Tokamak elongation – how much is too much? Part 1. Theoryjungpyo/journal_paper/Freidberg_JPP... · 2017. 6. 11. · Tokamak elongation – how much is too much? Part 1. Theory 5

J. Plasma Phys. (2015), vol. 81, 515810607 c� Cambridge University Press 2015doi:10.1017/S0022377815001270

1

Tokamak elongation – how much is too much?Part 1. Theory

J. P. Freidberg1, A. Cerfon2 and J. P Lee1,2,†1Plasma Science and Fusion Center, MIT, Cambridge MA, USA

2Courant Institute of Mathematical Sciences, NYU, New York, NY, USA

(Received 26 August 2015; revised 1 November 2015; accepted 2 November 2015)

In this and the accompanying paper, the problem of the maximally achievableelongation in a tokamak is investigated. The work represents an extension ofmany earlier studies, which were often focused on determining limits due to(i) natural elongation in a simple applied pure vertical field or (ii) axisymmetricstability in the presence of a perfectly conducting wall. The extension investigatedhere includes the effect of the vertical stability feedback system which actually setsthe maximum practical elongation limit in a real experiment. A basic resistive wallstability parameter, � ⌧w, is introduced to model the feedback system which althoughsimple in appearance actually captures the essence of the feedback system. Elongationlimits in the presence of feedback are then determined by calculating the maximum against n = 0 resistive wall modes for fixed � ⌧w. The results are obtained by meansof a general formulation culminating in a variational principle which is particularlyamenable to numerical analysis. The principle is valid for arbitrary profiles butsimplifies significantly for the Solov’ev profiles, effectively reducing the 2-D stabilityproblem into a 1-D problem. The accompanying paper provides the numerical resultsand leads to a sharp answer of ‘how much elongation is too much’?

1. IntroductionIt has been known for many years that tokamak performance, as measured

by pressure and energy confinement time, improves substantially as the plasmacross-section becomes more elongated. There are, however, also well known limits onthe maximum achievable elongation, which arise from the excitation of n = 0 verticalinstabilities. When designing the next generation reactor scale tokamak experiments(Najmabadi et al. 1997, 2006; Aymar, Barabaschi & Shimomura 2002; Sorbom et al.2015), where high performance is critical, it is thus important to be able to accuratelypredict the maximum achievable elongation as a function of inverse aspect ratio"= a/R0, where a is the minor radius of the device, and R0 is the major radius.

The inverse aspect ratio is a particularly important parameter since different reactordesigns have substantially different values for this quantity. Specifically, standard andhigh field tokamak reactor designs have 1/"⇠ 3–4.5 (Schissel et al. 1991; Hutchinsonet al. 1994; Najmabadi et al. 1997, 2006; Aymar et al. 2002; Sorbom et al. 2015),

† Email address for correspondence: [email protected]

mailto:[email protected]

2J.

P.

Freidberg,

A.

Cerfon

and

J.

P.

Lee

while

sphericaltokam

akshave

smaller

1/"⇠

1.5

(Sabbaghet

al.2001;

Sykeset

al.2001).Since

optimized

plasma

performance

andcorresponding

minim

izedcostdepend

stronglyon

,it

isim

portantto

havean

accuratedeterm

inationof=

("

)and

thisis

thegoal

ofthe

presentand

accompanying

paper.The

problemof

determining

=("

)has

receivedconsiderable

attentionin

paststudies

but,as

discussedbelow

,there

isstill

anim

portantgap

inour

knowledge.

Toput

theproblem

inperspective,

we

notethat

earlierstudies

generallyfall

intoone

ofthree

main

categories,each

oneproviding

valuableinform

ation,but

notthe

whole

story.These

aresum

marized

asfollow

s.The

firstclass

ofstudies

involvesthe

conceptof

‘naturalelongation’.

Here,

thetokam

akis

imm

ersedin

adeliberately

simple

externalpoloidal

magnetic

field,usuallya

purevertical

field.The

plasma

isexpected

tobe

stableagainst

n=0

verticalinstabilities

sinceone

verticalposition

isthe

same

asany

other.In

thesestudies

(Peng&

Strickler1986;

Roberto

&G

alvão1992)

thefree

boundaryequilibrium

surfaceis

calculated(i.e.

theelongation

and

thetriangularity

�are

determined)

fora

rangeof"

fromw

hichitis

straightforward

toextractthe

desired=

("

).Excellentphysical

insightis

providedby

thesecalculations.

Evenso,

them

aximum

predictednatural

elongationsare

substantiallysm

allerthan

thoseachieved

inexperim

ents.The

reasonis

thatexperim

entshave

feedbacksystem

sthat

canstabilize

n=0

verticalinstabilities,

therebyallow

inglarger

’sthan

thosepredicted

bym

arginallystable

naturallyelongated

configurations.The

secondclass

ofstudies

involvesthe

calculationof

thecritical

normalized

wall

radiusb/a

foran

idealperfectly

conductingconform

alw

allrequired

tostabilize

adesired

elongation;that

is,the

analysisbasically

determines

b/a=

f(",).

Here

too,the

studies(W

esson1978)

(andreferences

therein)provide

valuableinsight.

Specifically,the

studiesdeterm

inethe

maxim

umb/a

thatm

ightpotentially

beable

tobe

stabilizedby

feedback.H

owever,

theanalysis

doesnot

actuallypredict

whether

ornot

apractical

feedbacksystem

canbe

constructedto

providestability.

Equallyim

portant,froma

practicalpoint

ofview

,theactual

valueof

b/a

doesnot

varym

uchfrom

experiment

toexperim

ent.The

thirdclass

ofstudies

consistsof

detailed,engineeringlevel

designsthat

predictthe

maxim

umelongation

andw

hichinclude

many

effectssuch

asplasm

aprofiles,

realgeom

etry,safety

margins

andm

ostim

portantly,the

engineeringproperties

ofthe

feedbacksystem

.Such

studies(K

esselet

al.2006)

arevery

realisticand

areexactly

what

isrequired

todesign

anactual

experiment.

