Active control of 2/1 magnetic islands in a tokamak

PHYSICS OF PLASMAS VOLUME 5, NUMBER 5 MAY 1998

Active control of 2/1 magnetic islands in a tokamak *G. A. Navratil,†,a) C. Cates, M. E. Mauel, D. Maurer, D. Nadle, E. Taylor, and Q. XiaoDepartment of Applied Physics, Columbia University, New York, New York 10027

W. A. Reass and G. A. WurdenLos Alamos National Laboratory, Los Alamos, New Mexico 87545

~Received 21 November 1997; accepted 11 February 1998!

Closed and open loop control techniques were applied to growingm/n52/1 rotating islands inwall-stabilized plasmas in the High Beta Tokamak-Extended Pulse~HBT-EP! @J. Fusion Energy12,303~1993!#. HBT-EP combines an adjustable, segmented conducting wall~which slows the growthor stabilizes ideal external kinks! with a number of small~6° wide toroidally! driven saddle coilslocated between the gaps of the conducting wall. Two-phase driven magnetic island rotation controlfrom 5 to 15 kHz has been demonstrated powered by two 10 MW linear amplifiers. The phaseinstability has been observed and is well modeled by the single-helicity predictions of nonlinearRutherford island dynamics for 2/1 tearing modes including important effects of ion inertia andfinite Larmor radius, which appear as a damping term in the model equations. The closed loopresponse of active feedback control of the 2/1 mode at moderate gain was observed to be in goodagreement with the theory. Suppression of the 2/1 island growth has been demonstrated using anasynchronous frequency modulation drive which maintains the inertial flow damping of the islandby application of rotating control fields with frequencies alternating above and below the naturalmode frequency. This frequency modulation control technique was also able to prevent disruptionsnormally observed to follow giant sawtooth crashes in the plasma core. ©1998 American Instituteof Physics.@S1070-664X~98!95705-6#

aniz

e

etnt

ntossv

oesrs

heex

b-edti

lyervel-aces

ee

dan-

Thee in

f a. Aide

isldex-wnect-ts,

eg-of

lts

I. INTRODUCTION

Economically attractive, steady state fusion power-pldesigns based on advanced tokamak physics emphascombination of three important features:~i! high beta,~ii !large and well-aligned noninductive bootstrap current to pmit economic, steady state operation, and~iii ! good confine-ment. The prospects for improved confinement in magncally confined toroidal fusion plasmas has made substaprogress moving from the H~high! mode of the early 1980sto the VH ~very high! mode in the early 1990s to the receuse ofE3B shear to suppress turbulence in the plasma cand reduce the level of ion transport to neoclassical valuemost of the plasma volume.1 However, high beta plasmawith well-aligned bootstrap current require operation at leels of bN51028baB/I p well above the beta limit for thelow-n ideal kink mode.2,3 The most promising approach tstabilize the low-n ideal kink mode is the use of a closfitting conducting wall which, if perfectly conducting, habeen predicted to improve the no-wall beta limit by factoof more than 3.2,3 Experimental studies have shown that tno-wall beta limit can be exceeded by modest factors inperiments on DIII-D4 and on HBT-EP~High Beta Tokamak-Extended Pulse!,5 however, slower growing modes are oserved to set a lower beta limit in these wall-stabilizplasmas. Since these residual modes have much slowerscales than the ideal magnetohydrodynamic~MHD! time

*gTuaI1-5 Bull. Am. Phys. Soc.42, 1875~1997!.†Invited speaker.a!Electronic mail: [email protected]

1851070-664X/98/5(5)/1855/9/$15.00

Downloaded 08 Oct 2001 to 128.59.51.207. Redistribution subject to AI

te a

r-

i-ial

rein

-

-

me

scale for growth, they are in principle able to be activecontrolled. The two most important of this class of slowgrowing modes are the resistive tearing mode which deops large magnetic islands resonant on magnetic surfinside the plasma, and the resistive wall mode6 ~RWM!which has a growth rate slowed to the resistive wall timconstant and is an external mode~since its resonant surfaclies in the vacuum region outside the plasma edge!. Thispaper concentrates on experiments to control the mostgerous of these internal modes: them/n52/1 resistive tear-ing mode and its associated magnetic island structure.2/1 tearing mode has been found to play an important rolthe tokamak current disruption process.

