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 poloidal and toroidal mode numbers m=2 and n=1 at DIII-D. The analysis of 22,5 discharges shows that more than 18% of disruptions are due to locked or quasi- stationary modes with rotating precursors (not including born locked modes). A locked mode stability parameter, formulated by the plasma internal inductance l i divided by the safety factor at 95% of the toroidal flux q 95, is found to exhibit significant predictive capability over whether a locked mode will disrupt or not, and does so up to hundreds of milliseconds before the disruption. Within 2 ms of the disruption, a parameter that measures the shortest distance between the island separatrix and the unperturbed last closed flux surface, referred to as d edge, might surpass l i /q 95 in its ability to discriminate disruptive and non- disruptive locked modes. d edge is also seen to correlate with the duration of the locked mode better than any other parameter that was considered. Within 5 ms of a locked mode disruption, average behavior shows exponential growth of the n=1 perturbed field, which might indicate exponential growth of the m/n=2/1 locked mode. Surprisingly, even assuming the aforementioned growth, disruptivity shows little dependence on island width up to 2 ms before the disruption. Timescales associated with the mode evolution are also studied and inform necessary response times for disruption avoidance and mitigation techniques. Observations of the evolution of β N during a locked mode, the effects of poloidal beta on the saturated width, and the reduction in Shafranov shift during locking are also presented. 1
(Motivation) Reactor tokamaks have to run without disruptions 1. Locked modes are a significant cause of disruptions in many tokamaks [DIII-D, JET, ITER?] 2. Confidence in locked mode disruption avoidance relies on understanding the physics of locked mode disruptions 3. To prevent locked mode disruptions, we can: 1. Avoid locked modes (reduce field errors, eliminate seeds) 2. Operate where locked modes are unlikely to disrupt 3. Understand disruption physics well enough to drive plasma out of disruptive state (e.g. ECCD in island O-point, I p ramp-downs, others?) *Sub-bullets 3.2 and 3.3 are the focus of this work 2
Definitions IRLM = Initially Rotating Locked Mode (or locked mode with rotating precursor) Disruptive IRLM = IRLM that immediately precedes a disruption Non-disruptive IRLM = IRLM that decays or spins up prior to the I p ramp-down 3
Example of an initially rotating locked mode (IRLM) 1. m/n = 2/1 rotating mode Source of rotating mode not studied 2. Mode locks 1. Time between f = 2 khz and locked referred to as slow-down time (a) B p (G) of 2/1 rotating NTM 4 3 2 159428 Disruption 4 3 2 1 1 Survival Time 18 2 22 24 B r (G) of 2/1 LM 3. Exists as locked mode 1. Few to thousands of milliseconds 2. Referred to as survival time for disruptive IRLMs 4. Disrupts or ceases to be a locked mode (decays or spins up) (b) Frequency (khz) 6 5 4 3 2 1 Locking Slow Down Time 6 5 4 3 2 1 1 1 18 2 22 24 Time (ms) Frequency (Hz) 4
Disruptions and IRLM global statistics 5
(a) 5993 More than a quarter of high β N disruptions are due to IRLMs 1555 Survey of 22511 plasma discharges 1178 121 121 4641 Shots with IRLMs non-disruptive IRLMs disruptive IRLMs shot disrupted not due to IRLMs Shots without IRLMs disruptions without IRLMs normal discharge (b) 5742 11416 Survey of 16123 discharges of N >1.5 1154 14 2432 117 Shots with IRLMs non-disruptive IRLMs disruptive IRLMs shot disrupted not due to IRLMs Shots without IRLMs disruptions without IRLMs normal discharges 2/1 rotating NTMs 2/1 rotating NTMs Study performed on shots 122 to 159837 (25 to 214) 28% of all disruptions in shots with peak β N >1.5 are due to IRLMs, compared with 18% for all peak β N Born locked modes not considered in this work 6
