Stability of a Taylor-relaxed cylindrical plasma separated from the wall by a vacuum layer

(1)

Stability of a Taylor-relaxed cylindrical plasma separated from

the wall by a vacuum layer

Citation for published version (APA):

Schuurman, W., & Weenink, M. P. H. (1988). Stability of a Taylor-relaxed cylindrical plasma separated from the

wall by a vacuum layer. (EUT report. E, Fac. of Electrical Engineering; Vol. 88-E-207). Eindhoven University of

Technology.

Document status and date:

Published: 01/01/1988

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Stability of a Taylor-Relaxed

Cylindrical Plasma Separated

from the

Wall by a Vacuum Layer

by

W. Schuurman and M.P.H. Weenink

EUT Report 88-E-207 ISBN 90-6144-207-9 November 1988

(3)

ISSN 0167- 9708

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering Eindhoven The Netherlands

STABILITY OF A TAYLOR-RELAXED CYLINDRICAL PLASMA SEPARATED FROM THE WALL BY A VACUUM LAYER

by

w.

Schuurman

and

M.P.H.

Ween ink

EUT Report 88-E-207 ISBN 90-6144-207-9

Eindhoven

November 1988

(4)

association agreement of Euratom and the "13tichting Voor Fundamenteel

Onderzoek der Materie" (FOM) with financial support from the

"Nederlandse Organisatie voor Wetenschappelijk Onderzoek" (NWO)

and Euratom.

This report Was previously published as "Rijnhuizen Report 88-177".

CIP-GEGEVENS KONINKLIJKE B1BLIOTHEEK, DEN HAAG

Schuurman,

w.

Stability of a Taylor-relaxed cylindrical plasma separated from the wall by a vacuum layer / by W. Schuurrnan and M.P.H.

Wee~ink.

-

Eindhoven: Eindhoven University of Technology,

Faculty of Electrical Engineering. - Fig. - (EUT report,

ISSN 0167-9708; 88-E-207)

Met

lit.

opg., reg.

IS8N 90-6144-207-9

S1SO 535 UDC 537.5 NUG1 812

(5)

ABSTRACT

The minimum energy principle of Taylor is extended to the case where a vacuum layer is situated between the weakly resistive plasma and the per-fectly conducting wall. The second variation is investigated for periodic

perturbations (m and/or

k~O)

as well as for symmetric perturbations (m-k-O).

A necessary and sufficient criterion for stability is derived by means of

the Rayleigh ratio method and turns out to be identical with the result of

the resistive mode analysis. Stability diagrams

A

vs k are presented. A new instability region at m-I is found at negative values of k, already present when .\ is small. The symmetric mode becomes more unstable as the vacuum layer grows thicker.

Schuurman, W. and M.P.H. Weenink

STABILITY OF A TAYLOR-RELAXED CYLINDRICAL PLASMA SEPARATED FROM THE WALL

BY A VACUUM LAYER.

Faculty of Electrical Engineering, Eindhoven University of Technology,

The Netherlands, 1988.

EUT Report 88-E-207

Address of the authors:

Prof.dr. M.P.H. Weenink,

Faculty of Electrical Engineering,

Eindhoven University of Technology,

P.O. Box 513,

5600 MB

EINDHOVEN,

(6)

Contents 1. 2. 3. 4. 5. Abstract Introduction

The relaxation model

Derivation of the first and second variations Stability of the symmetric equilibrium

A) with respect to periodic perturbations

(m and/or k ~ 0)

B) with respect to symmetric perturbations

(m

=

k

=

0)

Discussion and numerical results Acknowledgement References Figures Appendix A Appendix B Appendix C i i i 1 3 5 10 10 15 20 23 24 26 37 38 40

(7)

1 . I NT[{ODUCT lUN

Taylor' 5 variational principle (1974) minimizes in the cylindrical

approximation the magnetic energy of a toroidal plasma column enclosed by a

perfectly conducting metal wall. The minimization is subj ected to the

constraints of constant

longitudi~al

flux and magnetic helicity K - JA-B dT

(Lagrange multiplier -

A/2).

These constraints are assumed to be valid on a

timescale intermediate between the ideal MHD timescale and the diffusion timescale. The minimizing magnetic field satisfies the Euler equation

v

X

B

AB ,

(1)

and its Bessel solution has among others the merit of correctly demon-strating the reversal of the longitudinal field component, as observed in reversed field pinch experiments.

A minor shortcoming of this theory is its inability to describe plasma

equilibria with a finite pressure, and equilibria with a current density

that falls off rapidly towards the metal wall. To the latter problem a

number of solutions have been proposed (ORTOLANI,

HASEGAWA,

SCHOENBERG

(1984». In some of these solutions, the role of a vacuum layer between the

plasma and the wall is acknowledged without, however, incorporating this

layer into the variational principle. KONDOH (1985) simulated the layer of

enhanced resistivity near the wall by generalizing the helicity invariant

to

ff(~)A.B

dr, where

~

is the poloida1 magnetic flux and

f(~)

a function

gradually decreasing from 1 at the centre of the plasma to 0 at the wall.

On the contrary, in our work f(~) steps down from 1 inside the plasma to 0

in a vacuum layer surrounding the plasma. This choice enables us to treat a

free-boundary problem, in which the position of this boundary is varied

along with the other quantities in the variational principle. The total

magnetic energy (plasma+vacuum) is minimized subject to the plasma helicity

and other constraints, necessary to let the occurring boundary terms in the principle vanish.

The first variation of the energy, put equal to zero, yields the

equilibrium states. This part of the analysis has recently been published

elsewhere (CENTEN et al., 1986) and is therefore dealt with rather briefly.

The accent lies on the full investigation of the stability of the

equi-librium states by means of the minimization of the second variation derived

(8)

In Section 2 the model is described, after which the first and second variation are derived in Section 3. The accessorial problem (minimization

of the second variation of the magnetic energy) is solved for periodic (m

and/or

k~O)

and symmetric modes

(m~k-O)

in Section 4. Finally, in Section 5,

(9)

2. THE RELAXATION MODEL

We consider a toroidal plasma surrounded

by

a vacuum layer and a

perfectly conducting metal wall with circular cross-section (radius b). The plasma boundary is allowed to move during the relaxation. In the variational principle, both the vector potential and the position of the plasma edge are varied and acquire stationary values at minimum total magnetic energy of the configuration. Constraints needed in this model are the conservation of the toroidal fluxes in the plasma and in the vacuum

region,

and of the magnetic helicity of the plasma. The plasma-vacuum

interface is assumed to be an ideal conductor, so that the peloidal vacuum

flux

is also unchanged during the

relaxation. The

latter invariant,

however, can be omitted if a surface current at the plasma boundary is absent.