II. MODE CONTROL SYSTEMS AND BASICPARAMETERS OF HBT-EP

The approach to mode control in HBT-EP consists ocombination of active and passive stabilization techniquesten-segment, adjustable conducting wall is used to provpassive stabilization of the low-n ideal kink mode. Activecontrol of the residual slower growing 2/1 tearing modeeffected by applications of rotating helical magnetic fieperturbations generated by highly modular saddle coilsternal to the vacuum vessel. This configuration is shoschematically in Fig. 1. The toroidal plasma has an aspratio, R/a56 with R50.92 m. The passive stabilizing conducting wall consists of ten, 1-cm-thick aluminum segmeneach of which covers 26° of toroidal angle. These wall sments can be varied in radial position to study the effectwall proximity on kink mode stabilization and these resu

5 © 1998 American Institute of Physics

P license or copyright, see http://ojps.aip.org/pop/popcr.jsp

-2/thfoe°da

mn2u

an

rtootinh

e

are

ndls.esa

puternse

theheld. 1,thebeedple

m-on-

atu-ter

ten

ot

EP.ivingthe

1856 Phys. Plasmas, Vol. 5, No. 5, May 1998 Navratil et al.

have been published previously.5,7,8 In the experiments reported in this paper on active mode control of the internaltearing modes, the wall segments were positioned less10% of the plasma minor radius from the plasmas edgemaximum passive stabilization of the ideal time scale mod

A set of m52 saddle coils is located at four of the 10wide gaps in the conducting segments. Each of these sacoils is 6° wide in toroidal angle and positioned in poloidangle so as to couple optimally to anm52, n51 helicalfield. The saddle coils project a radial field into the plasthrough 5° wide quartz toroidal gaps that allow efficient peetration of magnetic perturbations with frequencies up tokHz. This highly modular configuration only covers abo3% of the toroidal surface surrounding the plasma.

Two of these saddle coils are connected in seriesdriven with a 10 MW linear amplifier to provide a sin~vt1d!response field and the other two coils are connected in seand driven with an independent 10 MW linear amplifierprovide a cos~vt1d! response field; together these coils prvide a two-phase, quadrature winding to produce a rotamagnetic perturbation, as shown schematically in Fig. 2. T10 MW linear amplifiers used in these experiments havbandwidth greater than 25 kHz and can deliver6600 Athrough each of the nine-turn saddle coils. The phaseamplitude information from the rotating 2/1 island structuin the plasma is obtained from a set of sin 2u and cos 2uRogowski coils which are physically remote~as shown inFig. 1 and Fig. 2! from the saddle coil response fields ahave very low direct pickup from the driving saddle coiFor closed loop, active mode control experiments, thsin 2u and cos 2u signal measurements are digitized bydigital signal processor~DSP! at 100 kHz and in real time~D

FIG. 1. Schematic of the HBT-EP control coil configuration showing26° wide conducting wall segments with 6° widem52 saddle coils locatedat four of the 10° wide gaps in the conducting wall segments. A sin 2u andcos 2u Rogowski coil quadrature detector is located to be physically remfrom the saddle coils.


1anr

s.

dlel

a-0t

d

ies

-gea

nd

e

t510 ms! phase shifted and gain adjusted to drive the instages of the two 10 MW linear amplifiers. These high powamplifiers provide the sine and cosine quadrature respofields that are fed back to the rotating 2/1 islands inplasma. Shown in Fig. 3 is the relative amplitude for tn561, 62, and63 modes of the perturbed magnetic fiecreated by a single pair of the saddle coils shown in Figwhose relative poloidal orientation is chosen to maximize2/1 helicity. The 2/1 helicity component is computed tothe largest, and for positive helicity modes, this is followby the 2/2, and 2/3 modes, which we do not expect to coustrongly to the plasma.

The basic parameters of the HBT-EP tokamak are sumarized in Table I. For these experiments on 2/1 mode ctrol, the edgeq value was maintained nearq;2.5 and held tovalues less than 3. This type of plasma produced strong nrally occurring 2/1 mode activity that began about 3 ms af

e

FIG. 2. Schematic of the closed loop system for mode control on HBT-A digital signal processor produces phase and gain adjusted output drtwo 10 MW linear amplifiers which provide a sine and cosine phase forsaddle coil generated rotating 2/1 applied field.

FIG. 3. Relative amplitude of the mode spectrum forn561, 62, and63and 0<m<6, for a single pair of the saddle coils~sine or cosine phase!shown in Fig. 1.


mrgwgnenrio

3ed

le

nrom,

be

rif

f

pr

d

ur

theionich

-

at

e

ngon-

de

ceiondeck-g-of

ongeldaf-mi-

li-ofnts

-upy atheseof

t asatu-

t 5Hz

he

1857Phys. Plasmas, Vol. 5, No. 5, May 1998 Navratil et al.

plasma formation and which always led to a hard plasdisruption. The typical disruption sequence included lagrowth of the 2/1 islands immediately following a large satooth crash at about 7 ms after plasma formation resultinprompt~,100ms! loss of central temperature and significadensity loss followed by a collapse of the plasma currover the following 0.5 ms. These plasmas provided a peof relatively constant amplitude saturated 2/1 activity fromto 7 ms and then presented a strong disruption challengtest the robustness of any mode control scheme appliethe discharge.

