statistical analysis of m/n = 2/1 locked and quasi

Statistical analysis of m/n = 2/1 locked andquasi-stationary modes with rotating precursors atDIII-D

R. Sweeney1, W. Choi1, R.J. La Haye2, S. Mao3,+,K.E.J. Olofsson1,*, F.A. Volpe1, and the DIII-D Team1 Columbia University, New York, NY 100272 General Atomics, San Diego, CA 921213 University of Wisconsin, Madison, WI 53706+ Present address: Stanford University, Stanford, CA 94305∗ Present address: General Atomics, San Diego, CA 92121

Abstract.A database has been developed to study the evolution, the nonlinear effects on

equilibria, and the disruptivity of locked and quasi-stationary modes with poloidaland toroidal mode numbers m = 2 and n = 1 at DIII-D. The analysis of 22,500discharges shows that more than 18% of disruptions are due to locked or quasi-stationary modes with rotating precursors (not including born locked modes). Aparameter formulated by the plasma internal inductance li divided by the safetyfactor at 95% of the poloidal flux, q95, is found to exhibit predictive capabilityover whether a locked mode will cause a disruption or not, and does so up tohundreds of milliseconds before the disruption. Within 20 ms of the disruption,the shortest distance between the island separatrix and the unperturbed lastclosed flux surface, referred to as dedge, performs comparably to li/q95 in its abilityto discriminate disruptive locked modes. Out of all parameters considered, dedge

also correlates best with the duration of the locked mode. Disruptivity followinga m/n = 2/1 locked mode as a function of the normalized beta, βN , is observedto peak at an intermediate value, and decrease for high values. The decrease isattributed to the correlation between βN and q95 in the DIII-D operational space.Within 50 ms of a locked mode disruption, average behavior includes exponentialgrowth of the n = 1 perturbed field, which might be due to the 2/1 locked mode.Surprisingly, even assuming the aforementioned 2/1 growth, disruptivity followinga locked mode shows little dependence on island width up to 20 ms before thedisruption. Separately, greater deceleration of the rotating precursor is observedwhen the wall torque is large. At locking, modes are often observed to alignat a particular phase, which is likely related to a residual error field. Timescalesassociated with the mode evolution are also studied and dictate the response timesnecessary for disruption avoidance and mitigation. Observations of the evolutionof βN during a locked mode, the effects of poloidal beta on the saturated width,and the reduction in Shafranov shift during locking are also presented.

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Statistical analysis of m/n = 2/1 locked and quasi-stationary modes with rotating precursors at DIII-D 2

1. Introduction

Rotating and non-rotating ("locked") neoclassical tear-ing modes (NTMs) are known to degrade confinementand cause disruptions in tokamak plasmas under cer-tain plasma conditions, and thus represent a concernfor ITER [1].

NTMs can rotate at the local plasma rotationvelocity, apart from a small offset of the order ofthe electron or ion diamagnetic velocity [2]. Inpresent tokamaks with strong torque injection, thiscorresponds to rotation frequencies of several kHz.Magnetic islands can also steadily rotate at frequenciesof the order of the inverse resistive-wall time (tensof Hz, typically), if a stable torque balance can beestablished at that frequency. In this case, they arecalled Quasi Stationary Modes (QSMs) [3].

There are also NTMs that do not rotate at all,called Locked Modes (LMs). Some of these arethe result of an initially rotating NTM ("rotatingprecursor") decelerating and locking to the residualerror field. The deceleration might be due to themagnetic braking experienced by the rotating island inits interaction with the eddy currents that it induces inthe resistive wall [4]. Other modes are "born locked",i.e. they form without a rotating precursor, as a resultof resonant error field penetration.

It will be important to understand the onset,growth, saturation, and stabilization of all thesecategories of rotating and non-rotating NTMs, in orderto maintain good confinement and prevent disruptionsin ITER. Here we present an extensive analysis ofQSMs and LMs with rotating precursors, which wewill sometimes refer to as "initially rotating lockedmodes", or IRLMs. The analysis was carried overapproximately 22,500 DIII-D [5] plasma discharges,and restricted to poloidal/toroidal mode numbersm/n=2/1, because these are the mode numbersthat are most detrimental to plasma confinement inDIII-D and most other tokamaks [6]. QSMs andLMs of different m/n (for example 3/2, occasionallyobserved at DIII-D) and LMs not preceded by rotatingprecursors are not considered here and will be thesubject of a separate work.

Previous works have described classical [7, 8]and neoclassical [9, 6, 10] tearing modes (TMs),and the torques acting on TMs [4]. Effects ofTMs on confinement are described theoretically [11].Summaries of error-field-penetration locked modes inDIII-D [12], disruption phenomenology in JET [13],and disruption observations across many machines [14]have been reported.

Statistics regarding the role of plasma equilibriumparameters, and MHD stability limits on disruptionshave been reported at JET [15], and one work includesa detailed accounting of the instabilities preceding the

disruptions [16]. A comprehensive study of disruptivityas a function of various equilibrium parameters wasconducted at NSTX [17] followed by a work ondisruption prediction including n = 1 locked modeparameters [18]. A study of various disruption typeson JT-60U, including those caused by tearing modes,has been reported [19]. A brief study of disruptivityas a function of the safety factor and the normalizedplasma beta on DIII-D has been reported [20].

Machine learning approaches to disruption predic-tion have shown rather high levels of success at ASDEXUpgrade [21, 22], JET [23], and JT-60U [24], and a pre-dictor tuned on JET was ported to ASDEX Upgrade[25]. Similarly, neural networks have been used for dis-ruption prediction at TEXT [26], and ADITYA [27],and for predicting ideal stability boundaries on DIII-D [28]. Discriminant analysis for disruption predictionwas tested at ASDEX Upgrade [29].

A statistical work on MAST investigated thedifferences between disruptive and non-disruptive LMsas a function of normalized plasma beta and the safetyfactor [30]. Recently a locked mode thermal quenchthreshold has been proposed based on equilibriumparameters from studies on JET, ASDEX Upgrade,and COMPASS [31]. The automatic detection oflocked modes with rotating precursors makes thedisruption statistics presented here unique. Inaddition, this work includes basic observations oflocked modes and their effects on equilibria.

During the first discharges in this database, DIII-D was equipped with one poloidal and four toroidalarrays of Mirnov probes and saddle loops [33]. Duringthe time spanned by the database, additional sensorswere added for increased 3D resolution [34]. However,a limited set of six saddle loop sensors (external to thevessel) and three poloidal sensors (inside the vessel) areused for simplicity, and for consistency of the analysisacross all shots, spanning the years 2005-2014.

An example of the 2/1 IRLMs considered here isillustrated in figure 1. The poloidal field amplitudeof the rotating precursor is detected by the toroidalarray of Mirnov probes around 1800 ms. The modesimultaneously grows and slows down until it locks at1978.5 ms. Due to the finite time binning used in theFourier analysis during the rotating phase, the rotatingsignal is lost at low frequency, and is instead measuredby a set of large saddle loops (ESLDs: external saddleloops differenced). The response of the saddle loopsincreases when ω < τ−1

w , where τw ∼ 3 ms is thecharacteristic n = 1 wall time for DIII-D. As shownin Fig.1, it is not uncommon for the amplitude of anIRLM to oscillate due to minor disruptions, and togrow prior to disruption (this will be investigated insection 6.2).

Three interesting results of this work are intro-


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Figure 1. Example of an initially rotating locked mode (IRLM).The black trace presents a fast rotating 2/1 NTM, as measuredby the set of Mirnov probes and analyzed by eigspec[32]. At thetime of locking (1978.5 ms), the low frequency mode is detectedby the ESLDs, shown in blue. The slow-down time is the timetaken for a mode rotating at 2 kHz to slow and lock; survivaltime is the duration of an IRLM that ends in a disruption. Afactor of 0.5 has been multiplied to the Mirnov probes signalto account for the eddy currents in the wall during fast moderotation. A factor of 2 has been multiplied to the ESLD signalto obtain the peak radial magnetic field from the measurementaveraged over the large ESLD area.

duced now. First, it will be shown that the m/n = 2/1island width cannot be used to distinguish disruptivefrom non-disruptive IRLMs 20 ms or more ahead ofthe disruption time. Similarly, the island width showslittle correlation with the IRLM survival time.

Second, the plasma internal inductance dividedby the safety factor, li/q95, distinguishes IRLMs thatwill disrupt from those that will not. The predictivecapability of li/q95 might be related to the energyavailable to drive nonlinear island growth.

Finally, a spatial parameter which couples theq = 2 radius and the island width, referred to asdedge (see section 5 for definition), also distinguishesdisruptive from non-disruptive IRLMs well within 20ms of the disruption. It also correlates best with theIRLM survival time. The predictive capability of dedge

is believed to be related to the physics of the thermalquench.

The paper is organized as follows. Section 2explains the method of detection of disruptions, ofrotating tearing modes, and of LMs. Section 3 providessome general statistics of IRLM occurrences in DIII-D.Section 4 quantifies the timescales of interest beforelocking. Section 5 investigates the time availableto intervene before an IRLM causes a disruption.Section 6 discusses the width and phase behavior atlocking, and the exponential growth of the n = 1field before the disruption. Section 7 details theinterdependence between IRLMs and plasma β (β =〈p〉/(B2/2µ0) where 〈p〉 is the average pressure and Bis the average total field strength). Section 8 decouplesthe influence of ρq2, q95, and li on IRLM disruptivity,and investigates the effectiveness of li/q95, the islandwidth, and dedge as disruption predictors. A discussionsection follows which offers possible explanations of thephysical relevance of li/q95 and dedge. Finally, twoappendices are dedicated to the mapping from radialmagnetic field measurements to the perturbed islandcurrent, and from the perturbed current to an islandwidth.

2. Method

2.1. Detection of disruptions

To categorize disruptive and non-disruptive modes, aclear definition of disruption is needed. The plasmacurrent decay-time is used to differentiate disruptiveand non-disruptive plasma discharges. The decay-timetD is defined as the shortest interval over which 60%of the flat-top current is lost, divided by 0.6. In caseswhere the monotonic decrease of Ip extends beyond60%, the entire duration of the current decrease isused, with proper normalization. The disruption timeis defined as the beginning of the current quench,which is usually preceded by a thermal quench, a fewmilliseconds prior.

The criterion tD <40 ms used to identify DIII-Ddisruptions was formulated as follows.

A histogram of all decay-times is shown in figure2, and features of the distribution are used to definethree populations. The first group peaks near tD=0and extends up to tD=40 ms. These are rapid lossesof Ip and confinement, quicker than typical energy andparticle confinement times. Discharges in this groupare categorized as major disruptions, either occuringduring the Ip flat-top, or occurring during a partialcontrolled ramp-down of less than 40% of the flat-topvalue. The sudden loss of current during the partialramp-down cases must be fast enough to normalize toan equivalent 40 ms or less full current quench. It isworth noting that of the 5,783 disruptions detected,


666 occurred within the first second (in the ramp-upphase), none of which were caused by a 2/1 IRLM.At the opposite limit, group iii, with tD > 200 ms,contains mostly (88 ± 4%) non-disruptive discharges,in which the plasma current decays at a steady ratefor at least 80% of the ramp-down. Population iihas tD in the range 40 ms < tD < 200 ms, andmostly consists of shots that disrupted during thecurrent ramp-down with “long decay” times relativeto population i disruptions.

