J. Plasma Fusion Res. SERIES, Vol.5 (2002) 278-282

Analytic Modeling of the Feedback Stabilization ofResistive Wall Modes
PUSTOVITOV Vladimir D.
Russian Research Centre "Kurchatov Institute", Moscow 123182, Russia(Received: ll December 20Ol / Accepted: 15 May 2002)

Abstract
Feedback suppression of resistive wall modes (RWM) is studied analytically using a model based ona standard cylindrical approximation. Optimal choice ofthe input signal for the feedback, effects related

to the geometry of the feedback active coils, RWM suppression in a configuration with ITER-like double

wall, are considered here. The widespread opinion that the feedback with poloidal sensors is better thanthat with radial sensors is discussed. It is shown that for an ideal feedback system the best input signal
would be a combination of radial and poloidal perturbations measured inside the vessel.

Keywords:
resistive wall mode (RWM), cylindrical approximation, feedback suppression

1. Introductionof magnetic sensors was earlier studied numerically[2,3,5,6]. The conclusion, stated in ref. 12) andResistive wall modes (RWM) can limit achievablebeta below acceptable level in advanced tokamaks withconfirmed in refs. t3,5,61, was that the feedback systemwith sensors measuring the poloidal field is much better
than that with the radial-field sensors. With solid proofs,

low internal inductances [-3]. Experiments in the DIII-

D tokamak show two directions for stabilizing theRWM in high-beta tokamaks: toroidal plasma rotation

including experimental confirmation [5], the conclusion
cannot raise any doubt. But some basic questions remainunanswered. For example, it follows from divB = 0 thatpoloidal and radial components of the plasma-produced
helical magnetic perturbation must be (approximately)
equal. Why then 'sensors measuring the poloidal field
perturbations are superior to radial sensors' [3]? Is it ageneral rule or a property of the considered feedback
schemes? The problem is analyzed here analyticallyand active feedback control of the mode using magnetic

coils. At the presence ofthe uncorrected resonance error
field, the rotation alone cannot completely suppress theinstability, urging the need of the feedback suppression
for achieving high 0 tal.
Simulations show that a feedback control of RWM
can significantly raise the n = I ideal MHD beta limit,
up to B1 = 5, doubling the no-wall limit [3]. Though
theory was always optimistic, it took several years ofusing a model based on a standard cylindricalstudies in DIII-D [1,4] before it was reported [5] that, inapproximation.

experiments with active control, it became possible tosustain a discharge at pressures approaching twice theno-wall limit. This success in 2001 has been explained[5] as resulted owing largely to an extensive new set ofmagnetic sensors installed inside the vacuum vessel.
The problem of proper positioning and orientation

2. General Equations in Gylindrical
Approximation
Feedback stabilization of RWM is often
analyzed in cylindrical approximation [6-12]. Such@2OO2 by The Japan Society of Plasma

Corresponding author's e-mail: pustovit@ nfi.kiae.ru

Science and Nuclear Fusion Research278

Pustovitov V.D., Analytic Modeling of the Feedback Stabilization of Resistive Wall Modes

analysis is in a good qualitative agreement with theobserved feedback characteristics in DIII-D tokamak[6]. The use of cylindrical model is also justified by+=-1r+comparison of the model predictions with results oftoroidal computations [9].
The perturbed magnetic field must be calculatedr)-

2K lo x2*2rc+ loll-*,*)

0)

For plasma-facing wall, this gives |1 through rb'y'bp on
the outer plasma surface, x = alrw. The radial magnetic
field is a continuous function, the same is true for bi if
surface currents are absent. Then at x = alr* the left

with account of currents in all active and passive
elements and with proper matching at the dividing

hand side of eq. (7) is expressed through b, in thesurfaces: plasma boundary, walls, and active coils. In
the model, the surfaces are coaxial cylinders.
In the vacuum gaps between the dividing surfaces,plasma.

In the plasma, D. must be found from MHD
equations. It is known that, in cylindrical geometry, they

the perturbed magnetic field can be described by its

are reduced to the equation [13] (see also ref. [12] andradial component:

references therein)b,=2 boQ)exp(iko- in(+ yt).

v2-[ir\ a)mi'(rl'r' v-'tpv,- Fl=-vittt+ p,""# ttt.