III. ROTATING ISLAND MODEL EQUATIONS

To analyze our results we use the nonlinear, singhelicity model which has been built up over many years9–14

to explain the interaction of external magnetic perturbatioand a saturated tearing mode which appears as a set ofing magnetic islands on the resonant surface in the plasNormalizing the mode amplitude,b, and rotation frequencyV, the simplified set of coupled dynamical equations forbandV used to model the mode control experiments descriin this paper are given by

db

dt5g1~12Ab!Ab1g2

bd

Ab2g3

~V21!2

b, ~1!

dV

dt52h1~V21!1h2bqb2h3

Vv0twall

m21~Vv0twall!2

b2,

~2!

where,g1 ,g2 ,g3 ,h1 ,h2 , andh3 are constants. WithWsat de-fined as the saturated island width,b[(W/Wsat)

2; andV[v/v0, wherev0 is the natural island rotation frequency~whichwe measure to be primarily toroidal and in the electron ddirection!. The dynamical equation for island rotation@Eq.~2!# is phenomenological as used in Ref. 9, and the sizethe constantsh1 , h2, and h3 depend upon the fraction oplasma rotating with the island. Reference 14 presentsalternate treatment of island momentum~not used here!which assumes the mass rotating with the island to beportional to the island width.

If we further normalize time in units ofv0, the values ofthe constant coefficients in Eqs.~1! and ~2! are dimension-less. For example, the first terms in Eqs.~1! and~2! describethe Rutherford island growth term and a simplifieamplitude-independent relaxation of island rotation tov0.Defining r s as the minor radius of the resonant tearing s

TABLE I. Typical plasma parameters for HBT-EP.

Major radius,R0 0.92–0.97 mMinor radius,a 0.15–0.19 mPlasma current,I p <25 kAToroidal field,BT <0.35 TPulse length 10 msElectron temperature,Te <80 eV averageDensity,ne ;131019 m23


ae-inttd

toto

-

stat-a.

d

t

of

an

o-

,

-

face,t res as the measured Rutherford growth rate, andtv asthe measured rotational relaxation time,g152(r s /Wsat)/t resv0;0.1, andh151/tnv0;10.

The second terms in Eqs.~1! and~2! are the direct driveand the quadrature drive, respectively, which describeinteraction between the applied rotating helical perturbatfield and the rotating 2/1 islands. The direct drive term wheither increases or decreases the mode amplitude,bd , is pro-portional toI coil cos(d) and the quadrature drive term is proportional toI coil sin(d) with

d[d01E0

t

dt8~V2Vcoil!,

whered is the phase relation between the rotating islandfrequencyV and the externally applied field,Vcoil. The di-mensionless coefficients are approximately~see Refs. 9–14!g2'2g1 /(r sD8), and h2'(4m/pa)(VA /qRv0)2

3@bsat/Bu(a)#2;10, whereVA /qR is the characteristic Al-fven frequency, anda <1 represents the fraction of thplasma free to rotate with the island.

The third terms include the most important dampiterms needed to fit the observed behavior of the active ctrol experiments in HBT-EP. In Eq.~2! this damping is awall drag term,h3'h2(Bt /Bu)2;0.1, which is small sincev0twall;80 for HBT-EP withv0;63104 s21. In Eq. ~1!,this is a damping or stabilizing term on the mode amplitufound by including ion inertia and finite Larmer radius~FLR!effects, and it is quadratic in the rotation velocity differenbetween the island and background plasma fluid. Thisinertial stabilizing effect diminishes the mode amplituwhenever the island moves faster or slower than the baground plasma toroidal rotation velocity. Defining the manetic shear,S5r sq8/q, Ref. 14 estimates the magnitudethis term for collisionless plasmas asg35g1(4/S2)3(R/r s)

2(r s /Wsat)3(v0qR/VA)2/(r sD8). Depending upon

the saturated island size,r s /Wsat, g3 /g1 can range from;0.1 to approximately unity. Previous experimentsCompass-C15 suggested that this ion inertial flow dampinled to a reduction in island amplitude when a static error fiwas applied. We also find this term to be significant infecting the transient island evolution in response to dynacally applied external perturbations.

