Jay Kappraff.

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Courant Institute of
Mathematical Sciences
Magneto-Fluid Dynamics Division

The Stability of a Class of
Bifurcated, Magnetohydrodynamic
Free Boundary Equilibria

Jay Kappraff

AEG Research and Development Report

Plasma Physics
June 1974

New York University


New York University

Courant Institute of Mathematical Sciences

Magneto-Fluid Dynamics Division

MF-79 COO-3077-60


Jay Kappraff
June 1974

U. S. Atomic Energy Commission
Contract No. AT(ll-l)-3077




I. Introduction ^

II. Equilibrium °

III. Stability 1^

IV. Bifurcated Equilibrium 35

V. Bifurcated Stability ^1

Appendix 53

References ™


One model for studying the dynamics of a contained
plasma is called ideal magnetohydrodynamics . The plasma is
assumed to be described by pressure, velocity, and density.
The plasma is assumed to be in local thermodynamic equilib-
rium, and to be governed by the laws of conservation of mass,
energy, and momentum. Lorentz forces, J x B, are present
but no electric forces are included. Forces arising from
the electric field are ignored since density is assumed to
be zero in a plasma. Finally, the plasma is assumed to be

perfectly conducting, i.e., E + u x b = 0, so that B is de—

% 'V 'b a.

termined by the Faraday law, B, = -V x e = Vx(u x b) . In

%t ^ % ^ % %

this model, displacement current is ignored and J = V x b.

'\, % '\^

In this paper, we study the equilibrium and stability
of a system consisting of a plasma column of length H, ap-
proximately a cylinder of radius r , surrounded by a vacuum
region, all enclosed within an outer wall approximately a
perfect conducting cylinder of radius, pr . Vacuum magnetic
fields B"'"^ and B°^ are present both in the plasma and
vacuum regions, while surface currents are present on the
plasma vacuum interface.

In equilibrium, the stress on the plasma vacuum inter-
face from the plasma equals the stress from the external
magnetic field. In this study, the plasma fluid pressure, P;

Is constant, characterized by a dimenslonless parameter, B,

referred to as the plasma beta and defined as the

ratio of the plasma fluid pressure to the total

pressure both fluid and magnetic. Furthermore,

we assume that the limiting values of B and

B° on the plasma vacuum Interface are tangential to the

surface, i.e. n • B = 0. Thus the condition for equilibrium

is [p + ^ B ] = 0, where [ ] denotes the magnitude of the
discontinuity of a function across a surface. This equilib-
rium is called a free boundary equilibrium.

We study the stability of this equilibrium by following

the analysis of Bernstein, et al. (Ref. 1), modified by Lust,

et al. (Ref. 2). The dynamic equations of the plasma system

are linearized by expressing the velocity, pressure, and

magnetic field as the sum of their values at equilibrium,

u = 0, B , p , and their perturbed values, or first vari-
*^o ' %o' '^o' ^

ations, u-, , B, , p, , retaining only those terms linear in the

perturbations. When the Lagrangian displacement, ^, is

introduced, where u = 5, , the linearized equations can be

rewritten as a formally self-adjoint linear operator, L,

mapping E, into its second derivative, i.e. E, = LC- If this

equation is multiplied by E,. and Integrated over the entire


system, the result is an expression for conservation of
energy. The second variation of the potential energy, 6W,
is expressed here by the quadratic form-(C, L^) •

The system is called stable if C does not grow exponen—
tially in time in an L^ sense. If 5W > for all E, , then
the system is stable in this sense. The system is called
unstable if there exist C that grow exponentially in an L„
sense, which is the case if there exist ^ for which 6W < .

In analogy with the theory of linear operators, we
expect stationary states of 6W to occur for eigenvalues of
L. Thus, in keeping with the above definition, we will con-
sider the system stable if all eigenvalues are non^iegative ,
whereas the system is considered unstable if there exist
negative eigenvalues.

In this study, we will show that for a certain class of

equilibria characterized by p and other parameters, there

exists a value of 6 referred to as beta critical, 3 , such


that for all 6 _ L - 1 to insure
regularity. For our study we consider fields consisting of
a small quantity of L = 2 component whose ratio to the theta
pinch field is characterized by a small parameter c, and an

even smaller amount of L = 4 field, whose ratio to the theta

pinch field is characterized by Xc .