In a 1D study, IRLMs appear most often in intermediateβ N plasmas.3.25.2 Fraction of shots w. IRLM No. shots w. IRLM No. shots.15.1.5 1 1 1 1 1 2 3 4 Peak N Low occurrence at β N > 4.5 might be explained by observed conditions of q 95 > 7 and T NBI > 6 NM in most of these shots 3D study of IRLM rate of occurrence vs. β N, T NBI, and ρ q2 might be more informative 7
8 IRLM Timescales
66% of 2/1 NTMs rotating at 2 khz will lock in 45 ± 1 ms Indication of time available to prevent locking (a) No. Occurrences (b) 25 2 15 1 5 5 1 15 2 25 3 Slow Down Time (ms) 3 Larger T wall results in shorter slowdown time Slow Down Time (ms) 25 2 15 1 5 2 4 6 8 1 12 14 Torque at f = 2 khz (a.u.) 9
Disruptive IRLMs most frequently survive 27 ± 6ms 66% of disruptive modes terminate between 15 to 11 ms Time available to avoid disruption after locking No. Occurrences 12 1 8 6 4 2 Distribution of 111 LM/QSM Events 5 1 15 2 25 3 Survival Time (ms) 1
Degradation of β N throughout IRLM evolution 11
On average, β N continually decreases throughout IRLM lifetime (a) N at locking (b) Percent change in N (disruptive) (c) Percent change in N (non dis.) 4 Disruptive Dis. (final) modes Dis. avg. 5 Locking 5 Locking Saturation Saturation 3 Non dis. avg. End End 2 1 5 5.5 1. 1.5 2. 2.5 3. 3.5 N at rotating onset 1.5 1. 1.5 2. 2.5 3. 3.5 N at rotating onset (Left) β N tends to decrease between rotating onset and locking Purple preceded by former locked mode 1 (Middle) Disruptive IRLMs decrease β N by up to 8%.5 1. 1.5 2. 2.5 3. 3.5 N at rotating onset (Right) β N decreases by a smaller amount during lifetime of nondisruptive IRLMs 12
IRLMs change the 2D equilibrium shape by reducing the Shafranov shift Disruptive Non-disruptive Disruptive avg. Δ R (cm) over first 2 ms of mode 2-2 -4 -.6 -.4 -.2..2 Δ β p over first 2 ms of mode ΔR /Δβ p 4 cm Decrease of m/n=1/ shaping might affect toroidal coupling of m/n=2/1 with other n=1 perturbations 13
Decreasing IRLM disruptivity at high β N observed in 1D, partially attributed to coincident high q 95 Peak β N 5 4 3 2 1 Dis. modes Non-dis. modes 1.2 1. β4.8 β3 β2.6 β1.4 β.2. IRLM disruptivity β β1 β2 β3 β4 4 3 2 1 Distribution (%) β β1 β2 β3 β4 2 3 4 5 6 7 8 q 95..5 IRLM dis. -.2 3 4 5 q 95 3. 3.5 4. 4.5 5. 5.5 q 95 (Left) 1D projection of IRLM disruptivity vs. β N shows decreasing disruptivity with increasing β N (Middle) IRLM disruptivity decreases with increasing q 95 (Right) Percent distributions in q 95 show high betan bins (purple and green) have less low q 95 discharges 14
IRLM amplitude and angular position behavior 15
From rotation at 2 khz to 5 ms post locking, the island width usually does not change within error Island widths not validated to better than ± 2 cm (conservative error bar) ~3 small rotating islands grow significantly 16
Angular distribution shows preferred angle of locking Cumulative distribution of angular positions during locked, and quasi-stationary phase Shots 4 3 2 Related to residual EF or viscous drag? Disruptive Non-Disruptive Intrinsic EF 1 Peaks offset from intrinsic EF by -35, which is in the normal plasma rotation direction -1 1 Angle (deg) I p, ω 17
Saturated width scales with β p /(dq/dr), indicating at least partial drive from bootstrap deficit w/r at saturation w/r at saturation.3.2.1..3.2.1 Disruptive Non-disruptive q 95 < 4 q 95 > 4...4.8.12 β p / (dq/dr) at saturation (m) Expected from steady-state Modified Rutherford Equation IRLMs occurring at low q 95 (top) correlate better than those at high q 95 (bottom) 18