IV. DRIVEN TOROIDAL ROTATION OF 2/1 ISLANDS

To benchmark the model described in Sec. III for appcation to the control coil geometry of HBT-EP, a seriesopen loop frequency ramp-up and ramp-down experimewas carried out. The results of a typical frequency rampexperiment are shown in Fig. 4. In this example we applsinusoidal current to the saddle coils and linearly advancefrequency from 2 to 15 kHz over a period of 2 ms. The phadifference between the applied magnetic field and the fieldthe 2/1 rotating island is shown in Fig. 4 and indicates thathe rising frequency of the applied field approaches the nral frequency of the island rotation~;8 kHz!, the mode de-celerates and ‘‘locks’’ to the rotating applied field at aboukHz. The 2/1 island is then accelerated up to about 14 ktoroidal rotation frequency at which time the lock is lost. T


ea

ardx-

preidren

tok,issityvelseion

ler-ccel-ncyeroto5

aridtheus

n

pflymaley aur-din

n ingowyste.m-pli-rm

ith

ui-se

ld2/1ckgetheilllt-id

haseof

uc-Pns

e

che


phase difference between the driving field and mode fislowly advances from negative values to positive valuesthe torque demands on the driving field changes from reting the rotation to accelerating the rotation as was reportesimilar experiments on the Divertor Injection Tokamak Eperiment~DITE!.10

During the frequency ramp-up and frequency ramdown experiments, the local ion flow velocity was measuwith a Langmuir Mach probe which measures the ion fluflow velocity near the 2/1 island. The results of this measument are shown in Fig. 5 for both frequency ramp-up a

FIG. 4. Frequency ramp-up experiment from 2 to 15 kHz showing~a! theapplied and plasma 2/1 mode field,~b! the phase of the 2/1 mode in thplasma relative to the phase of the applied field, and~c! the frequency of theapplied field and the 2/1 mode in the plasma as a function of time.

FIG. 5. Local measurement of the ion flow velocity with a Langmuir Maprobe during a frequency ramp-up and a frequency ramp-down experim


lds

d-in

-d

-d

ramp-down discharges. The ion fluid is normally observedhave a toroidal flow velocity near zero prior to mode locwhich is consistent with the picture that the ion fluidnearly stationary due to the large charge exchange viscoon neutrals near the plasma edge, while the mode trawith the electron fluid in the electron drift direction. In thcase of the frequency ramp-up experiment of Fig. 4, thefluid flow velocity drops to21 kHz when the mode fre-quency locks on to the applied field at 5 kHz and is acceated to a value somewhat above zero as the mode is aerated to near 14 kHz. The reverse is seen with the frequeramp-down case: The ion fluid is accelerated from near zwhen the mode locks on to the applied field at 14 kHzsomewhat below21 kHz as the mode is decelerated tokHz. After mode lock is lost, the ion fluid returns to nezero toroidal velocity in a few tenths of a ms. The ion fluacceleration rate is seen here to be only about 20% ofmode acceleration rate, which is consistent with previoobservations of this effect in Compass-C15 with applicationof a static perturbation field and in the JAERI FusioTorus-2M ~JFT-2M!16 with a rotating perturbation field.

At the time the external field is applied in the ramp-uexperiment of Fig. 4, the mode rotation frequency is briedriven up to a large value and then relaxes back to its norequilibrium value prior to mode lock to the applied field. Wnote that this kick upward in frequency is accompanied bdecrease in the mode amplitude. A similar event occurs ding the loss of mode lock with a rapid drop to 7 kHz anreturns to near 15 kHz which is accompanied by a dropmode amplitude. This effect has been modeled as showFig. 6 both with and without the ion inertia flow dampinterm in the model equations. In the case without the fldamping term included in the model, the mode is alwafound to lock on to the applied field and smoothly acceleraIn the case including the flow damping term, the phenoenon of a frequency excursion with associated mode amtude reduction could be simulated, indicating that this temay play an important role in the island dynamics.

V. STUDY OF THE PHASE INSTABILITY

When the phase of an applied external field rotating wa 2/1 island is maximally stabilizing in Eq.~1!, the phase ofthe mode relative to the applied field is in an unstable eqlibrium and is subject to an unstable growth of this phadifference called the phase instability.9 This instability canbe produced in HBT-EP by applying an external 2/1 fieperturbation rotating at a frequency near to the naturalisland rotation frequency. The islands will immediately loon to the applied field and grow in amplitude to a new larsaturated level. If at a predetermined time the phase ofapplied field is suddenly advanced by 180°, the islands wnow see an applied external field which is stabilizing, resuing in a reduction in the mode amplitude and in a rapadvance in the phase of the islands in response to pinstability as the islands again move into a conditionphase lock and amplitude growth. This technique for inding the phase instability has been employed in HBT-E17

and is shown in Fig. 6 together with two model simulationt.


the time


FIG. 6. Model calculation for a frequency ramp-up experiment with and without inclusion of the flow damping term in the model equations showingevolution of the applied 2/1 mode frequency and the applied 2/1 mode amplitude.