(1)

Here r, 0 and z = RC are the usual cylindrical coordinates (2nR is the total length of the system), 7is thewhere rf = yr(r)exp(im? - in( + yt), the perrurbed magnetic field is given by E = Y ry x 2,2 is the unit vector
along the z-axis of the cylindrical coordinate system r,e, z= R(, p = (B/r)[m - nq(r)], B;(r) is the equilibrium
poloidal freld, q is the safety factor, p is the plasma density, j is the z-component of the current density, Vr = V

growth rate, , is the time.
At the resistive wall, considered as a thin shell r =r*, two conditions must be satisfied:[r-]=0,[rui]=?Bo.

(8)e)

Here [X] = Xl'-**lis a jump in the function across theI rw-0
wall, the prime is the radial derivative, i = TT*, r* =- 2t2:v).

respectively, its conductivity and thickness, B1 = b*(r*)
is the amplitude of the ft th harmonic of the perturbation
at the wall. More details are given in ref. [2].
The contributions of inner and outer sources to b,

This is the simplest equation for the magnetic
perturbation in the plasma. But even the utmost
simplification does not help to obtain a general
expression for b,: analytical solutions of eq. (8) are
known for uniform or parabolic [14] current profiles
only. Therefore, instead of eq. (7), another way of

have different l'adial behavior:prescribing ,l-o is needed in a general case.

lhcnd*r* is the time constant of the wall, o* and d- are,bi = 3'; t-*-r ,bT'= B3'x*-t ,Equation (8) implies that parameters Q are(3)determined by the current density. If its profile is given,
Q can be prescribed as some constants characterizing
this particular profile. These constants can be evaluated

where x = rlrwr K= lkl, and B;'and Bf;ut are the values ofbf and bt"' = bt - b'f at r - rw, respectively. These expressions, with continuity at r - r*, describe also thefrom experimental data or numerical results. We need toknow what happens with 1o when the feedback isswitched on. It seems natural to assume that thefeedback field does not change the current density. If so,
the solution of eq. (8) and the shape of b,(r) in theplasma would remain unaffected by the feedback, and|1 can be considered as fixed constants independent of
the feedback field.
This conclusion is justified by results of toroidal
calculations [2,3] showing that the feedback does notstrongly modify the perturbation in the plasma. This was

self-field of the wall in the inner (r < r*) and outer regions. Thus, at the outer side of the wall
rbtrl- --(r+ l) Bo +2rBi't,.^^trw+u(4)where Bf*' = bfu'|,**o is the part of Bo due to the field
produced by all sources outside the wall: by other walls
at r > rw, if any, and by the feedback system.

With eq. ('1) and eq. (2) lead to(i - f) Bo= 2rcBi*'(s)with Q defined as

fo: - Q b'o / b o)1,

confirmed in DIII-D experiments and related numerical*_o-(r + 1) .

analysis, and the observed invariance of the mode
structure during the feedback process was called mode

(6)

In the vacuum gap inside the shell, r < r*,rigidity [6].
The system eq. (5) with constant 1-o is a starting279

Pustovitov V.D., Analytic Modeling of the Feedback Stabilization of Resistive Wall Modes

point for analysis. It must be supplemented by an

the same helical harmonic as the plasma-generated
perturbation, and nothing more. This would require
stellarator-like helical windings. In practice, thecorrection coils consisting of frame-like rectangularsegments [1-9] are always considered for tokamaks.

algorithm prescribing the feedback-produced magnetic
field as a function of some input signal combined of themagnetic signals measured outside the plasma.

Such coils, called below conventional, allow the desired
RWM suppression in theory and in DIII-D experiments,
but they are not optimal: RWM has a helical structure
while the rectangular active coils are aligned toroidally.
Non-optimal active coils generate side-band harmonics,

3. One Resistive Oall, ldeal Feedback
If only one resistive wall is present, the external
field in eq. (5) is the field produced by the feedback
system, Bt*'= Bf,. For a single unstable mode k = m ag.
(5) gives
V = f^+2pBt^ | B^,

different from the intrinsically unstable principle
harmonic. This harmonics can contribute to themeasured input signals 1, affecting the dispersion(9)

where p = lml, l^ > 0 is the no-feedback growth rate,and B^ is the amplitude of the mode.
RWM is suppressed when / < 0. Assuming asimple proportional control

Bt^=- Kx I ,relation eq. (9) through eq. (10) or another feedback
algorithm.

