J. Plasma Fusion Res. SERIES, Vol.5 (2002) 4249

Resistive Wall Mode Control on the Dlll-D Device
OKABAYASHI Michio, BIALEK Jamesr, CHANCE Morrell. CHU Mingr. FREDRICKSON Eric,
GAROFALO Andrear, HATCHER Ronald, JENSEN Torkil2, JOHNSON Larry, LA HAYE Robertr,
LAZARUS Edward3, MANICKAM Janardhan, NAVRATIL Geraldl, SCOVILLE John Timz,STRAIT Edward2. TURNBULL AIan2. WALKER Michael2 and DIII-D Team2
Princeton Plasma Physics Laboratory, Princeton, New Jersey' Columbia University, New York, New York
2

General Atomics, P.O. Box 85608, San Diego, Califurnia 92186-5608
3

Oak Ridge National Laboratory, Oak Ridge, Tennessee(Received: 11 December 2001 / Accepted:24 May 2OO2)

Abstract
High beta plasmas with twice of the no-wall ideal kink limit have been achieved in DIII-D device bysustaining the plasma rotation. The most important element is the control of the resonant phenomena of
residual error field which excites the stable resistive wall mode near the marginally stable condition(Resonant Field Amplification).

Keywords:
external kink. resistive wall mode. ideal MHD. error field

1. lntroduction

DIII-D have shown the feasibility of both feedback

The external kink mode has been considered as oneof most dangerous obstacles which may hinder us fromachieving commercially-attractive fusion reactors. The
efficacy of the conducting shell has been demonstrated

stabilization [8] and rotational stabilization [9].
On the DIII-D device, it has been discovered thatboth magnetic feedback and rotation stabilization
successfully function in a synergetic manner to reducein early tokamaks, and RFPs. The presence of a

the RWM amplitude, maintain the plasma rotation, andestablish a high p* = Bl(Brlalp) configuration (where theconducting shell reduces the growth rate and the shell-

B is the ratio of plasma pressure to the toroidal magnetic
pressure). A key factor is the discovery of excitation of
stable RWMs by a residual error field [9,10] (Resonant

stabilized plasmas transiently have achieved higher beta

[1-3]. Since the wall with finite resistivity loses the fluxconserving stabilizing force in time, the external kinkinstability is branched into the resistive wall mode

Field Amplification), predicted by Boozer [11]. Near themarginal condition, the residual error field excites the(RWM) [4]. Since the RWM mode exhibits astable RWM in a helically-resonant manner, causing the
growth of the RWM resulting in a strong drag on theplasma rotation due to electro-magnetic torque [12,13].
Feedback which senses the mode is found not only to
reduce the mode amplitude, but also compensate theresonant component of the enor field. When the plasma

sufficiently slow growth rate, various approaches are
possible to suppress or control the mode amplitude.
Over the last decades, the magnetic feedback
stabilization has been considered as a possible cure evenin reactor oriented devices [5]. Recently a scheme withthe plasma rotation through kinetic dissipation has been

rotation is maintained above a critical value withproposed as an alternative [6,7]. These approaches areshown schematically in Fig. l. Recent experiments in
Corresponding author' s e-mail : mokabaya@ pobox.pppl. gov
42@2002 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Okabayashi M. et aI., Resistive Wall Mode Control on the DIII-D Device

over so long a time scale (> Alfv6nic time) seems likean over simplified hypothesis. This seemingly puzzling

Feedback Scheme

character based on conventional ideal MHD predictions
has necessitated series of experimental, theoretical, andnumerical studies.
Tcw

To address the issue of the mode structure under

Higher Gainthe influence of finite resistivity, theoretical andexperimental efforts have been made during last a few
years. A numerical study has been carried out with the

0.0

-2.0

GATO and DCON plasma stability code combined with
the VACUUM code [7,18]. In these analyses, theresistive flux loss on the wall and its impact back to theplasma surface is treated in a self-consistent manner.
The results indicate that the mode structure inside theplasma remains largely intact and that the eddy current(b) Rotalion Stabilizatlon Scheme

Plasrn_Dis-sipation
Parameter

asrd = 0

pattern on the wall is not significantly modified
18,12,17,181. On the DIII-D device, the mode strucrure

has been studied with both a high resolution ECE
spectrometer and two toroidally-separated soft x-ray
arrays. The results indicate that the RWM evolves intime without creating any noticeable magnetic islands
even when the mode amplitude observed outside thevessel reaches to the order of 5-10 gauss [8,10,13]. The
slow time evolution of the observed mode inside the