Note that while the stringent threshold of tD <40 ms will prevent falsely categorizing non-disruptivedischarges as disruptive, it may also categorize somedisruptions with slightly longer decay times into groupii. However, the thresholds are chosen to protectagainst false positives better than false negatives;they are chosen to compromise missing a number ofdisruptive shots in exchange for ensuring the validityof all disruptive discharges. In the remainder of thiswork, we will focus on groups i and iii only.

In a manual investigation of 100 discharges ingroup i, 85 disruptions occurred during the currentflat-top, and 15 disruptions occurred during thecurrent ramp-down phase. Therefore, the majority ofdisruptions studied in this work (i.e. 85 ± 4%) occurduring a current flat-top.

In section 3, disruptivity will be studied over theentire database, including shots that did not containIRLMs. We will refer to disruptivity in this contextas global disruptivity (i.e. the number of disrupteddischarges divided by all discharges).

In all sections following section 3, disruptivity willbe studied on shots that contained IRLMs only. Inmost cases, we will be interested in studying whatdifferentiates a disruptive IRLM from a non-disruptiveIRLM, and therefore we define IRLM disruptivity asthe number of disruptive IRLMs divided by the totalnumber of IRLMs (where the total is the sum ofdisruptive and non-disruptive IRLMs). In one case,it will be useful to discuss IRLM shot disruptivity,which is the number of disruptive IRLMs divided bythe total number of discharges with IRLMs (note that anon-disruptive discharge can have many non-disruptiveIRLMs, making this distinction non-trivial).

2.2. IRLM Disruptivity during current flat-tops

For all IRLM disruptivity studies, disruptions thatoccur during Ip ramp-downs are limited to 15 ± 4%of the studied set. Ip ramp-downs are characterizedby major changes of the plasma equilibrium, and areexpected to greatly impact the locked mode evolution.Namely, key parameters such as q95, li, and ρq2evolve during an Ip ramp-down, complicating theinterpretation of their effect on IRLM disruptivity.Moreover, flat-tops will be longer and longer in ITER

and DEMO, and thus disruptivity during ramp-downwill become less and less important. Eventually, ina steady-state powerplant, only flat-top disruptivityshould matter.

Out of 1,113 shots which disrupted due to anIRLM, 105 contained an additional IRLM distinct intime from the final disruptive one. As these additionalIRLMs decayed or spun-up benignly, yet occurred inplasmas that ultimately disrupted, they are consideredneither disruptive nor non-disruptive, and are excludedfrom the IRLM disruptivity studies. Similarly, asmall number of discharges disrupt without an IRLMpresent, but contain an IRLM 100 ms before thedisruption or earlier. In these cases, it is not clearwhether the IRLM indirectly caused the disruption ornot, and therefore these cases are also excluded.

2.3. Detection of rotating modes of even m and n=1

Detection of a rotating mode is performed intwo stages. For each shot, the signals froma pair of toroidally displaced outboard midplanemagnetic probes are analyzed by the newspec Fourieranalysis code [33], in search of n=1 activity.Genuine n=1 magnetohydrodynamic (MHD) activityis distinguished from n=1 noise by searching for bothan n=1 amplitude sustained above a chosen threshold,as well as requiring the corresponding n=1 frequencyto be "smooth". An adaptive threshold is used toaccept both large amplitude, short duration modes, aswell as small amplitude, long duration modes. Oncedetected, this activity is analyzed by a simplifiedversion of the modal analysis eigspec code, based onstochastic subspace identification [32]. Here eigspecuses 9 outboard mid-plane and 1 inboard mid-planeMirnov probes to isolate n = 1 and determine whetherm is even or odd. Selecting modes of even m andn = 1 rejects n = 1 false positives due to 1/1 sawtoothactivity, and other odd m and n = 1 activity. Modeswith n=1 and evenm are predominantly 2/1. Providedq95 is sufficiently high, they might in principle be 4/1 or6/1 modes, but these modes are rare. Manual analysisof 20 automatically detected modes of evenm and n=1found only 2/1 modes.

2.4. Detection of n = 1 locked modes

Locked modes are detected using difference pairsof the integrated external saddle loops (ESLDs).A toroidal array of six external saddle loops isavailable. Differencing of loops positioned 180◦ aparttoroidally eliminates all n = even modes, including theequilibrium fields. A least squares approach is thenused to fit the n = 1 and n = 3 toroidal harmonics[33]. This approach assumes that the contributions ofodd n ≥ 5 are negligible.


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Figure 2. The distribution of plasma current decay time,roughly split into three populations. Panel (a) shows the non-disruptive discharges with decay times greater than 200 ms;panel (b) further distinguishes the remaining population intomajor disruptions (< 40 ms, consisting of > 80% flat-topdisruptions), and disruptions with longer decay times, whichare predominantly disruptions during ramp-down. Note thatthe vertical axis on panel (a) is interrupted to better show thefeatures in the distribution.

Each pair-differenced signal is compensated forpickup of the non-axisymmetric coils, using a combi-nation of analog and digital techniques. The accuracyof the coil compensations was assessed using vacuumshots from 2011-2014. Residual coil pickup peaks at∼ 3 G, but a conservative threshold of 5 G is chosenfor identification of LMs to avoid false positives. SmallLMs that produce signals less than 5 G are not consid-ered in this work.

Analog integrators are known to add linear driftsto the saddle loop signals. In addition, n =1 asymmetries in the plasma equilibrium can alsoproduce background noise.

A simple yet robust algorithm was developedto subtract this background. The algorithm works

on the principle that times exist during which it isimpossible for a LM to exist, and the n=1 "lockedmode signal" at those times must be zero. Thesetimes include the beginning and end of every shot,and times at which m/n = even/1 modes are known,from Mirnov probe analysis, to rotate too rapidly tobe QSMs or LMs. As LMs cause a significant decreasein βN = βaB/Ip (as will be discussed in Section 7),the time when βN is maximized is also highly unlikelyto have a coincident LM, and therefore, this time isalso used. A piecewise linear function with nodesat each identified "2/1 LM free" time is fit to eachESLD signal independently and subtracted to producea signal with minimal effects from integrator drift andnon-axisymmetric equilibrium pickup.

Fifty shots automatically identified to have LMswith n = 1 signals in excess of 5 G were manuallyanalyzed. This analysis confirmed that in most casesthe automatic identification was accurate, with apercentage of false positives for LMs with rotatingprecursors of < 4%. The identification of locked modeswithout rotating precursors (born locked modes), onthe other hand, exhibited a percentage of false positives> 30%. Greater accuracy is achieved for LMswith rotating precursors because the fast rotatingprecursor provides a LM free background subtractionjust prior to locking. In addition, locked modes withrotating precursors require two subsequent events: theappearance of a rotating m/n = even/1 tearing modefollowed by an n = 1 locked mode. Due to the highpercentage of false positives in the identification ofborn LMs, they are not considered in this work, butwill be the topic of future work.

During the locked phase, no poloidal harmonicanalysis is performed. It is assumed that the confirmedm/n = even/1 rotating mode present immediatelybefore locking is likely a 2/1 mode, and upon locking,the mode maintains its poloidal structure. In addition,it is assumed that the locked n = 1 signal measuredby the ESLDs is predominantly due to the 2/1 mode,such that this field measurement can be used toinfer properties of the mode. A set of 63 disruptiveIRLMs, occurring in plasmas with |BT | > 2 T, wereinvestigated using the 40 channel electron cyclotronemission (ECE) diagnostic [35] to validate theseassumptions. Only plasmas with |BT | > 2 T areconsidered to ensure that ECE channels cover a plasmaregion extending from the core through the last closedflux surface on the outboard midplane. Among these63 IRLMs, 26 exhibited QSM characteristics, makingfull toroidal rotations, allowing the island O-point tobe observed by the toroidally localized ECE diagnostic.In all 26 cases, a flattening of the electron temperatureprofile is evident at the q = 2 surface, and no surfaceswith q > 2 show obvious profile flattening, suggesting


that higher m modes are not present, or are too smallto resolve with ECE channels separated by ∼ 1−2 cm.

As the gradient in the electron temperature tendsto approach zero near the core, confirming the presenceor absence of a 1/1 mode is difficult. Despite the 1/1mode existence being unknown, island widths derivedfrom the radial field measured by the ESLDs, wherethe n = 1 signal is assumed to be a result of the2/1 mode only, are calibrated to within ±2 cm withthe flattened Te profiles as measured by ECE (seeAppendix B). We conclude that in the majority ofcases, the locked modes are 2/1 and the inferred islandwidths are reasonably accurate. The minority of caseswhere this is not true are not expected to affect thestatistical averages presented in this work.

The locked mode analysis includes a check for theexistence of a q = 2 surface, which is a necessarycondition for the existence of a 2/1 IRLM. In 114disruptive discharges, reconstructed equilibrium dataare absent for 80% or more of the locked phase, duringwhich time the existence of a q = 2 surface cannot beconfirmed. The majority of these omitted dischargeshave locked phases lasting less than 20 ms, which isthe time-resolution of equilibrium reconstructions, andtherefore no equilibrium data exist after the modelocks (note that although 20 ms is a relatively shorttimescale, significant changes in the equilibrium areexpected upon mode locking). All 114 dischargeswere manually analyzed, and approximately 40% wereidentified as vertical displacement events or operatorinduced disruptions, such as discharges terminatedby massive gas or pellet injection. About 15% lacknecessary data to manually identify the cause of thedisruption. The remaining 53 disruptive dischargesare considered valid 2/1 IRLM disruptions. Thesedischarges are not included in the majority of thefigures and discussion herein, but will be discussed inthe disruption prediction section (section 8.5) as theyare expected to modestly decrease the performance ofthe predictors.

2.5. Perturbed currents associated with the islands

The mode amplitudes will sometimes be reported interms of the total perturbed current carried by theisland δI. δI is a quantity that is local to the q = 2surface, and its calculation accounts for toroidicity.The wire filament model used in [36] was shown toreproduce experimental magnetics signals well and wasadapted for this calculation. An analytic version ofthis model was developed which simulates the islandcurrent perturbation with helical wire filaments thattrace out a torus of circular cross-section. The torushas the major and minor radii of the q = 2 surfaceinformed by experimental EFIT MHD equilibriumreconstructions [37], which use magnetics signals and

Motional Stark Effect measurements [38] to constrainthe reconstruction. An analytic expression is foundfor δI as a function of the experimental measurementof BR from the ESLDs, and R0 and rq2 from EFITreconstructions. The model and the resulting analyticexpression are detailed in Appendix A.

