ 3x, 3x . Sx, kj J k oj k J which by virtue of Eq. (7-34) gives 3x . 3x oi nJ1 , ki so that .. 3x . 3 1 3 3 oi l_ . (1 .,. 3x, 3x, 3x . * J ki 3x . " W-50; k k oi oi We also need the Eulerian counterpart of Jï «V , By virtue of Eqs. (7-4 3) and (7-34) B. -L. . 1 . A.. B . v-*-.. i fi..B ,J-.»Bi -J-. (7-57) l 3x. j2 ij ik oj 3x J IJ oj 3x , J ok 3x , l J ok ok ok By means of the Eqs. (7-56) and (7-57) we may transform Eq. (7-54) to: .103. dv. p TT + Ji^- _JB-^-B; =0 » odt 3x. Z iBx.i or dv P -,v + V(P + \ R2) " B*7B " ° » (7-58) which is the correct equation of motion. This proves that the Lagrangian density (7-51) is the appropriate expression. As in Eq. (7-12) the only step to be taken to get the Hamiltonian density is to introduce a generalized momentum density corresponding to the generalized coordinate x. Such a quantity was already introduced in Eq. (5-30), but we need here the Lagrangian counterpart: TT. (x ,t) = 4^- = P *• • (7-59) l ^o 3x. oi l We then obtain the Hamiltonian density Tt (x. ., x., IT., x , t) = n. x. - * 1J 1 1 O 11 tr2 . Po .1 _ + r + ^r x. . x.. B . B . , (7-60) Po (Y-1)JY" 1J oj ok ' which again corresponds with the Eulerian expression introduced in Eq. (5-32). The Hamiltonian equations of motion now read: i 3i7. , l \ 'X alTT 1777 - 377 * (7_61) 3 OJ 1J 1 Substituting (7-60) into the first equation gives us back the definition (7-59) of it, whereas substitution of the expression (7-60) into the second equation, of course, gives us the equa tion of motion (7-54) in Lagrangian form again. Since TR. does not explictly depend on time we have by virtue of Eq. (7-59) : .104. dl = dT lnldTo " J fexi + i^T xij + Ï7T *i)dTo i IJ i " 1 ix x • + •; ^ "; + 7 if. )dx J 3x. 1 3x.. 3x . 3ir. 3ir. ï o i ij oj i x J L 3x. 3x . 3x. . x 3ir. iJ o 1 oj xj x - f(- *.x. + x.ft.)dT = O . J XX X X O Hence, we recover the energy conservation law (5-47). REFERENCES 1. H. Goldstein, Classical Mechanics (Addisori-Wesley, Reading, 1950) . 2. W.A. Newconib, Nucl. Fusion, 1962 Suppl., Part 2 (1962) 451. "Lagrangian and Haitiiltonian Methods in Magnetohydrodynamics". 3. W.A. Newconib, Lecture Notes on Magnetohydrodynamies (unpub lished) . .105. VIII. LINEARIZED IDEAL MHD A. INTRODUCTION For many purposes it is desirable to have a deeper insight in the dynamics of the plasma than is obtained from a study of the non-linear equations. This contradictory statement may be clarified by pointing out the extreme limitations posed by present-day mathematical knowledge about non-linear partial differential equations. Once the system has been linear ized many more techniques become available and, consequently, a much better grasp of the problem is obtained. Of course, one would always be hindered by a bad conscience if there were no physical conditions where linearization is appropriate. It is our gocd fortune that we are interested in studying the be havior of confined plasma for thermonuclear purposes. Here, it is imperative for the eventual success of the project that the plasma is confined in an equilibrium state that lasts for a period that is much longer than typical time-scales occurring in the dynamics of the plasma (e.g., the Alfvén transit time of the machine). For those systems the approximation of a static equilibrium of the plasma is quite appropriate. In this context, three kinds of problems may be adequately treated with the equations of ideal MHD. First of all, one needs to know the equilibrium state of a realistic configuration. (Here, toroidal systems are the most important ones). This problem is still a non-linear one, but it may be solved for quite real istic geometries due to the special properties of the non-iinear equations of static equilibrium. Next, the problem of stability .106. with respect to small oscillations about the equilibrium state of such a configuration needs to be studied. Indeed, if one could only show that static equilibria are possible, but that they are all unstable, fusion by means of magnetically confined systems would be impossible. Finally, it is of in terest both from a purely scientific point of view and also for practical purposes (like wave-heating, feed-back stabili zation, and diagnostics) to obtain the different waves of the system. Of course, the latter two problems are intimately connected, so that an understanding of the wave dynamics greatly facilitates the study of the stability properties as well. For all these problems the study of the linearized system is quite adequate and it leads to many interesting problems. Our starting point will be the ideal MHD equations in the Eulerian form: Eqs. (3-1)-(3-4). The Eulerian descrip tion is most adequate for the present problem since one of the main complications of the analysis is the presence of an outer vacuum region (model (2) of Sec. Ill C), where a Lagrangian de scription is not available. Consequently, in a Lagrangian de scription one always needs to connect to Eulerian variables at the plasma boundary. To avoid these problems we have chosen for the Eulerian description. Of course, this choice is largely a matter of taste. Let us first restate the complete set of non-linear differential equations and boundary conditions for a plasma surrounded by a vacuum region, which in turn is enclosed by a conducting wall (see Sec. Ill B). The plasma is described by the variables %, jg, p, and p satisfying the following equations: .107. 3v p -^- = - pv7v - Vp + (tfxB) x B , (8-1) 3B -^ = V x (vxB) , V-B = O , (8-2) |^ = - v-Vp - Tp7-v , (8-3) l£. = - V(pv) . (8-4) The vacuum is described by the variable Ö satisfying the equations VxB = 0 , V-I = 0 . (8-5) At the plasma-vacuum interface the following boundary condi tions are imposed: [p + ^ B2I= 0 . (8-7) At the conducting wall the vacuum magnetic field is subjected to n-§ = 0 . (8-8) It may appear less obvious at first sight that the plasma vari ables are also subject to boundary conditions. These are usual ly quite obvious when the geometry is specified. E.g., in a torus one would specify regularity of the variables at the magnetic axis, and periodicity the short and the long way around the torus. The latter conditions are also to be imposed on the vacuum field ^. Instead of a vacuum it is sometimes also of interest to consider an external region which is also a plasma but with different magnitude of the variables (e.g. a low-density force- free plasma), so that still jump conditions need to be applied .108. at a fluid-fluid interface. Of course, the equations (8-5) are then replaced by equations analogous to Eqs. (8-1)-(8-4) for the variables y, £, p, and p. At the fluid-fluid interface the boundary conditions (8-6) and (8-7) should be supplemented with n.JvH = 0 . (8-9) At the wall we get in addition to Eq. (8-8): Jfï " 0 • (8-10) Notice that a tenuous plasma with p = p = j = 0 is different from a vacuum because Faraday's law (8-2) still implies the picture of frozen field lines. B. LINEARIZED EQUATION OF MOTION Consider now a static equilibrium, so that v = 0 and 3/3t = 0. The Eqs. (8-1)-(8-8) then lead to the following equi librium equations: - for the plasma region: Vp -j xB , j - VxB, 7-B - 0 , (8-11) - for the vacuum region: V x J « 0 , 7*| » 0 t (8-12) - at the plasma-vacuum interface: n • B -ti'S -0 , lip * TT *2\- 0 , }* - n xÏÏiJ , (8-13) - at the wall: n •% - 0 . (8-14) The third relation of Eq. (8-13) for %* is not really a restric tion on the kind of jumps one may allow. It simply tells us .109. what the magnitude is of the surface current associated with the arbitrary jumps in the tangential magnetic field components. For a plasma-vacuum system the current lines are alsn parallel to the plasma-vacuum interface, so that no x Vpo « 0 (8-15) there. Strictly speaking, this relation needs not to be satis fied at a fluid-fluid interface (it does not follow from the jump conditions for contact discontinuities derived in Sec. V B), but it is usually the roost realistic choice. The system of Eqs. (8-11)-(8-14) is far from a trivial problem, in particular because of the non-linear pressure balance equation (8-11). However, for simple geometries like slabs and straight circular cylinders the solutions are easily obtained. For more realistic geometries like toroidal configurations the equations lead to a non-linear elliptic partial differential equation for the poloidal flux function (the so-called Grad- Shafranov equation) for which quite accurate numerical solution techniques exist. These will be considered in a later chapter. For the present purpose we will imagine that the equations (8-11)-(8-14) are solved so that p , B , and B are known. It should be noticed that the Eqs. (8-11)-(8-14) do not uniquely determine these solutions so that a lot of freedom is left to choose particular equilibria. Next, perturb this static equilibrium by a displacement vector field £<£*t) so that X-dT-DT (8-16) .110. to first order. Notice that £ is similar to the variable in troduced in Sec. VII B, Eq. (7-27), except that we use here the same symbol for the Eulerian variable. Also, C is now considered to be small, for expansion purposes even infinitesimally small. The perturbed variables g, p, pr and fi are now written in Eulerian from (i.e. perturbed quan tities at the unperturbed position): B « B + 6B , p - p + 6p , ° (8-17) p - Po • 6p , B • 8 * 6$ . Like in the discussion on holonomic constraints in the previous chapter, we again treat the equation of motion (8-1) on a dif ferent footing than the remaining equations (8-2)-(8-4). In serting the expressions (8-16) and (8-17) into the latter equa tions we find that they are easily integrated. E.g., since £ does not depend on time. Consequently, to first order in k : v x } <8_18) «I X <* * l0 = ft ' r V B *P $ "k'*V0 " *o 'S * * (8-19) *P %-V- where we have introduced the symbols Q and * for the Eulerian perturbations of the magnetic field and the pressure, respec tively.* Inserting the above expressions into the equation of motion (8-1) and keeping only first order contributions leads to the famous formulation of the force-operator equa tion of linearized MHD: where Hence, in linearized ideal MHD only one vector £(r,t) appears as a variable,in contrast to the variables v, B, p, and p needed in non-linear MHD. In addition to the linearity, this is a very significant simplification. Also notice that the perturbation of the density does not appear so that Eq. (8-20) may be dropped in the linear analysis. It is also of interest to obtain the equation of motion for incompressible plasmas. As in Sec. Ill B, Eqs. (3-5)-(3-8), the equation for incompressible fluids is obtained by taking the limit y •*• • and 7«£ -»• 0 such that IT = - -yp v*£ - £*VP remains finite. Notice that it is only the Lagrangian part - yp V»£ of the perturbation of the pressure that should be handled with care in the limit. This procedure gives: * Sorry, even the Greek alphabet is finite. In chapter II the symbol n was used for the anisotropic part of the pressure tensor, in chapters V^and VII the synfcol ir was used for the momentum vector, whereas here the scalar it . is used to indicate the Eulerian perturbation of the pressure. Likewise, the scalar Q was used in chapter II for the generated heat, whereas here the vector Q indicates the Eulerian perturbation of the magnetic field. .112. »•* F(£) = -Vit + (Vxfl ) x O + (VxQ) x B (8-22) 'I = (8-23) Again, the subsidiary condition (8-23) needs to be supplied in order to be able to solve for the four variables E and ir. For the vacuum we introduce the magnetic field per turbation Q satisfying v x§ » 0 , V-§ - 0 , (8-24) and the boundary condition j(i*Q • 0 at the conducting wall. (8-25) Notice that Q is not defined as in Eq. (8-18) since there is no displacement vector defined in the vacuum. C. BOUNDARY CONDITIONS Next, we need to linearize the boundary conditions (8-6) and (8-7) to connect the plasma variable £ with the vacuum vari able Q. In the linearization of the boundary conditions one needs to supplement the perturbation of the plasma variables given in Eq. (8-18) and (8-19) with the change due to the fact that the boundary conditions are to be satisfied at the perturbed, bound ary. Also, we need an expression for the normal to the perturbed boundary. An expression for the perturbation of the normal is most easily obtained f row the kinematic relation (7-36) which gives the change of a line-element moving with the fluid: di " dV (* * v *>• (8"26) In this relation the differentiation with respect to the Lagran- gian coordinate g has been replaced by the differentiation .113. with respect to the Eulerian coordinate x, which is correct to first order since the difference between the two descriptions is of higher order. From this expression we now have 0 = n-di * dl *(I • 7 E)-(n + n, ) •v di *n + di *V£-n + d*. *n, = dZ *(V £«i ll> 11-. v where 3 Hence, ^ ' - < $>'20 * *' ' £)* But dl may have any direction in the perturbed , surface unperturbed surface so that \ * yn . unperturbed Since !n| = 1, we have n «n. = 0 so surface that y = xx • (V£) «n . This provides us with the required perturbation of the normal: n. - - (v"£)-n + n n '(Vp'ti . (8-27) Evaluating B leads to an extra term E-VB due to the fact that B is to be taken at the perturbed boundary: (B) * (B + Q + £«VB ) . (8-28) A. "to Inserting the Eqs. (8-27) and (8-28) into the boundary condition (8-6) gives 0 - ft-* - feo - (7#-*o+ «o v(7*),|lJ,(*<> * *+ *"yW -Ho'7 X (W * «o'« • where use has been made of Eq. (8-13). This relation is automat ically satisfied by virtue of the definition (8-18) for Q. How ever, the same derivation also applies for the equation n*£ - 0 which now gives the required relation between £ and Q: n 'Vx(£ x S )- n '0 . (8-29) .114. That this boundary condition in fact depends on the normal component of £ maybe shown by one of those tedious vector manipulations that abound in this field: B -7(n «O - n •£ n • (VB )-n » n *Q . (8-29)» For explicit calculations this form is to be preferred as it gives directly the relation between n -C and n -0. To evaluate the pressure jump condition (8-7) at the perturbed boundary we need an expression for (p) analogous to Eq. (8-28): (P) - (p • , + 4.7p ) - Here, the left-hand side is just the Lagrangian perturbation of the total pressure. For a plasma-vacuum system the equation of motion (8-21) for |, the equations (8-24) and (8-25) for £, and the boundary conditions (8-29) and (3-31) connecting * and Q at the plasma-vacuum interface constitute a complete set of equations by means of which waves and stability properties may be inves tigated. For a plasma-plasma system some extra care in the use of the boundary conditions is needed. In that case Ö = ?x(cx6 ) so that the boundary condition (8-29) is to be replaced by the .US. linearized version of Eq. (8-9) : n • I = n * f . (8-32) For the pressure balance equation one has tc "*j?d pressure terms of the exterior fluid to the boundary conJ-tion (8-31). One may then be tempted to infer from the continuity of the Lagrangian perturbation of the total pressure that the RHS of the boundary condition should be just the same expression as the LHS of Eq. (8-31) with £, Q, p, and B replaced by ?, Q, p, and 6. In fact, such a regretteble mistake has been made in the V literature*. The point is that although - yp v.| * B • (Q + £* £Q) is the Lagrangian perturbation of the total pressure of the inner fluid, and - yp ?'\ + 8 ' (Q + |"v£ ) is t^ie Lagran gian perturbation of the total pressure of the exterior fluid, the two pressures are not evaluated at the same position since the tangential components of £ are not continuous. For the sake of symmetry between inner and exterior fluid it is therefore to be preferred to express the perturbation at the perturbed boundary at the posi tion r + (n *£)n since the normal *vO '-O 'S ^O component of £ is continuous. The ex pression for the perturbation of the pressure and the magnetic field at that position read: * J.P. Goedbloed, Physica 53 (1971) 412. Fortunately, the error in the boundary condition applied in this paper vanishes for the cases considered, viz. plane slab and cylindrical geometry. For toroidal systems the error would not have cancelled. .116. AD * <5p + n •£ n -Vp , (8-33) so that MP + £ B*> = - YPOV.^ • Bo-§ - kt . 7PQ • no.$ n^v I BJ , (8-34) where £ = £ - n •£ n . We will neglect the term £ *?P by virtue of Eq. (8-15). The boundary condition then becomes: - YP V-E + B «Q + n •£ n -7 J B2 --yp 7»| + g »Q + n • I n «V - g2 , (8-35) which is nicely symmetric now. Since this boundary condition also applies to fluid-vacuum systems when we put p =0, the form (8-35) is actually to be preferred over (8-31). For a fluid- fluid interface the two boundary conditions (8-32) and (8-35) may be combined to give I n • £ M which shows that the specific value of n •£ scales out of the pro blem (as it should because the problem is linear). From now on we will drop the subscript o and denote the equilibrium quantities simply by B, p, p, n, and S because no con fusion is possible with the perturbations which are denoted by the different symbols £, Q, t, and Q, respectively. .117. D. SELF-ADJOINTNESS OF THE FORCE-OPERATOR Consider two vector fields £(£,t) and rt(£ft) de fined over the plasma volume fdx" (the superscript p denotes the plasma and the vacuum will be indicated by the superscript v) , not necessarily satisfying the ideal MHD equation of motion (8-22). These vector fields will be connected to two vector fields Q(£,t) and R(j£,t), defined over the vacuum volume fdt , that do satisfy the vacuum equations by means of the boundary conditions (8-29) ünd (8-31) , so that we have: v J- : 7 *§ * 0 , *-J- o , (8-37) 7 x ft - o , 7«ft s 0 , : *'• 7 x (8-39) - vv*'Ji * S*(R + r7V -?•$ •«•*?> » I s 7x^x?> » on \ iav (the wall): n«3 = 0 , n.» - 0 . (8"40> Let us now define an inner or scalar product of the two vector fields £ and JQ: Restrict the functions £(£,t) to be considered to have a finite norm: | j £ 11 < » . The function space thus obtained is a Hubert space, which is a space of infinite dimen sionality. To be specifier if £(r,t) is written as £.(x.,t) we have an infinite set of functions values labelled by the discrete label i that takes the values 1, 2, and 3, and three continuous labels x. which run over the pertinent intervals corresponding with fdt . In this context, the time variable t is simply considered as a parameter. The formal properties of the linear vector space defined above are the following ones: (1) For any two elements £ and n belonging to the space also a£ + 6n belongs to the space, where a and 8 are any two complex scalars. (2) The scalar product is linear with respect to the right-hand side element: whereas <£, nn>* = < rj, £> , so that (3) The norm of an element £ is non-negative: miiio, where equality only holds for the zero-element. (4) The space is complete: The limit element £ of a Cauchy sequence [L \ > i.e. a sequence for which lim || £ - 5 || =0, also belongs to the space: I !%! | - Ho | UnM <- • T .119. (5) Conversely, separability should also hold: To each element E, a corresponding Cauchy sequence \ £ } can be found such that lim i*.n • urn • These properties make the linear vector space a Hubert space. For linearized ideal MHD the properties (l)-(3) are obvious from the definitions above, whereas the properties (4) and (5) which actually need to be proved are simply assumed to be true. As we shall see, property (4) is extremely important in connection with the occurrence of so-called continuous spectra. Property (5) provides the basis for approximating functions by finite sets of known functions, which is especially important in numerical applications. The idea of the relations (8-37)-(8-40) is to continue the function 5 into the vacuum by means of the magnetic field variable 0, and likewise to continue n by means of ft, by match ing something like the function value and the normal derivative at the plasma vacuum interface. This is schematically indicated in the figure. It is a very remark able property of ideal MHD that •fc. r only two conditions need to be satisfied to connect two vector fields £ and Q. Hence, it appears like we are dealing only with ordinary second order differ ential equations. The reason behind this is the extreme aniso- tropy of ideal MHD as regards motion inside and across the magnetic surfaces, to the study of which we will turn later on. T .120. Thus, we have obtained a definition of the scalar product that involves an integration over the plasma volume only. The physical significance of this is the following. The kinetic energy of the perturbations may be written as: K = I jpv2 drP % \ Jp£2 dTP = <£, £> . (8-43) Since t merely plays the role of a parameter, this implies that the vector field £(r,t) may be chosen to belong to the same class of functions as £(r,t) . In other words, the physical significance of restricting the consideration to displacement vector fields |(£,t) that are bounded in norm is that they provide the plasma with a finite amount of kinetic energy. Returning now to the discussion of the force-operator £(|), one extremely important property of this operator is that it is self-adjoint or hermitian; **>• p"1 W " Hence: • TJVS($*7P + YP7*4 ~ %'& do - \ Jfav-rj 7.| * §-R * Vr,($-vp) • (VxB) -p$ By the application of the boundary condition (8-31) the first term on the RHS may be transformed: • - ?ƒ*•« «•* rï7 1 do -ih'zWdo> <8"46> where the latter expression derives from the fact that, since |p + j B21 « 0, the tangential component of the jump of the derivative vanishes as well: t«|[v(p + ^ B2) J = 0. Next, trans form the term "" ?.]£*# $"Q^o: Introduce vector potentials in the vacuum: Ö = V x £, £ = V x £. The boundary condition (8-29) then gives n-VxCnxS) » n*7x£ , so that C » r,x§ • 7$ and nxC * nx(nxB) • nxV* , .122. Choosing the gauge such that nx7* = 0, we have (which is precisely the boundary condition (2-33) of Bernstein e.a.). Thus, we have: ~ rfn-n 5*0 do = — f nxC-Ö do • yJVS"Vx$ d°= • H(7x^)x?*"do = £ f7.r(?xS)xe] dxv • \ I l£'****b - **?•**?] dtv -4" Ivx?-7xsdTV = -1 JH dtV • (8-47) where a minus sign appears in the conversion of the term I da to the volume term J dtv because the latter volume is situated outside the piasma-vacuum interface. The contribution of the integral over the outer conducting wall could be added for free since it vanishes by virtue of the boundary condition (8-40). The term with 7 x 7 x A vanished due to the vacuum equation (8-37). Most of the terms are now symmetric in the variables except for the last two terms in the volume integral of Eq. (8-45). To establish the symmetry of this part requires another page of boring algebra. One may then prove by using the equilibrium equation 7p = j x B and a lot of vector manipulations that (|.7p) V-rj - (rj-vp> ?•£ * 4' (jrjx^ - £xR) * V.(Bj>r,X|) . (8-48) Hence, #7 + P " T [ ^'Z> £ P (?xB)'(nxp)l dT - - 7 J H V'P + <*>*>• W +v '| H'VP+^^).(^XR)] dip -123. - jh'* *•$**da • (8_49) where the latter integral vanishes by virtue of n*B = 0. Combining the results of the expressions (8-45)-(8-47) and (8-49) now gives a completely symmetric expression in £ and n# Q and R, ö an& R: <£, P_1 F(|)> + + •-if [™<*-JG>V£ «•* i '-ja(j|-vp) + + P J '^(^P) \ (VxB).^»^ + £XR)j dT -H« «"I «-ÏVCP * J B2)]}da - IJj.Jg dv - <£. P~ |(n)> , q-e.d. (8-50) E. HAMILTON'S PRINCIPLE From the non-linear expressions (5-40), (5-41), and (5-45) for the total energy H, H = J (ipv2 + _P_ + I B2) dT , (8-51) p^v 2 Y-l 2 one might derive the total energy of the perturbations. This would be a second order quantity in £. To that end the perturbed quantities B and p should be developed to second order in £. For that purpose the Lagrangian representation given in Eqs. (7-42)- (7-47) would be quite adequate because the variables B, p, and p are exactly integrated, so that only a Taylor series expansion in terms of £ of these expressions would be needed. Inserting these expressions into Eq. (8-51) would lead to the result that the zeroth order just gives a constant that can be subtracted, whereas the first order vanishes by virtue of the equilibrium equations (8-11)-(8-14). To get the second order expression for H a lot of additional algebra would be required. .124. However, we may obtain the second order expression for the energy by simpler means. To that end, we employ Eq. (5-59) for the conservation of the total energy of the plasma- vacuum system: dH dw dK ,„ ^„^ — • + = n0 . S?