hmtiad2hid

nr-tiohe

r

n-iseo

lat

JE-tu

oo

ssigith

is-ple

rela-rnal

the

pli-sed

n

dher

outck

aton

o ais

glly

e.

using Eqs.~1! and ~2!, one with and one without the ioninertial sheared flow damping term. Immediately after tphase change, there is a large decrease in the mode atude which we are unable to account for if the ion inerstabilizing term is not included. As the island’s phase avances to reestablish a phase lock with external rottingfield, a large instantaneous frequency increase occurs wcauses a significant decrease in mode amplitude inducethis stabilizing effect. The coefficients used in Eqs.~1! and~2! are largely determined by experimental measuremeThe coefficientsg1 and h1 are determined by the unpeturbed growth and/or decay of the island’s size and rotawhen the external rotating magnetic perturbation is switcoff. The coefficientsg2 , h2, andh3 are determined by thecoil and wall geometry. This leaves a single parameter,g3, toadjust until the observed decrease in island width is repduced in the simulation. In Figs. 6 and 7,g3 /g1;1, as ex-pected for large islands.

VI. ACTIVE FEEDBACK CONTROL

By applying a rotating external 2/1 field which is maitained in a stabilizing phase relation with the mode, itpossible to use a closed loop system to carry out active feback suppression of the 2/1 island amplitude as was demstrated in TO-118 and DITE.10 In the work reported here wecarry out similar experiments, but use the highly modusaddle coil set described in Sec. II and also implemenfeedback scheme similar to that proposed for use on~Joint European Torus!13,19using a fast digital signal processor. As discussed earlier in reference to Fig. 2, the quadradetection scheme on HBT-EP measures a sin 2u and cos 2usignal giving a phase and amplitude reference for closed lactive feedback. These sin 2u and cos 2u signals are digitizedat a 100 kHz rate and processed by a digital signal proceto generate a phase-shifted wave form that drives the hpower amplifiers and applies a 2/1 rotating applied field w


epli-

-/1chby

ts.

nd

o-

d-n-

raT

re

p

orh

a predetermined phase shift with respect to the rotatinglands. The feedback control algorithm implements a simrotation matrix:

F I cos~ t1Dt !

I sin~ t1Dt !G5GFcos~d! 2sin~d!

sin~d! cos~d!GFBcos 2u~ t !

Bsin 2u~ t !G . ~3!

This scheme is designed to maintain a constant phasetion between the detected mode phase and applied exterotating field.

We can model the expected equilibrium response of2/1 islands by setting the time derivatives to zero in Eqs.~1!and ~2!. Assuming thatV v0twall@1we solve the followingcoupled nonlinear equations for the expected mode amtude and frequency as a function of phase angle and cloloop system gain,G:

05g1~12Ab!Ab1g2G cos~d!Ab2g3

~V21!2

b, ~4!

052h1~V21!1h2G sin~d!b2h3b2

Vv0twall. ~5!

Assuming moderate system gain the variation expected iVandb are plotted in polar form for variations ind in Fig. 8.The mode is reduced in amplitude ford50° and driven tolarger amplitude ford5180°. For phase angles of 90° an270° the amplitude is little changed, but we expect a higfrequency atd590° and a lower frequency whend5270°. Ifthe normalized mode frequency is reduced to values of ab0.5 by the active control loop, the mode is predicted to loto f 50 in the lab frame. This mode locking is most likelyabout 230° and has been observed in the experimentsHBT-EP when the gain is increased. The effect is due tnonlinear interaction between the mode amplitude whichdirectly proportional to the torque applied by the drivinfield and the ion inertial damping which scales quadraticawith the frequency and inversely with the mode amplitud


history ofthe model


FIG. 7. Measurement of the phase instability when the toroidal phase of a rotating magnetic perturbation is suddenly advanced by 180°. The timethe measured amplitude and phase of the 2/1 island is shown compared with simulation calculations with and without the flow damping term inequations.

ggli

,ndm

lyinrg1

librth

n in

see-ent

eain-

cyob-s.

rvals

tor toheatthe

fiedthe100tonte,

thanasncyrior

emcted

Closed loop experiments were carried out ford50° and180° on a single discharge with the stabilizing phase anapplied for 1 ms followed by the destabilizing phase anapplied for the following 1 ms. These results are shownFig. 9 showing the time history of the applied phase angled,the amplitude of the sine and cosine phase of the 2/1 islaand the frequency of the island rotation. During the 1application of the stabilizing phase angle, the frequencyobserved to rise from about 7 to 10 kHz with a relativeconstant amplitude. After the transition to the destabilizphase angle, the mode amplitude is observed to grow laand the frequency of the mode is observed to decline fromto 7 kHz in agreement with the expectations of the equirium model of Eqs.~4! and~5!. If the data are averaged ove0.1 ms intervals and plotted against the predictions of

FIG. 8. Results of the equilibrium model calculations Eqs.~4! and ~5! as afunction of phase angle for moderate gain.


leen

s,sis

ger0-

e

model, we find reasonably good agreement as also showFig. 9.