3. Incidence of locking and global disruptivityon DIII-D

To motivate the importance of study of these m/n =2/1 modes, figure 3 shows how often initially rotating2/1 locked modes (IRLMs) occur in DIII-D plasmas.When considering all plasma discharges, 25% containa 2/1 rotating NTM, 41% of which lock. Shots withIRLMs end in a major disruption 76% of the time(using only the red and green portions of figure 3a,and excluding the blue portion). Approximately 18%of all disruptions are a result of an IRLM, in goodagreement with the ∼ 16.5% reported on JET [16], andthis statistic rises to 28% for shots with peak βN > 1.5(figure 3b). The correlation between high βN and rateof occurrence of IRLMs will be detailed in section 7.2.

The blue slices show the number of IRLMsexcluded from the disruptivity studies, which consist ofIRLMs in type ii disruptions (long decay disruptions),IRLMs that terminate during a non-disruptive currentramp-down, or IRLMs that cease to exist prior toa major disruption. The “other discharges” do notcontain IRLMs, and include long decay disruptions andnon-disruptive discharges.

While the vast majority of rotating NTMs lockbefore causing a disruption, there were approximately23 instances of the rotating 2/1 mode growing largeenough to disrupt before locking.

4. Timescales of locking

In this Section we present two timescales indicativeof the time available for intervention before locking.These timescales are useful for disruption avoidanceand mitigation techniques.

Figure 4 shows the duration of all rotatingm=even, n=1 modes that locked. A broad peak existsbetween 50 and 400 ms. The rotating duration candepend on several different factors, such as the plasmarotation frequency, applied Neutral Beam Injection(NBI) torque, the island moment of inertia, and viscoustorques. The spread of values gives an indication of thetime available to prevent locking, if an intervention istriggered upon rotating mode detection.

The time taken for a mode rotating at 2 kHzto slow down and lock is referred to here as theslow-down time. The threshold of 2 kHz was chosenempirically, as modes that decelerate to this frequency


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Figure 3. (a) Color pie chart surveying all plasma discharges,showing the fraction of discharges with disrupting and non-disrupting initially rotating locked modes (IRLMs), as well asdisruptions without IRLMs. Overplotted as a hatched regionare the discharges with rotating 2/1 NTMs. (b) Same pie chartas (a), but for discharges with peak βN > 1.5. Note that thereis an overlap of 23 shots between the hatched rotating NTM andthe purple disruption regions.

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are often observed to lock. It is probable that at thisfrequency, the decelerating wall torque is stronger thanthe accelerating viscous torque, and causes the modeto lock. Figure 5a shows 66% of of slow-down timesbetween 5 and 45 ms, with the peak of the distributionat 17±10 ms; this is an indication of the time availableto prevent locking if measures are taken when the modereaches 2 kHz.

Figure 5b shows that modes which experience

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Figure 5. (a) The time taken for a rotating m/n = 2/1 modeto slow from 2 kHz to locked, as measured by eigspec and ESLDsrespectively. (b) A correlation is observed between the measuredslow-down time and electromagnetic torque between the modeand the wall. The torques are calculated by equation 1, wherethe perturbed magnetic field is taken when the mode is rotatingat 2 kHz, and ωτw is set to 1, representing the maximum ofthe frequency dependent steady-state wall torque. The pointsand error bars are the mean and standard deviations of each binrespectively. Note that about 5% of the events lie beyond 300ms, and are not plotted (in either panel).

a larger wall torque generally slow down quickerthan those with smaller wall torques. At lowelectromagnetic torque, the spread of slow-down timesin figure 5b suggests that other effects, such as the NBItorque, also become important.

The toroidal electromagnetic torque between therotating mode and the wall Tφ,w is expressed as follows[4],

Tφ,w = R0(2πrsBrs)2

µ0n/m

(ωτw)(rs+/rw)2m

1 + (ωτw)2[1− (rs+/rw)2m]2 (1)

wherem and n are the poloidal and toroidal harmonics,rs is the minor radius of the q = 2 surface, rs+ =rs + w/2 with w being the island width, Brs is theperturbed radial field at the q = 2 surface, rw is theminor radius of the resistive wall, and ω is the rotationfrequency of the NTM co-rotating with the plasma.This form of the electromagnetic torque comes froma cylindrical approximation. The maximum of thistorque occurs at the rotation frequency where ωτw = 1,and is the quantity plotted on the horizontal axis offigure 5b.


Distribution of 993 LM/QSM Events

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Figure 6. A histogram of the survival time, defined as theduration of a locked mode that ended in a disruption. Less than2% of events survive for more 3000 ms.

5. Time between locking and disruption

The survival time is defined as the interval betweenlocking and disruption (note this is only defined fordisruptive IRLMs). Figure 6 is a histogram of thesurvival times of all disruptive IRLMs. Two groups canbe observed, peaking at less than 60 ms and at 270 ms.The first group consists of 55 large rotating modes thatlock and disrupt almost immediately, as opposed to thelatter spread of LMs that reach a meta-stable statebefore disrupting. These short-lived modes, thoughdangerous and undesirable, could not be studied withthe automated analysis as necessary equilibrium datado not exist. These discharges are the same omitteddischarges mentioned previously in the end of section2.4. Recall that these discharges are not included inthe majority of the figures and discussion herein, butwill be discussed in the disruption prediction section(section 8.5) as they are expected to modestly decreasethe performance of the predictors.

While 75% of the population (excluding the 55transient modes) survive between 150 to 1010 ms,the most frequent survival time is 270 ± 60 ms, anindication of the time available to avoid disruptionwhen a mode locks.

Gaining predictive capability over how long adisruptive locked mode is expected to survive mightguide the best course of action to take, e.g. whetherto stabilize the mode, or directly deploy disruptionmitigation techniques.

Figure 7 shows the survival time plotted againstthe poloidal beta βp, the distance dedge, and theperturbed island current δI. dedge is a quantity thatmeasures the shortest distance between the islandseparatrix and the unperturbed plasma separatrix:dedge ≡ a − (rq2 + w/2) where a is the minor radiusof the unperturbed plasma separatrix, rq2 is the minorradius of the q = 2 surface, and w is the island width.

In figure 7, the survival time shows somecorrelation with βp and dedge, but not with δI. The

Parameter Correlation with tsdedge 0.47ρq2 -0.42li/q95 -0.39q95 0.36βp 0.34

dq/dr(rq2) -0.15li -0.11w 0.10δI -0.01

Table 1. Correlations of various parameters with the IRLMsurvival time ts. The parameters are ordered in the table bythe absolute value of their correlation coefficient. Negativecorrelation means that a linear relationship with a negative slopeexists between the parameters.

correlation coefficients for these, and other parametersare listed in table 1. The lack of correlation betweensurvival time and δI is consistent with the lack ofcorrelation with the island width w. This suggeststhat large islands will not necessarily disrupt quickly,but islands which extend near to the unperturbedseparatrix (i.e. islands with small dedge) tend todisrupt quickly. The correlation of dedge with survivaltime suggests that it is pertinent to the physics ofthe thermal quench. Other works have found thatparameters similar to dedge appear to cause the onsetof the thermal quench [39, 40, 41] (see section 9 fordetails).

The best correlations in the table are consideredmoderate (i.e. moderate correlations are in the rangerc = [0.4, 0.6], where rc is the correlation coefficient).The correlations in table 1 do not provide significantpredictive capability. A macroscopic timescale like thesurvival time likely depends on many variables, andon the nonlinear evolution of the plasma under theinfluence of the locked mode. Note that 15±4% of thedisruptive IRLMs in this survival time study terminateduring a plasma current ramp-down, and might havesurvived longer, had they not been interrupted.

6. Mode amplitude and phase evolution

6.1. Distributions of IRLM toroidal phase at locking

As a rotating n = 1 mode is slowing down and aboutto lock, it tends to align with existing n = 1 fields. Inmost cases, this will be the residual error field, definedas the vector sum of the intrinsic error field and theapplied n = 1 error field correction. Figure 8 shows ahistogram of all locked mode phase data, normalized bymode duration and total number of modes in the givenset: each mode contributes a total of 100/N , whereN is the total number of disruptive or non-disruptivemodes; and further normalized by the binsize of 5


(a)

0.0 0.4 0.8 1.2βp 50 ms post-locking

0.0

1.5

3.0Su

rviv

al T

ime

(s)

(b)

0 2 4 6 8 10 12 14dedge (cm) 50 ms post-locking

(c)

0 5 10 15 20 25 30δ I (kA) 50 ms post-locking

Figure 7. Survival time shows some dependence on (a) βp and (b) dedge. (c) No correlation is found with δI or similarly with theisland width w (see table 1).

degrees. A clear n = 1 distribution is observed infigure 8a (left-hand helicity discharges), with a peakat ∼ 125◦ for disruptive modes and ∼ 110◦ for non-disruptive modes. Figure 8b shows right-hand helicitydischarges. The disruptive distribution shows an n = 1component, though an n = 2 component is also clearlyvisible. The non-disruptive distribution shows a strongn = 2 component. The presence of n = 2 might bedue to occurrences of both over and under correctionof the intrinsic error field. Alternatively, the n = 2distribution might arise from the presence of bothlocked and quasi-stationary modes. Locked modes areexpected to align with the residual, whereas quasi-stationary modes are expected to move quickly pastthe residual, spending the most time in the anti-alignedphase. Similar analysis on subsets of these data showsconsistent results, suggesting that these distributionsare not specific to a certain experimental campaign.

The intrinsic n = 1 error field in DIII-D has beencharacterized by in vessel apparatuses [42, 43] and theerrors attributed to poloidal field coil misalignmentsand ellipticity, and the toroidal field buswork. It isfound that the intrinsic error is well parameterizedby the plasma current Ip and the toroidal field BT ;"standard error field correction" in the DIII-D plasmacontrol system calculates n = 1 correction fields basedon these. The majority of DIII-D plasmas are run withIp in the counter-clockwise direction and BT in theclockwise direction when viewed from above, referredto as the "normal" directions. Taking ranges for Ipand BT expected to encompass the majority of left-hand helicity discharges (i.e. Ip = 0.8 to 1.5 MA andBT = −1 to -2.1 T), the standard error field correctionalgorithm applies correction fields between −164◦ and−110◦. The preferential locking angles shown in figure8a might be due to a residual EF, resulting from thevector addition of the intrinsic EF and an imperfectcorrection field.

Figures 8c-d show how residual fields can arisefrom changing intrinsic and/or correction fields.

Figures 8e-f illustrate how the distributions in 8a-bmight look if the residual is reproducible, or not. In theformer case, a narrow, peaked distribution is expected,whereas in the latter case, a broad, flat distributionis expected. This might explain the distributionsin figures 8a-b. In addition, quasi-stationary modesare also present in these data, and contribute to thebroadening.

6.2. Change in n = 1 field at locking and growthbefore disruption

Figure 9 shows the change in mode width as the NTMslows from rotating at f = 2 kHz to 50 ms after locking.To within the ±2 cm errors on the width estimate frommagnetics, a significant growth or decay at locking isnot observed for the majority of modes. Only ∼10%of locked modes appear above the top diagonal dashedline, indicating a growth at locking beyond error.

