lecture notes on ideal magnetohydrodynamics
TRANSCRIPT
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 . I n t r o d u c t i o n 1
I I . D e r i v a t i o n of macroscop ic e q u a t i o n s 4
A. Boltzmann e q u a t i o n 4
B. Moments o f t h e Boltzmann e q u a t i o n 6
C. Two-f lu id e q u a t i o n s T 12
D. O n e - f l u i d e q u a t i o n s 13
I I I . The model of i d e a l MHD 23
A. I n t r o d u c t i o n 23
B. D i f f e r e n t i a l e q u a t i o n s 24
C. Boundary c o n d i t i o n s 27
D. Equa t ion of s t a t e 29
IV. C h a r a c t e r i s t i c s 33
A. P a r t i a l d i f f e r e n t i a l e q u a t i o n s i n two
independen t v a r i a b l e s 33
B. C h a r a c t e r i s t i c s i n i d e a l MHD 36
V. C o n s e r v a t i o n laws 49
A. C o n s e r v a t i o n form of the i d e a l MHD e q u a t i o n s 49
B. Shocks 51
C. Globa l c o n s e r v a t i o n laws 56
D. Energy c o n s e r v a t i o n f o r models 2 and 3 59
VI . An example : Dynamics of t he screw pinch 63
A. Pinch exper imen t s 63
B. Mixed i n i t i a l - v a l u e bounda ry -va lue problem — 66
C. F i e l d - l i n e t o p o l o g y 73
D. Reduct ion of t h e plasma e q u a t i o n s 77
E. C i r c u i t e q u a t i o n s 79
F . S o l u t i o n of t he problem 82
G. F lux and energy c o n s e r v a t i o n 86
V I I . Lagrang ian and Hami l ton ian f o r m u l a t i o n s of i d e a l
MHD 92
A. Summary of some concep t s of c l a s s i c a l
mechanics 92
B. Kinemat ic c o n s i d e r a t i o n s — — - — - — - - 96
C. Lagrange and Hamilton e q u a t i o n s of motion 101
VIII. L i n e a r i z e d i d e a l MHD 105
A. I n t r o d u c t i o n 105
B. L i n e a r i z e d e q u a t i o n of motion 108
C. Boundary c o n d i t i o n s — 112
D. S e l f - a d j o i n t n e s s of the f o r c e - o p e r a t o r 117
E. Mamil to . i ' s p r i n c i p l e 123
IX. S p e c t r a l t heory 128
A. Mathemat ical p r e l i m i n a r i e s 128
B. R a y l e i g h - R i t z v a r i a t i o n a l p r i n c i p l e 132
C. I n i t i a l v a l u e problem 135.
D. S t a b i l i t y . The energy p r i n c i p l e 138
E. o - S t a b i l i t y 145
X. Waves i n p l ane s l a b geometry 149
A. Waves i n i n f i n i t e homogeneous p lasmas 149
B. The c o n t i n u o u s spec t rum fo r inhomogeneous
media 158
C. Damping of Alfvén waves 170
D. S t a b i l i t y of p l a n e f o r c e - f r e e f i e l d s . A t r a p 191
XI. The d i f f u s e l i n e a r p inch 204
A. E q u i l i b r i u m model 204
B. D e r i v a t i o n of t he Hain-LÜst e q u a t i o n 208
C. E q u i v a l e n t sys tem of f i r s t o r d e r d i f f e r e n t i a l
e q u a t i o n s 216
D. Boundary c o n d i t i o n a t t h e plasma-vacuum
i n t e r f a c e 219
E. O s c i l l a t i o n theorem 222
F . Newcomb's marg ina l s t a b i l i t y a n a l y s i s . Suydam's
c r i t e r i o n 2 33
G. Free -boundary modes 242
H. F ixed-boundary modes - 247
I . o - s t a b l e c o n f i g u r a t i o n s 251
X I I . Sharp-boundary h i g h - b e t a tokamaks 254
A. I n t r o d u c t i o n
B . E q u i l i b r i u m 260
C. Vacuum f i e l d s o l u t i o n fo r the c i r c l e 267
D. V a r i a t i o n a l p r i n c i p l e f o r s t a b i l i t y 273
E. Numerical s o l u t i o n f o r c i r c u l a r cross-sect ions 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 h e u r i s t i c der iva t ion (chapter I I ) in order to render
some c r e d i b i l i t y to the equat ions and a l so to obta in some
understanding of the domain of v a l i d i t y of the idea l MHD de
s c r i p t i o n . S t r i c t minds may skip t h i s chao t ic exposi t ion and
s t a r t reading a t chapter I I I .
The MKSA system of u n i t s has been chosen for the
next chapter , whereas s t a r t i n g with chapter I I I u w i l l be
put equal to 1 for convenience. The only opera t ion needed to
re tu rn t o the conventional systems of un i t s i s then to d ivide
B2 by v (MKSA system of un i t s ) or 4TT (Gaussian system of
u n i t s ) .
.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
3 3 f (£,^,t) a r d 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
f ields and the averaged in te rna l f ie lds or iginat ing from the
long-range in te rpa r t i c l e in te rac t ions . The PHS of Eq. (2-1) gives the
rate of change of the d is t r ibut ion function due to short-range
in te rpa r t i c l e in te rac t ions , which are somewhat a r b i t r a r i l y
called co l l i s ions . Neglect of these col l i s ions leads to the
Vlasov equation;
3f 3f q 3f
jr + i-jf * IT<I***V'TT - °- < 2 - 2 >
A closed system of equations i s 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 d i s t r i -
. 5 .
bution funct ion . The zeroth moment gives the number of p a r t i
c les of spec ies a per u n i t volume:
v*-0 5 K(«-t)d3v' (2_3)
whereas the first moment gives the average velocity:
i r 3 u (r.t) * v = — ; —r- i vf {r,y,t)d v. ^a ^ ' -u n (£.t) J \, a 't'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 V a ^ ' 0 ' (2"5)
a.
a, a
Since a l l charges and cu r r en t s in the plasma are supposed to
be f r ee , p o l a r i z a t i o n and magnetizat ion e f f e c t s are n e g l i g i b l e
so tha t Maxwell's equat ions only involve % and £ . In the r a t i o
na l ized MKSA system of u n i t s we then have: 9 B
*** " " ST • ( 2 " 7 )
3E VxB - y J + — T7 , (2 -8 )
V<E - i / e , (2-9) 'u 0
7«B * 0 , (2-10)
-1 /2 where c= (e y ) . o o
* Average q u a n t i t i e s of a function g ( r , ^ , t ) a re defined as
5(«'fc) ' n T r f - t r l 9 ^ ^ ' ^ **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 a e
.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. I t suffices to l i s t 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
f c d3v = 0 ( i n c l u d i n g 0 = a ) . (2-12)
Also, momentum and energy are conserved for collisions between
like part icles:
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 + K x C g a d 3 v = °» ( 2 " 1 5 )
! K v 2 C a B d 3 v + 1 K v 2 Ct a
d 3 v B °* (2~16)
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 _ J i d 3 v , _ £ ( d e f . ( 2 . 3 ) ) i
df v.?-SLd3v . V.(n oU a ) (def . ( 2 - 4 ) ) ,
q 3f J q d i
^l*KxV "^rdiy " ° ( i n t e g r a t i n g by p a r t s ) , m % 'Kt *& 3 v
o t»
J C d3v - 0 (summing Eq. (2-12).
Consequently,
3n T T * ' ^ V ^ ' °' (2-17)
which i s the c o n t i n u i t y equation for p a r t i c l e s o f s p e c i e s c.
Mul t ip ly ing Eq. (2-1) by v and i n t e g r a t i n g over
v e l o c i t y space r e s u l t s in the f o l l o w i n g terms:
3f
1 nV3v - ^vs.^ r 3 f r
V'—— yd3v • V» yyf d3v - 7 - (n vv)
(where averages are def ined i n agreement w i t h Eq. ( 2 - 4 ) ' ) /
f q 3f q n J -2-(E+yxB).T-SLvd3v - - -SL_2.(E+U x B ) •> m *v» *v *»# 3 v <v m <v 'vet <u
a ^ a
J C a * d 3 v - I C a e ^ d 3 v " * » > • Hence, the f i r s t moment o f Eq. (2-1) g i v e s
r - (n m u ) + V*Utn yy) - n q (E+u xB) - J C .in vd 3 v ,
(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 n a 1 ,
r q 3f n q [ _ £ ( E + v x B ) . a 2 d 3 v = _ 2-2-^E.u ,
a 'v a
f C v 2 d 3 v = f c 0 v 2 d 3 v ( 3 i * a ) . J a J a p
Multiplying these terms by y ma gives
— (n -z-m v 2 ) + V«(n T-m v 2 v ) - n q E-u = C . rin v 2 d 3 v , 3t a 2 a s a 2 a ^ a a'v ^a J a£> 2 a
( 2 - 1 9 ) 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 i t 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)
where P ( r , t ) = •=• n m v 7 7 ti-??\
. 1 0 .
and ir ( r , t ) i s the p a r t due to the anisot ropy of the d i s
t r i b u t i o n funct ion. Likewise, the random v e l o c i t y p a r t of the
s c a l a r v3" occurr ing in the energy equat ion (2-19) gives rise
to a q u a n t i t y r e l a t e d to the mean k i n e t i c energy of p a r t i c l e s
in the frame moving with v e l o c i t y u .which we define to be
a
m
the temperature T
T (r.,t) = * I v « 2 fr t
( M » t ) d 3 v r <2-23> a * 3 k n a ( ^ , t ) J a a * %
where k i s Boltzmann's cons tan t . Not ice , P « n kT (2-24)
o a a
F i n a l l y , the random ve loc i t y p a r t of the vec to r v2v occurring
in Eq. (2-19) gives r i s e to a q u a n t i t y
Wfrt} - K V a 2 ^ ' (2-25)
which i s the heat flow by random motion of the p a r t i c l e s of
spec ies a.
The c o l l i s i o n terms may a l so be s imp l i f i ed by
transforming to the moving frame \ - From Eq. (2-12) i t follows
t h a t only the random p a r t c o n t r i b u t e s to the RHS of Eq. (2-18):
fc 0m yd3v - f C .m v 'd3v = R , (2-26)
which i s the mean momentum t r a n s f e r from p a r t i c l e s of spec ies S
to p a r t i c l e s of spec ies o. By the use of the same r e l a t i o n we
f ind t h a t the RHS of Eq. (2-19) may be w r i t t e n as
J °asim v 2d 3v- fc flm u »v'd3v + fc _m v , 2 d 3 v
• u «R + Q , -va ^a o
. 1 1 .
where
Q _ f c „ -Ln v , 2 d 3 v , ( 2 - 2 7 ) o = J a3 2 a a
which i s the generated h e a t i n the system of p a r t i c l e s o due
to c o l l i s i o n s with p a r t i c l e s 6.
S u b s t i t u t i n g the d e f i n i t i o n s (2-21)-(2-27) the equa
t i ons for momentum and energy conservat ion take the form
—(n m u ) + V«(n m u u ) + 7*P - n q (E+u xB) - R , (2-28)
Tr(yn„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 s impl i f i ed by us ing the
con t i nu i t y equation (2-17) to remove con t r ibu t ions 3n / 3 t ,
whereas the energy equat ion (2-29) may ba s impl i f i ed by
removing the bulk k i n e t i c energy p a r t by means of both Eq.
(2-17) and (2-28) . Defining the Lagrangian de r iva t i ve along
flow l i n e s u a ,
the th ree moments of the Boltzmann equat ion then take the form
dn
7T * V « B * °' (2_31)
n m - ~ + 7'? - n q (E+u xB) - R , (2-32)
dT
7 na
k d t - + Ja;^a + V ' S a = Qcc' < 2 " 3 3 >
.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. - " <L + («<-£»>'£ • (2-35)
The t w o - f l u i d moment e q u a t i o n s t h e n r e a d :
dn •— + n V-u » 0 , de e ^e '
da.
dt l T>I
n m -r-S- + V*P + en (E+u xB) • R, e e d t 've e 'v *\<e *v *v
n im i IT * 7 ' £ i ' Z e n i ( ^ i x ^ " " *'
dT
(2 -36)
(2 -37)
I n e k d T + 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
. 1 3 .
whereas the i s o t r o p i c p a r t s o f the pres sure t e n s o r s read :
Pe " ne
k T e ' Pi = n ikV ( 2 _ 3 9 )
We would have produced a c l o s e d s e t of t w o - f l u i d
equat ions i f the a n i s o t r o p i c parts ir and £ . o f the pressure
t e n s o r , the heat conduct ion terms h and h . , the momentum
t r a n s f e r R, and the heat production Q were known in terms of
the macroscopic v a r i a b l e s n , n . , u , u. , T , and T . . A way r e l ^e "^i e i
t o e f f e c t t h i s i s t o s imply put a l l t h e s e terms equal to zero.
This procedure may be hidden i n a long s t o r y about large and
small parameters , but t h i s i s a c t u a l l y what i s done to g e t
the t w o - f l u i d equat ions o f plasma theory .
D. ONE-FLUID EQUATIONS
The o n e - f l u i d equations o f magnetohydrodynamics are
produced by combining the p a i r s o f equat ions ( 2 - 3 6 ) - ( 2 - 3 8 )
by means of e x p r e s s i o n s for the t o t a l mass d e n s i t y P , the
c e n t e r o f mass v e l o c i t y v f the charge d e n s i t y t , and the cur
rent d e n s i t y j :
-v
p = n m + n . m . . e e i l
X s ( n e m e2e + n i m i S i ) / p ' (2 -40)
T = - en • Zen . , e l
j = - en u + Z e n . 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):
. 1 4 .
Z p - ( m . / e ) x Zo
n = \ e ra.+Zm * m.*Zm
i e i e
P + ( m / e ) T p
n . =i -\, > 1 ra.+Zm *V m.+Zm
i e i e
Z e p v - m . j m .
ZeP-m.T Zep
^ e ^ -v e . 'ui ep+m t *\« ep ^
( 2 - 4 1 )
where the approximations on the RHS are due to the assunvtion
of q u a s i - n e u t r a l i t y :
n - Zn. << n or m.T << ep. (2-42)
Quas i -neu t r a l i t y i s a good approximation for the study of
plasma phenomena with a sca le length L such t h a t
L >> *D . (2-43) 1 /2 where the Debye length i s defined as X = (e kT/e2n ) =
1 /2 = v.u ^A> ~i where u> = (n e 2 / e m ) ' . For a thermonuclear t h , e pe pe e o e
plasma with n = 10 cm , T « 10 °K, B = 10 Wb/m2 = 10s gauss -4 12 -1 we have X_ = 7x10 cm and io = 6x10 sec . so t h a t t h i s D pe
condit ion i s e a s i l y s a t i s f i e d for the global phenomena we want
to s tudy.
Mult iplying the p a i r of Eqs. (2-36) by the masses
and adding them gives the equation of mass conservat ion:
§f + v-(pv) - 0, (2-44)
whereas multiplication by the charges and subtraction
results in the equation of charge conservation;
. 1 5 .
| f • 7- j » 0 . (2-45)
Likewise , adding the p a i r o f equat ions (2-37)
w h i l e us ing the approximations on the RHS of Eqs. (2-41)
r e s u l t s i n the equat ion of n o t i o n :
3v a m. PaT * «>X*VX * - £ - L J*7J * v * p - TE - jxB - 0 , (2-46)
where P = Pe + £ i - N o t i c e t h a t t h i s equat ion transforms to the
usual Navier-Stokes equat ion o f hydrodynamics i n the case t h a t
e l e c t r i c and magnetic e f f e c t s are absent .
M u l t i p l y i n g the p a i r o f Eqs. (2-37) by the
quotient charge/mass and s u b t r a c t i n g r e s u l t s i n an equat ion for
the rate of change o f the current d e n s i t y , which i s known under
the name g e n e r a l i z e d Ohm's law:
- 1 + V-f-rvv • iv + vi - ~ - ( l - z—•> —i 3 3 - —(V»P - Z — V'P.;
+ _ L ( 1 - Z - * ) i x B - ^-2.(E+vxB) - - ^-(1+Z-^)R . e i e i e x
(2 -47)
The term with ixg i s known as the Hal l term.
F i n a l l y , adding the equat ions (2-38) r e s u l t s i n
the heat balance equat ion:
„ „ i e t Z m p . - m . p
nt tn. + p . V « ( — j ) - p „ ' . # ( 7 T 7 J > • r :Vu • r . : 7 u , • V»h =
- 7 ^ r ( l * Z - S . ) j - R ( 2 - 4 8 )
.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 s t i l l
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 rat io of electrons and ions is small:
m <<m., (2-49)
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:
. 1 7 .
3v PTT + py'y + V P - T? - i x B = ° . (2-54)
r ^ + V-(TVV-MV+VJ) - ^ £ - ( E + v x B ) = - ^ £ n j , (2 -55) e i e i
2 | E + 2 v . 7 p + 2 p v . v » r ) j 2 . (2 -56) 2 3 t 2 ^ K 2 F , \ . J s
Toge the r w i t h t h e Maxwell e q u a t i o n s (2-7) and (2-8) and t h e
mass and charge c o n s e r v a t i o n e q u a t i o n s (2-44) and (2-45)
t h e s e e q u a t i o n s c o n s t i t u t e a c l o s e d s e t of e v o l u t i o n e q u a
t i o n s f o r t he v a r i a b l e s p ( r , t ) , T ( r , t ) , y ( r , t ) , j ( r , t ) ,
p ( £ , t ) , g ( £ , t ) , and J J ( £ , t ) . The Maxwell e q u a t i o n s (2-9) and
(2-10) may then be c o n s i d e r e d a s i n i t i a l c o n d i t i o n s on E and
B s i n c e they remain s a t i s f i e d i f they a r e i n i t i a l l y s a t i s f i e d ,
by v i r t u e of t he Eqs . ( 2 - 7 ) , ( 2 - 8 ) , and ( 2 - 4 5 ) .
Although the sys tem ( 2 - 7 ) - ( 2 - 1 0 ) , (2-44) - ( 2 - 4 5 ) ,
( 2 - 5 4 ) - ( 2 - 5 6 ) i s m a t h e m a t i c a l l y c o n s i s t e n t , t h e corresponding
p h y s i c a l problem i s q u i t e c r a z y . Comparing t h e o r d e r s of
magni tude of the terms i n t he g e n e r a l i z e d Ohm's law, i t turns
out t h a t t he terms 3 j / 3 t and v , ( T V V + J V + V J ) a r e much s m a l l e r r\, ' W '\i\ *V\»
than the remain ing t e r m s . Numerical computat ion of t he e v o
l u t i o n of the c u r r e n t d e n s i t y by means of Eq. (2-55) would
be v i r t u a l l y i m p o s s i b l e . The terms may be n e g l e c t e d i f a
c o n d i t i o n i s met t h a t i s s l i g h t l y more r e s t r i c t i v e than t h e
q u a s i - n e u t r a l i t y c o n d i t i o n ( 2 - 4 3 ) , v i z . :
L >> c/u - K'c/v . . (2-57) pe D ch,e
. 1 8 .
Th i s c o n d i t i o n i s e a s i l y s a t i s f i e d ( fo r t he example g iven - 3 below Eq. (2-43) one f i nds c/u = 5x10 cm). However, t h e pe
n e g l e c t of t h e s e te rms changes the ma thema t i ca l n a t u r e of t h e
sy s t em, s i n c e now we no l o n g e r have an e v o l u t i o n e q u a t i o n fo r
j . Th i s does n o t p r e s e n t a r e a l problem because we o b t a i n an
a l g e b r a i c r e l a t i o n between j and E i n s t e a d by which we may
e l i m i n a t e j from t h e problem. Th i s r e l a t i o n i s p r o p e r l y •v
c a l l e d Ohm's law.
Summarizing:
Under t h e c o n d i t i o n s ( 2 - 4 3 ) , ( 2 - 4 9 ) - ( 2 - 5 3 ) , (2-57) t h e moment
e q u a t i o n s of t he Boltzmann e q u a t i o n t o g e t h e r w i t h Maxwel l ' s
e q u a t i o n s form the c l o s e d s e t of r e s i s t i v e MHD e q u a t i o n s fo r
t h e macroscop ic v a r i a b l e s p , T , v , p , Jg, and B: | | + V.(pv) = 0 , ( c o n t i n u i t y ) < 2 - 5 8 >
f l • ' • j " ° ' ( cha rge) (2-59)
dv. p — t 7 p • TE - jxB = 0 , (momentum) (2 -60)
2 ? ! * 2 ^ " V p * 2 p V *^ " n ^ 2 ' < i n t e r n a l e n e r gy) (2-61)
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
. 1 9 .
and i n i t i a l l y the following condi t ions need to be s a t i s f i e d : 7-E = T/C , (2-65)
% o V-B - 0. (2-66)
Instead of cons ider ing (2-59) as the evo lu t ion equat ion for
T, one a lso could drop i t s ince i t i s a consequence of (2-63)
and (2-65) . Then, we do no t have an evo lu t ion equat ion for T ,
but we wr i t e T = e 7*E ins t ead so t h a t x may be e l imina ted o ^
from the equa t ions .
Next, we turn to n o n - r e s i s t i v e MHD. Res i s t i ve
e f f e c t s may be considered n e g l i g i b l e ( e . g . , compare the term
nj with vxB) i f
R„ s v vL/n >> 1 . ( 2 - 6 7 ) M 0
Here, IL. i s c a l l e d the Magnetic Reynolds number i n analogy
with the hydrodynamic Reynolds number R = vL/v, whichmeasures
the importance of viscous e f f e c t s . This t u rn s Eq. (2-64)
i n t o Ohm's law of i d e a l MHD: E + vxB » 0 . (2-68)
This assumption changes the cha rac t e r of Eq. (2-64) from one
t h a t determines } i n t o one t h a t expresses E in terms of v and *\»
B. We then need another equation determining j . Le t us take
Maxwell's equat ion (2-63) for t h a t purpose . That equat ion then
changes from an evolu t ion equation for Jjj (which i s no longer
needed s ince £ i s now considered as a known q u a n t i t y from Eq.
(2-68)) in to an expression for j .
The i n t e r p r e t a t i o n we j u s t gave of Maxwell's
equation (2-63) i s somewhat confusing. I t i s r.ore cons i s
t e n t , phys ica l ly as wel l as mathemat ica l ly , to ge t r i d of
.20.
the displacement current altogether by realizing that we are
dealing with flows that satisfy
v2/c2 < < 1. (2-69)
Thus , we r e t u r n t o t h e s o - -^1 l e d pre-Maxwel l e q u a t i o n s ,
c h a r a c t e r i z e d by t h e f a c t tha*" Eq. (2-6 3) i s r e p l a c e d by
Ampere 's law (as Ampere knew i t ) :
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 7. ~ - - e -2- 7-(vxB) * 0 3t a 3c o 3t <\. "~
i n g e n e r a l . C l e a r l y , f o r ma thema t i ca l c o n s i s t e n c y someth ing
more i s needed t o r e s t o r e t h e peace in t he s y s t e m . The b e s t
way of f i n d i n g a c o n s i s t e n t s e t of e q u a t i o n s i s t o app ly an
o r d e r i n g i n t h e s m a l l p a r a m e t e r v 2 / c 2 . One t h e n f i n d s t h a t
t h e term TE i n t h e momentum e q u a t i o n i s an o r d e r s m a l l e r
t h a n t h e o t h e r te rms s o t h a t i t may be d ropped . A f t e r t h i s ,
a l l e q u a t i o n s a r e o f t h e same o r d e r , e x c e p t t h e cha rge conser
v a t i o n e q u a t i o n (2-59) which i s one o r d e r in v 2 / c 2 s m a l l e r .
In o t h e r w o r d s , t o l e a d i n g o r d e r i n t h e s m a l l p a r a m e t e r vVc 2
t h e charge c o n s e r v a t i o n e q u a t i o n may be d ropped . P o i s s o n ' s
e q u a t i o n (2-65) may be used t o c a l c u l a t e T , b u t s i n c e i t does
n o t occur in any of t h e o t h e r e q u a t i o n s , i t may be dropped
as w e l l . The r e s u l t i n g s e t of e q u a t i o n s i s a m a t h e m a t i c a l l y
c o n s i s t e n t s e t , which en joys the p r o p e r t y of b e i n g G a l i l e a n
i n v a r i a n t .
In c o n c l u s i o n , in n o n - r e s i s t i v e i d e a l MHD r e s i s
t i v i t y , d i s p l a c e m e n t c u r r e n t , and s p a c e cha rge e f f e c t s a r e
.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 Univers i ty , NY0-6486-I, New York, 1956).
2. A.A. Blank, K.O. F r i e d i c h s , and H. Grad, Notes on Magneto-
hydrodynamics, V, "Theory of Maxwell's Equations without
Displacement Current"
(New York Univers i ty , 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).
. 2 3 .
I I I . THE MODEL OF IDEAL MHD
A. INTRODUCTION
Why should a modem t h e o r e t i c a l p h y s i c i s t be
i n t e r e s t e d in idea l MHD? Remember: No quantum ef fec ts are
taken in to account, n e i t h e r are r e l a t i v i s t i c cor rec t ions
considered; in the de r iva t ion of the preceding sect ion a l l
k i n e t i c e f f ec t s were removed, whereas f i na l l y the neg lec t
of the displacement cu r ren t even removed electromagnet ic
waves from the system. In o ther words, we have moved back
ward in time to a period p r i o r t o , subsequently 1925
(Schrödinger equa t ion ) , 1905 (specia l theory of relativity) ,
1872 (Boltzmann e q u a t i o n ) , and f ina l ly 1865 {Maxwell equa
tions) . All i n t e r e s t i n g modern physics seems to have been
removed from the system so t ha t we wind up with a completely
c l a s s i c a l f i e l d t h a t could have been s tudied more than 120
years ago.
Never the less , th ree important reasons may be l i s t e d
t h a t should be s u f f i c i e n t ground for i n t e r e s t in t h i s f i e l d :
- Ideal MHD i s the s imples t physical theory t h a t s t i l l makes
sense in the context of confinement of plasmas for purposes
of nuclear fusion. In p a r t i c u l a r , i t i s the only theory so
far t ha t takes proper account of the global geometry of
closed magnetic confinement systems.
- The l i n e a r i z e d equat ions of idea l MHD may be ca s t in a form
tha t i s s u i t a b l e for s p e c t r a l a n a l y s i s . In p a r t i c u l a r , the
system can be described by means of s e l f - ad jo in t l i n e a r
. 2 4 .
sidered as the prototype of a theoret ical model in physics
i s to a large extent due to the same fact. Hence, l inearized
ideal MHD can be endowed with a l l the mathematical respect
ab i l i t y one wishes to have.
- Recent developments in computing, specif ical ly computations
in hydrodynamics, make i t possible to solve the ful l non-lin
ear i n i t i a l value problem for r e a l i s t i c geometries. Since
non-linearity plays an essent ia l role here, qual i ta t ive new
physics i s 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
-di f ferent ia l to di f ferent ia l equations.
- The n e g l e c t of d i s s i p a t i o n has changed t h e sys tem from a
n o n - c o n s e r v a t i v e t o a c o n s e r v a t i v e sys t em.
We s t i l l have one major ma themat i ca l c o m p l i c a t i o n i n t h e e q u a
t i o n s v i z . n o n - l i n e a r i t y . Th i s c o m p l i c a t i o n w i l l be removed i n
a l a t e r c h a p t e r when we l i n e a r i z e t h e e q u a t i o n s .
B. DIFFERENTIAL EQUATIONS
As s t a t e d i n t h e i n t r o d u c t i o n we may j u s t as w e l l
p o s t u l a t e the e q u a t i o n s of i d e a l MHD r a t h e r than t r y t o g ive
a comple te ly s a t i s f a c t o r y d e r i v a t i o n from f i r s t p r i n c i p l e s .
T h e r e f o r e , l e t us no l o n g e r worry about t h e domain of v a l i d i t y
and j u s t s t a t e t h e model and s t a r t work ing w i t h i t . C o n s i d e r
a p e r f e c t l y c o n d u c t i n g , i d e a l and c o m p r e s s i b l e f l u i d i n t e r -
. 2 5 .
a c t i n g wi th a magne t i c f i e l d . The e v o l u t i o n of t h e 8-dimen-
s i o n a l s t a t e v e c t o r v ( £ , t ) , J j j t r^ t ) , p ( ^ , t ) , p( j r , t ) i s
d e s c r i b e d by t h e e q u a t i o n of motion fo r v , F a r a d a y ' s law f o r jg,
an e q u a t i o n of s t a t e f o r p , and the c o n t i n u i t y e q u a t i o n for p .
In the E u l e r i a n p i c t u r e :
p3^ = - *rvx. - 7P + <VxS> x I > ( 3 - D
— = Vx(vxB) , VB = 0 , ( 3 - 2 )
3P — = - v-7p - Y P 7 - v , ( 3 - 3 )
H = " V ( p v ) • (3-4)
In t h e Lagrangian p i c t u r e :
dv p _ i = - vp + (VxB) x B , (3 -1 ) '
a t 'v *v
dT = $* v * " I "'* ' v ' l = ° ' ( 3"2 )'
f f = " YP?-v , ( 3 - 3 ) '
ft - " PV.V . ( 3 - 4 ) '
Here , we have s u b s t i t u t e d Ampere 's law (2-75) i n t o t h e e q u a t i o n
of motion and Ohm's law (2-76) i n t o F a r a d a y ' s l aw.
We have i n t r o d u c e d the r a t i o of s p e c i f i c h e a t s y , which
fo r monoatomic gases has t h e va lue 5 / 3 . Although we r e s t r i c t
. 26 .
the ana ly s i s t o roonoatomic gases ( fu l ly ionized plasma!)
we w i l l wr i t e Y for g e n e r a l i t y . (See d iscuss ion in Sec. I l l
D).
The equation for incompressible f l u id s may be found
h e u r i s t i c a l l y from the Eqs. (3-1)- (3-4) in the l i m i t y -*• m,
V-VH-0, such t h a t dp/dt = - TPV 'V remains f i n i t e . The
l a t t e r r e l a t i o n may be dropped from the equations s ince i t
merely t e l l s us what the magnitude of the quan t i ty dp/dt i s . To
make up for the missing r e l a t i o n the c o n s t r a i n t V«v = 0 needs
then to be added t o the equa t ions , so t h a t the equations for
incompressible i dea l MHD read:
dv pdT " ' Vp + ( V x S } 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-
. 2 8 .
u i a r shape of t h e w a l l then comple t e ly d e t e r m i n e s t h e problem.
At t h i s p o i n t one might even r e s o r t t o t h e computer t o p r o v i d e
us w i t h s o l u t i o n s .
(2) PLASMA SURROUNDED BY VACUUM
Another p o s s i b i l i t y i s n o t y e t c o v e r e d , v i z . t h e
plasma may be i s o l a t e d from t h e w a l l by a vacuum (a u s e f u l
model fo r c o n f i n e d p l a s m a s ) . The f l u i d v a r i a b l e s a r e n o t
d e f i n e d i n t h e vacuum and t h e magne t i c f i e l d i s de te rmined by
vx6 = 0 , V ' | = 0 , (3-13)
s u b j e c t t o t h e boundary c o n d i t i o n
n-Ê = 0 (3-14)
a t t h e w a l l . Here , vacuum v a r i a b l e s a r e d i s t i n g u i s h e d from
f l u i d v a r i a b l e s by h a t s . At t h e plasma-vacuum i n t e r f a c e we
now may admit jumps i n some of t he v a r i a b l e s , v i z . i n p , p , and
the t a n g e n t i a l components o f B:
[ + J- B«] - 0 , (3-15)
« • i a - ° . (3-16)
; X W 'J* (3-17)
where jumps are ind ica ted by the no ta t ion |[fj = f - f. The
surface cur ren t densi ty j * i s obtained in the l i m i t of a
surface l ayer of thickness <5 with cu r ren t dens i ty i , when the
l im i t s 6 •*• 0 and 1 •> » are taken in such a way t h a t j * = j<5
remains f i n i t e . Notice t h a t j * has the dimension of cu r ren t
densi ty times length .
A spec i a l case i s the s t a t i c equ i l ib r ium problem ,
which i s completely posed by the equat ions (3-9) , (3-10) , (3-13)-
(3-17).
.29.
(3) EXTERNAL COILS
Fina l l y , we may a l so consider a t ime-dependent
boundary-value problem, where the wall i s not p a s s i v e , l i k e
in the previous two cases , but c a r r i e s a surface c u r r e n t
J* 11 ( r , t ) which forces o s c i l l a t i o n s onto the plasma. This
wall may have gaps so t ha t the system i s not i s o l a t e d from
the outs ide world. In that case we have the following
boundary condi t ions a t the w a l l :
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 y EC /C , where C is the specific heat at con-r ' 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
. 3 0 .
thermodynamic var iables . In part i cu lar , the evolut ion equa
t ion for the pressure rea l ly ar i ses as a consequence of our
having chosen a part icular equation of s ta te for the ionized
gas, v i z .
P * f ( s ) p T , (3-20)
where s is the entropy per unit mass. Eq. (3-3) then obtains
for adiabatic processes where
f f - 0, (3-21)
so that
# H - »«>"T"1 - ? • •>. <3-22' where c i s the ve loc i ty of sound. The e x p l i c i t dependence of
the function f i s given by
f - A exp ( s / c v ) (3-23)
where A i s a constant. Another convenient thermodynamic varia
ble i s 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 P77 = " <*-1> P V e ~ CTT-D eVp-Bx(VxB) , (3-26)
a t *\# *\» dS __ = B*7V - BV-v , V«B = 0 , (3-27)
| f = - (Y- l ) eV-v , (3-28)
| | = - pV-v . (3-29)
For t he cho ice v , B, p , and s one o b t a i n s ;
p d t " " 7 p " £ x ( V x £ } ' (3 -30)
— - B-Vy - BV-y , V-B = 0 , (3-31) *1 dt
dt = " YpV'X ' (3-32)
j f - 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
i t ly . 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:
. 3 2 .
- c2Vp - — pVs - Bx (VxB) , ( 3 - 3 4 ) Cp
S*V Ï " §V*X • v ' § " O . (3-35)
~ P7-V » ( 3 - 3 6 )
O • ( 3 - 3 7 )
where c2 = c 2 ( s , p ) i s again obtained from the Eqs. (3-20) and
(3-23) .
pdT =
!?-d t
dt
d£ dc
.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 ,<p ,x,y). (4-1) xx xy yy x y
The Cauchy problem c o n s i s t s i n f i n d i n g t h e s o l u t i o n $ away
from the boundary $ (x ,y ) = c o n s t a n t = $ , when, e . g . , ty and 4»
a r e s p e c i f i e d on <|> .
W r i t i n g
£ = * 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 p e r t i n e n t Cauchy problem i s t o de te rmine E and n away from
the boundary , when they a r e g iven on <f> .
I n t r o d u c e c o o r d i n a t e s $,x i n s t e a d of x , y , e . g .
o r t h o g o n a l ones so t h a t V<j>»Vx = 0 . (Example : p o l a r c o o r d i n a t e s
<t> = r , x = 9 ) . The boundary d a t a may then be exp re s sed as
0 0 0 0
( 4 -3 )
n * * « » x A ) * n (x ) • 0 0 0 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:
C U . x ) - C ( * B , x „ ) + ( • - • „ ) | f - • (x~x ) | f -o o o 3 © O 3 Y o o
n U , x > - n (* , x ) + ( • - • „ ) I J - + ( x " x ) If" 0 0 0 3 $ O 3 X
+
• * -
(4-4)
Here , £ * y o ' x o * ' 3n 3 F
n (<t> , x ) # T — r and ^ i - a r e known o o 3X Q 3x0
from the boundary c o n d i t i o n s ( 4 - 3 ) , so t h a t we wish t o i n v e s -
can be c a l c u l a t e d . t igate under which circumstances 77- and T T 9m 3 6
0 Y o
Once the l a t t e r two d e r i v a t i v e s a r e known the h i g h e r o r d e r
ones i n t h e e x p r e s s i o n (4-4) may be found by s u c c e s s i v e
d i f f e r e n t i a t i o n s of t h e o r i g i n a l e q u a t i o n (4-2) , so t h a t t he
problem c?n be s o l v e d .