We assume long helical wavelengths, and introduce a

second small dimenslonless parameter, e, defined as the

plasma column radius times the helical wavenumber, i.e.,

e = kr . We are then able to facilitate the mathematical

analysis by expanding all geometrical and physical quantities
as power series in e and c, and carrying out a perturbation
analysis of these expansions, as first c and then c is made

small. First, e is made small enough to neglect all terms


smaller than order e . In the subsequent analysis, e is

considered fixed, leaving c as the remaining arbitrary, small

parameter. Then c is made small enough to retain only those
leading order terms of significance to the analysis which
follows .

Again, all terms past the leading order terms are
assumed to be negligible. We emphasize that although we
have not attempted to justify this assumption, our results
depend strongly on its validity.

As mentioned above, the plasma pressure is characterized
by the plasma beta. If the magnetic field in the plasma
region were constant, since the fluid pressure is constant,
the pressure would be effectively defined by specifying
3. However, in our system, the magnetic field is only
approximately constant, and correspondingly we define an
approximate, constant B, which merely characterizes rather

than defines the plasma pressure. For our study, 6 will be

considered small, of order c , appropriate to an experimental

device called a Stellarator.

The displacement, Cj is expanded in a Fourier series of
terms of the form c ^r"^ exp [KnG - £z)], where n and I
are the azimuthal and axial wavenumbers respectively, n takes
on only Integer values, m >_ n - 1 to insure regularity, and
the coefficients c . are expressed as power series in e
and c, as mentioned above.

For any value of n, and for all small betas, we will
show that there exists a family of unstable eigenstates.

These instabilities were also observed by Freidberg (Ref. 3)
at large beta, and are called "kink instabilities." Since
large values of n have not been observed in plasma systems,
we will show that kink instabilities can be omitted by
assuming n - kz being

the helical angle Introduced in the previous section. Thus

r, 0, and w are related to the cylindrical coordinates

r, (J), and z by ^ ^










+ z)

2 2 2

where a = 1 + k r .

Since, from Maxwell's equations, the fields are diver-
gence free, we can express the helically symmetric vacuum

field in terms of a flux function, ijj(r,0), such that

B-V(|; = (1)

and B = Vijjx^+Bcj , (2)

% a, a w

where in view of Maxwell's equation, J = V x B =

'\. % a,

B = ^
oj a

for C constant, and ijj satisfies

Explicitly, (2) becomes

B = i ^

B = - i^
^e a 3r

B = i
w a

For reasons described in the previous section, we wish

to expand B in the small parameter helical wavenumber, k. In

order to proceed non-dimensionally , we introduce the small
parameter, e, defined as

specified precisely later.

Following an idea of Weitzner (Ref. 7 ) , we scale r and
4; by setting r = rr and ijj = er i|j and taking c to be
independent of e so that

Br'?.') =t!f-^

B.(?,6) = -e ii- + 0(e3)
o ar

B,^ = c(l - e^ |-) + 0(eS
Equivalently , in cylindrical coordinates


B,(?,0) = £(C? - ^) + 0(e3) (4)

only, and r Is the position vector In this plane only. In
which case {k) and (5) hold, and the magnetic field,
expressed by (4), has the concise representation

B = z[c + e^(i'-Vii;-Cr-r)]+0(e^)-ez x [V(4j'- \ r-r)+0(£^)] (7)

In (7) J we may extend V to the derivative In the full
r,(f),z space, provided r Is still the position vector In the
r,(|) plane. While the various representations seem peculiar
at present, they are particularly appropriate to facilitate
the various transformations of coordinates which will be
described In detail at the end of this section.

A free boundary problem Is usually solved by giving the
external field sources and determining the plasma-vacuum
interface and fields. However, here we employ an inverse


procedure In which we specify a flux function for the plasma
region satisfying (5)- The flux function for the vacuum
region is then determined, satisfying (5) and the boundary
conditions for free boundary equilibria, namely, that the
flux function be constant at the plasma vacuum interface as
required by (1), and be chosen so as to achieve a pressure
balance across the interface.