It has been observed across the database thatthe radial field BR measured by the ESLDs tendsto grow before disruption. This growth is distinctfrom the dynamics of the locking process, as it oftenoccurs hundreds of milliseconds after locking. Figure10a shows the BR behavior before disruption for fiverandomly chosen disruptive IRLMs. A general periodof growth occurs between ∼ 100 and 5 ms prior tothe disruption, followed by a sharp rise in BR withinmilliseconds of the current quench, marked by thetransient rise in Ip. Although interesting in its ownright, we will not investigate the details of the BR spikeoccurring near the current quench, as we are interestedin the dynamics leading up to the thermal quench.

Histograms in radial field BR are plotted in figure10b, at times before disruption, to study the averageevolution. The BR field in a single case often followsa complicated trajectory, as evidenced by figure 10a,but these histograms reveal global trends. In Figure10b, the centers of the distributions shift to highervalues of BR as the disruption is approached. Themedian is chosen to be the representative point in the


(a)

Left−hand helicity

−180−120 −60 0 60 120 180Angle (deg)

0.00.10.20.30.40.50.60.7

Dis

tr. P

erc.

/deg

ree

(b)

Right−hand helicity

−180−120 −60 0 60 120 180Angle (deg)

DisruptiveNon−Disruptive

Bx

By

Intrinsic

ResidualBx

By

Residuals

Bx

By

Average Residual, reproducible

Bx

By

Average Residual, not reproducible

(c) (d)

(e) (f)

Correction

Figure 8. The normalized phase distribution of all lockedmodes. Each mode contributes a total of 100/N across all bins(N is number of disruptive or non-disruptive IRLMs). A binsizeof 5 degrees was chosen as a compromise between being largeenough to have sufficient statistics in one bin and fine enoughto show features of interest. The angle is where the radialfield is largest and outward on the outboard midplane. (a)Left-hand helicity (i.e. normal Ip and BT , or both reversed)with 980 disruptive and 1029 non-disruptive IRLMs. (b) Right-hand helicity plasma discharges (i.e. either Ip or BT reversed).Only 130 disruptive and 204 non-disruptive IRLMs here. (c-f)Illustrations to explain distributions. (c) The residual EF is thedifference between the intrinsic EF and the applied correction.(d) Due to changes in the intrinsic and/or the correction, theresidual EF can change. (e) An average residual can be definedby averaging over several shots and times. A small standarddeviation in its amplitude and phase (illustrated by the smallcircle) are indicative of high reproducibility of the residual EF.In that case, a narrow distribution is expected in Fig.a-c. (f)In the opposite limit, the residual EF phasor can point to anyquadrant, and a broad, flat distribution is expected

analysis that follows. The median is the point on thedistribution which divides the area under the curveequally.

The median is plotted in figure 10c as a functionof time (note the vertical axis is logarithmic). Themedian major radial field is about 7 G at saturation(i.e. 50 ms after locking). Later in the lifetime of themode, and approximately 50 ms prior to the disruption,the median grows consistent with an exponential fromthe saturated value, reaching a final value of > 11

0 2 4 6 8Rotating island width (cm)

0

2

4

6

8

Lo

cked

isla

nd

wid

th (

cm)

Figure 9. The mode width before and after locking,as calculated from the Mirnov probe array and ESLDmeasurements respectively. The rotating mode width isevaluated when mode rotation reaches 2 kHz; locked mode widthevaluated at 50 ms after locking to allow decay of shieldingcurrents in the wall. The solid line shows where the widths areequal, while the dashed red lines quantify the conservative ±2cm error bar on the island width estimates.

G at ∼ 5 ms before the disruption. This time isapproximately coincident with the onset of the thermalquench. From the slope of the dashed line, in figure10c, an exponential e-folding time for BR in the rangeτg = [80, 250] ms is found.

The present analysis is not sufficient to discernbetween possible sources of this increased BR. Out ofthe 26 IRLMs for which 2/1 modes can be clearly seenon ECE (see discussion in section 2.4), less than 10of these provide a view of the island O-point duringthe disruption. One such case is shot 157247 shownin figure 11 which disrupted at 3723 ms. A significantflattening of the Te profile is seen at each time slice,confirming that a 2/1 island with w > 5 cm is present.The solid horizontal bars show the calculated islandposition (from EFIT) and width (derived from δI, seeAppendix B). The solid line in figure 11b shows thetoroidal location of the island O-point on the outboardside in the midplane, and it is seen to intersect thelocation of the ECE diagnostic (dashed horizontal) att ≈ 3660 ms. The vertical dashed lines show thetimes of the profiles in figure 11a. Note that theworst toroidal alignment for O-point viewing occursfor φLM = −99◦, where the X-point is aligned with theECE. The horizontal bar at 3710 ms (green in onlineversion) looks to over predict the island width by ∼ 3cm, though the island toroidal alignment with the ECEis intermediate between the O and X-points, where theflattened region is expected to be smaller. EFIT datado not exist for the last two timeslices.


05

10152025

B1 R (G

)

-40-20020406080100Time before disruption (ms)

0.0

0.5

1.0

1.5

I p (M

A)(a)(a) 124865

137232156688150741131089

No. of Disruptive IRLMs

0 5 10 15 20 25B1

R (G) at Δt before disruption0

50

100

150

B1R (G) of median

050100150200Time before disruption (ms)

10

100 ms40 ms8 ms

5

15(b) (c)

Figure 10. (a) n = 1 radial field and plasma current traces from five randomly chosen disruptive IRLMs. (b) Histograms of BR

for all disruptive IRLMs at times approaching disruption. (c) The medians of histograms at six time-slices (three shown in (b), andthree not shown) undergo growth consistent with exponential within 50 ms of the disruption, as shown by the linear trend on thesemi-log plot. From the slope of the dashed line, an e-folding time in the range τg = [80, 250] ms is estimated.

In all shots where magnetics predict O-pointalignment with ECE during the disruption, a clearflattening of the Te profile like that shown in figure11 is observed. We conclude that the assumption ofthe presence of an m = 2 island during the disruptionis accurate, and in the shots where ECE data areavailable, the predicted island widths are reasonable.

If we assume that the increase in the n = 1field measured by the ESLDs is due to growth ofthe 2/1 island, we can estimate how much the islandwidth would increase. We assume a constant BT , R,and dq/dr|q2 during the ∼ 100 ms of growth, wherethe field increases from 7 to 11 G. Therefore, fromequation 17, we expect the island width to increaseby√

11/7 ≈ 1.25. With an average saturated widthfor disruptive islands of ∼ 4−6 cm (this will be shownin section 8), a disruptive island would be expected togrow ∼ 1−1.5 cm during the exponential growth. Witha channel spacing of 1-2 cm for ECE measurements,and with less than 10 disruptive IRLMs whose O-pointsare well aligned with the ECE diagnostic during thistime interval, validating this small change in islandwidth would be difficult.

As the poloidal harmonic of this exponentiallygrowing n = 1 field is unknown, in principle itis possible that the 2/1 island is unchanging, whilean m = 1 or m = 3 instability is appearing andgrowing. Coupling of the 2/1 and 3/1 modes has beenobserved on ASDEX-Upgrade [44] and investigated innumerical studies [45], while coupling of the 2/1 andm/n = 1/1 has been observed on TEXTOR [46], theRTP Tokamak [14], and studied in simulations [41].Although m = 4 might be a candidate for this growthin some shots, it will be shown in section 8 that morethan half of the disruptive discharges have q95 < 4, soit could only explain a minority of cases. Similarly,out of 13 IRLMs that occur in plasmas with q95 < 3,

157247

Time (ms)3660+20+50+61+61.6

195 200 205 210 215 220R (cm)

0.0

0.2

0.4

0.6

0.8

1.0

T e (k

eV)

ECE location

3640 3660 3680 3700 3720Time (ms)

-500

50100150200

φLM

O-p

oint

(deg

)

(a)

(b)

Figure 11. (a) Electron temperature Te profiles from ECEprior to an IRLM disruption show a clear flattening at the q = 2surface. Horizontal bars show the automated estimation of islandposition and width (note the vertical position of the bars ischosen for visual purposes only). Two error bars are shown onthe gray profile, and are representative of all Te measurementerrors. (b) The toroidal position of the island O-point on theoutboard side in the midplane. The horizontal dashed line showsthe toroidal location of the ECE diagnostic, and the verticaldashed lines show the time slices from (a).


at least 7 show a clear disruptive growth, in which them = 2 or m = 1 modes are the only candidates for thisgrowth.

Note that although we have chosen to characterizethe final disruptive growth prior to the final thermalquench here, a single mode might undergo multiplegrowths and minor disruptions, as shown in figure 1.

7. Interdependence of locked modes and β

7.1. Effect of locked mode on β and equilibrium

A common sign of the existence of a LM is a reductionin plasma β. Here we investigate βN = β(aB/Ip) as ithas been shown to affect NTM onset thresholds [47] (ais the plasma minor radius and B is usually taken tobe the toroidal field on axis).

Figure 12a shows βN at locking as a function ofβN at mode onset. Raw data for disruptive modes areshown in red and purple. The purple disruptive modesare preceded by an earlier LM, while the red are not.The majority of the red points are observed to lie belowthe dashed diagonal. This observation is reiterated bythe light blue points, showing the average and standarddeviation also lying predominantly below the diagonal,meaning that βN decreases from NTM onset to locking.

Notice that a significant population of purplepoints lie above the dashed line for βN at rotating onset< 1.5. The rotating phase of these IRLMs begins withfrequency f = 0 (locked), at which point they spin up.In these cases, βN has suffered a large degradation fromthe previous locked mode. In most of these cases, theβN at the following locking is greater than the degradedβN at the time of spin up, and therefore the points lieabove the dashed line.

The light blue and black averages from figure 12aare replotted in figures 12b and 12c in the form ofpercent changes in βN at each stage, relative to the βNat rotating onset. These plots show how βN changesfrom the rotating onset to locking, to βN saturation,and to mode termination. The time of βN saturationis taken to be 200 ms after locking (found empirically,in approximate agreement with twice the typical DIII-D energy confinement time, τE ≈ 100 ms), and thetime of mode termination is taken to be 50 ms beforedisruption or disappearance of the mode. Except forthe initial phase with βN at rotating onset < 1.5, acontinuous decrease in βN is observed during successivephases of the IRLM.

On average, the disruptive modes cause a largerdegradation of βN than non-disruptive modes duringall three phases. For βN at rotating onset > 1.6,disruptive modes cause 20-50% reduction during therotating phase, 50-70% reduction by βN saturation,and 50-80% reduction within 50 ms of the disruption.

-0.6 -0.4 -0.2 0.0 0.2p over first 200 ms of mode

-4

-2

0

2

R 0 (c

m) o

ver f

irst 2

00 m

s of

mod

e

Disruptive Non-disruptive Disruptive avg.

Figure 13. Both disruptive and non-disruptive modes show alinear dependence with ∆R0/∆βp ∼ 4 cm. A number of outliersare produced in the non-disruptive population by significantchanges of plasma shape (e.g. diverted to wall-limited plasma).