Transform t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n (4-2) t o
$ 1 X c o o r d i n a t e s by writing > T 7 + X a . e t c . X 3$ x ox
This g ive s
< A * x + B * y > I f + ( B * x + C*y> U - F " ( A *x + V If - (B*x + C*y> f j '
4 ü _ é ia * y 3<|> * x 3 $ -X | i •
( 4 - 5 )
y ax x* f *
»x Consequen t ly / t h e d e r i v a t i v e s 7-7 and 7-7 may be de te rmined from
dm a <p
Eq. (4-5) i f t h e d e t e r m i n a n t of t h e c o e f f i c i e n t s on t h e l e f t
hand s i d e does n o t v a n i s h . The c o n d i t i o n t h a t t h e determinant
v a n i s h e s ,
A* . + Bó B$ + C<* Y x Y y x y
- A* 2 - 2B* * - c* 2 - 0 , ( 4 - 6 ) A x y y
. 3 5 .
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
ÈL = - . ! * . B± VB2-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: ^ •"- <j, *
" AC > B2: the characteristics are complex, the equation is of
an elliptic type, example: ^ + ^ * 0 . xx yy
In the following, we shall mainly be concerned with
hyperbolic equations. Cauchy initial conditions (the variable
y becomes t) may then be considered appropriate if the boundary
is not a characteristic. For the example of the wave equation _1_
XX C 2 T t t *-„ - — K. - ° .
the characteristics are given by — = i c. d t
* p « C « . llK€
The init ial data propagate along the characteristics.
In spaces of higher dimension than 2 i t is not sufficient for
the well-posedness of the Cauchy problem that the boundaries
are not coincident with a characteristic. One has to demand
. 3 6 .
in a d d i t i o n t h a t they a re s p a c e - l i k e * . In p h y s i c a l problems
i n i t i a l d a t a a r e u s u a l l y given a long s p a c e - d i r e c t i o n s * * , so
t h a t t h i s does n o t p r e s e n t a r e s t r i c t i o n .
F i n a l l y , i t i s u s e f u l to d i s t i n g u i s h two c o n c e p t s :
t h e domain of i n f l u e n c e of I , which i s t h e r eg ion i n t h e x - t
d6~»..,o« p l a n e where t h e i n f l u e n c e o f
t he i n i t i a l d a t a I i s f e l t ,
ie^»;^o< and t h e domain of dependence
of t h e s p a c e - t i m e p o i n t P ,
which i s t h e r eg ion which i n f l u e n c e s t he b e h a v i o r a t P .
N o t i c e t h a t i t does n o t m a t t e r whe the r t h e c o e f f i
c i e n t s A, B, and C depend on £ and n a s w e l l , so t h a t t h e
method of c h a r a c t e r i s t i c s a l s o works f o r n o n - l i n e a r e q u a t i o n s ,
s p e c i f i c a l l y q u a s i - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s .
B. CHARACTERISTICS IN IDEAL MHD
We g e n e r a l i z e t he p r e c e d i n g d i s c u s s i o n t o p a r t i a l
d i f f e r e n t i a l e q u a t i o n s i n more than two independen t v a r i a b l e s
and a l s o more than two dependent v a r i a b l e s , in p a r t i c u l a r t h e
e q u a t i o n s of i d e a l MHD. We wish t o show t h a t t h e s e e q u a t i o n s
a r e symmetr ic h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s , where
the n o n - l i n e a r i t y i s on ly of a q u a s i - l i n e a r n a t u r e .
* The reason t h a t we have t o demand s t r o n g e r c o n d i t i o n s in
s p a c e s of h i g h e r dimension than 2 i s t he f a c t t h a t t h e spat ia l
p a r t by i t s e l f now c o n t a i n s an e l l i p t i c o p e r a t o r , so t h a t
Cauchy ' s problem i s i l l - p o s e d i f we c o n s i d e r t i m e - i n d e p e n
d e n t s o l u t i o n s .
** An e x c e p t i o n i s the e x c i t a t i o n of waves by t i m e - d e p e n d e n t
f o r c i n g terms a t the boundary of t he p lasma. In t h a t ca se data
a r e given on t i m e - l i k e b o u n d a r i e s .
. 3 7 .
The e q u a t i o n s of i d e a l MHD a r e p a r t i a l d i f f e r e n t i a l
e q u a t i o n s w i t h r e s p e c t t o t h e v a r i a b l e s r , t . Consequen t ly ,
c h a r a c t e r i s t i c s w i l l be 3-d imens iona l mani fo lds
C>(rft) = c o n s t a n t = 6 (4-8)
in 4 -d imens iona l s p a c e - t i m e r , t . These mani fo lds may be
v i s u a l i z e d as b e i n g g e n e r a t e d by the motion of s u r f a c e s i n
o r d i n a r y 3 -d imens iona l space r . L e t u s apply t he same t e c h
n i q u e s as i n the p r e v i o u s s e c t i o n t o de t e rmine when a ( r , t ) =
4 i s a c h a r a c t e r i s t i c . o
Assume t h a t boundary d a t a f o r v ( r , t ) , B ( r , t ) , p ( r , t ) ,
p ( r , t ) a r e g iven on è . [Not ice t h a t t he i n i t i a l va lue problem
w i l l co r respond t o g i v i n g v ( £ , 0 ) , B ( r , 0 ) , p ( £ , 0 ) , and p{ r , 0 )
on t h e domain of i n t e r e s t i n o r d i n a r y 3 - s p a c e . In o r d e r fo r
t he i n i t i a l va lue problem t o be w e l l - p o s e d o r d i n a r y 3-space
shou ld n o t be a c h a r a c t e r i s t i c . H e r e , we c o n s i d e r t h e opposite
case t h a t d a t a a r e g iven on a c h a r a c t e r i s t i c , so t h a t t he
Cauchy problem i s n o t w e l l - p o s e d ] . Like in s e c t i o n IV A we con
s i d e r é a s a c o o r d i n a t e and i n t r o d u c e a d d i t i o n a l c o o r d i n a t e s
X,a, and T , so t h a t 4 -space r , t i s covered by t h e c o o r d i n a t e s
4>,X,0/ and T . The d a t a may then be w r i t t e n as v (* ,x ,a f") = y <x .a T ) e t c . , (4-9) < o o o o ^o o o o
where x r° i a " d T p a r a m e t r i z e t h e mani fo ld * = <J> . S ince t he 0 0 0 O
function v (x ,a ,t ) is a known function, the derivatives *\»o o o o
3v /3x , 3v /da , and 9v /3T may also be considered to be known, "o o ^o o ^o o
We wish to investigate under which conditions solutions
v(<f>,X/ö,t), B(ó,x,o,-t) , pUarOfT), and p (<j>, x , a, T) away from the
boundary * = <j> may be obtained, or rather may not be obtained
since then 6 = 4 is a characteristic. Write the variables in o
terms of a power series;
. 3 8 .
3v 3v ? ( * . X . O , T ) - v o ( x o , O o , t o ) * ( 4 - ^ ) ^ • ( X - X 0 ) ^ " •
o o
Sv 3v • ( ( T - 0 ) _ i _ + ( T - T ) J L . + , ( 4 -10 )
0 3a ° 3To o
l ikewise for B, p , and p .
As in the previous s e c t i o n , we may consider the problem t o be
so lvab le i f 3v/3$ , 3B/3$ , 3p/3<fr , and 3p/3<fr can be con-
s t r u c t e d , s ince the o the r f i r s t o rder d e r i v a t i v e s are found
from the boundary da ta ( 4 - 9 ) , whereas the higher o rder ones
may be obtained by subsequent d i f f e r e n t i a t i o n s of the o r i g i n a l
p a r t i a l d i f f e r e n t i a l equa t ions .
For convenience, l e t us denote the unknown d e r i v a
tives with r e spec t t o <j> with a prime:
\ - a* ' \ - n ' p " 3 * ' p ' »• ' l '
The d i f f e r e n t expressions occur r ing in the MHD equat ions nay
then be wr i t t en a s : 3v 3v 3v
V - v • Vifi 'v' + Vx'T— + VO'T— + V f — , <v r <\, 3x 3° 3 T
3v 3v 3v 3v (4-12)
IT " *t I * Xt 3^ + °t 17 + Tt 77 '
dy 3v 3v -:— * (ó • v 7 ó ) v ' + (Y + V * 7 Y ) IT" * (.0 * V 'Vo )^ - + <jt t * "v * t *v 3x t "v "'da
• < t t + y - V T ) j f , e t c .
Next, define the following q u a n t i t i e s :
S = v*» (4-13)
u = -<t>t - v / v $ .
Here, jr» i s the normal to the space -pa r t of the c h a r a c t e r i s t i c
($ can of course be chosen such t h a t |v$ | = 1 so t h a t JJ has
u n i t l e n g t h ) , and u i s the c h a r a c t e r i s t i c speed, i . e . the
.39 .
normal v e l o c i t y of the c h a r a c t e r i s t i c $ measured wi th r e s p e c t
to the f l u i d which roves with v e l o c i t y v .
For reasons t h a t w i l l soon become c l e a r we w i l l
not s t a r t from the Eqs. ( 3 - 1 ) - ( 3 - 4 ) , bu t r a t h e r from the Eqs.
(3-26)-(3-29) in terms of the b a s i c v a r i a b l e s v , J3, e , p .
I n s e r t i n g the express ions(4-12) and w r i t i n g the primed (un
known) v a r i a b l e s on the l e f t -hand s ide and the known v a r i a b l e s
on the r igh t -hand s i d e , we ob t a in :
-puy* + (y-1) ripe' + ( Y - D J?e p *
-uB ' - n.Bv' + Bn-y' = ,
- u e ' + (y -De n«y' = , (4-16)
- u p ' + 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<Y-De . < 4 " 1 8 )
This p a r t i c u l a r choice symmetrizes the matr ix on the LHS as
we w i l l s ee .
Let us choose B along the z-axis and JJ in the x, z
p lane :
B - (0,0,B) , £ - ( n x , 0 , n z j . (4-19)
n*BB' + nB-B' (4-14)
(4-15)
. 4 0 .
Furthermore, introduce the Alfvén speed
b = B / \ T P , (4 -20)
and the sound speed
c S \ /YP7P. (4 -21)
The system o f Eqs . ( 4 - 1 4 ) - ( 4 - 1 7 ) may then be w r i t t e n as
0
-u
0
-n z b 0
-n„b
n b 0 x
n x c 0
n c 0 x
0
0
-u
0
0
0
n 2 C
n c z
-n z b n b x
0
0
- u
0
0
0
0
- n z b 0
0
0
- u
0
0
0
0
0
0
- u
0
0
n c x
n c z
0
0
0 (4
where n2 = Bn/5, n x = y i - u T / B ) 2 .
C h a r a c t e r i s t i c s are obtained when the determinant
o f the LHS o f Eq. (4-22) van i shes so that s o l u t i o n s cannot be
propagated away from the manifold <fr - $ . This c o n d i t i o n may
be w r i t t e n as
A - zbr «* ("2 " b 2 ) [u1* - (b 2 + c 2 ) u 2 + b2 c 2 ] - 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
. 4 1 .
values are assigned to the va r i ab le s v, B, e , and P in
3-dimensional space a t t = 0, i s wel l -posed.
Before we go on t o d iscuss the s ign i f i cance of the
so lu t i ons obta ined above, i t i s i n s t r u c t i v e to see what hap
pens if we choose as b a s i c va r i ab l e s v , B, p , and s . The Eqs.
(3-30) to (3-33) now lead t o
-puv' + np ' - n«B B.' + nB«B* = ,
-uB' - n-By'+ Bn»v' = ,
-up* + Ypn-v' = ,
-us
'X,
f __ _ _ _
(4 -24 )
(4 -25 )
(4 -26 )
(4 -27 )
Mult iplying Eq. (4-25) by \fij and Eq. (4-26) by 1/c to ge t
compatible dimensions and i n s e r t i n g the express ions (4-19) for
B and JJ we ge t ,
f-u 0
0 - u
0 0
-n b 0 z
0
0
- u
0
-n z b
-n z b 0
n b o
n c x n z C
0
0
- u
0
0
0
0
- n z b
0
0
-u
0
0
0
n b x
0
0
0
0
- u
0
0
n c x
n z c
0
0
0
- u
0
0
0
0
0
0
0
0
- u
(p B ' ,H x
VP By»
^ B ; (4 -28)
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 = + b n (4-30)
u = us = Ml C b 2 + c2) " K ( b 2 + c2)2_ 4 b n c 2 ] 1 / 2 } 1 / 2 »
(4-31)
u - u f = ± { | ( b 2 + c2) + I [ ( b 2 + c2)2 - 4 b ; c 2 ] 1 / y / 2 .
(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
t ic waves. Notice the following properties of the characteristic
speeds:
0 = 1UJ i i u J 1 IUA( i ( u
f ( < - ' (4-33)
. 4 3 .
In p a r t i c u l a r , i f n / / jS:
|u | * min (b,c) , |u A | - b , | u f | = max ( b , c ) , (4-34)
and i f n J. B:
= 0 , | u | = (b2 +c*)l/1 . <4~35> | u s | = |uK , . , , . f
I t should be no t i ced t h a t the equat ions of i dea l MHD are
general enough t h a t they contain the equat ions of gas dynamics
as a s p e c i a l c a se . I f b = 0 (no magnetic f i e ld ) the slow and
Alfvén waves disappear and the f a s t magnetoacoustic wave
degenerates i n to an ordinary sound wave. Another l i m i t of
i n t e r e s t i s the case of incompressible plasma (c •* • ) . In
t h a t case the speed of the fas t magneto-acoustic wave d isappears a t
i n f i n i t y ( ins tantaneous propagation) , whereas the slow rrtristo-
acous t i c speed and the Alfvén speed coincide. The waves themselves
do not ooincide, of course, because their phys ica l p r o p e r t i e s
(po la r i za t ion e . g . ) are d i f f e r e n t .
Let us now look a t the s p a t i a l p a r t of a cha rac
t e r i s t i c a t a c e r t a i n time t = t , the s o - c a l l e d ray s u r f a c e .
This may be considered as a wave f r o n t , i . e . a sur face across
which d i s c o n t i n u i t i e s may occur , emit ted a t time t = 0 from
the o r ig in x = y = z = 0. E . g . , in the case of van ish ing ly
small magnetic f i e l d (b = 0) a c h a r a c t e r i s t i c manifold
would j u s t be the spher i ca l wave f ron t x2 + y2 + z2 = c 2 t 2 .
Dropping the z-dependence one may v i s u a l i z e t h i s in x, y , t
space as a cone through the po in t »*
x = y = t = 0. The c i r c u l a r (in case C_f*«
the z-dependence i s kept : spher ica l )
i n t e r s e c t i o n of t h i s cone with the
. 4 4 .
plane t = t then const i tutes the ray surface at t = t • For
the MHD case we get of course more complicated f igures, in
par t icu lar because the medium i s anisotropic and the coef
f ic ients of the pa r t i a l d i f ferent ia l equation are not con
s t an t .
To get the ray surface we f i r s t of a l l compute from
Eqs. (4-29)-(4-32) the distance ut which a plane wave-
front t ravels along n af ter 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 th i s i s the
extremely high-8 regime: g > 2-y) :
This i s not the ray surface, but the so-called reciprocal
normal surface, [of course, everything is symmetric around
the direct ion 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 = l b . 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
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
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
> - o
r a y s u r d t c e
that on a character is t ic manifold relat ions between the values
. 4 6 .
v ' , B ' , e ' , and p' e x i s t . The meaning of t h i s i s t h a t we may
cons ider these q u a n t i t i e s as d i s c o n t i n u i t i e s o f the flow that
are propagated along w i th the c h a r a c t e r i s t i c s . In the c a s e
t h a t the primed v a r i a b l e s r e p r e s e n t d i s c o n t i n u i t i e s ,
v' i 3v/3<fr >> 3v/3x» e t c . , s o that we may j u s t n e g l e c t the
RHS of Eq. ( 4 - 2 2 ) . We then f ind the f o l l o w i n g r e l a t i o n s h i p s
which c h a r a c t e r i z e the kind o f wave motion:
1) Entropy d i s turbances (u = 0) :
v* = v ' = v* = 0 , x y z
B» = B» . B ' = 0 , ( 4 - 3 6 ) x y 2
e'/e = - P ' / P = s ' /C * 0 , p '= 0 .
[Fr iedr ichs has p' = 0 because h i s momentum equat ion i s d i f f e r
e n t ] . Hence, the only p e r t u r b a t i o n s that, occur are i n the
thermodynamic v a r i a b l e s . This e x p l a i n s the name o f t h e s e d i s t u r
bances .
2) Alfvén waves (u - u . ) :
These are pure ly t r a n s v e r s e waves where v and E are
perpendicu lar t o the p lane through n and Bs
v^ = v^ = 0 , v j * 0 ,
B x - 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 _ s , i
) i
These waves are p o l a r i z e d in the p lane through n and
B:
. 4 7 .
v , = ^ u2
x 2
n_ u^ -b* z Y
B! -n.. u b B' n2 - . . „ « ,
^P n_ u2-b2 * ' VP " u ^ b 2 ^ v z ' By " ° ' z u UP
_ up 5f ' , ' » = s^v;-p'= £ * . » • (4 -38 )
The polarizations of the fast and slow magnetoacoustic wave
are perpendicular to each other as indicated in the figure.
This difference ar ises through the factor u2 - b2 which i s
posit ive for the fast wave and negative for the slow wave.
Notice tha t for a l l these perturbations the equation
V-B = 0 leading to n*B' = 0 does not have to be considered
separately because i t i s an automatic r e su l t of Eq. (4-15).
In case n»B = 0 the root u = 0 i s sixfold degenerate.
The Eqs. (4-14)-(4-17)then resu l t in the following two condi
tions :
n .v ' = 0,
p ' + | - B / = 0 (or p ' + | 3 2 ' = 0) ,
whereas now we also have as an independent condition:
(4-39)
(4-40)
(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.
. 4 9 .
V. CONSERVATION LAWS
A. CONSERVATION FORM OF THE IDEAL MHD EQUATIONS
A q u a s i - l i n e a r s y s t e m o f p a r t i a l d i f f e r e n t i a l
e q u a t i o n s i s s a i d t o b e i n c o n s e r v a t i o n form i f a l l t h e terms
can b e w r i t t e n a s d i v e r g e n c e s o f p r o d u c t s o f t h e d e p e n d e n t
v a r i a b l e s (where i n t h e d i v e r g e n c e a l s o t h e t i m e - d e r i v a t i v e
i s i n c l u d e d ) :
_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 + P v v y + Vp + Bx(vxB) - 0 , p - ( r D p e , ( 5 - 1 )
3B T - - Vx^xjg) = 0 , 7 - 3 = 0 , ( 5 - 2 )
^- + ( y - l ) e V-v + v-Ve = 0 , ( 5 - 3 )
j& + V . ( p v ) = 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)
. 5 0 .
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 t h a t the magnetic terms then have the requi red p r o p e r t y .
From Eq. (5-1) we find by using Eq. (5 -4 ) :
£ ( p v ) - 7-(pvv + (p + | B 2 ) I - flg] - 0 , (5-11)
which i s the conservation form of the momentum equa t ion .
From the Eqs. (5-2) and (5-9) we have
SB — - 7- (Bv - vB) = 0 , (5-12)
which i s the conservation form of Faraday ' s law.
F i n a l l y , the evolut ion equation (5-3) for the i n t e r n a l energy
cannot be brought i n t o conservat ion form for the obvious reason
t h a t i t conta ins only p a r t of the energy which can be converted
i n t o o t h e r forms of energy. We the re fo re need a conservat ion
equation for the t o t a l energy d e n s i t y . This i s ob ta ined by adding
the con t r ibu t ions of the k i n e t i c energy, magnetic energy and
i n t e r n a l 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):
. 5 1 .
^ - ( j p v 2 + pe + I E 2 ) + v [ ( | p v 2 + pe • p + B2)v - v-BB] - O
( 5 - 1 3 ) ( c h e c k ! )
The equations (5-11), (5-12), (5-13) together with Eq. (5-4)
constitute the conservation form of the ideal MHD equations.
B. SHOCKS
Cons ider t he o n e - d i m e n s i o n a l flow of gas where sound
waves a r e e x c i t e d . The c h a r a c t e r i s t i c s a r e s t r a i g h t l i n e s i n
t h e t - x p l a n e : d x / d t = +. c . Suppose now t h a t we suddenly
inc rease the p r e s s u r e , so t h a t t h e sound speed i n c r e a s e s . In t h e
t - x p l a n e t h i s means t h a t t h e s l o p e of t h e c h a r a c t e r i s t i c s
d e c r e a s e s . T h e r e f o r e , we may a r r i v e
a t t h e s i t u a t i o n where the c h a r a c t e r - n
i s t i c s c r o s s , i . e . i n f o r m a t i o n o r i
g i n a t i n g from d i f f e r e n t s p a c e - t i m e
p o i n t s a c c u m u l a t e s . Consequen t ly ,
g r a d i e n t s i n t he macroscop ic v a r i a
b l e s b u i l d up u n t i l the p o i n t t h a t t h e i d e a l i z e d model b r e a k s
down and d i s s i p a t i v e e f f e c t s due to t h e l a r g e g r a d i e n t s have to
be taken i n t o a c c o u n t . E v e n t u a l l y , a s t e a d y s t a t e w i l l be reached
where n o n - l i n e a r and d i s s i p a t i v e e f f e c t s c o u n t e r b a l a n c e each
o t h e r : a shock-wave has been c r e a t e d .
Without s p e c i f y i n g the k ind of d i s s i p a t i o n , one may
a r r i v e a t the s o - c a l l e d shock r e l a t i o n s t h a t r e l a t e v a r i a b l e s
on e i t h e r s i d e of t he p r o p a g a t i n g shock f r o n t . The idea i s t h a t
t h e i d e a l model b r e a k s down i n s i d e a l a y e r of i n f i n i t e s i m a l
t h i c k n e s s ó ( i . e . , a t h i c k n e s s p r o p o r t i o n i l t o the d i s s i p a t i v e
c o e f f i c i e n t t h a t i s assumed to be v a n i s h i n g l y s m a l l ) , bu t i t
. 5 2 .
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:
l i m ( Vf d i = n J j f - d i = n ( f 2 - 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£ '
where (df/<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 | f d £ . - u \ | f d t - - u ï f T l . (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 i t 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
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 f p v j + n * i T P ^ • (P + \ B2) I - £ £ ] = 0 , (5 -16)
- u Ï B l - r [ Bv. - v B l = 0 , r l B ] = 0 , ( 5 -17 )
- u i j pv2 + pe + j B 2 ! + n- I ( | pv2 + pe + p + B2)v - v'BBÜ - 0 ,
( 5 - 1 8 )
- u l 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
-\T7
- ^
.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)
<n.? - u)TnxBl - n-BlnxvJ = 0 , (5-24)
(%'Z - U ) Ï | P V 2 - P e + | B 2 l + p . v Up + 1 B 2 ] - n - B t v - B l l = 0 ,
(5-25)
(Jj-v - u)üpl = 0 . ( 5 _ 2 6 )
We w i l l not continue the d iscuss ion of these shocks
fu r the r . Our i n t e r e s t here i s the case n/v. - u = 0, which was
not so i n t e r e s t i n g in the context cf characteristics, but which is
quite important in the context of shocks. Usual ly , when n-v - u = 0
one does not speak about a shock s ince the f ront i s j u s t carried
with the f l u id v e l o c i t y , but one c a l l s t h i s a contac t discontinuity.
From the Eqs.(5-20)-(5-26) i t i s c l e a r t h a t i n t h i s 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 + | B 2 1 = 0 , C 5-28)
whereas a l l o ther v a r i a b l e s , i . e . v , B , and p may display
a r b i t r a r y jumps. The jump of the t angen t i a l f i e l d component
B , in p a r t i c u l a r , would give - i s e to a surface cur ren t density
of a r b i t r a r y 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.
. 5 6 .
C. GLOBAL CONSERVATION LAWS
Let us c o n t i n u e w i t h t h e d i s c u s s i o n of t he c o n s e r v a
t i o n l a w s . In o r d e r t o u n d e r s t a n d t h e p h y s i c a l meaning of the
d i f f e r e n t t e r m s , d e f i n e t h e f o l l o w i n g q u a n t i t i e s :
- momentum d e n s i t y it = py , (5-30)
- s t r e s s t e n s o r T = pvv + (p + - B2) I - BB , (5-31)
tfi = vB - Bv , (5-32)
- to ta l energy density 'K = •=• pv2 + pe + y B2 , (5-33)
- energy flow U = (^ pv2 + pe + p)v + B2v - vBB . (5-34)
The e q u a t i o n s ( 5 - 1 1 ) - ( 5 - 1 3 ) , and (5-4) may then be w r i t t e n
— + V«T - 0 , ( c o n s e r v a t i o n of momentum) (5 -35)
— + y.jj, = o , ( c o n s e r v a t i o n of f lux) (5-36)
— + 7-U * 0 , ( c o n s e r v a t i o n of ene rgy) (5-37)
v r + V*ir = 0 , ( c o n s e r v a t i o n of mass) . ( 5 - 3 8 ;
These a r e t h e e v o l u t i o n e q u a t i o n s f o r *, £,*>*•, and p i n c o n s e r
v a t i o n form, where i t shou ld be n o t i c e d t h a t t h e q u a n t i t i e s
a p p e a r i n g i n the d i v e r g e n c e t e rms can a l l be e x p r e s s e d i n terms
of TT, jg/K , and p , so t h a t t h e s e f o u r v a r i a b l e s c o n s t i t u t e a n o t h e r
b a s i c s e t of v a r i a b l e s t o d e s c r i b e i d e a l MHD.
The s t r e s s t e n s o r J£ i s composed of t h e Reynolds s t r e s s
t e n s o r p y y , t h e i s o t r o p i c p r e s s u r e p i , and t h e magnet ic p a r t
=• B2I - B | of Maxwel l ' s s t r e s s t e n s o r . In a p r o j e c t i o n based on
y , t h e on ly non v a n i s h i n g c o n t r i b u t i o n t o t h e Reynolds s t r e s s i s a
p o s i t i v e s t r e s s ( p r e s s u r e ) pv2 a l o n g jr. I n a p r o j e c t i o n based on
B the r ema in ing p a r t of t he s t r e s s t e n s o r may be w r i t t e n as
. 5 7 .
p • ^ B2 O O \ 2
l O p • y B2 O
O O -W «so that the £ - f i e l d g ives a pos i t ive s t r e s s (pressure) in
d irect ions perpendicular to £ and a negative s t r e s s (tension)
in direct ions para l l e l 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 t h i s symbol because i t appears to have
no d irec t physical meaning (at l e a s t we do not know of any) -
The only reason we wrote the flux equation in t h i s way i s the
fact that one obtains the jump conditions of Sec. V B roost
e a s i l y . 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
t ion for B. For that reason the previously exploited fonn of Fara
day's law with the term vx(v x £) appearing i s to be preferred over
that of Eq. (5-36) .
The dif ferent terms appearing in the to ta l magnetic
energy density o*. may be grouped in two parts :
»» = >*+ *r , (5-39)
where OC i s the k ine t i c energy density;
rK 3 y pv2 , (5-40) and « i s 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)
Cons ide r now a plasma su r rounded by a p e r f e c t l y
c o n d u c t i n g w a l l (model (1) of S e c . I l l C ) , s o t h a t bo th
v n = 0 and n*B = 0 a t the w a l l . Def ine t h e fo l l owing quan
t i t i e s :
- t o t a l momentum: " = { j d t , (5-43) ^
- t o t a l f l u x through a s u r f a c e O bounded by a c lo sed curve I on
t h e w a l l : <T = ^B-ndO , (5-44)
" t o t e l e n e r g y : H = J * d T ^ (5-45)
- t o t a l mass : M = \ pdx. (5 -46)
By app ly ing Gauss ' theorem t o Eq. (5-35) we f i n d :
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)
s i n c e v_» £ a n d d £ a r e t a n g e n t i a l to t h e w a l l . Hence, f lux c a n
n o t l e ave o r e n t e r t h e v e s s e l .
Applying Gauss ' theorem again t o Eq. (5-37) g ives
H = - U«Udt = - i u - n d i - 0 , (5-49)
which s t a t e s t h a t t o t a l energy i s c o n s e r v e d .
. 5 9 .
S i m i l a r l y , Eq. (5-38) g i v e s
M » - W*irdT » iw-ndo » 0 , (5 -50)
so tha t the t o t a l mass i s c o n s t a n t .
F i n a l l y , one of the most inqportant c o n s e r v a t i o n laws
of a p e r f e c t l y conducting f l u i d i s obta ined by i n t e g r a t i n g
Faraday's law over a s u r f a c e moving wi th the f l u i d . The r e s u l t
i s tha t the f l u x through a contour moving w i th the f l u i d i s
c o n s t a n t :
• = J B-jjda = c o n s t a n t . (5 -51) c
This theorem i s proved i n S e c . VII B, Eq. ( 7 - 2 4 ) . The i n t u
i t i v e p i c ture a s s o c i a t e d wi th t h i s c o n s e r v a t i o n law i s t o say
t h a t the f i e l d l i n e s are frozen i n t o the f l u i d (Alfvén) . Indeed,
in i d e a l MHD the concept o f magnetic f i e l d l i n e s o b t a i n s more
p h y s i c a l r e a l i t y than i t even had in the o l d days o f Faraday.
D. ENERGY CONSERVATION FOR MODELS 2 AND 3
In the prev ious s e c t i o n we proved energy c o n s e r
v a t i o n of the n o n - l i n e a r system of i d e a l MHD equat ions f o r
model 1 (plasma e n c l o s e d by a w a l l ) . In a l a t e r chapter (Sec .
VIII E) we w i l l need the law o f conservat ion of t o t a l energy
for a plasma-vacuum system (model 2 ) . The g e n e r a l i z a t i o n to
model 2 i s straightforward. The t o t a l energy for plasma and
vacuum i s
H . [ * * dTP +\>7? d t V , ( 5 - 5 2 )
where
* P = i pv2 • pe • \ B2 , * V . l S2 . (5-53)
In the time dependence o f these e n e r g i e s one needs t o take
i n t o account the ra te o f change of the volume e l e m e n t s . In
Sec. VII B (Eq. (7 -23) ) we w i l l der ive that
. 6 0 .
^ (dT) - 7-vdT , ( 5 - 5 4 )
so that
J M * - - ! ^ «•(*£<«> \ (2^L + v.yijt + r j ( v . v ) d
St ^ -v.
1-TT- d x • fa v-nda (5-55)
Although v is only defined in the plasma so that Eq. (5-54) i s
only valid there, the latter result obviously also applies
to the vacuum as i t merely tells us that the rate of change of
the energy i s 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 p y 2 + p e + p + B 2 )?*£d o
- - f a p r t d o - 5 ( p + \ B2>redo»
so that
* — " - [(P • J B2>v-n.do. (5-56)
For the vacuum contribution we find
*ir " B# ^7 - - S*v*f - - v-(fxB) + e-7xS - - 7-(ËXS) ,
so that
dHv faaiv . v f m v
. 6 1 .
= \ ÊxE-nda - \— B 2 v n d a
To remove the e l e c t r i c f i e l d from t h i s express ion we need to
apply Faraday's law j u s t ou t s ide the surface of the i d e a l l y
conducting plasma. This gives a t the plasma-vacuum i n t e r
face: nxl = n-vB . (5-57)
Hence,
~r = \B 2y-ndo - \ \ B 2 v n d a = \ | B 2 v n d a . (5-58)
Combining the Eqs. (5-56) and (5-58) and us ing the jump
condi t ion (5-28) we obta in
| f - J I p + \ 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 ext to the wall, i.e. B = 0 , the rate of change of the energy
is given by
.int f c - j - w a 1 1 f r. • * J wall ,. ,_. = - l S•n do = - E*i* ,, do , (5-60) t J "v 'v. J x rwall * \ " /
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.
. 6 2 .
REFERENCES
1 . K.O. F r i e d r i c h s and H. Kranze r , Notes en Ma<^etohydrodynanics,
V I I I . " N o n - l i n e a r wave m o t i o n " . (New York u n i v e r s i t y , NYO -
6486 - V I I I , New York, 1 9 5 8 ) , S e e s . 6 - 9 .
2 . W.A. Newcomb, Notes on Magnetohydrodynamics ( u n p u b l i s h e d ) .
3 . T . J .M. Boyd and J . J . Sanderson , Plasma Dynamics (Nelson ,
London, 1 9 6 9 ) , Chap t e r 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 i s the capaci ty and V the vol tage over the
capac i tor . The quest ion then a r i s e s how t h i s e l e c t r o s t a t i c
energy can be optimally converted in to magnetic f i e ld energy
needed for the confinement of plasma.
The s imples t scheme i s undoubtedly the l i nea r 6-pinch
where the primary cur ren t I i s created by dischargina the
capaci tor over a s ing le turn co i l surrounding a straidit cy l in
d r i ca l plasma column of c i r c u l a r c r o s s - s e c t i o n . The induced
plasma cu r ren t I w i l l mainly flow on the plasma surface i f
the plasma i s well conducting. This surface current w i l l c rea te
a drop in the longi tud ina l B z - f i e l d , which produces an inward
pinching force on the plasma column. [Vfe assume t h a t a b i a s -
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 t o t a l pressure p + - — ( B 1 1 ^ ) 2 of the plasma. The r e s u l t i n g h o t
2UQ z
plasma would be wel l -confined i f a t r i v i a l e f f e c t did not
occur , v i z . rapid flow out of the ends of the 8-pinch. This
r e s u l t s in loss of the plasma. These end losses opera te on
the usee t ime-sca le and, t h e r e f o r e , block the road to con
t r o l l e d fusion. To avoid the endloss problem one could c lose
the plasma onto i t s e l f by e x p l o i t i n g a t o r o i d a l v e s s e l . How
ever , the t o ro ida l 8-pinch rap id ly expands due to the curva
ture of the B - f i e l d (which i s then t o r o i d a l ) . This lack of
equi l ibr ium a l so leads to plasma losses on the visec t ime-
s c a l e .
A configurat ion curing both defects i s the t o r o i d a l
z-pinch, where the t o r o i d a l cu r ren t I on a ree l -shaped p r i -
mary z -co i l induces a long i tud ina l cu r ren t I in the plasma.
This t o ro ida l plasma cu r ren t produces an ex t e rna l po lo ida l
magnetic f i e l d B which again pinches the plasma. However,
t h i s configurat ion i s v i o l e n t l y uns table
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 i s los t on the ijsec time-scale. Also notice
that in the case of a z-pinch the energy i s already not o p t i
mally used because the flux st icking through the inner hole
of the torus i s to be considered as l o s t , i . e . not used for
plasma confinement.
A combination of the two, avoiding endloss due to
open-ended systems and i n s t a b i l i t i e s due to the absence of a
s tab i l iz ing toroidal f i e^ i , is the toroidal screw pinch. Here,
both I . and I are switched simultaneously, so that the plasma 6 Z
e x p e r i e n c e s an inward fo rce c o n s i s t i n g of the two components
j „ B and j B„. The c u r r e n t and f i e l d l i n e s a r e helices wound 6 z z 6
on n e s t e d t o r o i d a l s u r f a c e s , t h e s o - c a l l e d magnet ic s u r f a c e s .
The magne t i c c o n f i g u r a t i o n i s s i m i l a r t o t h a t of a tokamak,
the d i f f e r e n c e b e i n g tha t he t o r o i d a l f i e l d i s c r e a t e d by a
p u l s e d p o l o i d a l c u r r e n t i n t h e case of a screw p i n c h , whereas
i t i s q u a s i - s t a t i o n a r y i n t he ca se of a tokamak. Both c o n f i
g u r a t i o n s may be i n s t a b l e e q u i l i b r i u m due t o t h e l a r g e t o r o i -
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 t h i s provides the b e s t i l l u s t r a t i o n
of f lux and energy conservation in a dynamical system con
s i s t i n g of two c i r c u i t s and a plasma with two time-dependent
magnetic f i e ld components.