The mathematical formulation becomes as follows. We minimize the total magnetic energy

W B B2

J

*

dr

V

0 P

+J

V v B2 V

dr

21' o

with respect to variations in V , A and A . V

P P v P

(2)

and V are the volumes of

v

and A

are the

p v

the plasma and the vacuum region, respectively,

corresponding magnetic vector potentials. The conserved quantities are:

I' K -

J

A B

dr ,

0

V P P magnetic plasma helicity

P

toroidal plasma flux

_1/>pt

J

B

· dS ,

S P

pp

1/>vt

J

B • dS ,

S

v

toroidal vacuum flux

vp

1/>vp

J

B

• dS .

S

v

poloidal vacuum flux

vt

where S , S

pp

vp

and Svt are the poloidal cross-section of the plasma,

(3)

(4)

(5)

(6)

the

poloidal cross-section of the vacuum, and the toroidal cross-section of the

vacuum (e.g. in the median plane), respectively.

(10)

The toroidal geometry calls for an additional constraint, the flux

through the hole of the torus (REIMAN, 1981) ,

B • dS

f

C

mt

A • de ,

v

(7)

where Sh is the cross-section of the hole of the torus and C is a contour

mt

on the metal wall encircling the hole.

Furthermore, there are a number of boundary condi tians. First I we

have the continuity of the magnetic pressure across the plasma boundary r :

p

1

2"

_o B2 p (r ) p B2 v (r ) p

The vector potential can be taken continuous across the boundary r : p

A

(r ) -

A

(r ) .

p p

v

p

(8)

(9)

Finally, since the plasma boundary and the metal surface are flux

surfaces, we have n (r ) p p n (r ) • B (r ) - 0 . m rn v m (10) (11)

(11)

3.

DERIVATION OF THE FIRST AND

SECONlVARIATICiNS--Of the four invariants defined in Eqs (3) - (6) needed in the

mini-mization of W

B, only the helicity K is taken account of by means of a

Lagrange multiplier (-

A/2).

The remaining three enter into the conditions

on the free and rigid boundaries. We therefore minimize the generalized energy functional

f

8 2

fA'

F

5

-E.. dT +

f

21"V dT -

A B

dT

(12)

21"

_V

_P P V 0 V 0 0

p

v

p

*

Note

that the

integrals

in Eq. (12)

have to be read as

f~

dT

etc.

21"

0

The vector variables A, B, r are decomposed into equilibrium parts A ( )'

P P v

B

p(v)

(index p indicating the plasma, index v the vacuum), aer , and the

perturbations

5A ( )' 5B ( )'

5a.

The latter vectors are Fourier-analysed in

p v

i(m8+kz)

Band z, and modes (m,k) proportional to e are considered separately. There is some freedom in the definition of the displacement oa of the plasma-vacuum boundary.

In

Appendix A, we prove that we may conveniently choose oa in the radial direction:

5 (8

_a _,z

)

_-

5

_ae

i(m8+kz)

_e

r (13)

In order to evaluate the first and second variation of F, we

deter-mine the energy difference LlF -

F[A+5A]-F[A] between the perturbed state

and the equilibrium state in toroidal geometry:

I" _o

t.F

-2rrR

2rr

r

J

z-O

8-0

a+oa(8,z)

J

r-O

A2 (A +5A )·(B +5B )}rdrd8dZ +

p

2rrR

2rr

+

r

f

b

f

(14)

Z-O

8-0 r-a+5a(8,z)

2rrR

2rr

r

I

z-o

8-0

a

IH

r-O

B2_

~

A .B }rdrd8dz

-p 2 P P

2rrRt 2rr

f f

z-O 8-0

b

f

1 8

2 rdrd8dz .

2

v

r-a

(12)

I,

The terms are re-arranged as follows:

a+oa(8,z)

+

f

a

b

{ B -oB

_p _p

- A (A -oB +B -oA )}rdr +

₂ _P _P _P _P

(oB )

2 P

A oA -oB }rdr +

2 P P

+

f

a+oa(8,z)

(B oB )rdr

-v v

f

t

B~

rdr +

f

t

(OB

V)2

rdr d8dz .

a+Oa(8,z)

b

I

a a

(15)

The first integral

I,

in Eq. (15) is partially integrated:

21rR

21r

r

f

z-O

8-0

a+oa(O, z)

f

o

5A -(VxB -AB )rdr +

p

a+5a(O,z)

+

f

o

a+5a(8,z)

f

5A -(VxB -AB )rdr+(a+5a(O,Z)){B

(a+5a(O,z))

p

pz

o

where the result that OA

p8

and 5Apz can be considered real has been

(13)

Functions of a+oa in the integrand are developed in Taylor series

around r=a. Derivatives like B' (a), A' (a) are eliminated with the aid of

pz

VxB - AB and B -

VxA,

and Egs (C5,6,lO,11) are applied. I, then becomes

2"R

2"

a+6a(9,z)

J

t

J

I, -

J

6A o(VXBp-AB )rdrd9dz+2,,2R (6a) 2{-2l (B 2 ._B2)

-P

P

t

po

pz

a

°

(17)

In the second integral 12 between square brackets in Eg. (15), B (r)

p

and A (r)

p

are developed in Taylor series in r-a in the interval

(a,a+oa).

Then the r-integration is performed first. The 8-z-integrations follow,

with proper account of the cos (me+kz) -dependences in the integrand. The

result is

2"

a+6a(e, z)

J

{t

B~(r)-

I

A

_p

(r)oB

_p

(r)}rdrd9dz

o

°

a

- 2,,2Rt{-41 B

2 p

(a) - -2

1 B

2 p

.(a) + -4

A

(A

B -A

B )

+

o

p9 p9

pz pz a

+

(18)

(14)

b

J

_{{5Av ·VXBv}rdr +}

a+5a(8,z)

J

{V.