Nevertheless,

theseare

pointdesigns

whose

main

goalsare

focusedon

aspecific

machine

ratherthan

providingscaling

insightin

theform

of=

("

).The

presentanalysis

attempts

tofill

anim

portantgap

inour

knowledge,

namely

determining

them

aximum

elongationas

afunction

ofinverse

aspectratio

includingthe

constraintsarising

fromthe

feedbacksystem

.The

calculationis

largelyanalytic

combined

with

some

straightforward

numerics.

Inour

model,

theplasm

ais

assumed

tobe

up–down

symm

etricand

ischaracterized

bythe

following

parameters:

inverseaspect

ratio"=

a/R

0 ,elongation

,triangularity

�,poloidal

beta�

p ,norm

alizedw

allradius

b/a

andan

appropriatelydefined

parameter

representingthe

capabilitiesof

thefeedback

system.

Defining

thisfeedback

parameter

andincluding

itin

theanalysis

isthe

main

newcontribution

ofthe

research.A

simple

butreliable

definitionof

thefeedback

parameter

isbased

onthe

following

two

observations.First,

practicalfeedback

isfeasible

when

thegrow

thrate

�of

then=

0vertical

instabilityis

small.

Sincethe

plasma

issurrounded

bya

finiteconductivity

wall,

theinstability

ofinterest

isactually

aresistive

wall

mode.

The

Tokamak elongation – how much is too much? Part 1. Theory 3

second observation is that, if the feedback coils are located outside the resistivewall as they usually are, then an effective feedback system must have a rapid fielddiffusion time ⌧w. This is important because once an unstable plasma motion isdetected the feedback response fields must quickly diffuse through the wall in orderto reach the plasma. In other words ⌧w should be short.

In principle, it is possible to trade off growth rate in favour of response time, orvice versa. However, the overall effectiveness of the feedback system is dependentupon the combined smallness of � and ⌧w. The conclusion is that a simple parameterthat takes into account the feedback system is the product of these two quantities:

� ⌧w = feedback capability parameter. (1.1)

A good way to think about this parameter is as follows. In designing a newexperiment, the properties of the feedback system depend mainly on the geometry ofthe vacuum chamber, the availability of fast-response power supplies, the maximumfeedback power available, the number of feedback coils and the sensitivity of thedetectors. These represent a combination of engineering and economic constraintsthat should not vary much as " changes. Consequently, the engineering value of � ⌧wis a good measure of the feedback capability. It represents the maximum value of theresistive wall growth rate that can be feedback stabilized.

Now, consider the design of a new experiment with " as a parameter over whichoptimization is to be performed. For a fixed " the plasma elongation should beincreased until the growth rate of the n = 0 resistive wall mode is equal to the valueof the feedback capability parameter given in (1.1). In this process, the triangularity� may also be adjusted to find the maximum allowable . This resulting is theelongation that maximizes performance for the given ". By repeating the procedurefor different ", it is then possible to determine the optimum aspect ratio, including theeffects of transport, heating, magnetic design, etc., that results in the peak value ofmaximum performance and corresponding minimizes cost. In our analysis, practicalengineering values of � ⌧w are chosen by examining the data from existing experimentsas well as ITER (Aymar et al. 2002).

Based on this discussion, we can state that the goals of the analysis are to determinethe maximum allowable elongation and corresponding triangularity � as a functionof inverse aspect ratio " subject to the constraints of fixed poloidal beta �p, normalizedwall radius b/a and feedback parameter � ⌧w; that is, we want to determine

= ("; �p, b/a, � ⌧w)� = �("; �p, b/a, � ⌧w).

�(1.2)

This paper presents the analytic theory necessary to determine these quantities. Theanalysis is valid for arbitrary plasma profiles. An additional useful result presentedas the derivation progresses is an explicit relationship between vertical stabilityand neighbouring equilibria of the Grad–Shafranov equation, a relationship longbelieved to be true but, to the authors’ knowledge, never explicitly appearing in theliterature.

The second paper (Lee et al. 2015) in the two part series presents the numericalresults. For numerical simplicity, the results are obtained for the Solov’ev profiles(Solov’ev 1968) although with some additional equilibrium numerical work it ispossible to include arbitrary profiles. One may expect Solov’ev profiles to providereliable physical insight since the qualitative properties of axisymmetric MHD modes

4 J. P. Freidberg, A. Cerfon and J. P. Lee

are thought to not depend sensitively on the details of the current profile (Bernardet al. 1978). The overall conclusions are that the maximum elongation (1.1) decreasessubstantially as " becomes smaller and (1.2) is substantially higher than that predictedby natural elongation calculations, much closer to what is observed experimentally.The maximum elongation is weakly dependent on �p and also does not vary muchwith b/a primarily because this quantity itself has only limited variation in practicaldesigns. There is, however, a substantial increase in maximum elongation as � ⌧wincreases. As expected, feedback is very effective in increasing elongation and overallplasma performance. A slightly more subtle effect is the value of the optimumtriangularity corresponding to maximum elongation, which is noticeably smaller thanthat observed experimentally in high-performance plasma discharges. The reasonis presumably associated with the fact that maximum overall performance dependson turbulent transport, MHD ballooning/kink modes as well as MHD axisymmetricstability. Although the stored energy has been shown to increase with increasingtriangularity, which helps explain the data, it is unfortunate that the current empiricalscaling laws for transport and ballooning/kink stability do not explicitly include thiseffect.

The analysis is now ready to proceed.

2. EquilibriumThe equilibrium of an axisymmetric tokamak is described by the well-known Grad–

Shafranov equation (Grad & Rubin 1958; Shafranov 1958)

�⇤ = �µ0R2 dpd

� 12

dF2

d . (2.1)

Here, (R, Z) = poloidal flux/2p and p( ), F( ) are two free functions. The �⇤operator, plus the magnetic field B = Bp + B�e� and current density J = Jp + J�e� aregiven by

�⇤ ⌘ R@

@R

✓1R@

@R

◆+ @2

@Z2(2.2a)

B = 1R

r ⇥ e� + FR

e� (2.2b)

µ0J = dFd

Bp � 1R�⇤ e�. (2.2c)

For the study of n = 0 resistive wall modes, the essential physics is captured byconsidering up–down symmetric equilibria, and this is the strategy adopted here. Also,significant analytic simplicity follows by choosing the free functions to correspondto the Solov’ev profiles (Solov’ev 1968) for which FF0( ) = C1 and p0( ) = C2,with primes denoting d/d . Only two constants, C1 and C2, are needed to specifyan equilibrium.