^ dt dt dt (ft—K° 3Z; These espressions will now be used to second order, so that W, K, and H will be quadratic forms in £ and £. This fact will not be indicated by further indices. From the expressie» (8-4 3) for the kinetic energy K we have where we have used the self-adjointness property of P. Inte grating the expression (8-53) leads to the required result: Here, the integration is carried out over the plasma volume only. The intuitive meaning of Eq. (8-54) is clear: The raise in the potential energy due to the perturbation is just the work done against the force F to displace the plasma by an amount £ (where the factor 1/2 appears as a result of the fact that the full force is only obtained when the displacement reaches its final amplitude). Although the expression (8-4 3) for the kinetic energy K and (8-54) for the potential energy W are quite attractive for analytical purposes, it is a little strange that the vacuum variable Q does not appear explicitly in them. One always has to remember the additional information that £ carries with it a continuation .125. Q into a vacuum that satisfies the equation (8-24) and the boundary conditions (8-25), (8-29) , and (8-31) . In particular, the last boundary condition is a complicated one which we would like to dispose of. One may transform the expression for the potential energy W into one that explicitly exhibits its dependence on the vacuum variable Ö and also remove the complicated bound ary condition (8-31) by identifying £ and n, and Q and £ in the syninetric form (8-50) from which the self-adjointness of the operator £ was proved. This gives: where W? V 2 + 7 + + P U1 " \ I [>P( *P <£**P> *£ f (Vxfl).(i£xQ)]dT , (8-56) g = 7 x <£xB) , ws[ej- ij(n*S)2 S'tv(p + 7 B2)Ï d0 » (8-57) «v«l =iJ> d*v- <8-58> This shows that the work done against the force F actually leads to an increase of the potential energy Wp of the plasma proper, the potential energy W of the plasma-vacuum surface, and the potential energy Wv of the vacuum, [it should be no ticed that the distinction between the potential energy of the plasma proper and the potential energy of the surface is rather arbitrary because one could extract different surface contribu tions from the plasma energy. What is not arbitrary is the dis tinction between Wp + wS on one side and Wv on the other] . .126. We may now state the linearized version of Hamilton's principle: The evolution of the perturbation ï^(r,t) j r € f dxP; Q(£, t) jre j dtv ] is such that ii 6 Jdc L<$. £. V^« S> - ° . <8"59> t, where L = K - W = n*7 x (SxB) = n*Q at the plasma-vacuum interface, (8-61) n*Q =0 at the wall. (8-62) Thus, we have absorbed the complicated condition (8-31) in the form of the Lagrangian. Carrying out the minimization of the expression (8-59) would lead to the following Euler equation: 32< p F <£) = p— for re(dT , (8-63) 3t2 where F(|) = v(|-vp • ypv.|) + (7xB) x p. + (7xjg) x B , 7x0-0, 7-Q « 0 for r e fdtV , (8-64) - Yp7«£ + B«(Q • £'7B) = 6»(Q + E*7B) for r c f do. (8-65) In addition, the boundary conditions (8-61) and (8-62) should be satisfied. Of course, these equations are just restatements of the equations (8-21), (8-24), and (8-31), respectively. They are duplicated here for the purpose of comparing the integral and the differential forms of the problem. Clearly, the varia- T .127. tional formulation of Eqs. (8-59)-(8-62) is fully equivalent to the differential equation formulation of the Eqs. (8-63)- (8-65). REFERENCES 1. I.B. Bernstein, E.A. Frieman, M.D. Kruskal, and R.M. Kulsrud, Proc. Roy. Soc. A244 (1958) 17. "An energy principle for hydromagnetic stability problems". 2. K. Hain, R. Lust, and A. Schltiter, 2. Naturforsch. 12_ (1957) 833. 3. B.B. Kadomtsev, "Hydromagnetic stability of a plasma" in Reviews of Plasma Physics, vol II, ed. M.A. Leontovich (Consultants Bureau, New York, 1966) p. 153. .128. IX. SPECTRAL THEORY A. MATHEMATICAL PRELIMINARIES The spectral problem of linearized ideal MHD arises from a study of the equation of motion (8-63) when one con siders normal mode solutions with an exponential time-depen dence exp C-iuit) , so that Here, we have eliminated the exponential time-dependence so that £ = £(r) from now on, unless stated otherwise. The spec trum of the operator p £ consists of the collection of eigen values tjj2. An important property of the eigenvalues follows from the seif-adjointness of the operator p F. Let E be the eigen- - co2: function belonging to the eigenvalue n p"1 F(£ ) = -a.2 £ . Then, the complex conjugate equation reads: °u a.n -o t»n n ^n Multiplying the first equation with £* and the second with £ 'Sn ^n subtracting and integrating over fdtP yields: 2 2 0 = (u-(u*n - u-n *) <£,£>" n ^n , so that co2 = u,2* . (9-2) n n In other words: the eigenvalues io2 are real, so that the spec- n c trum of the operator p-1 F considered in the complex u-plane is confined to the real and imaginary axes. For eigenvalues on the imaginary axis (id2 < o) the exponential time-dependence of d\e normal modes becomes exp (iiot) = exp (yt) , where y = iw > 0. (One may always choose this to be true because the eigenvalues occur .129. in pairs). These solutions grow exponentially in time and are therefore called exponential instabilities. The conditions under which such instabilities occur are analyzed in Sec. IX D. The spectral problem associated with partial differ ential equations like Eq. (9-1) is just a generalization of the methods used in linear algebra of finite dimensional vector spaces. There, the eigenvalue problem arises in the studies of finite N xN matrices L.. (i, j = 1, 2, — N): N Yl Lij Xj = Xxi' °r k'* = X*' (9"3) j = l The eigenvalues are found from the condition Det (L.. - A6..) =0 , (9-4) where substitution back into Eq. (9-3) yields the aigenvectors x . Another formulation of the same problem is obtained by the construction of quadratic forms; N N / N * • C C x* L x YL *\ . (9-5) i = l j-1 1 1J J ' i=j x Finally, a third formulation arises from the consideration of the inhomogeneous equation where a is a known vector. Here, the Fredholm alternative states I\I ——.————————_____— that either the homogeneous equation (9-3) has a solution,so that \ coincides with one of the eigenvalues, or the inhomogeneous equa tion (9-6) has a solution, so that X is outside the spectrum of eigenvalues of the matrix L. In the generalization of these ideas to infinite dimen sional Hilbert space associated with the operator p~ F two kinds of mathematical problems are encountered. The first one is the .130 fact that the operator p £ is a differential operator and, therefore, an unbounded operator. Here, bounded operators U have the property that |!Ux|l £ M l|x|| (9-7) for all x c H.S., where M is some constant. Differential opera tors do not have this property. Operating on a bounded (square integrable) sequence of functions in Hubert space they may produce a sequence that is unbounded and, therefore, leads out side Hilbert space. [Example: d/dx tranforms the sequence sin mrx into the diverging sequence nn sin nirx] . One usually tries to avoid this problem by transforming it to one that involves com pletely continuous or compact operators. These operators have the opposite property. They transform a sequence of bounded functions into one that converges in the mean. For these operators the theory of infinite dimensional Hilbert space is completely analogous to that of the finite dimensional vector spaces of linear algebra. In the case of differential operators this implies that one tries to invert the operator so that one has to study an integral opera tor involving Green's functions which frequently do have the re quired property of compactness. Another, more serious problem is the existence of a third class of operators where the above trick does not work, viz. that of bounded operators that are not compact. [Example: the operator of multiplication by x] . Those operators may give rise to a continuous spectrum, which is roughly speaking the collection of "improper eigenvalues" for which the eigenvalue equation is solved, but not by functions that belong to Hilbert space. In the mathematical discription one then has the option .131. of either sticking to the notion of Hilbert space by intro ducing the concept of approximate spectrum where sequences are considered that do not converge (the approach of von Neumann in his treatment of spectral theory for quantum me chanics) , or one may consider wider classes of elements than those that belong to Hilbert space, viz. distributions (the approach of Dirac, perfected by L. Schwartz) . One could say that the diverging sequences of functions, that are considered in the first approach, converge to elements outside Hubert space which are the distributions considered in the second approach. Having all these words available now we may give the following generalization of the ideas of linear algebra ex pressed in the equations (9-3)-(9-6). The spectrum of a linear operator L is obtained from the study of the inhomogeneous equa tion (L - A) x = a , (9-8) where a is a given element in Hilbert space and we look for solutions x = (L - X)"1 a . (9-9) For complex A the following possibilities arise: (1) (L - A) does not exist because (L - A ) x = 0 has a solution: A belongs to the point or discrete spectrum of L, (2) (L - A) exists but is unbounded: A belongs to the continuous spectrum of L, (3) (L - A) exists and is bounded: A belongs to the resolvent set of L. (see: B. Friedman, Principles and Techniques of Applied Mathemat ics, p. 125) . Thus, a complex value of A either belongs to the spectrum or .132. to the resolvent set, so that one may say that the spectrum of L consists of the collection of A*s where the so-called resolvent operator R. E (L - A) misbehaves. B. RAYLEIGH-RITZ VARIATIONAL PRINCIPLE In Sec. VIII E we have derived two formulations of the linearized equations of ideal MHDf viz. a differential and an integral formulation. Correspondingly, the spectral problem also takes two forms, viz. a normal mode analysis by means of the differential equation (9-1) [which should be supplemented with the Eqs. (8-64) and (8-65) for the vacuum region if such a region is present] and a variational principle based on the quadratic forms defined in Sees. VIII D and E. These two formu lations constitute the infinite dimensional generalizations of the Eqs. (9-3) and (9-5) for the finite-dimensional vector spaces. The variational formulation may be stated as follows: Eigenfunctions of the operator p P are obtained for functions £ for which the functional 1 W 2r, ^-p" zty> W "Mil — (9-10) becomes stationary. Here, besides the potential energy W[E] defined in Eq. (8-53) and the kinetic energy K[|] defined in Eq. (8-4 3) another quadratic form has been introduced that is quite useful in the present context, viz. the virlal: I W s<*' ^ ' I UN2 - (9-1D .133. The stationary values of the functional a2[£] are the dis crete eigenvalues u2. The collection of all these eigenvalues constitutes the discrete spectrum. The Rayleigh-Ritz principle is extremely useful for the approximation of eigenvalues by means of finite-dimension al subspaces of Hilbert space. Here, one selects a suitable class of square-integrable functions £ n-. , n?/ •• nw} which are used as trial functions in the expression (9-10) . The linear combination of these functions that minimizes the func tional Ü2 then constitutes an approximation for the lowest eigenvalue u2 , where the minimum value of ü2 is always larger than the actual eigenvalue u2. An approximation to the N lowest eigenvalues may be obtained as follows. Choose the n's to be orthonormal: < n , n > = 6 . (9-12) ^i ^n mn Since these functions are supposed to be known one may compute the matrix elements w = < n , p~x p(n )> . (9-13) Writing N JO = Z an n , (9-U) •v» n»~*l n ^n one then obtains the following approximation: N N 2 2 a* W a tl-1 Hence, the problem again boils down to the finite dimensional one of Eq. (9-5), viz. the simultaneous diagonalization of the two quadratic forms wfn] and 1^1. Since the n's have been .134. chosen to be orthonormal the diagonalization of I [r^j has been obtained already. Consequently, the eigenvalues u2 and the eigenfunctions ri of the natrix W are approximations to the lowest N eigenvalues u2 and eigenf unctions £ of the operator p £. Of course, the accuracy of the approximation depends on the choice of the basis functions n. The equivalence of the variational problem (9-10) with the eigenvalue problem (9-1) is easily proved. Let u2 = a2 [^] be a stationary value of the functional (9-10) , so that 2 < 1 > 1 5ft2 - ^.P"5 ^(|)Ï-Ï <£.£1-Ï > + 2 (always using the Hermitian property of p £ !). Then, — « • -" / so that<5|, (P_1 F(|) + cu2,|)> = 0 . But, since 5£ is arbitrary, this is equivalent to p"1 |(jp - - w2| , q.e.d. For plasma-vacuum systems it is again useful to extend the variational principle so as to explicitly exhibit the depen dence on the vacuum variable Q: ft Tl» Q] • » (9-16) where Wp, WS, and WV are defined in Eqs. (8-56)-(8-58) , and | and Ö should satisfy the boundary conditions (8-61) and (8-62). This formulation ir equivalent to the normal mode equation (3-1) supple mented with the equations (8-64) and (8-65) for the vacuum vari able Q. Again, notice that the boundary condition (8-65) has to .135. be explicitly considered in the noma! node analysis, whereas it is automatically taken care of in the variational formula tion of Eq. (9-16). C INITIAL VALUE PROBLEM According to the exposition given in Sec. IX A in connection with the Eqs. (9-8) and (9-9) the third, and most general, approach to the spectrum of the linear operator p £ is to consider the inhomogeneous problem 1 2 (p" F + a, ) | = X , (9-17) where X is a known vector. Our task is then to construct the resolvent operator (p £ + ID2) and to study its behavior for complex values of w2. In order to see how this is connected with physics, consider the initial value problem. We define the Laplace transform of £(rr-t) in the complex w-plane: l<£Ju> £ I k{Z'>l) elUt dt • (9_18) 0 so that Eq. (3-63) takes the form: 2 c ~ 1 ,, , * x I * lWt - -. (^ = J 77 e dt = " u" £ (TT " lu^> e iui t i t-*"» O (9-19) Writing u = o + iv we then get for v > 0 (P_I £ + "2> I <**«> - *<*$£<*) " 4i<£> =- *' ««"ZO) where the vector X of Eq. (9-17) thus turns out to be the func- tion of initial displacement £.(r) and initial velocity f,.(r) defined in the RHS of Eq. (9-20) . In order to find the response £(r;t) to a certain initial perturbation :<, one then first has to invert Eq. (9-20) to find the Laplace transformed variable % in terms of X: .136. |(r;u>) = (p-1 F + U2)'1 Xtr;u>) , (9-21) and next perform the inverse Laplace transformation; iv +" * t \ 1 f -2/ v -1Wt , l v -» O c It is clear that for the inverse si»;p «< eonvtr- 'K"*. ,/,/// r-^ / / / / >">•/ f//////© transform more is needed than just v > 0 because £(r;w)may not exist W%J» for certain values of u or it may be singular. According to the discussion above, it is precisely when w belongs to the spectrum of the operator Q~ £ that we may expect trouble with Eq. (9-21). If u is a point eigenvalue the operator (p F + u>2) simply does not exist, whereas for improper eigenvalues (i.e. u in the continuum) the operator (p F + u2) is unbounded. Before we know where to place the integration contour C for the inverse Laplace transform we, therefore, have to know the spectrum. Here, we get substantial help from the fact that p F is Her- mitian so that the eigenvalues (including the improper ones) have to be real {Eq. (9-2)), so that the spectrum is confined to the real and imaginary axes of the complex u-plane. In fact, we would be completely lost if the operator £ were not self- adjoint because a general theory of non-Hermitian operators does not exist. Further help comes from a conjecture by H. Grad that the continuous spectrum of ideal MHD should be confined to posi tive w2, i.e. occur only on the stable side. Although this has not .137. been proved it is sufficiently plausible, also because all continua found so far (including those for the case of a general axi- synwetric toroidal system) confim it, so that we may delay worries about this point to future investigation. (Hov/ever, see footnote on p. 146.) We then conclude that the integration contour must be placed above the largest point eigenvalue v of p F, max '^ i.e. v > v , the most unstable eigenvalue: o max ^ IV © u c > V max r +-* -6" i > In other words, the class of permissible funccions t(r;t) is restricted to functions of exponential order exp (v t) where o v is larger than the largest growth rate of the system. In the picture above we have schematically indicated our knowledge so far of the spectrum of ideal MHD, which will be analyzed in more detail in a later chapter. One finds two pairs of continua on the real axis, whereas point eigenvalues can occur almost every where on the real a-axis (including inside the continua) and -v also at a limited part max £ v £ v of the imaginary \j-axis. Of course,it is extremely difficult to obtain the ex plicit time-dependence of £(r;t) in situations of practical interest so that one usually restricts the study to time-asynptotic solutions. It is clear that for t ->- <*> one wishes to deform the integration contour in the inverse Laplace transform to the lower half of the w-plane in order to exploit the smallness of the ex ponential factor exp (-iut) in Eq. (9-22). For this advantage one must pay in the form of a study of the analytic continuation of ' about the occurring poles (pointeigenvalues) and branch points of \ (associated with the continuous spectrum). The branch point .138. singularities lead to different branches of the complex func tion |(r;«) so that the inverse Laplace transform contour may be moved to another Riemann sheet where it could pick up poles. Such poles could not correspond to point eigenvalues since these are confined to the real and imaginary axes of the principal branch of %, but they may be significant physically. We will continue the analysis of the initial value problem in Sec. X C where we consider the explicit example of aninhomogeneous slab. D- STABILITY. THE ENERGY PRINCIPLE Let us consider a pair of discrete normal modes 2 2 exp(-ioint)andexp(iui t) belonging to the same eigenvalue u = u> . If we neglect all other modes, e.g. by preferentially exciting this one pair of modes, the solution of the initial value prob lem given in Eq. (9-20) may be easily completed. Since 0_1 VV - - < k , (9-23) the resolvent operator would be simply given by (p"1 F + u.2)"1 = (o>2 - u2)'1 . (9-24) <\» n Hence, the discrete eigenvalue w2 gives rise to two poles u = + u n r — n which, by virtue of Eq. (9-2), are situated on either the real axis or the imaginary axis of the complex w-plane. Clearly, for u2 = u2 the resolvent operator does not exist, but everywhere else in the complex w-plane it is now defined (of course, when we ignore the rest of the spectrum). We may now integrate Eq. (9-22) by deforming the contour around the two poles u2 = + u . Shifting the straight part of the contour to v = - - so that exp (-iwt) vanishes exponentially fast the only contribution that remains will be the two residues picked up at the poles. By .139. means of Cauchy's integral formula we then find: C(u*ui)(t«)n — un ) lm t -lu t K%i{s>+ 4i<^i* n + r^i<«) -4i<€>]e n 2i u (9-25) (where one should notice that the contour C deformed around a pole has just the opposite sense of a Cauchy contour). Writing us = a + iv , we either have v = 0 cr a = 0. If \> =0 the poles n n n n n n ^ are situated on the real axis so that -1 S(r;t) = £. (r) cos a t + £.(r)a sin a t , (9-26) which is a stable undaniped oscillation excited by an initial displacement £.(r) or an initial velocity £.(r) or by a combi- nation of both. If a =0 the poles are situated on the imarri- n ^ nary axis and we have -1 £(r;t) = L. (r) cosh v t + £.(r)v sinh v t (9-27) Since both cosh (v t) and sinh (\> t) eventually grow as exp (v t) this is called an exponential instability. Again, it may be excited by initial displacements or velocities. Q e • -t .140. % f Q> -v. The important feature here is that true normal modes, i.e. discrete eigenvalues, are either oscillatory or exponen tially growing, but never damped. This is the real simplifying feature of ideal, i.e. conservative, MHD which is expressed by the self-adjointness of the force-operator. As a consequence, stability studies may be simplified considerably as compared to the analysis needed in dissipative systems. If the equilib rium is described by a set of parameters a.., — a (basically expressing the pressure and magnetic field distribution), in general marginal states would be defined by the condition lm u (ctlt--cxH) = 0 , (9-28) where the components of ^ are the wave numbers labelling the different modes. However, in ideal MHD this condition may be replaced by the much simpler one «2 (ar— aN) - 0 , (9-29) i.e. transfer of stability to instability takes place via the origin w = 0 of the complex w-plane. Stability may then be studied by means of a marginal mode analysis which seeks to es tablish the locus in parameter space a,, — <*M where the mar ginal equation of motion |(£) - 0 (9-30) is satisfied. The variational counterpart of this equation, viz. the marginal form of Rayleigh's principle (9-10) is known under the name energy principle. This principle states that an equi- .141. librium is stable if (sufficient) and only if (necessary) w[|] > 0 (9-31) for all displacements £(r) that are bound in norm and satisfy the boundary conditions. Here, £ is again meant in the extended sense of carrying a continuation $ into the vacuum if a vacuum region is present. The advantage of the energy principle over the mar ginal stability analysis by means of Eq. (9-30) is that one may use trial functions in Eq. (9-31) to test for stability. Thus, if one has a good physical intuition one may be able to design a trial function that shows right away that the system is unstable by picking up the prcper part of the driving energy of the instability. Also, one may.formalize this approach by testing with a finite class of trial functions that may be con sidered as a subspace of the Hilbert space of the system. One may also replace the normalization | |Ê[ | =1, where the norm is defined in Eq. (8-42), by another normalization condition, e.g. by normalizing only one of the components of £ if that would simplify the analysis. The only limitation in the choice of the normalization of the trial functions is that the original norm ||l|| should remain finite (see Sec. X D). Of course, in the process of dropping the proper normalization of the Hilbert space one loses the possibility of calculating the actual growth rates of the instabilities. Intuitively clear as the energy principle may seem, its proof is actually not quite straightforward. If the operator F would only allow for discrete eigenvalues satisfying p"' W " * < Sn ' (9-32) .142. it would be reasonable to assume that the set \£ \ constitutes a complete basis for the Hilbert space. In that case the eigen- functions ^£n could be chosen to be orthonormal: <£ ,£ > - 6 . (9-33) *m ^n ran An arbitrary t could then be expanded in eigenfunctions: n=l so that n=l Hence, if we could find a £ for which W < 0 at least one eigen value u2 < 0 should exist. Such an eigenvalue would correspond with an exponential instability. This proof was given in the original paper by Bernstein e.a. before it was known that ideal MHD systems as a rule have a continuous spectrum that usually also extends to the origin w2 = 0. The latter fact implies that: the simplicity of the marginal stability is spoiled and a lot more care is needed to establish necessity of the energy prin ciple. It is likely that a correct proof may be given which properly incorporates the continuous spectrum, but it is certain that such a proof will be quite involved. A correct proof of both the necessity and the sufficiency of the energy principle without invoking the assumption of a com plete basis of discrete eigenvalues, but also avoiding an anal ysis of the continuous spectrum, has been given by Laval, e.a.. (Nuclear Fusion 5_ (1965) 156). The proof is based on energy conservation, H«K+W,H«0, (9-36) and the virial equation Ï " <4-4>" " 2 The proof of sufficiency is actually quite simple: Sufficiency: If w["^] > 0 for all £ one cannot find a motion n (t) such that the kinetic energy K[n(t)] grows without bound. Proof. W=H-K>OrH finite. Hence, unbounded growth for K would violate energy conservation. [Notice that we exclude so-called linearly growing instabilities where £ ^ t and I ^ t2]. The proof of necessity is more involved: Necessity: If a function n exists such that W[nJ < 0, the system will exhibit an unbounded motion l(t). Proof. (1) w[rt] < 0. Choose as initial data £<0) = tj, £(0) = 0. From Eq. (9-36) H(t) =H{0) =W(0) + K(0) =W[jQ] < 0, so that I(t) = 2K - 2W = 4K - 2H >. - 2H(t) > 0. Hence, I grows without limit as t +• and I grows at least like - Ht2. As a result £ grows at least linearly in t. This simplified version of the proof is due to Kruskal. Laval e.a. gave a sharper version by also estimating the growth rate: (2) W[T,] < 0. Define X = - W[n.]