(5AvXB) }rdrjd8dZ

a+5a(8,z)

-(a+5a(8,z»)B

(a+5a(8,z»)5A 8 (a+5a(8,z»)cos(m8+kz) +

vz

v

+ (a+5a(8,z»)B 8(a+5a(8,z»)5A

(a+5B(8,z»)cos(m8+kz)-jd8dZ +

v

vz

b

J

Z~O 8~0

a+5a(8,z)

(5A .VxB )rdrd8dz .

v v

(19)

As in the first integral I, in Eq. (16), functions of a+5a are

ex-panded in power series around r=a. Derivatives of B

v8' Bvz' AvO' and Avz

are eliminated with VxB -

0 and B - VxA, and Eqs (C7,8,12,13) are applied.

We obtain the result

(20)

The fifth integral Is in Eq. (15) is reduced in the manner of 12 and

the result can most easily be found by accepting Eq. (18), changing the

indices p to

v

and putting A-O ,

(21)

We substitute the integrals in Eqs (17), (18), (20) and (21) into

Eq. (15), leaving the third and the sixth integral in Eq. (15) unchanged.

(15)

First and second-order terms are collected and the following results are obtained:

I. First variation for cylindrically symmetric equilibria with perturbation

II. .

i(mO+kz)

proportional to e : !l

of

o

2"R

2"

r

f

a+6a(O. z)

f

z~O 0-0

r-O

2"R

2"

+

r

f

b

f

z-O

B-O r~a+oa(B.z)

Euler equations:

VxB

P

AB .

p'

Second bation

VxB

_v

- 0

variation for cylindrically

.

i(mB+kz)

proportional to e :

2"R

2"

a r ,

V

P r , V v

(22)

(23)

(24)

symmetric equilibria with

pertur-r

f

z~O e~o

f

{~(5Bp)2

-

t

5A

p

'5B

p

}rdrdBdZ +

r~O

+

2"R

2"

r

f

z-O

B-O b

f

t(5B

_v

)2 r drdBdZ

r-a

+ ,,2R {B2 (a) _ B2 (a)}(5a) 2 .

t vz pz

(25)

(16)

4. STABILITY OF THE SYMMETRIC EQUILIBRIUM

a) Stability with respect to periodic perturbations (m and/or k#O).

First, we determine the perturbed quantities in the vacuum region. When we write out the basic equations for 6B

v

'1

x

5B

v

o .

'1 • 5B - 0 • v

in cylindrical coordinates with a dependence proportional to find for the radial part of the z-component the equation

5B"

+

1 5B'

vz r vz

m

Z

- (k

z

+

_?

)

5B - 0 . vz

The total solution thus becomes

5B vr 5B vz A, I m (kr) + AzK m (kr) • (26)

(27)

i(m8+kz) e , we (28)

(29a)

(29b)

(29c)

where the prime now denotes differentiation with respect to the argument kr. The perturbation of the vector potential, SA, follows from

5B - '1

x

5A

v v

(30)

Since it contains an arbitrary gauge we can leave the value of 5A (r) vr

open. It drops out in what follows so that we do not actually use the gauge

freedom. The components of

Eq. (30)

lead to

SA (r) vz i k 5B vr r k:

f

_{5BvO (s)ds} b r r im

+

_r

-f

5B_vO(s)ds + ik

f

SAvr(s)ds b b r

f

SA (s)ds vr b (31a) (31b)

(17)

Equation (3Ib) satisfies the boundary condition oA (b) vz we fulfil the boundary condition oAve(b) ~

0

by putting

- O. In Eq. (3Ia) oB _vr(b) - 0, which

yields one relation between the integration constants AI. A2:

A,l' (kb) _m

+

A2K' (kb) _m ~ 0 . (32)

Next we consider the second variation 62_{F derived in Section 3,}

Eq. (25). Since the perturbation 6A in the plasma is arbitrary, we must find the minimum value of 62F (accessorial variational problem). To this end we define the Rayleigh ratio R

A:

(33)

RA ~

---J(VX6A)2dT

+

J(VX6A)2 dT

p

V

where P and V are shorthand notations for V and V respectively.

p v

Here we have assumed a cylindrical geometry with energies measured

per unit length in the z-direction (dr - rdrdB). We denote the denominator in Eq. (33) (the norm), by c 2 , so that the stability condition becomes

(34)

At the minimum of 52F, RA will reach a maximum, which we call

~

(we

take A positive; negative values of A could be taken into account by using

absolute values of A, Q and R

A). According to Eq. (34), the stability

criterion can now be written as

Q

>

A • (35)

We use the shorthand notation y for 6A and g - Vxy for 6B. The maximum value of RA is determined by putting 6R

A ~ 0:

J(VXy)oydT -

~{B~z(a)

-

B~z(a)}(6a)2

p

J

(Vxy) 2dT

+

P J(vxy)2 d r V

o ,

(18)

or

2

f

(Vxy) oydT - 2;

{B~z

(a) -

B~z

(a) }(oa) 2

p

{f(Vxy

)2

d

T

P

+

f

(Vxy) 2 }dT

V

The last term between square brackets can be reduced as follows

of ;

(OB

_v

)2dT - f(vxoy)ovxoYdT

f(vxvxy)ooydT

+

f

{oYX(VXy)}odS

V

V V S _p,S _v

In the vacuum we have VxVxy - 0 and oy - 0 on the wall at r - b

Therefore, we obtain

of

t

(oBV)2dT - - f{oyx(vxy)}odsa .

V

We substitute this into Eq. (36) and mUltiply with c 2 . The result is

J 2 (Vxy) oOydT

P

- 27r{B2 (a) - B2 (a)}oao 2a

+

J(OYXy)odS

_

A

vz pz a 2

"

f

(VXVxy) oOydT

P

+

Jvo{OYX(VXY)}dT

P

Equation (37) leads to the two following conditions.

- 0 .

(36)

(19)

T. The Euler equation in the plasma:

Vxvxy - oVxy - 0 . (38)

II. Surface condition on the plasma boundary:

f

(6y xy -

l

6y x(Vxy

).1

6y x(Vxy )) .dS - 2" {B2 (a) _B2 (a)}6a6 2a - 0 •

p p o p

p

0

v

a

A

vz

pz

(39)

where the indices p, v now denote values at the plasma side and at the

vacuum side of the plasma boundary respectively.