The Grad–Shafranov equation can now be conveniently normalized in terms of twoequivalent constants, 0 and A, one of which ( 0) scales out entirely from the finalformulation. The full set of normalizations is given by

R = R0X (2.3a)Z = R0Y (2.3b)


= 0 (2.3c)

µ0dpd

= � 0(1 � A)

R40

(2.3d)

FdFd

= � 0AR2

0. (2.3e)

Note that the normalized flux has an upper case italic font. In terms of thesenormalizations, the Grad–Shafranov equation plus the critical field quantities neededfor the analysis reduce to

XX � 1X X + YY = A + (1 � A)X2 (2.4a)

(R20/ 0)Bp = 1

X(� YeR + XeZ) (2.4b)

(R30/ 0)µ0J� = � 1

X⇥A + (1 � A)X2⇤ . (2.4c)

The boundary conditions require regularity in the plasma and (SP) = 0 with SP, theplasma boundary. This implies that (X, Y) < 0 in the plasma volume. Typical valuesof the free constant A in decreasing order are as follows

A =

8>><

>>:

1 p( ) = 0 (force free)0 F( ) = R0B0 (vacuum toroidal field)

� (1 � ")2

1 � (1 � ")2J�(1 � ", 0) = 0 (inner edge current reversal).

(2.5)

The task now is to find a solution for (X, Y). Exact solutions to the Grad–Shafranovequation for Solov’ev profiles have been derived by a number of authors (Zheng,Wootton & Solano 1996; Weening 2000; Shi 2005). Here, we follow the formulationof reference (Cerfon & Freidberg 2010; Freidberg 2014). Specifically, an exactsolution to (2.4) can be written as

(X, Y) = X4

8+ A

✓12

X2 ln X � X4

8

◆+ c1 1 + c2 2 + c 3 + c4 4

+ c5 5 + c6 6 + c7 7 (2.6a) 1 = 1 (2.6b) 2 = X2 (2.6c) 3 = Y2 � X2 ln X (2.6d) 4 = X4 � 4X2Y2 (2.6e) 5 = 2Y4 � 9X2Y2 + 3X4 ln X � 12X2Y2 ln X (2.6f ) 6 = X6 � 12X4Y2 + 8X2Y4 (2.6g) 7 = 8Y6 � 140X2Y4 + 75X4Y2 � 15X6 ln X + 180X4Y2 ln X � 120X2Y4 ln X.

(2.6h)

The free constants cj are chosen to match as closely as possible the well-knownanalytic model for a smooth, elongated, ‘D’ shaped boundary cross-section SP givenparametrically in terms of ⌧ by Miller et al. (1998)

X = 1 + " cos(⌧ + �0 sin ⌧ ) (2.7a)Y = " sin ⌧ . (2.7b)


FIGURE 1. Geometry of the plasma equilibrium.

Here, " = a/R0 is the inverse aspect ratio, is the elongation and � = sin �0 is thetriangularity. The geometry is illustrated in figure 1.

In practice the cj are determined by requiring that the exact plasma surface = 0,plus its slope and curvature match the model surface at three points: (i) the outerequatorial point, (ii) the inner equatorial point, and (iii) the high point maximum.Taking into account the up–down symmetry we see that the requirements translate to

(1 + ", 0) = 0 outer equatorial point, (2.8a) (1 � ", 0) = 0 inner equatorial point, (2.8b)

(1 � �", ") = 0 high point, (2.8c) X(1 � �", ") = 0 high point maximum, (2.8d)

YY(1 + ", 0) = �N1 X(1 + ", 0) outer equatorial point curvature, (2.8e) YY(1 � ", 0) = �N2 X(1 � ", 0) inner equatorial point curvature, (2.8f ) XX(1 � �", ") = �N3 Y(1 � �", ") high point curvature, (2.8g)

where for the given model surface,

N1 =

d2XdY2

�

⌧=0= � (1 + �0)

2

"2(2.9a)


FIGURE 2. Flux surfaces for NSTX using the Solov’ev model. The value of A has beenchosen such that J�(1 � ", 0) = 0.

N2 =

d2XdY2

�

⌧=p

= (1 � �0)2

"2(2.9b)

N3 =

d2YdX2

�

⌧=p/2= �

" cos2 �0. (2.9c)

The determination of the cj has been reduced to finding the solution to a set ofseven linear, inhomogeneous, algebraic equations; a very simple numerical problem.Hereafter, we assume that the cj have been determined, thereby completely definingthe equilibrium solution. A typical set of flux surfaces corresponding to the high �,tight aspect ratio NSTX spherical tokamak is illustrated in figure 2, where A has beenchosen so that J�(1 � ", 0) = 0. The surfaces look quite reasonable and, as expected,exhibit a large shift of the magnetic axis.

