Influences of multiple low- n modes on n = 1 resistive wall mode identification andfeedback controlY. In, J. Kim, J. S. Kim, A. M. Garofalo, G. L. Jackson, R. J. La Haye, E. J. Strait, M. Okabayashi, and H.Reimerdes Citation: Physics of Plasmas (1994-present) 15, 102506 (2008); doi: 10.1063/1.2999526 View online: http://dx.doi.org/10.1063/1.2999526 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/15/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Erratum: “Adaptive stochastic output feedback control of resistive wall modes in tokamaks” [Phys. Plasmas13,092508 (2006)] Phys. Plasmas 14, 109901 (2007); 10.1063/1.2754622 Adaptive stochastic output feedback control of resistive wall modes in tokamaks Phys. Plasmas 13, 092508 (2006); 10.1063/1.2355451 Model-based dynamic resistive wall mode identification and feedback control in the DIII-D tokamak Phys. Plasmas 13, 062512 (2006); 10.1063/1.2214637 Adaptive optimal stochastic state feedback control of resistive wall modes in tokamaks Phys. Plasmas 13, 012512 (2006); 10.1063/1.2161168 Feedback stabilization of nonaxisymmetric resistive wall modes in tokamaks. II. Control analysis Phys. Plasmas 7, 4143 (2000); 10.1063/1.1290481

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Influences of multiple low-n modes on n=1 resistive wall mode identificationand feedback control

Y. In,1 J. Kim,1,a J. S. Kim,1 A. M. Garofalo,2 G. L. Jackson,2 R. J. La Haye,2

E. J. Strait,2 M. Okabayashi,3 and H. Reimerdes4

1FAR-TECH, Inc., 3550 General Atomics Court, San Diego, California 92121, USA2General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA3Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, New Jersey 08543-0451, USA4Columbia University, 2960 Broadway, New York, New York 10027-6902, USA

Received 31 July 2008; accepted 19 September 2008; published online 22 October 2008

It is well known in theory that even after the n=1 resistive wall mode RWM is suppressed, theother low-n modes, such as n=2 or 3, can appear sequentially, as increases. In recent DIII-Dexperiments J. L. Luxon, Nucl. Fusion 42, 614 2002, we found such an example that supportsthe theoretical prediction: while the n=1 mode was suppressed, an n=3 mode grew dominant,leading to a collapse. The n=1 RWM suppression was likely due to a combination of rotationalstabilization and n=1 RWM feedback. The multiple RWM identification was performed using anexpanded matched filter, where n=1 and n=3 RWM basis vectors are simultaneously considered.Taking advantage of the expanded matched filter, we found that an n=3 mode following anedge-localized-mode burst grew almost linearly for several milliseconds without being hindered.This n=3 mode appeared responsible for the collapse down to the n=3 no-wall limit, as well asfor a drop in toroidal rotation. A preliminary analysis suggests that the identity of the n=3 modecould be related to the n=3 RWM possibly the first observation in tokamak experiments, while theimpact of the n=3 mode was not as destructive as that of n=1 RWM. A numerical postprocessingof Mirnov probes showed that the n=2 mode was also unstable, consistent with the theoreticalprediction. In practice, since the presence of an n=3 mode can interfere with the existing n=1 RWMidentification, multiple low-n mode identification is deemed essential not only to detect n1 mode,but also to provide accurate n=1 RWM identification and feedback control. © 2008 AmericanInstitute of Physics. DOI: 10.1063/1.2999526

I. INTRODUCTION

While significant progress has been made for the n=1resistive wall mode RWM identification and control formore than a decade,1–6 it is not surprising in theory that n1RWMs can appear even after the n=1 RWM is suppressed.Specifically, as increases, the ideal magnetohydrodynamicMHD no-wall stability limits for n=1, 2, and 3 modes areexpected to be encountered sequentially.7 Here, is the ratioof plasma kinetic pressure to externally imposed magneticfield pressure. A normalized , i.e., N, which is defined as / Ip /aB % m T /MA, indicates how close the plasmacondition is with respect to an ideal kink limit. Here, Ip is theplasma current in MA, a is the minor radius of the plasmacolumn in meters, and B is the vacuum magnetic field onaxis in Tesla. The stability gaps between the ideal MHDno-wall and ideal-wall -limits of n1 RWMs are predictedto be narrower than the stability gap of the n=1 RWM. Thus,one of the experimental expectations is to identify the n=2or 3 RWMs requiring n=1 wall-stabilized plasmas. In thisletter, we report an experimental observation in the DIII-Dtokamak that is consistent with such a theoretical predictionin high-, high-torque plasmas. Since the presence of n1modes can compromise the accuracy of the n=1 RWM iden-tification typically based on integrated odd-component sig-

nals, it is imperative that a multiple low-n mode identifica-tion be implemented even for more accurate n=1 RWMidentification and feedback control. It is to be noted that thediscussion of the n1 RWM identification would be mean-ingful on top of the n=1 mode consideration, in that the n=1RWM becomes more readily unstable than any other low-nRWM for a given . The discussion of the low-n modes inthis paper is limited to n=1, 2, and 3 modes, unless other-wise specified.