/l[nJ > 0. (9-38) We prove that there exists a ^(t) growing at least as exp(/xt). * Choose as initial data £(0) - rj, j^(0) = /AQ (i.e., in contrast to case (1), we excite the motion with the proper relationship between £ and £ for a normal mode that grows exponentially). Consequently, H(t) - H(0) - K[^(0)j + W[£(0)] * Xl[r,] + W[nJ - 0. From Eq, (9-37) we then have: • 14s . I(t) - 2K - 2W = 4K - 2H « 4K(t) > 0 , (S-39) whereas Sdiwartz inequality gives i2 I(t)/I(t) i I(t)/i(t) , tn[l(t)/I(0)] <_ 4n[l(t)/I(0)] = in[l(t)/2-/Xl(0)] , I(t)/I(0) < I(t)/2VTl(0), I(t)/I(t) 2 2VI, in[l(t)/I(0)] >. 2V^t , I(t) > 1(0) exp (2 fit) . Consequently, ,£ grows at least as exp (/xt), q.e.d. One may also prove the following theorem. Theorem. If the ratio - w[i]/l'|] has a smallest upper bound * >_ *[.$] = " w[£]/l[;/j for all | , then Kt) cannot grow faster than exp (V7t). Proof. I(t) - 2K(t) - 2W(t) - 2H(t) - 4W(t) < 2H(t) 4 4A I(t) . Hence, Kt) - 4A I(t) <_ 2H(c) - 2H(0) . Consequently, Kt) grows at most like exp {7^T.-:< and Kt) cannot grow faster than '.v.< ;VAt) , q.e.d. We have given all these proofs here because they naturally lead to an extension of the stability concept to be introduced in the next section. .145. E. o-STABILITY For thermonuclear confinement of plasma the stabi lity concept used above may be relaxed. One is not really inter ested in whether the plapma is stable, but one is interested in whether or not one can confine plasma long enough to ob tain fusion. For example, if the worst instability would behave like: I A a -*- t where a is the radial dimension of the plasma vessel and T is the characteristic confinement time needed for fusion, one would call this configuration stable for all practical purposes. One could also take T to be another time-scale, e.g. the time-scale for which one believes that the ideal MHD model is a valid description, or one may choose r to be the time-scale for the decay of the external currents used for the magnetic confinement of the plasma. For all these purposes one may allow perturbations that grow at most like exp {at), where a = 1/T. We shall call equilibria o-stable if they do not manifest growth faster than exp (ct) . Except for practical purposes the concept of o-stabi- lity is also useful for analytical purposes. We will show in a later chapter that the continuous spectrum always reaches the origin u> = 0 and frequently it carries with it infinitely many .146. point eigenvalues that accumulate at the edge of the continuum. Hence, the marginal point u2 = 0 is a highly singular point in the spectrum so that the supposed simplicity of a marginal stability analysis (as compared to calculating actual growth rates) often turns out to be illu sory. In contrast, a o-stability analysis avoids these diffi culties by staying on the unstable side of the spectrum -*t M—x-mwj——^M )i ), i which consists of point eigen values only. (At least that is Grad's conjecture to which no exceptions have been found yet)*. This is of particular importance for numerical stability studies where one wishes to avoid the occurrence of singularities as much as possible. Since we are dealing now with point eigenvalues only, we may define an equilibrium to be o-stable if no point eigen values u2 < -a2 exist, and o-unstable if such eigenvalues do exist. A o-marginal stability analysis then seeks to find the ff-stability boundary in parameter space replacing Eq.(9-29) by "l ((V a2'"" "N5 • * °2' (9-42) This problem may be studied by means of the o-marginal equation of motion: FCT(|) £ $<£> - po2| » 0 , (9-43) * Here, one should actually exclude perturbations characterized by infinitely large mode numbers since these may lead to dense sets of unstable point eigenvalues in certain cases. The closure of these sets then formally con tains a continuous spectrum. See G.O. Spies, Phys. Fluids j£ (1976) 427. .147. where the force F available for driving a a-instability is reduced by the amount pazg with respect to the force F for driving an instability under the usual definition (i.e. a O-instability). The variational form of this problem is the modified energy principle which states that an equilibrium is o-stable if and only if W°U1 H WUI + °2 l[£] > 0 . (9-44) for all square-integrable displacements £ that satisfy the boundary conditions. Clearly, the amount of negative potential energy available for driving a c-instability is reduced by a2I[£] as compared to that available for driving an ordinary instability. Comparing the Eqs. (9-43) and (9-44) with the normal mode equations (9-1) and (9-10) one observes that their formal structure is the same. One might even wonder whether the whole concept of c-stability does not boil down to a normal-mode analysis. This is not the case, the important difference being that in a normal-mode analysis the eigenvalue u has to be determined, whereas in a o-stability analysis a is simply a pre-fixed parameter. Hence, the problem is of the same nature as a stability analysis by means of the energy principle, al though the equations are more complicated (i.e., they have more terms). The latter complication (which is unimportant for numerical applications anyway) is more than offset by the absence of the singularities associated with the continuum at u2 = 0. The proof of the modified energy principle can be given in complete analogy with that of the ordinary energy principle given in the previous section. Sufficiency is proved by writing W°[£] - H - (K - a21) > 0 for all £, H finite, so that for a o-instability, where K-o2I grows without bound, energy conservation would be violated. The necessity of the modified energy principle implies that a o-unstable motion £(t) can be found if one knows a function n such that W [n] < 0. This is an immediate consequence of the proof of necessity of the ordinary energy principle. Like in Eq. (9-38) define V = - W°[rj]/l[r,] = - W[rj]/l[nJ - o2 = A - a2 > 0 . Then, * - " WLQ]/I[J&] - M + o2 > a2 , so that |(t) grows at least as exp (/Xt) = exp (V^+ah,) and the equilibrium is, therefore o-unstable. REFERENCES 1. B. Friedman, Principles and Techniques of Applied Mathe matics (Wiley & Sons, New York, 1956). 2. I.B. Bernstein, E.A. Frieman, M.D. Kruskal, and R.M. Kulsrud, Proc. Roy. Soc. A224 (19 58) 1; "An energy principle for hydronac,netic stability problems". 3. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, 1961). 4. G. Laval, C. Mercier, and R.M. Pellat, Nuclear Fusion 5 (1965) 156; "Necessity of the energy principles for magnetostatic stability". 5. J.P. Goedbloed and P.H. Sakanaka, Phys. Fluids 17_ (1974) 908;"New approach to magnetohydrodynamic stability". 6. G.O. Spies, Elements of nagnetohydrodynamic stability theory (Courant Institute of Mathematical Sciences, New York, MF-86, 1976). .149. X. WAVES IN PLANE SLAB GEOMETRY A. WAVES IN INFINITE HOMOGENEOUS PLASMAS As a preliminary to the study of waves and instabilities in inhomogeneous systems, let us first study the normal ipodes of an infinite homogeneous plasma. Taking B in the z-direction the equilibrium state is specified as follows: I = (0, 0, B) , (10-1) B, p, p constant . Since Vp = 0 and V x B = 0 the normal mode equation (9-1) with the force-operator defined as in Eq. (8-21) becomes: P"1 £(£) = (YP/P)7V.| + p-1 (Vxo) x £ where 1/2 c = (YP/P)1/2 and b = B/p , in agreement with Eqs. (4-20) and (4-21). From now on we will consistently write o2 instead of u2 to indicate that the eigenvalues v;e are looking for are real. [Of course, one should n'.t confuse this notation with that of o-stability of the previous section where transition from o-stability to o-instability takes place at the prefixed value of the eigenvalue parameter u 2 = -a2. Also, notice the unfortunate difference of the sign'.]. .150. Since all equilibrium quantities are constant we may write £(r) as a Fourier integral (or a Fourier series if one considers a finite box) of plane wave solutions: £(r) = (2ir)"3/2 \\\ £<£) exp (ik-r) d'k (10-3) We may then study the modes £(k)expi(k.r - at) separately by making the substitution V •+• ik in Eq. (10-2). This gives: p'1 F(|) = - c2kk.| - b x^k x fkx(bx|)]> • -&2 + c2)feVÏ " 15'M'*? - #•? " &"?> - " °2I (10-4) Defining L.£ = p-1F(£), this may be written as a problem in j^ *\. >\J a. linear alaebra: --H^ki (10-5) where 2 fe - - In components: -k2(b2+c2) - k2b2 -k k (b2+c2) X Z x y -k k (b2+c2) -k2(b2+c2> - k2b2 -k k c2 x y y z y z - „2 -k k c2 -k k c2 l X Z y z T./ \ (10-6) .151. Solutions are obtained by setting the determinant of the LHS zero. This gives: 2 2 2 2 2 2 2 2 2 2 2 (o - k b ) [o" - K K2 = k2 + k2 + k2 , k2 = k2 x y z f/ z Consequently, we obtain three solutions a2 = o\ E k2b2 , 4kjb2c2 a2 K2(b2+c2 1 ±1 (10-8) -°l,i 4 > /• K2(b2+C2)2. which are the frequencies of the Alfvén-waves and the fast (+) and slow ( - ) magnetoacoustic waves» respectively. Comparing the expressions (10-8) with the expressions (4-23) for the characteristic speeds of the same waves, obtained from the non-linear equations, it is clear that there is a close correspondence between the characteristic speed u and the angular frequency (o and between the normal n to a characteristic and the wavevector k. This correspondence is given by the transformation o/K , £ - k/K (10-9) .152. We may now also express the characteristic coordinate $ in terms of u and k: * = K (k'Z ~ ct> • (10-10) Apart from a constant factor, this is just the phase of the plane wave exp 1 (k.r-ot) . It should be noticed that we have lost Lhe entropy disturbances in the linear theory. This is due to the fact that these disturbances are not expressible in terms of the displacement £. [They simply move with the fluid]. We may also compute the corresponding eigenvectors £ and the associated magnetic field perturbation Q and the pressure perturbation TT by substituting the expressions (10-8) into Eq. (10-6) and using the relations (10-11) ~. = - YPV'I = - ipc k-| . Without loss of generality the k-vector may be chosen to lie in the x-2 plane, so that k = 0. We ther. obtain the following expressions for the Alfvén eigenmodes; \ = *z " ° ' S ' ° ' \ * *Z = 0 ' S " " * VPK//b ^y > (10-12) TT = 0 , .153. in perfect agreement with the Eqs. (4-37). Likewise, we find for the slow and fast magnetoacoustic eigennodes in agreement with Eq. (4-38) : = *y ° > ê, =asf (^/kx)Ix , % - ° • kx *x + kz *. • ° . K - - i ^kx^x' (10-13) 7 = - ioc kx.(l • asf kj/kj)?x , where 2 2 2 as, f = 1 - K b /as, f. , so that as —< 0 •, aff —> 0 . (10-14) The latter factor has a different sign for the slow and the fast modes, so that the spatial orientation of £ with respect to B and 5 is different for the two modes. The Alfvén waves are transverse waves, both as regards £ and as regards Q, whereas the pressure is unaffected by them. The nagnetoacoustic waves do affect the pressure and they have both transverse and longitudinal components. Putting all three waves together gives the interesting effect that £ , E , and £- form an orthogonal triad in space. This is very satisfactory as it indicates that arbitrary displacements can be decomposed in the three different eigenmodes. The ideal MHD waves display a strong anlsotropy as is clear from a consideration of the phase velocity of the plane -154. waves: v ok/K' (10-15) If we call 6 the angle between k and B we find that v . = a/K = f(0) , but it does not depend on K. Such waves are called non-dispersive as a plane wave packet constructed from them may propagate without distortion. The group velocity of such a packet gives the flow of energy: 'vgy r = 3o/3k^ (10-16) For the Alfvén waves we get the interesting result that the energy flow is always along the magnetic field: v . = b (10-17) For the magnetoa^oustic waves things are more complicated. This is best illustrated by plotting o2 as a function of k., while // keeping k fixed, and vice versa: A lfv«M JU. liatA While the group velocity in the parallel direction 3c/3k > 0 for all three kinds of waves, the group velocity in the perpendicu lar direction 3o/3kA displays a characteristic difference for the three waves: 3o/3kL > 0 for the fast waves , 3a/3kv = 0 for the Alfvén waves , (10-18) Zo/dkL < 0 for the slow waves . Hence, the energy propagation of a slow wave packet in the perpendicular direction is antiparallel to the propagation of the wave packet itself'. The two diagrams of the reciprocal normal surface (p. 44) and the ray surface (p. 45) derived in Sec. IV B may now be interpreted in terms of the concepts of phase and group velocity. The reciprocal normal surface is simply a plot of the tip of the vector v. for different angles of propagation 6, whereas the ray surface gives the similar plot for the vector y . Clearly, the slow waves behave the least classical of all t-gr three waves. This fact will return in the discussion of the spectrum of inhomogeneous media. For the discussion of inhomogeneous media it is useful to return to a description where k has three components k , k , k , where k is in the direction of the magnetic field, and k Z Z X and k are in the perpendicular directions. In the next section the x-axis will be chosen as the direction of inhomogeneity. It .156. is therefore instructive to also plot a1 as a function of K while keeping k,, and k fixed: Vr • *» ?>«« A ^,fc$^vx>o • x * *, ; l i If we consider a slab of finite extension In the x-direction by putting conducting platec at x = ±a, the wavenumber k is quantized: k = n:r/2a. Here, n is the number of nodes of the eigenfunction £ in the x-direction. Such a number to label the point eigenvalues still makes sense in an inhomogeneous medium, when the equilibrium quantities vary in the x-direction. The essential features of the three discrete spectra of point eigenvalues labelled by n are: (1) The point eigenvalue a* = k* b2 of the Alfvén point spectrum is infinitely degenerate. (2) The slow wave point eigenvalues have an accumulation point for k •+ » (or n + ») : o2 = k2b2cz/(b2+c2) . (10-19) (Notice the notation with the subscript s indicating the slow modes themselves and S the accumulation point). .157. (3) For large wavenumbers k •* °° the fast wave point eigenvalues behave as cr2 % k2(b2 + c2) - o2, = » , (10-20) f •*• x F so that °° is an accumulation point of the fast wave point spectrum. These three facts turn out to be the basic ones for the discussion of the inhomoge.ieous case, where the infinite degeneracy of the Alfvin point eigenvalues is lifted by the appearance of a continuum of improper Alfvén modes instead and the accumulation point of the slow point spectrum is spread out in a continuum of improper slow modes. The two values of a2 denoted by a2 and a2 , where the slow and the fast modes emerge in the diagram above, have been the source of some controversies in the development of the spectral theory for one-dimensional inhomogeneous configurations. Their values are given by: 4k2b2c2 kI(k! 2 1 ±\/l - (10-21) •i.n 4 • ° > k2(b2+c2)2 J where k2 = k2 + k* . [Notice the notation K for the total wavenumber and k for the wavenumber in the plane perpendicular to the x-direction, which will become the direction of inhomogeneity in the next section. There, K loses its meaning, but k can still be defin ed . ] The role of these two special values for the discussion .158. of the spectra will be discussed in Sees. X B and XI E. The following sequence of inequalities is useful: 2 2 0 1 ° S 1 ° S 1 o\ 1 o\ < o^<_ o\ <_ a* = . . (10-22) We now have obtained a clear separation of the three discrete subspectra for homogeneous media. Our next task is to trace these spectra when inhomogeneity is added to the system. B. THE CONTINUOUS SPECTRUM FOR INHOMOGENEOUS MEDIA Consider a slab of plasma, infinite in the y- and z-directions, and contained between two ideally conducting plates at x = x, and x = x~. The equilibrium is assumed to vary in the x-direction: B = (0, B (x), B (x)), p = p(x), p = p(x), u y z. (10-23) where the pressure balance equation A €i (8-10) leads to the only restriction that * has to be made in the possible choices of the functions B (x), B (x), and p(x): y z (p + \ B2)' 0 . (10-24) Here and in the following primes denote differentiation with respect to x. Again, we study modes satisfying the normal .159. mode equation (9-1): e"1 *(£> = " u2| * (10-25) Here, £(£) niay bs decomposed in Fourier components for the two homogeneous directions: W * 7^£fx;Vkz) exp (iV + ikzz) dky dkz -<10-26> We will now study separate Fourier components x ex Ic + ik z For £k k ^ ' P^ vy z ^ * convenience in notation we will y' z y drop all the decorating symbols and indicate the amplitude of a Fourier component simply as |(x). The most convenient form of Eq. (10-25) is obtained after projecting all the occurring vectors on the three unit vectors ev, e^ and e^: e = e , Si " Sxe,v/B * (0' Bz' " V/B ' (10-27) ey/ = B/B - (0, By, Bz)/B . [it is important not to confuse projections liVe this one with orthogonal coordinate systems, as is sometimes done in the literature. The point is that, in general, one cannot find coordinates a, S, y such that £v = Va/jVaj, .160. ex= VB/ J VB | , e t/ = ?Y/|VY|. The existence of such coordinates would imply B = hVy, where h is some scalar field. Hence, ^ = V x B = Vh x 7-y = — Th x B, so that j/7 = 0. In plane slab geometry the latter condition implies that B should be unidirectional. Only for such trivial fis Ids can one find an orthogonal coordinate system based on the field lines]. In this projection the part of the gradient operator that acts on the perturbation £ (x) can be written as 7 " is, a7 + *UL6 + •€*if » (10"28) where g - g(x) = - ie^-V - (kyBz - kzBy)/B , f = f(x) = - ie «V = (k B • k B )/B , ^f y y z z and one should remember that the directions of the unit vectors e^and e,, vary with x when JjJ is not unidirectional: where We also project ^ on the three unit vectors: .161. « E *v£ - ^ . n = ie,-E = i (B C - B g )/B , (10-30) •>»•»• 't z y yz 5 = ie •£ = i(B C + B £ )/B , <\,// % yv vy z z where the factors i have been inserted in such a way that one has to deal only with real functions £, r\, and Z, in the final analysis. By the use of this projection the normal roode equation (10-25) may be written as: J-(b2+c2) d _f2b2 «L (b2 2) dx dx dx° &AU\ /'\ 2 2)_ 2 2 _ 2 2 :i^'>5 -02 (b +c f b fgc = -a (10-31) -fc2 -fgc: -f2c2 dx where p has been chosen constant for convenience, so that it can be pulled under the derivative d/dx. [otherwise, we would have to write p_1d/dx(yp + B2)d/dx, etc.]. Notice that the operator p"1 £ now depends on x through b2(x), c2(x), g(x), and f(x) . Apart from this important difference the expression (10-31) is completely analogous to that for infinite homogeneous plasmas. The dispersion equation (10-6) may be recovered by writing d/dx = ik . Let us now reduce the matrix equation (10-31) to a .162. single second order differential equation in £ by eliminating n and s by means of the second and third component, which are algebraic in n and S: 2 2 2 2 2 n = g[(b ^c )a -f2b c ]^, D (10-32) _ fc2(q2-f2b2) > *" s » D where D = D(x;a2) = a1» - k2(b2+c2)o2 + k2f2b2c2. (10-33) Substituting these expressions into the first component gives us the required second order differential equation: 2 2 2 [| £']' + (a - f b K » 0 t (10-34) where N - N(x;02) = (a2 - f2b2) [(b2+c2)a2 - f2b2c2] . (10-35) This equation has to be solved subject to the boundary conditions S(xj) - ?(x2) - 0 . (10-36) It is clear that the factor N/D in front of the highest derivative of the differential equation will play an important .163. role in the analysis. We may write this factor in terms of the four a's that were already introduced for the case of a homogeneous plasma: [a2-a2(x)] [a2-a2U)] £ = (b2+C2) (10-37) [a2-a2(x)j [o2-c£(x)] where o2(x) E f2b2, b2c2 a2(x) H f' b2+c2 (10-38) 2 2 2 2 2 b c (x) H i-k (b +c ) 1 ± 4 i- '1,11 k2 (b2+c2)2 Notice that all four o's depend on x through f2 (x) , b2(x) , and c2(x). When a problem has been reduced to a non-singular ordinary second order differential equation it may be considered to have been solved, because one can always obtain the explicit answers numerically to any degree of accuracy one would be interested in. The essential problem left is, therefore, a proper treatment of the singularities occurring in Eq. (10-34). This leads to a consideration of the continuous spectra. Let us assume that the equilibrium quantities are 2 2 chosen such that the functions a (x) and aG(x) defined in Eq. (10-38) are well-separated and monotonically increasing profiles: 0 a u o* i<> o;' --^(«V- W »»--*tVfc* ^i • In addition, we assume that the sets {a*(x)} and {a2 (x)} do not overlap with (a2(x)} and {c2(x)}. [This assumption is not necessary as we shall see. Here it is only made in order not to have to worry about the significance of these frequencies at this point in the analysis]. We prove that the collection of 2 2 frequencies o c{o^(x) |xj <_ x <_ x2> and a fe{a|(x) [xj <_ x <_ x2> constitutes the continuous spectrum, i.e. the set of improper _i eigenvalues of the operator p F. We will concentrate on one continuum, e.g. the Alfvén continuum, so that a2€ {crjf (x) }. The monotonically increasing profile a2 = 0j[(x) may be Inverted to give a monotonically increasing profile x = x (a*): i i *»ci i «#o *. X#n <*•-«•;<*> XA . X*«0 .165. At a singular point x = x (a2) when o? = af(x) the function A A N(x,-o2) vanishes. We may now expand around this singularity: N(x;o2) - N(x;x (o2)) £ a [x - x (o2)] = as , (10-39) where 2 s = x - *A(<* ) , and a is a constant factor depending on the equilibrium functions at s =0. Close to the singularity the differential equation (10-34) then reduces to (s O' ~ B s £ » 0 , (10-40) where 3 is another constant factor depending on the equilibrium functions at s = 0. Frc:?. this equation the behavior of £ close to the singularity may be found by series expansion. Substituting the leading order term s11 ir.to Eq. (10-40) gives rise to the 2 indicial equation n = 0r so that the indices are equal; n = n2 - 0. As is well known from the theory of ordinary second order differential equations this implies that one of the two independent solutions contains a logarithmic function: s 2 r Ct u(s?a ) ("small" solution) \ (10-41) 3 2 2 L C2 u(s;o ) en |s| + v(s;o ) ("large" solution) , . i<»6 . where u(s;o ) and v(s;,7 ) are analytic functions of s: u, v ^ a + bs + ... The interval (Xj,x~) contains only one singular point for a fixed value of a2 so that the general solution may be written as 5 - [A,u + B,(u tn is[ + v)| H(s) + TAOU + B (u In [s| + v)] H(-s) , (10-42) where H(S) is the Heaviside function, and we still have to determine the values of A,, 3^, Aj and B2. Of course, for a non-singular second order differential equation the solution should be continuous so that A, = A_ and B = B_. We now prove that for the singularity under consideration only the large solution has to be continuous whereas the small one may jump: A1 / A2 , Bj^ = B2. To that end, write Eq. (10-34) as (PC1)' -QC - 0 , (10-43) where P(x;a2) = N(x;a2)/D(x;a2) * s , Q(x;a2) = - (a2 - f2s2) % s . Substitution of a small solution £ = uH(s) leads to the following expressions, successively: C' - u'H(s) + u6(s), PC' - Pu»H(s) + PuS(s) = Pu»H(s) , (pr')« = (Pu')»H(s) + Pu'5(s) = (Pu')'H(s) , (PC')' - QS - [CPu')1 - Qu] H(s) = 0 , by virtue of the fact that u(s) is a solution of Eq. (.10-43) . Here, we have wade use of such properties as H*(s) = 6(s) and s5 (s) = 0. Consequently, AjU H(s) is a solution of Eq. (10-43) but, likewise, A2u H(-s) is also a solution, where Aj and A2 are totally unrelated. Performing a similar analysis for the large solution it turns out that the term u £n|s|H(s) produces a 5-function contribution that does not vanish so that B, = B, has to be satisfied. The general solution to Eq. (10-43) may now be written as i = Au + BuH(x - xA) + C[u injx - x | + v] . (10-44) Due to the fact that we have now three (rather than the usual two) constants available the two boundary conditions (10-36) may always be satisfied for c2€{af(x)} so that there is a singular point on the internal (x^, X2). The improper eigenfunctions for an Alfvén continuum mode may then be written as: 2 2 x-xA(o ) v^o ) 2 2 } 2 2 -:A(x;3 ) = C(o ) {in £-r-Z\ - -V-rr "(x;o ) H(x.