The magnetic field br = Y Vt x f produced by aconventional feedback system with correction coils
similar to those used in DIII-D has a property Vr(e) =-rftee). Accordingly,(10)

we compare the feedback stabilization with input signals

l from the radial probes in the mid-plane (1.), and frompoloidal probes measuring -bs(r,0) inside and outside
the vessel (ft' and 1f").

An ideal feedback system would create only

br,1e; = bt,ee) ,(13)brr1el = btr(-o) .

(14)

It is known that even in the case of DIII-D (six active

necessary (m,n) harmonic. In this casecoils each covering a 60-degree toroidal arc) the feed-I,=B^'The values 1$' and lfut are expressed through B. usingback field can be modeled by a single harmonic in toroidal angle EMBED [1]. Thus, the toroidal discretenesscan be disregarded in the analysis.
A real feedback system in addition to necessary

eqs. (l)-(3) and divB = 0.

(m,n) harmonic generates a number of side-bandI'; = (l + f^l1t) B^,

I3'= I't -i n.tp .

(1l)

It follows from eq. (9) that any choice from eq.

harmonics. Conventional feedback has a property eq.

(11) would allow RWM suppression with algorithm eq.
(10). For example, for I = I, the stability criterion is( l3), so that bt, spectrum is symmetric. bt* = bt-*.
Therefore, even if a system is somehow optimized forstabilizing (m,n) mode, at least (-m,n) side-band
harmonic in b| must be taken into account. With two
harmonics of the field from active coils, eq. (9) must be

K>Ko=f^l(2lt).

(r2)

The largest ofthe three signals in eq. (11) is ljn. Larger
signal can be detected easier, and feedback stabilization
can start earlier. This is an advantage of the feedback
with poloidal sensors inside the vessel. But it hardly canconsidered together withV=f ^+2pB:.lB_-.

explain the observed essential superiority [2,3,5,6] ofthis system over that with radial sensors, since I differsfrom Ij" by a constant multiplier only. This difference
results in larger gain K for a system with radial sensors,in other respects both feedback systems are equivalent.
This is true for the ideal case. But relations eq. (l 1)
may not be valid for a real feedback.

(1s)

Instability of a single mode means that without feedback/i->0, but 11.<0.
If the feedback field is described by two harmonics,(m,n) and (-m,n), and Br-^= Bf^,

i-t' I"
I-=B-.*B--=28-+'r -m'--m--^l_^'
V_(16)

where

4. One Wall, Gonventional Feedback, Radial
VSensors",=0'51f--+ f^) '(r7)With eq. (16) and eq.(10), eq. (9) turns into quadratic

In the ideal case, the feedback system must produce
280

Pustovitov V.D., Analytic Modeling of the Feedback Stabilization of Resistive Wall Modes

equation for i:1,2feedback, while for the walls inside, B"*t also includes afield produced by all walls screening a given wall from+2iQltK - V")f^ l* -41tK ir.,=0. (18)the active coils. The radial magnetic field is a

For stability, both roots of (18) must be negative. Thisrequires 1",<(l andK t fo(t -0.5f^1V,,).

continuous function described by eq. (3) between thewalls.

For a plasma column surrounded by two resistive
walls of radii 11 and 12 the dispersion relation is [12](1e)The necessary gain becomes infinitely large when i'".
-+ 0 because the measured input signal 1, vanishes at i=i'",. When i,", becomes positive, the unstable (m,n) modecannot be stabilized by the conventional feedback described by (10) with 1= 1".

V'-V(f*-wLr')-21tLf (w - l)(Ko + BllB)= Q .