It is not unusual to observe the presence of the n1modes in high- plasmas. For example, in DIII-D, varioustoroidal mode spectra, including n=3 mode, are frequentlyobserved during edge-localized modes ELMs.8 Anothersystematic study showed the existence of n=2 resonant fieldamplification RFA in high- plasmas.9 Also in NSTX, then=2 or 3 modes are often observed, accompanying the n=1RWM.1,10 However, such n1 modes have not been attrib-uted to the n1 RWM, but to the sidebands of the n=1mode. In JT-60U, both current-driven and pressure-drivenn=1 RWMs resulted in thermal quenches accompanied byn1 modes,11 which were correlated with the n=1 growthrate changes. As a result, this letter is uniquely poised todiscuss the experimental observation of the n1 mode,which could be referred to as the n1 RWM under the n=1wall-stabilized plasmas in tokamaks.aPresent address: National Fusion Research Institute NFRI, Korea.

PHYSICS OF PLASMAS 15, 102506 2008

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II. EXPERIMENTAL OBSERVATIONS

In DIII-D, the main RWM diagnostics consist of a set ofmagnetic loop and probe pairs,12 in which one magnetic sen-sor is hard-wired in antiseries with the other counterpart sen-sor displaced in 180° in the toroidal direction primarily inorder to eliminate the axisymmetric component n=0. Thus,all the odd components n=1,3 , . . . are contributing to themeasured signals, while all the even components n=0,2 , . . . are being eliminated. Since the dominant RWMcomponent is typically the n=1 that occurs below the no-wall stability limits of n1 modes, no complication is usu-ally involved in the mode identification based on such odd-component based magnetic sensor signals. Nonetheless, toprovide an accurate n1 RWM-relevant mode identificationin high- plasmas, a new approach has been developed andis evaluated here. Specifically, a new set of magnetic sensorshas been configured for the n=2 mode identification inDIII-D, while an advanced algorithm has been developed toextract the n=3 mode from the existing odd-componentbased magnetic sensor signals. Considering that the prelimi-nary result of the n=2 mode detection is rather noisy, a nu-merical postprocessing of Mirnov probes has been used forn=2 mode analysis. It was based on a numerical integrationof dB /dt measured with a set of eight midplane magneticprobes with the n=0 component removed. Considering basisvectors for n=1, 2, and 3 simultaneously, each mode ampli-

tude and phase are calculated. Meanwhile, the existence ofthe n=3 component may influence the accuracy of the stan-dard n=1 RWM identification, because all slowly evolvingn=odd components contribute collectively to the existingmagnetic sensor signals for RWM feedback in DIII-D.

Figure 1 shows the time evolution of the high-, high-torque experiments, where the n=3 mode appears respon-sible for the collapse and rotation drop Figs. 1a, 1c,and 1e. In this discharge, the ELM bursts are almost al-ways accompanied by various nonaxisymmetric modes, suchas the n=1 and n=3 components Figs. 1b and 1e. Basedon the expanded matched-filter analysis13 as explained in thesupplement,14 the magnitudes of the ELM-driven n=1 andn=3 components are up to 30, and 15 G, respectively Fig.1e, while the n=2 components are up to 18 G Fig. 1g.The n=1 mode identification comparison with and withoutthe n=3 basis vectors are shown in the blue and cyan tracesin Fig. 1d, respectively. At the time of the RWM experi-ments, the feedback coils were configured to deliver thecounteracting n=1 fields based on the measured magneticfields from four pairs of the midplane poloidal sensors with-out including the n=3 basis vectors. Thus, the difference ofthe two traces is primarily attributable to the n=3 compo-nent, whose mode amplitude is shown in the red trace ofFigs. 1e and 1g. It is clear that without taking into ac-count the multiple low-n modes, the odd-component based

βN

βNn=1 no wall

~ 4 li

Shot 122929(a)

Photodiode (au) (b)

Plasma rotation (km/s) at ρ ~ 0.7 (c)

Matched filtered δBn=1 (G) (d)n=1 bases onlyn=1 and n=3 bases

1.62 1.64 1.66 1.68 1.70Time (s) Time (s)