The one final piece of information required for the resistive wall stability analysis isthe value of poloidal beta �p. In general �p and the toroidal current I can be expressedin terms of the free constants A, 0. The advantage of the normalizations introducedby (2.3) is that �p is only a function of A but not 0. This can be seen by notingthat the average pressure and toroidal current can be written as,


(i) Average pressure:

p ⌘

Zp dr

Zdr

= �2p 20 (1 � A)

µ0R0V

Z X dX dY (2.10a)

V =Z

dr = 2pR30

ZX dX dY. (2.10b)

(ii) Toroidal current:

µ0I =I

=0Bpdl =

Z

60µ0J�dS = � 0

R0

Z

60

1X

[A + (1 � A)X2] dX dY, (2.11)

where Bp is the magnitude of the poloidal magnetic field. We now define �p as

�p ⌘ 2µ0pB2

p(2.12a)

Bp =

I

=0Bp dl

I

=0dl

= µ0ILP

(2.12b)

LP =I

=0dl. (2.12c)

Substituting for p and I yields

�p = �4pR0L2P

V

(1 � A)

Z

60 X dX dY

⇢Z

60

1X

[A + (1 � A)X2] dX dY�2 . (2.13)

Since is a function of A but not 0, it follows that, as stated, �p =�p(A, ",, �).Note also that the volume V and poloidal plasma circumference LP are purelygeometric factors that can be calculated from the solution given by (2.6) once thecj have been determined. Furthermore, with our normalizations, the ratio R0L2

P/Vis independent of R0 and 0; that is R0L2

P/V = f (A, ", , �). This completes thediscussion of the equilibrium.

3. Resistive wall stability formulationIn this section we derive a variational principle describing the stability of n = 0

resistive wall modes in a tokamak. The analysis starts from the ideal MHD normalmode equation for plasma stability. To obtain the variational principle we will need todecompose the volume surrounding the plasma into three regions: an interior vacuumregion, a thin wall and an exterior vacuum region (Haney & Freidberg 1989). Thefinal variational principle is expressed in terms of three perturbed flux functions forthe (i) plasma, (ii) interior vacuum region, and (iii) exterior vacuum region. To obtainthis variational principle use is made of two natural boundary conditions.


The final step in the formulation is to derive relationships between the two perturbedvacuum fluxes and the perturbed plasma flux, a task accomplished by the applicationof Green’s theorem. The analysis is somewhat simplified by focusing on up–downsymmetric equilibria and considering only vertical-type displacements, which arethe most dangerous experimentally. The final form of the variational principle isstraightforward to evaluate numerically and, most importantly, directly takes intoaccount the feedback constraint � ⌧w = constant as described in the introduction.

The variational formulation, when implemented numerically, allows us to determinethe maximum elongation and corresponding triangularity as a function of aspect ratio.

3.1. The basic ideal MHD stability equationsThe starting point for the analysis is the ideal MHD linear stability equations for theplasma given by

!2⇢⇠ + F(⇠) = 0, (3.1a)F(⇠) = J1 ⇥ B + J ⇥ B1 � rp1, (3.1b)

B1 = r ⇥ (⇠? ⇥ B), (3.1c)µ0J1 = r ⇥ r ⇥ (⇠? ⇥ B), (3.1d)p1 = �⇠? · rp � � pr · ⇠ , (3.1e)

where ⇠ is the plasma displacement vector and quantities with a 1 subscript representfirst order perturbations (see for instance Freidberg (2014) and references therein). Forresistive wall modes, the inertial effects can be neglected because the correspondinggrowth rates are very slow compared to the characteristic MHD time a/VTi where VTiis the ion thermal speed. Thus, the plasma behaviour is described by

F(⇠) = 0. (3.2)

In other words, referring to the general ideal MHD formulation of the energy principleonly �W(⇠ ⇤, ⇠) is needed to describe the plasma behaviour, and !2K(⇠ ⇤, ⇠) can beignored.

Next, form the total energy integral �W in the usual way:

�W = �12

Z⇠ ⇤ · F(⇠) dr = 0. (3.3)

Using standard analysis (Sykes et al. 2001) we can rewrite (3.3) as follows

�W(⇠ ⇤, ⇠) = �WF(⇠ ⇤, ⇠) + BT = 0, (3.4a)

�WF(⇠ ⇤, ⇠) = 12µ0

Z[|Q|2 + �µ0p|r · ⇠ |2 � µ0⇠

⇤? · J ⇥ Q + µ0(⇠? · rp)(r · ⇠ ⇤

?)] dr,

(3.4b)

BT = 12µ0

Z

SP

(n · ⇠ ⇤?)[B · Q � �µ0pr · ⇠ � µ0⇠? · rp] dSP, (3.4c)

where Q = r ⇥ (⇠? ⇥ B), n is the outward unit surface normal vector, and the surfaceintegral in BT is evaluated on the plasma surface SP given by = 0. This form isgeneral, and in particular is valid for n = 0 modes in a tokamak with arbitrary profiles.


Many researchers have long believed that n = 0 stability is closely related to theproblem of neighbouring equilibria of the Grad–Shafranov equation. This is indeeda correct belief, although to the authors’ knowledge, a derivation of this connectionhas not appeared in the literature. We have derived a relationship which explicitlyshows this connection. The analysis is somewhat lengthy and is given in appendix A.The final result is a simplified form of �WF valid for up–down symmetric tokamaksundergoing vertical-like displacements; that is displacements for which ⇠Z(R, Z) =⇠Z(R, �Z) and ⇠R(R, Z) = �⇠R(R, �Z), (e.g. ⇠Z = ⇠0 = constant, ⇠R = 0). Becauseof up–down symmetry, these displacements exactly decouple from the horizontal-likedisplacements for which ⇠Z(R, Z) = �⇠Z(R, �Z), ⇠R(R, Z) = ⇠R(R, �Z). In addition,it is shown in appendix A that the most unstable displacements are incompressible;r · ⇠ = 0 Note also that for the n = 0 mode in an up–down symmetric tokamak, itfollows that ⇠ is purely real.

The simplified form of �WF is conveniently expressed in terms of the perturbedplasma flux which is related to the displacement vector by

= �⇠? · r . (3.5)

The resulting form of �WF, valid for arbitrary equilibrium profiles, is given by

�WF = 12µ0

Z (r )2

R2�✓

µ0p00 + 12R2

F200◆ 2�

dr + 12µ0

Z

SP

✓µ0J�R2Bp

2◆

dSP,

(3.6)

where, as before, a prime denotes a derivative with respect to the poloidal fluxfunction, i.e. d/d . Note that �WF is already written in self-adjoint form.

The connection to Grad–Shafranov neighbouring equilibria is now apparent. Thedifferential equation in obtained by setting the variation in �WF to zero is identicalto the neighbouring equilibrium equation obtained by letting ! + in theGrad–Shafranov equation and setting the first-order contribution to zero.