A similar experiment can be carried out for the phaanglesd5270° and 90° where we expect a frequency dcrease and frequency increase. The results of this experimare shown in Fig. 10. During the initial application of thfrequency decrease phase angle, we see the frequency mtained at about 7 kHz. After the transition to the frequenincrease phase angle, the island rotation frequency isserved to increase to about 9 kHz over a period of 1 mThese results are also plotted averaged over 0.1 ms inteagainst the model predictions showing good agreement.

When the gain of the feedback system is increasedachieve a larger suppression of the mode amplitude oattempt to control the 2/1 island size immediately prior to tnormal strong growth phase prior to disruption, we find thphase accuracy of the response field is degraded andclosed loop system is no longer able to maintain a specistabilizing phase angle. This effect appears to be due tolarge phase angle delay inherent in the system using akHz digitization rate and a 10ms delay in computing the nexwave form update. The zero-order hold digital delay usedthe output combined with the 100 kHz data acquisition raresults in a frequency-dependent phase delay of more50°. Given the higher growth rate for the phase instabilitythe system gain is increased as well as the large frequechanges which can occur as the mode amplitude grows pto disruption, the 100 kHz digital signal processing systwas unable to maintain phase lock on the mode. As expe


ceemthis

andonea

tioro-sththdeain

rei

udf

qs.usex-ofofeewdelyingandwer

d-areionvere-thea

inden

und

a-rge

n i ilib-


in light of the difficulties in maintaining a phase referenfor the applied external rotating 2/1 field, tests of the systat high gain during the period of large 2/1 island growimmediately prior to disruption showed no effect on the druption timing or severity.

VII. FREQUENCY MODULATION STABILIZATION

Since we have observed in the frequency rampphase instability model benchmark experiments describeSecs. IV and V, a significant role played transiently by iinertia flow stabilization in reducing the island amplitudexperiments were carried out to induce this effect overextended period. The use of time averaged flow stabilizato control the average mode amplitude was originally pposed by Kuritaet al.,20 who simulated the effect of modulation of an applied external rotating resonant field whofrequency was modulated alternately above and belownormal mode rotation frequency. Since the response ofrotating island to such an external field is to increase orcrease the rotation frequency in order to establish a phlock on the applied field, this should result in a reductionthe mode amplitude during the periods where the (V21)2

ion inertia flow damping term in Eq.~1! becomes larger. Bychoosing the duration of the alternating high and low fquency periods to be sufficiently short so that mode lockavoided, a time average reduction in the 2/1 island amplitshould be obtained. Shown in Fig. 11 are the results o

FIG. 9. Application of stabilizing~0°! phase and destabilizing~180°! phaseeach for 1 ms compared with the results of the equilibrium model showFig. 8.


-

din

,nn-

eee-

se

-sea

simulation of this effect using parameters in the model E~1! and ~2! which were benchmarked against the previofrequency ramp, phase instability, and active feedbackperiments. Assuming a normal island rotation frequency10 kHz, a 2/1 rotating field was applied with a frequency15 kHz alternating with 5 kHz every 0.2 ms. We can sfrom the results of the simulation, that if the ion inertia flostabilization term is not included in the model, the morapidly locks on to the applied field and grows to relativelarge amplitude. On the other hand, when this stabilizterm is included, the islands do not achieve phase lockthe mode amplitude is observed to saturate at a much loaverage amplitude.

The frequency modulation stabilization experiment moeled above was carried out in HBT-EP and the resultssummarized in Fig. 12. Shown in the figure is the applicatof the alternating 5 and 15 kHz externally applied 2/1 drifield together with the response of the island rotation fquency. As the frequency of the applied field alternates,rotating island toroidal velocity is strongly modulated andphase lock with the applied field is avoided. Also plottedFig. 12 is the (V21)2 ion inertia flow damping term baseon the actual mode rotation frequency. During times whthe values computed for (V21)2 are relatively large com-pared with the measured 2/1 island amplitude they are foto correlate with a reduction in mode amplitude.