No Rotation
Trw
0.0

-2.0
1.0

NoryallLimit

plasma coincides well with the flux time history
observed outside the vacuum vessel. In addition, theradial flux measured at above/below mid-planecompared with the flux evolution at the mid-plane

Fig. 1 Schematic diagrams of RWM stabilization. (a) Rotational stabilization without magnetic feedback.
(b) Magnetic feedback stabilization with a control

indicates that the mode behaves as one large rigid

logic "smart shell". ft/p,i'*"rr = 1.0 corresponds tothe no-wall limit and Au/Bt"*"' = 2.0 corresponds
to the ideal-wall limit.

displacement with a helicity conserved.
Bondeson and Ward [6,7] proposed that the RWM

can be stabilized by the dissipation due to plasma
rotation if the plasma rotation is above a critical value,
typically, a few percent of Alfv6nic velocity. DIII-D
experiments have revealed that rapid RWM growth iscoincided with sharp decrease of plasma rotation

sufficient angular momentum input sustained, the RWM
is stabilized, u'hich is consistent with a mechanism as
discussed by Bondeson and Ward [6,7].

The improvement via maintaining the plasma
rotation is obtainable in wide rage of plasma parameterssuch as quasi-steady state high Bp discharges [4]. The

112,19,201. With higher angular momentum injection [8-l0l, the piasma rotation was increased while holding thetotal stored energy constant at a plasma pressure justabove the no-wall beta limit. When the plasma rotation

study with the MARS simulation code by [5,16]
supports the observed feedback sensor preference.

was increased by 207o, the RWM onset was delayed by
100-200ms. However, the RWM eventually grew andled to the beta collapse, indicating that the higher rotation alone was not sufficient for complete stabilization.

In this paper, we have summarized the recent
progress of RWM control on the DIII-D device.

2. Resistive Wall Mode and Error Field
Amplification
2.1 Mode characteristics of RWM

Nonetheless, this experimental result suggests thatplasma rotation is a major factor in achieving RWM
control.

The RWNI is assumed to have a global helicaldisplacement extending from the core plasma to beyond
the vacuum vessel. The existence of ideal kink nature

2.2 Resonant Field Amplification (RFA)
Boozer [l] proposed that when a plasma+JOkabavashi M. et al.. Resistive Wall Mode Control on the DIII-D Device

Experimentally it was demonstrated [9] that when BN
approaches the marginal condition, the applied pulsed
error field can induce the n=1 helical response (Fig. 2).

approaches the marginal stability condition, namely, theno-wall beta limit against the external kink, theamplitude of the plasma distortion to a resonant
component of residual external perturbation (such as
error field) increases inversely proportional to a torqueWith BN above the no-wall beta limit, Bf,o*uil, theparameter, plasma toroidal rotation. This effect has been

Brcwall. After the pulsed field was turned off, the mode

distortion is larger compared with the response at BN =decayed at the rate of llt., indicating that the modetermed resonant field amplification (RFA).

excited did belong to marginally stable regime.
Figure 3(a,b) shows the RFA amplitude and phase forvarious BN summarized from the pulsed field operation.
Here, the RFA amplitude is defined as the helical n=1
flux due to the plasma perturbation, namely the total
observed flux subtracted by the applied external field.

t{:}s4r3t 105439 105444

52*d

The most important result is that there is a finite phaseshift between the applied field and the excited mode.

Szooo

o
q
3
.E

The phase shift is 20 degrees at BN=Bfirwdl and increases
to 90-120 degrees at higher By. This toroidal phase shiftof the plasma response should be taken into account for

n=l Br at Wall (plasma response only)E)

feedback operation. When a larger extra n=1 field isapplied, the plasma rotation gradually decreases andonce the velocity is below a critical value, the RFA
becomes too strong, leading to rotational collapse. This
indicates that the rotational stabilization requires a
critical rotational velocity, which corresponds to =6kHz
for the present experimental condition. This value is not
far from a critical velocity estimate given by [6,7] asTime{m}Fig.2 Stable resistive wall mode excited with pulse
field. The larger amplitude of RFA is induced with
B*/pno*"tr > 1.0. Decay of the mode after the pulse
field indicates that the plasma condition belongs
to the stable regime. (a) 0" vs. time; (b) Nonaxisymmetric field correction current vs. time.
(c) The plasma response observed on the external
68, saddle loop.