(o ) - x) + X1~XA u-^cr) A >:-x.(o2) v,(a2) „ 1 2 2 + { £n ^-r-rr ~ -W > "(^;° ) H(x-x (a )) + v(x;o')j , 2 x2~x (o ) u2^° ' (10-45) 2 2 where u1(a ) = u(x.; a ) , etc. .168. The factor C(g2) may be fixed by "normalizing" the eigenfunctions according to 2 2 2 2 Likewise, one obtains improper eiger.functions ^(x; a2) for a2£o*{x) . Therefore, we have "solutions" satisfying the boundary conditions for any *- x o2e{o* (x) |x <_ x <_ x } and a2€{oi(x) |x f x _< x }, q.e.d. Although this establishes the existence of two continuous spectra, the most characteristic part of the eigenfunctions is not yet obtained. Actually, if we restrict the analysis to the radial part of the eigenfunction we could not even prove that we have "improper" eigenfunctions because the singularities £n|s| and H(s) are square integrable. The dominant non-square integrable part of the eigenfunction resides in the tangential components n and z,, which follow from the application of Eq. (10-32). Since n * we find for the dominant non-square integrable part of the eigenfunctions: .169. C n + X(o2) Ó(X-X (J2)) , c % o , A $ ° > A $ 5*-V. 2 \ A4 A "v* x-x A (o^) ? + x( 2) 5 x x (aZ)) cs £* ° » %s ^£ ° » ^s ï ^-^ ^TTZ2 ° ( - s * x-xs(o ) (10-46) where X(a2) ie a function involving the boundary data of u and v. Therefore, the continuum 1A modes are characterized by k k a non-square integrable tangential component IX -»x x^ X,lG') perpendicular to the X. magnetic field for the Alfvén modes and a non-square integrable parallel tangential component for the slow modes. This shows the extreme anisotropy of ideal MHD waves as regards motion inside and across magnetic surfaces. This property remains true for cylindrical and toroidal geometries. As regards the zeros of the function D(x;a2): It can be shown that these singularities are only apparent, i.e. they do not lead to non-square integrable solutions. The proof will be given for the similar cylindrical, problem in Sec. XI C. In conclusion: The spectrum of an inhomogeneous plasma slab schematically looks like: n * n—vw- -AAA»—x »H*t -*.*•' i O » s\»« -*»»t .170. The sets {G*} and (cri.} are not part of the spectrum. They only act as a kind of separators of the three subspectra. The separation of these subspectra only obtains if the inhomogeneity is not too strrag. C. DAMPING OF *LFVËN WAVES We wish to complete the solution of the initial value problem given in Eq. (9-22), i.e. , -lint , M>^ = 2^ J i{Va ) e doj (10-47) by explicitly constructing the resolvent operator (p F +U2) for a special case. For the plane inhomogeneous slab model of Sec. X B the inhomogeneous equation relating ^(^ ; u) to the initial data £ = iu^. (^) - £. (£) is obtained by just adding the vector X to the RHS of Eq. (10-31): />2+c2>£-f2b2+"2 £*&+* ^ \/M IA 2 - gO> +c2)-i- - g2(b2+c2) - f2b2 + o>2 - fgc2 = Y dx - fgc' - f2C2 +Ü)2 \ dx I W (10-48) Here, the initial data are also projected as indicated in Eq. (10-30): £ = Xe - iY^ - iZ^ // It is interesting to notice the difference in the study of the initial value problem by means of the Laplace .171. transform when w is complex and the study of the continuous spectrum of the previous section when u was taken to be real. In a way these two approaches are complementary and correspond to the two methods for the study of the continuous spectrum mentioned in Sec. IX A. If one stays on the real o-axis the occurrence of singularities forces one to introduce distributions (5-functions) in the theory. In the Laplace transform method, on- the other hand, one stays away from the real axis and in the end one just takes the limit that w approaches the spectrum. In principle the problem is posed by the equations (.10-47) and (10-48) . However, we have seen that the spectrum of the plane inhomogeneous slab consists of the Alfvén and slow continua (a*(x)} and ( For the study of the Alfvén continuum there is no need having a varying direction of B. We will therefore take the field to be unidirectional, so that the functions f and g become constant wavenumbers: .172. f = k , g - kL (10-A9) Next, we consider a low 8 plasma (0 = 2p/B2), so that (10-50) This assumption separates the slow and the Alfvën modes: a2 % k2 c2 % a2 << a2 = k2 b2. In order to separate off the influence of the fast modes we concentrate our study on nearly perpendicular propagation: k// < • ki X k . (10-51) 2 o so that aj^kj, b^^jjfek^ i t H *•«' to leading order '• I = Z = 0, so that Fq. (10-48) simplifies to \ fb2f- k* b2 • „2 k b' \ dx dx - kj, b/ k* b' - k' b^ + dj V dx •/ n / Only transverse motion need? to be studied. In this equation we have kept terms of unequal order in k//f and k± because large terms cancel upon eliminat-ion of n. After elimination we keep terms of comparable order only resulting in the following .173. equations: - JL [(u2 _ 02) |.]t + (u2 . 02) I = x + 1. Y, t (10-53) k ~ = _ I?.._JL n = " rk 5'-—-r , (10-54) «.2 2 k2bK' where a* = a* (x) = k2„ b2(x) . Defining 2 2 2 2 P(x;U ) = - (a, - a )/k , Q(".;u2) = " (a)2 - oj) , (10-55) R(x; u) = X + Y'/k Eq. (10-5 3) may be written as (P V)1 - Q X - R • (10-56) The solution of an equation of this type is obtained by means of the Green's function G(x,x';w2) which satisfies the equation r 3Gf x x' ) 1 a P(x) r^ - Q(x) G(x,x') = 6(x-x') , (10-57) 3x LrvJW He J and the boundary conditions 2 2 G(x = x ,x';u ) = G(x=x2,x' ;u) ) - 0 . (10-58) Integrating the differential equation (10-57) we find that the Green's function itself is continuous but the first derivative .174. displays a jump at x = x' : (10-59) Ï ti 3X U X X The solution of EG. (10-57) then reads: f(x;iij) = G(x,xf;oi2) R(x';u)dx' , (10-60) which gives the inversion in terms of an integral operator, xii£ i ru iOi.iG^ en 3 cu3 equation (10-57) allows for a GU.«';0') unique solution for the Green's function when the homogeneous equation does not have a non-triv: _-l >< ^4^ solution (Fredholm alternative:. Proper and improper solutions :>t •for ff'-sO.'- the homogeneous equation occur *• X for values of .J2 inside the sp which is confined to the real o2-axis, so that we certainly have a unique Green's function »-x » W) for complex v».luas ot w on the Laplace contour. The procedure is then to construct the Green's function for ^cr.clc:: vl'.:^: of J2 ':h3re evister.re is guaranteed and to defcrzi the contour in such a way that the spectrun is approached. The symmetric expression ror G^x,x!;ur) is found in .175. 2 terms of the solutions $(X;M2) and ^(x;w ) of the homogeneous equation satisfying the left and right boundary conditions, respectively: (P •')' - P $ = 0 , In terms of these functions one finds for the Green's function: r(x,x»;u2) 2 G(xrx';w ) = » (10-62) A(ui2) where 1 2 z 2 rU.x ^ ) E *(x, ;u ) ^(x>;w ) = A Here, we have introduced the notation x< = inf (x,x') , x> E sup (x,x') . The expression inside the square brackets in the definition of A is recognized as the Wronskian. Ey means of Eqz. (10-61) one proves .176. IA = P» ( - so that A y A(x) . For eigenfunctions the solution of the homogeneous equation satisfies both left and right boundary conditions, so that Let us again specify the profile a* = o2(x) to be monotonically increasing on th interval (x.,x2), as in Sec. X B, and co...l.ru't 'che -inverse profile x. = x (a2) . E.g., for a simple linear prof Me che explicit functions would read: 2 2 i <£» w*» 0*tx} f a1 \ tf. i i X *,«') In the previous section we expanded around the singularity 2 2 x « x.(a ) of Eq. (10-63) in terms of the variable s = x-xA(a ) Here, w2 is complex so that the corresponding singularity of .177. Eq. (10-61) occurs in the complex z-plane for z = z.du2) where 2 2 z (w ) is the analytic continuation of xA(o ) . For the linear profile the explicit expression for z. (w2) would be (10-64) A o o A introducing a complex variable 5 replacing s, 2 2 5 - ;(X;(Ü ) = x - ZA(w ) , (1C-65) © V-rt the solutions^ and if> of the equations (10-61) may be expressed as a linear -i—•x combination of the functions u(0 (10-66) u(s)£n C + v(c) , where u(C) and v(0 are the analytic continuations of the functions u(s) and v(s) introduced in Eq. (10-41), which may be written as a power series in ?: u,v ^ a + b; + .... Hence, f É(X,-W2) V.(U2) • (C) =«tt l u(c;u»z) + v(?;u.,.,2z)' , 2 2 L 5L(u) ) u^a, ) 2 2-2» n (10-67) S(x;u ) v2(u ) IP CO in u(c;u*.,..2 ) + v(c;u>*..,2- ) . 2 2 0» > u2(ui ) Substituting these expressions into Eq. (10-62) provides us with the formal solution of the Green's function: .178. 2 G(x,x';W ) = 2 2 2 2 2 - u/-a (x,) v (u ) i ^ r U) -0 (x ) V_(oü ) " 2 2 in -i- _ |U(X< ;o )+v(x ;« )[. Scn- u(x :CJ2)+V(X :K2)1 u^w-) J 2 u -pA 2 «•2(<,> ) J 2 2 v (O V (ÜJ2) A2 2 (10-68) ï.n 2 2 a)—a Al ^(w ) U2(OJ ) Here, the logarithmic express ion in terms of <; has been converted into the nort transparent form in terms of a2 - af(x) by means of the relation ,2 _ 2 ? 2 (w2) (10-69) x - A = - (• ° Au>)/° / . which is, strictly speaking, only valid for the linear profile. However, for an arbitrary monotonically increasing profile Eq. (10-68) is also valid if 'he functions u and v are redefined such that the expression for the basic solutions are written as u(u2 - c2(x)) , u(u2 - a2 (x)) £n(w 2 - a2(x)) + v(x;u2) (10-70) instead of Eq. (10-66). Clearly, for the derivation of the expression (10-68) of the Green's function no other property has been uüd then the fact that crjl(x) is a monotonie function and that the slow continuum is far away so that we are dealing with only one singularity at a time. For the completion of th<* initial value problem we .179. now need to study the behavior of the Green's function when u approaches the spectrum. K«=> have already seen that the zeros of the denominator A(di2) represent the discrete spectrum. The continuous spectrum arises as a result of the multivaluedness of the logarithmic terms appearing in both T (x,x' ; In order to make a logarithmic function ?-n z single- valued one may cut the z-plane along any curve starting at the branch point z = 0 and extending to m. Let us choose the . K\ © negative real axis as a branch cut. Along '* this branch cut one may write: ui ->- x Him in z = in | z l ± iri y-0± ' (on the principal Riemann sheet n = 0) , where + iri is the value immediately above the branch cut and- iri immediately below. If one wishes to deform a contour across a branch cut one moves to another Riemann sheet of the logarithmic function. These sheets are labelled by n and the logarithmic function increases by an amount 2fri every time one encircles the branch point and moves to the next Riemann sheet. Therefore, the general expression for the logarithmic function when ;ppreaching the real axis .180. may be written: £.int Zn z Jtn|x| ± iri H(-x) + 2mri (10-71) y-0- where the jump of the Heaviside function occurs at the branch point. Accordingly/ for complex values of w = a + iv one may write for a logarithmic expression of the type £n [(w2 - a 2)/(w2 p •i>] when approaching the real axis: 2_„2 2 2 Ö = o -o: £im. £n JLn ± sg(a)iTr[H(a-aa) - H(o-a ) V*0- 2 „2 2 2 a a + H(a+a ) H(a+o ) + 2mri . a e (10-72) Hence, assuming a| > a2: (nTS) Hi ^yUw*- -0 t w4 * mo (1iB Ui W- X : iCdrteVipo'n-t Here, we have indicated how one moves from the principal sheet to the n • 1 and n = -1 sheets when crossing the branch cuts. On the basis of the expression (10-70) we find that .181. the function r(x,x';tü ) has branch points a* = o^{x<) , = a?(xj, , and a* _, whereas the function A(uz) only has alA> A > alAl A2 branch points at a?, and cr* . One may connect these branch poirts as follows: •*wwvK- -*.ff rcx.x-, w'-'i nv TU ff nw • •• c AL^) For the Green's function G = r/A, these branch points should be joined, one may d : this by choosing the branch cuts for A dif ferently, so that the Lap"ace contour C may be deformed to a contour C* as follows (see Re£.*): C £*., X\ to"-") This clearly shows that the contribution of the continuous spectrum is due to the jump in the logarithmic function along the branch cuts. Let us now calculate the typical contributions of the spectral cuts to the solution» of the initial value problen. Take .182. special initial data: £. (x) ^ 0, r, (x) = n(x) = rj.(x) = 0. The solution of the initial value problem can then be written from the Eqs. (10-47), (10-54), and (10-60) as: ** i{iit 2 E(x;t) - ^- I du -^— e' \ dx' r(x,x';(i) ) E.(x'), n(xit) = - £ -^ £(x;t) . (10-73) From Eq. (10-72) one then finds as the typical contribution from a jump of the logarithmic function at some frequency a : E(t) % \ io e~IOt H(o-o )do i a. C c r -iat f -iat = 1 È H(a-a )da + \ a 6 (o-a ) da . > t a J t a Asymptotically, the first integral may be neglected because the rapidly oscillating integrand kills this contribution for large t. Thus, we are left with terras like: £(t) * o,, e^V/t , 10 1 n(t) -v - i(aao;/k) e" » , (10-74) Consequently, the continuous spectrum gives rise to oscillatory normal components that are damped like t , but the tangential components execute undamped oscillations where each point oscillates with .183. its own local Alfvên frequency. As time goes en the factor exp(-ia t) gives rise to an ever more fluctuating spatial structure of the motion, finally resulting in completely uncoordinated oscillations. In contrast to the situation just described another kind of motion exists that does display coherent oscillations. To exhibit this let us start with a profile oj- (x) that has a step discontinuity at some value of x, say in the middle of the slab at x = x »-X o - "2 ^xl + x2^ * Tne singularities of 2 2 the continuous spectrum a < a < 0z are ncw a^ concentrated r Al — — A2 in the point x = x . This gives rise to a special mode which is called a surface mode. It may be found from the homogeneous equation corresponding to Kq. (10-53): °\) 5 2 _ ol) (10-75) k2 l •]• - <• where cr*(x) = o* H(x -x) + a* H(x-x ). On the left and right A Al O AZ O intervals x^ <_ x < x and x < x _< x, this equation reduces to ?" - k2 5 - 0 , having the solutions exp{kx) and exp(-kx), when a2 J al. and t2 ¥ a^2' respectively. The solution £ = sinh[k(x-x )] satisfying the left-hand boundary condition may be combined with the .184. solution S2 = sinh [k(x2-x)] satisfying the right-hand boundary condition to form a cusp-shaped perturbation which is an eigenfunction of the system. That this is so may be seen by applying the proper boundary condition to join £^ to £2' This condition is found from Eq. (10-75) by integrating across the jump: 2 2 -{.»->,) {{-0. öA 2_ ) a = o , or = 0 l^-'Vh- (10-76) This condition is fulfilled for o2 = o2 4(a*+o* ), which is 2' Al A2' the eigenfrequency of the cusped surface wave. Let us now remove the degeneracy of the step and introduce a genuine continuum by smoothing out the discontinuity, This we do by replacing the step by a linearly increasing profile % between x = - -_- a and x = ^ a, r,v Al ff» where we have fixed x = 0. For simplicity, we also take x^ •*• -» and x„ -*• +<*>. The spectrum of the system » % -a 0 a then changes as follows: X dUtcrdc * X X —*• *r -* ö*x -<*«. "t> ff*. Notice that for the stepped and the continuous profile there are also infinitely many discrete A]fven modes with eigen- frequencies a - ± o. and a = ± crA_. These are localized on the left and the right homogeneous intervals, respectively. That this is so may be seen from Eq. (10-75) by pulling out the factor a2- a1 which is constant on the homogeneous A intervals: 2 2 (02 . Hence, for a2 = al, on the left Al homogeneous interval £ may be A -*-% chosen arbitrarily. Each choice of A, this function is a proper Alfvén 2 2 eigenfunction. Likewise, for o =o 2 on the right interval. Here we wish to concentrate however on the influence of the inhomogeneity. In particular, we want to see what happened to the surface wave by the introduction of the linearly increasing profile. Does the appearance of a continuous spectrum imply that all of a sudden the coherent oscillations of the surface wave have disappeared to make place for the kind of chaotic response expressed by Eq. .186. (10-74)? This is hard to believe. We already noticed that the discrete spectrum comes about from the poles of the Green's function, i.e. the zeros of the dispersion function Ma2). Let us, therefore, study the expression A(w2) for the present case. To that end, we need the explicit solutions <(i and ty to the homogeneous equations (10-61) on the three intervals (-°°,-a), (-a,a) , and (a,00). The virtue of the choice of a linear profile on (-a,a) is that the homogeneous equation for this interval may be written as 2 d d* u> -02(x) t _£ _ k25(f - o , C 5 - 2a - t (10-78) so that we obtain modified Bessel functions of complex argument as solutions: Vk° = 1 + i when Y fc«577 is Euler's constant. Consequently, the following solutions are obtained: e f C2 D2 e (-"»-•) • - { Al I0(kc) • Bx Ko(kc) * - | A2 I0(kC) + B2 Ko(k!;) (-a,a) 11 ^a'"^ ' .187. The constants A, -, B. _, c. _, and D are fixed by equating functions and first derivatives at the boundaries of the intervals. For the calculation of A(u»2) we actually only need to compute A _ and B , because A (ID2) is independent of x so that we may choose to evaluate it in the inhomogeneous layer. The solutions ka k5 ) + • - k5l «" [[K0< i ^(k^j^CkO -[i^Ck^-i^k^)]^*?)} , -ka |[KO(U2) -Kl(kc2)]lo(kO ~[VkC2) • I^)]*^)}. « - - k?2 e (10-80) where •i)2,.2a(ü)2.02i^)/(022_02i) Inserting these solutions into the dispersion function we find I k K t " "l^^V^l* " i< 51)][ 0(l C2) -K^k^)] - [^(k^) + ^(k^)] [l0(k?2) + I,(k?2)] } , (10-81) where C is a constant that is unimportant for the present purpose. To obtain Eq. (10-81) we have used the property z[l0(z)K,(z) + Ij (z)K0(z)] = 1. The dispersion equation .188. A(w2) - O (10-82) gives right away the two solutions z, - 0 and x, - 0 corresponding to the two discrete eigenvalues a2 = al, and o2 = cr^2* Let us now investigate whether some more solutions exist, hopefully corresponding to the surface wave solution of the step function model. To that end we study a situation where the continuous profile model is close to the step function model, i.e. a is considered to be small. Since the other intervals are infinite the only scale to compare a with is the perpendicular wavelength k . Hence, we assume k a << 1 and expand Eq. (10-82) in orders of ka. By means of the expansions (10-79) of the Bessel functions we find to leading order: in **2 + l ( l + i T \ r f> • ° • or u2 " aA2 °A2 ' aAl An 0. (10-83) w2-°Al 2ka Lu[2 ^" °A1 "2 " 'A2 Let us now study this expression in the neighborhood of the real axis so that v << a. We then have from Eq. (10-72) for o in the range of the continua: 2 ff 2 "2 * °A2 ° " ?- a?, - a A2 + sg(o)sg(v)Tti • 2niri + 2ivo A2 • Al 2 7 ^ o2 -a2 U - °A1 Al .189. where the last term m:ny be dropped again as it is small compared to the other imaginary contributions. This gives: °2 - -h •L - °ii °2 - 5<»ii • "IP In 2 ° - «ii ka (.» - „*>(.» - ,»2) • ,, ,,-•, •• •vo<0^'°"H°2' W*(°2'°"* „ + sg(o)sg(v)iri + 2mri + l = 0 2 2 z 2 2 ka (a - o kl) (o - oj^) (10-84) The real and imaginary parts of this dispersion equation give the roots we are looking for: 0=±°oE±\/è(ail + °A2>' a2 - a2 v = v = - i «ka [sg(v)sg(o) + 2n] — — . (10-85) o o o This seems to give a satisfactory generalization of the surface mode as it reduces to u = a for a=0. If a / 0 a "mode" is obtained which has a small imaginary part to the "eigenfrequency". We have put quotation marks here because we have proved already that in ideal MHD normal modes cannot have complex eigenvalues. On the other hand/ we have obtained a genuine pole of the Green's function, which certainly will influence the response to the initial data. For n = 0 the expression for v in Eq. (10-85) gives a contradiction, so that no solutions are found on the principal Riemann sheet, corresponding to the fact that .190. complex eigenvalues do not exist in ideal MHD. For n = 1 and n = -1, however, we find two poles with ka(0 )/o (10-86) i * i2 " Al We may now deform the Laplace contour across the branch cuts so that the contributions of the complex poles on the neighboring Riemann sheets are picked up: Gcss*sv^>s*ib ^ftwv/v/^/v^ ii i' • » .i ¥l.» n«-i »1=0 «1» I «!» Ignoring the contributions of the branch cuts corresponding to the continuous spectrum (and also the contribution of the branch points which are simultaneously poles corresponding to the degenerate Alfvën modes), we find asymptotically for large t for the contributions of these poles: cff\ n -1-1,1 lM - "lot „ if. 10) -lut w we o/\, to»e o, o e -hjt -io t Likewise, n(t) "v e e Hence, we have found a "mode" that is exponentially damped. Since the pole is not on the principal branch of the Green's function tiiere is no contradiction with the general proof that complex eigenvalues do not occur for self-adjoint linear operators. On the other hand, it is clear that the present "mode'* of the plasma is of physical interest as it represents a coherent oscillation of the inhomogeneous system. In contrast to the chaotic response produced by the branch cuts of the continuous spectrum this "mode" constitutes a very orderly motion. The plasma as a whole oscillates with a definite frequency that cannot be distinguished from a true eigenmode during times x << v . "Modes" like these occur in many branches of physics and, accordingly, they have received many different names, like quasi-modes, collective modes, virtual eigenmodes, resonances, etc. The damping is completely analogous to the well-known phenomenon of Landau damping in the Vlasov description of plasmas. Landau damping is due to inhomogeneity of the equilibrium in velocity space. Damping of Alfvén waves is due to inhomogeneity of the equilibrium in ordinary space. D. STABILITY OF PLANE FORCE-FREE FIELDS. A TRAP In the previous sections we considered a plasma slab with a unidirectional magnetic field of variable strength. Let us now turn to the opposite case, a magnetic field of constant magnitude but varying direction. The simplest case to treat .192. is a force-free field: / VxB * aB , (10-87) \ > X where we take a - constant.Again, we will consider a low (* plasma so that the pressure will be neglected. In components, Eq. (10-87) reads: ttB B = oB K ' " y • y z » (10-87)' which can easily be integrated: B • B(0, sin ax, cos ax), B • constant. (10-88) This represents a field with a uniformly varying direction. Let the plasma be confined between two perfectly conducting plates at x = x and x = x . We wish to investigate the stability of this configuration. Again, we decompose £(r) in Fourier components as Indicated in Eq. (10-26) and we study the stability of the separate modes. The stability may be studied by means of the expression (8-56) for the fluid energy, which for the present problem simplifies to .193. W = I 1 (82 + al'i* x 3)dx' (10-89) where we have normalized W with respect to the area in the y-z plane. Following Schmidt (Physics of High Temperature Plasmas, p. 141) we minimize this expression using the vector potential A, A* S=Vx£'£-£x£' (io-90) so that W 3 4- \ [CxA)2-aA*. VxA 1 dx . (10-91) L J <\, -v. \ According to Sec. IX D we may minimize W subject to some convenient normalization, for which we choose: -TT- \ A* . 7 x A dx = constant . (10-92) The proper way to minimize W subject co the.constraint (10-92) is to minimize another quadratic form W, W - i- \ [( V x A)2 - (X • a) A* . V x A] dx, (10-93) where the constraint is absorbed by means of an undetermined Lagrange mul tiplier \. .194. Since V. [A*x(VxA)] = 7 x A* . VxA-A*.VxVxA, (10-9 A) we may integrate the expression for W by parts: (10-95) *V 1 r ^ ? 1 ƒ W = -=- [ A* x (V x A) . n "I + JLA*. [v X V X A - U+a) Vx A1 dx - The boundary term vanishes by virtue of the boundary conditions B.n = 0 and ^.n = 0. Consequently, for arbitrary A* the ru quadratic form W is minimized by solutions of the Euler-Lagrange equation: VxVxA-(X+a)VxA = 0, (10-96) which may be written as another force-free field equation for the perturbations: VxQ=aQ , asA+a. (10-97) Eq. (19-97) is an eigenvalue equation, where a is determined by imposing the boundary condition JJ.Q = 0 at x = X- and x = X2« Inserting such a solution into the expression (10-95) for I. gives W - 0, so that W • V • H 4- A* . V x A dx - (a - a) 4- I A* . 