This equation and boundary conditions constitutes a
Cauchy problem for which the Cauchy-Kowalewski theorem guar-
antees a solution for ij;, the plasma vacuum interface, and
the outer flux surface in some neighborhood of the plasma
vacuum interface as e approaches zero for small but finite c ,
In general, a singularity in the solution appears within the
vacuum region for finite c. However, as c approaches zero,
the singularity moves beyond the outer flux surface. We are
then able to justify our expansion techniques by employing a
known existence theorem for free boundary equilibria
(Ref. 8). Once we have solved the inverse problem, this
theorem can be used to solve the direct problem in which the
outer flux surface is taken as the initial surface for the
exact solution of the free boundary equilibrium and the
plasma vacuum interface determined. It is expected that
this interface will be close to the one computed for the
inverse problem.

We proceed to construct the equilibrium using this
inverse procedure.


We take the flux function for the plasma region to be
_2 _2 _^

(Jj^"^ = y( I ec |- cos 29 + \ec^ r cos ^19) (8)

which Is a solution of (5) with the constant y replacing ?.

For the subsequent analysis, 41 will be written without
primes .

From (7), we determine the field In the plasma:

B^"^ = y[z + ecr(sln 29r + cos 29$)

? -3 - (9)

- 4Xec r (sin 49r + cos 49(())]

The radius of the plasma-vacuum Interface is expanded

In a Fourier series:

r = 1 + a2 cos 29 + a^^ cos 49 + ag cos 69 + ... (10)

From the condition that ij;''"'^ be constant at the plasma

surface, we solve for a2, a^, and ag order by order In c.

The result is

-2 = 1^ ^It- ! ^)^^

a^^ = (^ - X)c^ (11)

For the vacuum region, since \p must also satisfy (5),

we can with all generality, set

_2 2 _-2

ijj^^ = C(?— +ec^ log r - Cp £cr cos 29 + c_2 ecr cos 29

p ^ p _-4

+ c^ ec r cos 49 + c_^ ec r cos 49 + ...) (12)

We take c^ = so that the net current in the ^ree
boundary is zero. The inclusion of c^ is possible, but
experience suggests that it leads to less stable equilibria.


Since the plasma pressure supported by the field, at
equilibrium, is just [ p— ], the plasma beta is

a - P _ [B^]

p + B^ ^" b2 °^t

and since, from (6)

[b2] = ^2 _ ^2 ^ Q(^2^^

6 = (1 - i^)'^) (1 + Oie^) .
Without serious error, we define

3=1- (^)^ . (13)

The remaining boundary conditions for free boundary
equilibria are,

,OUt/-v , ,

\li (r) = constant



[B ] = constant

at the plasma surface.

These conditions, along with (6), (10), (11), (12), and

(13) and the assumption that plasma beta, 3, is small of

order c , enable us to solve order by order in c for the

coefficients Cp, c „, c^y and c_2, of (12). After a long


= £ c^
^2 2 ~ T~


Cj, = \c + 0(c)




Replacing these coefficients in (7), we have for the
outer vacuum fields,
B°^^ = c (z+cer(sln 2er+cos 26$)- |^ er(sin 2er+cos 29^)

+ e^l f (sin 2er - cos 20$)-4Xc^ r (sin 4er+cos 4ei)

+ (higher order terms In e and c) . (15)

Thus, the inverse procedure has yielded the desired
vacuum magnetic fields.

In addition, we see from (10) that the equilibrium con-
figuration of the plasma surface is that of a cylinder with
small helical corrugations of width, order c, with respect
to the characteristic radius, r , of the cylinder.

We conclude this section by giving some useful geomet-
rical preliminaries, associated with the several coordinate
transformations Introduced earlier in this section.