0.2sglshown later.

(b) Observed Toroidal Phase Shift

Observed Amplitude

v.vffi#

Mode

Toroidal

Mode

;ke

Amplilude

0.0 t-

0.6

*x

Phase

shn

Mode

Amplitude
(au)

)R

\

(degrees)*tq IQ
1,0

-180

't.6

1.6

1.0

pgppo-watt
0.25

Yx

Biy'Pil'"dl

Estimated Amplitude^v.v- ^

*S***
i J$-:(d) Estimated Toroidal Phaso Shift

Mode

Toroidal

f*-

Phase

rk*I

shift
(degrees)

0.0

x
x
x

-180.0
0.6

1.0

..

1.6

Piy'Pft**t"Fig. 3 RFA amplitude and toroidal phase (a) and (b): Experimental results. (c) and (d): Estimated values by eq. (4) using
the observed rotation velocity. The amplitude scale is in arbitrary unit.
44

Okabayashi M. et al., Resistive Wall Mode Control on the DIII-D Device

3. Hardware for RWM Control on Dlll-D

coil current. T'he mode control logic works best with

serve as the active feedback actuator. Present active
coils are toroidally located in phase to the saddle flux
loops. Coils located 180 degrees apart are also paired inanti-series so that only z=odd components are produced
and are energized with current power supplies up to 5kA
with DC-l00Hz capability. A coil current of 1kA
produces l3 gauss radial field on the vacuum vessel. As
shown later, about one half of the maximum current isused for error field correction. It should be noted thatthe applied active field does not have any helicity
preference, since only one layer of coil exists in thepoloidal direction.
The possible performance of these sensors alongwith control logic has been analyzed with the VALEN
code [21] with present coils and possible future upgrade
coil locations. Best performance is obtained with the 6,8o

poloidal field sensors inside the vacuum vessel.
On DIII-D, various sensors have been installed

Mirnov loops located inside the vacuum vessel and theperformance with 68, saddle flux loop located outside

inside and outside the vessel. Figure 4 shows aschematic diagram of sensor and active coil location.

the vacuum vessel is predicted to be less effective. The

DeviceThe power supply required for magnetic feedback
is rather modest, since the RWM growth time is of theorder of the resistive wall skin time ?*, which is far
slower than ttre Alfv6nic time. The sensor geometry
depends on the choice of control logic. The smart shell
scheme is to control the total flux (including the flux
supplied from rhe coil) by compensating the helical flux
leakage of the wall and to build the virtual ideal shell on
the wall [5]. The smart shell approach works best with
6-8" saddle loop, which detects the flux decay over theshell surface. Another scheme is the mode control logic

using the signal originating only from the plasma
surface displacement and without coupling to the active

Each saddle loop covers 60 toroidal degrees. The input
signal to the ft:edback system is made by combining apair of sensors located at toroidally opposed angles, toproduce only n = odd components.
The activt: coils on DIII-D device have two roles:
one is to comp,ensate the error field and the other is tolnternal
68"Loops

External
0Br Loops

ActlveCoilInternal
6BpLoops

VALEN ccode also predicts that the addition of coils
located above/below the midplane with 68o sensor
operation should be able to stabilize the RWM up to avalue of BN that is 90Vo above the Bflo *dr and l\Vo below
the pi$eat*att [mit. These results are consistent with otherstudies by Liu [16] and Chu [7,18].

4. Experimental Results of RWM Control
4.1 Comparison of various feedback sensors
The comparison of 58, and 6,8o located inside thevessel is shown in Fig. 5. The advantages of 68, sensor
is the rejection of 58, component produced by either theactive coil or the eddy current excited by the active coil.
In this series of experiments, the discharge loses thehigh beta period at l380ms. The 6^8, sensor operation
extended the discharge to l580ms compared to l440ms
with the 68. sensor. Since the q-edge was decreasing
toward 3, the longer duration means that the discharge
faces stronger external kink onset. The requested coilcurrent amplitude and phase of n=l pattem are shown in

Fig.5(c,d).