Of course a further, complete loss of βN occurs atdisruption (not shown).

With the reduction in βN in figure 12, andassuming constant Ip and BT (which is accurate formost modes occurring during the Ip flattop), a similarreduction in βp = 〈p〉/(B2

θ/2µ0) is expected as well.The reduction in βp causes a reduction in the Shafranovshift. Figure 13 shows the linear dependence of R0on βp during the first 200 ms after locking. A linearfit to the disruptive data provides a shift ratio of∆R0/∆βp ≈ 4 cm. Most modes reduce the Shafranovshift by 0-3 cm. This reduction of the 1/0 shapingreduces toroidal coupling of the 1/1 and 2/1 modes,the 2/1 and 3/1 modes, and other m/n and (m+ 1)/nmode combinations [48]. The larger scatter in the non-disruptive population can be explained at least in partby significant shape changes, such as the transitionfrom diverted to wall limited. For this reason, thelinear fit is performed only on the disruptive data.

7.2. Effect of β on the saturated width, IRLM rate ofoccurrence, and disruptivity

As NTMs are the result of helical perturbations to thebootstrap current [10], we expect the island width todepend on terms that drive the bootstrap current. TheModified Rutherford Equation (MRE) has been shownin previous works to describe 3/2 [6, 49] as well as 2/1[50, 51] tearing mode saturation well. The saturatedmodes of interest here have an average width w ≈ 5 cmand a standard deviation σ ≈ 2 cm. Being that smallisland terms in the MRE start to become small forw > 2 cm (assuming wpol ≈ 2 cm [10]), small islandeffects are ignored. A steady state expression of the


Figure 12. (a) The time evolution of βN between onset of the rotating mode and mode locking. The blue and black points showthe average and standard deviation of the disruptive (red and purple) and non-disruptive (raw data not shown) populations. Thepurple disruptive IRLMs are preceded by another LM (red are not). (b) The percent change in βN during each phase of disruptiveIRLMs, as compared with the βN at time of rotating onset. (c) Same as (b), but for non-disruptive IRLMs.

MRE (i.e. dw/dt→ 0) is given as follows,

0 = r∆′(w) + αε1/2 LqLpe

βper

w+ 4

(wvw

)2(2)

where ∆′(w) is the classical stability index, α is anad hoc accounting for the stabilizing effect of fieldcurvature (α ≈ 0.75 for typical DIII-D parameters),ε = r/R is the local inverse aspect ratio, Lq =q/(dq/dr) is the length scale of the safety factor profile,Lpe = −pe/(dpe/dr) is the length scale of the electronpressure profile (Lpe > 0 as defined), βpe is the electronpoloidal beta, and wv is the vacuum island widthdriven by the error field (wv ∼ 1 cm in DIII-D).Note that usually a cosine term appears on the terminvolving wv [52], but it is set to unity here, whichassumes the most destabilizing alignment of the lockedmode with the error field.

We take ∆′(w) to be of the form ∆′(w) = C0/r−C1w/r

2 [8]. With this definition of ∆′(w), equation2 is a nontrivial cubic equation in w. However,an approximate solution was found by approximating(wv/w)2 ≈ awv/w− b (with a = 0.45, b = 0.045, foundto be accurate to within 15% over the domain wv/w =[0.12, 0.3]). The resulting equation is quadratic, ofstraightforward solution. The approximate saturatedwidth expression is given by,

2wsat

r =(C0−4bC1

)+[(

C0−4bC1

)2+ 4

C1

(αε1/2 Lq

Lpβp + 4awv

r

)]1/2 (3)

The data in figure 14 show the normalized islandwidth as a function of βp/(dq/dr). The parametersLp and ∆′(w) are difficult to acquire with automatedanalysis, and therefore are not available in thedatabase. Therefore, a direct fitting of equation 3 is notappropriate, but the conclusion that wsat/r increases

0.0

0.1

0.2w

/r at

sat

urat

ion

0.00 0.04 0.08 0.12p / (dq/dr) at saturation (m)

0.0

0.1

0.2

w/r

at s

atur

atio

n

q95 < 4

q95 > 4

(a)

(b)

Disruptive Non-disruptive

Figure 14. (a) The normalized island width at saturation asa function of βp/(dq/dr) for discharges with q95 < 4. (b) Sameas (a) for discharges with q95 > 4. Only islands with w < 9 cmare shown here.

with βp/(dq/dr) is evident. A correlation of rc = 0.55is found between wsat/r and βp/(dq/dr) for plasmaswith q95 < 4. Plasmas with q95 > 4 show a weakercorrelation with rc = 0.36. This correlation suggeststhat locked modes in DIII-D are driven at least in partby the bootstrap current.

A one dimensional study of IRLM frequency(or prevalence) versus βN appears to suggest thatintermediate βN shots are the most prone to IRLMs,as shown in figure 15. The fraction of shots containingan IRLM increases with βN up to 2.75.


0.05

0.10

0.15

0.20

0.25

0.30

110

1001000

Fraction of shots w. IRLMNo. shots w. IRLMNo. shots

1 2 3 4Peak βN

Figure 15. The shot-wise rate of occurrence of IRLMs (numberof shots with IRLMs / total number of shots) as a function ofthe maximum βN achieved during the shot. Blue bars (on top)are formed by the quotient of the white and gray (on bottom).Note the logarithmic axis for the lower axis.

The fraction of shots with IRLMs decreasessignificantly for βN ≥ 4.5. A manual investigation ofthe 38 shots in the βN = 4.75 bin reveals that thesedischarges often have (1) q95 ≥ 5, or (2) high neutralbeam torques of T ≈ 6 Nm, or both. In both cases, alow occurrence of IRLMs might be expected due to aweak wall torque (see equation 1): (1) the q = 2 surfaceis far from the wall in discharges with high q95, and(2) shots with high injected torque likely exhibit highplasma rotation, where the wall torque goes like ω−1,and is therefore relatively small. A three-dimensionalanalysis of IRLM frequency versus βN , ρq2, and NBItorque might be more informative. These data are notpopulated for all shots in the database, so this analysisis reserved for future work.

The IRLM shot disruptivity as a function of βNand q95 is plotted in figure 16. The IRLM shotdisruptivity is defined as the "number of shots withdisruptive IRLMs" divided by the "number of shotswith IRLMs". Considering IRLM shot disruptivity asa function of βN only, the histogram at the right offigure 16a shows a peaked distribution, with the highestvalues for βN ∼ 1.5− 3. Similar results were obtainedfor the global disruptivity as a function of βN at NSTX[17] (the NSTX study is not limited to locked modes,and disruptivity is normalized by the amount of timespent at the given βN value). Reduced locked modedisruptivity at high βN was also observed at MAST[30].

The IRLM shot disruptivity dependence on βNcan be explained in part by the reduced number ofdischarges with q95 < 3.5 when βN is high. To see this,we bin the data in figure 16a into five βN windows,

denoted β0, β1, . . . β4. First, note that figure 16bshows a general decrease in IRLM disruptivity as q95is increased, across all βN windows. Next, figure16c shows the percent distribution of q95 values ineach of these βN windows. The two highest valuesof βN (purple and green) have the lowest percentage ofdischarges with q95 < 3.5, and the highest percentageof discharges with q95 between 4.5 and 5. Thisdistribution of q95 reduces the number of disruptivedischarges, with the net result that the higher βNdischarges appear less disruptive.

As q95 is shown here to affect IRLM shotdisruptivity, fixing q95 removes one variable, and allowsus to more closely inspect the dependence of IRLMshot disruptivity on βN . Figure 16b shows IRLMshot disruptivity across five windows in βN (denotedby color), for five values of q95 (separated by verticalgray lines). In all q95 windows, there is either nosignificant dependence on βN , or a lower IRLM shotdisruptivity at higher βN , particularly in the window3.5 < q95 < 4.5. Although larger islands are expectedat higher βN , it will be shown in the next sectionthat IRLM disruptivity depends very weakly on islandwidth (refer to figure 18), which in turn might explainthe weak dependence on βN . It should be noted thatonly q95 is fixed here. Other parameters that arecorrelated with βN (e.g. NBI torque, ion and electrontemperatures, and possibly others) are not fixed, andmight affect the apparent IRLM disruptivity scaling.

8. IRLM disruptivity

8.1. Decoupling the effects of ρq2, q95, and li onIRLM disruptivity

Figure 17a shows the normalized radius of the q = 2surface ρq2 and q95 at the mode end, defined here to be100 ms prior to the termination of the mode. The dataare from equilibrium reconstructions constrained byboth the external magnetics, and the Motional StarkEffect (MSE) diagnostic. On the right and below figure17a are histograms of IRLM disruptivity as a functionof ρq2 and q95 respectively. The q95 histogram showsthe expected result that lower q95 is more disruptive,as was also seen in figure 16b. The total disruptivity(i.e. not limited to IRLM disruptions) is observed toincrease as q95 is decreased in DIII-D [20]. This isin agreement with, but not limited to, observationsfrom JET [15], NSTX [17], and MAST [30]. Thehistogram in ρq2 has a qualitatively similar shape, withthe highest IRLM disruptivity at large ρq2, and thelowest at small ρq2.

The raw data show an expected correlationbetween ρq2 and q95. In a circular cross-section,


2 3 4 5 6 7 8q95

0

1

2

3

4

5Pe

ak β

N

0.0 0.5IRLM shot dis.

IRLM shot disruptivity

3 4 5-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2Distribution (%)

3.0 3.5 4.0 4.5 5.0 5.50

10

20

30

40Dis. shots Non-dis. shots

(a) (b) (c)

q95q95

β0β1β2β3β4

β0β1β2β3β4

β0β1β2β3β4

Figure 16. (a) The raw data for the highest achieved βN as a function of q95. The histogram on the right shows the one-dimensional IRLM shot disruptivity as a function of peak βN . Windows in β are labeled, and the binning for figures (b) and (c) areshown in gray. (b) IRLM shot disruptivity as a function of q95 for the binned data in (a). (c) The distribution of each βN bin inq95 in percent.

cylindrical plasma, q95 can be defined as follows,

q95 = 2ρ2q2

[1 + Iout

Ienc

]−1(4)

where Ienc is the total toroidal current enclosed by theq = 2 surface, and Iout is the total toroidal currentoutside of the q = 2 surface. Note that the quotientIout/Ienc behaves similar to the inverse plasma internalinductance l−1

i : when Iout/Ienc is large, li is small,and vice versa. The lack of one-to-one relationshipbetween ρq2 and q95 in the raw data of figure 17amay therefore be attributed in part to variations inIout/Ienc. Toroidal geometry and plasma shaping mayalso introduce variations in the relationship betweenq95 and ρq2.