The solution of the Euler equation (38) with

y

proportional to ei(mO+kz) is

well known:

6B r

where D is an integration constant.

(40a)

(40b)

(40c)

The Euler equation (38) can be combined with g -

Vxy

to give after integration:

y

1 g

+

Vx

o

x

( )

i(mO+kz)

X

r e

.

From Eqs (40) and (41) then follows:

SA r 6A _z - v ~ z

(41)

(42a)

(42c)

(20)

In the complete solution appears a set of six constant parameters I viz.

four integration constants A}, A'2 I x(a) I and D, and the two undetermined

quantities

SA

(a)

vz (see

Eq.

(31» and the amplitude of the radial pertur-bat ion 6a. As we will see there are six boundary conditions, homogeneous in the six constants. The resulting 6x6-determinant, put equal to zero, yields

the relation aU) we are looking for. Then aU) ~ A gives us the critical

value of A above which the plasma becomes unstable. The six boundary

conditions are (32), (CS,7,10,12) [combined with the solutions (31), (42) and (B2 ,4)

1

and the surface condition (39). The latter condition can be simplified by noting that

Eq.

(C14) holds for all variations, not only for those that maximize

R

A

.

By taking the variation of

Eq.

(C14),

Sy(a)

(43)

we conclude that

Sy

(a)

x

y (a) - 0 in

Eq.

(39). Since S - 2na we obtain:

p p a

B (a)g (a)

+

B .(a)g .(a)

-pz pz p" p"

- B (a)g (a) - B .(a)g .(a) -

2~

{B2 (a) - B'

(a)}sa

vz vz Vu Vu ""a zv zp

o .

(44)

The 6x6-determinant

la

ij

l

mentioned above consists of the following

elements:

a" - I~(kb); a'2 - K~(kb); a'3

.'!'..

a'

a2S - 0; a26 - B (a); pz aS4 =- 0; ass ka m

o

J~{(a2_k2)l,a},

(a2 _k2)1, , a36 - -B p8 (a); B (a); vz

(21)

_

~

{B2

(a) -

82

(a)} .

2Aa

vz

zp

(45)

Fully written out, the determinant put equal to zero assumes the form

C>

{B2

(a) ZAa

vz

• {I' (kb)K'(ka) - I'(ka)K'(kb)}

m m m m

o .

(46)

This relation between ex and A will be investigated numerically in Section 5, for the case in which the zero order surface currents vanish.

b) Stability with respect to symmetric perturbations (m-k-O).

Symmetric modes differ from helical modes in their behaviour with respect to helicity conservation. K must be conserved up to all orders of

the

perturbations.

In

the accessorial problem

(minimization of

S2F)

(22)

condition that is both necessary and sufficient. The periodicity in

e

and z

of the helical modes guarantees the vanishing of

oK

since it is a volume

integral, linear in the variations 6A,

oS.

In the symmetric mode m=k-O.

however, the constraint

oK

= 0 must be taken into consideration explicitly.

For a moving plasma, K-conservation means

a+oa

J

(A +oA )o(B +oB )rdr

p p p p

o

or,

with

an expansion in

Sa:

a a

J

A °B rdr p p

o

J

(A ° oB

P

o

+ oA °B + oA °oB )rdr + aoa{(A +oA )o(B +OB)} +

p p p p p p p p r-a

+ 21 (oa)Z{rA oB }' -

a .

p p r-a

Up to first order we find:

a

J

(A °oB + B oM )rdr + aoaA (a)oB (a)

p p p p p p

o .

a

(47)

(48)

Partial integration of the first of the two integrals in Eq. (48),

combined with the relation oA(a) ~ oaxBp(a), yields the invariant

c:

a

c

=

f

o

B 00A

dr -

a .

p

(49)

Before solving the accessorial problem, as in case a). we determine

the perturbed symmetric fields in the vacuum region:

oB _vr

_{- a}

SA _vr

a

,

oBve cdr SAve Cz r2_b 2 2 r

oB

Cz SA

-

c, In

.E.

(50) vz vz b

(23)

Next we turn to the accessorial problem, i.e. the minimization of

52F* -

f

H

p

5A ·5B

}dr

+

f.1

(5B

)2dr

+

p p V 2 v

{ B2 (a) - B2 (a)}(5a)2 -

_vz

_pz

VA

J B ·5A dT •

_p

p

(51)

where energies are again taken per unit length in the z~direction (dT is

now rdrdO). and the constraint C has been incorporated with a Lagrange

multiplier -VA. The stability condition can now be put in the form

by the definition

p V

(5B )2dT .

v

and the introduction of a Rayleigh ratio R

A

:

f

(5A .Vx5A +

p

P

2vB

P

·M )

P

~{B~z

(a) _ B2

pz

(a)

}(5a) 2

P

RA

f

(Vx5A )2dT

+

f

(Vx5A )2dT

P

V

v

1

To determine the maximum of R

A, which we call

_""

we put 5R

A

0:

1

5R -

_A

~

_c

2" (B2 _B2 ) 5a52a ] _

_). _vz _{pz a}

1 c4

f

_ p

J{Y'VXY+2VB 'Y}dT -

p

~

A

{B2 (a)-B

vz

2 pz

(a)}(5a)2]

f

_

5 J (Vxy)2dT + 5 J (Vxy)2dT] - 0 .

p V

(52)

(53)

(54)

(24)

We multiply Eq. (55) by c',

introduce 1 and perform a few partial

a

integrations. Thus we arrive at

I.

The Euler equation in the plasma VxVxy

p

aVXy -

p

avB .

p

(56)

II. Surface condition on the plasma boundary,

identical to Eq.

(39) .

The Euler equation constitutes two coupled equations for the magnetic field

perturbations oBe and oB

_z

' both functions of r only. The solution reads:

avB

oBe

AJ, (ar)

-

_a->.

0

J,(>.r)

,

avB

oB

- AJ (ar)

-

_a->.

0

J (>.r)

(57a,b)

z

0 0

From oB - vxoA follows

1 avB

DAe

- AJ,(ar)

0

J, (>.r)

>.(a->.)