(23)

Here B> = b^(r) is the perturbation amplitude at the firstwall, 81 is the part of 82 created by the feedback system,

5. One Wall, Conventional Feedback,Internal Poloidal Sensors
If the feedback field is described by two harmonics
with Bt-= 3f^,

It'\r
Fr:-R-n\-'ltl
lr*/ l'-r-'--'d7-f_*'

Ks is defined by eq. (12),w =l + xltrltr,

and x2: r2lrr. One-wall case eq. (9) corresponds to theIt follows from eq. (23) that, with radial sensors(20)

and proportional control eq. (10), the condition of theRWM suppression is again eq. (12), but now it is valid

for modes only withr:) = - 2K (v + tt)(f^ - r:^). QI)

f^-wLl< O.
(2s)
In the opposite case, f^ - wA,f > 0, suppression ofThe growth tate i becomes negative when K >RWM is impossible. If this is true for the ideal feedback, it must be even more true for the conventional
feedback with radial sensors.

max{K1,K2}, whereK,=r-etry K2=+\'

Q4)

limit w -+ - in (23).

With this input signal and proportional control (10), eq.
(9) tums into(v - r^)(/ -a,f =2Hl(x'N -D,(22)

Ks is given by eq. (12) and i", by eq. (17).

7. Summary

This scheme is much better than the feedback withradial sensors since it allows suppression of RWM
without restrictions on plasma parameters, and smaller

The analysis gives a natural explanation of theobservation that, for the RWM feedback control,'sensors measuring the poloidal field perturbations aregain is required.

superior to radial sensors' [3].
The poloidal component of the perturbation inside
the vessel is always larger than the radial one, see eq.
(ll). Therefore, internal poloidal sensors are better for
the RWM feedback control since a measured (input)
signal must be above some detection level. From thisviewpoint, the best input signal would be a combination
I[" + 1,.

6. Two Walls, ldeal Feedback, RadialSensors
Predictions are quite encouraging for tokamaks
with a single resistive wall. However, the ITER isdesigned with two separate resistive walls (p. 2593[5]). So, it is natural to analyze the difference between

An ideal feedback system could be effective inthe one-wall and double-wall cases.

suppressing RWM with either 1, or 1j,', just different

The second wall increases the total resistive decay
time decreasing thereby the RWM growth rate. Also, thesecond wall acts as a screen affecting penetration of thegains K would be needed. But realistic feedback systemgenerates side-band harmonics that influence themeasurements. As a result, the conventional feedback
system with radial sensors cannot suppress RWM whenV", > O. This value eq. (17) is determined by plasma

field from the active coils to the first wall. This canweaken the stabilizing influence of the feedback system.

If there are several resistive walls, eqs. (5) and (6)must be applied successively to each wall. For theoutermost wall, B"^t is the field produced by theparameters and can vary during the discharge evolution.

It can lead to increase in the necessary gain eq. (19)28r

Pustovitov V.D., Analytic Modeling of the Feedback Stabilization of Resistive Wall Modes
EPS 200 I /fin/pdf/P4.008.Pdf

above the permissible level. That may be a reason of theloss of the control observed in DIII-D experiments withradial sensors [1]. The weak point of such a feedback is

16l M. Okabayashi et al., Phys. Plasmas 8' 2071
(2001).

l7l R. Fitzpatrick, Phys. Plasmas 8, 871 (2001).

aggravated, even for the ideal feedback, when a second
resistive wall is present.

[8] Hogun Jhang, S.H. Ku and Jin-Yong Kim, Phys.
Plasmas 8, 3107 (2001).

However, with internal poloidal probes thel9l A. Bondeson et al., to be published in Nucl. Fusion

conventional feedback allows suppression of RWM
without fail and at rather modest gains, see eq. (22)' The
difference between two cases, related to properties eqs.
(13) and (14), may be a reason of the dramatic
improvements in active control in DIII-D experiments in42 (2002\.

tl0l A.B. Mikhailovskii and B.N. Kuvshinov, Plasma
Phys. Rep. 22,172 (1996).

tlll M. Okabayashi et al., Nucl. Fusion 38, 1607
(1998).

2001 [5].

tl2l V.D. Pustovitov, Plasma Phys. Rep. 27, 195(2001).

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tl3l J.A. Wesson, Tokamaks (Clarendon Press, Oxford,

tll A. Garofalo er al., Nucl. Fusion 40,l49l (2000).
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[14] A.B. Mikhailovskii, Instabilities in a Confined
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//www. iaea.org/programmes/ripc/physic s

t5l L.C. Johnson et al., http://www.cfn.ist.utl.pt/282