Empirically, we find a high correlation (rc = 0.87)between li/q95 and ρq2, suggesting a relationship of theform,

li/q95 = αρq2 + c (5)

where α = 0.67 ± 0.01, and c = −0.23 ± 0.01. Thisequation suggests that ρq2 specifies li/q95, or viceversa. We begin by studying the effects of q95, li,and ρq2 on IRLM disruptivity individually. Then, weinvestigate IRLM disruptivity as a function of ρq2 andli/q95, which despite the high correlation between thetwo, reveals that li/q95 distinguishes disruptive IRLMsbetter than ρq2.

First, to decouple the effect of q95 and ρq2on IRLM disruptivity, we fix one and study thedependence on the other in figure 17. Weapproximately fix ρq2 by considering only data thatlie in small windows of ρq2 denoted ρ1, ρ2, ρ3, andρ4 (ρ4 covers a relatively large window in ρq2 asdata become sparse, but IRLM disruptivity appears

constant throughout the window). IRLM disruptivityas a function of q95 in the ρq2 windows is shown infigure 17b. Neither the trace corresponding to ρ1 norρ2 show a significant trend with IRLM disruptivitybeyond error. Both ρ3 and ρ4 show a possible decreasein IRLM disruptivity for q95 > 5.5, but appearconstant for q95 < 5.5.

The same study is performed on ρq2 by choosingwindows in q95 (or safety factor) denoted SF1, SF2,and SF3 (figure 17c). The three safety factor windowsagree within statistical error on an increasing lineartrend, with < 20% IRLM disruptivity for ρq2 < 0.7,and > 80% for ρq2 > 0.85.

From equation 5, we see that fixing ρq2 andvarying q95 (as in figure 17b) implies a variation of li aswell. The weak or absent trend in IRLM disruptivityas a function of q95 might be explained by competingeffects of q95 and li.

Similarly, equation 5 shows that fixing q95 andvarying ρq2 (as in figure 17c) also implies varying li.The strong dependence of IRLM disruptivity on ρq2thus suggests that either ρq2, li, or both have a strongeffect on IRLM disruptivity.

To help isolate the individual effects of ρq2, q95,and li, we compare how well they separate disruptiveand non-disruptive IRLMs single-handedly.

In order to quantify separation of the distributionsin one-dimension, we employ the BhattacharyyaCoefficient (BC) [53]. This coefficient was developedto measure the amount of overlap between twostatistical distributions, and is commonly used inimage processing, particularly for measuring overlapof color histograms for pattern recognition and targettracking [54]. For two discrete probability distributionsp and q parameterized by x, the Bhattacharyya


SF1 SF2 SF3

0.4

0.5

0.6

0.7

0.8

0.9 ρ

q2 a

t mod

e en

d

2 3 4 5 6 7 q95 at mode end

0.0

0.5

1.0

IRLM

dis

. 0.0 0.5 1.0IRLM dis.

IRLM disruptivity

3 4 5 6 7q95

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2IRLM disruptivity

SF3SF2

SF1

0.4 0.5 0.6 0.7 0.8 0.9(a) (b) (c)

DisruptiveNon-disruptive

ρq2

ρ1ρ2ρ3

ρ4

ρ1

ρ2

ρ3ρ4

Figure 17. (a) The relationship of disruptive and non-disruptive IRLMs in q95 and ρq2 space. One-dimensional IRLM disruptivityhistograms in q95 and ρq2 are shown on the bottom and right. Bins in ρq2 and safety factor are shown in gray, for use in (b) and(c). Note that only the binning intervals are shown, and not the explicit bins (i.e. the dashed and solid gray lines are unrelated).(b) IRLM disruptivity in windows of ρq2 as a function of q95. (c) IRLM disruptivity in windows of q95 as a function of ρq2.

Coefficient is given by,

BC(p, q) =∑x∈X

√p(x)q(x). (6)

The BC metric varies over the range 0 ≤BC(p, q) ≤ 1, where a value of 1 indicates that p andq are identical and perfectly overlapping. A value of 0implies they are completely distinct (no overlap)

Figure 18 shows the 1D separation of disruptiveand non-disruptive modes for six parameters at 100 msbefore mode termination, and reports the BC value foreach. Of the three interdependent parameters (i.e. ρq2,q95, and li), ρq2 is observed both visually, and by theBC = 0.70 value to best separate the two populations.The parameters q95 and li produce less separation withcoefficients of BC = 0.85 and BC = 0.88 respectively.All BC values reported in figure 18 have an error barof ±0.04.

Disruptivity is found to scale strongly with plasmashaping in NSTX [17]. The effects of shaping on IRLMdisruptivity in DIII-D will be included in a future work.

8.2. Decoupling effects of ρq2 and li/q95 on IRLMdisruptivity

In the previous section, it is assumed that li/q95 andρq2 are effectively equivalent, as suggested by equation5. However, here we show that li/q95 has a strongereffect on IRLM disruptivity. Figure 19 shows alldisruptive and non-disruptive IRLMs plotted in the2D space of li/q95 and ρq2 at 100 ms prior to modetermination.

The data in the region where ρq2 = [0.7, 0.8]show a clear vertical separation, rather than a

0.5 0.6 0.7 0.8 0.9 1.0q2 at mode end

0.2

0.3

0.4

l i/q95

at m

ode

end


Figure 19. Investigation of li/q95 as a function of ρq2 across alldisruptive and non-disruptive IRLMs. Mode end is 100 ms priorto mode termination. The horizontal blue line at li/q95 = 0.28shows approximately where IRLM disruptivity transitions fromlow to high. The high correlation between li/q95 and ρq2 isevident from the good clustering of the data along the linespecified by equation 5.

horizontal one. For instance, choosing a thresholdof li/q95 < 0.28 as the definition of a non-disruptiveIRLM results in 7% mis-categorized disruptive IRLMs,and 24% mis-categorized non-disruptive IRLMs. Tocapture a similar number of correctly identified non-disruptive IRLMs using ρq2, a value of ρq2 <0.78 is used and results in 14% mis-categorizeddisruptive IRLMs, and 25% mis-categorized non-disruptive IRLMs. This confirms the result of figures18b-c that li/q95 categorizes disruptive and non-disruptive IRLMs better than ρq2.


Distribution (%)

(a)

0 5 10 15 20 25 30dedge (cm)

0

10

20

30

40(c)

0.4 0.5 0.6 0.7 0.8 0.9 1.0ρq2

(b)

0.2 0.3 0.4 0.5li/q95

(d)

2 3 4 5 6 7 8 9q95

0

10

20

30

40(e)

0.5 1.0 1.5 2.0 2.5li

(f)

0 2 4 6 8w (cm)

Dis. modes Non-dis. modes

BC = 0.71BC20 = 0.61

BC = 0.70BC = 0.65

BC = 0.85 BC = 0.88 BC = 0.95BC20 = 0.98

Figure 18. Disruptive and non-disruptive IRLM distributions are shown as a function of six parameters to reveal the dominantclassifier. Amount of overlap between distributions is quantified by the Bhattacharyya Coefficient (BC). A BC value of 0 indicatesno overlap, and a value of 1 indicates complete overlap. All BC values have an error bar of ±0.04. Solid curves are evaluated at100 ms before mode termination, while the dotted red in (a) and (f) are evaluated 20 ms before the disruption, with correspondingBC20 values (the BC20 values of all other parameters, not shown, are similar to their reported BC values). Not shown, dq/dr|q2produces a value of BC=0.89.

To reduce mis-categorized disruptive modes, athreshold of li/q95 < 0.25 can be used which producesonly 3% mis-categorized disruptive IRLMs. However,the mis-categorized non-disruptive IRLMs increase to42%.

Similar empirical observations of a critical bound-ary in li and q95 space are reported for density limitdisruptions in JET [13], for current ramp-down disrup-tions in JT-60U [19], and for “typical” disruptions inTFTR (see figure 6 in [55]). Theoretical interpretationsof li/q95 will be presented in section 9.

Note that ∼ 17 of the 53 disruptive dischargesomitted due to lack of equilibrium data likely fall belowthe li/q95 threshold, but are not included in figure19, nor the categorization percentages reported in thissection. The impact of these omitted discharges isassessed in section 8.5.

8.3. dedge discriminates disruptive IRLMs within 20ms of disruption

In this subsection and subsection 8.4, it is assumedthat the exponential n = 1 growth in BR (figure 10c)is due to growth of the 2/1 island. It will be shownhere that dedge is as effective as li/q95 at discriminatingdisruptive IRLMs within 20 ms of the disruption. Apossible physical interpretation will be given in section

9.Figure 20 shows the island half-width as a function

of the distance between the q = 2 surface and theunperturbed plasma separatrix, a − rq2. The shortestperpendicular distance from the black dashed linerepresenting the unperturbed last closed flux surface(LCFS) to a given point is what we have called dedge =a−(rq2 +w/2). In other words, a point appearing nearthe solid black line represents an island whose radialextent reaches near the unperturbed LCFS. Due to theperturbing field of the locked mode, we expect kinkingof the LCFS. This kinking of the LCFS is not accountedfor in our calculation of dedge.

dedge is shown to separate disruptive and non-disruptive modes as effectively as ρq2 at 100 ms priorto disruption in figure 18 (recall the ±0.04 error baron all BC values). Within 20 ms of the disruption,dedge separates the two populations better with a BCvalue of 0.61 (compare the dotted red and solid blackdistributions for dedge in figure 18a). The betterseparation is due to the exponential growth of then = 1 field that occurs in the final 50 ms beforedisruption: that growth of BR, and thus of w, decreasesdedge without affecting ρq2. Note that evaluating dedgeat 20 ms prior to the disruption implicitly attributesthe n = 1 exponential growth to a growing 2/1 island,but the validity of this assumption does not affect the


discrimination ability of dedge.Choosing a threshold of dedge > 9 cm evaluated

20 ms before the disruption (blue dashed line in figure20), we find 8% mis-categorized disruptive IRLMs, and28% mis-categorized non-disruptive IRLMs. This iscomparable with the 7% and 24% found for li/q95 <0.28, though li/q95 is evaluated 80 ms earlier (thesethresholds on dedge and li/q95 correctly categorize 195and 194 non-disruptive IRLMs respectively). A moreconservative threshold of dedge > 11 cm produces3% mis-categorized disruptive IRLMs, and 58% mis-categorized non-disruptive IRLMs.

8.4. Weak dependence of IRLM disruptivity on islandwidth

The island width alone is found to be a poordiscriminator of disruptive IRLMs at times 20 msbefore the disruption or earlier (figure 18f). From thetime of saturation until∼ 20 ms prior to the disruption,disruptive and non-disruptive island widths are similar,with a slight tendency for non-disruptive modes tobe larger at the earlier times. IRLM disruptivity asa function of the island half-width is shown in thehistogram on the right of figure 20. For the majorityof islands with half-widths between 2 and 5 cm, IRLMdisruptivity does not change significantly, and mightbe constant within statistical error.

The region below the curved dashed line in figure20 shows the approximate mode detection limit dueto typical signal-to-noise-ratio discussed in section2.4. Modes appearing in this region were once abovethe detection limit, and are still measured due tothe asymmetry in onset and disappearance thresholdsused in the analysis. It is possible that undetectedmodes in this region might affect the resulting IRLMdisruptivity.