Prior to disruption, IRLM might grow exponentially (n=1 field does) B R (G) 25 2 15 1 5 1.5 124865 137232 156688 15741 13189 15 1 No. of Disruptive IRLMs 18 ms 1 ms 4 ms 12 ms 8 ms 15 1 B R (G) of median Exponential I p (MA) 1..5 5. 1 8 6 4 2-2 Time before disruption (ms) -4 5 1 15 2 25 B R (G) at t before disruption 5 2 15 1 5 Time before disruption (ms) (Left) Five shots with IRLMs show a general trend of increasing n=1 field within 1 ms of disruption (Middle) Distributions of n=1 field shift toward higher field as disruption is approached (Right) Median of distributions grows exponentially in last 5 ms Poloidal analysis necessary to conclude growth of m/n=2/1 19
IRLM disruptivity 2
IRLM disruptivity scales strongly with ρ q2 (at fixed q 95 ), and weakly with q 95 (at fixed ρ q2 ) ρ q2 at mode end IRLM dis..9.8.7.6.5.4 1..5. ρ1 ρ2 ρ3 ρ4 SF1 SF2 SF3 Disruptive Non-disruptive 2 3 4 5 6 7 q 95 at mode end..5 1. IRLM dis. 1.2 1..8.6.4.2. -.2 ρ4 IRLM disruptivity Flat ρ1 ρ2 ρ3 3 4 5 6 7 q 95 SF2 IRLM disruptivity Gradient SF1 SF3.4.5.6.7.8.9 ρ q2 A 1D study (blue histograms) suggests both ρ q2 and q 95 are important Fixing ρ q2 shows that q 95 alone has a weak effect on IRLM disruptivity Fixing q 95 shows that ρ q2 has a strong effect on IRLM disruptivity Are there any hidden variables here? 21
In DIII-D operational space, ρ q2 is related to l i and q 95 where α =.67 ±.1 and c = -.23 ±.1 Empirically, find extra high correlation of r c =.87 What is the effect of l i in the previous study? Does l i, q 95, or ρ q2 (l i /q 95 ) determine IRLM disruptivity? We will study how well a single parameter separates disruptive from non-disruptive IRLMs 22
Panels (24) and (25) inform investigation of l i /q 95, and we find it performs better than ρ q2 l i /q 95 at mode end.4.3.2 Disruptive Non-disruptive Separation between disruptive and non-disruptive IRLMs is predominantly vertical Also, high correlation observed between l i /q 95 and ρ q2.5.6.7.8.9 1. ρ q2 at mode end DIII-D ITER baseline 23
Bhattacharyya Coefficient 1 (BC) used to measure separation between two probability distributions For two discrete probability distributions p and q parameterized by x, over the parameter space X, the BC value is given by, BC= means p and q are completely separate BC=1 means p and q are identical (completely overlapping) 1 D. Comaniciu et al 2 Comput. Vision Graph. 2 24
Of the parameters considered, d edge and l i /q 95 separate disruptive from non-disruptive IRLMs best 4 3 Distribution (%) BC =.71 BC ** =.61 4 3 Dis. modes Separation BC =.64 Non-dis. modes 4 BC =.7 3 2 2 2 1 1 1-5 5 1 15 2 25 d edge (cm).2.3.4.5 l i /q 95.4.5.6.7.8.9 1. ρ q2 4 3 4 BC =.85 3 BC =.88 4 3 BC =.95 BC ** =.98 2 2 2 1 1 1 2 3 4 5 6 7 8 9 q 95.5 1. 1.5 2. 2.5 l i 2 4 6 8 w (cm) Nearly indistinguishable 25
Assuming exponential growth of 2/1 mode, d edge becomes predictive within 2 ms of disruption Separation 4 w/2 at mode end (cm) 3 2 1 LCFS d edge =9 cm Undetectable a plasma minor radius r q2 minor radius of the q=2 surface w island width 5 1 15 2 25 a r q2 (cm) at mode end..5 1. IRLM dis. Late predictive capability might suggest d edge is relevant to thermal quench physics Even assuming exponential island growth, island width still hardly affects IRLM disruptivity (blue histogram) 26
Parameters predictive of IRLM disruptions 27
Performance of l i /q 95 and d edge can alternatively be quantified by percent missed disruptions and false alarms Parameter threshold % missed disruptions % false alarms l i /q 95 <.28 18 1 l i /q 95 <.25 11 15 d edge > 9 cm 2 12 d edge > 11 cm 12 2 l i /q 95 also provides at least 1 ms of disruption warning, whereas d edge only provides 2 ms Although d edge is less favorable for disruption prediction, it might still be relevant to the physics of the disruption 28