We wish t o answer the following ques t ion : a t t = 0
two charged capaci tors C and C„ of vol tage V and Vn a re Z o Z 6
switched to the 9- and z - co i l surrounding the plasma. What i s
the r e s u l t i n g plasma motion and how are the ava i l ab l e e l e c t r o
s t a t i c energies -s- C VJ; and ~- CQ v | converted i n to magnetic i. Z Z t o o
f i e l d e n e r g i e s | ~— B* d i and \ ~— B~ d i ? [ in t h i s c h a p t e r
\i i s w r i t t e n aga in t o f a c i l i t a t e comparison wi th e x p e r i m e n t a l
d a t a ] .
B. MIXED INITIAL-VALUE BOUNDARY-VALUE PROBLEM
In the p r e s e n t problem energy and f l u x c o n s e r v a t i o n
p l a y t h e i m p o r t a n t r o l e . To s t r i p t he problem of u n n e c e s s a r y
c o m p l i c a t i o n s , we t h e r e f o r e n e g l e c t t h e e f f e c t of t h e p r e s s u r e
and t h e d e n s i t y on the plasma dynamics . Comparing te rms in t h e
momentum equa t ion (5-1) we s ee t h a t t he n e g l e c t o f t h e p r e s s u r e
i m p l i e s t h a t we c o n s i d e r a low B p la sma , B = 2n p /B 2 << 1 , which
i s c e r t a i n l y a v a l i d a p p r o x i m a t i o n . On the o t h e r h a n d , t h e
n e g l e c t of the plasma d e n s i t y i m p l i e s t h a t we c o n s i d e r v e l o c
i t i e s much s m a l l e r t h a n t he Al fv in v e l o c i t y , v « b , which i s a
r a t h e r poor assumption fo r t y p i c a l p inch i m p l o s i o n s . F o r m a l l y ,
we may j u s t i f y t h i s assumpt ion by c o n s i d e r i n g a slow compress ion
expe r imen t where t he d e n s i t y i s s m a l l enough for t h e Alfvén
t r a n s i t t ime t o exceed the t i m e - s c a l e of the compres s ion . Under
t h e s e assumpt ions the i d e a l MHO e q u a t i o n s t o be u s e d reduce t o
. 6 7 .
(V x B) x B = O ,
5B
3T = V X ^ X V •
(6 -1 )
(6 -2 )
The f i r s t equa t ion t e l l s us t h a t t h e c u r r e n t i s everywhere
p a r a l l e l t o t he magne t ic f i e l d so t h a t no magne t i c f o r c e s
o c c u r i n the i n t e r i o r of t he p lasma. The second e q u a t i o n im
p l i e s t h a t we c o n s i d e r the i d e a l i z e d Ohm's law
- 0 (6 -3) E + v x B "\i ^ K
t o be v a l i d t h roughou t t h e e n t i r e p lasma.
We i d e a l i z e t h e two c o i l s to be one copper s h e l l
c l o s e l y f i t t i n g t h e plasma v e s s e l wi th a p o l o i d a l c u t
(e.g. by the plane $ = 0) over which the voltage V i s applied and a toroidal
c u t ( e . g . by the p l a n e 9 = 0 , t he curves l a b e l l e d 3 and 3 ' in t h e
p i c t u r e ) o v e r which t h e v o l t a g e VQ i s a p p l i e d . The geometry of
t he t o r o i d a l v e s s e l i s f i x e d by p r e s c r i b i n g the major r a d i u s R
and t h e minor r a d i u s a of t h e t o r u s . Tc s imp l i fy t h i n g s we
c o n s i d e r the i n v e r s e a s p e c t r a t i o
E = a/R of t h e t o r u s to be a smal l
p a r a m e t e r e << 1 , and we on lv keep
l e a d i n g o r d e r e f f e c t s . For t h e
p r e s e n t a n a l y s i s t h i s i m p l i e s t h a t
t h e on ly genuine t o r o i d a l e f f e c t
c o n s i d e r e d i s the ; o u p l i n g of the
L T*.
r* t o r o i d a l c u r r e n t I t o the induced plasma c u r r e n t I by t he
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 B Q ) ' + Bz B z ' = 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 ) a t 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
e lec t r ic 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 6 ( t ) = £ £'d£ * 2 l T a Ë Q ( 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. I t was experimentally
.69.
observed that the pinching of the plasma column in a screw
pinch i s not perfect , i . e . the dense plasma i s not swept up
completely by the inward motion of the f ie ld , but a low den
s i ty plasma i s l e f t behind. This tenuous plasma is hot enough to
permit currents to flow there which create a magnetic f ield
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 f ield strongly influences the equilibrium and
s t ab i l i t y properties of the screw pinch, whereas the dynamics
i s also strongly influenced as we will see. Without going into
de ta i l 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)
a t the wall i s thought to originate from the creation of hot
plasma that instantaneously st icks to the inward moving f ield
l i nes .
The equations (6-8) and (6-9) s t i l l have another
defect that has to be removed before we can s t a r t solving the
problem. If we assume that the dense plasma i n i t i a l l y f i l l s
up the whole tube and that the bias field B (t=0) is purely
toroidal , Eq. (6-8) t e l l s us that the e l ec t r i c f ield Ë (t=0) ^ z
i s 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 i s considered tD be
infini tesimal . To balance the e l ec t r i c field a surface current
j*(0) i s created that produces a jump in the poloidal f ield Li
. 7 0 .
given by Eq. (5-29): 3*(0) = B (0). Likewise, we may assume
tha t the toroidal f ield displays a discontinui ty produced
by a possible i n i t i a l imbalance between the poloidal e l e c t r i c
f ie ld Ea(0) and the term v(0) B, (0) so that we have j*<0> =
B (0) - B (0). As the plasma moves inward the surface of z zp
discontinuity also moves inward. At th i s surface pressure
balance expressed by the jump condition (5-28) has to be
sa t i s f i ed . Since we also neglect the pressure of the "dense"
i n t e r i o r plasma th i s gives:
B* ( t ) = B 2 ( r ( t ) , t ) + B 2 ( r ( t ) , ' t ) a t r - r ( t ) . ( 6 - 1 0 ) zp z o 6 o o
The function r (t) wi l l only be known af ter we have solved
the problem so tha t v ( r , t ) i s known.
To recap i tu la te : The equations (6-4)-(6-6) are val id
in both the inner region 0 £ r < r (t) , where Bfl = 0, and the O 8
exte r io r region r (t) < i± a, where the force-free f ield has
both B and B components. At the surface of discontinuity
r = r (t) Eq. (6-10) re la tes the f ie lds on both s ides . Notice
tha t the inner plasma i s considered to be a "dense" plasma with
a pressure and a density much larger than that of the low den
s i ty ex te r io r plasma, but s t i l l small enough to be neglected
in the momentum equation. P P
Schematically: »
a i
c n n
777777, /
• u nwOu)
-*• 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:
- V*1
rjLo\a r„{\) *- r
Given the i n i t x a l d a t a 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) : V ( t ) = V„ (t ) = 0. At t h i s moment the e l e c t r o s t a t i c z c 6 c
energy of the capac i tor banks i s fu l ly converted i n t o magnetic
f i e l d energy.
The boundary data v ( t ) , B (t) , B (t) w i l l not be
forced onto the system, but they are determined by the c i r c u i t
equa t ions , which in turn are coupled to the plasma equat ions
by v i r t u e of Eqs. (6-8) and (6-9) . Therefore , the complete p r o b
lem cons i s t s in simultaneously so lv ing the plasma and c i r c u i t
equa t ions . Of course , the c i r c u i t s are descr ibed by Maxwell 's
equat ions as w e l l , but one would not r e a d i l y give up the sim-
. 7 2 .
p l i c i t y of network analysis for t ha t par t of the problem. On
the other hand, a c i r c u i t - l i k e description of the plasma i s
not adequate. E .g . , we wi l l see t ha t a concept l ike s e l f - i n
ductance, which i s very useful in c i r c u i t theory, loses
most of i t s sense for a plasma in motion. This i s due to the
non-l ineari ty of the plasma equations. The c i r c u i t equations
wi l l be derived in Sec. VI E.
The problem we have s ta ted above i s a mixed i n i t i a l -
value boundary-value problem. I t has the special pecul ia r i ty
that the i n i t i a l data are contained in the boundary data,
because B (0) i s the only variable that has to be known zp
over the cross-section of the cylinder a t t = 0 . I t s value
follows from the boundary data by
applying Eq. (6-10) a t t = 0. If we
consider for th is problem what the
charac te r i s t i cs a re , we see from
the Eqs. (4-29)-(4-32) t ha t the x
charac te r i s t i c speeds e i the r vanish or blow up. In fac t , an
analysis of the Eqs. (6-4)-(6-6) similar to the one of Sec.
IV B reveals that only three cha rac te r i s t i c speeds remain
for th i s simplified problem: the contact disturbance with
u - u and perturbations v' - 0 and B'/B* = - B /B . f 0, and 0 9 Z Z 9
the fast wave with u = u , = + » and perturbations v' / 0 and Bó = Bl = 0. Together, these charac te r i s t i cs propagate the
tf Z
boundary data from r = a inward. If we apply a radial veloci ty
perturbation at the wall the fas t wave instantaneously se t s
up a velocity prof i le over the en t i r e cross-section of the tube,
whereas the f ield perturbations are jus t carried inward as a
— b o u n d a r y A a\ a.
•*• y
\'\eS A<tA
.73.
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 app l i ca t ion of Eqs. (6-3)
and (6 -9 ) . Therefore, we w i l l have to reduce the number of
plasma equa t ions . This can be done by in t roduc ing a new v a r i
able descr ib ing the f i e l d l i n e topology. At the same time we
w i l l use the opportuni ty to dwell on some of the consequences
of f lux conservat ion s ince these p r o p e r t i e s have a much wider
v a l i d i t y than the problem we are p re sen t ly t r e a t i n g .
Let us cu t the torus (a cy l inder of radius r a c t u a l
ly) in two ways: a t r ansve r se and a l o n g i t u d i n a l cu t as i n d i
cated in the f igu re . With these two c r o s s - s e c t i o n s two f luxes
may be a s soc ia t ed :
the f lux the long way
r * ( r , t ) = 2-n f B ( r , t ) r dr , (6-11) z J z o $ 8 the flux the s h o r t way
r 4>e(rft) = 2TTR J BQ(r,t) dr. (6-12)
o
Ca lcu la t ing the r a t e of change of these fluxes from Fa raday ' s law
as expressed by the Eqs. (6-5) and (6-6) we f ind:
. 7 4 .
a* r 3Bf
at
3$
3t J
2itR | ~ dr * - 2irR
3B o r
at dr =
J(v BQ)'dr = o
r - 2w J (r v B ) ' d r
2irR v Be - - v *6 ,(6-13)
2irr v B * - v • , z z (6-14)
so that fluxes through contours moving with the fluid remain
constant:
d<f> d$ 8 _ z
dt dt (6-15)
in agreement with Eq. (5-51).
Another consequence of the Eqs. (6-13) and (6-14) i s
that one may define a local ( i .e . depending on r) variable
q(r , t ) = dif>, dT
rB (6-16)
anK
1 Ö|.WR
which is extremely useful for the description of field l ine
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
straight lines and q measures the
pitch of the field l ines;
q - d*/d9
along a field line (where $ is the
toroidal angle) . Traditionally, the name safety factor has been
in use for the parameter q because s tabi l i ty with respect to
internal kink modes in tokamaks requires q(axis) > 1, whereas
s tabi l i ty 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 ) :
1 C q < 0 l U r
. 75 .
2ar 2B , . i . ( 6 -17 ,
t o r
Th i s i s t h e r eason f o r t h e use o f q i n c o n n e c t i o n w i t h e x t e r n a l
k ink mode s t a b i l i t y c r i t e r i a , because t h e s e modes a r e d r i v e n
by the toroidal current. For s t a b i l i t y , It o r t a ) s h o u l d n o t s u r
p a s s a c r i t i c a l v a l u e , c a l l e d t he K r u s k a l - S h a f ranov l i m i t viiich
c o r r e s p o n d s p r e c i s e l y w i t h q ( a ) = 1 . Th2 f a c t t h a t q = 1 a l s o
c o r r e s p o n d s t o a t o p o l o g y where t he f l u i d l i n e s c l o s e on them
s e l v e s a f t e r one r e v o l u t i o n t h e s h o r t way and one r e v o l u t i o n
t h e long way around t h e t o r u s h a s been t h e s o u r c e o f much con
fus ion i n t h e l i t e r a t u r e . The p o i n t i s t h a t t h e l a t t e r f a c t
has n o t h i n g t o do w i t h e x t e r n a l k i n k mode s t a b i l i t y in a g e n
u i n e t o r u s because t h e l i n e a r r e l a t i o n s h i p be tween q and I
e x p r e s s e d i n Eq. (6-17) does not hold there, whereas t h e i n t e r p r e t a
t i o n of q a s a t o p o l o g i c a l p r o p e r t y of t h e f i e l d l i n e s reir.ains
v a l i d i n t o r o i d a l geomet ry . The r e a d e r s h o u l d be warned i n a d
vance t h a t s t a t e m e n t s t o the o p p o s i t e a r a encountered i n t h e
l i t e r a t u r e .
Some more c o n c e p t s t h a t a r e f r e q u e n t l y e n c o u n t e r e d :
- R a t i o n a l s u r f a c e : T h i s i s a s u r f a c e ( in t h i s ca se a c y l i n d e r
of a c e r t a i n r a d i u s ) where t h e f i e l d l i n e s c l o s e upon them
s e l v e s a f t e r M r e v o l u t i o n s t he s h o r t way and N r e v o l u t i o n s
the long way around t h e t o r u s :
q = N/M , (6-18)
I f q i s i r r a t i o n a l the f i e l d l i n e s GO not close ax t h e m s e l v e s and
j u s t cover a magne t i c s u r f a c e e r g o d i c a l l y .
- R o t a t i o n a l t r a n s f o r m : This q u a n t i t y has been used t radi t ional ly in
connec t ion wi th s t e l l a r a t o r s . I t j u s t measures t he ang le i
.76.
tui
® 0. (iota) over which a field line
proceeds after one revolution
the long way around the torus:
l = 2 i r /q ( 6 - 1 9 )
Hence, the use of i i s fully
equivalent to tha t of q to de
scr ibe f ie ld l ine topology.
- Normalized inverse pitch of the f ie ld l i ne s ;
\i = 1 /q R • ( 6 - 2 0 )
This synbol expresses the sane property as q. I t has been, used e x t e n
s i v e l y i n screw p i n c h l i t e r a t u r e f o r s t r a i g h t c y l i n d e r s . A
d i s a d v a n t a g e of t h i s v a r i a b l e i s t h a t i t h a s nc geometrical
meaning i n genu ine t o r o i d a l geomet ry . For t h i s r e a s o n t h e use
of q i s t o be p r e f e r r e d .
R e t u r n i n g now t o our d i s c u s s i o n o f t h e dynamics o f t h e
p lasma , the most i m p o r t a n t p r o p e r t y of q s t i l l r e m a i n s t o be
e x p l o i t e d . The r a t e of change of q may be c a l c u l a t e d by u s i n g
Fa raday ' s law ( 6 - 5 ) , (6-6) :
»„ „ 3B rB 3Ba , rB
a t RB„ at 2 at RB0 V ^ 2 v V q
0 RB e e e
Hence, moving w i t h t he f l u i d we f i n d
dq - 19. ^ • « T * = 7T + v q - 0 , d t 3t
( 6 - 2 1 )
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 c a l c u l a t e d i f we know v { r , t ) .
For force- f ree f i e l d s i n a cy l inder the reduct ion of
the number of plasma equat ions i s obtained by the fac t J i a t the
Eqs. (6-4) and (6-16) imply t h a t both f i e l d 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
\/ q2 + r2 / R2 L J q 2 + r 2 / R 2
% A(t) [l - \ r2/R2q2 - J(r/R2q2) dr \
B (r.t) - A(t) \LJ^ZSL exp [- [ - L & L - d r 1 9 Vq2+r2/R2 L J q2+r2 /R2 (6 -23)
A ( t ) ( r / R q ) [ l - \ r 2 / R 2 q 2 - j " ( r / R 2 q 2 ) dr ] %
as may e a s i l y be v e r i f i e d by s u b s t i t u t i n g these express ions
i n t o Eq. (6 -4 ) . Here, A i s an i n t e q r a t i o n cons tan t and the
approximation»on the RHS r e s u l t from the order ing q ^ 1, e << 1.
Clear ly , wo have succeeded in reducing the number of v a r i a b l e s
needed to descr ibe the fo rce- f ree f i e l d 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 3 Bz
(6-6) for -r- - and -r— and finally substituting the (unap-31 9 C
proximated) e x p r e s s i o n s (6-23) for Bfl and B : o z
. q 2 - r 2 / R 2
v « » + _i v t r q i + r Z / R 1
( 6 - 2 4 )
- JL [x + 2 -Jllï .. - r 2 / R 2 ( r g 2 ' + 4 r 2 / R 2 n y . Q . r2 q 2 + r 2 / ' ( q 2 + r 2 / R 2 ) 2
This would be a h o r r i b l e n o n - l i n e a r equat ion i n v ( r , t ) and
q ( r / t ) i f we d id n o t have our smal l parameter e a v a i l a b l e .
J u s t dropping small terms in e we g e t :
v " + i v ' - - L v - 0 . (6-25) r r 2
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 ) f o l l o w s from dr ( t ) / d t = v ( r ) ~ ( r / a ) v ( t ) , s o o o o o
t h a t t
r Q ( t ) - a exp [ j ( v ( t ) / a > d t ] . (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 + r 2 / q 2 ( r )R2] B 2(r( t ) ) = B2 (t) . (6-30) 6 o z o v o O J E ° zp
These expressions determine the value of A(t) to be used in
Eq. (6-2 3) i f the integral i s taken from r (t) :
A(t) = B (t) = ( a 2 / r 2 ( t ) ) B (0) . zp o zp ( 6 - 3 1 )
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 coi ls to the plasma is perfect , and
since we wi l l neglect s t ray inductances and res i s t ive losses in
the supply cables, the equivalent z-and 9-circuit look l i k e :
1 1 — > 1 * I . p
f. i =T f. 1 ' 1
E , * vB e =o
f. i =T f. p i A * •» a
.80.
Here, the self-inductance L T couples the primary toroidal
current I to the induced plasma current I , whereas the z zp
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 V * - - I T - - L T < Ï T * T ! * > • <6-'2)
where L T i s the s e l f - i nduc t ance of the t o r u s , which i s j u s t
a geometr ical f ac to r which does no t depend on the d i s t r i b u t i o n
of the plasma cu r ren t to leading o rder 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 t ha t the equat ion for the z - c i r c u i t becomes:
d2V , d I C £ + _L v = . _ ^ 2 . . ( 6 _ 3 5 ) z d c i LT z dt
Since the s e l f - i nduc t ance L_ has served i t s purpose in p r o v i d
ing us with a p i c t u r e of how the t o r o i d a l c u r r e n t i s induced
i n t o the plasma, we w i l l now push the o rde r ing in the inve r se
aspec t r a t i o to i t s very l i m i t and n e g l e c t the term V /L
a l t o g e t h e r . This i s allowed i f I»T i s much l a r g e r than t y p i c a l
i n t e r n a l s e l f - i nduc t ances of the plasma. We w i l l see t h a t these
are given by l , ^ y u R, so t h a t we assume L_>> I . Formally, Z £ 0 * Z
t h i s assumption i s j u s t i f i e d by t ak ing e small enough. However,
i t i s c l e a r t h a t t h i s assumption i s a r a t h e r pcor one for p r a c
t i c a l purposes due to the weak ' logar i thmic growth of L T wi th e .
The value of V i s r e l a t e d to the boundary values of
. 8 1 .
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 relat ions intq Eq. (6-35) and neglecting the
term with LT leads to the equation for the z-circui t :
dT ^ V - Az *e • A2 =- TT • ( 6 " 3 7 )
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 Ifl • (6-38) d t C9 -9
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 B means of z
l« " — Y P*d* s — B > ( 6 -39 ) o t o r 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
. 8 2 .
C l e a r l y , t h e two c i r c u i t e q u a t i o n s a r e n o n - l i n e a r and , more
o v e r , t h e y a r e coup l ed t h rough t h e o c c u r r e n c e of v i n b o t h
e q u a t i o n s .
F . SOLUTION OF THE PROBLEM
We have s o l v e d t h e p lasma e q u a t i o n s a l r e a d y , i . e . we
have e x p r e s s e d v ( r , t ) and q ( r , t ) i n t e rms of t h e boundary d a t a
v ( t ) and q ( t ' ) by means of E q s . (6-22) and ( 6 - 2 6 ) . To d e t e r m i n e
v ( t ) and q ( t ' ) we have t h e two c i r c u i t e q u a t i o n s (6-37? and
( 6 - 4 0 ) . From t h e s e two e q u a t i o n s one e a s i l y f i n d s an e v o l u t i o n
e q u a t i o n f o r q ( t ) :
* £ = (A9 - V < • (6"41)
whereas the evolutior equation for v(t) i s found by s u b s t i t u t
ing the expressions 6-23) and (6-31) into Eq. (6-40) , while
using Eq. (6-26) and neglecting terms of order e 2 :
£ -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:
v(t) m ~ 2 aü) cot8 («t + a) f (6-^3)
where the phase angle a i s determined
by the i n i t i a l veloci ty :
a = - arccotg (2v(0)/aw), (6-44)
and the frequency u i s determined by the constant kQ of 'rhe
e-c i rcu i t :
. 8 3 .
u> = ( 2 A Q / a ) 1 / ? = U q C g ) 1 / 2 - ( 6 - 4 5 )
Here, we have int roduced a se l f - induc tance l i k e q u a n t i t y for
the 9 - c i r c u i t :
*0 = 7 ^ . a 2 / R • (6-46)
6 2 o
Since Eq. (6-41) i s a l i n e a r equation in q the so lu t i on
i s e a s i l y found a f t e r s u b s t i t u t i n g v( t ) of Eq. (6 -43) : q"(t) = q"(0) [ cos (lot + a ) / c o s a ] X , ( 6 - 4 7 )
where the parameter A i s determined by the r e l a t i v e d i f f e rence
of the cons tan t s A„ and A of the c i r c u i t s : 6 Z
Afl " A „ ( £ f l C f l ) " 1 " < ^ C , ) _ 1
\ = - L 5- = — L i 5-5 . (6-48)
<Ve> Here, we have int roduced an a d d i t i o n a l s e l f - i nduc t ance l i k e
q u a n t i t y for the z - c i r c u i t :
£ = \ u R . (6-49) z 2 o
C l e a r l y , a n a t u r a l frequency u> appears in these s o l u t i o n s which
i s e n t i r e l y determined by the G-c i r cu i t .
I t i s i n s t r u c t i v e to consider the time-dependence of the
components corresponding to Eq. (6-47):
B" ( t ) = B (0 ) s i n ( u t + a ) / s i n a , ( 6 - 5 0 ) z z
B ( t ) - B ( 0 ) S i n ( a ) t + a> / s i r t° | . (6-51) [cos(ut+a)/cosaj
From these s o l u t i o n s i t i s c l ea r tha t the e - c i r c u i t o s c i l l a t e s
harmonically with a frequency u> = u> = U C.) ' . The reason 8 0 9
i s c l e a r : the z - f i e l d i s very l a rge and hardly a f fec ted
by the small plasma c u r r e n t I , so t h a t to leading order the 9p
. 8 4 .
8-circui t merely sees a vacuum f ie ld and, consequently, o s c i l
l a t e s with a frequency determined by l., which i s jus t the
vacuum self-inductance of the 6-coi l . On the other hand, the
6-field i s en t i re ly determined by the plasma current I , so zp
that the z -c i r cu i t i s strongly affected by the i.on-linear
plasma dynamics. Consequently, the z - c i r cu i t displays anhar-
nonic time-dependence.
There i s one case in which the z - c i r cu i t also displays
a harmonic time-dependence. This i s the case X = 0, which may
be writ ten as a so r t of resonance condition between the two
c i r c u i t s : « e = wz , ( 6 -52 )
where „e , U ^ ) " 1 7 2 - «", s " « V * 1 ' * •
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
. 8 5 .
fo rce - f ree f i e l d i s s a t i s f i e d . An equ iva l en t c i r c u i t r e p r e
s e n t a t i o n for the general case X f 0 does not e x i s t . There i s
no way around so lv ing the fu l l s e t of equa t i ons . Furthermore/
even in the resonant case the i n t e r p r e t a t i o n of the q u a n t i t y
l as the i n t e r n a l s e l f - induc tance a s soc i a t ed with the plasma z c u r r e n t I i s extremely doubtful as we w i l l see in the follow-
zp J
ing s e c t i o n .
If the resonant condi t ion (6-52) i s not s a t i s f i e d the
z - c i r c u i t does no t follow the o s c i l l a t i o n of the e - c i r c u i t and,
consequent ly , shear of the f i e l d l i n e s i s produced:
q0 ( -0 W * = » - W t r . l
qlo")
c rea t ion of shear in the
non-resonant case (X f 0)
-*• r
q ( r , t ) = q ( 0 ) [ { l - ( r V a " ) s i n 2 (u,t + a))/cos\]Xf] (6-53)
This so lu t ion develops a pathology a t the time u t + a = TT/2 ,
when v •+ 0 . Depending on whether A i s p o s i t i v e o r nega t ive
q ( t ) e i t h e r goes to zero or blows up. The reason for the s ingu
l a r i t y appears to be the mismatching of what would be the natural
frequencies of the two c i r c u i t s / which jauses t r oub l e a t the
end of the inward motion. Since the z - c i r c u i t cannot follow the
o s c i l l a t i o n of the e - c i r c u i t , the vol tage V (t) given by Eq.
(6-8) e i t h e r lags behind or runs ahead of the evolu t ion of
. 8 6 .
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) t o 9° t o z e r ° o r t o b l o w u p .
(This i s c a l l e d s e l f - c r o w b a r r i n g of t h e p l a s m a ) . Hence, i f
X ^ 0 t h e i d e a l MHD model b r e a k s down a t t h e end of t h e
compres s ion .
G. FLUX AND ENERGY CONSERVATION
In t h i s s e c t i o n we wish t o s t u d y t h e consequences o f
f l u x and ene rgy c o n s e r v a t i o n f o r t h e two c i r c u i t s and t h e two
magne t i c f i e l d components s e p a r a t e l y . We have a l r e a d y seen i n
Eq. (6-15) t h a t t he f l u x e s t r a p p e d i n a c o n t o u r moving w i t h
t h e f l u i d remain c o n s t a n t . However, i t i s of more i n t e r e s t
h e r e t o i n v e s t i g a t e t h e r a t e of change of t h e t o t a l f l u x i n
t h e t u b e by t h e i n f l u x of m a g n e t i c f i e l d from t h e b o u n d a r i e s .
From Sqs. (6-13) and (6-14) i t i s c l e a r t h a t , a s fa r as fluxes
a r e c o n c e r n e d , i t makes s e n s e t o a s s o c i a t e B_ w i t h the 6-ci rcui t , 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 * / 3 t • I T I I = 0
z " 31 / 3 t zp
3* / a t u t i i - _ z
where the reason for the use of quotation marks will become
clear below.
( 6 - 5 4 )
( 6 - 5 5 )
.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<J~/at = V . 0 l Z
Inserting this expression into Eq. (6-54) we obtain:
u V v B" " L •• = 2_5 = - M Ra &- = i- p R
z 2Tra 3 B e / 3 t ° 3Be/8t 2 ° 1 + Atg2(u)t + a) ( 6 _ 5 6 s
where we have s u b s e q u e n t l y s u b s t i t u t e d t he E q s . ( 6 - 3 6 ) , ( 6 - 8 ) ,
(6-43) , and (6-51) . C l e a r l y , t h e z - c i r c u i t s e e s a s e l f - i n d u c t a n c e
"L " o f t h e plasma t h a t changes in t i m e . Th i s change i n t ime i s
o n l y known a f t e r t h e problem has been s o l v e d . Also n o t i c e t h a t
"L " ^ l , as g iven by Eq. (6-49) e x c e p t f o r t he r e s o n a n t case
X = 0.
For t h e s e l f - i n d u c t a n c e o f t h e p lasma as seen by t he
e - c i r c u i t we have from Eq. (6-55) :
u V v" B , 2
9 2*R 3B / a t ° R 9B / 3 t 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. I t is well-known
that the self-inductance of a current-carrying conductor is
properly defined in terms of the total current flowing through
. 8 8 .
the conductor and the magnetic energy of the f i e l d c rea ted by
t h a t c u r r e n t according to the d e f i n i t i o n
o Accordingly, the i n t e r n a l s e l f - induc tances of the plasma
should be defined as fol lows:
\ Lz l2zp • we - I a t Be dT ' ( 6 " 5 9 )
r plasma ™
plasma o
For W and I we have the following exp re s s ions :
X .P • T 7 B e ( t ) •
so t h a t the i n t e r n a l s e l f - i nduc t ance of the plasma a s s o c i a t e d
with the B -component i s given by: ö
Lz " \ wo R ( 1 " r o / a l t ) = " L z " * lz • ( 6 " 6 1 )
Hence, the apparent s e l f - i nduc tance "L " as seen by the z-circuit
i s a t l e a s t twice as l a rge as L as c a l c u l a t e d from energy z
c o n s i d e r a t i o n s .
For VJ and I a + I . we have the following e x p r e s s i o n s : Z o üp
„ . i i l i rB2 r d r „ilMlj2it) J Z <v y z v ' z y 0 0
! + j , l l « B ( t ) * l l i B ( t ) , 6 6p ]iQ zp * M& z
which gives the fol lowing express ions for L :
Again, the expected r e s u l t i s obta ined due to the fac t t h a t the
z - f i e l d approximately I s a vacuum f i e l d . (This r e s u l t i s only
t r ue to l ead ing order in e ! ) .
. 8 9 .
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 _ ( B2 + B 2 ) ( 6 - 6 3 )
2u 8 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 ' d T = _ 4 7 T 2 R \ l ( r v v J ) . r d r = - 8TT2R a V/ÜT. <6 _ 6 5 )
3t 3t J J r
o
This rate of change of energy is due to the flow of energy
from the c i rcui ts 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 S . n do = - - L f ExB-n do = ^ (ÊQB - f Ba> -3 t J ^ ^ y j ' b ^ ^ u 0 z z 9
o o
• ve h ~ vz JZp
= £ (T ce ve) + £ (l Cz v22) • (6-67)
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
V. 1 = ( 4 I T 2 R / M ) a v B2 , ( 6 - 6 8 ) u V O Z
- V I = ( 4 * 2 R / M ) a v B 2
z zp o 9 '
. 9 0 .
so t h a t t h i s seems t o i n d i c a t e t h a t t h e z-component of t h e
magne t i c f i e l d i s a s s o c i a t e d w i t h t h e 0 - c i r c u i t and t h e
8-component w i t h t h e z - c i r c u i t .
L e t us examine w h e t h e r t h e l a t t e r s t a t e m e n t makes
s ense w i t h r e s p e c t t o t h e change i n t ime of t h e s e p a r a t e
magne t ic ene rgy components :
3Wz 2ir2R 3 r „ 2 ., 4TT2R - ^ 2TT2R 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 _ 6 9 )
0 2ir2R 3 r „ 2 , An2R - =? 2*2R f n2i -rr » — \ B* r dr » a v B* v Bz' 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 i n f l u x from t h e 0 - c i r c u i t , and v i c e v e r s a f o r t h e B -component and the z -c i rcu i t , but t h e r e i s a f low o f ene rgy F
0
from t h e z - c i r c u i t t o t h e B -component :
3W - ~ » - T- ( T Cfl V2) + F , (6 -71) 3t 3 t 2 6 0
3W _Jt 3 - i ( i C V2) - F , (6 -72) 3t 3 t v2 z z' *
where 2 a
F = ^-^ f v B z ' r dr . ( 6 -73 ) p J z
0 O
For t h e r e s o n a n t c a s e (A = 0) we f i n d from t h e s o l u t i o n s o b
t a i n e d i n Sees . VI D and F:
F . " 2 ^ - a v B? , (6 -74 ) yo 6
so t h a t 3WQ/3t = 2F - F = F . C o n s e q u e n t l y , t h e e n e r g y i n f l u x 6
from t h e z - c i r c u i t i n to the p lasma i s e q u a l l y dev ided between t h e
i n c r e a s e of m a g n e t i c ene rgy of t he B -component and flow of ene rgy
t o t he B -component . The conversion of e l e c t r o s t a t i c ene rgy of t h e Z
c a p a c i t o r banks t o m a g n e t i c ene rgy of t h e p lasma t u r n s o u t t o be
. 9 1 .
much more t o the advantage of the l a rge z-component of the
magnetic f i e l d than t o the small e-component.
As a r e s u l t of t h i s e f f e c t the s e l f - i nduc t ances Lfl
and L as def ined in Eqs. (6-59) and (6-60) lose t h e i r mean-
i n g , whereas the apparent s e l f - i nduc t ances "LQ" and " I ^ " seen
by the c i r c u i t s can only be c a l c u l a t e d a f t e r the so lu t i on t o
the complete problem has already been obtained. We r e p e a t : The b a s i c
reason i s t h a t f lux conservat ion holds for the two components
B and B s e p a r a t e l y , whereas energy conservat ion does n o t . 9 2 *
Diagram of the r i ch and the
poor c o u n t r i e s .
During compression you l o s e
your energy, dur ing expansion
you gain i t back. Morale: Don't
l e t yourse l f be squeezed. F igh t
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".
T t ( 2 C e v e )
3W z
9 y 9 3 W e <* i I f v2\-* 9 3 W e ~ 7 t ( 2 C * V at
. 9 2 .
V I I . LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF IDEAL MHD
A. SUMMARY OF SOME CONCEPTS OF CLASSICAL MECHANICS
One of t h e roost power fu l and b e a u t i f u l p a r t s of
c l a s s i c a l p h y s i c s a r e t h e L a g r a n g i a n and Hami l ton ian f o r
m u l a t i o n s o f c l a s s i c a l m e c h a n i c s . In p a r t i c u l a r , t h e formu
l a t i o n of a L a g r a n g i a n from which t h e e q u a t i o n s of motion
can be d e r i v e d by means of H a m i l t o n ' s p r i n c i p l e i s one of
t h e most c o n c i s e d e s c r i p t i o n s o f dynamica l sys tems . One may
c o n s i d e r a b r a n c h of p h y s i c s t o have become p a r t o f t h e
c l a s s i c a l c u r r i c u l u m i f one s u c c e e d s i n c o n s t r u c t i n g t h e a p
p r o p r i a t e L a g r a n g i a n . Fo r i d e a l MHD t h i s was accompl i shed
by Newcomb i n a p a p e r o f 1962 ( N u c l e a r F u s i o n , S u p p l . 2_, 1962,
4 5 1 ) .
L e t us f i r s t c o l l e c t a few p e r t i n e n t c o n c e p t s and
formulas from c l a s s i c a l m e c h a n i c s . For a c l a s s i c a l dynamical
sy s t em t h e L a g r a n g i a n L may be d e f i n e d as t h e d i f f e r e n c e of
t he k i n e t i c and p o t e n t i a l e n e r g y :
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 pr inciple then s t a t e s tha t the notion of the
system from time t . to time t - i s such tha t the l ine in tegral *k
x z
J L d t i s an extremum:
« j " L ( q k , q k > t ) d t - 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) :
<k " % • K " - wk • (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, v iz . tha t
the corresponding generalized momentum i s a constant of the
motion.
For continuous systems and f ields the motion i s not
described by a discrete s e t 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 f ie ld ^j(x, t) . (For a general f i e ld
n could have more than three components. However, for our
purpose a vector f ield in 3-space suffices as we wi l l s ee ) .