Observe that there is a boundary term arising from several integrations by partsin the derivation. This term is often zero since J�(S) = 0 on the plasma surface formany realistic profiles. However, it is non-zero for the Solov’ev profiles since theedge current density is finite and is, in fact, the main drive for vertical instabilities. Tocompensate this difficulty note that p00 = F200 = 0 for the Solov’ev profiles, implyingthat the volume contribution to �WF reduces to that of a vacuum field. In otherwords, the perturbed toroidal current density in the plasma is zero for the Solov’evprofiles. This simplification greatly outweighs the difficulty of maintaining the surfacecontribution. Specifically, it will ultimately allow us to make use of an analytic formof the Green’s function for the plasma region when evaluating �WF, a mathematicaladvantage that does not apply to general profiles.

The task now is to evaluate and simplify the boundary term BT, recasting it in aform that is automatically self-adjoint.

3.2. The inner vacuum regionThe first step in the simplification of BT in (3.4) focuses on the inner vacuum regionbetween the plasma and the resistive wall. By making use of incompressibility andrewriting (3.5) as n · ⇠? = /RBp where n is the unit normal vector to the plasmasurface, we see that the boundary term reduces to

BT = � 12µ0

Z

RBp(B · Q � µ0⇠? · rp) dSP. (3.7)


Next, assume that the perturbed magnetic field in the inner vacuum region is alsowritten in terms of a flux function i. This is convenient because the previous analysisfor the plasma region in terms of is directly applicable to the inner vacuum regionby simply setting the equilibrium J = p = 0 in the vacuum region.

Now, the jump conditions across SP require that (for incompressible displacementsand no surface currents) (Sykes et al. 2001)

i|SP = |SP, (3.8a)

B · Bi|SP = (B · Q � µ0⇠? · rp)|SP, (3.8b)

with B the equilibrium vacuum field in the inner vacuum region. At this point, the firstnatural boundary condition is introduced into the formulation by substituting (3.8) into(3.7)

BT = � 12µ0

Z

SP

i

RBp(B · Bi) dS. (3.9)

Continuing, in the inner vacuum region the total (i.e. equilibrium plus perturbation)magnetic field can be written as

BTot = 1R

r( + i) ⇥ e� + F( + i)

Re�. (3.10)

Since F( + i)= R0B0 = constant in a vacuum region it follows that the perturbationFi = (@F/@ ) i = 0. Thus the perturbed magnetic field in the inner vacuum regionfor an n = 0 perturbation is given by

Bi = 1R

r i ⇥ e�, (3.11)

from which it follows that

B · Bi|SP = 1R2

(r · r i)|SP = Bp

R(n · r i)|SP . (3.12)

Here we have used the equilibrium continuity relation B(SP) = B(SP) which is validwhen no surface currents are present. The boundary term can thus be written as

BT = � 12µ0

Z

SP

1R2 i(n · r i) dSP. (3.13)

The last part of this first step is to recognize that the magnetic energy in the innervacuum region (with subscript i) can be written as

�WVI = 12µ0

Z(Bi)

2 dr = 12µ0

Z(r i)

2

R2dr, (3.14)

with i satisfying�⇤ i = 0. (3.15)


Using (3.15) we can easily convert (3.14) into two surface integrals, one on the plasmasurface SP and the other on the inner wall surface SW ,

�WVI = 12µ0

Z(r i)

2

R2dr = 1

2µ0

Z

SW

1R2 i(n · r i) dSW � 1

2µ0

Z

SP

1R2 i(n · r i) dSP.

(3.16)Here and below, n always refers to an outward pointing normal. This relation allowsus to write (3.13) as

BT = �WVI � 12µ0

Z

SW

1R2 i(n · r i) dSW . (3.17)

3.3. The outer vacuum regionThe boundary term can be further simplified by introducing the magnetic energy inthe outer vacuum region. In analogy with (3.14)–(3.16) we can write the outer vacuumenergy (with subscript o) as

�WVO = 12µ0

Z(Bo)

2 dr = 12µ0

Z(r o)

2

R2dr

= 12µ0

Z

S1

1R2 o(n · r o) dS1 � 1

2µ0

Z

SW

1R2 o(n · r o) dSW . (3.18)

The contribution at the surface at infinity, S1, vanishes because of the regularityboundary condition far from the plasma. Thus, (3.18) reduces to

�WVO + 12µ0

Z

SW

1R2 o(n · r o) dSW = 0. (3.19)

This expression is now added to (3.17) leading to

BT = �WVI + �WVO + 12µ0

Z

SW

1R2

[ o(n · r o) � i(n · r i)] dSW . (3.20)

In (3.20), it is understood that the i terms are evaluated on the inner surface of thewall while the o terms are evaluated on the outer surface.

Equation (3.20) is in a convenient form since in the limit of a thin wall, the surfaceterms are ultimately transformed into a simple set of jump conditions arising from thesolution for the fields within the resistive wall.

3.4. The fields within the resistive wallThe perturbed fields within the resistive wall are determined as follows. We write

Bw = r ⇥ Aw (3.21a)Ew = ��Aw � rVw = ��Aw (3.21b)

Jw = �Ew. (3.21c)

Here, � is the wall conductivity, the scalar electric potential Vw has been set to zeroas the gauge condition and all quantities are assumed to vary as Q(r, t)= Q(r) exp(� t).Ampere’s law then can be written as

r ⇥ r ⇥ Aw = �µ0��Aw. (3.22)


Now, form the dot product of (3.22) with e�/R and define Aw · e� = w/R. A shortcalculation for n = 0 symmetry yields

r2 w � 2R

r w · rR = µ0�� w. (3.23)

It is at this point that the thin wall approximation is introduced in order to obtainan analytic solution for w. Assume that the wall thickness is denoted by d whilethe minor radius of the wall at Z = 0 is denoted by b ⇠ LW/2p, where LW is thewall circumference. The thin wall approximation assumes that d/b ⌧ 1. The thin wallordering is then given by

µ0� �bd ⇠ � ⌧w ⇠ 1 (3.24a)

n · r ⇠ 1d

(3.24b)

t · r ⇠ 1b. (3.24c)

The physical interpretation is as follows. The growth time � �1 ⇠ µ0�bd ⇠ ⌧w isthe characteristic diffusion time of magnetic flux through a wall of thickness d andconductivity � into a vacuum region of thickness b. The unit vector n is the outwardnormal to the wall and the ordering n · r ⇠ 1/d implies rapid variation across thewall. Similarly, the unit vector t is tangential to the wall in the poloidal direction.The ordering t · r ⇠ 1/b implies that tangential variation is slower (of the scale ofthe device size) than normal variation.