We define rectangular coordinates corresponding to the
r, 9 coordinates defined in this section, as follows:

X = r cos 9 = — cos kz + ^ sin kz
o o

y = r sin 9 = — sin kz + ^ cos kz
^o ^o

and extend these coordinates from the x , y -plane to a

t I I

transformation of x,y,z-space to x ,y ,z -space as follows:

r X = X cos kz + y sin kz

r y = -X sin kz + y cos kz (l6 )


These relations emphasize that, by studying long
helical wavelengths, we are assuming slow z dependence
relative to x or y dependence. If we define the corre—
spending system of orthogonal vectors x , y , z as

X = X cos kz + y sin kz

y = -X sin kz + y cos kz , .

z = z

I I t

the X and y vectors are not Independent of z , In fact

9x' -' , 8y' ^'

— r = y and -S- = - x
8z 8z

so that when we Introduce the standard polar coordinates,

II ft

r , 9 , In the x , y plane, previously referred to as

r, 6, and the unit polar vectors r , 6 , we see that

^= ^=
3r 3r

/^ » . •

9r , §' _ 9r

9z' 90*

96* _ C;' _ 36*
r - - r - jT

9z 96

With the above vector identities, we may rewrite any

vector relation given in terms of x,y,z or r,4),z in terms of

t « » t t I

X ,y ,z or r ,6 ,z . For example, it is easy to show that

ip' 9 ^e' 9 ^ -\ ^ ^n

— - r — r + -r — r + ez ( — r r)

9r r 96 9z



'"^ '^ r 9 6 90 36* 9r 9z ae'

+ (1, (ilLlrl. 9t;')),- (19)

r 9r 96

» I I
If we specify a surface in r ,0 ,z -space by

« I t I

r = r (0 ,z ), then the parametric representation of this

surface is

r((l),z) = r(4),z)r + zz
or, ,

» f t T » 2 '^ '

r(e ,z ) = r^r (0 )r + r^ -— z
% o o e

9^ . ^^
Two tangent vectors to this surface are — p and — , ,

9z 90
so that a normal vector to the surface is:

- ^ _1 %^ ^ e_ ^ ^ j,'_ 0_ 9r ^ 9r^ ^' (20)

'^ rr96 ^o9z r98 9z

The element of surface area is:

I 9r ^r I t »
dS = |(-^ X ^)|d6 dz
90 9z

Thus, at the' plasma— vacuum interface:

dS = f^ d6'dz'r' \n\ [l + 0(e^)) (21)

It is easy to show that although the set of unit

t I T

vectors x ,y ,z differ from x,y,z, the corresponding sets
of polar vectors r ,0 ,z and r,(}),z are identical. Thus,
any vector equation written in terms of r,$,z, can be re-
written identically in terms of r ,6 ,z , and for the re-
mainder of this paper, the primes will be omitted, and all
vectors will be referred to the r,8,z system and the coor-
dinates referred to, without primes, as r,0,z. Thus,


equation (9) can be rewritten as:

B^*^ = YZ + ecr(sln 26? + cos 299)

(9 )
- 4Aec^r^(sin 46f + cos 499)

Also (10), (20), and (21) can be rewritten, dropping

terms of order, 0(c) and 0(e ), as:

r = 1 + I cos 26 + (j^ - A)c^cos 46 do')

n = r + (c sin 26 + c^(| - 4X) sin 466)+ 0(£)z (20)


r ,

dS = j^ |n| rd64)z (2l)

A final useful preliminary computation that follows

from (6), (8), (12), (l4), and (20) Is:

2 2

fi.ylO = ^2^2(- c_B (^_j,-3) _ cB(r+r"3) cos 29] (22)


[B^] = cec6(-r^ + r"^) cos 26 . (22*)



We now commence with the stability analysis described
in the Introduction, in which we study the second variation
of the potential energy. The second variation of the poten-
tial energy, 6W, consists of three terms, a volume integral,
6wP, over the plasma region; a surface integral, 5W , over

the plasma— vacuum interface; and another volume integral,

6W , over the vacuum region. The system is considered stable

if the sum of the three contributions is Dositive for all ad-
missible plasma displacements, 5(r,9,z), and disturbed scalar
potentials of the vacuum fields, $(r,0,2), where $ and C are

related at the interface, and t— = on the bounding outer


surface, assumed to be a perfect conductor.

For this study, as we mentioned above, we are
interested in determining a critical beta and a
corresponding, marginally stable state to serve as the
starting point for bifurcation. Thus, we need only
determine the lowest eigenvalues and associated eigen—
functions of 6W. Therefore, it is sufficient to determine
£ and $ so as to minimize the two volume integrals, holding
n • C fixed on the plasma vacuum interface. As a result, £
and $ satisfy Euler variational equations, which, in effect,
reduce the volume integrals to surface integrals over the
plasma vacuum interface.