Most significantly, when this frequency modulation stbilization technique is applied at the end of the discha

nFIG. 10. Application of frequency downshift~270°! phase and frequencyupshift ~90°! phase each for 1 ms compared with the results of the equrium model shown in Fig. 8.


d 15 kHz


FIG. 11. Simulation of the effect on the mode amplitude and mode frequency in response to frequency modulation of the applied field with 5 anapplied for 0.2 ms alternately. The simulation illustrates the result of including the effect of flow damping in the model equations.

adano

ila. Il sthsthe

hoaboth

hoin

la-ra-sec-ternt.at

en-esehesta-

liefree

ilarnse

when the development of very large sawteeth normally leto a hard disruption, the hard disruption was preventedthe discharge lifetime was extended. These results are shin Fig. 13 for two discharges with edgeq;2.5, one with thefrequency modulation stabilization applied and a very simreference shot without frequency modulation stabilizationthe case of the reference shot 16 389, we see that centrax-ray emissivity rises in time until a very large sawtoocollapse occurs at about 6.6 ms with an associated loshalf the line integrated plasma density, and disruption ofplasma current which begins to collapse immediately. Whthe frequency modulation stabilization was applied in s16 411 for a 3 msperiod beginning at 4.5 ms, we see ththis discharge also suffers a large sawtooth collapse at athe same time as the reference shot 16 389, however, incase the plasma density and plasma current continue sing little or no effect. This appears to be due to preserv

FIG. 12. Results of experiment using frequency modulation of the appfield with 5 and 15 kHz applied for 0.2 ms alternately. The squaredquency modulation (V21)2 ~referenced against a 0.4 ms boxcar averagfrequency! is compared with the mode amplitude.


sd

wn

rnoft

ofent

tutisw-g

the integrity of the outer flux surfaces by frequency modution control of the 2/1 island size. The plasma core tempeture begins to recover after the sawtooth collapse and aond large sawtooth even occurs, again without loss of ouflux surface confinement or disruption of the plasma curre

The frequency modulation stabilization is turned off7.5 ms, and almost immediately~&0.1 ms! the plasma suf-fers a hard disruption of the current and loss of plasma dsity. In contrast to the usual disruption sequence for thplasmas which initiate with a large central collapse of tsoft x-ray emissivity, followed by growth of the 2/1 islandand then current disruption, in this frequency modulation s

d-d

FIG. 13. Disruption control demonstration of frequency modulation~FM!suppression. A disruptive reference shot 16 389 is compared with a simshot 16 411 with FM suppression applied which does not disrupt in respoto several large sawteeth events.


ctuthceaf-

th

d-b

ws

ve

sylaiil

iowir

abl

hnthe

onnr

aresn

linntaveisidn

ly

l ooaaninsmtr

res-re-ialbyolatu-the

deisolob-ore.l-enttion

rgyullyN.

tt.

ma

ma

ys.

l

l,

h

EE


bilized plasma the current and density loss disruption aally precedes the central soft x-ray collapse, indicatingdisruption event was initiated by loss of outer flux surfaconfinement shortly after the frequency modulation wswitched off. It has also been observed that application opredominantly 2/21 nonresonant helicity with the same amplitude and modulated frequency shows no coupling to2/1 mode and no measurable effect on the plasma.

VIII. SUMMARY AND DISCUSSION

Closed and open loop control techniques were appliegrowing m52, n51 rotating islands in wall-stabilized plasmas in the HBT-EP tokamak. The approach takenHBT-EP combines an adjustable segmented conducting~which slows the growth or stabilizes ideal external kink!with four highly modular~6° wide toroidally! saddle coilslocated between the gaps of the conducting wall and coing only about 3% of the plasma surface. The successthese experiments provides a basis for consideration oftems with greater modularity and localization. The instaltion of extensive internal coils sets in larger tokamakstechnically difficult and the higher frequency response coneeded for control of internal modes prevents their locatbehind passive stabilization structures needed to deallow-n ideal kink modes. The option of a highly modulaconfiguration in conjunction with a segmented passive stlizer may allow the possibility of active control of internaand external modes on medium scale experiments sucDIII-D and the National Spherical Tokamak Experime~NSTX!21 as well as even in larger scale devices like tInternational Thermonuclear Experimental Reactor~ITER!.22

In this paper we described experiments which demstrate two-phase island rotation control from 5 to 15 kHz athe observation of the phase instability, both of which awell modeled by the single-helicity predictions of nonlineRutherford island dynamics for 2/1 tearing modes. Thmodel equations include important effects of ion inertia aFLR which appear as a quadratic flow damping term scalike (V21)2 for the mode amplitude. We find this to be aimportant term in modeling the observations of driven rotion and the phase instability. Induced ion flow was obserwith local Langmuir Mach probes during the two-phaseland rotation drive experiments with the rate of ion fluacceleration equal to about 20% of the rate of the islaacceleration. The normal ion fluid flow velocity is nearzero and rotation rates somewhat larger than21 kHz weremeasured relative to the island rotation direction.