,

a

lAJ

avB

oA

_z

_a

(ar)

0

J (>.r) + k .

(58a,b)

0

>.(a->.)

0

A and k are constants that have to be determined from the boundary condi~

tions. As in the periodic case there are altogether 6 constants, A, V, k,

ell C2 and 8a. The corresponding 6 homogeneous equations determining these

constants are

(39),

(C5,7,10,12) and

(49).

They lead to a 6x6-determinant

with elements b

ij

:

b"

- aa{J (>.a)J (aa)+J,(>.a)J (aa)}; h"

0 0 0

aa { '

2}

-J,(>'a); b'5 - -J (>.a); b,. - -

~

B (a)-B

(a) ;

o a a vz pz

b 2, - J,(aa); b 22 - J,(>.a); b 23 - h 2• - b 25 - 0; b 26 - J (>.a);

_o

b 3, - J (aa); b 32 - J (>.a); b 33 - 1; b 3• - b 35 - 0; b 36 - -J,(>'a);

_o ₀

(25)

b61 - _a-AA {Jo(Aa)JI(aa) 1);b46 -B I B (a); vz o

b

1

In -; hS5 - 0; hS6 = -a

B

o (59)

The determinant

Ibijl

put equal to zero yields the required function

a(A). According to Eq. (52) the marginal stability is determined by putting aU) ~ A.

(26)

5. DISCUSSION AND NUMERICAL RESULTS

In the absence of zero-order surface currents, the marginal form of

Eq. (46), when

Q

is replaced by A, is also found from a resistive mode

anal-ysis

(c,'

-method, see e.g. GIBSON and WHITEMAN,

(1968». The differential

equations for the fields are the same in both cases. So, in order to prove the equivalence of the two theoretical approaches. we only have to show that the boundary conditions used are equivalent. Firstly. the continuity

of SB on the plasma surface. used in the resistive mode analysis. also

r

holds in the variational method. This follows from the continuity of B on

n

the surface, where

(60)

Since the zero-order field components are continuous, the continuity

of 6B

immediately follows. Secondly, the pressure balance at the perturbed

r

plasma surface has to be fulfilled:

B2 (a+6a) _ B2 (a+6a) .

p

v

(61)

Expansion in 6B and 6a yields the condition

[2B e(a) 6B

e

(a) + 2B

(a) 6B (a) + 6a(B2

e

' (a) + B2 '(a)}] 1 - [ ...

J •

(62)

o oz Z 0 oz p vac

The first and second terms on both sides cancel due to Eq. (44),

(pressure balance at the unperturbed boundary). The third terms are also

found to cancel when the zero-order fields are substituted. Thus. we see

that the pressure balance indeed holds, which establishes the equivalence

we set out to prove. It is satisfactory that the agreement between the resistive mode analysis and Taylor's variational principle, already present

without a vacuum layer, still exists in the presence of such a layer.

It has been known for some time that a force-free plasma with a free

boundary is intrinsically unstable (e.g. KRUGER (1967) for ideal MHD plasma,

MILLER (1985) for a slightly resistive plasma). In the framework of our

variational analysis we can deduce the instability as follows. The

equilib-rium of a force-free plasma surrounded by a vacuum layer can be represented

by the equation

(27)

j - ).(r)

B ,

(63)

where the function A(r) steps down from a constant value A in the plasma to zero in the vacuum. The second variation of the energy S2W can be expressed in the perturbation

roB :

r b

1. I

--.,----;-r

.,,---,,-20 m2 ₊_k2_r2

).'(krBe'mB)

.

B

k B

Z }

(r6B )2] dr .

m

e

+

r z

r

(64)

The contribution to 52W due to the jump .6.)" at the plasma-vacuum interface

then goes like

a3 ~). 1

+

k" a 2 (6 Br) a

2 "-'-.-;;:2:::B-;(-7)"'--:(--;-)"'/-="2""(--;-) a . r

A

D

a Bz a aB a

s

(65)

for a singular point r chosen to be near a, Since fj,),. is negative and

s

). . 2B

e

(a)B

z

(a)/aB 2 (a) is positive for the Bessel-function solution, .s2W

can be arbitrarily large and negative by having r

just inside the plasma

s

boundary (provided

~O),

implying instability.

We start the numerical results with a presentation of a: VB ..\ plots

according to Eq. (46) for various values of

P -

b/a. We restrict our analysis

to a model without zero-order surface currents on the plasma-vacuum

inter-face. Equation (46) then reduces to

I' (kb)K' (ka)

m m

I'(ka)K'(kb)

m

- H(a)V(),) , (66)

I (ka)K' (kb)

m m

I' (kb)K (ka)

m m where H(a) ~ (67)

and

(68)

(28)

The case {J=1 is outstanding here because it describes the stability of a plasma without a vacuum layer (Taylor). The degeneracy of this case is

apparent from Eq. (66) and leads to the straight horizontal lines H-O and

vertical lines V-O. In Fig. 5.1a we have plotted the lines H-O,

V~O

for m-O

and

Ikl-

0.50, 1.25 and 2.00. Fig. 5.1b shows these lines for m-l and k--l.OO,

+0.50, 1.20 and 1.70. For

P~l

the degeneracy is removed, as can be seen by

comparing Figs 1 with Figs 2 (P-1.0l). For m-O Figs 3a, 4a, Sa show the a

vs >. curves for

P~1.10,

1. 50 and 3.00 and 4 typical values of k. The

cor-responding curves for m-l (but p-10.00 instead of 3.00) are presented in

Figs 3b, 4b, 5b.

The transition stable-unstable can appear in two ways. First," a: can change sign tantamount to a transition from stability to instability

according to Eqs (34) and (35). Note that in this transition 02W jumps from

~

to

-~

so that the ordering in small perturbations (oW etc) breaks down.

Since we do not have an exact calculation of the perturbation in the

generalized magnetic energy we are forced to accept the change in sign of a: as an indication of a transition stable-unstable. Secondly, a:(A) may cross

the line of marginal stability '"

=

>. (for convenience also indicated in

Figs 2 - 6) .