In t h i s case the Lagrangian i s an in tegra l over a l l ava i l
able space of the Lagrangian density Ad :
L = J * dx , (7-7)
where H now becomes a function not only of the continuous se t
of generalized coordinates n*t but also of the pa r t i a l der iv
a t ives of TK with respect to x. and t :
* - * ( n . . , *i> n i ' xi» fc) • (7-8)
Here, we have adopted the notation
j
Hamilton's pr inciple now takes the form
6 j d t j&dt - 0 , (7-10) S
where the variat ion i s to vanish at the endpoints t . and t~ and a t the spa t i a l boundaries over which the volume integrat ion i s
taken. The d i f fe ren t ia l equations corresponding to th i s var ia
t ional problem now become p a r t i a l d i f fe ren t ia l equations with
x. and t as independent var iab les :
. 9 5 .
dt 8n. • 3x. 3n-• 3n.
Hence, ins t ead of the n o rd inary d i f f e r e n t i a l equat ions we
had in po in t mechanics (k = 1 , 2, n) , we have fewer but
p a r t i a l d i f f e r e n t i a l equat ions in cmtinuum mechanics (i = 1,2,3).
Again, one may cons t ruc t genera l ized momentum den
s i t i e s :
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 equat ions corresponding to Eq. (7-11) a r e :
11; = ^ „ » 1 d TT .
• _ y _L_ a * _ JO. • ( 7 _ l j ) 11 i " T Sx . 3 D - • a n .
3 j 1J 1
I f ^ does no t e x p l i c i t l y depend on time the t o t a l Hamiltonian
i s conserved: ~ » 0 . (7 -16 )
a t
This i s the form energy conservat ion takes in the Hamiltonian
formulation.
A p a r t i c u l a r example of a continuous system i s presented
by the propagation of sound in a gas . In t h a t case , the genera l ized
coordinates n(>:,t) can be taken to be the displacement of the
gas . In the case of i d e a l MHD our f i r s t ques t ion thus becomes
what to use as genera l ized c o o r d i n a t e s .
.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 p d7 + 7 ( P + I B2) " %'™ = ° » ( 7 _ 1 7 > d * d t B*7v - BV«v , V-B - 0 , (7 -18)
d£ = - YP V-v , (7 -19 ) dt
f f - " P *'% • (7-20)
Our program consists in formulating a Lagrangian density such
that the equation of motion (7-17) i s obtained as Lagrange's
equation corresponding to the var ia t ional problem expressed by
Hamilton's p r inc ip le . Also, a Hamiltonian density i s to be
constructed such tha t Eq. (7-17) i s jus t obtained as the Hamil
ton i an equation of motion.
We have to address two questions f i r s t : What to take
as generalized coordinates? Which role are the three additional
equations (7-18)-(7-20) to play in th i s formulation?
Let us s t a r t with the l a t t e r problem. To understand
the meaning of the three evolution equations for B, p and p
r eca l l the discussion in Sec. V C, where we derived global
conservation laws for Jg, ^ ( y e s , what in th i s chapter wi l l
become the Hamiltonian dens i ty) , 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 f luid. Without
.97.
proof we state the required equations:
•h <d*> - d * - ( 7 * > • (7_21)
± (d4) - - C?v)'d£ • (V-v)do , (7-22)
-*- (dr) « (V-y)dT . (7-23) at "v
These equat ions form the k inemat ic b a s i s for f l u i d mechanics.
From the second r e l a t i o n and Eq. (7-18) we may c a l
c u l a t e the r a t e of change of the l o c a l f lux B*d<j through a
surface element moving with the f l u i d : d d £ d
dt {l'dV "ST * d * + * ' d t ( d ^ } (7-24)
- <$- 7 x - i *•;&>•** + $•[-*?• ds + *'x,d%] - °-This i s the well-known r e s u l t , of c e n t r a l importance in i d e a l
MHD, t h a t the f lux through a surface moving wi th the f l u i d i s
conserved. From the r e l a t i o n (7-23) we may c a l c u l a t e the r a t e
of change of the l o c a l mass in a volume element moving wi th the
f l u i d :
-r- (pdx) » -T— dt + p— (d t ) » - pV'vdT + pV«vdt * 0 . at a t at ^ «v
(7-25)
This i s the l o c a l counter p a r t of Eq. (5 -43) : the mass in a v o l
ume element moving wi th the f l u i d i s conserved. F i n a l l y , Eq. (7-19)
for the evolu t ion of the p r e s su re may be use fu l ly combined with
Eq. (7-20) to prove t h a t
£ ( P P " Y ) » 0 , (7-26)
which i s of course no th ing e l s e than a r e s t a t emen t of the equa
t ion for entropy conserva t ion (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)
. 9 9 .
where X
i j 9x ( 7 - 3 0 )
o j
The l a t t e r m a t r i x c o n n e c t s t h e p o s i t i o n s x of t h e f l u i d e l e
ments a t t ime t t o t h e i r i n i t i a l p o s i t i o n s x . The J a c o b i a n
of t h e t r a n f o r m a t i o n from x t o x i s then j u s t t h e d e t e r m i n a n t
of x. . :
J = D e t ( x . . ) = r e . , . e . x . . x. x „ , i j 2 ik£ jmn t j km £n
where c . ., i s t h e L e v i - C i v i t a pseudo t e n s o r :
(7-31)
ijk
1 if ijk is an even permutation of 123
- 1 if ijk is an odd permutation of 123
0 i f i = j , o r j = k , o r i = k
( 7 - 3 2 )
(We a d o p t e d t h e summation c o n v e n t i o n t o sum ove r e q u a l indices) .
D e f i n i n g t h e c o f a c t o r A. . as t h e d e t e r m i n a n t o b t a i n e d from
(x. .) by t a k i n g o u t t he i - t h row and t h e j - t h column, e . g .
x 23
X l l X12 X31 X32
e .
, we have t h e f o l l o w i n g i d e n t i t i e s :
3 J i j 2 ik£ jmn km In 3x. .
i j
J 5. . = A, . x, . , i j k i k j
3A. . — U . - 0 . 3x 0 j
( 7 - 3 3 )
( 7 - 3 4 )
( 7 - 3 5 )
The l a t t e r r e l a t i o n s a r e s u f f i c i e n t t o p r o v i d e t h e i n
t e g r a t e d form of the k i n e m a t i c r e l a t i o n s ( 7 - 2 1 ) - ( 7 - 2 3) :
dx. = x . . dx . , ( 7 - 3 6 )
do. • A. . do . 1 1J Oj
( 7 - 3 7 )
di = J dT . o ( 7 - 3 8 )
.100.
From the conservation laws (7-24)-(7-26) we ha^'e:
PP_Y - PoP0"Y » (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* V o $ ) / J • ( 7 ' 4 7 )
I»
To recap i tu la te : We have shown tha t the Eqs. (7-18)-(7-20) may
be considered as holonomic const ra ints by exp l i c i t l y in tegra
ting them to obtain the Eqs. (7-42)-(7-44). In th i s form B, p ,
and p are given as functions of x , t . I t 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) 1 0 1 O
as Lagrange's equation.
. 1 0 1 .
C. LAGRANGE AND HAMILTON EQUATIONS OF MOTION
Hamil ton 's p r i n c i p l e s t a t e s t h a t the evolu t ion
of the system i s such t h a t
6 f d t [ g . ( x . . , x. , x. , x , t)dx - 0 , (7-48) J J I J l l O 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)):
^ = \ P v2 , UT= p/(Y-l) + | B2. (7-49)
A minor modification is needed to account for the fact that
the Lagrangian is defined as the integral over the initial
volume T : 0 f L - J*dT
- S*'dT - 5 c X - * T ) d T - J ( * - * T ) (p Q /p)dT o . (7-50)
Hence, we p o s t u l a t e
po L2 x ( T - D P 2 p l ' ( 7 5 1 )
o r , in terms of the proper genera l i zed coordina tes u 1 Po 1 a, - a- p x - - — x. . x., B . B . . (7-52)
(Y-1)JY"1 J ° J ° Lagrange 's equa t ion ' corresponding to the v a r i a t i o n a l
problem (7-48) r e a d s :
JL p . + £ _ ! _ - I X . . M. . o . (7-53) dt «x. Y x . ï x . . 3 x ,
.102.
Inserting the expression (7-52) gives:
P X. + A T ( T T ) 1 Oj 1 ]
+ ? 3 T T ( l k "TTT "k£ \ m V Bom " 7 x i k B oj Bok} = ° > i o j I J J
w h i c h by t h e u s e o f E q s . ( 7 - 3 3 ) , ( 7 - 3 5 ) , a n d ( 7 - 4 5 ) b e c o m e s :
p V. + Z [A. - r r — ( -^ + - V x D x. B B ) *o l . L i j 3x . jY 2J2 K-* K™ oa ora
- B . T-2— d x-, B . )] = 0 . ( 7 - 5 4 ) oj 3x . J ik ok J
J o j
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 */ M O
F u r t h e r m o r e , we n e e d t o c o n v e r t d e r i v a t i v e s w i t h r e s p e c t t o x
' *• ^o
to derivatives with respect to x. This is done as follows:
3x . 3x, 3x • . o i k _ o i
I J 3x, 3x . Sx, k j J k oj k J
w h i c h b y v i r t u e o f Eq . ( 7 - 3 4 ) g i v e s 3x . n o i 1 , 3x J k i
s o t h a t .. 3x . 3 1 3 3 o i l_ . (1 . , .
3x, 3x, 3x . * J k i 3x . " W - 5 0 ; k k o i o i
We a l s o n e e d t h e E u l e r i a n c o u n t e r p a r t o f Jï «V , By v i r t u e o f E q s .
(7-4 3) and ( 7 - 3 4 ) B. - L . . 1 . A . . B . v - * - . . i fi..B , J - . » B i - J - . (7-57)
l 3x. j 2 i j i k o j 3x J I J oj 3x , J ok 3x , l J ok ok ok
By means o f t h e E q s . ( 7 - 5 6 ) a n d ( 7 - 5 7 ) we may t r a n s f o r m E q . (7-54)
t o :
. 1 0 3 .
dv. p T T + J i ^ - <p + 7 B*> _ J B - ^ - B ; = 0 »
o d t 3 x . Z i B x . i or dv
P -,v + V ( P + \ R 2 ) " B * 7 B " ° » (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 - 5 9 ) l ^o 3x. o i
l
We then obtain the Hamiltonian density
Tt ( x . . , x . , I T . , x , t ) = n. x . - * 1 J 1 1 O 1 1
tr2 . P o . 1 _ + r + ^ r x . . x . . B . B . , ( 7 - 6 0 )
Po ( Y - 1 ) J Y " 1 J o j 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 _ 6 1 )
3 OJ 1 J 1
Substituting (7-60) into the f i rs t 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) :
. 1 0 4 .
d l = dT lnldTo " J fexi + i^T xij + Ï7T *i )dTo i I J i
" 1 ix x • + •; ^ "; + 7 if. ) d x J 3 x . 1 3 x . . 3x . 3ir. 3ir. ï o
i i j o j i x
J L 3x. 3x . 3x. . x 3ir. i J o 1 o j xj x
- f ( - * . x . + x.ft.)dT = O . J X X X X O
Hence, we recover the energy c o n s e r v a t i o n law ( 5 - 4 7 ) .
REFERENCES
1 . H. G o l d s t e i n , C l a s s i c a l Mechanics (Addisori-Wesley, Reading,
1950) .
2 . W.A. Newconib, N u c l . Fus ion , 1962 S u p p l . , P a r t 2 (1962)
451 .
"Lagrangian and Haitiiltonian Methods i n Magnetohydrodynamics".
3 . W.A. Newconib, Lecture Notes on Magnetohydrodynamies (unpub
l i s h e d ) .
. 1 0 5 .
V I I I . LINEARIZED IDEAL MHD
A. INTRODUCTION
For many pu rposes i t i s d e s i r a b l e t o have a d e e p e r
i n s i g h t i n t h e dynamics o f t h e plasma t h a n i s o b t a i n e d from a
study of the non-linear e q u a t i o n s . T h i s c o n t r a d i c t o r y s t a t e m e n t
may be c l a r i f i e d by p o i n t i n g o u t t h e ex t reme l i m i t a t i o n s
posed by p r e s e n t - d a y ma thema t i ca l knowledge a b o u t n o n - l i n e a r
p a r t i a l d i f f e r e n t i a l e q u a t i o n s . Once t h e s y s t e m h a s been l inea r
i z e d many more t e c h n i q u e s become a v a i l a b l e and , c o n s e q u e n t l y ,
a much b e t t e r g r a s p of t h e problem i s o b t a i n e d . Of c o u r s e , one
would always be h i n d e r e d by a bad c o n s c i e n c e i f t h e r e were no
p h y s i c a l c o n d i t i o n s where l i n e a r i z a t i o n i s a p p r o p r i a t e . I t i s
o u r gocd f o r t u n e t h a t we a r e i n t e r e s t e d i n s t u d y i n g t h e b e
h a v i o r of con f ined plasma f o r t h e r m o n u c l e a r p u r p o s e s . H e r e , i t
i s i m p e r a t i v e f o r t h e e v e n t u a l s u c c e s s of t h e p r o j e c t t h a t t h e
p lasma i s conf ined i n an e q u i l i b r i u m s t a t e t h a t l a s t s f o r a
p e r i o d t h a t i s much l o n g e r than t y p i c a l t i m e - s c a l e s o c c u r r i n g
i n t h e dynamics of t h e p lasma ( e . g . , t h e Alfvén t r a n s i t t ime
of t h e m a c h i n e ) . For t h o s e sys tems t h e app rox ima t ion of a
s t a t i c e q u i l i b r i u m of t h e plasma i s q u i t e a p p r o p r i a t e . In t h i s
c o n t e x t , t h r e e k i n d s of problems may be a d e q u a t e l y t r e a t e d w i t h
t h e e q u a t i o n s of i d e a l MHD. F i r s t of a l l , one needs t o know
the e q u i l i b r i u m s t a t e of a r e a l i s t i c c o n f i g u r a t i o n . (Here ,
t o r o i d a l sys tems a r e t h e most i m p o r t a n t o n e s ) . T h i s p rob lem
i s s t i l l a n o n - l i n e a r o n e , b u t i t may be s o l v e d f o r q u i t e r e a l
i s t i c g e o m e t r i e s due t o t he s p e c i a l p r o p e r t i e s of t h e non-iinear
e q u a t i o n s of s t a t i c e q u i l i b r i u m . Nex t , t he p rob lem of s t a b i l i t y
.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 -^- = - p v 7 v - Vp + (tfxB) x B , ( 8 - 1 )
3B - ^ = V x (vxB) , V-B = O , ( 8 - 2 )
| ^ = - v-Vp - T p 7 - v , ( 8 - 3 )
l£. = - V ( p v ) . (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
t ions are imposed:
[ p + ^ B2I= 0 . (8-7)
At the conducting wall the vacuum magnetic field is subjected
to
n-§ = 0 . (8-8)
I t may appear less obvious a t f i r s t s ight tha t the plasma v a r i
ables are also subject to boundary condit ions. These are usual
ly qui te obvious when the geometry i s specif ied. E .g . , in a torus
one would specify regulari ty of the variables a t the magnetic axis,
and periodici ty the short and the long way around the torus . The
l a t t e r conditions are also to be imposed on the vacuum field ^.
Instead of a vacuum i t i s sometimes also of in t e res t
to consider an external region which i s also a plasma but with
different magnitude of the variables (e.g. a low-density force-
free plasma), so that s t i l l jump conditions need to be applied
.108.
a t a f lu id- f lu id in te r face . Of course, the equations (8-5)
are then replaced by equations analogous to Eqs. (8-1)-(8-4)
for the var iables y, £, p , and p. At the f lu id - f lu id in ter face
the boundary conditions (8-6) and (8-7) should be supplemented
with
n . J v H = 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 x B , j - V x B , 7-B - 0 , (8-11)
- for the vacuum region:
V x J « 0 , 7 * | » 0 t (8-12)
- a t the plasma-vacuum in te r face :
n • B - t i ' S - 0 , lip * TT *2\- 0 , }* - n xÏÏiJ ,
(8-13)
- a t the wal l :
n •% - 0 . (8-14)
The th i rd re la t ion of Eq. (8-13) for %* i s not rea l ly a r e s t r i c
tion on the kind of jumps one may allow. I t simply t e l l s us
.109 .
what the magnitude i s of the su r face c u r r e n t a s soc i a t ed with
the a r b i t r a r y jumps in the t a n g e n t i a l magnetic f i e ld components.
For a plasma-vacuum system the c u r r e n t l i n e s are alsn p a r a l l e l
t o the plasma-vacuum i n t e r f a c e , so t h a t
no x Vpo « 0 (8-15)
t h e r e . S t r i c t l y speaking, t h i s r e l a t i o n needs not to be s a t i s
f ied a t a f l u i d - f l u i d i n t e r f a c e ( i t does no t follow from the
jump condi t ions for con tac t d i s c o n t i n u i t i e s der ived in Sec.
V B ) , but i t i s usual ly the roost r e a l i s t i c cho ice .
The system of Eqs . (8-11)-(8-14) i s fa r from a t r i v i a l
problem, i n p a r t i c u l a r because of the n o n - l i n e a r p re s su re balance
equat ion (8-11) . However, for simple geometries l i k e s l abs and
s t r a i g h t c i r c u l a r cy l inde r s the s o l u t i o n s are ea s i l y ob ta ined .
For more r e a l i s t i c geometries l i k e t o r o i d a l conf igura t ions t h e
equat ions lead to a n o n - l i n e a r e l l i p t i c p a r t i a l d i f f e r e n t i a l
equat ion for the po lo ida l f lux funct ion (the s o - c a l l e d Grad-
Shafranov equation) for which q u i t e accura te numerical s o l u t i o n
techniques e x i s t . These w i l l be considered in a l a t e r c h a p t e r .
For the p r e s e n t purpose we w i l l imagine t h a t the equat ions
(8-11)-(8-14) are solved so t h a t p , B , and B are known. I t
should be no t i ced t h a t the Eqs. (8-11)-(8-14) do not uniquely
determine these so lu t i ons so t h a t a l o t of freedom i s l e f t t o
choose p a r t i c u l a r e q u i l i b r i a .
Next, pe r tu rb t h i s s t a t i c equ i l ib r ium by a displacement
vec to r f i e l d £<£*t) so t h a t
X - d T - D T ( 8 - 1 6 )
.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 discuss ion on holonomic constraints in the previous
chapter, we again t rea t the equation of motion (8-1) on a d i f
ferent footing than the remaining equations ( 8 - 2 ) - ( 8 - 4 ) . In
sert ing the expressions (8-16) and (8-17) into the l a t t e r equa
t ions we find that they are e a s i l y integrated. E . g . ,
s ince £ does not depend on time. Consequently, to f i r s t order
in k :
«I X v x <* * l0} = ft ' <8_18)
*P $ "k'*V0 " r*o V 'S B * * (8-19)
*P %-V-<P0*> • (8-20)
.111.
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 l i n e a r i z e d i d e a l MHD only one vec tor £ ( r , t ) appears
as a v a r i a b l e , i n c o n t r a s t to the v a r i a b l e s v , B, p , and p needed
in non - l i nea r MHD. In a d d i t i o n to the l i n e a r i t y , t h i s i s a very
s i g n i f i c a n t s i m p l i f i c a t i o n . Also no t i ce t h a t the p e r t u r b a t i o n
of the dens i ty does not appear so t h a t Eq. (8-20) may be dropped
in the l i n e a r a n a l y s i s .
I t i s a l so of i n t e r e s t to ob ta in the equat ion of motion
for incompressible plasmas. As in Sec . I l l B, Eqs. ( 3 - 5 ) - ( 3 - 8 ) ,
the equat ion for incompressible f l u i d s i s obta ined by tak ing the
l i m i t y •*• • and 7«£ -»• 0 such t h a t IT = - -yp v*£ - £*VP remains
f i n i t e . Notice t h a t i t i s only the Lagrangian p a r t - yp Vȣ of
the pe r tu rba t ion of the p re s su re t h a t should be handled with
care in the l i m i t . This procedure g i v e s :
* 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
'I =
»•* (8-22)
(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 " d V (* * v *>• (8"26) In this relation the differentiation with respect to the Lagran-
gian coordinate g has been replaced by the differentiation
. 1 1 3 .
with r e spec t to the Eu l e r i an coord ina te x, which i s c o r r e c t t o
f i r s t o rde r s i nce the d i f f e r ence between the two d e s c r i p t i o n s
i s of h igher o r d e r . From t h i s express ion we now have
0 = n - d i * dl *(I • 7 E)-(n + n, )
•v di *n + di *V£-n + d*. *n,
= dZ * ( V £ « i l l > 11-.
Hence, ^ ' - <v$>'20 * *' w h e r e ' 3 £ ) * But
d l may have any d i r e c t i o n in the per turbed , sur face unperturbed s u r f a c e so t h a t \ * yn .
unperturbed sur face
Since !n| = 1, we have n «n. = 0 so
t h a t y = xx • (V£) «n . This provides
us wi th the r equ i red p e r t u r b a t i o n of the normal:
n. - - (v"£)-n + n n '(Vp'ti . (8-27)
Eva lua t ing B leads t o an e x t r a term E-VB due to the fac t t h a t
B i s to be taken a t the pe r tu rbed boundary:
(B) * (B + Q + £«VB ) . (8-28) A. "to
I n s e r t i n g the Eqs. (8-27) and (8-28) i n t o the boundary cond i t i on
(8-6) g ives
0 - ft-* - feo - ( 7 # - * o + «o v ( 7* ) , | lJ , (*<> * *+ *"yW
-Ho ' 7 X ( W * «o'« •
where use has been made of Eq. (8-13). This relation is automat
ically satisfied by virtue of the definition (8-18) for Q. How
ever, the same derivation also applies for the equation n*£ - 0
which now gives the required relation between £ and Q:
n 'Vx(£ x S )- n '0 . (8-29)
.114.
That this boundary condition in fact depends on the normal
component of £ maybe shown by one of those tedious vector
manipulations that abound in this field:
B -7(n «O - n •£ n • (VB )-n » n *Q . (8-29)»
For explicit calculations this form is to be preferred as it
gives directly the relation between n -C and n -0.
To evaluate the pressure jump condition (8-7) at the
perturbed boundary we need an expression for (p) analogous to
Eq. (8-28):
(P) - (p • , + 4.7p ) - <P0 - >P0V-$)r . (8-30) <t. 'to r«o
which i s j u s t the f i r s t order express ion for the Lagrangian
p r e s s u r e . I n s e r t i n g the Eqs. (8-28) and (8-30) i n t o Eq. (8-7)
and using the equi l ibr ium equat ion (8-13) leads to the second
boundary condi t ion r e l a t i n g £ and Q:
- YP v»E + B '(Q + £'VB ) = 8 *(Q + £-VfJ • (8-31)
Here, the le f t -hand s i d e i s j u s t the Lagrangian pe r tu rba t i on
of the t o t a l p r e s s u r e .
For a plasma-vacuum system the equat ion of motion
(8-21) for | , the equat ions (8-24) and (8-25) for £, and the
boundary condi t ions (8-29) and (3-31) connecting * and Q a t the
plasma-vacuum i n t e r f a c e c o n s t i t u t e a complete s e t of equat ions
by means of which waves and s t a b i l i t y p r o p e r t i e s may be inves
t i g a t e d .
For a plasma-plasma system some e x t r a care in the use
of the boundary condi t ions i s needed. In t h a t case Ö = ?x(cx6 )
so t h a t the boundary condi t ion (8-29) i s t o be replaced by the
. U S .
l i n e a r i z e d vers ion of Eq. (8-9) :
n • I = n * f . (8-32)
For the p ressure balance equation one has t c "*j?d p ressure
terms of the e x t e r i o r f lu id to the boundary conJ- t ion (8-31) .
One may then be tempted t o i n f e r from the con t inu i ty of the
Lagrangian p e r t u r b a t i o n of the t o t a l p ressure t h a t the RHS of
the boundary condi t ion should be j u s t the same expression as
the LHS of Eq. (8-31) with £ , Q, p , and B replaced by ?, Q, p ,
and 6 . In f a c t , such a regretteble mistake has been made in the
l i t e r a t u r e * . The p o i n t i s t h a t although - yp v . | * B • (Q + £*V£Q)
i s the Lagrangian pe r tu rba t ion of the t o t a l p res su re
of the inner f l u i d , and - yp ?'\ + 8 ' (Q + | " v £ ) i s t^ i e Lagran
gian pe r tu rba t ion of the t o t a l p ressure of the e x t e r i o r f l u i d ,
the two p ressu res are not evaluated a t the same pos i t ion s ince
the t angen t i a l components of £ are not cont inuous. For the sake
of symmetry between inner and e x t e r i o r f lu id i t i s therefore to be
p re fe r red to express the pe r tu rba t ion
a t the per turbed boundary a t the p o s i
t i on 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.
. 1 1 6 .
AD * <5p + n •£ n -Vp , (8 -33)
so t h a t M P + £ 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.
J-v : 7 * § * 0 , * - J - o , 7 x ft - o , 7«ft s 0 ,
: * ' •7 x <JW • n » Q »
r 7 x c^) a rS t
D. SELF-ADJOINTNESS OF THE FORCE-OPERATOR
Consider two vector f i e l d s £ ( £ , t ) and rt(£ ft) de
fined over the plasma volume fdx" (the superscript p denotes
the plasma and the vacuum w i l l be indicated by the superscript
v) , not necessar i ly s a t i s f y i n g the ideal MHD equation of motion
(8-22) . These vector f i e l d s w i l l be connected to two vector
f i e l d s Q(£,t) and R(j£,t) , defined over the vacuum volume fdt , t h a t
do s a t i s f y the vacuum equat ions by means of the boundary cond i t ions
(8-29) ünd (8-31) , so that we have:
(8-37)
(8-38)
(8-39) - vv*'Ji * S*(R + r7V - ? • $ •«•*?> » I s 7 x^x?> »
on \ iav (the w a l l ) : n«3 = 0 ,
n.» - 0 . ( 8 " 4 0 >
Let us now define an inner or sca lar product of the
two vector f i e l d s £ and JQ:
<k' r = I J V # s p d T P • <8-41> where the integration is over the plasma volume only. The equilib
rium density p has been absorbed as a weight function in the
definition of the inner product for reasons that soon will be
come clear. By means of this definition of the inner product one
may also define the norm of the vector field £(£,t):
.118.
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:
m i i i o , where e q u a l i t y only holds for the zero-e lement .
(4) The space i s complete: The l i m i t element £ of a Cauchy
sequence [L \ > i . e . a sequence for which lim | | £ - 5 | | = 0 ,
also belongs to the space:
I !%! | - H o | U n M < - •
T
.119.
lim
(5) Conversely, separability should also hold: To each element
E, a corresponding Cauchy sequence \ £ } can be found such that
i*.n • urn • These propert ies make the l inear vector space a Huber t space.
For l inearized ideal MHD the properties ( l )-(3) are obvious
from the definit ions above, whereas the propert ies (4) and
(5) which actually need to be proved are simply assumed to be
t rue . As we shal l see, property (4) i s extremely important in
connection with the occurrence of so-cal led continuous spectra.
Property (5) provides the basis for approximating functions by
f in i te sets of known functions, which i s especially important
in numerical appl icat ions.
The idea of the relat ions (8-37)-(8-40) i s to continue
the function 5 into the vacuum by means of the magnetic f ie ld
variable 0, and likewise to continue n by means of ft, by match
ing something like the function value and the normal derivative
a t the plasma vacuum inter face .
This i s schematically indicated
in the figure. I t i s a very remark
able property of ideal MHD tha t
only two conditions need to be
sa t i s f ied to connect two vector
fields £ and Q. Hence, i t appears
like we are dealing only with ordinary second order differ
en t i a l equations. The reason behind th i s is the extreme aniso-
tropy of ideal MHD as regards motion inside and across the
magnetic surfaces, to the study of which we w i l l turn l a t e r on.
•fc. r
T
.120.
Thus, we have obtained a defini t ion of the scalar
product that involves an integrat ion over the plasma volume
only. The physical significance of th i s i s the following. The
k ine t i c energy of the perturbations may be writ ten as:
K = I jpv 2 drP % \ Jp£2 dTP = <£, £> . (8-43)
Since t merely plays the role of a parameter, t h i s implies
tha t the vector f ie ld £ ( r , t ) may be chosen to belong to the
same class of functions as £ ( r , t ) . In other words, the physical
significance of r e s t r i c t i n g the consideration to displacement
vector f ields | ( £ , t ) tha t are bounded in norm i s that they
provide the plasma with a f in i t e amount of k ine t ic energy.
Returning now to the discussion of the force-operator
£ ( | ) , one extremely important property of this operator i s that
i t i s se l f -adjoint or hermitian;
**>• p"1 W " < p _ 1 ZW> ^ • <8"4A)
Notice tha t , s t r i c t l y speaking, with our definit ion of scalar
product i t i s not the operator £ but the operator p~l F that i s
se l f -adjoin t . Thus, having defined a Hilbert space for th i s
problem, the very f i r s t operator tha t we may want to study i s
an operator that enjoys the property of being sel f -adjoint . But
th is immediately provides the theory with a mathematical basis
of equal strength as that of non - r e l a t i v i s t i c quantum mechanics.
In pa r t i cu l a r , we are automatically led to the spectral theory of
hermitian operators in Hilbert space and we are home! But l e t us
not be carried away before the work i s done. Notwithstanding
many attempts to find a shorter path to the property expressed by
Eq. (8-44), the exp l i c i t proof remains a lo t of cumbersome
. 1 2 1 .
vec to r manipulat ions with l i t t l e beau ty . Unfo r tuna t e ly , we
need some of the in te rmedia te r e s u l t s in l a t e r s e c t i o n s . There
fo re , we w i l l j u s t reproduce the proof h e r e .
From the e x p l i c i t form of Eq. (8-21) of the fo rce -ope
r a t o r F we have:
%'Zlv ' <K7(4-7p+ v&'V - <7x$>*$+ (7*e>x ö - ' * U ^ * 7 P
+ yp*'V * (%*V x $1
Hence:
• T J V S ( $ * 7 P + YP7*4 ~ %'& do
- \ Jfav-rj 7 . | * §-R * Vr,($-vp) • (VxB) - p $ <KP. (8-45)
By the app l i c a t i on of the boundary condi t ion (8-31) the f i r s t
term on the RHS may be transformed:
• - ?ƒ*•« «•* r ï 7 < p + ? B 2 > 1 do -ih'zWdo> <8"46> where the l a t t e r express ion de r ives from the f ac t t h a t , s i n c e
| p + j B21 « 0, the t a n g e n t i a l component of the jump of the
d e r i v a t i v e vanishes as w e l l : t«|[v(p + ^ B2) J = 0 . Next , t r a n s
form the term "" ? . ]£*# $ " Q ^ o :
In t roduce vec tor p o t e n t i a l s in the vacuum: Ö = V x £ , £ = V x £.
The boundary condi t ion (8-29) then gives
n-VxCnxS) » n*7x£ ,
so t h a t
C » r,x§ • 7$
and nxC * nx (nxB) • nxV* ,
. 1 2 2 .
Choosing the gauge such t h a t n x 7 * = 0, we have
(which i s p r e c i s e l y the boundary condi t ion (2-33) of Bernstein
e . a . ) .
Thus, we have:
~ r f n - n 5*0 do = — f nxC-Ö do
• yJVS"Vx$ d ° = • H ( 7 x ^ ) x ? * " d o
= £ f7.r(?xS)xe] dxv
• \ I l£'****b - **?•**?] dtv
-4" I v x ? - 7 x s d T V = -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,
" T [ ^'Z> £#7P + (?xB)'(nxp)l dTP
- - 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 a n& R:
<£, P_1 F(|)>
• - i f [™<*-JG>V£ + « • * + i '-ja(j|-vp)
+ J '^(^P) + \ ( V x B ) . ^ » ^ + £XR)j dTP
- H « «"I «-ÏVCP * J B2)]}da - IJ j . Jg d v
- <£ . P~ | ( n ) > , q - e . d . (8 -50)
E. HAMILTON'S PRINCIPLE
From the n o n - l i n e a r e x p r e s s i o n s ( 5 - 4 0 ) , ( 5 - 4 1 ) , and
(5-45) fo r t h e t o t a l ene rgy 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 n ,„ „^ — • + = 0 . (ft—S?^ dt dt dt K° 3 Z ;
These espressions w i l l now be used to second order, so that
W, K, and H w i l l be quadratic forms in £ and £. This fact
w i l l not be indicated by further ind ices . From the expressie»
(8-4 3) for the k i n e t i c energy K we have
where we have used the se l f -adjo intness property of P. In te
grating the expression (8-53) leads to the required resu l t :
Here, the integrat ion i s carried out over the plasma volume
only . The i n t u i t i v e meaning of Eq. (8-54) i s c lear: The raise
in the potent ia l energy due to the perturbation i s jus t the
work done against the force F to displace the plasma by an
amount £ (where the factor 1/2 appears as a re su l t of the fact
that the f u l l force i s only obtained when the displacement
reaches i t s f inal amplitude).
Although the expression (8-4 3) for the k i n e t i c energy
K and (8-54) for the potent ia l energy W are quite a t tract ive
for analyt ical purposes, i t i s a l i t t l e strange that the vacuum variable
Q does not appear e x p l i c i t l y in them. One always has to remember
the additional information that £ carries with i t 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 i t s 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?U1 " \ I [>P(V*P2 + <£**P> 7*£ + f + (Vxfl).(i£xQ)]dTP, (8-56)
g = 7 x <£xB) ,
w s [ e j - i j ( n * S ) 2 S ' t v ( p + 7 B 2 ) Ï d 0 » (8-57)
« v « l = i J > 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, [i t 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 s t a t e the l i n e a r i z e d ve rs ion of Hamil ton ' s
p r i n c i p l e : The evolu t ion of the p e r t u r b a t i o n ï^ ( r , t ) j r € f dx P ;
Q(£, t) j r e j d t v ] i s such t h a t
i i
6 J d c L < $ . £ . V^« S> - ° . < 8 " 5 9 > t,
where L = K - W
= <i- *> - w PUi-w SU]-wv[§]- (*-w
I f the Lagrangian i s expressed as on the l a s t l i n e , the v a r i
ab les £ and Q should be sub jec ted t o the n a t u r a l boundary con
d i t i o n s (8-29) and (8-25) , which we r e s t a t e for convenience:
n*7 x (SxB) = n*Q at the plasma-vacuum interface, (8-61)
n*Q = 0 a t the w a l l . (8-62)
Thus, we have absorbed the complicated condi t ion (8-31) i n the
form of the Lagrangian.
Carrying out the minimizat ion of the express ion (8-59)
would lead to the following Euler equa t ion :
32< F <£) = p — for r e ( d T p , (8-63)
3t2
where F ( | ) = v ( | - v p • y p v . | ) + (7xB) x p. + (7xjg) x B ,
7 x 0 - 0 , 7-Q « 0 for r e fd t V , (8-64)
- Yp7«£ + B«(Q • £'7B) = 6»(Q + E*7B) for r c f do. (8-65)
In addition, the boundary condi t ions (8-61) and (8-62) should be
s a t i s f i e d . Of course , these equat ions are j u s t r e s t a t emen t s of
the equat ions (8-21) , (8 -24) , and (8 -31) , r e s p e c t i v e l y . They
are dup l i ca t ed here for the purpose of comparing the i n t e g r a l
and the d i f f e r e n t i a l forms of the problem. C l e a r l y , the v a r i a -
T
. 1 2 7 .
t i o n a l f o r m u l a t i o n o f E q s . ( 8 - 5 9 ) - ( 8 - 6 2 ) i s f u l l y e q u i v a l e n t
t o t h e d i f f e r e n t i a l e q u a t i o n f o r m u l a t i o n of t h e E q s . ( 8 - 6 3 ) -
( 8 - 6 5 ) .
REFERENCES
1 . I . B . B e r n s t e i n , E.A. Fr ieman , M.D. K r u s k a l , and R.M.
K u l s r u d , P r o c . Roy. S o c . A244 (1958) 1 7 . "An ene rgy
p r i n c i p l e f o r h y d r o m a g n e t i c s t a b i l i t y p r o b l e m s " .
2 . K. H a i n , R. L u s t , and A. S c h l t i t e r , 2 . N a t u r f o r s c h . 12_
(1957) 8 3 3 .