(e)δB

n=1δB

n=3(G)

Ωφ(km/s) (f)

Core: ρ ~ 0.1Middle:ρ ~ 0.7Edge: ρ ~ 0.9

1.664 1.668 1.672

(g)δBn=1

(G)

δBn=2

δBn=3

01234

0

123

050

100150

0255075

0

20

40

400

200

0

200

400

0

10

20

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40

FIG. 1. Color Time evolution of the n1 modes responsible for the plasma rotation drop and collapse. Shown are the time traces of a N, and Nn=1 no-wall

the estimated n=1 no-wall N - limit=4i; b upper plane view photodiode; c plasma rotation at =0.7, where is the normalized radius; d matched-filtered n=1 mode amplitudes with and without the n=3 basis vectors; e n=1 and n=3 mode amplitudes based on the expanded matched filter; f carbonplasma rotation at =0.1, 0.7 and 0.9; g expanded view of the n=1 and n=3 mode amplitudes with the numerically extracted n=2 mode amplitude fromMirnov probes. The time frames in f and g are denoted in c and e, respectively.

102506-2 In et al. Phys. Plasmas 15, 102506 2008

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mode identification cyan trace of Fig. 1d would not beappropriate even for the n=1 RWM identification, in that thegenuine n=1 RWM did not grow but remained rather fluc-tuating in amplitude but finite blue trace of Figs. 1d, 1e,and 1g. The fluctuating finite n=1 mode is likely due toresonant field amplification RFA in combination with the“imperfect” n=1 active feedback control. On the other hand,the ELM-driven n=3 mode grew unstably without being hin-dered e-folding growth time: 3 ms, while the n=2 modewas also observed to grow and then decay in a few millisec-ond scale Fig. 1g. As a note, the n=1 coil configurationsusing the in-vessel control coils in DIII-D have various side-band spectra, in which the n=3 and 5 components are calcu-lated to be as high as 10% and 20%, respectively.15 However,as the influences of the in-vessel coil currents on the mag-netic sensors are subtracted out prior to the RWM identifica-tion, the n1 components of the control coils would notaffect the active feedback process unless the plasma re-sponses to n1 modes become significant.

Interestingly, the rotation in the outer radius of plasmacolumn e.g., =0.7, and 0.9 in Fig. 1f appeared to de-crease slightly earlier than the ELM burst at t=1666 ms,while the core rotation e.g., =0.1 in Fig. 1f was ratherinsensitive to the ELM burst. The rotation profile near then=2 and 3 mode onsets is similar to what was observed atthe RWM onset where the edge rotation profile, not corerotation profile, appeared correlated with the n=1 RWMonsets e.g., Fig. 5 of Ref. 16. However, taking into accountthe charge exchange recombination CER diagnosticintegration time of 2 ms i.e., sampled at t=1664,1666,1668 ms, etc., the edge rotation drop at t=1666 ms is likely due to the ELM bursts occurring betweent=1666 and 1668 ms, rather than preceding the ELM burst.Similar caution needs to be taken to interpret the collapsenear t=1.67 s in Fig. 1a, where the real-time EFIT recon-struction was made at every 10 ms with 5 ms magneticsintegration time. As a note, the equilibria used for the de-tailed stability study were manually reconstructed using 1or 2 ms magnetics integration time.

More interestingly, there is a plasma rotation directionchange from counter-Ip to co-Ip direction at =0.9 near t=1666 ms, simultaneous with the growth of the n=2 and 3modes.

III. STABILITY ANALYSIS

The n=2 and 3 modes grew on the time scale of char-acteristic resistive wall time e.g., 3–5 ms for DIII-D12 andthe plasma stayed above the estimated n=1 no-wall limiteven after the collapse. Initially, the relatively benign collapse and quick rotation recovery were thought to besolely due to the synergistic effect of the n=1 active feed-back control with the plasma rotation.17 However, a moredetailed study showed that the n=2 and 3 modes, togetherwith the residual n=1 mode, are responsible for the plasmaperformance degradation, as presented in this letter. The de-tailed stability analysis was done at two time slices at t=1650 ms and 1662 ms, which were the closest to the timeof interest i.e., t=1666 ms with the Thomson scattering di-

agnostic signals necessary for an equilibrium reconstructionincluding kinetic data. Figure 2 shows the profiles of pres-sure, safety factor, and toroidal rotation. Each solid curve isfor t=1662 ms, which is right after an ELM burst. While thepressure profile solid curve shown in Fig. 2a is basedprimarily on the experimental measurements, the otherneighboring profiles either dashed or dash-dot curves are asubset of the scaled pressure profiles used for stability calcu-lation. In the toroidal rotation profile and the expanded viewof the edge pressure profile Figs. 2b and 2c, respec-tively, a dashed curve, which shows the pre-ELM character-istics at t=1650 ms, is added for comparison. The edge ro-tation in the post-ELM phase is significantly lower than inthe pre-ELM phase. Also the edge pressure gradient in thepost-ELM phase is weaker than in the pre-ELM phase, inwhich the edge bootstrap current is playing a critical role todetermine the stability.