The familiar stability diagrams >. vs k are obtained from the a vs >.

curves by putting a: either equal to A or equal to O. The results are seen in

Fig. 6a (m-O) for p-l.02, 1.10, 1.15 and 1.20 and in Fig. 6b (m-l) for p-l.10,

1.50, 3.00, 5.00 and 10.00. Stable and unstable regions are designated by the

letters S and I respectively. Transition lines Q=A are drawn fully, the

a-O

transition is dotted. For comparison, the Taylor curve (~-1) is also shown

(dashed line). The instability of the configuration at m-l according to

Miller manifests itself by the new curve at negative values of k, causing

instability at small A.

The special case m-k-O is also treated without zero-order surface

currents. Equation (59) then reduces to a 3x3 determinant with elements

J,(aa); d '2 - J,(Aa); d

'3 ~

J (Aa) ln

_o

P;

aa{J (aa)J (>.a) + J,(aa)J,(>'a»; d 22 - Aa{J

₀ ₀ ₀

2 (Aa) + J;(>.a»;

J2(>.a)

o

(29)

Its numerical evaluation is shown in Fig. 7 for three values of

p:

fi -

1.00 (Taylor), 1.25 and 1.50. If we move from left to right, the first crossing of a particular Q: vs ..\ curve with the line cr -

>.

determines the

critical value of A at which the instability sets in. This critical value if plotted vs {3 in Fig. 8. We observe that even this mode becomes more

unstable as the vacuum layer grows thicker.

Acknowledgement

The authors are greatly indebted to Dr. H.J.L. Hagebeuk and Mr. L.L.M.M. Verhoeven for their assistance with the numerical calculations. This work was partly performed under the Euratom-FOM association agreement with financial support from NWO and Euratom.

(30)

References

Centen, P. and M.P.H. Weenink, W. Schuurrnan

Minimum-energy principle for a free boundary, force-free plasma.

Plasma Phys. & Controlled Fusion, Vol. 28(1986), p. 347-355.

Gibson, R.D. and K.J. Whiteman

Tearing mode instability in the Bessel function model.

Plasma Phys., Vol. 10(1968), p. 1101-1104.

Hasegawa, A. and Y. Kodama, R. Gruber, S. Semenzato Study of reversed field pinch with surface current.

J. Phys. Soc. Japan, Vol. 53(1984), p. 1316-1325.

Houten, M.A. van and M.P.H. Weenink,

w.

Schuurman

On the second variation of a minimum energy principle for a force free plasma with a free boundary.

In: Proc. 12th European Conf. on Controlled Fusion and Plasma

Physics, Budapest, 2-6 Sept. 1985. Contributed Papers, Part 2. Ed. by L. Pocs and A. Montva1.

Petit-Lancy-{Switzerland): European Physical Society, 1985.

Europhysics Conference Abstracts, Vol. 9F, Part 2. P. 382-385.

Kondah, Y.

An energy principle for axisymmetric toroidal plasmas.

J. Phys. Soc. Japan, Vol. 54(1985), p. 1813-1822.

Kruger, J.G.

Bijdrage tot de studie van stabi1iteitsprob1emen in de

kinetische plasmatheorie.

Verh. Kon. Vlaarnse Acad. Wet. Lett. & Schone Kunsten Belgie

Kl. Wet., Vol. 29 (1967), No. 97.

This papep by Kpugep is coveped fpom the gpeatest papt by the

foLLowing papeps by Kpugep and CaLLebaut.

Kruger, J.G. and D.K. Callebaut

Energy principle for gravitational and magnetodynamic stability with application to force free fields.

Mem. Soc. R. Sci. Liege, Seme serie, Vol. 15(1967), p. 175 et seq. Kruger, J.G. and D.K. Callebaut

On the least stable mode in cylindrical systems.

Z. Naturforschung, Teil A, Vol. 23a(1968) , p. 1357-1361.

Kruger, J.G. and D.K. Callebaut

Relations between adiabatic and incompressible (non-adiabatic) systems and their stability.

Z. Naturforschung, Teil A, Vol. 25a(1970) , p. 88-100. Kruger, J.G. and D.K. Callebaut

Variational principles for compressible and incompressible systems.

Z. Naturforschung, Tei1 A, Vol. 25a(1970) , p. 1097-1100.

Miller, G.

Error magnetic fields in a cylindrical plasma: Stability with zero pressure.

(31)

Ortolani, S. and R. Paccagnella, E. Zilli

Characteristics of magnetic field profiles in a reversed field pinch.

In: Contributed Papers Int. Conf. on Plasma Physics, Lausanne,

27 June-3 July 1984. 3rd Joint Conf. of the 6th Kiev Int. Conf. on

Plasma Theory and 6th Int. Congress on Waves and Instabilities in

Plasmas. Ed. by M.Q. Tran and M.L. Saw1ey.

Centre de Recherches en Physique des Plasmas, Ecole

Poly technique

Federale de Lausanne, 1984. P. 150.

Reiman, A.

Taylor relaxation in a torus of arbitrary aspect ratio and cross section.

Phys. Fluids, Vol. 24(1981), p. 956-963.

Schoenberg, K.F. and R.W. Moses, Jr., R.L. Hagenson

Plasma resistivity in the presence of a reversed-field pinch dynamo.

Phys. Fluids, Vol. 27(1984), p. 1671-1676.

Taylor, J.B.

Relaxation of toroidal plasma and generation of reverse magnetic

fields.

(32)

• aa

aa

•

H=O

,

{J =

1.00 V=O m = 0 ka=±O.50

,

_,

• ,

Aa

• aa

_,

a.

•

H=O

,

{J

= 1.00 V=Q _m₌₀ kS=±1.25

, ,

_•

,

Aa

,

aa

H=O

• ,

{J

= 1.00 V=O _m₌₀ ka=±2.00

, ,

,

_•

Aa

aa

H=O

,

V=Q

{J

= 1.00 m = 1 ka=-1.0Q

,

_,

Aa

•

H=O

{J

= 1.00 V=O m = 1 ka

=

0.50

, ,

_•

Aa

,

H=Q

{J

= 1.00 v=o _{m =}

,.

_ka=_1,20

, ,

3

• ,

Aa

•

H=O

,~---~--~~---{J

= 1.00 V=O m

=

1 ka= 1.70 , L -____ L -_ _ _ _ L-, __ -LL-____ L-__ ~ o 2 3 .