7 x A dx 2 •S-^SL 4" f A* . 7 x V x A dx - -SL^-2- * f (Vx A) dx . (10-98) f ,195. Hence, the system appears to be unstable if the equation (10-97) has a eigenvalue a such that 0 < a < o . (10-99) It should be noticed that we could have dropped the normalization (10-92) altogether. The resulting Euler equation would have been VxQ-oQ = 0. (i0-100) We would then determine the eigenvalue a for which this equation has a solution satisfying the boundary conditions. Inserting this solution into V7 would give W = 0, so that v/e wou2d find the stability boundaries directly in terms of ot. Clearly, such an approach corresponds to a study of the marginal equation of motion £(£) = 0. That this is so may be shown by a similar integration by parts as above: W-4-(A*. (V X V X A - CIV X A)dx - 4" U*. [Bx(VxQ - «Q)ldx , (10-101) so that the Euler-Lagrar.ge equation is B x (V x Q - »Q) « 0, .196. which is just the margin?.], equation of motion for p = 0 (see Eq. (8-63)). This equation is equivalent to V x Q - aQ-Bx , (10-102) where x is an unknown perturbed quantity. Taking the divergence and using V.Q = 0 leads to B . Vx = 0 . The operator B.V is algebraic in this case. It may vanish only at isolated points where x would be a 5-function. This is not a permissible perturbation, however, so that x = ° and we are led again to Eq. (10-100). Let us now continue with the study of the Euler equation (10-97) and find out whether the condition (10-99) can be satisfied for the slab model, in this model the Euler equation may be reduced to an ordinary second order differential equation in the normal component of Q so that we may find an explicit stability criterion. To that end we again exploit the projection (10-28), (10-30) and write 7 " e,, ~— + i-Sei+ ife/y » »W OX ° 'V* «V» (10-103) Using this projection one should again (as in Sec. X B) take care of the fact that the unit vectors £j_ and ej are x-dependent, so that .197. e^ (O, B'/B, - B'/B) - - ct(0, B /B, B /B) - - a e„ , 3x (10-104) -^- e,, = (O, B^/B, BVB) - o (O , B^B, - By/B) = oet. Furthermore, we have from the equilibrium equation (10-87)' f' - ag , g' « - af . (10-105) Exploiting these relation»gives for the projected components of Eq. (10-97): - f R + gS = 3f Q, - f Q - S' - 0, (10-106) gQ + R' = 0 . One easily shows from the Eqs. (10-106) that V.Q = Q' +gR+fS=0, (10-107) which together with the first line of Eq. (10-106) gives the expression for R and S in terms of Q: R y (gQ' *of Q) , , (10-108) S - - p- (f Q' - QgQ) . .198. Inserting these expressions into the last line of Eq. (10-106) yields the required second order differential equation for Q: Q" + ( o2 - k2)Q = 0 . (10-109) The solution which vanishes for x = x. and x = x? reads: Q - sin Va2 - k2 x , (10-110) where Va - k = nir /a , a H x«, - x, . *2 "1 Hence, the instability criterion (10-99) is fulfilled for 2 2 c^t2 » .k2 +_,_ n TT <^ a2 or 2 2 (k/a) + (nir/oa) < 1 • (10-111) This gives an unstable region in the k/a - aa plane as indicated. Moving to the right in the shaded area subsequently n = 1, n = 2, ... become unstable. Marginal nodes (for which a = a) are distinguished by the number of nodes n - 1 of Q on the interval (x., x~) . Notice that in the long wavelength .199. limit k = O every time aa increases with ira, i.e. every time the oia magnetic field has changed its direction r\:t.\. p: i.a/ï unstable by 180°, a mode with one more node becomes unstable. This appears to be a perfectly reasonable result: a long wavelength instability driven by the current which has to surpass a certain critical value given by aa - IT. Let us double-check the result obtained by rederiving it from a formulation in terms of £ rather than Q. To that end, the projections Q, R, and S of the variable Q = Vx(SxB) are written in terms of the projections %, n, and X, of the variable Q = i B f £ , - i B( a 5 - fn ) , (10-112) - i B( V + gn) .200. Inserting these expressions into Eq. (10-89) gives x2 W = ^-B2 f [f2 l2 + (a? - fn)2 + U* +gn)2-aV + 2afCn] d: xl x2 2 2 2 2 2 » -i- B j [f (c + n ) • W + gn) ] dx > 0 . (10-113) xl Hence, the slab is trivially stable'. We may obtain the minimizing perturbations by rearranging terms: x2 W " 4" B2 J E f2(ef2/k2 +C2) • (kn • g£'/k)2] dx, xl so that W is minimized for perturbations that satisfy kn + g C'/k = 0, (10-il4) (f2S'>' - k2 %, = 0. (10-115) One easily checks that the latter equation is equivalent to Eq. (10-109) for a » a. There is no mistake in the algebra'. To see what went wrong let us plot the eigenfunctions € corresponding to the eigenfunctions Q shown above. Writing f = k cos(ax - 8) , we find: l r * _SL . sin(mrx/a) nn ,1M 5 ifB ikB sin(ax-8) (10-116) .201. Hence, if a solution Q exists such that W as given in Eq. (10-98) is negative, aa > IT and £ develops a singularity. f\ s » For every zero that is added in Q at least one zero is added to the function f because f oscillates faster than *- «a or at least as fast as Q. It is clear that these singularities are of such a nature that the norm II? II = /U2 + n2 + S2)pdx = f{$2 + g2C2A2)dx - «. Hence, the trial functions Q used in deriving the stability criterion (10-111) do not correspond to permissible displacements £. However, one may save the nice stability diagrams we obtained for another purpose. Observe that apparently a reservoir of energy is available that could drive instabilities if the associated displacement £ only were realizable. Such is the case if we allow a small amount of resistivity in the system so that the relation Q = iBf£ of ideal MHD has to be replaced by one that has extra terms proportional to the resistivity. These terms limit the amplitude of the displacement £ at the singularity (and, therefore, also the current that is .202. flowing there). As a result, the unstable energy reservoir is tapped so that resistive instabilities develop. Such instabilities are called tearing modes. Icial MHD instabilities of force-free fields may develop in cylindrical geometry. There, the variable Q may oscillate just a little faster than the function f in certain regions of the k/a - aa plane. This has been shown by Voslamber and Callebaut by a careful analysis taking proper care of the singularities. REFERENCES 1. T.H. Stix, The Theory of Plasma Waves (McGraw Hillr New York, 1962). 2. D. voslamber and D.K. Callebaut, Phys. Rev. 128 (1962) 2016. "Stability of force-free magnetic fields". 3. G. Schmidt, Physics of High Temperature Plasmas (Academic Press, New York, 1966) . 4. E.M. Barston, Annals of Physics ^£ (1964) 282. "Electrostatic oscillations in inhomogeneous cold plasmas". 5. Z. Sedlacek, J. Plasma Physics 5 (1971) 239. "Electrostatic oscillations in cold inhomogeneous plasma". 6. J.P. Goedbloed and R.Y. Dagazian,Phys. Rev. A4 (1971) 1554. "Kinks and tearing modes in simple configurations". 7. J. Tataronis and W. Grossmann, Z. Physik 261 (1973) 203. "Decay of MHD waves by phase mixing". 8. L. Chen and A. Hasegawa, Phys. Fluids _17 (1974) 1399. "Plasma heating by spatial resonance of Alfvén wave". 9. J. Tataronis, J. Plasma Phys. ^_3 (1975) 87. "Energy absorption in the continuous spectrum of ideal MHD". .203. 10. J.P. Goedbloed, Phys. Fluids ^8 (1975) 1258 "Spectrum of ideal magneto-hydrodynamics of toroidal systems". 11. W.A. Newcomb, Lecture notes on magnetohydrodynamics (unpublished). 12. A.E.P.M. van Maanen-Abels, Rijnhuizen Report 78-115 (1978). "Solution of the initial value problem and energy re distribution for electron and Alfvén waves in inhomo- geneous plasmas". .204. XI. THE DIFFUSE LINEAR PINCH A. EQUILIBRIUM MODEL For the study of confined plasmas the diffuse linear pinch is one of the most useful models. It is also probably the most widely studied model in plasma stability theory. Since we have obtained a basic understanding of the spectrum of inhomogeneous one-dimensional systems, the analysis of the diffuse linear pinch can now be undertaken with more fruit than was possible 20 years ago when this configuration was first investigated. Also, we will consider this configuration as a first approximation to toroidal systems, where the addition of a second direction of inhomogeneity leads to partial differential equations and, therefore, to tremendous complications in the analysis. For these systems a coherent picture of the spectrum of waves and instabilities is still non-existent. Consider a diffuse plasma in an infinite cylinder of radius a and surrounded by a vacuum field j| enclosed by a perfectly conducting wall at r * b. In the plasma region 0 < r < a the equilibrium is characterized by the profiles p(r), B_(r), J* and B (r).They are restricted to satisfy SP>. one differential equation, viz. 2 [p(r) • ± B (r) ]• • »!(r)/r - 0, (li-D .205. so that we may choose ta.'O profiles arbitrarily. At the plasma surface r = a surface currents produce jumps in the variables p, B-o , and B z which are restricted tc satisfy pressure balance: 12 "> 1 .2 «*» F, * T (BL + B; ) - -r B four of the five parameters p , B , n~* B ,and B may be chosen O ZO 0O ZO uO at will. Having fixed these constants the vacuum field solutions B 'r) and B.(r) on a < r < b are determined: Z o — — Bz(r) - B*> • B9(r) " B8o a/r • In a problem like this it is always important to enumerate the amount of freedom left to choose particular equilibria. To remove some of the freedom in the choice of parameters we normalize all occurring lengths with respect to r = a and all occurring magnetic fields with respect to B z (r=a) = B o . The constants a and B„o should not be considered as free parameters. They just establish the scale of the equilibrium, introducing *-.h<* parameters 2 = 2 p_/B , v a = Bn /B , p a s B, /B , (11-i) <> ° O ' 'o Go O O Öo 7.o the pressure balance equatior (11-2) may be written as 1 + S + p2a2 - (1 • y2a2)B2 /B2 , o o o zo o ' .206. so that on a < r < b 2 2 - 2 1 B2(r) 1 + 6 + y a B (r) o o Q *K * *y ~2 2 1 + y a* 2 2 o 1 • y a (11-5) In the plasma region 0 <_ r < a B,/fcl we may consider the profiles 2 p(r)/B and Bz(r)/B to be arbitrary, except for their values at r/a = 1 which should -*- »•/« be •= 8 and 1, respectively. V&. Fixing these profiles and the two parameters 0 and £ a then completely determines the equilibrium. The profile P/Btl B„(r)/Bo o^ is found by integrating Eq. (11-1) in 41». «7a outward direction starting *>/» from the value B„(r=0)=0. This integration also determines the value of w0a, which is therefore not a free parameter. Consequently, if the scale factors a and B are removed the dimensionality of parameter space is established by: (1) Two arbitrary profiles [Bz(r) - B ]/B and [p(r) - P0]/B^ which vanish at r/a = 1. .207. (2) Two arbitrary constants 8 and y a determining the jumps at r/a = 1. (3) The wall position b/a. Three cases are of special interest: - Sharp boundary models where B (r)/BQ = 1, Be(r) = 0, and p(r) = p on the plasma interval, so that 8 , u a, and b/a completely determine the equilibrium. This model will be used for the study of external kink modes (Sec. XI G). - Diffuse models with the wall at the plasma, so that the 2 choice of the profiles B (r)/B and p(r)/B_ fixes the c zoo equilibrium. This model will be used for the study of internal instabilities of the plasma (Sec. XI H) . - Diffuse models with no jumps at r/a = 1 (i.e., 6Q = 0 and u a - v a), so that the plasma profiles join smoothly onto the vacuum profiles. This is the most realistic choice for the equilibrium. If a cylinder of length 2TTR is considered as a first approximation to a torus of major radius R, it is convenient to replace the parameters u a and y a by the safety factors q and q : q0 - e/yoa , qQ - tllQa , where e = a/R is the inverse aspect ratio of the equivalent torus. .208. B. DERIVATION OF THE HAIN-LÜST EQUATION Our starting point is the equation of motion FU) = p f- , (11-6) n2 where F(0 = - 7ir - B x (Vx Q) - Qx(7 x B) , Q=-Vx(BxO» ir=-7p7. ?-?Jp . Because of the symmetry we may study normal mode solutions of the form ,rt, t | <».•.».«> -(trirt The subscripts m and k will again be dropped in the following analysis. For these separate modes the equation of motion may be reduced to an ordinary second order differential equation in terms of the component £ (r) . This equation was first derived by Hain and Lust in 1958. Like in the analysis of the plasma slab we exploit a projection based on the field lines: s ; H *, fc !U <0,Bz,-B9)/B, %/i- (0,B9,Bz)/B. (11-8) In this projection the gradient of a perturbed scalar quantity may be written as .209. 7 - *v h+ «iig + *•"' (11_9Ï where g = (roBz/r - kBg)/B = G/B, f s (mB9/r + kBz)/B - F/B. The use of the symbols G and F instead of g and f will prove slightly more convenient later on in the analysis. The representation (11-9) for the gradient operator should be used with care. We recall that in the analogous projection for plane slab systems with shear (Egs. (10-28) and (10-103)) the gradient operator could be used also for computing divergencies and curls if one properly accounted for the dependence of the unit vectors on the normal coordinate x (the direction of inhomogeneity). Here, the situation is basically different since the unit vectors e and eQ of the cylindrical coordinate system do not depend on the normal coordinate r but on the ignorable coordinate 6: g-y e = e», 3 13 alT f-e ~ _e,r' Hence' one should add a term e~— g-r- to the represent ation (11-9) of the gradient operator if one wishes to compute divergencies and curls of perturbed quantities. The projection of the displacement vector is denoted as * = Sv ' k = ^r' n i = i(B )/B - ej.' k zSe " Vz ' (11-10) s i(B + * i *r h - e*e VZ)/B. .210. In terms of these variables we have $ = ir - " P'C - YpV- * • (11-11) 7.5 * (rO'/r + gn + f^ • Inserting these expressions into Fq. (11-6) and adding a little algebraic effort, using the equilibrium equation (11-1), leads to the following formulation of the spectral problem: d YP»B2 ' ^B9B; I dr r dr , |r2j , r — g-1-1- 1 -2k '&?1W M dr * r 1 r I 1 c " ' 1 D 2d - ' -»2 2w2*2 2 2 ! "8 (YF+B )-T-• r 2k — . g^(YP+B^)-f B -fgYP •-pu TT; . dr 1 *_ j 2 1-fY -f p T dr "fgYP Y C' (11-12) Apart from the occurrence of a few factors r, this is a symmetric formulation. Notice that the matrix of Eq. (11-12) is analogous to that of Eq, (10-31) for the plane slab except for the occurrence of the three additional terms that have been put inside boxes. These terms are algebraic so that they cannot change the continuous spectrum of the system. Change is meant here in the sense of adding or taking away continuum eigenvalues. This is so because the continuous spectrum is associated with the singularities caused by the zeros of the factor in front of the highest derivative. For the discrete spectrum the additional tarms are quite important as they create the possibility of instabilities driven by the .211. curvature of the magnetic field, which is due to the poloidal field component B . As we have seen in the previous section, instabilities dc not occur for the plane slab. (The proof of Sec. X D may easily be extended to cover arbitrary fields and pressure gradients). Likewise, instabilities do not occur for the straight 6-pinch (BQ = 0). The typical structure of Eq. (11-12), with lower order differential equations for the tangential components n and S, allows us again to reduce the system to a single second order differential equation by expressing the tangential components in terms of the radial variable X = r£: 2 2 2 2 2 2 G [(YP+B )pü> - YpF lrx'+2kB (B pa. - YpF )X T> - 6 r2BD (11-13) 2 2 YPF [ (pu, - F )rX' + 2kBflGX ] ** ' ' • • • !• r2BD where 2 4 2 2 2 2 2 2 2 2 2 D = p a. - (m /r + k ) (y p+ B )pu + (m /r +k )ypF . Substituting these expressions into the first component of Eq. (11-12) gives the Hain-Lust equation: 2 [i>f[}(».'-'V(i- _ü£L„W -YPr , r r D f 2kB6G 0 7 ? l + 1 Y~ ((YP + B^pu - YP* )V*]x- 0 , (11-U) .212. where 2 2 2 2 2 N= (PÜI - F )((Yp + B )po) - YPF ) Comparing this equation with the corresponding equation (10-24) for the plane slab, it is clear that the additional terms caused by the curvature of the poloidal field complicate the equation considerably. For the case that B = 0 (0-pinch) these terms disapDear and we obtain a problem of equal complication as the plane slab. It also follows then directly that the linear 6-pinch is stable. Appropriate boundary conditions for Eq. (11-14) are X(0) - x(a) = 0 Ui-15) if the wall is at the plasma (b = a). If b / a the boundary condition at r = a is a rather complicated expression. It will be derived in Sec. XI D. For the purpose of the analysis we will abbreviate Eq. (11-14) as follows: 2 2 [P(r;u )X'l' - Q(r;tu )x - 0 , (11-16) where P(r;u>2) = S/CrD) , N - N(r;u2) = p2 D = D(r;u2) = p2[u>2 - Oj(r)] [u2 - o^ (r) ] The expressions for o*, o*, and o* ^ are completely analogous to those of Eq. (10-38): 2 *> YP FVP , (11-17) Yp + B' 2 1,2,2,2., D2, 4ypr 1±\ 1 " 7 7 2 2 2 /P. Uj nï y (« /r + k )(YP + B ) (mZ/r 4k")(YP + BV For a fixed radius r the four frequencies are ordered as follows: 2 2 2 2 0 £0 £ (11-18) * % * °I * °A II - ' —i——x - >•- * . h-*- Kotice that the collection of frequencies {cd_(r)} for the whole interval (0,1) stretches out to -infinity because a* (r -+ 0) -+ °°. At this point we may refer to the analysis of Sec. X B and conclude that the diffuse linear pinch has also two continua, the Alfvén continuum (a*(r)) and the slow continuum {cr*(r)}. The proof that the sets (a* IX (r)} do not constitute singularities and, therefore, do not lead to .214. continuous spectra will be given in the next section. The Hain-Lust equation is the basic equation for those spectral studies in ideal MHD which have direct relevance for plasma confinement in realistic geometries. At this level it is instructive to compare the problem with corresponding spectral problems in quantum mechanics. Here, the normal mode 2 equation F(£) = -p<»> £ should be compared with the Schrodinger equation Hip = Ety, which for a particle in a potential field V(r) becomes h2 [ - " A + V(r)l *(r) = E *(r). (11-19) One-dimensional problems are obtained for a potential that is spherically symmetric, like the H-atom where V = V(r) . In that case one vrites the wave function as a superposition of spherical harmonics which may be studied separately: * (r,e,*) - R(r) Y™<6,*), (11-20) in much the same way as we may study the separate Fourier components (11-7) for the case of the diffuse linear pinch. Inserting the expression (11-20) into Eq. (11-19) leads to a second order differential equation for the radial wave function: -li(rR)- [Üijil • -^-(V(r) - E)]R - 0. (11-21) dr r ti This is the equation that should be compared with the .215. Hain-LÜst equation. It is clear that the spectral problem fcr the diffuse linear pinch is a much more complicated problem than the determination of the energy levels of the hydrogen atom and even more complicated than the general problem of scattering of particles in an arbitrary one-dimensional potential field. In that case the only profile that enters is V(r) whereas in the Hain-LÜst equation three profiles p(r), B_(r), and B (r) occur. Also, the Hain-Lust equation reflects the fact that it was derived from a vector equation with three components £, n, and r, in that the eigenvalue u2 is scattered through the coefficients P and Q of Eq. (11-16) in a most complicated way. Eq. (11-21) is a simple differential equation of the Sturm-Liouville type where the linear occurrence of the eigenvalue E in the second term of the equation guarantees monotonicity with the number of nodes of the radial eigenfunction R(r). Eq. (11-14) is not of such a simple type so that the dependence of u2 on the number of nodes of xir) is much more complicated. Nevertheless, guided by the analogy, we will show in Sec. XI E that certain monotonicity properties still exist for the Hain-Lust equation. The vector character of ideal MHD is reflected in the occurrence of three subspectra. The general structure of each of these subspectra is similar to the complete spectrum of quantum mechanical systems. If one fixes the quantum numbers m and I for the H-atom one finds a discrete spectrum .216. of bound states for E < 0 clustering at E = 0, which is the edge of a continuum of free states for E > 0. fr«« * K K |I.>TH>«^M M HD •. c.p. c.p. e.p. CO" CO Slo») Alt"!* -«-..St Likewise, for the diffuse linear pinch the Alfvén and slow subspectra consist of discrete modes that may cluster (there is a condition for this to be so) at the edge of the continua (CTM and {a*}, whereas the fast subspectrum accumulates at ij2 = <*>. C. EQUIVALENT SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS A second order differential equation can also be written as a system of two first order equations. This turns out to be quite illuminating for this case. Rather than just rewriting Eq. (11-14) in terms cf the variables x and x'» we introduce a variable that has physical significance, viz. the 1 2 perturbation II of the total pressure p + -j B : n •n • B . Q. (11-22) [This is the Eulerian pressure nE which is related to the ' 2 2 i Lagrangian pressure nL by HE - IÏL + B0x/r J . .217. Inserting the expressions (11-11) and (11-13) into Eq. (11-22) gives 2 + 2 + 2 2 2 °" " T5*' " ~f T— \(YP B )pu -TPF }]X . (11-23) r r D Notice that all terms with radial derivatives occurring in the Hain-Lüst equation (11-14) appear in the expression for 2 2 Ü, apart from a factor B'/r which is due to the fact that we work here in terms of the Eulerian pressure rather than the Lagrangian pressure. Writing the function Q(r;w2) of Eq. (11-16) as Q(r;cu2) = -V- V/D - (W/D)' , (11-24) where U s (pu.2 - F2)/r - (B2/r2)' , V = -4(k2B2/r3)(B2pu2 - YPF2), 2 2 2 2 W = 2(kB9G/r ) [ we may transform the Hain-Lust equation into the follov.'ir.c Dair of first order differential equations: .218. »\ N « O, (U-25) r ïï' -c where 2 C = W - 2 B.Vrft B2 B = -2 -| P2"4 + 2* 4 n £ = -K[ü/r + 2(B2/r2)'/r + V/rD] - (W - 2DB2/r2)2/D 2 2 , p2 = -N 'P. - F , i ft\j *e 2 4+ . i I< +R2, 2 2 41 2 \ 2/ YPF ] . r5 Hrl \r / J-4 r7 p« +4 _-[(1fp + B)p« - This formulation, which is due to Appert et al., shows directly the two continua a2 ~ {oM and a2 = (aj) originating from the zeros of the factor N in front of the derivatives. But the real virtue of this formulation over that in terms of the second order differential equation is that the singularities D = 0 are immediately seen to be apparent ones as nothing singular shows up for D = 0 in this formulation. To prove this fact from the Hain-Lust equation requires a considerable amount of algebra. It is interesting to consider the numerical problem of solving Eq. (11-25) by means of a shooting method, i.e. the spatial initial value problem. To that end we invert the matrix in front of the derivatives (in this case a number) and get: .219. IJ U -«/IJ Giving initial data x and 1 at a certain aoint, we then ^ o o * calculate x' and IT', from which we obtain, new initial data o o X, and IK, and so forth. Clearly, the only difficulty which may arise is the occurrence of N = 0 singularities due to the slow and Alfvén continua. For D = 0 no problem turns up. D. BOUNDARY CONDITION AT THE PLASMA-VACTTÜM INTERFACE If there is an external vacuum region surrounding the plasma column the right part of the boundary conditions (11-15) should be replaced by the proper boundary conditions determining the amplitude and the normal derivative of the function v (r = 1), v/hich describes the freely moving plasna surface. This problem turns up when we want to investigate free boundary modes (external kink modes). The appropriate boundary conditions were derived in Sec. VIII C, Eqs. (8-31) and (8-29): "„ a. i, % •v, \, *;••>»% n , 7 x (£ x B) = n . Q . (11-118) .220. Here, the LHS of Eq. (11-27) Is the Lagrangian perturbation of the total pressure, so that this equation may be transformed by means of Eq. (11-22) to: 2 2 n - (Bj/r )X - SeQe+ BzQz - (B^/r )X , (11-29) wherell is given by Eq. (11-23). The second boundary condition is easily transformed to X » - i(r/f)Qr , (11-30) where F = roSQo/a + kB^ . The equations (11-29) and (11-30) determine the plasma variables II(or x') and x at the plasma surface completely if the vacuum solutions are known. This part of the problem can be carried out explicitly since the solutions in the vacuum are Bessel functions as we shall see. From the vacuum equations Vx£-0 , V.CJ-O, (11-31) we obtain the tangential components of § in terms of the radial component: £ (r % ' -TT2-m +k r2 5r>' • kr - (11-32) m +k r so that .221. m +k r The radial component satisfies the second order differential equation 2 2 2 2 2 (rör)" *-£- Wo Q -v I'(kr) , K*(kr). r m m Tho solution Q on tne interval (a,b) satisfying the boundary condition Q £b) = 0 nay then be written as Q = I'(kb) K'(kr) - K'(kb) I»(kr). (11-34) r m m mm Inserting this solution into Eq. (11-29) and dividing this equation by Eq. (11-30) leads to a single boundary condition to be satisfied for the ratio IT A : /M B?fa)-B?