Conclusions 1. In DIII-D, 18% of disruptions are due to IRLMs, and this percentage rises to 28% for β N > 1.5 2. The survival time is most often 27 ms, and exhibits moderate correlation with d edge 3. Disruptive IRLMs degrade β N by 5-8%, while non-disruptive IRLMs degrade β N less 4. Scaling of w/r with β p /(dq/dr) suggests that IRLMs in DIII-D are driven at least in part by bootstrap deficit 5. The n=1 radial field grows exponentially within 5 ms of the disruption, and might imply exponential growth of m/n=2/1 6. l i /q 95 is found to be predictive of IRLM disruptions 1 ms prior l i /q 95 <.25 yields 11% missed disruptions and 15% false alarms 7. Assuming exponential growth of m/n=2/1, d edge becomes predictive within 2 ms of the disruption 29
3 APPENDIX
31 Disruption Identification
Plasma current decay time used to identify disruptions Current quench Decay time is the duration of the current quench 1 Figure: D. Shiraki et al. 215 NF 1 T.C. Hender 21 IAEA Proceedings 32
Decay-time used to differentiate disruptive from nondisruptive discharges Three groupings i. t d < 4 ms Disruptive ii. 4 ms < t d < 2 ms Intermediate iii. t d > 2 ms Non-disruptive 5 shots manually analyzed in populations i and iii, confirmed that: No false positives in major disruptions (i.e. calling non-disruptive shot disruptive) No false negatives in non-disruptions (i.e. calling disruptive shot non-disruptive) (a) 1 Number of I p Decay Events (b) Number of I p Decay Events 8 6 4 2 25 2 15 1 5 i ii 22511 Events iii. non disruptive discharge completion 3 6 9 12 15 Decay Time (ms) i. major disruptions ii. disruption occuring during rampdown 5 1 15 2 25 3 Decay Time (ms) 33
l i /q 95 might be related to the potential energy to drive nonlinear tearing growth Assuming dj/dr monotonically decreasing, and q monotonically increasing, potential energy can be expressed as follows [Sykes PRL 8],!" =!!!!!/!!!"!" 2!!!!!!!!" +!!!!!!/!!"!"!! 2!!!!!"!! Recall l i /q 95 αρ q2 + c, and therefore l i /q 95 determine limits of integration As l i determines profile peaking, and q 95 ~1/I p, dj/dr is expected to depend on l i /q 95 34
d edge might be related to the physics of the thermal quench; other works that have also observed this Experimental results from Compass-C find locked mode disruptions occur when inequality that is similar to d edge is satisfied [Hender NF 92] Massive gas injection simulations using NIMROD find the thermal quench is triggered when m/n=2/1 island intersects the radiating edge [Izzo NF 6] Stochastic layer exists inside the unperturbed LCFS [Evans PoP 2, Izzo NF 8], which could stochastize the m/n=2/1 island when d edge sufficiently small 35
Analytic current filament model used to map from BR at the external saddle loops (ESLDs) to perturbed current δi Informed by n=1 signal from ESLDs and rq2 and R from EFIT constrained by motional stark effect Find the following equation for δi, where α(r) is a quadratic function of R 36
Definition of δi, and mapping to island width w δσ (ka/rad) δσ (ka/rad) 1-1 2 1 O-point O-point δi h δi X-point X-point (a) (b)..5 1. 1.5 2. 2.5 3. Poloidal angle (rad) δi h assumes that a deficit of current exists in O-point, and excess in X-point δi assumes that there is only current deficit, which is greatest at the O-point Note that both δi and δi h produce the same nonaxisymmetric field Where δi h is in Amps, r q2, R, and dq/dr are in meters, B T is in Tesla, and c=8/15 37
View of q=2 surface during a disruption T e (kev) 1..8.6.4 157247 Time (ms) 366. 368. 371. 372. 3722.5 Estimated position and width of islands appear reasonable.2 LM. 2 25 21 215 22 R (cm) 2 15 1 ECE location 5 364 366 368 37 372 Time (ms) 38
ρ q2 increases through lifetime of IRLM ρ q2 at mode end.9.8.7.6 Disruptive Non-disruptive evolution of ρ q2 from locking to 1 ms before mode end majority of disruptive IRLM move outwards.5 ρ q2 at mode end.9.8.7.6 Non-dis. ramp Dis. ramp.5.3.4.5.6.7.8.9 ρ q2 at Locking 39
4 Modified Rutherford Equation (MRE)
LM disruptivity increases above density limit 2. Disruptive Non-disruptive JET density limit l i at mode end 1.5 1..5 2 3 4 5 6 7 q 95 at mode end Disruptivity strongly increases past JET density limit 41