Aa

Fig. la. a vs A plots for m

=

0, Fig. lb. a vs A plots for m

=

1,

B

= 1.00 and

IkaJ

= 0.50, 1.25,

2.00

8

= 1.00 and ka=-1.00,0.50, 1.20,1.70

(33)

a. a. a.

•

• ,

•

• ,

'V

,

_,

,

•

• ,

• /

"

•

• ,

:V

,

"

• ./

/

7 /

V

/

[7

v

/

v

/3:1.01 m=o ka=O.50

•

• "

A. V

~

V

I~

'"

1.01 m,..o ka= 1.00

"

Aa

v

/

V

{) '" 1.01 m ; Q ka .. 1.50

"

_.\ II

V

L

V

/~ = 1.01 m",o ka", 2.00

"

Aa

Fig. 2a. a vs

A

plots for m

=

0,

B

=

1.01 and ka

=

0.50,1.00,1.50,2.00

...

f3 ".

1.01 m = 1

...

ka=I.20

..

,

'.'

..

,

..•

'.'

..

,

M

,.,

A.

..

,

".

,.,

,

..

,

.

,

..

,.,

...

,.'

f3

= 1.01 m =1

,

..

ka=I.70

..

,

2.50 2.16 3.02 3.U

,."

3.1" A. Fig. 2b.

a

vs

A

plots for

m 1,

S

= 1. 01 and

(34)

-"

...

a. a.

'"

_..•

..•

..,

'., ~ =1.10 m =1 ka .... OO ka =0.50

• "

,

..

,.,

..,

...

'"

'.'

A.

,i.

"

'"

a.

"'

.

,

••

...

"

••

..

,

f3

'"'.10 m

.,

ka=O-so

"

...

..

,

...

A.

A. "

..,

a. _{a •}

...

..

.

...

.

,

...

,

..

I~

'"

1.10

_'.'

moo ka", 1.50

...

..

,

"

...

..,

..•

...

.

..

'"

. h "i •

"

...

a. <to

..

,

...

_...

...

..

.

..

_..

.

,

...

..

,

() '" 1. to

...

m =1

...

ka= 2.50

..

.

"

'.'

...

.

.. ..

,

...

.

..

...

.

..

A.

Fig. 3a.

a

Vs

A

plots for

Fig. 3b.

a

vs

A

plots for

m ~

0,

S

~

1.10

and m

1 ,

S

~

1. 10

and

(35)

'.'

a. a.

_"

'.0 '"

...

'.' LO 0.'

,.

•..

..

,

A.

a. Aa

"

r---~---~~---~----'" a.

,

..

, ,

,

..

...

ka=1.70

,

..

,

..

,

..

"

2.$0 2.'. 3.01 3.21 ,.~ s.ao

A.

,..

a. n. u

...

,

..

,

..

,

,.

,

..

,

..

...

,

..

...

~ :1.50 >0. m

.

,

..

ka 3: <1.00 '

..

...

_,

.•

_,

.. ...

_,.,

_,

..

_,

..

_,

..

A.

Fig. 4a. a vs

A

plots for

m = 0,

S

= 1.50 and ka = 0.50,1.00,1.50,2.00

Fig. 4b. a vs

A

plots for

m

=

1,

S =

1.50 and

(36)

aa aa

,

.•

"

'.'

,

.•

,

..

.

..

M

'.'

.

..

.

..

"\a "\a

"

a.

•

a •

.

..

,

..

.

,

...

,

..

,

..

,

..

~ ;; 10.00 m

.

,

,.,

k • • O.70

• ,

..

"

,

..

,.,

,

.•

..,

...

.

..

..,

,,\. ,la

"

,

..

aa _aa

,.,

...

_•.

_.

...

..

,

•.

,

...

,

..

,

..

,.

,

..

,

..

,

..

• "

'"

,

..

,

..

..•

,.,

'.'

...

.

..

A.

,I.

"

...

aa _aa

'.'

...

'"

...

'-'

,.

,

'.'

,

..

','

,.

,

_~:: 10.00

,

..

m

.

,

..

ka =4.00 ,.

"

".

,

.•

,

..

.

.•

..

,

.

..

.,.

.

..

"\a

A.

Fig. So. rt vs )., rIots for

:--'ic: •

5b. a vs

A

plots for m ~ _Of

_G

~ 3.00 and

'"

~ 1 ,

_S

~ 111.

or'

an(;

(37)

Aa

_50 _ 4 5 _ 4 0 . 3 5 . 3 0 - 2 5 . 2 0 _1.5 .10_0.5 0 05 ' 0 15 20 25 30 35 40 4 5 5 0

~i""'"

""'" _

I

7~~;

•. 0 ~",,,,,,,,,,,,,,", . . .

= ... ,,,,,,,,,,,, ...

~

•.

0 3. 3.5 3. ,

5

3.0

2.'

~

...

== ...

====== .... ========================j2 .•

2.0 2",0

5

m=O {3=1.02 LO

•••

,L--L __ L-~ __ L-~~~~~~~~ __ ~~ __ -'-~~ __ ~L~ _ _ ~~ _ _ ~-J, -5.0 -4.5 _4.0 -3.5 -3.0 _2.5 -2.0 _1.5 -'.0.0.5 o 0.5 1.0 1.5 2.0 2.5 3.0 3.15 4.0 ".5 5.0

ka

Fig. 6al. A vs ka plot for m

o

and

S

1.02

_5.0 _ 4. S _4.0 _3.5 .3.0 -2·5 .2.0 _1.5 _1.0 -05

,

0.'

,.,

L, 24

,.

_{3.' 3.'}

•••

..,

'.0

,.,

_...

6.0

.-••

..

_•

'.0

_I

' . 0

..

,

_{•• 5} '.0 ....

_---

--

•. 0 3 .• _{3 .•} 3. 0

5

3.0 5 2.6 2.0 2.0 LS L '

5

m=O LO {3=1.10 LO

,

..

0 . ' 0 0 -5.0 - ... 5 .4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.' LO U 2.0

...

3.0

...

'.0

...

'.0

ka

(38)

-"

• 4.5 -4.0 _3.5 _3.0 _2.5 .2.0 -1.5 _1.0 _0' 0 0.' '.0

...

'.0

...