(a) ?2, . T (ka>K ' (kb) - K (ka)T'(lcb) f : \ _ 'J ' -• _ F (a ) m m m m \yj 1 ' ka 1' (ka)K' Ub> - K' (ka) I ' (kfa) r=a a m m m m (11-35) whereü may be expressed in terms of x' an<* X by means of Eq. (13-23) . The replacement of the two boundary conditions (11-20) and (11-30) by the single boundary condition (11-35) reflects .222. the fact that for homogeneous second order differential equations the choice of the amplitude of the eigenfunction x does not influence the value of the eigenvalue w2. Thus, Eq. (11-35) just corresponds to normalizing the eigenfunctions such that x(a) = 1* Obviously, if x(a) happens to vanish one should not divide by it, but one should then take a different normalization. This case would correspond to a situation where there is already an eigensolution if the vacuum is absent, i.e. when the wall is at the plasma (b = a). If one wants to numerically solve the Hain-Lüst equation (11-14) or the equivalent system of first order differential equations (11-25) one usually exploits a shooting method. One chooses a value of the parameter w2 and integrates in outward direction, starting with the value X - 0 at r = 0. One keeps changing the value of u2 until (JI/X) reaches the value prescribed by the RHS of Eq. (13-35) so that u2 becomes an eigenvalue. If no vacuum region is present (b=a) one follows the same method but now one integrates until x(r) goes through zero at r = a. For this procedure to be useful a guiding principle should exist on how to change the parameter a)2 in such a way that the solution for the next try is closer to satisfying the boundary condition at r = a than it was in the previous run. Such a principle is provided by the oscillation theorem to which the next section is devoted. E. OSCILLATION THEOREM We now wish to study the behavior of the eigenfunctions .223. of the Hain-Lust equation (11-14) on the plasma interval, where we restrict the analysis to fixed-boundary modes for which the boundary condition (11-15) should be satisfied. [The generalization of the discussion of the present s3ction to plasma-vacuum systems is straightforward] . As mentioned above we would like to know the qualitative behavior of the solutions X of Eq. (11-14) as a function of the eigenvalue parameter OJ2 . The kind of qualitative behavior we wish to obtain is exemplified by the classical Sturm-Liouville system which is described by the non-singular second-order differential equation (PX')' - (Q - XR)x = 0 , (11-36) where \ is the eigenvalue parameter and P = P(x) > 0, Q = Q(x) , R = R(x) . Let x and X be two linearly independent solutions of Eq. (11-36) for a fixed value of \. Denote two linear combinations of these two solutions by 2 X, - a.xCi) + a,X< > (1) (2) xh =bx • b,x b 1 2 If b2/b| / a2/a] those solutions are linearly independent, i.e. their Wronskian '< X' - X'x, noes not vanish at the interval a Li ao under consideration. Sturm's separation theorem (Ince, p. 223) now states that the zeros of these solutions separate each .224. other, i.e. if x and x are consecutive zeros of x then xv 12 at vanishes once in the open interval (x , x ). Proof: Suppose y, does not vanish on (x ,x ). Then, x and x? are consecutive zeros of the finite function x /x. . Hence. a b d/dx(X /X,) must vanish at least once on a b the interval. However, d XaXb " xa Xb MxJ x2 cannot vanish because that would imply that the Wronskian vanishes somewhere. This contradiction proves that xb must vanish at least once. It cannot vanish more than once since then we could interchange the roles of X and X. and get again a contradiction. As far as the oscillatory properties of Eq. (11-36) are concerned one could say that all solutions oscillate equally fast if X is kept fixed. If we now consider solutions of Eg. (11-36) for different values of A, we may compare their oscillatory behavior by means of Sturm's fundamental oscillation theorem (Ince, p. 224) stating the following: Let x and x be two consecutive zeros of the function X satisfying (PXJ)' - (Q - \R)X1 - 0 . (11-37) [in other words, A. would be an eigenvalue if (x ,x ) would be the interval corresponding to the physical problem, i.e. in .225. our case the interval (0,a)]. Then, the solutions y of the equation x (PX2')' - (Q - 2R)x2 = 0 , (11-38) Here oscillate faster than x if X2 ' *i* ' by faster oscillating is meant that the solution x2 that vanishes at the left end-point x = x. vanishes at least once on the interval (x^x.) . Proof: Multiply Eq. (11-37) by y and Eq. (11-38) by x,» integrate over(x.,x ) and 2 *-* subtract: X $[X2 Suppose the solution x3 which vanishes at x = x. does not vanish in the open interval (x.,x_). Then, the RHS of Eq. (11-39) is negative, whereas the LHS is positive. This contradiction proves that X? has to vanish at least once on the open interval (x. ,x2). Sturm's oscillation theorem gives us the behavior of the solutions of Eq. (11-36) on any svbinterval (x.,x_) of the interval (0,a) which we want to sttidy. Systems like that of Eq. (11-36) which have the property that the solutions oscillate faster upon increasing the eigenvalue parameter \ we will call * .226. Sturmian, whereas systems that have the opposite property (e.g. Eq. (11-36) when the sign of X is reversed) we call anti- Sturmian. An immediate consequence of these properties is that we can label k»»"t\- £t>»r««v «.»i different discrete modes by just counting *• X the nattier of (*i>'*ty ^o") nodes on the complete *t >»T Mil «k VI •"IK- ~ interval (0,a). «=« M=» /III If the system v\*l «I 1 rt:C w *\ A H H is Stumdan the avit- iW»wAvi eigenvalue X will be an increasing function of the number of nodes n, whereas for anti-Sturmian systems X decreases as a function of n. The classical example for the first kind of behavior is the vibrating string, described by the equation ci2a2£/3X2 = S25/3tz = - w2^ having the eigenvalues w2 = n2ir2a2/a2. Examples of the second kind of behavior are less familiar, but the ideal MHD equations will provide some. Turning now to the Hain-Lust equation (11-14) it is immediately clear that it is not an equation of the simple Sturm-Liouville kind as Eq. (11-36). Nevertheless, we may ask the question whether it still has the Sturmian property. It is clear that in order to prove such a property we certainly have to exclude those regions of w2 where the factor N/D develops .227. zeros (N = 0) corresponding to the continuous spectrum. Moreover, it turns out that we also have to exclude the regions of u2 where N/D becomes infinite (D = 0). Let us then study the monotonicity properties, if any, of the discrete spectrum of Eq, (11-14) for values of w2 outside the continua {o2} and {cj} and also outside the ranges - Recall the proof of the self-adjointness of the operator p~ F in Sec. VIII D. In particular, let us exploit the expression (8-45) for the inner product of two vectors I, and n. Since the volume part of that expression was proved to be symmetric, we only need to keep the surface contributions in the following relations: = 2"J""S(5*7p + ypV'S " 5'8)d0 " I\!T£(£'VF + YpV*3 " l'Vda * (11-40) whero the expressions in the brackets on the second line just turn out to produce the perturbation II of the total pressure as defined in Eq. (11-22). Therefore, we may write f .228. l l <^0~ T(P> - L r -« r»rx L J r=ri ffL[rT,r ?D(rVir *2 ~ 'LFerTD ^Vl"2 (ll"41) r=r J r=r L Jr=r l -L «J i Let us now consider two solutions £ and n of the normal mode equation corresponding to different values u>2 and ÜJ 2 of the eigenvalue parameter w2 : (11-42) 2 |(Q) - - pu22 rMi ' but not necessarily satisfying the boundary conditions (11-15). Then, the LHS of Eq. (11-41) transforms to 2 (m2 - «ƒ ) < £,ri> . Consider a subinterv?l (r-,r2) of the complete interval (o,a) bounded by two consecutive zeros of the radial component £ of £ (actually, of r £ to also include the case r = 0) . v % r X Let u;2) be close to w2 so that n is close to £ and <£,n> > 0. We may also choose n such that rn_ vanishes at r = r.. We now r\j ~ i wish to find out whether or not rnr has another zero on (r ,r ), .229. i.e., we want to investigate whether rn oscillates faster or slower than r£ for a given difference of cu* and us2. Under the mentioned conditions all that remains of Eq. (11-41) is 2 2 (o.2 - «1 ) Let rE > 0 on the open interval (r-,r ) so that (rE )'(r=r ) < 0 If N/D > 0 and CJ| - w* > 0 this implies that rn (r=rj < 0 so that rn oscillates faster than r£ (Sturmian behavior). If, on the other hand, N/D < 0 rn will oscillate slower than r5 (anti-Sturmian behavior). Consequently, the discrete spectrum outside the ranges 2 {oj?}, {CJ|}, {Oj}, and io } is Sturmian for N/D > 0, i.e. the eigenvalue UJ2 increases with the number of nodes of the eigenfunction r£ on the complete interval (0,a) . For N/D < 0, the discrete spectrum is anti-Sturmian. Therefore, the behavior of the discrete spectrum changes from Sturmian to anti-Sturmian every time w2 crosses one of the four mentioned regions: *- Stornilm < *mti> )Ur«t«i The regions {a|} and (olj) thus turn out to act as separators of the discrete spectra, where non-monotonicity nay occur. .230. Unfortunately, we have already seen that the range (aiT) for the diffuse linear pinch stretches all the way up to °°, so that not much can be proved about the fast subspectrum, except that it eventually has to become monotonie fcr large values of n since a2 = a£ = » is a cluster point of the fast subspectrum. One example of anti-Sturmian behavior we have already encountered when studying the homogeneous slab model in Sec. X A. From the picture of a" versus k for fixed k.. and k it f x " y is clear that o2 decreases on the slew wave branch if k , i.e. n, increases. Hence, the slow discrete subspectrum above the accumulation point o2 is anti-Sturmian, in agreement with the result obtained above. Another important property following from Eq. (11-43) concerns the orthogonality of the eigenfunctions of the dj screte spectrum. If r£ and rn both satisfy the left and the right- hand boundary conditions (11-35) the RHS of Eq. (11-43) vanishes, so that *£»£" = ° for ui * U2Z " (11-44) Hence, the discrete eigenfunctions form an orthogonal set, which may also be normalized to obtain an orthonormal set. It should be mentioned that a Sturmian branch of the slow and Alfvén subspe>ct>-a are foreseen in the proof above, but it still has to be shown that such branches actually exist. This fact will be obvious, however, when we consider instabilities occurring for values of m and k such that F = mBQ/r + kB„ .231. vanishes at some point in the interval (O.a). According to Eq. (11-17) in that case the continu a {all and ioi) stretch out to w* = 0, so that the mere existence of instabilities indicates that at least one of the Alfvén or slow branches of the discrete spectrum has become Sturmian. It is comforting that in any case the function N/D never changes sign on the unstable side of the spectrum, so that unstable modes are always Sturmian. This is also in agreement with our intuition that moving the wall inward does not increase the growth rate of an unstable mode, which would be the case if the unstable side of the spectrum were anti-Sturmian. Since the unstable side of the spectrum is non-singular we immediately obtain a stability theorem for o-stability of the diffuse linear pinch. To that end we should realize that the g-marginal equation of motion (9-43) for the diffuse linear pinch is obtained from the Hain-Lust equation (11-16) by just replacing w2 by -a2: 2 2 [P(r ; - a )X'] ' - Q(r ; - a )x = 0 , (11-4 5) where X(0) = X(a) = o . (11-46) The one-dimensional modified energy principle corresponding to this equation could be derived from Eq. (9-44) by a tedious analysis, similar to the one leading to the Hain-Lust equation. Hov/ever/ here v/e may pose it directly as that functional which .232. produces Eq. (11-45) as the a-Euler equation: a 2 2 2 2 W°[xj *L \ [P(r ;- o )x' + Q(r ;- o )X jdr , (11-47) where L is the length of the plasma column. Suppose now that we integrate Eq. (11-45) starting from the left end point r = 0 where we satisfy the boundary condition x = 0. If the solution X(r) thus obtained does not develop a zero in the open interval 0 < r < a, our oscillation theorem asserts that a discrete eigenvalue w2 < -a2 does not exist, so that the system is c-stable. On -XC-c^ the other hard, if the solution x(r) vanishes L \ -*• r *-r somewhere on the open interval 0 < r < a, a C-stable discrete eigenvalue ai2 < -a2 does exist for which both boundary conditions (11-46) are satisfied. This result could also have been obtained from Eq. (11-47) where it just coincides with Jacobi's minimization condition from the calculus of variations (see, e.g., Smirnov), We then have the following theorem for a-stability of the diffuse linear pinch. Theorem. For specified values of m and k, the diffuse linear pinch is a-stable if, and only if, the non-trivial solution x of the a-marginal equation of motion (11-45) that vanishes: at r = 0 does not have a zero .in the oper. interval (0,a) . .233. The wording of this theorem has been borrowed from the similar theorem of Newcomb for the theory of marginal stability (in the usual sense) of the diffuse linear pinch. Since there the singularities associated with the contir.ua at u2 = 0 have to be properly accounted for, the marginal theory in the usacl sense is much rors complicated than the corresponding thecry for s-stability. VJe will trest Uev-'coiria's theory in the next section. F. NEKCOKB'S BRBGINAT. STABILITY ANALYSIS. SUYD&N'S CRITERION For the study of stability in the usual sense we may start from either the marginal equation of motion (9-30) or the energy principle (9-31). For the diffuse linear pinch both may be obtained from the analysis presented above by setting a2 = 0. The Euler equation corresponding to marginal stability is obtained from the Hain-Lust equation (11-14) by inserting e*)1 = 0: ^K'-ur ZkB„G X a 0 2 r |y+k2r J L &-r - rim-+k~ r-) (11-43) This equation is of the form [A( rO ' ] ' " Br' - 0 , •which is equivalent to (Ar2?,')» - (Br2 - A»r)r, - 0 . .234. Therefore, the marginal equation may be written as: g o * Br" - A'r B2 \« 4k2rB? /Bj_|. _ 4k^ri; + ^ / 2kBeG y I rf2 y rF~ + r"' 2 2 U / 0 +k2r2 \al+k2T2l \a2+k2Tz) 2 2 2 2 2 2 r(n.Be/r-kBz) f r 2 2 2 2 2 2 3 2k r B +k r -l 2k r (n»Bfl/r-kB ) p. + rF2 2 z_ F ^ m2+k2r2 m2+k2r2 (m2+k2r2)2 (11-51) The equivalent one-dimensional form of the energy principle may be written as 2 2 W[e] - *t [ (foC • g05 )dr , (11-52) where L is the length of the plasma column. Since Eq. (11-49) is just the Hain-Lust equation for w2 = 0 we obtain from the oscillation theorem of the previous section directly Newcomb's stability theorem for the case that ths interval (0,a) contains no singularity F = 0: Theorem: For specified values of m and k such that F = mB_/r + >;3_ ¥ 0 en the interval (0,r.), the diffuse linear pinch is stable if, and only if, the non-trivial solution r£ of the marginal equation of motion (11-49) that vanishes at r - 0 does not have a zero in the open interval (0,a). It is clear, however, that the singularities F = G present a considerable complication as compared to the corresponding a-stability theorem. These singularities are just the left end points of the Alfvën and slow continua iol) and {cr|} which extend to w2 = 0 if the interval (0,a) contains a point where F = 0. To establish the significance of the singularities F = 0 for the marginal stability analysis, let us investigate the behavior of the solutions to the marginal stability equation (11-49) in the neighborhood of such a singularity. In terms of the normalized inverse pitch of the field lines, the parameter V s Be/rBz (11-53) that was introduced in Eq. (6-20), the expression F may be written as F - (k • ynOB, . (11-54) This shows that the singularities occur for a k + pm 0 f i.e. for values of the wavenumbers IT. and k such that the .236. tangential wavevector is perpendicular to £. In that case the phase of the perturbation is constant along the field lines at the position r = r of the singularity. Let us expand all quantities in terms of the variable s 2 r - rs . (11-55) We then have: F ^ mB vi's , m2 + k2r2 -v. m2(l + y2r2) , so that TH2 U'2 2u2r2 f sZ p o £ , \ 2 • *0 * —TT ' * (n-56) l+p2r2 ° l+w2r2 Consequently, close to the singularity the Eu.Ier equation (11-49) reduces to (s2£')' - where z The solutions of the equation (11-57) are s and s , where n and n ara the roots of the indicial equation 1 2 n(n + 1} - a = 0, so that n. , - - -T i T\/l + *a • (11 - •> S J Depending on whether 1 + 4ra is positive or negative the .237. indices are real or complex. For 1 + 4a < 0, when the indices arc complex, the real solutions to the Euler equation are oscillatory: -— + iw - —-iw 1 f = s + s =2s cos(w in s) , (11-59) --+ iw -- -iw 1 2 52 = i(s - s ) = - 2s sin (w An s) , where w= | /- (1 + 4a) . For s -*• 0 these solutions oscillate infinitely rapidly, whereas their amplitude also blows up. For 1 + 4a > 0, when the indices are real, the two solutions may be written as 5, * •- . (11-60) ^ * » ' . 1 1 r ' 1 where n = - ~ + •£ VI + 4a > = - 4 - i Vl + 4a' < - i. s 2 2 2' "* Hence, the large solution £ always blows up at s =n , ./he re as the " small" solution may or may not blow up depending on whether the square root is smaller or *• r larger than 1. .238. It is clear from our oscillation theorem that the oscillatory solution (11-59) will correspond to instabilities. This may also be proved directly from the expression (11-52) for the energy. Inserting the solution of the Euler equation (11-49) into W and integrating by parts one obtains for the contribution to the energy of a subinterval (r-w^) of the complete interval (0,a): 1 f2 ( ^LW(rl*r2} = J (f5'2 + 8S2)dr - j[fef2 • 5(f5,),]dr - 'Jlï J (11-61) If one now chooses a subinterval (rwr2) which is slightly larger than the distance between two consecutive zeros of a solution to the Euler equation, one may split the interval ^rl'r2^ into two subintervals (r,,rg) and (r0,r2) such that an Euler solution £ which vanishes at r = r 3 1 > r does not vanish again on (r^r/J and a solution E,. which vanishes at r s r, does not vanish a second time at (r0,r2). At r = r3 the amplitudes of the two solutions may be chosen equal. By applying Eq. (11-61) to a solution r and composed of £a on ^i> Q) Sb on (r ,rJ, one then obtains: ^W - [*«.«; K'-r0) - [f5b5£](r-ro> [Hill ~ tl )](r-r ) < 0 , (11-62) .239. so that the contribution to the energy of that subinterval is negative. By choosing the trivial Euler solution r = 0 on the remainder of the interval one then shows that the total energy W(0,a) < 0. The condition 1 + 4a > 0 which is necessary for the absence of the oscillatory solutions (11-59) was derived first by Suydam and is, therefore, known as Suvdam's criterion: . + J. rB 2 (Hi) 2 > 0 . (11-63) 8 z u Its violation implies the existence of highly localized in stabilities close to a singular surface where k + ym = 0. These instabilities are so-called flute modes which interchange the magnetic field lines without appreciable bending. Their impor tance, however, does not reside in this fact but may be ob tained from the application of the oscillation theorem of the previous section. Let Suydam's criterion be violated, so that the marginal equation of motion has solutions that oscillate infinitely rapidly, i.e. solutions with node number n -»• °° are unstable. Then, the oscillation theorem asserts that a global n = 0 solution to the full equation of motion exists « for which the' growth rate \ •'i[t -,.,2 is larger than that of all the higher node solutions. In other words.' violation of \lft? Suydam's criterion implies the existence of a global_n = 0 Cj\oO<»\ S«-jJk»«. r*oJl« instability. This instability .240. may also be global in the azimuthal direction (e.g. m = 1) if the mode number k may be chosen such that k + ym = 0 some where on the interval (0,a). Hence, Suydam's criterion provides a first test of stability which is quite significant. Clearly, the violation of Suydam's criterion (11-63) is the condition that the marginal point u2 = 0 is an accumu lation (cluster) point of the unstable side of the discrete spectrum: One may prove that similar accumulation points may occur on the stable side of the spectrum where e.g. the function a£ (x) has a minimum so that a£ (r) ^ a£ + c(r-r )2. This A "\» Ai s i leads to the same type of equation as Eq. , t..S^y (11-57), where the coefficient a is of £ course different. One may then derive -*• r similar conditions as Suydam's criterion to test whether the point is an accumulation point or not. Let us now assume that Suydam's criterion is satisfied so that the indices are real and the marginal solutions are those n of Eq. (11-60). The reason that we have called the solution s s n£ "small" and the solution s large is the fact that the energy contribution of the first solution is finite, whereas it is infinite for the second one. This is seen from Eq. (11-61) by applying it to a subinterval (r., r_) which is bounded by the singularity r , e.g. r2 = r . Then r *f .1 2tl+1 [ 55' ]r.r * s .241. which vanishes for n >- y (the "small" solution) but blows up for n <- •=• (the large solution). Hence, testing for stability while keeping the energy of the perturbations finite implies that we have to impose an additional internal boundary condition, viz. that £ be "small" at a singularity. At the singularity we may also allow jumps in the"small" solu tion by an argument similar to that of Sec. X B. Also, one notices that such jumps do not contribute to the energy: n+2 n 1 n fCsH(s) [£sH(s)]' 2n + 1 = s [nH(s) + s6(s)] H(s) + 0. Therefore, the interval (0,a) may be split into two independent subintervals (0,r ) and (r ,a) which may be tested separately for stability. Of course, in case there is more than one singu larity, there will be more than two independent subintervals. Consider a solution g of the Euler equation (11-4 9) CI which vanishes on the left interval (0,r ), is "small" tc the s right of the singularity r = r and vanishes once in the in terval (r ,a). Such a solution may be joined at a point r in between the singularity r and the zero point of g to another solution a E. which vanishes at the right end point *. r r = a, but does not vanish in the open interval (r ,a). The energy of the Euler solution consisting of £ = 0 on (0,r ) , £ = £ on (r ,r ) , and g = £. on (r ,a) may be shown to be negative by a completely analogous argument as that used in the derivation of Eq. (11-62). Hence, on independent subintervals the "smallness" of a solution should be counted as a zero, so .242. that for stability a solution that is "small" at the singula rity should not vanish somewhere in the interval. Thus, we obtain Newcomb's theorem for the case that the interval (0,a) contains one singularity F = 0 at r = r . I Theorem. For specified values of m and k such that \F £ mB ,/r + kB^ = 0 at some coint r = r of the interval (G,ai , i XJ Z * S I I the diffuse linear pinch is stable if, and only if, (1) Suydam's t criterion (11—53) is satisfied at r = r ; (2) the non-trivial solution £ of the marginal equation of motion (11-49) that is "small" to the left of r = r does not vanish xn the open inter val (0,r ); (3) the non-trivial solution r that is "small" to s R the right of r = r does not vanish in the open interval (r ,a). It is clear that the existence of singularities com plicates the marginal stability analysis considerably. Therefore, for numerical studies a o-stability analysis is certainly to be preferred. For analytic studies the presence of singularities often facilitates the construction of explicit analytic solutions by means of series expansion. However, the number of cases that may be treated this way is quite limited. G. FREE-BOUNDARY MODES In this section we continue the discussion of the boun dary value crohl'^n rxjsed v:v the Hain-Lust ecruation (11-44) .subject to the boun- darv conditions x (0) = 0 and (n/x)__ „ as given by Eq. (11-35) . We wish to study this problem for a sharp-boundary plasma where the current is confined to the plasma surface r = a (skin current model) . This model provides a vnry useful first approximation to the study of external kink modes, which are the most dangerous instabilities occurring in a cylindrical plasma column. Here, most .243. dangerous is meant in the sense of affecting the bulk of the plasma and having large growth rates. (For typical densities of high-B pinches they exponentiate on the usee time-scale) . For the sharp-boundary skin-current model the equi librium quantities for the interior of the plasma column are those ot -Ëi—r a homogeneous 5-pinch: B. - 0, B = B ,p = P , 1 8 z o o J . -»- »• (11-64) P = T BoBo ' where B , p , and B are constants. For this part of the con- o o o figuration one may again define the Alfvén speed and the sound speed: (11-65) bo " \/Bo/po ' Co 5 fö > which are related to each other by the value of 8 : J o 2 2 (11-66) co/bo = -2L Bo Y. For the interval 0 <_ r < a the Hain-Ltlst equation may be simplified to 2 2 I (a' k b ) = 0. o •> i o 2 2 2 , (k~- o-/b-)Ckfc - o /c) l o o z9 - m + 2 2 2 2 2 r (11-67) k - a /b"" - a /c o o From this equation one obtains first of all the discrete spectrum of Alfvén % aves with frequency a2 = a£ = k2b2. This spectrum again consists of infinitely many discrete modes for which the .244. eigenfunction x has a completely arbitrary radial dependence/ as is evident from Eq. (11-6 7). They propagate with the Alfvén speed b along the axis of the cylinder. For the discussion of the external instabilities they may be discarded because they are stable. For o2 ? k2b« Ec3* (H-67) may be solved in terms of Bessel functions: X = r I'(k*r), (11-68) where (k2- o2/b2)(k2- 02/c2) k* = ~~2 TTl T7~2 k - o /bo - a /co Here, the virtual wave number k* has been introduced. For a number of important cases, e.g. a2 « k2b2, k2c2 (i.e., also for the marginal modes), k* £ k. From the expression (11-68) one may obtain the internal modes of a 0-pinch column by eli minating the surface currents at r = a and putting the wall at the plasma (b = a). The boundary condition x(b) =0 then gives the result k*a » j' • (11-69) mn 4-Vi where Jim' n is the n - zero of the Bessel function J'(x)m . From this expression one obtains the dispersion equation for the slow and fast magneto acoustic waves in a homogeneous e-pinch; ,*. 