8.5. IRLM disruption prediction

Thus far we have discussed percent mis-categorizations,separation of disruptive and non-disruptive distribu-tions measured by the BC coefficient, and IRLM dis-ruptivity all at specific points in time. Although use-ful for understanding the physics of IRLM disruptions,single time-slice analysis is not sufficient for disruptionprediction. Prediction during the locked phase requiresestablishing a parameter threshold that will never beexceeded by a non-disruptive IRLM, but will be ex-ceeded by a disruptive IRLM at some point before thecurrent quench.

We consider li/q95 and dedge separately as IRLMdisruption predictors. These predictors are intendedto be used only in the presence of a detected lockedmode. Table 2 shows the percent missed disruptionsand percent false alarms for the given thresholds, and

q2

0.5 1.0IRLM dis.

Non-disruptive

Undetectable

Figure 20. The assumption that the 2/1 island dominatesthe disruptive exponential growth is implicit in this figure, andtherefore this figure is exploratory. The disruptive data are from20 ms before disruption, and the non-disruptive data are from100 ms before mode termination. The horizontal axis quantifiesthe distance of the q = 2 surface from the unperturbed lastclosed flux surface (LCFS). The shortest perpendicular distancefrom the unperturbed LCFS (black solid) to a given point isdedge (i.e. dedge = a − (rq2 + w/2)). The blue dashed line iswhere dedge = 9 cm. IRLM disruptivity as a function of islandhalf-width is shown in the blue histogram.

for the given warning times. The percent misseddisruptions is defined as the number of disruptiveIRLMs that do not exceed the threshold within Xms of the disruption, divided by the total numberof disruptive IRLMs. The percent false alarms isdefined as the number of shots where at least onenon-disruptive IRLM exceeds the threshold at anytime during its lifetime, divided by all issued alarms.Note that a single non-disruptive discharge can havemultiple non-disruptive IRLMs, but only one alarm.

If all IRLMs are considered disruptive, no IRLMdisruptions are missed, but the percent false alarmsincreases to 25 ± 2%. This case is shown asthe third row of table 2, labeled “none”, as noadditional condition is required before issuing analarm. Declaring an IRLM disruptive when it locksprovides a warning time that depends on the survivaltime of the IRLM. The distribution of survival times,and therefore also of warning times for this disruptionprediction criterion, are shown in figure 6.

Despite the seemingly short warning timescale, 20ms is a sufficient amount of time to deploy massive gasinjection in DIII-D [56].

Out of the 53 disruptive discharges omitted dueto insufficient equilibrium data (see section 2.4), 17of them might contribute to increasing the misseddisruption percentages associated with the li/q95criterion reported in table 2 by up to 3%, withnegligible effect on the percent false alarms. Also, outof the remaining 53−17 = 36 disruptive discharges that


Condition (%) Missed (%) Falsewith IRLM disruptions alarms

with warning with warning100ms 20ms 100ms 20ms

li/q95 > 0.28 7± 1 5± 1 11± 1 10± 1dedge < 9 cm 6± 1 4± 1 12± 1 12± 1None 0 25± 2

Table 2. IRLM disruption prediction statistics for li/q95, dedge,and no prediction parameter. The two thresholds are showngraphically by the dashed blue lines in figures 19 and 20 (notethat figures 19 and 20 are evaluated at single time slices, whereasthese statistics are evaluated over appropriate time intervals).These prediction criteria are intended for use in the presence of adetected IRLM only. The "None" condition shows the disruptionstatistics for the case where all IRLMs are considered disruptive(see text for warning times for this case). Note that dischargesomitted by the automated analysis might increase the misseddisruption percentages by up to 3% (see text for details).

satisfy li/q95 > 0.28, 32 exhibit survival times around20 ms, and thus the warning time for these modes is nolonger than 20 ms. The increase in missed disruptionscan be reduced to 1.5% by considering discharges withq0 < 1.6 only, as 9 of the 17 discharges contributingto the increased percent missed disruptions satisfy thiscriterion (only 40 discharges with 2/1 IRLMs in theentire database satisfy this criterion). The effectsof these omitted discharges on the dedge predictioncriterion are expected to be similar to those justdiscussed for the li/q95 prediction criterion.

8.6. ρq2 evolution

The evolution of ρq2 between locking and mode end,provides insight into the evolution of both li/q95 anddedge. Figure 21a shows that ρq2 tends to increaseby ∼ 5% between locking and mode end. This isnoticeable by the clustering of points above the dasheddiagonal. Equation 5 suggests that an increase in ρq2at fixed q95 implies an increase in li. As Ip, BT , andplasma shaping are fixed via feedback in most DIII-D discharges, the assumption of constant q95 is veryreasonable. The increase in li during locked modesagrees with earlier works [12, 13, 14].

9. Discussion

The physical significance of li/q95 might be related tothe potential energy available for tearing growth (i.e.to the island width dependent classical stability index∆′(w)).

A one-dimensional simulation including bothMHD evolution and transport has shown that theconstraint qmin > 1 enforced by the sawtoothinstability has a significant effect on the currentprofile, resulting in a steepening of the current gradient

0.3 0.4 0.5 0.6 0.7 0.8 0.9q2 at Locking

0.5

0.6

0.7

0.8

0.9

q2 a

t mod

e en

d


Figure 21. The evolution of ρq2 from locking to modeend (100 ms prior to mode termination) for IRLMs whichterminate predominantly during the Ip flat-top. The diagonalline represents unchanging ρq2. The horizontal line is at ρq2 =0.75, and marks an approximate transition from low to highIRLM disruptivity.

2 3 4 5 6 7q95 at mode end

0.5

1.0

1.5

2.0

l i at

mod

e en

d

JET density limitli /q95 = 0.28 threshold


Figure 22. All disruptive and non-disruptive IRLMs shown inli and q95 space. The JET density limit shown in blue defines theapproximate lower bound of the disruptive locked modes in DIII-D remarkably well. The cyan dashed line is the li/q95 = 0.28value again, first seen in figure 19 to divide the populations well.

between the q = 1 and q = 2 surfaces as theedge q value is decreased [57]. A steepened currentprofile on the core-side of the 2/1 island is classicallydestabilizing, and thus, ∆′(w) is shown to increase asthe edge q decreases.

A three-dimensional simulation [58] later corrob-orated the classically destabilizing phenomenon foundin [57]. Citing these works, a study followed [55], where∆′(w = 0) stability was investigated as a function ofthe current profile shape, with fixed axial and edge qvalues. In that study, monotonically decreasing currentprofiles with q0 ∼ 1 were varied in search of a ∆′ stableprofile, and a limit in li and q95 space was found where


no stable solution exists (see figure 6 of [55]). Thatlimit is similar to the empirical IRLM disruption limitshown by the cyan dashed line in figure 22. These threeworks together [57, 58, 55] provide a theoretical basisfor the hypothetical connection between li/q95 and ∆′.

li/q95 might be responsible for the pre-thermalquench growth shown in figure 10, and it might alsoinfluence the evolution of MHD during and after thethermal quench.

Separately, it is interesting that at high density,a high value of li is itself unstable, causing radiativecontraction of the temperature and current profiles[13], which further increases li. Since q95 isapproximately fixed via feedback, increasing li impliesincreasing li/q95.

Alternatively, the physical significance of li/q95might be related to an excess of radiative losses inthe island, overcoming the Ohmic heating, resultingin exponential growth [59]. Figure 22 shows goodseparation of disruptive and non-disruptive IRLMs inthe li and q95 space used in reference [13] to identifydensity limit disruptions. Although the plasmasdiscussed herein are expected to be far from the densitylimit [60] as a result of the degraded confinement,they are likely near the radiative tearing instabilitylimit which is theorized as the fundamental mechanismcausing the density limit [61]. This radiative tearinginstability limit is expected qualitatively to scale withli/q95.

li/q95 and dedge are both good IRLM disruptionpredictors and are both correlated with ρq2 (theformer by equation 5, and the latter by definition).Although this does imply some common underlyingphysics, li/q95 and dedge differ from ρq2 by capturinginformation on the q-profile shape and the island widthrespectively. Both discriminate disruptive from non-disruptive IRLMs better than ρq2 alone, and do soby leveraging different physics. We conjecture thatli/q95 is a proxy for ∆′(w), as discussed earlier inthis section and supported by earlier publications.The success of the dedge parameter might suggest aposition-dependent critical 2/1 island width, wherecriticality depends on proximity to the unperturbedlast closed flux surface. Alternatively, if the n = 1field is the result of multiple islands and not just the2/1 island, as implicitly assumed in the definition ofdedge, then island overlap might result from n = 1islands of m ≥ 2 becoming closer and larger as dedgedecreases. Island overlap is known to cause stochasticfields [62], and their effect on confinement might besufficient to induce a disruption. Commonalities anddifferences between li/q95 and dedge will be the subjectof future work.

Other works have found parameters similar todedge to be relevant to the onset of the thermal

quench, and are briefly listed here. In reference [40], astable Alcator C-Mod equilibrium is used as an initialcondition for a numerical simulation of massive gasinjection. It is found that by uniformly distributing ahigh-Z gas in the edge, a 2/1 island is driven unstableand upon intersecting with the high-Z gas (i.e. whendedge becomes small), the 2/1 grows rapidly followed bya 1/1 tearing mode that leads to a complete thermalquench. Similarly, a limited plasma is simulated inreference [41] with a current profile unstable to the2/1 island, and the plasma current is ramped up todrive the 2/1 island towards the plasma edge. It isfound that when the 2/1 island comes into contactwith the limiter or a cold edge region (i.e. in bothcases, when dedge is small), a rapid growth of the2/1 ensues followed by a 1/1 kink displacement of thecore, ending in a disruption. Similarly, experimentalresults of disruptions induced by EF penetration modeson COMPASS-C [39] are attributed to exceeding athreshold of w/(a− rs) > 0.7, where w and rs are the2/1 island width and minor radius. This observationis thought to be due to the 2/1 island interacting witha 3/1 island.

Large 3-D fields at the plasma separatrix causeradial deformations of the edge plasma, leading toedge flux-surfaces (like the q = 2 when dedge is small)intersecting the vessel [63]. Further, some workssuggest the existence of a stochastic layer within theLCFS [64, 65], which might facilitate stochastizationof the 2/1 island when dedge is sufficiently small.

Separately, the radiation drive of tearing modesis sensitive to edge proximity [59], causing islandgrowth which reduces dedge, and could then lead to thethermal quench through one of the above mechanisms.Again, recall that our interpretation of dedge at 20 msprior to the disruption requires the exponential growthobserved in the n = 1 BR signal to be due to the 2/1island growth, which is plausible but not confirmeddue to lack of poloidal harmonic analysis in the lockedphase. Regardless of the theoretical interpretation,dedge is a useful IRLM disruption predictor within 20ms of the disruption for DIII-D.