3 . B.B. Kadomtsev, "Hydromagne t i c s t a b i l i t y o f a p lasma"
i n Reviews of Plasma P h y s i c s , v o l I I , e d . M.A. L e o n t o v i c h
( C o n s u l t a n t s Bureau , New York , 1966) p . 1 5 3 .
. 1 2 8 .
IX. SPECTRAL THEORY
A. MATHEMATICAL PRELIMINARIES
The s p e c t r a l problem of l i n e a r i z e d i d e a l MHD a r i s e s
from a s tudy of the e q u a t i o n of mot ion (8-63) when one c o n
s i d e r s normal mode s o l u t i o n s w i t h an e x p o n e n t i a l t i m e - d e p e n
dence exp C-iuit) , s o t h a t
Here , we have e l i m i n a t e d t h e e x p o n e n t i a l t ime-dependence so
t h a t £ = £ ( r ) from now on , u n l e s s s t a t e d o t h e r w i s e . The s p e c
t rum of the o p e r a t o r p £ c o n s i s t s of t h e c o l l e c t i o n of e i g e n
v a l u e s tjj2.
An i m p o r t a n t p r o p e r t y of t he e i g e n v a l u e s fo l lows from
t h e s e i f - a d j o i n t n e s s of t h e o p e r a t o r p F . Le t E be t h e e i g e n -
f u n c t i o n b e l o n g i n g t o the e i g e n v a l u e
p " 1 F(£ ) = -a.2 £ .
- co2: n
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
s u b t r a c t i n g and i n t e g r a t i n g o v e r f d t P y i e l d s :
= ( u - 2 - - 2
so t h a t
0 = (u* - u 2*) < £ , £ > , n n " n ^n
co2 = u,2* . (9 -2 ) n n
In o t h e r words : the e i g e n v a l u e s io2 a r e r e a l , so t h a t t h e s p e c -n c
t rum of t he o p e r a t o r p -1 F c o n s i d e r e d in t h e complex u - p l a n e i s
c o n f i n e d t o t h e r e a l and imag ina ry a x e s . For e i g e n v a l u e s on the
imag ina ry a x i s (id2 < o) t h e e x p o n e n t i a l t ime-dependence of d\e
normal modes becomes exp (iiot) = exp (yt) , where y = iw > 0. (One
may always choose t h i s to be t r u e because t h e e igenvalues o c c u r
. 1 2 9 .
in p a i r s ) . These solutions grow exponentially in time and
are therefore called exponential i n s t a b i l i t i e s . The conditions
under which such i n s t a b i l i t i e s occur are analyzed in Sec. IX D.
The spectral problem associated with pa r t i a l d i f fe r
en t i a l equations l ike Eq. (9-1) i s jus t a generalization of the
methods used in l inear algebra of f in i t e dimensional vector spaces.
There, the eigenvalue problem arises in the studies of f i n i t e N xN
matrices L . . ( i , j = 1, 2, — N): N
Yl L i j Xj = X x i ' ° r k'* = X*' ( 9" 3 )
j = l
The e i g e n v a l u e s a r e found from the c o n d i t i o n
Det ( L . . - A6. . ) = 0 , ( 9 - 4 )
where s u b s t i t u t i o n back i n t o Eq. (9-3) y i e l d s t he a i g e n v e c t o r s
x . Another f o r m u l a t i o n of t h e same problem i s o b t a i n e d by t h e
c o n s t r u c t i o n of q u a d r a t i c forms; N N / N
* • C C x* L x YL *\ . (9 -5) i = l j-1 1 1 J J ' i = j x
Final ly , a third formulation arises from the consideration of
the inhomogeneous equation
where a is a known vector. Here, the Fredholm a l te rna t ive 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 £ i s a differential operator and, therefore,
an unbounded operator. Here, bounded operators U have the
property that
|!Ux|l £ M l | x | | (9-7)
for a l l x c H.S., where M i s some constant. Differential opera
tors do not have this property. Operating on a bounded (square
integrable) sequence of functions in Huber t space they may
produce a sequence that i s 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 t r i e s to
avoid th i s problem by transforming i t to one that involves com
plete ly 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 in f in i t e dimensional Hilbert space i s completely analogous to
that of the f in i te dimensional vector spaces of l inea r algebra.
In the case of d i f ferent ia l operators th i s implies that one t r i e s
to invert the operator so that one has to study an in tegra l opera
tor involving Green's functions which frequently do have the r e
quired property of compactness.
Another, more serious problem i s the existence of a
third class of operators where the above t r ick does not work,
viz . that of bounded operators tha t are not compact. [Example:
the operator of multiplication by x] . Those operators may give
r i se to a continuous spectrum, which i s roughly speaking the
collection of "improper eigenvalues" for which the eigenvalue
equation i s solved, but not by functions that belong to Hilbert
space. In the mathematical discr ipt ion one then has the option
. 131 .
of e i t h e r s t i c k i n g to the notion of Hi lbe r t space by i n t r o
ducing the concept of approximate spectrum where sequences
are considered t h a t do not converge (the approach of von
Neumann in h i s t rea tment of s p e c t r a l theory for quantum me
chanics) , or one may consider wider c lasses of elements than
those t h a t belong to H i lbe r t space , v i z . d i s t r i b u t i o n s (the
approach of Dirac , per fec ted by L. Schwartz) . One could say
t h a t the d iverging sequences of funct ions , t h a t are considered
in the f i r s t approach, converge to elements ou t s ide H u b e r t space
which are the distributions considered in the second approach.
Having a l l these words ava i l ab le now we may give the
following gene ra l i za t ion of the ideas of l i n e a r a lgebra ex
pressed in the equat ions ( 9 - 3 ) - ( 9 - 6 ) . The spectrum of a l i n e a r
ope ra to r L i s obtained from the study of the inhomogeneous equa
t ion
(L - A) x = a , (9-8)
where a i s a given element in Hi lbe r t space and we look for
so lu t ions
x = (L - X)"1 a . (9-9)
For complex A the following p o s s i b i l i t i e s a r i s e :
(1) (L - A) does not e x i s t because (L - A ) x = 0 has a solution:
A belongs to the po in t o r d i s c r e t e spectrum of L,
(2) (L - A) e x i s t s but i s unbounded: A belongs to the continuous
spectrum of L,
(3) (L - A) e x i s t s and i s bounded: A belongs to the r e so lven t
s e t of L.
( see : B. Friedman, P r i n c i p l e s and Techniques of Applied Mathemat
i c s , p . 125) .
Thus, a complex value of A e i t h e r belongs to the spectrum or
.132.
to the resolvent s e t , so that one may say that the spectrum
of L consists of the collect ion 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 l inearized equations of ideal MHDf v iz . a d i f fe ren t ia l and
an in tegra l formulation. Correspondingly, the spect ra l problem
also takes two forms, v iz . a normal mode analysis by means of
the d i f ferent ia l equation (9-1) [which should be supplemented
with the Eqs. (8-64) and (8-65) for the vacuum region if such
a region i s present] and a var ia t ional pr inciple based on the
quadratic forms defined in Sees. VIII D and E. These two formu
lat ions const i tute the in f in i t e dimensional generalizations of
the Eqs. (9-3) and (9-5) for the finite-dimensional vector
spaces.
The var ia t ional formulation may be s ta ted as follows:
Eigenfunctions of the operator p P are obtained for functions
£ for which the functional
2 r , ^ - p " 1 zty> WW "Mi l — (9-10)
becomes s ta t ionary. Here, besides the potent ia l energy W[E]
defined in Eq. (8-53) and the k inet ic energy K[|] defined in
Eq. (8-4 3) another quadratic form has been introduced that i s
quite useful in the present context, v iz . the v i r l a l :
I W s < * ' ^ ' I UN2 - (9-1D
.133.
The s ta t ionary values of the functional a2[£] are the d i s
cre te eigenvalues u 2 . The collection of a l l these eigenvalues
cons t i tu tes the d iscre te spectrum.
The Rayleigh-Ritz pr inciple i s extremely useful for
the approximation of eigenvalues by means of f ini te-dimension
a l subspaces of Hi lber t space. Here, one se lec t s a su i t ab le
c lass of square-integrable functions £ n-. , n ? / •• nw} which
are used as t r i a l functions in the expression (9-10) . The
l inea r combination of these functions that minimizes the func
t iona l Ü2 then cons t i tu tes an approximation for the lowest
eigenvalue u2 , where the minimum value of ü2 i s always larger
than the actual eigenvalue u 2 .
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 ^ n
n»l
one then obtains the following approximation:
N N
2 2 a* W a
t l - 1
Hence, the problem again boi l s down to the f i n i t e dimensional
one of Eq. (9-5), v i z . 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 equ iva lence of the v a r i a t i o n a l problem (9-10) wi th
the e igenvalue problem (9-1) i s easi ly proved . Le t u2 = a2 [^]
be a s t a t i o n a r y value of the func t iona l (9-10) , so t h a t
5ft2 5 Ï-Ï 1-Ï 1 ^ 2.2L- = 0 - 2 < ^ . P " 1 ^ ( | ) > <£.£> + 2 <l>o~1Zty><6$.l>
(always us ing the Hermit ian p roper ty of p £ ! ) .
Then, — « • - " /
so t h a t < 5 | , ( P _ 1 F ( | ) + cu2,|)> = 0 .
But, since 5£ is arbitrary, this is equivalent to
p"1 |(jp - - w2| , q.e.d.
For plasma-vacuum systems i t i s again useful t o extend
the v a r i a t i o n a l p r i n c i p l e so as t o e x p l i c i t l y e x h i b i t the depen
dence on the vacuum v a r i a b l e Q:
ft Tl» Q] • » (9-16)
where Wp, WS, and WV are defined in Eqs. (8-56)-(8-58) , and | and
Ö should satisfy the boundary conditions (8-61) and (8-62). This
formulation ir equivalent to the normal mode equation (3-1) supple
mented with the equations (8-64) and (8-65) for the vacuum vari
able Q. Again, notice that the boundary condition (8-65) has to
. 1 3 5 .
be e x p l i c i t l y c o n s i d e r e d i n the n o m a ! node a n a l y s i s , whereas
i t i s a u t o m a t i c a l l y t aken c a r e of i n t h e v a r i a t i o n a l fo rmula
t i o n of Eq. ( 9 - 1 6 ) .
C INITIAL VALUE PROBLEM
Accord ing t o t h e e x p o s i t i o n g iven i n S e c . IX A i n
connec t ion w i t h the E q s . (9-8) and (9-9) t h e t h i r d , and most
g e n e r a l , approach to the spectrum of the l i n e a r o p e r a t o r p £
i s t o c o n s i d e r t h e inhomogeneous prob lem
( p " 1 F + a,2) | = X , ( 9 - 1 7 )
where X is a known vector. Our task is then to construct the
resolvent operator (p £ + ID2) and to study its behavior
for complex values of w2. In order to see how this is connected
with physics, consider the initial value problem. We define the
Laplace transform of £(rr-t) in the complex w-plane:
l < £ J u > £ I k{Z'>l) elUt dt • (9_18)
0
so t h a t Eq. (3-63) t a k e s t he form: 2 c
iui t i t-*"» ~ 1 , , , * x I * l W t - - . ( ^ = J 7 7 e dt = " u" £ ( T T " lu^> e O
(9-19) W r i t i n g u = o + iv we then g e t fo r v > 0
( P _ I £ + " 2 > I <**«> - *<*$£<*) " 4 i < £ > =- *' ««"ZO)
where t h e v e c t o r X of Eq. (9-17) thus t u r n s o u t t o be t he func-
t i o n of i n i t i a l d i s p l a c e m e n t £ . ( r ) and i n i t i a l v e l o c i t y f , . ( r )
d e f i n e d in t he RHS of Eq. (9-20) . In o r d e r t o f i nd the response
£ ( r ; t ) to a c e r t a i n i n i t i a l p e r t u r b a t i o n :<, one then f i r s t h a s
t o i n v e r t Eq. (9-20) t o f i nd t h e Lap lace t r ans fo rmed v a r i a b l e
% in terms of X:
.136.
|(r;u>) = (p-1 F + U2)'1 Xtr;u>) , (9-21)
and next perform the inverse Laplace transformation; i v +"
* t \ 1 f -2 / v -1Wt ,
l v - » O
W % J » c
I t i s c l e a r t h a t fo r t h e i n v e r s e si»;p «< eonvtr- 'K"*. , / , / / / r-^ / / / / >">•/ f / / / / / / ©
transform more i s needed than jus t
v > 0 because £(r;w)may not ex i s t
for certain values of u or i t 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
. 1 3 7 .
been proved i t i s s u f f i c i e n t l y p l a u s i b l e , a l s o because a l l
continua found so far (including those for the case of a general a x i -
synwetric toroidal system) conf im i t , so that we may delay worries about
th is point to future invest igat ion. (Hov/ever, see footnote on p . 146.)
We then conc lude t h a t t h e i n t e g r a t i o n c o n t o u r must
be p l a c e d above the l a r g e s t p o i n t e i g e n v a l u e v of p F , max '^
i . e . v > v , t he most u n s t a b l e e i g e n v a l u e : o max ^
IV
u c >
V max
r i
>
©
+-* -6"
v
In other words, the class of permissible funccions t(r;t) is
restricted to functions of exponential order exp (v t) where o
i s l a r g e r than t h e l a r g e s t growth ra te of the system. In the
p i c t u r e above we have s c h e m a t i c a l l y i n d i c a t e d ou r knowledge
so far of the spectrum of i d e a l MHD, which w i l l be a n a l y z e d i n
more d e t a i l i n a l a t e r c h a p t e r . One f i n d s two p a i r s of c o n t i n u a
on the real axis , whereas point e i g e n v a l u e s can o c c u r a l m o s t e v e r y
where on the r e a l a - a x i s ( i n c l u d i n g i n s i d e t h e c o n t i n u a ) and
also a t a l i m i t e d p a r t - vm a x £ v £ v of t h e imag ina ry \ j - a x i s .
Of c o u r s e , i t i s e x t r e m e l y d i f f i c u l t t o o b t a i n t h e e x
p l i c i t t ime-dependence of £ ( r ; t ) i n s i t u a t i o n s of p r a c t i c a l
i n t e r e s t so t h a t one u s u a l l y r e s t r i c t s t he s t u d y to time-asynptotic
s o l u t i o n s . I t i s c l e a r t h a t f o r t ->- <*> one wi shes to deform t h e
i n t e g r a t i o n c o n t o u r in t he i n v e r s e L a p l a c e t r a n s f o r m t o t h e lower
h a l f of t h e w-plane in o r d e r t o e x p l o i t t he s m a l l n e s s o f t h e e x
p o n e n t i a l f a c t o r exp ( - i u t ) i n Eq. ( 9 - 2 2 ) . Fo r t h i s a d v a n t a g e
one must pay in t h e form of a s t u d y of the a n a l y t i c c o n t i n u a t i o n
of ' abou t the o c c u r r i n g p o l e s ( p o i n t e i g e n v a l u e s ) and branch points of
\ ( a s s o c i a t e d w i th the c o n t i n u o u s s p e c t r u m ) . The b r anch p o i n t
. 1 3 8 .
s i n g u l a r i t i e s l e a d t o d i f f e r e n t b r a n c h e s of t h e complex f u n c
t i o n | ( r ; « ) s o t h a t t h e i n v e r s e L a p l a c e t r a n s f o r m c o n t o u r may
be moved t o a n o t h e r Riemann s h e e t where i t c o u l d p i c k up p o l e s .
Such p o l e s c o u l d n o t c o r r e s p o n d t o p o i n t e i g e n v a l u e s s i n c e t h e s e
a r e c o n f i n e d t o t h e r e a l and imag ina ry axes o f t h e p r i n c i p a l
b ranch o f %, b u t t h e y may b e s i g n i f i c a n t p h y s i c a l l y .
We w i l l c o n t i n u e t h e a n a l y s i s of t h e i n i t i a l v a l u e
p rob lem i n S e c . X C where we c o n s i d e r t h e e x p l i c i t example
of aninhomogeneous s l a b .
D- STABILITY. THE ENERGY PRINCIPLE
L e t us c o n s i d e r a p a i r o f d i s c r e t e normal modes
e x p ( - i o i n t ) a n d e x p ( i u i t ) b e l o n g i n g t o t h e same e i g e n v a l u e u2 = u>2.
I f we n e g l e c t a l l o t h e r modes, e . g . by p r e f e r e n t i a l l y e x c i t i n g
t h i s one p a i r of modes, t h e s o l u t i o n of t h e i n i t i a l v a l u e p r o b
lem g i v e n i n Eq. (9-20) may be e a s i l y c o m p l e t e d . S i n c e
0 _ 1 VV - - < k , (9-23) t h e r e s o l v e n t o p e r a t o r would be s imp ly g iven by
( p " 1 F + u . 2 ) " 1 = (o>2 - u 2 ) ' 1 . ( 9 -24 ) <\» n
Hence, t h e d i s c r e t e e i g e n v a l u e w2 g i v e s r i s e t o two p o l e s u = + u n r — n
wh ich , by v i r t u e of Eq. ( 9 - 2 ) , a r e s i t u a t e d on e i t h e r t h e r e a l
a x i s o r t h e i m a g i n a r y a x i s o f t h e complex w - p l a n e . C l e a r l y , f o r
u 2 = u 2 the r e s o l v e n t o p e r a t o r does n o t e x i s t , b u t everywhere
e l s e i n the complex w-p lane i t i s now d e f i n e d (of c o u r s e , when
we i g n o r e t h e r e s t o f t h e s p e c t r u m ) . We may now i n t e g r a t e Eq.
(9-22) by deforming t h e c o n t o u r around t h e two p o l e s u2 = + u .
Shifting the s t r a i g h t p a r t o f t he c o n t o u r t o v = - - so t h a t exp
( - iw t ) v a n i s h e s e x p o n e n t i a l l y f a s t t h e on ly c o n t r i b u t i o n t h a t
r emains w i l l be t h e two r e s i d u e s p i c k e d up a t t h e p o l e s . By
means of Cauchy ' s i n t e g r a l formula we then f i n d :
. 1 3 9 .
C ( u * u i ) ( t « ) — u ) n n
lm t - l u t
K%i { s> + 4 i<^i* n + r^i<«) -4i<€>]e n
2i u
(9-25)
(where one shou ld n o t i c e t h a t t he c o n t o u r C deformed around a
p o l e has j u s t the o p p o s i t e s e n s e of a Cauchy c o n t o u r ) . W r i t i n g
us = a + i v , we e i t h e r have v = 0 cr a = 0 . I f \> = 0 t h e p o l e s n n n n n n ^
a r e s i t u a t e d on the r e a l a x i s s o t h a t
- 1 S ( r ; t ) = £. ( r ) cos a t + £ . ( r ) a s in a t , (9 -26)
which i s a s t a b l e undaniped o s c i l l a t i o n e x c i t e d by an i n i t i a l
d i s p l a c e m e n t £ . ( r ) o r an i n i t i a l v e l o c i t y £ . ( r ) o r by a combi-
n a t i o n of b o t h . I f a = 0 the p o l e s a r e s i t u a t e d on t h e imarri-n ^
n a r y a x i s and we have
- 1 £ ( r ; t ) = L. ( r ) cosh v t + £ . ( r ) v s i nh 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
. 1 4 0 .
<t> % f
V.
Q> -v.
The important feature here i s that true normal modes,
i . e . d iscrete e igenvalues, are e i ther o s c i l l a t o r y or exponen
t i a l l y growing, but never damped. This i s the real simplifying
feature of i d e a l , i . e . conservative, MHD which i s expressed by
the se l f -adjo intness of the force-operator. As a consequence,
s t a b i l i t y s tudies may be s impl i f ied considerably as compared
to the analys is needed in d i s s ipat ive systems. I f the e q u i l i b
rium i s described by a s e t of parameters a.., — a (bas ical ly
expressing the pressure and magnetic f i e l d d i s t r i b u t i o n ) , in
general marginal s tates would be defined by the condition
lm u ( c t l t - - cx H ) = 0 , ( 9 - 2 8 )
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 ( a r — 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)
i s s a t i s f i e d . The variat ional counterpart of th i s equation, v i z .
the marginal form of Rayleigh's pr inc iple (9-10) i s known under
the name energy pr inc ip le . This principle s t a t e s 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)
. 1 4 2 .
i t would be reasonable to assume that the se t \ £ \ const i tutes
a complete basis for the Hilbert space. In that case the eigen-
functions £ could be chosen to be orthonormal: ^n <£ ,£ > - 6 . ( 9 - 3 3 )
*m ^n ran
An arbitrary t could then be expanded in eigenfunctions:
n=l so that
n=l
Hence, i f we could find a £ for which W < 0 a t l eas t one eigen
value u2 < 0 should ex i s t . Such an eigenvalue would correspond
with an exponential i n s t a b i l i t y . This proof was given in the
or iginal paper by Bernstein e .a . before i t was known that ideal
MHD systems as a rule have a continuous spectrum that usually
also extends to the origin w2 = 0. The l a t t e r fact implies that:
the simplicity of the marginal s t a b i l i t y i s spoiled and a l o t
more care i s needed to establ ish necessity of the energy p r in
c ip l e . I t i s l ike ly that a correct proof may be given which
properly incorporates the continuous spectrum, but i t i s cer ta in
tha t such a proof wi l l be quite involved.
A correct proof of both the necessi ty and the sufficiency
of the energy principle without invoking the assumption of a com
ple te basis of discrete eigenvalues, but a lso avoiding an anal
ys is of the continuous spectrum, has been given by Laval, e.a.. (Nuclear
Fusion 5_ (1965) 156). The proof i s based on energy conservation,
H « K + W , H « 0 , (9-36)
and the v i r i a l equation
Ï " <4-4>" " 2 < i4 > + 2 < ^ » ï > - " - 2W . (9-37)
.143.
The proof of suff ic iency i s a c tua l l y q u i t e s imple:
Suff ic iency: If w["^] > 0 for a l l £ one cannot find a motion
n (t) such t h a t the k i n e t i c energy K[n(t)] grows without bound.
Proof. W = H - K > O r H f i n i t e .
Hence, unbounded growth for K would v i o l a t e energy conserva t ion .
[Notice t h a t we exclude s o - c a l l e d l i n e a r l y growing i n s t a b i l i t i e s
where £ ^ t and I ^ t 2 ] .
The proof of neces s i t y i s more involved:
Necess i ty : I f a function n e x i s t s such t h a t W[nJ < 0, the system
w i l l e x h i b i t an unbounded motion l ( t ) .
Proof.
(1) w[rt] < 0. Choose as i n i t i a l d a t a £<0) = tj, £(0) = 0.
From Eq. (9-36) H(t) =H{0) =W(0) + K(0) =W[jQ] < 0,
so t h a t I ( t ) = 2K - 2W = 4K - 2H >. - 2H(t) > 0 .
Hence, I grows without l im i t as t + • and I grows a t l e a s t l i k e
- Ht 2 . As a r e s u l t £ grows a t l e a s t l i n e a r l y in t .
This s imp l i f i ed vers ion of the proof i s due to Kruskal .
Laval e . a . gave a sharper vers ion by a l so es t imat ing the growth
r a t e :
(2) W[T,] < 0. Define X = - W[n.]/l[nJ > 0. (9-38)
We prove t h a t t he re e x i s t s a ^ ( t ) growing a t l e a s t as e x p ( / x t ) . *
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 ,
whereas Sdiwartz i n e q u a l i t y g i v e s
i 2 < c > - 4 < i » i > 2 1 4 < I . I > < $ > $ > = 4 I ( c ) K ( t )
S i n c e 1(0) - 2*X <n,ri> = 2>n: 1(0) > 0 ,
( S - 3 9 )
K t ) I ( t ) .
( 9 - 4 0 )
( 9 - 4 1 )
we have from Eq. (9-39) t h a t I ( t ) > 0 f o r t > 0 , s o t h a t we may
d i v i d e i n e q u a l i t y (9-40) by I (t) I ( t ) , g i v i n g s u b s e q u e n t l y :
I ( t ) / I ( t ) i I ( t ) / i ( t ) ,
t n [ l ( t ) / I ( 0 ) ] <_ 4 n [ l ( t ) / I ( 0 ) ]
I ( t ) / I ( 0 ) < I ( t ) / 2 V T l ( 0 ) ,
I ( t ) / I ( t ) 2 2 V I ,
i n [ l ( t ) / I ( 0 ) ] >. 2V^t ,
I ( t ) > 1(0) exp (2 f i t ) .
= i n [ l ( t ) / 2 - / X l ( 0 ) ] ,
C o n s e q u e n t l y , ,£ grows a t l e a s t as exp ( / x t ) , q . e . d .
One may a l s o p rove t h e f o l l o w i n g t h e o r e m .
Theorem. I f t h e r a t i o - w [ i ] / l ' | ] h a s a s m a l l e s t uppe r bound
* >_ *[.$] = " w [ £ ] / l [ ; / j f o r a l l | ,
t hen K t ) c a n n o t grow f a s t e r t han exp ( V 7 t ) .
Proof .
I ( t ) - 2K(t) - 2W(t) - 2H(t) - 4W(t) < 2H( t ) 4 4A I ( t ) .
Hence, K t ) - 4A I ( t ) <_ 2H(c) - 2H(0) .
C o n s e q u e n t l y , K t ) grows a t most l i k e exp
{7^T.-:< and K t ) c a n n o t grow f a s t e r t han
'.v.< ;VAt) , q . e . d .
We have g iven a l l t h e s e p r o o f s
h e r e because they n a t u r a l l y l e a d to an
e x t e n s i o n of t h e s t a b i l i t y c o n c e p t t o be
i n t r o d u c e d i n the n e x t s e c t i o n .
.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
-*t M — x - m w j — — ^ M )i ), i
unstable side of the spectrum
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 - o 2 = A - a 2 > 0 .
Then,
* - " W L Q ] / I [ J & ] - M + o 2 > a 2 ,
so t h a t | ( t ) grows a t l e a s t 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).
. 1 4 9 .
X. WAVES IN PLANE SLAB GEOMETRY
A. WAVES IN INFINITE HOMOGENEOUS PLASMAS
As a p r e l i m i n a r y t o t h e s t u d y of waves and i n s t a b i l i t i e s
i n inhomogeneous s y s t e m s , l e t us f i r s t s tudy t h e normal ipodes
of an i n f i n i t e homogeneous p l a s m a . Taking B i n t h e z - d i r e c t i o n
t h e e q u i l i b r i u m s t a t e i s s p e c i f i e d a s f o l l o w s :
I = ( 0 , 0 , B) ,
(10-1)
B, p , p c o n s t a n t .
S ince Vp = 0 and V x B = 0 t h e normal mode e q u a t i o n (9-1) w i t h
t h e f o r c e - o p e r a t o r de f ined a s i n Eq. (8-21) becomes :
P " 1 £ ( £ ) = ( Y P / P ) 7 V . | + p - 1 (Vxo) x £
where
c = ( Y P / P ) 1 / 2 and b = B / p 1 / 2 ,
i n agreement w i th Eqs . (4-20) and ( 4 - 2 1 ) .
From now on we w i l l c o n s i s t e n t l y w r i t e o2 i n s t e a d of u 2 t o i n d i c a t e
t h a t t h e e i g e n v a l u e s v;e a r e look ing f o r a r e r e a l . [Of c o u r s e , one s h o u l d
n ' . t confuse t h i s n o t a t i o n w i t h t h a t of o - s t a b i l i t y of t h e p r e v i o u s
s e c t i o n where t r a n s i t i o n from o - s t a b i l i t y t o o - i n s t a b i l i t y t a k e s
p l a c e a t t he p r e f i x e d v a l u e of t h e e i g e n v a l u e p a r a m e t e r
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 s t u d y t h e modes £ ( k ) e x p i ( k . r - a t ) s e p a r a t e l y by
making t h e s u b s t i t u t i o n V •+• i k i n Eq . ( 1 0 - 2 ) . Th i s g i v e s :
p ' 1 F ( | ) = - c 2 k k . | - b x ^ k x f k x ( b x | ) ] >
• -&2 + c2)feVÏ " 15'M'*? - # • ? " &"?> - " °2I (10-4)
D e f i n i n g L .£ = p - 1 F ( £ ) , t h i s may be w r i t t e n as a p rob lem i n j ^ *\. >\J a .
l inear alaebra:
- - H i ^k (10-5)
where
fe - - < b * • C 2 ) k k - ( J c - f c ) 2 I + Jc.fcCJtJj • bfc)
I n components :
-k2(b2+c2) - k2b2
X Z
-k k (b2+c2) x y
-k k c2
X Z
-k k (b2+c2) x y
-k2(b2+c2> - k2b2 -k k c2
y z y z
-k k c2
y z
- „2
l T . / \
(10-6)
.151.
Solutions are obtained by setting the determinant of the LHS
zero. This gives:
(o 2 - k2b2) [o" - K2<b2+C
2) C2 • k2 K2b2c2] » 0 , (10-7)
where
K2 = k2 + k2 + k2 , k2 = k2
x y z f/ z
Consequently, we ob ta in t h r e e s o l u t i o n s
a2 = o\ E k2b2 ,
a2-°l,i 4 K 2 ( b 2 + c 2 > 1 ±1 / •
4kjb2c2
a2-°l,i 4 K 2 ( b 2 + c 2 > 1 ±1 / • K 2 ( b 2 + C 2 ) 2 .
(10-8)
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 ~ c t > • (10-10)
Apart from a constant factor, this is just the phase of the
plane wave exp 1 (k.r-ot) . I t 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, = a s f (^/kx)Ix ,
% - ° • k x *x + k z * . • ° . K - - i ^ k x ^ x ' (10-13)
7 = - ioc kx.(l • a s f k j / k j ) ? x ,
where
a = 1 - K2b2/a2 . , so that a < 0 , af > 0 . (10-14) s,f s,f s — • f —
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 o k / K ' (10-15)
I f we c a l l 6 the angle between k and B we f ind t h a t
v . = a/K = f(0) , but i t does not depend on K. Such waves are
c a l l e d non-dispers ive as a plane wave packet cons t ruc ted from
them may propagate wi thout d i s t o r t i o n . The group v e l o c i t y of
such a packet gives the flow of energy:
y = 3o/3k 'vgr ^ (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 th ings are more complicated. This
i s b e s t i l l u s t r a t e d by p l o t t i n g o2 as a funct ion of k., whi le
keeping k f ixed , and vice v e r s a : //
A l f v « M
JU. l i a t A <to»c"> i „ f.««A
.155.
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 d i rec t ions . In the next section
the x-axis wi l l be chosen as the di rect ion of inhomogeneity. I t
.156.
is therefore instructive to also plot a1 as a function of K
while keeping k,, and k fixed:
Vr • * » ?>«« A
^ , f c $ ^ v x > o • x * *, ; l i
If we consider a slab of f i n i t e extension In the x-direction by
putt ing conducting platec a t x = ±a, the wavenumber k is
quantized: k = n:r/2a. Here, n i s the number of nodes of the
eigenfunction £ in the x-direct ion. Such a number to label
the point eigenvalues s t i l l makes sense in an inhomogeneous
medium, when the equilibrium quant i t ies vary in the x-direct ion.
The essent ia l features of the three discre te spectra of
point eigenvalues labelled by n a re :
(1) The point eigenvalue a* = k* b2 of the Alfvén point spectrum
is inf in i te ly degenerate.
(2) The slow wave point eigenvalues have an accumulation point
for k •+ » (or n + ») :
o 2 = k 2 b 2 c z / ( b 2 + c 2 ) . ( 1 0 - 1 9 )
(Notice the notation with the subscript s indicat ing the slow
modes themselves and S the accumulation p o i n t ) .
. 1 5 7 .
(3) Fo r l a r g e wavenumbers k •* °° t h e f a s t wave p o i n t e i g e n v a l u e s
behave as
cr2 % k 2 ( b 2 + 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:
•i.n 4 k I ( k ! • °2>
where k2 = k2 + k* .
1 ±\/l -4k2b2c2
k2(b2+c2)2 J (10-21)
[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:
0 1 °2S 1 °2
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:
A €i
B = (0, B (x), B (x)), p = p(x), p = p(x), u y z.
(10-23)
where the pressure balance equation
(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 + \ B 2)' 0 . (10-24)
Here and in the following primes denote differentiation with
respect to x. Again, we study modes satisfying the normal
.159 .
mode equat ion (9 -1 ) :
e " 1 *(£> = " u 2 | * (10-25)
Here, £(£) niay bs decomposed in Four ier components fo r the
two homogeneous d i r e c t i o n s :
W * 7^£fx;Vkz) exp ( iV + i k z z ) d k y d k z - < 1 0 - 2 6 >
We w i l l now study sepa ra t e Four ie r components
£k k ^ x ' e x P ^ I cv y + i k
zz ^ * F o r convenience in no ta t ion we w i l l
y ' z y
drop a l l the decora t ing symbols and i n d i c a t e the ampli tude of
a Four ie r component simply as | ( x ) .
The most convenient form of Eq. (10-25) i s ob ta ined
a f t e r p r o j e c t i n g a l l the occur r ing vec to r s on the t h r ee u n i t
vec to r s e v , e ^ and e^ :
e = e ,
Si " S x e , v / B * ( 0 ' B z ' " V / B ' (10-27)
ey/ = B/B - (0 , B y , Bz)/B .
[ i t i s important not to confuse p r o j e c t i o n s liVe t h i s one
wi th or thogonal coord ina te systems, as i s sometimes done in
the l i t e r a t u r e . The point i s t h a t , in gene ra l , one cannot
f ind coord ina tes a, S, y such t h a t £ 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 + •€* i f» (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 <J> is the anyle between B and the z-axis. Hence, the quanHtie.'-
q and i may ue considered as the wavevec t'jrs in the perpendicular
and parallel directions, but they are functions of x in general.
Notice that the sum of g2 and f2 does not depend on x:
g2 + f2 - k2 - k2 + k2 . (10-29)
We also project ^ on the three unit vectors:
. 1 6 1 .
« E *v£ - ^ .
n = i e , - E = i (B C - B g )/B , •>»•»• ' t z y y z
(10-30)
5 = i e •£ = i (B C + B £ )/B , <\,// % v v z z y y z z
where t h e f a c t o r s i have been i n s e r t e d i n such a way t h a t one
has t o d e a l o n l y w i t h r e a l f u n c t i o n s £ , r\, and Z, i n t h e f i n a l
a n a l y s i s .
By t h e use of t h i s p r o j e c t i o n t h e normal roode e q u a t i o n
(10-25) may be w r i t t e n a s :
J - ( b 2 + c 2 ) d _ f2b2 «L ( b 2 2 )
dx dx dx° &AU\ :i^'>5 -02 ( b 2 + c 2 ) _ f 2 b 2 _ f g c 2
- f c 2
dx - f g c : - f 2 c 2
/ ' \
= - a 2 (10-31)
where p h a s been chosen c o n s t a n t f o r c o n v e n i e n c e , so t h a t i t
can be p u l l e d under t h e d e r i v a t i v e d / d x . [ o t h e r w i s e , we would
have t o w r i t e p _ 1 d / d x ( y p + B 2 ) d / d x , e t c . ] . N o t i c e t h a t t h e
o p e r a t o r p " 1 £ now depends on x th rough b 2 ( x ) , c 2 ( x ) , g ( x ) , and
f (x) . Apa r t from t h i s i m p o r t a n t d i f f e r e n c e t h e e x p r e s s i o n (10-31)
i s c o m p l e t e l y ana logous t o t h a t fo r i n f i n i t e homogeneous p l a s m a s .
The d i s p e r s i o n e q u a t i o n (10-6) may be r e c o v e r e d by w r i t i n g
d / d x = ik .