The analytic solution to (3.23) can now be obtained using the resistive wall analogof the ‘constant ’ approximation that arises in tearing mode theory (Furth, Killeen &Rosenbluth 1963). We define a tangential poloidal arc length l and a normal distances measured with respect to the inner surface of the resistive wall such that 0 < s < d.This implies that

n · r = @

@s(3.25a)

t · r = @

@l. (3.25b)

The thin wall expansion for w is then written as

w(s, l) = (l) + (s, l) + · · · , (3.26)

where / ⇠ d/b. After some analysis, we find that the first non-vanishingcontribution to (3.23) is given by

@2

@s2= µ0�� . (3.27)

This equation can be easily integrated leading to the following analytic solution for w

w(s, l) =✓

1 + µ0��s2

2+ c1 + c2

sd

◆ , (3.28)

where c1(l), c2(l) are two free, order d/b integration constants arising from the solutionto (3.27).


The critical information to extract from these solutions is the change in w and itsnormal derivative across the resistive wall. These values are given by

w(d, l) � w(0, l) =

✓↵d2

+ c2

◆⇠ d

b ⇡ 0 (3.29a)

n · r w(d, l) � n · r w(0, l) = @ w(d, l)@s

� @ w(0, l)@s

= ↵ . (3.29b)

Here, ↵= �µ0�d ⇠ 1/b is related to the resistive wall growth time. Observe that thereis a small, negligible, jump in the flux across the wall. There is, however, a finite jumpin the normal derivative, representing the current flowing in the wall.

The information in (3.29) is related to the inner and outer vacuum solutions byrecognizing that even though the wall is thin, it still has a small finite thickness. Theimplication of finite thickness is that there are no ideal infinitesimally thin surfacecurrents on either face of the wall. Therefore, across each face of the wall, theflux and its normal derivative must be continuous with the adjacent vacuum fields.Specifically, coupling from the wall to the vacuum regions requires that

Inner surface Outer surfaceh i(s, l) � w(s, l)

i

s=0= 0

h o(s, l) � w(s, l)

i

s=d= 0

hn · r i(s, l) � n · r w(s, l)

i

s=0= 0

hn · r o(s, l) � n · r w(s, l)

i

s=d= 0.

9>>=

>>;(3.30)

By combining (3.29) and (3.30), we obtain a set of jump conditions on the vacuummagnetic fields that takes into account the effects of the resistive wall. Simplesubstitution yields

o(d, l) � i(0, l) = 0 (3.31a)n · r o(d, l) � n · r i(0, l) = ↵ i(0, l). (3.31b)

This is the information required to complete the variational principle.

3.5. The final variational principleReturn now to the expression for the boundary term given by (3.20). By substitutingthe top relation in (3.31) we obtain an expression for the boundary term in the thinwall limit that can be written as

BT = �WVI + �WVO + 12µ0

Z

SW

i

R2[(n · r o) � (n · r i)] dSW . (3.32)

The second natural boundary condition is introduced into (3.32) by substituting thebottom relation from (3.31)

BT = �WVI + �WVO + ↵

2µ0

Z

SW

2i

R2dSW . (3.33)

Observe that the last term is now in a self-adjoint form.By combining (3.33) with (3.4) we finally obtain the desired variational principle

�W = �WF + �WVI + �WV0 + ↵WD = 0 (3.34a)


�WF = 12µ0

Z

VP

(r )2

R2�✓

µ0p00 + 12R2

F200◆ 2�

dr + 12µ0

Z

SP

✓µ0J�R2Bp

2◆

dSP

(3.34b)

�WVI = 12µ0

Z

VI

(r i)2

R2dr (3.34c)

�WVO = 12µ0

Z

VO

(r o)2

R2dr (3.34d)

WD = 12µ0

Z

SW

2i

R2dSW . (3.34e)

The variational principle is very similar in form to ideal MHD. When substitutingtrial functions, all that is necessary is to ensure that the perturbed fluxes arecontinuous across the plasma-vacuum interface and across the resistive wall. Thenormal derivative requirements are automatically accounted for by means of thenatural boundary conditions.

4. Summary

We have presented a general formulation of the n=0 resistive wall stability problem.The end result is a variational principle which, as shown in the accompanying paper, isquite amenable to numerical analysis. The key new feature introduced in the analysisis the presence of a feedback system. A simple but reliable measure of the effect offeedback is determined by calculating the maximum stable and corresponding � forfixed � ⌧w / ↵b⇤. Numerical results obtained with this formalism and implications forreactor designs are given in the accompanying paper.

Acknowledgements

The authors would like to thank Professor D. Whyte (MIT) for providingthe motivation for this work and for many insightful conversations. J. P. Leeand A. J. Cerfon were supported by the U.S. Department of Energy, Office ofScience, Fusion Energy Sciences under Award Numbers DE-FG02-86ER53223 andDE-SC0012398. J. P. Freidberg was partially supported by the U.S. Departmentof Energy, Office of Science, Fusion Energy Sciences under Award NumberDE-FG02-91ER54109.

Appendix A. Relation between MHD stability and neighbouring equilibriaA.1. Introduction

We present a derivation of the fluid contribution (�WF) to the total potential energy(�W) describing the n = 0 stability of a tokamak. The derivation is presented forarbitrary profiles and then simplified at the end for the Solov’ev profiles. Thederivation requires a substantial amount of analysis. However, the final result isquite simple and is shown to be directly related to neighbouring equilibria of theGrad–Shafranov equation.