The error field correction was set 2kA tocompensate approximately empirically-determined error

field. For the operation of both sensors, the amplitude
requested from the feedback stays at approximately 2kAlevel, which indicates that the initial estimated coil

ActiveColl
Fig. 4 Active coil geometry and sensor locations.

current was reasonable. However, 68o sensor shifted thefield direction of n=l about l0-20 degrees immediately
after the feedback was turned on. On the other hand, the68, sensor did not sense the need for the directional
shift. The slight increase of amplitude around 1400ms

Okabayashi M. et al., Resistive Wall Mode Control on the DIII-D Device

3.0
Fn
0.0

dt%n 10.0
(kHz)

3.0

0.0
5.0

Coil
2.0

amplitude

i

(kA)

0.0
50.0

Coil

toroidal

1200

1400

rollli8.l roorfit6{xTime(ns)

Time(ms)nw

Fig.6 Comparison with/without optimized error compensation. (1) Shot with no feedback (106530)
with pre-programmed non-optimized error correction current (with amplitude of 1kA and 0" =7 degrees). (2) Shot with feedback (106532) withpre-programmed non-optimized error correction
current (amplitude of 1kA and 0" = 7 degrees). (3)Shot with feedback shot (106534) with pre-programmed error correction current adapted fromthe feedback shot (106532) (a) 0* and estimatedlimit, (b) Plasma rotation frequency at g = 3
surface, (c) coil current amplitude for the n=1
component, (d) Coil current toroidal phase for the

Fig. 5 Comparison of 68, and 68o sensors located inside
the vessel. (1) Shot (106196) with no feedback, (2)Shot (106187) with the internal 68, sensor feedback, (3) Shot (106193) with the internal 68, sensor feedback (a) P* and /, (b) Rotation frequency at

e = 2 surface (c) Coil current amplitude for the
n=1 component (d) Coil current toroidal phase forthe n=1 component.

could not stabilize the mode, leading to the final
collapse. The fine detection of field direction and its

n=1 component.

adjustment seem to have been the crucial element forstabilizing the RWM.

to the results obtained with feedback (case 3).
The comparison of these three types of operations
indicates that (1) the feedback process can track the

4.2 High B1,1 achievement via error fieldcorrection by feedback
Once it was determined that the 68o sensor issuperior to other sensors, the long-duration, high B11,

error field with good accuracy, and as a consequence,
the n=l component of error field was determined withthe feedback process and (2) the use of the obtained
current as error field correction without feedback can
produce the high plasma rotation velocity and high bN
configuration. This observation along with discussion in
Sec. 2 on high fiq achievement with higher angularmomentum indicates that the higher rotation velocity
with better error field correction is the essence of thisdischarge was explored using the internal 68o sensor.

The target discharge was with a modest 1o ramp of
0.6MA/s. Without feedback the preprogrammed current
for error field correction was set to lkA with 0" =7 degrees (Fig. 6). The rotation velocity started todecrease rapidly, similar to the case in Fig. 6, due to theincrease of the RWM and the Bp decreased graduallyfrom l300ms (case l). When the feedback was appliedwith the same pre-programmed current, the feedback
increased immediately the coil current to 2kA andgradually up to 3-4kA level and the field direction was
shifted to Q" - 40 degrees. The high beta duration was

success.

5. Advancing the RWM Control Goncept
These experimental results indicate that magnetic

feedback stabilization and stabilization by plasma
rotation are not two distinct processes, but that both ofthem work together in a tightly coupled manner asshown in Fig. 7.

increased to 2000ms together with fiq increase to 3.0,
which is about twice of Blowatt and close to according toBidarwail GATO calculation (case 2). When the preprogrammed current was modified to match the coilcurrent obtained by feedback operation (case 2), theRecently, theoretical studies have been carried out

on the RWM process including plasma rotation by
several groups t22-251. The overall feedback process

time evolution of B1{ is identical to the feedback (case2), and the plasma rotation velocity is also very similar

can be described qualitatively with a cylindrical lumped

46

Okabayashi M. et al., Resistive Wall Mode Control on the DIII-D Device

model as was discussed in [26]. The main parameter is,L"s, which can represent the pressure balance and thenonnal magnelic field continuity at the plasma surface.

lrs61o+ C*p61*+ C"051"=0

respectively. This formulation is equivalent to the treatment of the kinetic energy and dissipation term with 6w
approach in the [24](1)(yr* + i{26r*)t(Kl rh + (yrw + idLrr*)(al r*)'where, suffixes p, w, c, and o correspond to plasma,passive wall, external (active or error field) coil, and+6We+ (6W!yr*+6Wfl/(yr* + l)=0

observation sensor respectively. The Cii are mutual inductances between these elements. 6/0, 61*, and 61.
correspond to the plasma skin current, the passive wall
eddy current, and the external coil current respectively.
The value, 1"1, includes the MHD mode displacement
gradient, ft, dofined just inside the plasma surface, andthe safety factor through f = * - nq. The formulation
can be expanded in order to include the dissipation of
the mode due to the plasma rotation and kinetic term for
low frequency limit (yco < l),where, the first and second terms represent the impact ofplasma rotation and dissipation, and the third term theideal MHD potential energy. The fourth term is for theeffect of finite wall resistivity.
When an external field such as error field exists,
the plasma mode response in the quasi steady state isgiven by
51p = (-C"p/Z"rr) 61"

a9)

(2).