Le t us now reduce t h e m a t r i x e q u a t i o n (10-31) t o 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:
n = g[(b2^c2)a2-f2b2c2]^,
D (10-32)
_ fc2(q2-f2b2) > *" s »
D
where
D = D(x ; a 2 ) = a1» - k 2 ( b 2 + c 2 ) o 2 + k 2 f 2 b 2 c 2 . (10-33)
S u b s t i t u t i n g t h e s e e x p r e s s i o n s i n t o t h e f i r s t component g i v e s
us t h e r e q u i r e d second o r d e r d i f f e r e n t i a l e q u a t i o n :
[ | £ ' ] ' + ( a 2 - f 2 b 2 K » 0 t (10-34)
where
N - N ( x ; 0 2 ) = (a 2 - f 2 b 2 ) [ ( b 2 + c 2 ) a 2 - f 2 b 2 c 2 ] . (10-35)
Th i s e q u a t i o n has t o be s o l v e d s u b j e c t t o t h e boundary c o n d i t i o n s
S(xj) - ? ( x 2 ) - 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:
£ = (b2+C2)
where
o2(x) E f2b2,
[a2-a2(x)] [a2-a2U)]
[a2-a2(x)j [o2-c£(x)] (10-37)
a2(x) H f' b2c2
b2+c2
'1,11 (x) H i-k2(b2+c2) 1 ± 4 i-
b2c2
k2 (b2+c2)2
(10-38)
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* o* i<>
o;' --^(«V-
W »»--*tVfc* ^ i
»»--*tVfc*
•
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
frequencies o2c{o^(x) |xj <_ x <_ x2> and a2fe{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*):
«#o <r=$
*.
i i
ci * » i
\ '
X #n < * • - « • ; < * > XA . X*«0
.165.
At a singular point x = x (a2) when o? = af(x) the function A A
N(x,-o2) van ishes . We may now expand around t h i s s i n g u l a r i t y :
N(x;o2) - N(x;x (o2)) £ a [x - x ( o 2 ) ] = as , (10-39)
where
s = x - *A(<*2) ,
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
indicial equation n2 = 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:
r Ct s u(s?a2) ("small" solution)
\ (10-41) L C2
3 u(s;o2) en | s | + v(s;o2) ("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) + TA OU + 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 .
S u b s t i t u t i o n of a small so lu t ion £ = uH(s) leads to the
following express ions , success ive ly :
C' - u'H(s) + u 6 ( 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 v i r t u e of t h e f a c t t h a t u ( s ) i s a s o l u t i o n of Eq . (.10-43) .
Here , we have wade use of such p r o p e r t i e s as H*(s) = 6 ( s ) and
s5 (s) = 0. Consequen t ly , AjU H(s) i s a s o l u t i o n of Eq. (10-43)
b u t , l i k e w i s e , A2u H(-s ) i s a l s o a s o l u t i o n , where Aj and A2
a r e t o t a l l y u n r e l a t e d . Pe r fo rming a s i m i l a r a n a l y s i s f o r t h e
l a r g e s o l u t i o n i t t u r n s o u t t h a t t h e t e rm u £ n | s | H ( s ) p roduces
a 5 - f u n c t i o n c o n t r i b u t i o n t h a t does no t v a n i s h so t h a t B, = B ,
has t o be s a t i s f i e d .
The g e n e r a l s o l u t i o n t o Eq. (10-43) may now be
w r i t t e n as
i = Au + BuH(x - xA) + C[u in jx - x | + v] . (10-44)
Due t o t h e f a c t t h a t we have now t h r e e ( r a t h e r t h a n t h e u s u a l
two) c o n s t a n t s a v a i l a b l e t h e two boundary c o n d i t i o n s (10-36)
may always be s a t i s f i e d f o r c 2 € { a f ( x ) } so t h a t t h e r e i s a
s i n g u l a r p o i n t on t h e i n t e r n a l (x^ , X2) . The improper
e i g e n f u n c t i o n s fo r an Alfvén cont inuum mode may then be
w r i t t e n a s :
x-xA(o2) v^o 2 ) -:A(x;32) = C(o2) {in £-r-Z\ - - V - r r } " ( x ; o 2 ) H ( x . ( o 2 ) - x) +
X1~XA u-^cr) A
>:-x.(o2) v , (a 2 ) „ 1 + { £ n ^-r-rr ~ - W > "(^;° 2) H(x-x (a2)) + v(x;o ' ) j ,
x2~x (o2) u2^° '
(10-45)
where u 1 ( a 2 ) = u ( x . ; a 2 ) , e t c .
.168.
The factor C(g2) may be fixed by "normalizing" the eigenfunctions
according to
<f;ACx;a2) , CA(x;a
2')> = ó<o 2-a 2')
Likewise, one obtains improper
eiger.functions ^(x; a2) for
a2£o*{x) . Therefore, we have
"solutions" satisfying the
boundary conditions for any
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
*- x
n * <a2-o2H' , ; * <a2-o2)C' ,
we find for the dominant non-square integrable part of the
eigenfunctions:
.169.
CA $ ° > nA $ 5*-V. x-x (o^) A
+ X(o2) Ó ( X - X 4 ( J2 ) ) , c % o ,
2 \ A A "v*
?c £ ° » % £ ° » s ï - ^TTZ + x ( ° 2 ) 5( x- xs ( a Z ) ) * s * s x-xs(o2)
(10-46)
where X(a2) ie a function involving the boundary data of u
and v.
Therefore, the continuum
modes are characterized by
a non-square integrable
tangential component
perpendicular to the
1A
k
IX X.
x ^
k
X,lG') -»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:
i O n * n—vw-
s\»«
-AAA»—x »H*t -*.*•' »
-*»»t
. 1 7 0 .
The s e t s {G*} and (cri .} a r e no t p a r t of t h e s p e c t r u m . They
on ly a c t as a k i n d of s e p a r a t o r s of t h e t h r e e s u b s p e c t r a . The
s e p a r a t i o n of t h e s e s u b s p e c t r a on ly o b t a i n s i f t h e inhomogenei ty
i s n o t t oo s t r r a g .
C. DAMPING OF *LFVËN WAVES
We wish t o complete t h e s o l u t i o n of t h e i n i t i a l v a l u e
p rob lem g iven i n Eq . ( 9 - 2 2 ) , i . e .
M>^ = 2 J i{Va , - l i n t , ) e doj (10-47)
by e x p l i c i t l y c o n s t r u c t i n g t h e r e s o l v e n t o p e r a t o r (p F +U2)
f o r a s p e c i a l c a s e . For t h e p l a n e inhomogeneous s l a b model of
S e c . X B t h e inhomogeneous e q u a t i o n r e l a t i n g ^ ( ^ ; u) t o t h e
i n i t i a l d a t a £ = i u ^ . (^) - £ . (£) i s o b t a i n e d by j u s t add ing t h e
v e c t o r X t o the RHS of Eq. (10-31):
/ > 2 + c 2 > £ - f 2 b 2 + " 2 £*&+* ^ \ / M IA
- gO>2+c2)-i-dx
- g 2 (b 2 +c 2 ) - f 2 b 2 + o>2 - fgc2
\ dx - f g c ' - f 2 C 2 +Ü)2
= Y
I W (10-48)
Here, the initial data are also projected as indicated in Eq.
(10-30): £ = Xe - i Y ^ - 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 (<J*(X)} and the fast discrete
spectrum, whereas there may also be some slow discrete modes
left that are not swallowed by the slow continuum. Consequently,
the solution of the full initial value problem consists of the
simultaneous evolution of all these modes. In order not to
get lost in formal generalities, let us concentrate on the
important features. To that end we will make some simplifying
assumptions to the effect that the three subspectra become
widely separated. We may then study the separate influence of
one subspectrum, in this case the Alfvén continuum.
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 b 2 . In order to separate off the
influence of the fast modes we concentrate our study on
nearly perpendicular propagation:
k / / < • k i X k .
so that aj^kj, b^^jjfek^2 o i t
<tf
(10-51)
H * • « '
0 t
Under these conditions there i s no paral le l motion
to leading order '• I = Z = 0, so that Fq. (10-48) simplifies
to
V
f b 2 f - k* b2 • „2 dx dx </ dx
k L b'
- k j , b / dx
k* b ' - k ' b^ + dj
\
• /
\
n /
l*\ (10-52)
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
. 1 7 3 .
e q u a t i o n s :
- JL [ ( u 2 _ 0 2 ) | . ] t + ( u 2 . 0 2 ) I = x + 1. Y , t (10 -53) k
~ = _ I?.._JL n = " r 5 ' - — - r , (10-54) k 2 b '
k « . 2 K 2
where a* = a* (x) = k2„ b 2 ( x ) .
De f in ing
P ( x ; U2 ) = - (a,2 - a 2 ) / k 2 ,
Q(".;u2) = " (a)2 - o j ) , (10-55)
R(x ; u) = X + Y ' / k
Eq. (10-5 3) may be w r i t t e n 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
a r 3
3Gf x x' ) 1 P(x) r - Q(x) G(x,x') = 6(x-x') , (10-57)
x L r v J W He J
and the boundary conditions
G(x = x ,x';u2) = G(x=x2,x' ;u)2) - 0 . (10-58)
I n t e g r a t i n g t h e d i f f e r e n t i a l e q u a t i o n (10-57) we f i nd t h a t t h e
G r e e n ' s f u n c t i o n i t s e l f i s c o n t i n u o u s b u t t h e f i r s t d e r i v a t i v e
.174.
d i sp lays a jump a t x = x' :
Ï ti 3X U X X (10-59)
The solution of E G . (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
unique solution for the Green's
function when the homogeneous
GU.«';0')
equation does not have a non-triv: _-l ><
solution (Fredholm alternative:.
Proper and improper solutions :>t
the homogeneous equation occur
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
for complex v».luas ot w on the
^ 4 ^
• for ff'-sO.'-
*• X
»-x » W)
Laplace contour . The procedure i s then to cons t ruc t the Green's
function for ^cr.clc:: v l ' . : ^ : of J 2 ' :h3re e v i s t e r . r e i s guaranteed
and to defcrzi the contour in such a way t h a t the spec t run i s
approached.
The symmetric express ion ror G^x,x ! ;ur) i s found in
.175.
terms of the solutions $(X;M2) and ^(x;w2) of the homogeneous
equation satisfying the left and right boundary conditions,
respectively:
(P • ' ) ' - P $ = 0 , <Kxx) = 0 , (10-61)
(?•')' - H - O , *(*2) - 0 .
I n te rms of t h e s e f u n c t i o n s one f i n d s f o r t h e G r e e n ' s f u n c t i o n :
r ( x , x » ; u 2 ) G ( x r x ' ; w 2 ) = » (10-62)
A ( u i 2 )
where
r U . x 1 ^ 2 ) E *(x, ; u z ) ^ ( x > ; w2 )
= <f(x;w2) Mx' ;u>2) H ( x ' - x ) + <J>(x';w2) *(X;ÜJ2) H ( X - X ' ) ,
A<u)2) = P(x;ui2) [MX;U> 2 ) ** (X;U> 2 ) - <f>'(x;w2) # ( x ; u 2 ) ] .
Here , we have i n t r o d u c e d the n o t a t i o n
x< = i n f ( 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 3x
= P» (<H' - *>) + P(**' ' - <J>' '4>)
- <KP*')' - *(P<tT)' = Q<^ - QiH = O ,
so that A y A(x) . For eigenfunctions the solution of the
homogeneous equation satisfies both left and right boundary
conditions, so that <f> = i|>. In that case Ma2) = 0. For that
reason, A(CD2) is called the dispersion function.
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 M e che explicit functions would read:
<J2(X) o A o'
r2 , i(aii *?,)
xA(a2) X + 0
(a2 - o*)/o 2 i
j(x1 + x2)
o A '
(10-63)
0*tx} f <£»
tf. \
i i
X
a1
*,«')
w*»
In the previous section we expanded around the singularity
x « x.(a2) of Eq. (10-63) in terms of the variable s = x-xA(a2)
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
z (w2) is the analytic continuation of xA(o2) . For the linear
profile the explicit expression for z. (w2) would be
A o o A (10-64)
introducing a complex variable 5 replacing s,
© V-rt
-i—•x
5 - ; ( X ; ( Ü 2 ) = x - Z A ( w 2 ) , (1C-65)
the s o l u t i o n s ^ and if> of the equa t ions
(10-61) may be expressed as a l i n e a r
combination of the func t ions
u ( 0
u(s)£n C + v(c) ,
(10-66)
where u(C) and v ( 0 are the a n a l y t i c con t inua t ions of the
funct ions u(s) and v(s) in t roduced in Eq. (10-41) , which may
be w r i t t e n as a power s e r i e s i n ? : u ,v ^ a + b ; + . . . . Hence,
f É(X,-W2) V . ( U 2 ) • (C) =«tt l
L 5L(u)2) u^a,2) u ( c ; u » z ) + v ( ? ; u z ) , . , . ,2 '
IP CO
2-» n
i n S ( x ; u 2 ) v 2 ( u 2 )
0»2> u 2 ( u i 2 )
(10-67) .,..2 u ( c ; u * ) + v(c;u>*) . ..,2-
Substituting these expressions into Eq. (10-62) provides us
with the formal solution of the Green's function:
. 1 7 8 .
G ( x , x ' ; W2 ) =
- u / - a 2 ( x , ) v (u 2 ) i ^ r in - i - _ | U ( X < ;o 2 )+v (x ; « 2 ) [ .
u ^ w - ) J
U ) 2 - 0 2 ( x ) V_(oü2) " Scn-
u - p A2 «•2(<,>2) J u(x :CJ 2 )+V (X : K 2 ) 1
2 2
ï.n A2
a)—a Al
v ( O V 2 (ÜJ 2 )
^ ( w 2 ) U2(OJ2)
(10-68)
Here, the logar i thmic express ion i n terms of <; has been
converted i n t o the nor t t r a n s p a r e n t form in terms of a2 - af(x)
by means of the r e l a t i o n
x - 2 A ( w 2 ) = - ( • ,2 _ °2
Au>)/°?/ . ( 1 0 - 6 9 )
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' ;<i)2) and
A(^2) . In order to make these logarithmic terms single-valued
one needs to cut the complex to-plane along branch cuts that
precisely correspond to the continuous spectra ±{a_(x)} as we
shall see.
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 '*
->- x t h i s branch cut one may w r i t e : ui
Him i n z = in | z ± iri y-0± ' l
(on the p r i n c i p a l Riemann sheet n = 0) , where + iri i s the value
immediately above the branch cut a n d - iri immediately below. If
one wishes to deform a contour across a branch cut one moves to
another Riemann sheet of the logar i thmic funct ion. These sheets
are l abe l l ed by n and the logar i thmic function i nc r ea se s by an
amount 2fri every time one e n c i r c l e s the branch point and moves
to the next Riemann sheet. Therefore, the general express ion
for the logari thmic function when ;ppreaching the r e a l ax is
. 1 8 0 .
may be w r i t t e n :
£.int Zn z y-0-
J t n | x | ± i r i H ( - x ) + 2 m r i (10-71)
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
when approaching the real axis:
•i>]
2 _ „ 2
£im. £n V*0- 2 „ 2
a
Ö = JLn
2 2 o - o :
2 2 a
± s g ( a ) i T r [ H ( a - a a ) - H ( o - a )
+ H ( a + a ) a
H ( a + o e ) + 2 m r i .
( 1 0 - 7 2 )
Hence, assuming a | > a2:
(nTS)
^yUw*--0 t
m o
w4 *
H i
Ui W-( 1 i B
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<) ,
al = a?(xj, al, and a* _, whereas the function A(uz) only has A> A > Al A2
branch points at a?, and cr* . One may connect these
branch poirts as follows:
•<r»i.nv -a, '»> _5»t TU
"»I
• * w w v K - -*.ff
Ok. ff*< «"*> "nw <*,, rcx.x-, w'-'i
•• 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
. 1 8 2 .
s p e c i a l i n i t i a l d a t a : £. (x) ^ 0 , r, (x) = n ( x ) = rj .(x) = 0 .
The s o l u t i o n of t he i n i t i a l v a l u e problem can t h e n be w r i t t e n
from t h e E q s . ( 1 0 - 4 7 ) , ( 1 0 - 5 4 ) , and (10-60) a s : **
E ( x ; t ) - ^ - I du - ^ — e'i{iit \ d x ' r ( x , x ' ; ( i )2 ) E . ( x ' ) ,
n ( x i t ) = - £ - ^ £ ( 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 ,
n(t) -v - i(aao;/k) e"10»1, (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
o
- "2 xl + x2^ * T n e singularities of
the continuous spectrum a2 < a2 < 0z a r e n c w 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):
»-X
k2 l °\) 5 • ] • - <• 2 _ ol) (10-75)
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 s o l u t i o n s exp{kx) and exp ( -kx ) , when a2 J al. and
t 2 ¥ a ^ 2 ' r e s p e c t i v e l y . The so lu t ion £ = s inh[k(x-x )] s a t i s f y i n g
the le f t -hand boundary condi t ion 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:
-{.»->,) {{-0. 2 2 ö _ A2 ) a = o ,
or
l^-'Vh- = 0 (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,
where we have fixed x = 0. For
s i m p l i c i t y , we a l so take x^ •*• -» and
x„ -*• +<*>. The spectrum of the system
then changes as fol lows:
r,% vAl
ff»
» % -a 0 a
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:
( 0 2 . < J 2 ) U " - k 2 0 A
0 . (10-77)
A A,
Hence, for a2 = al, on the left Al
homogeneous interval £ may be
chosen arbitrarily. Each choice of
this function is a proper Alfvén
eigenfunction. Likewise, for o2=o22
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 M a 2 ) . 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
d d* u>2-02(x)
t _£ _ k2 5 ( f - o , C 5 - 2a - t (10-78)
so that we obtain modified Bessel functions of complex argument
as solutions:
Vk° = 1 + i < k ° 2 + — •,* ' \ (10-79)
K (kc) - - (in \ kc + Y) I (kO + \ (kO2 + — ,
when Y fc«577 i s E u l e r ' s cons t an t .
Consequently, the following s o l u t i o n s are ob ta ined :
e f C2 D2 e (-"»-•)
• - { Al I 0 (kc ) • Bx Ko(kc) * - | A2 I 0(kC) + B2 Ko(k!;) (-a,a)
1 1 ^a '"^ '
.187 .
The constants A, - , B. _, c . _, and D are f ixed by
equat ing funct ions and f i r s t d e r i v a t i v e s a t the boundaries
of the i n t e r v a l s . For the c a l c u l a t i o n of A(u»2) we a c t u a l l y
only need to compute A _ and B , because A (ID2) i s
independent of x so t h a t we may choose to evaluate i t in the
inhomogeneous l a y e r . The so lu t i ons <J> and ty on ( -a ,a) r ead :
• - k5l «"k a[[K0<k 5 i ) + ^(k^j^CkO -[i^Ck^-i^k^)]^*?)} ,
|[KO(U2) -Kl(kc2)]lo(kO ~[VkC2) • I ^ ) ] * ^ ) } . -ka « - - k?2 e
(10-80)
where
• i ) 2 , . 2 a ( ü ) 2 . 0 2 i ^ ) / ( 0 2 2 _ 0 2 i )
I n s e r t i n g these s o l u t i o n s i n t o the d ispers ion function we
find
" " l ^ ^ V ^ l * " I i < k 5 1 ) ] [ K0 ( l t C 2 ) - K ^ k ^ ) ]
- [ ^ ( k ^ ) + ^ ( k ^ ) ] [ l 0 ( 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* L e t us n o w 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:
**2 l l i in T + \ (r + f > • ° •
or
An u 2 " aA2 °A2 ' aAl
w 2 - ° A l 2 k a L u 2 " °A1 " 2 " 'A2 [ ^ 0 . (10-83)
Let us now study t h i s expres s ion i n the neighborhood of the
r e a l a x i s s o tha t v << a. We then have from Eq. (10-72) for
o i n the range of the cont inua:
" 2 * °A2 2 7 ^
U - °A1
° 2 " ff?- a ? , - a2 A2
o2 - a 2
Al
+ sg(o)sg(v)Tti • 2niri + 2ivo A 2 • A l
.189.
where the last term m:ny be dropped again as it is small
compared to the other imaginary contributions. This gives:
In °2 - -h °2 - «ii
•L - °ii °2 - 5<»ii • "IP ka (.» - „ * > ( . » - ,»2 )
• , , , , - • , • • • v o < 0 ^ ' ° " H ° 2 ' W * ( ° 2 ' ° " * „ + sg(o)sg(v)iri + 2mri + l = 0
ka (a2 - o2kl)
z (o2 - oj^)2
(10-84)
The r e a l and imaginary p a r t s o f t h i s d i s p e r s i o n e q u a t i o n
g i v e t h e r o o t s we a r e l o o k i n g f o r :
0 = ± ° o E ± \ / è ( a i l + ° A 2 > ' a2 - a2
v = v = - i «ka [ s g ( v ) s g ( 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
i *ka(0i2 " Al )/o (10-86)
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
¥l.»
ii i '
• »
. i
n«-i
^ftwv/v/^/v^
»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 l M - "lot „ i f . 10) -lut w
w e o / \ , e t o » e o , o
-hjt -io t
L i k e w i s e , n ( t ) "v e e
Hence , we have found a "mode" t h a t i s e x p o n e n t i a l l y
damped. S ince t h e p o l e i s no t on t h e p r i n c i p a l b r a n c h of t h e
G r e e n ' s f u n c t i o n tiiere i s no c o n t r a d i c t i o n w i t h t h e g e n e r a l
p roof t h a t complex e i g e n v a l u e s do n o t o c c u r f o r
s e l f - a d j o i n t l i n e a r o p e r a t o r s . On t h e o t h e r hand , i t i s c l e a r
t h a t t h e p r e s e n t "mode'* of t h e p lasma i s o f p h y s i c a l i n t e r e s t
a s i t r e p r e s e n t s a c o h e r e n t o s c i l l a t i o n of t h e inhomogeneous
s y s t e m . In c o n t r a s t t o t h e c h a o t i c r e s p o n s e p roduced by t h e
b r a n c h cuts of t h e con t i nuous spec t rum t h i s "mode" c o n s t i t u t e s
a v e r y o r d e r l y mo t ion . The plasma as a whole o s c i l l a t e s w i t h a
d e f i n i t e f requency t h a t canno t be d i s t i n g u i s h e d from a t r u e
eigenmode d u r i n g t i m e s x << v . "Modes" l i k e t h e s e occur
i n many b r a n c h e s of p h y s i c s and , a c c o r d i n g l y , t h e y have
r e c e i v e d many d i f f e r e n t names, l i k e q u a s i - m o d e s , c o l l e c t i v e
modes, v i r t u a l e igenmodes , r e s o n a n c e s , e t c . The damping i s
c o m p l e t e l y ana logous t o t h e wel l -known phenomenon of Landau
damping i n t h e Vlasov d e s c r i p t i o n of p l a s m a s . Landau damping
i s due t o inhomogenei ty of t h e e q u i l i b r i u m i n v e l o c i t y s p a c e .
Damping of Alfvén waves i s due t o inhomogenei ty of t h e
e q u i l i b r i u m i n o r d i n a r y s p a c e .
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.
i s a force-free f ield:
VxB * aB , (10-87)
where we take a - constant.Again,
we wil l consider a low (* plasma
so that the pressure will be
neglected. In components, Eq.
K ' " ttBy • B y = oBz » (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
/ \
> X
(10-87) reads:
.193.
W = I 1 ( 8 2 + al'i* x 3 ) d x ' (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 = V x £ ' £ - £ x £ ' (io-90)
so that
W 3 4 - \ [ C x A ) 2 - a A * . 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* . V x A - A * . V x V x A , (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 + J L A * . [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:
V x V x A - ( X + a ) V x A = 0 , (10-96)
which may be w r i t t e n a s another fo rce - f r ee f i e l d equat ion fo r
the p e r t u r b a t i o n s :
V x Q = a Q , a s A + a . (10-97)
Eq. (19-97) i s an e igenvalue equa t ion , where a i s determined
by imposing the boundary condi t ion JJ.Q = 0 a t x = X- and
x = X2« I n s e r t i n g such a so lu t i on i n t o the expression (10-95)
for I. gives W - 0 , so t h a t
W • V • H 4 - A* . V x A dx - (a - a) 4- I A* . 7 x A dx
•S-^SL 4 " f A* . 7 x V x A dx - -SL^-2- * f (Vx A ) 2dx .
(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)
I t should be not iced t h a t we could have dropped the
normalizat ion (10-92) a l t o g e t h e r . The r e s u l t i n g Euler
equation would have been
V x Q - o Q = 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 i s j u s t the margin?.], equat ion of motion for p = 0 (see
Eq. ( 8 - 6 3 ) ) . This equat ion i s equ iva l en t t o
V x Q - a Q - B x , (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 = ° a n d
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 - S e i + i f e / 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
. 1 9 7 .
3x e ^ (O, B'/B, - B'/B) - - ct(0, B /B, B /B) - - a e„ ,
(10-104)
- ^ - e,, = (O, B^/B, BVB) - o (O , B^B, - By/B) = o e t .
Fu r the rmore , we have from t h e e q u i l i b r i u m e q u a t i o n (10-87) '
f' - a g , g' « - a f . (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' + g R + f S = 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 s o l u t i o n which v a n i s h e s f o r x = x . and x = x ? r e a d s :
Q - sin Va2 - k2 x , (10-110)
where
Va - k = nir /a , a H x«, - x, . *2 "1
Hence , t h e i n s t a b i l i t y c r i t e r i o n (10-99) i s f u l f i l l e d f o r
o r
^2 .2 _,_ n2 TT2 ^ 2 ct » k + < a
2 2 (k/a) + ( n i r / o a ) < 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
. 1 9 9 .
oia
r\:t.\. p: i.a/ï unstable
limit k = O every time
aa increases with ira,
i.e. every time the
magnetic field has
changed its direction
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* + g n ) 2 - a V + 2afCn] d:
xl
x2 » -i- B2 j [f2(c2 + n
2) • W + gn)2] 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 le t us plot the eigenfunctions
€ corresponding to the eigenfunctions Q shown above. Writing
f = k cos(ax - 8) , we find:
r * _SL . l sin(mrx/a) n n , 1 M 5 ifB ikB s i n ( a x - 8 ) (10-116)
.201.
f\ s »
*- «a
Hence, if a solution Q
exists such that W as
given in Eq. (10-98) is
negative, aa > IT and £
develops a singularity.
For every zero that is
added in Q at least one
zero is added to the
function f because f
oscillates faster than
or at least as fast as
Q. It is clear that these
singularities are of such a nature that the norm
II? II = / U 2 + 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),
and B (r).They are restricted to satisfy
one differential equation, viz.
[p(r) • ± B2(r) ]• • »!(r)/r - 0,
J* SP>.
( l i - D
. 2 0 5 .
so t h a t we may choose ta.'O p r o f i l e s a r b i t r a r i l y . At t h e p lasma
s u r f a c e r = a s u r f a c e c u r r e n t s p roduce jumps i n t h e v a r i a b l e s
p , B - , and B which a r e r e s t r i c t e d t c s a t i s f y p r e s s u r e b a l a n c e : o z
1 2 "> 1 .2 «*» F, * T (BL + B; ) - -r <K * B" )• ( i i - 2 ) o 2 do zo 2 9o zo
The subscript o indicates values at the plasma surface. Hence,
four of the five parameters p , B , Bn~* 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 — —
B z ( r ) - B*> • B 9 ( r ) " B8o a / r • < n~3>
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 (r=a) = B . The constants a and B„ should not be considered z o o
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 p r e s s u r e b a l a n c e e q u a t i o r (11-2) may be w r i t t e n a s
1 + S + p 2 a 2 - (1 • y 2 a 2 ) B 2 /B2 , o o o zo o '
.206.
so that on a < r < b
B2(r) 2 2
1 + 6 + y a o o ~2 2
1 + y a* o
BQ(r) - 2
In the plasma region 0 <_ r < a
we may consider the profiles
p(r)/B2 and Bz(r)/B to be
arbitrary, except for their
values at r/a = 1 which should
be •= 8 and 1, respectively.
Fixing these profiles and the
two parameters 0 and £ a then
completely determines the
equilibrium. The profile
B„(r)/B^ is found by o o
integrating Eq. (11-1) in
outward direction starting
B,/fcl
V&.
P/Btl
41».
1 *K * *y 1 • y 2 a2
( 1 1 - 5 )
-*- »•/«
*>/» «7a
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 z o o
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.
. 2 0 8 .
B. DERIVATION OF THE HAIN-LÜST EQUATION
Our s t a r t i n g p o i n t i s t h e e q u a t i o n of mot ion
F U ) = p f- , (11-6 )
n 2
where
F ( 0 = - 7ir - B x (Vx Q) - Qx(7 x B) ,
Q = - V x ( B x O » i r = - 7 p 7 . ? - ? J p .
Because of the symmetry we may study normal mode solutions
of the form
| <».•.».«> -(trirt<r).ï.k<r).f1>rt(r)>^'-,rt,--t'. (11-7)
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:
*, sfc ; !UH <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 + « i i g + * • " ' ( 1 1 _ 9 Ï
where
g = (roBz/r - kBg)/B = G/B,
f s (mB9/r + kB z ) /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' H e n c e' o n e 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 ej.' k = i(BzSe " V z) / B ' (11-10)
*s i *r h - i(Be*e + VZ)/B.
.210.
In terms of these variables we have
$ = <ifB£, -fB05)' + kBn, -(rBzC) Vr - mBn/r),
ir - " P'C - YpV- * • (11-11)
7.5 * (rO'/r + gn + f^ •
Inserting these expressions into Fq. (11-6) and adding a
l i t t l e algebraic e f f o r t , using the equilibrium equation (11-1) ,
leads t o the following formulation of the spectra l problem:
I dr r dr , |r2j , d YP»B2 ' ^ B 9 B ;
r — g-1-1- 1 -2k dr * r 1 r I
'&?1W 1 c " '
2 d 1 -D ' "8 (YF+B )-T-• r 2k — .
d r 1 * _ j
-»2 2 w 2 * 2
1-fY dr
g^(YP+B^)-f2B
"fgYP
-fgYP
-f2Yp
M 2 !
• - p u TT; .
TC'
(11-12) Apart from the occurrence of a few factors r , t h i s i s a symmetric
formulation.
Notice that the matrix of Eq. (11-12) i s analogous
to that of Eq, (10-31) for the plane s lab except for the
occurrence of the three additional terms that have been put
ins ide boxes. These terms are algebraic so that they cannot
change the continuous spectrum of the system. Change i s meant
here in the sense of adding or taking away continuum
eigenvalues . This i s so because the continuous spectrum i s
associated with the s i n g u l a r i t i e s caused by the zeros of the
factor in front of the highest der ivat ive . For the d i scre te
spectrum the addit ional tarms are quite important as they
create the p o s s i b i l i t y of i n s t a b i l i t i e s 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£:
G [(YP+B2)pü>2 - YpF2lrx'+2kB (B2pa.2 - YpF 2) X
T> - 6
r2BD
(11-13)
YPF [ (pu,2 - F 2 ) r X ' + 2kBflGX ] ** ' ' • • • !•
r2BD
where
D = p2 a.4 - (m 2 /r 2 + k2 ) (y p+ B 2 ) p u2 + (m2 /r2 + k 2 ) y p F 2 .
S u b s t i t u t i n g these e x p r e s s i o n s i n t o the f i r s t component of Eq.
(11-12) g i v e s the Hain-Lust equat ion:
[ i > f [ } ( » . ' - ' V ( i - _ü£L„W -YPr2,
r r D f 2 k B 6 G 0 7 ? l
+ 1 Y~ ((YP + B^pu - YP* ) V * ] x - 0 , ( 1 1 - U )
.212.
where
N= (PÜI2 - F2)((Yp + B2)po)2 - YPF2)
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:
[P(r;u2)X'l' - Q(r;tu
2)x - 0 , (11-16)
where
P(r;u>2) = S/CrD) ,
N - N(r;u2) = p2 <TP + B2) ' [J - a\ (r)] [ J - o* (r)] ,
.213.
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
Yp + B' FVP , (11-17)
2 1 , 2 , 2 , 2 . , D2, Uj n ï y (« /r + k )(YP + B )
1±\ 1 " 4ypr
7 7 2 2 2 (mZ/r 4k")(YP + B V
/P.
For a fixed radius r the four frequencies are ordered as
follows:
2 2 2 2 0 * % * °I * °A £0II £ - '
— i — — x - >•- * . h-*- <?x
(11-18)
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.
* K K |I.>TH>« M fr««
M H D •.
c.p. c.p.
Slo») Alt"!*
e.p. CO"
CO
-«-..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
i j 2 = <*>.
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 a n d x'» w e
introduce a variable that has physical significance, viz. the
1 2 perturbation II of the t o t a l 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
° " " T5*' + " 2 ~ f T— \(YP + B 2 ) p u2 - T P F 2 } ] 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),
W = 2(kB9G/r2) [<YP + B2)ou2 - ypF2 ] ,
we may transform the Hain-Lust equation into the follov.'ir.c
Dair of first order differential equations:
.218.
N r
ïï'
where
.Vr2
» \
-c
C = W - 2 Bft B2 B
= -2 -| P2"4 + 2* 4 n <YP + »2>P<-2 - YPF21 ,
« O, (U-25)
£ = -K[ü/r + 2(B2/r2)'/r + V/rD] - (W - 2DB2/r2)2/D
= -N 'P.2 - F 2 , i ft\j *e 2 4 + ., i p 2
I < + R 2 , 2 41 5 H l -4 7p« +4 _-[(1fp + B)p« -2 \ 2/ r r \r / J r
YPF2] .
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:
n - (Bj/r2)X - SeQe+ BzQz - (B^/r2)X , (11-29)
wherell i s 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 a t the plasma surface completely i f
the vacuum solutions are known. This part of the problem can be
carried out e x p l i c i t l y s ince the so lut ions i n the vacuum are
Besse l functions as we sha l l s e e . From the vacuum equations
V x £ - 0 , V . C J - O , (11-31)
we obtain the tangential components of § in terms of the radial
component:
% ' £ -TT2-2 (r 5r>' • m +k r
kr - (11-32)
m +k r
so that
. 2 2 1 .
m +k r
The radial component s a t i s f i e s the second order d i f fe rent ia l
equation
2 2 2 2 (rö r)" *-£- W o <rQr>'~ (-^V * k2> rQ = 0 , (11-33)
-n~ + k~r" r~
which has the modified Bessel functions as solutions:
Q -v I'(kr) , K*(kr). r m m
Tho solution Q o n t n e 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 m m
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' U b > - 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
X, - a.xCi) + a,X<2>
xh = b x ( 1 ) • b,x(2)
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 a o
under consideration. Sturm's separation theorem (Ince, p . 223)
now s ta tes that the zeros of these solutions separate each
. 2 2 4 .
o the r , i . e . i f x and x are consecut ive zeros of x then xv
1 2 a t vanishes once i n the open i n t e r v a l (x , x ) .
Proof: Suppose y, does not vanish on
(x ,x ) . Then, x and x ? a r e consecut ive
zeros of the f i n i t e function x /x. . Hence. a b
d/dx(X /X,) must vanish at least once on a b
the interval. However,
dMxJ XaXb " xa Xb
x2
cannot vanish because that would imply that the Wronskian
vanishes somewhere. This contradiction proves that x b 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
(PX2')' - (Q - x2R)x2 = 0 , (11-38)
oscillate faster than x if X2 ' *i* H e r e' 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.,x2) and
subtract:
$[X 2 <PXJ) ' - X l ( P x 2 ' ) ' J dx = [x2PXl« - X l P X 2 ' ] X 2 X l
= fX2Pxi ] ( x « ) " (X1 " V J x l x 2 d x ' (11-39)
*-*
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. ,x 2).
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
*
. 2 2 6 .
Sturmian, whereas systems t h a t have the oppos i te proper ty ( e . g .
Eq. (11-36) when the sign of X i s reversed) we c a l l a n t i -
Sturmian. An immediate consequence of these p r o p e r t i e s i s t h a t
we can label
different
discrete modes
by just counting
the nattier of
nodes on the
complete
interval (0,a).
If the system
is Stumdan the
eigenvalue X
k»»"t\- £t>»r««v «.»i
*• X
( * i > ' * t y ^ o " )
*t >»T Mi l «k VI
•"IK- ~
«=« M = » / I I I
v\*l « I 1 rt:C w *\
A H H
avit- iW»wAvi
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 -<j| an<^ ^°TT'' F o r t h o s e
values of w2 the Hain-Lust equation is non-singular, but the
way in which w2 appears in the equation makes it virtually
impossible to prove anything directly from the equation
itself. In such a case, the only hope to prove general
properties about the spectrum is to go back to first principles
and, in particular, to exploit the only property of the
original operator p F we have, viz. that it is self-adjoint.