⇤The precise definition of the wall diffusion time, ⌧w = µ0�dLW/2p with LW the wall circumference, willappear naturally during the discussion of the numerical analysis.


A.2. The starting pointThe starting point for the analysis is the expression for the total MHD potential energy�W written in the ‘standard’ form (see (3.4)) (Sykes et al. 2001)

�W(⇠ ⇤, ⇠) = �WF(⇠ ⇤, ⇠) + BT, (A 1a)

�WF(⇠ ⇤, ⇠) = 12µ0

Z[|Q|2 + �µ0p|r · ⇠ |2 � µ0⇠

⇤? · J ⇥ Q + µ0(⇠? · rp)(r · ⇠ ⇤

?)] dr,

(A 1b)

BT = 12µ0

Z

SP

(n · ⇠ ⇤?)[B · Q � �µ0pr · ⇠ � µ0⇠? · rp] dSP, (A 1c)

where Q = r ⇥ (⇠? ⇥ B). The quantity �WF is the fluid contribution while BT is theboundary term that will ultimately be related to the vacuum energy and the resistivedissipated wall power. Note also that for the n = 0 mode in an up–down symmetrictokamak we can assume that ⇠ is purely real.

The present analysis is focused solely on obtaining a simple form for �WF. Observethat even though our goal is to calculate the eigenvalue ! for resistive wall modes, wecan neglect the !2 contribution due to the plasma inertia. The reason is that resistivewall growth rates are much slower than ideal MHD growth rates, thus justifying theneglect of MHD inertial effects. This is the explanation of why only �W(⇠ , ⇠) and not!2K(⇠ , ⇠) is needed to describe the plasma behaviour. The eigenvalue ! will appearin the evaluation of BT.

A.3. IncompressibilityThe first step in the analysis is to examine the plasma compressibility term. As iswell known this is the only term in which ⇠k appears. The question is whether ornot a well behaved ⇠k can be found that makes r · ⇠ = 0 which, if possible, clearlyminimizes the plasma compressibility term.

To answer this question we have to make use of the following symmetries withrespect to the Z dependence of the equilibrium magnetic fields

BR(R, �Z) = �BR(R, Z) (A 2a)BZ(R, �Z) = +BZ(R, Z) (A 2b)B�(R, �Z) = +B�(R, Z). (A 2c)

Now, for the n = 0 mode, a general perturbation ⇠ can be always be written as thesum of an ‘even ⇠Z’ contribution plus an ‘odd ⇠Z’ contribution, which are completelydecoupled from one another because of the equilibrium symmetry. The perturbationsymmetries are as follows,

Even ⇠Z symmetry Odd ⇠Z symmetry⇠Z(R, �Z) = +⇠Z(R, Z) ⇠Z(R, �Z) = �⇠Z(R, Z)

⇠R(R, �Z) = �⇠R(R, Z) ⇠R(R, �Z) = +⇠R(R, Z)

⇠�(R, �Z) = +⇠�(R, Z) ⇠�(R, �Z) = �⇠�(R, Z)

⇠k(R, �Z) = +⇠k(R, Z) ⇠k(R, �Z) = �⇠k(R, Z)

r · ⇠?(R, �Z) = �r · ⇠?(R, �Z) r · ⇠?(R, �Z) = +r · ⇠?(R, �Z).

9>>>>>>=

>>>>>>;

(A 3)

The n = 0 modes of interest have even ⇠Z symmetry (e.g. ⇠Z = constant) correspondingto ‘vertical displacements’. The less interesting modes have odd ⇠Z symmetry (e.g.⇠R = constant) and represent ‘horizontal displacements’.


For the vertical displacements of interest, the most unstable modes are alwaysincompressible. To show this we note that for a minimizing incompressibledisplacement ⇠k must satisfy

B · r ⇠kB

= Bp · r ⇠kB

= Bp@

@l⇠kB

= �r · ⇠?, (A 4)

where Bp is the poloidal magnetic field and l is poloidal arc length. Thus, for ⇠k tobe well behaved (i.e. be periodic) it must satisfy the periodicity constraint

0 =I

@

@l

✓⇠kB

◆dl = �

I r · ⇠?Bp

dl = �I r · ⇠?

BZdZ. (A 5)

This constraint is automatically satisfied for the even ⇠Z symmetry displacement.The conclusion is that the plasma compressibility term in �WF is minimized by

choosing ⇠k to satisfy (A 4) leading to the result � p(r · ⇠)2 = 0. The remainder of�WF is only a function of ⇠?

A.4. Reformulating �WF in terms of the vector potentialThe evaluation of �WF is more conveniently carried out in terms of the perturbedvector potential A?1 rather than the plasma displacement ⇠?. The familiar relationbetween these two quantities is

A?1 = ⇠? ⇥ B ⇠? = B ⇥ A?1

B2. (A 6)

In general A?1 can be vector decomposed as follows

A?1 = f (R, Z)r + g(R, Z)Bp + h(R, Z)e�. (A 7)

Here, (R, Z) is the equilibrium flux function satisfying the Grad–Shafranov equation.We next make use of the constraint A?1 · B = 0 and introduce the perturbed flux (R, Z) = RA�1(R, Z) = Rh(R, Z). A short calculation then enables us to rewrite A?1as

A?1 = f r � F R2B2

pBp +

Re�, (A 8)

where F( ) = RB� . The basic unknowns in the problem are , f .Before proceeding with the evaluation of �WF, it is useful to list a number of

relations involving A?1 that enter the analysis. These are relatively straightforward toderive:

Q = r ⇥ A?1 = 1R

r ⇥ e� + rf ⇥ r +"

1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#e�,

(A 9a)

Q · A?1 =

R

"1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#� F

R4B2pr · r

+ 2 (Bp · rf ) � r · ( f Bp), (A 9b)