The key parameters of the q-profile shown in the dashedcurve of Fig. 2a are q0=3.1, qmin=2.2, q95=5.4, and qedge

=7.1. According to the kinetic EFIT equilibrium reconstruc-tion, the N was 3.76, which is approximately 4% higherthan in the real-time EFIT shown in Fig. 1a. Based on theideal MHD stability analysis without a wall using DCON,18

the equilibrium at t=1662 ms in the post-ELM phase waspredicted to be unstable for all the low-n modes i.e., n=1, 2,and 3. The DCON stability results were also checked to beconsistent with the stability results using GATO.19 Based onthe post-ELM kinetic equilibrium pressure profile shown inFig. 2a, the ideal MHD no-wall and ideal-wall limits wereinvestigated. Assuming that the shape of the pressure profileswould remain the same and that the pressure profile can belinearly scaled as increases, a series of equilibria wereconstructed and their stabilities were evaluated using the

Normalized Radius, ρ

Profiles of p, q and Ωφ on shot 122929 near t = 1662 ms

0.0 0.5 1.0ρ

Ωφ (krad/s)(b)

Post-ELMPre-ELM

0.85 0.90 0.95 1.00ρ

Edgepressure (kPa)

(c)

0

10

20

0.0 0.2 0.4 0.6 0.8 1.0

100

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(a)

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FIG. 2. Color online Profiles of pressure, safety factor, and plasma rota-tion. Shown are the profiles of a pressure and safety factor at t=1662 ms,b plasma rotation, and c detailed edge pressures from the region denotedin a. In a, the pressure profile solid curve is based on the kinetic EFIT

using the experimental measurements, whereas the other neighboring pro-files either dashed or dash-dot curve are a part of the scaled pressureprofiles adopted for stability study. In b and c, the dashed curves are theprofiles at t=1650 ms in the pre-ELM phase, while the solid curves are att=1662 ms in the post-ELM phase. The rotation measurement error bars arewithin 3–4 krad /s, which are not distinguishable on this figure, if plotted.

102506-3 Influences of multiple low-n modes… Phys. Plasmas 15, 102506 2008

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DCON stability code. Figure 3 shows the results of the idealMHD no-wall and ideal-wall limits. According to the DCON

calculations, the no-wall limits for n=1, 2, and 3 modes are2.73, 3.30, and 3.30, respectively. Also, the ideal-wall limitsfor n=1, 2, and 3 modes are 5.37, 4.14, and 3.85, respec-tively, as shown in Fig. 3. Meanwhile, the analysis of thecounterpart equilibrium at t=1650 ms in the pre-ELM phasenot shown indicated that both n=1 and n=2 modes wouldbe unstable, but that the n=3 mode would be marginallystable. In fact, the no-wall limits for all the low-n modes inthe pre-ELM phase were found to be higher by approxi-mately 5% e.g., 2.88 for the n=1 mode. Considering thatthe edge pressure and current density profiles would be criti-cal in terms of the kink and ballooning calculations, the edgeprofiles were systematically studied. Taking into account anextreme case with a flattened edge pressure profile notshown, we assessed the integrity of the stability calculationresults. While the stability calculation results for the n=1mode were not so sensitive to edge profile modifications, thevariation of no-wall N-limit for the n=2 and 3 modes wasup to 10%. However, when we excluded the extreme casewith the flattened edge pressure profile, the variation of then1 mode no-wall N-limits was reduced to 7%. Thus,we believe that the stability calculation trends for the n1modes are reasonable, although they are still more sensitivethan the stability limits of the n=1 mode to variations in theedge profiles.