3.0 3.' '.0

•••

.~ '.0 ' . 0

.-AS

•••

'.0

...

I

..

0

..

.

'

....

· ..

'

....

'.0

_S

....

-

---

_S

' . 0 3 .• 3 . ' 3.0 3 .0

I

_•••

'.0 . 0

...

_S

m=O

...

'.0 _f>=1.15 . . 0 0.' 0,

•

0 0 _5.0 -" .5 -4.0 _3.5 -3.0 -2.5 -2.0 -1.5 _1.0 .0·5 0 0.5 '.0

•••

'.0 2.5. 3.0 3.' '.0

...

'.0

ka

Fig. 6a3. A vs ka plot for m

o

and

8

1.15

.5.0 _ •. is _{- '.0}_3.5 .3.0 _2.5 .2.0 _1.5 -1.0 _0.5 0 0.' '.0

..

,

'.0

...

3.0

...

'.0

'.'

'.0 '.0 ' . 0 "-

.-Aa

...

• ,

'.0

I

' . 0

..

,

_•••

'.0

S

---

' . 0 3.' 3 .• 3.0

I

3.0

...

..,

'.0 • .0

...

m=O

...

S

13=1.20 • .0 0.' o .• 0 0 -!i.0 -" .5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.' '.0

..

,

'.0

..

,

3.0 3.' •• 0

..

,

'.0

ka

(39)

4 r - - r - - r - - r - - r - - r - - r - - r - - r - - r - r r - - r - ' - - ' - - ' - - ' - - ' , r 7 , - - , - - , - - , 4 ~ I~,'

A8

"

S

3

s

,.-.--

....

... I

3

...

...•....•...•••.•

2 2

s

1

m: 1

p=

1.10 ~5.0 _4.5 .4.0 -3.5 -3.0 .25 -2.0 _1.5 _1.0 _Q5 0 as 1.0 1.5 2.0 2S 3.0 3.5 4.0 4.5 5.00 ka Fig. 6bl. A vs k plot for m 1 and

S

= 1.10

4r--,---.--~--~--r---r--,---.---r~ •• ,---r--,r--,---r---r--~-,.--,r--,---,4

/~,','

.-s

3 2

I

...

_

...

'

'.

_'.

'

.

....

Aa

s

I

___ ---,3

... ...

2

s

m

=

1

p=

1.50 7 0

'.8

.5,0 -4.5 .4.0 -35 -3.0 -25 _2.0 .15 _1.0.05 0 os 1.0 1.5 2.0 2S 3.0 3.5 4.0 4.5 5 ka Fig. 6b2. A vs k plot for m 1 and

S

= 1. 50

(40)

4r--r--r--r--r--r--r--r--r--r~r--r--r--r--r--r--V--y--r-,,-.4

,/ I

'\

(

"

1 ~~/'

S

I

r:~:~...

I

"---=

3

...

3

-5

Aa

2 - - 2

5 I

1 - - 1 m=l

f3.

3.OO ~5.0 -4.5 .4.0 -3.5 .3.0 -2S .2.0 _15 .1.0 _QS 0 OS 1.0 15 2.0 2.5 3.0 3.5 4.0 4.S 5.00

ka

Fig.-6b3.

A

vs k plot for rn

=

1 and

S

=

3.00

2 - - 2

5 I

-

- 1 m= 1

f3=

5.00 ~5~~~-4~.5~-4~~~.~~~.~~~O-.~U~_~2~~~_~~-_~,.0~_~~~O~~~~-,~~~~~~2~~~~~~3~~~~~-4~.O~~~~~~o

ka

(41)

4r-~~--~-r--r-~~--~-r~r-~~--~-r--r-~~--~-r-.4

.

,If"~

/

.,

I , , "

(

'\::--~/

I

31- I

r··.

-

3

~---~\ ~

...

~

s

.1a

_ 2

s

I

-

_ 1 ~5.0 -4.5 -4.0 -3.5 _3·0 -25 -2.0 -1.5 -1.0 .05 0 0.5 1.0 1.5 2.0 25 3.0 3.5 4.0 45 5.0° ka Fig. 6b5.

A

vs

k plot for m 1 and

S

10.00

(42)

o

•

• • •

• {3

=1.00 m=o ka=o

• A8

{3=1.25

m=o ka=o

• Aa

•

'0

.

"

a e 10

Aa

Fig. 7. Stability diagrams for

m~k~ 0 and S~ 1.00,1.25,1.50 .tl.a s

.

.•

,

...

..

,

'.0

L._-''-_-' _ _ -'-_ _ ... _ _ ..L.._--I .. 0 _'.0

,

..

3.0

_{3.' {3}

'.0

Fig. 8. A vs S plot for m ~ k ~ 0 cr

(43)

Appendix A

If 6a(a) at the plasma boundary originally has three components in

the r-, 0- and z-directions, we perform a transformation to a system (A,

oa)

indicated by bars, such that

6a

e

(a) -

6a

z

(a) -

O. According to Eq. (CI4) of

the Appendix C:

and

6An(a) -

-B

(a)6a (a)

u

z

r

6A

(a)

z

6K

(a) -

0 ,

r -B

(a)6a (a)

z

r

(AI)

6K

(a) -

Bn(a)6a

(a) ,

(A2)

Z u r

where the equilibrium relation

B

(a) -

0

has been used.

r

The transformed

Ii.

differs from the original

A

by the gradient of a

single-valued function ¢ which is chosen such that

6K

(a) -

O. Therefore

r

6A (a)

r

ill.

8r

6A (a) -

6K

(a) - ik6¢

Z Z (A3)

Using Eqs (AI) and (A2) in Eq. (A3b,c) we find

im 6¢

+

B

(a)6a (a) - B (a)6a

(a)

a

z

r

z

r

(A4)

These two equations can be solved for 6a

r

(a) and 6¢ and thus determine

the transformation

oa

~

5a, oA

~ SA that was sought. The conclusion is that

in the main text we can take

6a

_e

(a) -

6a

z

(a) -

0, 6A

r

(a) - 0, although in

the two volumes 6A

~

O.

(44)

Appendix B

For further reference we give a listing of the equilibrium quantities

in the plasma and in the vacuum, and derive some useful relations between these quantities. In the case of a current 1* on the plasma surface, we have