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:
<J1»P"1J[(|>> " <k^~Xl^>
= 2 " J " " S ( 5 * 7 p + y p V ' S " 5 ' 8 ) d 0 " I \ ! T £ ( £ ' V F + Y p V * 3 " l'Vda *
( 1 1 - 4 0 )
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
. 2 2 8 .
<^0~lT(P> -<Z,o~lTill)>
L -« r»rx L r J r= r i
ffL[rT,r ?D ( r V i r *2 ~ 'LFerTD ^ V l " 2 (ll"41)
L Jr=r L «J r = r l - J r = r 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 :
| ( Q ) - - pu22ri
(11-42)
2 M '
but not necessarily satisfying the boundary conditions
(11-15). Then, the LHS of Eq. (11-41) transforms to
(m22 - «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 i s 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 ),
. 2 2 9 .
i . e . , we want to invest igate whether rn o sc i l l a t e s fas ter
or slower than r£ for a given difference of cu* and us2. Under
the mentioned conditions a l l tha t remains of Eq. (11-41) i s
(o.22 - «1
2)<J|^> - * L f n r pö ( r5 r ) ' ] (r=r2J. (11-43)
Let rE > 0 on the open in te rva l ( r - , r ) so tha t (rE )'(r=r ) < 0
If N/D > 0 and CJ| - w* > 0 t h i s implies that rn ( r = r j < 0 so
that rn osc i l l a t e s fas te r than r£ (Sturmian behavior) . If, on
the other hand, N/D < 0 rn w i l l o sc i l l a t e slower than r5
(anti-Sturmian behavior).
Consequently, the d i sc re te spectrum outside the ranges
{oj?}, {CJ|}, {Oj}, and io2} i s Sturmian for N/D > 0, i . e . the
eigenvalue UJ2 increases with the number of nodes of the
eigenfunction r£ on the complete in te rva l (0,a) . For N/D < 0,
the d iscre te spectrum i s anti-Sturmian. Therefore, the
behavior of the d iscre te spectrum changes from Sturmian to
anti-Sturmian every time w2 crosses one of the four mentioned
regions:
* - S t o r n i l m
< * m t i > ) U r « t « i
The regions {a|} and (olj) thus turn out to act as separators
of the d iscre te 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 a t some po in t i n the i n t e r v a l (O.a ) . According to
Eq. (11-17) in t h a t case the continu a {all and ioi) s t r e t c h
out to w* = 0, so t h a t the mere ex i s tence of i n s t a b i l i t i e s
i n d i c a t e s tha t a t l e a s t one of the Alfvén or slow branches of
the d i s c r e t e spectrum has become Sturmian. I t i s comforting
t h a t i n any case the function N/D never changes s ign on the
uns tab le s ide of the spectrum, so t h a t uns tab le modes are
always Sturmian. This i s a l so in agreement with our i n t u i t i o n
t h a t moving the wal l inward does not increase the growth r a t e
of an uns table mode, which would be the case i f the uns tab le
s i d e of the spectrum were an t i -S turmian .
Since the uns tab le s ide of the spectrum i s non-s ingu la r
we immediately obta in a s t a b i l i t y theorem for o - s t a b i l i t y of
the diffuse l i n e a r p inch . To t h a t end we should r e a l i z e t h a t
the g-marginal equat ion of motion (9-43) for the d i f fuse
l i nea r pinch i s obtained from the Hain-Lust equation (11-16)
by j u s t r ep lac ing w2 by - a 2 :
[P(r ; - a 2 ) X ' ] ' - Q(r ; - a2)x = 0 , (11-4 5)
where
X(0) = X(a) = o . (11-46)
The one-dimensional modified energy p r i n c i p l e corresponding
to t h i s equation could be derived from Eq. (9-44) by a tedious
a n a l y s i s , s imi l a r to the one leading to the Hain-Lust equa t ion .
Hov/ever/ here v/e may pose i t d i r e c t l y as tha t funct ional which
. 2 3 2 .
produces Eq. (11-45) as the a-Euler e q u a t i o n :
a
W°[xj *L \ [P(r ; - o 2 ) x ' 2 + Q(r ; - o 2 ) X2 j d r , (11-47)
where L i s t he l eng th of the plasma column.
Suppose now t h a t we i n t e g r a t e Eq. (11-45) s t a r t i n g
from the l e f t end p o i n t r = 0 where we s a t i s f y the boundary
condi t ion x = 0. I f the s o l u t i o n X(r) thus obta ined does not
develop a zero i n the open i n t e r v a l 0 < r < a , our o s c i l l a t i o n
theorem a s s e r t s t h a t a d i s c r e t e e igenvalue w2 < - a 2 does not
e x i s t , so t h a t t h e
system i s c - s t a b l e . On
the o ther ha rd , i f the
s o l u t i o n x(r) vanishes
somewhere on the open
i n t e r v a l 0 < r < a , a
d i s c r e t e e igenvalue
ai2 < - a 2 does e x i s t for which both boundary condi t ions
-XC-c^
L \ -*• r
C-stable
*-r
(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:
|y+k2r2 J Lr
(11-43) Th i s e q u a t i o n i s of t h e form
[A( r O ' ] ' " B r ' - 0 ,
•which i s e q u i v a l e n t t o
&-r - ^K'-ur ZkB„G
r i m - + k ~ r - ) X a 0
( A r 2 ? , ' ) » - (B r 2 - A»r)r , - 0 .
.234.
Therefore, the marginal equation may be written as:
< V ) ' " 8oC * ° • (11-49)
where
f = Ar2 « — ~ — , (11-50) ° o2+k2r2
g * Br" - A'r o
B 2 \« 4k 2rB? rF~ + r"'
/Bj_|. _ 4k^ri; + / 2kBeG y I rf2 y
U 2 / 02 + k 2 r 2 \al+k2T2l \a2+k2Tz)
rF 2 — (r 2Bf )» • r2
r (n .B e / r -kB z ) 2 f r2<m2B2 / r 2 -k 2 B* ) j •
m2+k2r2 * \ m2+k2r2 1
2k 2 r 2 B 2 +k 2 r 2 - l 2k 2 r 3 (n»Bfl/r-kB ) p. + r F 2 2 z_ F ^
m2+k2r2 m2+k2r2 (m2+k2r2)2
(11-51)
The equ iva l en t one-dimensional form of the energy p r i n c i p l e
may be w r i t t e n as
W[e] - *t [ ( f o C 2 • g 0 5 2 )d r , (11-52)
where L i s the length of the plasma column.
Since Eq. (11-49) i s j u s t the Hain-Lust equat ion for
w2 = 0 we o b t a i n from the o s c i l l a t i o n theorem of the prev ious
sec t ion d i r e c t l y Newcomb's s t a b i l i t y theorem for the case t h a t
t h s i n t e r v a l (0,a) con ta ins no s i n g u l a r i t y 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
k + pm a 0 f
i.e. for values of the wavenumbers IT. and k such that the
. 2 3 6 .
t a n g e n t i a l wavevector i s perpendicular t o £ . In t h a t case the
phase of the pe r t u rba t i on i s cons tant along the f i e l d l i n e s
a t the pos i t i on r = r of the s i n g u l a r i t y . Let us expand a l l
q u a n t i t i e s in terms o f the v a r i a b l e
s 2 r - r s . ( 1 1 - 5 5 )
We then have:
F ^ mB v i ' s , m2 + k 2 r 2 -v. m 2 ( l + y 2 r 2 ) ,
so t h a t
TH2 U ' 2 2u 2 r 2
fo £ , \ 2 s Z • *0 * — T T p ' * ( n - 5 6 )
l + p 2 r 2 ° l + w 2 r 2
Consequently, c lose t o the s i n g u l a r i t y the Eu.Ier equat ion
(11-49) reduces t o
( s 2 £ ' ) ' - <xi = 0 , (11-57)
where
z
The so lu t ions of the equat ion (11-57) a re s and s ,
where n and n ara the roo t s of the i n d i c i a l equat ion 1 2
n(n + 1} - a = 0, so t h a t
n. , - - -T i T \ / l + * a • ( 1 1 - •> S J
Depending on whether 1 + 4ra i s p o s i t i v e or negat ive the
.237 .
i n d i c e s a r e r e a l o r complex.
For 1 + 4a < 0, when t h e i n d i c e s a r c complex, t h e r e a l s o l u t i o n s
t o t he E u l e r e q u a t i o n a r e o s c i l l a t o r y :
1
(11-59)
1
-— + iw - —-iw f = s + s = 2 s cos(w in s ) ,
- - + iw - - - iw 52 = i ( s 2 - 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 ' where n = - ~ + •£ VI + 4a > s 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
larger than 1.
1 2' "*
= - 4 - i Vl + 4a' < - i.
*• r
.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 ( r l* r2} = J ( f 5 ' 2 + 8 S 2 ) d r - j [ f e f 2 • 5 ( f 5 , ) , ] d r
- 'Jlï J ( 1 1 - 6 1 )
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^ i n t o t w o 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
composed of £a on ^i>rQ) and Sb on (r ,rJ, one then obtains:
^ W - [*«.«; K'-r0) - [f5b5£](r-ro>
[Hill ~ tl )](r-r ) < 0 , (11-62)
.239.
so t h a t the con t r ibu t ion to the energy of t h a t s u b i n t e r v a l
i s nega t ive . By choosing the t r i v i a l Euler so lu t ion r = 0
on the remainder of the i n t e r v a l one then shows t h a t the
t o t a l energy W(0,a) < 0.
The condit ion 1 + 4a > 0 which i s necessary for the
absence of the o s c i l l a t o r y s o l u t i o n s (11-59) was der ived
f i r s t by Suydam and i s , t h e r e f o r e , known as Suvdam's criterion:
. + J. rB 2 (Hi) 2 > 0 . 8 z u
(11-63)
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 «
•'it for which the' growth rate \ [
-,.,2 is larger than that of
all the higher node solutions.
In other words.' violation of
Suydam's criterion implies the
existence of a global_n = 0
instability. This instability
\l ft?
Cj\oO<»\ S«-jJk»«. r*oJl«
.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
(11-57), where the coefficient a is of £
course different. One may then derive
similar conditions as Suydam's criterion
..S^y
-*• r
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 * . 1 2 t l + 1
[ f55' ] 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:
fCsH(s) [£sH(s)]'<v, sn + 2 [ns
n"1H(s) + sn5(s)]H(s)
= s2 n + 1[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
*. r
E. which vanishes at the right end point
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 i s
"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
t h e r i g h t of r = r does n o t v a n i s h i n t h e open i n t e r v a l ( r , a ) .
I t i s c l e a r t h a t t h e e x i s t e n c e of s i n g u l a r i t i e s com
p l i c a t e s t h e marg ina l s t a b i l i t y a n a l y s i s c o n s i d e r a b l y . T h e r e f o r e ,
fo r numer i ca l s t u d i e s a o - s t a b i l i t y a n a l y s i s i s c e r t a i n l y t o be
p r e f e r r e d . For a n a l y t i c s t u d i e s t h e p r e s e n c e of s i n g u l a r i t i e s
o f t e n f a c i l i t a t e s t h e c o n s t r u c t i o n o f e x p l i c i t a n a l y t i c s o l u t i o n s
by means of s e r i e s e x p a n s i o n . However, t h e number of c a s e s t h a t
may be t r e a t e d t h i s way i s q u i t e l i m i t e d .
G. FREE-BOUNDARY MODES
In t h i s s e c t i o n we c o n t i n u e t h e d i s c u s s i o n 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 ) _ _ „ a s 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 f i rs t approximation to
the study of external kink modes, which are the most dangerous
instabilit ies occurring in a cylindrical plasma column. Here, most
. 2 4 3 .
dangerous is meant in the sense of affecting the bulk of the
plasma and having large growth r a t e s . (For typical dens i t ies
of high-B pinches they exponentiate on the usee time-scale) .
For the sharp-boundary skin-current model the equi
librium quant i t ies for the in t e r io r
of the plasma column are those ot
a homogeneous 5-pinch:
B. - 0, B = B ,p = P , 8 z o o
-Ëi—r
J1 . - » - »•
P = T BoBo ' (11-64)
where B , p , and B are constants . For th is part of the con-o o o
figuration one may again define the Alfvén speed and the sound
speed:
bo " \/Bo / po ' Co 5 fö > (11-65)
which are related to each other by the value of 8 : J o
c 2 / b 2 = -L B Y . o o 2 o (11-66)
For the interval 0 <_ r < a the Hain-Ltlst equation may be simplified
to
(a ' 2 2 k b ) o •> i o 2 2 2
, ( k ~ - o - / b - ) C k f c - o /c) 9 l o o z - m + 2 2 2 2 2 r
k - a /b"" - a /c o o
I = 0.
(11-67)
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
k* = (k2- o2/b2)(k2- 02/c2)
~~2 TTl T7~2 k - o /b - a /c o o
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 i ' i s the n - zero of the Bessel funct ion J ' ( x ) . From Jmn m
t h i s express ion one ob ta ins the d i s p e r s i o n equa t ion for the
slow and f a s t magneto a c o u s t i c waves in a homogeneous e-pinch;
, * . ( k2 + j ' 2 / a
2 ) ( b 2 + c 2 ) a 2 + k 2 ( k 2 + j ' 2 / a 2 ) b 2 c 2 - 0 . (11-70) mn o o mn o o
This equat ion i s completely analogous to Eq. (10-7) for the
magneto acous t i c waves of a plane homogeneous s l a b .
.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)
P a2 = 2 m
o p i (k*a) o n
S 2 (mBD/a + kB ) 2 I ( k a ) K ' ( k b ) - K ( k a ) I , " - t u p + " z m IP jvj m_ \\t
. a 2 ka r (ka)KMkb) - K^(ka) 1^ (kbjS
(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, Ê < < S so that q = efi /B ^ 1. 8 z ^ z o
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 <k*a) * jmj ,
I ( k a ) K ' (kb) - K ( k a ) I ' (kb) < w a J » l * tu/.\-\*\ m m ram » \ol&) + ( b / a ) ' T» E X
i; <k.>K; (kb) - K; (ka)r; <kb) |.| (b/a)|.j _ (b/a)-|*|
(m j* 0)
(11-73) Inserting these approximations into Eq. (11-71) leads to the
following dispersion equation:
- 2 4 6 .
e-B a ' ~«
a-o q~ o
- ! m l ( b / a ) i m U ( b / 3 ) - ' m ! l
, ? — ~> i i . , . — . -> ! <. - q ~ - |m | + vm+icq)-
( b / a ) ^ m l - ( b / a ) " ' 1 " 1 - '
( 1 1 - 7 4 )
I n E q . ( 1 1 - 7 4 ) we h a v e n e g l e c t e d s m a l l t e r m s B a n d 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
0 2 'V e 2 B 2
a2p q 2 L o
j i m | ( i m j - 2 ) + i (2 *q" + m) 2 + 2 (£q + m)"
. ,2|mi (b/a) -1J
(11-75)
This rearrangement of the terms should reveal some of the phys
ical mechanisms a t work in this model: F i rs t , there is the kink
term which is only destabilizing when [m[ = 1. Then, there is a
stabil izing 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 las t term represents the stabil izing influence of the wall,
ranging from infinitely stabil izing when b/a = 1 to no effect
when b/a ** °°.
Since only jmj = 1 i s unstable we may res t r i c t the anal
ysis to that mode:
a?(n=-l) = a 2 o q 2
o
(4q - l ) ( i q - a 2 / b 2 )
1 - a 2 / b 2 • ( 1 1 - 7 6 )
. 2 4 7 .
Hence, the e x t e r n a l kink mode i s always
uns tab le for t h i s model in the region
a 2 / b 2 < Stq < 1 . ( 1 1 - 7 7 )
This leads t o the obvious remedy of
the e x t e r n a l kink mode to exclude i t by j u s t p r e s c r i b i n g the
geometry of the torus and the t o t a l plasma c u r r e n t I such t h a t
q - 2Tia2B / R I > 1 , ( 1 1 - 7 8 ) 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 a t the plasma and cons ide r the
boundary value problem posed by the Hain-Lüst equat ion (11-44)
with the boundary condi t ions (11-15) . In o rder to have a problem
t h a t can be solved ana ly t i c a l l y ,we fix the equ i l ib r ium p r o f i l e s
as fol lows:
B -v B (1 - a 2 r 2 ) , Z T» O
B3 % u r » o , (11-79)
P B [ i 6 • U 2 - M 2 ) r 2 ] ,
.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 impos ing p e r i o d i c i t y o v e r
2TTR and impose t h e low-B tokamak a p p r o x i m a t i o n , where
q = 1/yR^ 1 , (11-80)
and where we have t h e f o l l o w i n g sma l l p a r a m e t e r s a t 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 \L/f < < k , k a << 1 a, m ,
2 2 2 2 pui << m B / a . o (11-83)
The r e s u l t s we o b t a i n w i l l j u s t i f y t h i s c h o i c e .
Under t he a s s u m p t i o n s of E q s . ( 1 1 - 7 9 ) - ( 1 1 - 8 3 ) t h e
H a i n - L ü s t e q u a t i o n r e d u c e s t o t h e f o l l o w i n g form:
JL r È1 dr dr
1 / 2
\r2)x ( U - 8 M
where
A = 2 2D2 - . , 2 y m B [ 4k PLU
2 1 2 YB
2 , 2n2 Pu " ^ B m
2 1 2 1 7 p a . ' ( l + y Y B ) - k ; jyfiB
. JiL rB
] •
.249.
The solutions of this equation are Bessel functions:
= J (\jT r/a) , m
(11-8 5)
s o t h a t e i g e n v a l u e s a r e o b t a i n e d by e q u a t i n g / x w i t h t h e z e r o s
j o f t h e B e s s e l f u n c t i o n J ( x ) :
X = j en
(11-86)
From t h e l a t t e r c o n d i t i o n t h e d i s p e r s i o n e q u a t i o n i s o b t a i n e d :
> -1+Y0
k l + 2 ^ e
r 2 . 2 Y8 m 2 - 2
k, + v + i -2 - 2 1 - J * J
2 m y
1+jYB Jmn mn rB'
- 2
JYB
1+JY6
r2 f k 2 2 - 2 Ü2 f r 2
+ 2 m u
'mn r B 2 ' 0 , (11-87)
where we have i n t r o d u c e d d i m e n s i o n l e s s v a r i a b l e s
_ 2 - ' , 2 / B 2 \ 1 . 1 Z w = ( p a z / B z ) u ' 2 ; k//r s fya , y s ya
The two s o l u t i o n s of t h i s q u a d r a t i c e q u a t i o n may be w r i t t e n a s :
- 2 1+YB £ 2 + 2 y 6
i^e " l^re j ^
2 A 2 ~ 2
m* — 4 ^ 2m y u +
inn rB
r iT4 —2 + I * + 4m*_ _ y f y B ( l + YB) - 2 p '
J — y + — t —
& r B 2
k* +
2 m2 -2 -2 * J ^=- v {-^f— y +
mn I + JTB rB
1/2 (11-88)
.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"y s
( 1 1 - 8 9 )
j ï S d + ï S ) _ 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 max = - Y 2 & 2 ( m 2 / j ^ n ) y 4 [ l - { 1 + ( l / y B X p ' / r B 2 ) y2 } 1/2
(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
CD = 2 (m / j ) y (TT-TT.). max Jmn 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.
- 2 5 1 .
I . o-STABLE CONFIGURATIONS
On t h e b a s i s of t h e o - s t a b i l i t y theorem i t i s p o s s i b l e
to s y s t e m a t i c a l l y s e a r c h f o r o - s t a b l e c o n f i g u r a t i o n s w h i l e
t a k i n g a r e a s o n a b l e c h o i c e f o r o , e . g . one which c o r r e s p o n d s
to t h e msec t i m e - s c a l e . From a l a r g e nuirber of n u m e r i c a l runs
t h e f o l l o w i n g q u a l i t a t i v e p i c t u r e emerged . There a r e , b r o a d l y
s p e a k i n g , f o u r c a t e g o r i e s of d i f f u s e l i n e a r c o n f i g u r a t i o n s
t h a t a r e o - s t a b l e with respect to i n t e r n a l modes . A l l fou r o f them
a r e c h a r a c t e r i z e d by a m o n o t o n i c a l l y i n c r e a s i n g o r d e c r e a s i n g
q - p r o f i l e , r e p r e s e n t i n g s h e a r of t h e f i e l d l i n e s which t u r n s
o u t t o f a c i l i t a t e s t a b i l i t y . The q and j p r o f i l e f o r t h e s e
c o n f i g u r a t i o n s a r e t h e most c h a r a c t e r i s t i c ones t o d i s t i n g u i s h
t h e d i f f e r e n t c o n f i g u r a t i o n s :
tokamak pinch
As t h e c u r r e n t p r o f i l e i s b roadened t h e maximum a l l o w a b l e 6
for s t a b i l i t y in g e n e r a l i n c r e a s e s from a few per cent for tokamaks
t o some 40% for t h e r e v e r s e d f i e l d p i n c h . Except fo r t h e l a t t e r
c o n f i g u r a t i o n a l l o t h e r c o n f i g u r a t i o n s r e q u i r e q > 1 , e i t h e r on
a x i s when the q - p r o f i l e i s i n c r e a s i n g as in r. tokamak, o r a t t h e
. 2 5 2 .
wall when the q-profi le i s decreasing as in a screw pinch. For
more d e t a i l s : 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".
. 2 5 4 .
X I I . SHARP-BOUNDARY HIGH-BETA TOKAMAKS
A. INTRODUCTION
In the previous sections we analyzed one-dimensional
systems, i . e . systems in which there i s only one direct ion of
inhomogeneity. This leads to an ordinary second order d i f fe r
e n t i a l equation in the unknown E, (r) where r i s the coordinate
in the direct ion of ir.homogeneity. The dependence on the
homogeneous direct ions can be taken care of by means of a simple
Fourier decomposition. The problem of ult imate i n t e r e s t in CTR
i s to study the s t a b i l i t y of diffuse toro ida l systems. This
problem involves p a r t i a l d i f fe ren t i a l 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 direct ions of inhomogeneity. Only the dependence
on the ignorable toroidal angle 4» can be Fourier-decomposed in
a simple manner. Before we embark on th i s complicated problem
i t i s , 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
i s 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
e l l i p t i c p a r t i a l d i f ferent ia l equation known, v iz . Laplace's
equation in two dimensions. This involves complex analysis of
harmonic functions, which, together with the theory of ordinary
second order d i f fe ren t ia l equations, const i tu tes the main resource
for known useful techniques from applied mathematics. This i s the
mathematical motivation for the present chapter.
. 2 5 5 .
The physical motivation for the invest igat ion of
sharp-boundary high-beta tokamaks i s the question about the
maximum obtainable & in a toroidal plasma. As is known, the
value of B ^ 2P/Bi cons t i tu tes a f igure of merit for future
fusion reactors . I t indicates the amount of plasma producing
fusion energy contained by a certain magnetic f ield which i s
costly to produce. The value of 3 is limited both by the
requirement tha t equilibrium exist and also by s t a b i l i t y
considerations. As far as gross s t a b i l i t y i s concerned, the
most dangerous i n s t a b i l i t i e s are the current-driven exLernal
kink modes, which impose l imits on both the maximum & and the
maximum toroidal current . The present chapter i s jus t an in
vestigation on how B affects the Kruskal-Shafranov l imit
q > 1, expressed by Eq. (11-78). To leading order in the
inverse aspect r a t io e = a / R o t* ie s 7 s t e m may be considered as
a s t ra igh t cylinder. The next order in e, however, leads to
toroidal effects of ft which d i s t o r t the angular symmetry of
the magnetic f ield leading to the poss ib i l i t y of additional
i n s t a b i l i t i e s . Also, a l imit on the equilibrium arises through
the occurrence of a so-called second magnetic axis when 6
surpasses a certain c r i t i c a l value.
Consider a dense plasma
region tp of uniform pressure
confined by surface currents
flowing on the toroidal plasma
surface S. Surrounding the core
of plasma i s a vacuum region T
enclosed by a perfectly conducting
* - K
.256.
wal l . The wall wi l l be assumed to be far away so tha t i t has
no influence on the s t a b i l i t y . The equilibrium problem wi l l
not be a free-boundary problem in the usual sense where ex
ternal currents or a wall posit ion arc given and the shape
of the surface S i s found by solving the equilibrium equations.
Rather, we consider the inverse problem where the shape of S
i s given and the external f ie lds are what they come out to be.
We may then put the wall a t any posit ion consistent with the
calculated f ie ld d i s t r ibu t ion . In other words: the wall i s not
specified a t a l l in th i s problem, except that i t should be far
away.
Bacause of the toroidal symmetry we wi l l f ina l ly have
to deal with the unknowns on the poloidal cross-sect ion of the
torus only. Let us denote the poloidal cross-section of the
volumes TP and TV by the surfaces <r and av , respect ively , and
the cross-section of the surface S by the curve C. Sharp-boundary
theory and the high-beta tokamak ordering w i l l permit us to
formulate the s t a b i l i t y problem en t i re ly in terms of harmonic
functions on <r and av connected by Neumann or Dir ichle t condi
t ions on C. Let us, therefore, f i r s t introduce some geometric
propert ies and notations tha t are especial ly sui table for such
funccions.
Diraensionless rectangular
coordinates centered about the c i r c l e
R = V
x 3 (R - R ) /a , o y « Z/a , (12-1)
give r i se to a three-dimensional
coordinate system x, y, $ with the
following representation of the gradient operator:
. 2 5 7 .
y = J_ ( e _L- + e - J - + e - H êr )• (12-2) a \ x 3x i,y ?y -\.<? 1 + ex 34»
Because of the ax i a l symmetry i t i s convenient to s p l i t t h r e e -
dimensional vec tors (3-vectors) i n t o po lo ida l and t o ro ida l p a r t s :
U = U + U e , U = CJ , u , , 0 ) , (12-3)
where, in genera l , the components U and L' w i l l depend on
x, y , $. Four ie r a n a l y s i s i n the angle <{. and the high-p tokamak
o rde r ing w i l l permit us to even tua l l y e l imina te a l l t o r o i d a l
components of 3-vectors and a l s o a l l dependences on the t o r o i d a l
angle $, so t h a t the two components U (x,y) and U (x,y) are then x y
convenient ly grouped i n t o a 2 -vec tor
Ux S ( ü x > ü y ) . (12-4)
S i m i l a r l y , we int roduce a two-dimensional dimensionless g rad ien t
ope ra to r
* whereas a kind of dual ope ra to r V takes the p lace of the usual
cur l o p e r a t o r :
?* 3 (-4- , - ~-). (12-6) •i. oy ox
* Not ice : VX-7X = 0, whereas the two-dimensional Laplacian may be
w r i t t e n as
\ - \ • \ - < •*: = 4+ -A • (i2-?) To i l l u s t r a t e the power of t h i s no ta t ion (due to Now-
comb, Ref. 3) consider a s p e c i a l 3-vector V(x,y,4>) having the
f requent ly occurr ing proper ty
. 2 5 8 .
V = O , (7 x V ) . = O . ( 1 2 - 8 ) •V, <\, (p
Fourier analysis and the ordering then leads to the r e su l t
that BV /a^ can be neglected, so tha t we get the following
relat ionship for the corresponding 2-vector V (x ,y) : 3 V 3 V 'V ' 1
7 V = — + 2L = 0 JL ' * -» • 3 x 3 y U ' 3y
3V 3V Vx • V.L - V 1 - - -T2- = ° ' ( , 2 _ 9 )
< \ , "
which w i l l be r e c o g n i z e d as t h e Cauchy-Riemann c o n d i t i o n s f o r
V and V . The v e c t o r V. can t h e n be d e r i v e d from e i t h e r one x y ^J-
of t h e two c o n j u g a t e ha rmonic p o t e n t i a l s A o r B:
* V± = - i 7j_ A = 7AB, (12-10)
where
AXA = Ax B = 0 .
Th i s i s t h e b a s i s o f t h a t p a r t of t w o - d i m e n s i o n a l MHD t h a t can
be d e s c r i b e d by complex a n a l y s i s .
L e t us i n t r o d u c e l o c a l o r t h o g o n a l c o o r d i n a t e s X and v
on t h e curve C, where X i s an a n g l e - l i k e c o o r d i n a t e b a s e d on
t h e a r c l e n g t h i a l o n g C:
dft - a e d X , (12-11)
where e ^ L/2ira i s t h e f a c t o r of e l o n g a t i o n of t h e cu rve C as
compared t o a c i r c l e of r a d i u s a . Normal and t a n g e n t i a l d e r i
v a t i v e s a l o n g C a r e t h e n w r i t t e n a s
n . . 7 . - 3/3v ; t , . V. - e " 1 3/3X . (12-12)
In terms of t h e s e c o o r d i n a t e s on C t h e above Cauchy-Riemann
c o n d i t i o n s become
259 .
v e 3 A <) v
- i v = 4 ^ = " - TT ' (12-13) A 9v e !U
Gauss* theorem f o r 2 - v e c t o r s V, s a t i s f y i n g Eqs . (12-9) t hen
g i v e s
( V^daP = ( l V x A ! 2 d o P = \ V^.(A* VxA)da P
= e 4 A* - ^ dl = e I B* - P - dX . (12-14)
3v ) 3v
C l e a r l y , o u r aim i s t o r educe a l l t h e f o l l o w i n g c a l c u l a t i o n s t o
e x p r e s s i o n s of t h i s form so t h a t t h e problem w i l l be t h e e v a l u a
t i o n of o n e - d i m e n s i o n a l c o n t o u r i n t e g r a l s a l o n g C.
The shape o f t h e curve C w i l l be p r e s c r i b e d , e . g . i n
p o l a r c o o r d i n a t e s by g i v i n g t h e f u n c t i o n r = g ( 9 ) . The c o o r d i
n a t e s (x , y 0 ) o f a p o i n t on C a r e then g iven i n te rms of two
f u n c t i o n s of t he a r c l e n q t h c o o r d i n a t e A: x = x ( A ) = e ( 8 ) c o s i o o
y = y (A) - g (8 ) s inö o o
(12-15)
The e x p l i c i t form o f t he f u n c t i o n s x (A) and y (X) a r e 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 e J
A(0) = — J 'yg-(9') + (dg/d9') d9'. (12-16) 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 <c are found from expressions
in terms of the triad of unit vectors n, t, e. : •\. a T. v
K = a t . ?n . t , BC,. = R e . Vn . e ,
where we have added the f a c t o r s a and R t o 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 d i f f e r e n t i a t i o n wi th r e s p e c t t o the argument
X. Furthermore, e ' = 0 , so t h a t x' x" + y ' y " = 0. Using o o - * o o
these relations one finds:
7 A n x . £•- = _1_
e
3«x 3X e
X " o K - t . .
p *vi 7 A n x . £•- =
_1_ e
3«x 3X e * ;
»
3n * .
e >
(12-17)
where we have n e g l e c t e d h igher order terms i n e i n the d e r i v a
t i o n of ic .
B. EQUILIBRIUM
The e q u i l i b r i u m i s s p e c i f i e d as f o l l o w s :
On TP : p * cons tant , p • c o n s t a n t , (12-18)
B - B .ïA - (R B /R)e , B HBA(R-R )
o . . : , * 4. ij . i i; * i i j . <«-»«
0 , , ' i K B • 0 , 7 . 8 - 0 . (12-20)
. 2 G 1 .
The m a g n e t i c f i e l d i n t he plasma has t h e u s u a l 1/R dependence
c h a r a c t e r i s t i c for a c u r r e n t - f r e e r e g i o n . Eqs . (12-20) a r e
t he on ly p a r t i a l d i f f e r e n t i a l e q u a t i o n s t h a t have t o be s o l v e d
t o comple te t h e d e s c r i p t i o n of t h e e q u i l i b r i u m .
The h i g h - b e t a tokamak o r d e r i n g , l i k e t h e l o w - b e t a
tokamak o r d e r i n g , i s an o r d e r i n g i n t h e i n v e r s e a s p e c t r a t i o t ,
b u t t h e i m p o r t a n t d i f f e r e n c e i s t h a t t he dense i n t e r i o r p lasma
i s c o n f i n e d by a d i a m a g n e t i c w e l l i n t h e t o r o i d a l f i e l d r a t h e r
t han by t h e e x t e r n a l p o l o i d a l f i e l d . Th i s p e r m i t s us t o i n v e s t i
g a t e t h e e q u i l i b r i u m l i m i t a t i o n s a s s o c i a t e d w i th t h e a p p e a r a n c e
of a s e p a r a t r i x a t t h e plasma boundary as w e l l as t h e f i n i t e - S
m o d i f i c a t i o n s of t he e x t e r n a l k ink modes. In t h e h i g h - g tokamak
o r d e r i n g t h e q u a n t i t i e s a r e o r d e r e d a s f o l l o w s :
B /B -v, B \ / B T, 1 , B /B ^ e , $ o <p o p o
2 (12-2-) S = 2p/B -v e.
The consequences of t h i s order ing for the p ressure balance equa
t ion (12-19) a re as fol lows. On 3 the po lo ida l v a r i a t i o n of the
t o r o i d a l f i e l d i s determined by
B /B - B / B = R /R = R / ( R + ax ) = r 1 + zv. (X) 1 ~l . ( 1 2 - 2 2 ) •p o $ o o c o o - o
S u b s t i t u t i o n i n t o Eq. (12-19) g i v e s
SB2 * r l + F.K t\) 1~2 B2 = %:(J) * r l + z*. O.) " ]"2 B2
o '- o , _ o p ^ o • o
so t h a t to second o r d e r ;
B 2 ( . \ ) /B 2 = L - 5 2 /B 2 + F.il •" 2r.x (>.)"! p o o o • 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 Ë /B = 1 so that all the pressure is confined by the poloida] o o
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
a . = 23/c. mm
The range between the low-e tokamak end and the high-B tokamak
l i m i t i s more use fu l ly descr ibed by a parameter of u n i t range:
k 2 . ' 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 l i m i t .
We may now w r i t e
S ( X )
e B r t k o
4- \ / 4 - \ / i - 4 - *2 [i - *„<*>] • <12"26)
which shows t h a t the parameter k2 j u s t measures the amount of
v a r i a t i o n of the po lo ida l f i e l d going the sho r t way around the
t o r u s . Notice t h a t in the high-e tokamak o rde r ing , where k2 *> 1 ,
. 2 6 3 .
t h e r e l a t i v e v a r i a t i o n o f S (,\) g o i n g t h e s h o r t way a r o u n d
t h e t o r u s i s o f t h e o r d e r u n i t y . S i n c e t h e t o r o i d a l f i e l d i s
v i r t u a l l y c o n s t a n t , t h e b e h a v i o r o f t h e f i e l d l i n e s on t h e
s u r f a c e S i s l a r g e l y d e t e r m i n e d by Ê (A) . D e n o t i n g t h e d i s t a n c e
a l o n g a f i e l d l i n e by t h e symbol s , t h e e q u a t i o n o f a f i e l d l i n e
i s g i v e n by
B x ds = 0 ,
whe r e ds = ( 0 , aedX , Rd $) ,
s o t h a t
RB dé P
aeB^ dX
We may now g e n e r a l i z e t h e d e f i n i t i o n ( 6 - 1 7 ) f o r t h e s a f e t y f a c t o r
o f t h e f i e l d l i n e s i n a t o r u s . L o c a l l y , t h e n o r m a l i z e d p i t c h o f
t h e f i e l d l i n e i s dO/dA = a e B^/RÖ , s o t h a t t h e o v e r a l l i n c r e a s e
i n <j> a f t e r o n e r e v o l u t i o n o f t h e
f i e l d l i n e t h e s h o r t way a r o u n d
t h e t o r u s i s g i v e n by
1 - e If'."']"1-. 2TT \ \ eB /
Aft 2n
1 I "A 2 7 t R i T d ^ J p
(12-27)
vaq
C l e a r l y , as k2 -» 0 , so t h a t Ê (IT) •* 0 , the i n t e g r a n d of Eq. (12-27)
blows up so t h a t q -*<*>. We w i l l s ee t h a t t h i s s i n g u l a r i t y i s a
very s e n s i t i v e func t i on of t h e t o t a l c u r r e n t 1 4 . A t i n y d e c r e a s e
of I . may cause the s a f e t y f a c t o r t o jump from a modest va lue to
i n f i n i t y . I t i s c l e a r t h a t such a p a t h o l o g i c a l dependence on param
e t e r s i s no t i n agreement w i t h the g l o b a l d e s c r i p t i o n of e q u i l i b
rium and s t a b i l i t y one has i n mind when a p p l y i n g i d e a l MHD t h e o r y .