2 + j'2/ 2)(b2 + c2)a2 +k2(k2 + j'2/a2)b2c2 - 0. (11-70) (k mn a o o mn o o This equation is completely analogous to Eq. (10-7) for the magneto acoustic waves of a plane homogeneous slab. .245. Returning to the sharp-boundary model we may obtain the dispersion equation for free-boundary modes by just inser ting the solution (11-68) into the boundary condition (11-35): k2B_2 k*al_' (k*a) 2 2 , u 2 Sp (mBD/a + kB ) I (ka)K'(kb) - K (ka)I "-t a = 2 m + " z m IP jvj m_ \\t P o pi (k*a) .a2 ka r(ka)KMkb) - K^(ka) 1^ (kbjS o n (U-71) Notice that the dependence on o2 also occurs through the factor k*, so that this dispersion equation is a transcendental equation in a2. Many different limits may be studied for this equation, but the most important one is obtained for the tokamak approxima tion which consists of considering e linear pinch of length 2irR as a first approximation to a torus of major radius R. In that case, the wave number k is quantized: k = l/R , so that ka * tl << l. (11-72) Furthermore, Ê8 < < Sz so that ^q = efiz / Bo ^ 1. In view of Eq. (11-72) the arguments of all the occurring Bessel functions are small, so that we obtain the following approxima tions : k*a I' (k*a)/I I (ka)K' (kb) - K(ka)I' (kb) i; (m j* 0) (11-73) Inserting these approximations into Eq. (11-71) leads to the following dispersion equation: -246. -!ml e-B (b/a)imU (b/3)-'m!l , ? — ~> i i . , . —. -> ! a' ~« <. - q ~ - |m| + vm+icq)- a-o q~ o (b/a)^ml- (b/a)"'1"1-' (11-74) In Eq. (11-74) we have neglected small terms B and c~/• L in o agreement with the low-e tokamak ordering: q ^ 1 , 0 -v e2 . Rearranging terras Eq. (11-74) may be written as e2B2 02 'V 2 (£q + m)" jim|(imj - 2) + i (2 *q" + m)2 + . ,2|mi a2p q2 L (b/a) -1J o (11-75) This rearrangement of the terms should reveal some of the phys ical mechanisms at work in this model: First, there is the kink term which is only destabilizing when [m[ = 1. Then, there is a stabilizing term representing the average line-bending across the plasma boundary which disappears for modes that propagate perpen dicular to the direction of the field averaged across the surface layer at r = a (recall that q = » for r = a and q = q for r = a). The last term represents the stabilizing influence of the wall, ranging from infinitely stabilizing when b/a = 1 to no effect when b/a ** °°. Since only jmj = 1 is unstable we may restrict the anal ysis to that mode: (4q - l)(iq - a2/b2) a?(n=-l) = • (11-76) a2 o q2 1 - a2/b2 o .247. Hence, the external kink mode is always unstable for this model in the region a2/b2 < Stq < 1 . (11-77) This leads to the obvious remedy of the external kink mode to exclude it by just prescribing the geometry of the torus and the total plasma current I such that q - 2Tia2B /RI > 1 , (11-78) o z so that all modes £ = 1,2,— are stable. This condition is called the Kruskal-Shafranov limit. The limit imposed on the plasma currents by Eq. (11-78) is a quite important consider ation in the operation of Tokamaks. It is appropriate to repeat here the remark ir.ade in Sec. VIC that the fact that q = 1 corresponds to a topology with closed lines has nothing to do with the stability of the external kink modes. This is a purely accidental coincidence which disappears as soon as one introduces genuine toroidal effects in the theory (chapter XII). H. FIXED-BOUNDARY MODES We now put the wall at the plasma and consider the boundary value problem posed by the Hain-Lüst equation (11-44) with the boundary conditions (11-15). In order to have a problem that can be solved analytically,we fix the equilibrium profiles as follows: B -v B (1 - a2r2) , Z T» O B3 % ur»o , (11-79) 2 2 2 P B [i 6 • U -M )r ] , .248. where we have put % signs to indicate that we consider these expressions to follow from a series expansion in r where we have kept only the leading order terms. We simulate toroidal o o geometry by imposing periodicity over 2TTR and impose the low-B tokamak approximation, where q = 1/yR^ 1 , (11-80) and where we have the following small parameters at our disposal: 2 2 2 3 ^ (aa) ^ (ya) «* e a2/R2. (11-81) With these approximations the pitch of the magnetic field lines is approximately constant, so that we may introduce a parallel wavenumber k„ = k + urn, (11-82) which is constant. Let us now search for instabilities in the following regime of parameter space; 2 2 2 2 2 2 2 pui << m B /a . (11-83) \L/f < JL È1 1 / 2 2 dr r dr \r )x (U-8M where 2 1 y 2 m2 DB2 [- 4.,k 2 PLU 2 YB A = 2 . JiL , 2 2 2 1 2 1 7 ]• Pu "^Bn m pa.'(l+yYB)-k; jyfiB rB .249. The solutions of this equation are Bessel functions: = J (\jT r/a) , (11-8 5) m so that eigenvalues are obtained by equating /x with the zeros j of the Bessel function J (x): X = j (11-86) en From the latter condition the dispersion equation is obtained: 1+Y0 2 2-2 r . 2 Y8 m 2 m y -2 > - k, + v + 1i -2 - 2 - JJ* J rB' kl+2^e 1+jYB mn mn 2-2 JYB 2 22 Ür2 fkf r + 2mu 0 , (11-87) rB2' 1+JY6 'mn where we have introduced dimensionless variables _2 - ' ,2/B2\z z 1.12Z w = (pa /B )u' ; k//r s fya , y s ya The two solutions of this quadratic equation may be written as: 2 2 -2 2 2y6 2 A ~ 1+YB £ + m* —u 4 +^ 2m y i^e " l^re j^ rB inn 4 r iT —2 2 + I * + 4m*_ _y f yB(l + YB) - p' J — y + —t— k* + & rB2 2 m2 -2 -2 1/2 * J ^=- v {-^f— y + (11-88) I + JTB rB mn .250. Defining w = p'/rB2, we find two special values of the pressure gradient where the modes change character, viz. 1 y is V i + 4"ys (11-89) jïS d+ïS) _2 (1 + -j- Ye )" For IT 9 < ir < 0 the maximum growth rate occurs for k/y, £ 0 . These modes are called quasi-interchanges. Their maximum growth rate is given by -2 u 2 2 2 4 2 2 1/2 max = -Y & (m /j^n) y [l- { 1 + (l/yBXp'/rB ) y } (11-9C) For TT <_ TT the maximum growth rate occurs for k/x = 0. These modes are called pure interchanges. Their maximum growth rate is given by - & 11 CDma x = 2 (m /J jm n ) y (TT-TT.)1 . (11-91) Hence, for p' < 0 first quasi- interchanges become unstable, -2VDB? whereas only for p' < — pure interchanges become unstable rlvp+BM Clearly, Suydam's stability criterion (11-63) for a constant pitch magnetic field degenerates into the quasi-interchange stability condition p' > 0 and not into the pure interchange condition as one might have expected. -251. I. o-STABLE CONFIGURATIONS On the basis of the o-stability theorem it is possible to systematically search for o-stable configurations while taking a reasonable choice for o, e.g. one which corresponds to the msec time-scale. From a large nuirber of numerical runs the following qualitative picture emerged. There are,broadly speaking,four categories of diffuse linear configurations that are o-stable with respect to internal modes. All four of them are characterized by a monotonically increasing or decreasing q-profile, representing shear of the field lines which turns out to facilitate stability. The q and j profile for these configurations are the most characteristic ones to distinguish the different configurations: tokamak pinch As the current profile is broadened the maximum allowable 6 for stability in general increases from a few per cent for tokamaks to some 40% for the reversed field pinch. Except for the latter configuration all other configurations require q > 1, either on axis when the q-profile is increasing as in r. tokamak, or at the .252. wall when the q-profile is decreasing as in a screw pinch. For more details: see Ref. 12b. REFERENCES 1. E.L. Ince, Ordinary Differential Equations (Dover Publ., New York, (1956) . 2. V.L. Smirnov, A Course of Higher Mathematics, Vol. IV (Pergamon Press, Oxford, 1964) . 3. K. Hain and R. LÜst, Z. Naturforsch. 13a (1958) 936, "Zur Stabilitat zylindersymmetrischer Plasmakonfigurationsu mit Volumenströmen". 4. M.D. Kruskal and J.L. Tuck, Proc. Roy. Soc. A245 (1958) 222, "The instability of a pinched fluid with a longitudinal magnetic field". 5. B.R. Suydam, Proc. 2nd U.N. Intern. Conf. on Peaceful Uses of Atomic Energy, 31 (Columbia Univ. Press, New York, 1959) 1 "Stability of a linear pinch". 6. W.A. Newcomb, Ann. Phys. (N.Y) 1_0 (i960) 232, "Hydromagnetic stability of a diffuse linear pinch". 7. A.A. Ware, Phys. Rev. Lett. 12. (1964) 439, "Role of compressibility in the magnetohydrodynamic stability of the diffuse pinch discharge". 8. V.D. Shafranov, Sov. Phys. - Tech. Phys. 15 (1970) 175, "Hydromagnetic stability of a current-carrying pinch in a strong longitudinal field". 9. D.C. Robinson, Plasma Phys. JL3 (1971) 439, "High-6 diffuse pinch configurations". .253. J.P. Goedbloed and H.J.L. Hagebeuk, Phys. Fluids ^5 (1972) 1090. "Growth rates of instabilities of a diffuse linear pinch". H. Grad, Proc. Natl. Acad. Sci. USA 70_ (1973) 3377, "Magnetofluid-dynamic spectrum and low shear stability" . J.P. Goedbloed and P.H. Sakanaka, Phys. Fluids 1]_ (1974) 908, P.II. Sakanaka and J.P. Goedbloed, Phys. Fluids 17_ (1974) 918, "New approach to magnetohydrodynamic stability" . K.Appert, R. Gruber and J. Vaclavik, Phys. Fluids ]/7_ (1974) 1471, "Continuous spectra of a cylindrical magnetohydrodynamic equilibrium" J.A. Wesson, Nuclear Fusion 18 (1978) 87, "Magnetohydrodynamic stability of tokamaks". .254. XII. SHARP-BOUNDARY HIGH-BETA TOKAMAKS A. INTRODUCTION In the previous sections we analyzed one-dimensional systems, i.e. systems in which there is only one direction of inhomogeneity. This leads to an ordinary second order differ ential equation in the unknown E, (r) where r is the coordinate in the direction of ir.homogeneity. The dependence on the homogeneous directions can be taken care of by means of a simple Fourier decomposition. The problem of ultimate interest in CTR is to study the stability of diffuse toroidal systems. This problem involves partial differential equations in the unknowns £(r,e) , n(r,e) , and e(r,6) , where both the radius r and the poloi- dal angle e are directions of inhomogeneity. Only the dependence on the ignorable toroidal angle 4» can be Fourier-decomposed in a simple manner. Before we embark on this complicated problem it is, therefore, advisable to first acouire sane insight in a simplified toroidal system where the radial dependence is simple but the dependence on the poloidal angle represents the major complication. Such a system is obtained when we consider the toroidal extension of the theory of external kink modes developed in Sec. XI G. The application of the high-beta tokamak approximation here leads to the simplest elliptic partial differential equation known, viz. Laplace's equation in two dimensions. This involves complex analysis of harmonic functions, which, together with the theory of ordinary second order differential equations, constitutes the main resource for known useful techniques from applied mathematics. This is the mathematical motivation for the present chapter. .255. The physical motivation for the investigation of sharp-boundary high-beta tokamaks is the question about the maximum obtainable & in a toroidal plasma. As is known, the value of B ^ 2P/Bi constitutes a figure of merit for future fusion reactors. It indicates the amount of plasma producing fusion energy contained by a certain magnetic field which is costly to produce. The value of 3 is limited both by the requirement that equilibrium exist and also by stability considerations. As far as gross stability is concerned, the most dangerous instabilities are the current-driven exLernal kink modes, which impose limits on both the maximum & and the maximum toroidal current. The present chapter is just an in vestigation on how B affects the Kruskal-Shafranov limit q > 1, expressed by Eq. (11-78). To leading order in the inverse aspect ratio e = a/Ro t*ie s7stem may be considered as a straight cylinder. The next order in e, however, leads to toroidal effects of ft which distort the angular symmetry of the magnetic field leading to the possibility of additional instabilities. Also, a limit on the equilibrium arises through the occurrence of a so-called second magnetic axis when 6 surpasses a certain critical value. Consider a dense plasma region tp of uniform pressure confined by surface currents flowing on the toroidal plasma *-K surface S. Surrounding the core of plasma is a vacuum region T enclosed by a perfectly conducting .256. wall. The wall will be assumed to be far away so that it has no influence on the stability. The equilibrium problem will not be a free-boundary problem in the usual sense where ex ternal currents or a wall position arc given and the shape of the surface S is found by solving the equilibrium equations. Rather, we consider the inverse problem where the shape of S is given and the external fields are what they come out to be. We may then put the wall at any position consistent with the calculated field distribution. In other words: the wall is not specified at all in this problem, except that it should be far away. Bacause of the toroidal symmetry we will finally have to deal with the unknowns on the poloidal cross-section of the torus only. Let us denote the poloidal cross-section of the volumes TP and TV by the surfaces x 3 (R - Ro )/a , y « Z/a , (12-1) give rise to a three-dimensional coordinate system x, y, $ with the following representation of the gradient operator: .257. y = J_ (e _L- + e -J- + e-H êr )• (12-2) a \x 3x i,y ?y -\. Because of the axial symmetry it is convenient to split three- dimensional vectors (3-vectors) into poloidal and toroidal parts: U = U + U e , U = CJ , u,, 0), (12-3) where, in general, the components U and L' will depend on x, y, $. Fourier analysis in the angle <{. and the high-p tokamak ordering will permit us to eventually eliminate all toroidal components of 3-vectors and also all dependences on the toroidal angle $, so that the two components U (x,y) and U (x,y) are then x y conveniently grouped into a 2-vector Ux S (üx> üy). (12-4) Similarly, we introduce a two-dimensional dimensionless gradient operator * whereas a kind of dual operator V takes the place of the usual curl operator: ?* 3 (-4- , - ~-). (12-6) •i. oy ox * Notice: VX-7X = 0, whereas the two-dimensional Laplacian may be written as \ - \ • \ - < •*: = 4+ -A • (i2-?) To illustrate the power of this notation (due to Now- comb, Ref. 3) consider a special 3-vector V(x,y,4>) having the frequently occurring property .258. V = O , (7 x V). = O . (12-8) •V, <\, (p Fourier analysis and the ordering then leads to the result that BV^/a^ can be neglected, so that we get the following relationship for the corresponding 2-vector V (x,y): 3V 3V 'V'1 7 V = — + 2L = 0 JL ' *-»• 3x 3yy U ' 3V 3V V V 1 2 = (,2_9) x • <\,.L" - V - - -T - ° ' which will be recognized as the Cauchy-Riemann conditions for V and V . The vector V. can then be derived from either one x y ^J- of the two conjugate harmonic potentials A or B: * V± = - i 7j_ A = 7AB, (12-10) where AXA = Ax B = 0. This is the basis of that part of two-dimensional MHD that can be described by complex analysis. Let us introduce local orthogonal coordinates X and v on the curve C, where X is an angle-like coordinate based on the arclength i along C: dft - a e d X , (12-11) where e ^ L/2ira is the factor of elongation of the curve C as compared to a circle of radius a. Normal and tangential deri vatives along C are then written as n. .7. - 3/3v ; t,. V. - e"1 3/3X . (12-12) In terms of these coordinates on C the above Cauchy-Riemann conditions become 259. v e 3 A <) v - iv = 4^ = " - TT ' (12-13) A 9v e !U Gauss* theorem for 2-vectors V, satisfying Eqs. (12-9) then gives P 2 P ( V^da = (lVxA! do = \ V^.(A* VxA)da P = e 4 A* -^ dl = e I B* -P- dX . (12-14) 3v ) 3v Clearly, our aim is to reduce all the following calculations to expressions of this form so that the problem will be the evalua tion of one-dimensional contour integrals along C. The shape of the curve C will be prescribed,e.g. in polar coordinates by giving the function r = g(9). The coordi nates (x , y0) of a point on C are then given in terms of two functions of the arclenqth coordinate A: x =x(A)=e(8)cosi o o (12-15) y = y (A) - g(8) sinö o o The explicit form of the functions x (A) and y (X) are found by o •" o inserting the relationship 8 = 9 (A) which ma/ be found by nume rical inversion of the integral _1_ f » / 2 , . ,''.. . s2 A(0) = — J 'yg-(9') + (dg/d9') d9'. (12-16) e J o The stability properties tv;rn out to depend strongly on the principal curvatures of the magnetic surface S. These quanti ties may be expressed in terms of the functions x and y deter- .260. mining the shape of the curve C. The dimensionless poloidal and toroidal curvatures k' and K = a t . ?n . t , BC,. = Re . Vn . e , where we have added the factors a and R to make both < and o p K dimensionless quantities of order unity. To express these quantities in terms of the functions x and y notice that V^I c • i, - • <*;•*;>• ?x •e <*;•»;>• where priires denote differentiation with respect to the argument X. Furthermore, e' = 0, so that x' x" + y' y" = 0. Using oo-*oo these relations one finds: X" _1_ 3«x 7 n . = o K - t . . A x £•- e 3X e » p *vi *; (12-17) 3n * . e > where we have neglected higher order terms in e in the deriva tion of ic . B. EQUILIBRIUM The equilibrium is specified as follows: On TP : p * constant , p • constant, (12-18) B - B .ïA - (R B /R)e , B HBA(R-R ) o.. : , * 4. ij . i i; * i ij. <«-»« 0,,'i KB • 0 ,7.8-0. (12-20) .2G1. The magnetic field in the plasma has the usual 1/R dependence characteristic for a current-free region. Eqs. (12-20) are the only partial differential equations that have to be solved to complete the description of the equilibrium. The high-beta tokamak ordering, like the low-beta tokamak ordering, is an ordering in the inverse aspect ratio t, but the important difference is that the dense interior plasma is confined by a diamagnetic well in the toroidal field rather than by the external poloidal field. This permits us to investi gate the equilibrium limitations associated with the appearance of a separatrix at the plasma boundary as well as the finite-S modifications of the external kink modes. In the high-g tokamak ordering the quantities are ordered as follows: B /B -v, B\/B T, 1 , B /B ^ e , $ o 2 (12-2-) S = 2p/B -v e. The consequences of this ordering for the pressure balance equa tion (12-19) are as follows. On 3 the poloidal variation of the toroidal field is determined by B /B - B /B = R /R = R /(R + ax ) = r 1 + zv. (X) 1 ~l . (12-22) •p o $ o o co o - o Substitution into Eq. (12-19) gives 2 r 1 2 2 : r ] 2 2 SB * l + F.K t\),_ ~ B = % (J) * l + z*. O.) " " B o '- o op ^ o • o » so that to second order; 2 2 2 2 Bp (.\)/Bo = L - 5o /Bo + F.i• l •" 2r.x o (>.)"! .262. This still leaves an arbitrary constant a undetermined in the second order expression: B2 (X)/B2 = 2eB x (A) + ae2 , (12-23) p o o B2/B2 = 1 + 6 - ac2. (12-24) o o To determine the useful range of a notice that a limit is reached when Ëo /Bo = 1 so that all the pressure is confined by the poloida] field (low-B tokamak): a = 6/E . max Another limit is reached when the pressure is so high that the poloidal field develops a zero on the inside of the torus: 8 (TT) = 0 (high-$ tokamak limit). Since x (IT) = -1, this happens for am m. = 23/c. The range between the low-e tokamak end and the high-B tokamak limit is more usefully described by a parameter of unit range: k2. 'B'C, (0 < k2 < 1), (12-25) a + 2$ f e — — where k2 = 0 corresponds to the low-p tokamak and k2 = 1 to the high-3 limit. We may now write S (X) 2 <12 26) e Brt 4-k \/4- \/i - 4- * [i - *„<*>] • " o which shows that the parameter k2 just measures the amount of variation of the poloidal field going the short way around the torus. Notice that in the high-e tokamak ordering, where k2 *> 1, .263. the relative variation of S (,\) going the short way around the torus is of the order unity. Since the toroidal field is virtually constant, the behavior of the field lines on the surface S is largely determined by Ê (A) . Denoting the distance along a field line by the symbol s, the equation of a field line is given by B x ds = 0 , whe re ds = (0, aedX , Rd $) , so that RB dé aeB^ dX P We may now generalize the definition (6-17) for the safety factor of the field lines in a torus. Locally, the normalized pitch of the field line is dO/dA = a e B^/RÖ ,so that the overall increase in field line the short way around the torus is given by vaq Aft 1 I 1"A e 1 d - If'."']" -. 2n 27tRiTJ ^ p 2TT \ \ eB / (12-27) Clearly, as k2 -» 0, so that Ê (IT) •* 0, the integrand of Eq. (12-27) blows up so that q -*<*>. We will see that this singularity is a very sensitive function of the total current 14. A tiny decrease of I. may cause the safety factor to jump from a modest value to infinity. It is clear that such a pathological dependence on param eters is not in agreement with the global description of equilib rium and stability one has in mind when applying ideal MHD theory. In fact, it is not clear at all whv the rotational transform of .264. field lines should play such an important role in the gross MHD stability of plasmas. The source of the confusion seems to be circular zero-6 limit: 2 aB 2ira B ° ° = q* , (12-28) 4 Rl RI P $ where we show the coincidence of two quantities, one (g) mea suring the rotational transform of field lines, and the other (q*) measuring the total toroidal current flowing in the plasma. Since the external kink mode is driven by the current, let us generalize the latter definition to apply to arbitrary B and non-circular cross-sections in the high-B tokamak ordering: P r = ( jAdo = I B . di, = ae \ B dA , so that a L tx B (A) -1 n * = -P. -d x RI 'u eB (12-29) 1 - A [+f o f' This definition of the fundamental parameter measuring the total current immediately cures the defect of the original definition of q. Notice that for low 6 when B becomes approximately con stant q* £ q/e, so that we also have (purposely) introduced a discrepancy between the two parametars at low 6 and non-circular cross-sections. This definition of q* turns out to be the better choice when describing kink-mode stability. Next, let us measure the plasma g in units of q*. This leads to the definition of the poloidal 6: .2C5. 2 eS * (12-30) 2TT I p From the Eqs. (12-26) and (12-29) the significance of the parani eter y' is now seen to be just another way of fixing ESD: i, 2 r i p. * The critical value of the poloidal B is reached when k2 = 1: r 1 + x (X) dX 1-! 