More precisely,

6W = 6W^ + 6W^ + 5\i^


'6WP = I

dv { [rpCv-O^] + [Vx(? X B)f} , (23)

26W^ = 1 dS(n-5)^ n-V ^

= f dS(n-0^ n-y if-J- , (2i|)

26W^ = I dv(V$)2 , (25)

while on the Interface:

n-V$ = n-Vx(? X B°"^) (26)

where r In equation (23) Is the ratio of specific heats.

Provided that the Euler equations

= rpV(V-C) - Bx{Vx[Vx(C X B)]} , (27)

■AjO/a. 'x, '\i 'x, rx, i\,

= V-V$ (28)

are satisfied, we may rewrite equations (23) and (25) as

26WP = dS(n-C)[rpV-^ - B-Vx(5 x B)] (23)

26W^ = I dS$n'V$ (25)

The only negative term In 6W comes from equation (24)
[c.f. (23) - (25)]. In view of (22), we see that If C (and


n'C) are 0(1) In e, then unless all positive terms In equa—

tlons (23) and (25) are of order e or smaller, the system

Is automatically stable. In particular, we Infer that

V-C = 0(e) (29)


Vx(C X B) = 0(e) (30 )


If we expand ^ in a formal series In e
|(r,e,z) = _^Q(r,e,z) + e^(r,0,z) + £^^2^^'^'^) + ••• ^3l)

then, from equations (l8), (30), and (32):
9(re';(r,e,z)) 3?^

We may now rearrange terms among ^ and 5, In equation
(32), leaving C fixed so that

or the divergence of 5 restricted to the r,6 plane vanishes.
Equation (32 ) Is a more convenient form of equation (3 2).

Euler variational equation, (2 7), as

VJ^ = V[pr(y-C) - B-Q] = - Q-VB - B-VQ (33)

where. In view of equation (3 2 )

Vx(^ X B) = (B-V)^ - (^•V)B + O(e^) . (3^^)

We set

B = B + eB.

A, rv.O 'V-L

and find that

Q = Vx[(^^ X B^) + (F X eB,) + (eC. x B ) + 0{e^)-]


In view of equations (l8) and (32), to order e,
Q = ^(It - Ia)?. + e(B -V)?^ - (5^-V)eB^ - Ye(y-C.)i




- Q-VB - B-VQ = - e(Q^+Q, )-VB, - (B^+eB, ) • V (Q^+Q, )

+ 0(e3) + o(e^)S (35)

Since the latter expression is the gradient of a scalar
function, its curl must vanish. And, since the leading
order term in this expression is order e from equation (19);
the vanishing of the r and 6 components of the curl imply
that the coefficient of the z is of order e rather than e .

Thus, if we are interested only in the leading order terms


in e, all terms of order e are negligible. Since the lead-
ing order terms of equation (35) involve only the £ compo—
nent of £, we fix e at a value, small enough to neglect
terms of order e^ , and with no ambiguity, refer to E,^ as ?

ryjO %

for the remainder of this paper.

We proceed to expand | in terms of the other small
parameter, c. Since -B-VQ - Q'VB is the gradient of a
potential Q,, its curl vanishes, and from the z-component of

the curl, we solve order by order in c for the r,G compo—

r 9
nents C and 5 of E,. These components are 0(1) in e.

Again, from the vanishing of the r,e components of
Vx[-B'VQ - Q«VB], C^ is found to be 0(e) and hence contrib-

utes only terms of O(e^) to -B'VQ


not affect this analysis.

More precisely, we set

5 = £ + cS + c^Cp + •..

r\j r>^0 %1 '\,d.

? = ^o -^ ^^1 -^ ^% ^ ■'■ (36)

^ = ^o + ^'^1 -^ ^^^2 -^ •••
where the subscripts now refer to the order of c appearing
explicitly in the expression for E,, B, and Q.

In (36), B., 1 = 0,1,2 is given by (19), while from the
definition of Q,

Q. = I [(B. -V)?. . - (£. . •V)B ] (37)

1 3

Online LibraryJay KappraffThe stability of a class of bifurcated, magnetohydrodynamic free boundary equilibria → online text (page 1 of 3)