The closed loop response of active feedback controthe 2/1 mode at moderate gain was observed to be in gagreement with the theory when the phase angle of theplied field was varied relative to the 2/1 island. However,high system gain stable phase control was not achieved ahigher digitization rate and faster digital signal processsystem will be necessary. As a consequence the effectthe disruption process through control of the 2/1 island aplitude have not yet been demonstrated for this mode contechnique.


-e

sa

e

to

yall

r-ofs--ssnth

i-

as

-de

edg

-d-

d

fodp-td agon-ol

Experiments have been carried out which show suppsion of the 2/1 island growth using an asynchronous fquency modulation drive which maintains the ion inerteffect of flow damping. Originally suggested and modeledKurita et al.,20 we apply asynchronous external 2/1 contrfields at frequencies alternatively above and below the nral mode frequency. Previous experiments have not hadcapability to modulate the frequency of the rotating mosufficiently fast to observe significant damping from thflow stabilizing effect. The frequency modulation contrtechnique was also able to prevent disruptions normallyserved to follow giant sawtooth crashes in the plasma cBy preventing the uncontrolled growth of the 2/1 island folowing a large sawtooth as normally occurs, the confinemof the outer flux surfaces was preserved and the disrupwas prevented, allowing the plasma core to re-heat.

ACKNOWLEDGMENTS

This research is supported by Department of EneGrant No. DE-FG02-86ER53222. The authors also gratefacknowledge the technical support provided by M. Cea,Rivera, and E. Rodas.

1E. A. Lazarus, G. A. Navratil, C. M. Greenfieldet al., Phys. Rev. Lett.77,2714 ~1996!.

2C. Kessel, J. Manickam, G. Rewoldt, and W. Tang, Phys. Rev. Lett.72,1212 ~1994!; J. Manickam, M. S. Chance, S. C. Jardinet al., Phys. Plas-mas1, 1601~1994!.

3A. Turnbull, T. S. Taylor, Y. R. Lin-Liu, and H. St. John, Phys. Rev. Le74, 718 ~1995!.

4T. Taylor, E. J. Strait, L. L. Laoet al., Phys. Plasmas2, 2390~1995!.5T. Ivers, E. Eisner, A. Garafaloet al., Phys. Plasmas3, 1926~1996!.6J. P. Freidberg,Ideal Magnetohydrodynamics~Plenum, New York, 1987!.7D. A. Gates, Ph.D. thesis, Columbia University, 1993, Columbia PlasPhysics Laboratory Report No. 127.

8A. Garofalo, Ph.D. thesis, Columbia University, 1997, Columbia PlasPhysics Laboratory Report No. 130.

9E. Lazzaro and M. F. F. Nave, Phys. Fluids31, 1623~1988!.10A. W. Morris, T. C. Hender, J. Hugillet al., Phys. Rev. Lett.64, 1254

~1990!.11R. Fitzpatrick and T. C. Hender, Phys. Fluids3, 644 ~1990!.12G. Bosia and E. Lazzaro, Nucl. Fusion31, 1003~1991!.13G. D’Antonia, IEEE Trans. Nucl. Sci.41, 216 ~1994!.14A. I. Smolyakov, A. Hirose, E. Lazzaro, G. B. Re, and J. D. Callen, Ph

Plasmas2, 1581~1995!.15T. C. Hender, R. Fitzpatrick, A. W. Morriset al., Nucl. Fusion32, 2091

~1992!.16K. Oasa, H. Aikawa, Y. Asahiet al., Plasma Physics and Controlled

Nuclear Fusion Research 1994, Proceedings of the 15th InternationaConference, Seville~International Atomic Energy Agency, Vienna, 1995!,Vol. 2, p. 279.

17M. E. Mauel, E. Eisner, A. Garofaloet al., Plasma Physics and ControlledNuclear Fusion Research 1996, Proceedings of the 16th InternationaConference, Wu¨rtzburg ~International Atomic Energy Agency, Vienna1997!, Vol. 1, p. 731

18V. V. Arsenin, L. I. Artemenkov, N. V. Ivanovet al., Plasma Physics andControlled Nuclear Fusion Research 1978, Proceedings of the SeventInternational Conference, Innsbruck~International Atomic EnergyAgency, Vienna, 1979!, Vol. 1, p. 233.

19R. Savrukhin, D. J. Campbell, G. D’Antona, and A. Santagiustina, IETrans. Plasma Sci.43, 238 ~1996!.

20G. Kurita, T. Tuda, M. Azumi, and T. Takeda, Nucl. Fusion32, 1899~1992!.

21M. Ono and The NSTX Team, Bull. Am. Phys. Soc.42, 1988~1997!.22ITER-JCT and Home Teams, Plasma Phys. Controlled Fusion37, A19

~1995!.


Active control of 2/1 magnetic islands in a tokamak

Documents