2 (12-32) ••» > -4- -r-l V " o J p,crit 2 L 2TT J V which is seen to be a simple function of the cross-section alone which may easily be calculated for different choices of the cross- section: b/a = 1 b/a O .617 •10 .617 .673 .617 1 .422 .360 1.18 2.25 Clearly, as far as equilibrium is concerned, a triangularly shaped plasma pointing away from the major axis of the torus is about the bast choice for the cross-section. .266. Let us formalize the dependence on parameters one step further by introducing a normalized variable of average value unity that describes the variation of the poloidal field: 1 -I k2[l -x (X)] q*B (A) C (X) = (12-33) P E B rJl\ll-h*ll- ".(1|1" so that dl 2^fV» = 1 The equilibrium is completely fixed by prescribing the param eters eg and q*, which determine the parameters B/e and k2 through the equations (12-30) and (12-31) and the normalized poloidal field profile through Eq. (12-33). For a circular cross- section, where xo^ = cos 9, the 6UV") * different expressions may be evaluated in terms of the -«vk Wi complete elliptic integrals of the first and second kind: 1 d8 Ktfc2) 1 + Ik2 + 2 it' " "I 4 * » Vl-k2 sin2 I e' 1 2 2 2 2 . v- k sin j- 6 d9 * -| E(k ) 1 -i k + hS-f 4 Hence: eg - !>k/4E(k2)]2 , (12-34) P u eg . - IT2 /16 % .617 , (12-3 5) p,crit ^ ' .267, b (9) [>/2 E(k2)] 'W 1 - k2 sin2 ~ 6 , (12-36) P q = q* . 4 E (k2) K (k2)/*" . (12-37) This gives the following pictures: !>7,(. *-0 tt an o » a o poloidal field curves of constant q curves of constant q C. VACUUM FIELD SOLUTION FOR THE CIRCLE We did not pay any attention yet to the solutions of the partial differential equations for the vacuum because these solutions are not needed in the stability analysis. Never theless, they are of interest by themselves. Notice that the equations (12-20) are of the type (12-8) so that v/e nay introduce a harmonic potential $ from which the normalized poloidal field 2-vector 6 in the vacuum may be derived: b. 3 qft B,/eB (12-38) satisfying c* .6-0, (12-39) K = ° ,. j. so that K • - 7** • (12-40) .268. where _ v A x i> = 0 on o . (12-41) The latter equation has to be solved subject to the boundary con ditions * = 0 , |i- = S (A) on C , (12-42) where b U) is given by Eq. (12-33). This problem is ill-posed: Eq. (12-41) is an elliptic partial differential equation and the boundary conditions (12-42) specifying both function and normal derivative are of the Cauchy type. Consequently, unique: •.;;» and continuous dependence on boundary data is not guaranteed. This problem is connected with the occurrence of a separatrix in the vacuum beyond which the solution fails to be uniquely determined. However, if e6 is small enough so that the separatrix is far away, we still may obtain solutions in a large region. For a circular cross-section the above equilibrium problem may be solved explicitly by means of analytic continuation of the boundary data (see Ref. 2) . We have to solve ± J_ r JJL + _J_ _if*_ = o (12-43) V do subject to the boundary data on r = 1 2 2 2 * - 0 , Se(9) = -|^ = [ir/2 E (k )] y 1 - k sin ~ 8 . (12-44) Notice that we consider normalized radii here: r/a -»• r. Let us now Fourier-decompose the poloidal field at the boundary: CO Be(e) - 1 + jP tm cos m 8 , (12-45) tn=l where t - — • 6 The solution to the problem (12-43), (12-44) can then be written down immediately : Mr ,6) " inr+ YL üm ~— (rm~rm)cosn!e. (12-46) m=l Although this series solution formally solves the problem, it is actually of little use because it turns out to converge like a snail. A much better representation of the solution is ob tained as follows. Observe that both sides of the boundary condition y 1 v v (r _*L^)\ . ==1 1 ^+ P\ tt cosm0cos m0 == 2_s_- W ' l -4- ^ *+ — -5-^ cos9 k=\ ™ 2 E (k2) are analytic functions of 6. They remain so when 8 is replaced by the complex variable z: 9+ z = 9 - i Unr. This gives 1 + > t tos m9 cos(ir?. In r) + sin m5 sin(im Zn r)] '—- m ^ m=l TT 1 I o I o r .1 1 - -„- k*- + j k'-[cos 6 cos(i v.n r) + sin 0 sin(i -.n r) j , 2E(k<) or \ ' 1 T/^n - m. , . . rn - m. . -•. 1 + / i — c ^ (r +r ) c o s m e + i(r - r ) s m m 9 ; m= 1, 1 - -—V2 + — k2 f Or + -)rose + i(r- —) sinö 1 . 2 E (k") W Hence, the real part of this expression gives the poloidal field in the entire plane: .270. CO ) 1 , m -m. 'Ee<'-9» "'TT " l + +r )co s m 9 m=Z_l i"^^ 2 2 R e A + iB = 'A + + B 2 E (k ) V' 2 v/? E (k ) f (12-47) where i-j 12 1 12 1 A = 1 - -y- k + -i-k (r + -J-)cose , B = -±-k (r - —) sine . We could integrate Eq. (12-4 7) once more to obtain the flux function ty itself. This yields a solution in terms of incomplete elliptic integrals. However, we here wish to obtain only a specific detail of this solution, viz. the position x = x of the stagnation point of the separatrix. At this point £ ->- 0. The stagnation point is expected to occur on the inside of the torus, i.e. for 8 = IT, so that we look for zeros of the function 2 2 2 rbe(r,6 = ^) - [TT/2 E (k )] yi -k /n(r) , n (r) = 4r / (r + 1 ) . (12-48) Hence, the position of the stagnation point is given by n k" = (rs>' (12-49) or, in terms of eg / 2 n(r ) eB. TT S (12-49)' 16 E(n(rg)) For small values of e6 the stagnation point is far out so that r >> 1 and s <(({<"è(((({ti\ n(r ) << 1. This gives-. i -1 r % (eB ) (12-50) s p .271. 2 For large values of cB , when k •+ lf the separatrix hits the plasma surface so that r = 1 and n =1. This gives: e 3 . = TT2/16 = .617. (12-51) P.crit Now that we have obtained the exact solution it is interesting to return to Eq. (12-4 6) to investigate the conver gence of that series. One may calculate all the coefficients t explicitly by recursion.For the first coefficient we get t.--L u-^4 •* -L^si JLii^t-i.^,,,,. <12-52) 1 J '- k E(k ) P Likewise, t- ^ (E0 )2, t^ -\. (EB )3, etc. 2 p 3 p This seems to indicate that we may expand the flux function in terns of E8 : P Hr,e) % Unr + yEB (r - —)cosO, (12-53) which is the solution which appeared in Ref. 4. Hence, we obtain to leading order rb =1+ -z- r-& ' r + — ) cos 9, v 2 p r so that the stagnation po-'nt would occur at r *>. 2(FB )"L , (12-54) s p i.e. twice as far as the correct solution of Eq. (12-iO). The reason that this result is wrong is the following. The coeffi cients t ^ (E3 )', whereas rm ^ (:6 )~m, so that the>re is no m P S P justification for the neglect of the higher order terms. They .272. just provide a series of alternating terms that cancels the factor 2 in the expression (12-54) for the position of the stagnation point. One should be careful with asymptotic expansions 1 There is one more interesting aspect to the represen tation (12-46) of the vacuum flux function i|». Let us write ij; = In r + i^ - if» , (12-55) whe re « ill (r, 6) = ) t -T— r cos m 8 , m=l and ^ 1 _m tli (r , 8 ) - ) t -£— r *" co s m 9 . m=l m 2m Thus, the solution ij» for r >_ 1, which is due to the surface currents at r = 1, is represented by the solution of an equi valent problem where the f;'eid in the whole poloidal plane is represented by a potential it of an infinite series of multipole currents of strength t situated at r = 0 and a similar potential m ifi of an infinite series of multipole currents of the same strength at r = ». The joint effect of these multipole currents is the creation of a flux surface ty 0 at r = 1 and a poloidal field r3iji/3r = rb (e) given by Eq. (12-48) . The field i|; is an^ytic for r > 1 and can be represented in an integral form by means of Po^sson's integral formula. The field tji however, is not analytic for r > 1. One could represent i by means of Poisson's formula for r< 1 and then try to con- tinue this solution past the unit circle. This is not possible however, since the unit circle is densely covered with singular ities of the kernel of the Poisson integral formula. This is another w?.y in which the ill-posedness of the present problem appears in the analysis. -273. Finallv, it is interesting to notice the analogy of the appearance of a stagnation point in the solution of the flux function with the similar phenomenon in hydrodynamics known as the Magnus effect (see, e.g., Ref. 1, p.423}. Here, the flow produced by a cylinder which is rotating with angular velocity *: and which is situated in a steady flow of velocity v at infinity is represented by the stream function iji = - K9.nr - v(r — )cos9 . (12-56) * For an angular velocity K = v a separatrix appears at the rotating cylinder. For '. > v this separatrix moves away from the cylinder ; id we get a similar topology of the stream e6 woulc function as the flux function in ideal MHD. Here, •= D * correspond with V/K if t^e equation (12-54) were correct. At high 3 the additional terms in the flux function (12-46) pro duce a topology that is qualitatively different from \e. Magnus flow pattern for K <. v. D. VARIATIONAL PRINCIPLE FOR STABILITY The stability of the sharp-boundary configuration will be investigated by means of the Rayleigu-Ritz variational principle (9-16): -274. p + s + 2 * :$-K,: « ;y ^m (12-57) U = i : •: where 21 P W . - , \ V + Tp(^-0 dx , Q = 7x(Ex B), s 2T1 2 W r£ 1. J- ( „ . "v J-B (n .EE) dS , WV [j! - f ( f JT" , (12-58) The variables £ and Q are connected by means of the boundary condition (8-29) on S: n . Q - B . 7 (a . E) - (n . VB . n)n . E , (12- 59) whereas fi.Q should vanish at the conducting wall, i.e. at infinity in this case. Since the system is axially symmetric we may Fourier- decompose % into independent components £(R,Z)e . From now on we will exclusively study perturbations of this form without indicating this by further subscripts. The case l = 0 requires separate treatment. It will be excluded from the present analysis, We will exploit the high-g tokamak ordering to minimize the expression (il-57) for w2 order by order. This way we ./ill eliminate the longitudinal components £ and Q and obtain a 9 $ problem in terms of the transverse 2-vectors E j_ and Q x only. The connection between the 2-vectors £ ^and 0^ is obtained from the boundary condition (12-59). Since n . 7B . n - - l/(eaR) *(RB )/3X » .275. this condition may be written u B 3(RB ) 1 x A •ï - $ R ae 3 A. % £ eaR 3* T. 'i/ Exploiting the normalized poloidal field variable B defined in Eq. (12-33) this gives to leading order: -i ( -p=-eB }n. . Q. = (Lq* - ~ — b ) -^- n. £. . (12-60) Since p only appears in W we may separately minimize with respect to Q. One should tlien keep n.Q fixed at the plas ma- vacuum boundary, so that n.£ is fixed and, hence, 6 W- = 5W = 0. Introducing the vector potential £ so that Q = VzA, we find by a reduction similar to that of Sec. VIII D, Eq. (8-47) , but in reverse order: wv = 4" \ Q2dxV = -~ \ n -K B-QdS +4-1 A . 7 x7 x A d~V . (12-61) Clearly, if n.| is kept fixed at S this expression is minimized by 7 :-: 7 x A = 0 , or 7 y. Q = 0. (12-6 2) In components: e l , ' = i>'.q , ('X X -1 ^ if.Q , (12-63) o-j y .276. From these equations and Eq. (12-60) it emerges that it is expedient to exploit the following dimensionless variables: (q*/e2B )6\ -\, (q*/eB )Q, -x. (1/a) 5, • (12-64) 09 o *• *• The equations (12-63) provide the attractive property (12-9): V* . Qx = 0 , 7A . §x = 0 , (12-65) so that Qx may be derived from either one of the conjugate harmonic potentials $ or *: - i(q*/eB )Qj_= - i V* $ = Vx5. (12-66) The longitudinal component Ö also may be expressed in terms of 1 he potential ?: 2 (q*/e Bn)Q. - - i V. (12-67) Hence, the vacuum energy may be written as wV + )dx (12 68) - "*« $** • 51 so that .277. n V 2 r 2 • Et 4- B H1 = - B (K - ) + 2cpr 7 /a 'v "• Z - - "PP €K C t- - (B K + 2cpK )/a. PP t Hence, Ws = - -Re t (B2K + 2epK J(n,.?^)2dX . (12-69) J p p t i1 %*- This clearly shows all the important ingredients acting in a global instability. Below we will give a detailed discussion of these terms. We may now estimate the various orders of magnitude of the contributions to W by means of Eq. (12-64) . From Eq. (11-58) for Wp and I, Eq. (12-68) for Wv , and Eq. (12-69) for Wv it s v emerges that the expressions W and W are of second order as compared to the expressions Wp and I. Thi3 implies that the first two orders of W, depending on the positive definite quadratic forms Wp and I, have to cancel .if we are to wind up with a significanSeparatint g problemtransvers. e and longitudinal dependences in Vr and I, p r 2 2 2 W =^-\ B (V1. r - £_/R) + (iV/lTH 2 P + YP(V,. £. + £D/R + ias./R) ! dt , (12-70) if 2 2D 1 = 'T- \ °^ * C> we obtain to zeroth order: 0) 2 2(0) H(o, 4f.x- ^ > ^ üi ' • - f^ = — 7^ 77T^ r~ • (12-72) I (0, 4J(l(t(0)2,{(0)2)dtp Consequently, the perturbations are marginal and incompress ible in the poloidal plane: .2(0) = 0 , V . K^ - 0 , (12-73) whereas the longitudinal displacement V, remains undetermined in this order. The first order term ( 0) 2 P W(D = ± j yp(v 5 x ) dx (12-74) vanishes trivially by virtue of the zeroth order result. The first significant non-vanishing growth rate is indeed obtained in second order: 2 2 2 S(2) V(2) w < > = [WP< > + W + W ] /I<°\ (12-75) where «p(2)-i-i LW til)-ti0>""2*«I»X't?)2l*tp • (12-76) and Ws and w ' are given by the expressions (12-69) and (12-68), respectively. Minimization of w2 (21' with respect to |(1) is trivial: ( )/R 7A . % P = 4° • (12-77) Since £ only appears in I , the maximum growth rate is obviously obtained for C(0) = 0. (12-78) .279. Thus, .2„2 ( )2 V,P(2 ) . 1 12s- ( r ° dtP, (12-79) 2 R„ 2 J I(0) - i Aé0)2 dtP, (12-80) so that we have obtained a problem in terms of the leading order transverse displacement E,(0 ) , whereas both of the leading order components of the vacuum field Q, for consisting of the notation to be denoted as 5k and Q' still appear. Writing 2 2 a,2 = -r-£- + W t W , (12-81) Ro P , it is clear that, for w' < 0, the growth rate is maximized by minimizing the norm I subject to the constraint that n •£ be held fixed on S. This leads, by an analysis completely analogous to the one leading to Eq. (12-62) for the vacuum energy» to the result that £^ should be curl-free. Com bining this result with Eq. (12-52) gives v. . S(°} = 0 , V* C(0> = 0 , (12-S2) so that £ nay be derived from either one of the conjugate harmonic potentials xor!^: (JL )F^' - - iv* v - 7, Q - (12-83) Hence, by virtue of Eq. (12-14), 2 "> A 2 A 2 I ?r IT a B „ / „ ïïa B :p . li,„2 . HJJL,R ' IJ ", . 3v* d» . r^jR i -' i JI , - „.42;>v . dl p R c o o (12-84) .280. This completes the reduction of the problem to that of calculating harmonic potentials x or Q for the plasma and harmonic potentials t and 5 for the vacuum. Let us now eliminate all trivial scale factors by introducing dimensionless variables -2 , 2 .2/ 2D2. 2(2) to = (pa q*Ve BO)Ü> , W = (q*2/e22ir2 ea2R B2)W<2), (12-85) Ï i (l/2ir2e a4 R p)I(0). o Thus, Ü2 = W/T, (12-86) where we have from Eq. (12-84) WP - (Hq*)2I , I = JIM X* lv" dA ' (12-87) from Eqs. (12-69) and (12-83) Ws 2TT J p p p t e2 3X and from Eqs. (12-68), (12-66) and (12-67) Finally, the boundary condition (12-60) in terms of the potentials reads: ** /„ * i 9 C \ lx. which may be integrated once: .281. D $ • ?X. * = ~ i(aq*/eBo)E-V £ J.q* - i -^ -^ . (12-89) Using this boundary condition and the conjugate relations (12-66) and integrating by parts the expression for W may be reduced as follows: This completes the reduction, where we have automatically chosen for a description in terms of the potentials x a^d $ because of the simplicity of the boundary condition (12-89). Collecting the Eqs. (12-86)-(12-90) we may now state the stability problem in a very compact way: To second order the growth rates of external kink modes in a sharp-boundary high-beta tokamak of arbitrary cross-section are given by 2 {jlq*)2( * |X dA . II(62K + ep K ) |^| dX _if* |i dx _ M jA 3v ejpp pt 3X[ J 3v (JJ2 = f Jx*UIX A", (12-91) where F AlX * 0 on a , (12-92) V AA« = 0 on 0 , (12-93) ? = Px on c . ( 12 -9 4 ) The shape of the cross-section is prescribed: x = x (A), o o y^ - v^ U). This determines the curvatures o o «•p (X) and Kfct(A ) through Eq, (12-17). The only parameters for the stability .282. problem turn out to be c6 and Zq*. For the computations an auxiliary parameter k2, running from 0 to 1, is used. It is related to E6 through Eq. (12-31). For a prescribed cross-section and a given value of ES (k2) the poloidal field 6 (X) is found from Eq. (12-33). P Notice that both the plasma energy and the vacuum energy are positive definite, so that instabilities may only arise through the surface term. The first term of W is nega tive definite for convex cross-sections, whereas the second term is negative on the outside and positive on the inside cf the torua. The first term is the one responsible for external kink modes in low-& systems. One should not identify the second one as the only one responsible for ballooning modes in high-B sys tems. As we shall see, one can extract a similar contribution from the first term that is twice as large and of the same sign, so that the ballooning term becomes three times more effective. It remains to solve the Laplace equations (12-92) and (12-93) iit order to relate the normal derivatives v and * to X and J on C. Once this has been done,the expression (12-91) only contains line integrals along C involving the unknown func tions x(*) an<3 5(X) related to each other through the boundary condition (12-94), so that the final minimization of w2 is one with respect to x(*) only. That very last part of the problem has to be carried out numerically. To solve the Laplace equations (12-92) and (12-93) one may resort to two methods basically. In the first method one makes use of separable coordinates. This method is only appli cable for a restricted class of cross-sections, typically cir cular and elliptic ones. The second method employs Green' s theorem .283. relating a harmonic function and its normal derivative on the boundary curve c, which may have any shape now. For simplicity we choose for the first method and treat the case of a circular cross-section. For the general method the reader is referred to the literature CRefs. 6 and 7). E. NUMERICAL SOLUTION FOR CIRCULAR CR0S5-SECTIGNS The stability problem is treated by means of a Fourier analysis in the angle A giving rise to Fourier com ponents exp (imX) which are coupled due to the angular varia tion of the poloidal field B (A) and through the angular varia tion of the curvatures K (A) and < {A). The mode coupling through B„(A) originates both from the surface term W and from the vacuum term W through the boundary condition (12-94). Let us now complete the solution by specifying the shape of the cross-section to be circular, so that v - r , A « 9 , x " cose , y = sine , e= 1 , (12-9 5) o o < - l , < - cose . p t The noloidal field is determined as in Eq. (12-45): (12-96) a (?>) = ( f/2£f!;2)) \l ~ k2 s in2 -f" '•> - 1 + Y"^ t cos P ' V 2 fr? ra wh-^re the terms t^ should be calculated numerically. —s The expression in brackets in the surtacs energy W becomes .284. S2 K + eB * = S2 + eg cosö PP P t p p 2 ir 2 2 ('11 - k^ sin'sin 1 .h- 8e)) ++ EB COS8 4E2(k2) 2 p (l - i k2) + 3e6 cosO , (12-97) 4E2(k2) 2 p which shows the promised factor of 3. Hence, the poloidal variation of the equilibrium will couple the modes exp(ime) , exp(i(m+l) 8) , and exp(i (m-1) 8) through the surface term 3eg cos 0, but it will couple all the modes through the infinite series (12-96) for B_(e) which enters the boundary condition (12-94). The solutions of the Laplace equations (12-92) and (12-93) for a circular cross-section are easily obtained: 1/2 m X(r,6) - Yl' xm |mf rl leXp(ime) , m=~" (12-98) *(r,8) - ) 1 sg(m)[raf1/2 r~'mIexp(im8) , • * ID where the prime on the summation sign indicates that the m = 0 contribution should be left out. This component would require compressibility in the poloidal plane which we have shown to be absent in this order. The functions x(rre) and 5(r,8) ate represented by the infinite dimensional vectors £ = f x } and \ = ( 5m)/ which may now be considered as the unknowns. The vector | is related to the vector g through the boundary condition (12-94) : -2ï»5. ^m sg(m) |mt e = [Aq* - i(l + \ tM cos MG)i \ x„ ; Uj t Applying the operator \de» im sg(r.) jm| ' on both sides of this equation gives the wanted relation between | and x: Ó = F-x , (12 -99) vrnere 1/2 P = lq* sg(m)6mM + \ sg(mii)|mu| t, m u m U 2 ' | m-u| Substituting the expressions (12-98) into the norm I and the vacuum energy W we find that these are represented by the unit matrix: oo 1 =2T$X* Tr" de " Ë' xm =r* • (12-100) f *+ OB *v s - h ** "^ d9 = C' K - H • (12-ioD The surface energy becomes ws = - ±[ (S2* + ,e = 1 L':VC • sS(inu) !mu|1/2. - 4 1(62 + s8 cosö)e"i<1B"u)9dG m t 2* J P V - !5'^S'x . (12-102) Writing the eigenvalue oroblem as + -. • -• •i, 0 — , (12-103) we find for the explicit form of the matrix W: .286. W = Uq*)2ö - ^ sgdn-^lmpl1 2[(l-y k2) * •+ 3tS i , ,] 2 2 P mu trip 2 ^ <£ 2£ (k ) '' '"' co + 51 * i * i • (12-104) f—' mm ray Since the norm is represented by the unit matrix, the eigen values of the real symmetric matrix J[J are the required eigen- values w2 of the variational problem (12-91). Truncating the representation to some reasonable number of harmonics then leads to the completely standard numerical problem of cal culating the eigenvalues of a real symmetic matrix. The low est eigenvalue determines the stability of the system. Mar ginal eigenvalues are found by fixing e 8 and scanning in the parameter iq* until u2 =0 to a sufficient degree of accuracy. Alternatively, we may perform a o-stability anal ysis by the same technique except that we now scan until 2 2 w = -a to a suffcient degree of accuracy. In the limit e 8 •* 0 the mode compling disappears and we obtain from Eq. (12-104): W + [ in precise agreement with the low-0 straight cylindrical result of Eq. (11-74). For finite eB numerical solution is the only way to get solutions of Eqs. (12-103) and (12-104). The results of a marginal analysis are as shown below: (Vefh i «i •-If * , »* *-'f .287. There is one stable region that generalizes the Kruskal- Shafranov limit (11-78) for finite 8. In this region 1=1 is the most unstable mode since q* scales with i . Hence, we may put I = 1 to get the overall stability boundary. There is clearly an optimum value of q* which maximizes °/z in this region. For large q* (low current density) the sta bility is good but the equilibrium pressure which can be contained is small. At low values of q* (higher currents) the equilibrium conditions become less severe, but the stability now leads to a limitation in B- The optimum occurs where the equilibrium and stability curves intersect. This condition is given by q* = 1.7 , 3/e = .21 . (12-106) The harmonic structure of the marginal mode, which is m = 1 at eö - 0, is predominantly m = 2 for higher values of e 6 . The optimum value of q* given above is considerably larger than the low-0 stability limit q* = 1. Moreover, the resulting value of 3/e is quite small indicating a need to improve the configuration. This can be done by adding a Inyer of force-free currents outside the main plasma core and by shaping the cross-section. These effects are considered in detail in Ref. 7. RT.rCRENCES 1. G.K. Batcheior, An Introduction to Fluid Mechanics (Cambridge University Press, London, 196 7). 2, P.. Gajev/ski, Phys. Fluids 15_ (1972) 70, "Mngnetohydrodynamic equilibrium of an elliptic plasma cyli .288. W.A. Newcomb, Ann. Phys. (NY) jH_ (1973) 231, "Gyroscopic-quasielastic fluid systems". J.P. Fieidberg and F.A. Haas, Phys. Fluids 1£ (1973) 1909. "Kink instabilities in a high-B tokamak". J.P. Freidberg and F.A. Kaas, Phys. Fluids 17_ (1974) 440. "Kink instabilities in a high-B tokamak with elliptic cross-section". J.P. Freidberg and W. Grossmann, Phys. Fluids 18 (1975) 1494. "Magnetohydro dynamic stability of a sharp boundary model of tokamak". D.A. D'lppolito, J.P. Freidberg, J.P. Goedbloed, and J. Rem, Rijnhuizen Report 78-108 (1978), Phys. Fluids. 21 (1978) 1600, "High-beta tokamaks surrounded by force-free fields". VECTOR IDENTITIES .289, a* (fe*£) - £* (a*b) - fa* <£xfc) (1) a x (b^c) = a-cb-a-bc , (axb) x c = a'cb-b'ca (2) V x V(p = 0 (3) V • (Vxè) = 0 (4) V x (Vxa) = 77 • a - Aa (5) v • («a) = a* V*-MV • a (6) V x ((^a) = V$ x a+ $7 x a (7) a x (V*b_) = (Vb) • a - a • 7b (8) (axV) xb = (Vfe) • a.-aV-fe s x v(a-fe) = (?a> • fe + (vbj • a a* vfe + fe. va + a (7xbj + &X (7xê) uo) v • (afe) s 'a • vfe + fev • a (11) V • (axb) = b • Vxa- a • 7*b (12) 7 x (Sxbj = 7 • (fea - afe) = av • fe + te - ^a - fev • a - a • ^fe (13) rrr xr (14) 7 • adT - (Vxa) • nda » ia-du (stokes) (18) a- a*£ ! J, (8*V) x a do »