Maintain
PlasmaRotation*(4)

does not include the value p1q. However, we still can dis-cuss the qualitative behavior by relating B* to f with B*

where, the value, d)6, is the angular rotational frequency
and K, and o represents the strength of kinetic, anddissipation tenn relative to the plasma potential energy(a).

This simple cylindrical formulation is obtained for acurrent kink driven RWM withfiin <,f ( I where/=1
and.f='fl,rn correspond to current driven kink onset with
no-wall and with ideal wall respectively. The modelL"r = (Fo - 2lf +l + Kq + i a{zil(8. - 2tf -r + rcfi + i(3)= 2lf - 1, when the plasma condition approaches theideal MHD limit. With this definition, we preserve thefundamental criterion: /=l as the marginal stability condition for ideal kinks at PN - Bffi'*urr and the usual wallplasma separation determinesfi6 (<l) at 0" = pif"u'*u".

Reducing
Residual
Error Field

A similar relation is also used in 121,271.
Figures 3(c) and (d) show the estimated values forthe observed RFA using 0o = 1.3, a = 0.002, r = 0, andexperimentally observed plasma rotation. The absolutevalue of the mode amplitude can not be estimated fromI,
RWM/EFA Maonetic-\
Feddbacx-'z
SensorI

the cylindrical mode, mainly due to the limitations
imposed by the single pattem assumption in this simple
cylindrical model, causing a mismatch of field pattems
between the externally-applied and mode fields. The
estimated dependence of the toroidal phase shift versus,

0*/0f,i"tseems to be qualitatively consistent with theobservations. The large scattering on the estimatedYrwvalues of amplitude and phase can be attributed to toohigh of a sensitivity due to the resonant condition and to0,0

the uncertainty of the rotational frequency. However,this scattering itself may reflect the hidden parametersof the actual dissipation mechanism, of which details are
not included in this simple model.
For completeness in modeling the feedback scheme
shown in Fig. 8(a), the time behavior of the plasma-2.0

Fig. 7 Schematic of unified theme for RWM control bothwith feeclback and plasma rotation. (a) The performance with the unified theme and (b) Schematic

rotation must be included. Here, we will use the angularmomentum dissipation equation with the electromagnetic torque applied by the external field to thediagram.
47

Okabavashi M. et al.. Resistive Wall Mode Control on the DIII-D Device

mode on the plasma surface 122,23,25,28) expressedusing the lumped parameter approach [26] instead of theindicates that, although the RWM feedback process iscommonly used flux discontinuity on the plasma

significant roles on the process.

complex, a few fundamental parameters may playsurface.

7. Acknowledgments
This work is supported by U.S. Department of

rfldldt(a) = cii' 61p (co* 61* + cp. 61")t51p(0) 61"(0)l-r

Energy Grant DE-FG02-89ER53297 and Contracts DEAC03-99ER5 4463, DE-AC05-00OR22125, and DEAC02-76H03073. We are greatly thankful to physicists,(cp*cp")-lt2 (5)where, Cf,f' = 2r (alR)L*aGfB-p ?iB-ext), and time isengineering staffs, technical staffs involved in thisproject. In particular, we would like to express ournormalized to z*.

Here, zfB-r and tfr-"*t are Alfvdn times formagnetic field on the plasma surface produced byappreciation to DIII-D operation group.

610(r=0) and 61"(10) respectively.

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The shell boundary condition provides

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(6)

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6. Summary
We have made substantial progress in the control of

one of most dangerous MHD modes for practical
rectors, the resistive wall mode. Open loop operation
discovered that at B* = p;o*utt. the marginally stable
RWM responds in a resonant manner to the applied fieldwith the toroidal phase shift relative to the applied field
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Okabavashi M. et al.. Resistive Wall Mode Control on the DIII-D Device

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49