IV. DISCUSSION

Noting that the pedestal pressure in the pre-ELM phaseis higher than in the post-ELM phase, such higher no-wallstability limits in the pre-ELM phase are also consistent withthe results obtained from a recent systematic study of ideal

stability with transport barriers.20 This suggests that theRWMs are more likely to grow in the post-ELM phases thanin the pre-ELM phases, in that the ideal no-wall limits in thepost-ELM phases are lower. Such theoretical calculation re-sults are also consistent with the experimental observationsthat most of the RWMs correlated with ELMs occurred inthe post-ELM phases even in high-torque plasmas in whichthe plasma rotations supposedly well exceeded the rotationalthresholds for RWMs. On the other hand, it is equally impor-tant not to underestimate the impact of the ELM-inducedtoroidal mode spectra on the rotationally stabilized RWMplasmas. For example, when ELM-induced nonaxisymmetricmodes are present in high- plasmas, either resonant or non-resonant fields would be amplified or penetrating into theplasma, unless they are rotationally shielded. The unshieldednonaxisymmetric modes would lower the plasma rotation,interacting with the rotationally stabilized RWM directly orindirectly, and then leading to an unstable RWM. So far, it isnot clear whether the n1 mode was triggered by the n1RWM i.e., converted from n1 ideal kink mode, or it wasdue to the interaction of rotationally stabilized n1 RWMwith the ELM-induced n1 mode, escalating to an unstablen1 RWM.

Based on the multimode identification analysis, there areseveral interesting points to note. First, the n=1 RWM feed-back control did not suppress the n=1 mode completely andmight have been adding some n=1 field due to the overesti-mated n=1 mode identification by the amount contributedby the other n1 odd modes. Actually, the n=1 compo-nents did not decrease but remained finite, likely due to theRFA in combination with the overdriven n=1 components.Second, the n=3 mode grew up without being hindered untilthe N collapsed below the n=3 no-wall -limit, while then=2 mode also grew up and then damped away quickly. Asthe n=1 feedback control coils were configured not to de-liver an n=3 field, the origin of the n=3 mode is believednot to be in the external feedback control coils, but to be inthe plasma either from the n=3 RWM or due to the inter-action of the rotationally stabilized n=3 RWM with theELM-induced n=3 mode. It is also conjectured that theplasma rotation alone might not have been strong enough todampen the n=3 mode and that the rotation threshold of then=3 mode could be higher than that of the n=1 mode. Third,the N and rotation collapses are likely due to the n=3 mode,rather than solely due to the n=1 mode. Considering thatthere was no other collapse similar to the one shown neart=1666 ms, the n=1 mode might have been effectively con-trolled by the n=1 RWM feedback albeit with imperfectestimates, as well as by rotational stabilization. In fact, the was maintained to be below the n=3 no-wall -limit, butabove the n=1 no-wall -limit without further collapse,when the high neutral beam power was continuously in-jected. Also, when the n=3 mode disappeared, the rotationdid not decrease further, but it was instead followed by quickrecovery. It is fair to remark that the n=2 no-wall -limitwas not exceeded after the collapse, either. Meanwhile, itis possible that the n=2 mode might have been stabilizedquickly due to the plasma rotation. Taking into account thatmultiple RWMs were successfully stabilized by active feed-

1 2 32

3

4

5

6

Toroidal mode number, n

β NIdeal MHD no wall and ideal wall

βN limits based on DCON

2.73

5.37

3.30

4.14

3.30

3.853.76 (Experiments)

122929 at t = 1662 ms(right after an ELM burst)

No wall limitIdeal wall limit

FIG. 3. Color online Ideal MHD no-wall and ideal-wall -limits based onDCON. Based on a kinetic equilibrium reconstruction at t=1662 ms of theDIII-D discharge 122929, the ideal MHD no-wall and ideal-wall limits were calculated for n=1, 2, and 3 modes using the DCON stability code.The experimental value is also shown for comparison.

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back control in reversed field pinch e.g., EXTRAP T2R21

and RFX22, multiple low-n modes in high- tokamak plas-mas would not be an obstacle for ITER project but it ishighly desirable to have an accurate RWM identification andfeedback control system.

V. CONCLUSION

In conclusion, there was an experimental observation ofthe n1 modes that appeared responsible for the plasmarotation drop and collapse down below the n1 no-wall-limits but above the n=1 no-wall -limit during high-,high-torque plasmas in DIII-D. This appears consistent witha theoretical prediction about the n1 RWM in the n=1wall-stabilized plasmas, when an ELM burst lowers the n1no-wall limit. However, it is also possible that the n1mode was due to the interaction of a rotationally stabilizedn1 RWM with an ELM-induced n1 mode, leading to anunstable n=2 and 3 RWMs. Regardless of the origin of then1 modes, a multiple low-n mode identification is criticalnot only to diagnose the n1 modes, but also to enhance theaccuracy of the n=1 mode identification and feedback con-trol.

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of En-ergy SBIR under Contract Nos. DE-FG02-06ER84442, DE-FC02-04ER54698, DE-AC02-76CH03073, and DE-FG02-89ER53297. We would like to thank all the staff in theDIII-D Team for their successful operations of the machine.

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