In f a c t , i t i s n o t c l e a r a t a l l whv the r o t a t i o n a l t r a n s f o r m of
.264.
f ie ld l ines should play such an important role in the gross
MHD s t a b i l i t y of plasmas. The source of the confusion seems
to be c i rcular zero-6 l imi t :
2 aB 2ira B
° ° = q* , (12-28) 4 Rl RI P $
where we show the coincidence of two quan t i t i e s , one (g) mea
suring the rotational transform of f ie ld l i ne s , and the other
(q*) measuring the t o t a l toroidal current flowing in the plasma.
Since the external kink mode i s driven by the current , l e t us
generalize the l a t t e r def ini t ion to apply to a rb i t ra ry B and
non-circular cross-sect ions in the high-B tokamak ordering:
r = ( j A d o P = I B . di, = a e \ B dA ,
so that
n * = a L tx
[+f B ( A ) - P . - d x
eB o 1 - R I A 'u [+f B ( A ) - P . - d x
eB o
-1 (12-29) f'
This definit ion of the fundamental parameter measuring the to ta l
current immediately cures the defect of the or ig ina l definit ion
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 a t low 6 and non-circular
cross-sect ions . This def ini t ion of q* turns out to be the be t t e r
choice when describing kink-mode s t a b i l i t y .
Next, l e t us measure the plasma g in uni ts of q*. This
leads to the defini t ion of the poloidal 6:
. 2 C 5 .
eS
2TT I p
* 2 (12-30)
From t h e Eqs . (12-26) and (12-29) t h e s i g n i f i c a n c e of t h e paran i
e t e r y' i s now seen t o be j u s t a n o t h e r way of f i x i n g E S D :
i , 2 r i p. *
The c r i t i c a l va lue of t h e p o l o i d a l B i s r e a c h e d when k2 = 1 :
••» > - 4 - r - r - l V1" p , c r i t 2 L 2TT J V
+ x ( X ) dX ! o J
1-2 (12-32)
which i s seen t o be a s imp le f u n c t i o n of t h e c r o s s - s e c t i o n a l o n e
which may e a s i l y be c a l c u l a t e d f o r d i f f e r e n t c h o i c e s o f the c r o s s -
s e c t i o n :
b / a = 1 b / a
O .617
•10 . 617
.617
.673
1 .422
1.18
.360
2 . 2 5
C l e a r l y , as f a r as e q u i l i b r i u m i s c o n c e r n e d , a t r i a n g u l a r l y
shaped plasma p o i n t i n g away from t h e major a x i s of t he t o r u s i s
a b o u t the b a s t cho i ce f o r t h e c r o s s - s e c t i o n .
. 2 6 6 .
L e t us f o r m a l i z e t h e dependence on p a r a m e t e r s one
s t e p f u r t h e r by i n t r o d u c i n g a n o r m a l i z e d v a r i a b l e o f a v e r a g e
v a l u e u n i t y t h a t d e s c r i b e s t h e v a r i a t i o n of t h e p o l o i d a l
f i e l d :
1 - I k 2 [ l - x (X)]
C (X) = P
q*B (A)
E B rJl\ll-h*ll- " . ( 1 |1" (12-33)
so that
2^fV»dl = 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,
different expressions may be
evaluated in terms of the -«vk
complete el l ipt ic integrals
of the first and second kind:
where x^ = cos 9, the o 6UV") *
W i
2 it'
1 d8 " "I Ktfc2)
Vl-k 2 sin2 I e' 1 + I k 2 +
4 * »
hS-f1 . v - k2 s i n 2 j - 6 d9 * - | E ( k 2 ) 1 - i k 2 +
4
Hence:
eg - ! > k / 4 E ( k 2 ) ] 2 , P u
eg . - IT2 /16 % .617 , p , c r i t ^ '
(12-34)
( 1 2 - 3 5)
.267,
b (9) P
[ > / 2 E ( k 2 ) ] 'W 1 - k 2 s i n 2 ~ 6 , (12-36)
q = q* . 4 E (k 2 ) K ( k 2 ) / * " . (12-37)
Th i s g ive s t h e f o l l o w i n g p i c t u r e s :
*-0 t t an
poloidal field
!>7,(.
o » a o
curves of constant q curves of constant q
C. VACUUM FIELD SOLUTION FOR THE CIRCLE
We d i d n o t pay any a t t e n t i o n y e t t o t he s o l u t i o n s
of the p a r t i a l d i f f e r e n t i a l equat ions for t h e vacuum because
t h e s e s o l u t i o n s a re n o t needed i n t h e s t a b i l i t y a n a l y s i s . Never
t h e l e s s , they a r e of i n t e r e s t by t h e m s e l v e s . N o t i c e t h a t the
e q u a t i o n s (12-20) a r e of t h e type (12-8) so t h a t v/e nay i n t r o d u c e
a harmonic p o t e n t i a l $ from which the n o r m a l i z e d p o l o i d a l f i e l d
2 - v e c t o r 6 i n the vacuum may be d e r i v e d :
b . 3 qft B , / e B
satisfying
( 1 2 - 3 8 )
K = ° c * . 6 - 0 , so t h a t
,. j . (12-39)
K • - 7** • (12-40)
. 2 6 8 .
where _ v A x i> = 0 on o . (12-41)
The l a t t e r e q u a t i o n h a s t o be s o l v e d s u b j e c t t o t h e boundary cond i t i o n s
* = 0 , | i - = S (A) on C , (12-42)
where b U) i s g iven by Eq. ( 1 2 - 3 3 ) .
T h i s p rob lem i s i l l - p o s e d : Eq. (12-41) i s an e l l i p t i c p a r t i a l
d i f f e r e n t i a l e q u a t i o n and t h e boundary c o n d i t i o n s (12-42)
s p e c i f y i n g bo th f u n c t i o n and normal d e r i v a t i v e a r e o f t h e
Cauchy t y p e . C o n s e q u e n t l y , un ique : •.;;» and c o n t i n u o u s dependence
on boundary d a t a i s n o t g u a r a n t e e d . T h i s p r o b l e m i s c o n n e c t e d
w i t h t h e o c c u r r e n c e of a s e p a r a t r i x i n t h e vacuum beyond which
t h e s o l u t i o n f a i l s t o be u n i q u e l y d e t e r m i n e d . However, i f e6
i s s m a l l enough so t h a t t h e s e p a r a t r i x i s f a r away, we s t i l l
may o b t a i n s o l u t i o n s i n a l a r g e r e g i o n .
For a c i r c u l a r c r o s s - s e c t i o n t he above e q u i l i b r i u m
prob lem may be s o l v e d e x p l i c i t l y by means of a n a l y t i c continuation
of t he boundary d a t a ( see Ref. 2) . We have to s o l v e
± J_ r JJL + _J_ _if*_ = o (12-43) V do
s u b j e c t t o t he boundary d a t a on r = 1
* - 0 , S e ( 9 ) = - | ^ = [ir/2 E ( k 2 ) ] y 1 - k2 s i n 2 ~ 8 . (12-44)
N o t i c e t h a t we c o n s i d e r n o r m a l i z e d r a d i i h e r e : r / a -»• r . L e t us
now Four ie r -decompose t h e p o l o i d a l f i e l d a t t h e bounda ry :
CO
B e (e ) - 1 + j P tm cos m 8 , (12-45) tn=l
where
t - — • 6 <e) cos me de. m IT ) p
.269.
The solution to the problem (12-43), (12-44) can then be
writ ten down immediately :
Mr ,6) " i n r + YL üm ~ — ( r m ~ r m ) c o s n ! e . (12-46) m=l
Although this ser ies solution formally solves the problem, i t
i s actually of l i t t l e use because i t turns out to converge
l ike a sna i l .
A much be t te r representation of the solution i s ob
tained as follows. Observe that both sides of the boundary
condition
_*L\ = 1 ^ \ t c o s m 0 = s - y ' 1 v * — v
k=\ ™ 2 E ( k 2 ) (r ^ ) . = 1 + P t cos m0 = 2 _ _ Wl - 4 - ^ + - 5 - ^ cos9
are analyt ic functions of 6. They remain so when 8 i s 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
2E(k<) o r
I o I o r . 1
1 - -„- k*- + j k'-[cos 6 cos ( i v.n r ) + sin 0 s i n ( i -.n r ) j ,
\ ' 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 + - ) r o s e + i ( r - — ) s i n ö 1 . 2 E ( k " ) W
Hence, the real part of this expression gives the poloidal
field in the entire plane:
. 2 7 0 .
' E e < ' - 9 » " ' T T " l
CO
) 1 , m - m . + Z_ i " ^ ^ + r ) c o s m 9
m=l
2 E (k ) R e V' A + iB =
2 v/? E (k ) 'A + f 2
+ B2
(12-47)
where
i - j 1 2 1 1 2 1 A = 1 - -y- k + - i - k (r + -J-)cose , B = -±-k ( r - — ) s i n e .
We cou ld i n t e g r a t e Eq. (12-4 7) once more t o o b t a i n
t h e f l u x f u n c t i o n ty i t s e l f . T h i s y i e l d s a s o l u t i o n i n t e rms
of i n c o m p l e t e e l l i p t i c i n t e g r a l s . However, we h e r e wish t o
o b t a i n on ly a s p e c i f i c d e t a i l o f t h i s s o l u t i o n , v i z . t h e
p o s i t i o n x = x of t he s t a g n a t i o n p o i n t of t h e s e p a r a t r i x .
At t h i s p o i n t £ ->- 0. The s t a g n a t i o n p o i n t i s e x p e c t e d t o
o c c u r on the i n s i d e of t h e t o r u s , i . e . f o r 8 = IT, so t h a t
we look f o r z e ro s of t h e f u n c t i o n
r b e ( r , 6 = ^) - [TT/2 E (k 2)] y i - k 2 / n ( r ) , n ( r ) = 4r / (r + 1 ) 2 .
(12-48)
Hence, t h e p o s i t i o n of t h e s t a g n a t i o n
p o i n t i s g iven by
k" = n ( r s > ' o r , i n te rms o f eg /
eB. 2 n ( r )
TT S
16 E ( n ( r g ) )
(12-49)
( 1 2 - 4 9 ) '
For s m a l l v a l u e s of e6 t h e s t a g n a t i o n
p o i n t i s f a r o u t so t h a t r >> 1 and s
<(({<"è(((({ti\ i
n ( r ) << 1 . T h i s g i v e s - .
- 1 r % (eB )
s p ( 1 2 - 5 0 )
. 2 7 1 .
F o r l a r g e v a l u e s o f cB , when k 2 •+ l f t h e s e p a r a t r i x h i t s t h e
p l a s m a s u r f a c e s o t h a t r = 1 a n d n = 1 . T h i s g i v e s :
e 3 . = T T 2 / 1 6 = . 6 1 7 . ( 1 2 - 5 1 ) P . c r i t
Now t h a t we h a v e o b t a i n e d t h e e x a c t s o l u t i o n i t i s
i n t e r e s t i n g t o r e t u r n t o E q . (12-4 6) t o i n v e s t i g a t e t h e c o n v e r
g e n c e o f t h a t s e r i e s . One may c a l c u l a t e a l l t h e c o e f f i c i e n t s
t e x p l i c i t l y by r e c u r s i o n . F o r t h e f i r s t c o e f f i c i e n t we g e t
t . - - L u - ^ 4 • * - L ^ s i J L i i ^ t - i . ^ , , , , . <12-52) 1 J '- k E ( k ) P
L i k e w i s e , t - ^ (E0 ) 2 , t^ -\. (EB ) 3 , e t c . 2 p 3 p
Th i s seems t o i n d i c a t e t h a t we may expand t h e f l u x f u n c t i o n i n
t e r n s of E8 : P
H r , e ) % Unr + y E B (r - — ) c o s O , (12-53)
which i s t he s o l u t i o n which appea red in Ref. 4 . Hence, we o b t a i n
to l e a d i n g o r d e r
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 a s f a r as the c o r r e c t s o l u t i o n of Eq. ( 1 2 - i O ) . The
reason t h a t t h i s r e s u l t i s wrong i s the f o l l o w i n g . The c o e f f i
c i e n t s t ^ ( E 3 ) ' , whereas rm ^ (:6 ) ~ m , so t h a t the>re i s no m P S P
j u s t i f i c a t i o n f o r t he n e g l e c t of the h i g h e r o r d e r t e r m s . They
. 2 7 2 .
just provide a series of al ternating 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 interest ing 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 c o s 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 i s due to the surface
currents at r = 1, i s 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 inf in i te series of multipole
currents of strength t situated at r = 0 and a similar potential m
ifi of an infini te series of multipole currents of the same strength
at r = ». The joint effect of these multipole currents i s 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|; i s a n ^ y t i c for r > 1 and can be represented
in an integral form by means of Po^sson's integral formula. The
field tji however, i s 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 c i r c l e . This is not possible
however, since the unit circle is densely covered with singular
i t i e s 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.
- 2 7 3 .
F i n a l l v , i t i s i n t e r e s t i n g to n o t i c e t h e ana logy of
t he appea rance of a s t a g n a t i o n p o i n t i n t h e s o l u t i o n of the
f l u x f u n c t i o n w i t h t h e s i m i l a r phenomenon i n hydrodynamics
known as t he Magnus e f f e c t (see, e . g . , Ref. 1, p.423}. H e r e , t h e
flow p roduced by a c y l i n d e r which i s r o t a t i n g w i t h a n g u l a r
v e l o c i t y *: and which i s s i t u a t e d i n a s t e a d y flow of v e l o c i t y
v a t i n f i n i t y i s r e p r e s e n t e d by t h e s t r e a m f u n c t i o n
iji = - K9.nr - v ( r — )cos9 . (12-56)
*
For an a n g u l a r v e l o c i t y K = v a s e p a r a t r i x a p p e a r s a t t h e
r o t a t i n g c y l i n d e r . For '. > v t h i s s e p a r a t r i x moves away from
the c y l i n d e r ; i d we g e t a s i m i l a r topo logy of t h e s t r e a m
f u n c t i o n as t h e f l u x f u n c t i o n i n i d e a l MHD. H e r e , •= e 6D
w o u l c *
c o r r e s p o n d w i t h V/K i f t ^ e e q u a t i o n (12-54) were c o r r e c t . At
h igh 3 t he a d d i t i o n a l terms i n t he f l ux f u n c t i o n (12-46) p r o
duce a t o p o l o g y t h a t i s q u a l i t a t i v e l y d i f f e r e n t from \e.
Magnus flow p a t t e r n f o r K <. v .
D. VARIATIONAL PRINCIPLE FOR STABILITY
The s t a b i l i t y of the sha rp -bounda ry c o n f i g u r a t i o n
w i l l be i n v e s t i g a t e d by means of t he R a y l e i g u - R i t z v a r i a t i o n a l
p r i n c i p l e ( 9 - 1 6 ) :
- 2 7 4 .
2 -K, U =
* p : $ : + «s ; y + m i : •:
(12-57)
where
W . - ,
E ) 2 dS ,
\ V + T p ( ^ - 0 2 1 dxP , Q = 7x(Ex B),
Ws r£ 1 . J - ( „ . "v J - B 2 T 1 (n .E
WV [ j ! - f ( f J T " , (12-58)
The variables £ and Q a re connected by means of the boundary
condi t ion (8-29) on S:
n . Q - B . 7 (a . E) - (n . VB . n ) n . E , ( 1 2 - 59)
whereas fi.Q should vanish a t the conduct ing w a l l , i . e . a t
i n f i n i t y in t h i s c a s e .
Since the system i s a x i a l l y symmetric we may Four ie r -
decompose % i n t o independent components £(R,Z)e . From now
on we w i l l e x c l u s i v e l y s tudy perturbations of t h i s form wi thout
i n d i c a t i n g t h i s by fu r the r s u b s c r i p t s . The case l = 0 r equ i r e s
s epa ra t e t r e a t m e n t . I t w i l l be excluded from the p r e s e n t ana lys i s ,
We w i l l e x p l o i t the high-g tokamak o rde r ing to minimize
the express ion ( i l -57 ) for w2 o rde r by o r d e r . This way we . / i l l
e l im ina t e the l o n g i t u d i n a l components £ and Q and ob t a in a 9 $
problem in terms of the t r a n s v e r s e 2 -vec to r s E j_ and Q x only .
The connection between the 2 -vec to r s £ ^and 0^ i s
obta ined from the boundary cond i t ion (12-59) . Since n . 7B . n - - l / (eaR) *(RB )/3X »
. 275 .
t h i s condi t ion may be wr i t t en
u B 3 ( R B )
•ï1- $ R a e 3 A. % x £ A eaR 3 * T. 'i/
Explo i t ing the normalized po lo ida l f i e l d va r i ab l e B defined
in Eq. (12-33) t h i s gives to l ead ing o rder :
- i ( -p=- }n. . Q. = (Lq* - ~ — b ) -^- n . £. . (12-60) eB
Since p only appears in W we may separately minimize
ma-with r e spec t to Q. One should tlien keep n.Q fixed a t the p l a s
vacuum boundary, so t h a t n .£ i s f ixed and, hence, 6 W- = 5W = 0 .
In t roduc ing the vec tor p o t e n t i a l £ so t h a t Q = VzA, we f ind
by a reduct ion s imi l a r to t h a t of Sec . VIII D, Eq. (8-47) , but
in reverse o rde r :
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 ^ o-j y
if.Q , (12-63)
.276.
From these equations and Eq. (12-60) i t emerges tha t i t i s
expedient to exploit the following dimensionless var iables :
( q * / e 2 B ) 6 \ -\, ( q * / e B ) Q , -x. ( 1 / a ) 5, • ( 1 2 - 6 4 ) 0 9 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 ?:
(q*/e2Bn)Q. - - i V. (12-67)
Hence, the vacuum energy may be written as
wV - " * « $ * * • 51<V** + l P • 8 * ) d x • (12~68)
where Q and QA are related to the potentials 1 and y by Eqs.
(12.66) and (12-67) .
Next, let us reduce the expression for the surface
energy. By means of the definitions (12-17) for the principal
curvatures < and K of the surface S and by using the equilib
rium jump equation (12-19) we may write
so that
. 2 7 7 .
n • Et V 4 - B2 H = - r B 2 ( K - €K ) + 2 c p r 7 / a 'v "• Z -1- " P P C t -
- (B K + 2cpK ) / a . P P t
Hence,
Ws = - -Re t (B2K + 2epK J ( n , . ? ^ ) 2 d X . (12-69) J p p t i 1 %*-
This c l e a r l y shows a l l the important i n g r e d i e n t s act ing in
a global i n s t a b i l i t y . Below we w i l l give a d e t a i l e d d iscuss ion
of these terms.
We may now es t imate the var ious o rde r s of magnitude of
the con t r ibu t ions 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 i t s v
emerges t ha t the express ions W and W are of second o rder as
compared to the express ions Wp and I . Thi3 implies t h a t the
f i r s t two orders of W, depending on the p o s i t i v e d e f i n i t e
q u a d r a t i c forms Wp and I , have to cancel .if we are to wind up
with a s i g n i f i c a n t problem. Separat ing t r ansverse and l o n g i t u d i n a l dependences in Vr
and I ,
Wp = ^ - \ r B 2 ( V 1 . r - £_/R)2 + ( i V / l T H 2
+ Y P ( V , . £. + £D/R + i a s . / R ) 2 ! d t P , (12-70)
i f 2 2 D 1 = 'T- \ ° ^ * C>< iT » (12-7 1)
. 2 7 8 .
we obta in to zeroth order :
2 ( 0 ) H(o, 4 f . x - ^0)>2^ üi ' • - f^ = — 7 ^ 77T^ r ~ • ( 1 2 - 7 2 )
I (0 , 4 J ( l ( t ( 0 ) 2 , { ( 0 ) 2 ) d t p
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
W(D = ± j y p ( v 5(x0))2dxP (12-74)
vanishes trivially by virtue of the zeroth order result.
The first significant non-vanishing growth rate is indeed
obtained in second order:
w2<2> = [WP<
2> + WS(2) + W V ( 2 )] /I<°\ (12-75)
where
« p ( 2 ) - i - i L W ti l )-t i 0 >"" 2*« I»X't? ) 2 l*tp • (12-76)
and Ws and w ' are given by the express ions (12-69) and (21 (12-68), r e s p e c t i v e l y . Minimization of w2 ' with r e s p e c t t o
| ( 1 ) i s t r i v i a l :
7A . %(P = 4 ° ) / R • (12-77)
Since £ only appears in I , the maximum growth r a t e i s
obviously obta ined for
C ( 0 ) = 0 . ( 1 2 - 7 8 )
. 2 7 9 .
T h u s , . 2 „ 2
,P(2) . 1 12s- ( r ( ° ) 2 dtP, (12-79) 2 „ 2 J <v, l
V R
o
I ( 0 ) - i Aé0)2 d t P , (12-80)
so that we have obtained a problem in terms of the leading
(0) order transverse displacement E, , whereas both of the leading
order components of the vacuum field Q, for consisting of the
notation to be denoted as 5 k and Q' s t i l l appear.
Writing
2 2
a,2 = - r - £ - + W t W , (12-81) R P o ,
i t 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 x o r ! ^ :
(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 . l i , . HJJL, ' I , . 3* d» . r ^ j i -i I, „.42. d l „2 R J " 3v R ' J - ;>v p R c o
o ( 1 2 - 8 4 )
. 2 8 0 .
This completes the reduction of the problem to that of
calculat ing harmonic poten t ia l s x or Q for the plasma and
harmonic potent ia ls t and 5 for the vacuum.
Let us now eliminate a l l t r i v i a l scale factors by
introducing dimensionless variables
- 2 , 2 . 2 / 2 D 2 . 2 ( 2 ) to = (pa q * V e B O )Ü> ,
W = ( q * 2 / e 2 2 i r 2 ea 2 R B 2 ) W < 2 ) , ( 1 2 - 8 5 )
Ï i ( l / 2 i r 2 e a 4 R p ) I ( 0 ) . o
T h u s ,
Ü2 = W/T, (12-86)
where we have from Eq. (12-84)
WP - (Hq*)2I , I = J I M X* lv" dA ' (12-87)
from E q s . (12-69) and (12-83)
Ws
2TT J p p p t e 2 3X
and from Eqs . ( 1 2 - 6 8 ) , (12-66) and (12-67)
F i n a l l y , t h e boundary c o n d i t i o n (12-60) i n t e rms of t h e p o t e n t i a l s
r e a d s :
** /„ * i 9 C \ l x .
which may be integrated once:
. 2 8 1 .
D
$ • ?X. * = ~ i ( aq* /eB o )E-V £ J.q* - i - ^ - ^ . (12-89)
Using t h i s boundary c o n d i t i o n and t h e c o n j u g a t e r e l a t i o n s
(12-66) and i n t e g r a t i n g by p a r t s t h e e x p r e s s i o n fo r W may
be reduced as f o l l o w s :
T h i s comple t e s t h e r e d u c t i o n , where we have a u t o m a t i c a l l y
chosen f o r a d e s c r i p t i o n i n te rms of t he p o t e n t i a l s x a ^ d $
because of t h e s i m p l i c i t y of t h e boundary c o n d i t i o n ( 1 2 - 8 9 ) .
C o l l e c t i n g t h e Eqs . ( 1 2 - 8 6 ) - ( 1 2 - 9 0 ) we may now s t a t e
t h e s t a b i l i t y problem in a ve ry compact way: To second o r d e r
t h e growth r a t e s of e x t e r n a l k ink modes i n a s h a r p - b o u n d a r y
h i g h - b e t a tokamak o f a r b i t r a r y c r o s s - s e c t i o n a r e g iven by
{ j l q * ) 2 ( * |X dA . I I ( 6 2 K + e p K ) | ^ | 2d X _if* | i d x
_ M j A 3v e j p p p t 3X[ J 3v ( J J 2 = f
IX A, (12-91) J x * U " where
A l X * 0 on a F ,
A A « = 0 on V 0 ,
? = Px on c .
(12-92)
(12-93)
( 12 -9 4 )
The shape of t he c r o s s - s e c t i o n i s p r e s c r i b e d : x = x (A) , o o
y^ - v U ) . This determines the curvatures «• (X) and Kfc(A) o o p t
through Eq, (12-17). The only parameters for the stability
. 2 8 2 .
problem turn out to be c6 and Zq*. For the computations
an auxil iary parameter k 2 , running from 0 to 1, i s used. I t
i s related to E6 through Eq. (12-31). For a prescribed
cross-section and a given value of ES (k2) the poloidal
f ield 6 (X) i s found from Eq. (12-33). P
Notice that both the plasma energy and the vacuum
energy are pos i t ive de f in i t e , so tha t i n s t a b i l i t i e s may only
arise through the surface term. The f i r s t term of W i s nega
t ive def in i te for convex cross-sec t ions , whereas the second
term i s negative on the outside and posi t ive on the inside
cf the torua. The f i r s t term i s 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 sha l l see , one can ext rac t a similar contribution
from the f i r s t term tha t i s twice as large and of the same sign,
so tha t the ballooning term becomes three times more e f fec t ive .
I t remains to solve the Laplace equations (12-92) and
(12-93) ii t order to r e l a t e the normal derivatives v and * to
X and J on C. Once th i s has been done,the expression (12-91)
only contains l ine in tegra ls along C involving the unknown func
tions x(*) an<3 5(X) re la ted to each other through the boundary
condition (12-94), so tha t the final minimization of w2 i s one
with respect to x(*) only. That very l a s t par t of the problem
has to be carr ied out numerically.
To solve the Laplace equations (12-92) and (12-93) one
may resor t to two methods bas ica l ly . In the f i r s t method one
makes use of separable coordinates. This method i s only app l i
cable for a r e s t r i c t ed class of cross-sections, typical ly c i r
cular and elliptic ones. The second method employs Green' s theorem
. 2 8 3 .
r e l a t i n g a h a r m o n i c f u n c t i o n a n d i t s n o r m a l d e r i v a t i v e on
t h e b o u n d a r y c u r v e c , w h i c h may h a v e any s h a p e now. F o r
s i m p l i c i t y we c h o o s e f o r t h e f i r s t m e t h o d and t r e a t t h e c a s e
o f a c i r c u l a r c r o s s - s e c t i o n . F o r t h e g e n e r a l m e thod t h e r e a d e r
i s r e f e r r e d t o t h e l i t e r a t u r e C R e f s . 6 and 7 ) .
E . NUMERICAL SOLUTION FOR CIRCULAR CR0S5-SECTIGNS
The s t a b i l i t y p r o b l e m i s t r e a t e d by means o f a
F o u r i e r a n a l y s i s i n t h e a n g l e A g i v i n g r i s e t o F o u r i e r com
p o n e n t s e x p (imX) w h i c h a r e c o u p l e d due t o t h e a n g u l a r v a r i a
t i o n o f t h e p o l o i d a l f i e l d B (A) a n d t h r o u g h t h e a n g u l a r v a r i a
t i o n o f t h e c u r v a t u r e s K (A) a n d < {A) . The mode c o u p l i n g
t h r o u g h B„(A) o r i g i n a t e s b o t h f rom t h e s u r f a c e t e r m W a n d
f rom t h e vacuum t e r m W t h r o u g h t h e b o u n d a r y c o n d i t i o n (12-94) .
L e t u s now c o m p l e t e t h e s o l u t i o n by s p e c i f y i n g t h e
s h a p e o f t h e c r o s s - s e c t i o n t o b e c i r c u l a r , s o t h a t
v - r , A « 9 ,
x " cose , y = s ine , e = 1 , ( 1 2 - 9 5) o o
< - l , < - cose . p t
The n o l o i d a l f i e l d i s d e t e r m i n e d a s i n E q . ( 1 2 - 4 5 ) : ( 1 2 - 9 6 )
a (?>) = ( f / 2 £ f ! ; 2 ) ) \ l ~ k 2 s i n 2 -f" '•> - 1 + Y"^ t cos P ' V 2 f r ? 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 P P P t p p
ir 2 '1 - k2 s i n 2 h e) + (1 - k^ s in ' 1 . - 8) + EB COS8
4 E 2 ( k 2 ) 2 p
( l - i k 2 ) + 3e6 cosO , (12-97) 4 E 2 ( k 2 ) 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 i t will couple al l 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:
X(r,6) - Yl' xm |mf 1 / 2 r l m l e X p ( i m e ) , m = ~" (12-98)
* ( r , 8 ) - ) 1 s g ( m ) [ r a f 1 / 2 r ~ ' m I e x p ( i m 8 ) , • * 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) :
- 2 ï » 5 .
^m s g ( m ) |mt e = [Aq* - i ( l + \ t M cos MG)i \ x„ ; Uj t
Applying the operator \de» i m sg(r.) jm| ' on both sides of this
equation gives the wanted relation between | and x:
Ó = F-x , (12 -99)
vrnere
P = l q * s g ( m ) 6 m M + \ s g ( m i i ) | m u | 1 / 2 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 = 2 T $ X * Tr" d e " Ë ' xm = r * • (12-100)
f *+ OB
*v s - h ** "^ d9 = C ' K - H • (12-ioD The surface energy becomes
ws = - ±[ (S2* + ,e <JlfVde 2 7i j p p p t ' i r '
= 1 L ' : V C • s S ( i n u ) ! m u | 1 / 2 . - 4 1 ( 6 2 + s8 c o s ö ) e " i < 1 B " u ) 9 d G m t 2* J P V
- !5 ' ^ S ' x . (12-102)
Writing the eigenvalue oroblem as
— , (12-103)
+ -. • -•
•i, 0
we find for the explicit form of the matrix W:
. 2 8 6 .
W = U q * ) 2 ö - ^ s g d n - ^ l m p l 1 2 [ ( l - y k2) * •+ 3tS i , , ] mu trip 2 ^ <£ 2 £ 2 ( k 2 ) P ' ' '"'
co
+ 51 * i * i • f—' mm ray
( 1 2 - 1 0 4 )
S i n c e t he norm i s r e p r e s e n t e d by t h e u n i t m a t r i x , t h e e i g e n
v a l u e s of t h e r e a l symmetr ic m a t r i x J[J a r e t h e r e q u i r e d e i g e n -
v a l u e s w2 of t h e v a r i a t i o n a l p rob lem ( 1 2 - 9 1 ) . T r u n c a t i n g t h e
r e p r e s e n t a t i o n t o some r e a s o n a b l e number of h a r m o n i c s t h e n
l e a d s t o t h e comple t e ly s t a n d a r d n u m e r i c a l p rob lem of c a l
c u l a t i n g t h e e i g e n v a l u e s of a r e a l symmetic m a t r i x . The low
e s t e i g e n v a l u e d e t e r m i n e s t h e s t a b i l i t y o f t h e s y s t e m . Mar
g i n a l e i g e n v a l u e s a r e found by f i x i n g e 8 and s c a n n i n g i n
t h e p a r a m e t e r i q * u n t i l u 2 = 0 t o a s u f f i c i e n t d e g r e e of
a c c u r a c y . A l t e r n a t i v e l y , we may pe r fo rm a o - s t a b i l i t y a n a l
y s i s by t h e same t e c h n i q u e e x c e p t t h a t we now scan u n t i l
w2 = -a2 t o a s u f f c i e n t deg ree of a c c u r a c y .
In t h e l i m i t e 8 •* 0 t h e mode compl ing d i s a p p e a r s
and we o b t a i n from Eq. ( 1 2 - 1 0 4 ) :
W + [ < i q * ) 2 - |m| + (ra • * q * ) 2 ] « » mp W M
(12-105)
i n p r e c i s e agreement w i t h t he low-0 s t r a i g h t c y l i n d r i c a l
r e s u l t of Eq . ( 1 1 - 7 4 ) . For f i n i t e eB n u m e r i c a l s o l u t i o n i s t h e
on ly way t o g e t s o l u t i o n s of Eqs . (12-103) and ( 1 2 - 1 0 4 ) .
The r e s u l t s of a m a r g i n a l a n a l y s i s a r e a s shown be low:
(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 s t r u c t u r e of t h e m a r g i n a l mode, which i s m = 1 a t
eö - 0 , i s p r e d o m i n a n t l y m = 2 f o r h i g h e r v a l u e s of e 6 .
The optimum v a l u e of q* g iven above i s c o n s i d e r a b l y
l a r g e r than t h e low-0 s t a b i l i t y l i m i t q* = 1. Moreover , the
r e s u l t i n g v a l u e of 3/e i s q u i t e sma l l i n d i c a t i n g a need t o
improve t h e c o n f i g u r a t i o n . Th i s can be done by adding a I n y e r
of f o r c e - f r e e c u r r e n t s o u t s i d e t h e main plasma c o r e and by
s h a p i n g t h e c r o s s - s e c t i o n . These effects a r e c o n s i d e r e d i n d e t a i l
in Ref. 7 .
RT.rCRENCES
1. G.K. B a t c h e i o r , An I n t r o d u c t i o n to F l u i d Mechanics
(Cambridge U n i v e r s i t y P r e s s , London, 196 7).
2, P.. Gajev/ski , P h y s . F l u i d s 15_ (1972) 70 ,
"Mngnetohydrodynamic e q u i l i b r i u m of an e l l i p t i c plasma c y l i
. 2 8 8 .
W.A. Newcomb, Ann. Phys. (NY) jH_ (1973) 231,
"Gyroscop ic -quas ie las t i c f l u i d sys tems" .
J . P . F ie idberg and F.A. Haas, Phys. F lu ids 1£ (1973) 1909.
"Kink i n s t a b i l i t i e s i n a high-B tokamak".
J . P . Freidberg and F.A. Kaas, Phys. F lu ids 17_ (1974) 440.
"Kink i n s t a b i l i t i e s in a high-B tokamak wi th e l l i p t i c
c r o s s - s e c t i o n " .
J . P . Freidberg and W. Grossmann, Phys. F lu ids 18 (1975) 1494.
"Magnetohydro dynamic s t a b i l i t y of a sharp boundary model
of tokamak".
D.A. D ' l p p o l i t o , J . P . F re idbe rg , J . P . Goedbloed, and
J . Rem, Rijnhuizen Report 78-108 (1978),
Phys. F l u i d s . 21 (1978) 1600,
"High-beta tokamaks surrounded by fo r ce - f r ee f i e l d s " .
VECTOR IDENTITIES .289,
a* (fe*£) - £* (a*b) - fa* <£xfc)
a x (b^c) = a - c b - a - b c , (axb) x c = a ' c b - b ' c a
V x V(p = 0
V • (Vxè) = 0
V x (Vxa) = 77 • a - Aa
v • («a) = a* V*-MV • a
V x ((^a) = V$ x a + $7 x a
a x (V*b_) = (Vb) • a - a • 7b
(axV) x b = (Vfe) • a . - aV- fe
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
v(a-fe) = (?a> • fe + (vbj • a s a* vfe + fe. va + a x (7xbj + &X (7xê) uo)
v • (afe) s 'a • vfe + fev • a
V • (axb) = b • Vxa- a • 7*b
7 x (Sxbj = 7 • (fea - afe) = av • fe + te - ^a - fev • a - a • fe
rrr xr 7 • adT - <R a • tt<icr (Gauss)
a - a x £ s If ?xadT = J a x a ^
a - <J>£ s I W T - O <j)&do where £ i s a c o n s t a n t v e c t o r .
a = ^7i|i - iV<|> : ($&ty - i|iA$)dT = O (<j>7i|; - i|»V$) • &da (Green)
(Vxa) • nda » i a - d u (s tokes)
(11)
(12)
(13)
(14)
(15)
(16)
a- a*£ ! J, (8*V) x a do » <P d i x a. JJ
a * *£ J II axV$ da * p <td&
(17)
(18)
(19)
(20)