Q · B = 1R2

r · r + FR

"1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#+ F(Bp · rf ), (A 9c)

r · (e� ⇥ A?1) = � F R3B2

pr · r � Rf Bp · r , (A 9d)

⇠? · rp = �p0 , (A 9e)A?1 · (r ⇥ B) = �B2 , (A 9f )

with primes denoting d/d as in the main text. We are now ready to evaluate �WF

A.5. Evaluation of �WF

There are three terms appearing in �WF as given by (A 1) after setting r · ⇠ = 0. Using(A 9), these terms can be evaluated in a straightforward way. The magnetic energyterm is given by

(Q)2 = (r )2

R2+"

1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#2

+ (RBp · rf )2 + 2(RBp · rf )

"1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#. (A 10)

The current term is evaluated by making use of the equilibrium relation µ0J = F0B +µ0Rp0e� which leads to a fairly complicated expression

�⇠? · (µ0J ⇥ Q) = �✓

F0 + Fµ0p0

B2

◆(A?1 · Q) + µ0p0

B2 (B · Q)

= �✓

F0 + Fµ0p0

B2

◆(

R

"1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#

� F R4B2

pr · r + 2

R(RBp · rf ) � r · ( f Bp)

)

+ µ0p0 B2

(1R2

r · r + FR

"1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

#

+ FR

(RBp · rf )

). (A 11)

The remaining pressure term requires some minor algebraic manipulations. Theresult is

µ0(⇠? · rp)(r · ⇠?) = µ0r · [(⇠? · rp)⇠?] � µ0⇠? · r(⇠? · rp)

= �µ0r · (p0 ⇠?) + µ0p00 B2

[A?1 · (r ⇥ B)] + µ0p0

B2[A?1 · (r ⇥ B)]

= �µ0r · (p0 ⇠?) � µ0p00 2 + µ0p0FRB2

[r · (e� ⇥ A?1)] � µ0p0 R2B2

(r · r )

= �µ0r · (p0 ⇠?) � µ0p00 2 � µ0p0 R2B2

p(r · r ) � µ0p0F

RB2[f (RBp · r )]. (A 12)


The next task is to sum these three complicated contributions and simplify the result.The terms in the sum can be collected and written as follows

�WF(⇠ , ⇠) = 12µ0

Z[(Q)2 � µ0⇠? · J ⇥ Q + µ0(⇠? · rp)(r · ⇠?)] dr = 1

2µ0

ZU dr

(A 13a)U = U1(f , f ) + U2(f , ) + U3( , ). (A 13b)

Here, each Uj is a combination of algebraic and differential operators acting on theterms appearing in the arguments. After a short calculation, the first two terms on theright-hand side can be simplified, leading to

U1(f , f ) + U2(f , ) = (RBp · rf )2 + r · (F0f Bp)

+ 2(RBp · rf )

"1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

� F0 R

#. (A 14)

The divergence term integrates to zero over the plasma surface. The f dependenceof the remaining terms involves only (RBp · rf ). These terms are simplified bycompleting the square on (RBp · rf ). The fluid energy �WF is then minimized bychoosing

(RBp · rf ) = � 1R

r · r

F R2B2

p

!+ µ0J�F

R2B2p

+ F0 R

. (A 15)

Note that as for the incompressibility analysis, the periodicity constraint on (RBp ·rf )is automatically satisfied for the perturbation with even ⇠Z symmetry. The end resultis that

U1(f , f ) + U2(f , ) = �"

1R

r · r

F R2B2

p

!� µ0J�F

R2B2p

� F0 R

#2

, (A 16)

which is only a function of .The complete integrand U is now evaluated by adding (A 16) to U3( , ). After

another calculation, we obtain

U = (r )2

R2�

µ0p00 + F02

R2+ FF0µ0J�

R3B2p

! 2 � µ0r · (p0 ⇠?)

+ F0 R2

"r · r

F

R2B2p

!+ F

R2B2p(r · r )

#. (A 17)

After some further manipulations that make use of the equilibrium Grad–Shafranovequation, we find that the term on the second line of (A 17) can be rewritten as

F0 R2

"r · r

F

R2B2p

!+ F

R2B2p(r · r )

#

= r ·

FF0 2

R4B2p

r !

�

FF00

R2� FF0µ0J�

R3B2p

! 2. (A 18)


Equation (A 18) is substituted into (A 17). The divergence terms are converted intosurface integrals and then simplified using the equilibrium Grad–Shafranov equation.This leads to the final desired form of �WF

�WF = 12µ0

Z (r )2

R2�✓

µ0p00 + 12R2

F200◆ 2�

dr + 12µ0

Z

SP

✓µ0J�R2Bp

2◆

dS. (A 19)

There are three points worth noting. First, a simple variational analysis shows that thefunction that minimizes the volume contribution satisfies

�⇤ = � �R2µ0p00 + 12 F200� . (A 20)

This equation is identical to the perturbed Grad–Shafranov equation corresponding toneighbouring equilibria. Second, the surface contribution is non-zero only if there isa jump in the edge-current density. For many profiles, J� is zero at the plasma edgebut it is finite for the Solov’ev profiles. Third, there is a great simplification in thevolume contribution for the Solov’ev model in that p00 = F200 = 0. Thus the fluid energyfor the Solov’ev model reduces to

�WF = 12µ0

Z (r )2

R2

�dr + 1

2µ0

Z

SP

✓µ0J�R2Bp

2◆

dS, (A 21)

with satisfying�⇤ = 0. (A 22)

Assuming a solution to this equation can be found, then the volume contribution canbe converted into a surface integral. The fluid energy can then be written solely interms of a surface integral

�WF = 12µ0

Z

SP

✓1R2 n · r + µ0J�

R2Bp 2◆

dS. (A 23)

This form is actually valid for any equilibrium profile. The beauty of the Solov’evprofiles is that n · r on the plasma surface can be conveniently evaluated in termsof on the surface using Green’s theorem, since we know the analytic form of theGreen’s function corresponding to equation (A 22). For general profiles, this procedureis not as convenient since we do not know the Green’s function in a simple analyticform.

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