Summary and conclusions

Approximately 22,500 DIII-D plasma discharges wereautomatically analyzed for the existence of initiallyrotating 2/1 locked modes (IRLMs). The results of thisanalysis permits statistical analysis of timescales, modeamplitude dynamics, effects of plasma β and majorradius R, and disruptivity as a function of plasma andmode properties.

Timescales investigated suggest that a rotating2/1 NTM that will eventually lock rotates for ∼ 200ms, decelerates from f = 2 kHz to locked in ∼ 15


ms, and survives as a locked mode for ∼ 300 ms(these values all represent the most frequent value intheir respective histograms). These timescales provideinsight into how to respond to a mode locking event,whether with a disruption avoidance approach (suchas fast controlled shut-down, mode spin-up, or modestabilization) or a disruption mitigation system (suchas massive gas injection), depending on the responsetimes of different systems and approaches.

Prior to disruption, the median n = 1 perturbedfield grows consistent with exponential with an e-folding time between τg = [80, 250] ms, which mightbe due to exponential growth of the 2/1 IRLM.

The parameter li/q95 is shown to have a strongability to discriminate between disruptive and non-disruptive IRLMs, up to hundreds of millisecondsbefore the disruption. As an example, the criterionli/q95 > 0.28 in the presence of a detected IRLMmissesonly 7% of disruptions and produces 11% false alarms,with at least 100 ms of warning time. li/q95 mightbe related to the free energy available to drive tearinggrowth.

dedge performs comparably to li/q95 in its abilityto discriminate disruptive IRLMs. A threshold belowwhich IRLMs are considered disruptive of dedge = 9 cmproduces 4% missed disruptions and 12% false alarms,with at least 20 ms of warning time. dedge is alsoobserved to exhibit the best correlation with the IRLMsurvival time. dedge might be a fundamental trigger ofthe thermal quench, supported by similar observationsby other authors [39, 41, 40, 63].

Future work will attempt to validate thermalquench onset mechanisms in plasmas with lockedmodes, with the goal of a fundamental understandingof locked mode disruptions, and thus how to avoidthem.

10. Acknowledgements

This work was conducted in part under the DOEGrant de-sc0008520. In addition, this work issupported by the U.S. Department of Energy, Officeof Science, Office of Fusion Energy Sciences, usingthe DIII-D National Fusion Facility, a DOE Officeof Science user facility under awards, DE-FG02-04ER547611, DE-FC02-04ER546982, and DE-FG02-92ER541393. DIII-D data shown in this paper canbe obtained in digital format by following the links athttps:fusion.gat.comglobal.D3D_DMP.

The authors would like to thank E.J. Strait andR.J. Buttery for carefully reading the manuscript, andA. Cole, J. Hanson, E. Kolemen, M. Lanctot, L. Lao,C. Paz-Soldan, D. Shiraki, and R. Wilcox for fruitfulconversations that contributed to this work. Theauthors would also like to thank S. Flanagan and

D. Miller for their computer expertise in developingthis database, and M. Brookman for assistance inpreparing the electron cyclotron emission data.

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Appendix A - Mapping from radial field toperturbed island current

A model was produced to map from the value ofthe n = 1 major-radial magnetic field BR at theexternal saddle loops to the perturbed current carriedby a circular cross-section toroidal current sheet.This model was inspired by a similar technique thataccurately reproduced experimental magnetic signalsin DIII-D [36]. The current sheet has major radiusR and minor radius rq2, derived from equilibriumreconstructions using magnetics and Motional StarkEffect data. A shaping study was performed to testthe impact of ellipticity on the n = 1 external saddleloop measurement. The impact was found to be

Figure 23. (a) A 3D filament model of a 2/1 tearing modeused to map from the radial field at the external saddle loopsBR to a perturbed island current δI. This model, and figure,were inspired by [36]. (b)The radial field at the external saddleloops is shown in the color contour for a mode carrying 3.14 kAof n = 1 current. The thick black lines outline the six externalsaddle loops. The thin black lines define cells of equal area,which are used to sample the field to compute an average overthe loop.

relatively small, introducing corrections of less than5%. Hence, it was concluded that circular cross-sections are sufficient for modeling external saddle loopsignals for the sake of the present study.

The modeled current sheet is discretized into Nhelical wire filaments with 2/1 pitch giving a toroidalspacing of 360/N degrees. The currents are distributedamong the wires to produce a 2/1 current perturbationas shown by the colors in figure 23a (black representsno perturbed current and red represents maximumperturbed current).

The field is calculated by numerically integratingthe Biot-Savart Law along each wire. Figure 23bshows the resulting major-radial field BR as "seen"by the external saddle loops, due to a 2/1 currentdistribution. The color contour shows a clear n = 1field distribution. Although the external saddle loopshave complete toroidal coverage, they have only ∼ 25%poloidal coverage for the non-elongated, cylindricallyapproximated plasma.

A single measurement for each saddle loop isestimated by averaging the field at 100 sample points.Care was taken to ensure the area of each samplepoint is identical, which simplifies the flux calculation.The field samples are averaged, which is identical tocalculating the total flux and dividing by saddle looparea Asl:∑n

i=1 BR,iaiAsl

= a

na

n∑i=1

BR,i = 1n

n∑i=1

BR,i (7)

where Asl is the total area of one saddle loop, ai is theconstant sample area (i.e. ai = a), n is the number of


-1

0

1

r s (kA/

rad)

0.0 0.5 1.0 1.5 2.0 2.5 3.0Poloidal angle (rad)

0

1

2

r s (kA/

rad)

O-point X-point

O-point X-point

(a)

(b)

Ih

I

~~

Figure 24. (a) Half the perturbed island current δIh is shownby the blue shaded region. Here, it is assumed that the currentperturbation is sinusoidal about zero. (b) The total perturbedisland current δI is shown by the blue shaded region. Here, it isassumed that no perturbed current flows at the X-point.

0.1 0.2 0.3 0.4 0.5 0.6 0.7rq2 (m)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

B1 R (G

)

R0=1.6 m R0=1.65 m R0=1.7 m R0=1.75 m R0=1.8 m

Figure 25. The n = 1 major-radial field B1R measured at

the saddle loops as a function of the minor radius of the q = 2surface rq2.

sample regions per saddle loop, and BR,i is the major-radial field at the ith sample region.

The above averaging is done for each of the sixsaddle loops. The simulated signals are then pair-differenced, and a least squares fit is used to extractthe amplitude of the n = 1 component, in the sameway that experimental data are analyzed in section 2.4.

The island current, δI, is defined as the totalcurrent deficit, as seen in figure 24b. A seconddefinition of island current is useful for mapping toa cylindrical island width, being half the sinusoidalperturbed current δIh (see figure 24a). δIh andδI are related by the equation δI = πδIh. Thecurrent distribution in figure 24b can be seen as thedistribution in 24a, plus an axisymmetric current. Thenon-axisymmetric fields produced by 24a and 24b areidentical.

The n = 1 BR component in the external saddleloops, B1

R, was then studied as a function of R and

rq2 for fixed island current δIh as seen in figure 25.B1R is found to be quadratic in rq2 in agreement with

the cylindrical approximation. As R is increased, theleading order coefficient of the quadratic is seen toincrease as well. Defining the leading order coefficientα(R) (which note is a function of R), we find thefollowing relationship:

B1R = α(R)δIh[rq2]2 (8)

where B1R is in Gauss, δIh is in kA, rq2 and R are in

meters, and α has units of G/kA/m2. Evaluating thechange in α as a function of R, the dependence is foundto be well approximated by a second order polynomialα(R) = aR2 + bR + c where a = 19.15 G/kA/m4,b = −50.40 G/kA/m3, and c = 35.71 G/kA/m2.

With equation 8, experimental n = 1 measure-ments from the external saddle loops are mapped toδIh by a simple inversion of the equation and withmajor and minor radii provided by EFIT informed bymagnetics and Motional Stark Effect data:

δIh = B1R

α(R)[rq2]2 (9)

Appendix B - Mapping from perturbed islandcurrent to width

Although the perturbed island current δI is an intrinsicisland quantity that accounts for toroidicity, an islandwidth was desirable for some studies. We will use δIto map to an island width.

We solve for the radial field at the q = 2 surface Brby assuming cylindrical geometry, using the orderingkθ = m/r � kz = n/R, and assuming vacuumsolutions for the radial tearing eigenfunctions. Thenon-axisymmetric field B produced by the island isexpressed as,

B = ∇×Ψz (10)

where Ψ is the perturbed flux function which is of theform Ψ = ψ(r)eimθ (note that the toroidal wavelengthis assumed infinite here). We assume the perturbedcurrent in the island j to be a sheet current located atthe q = 2 surface expressed as,

j = σ sin(2θ)δ(rq2 − r)z (11)

where σ is constant with units (A/m), δ(...) representsthe Dirac delta function, and the poloidal harmonicm = 2 is implied. Invoking Ampere’s law where thetime derivative of E is neglected (the characteristicvelocity of the system v � c), and the equilibriumplasma current is neglected (i.e. searching for vacuumsolutions), we find

−∇2Ψ = µ0σ sin(2θ)δ(rq2 − r) (12)


Solving the cylindrical Laplacian both inside andoutside of the q = 2 surface, requiring the solutionsbe continuous across rq2, and defining the jump in theradial derivative with the radial integral of j, we find

Ψ = µ0σ

4 sin(2θ)

r2

rq2for r < rq2

r3q2r2 for r ≥ rq2

(13)

Using equation 10, the field at the q = 2 surfaceis then given by

Bq2 = µ0σ

2

[cos(2θ)r + sin(2θ)θ

](14)

We now want to replace σ with an expression forδI. We do so as follows (see figure 24a),

δIh ≡∫ π/2

0

∫ rq2+

rq2−

j · dA (15)

where dA = r dr dθ z, and rq2± = rq2 ± w/2. Aftersubstituting equation 11 and evaluating the integral,we find σ = δIh/rq2.

Finally, substituting this into equation 14, we have

Bq2(δI) = µ0δIh2rq2

[cos(2θ)r + sin(2θ)θ

](16)

The cylindrical island width is given by [66],

w = c

[16RBrq2

mBT dq/dr

]1/2

(17)

where c is a toroidal correction for island widthsmeasured at the outboard midplane, Br is theperturbed radial field at the q = 2 surface, q = 2 isthe local safety factor, m = 2 is the poloidal harmonic,BT is the toroidal field at the magnetic axis, and dq/dris the radial derivative of the safety factor evaluatedat rq2 on the outboard midplane (the cylindricalexpression for the safety factor q = rBT /(RBθ) wasused here). Substituting q = 2 and m = 2, and usingthe maximum of equation 16 for Br, we have

w = c

[16R

BT dq/dr

µ0δIhrq2

]1/2(18)

This expression for the island width was validatedusing the electron cyclotron emission diagnostic [35] toobserve the size of flattened regions in the temperatureprofile. Using nine islands, the toroidal correctionfactor is calibrated to c ≈ 8/15. Using equation 18,we expect a ∼ 25% statistical error on island widthestimates. This expression for the island width is usedfor all plots of w and dedge.

statistical analysis of m/n = 2/1 locked and quasi

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