Perturbative Determination of Plasma Microinstabilities in Tokamaks
A. O. Nelson, F. M. Laggner, A. Diallo, Z. A. Xing, D. R. Smith, E. Kolemen
PPerturbative Determination of Plasma Microinstabilities in Tokamaks
A. O. Nelson, F. M. Laggner, A. Diallo, Z. A. Xing, D. R. Smith, and E. Kolemen
1, 2 Princeton University, Princeton, NJ 08544, USA Princeton Plasma Physics Laboratory, Princeton, New Jersey 08540, USA University of Wisconsin-Madison, Madison, Wisconsin 53706, USA (Dated: 9 February 2021)
Recently, theoretical analysis has identified plasma microinstabilities as the primary mechanism responsible foranomalous heat transport in tokamaks. In particular, the microtearing mode (MTM) has been credited withthe production of intense electron heat fluxes, most notably through a thin self-organized boundary layer calledthe pedestal. Here we exploit a novel, time-dependent analysis to compile explicit experimental evidence thatMTMs are active in the pedestal region. The expected frequency of pedestal MTMs, calculated as a functionof time from plasma profile measurements, is shown in a dedicated experiment to be in excellent agreementwith observed magnetic turbulence fluctuations. Further, fast perturbations of the plasma equilibrium areintroduced to decouple the instability drive and resonant location, providing a compelling validation of theanalytical model. This analysis offers strong evidence of edge MTMs, validating the existing theoretical workand highlighting the important role of MTMs in regulating electron heat flow in tokamaks.Utilizing tokamak reactors to realize magnetic con-finement fusion holds the prospect of producing cleanand sustainable energy . This effort requires the es-tablishment of hot, dense plasma cores through a self-organized high-confinement regime (H-mode) character-ized by steep plasma gradients in a thin region called thepedestal . While only covering ∼
10% of the plasma ra-dius, the pedestal can be responsible for up to ∼
70% ofthe total plasma pressure and fusion performance and isthus essential for the successful optimization of tokamakdevices.A standard H-mode pedestal is characterized by twocompeting physics phenomena. First, strong velocityshear caused by variation in the radial electric field sup-presses transport in the pedestal by tearing apart tur-bulent eddies, allowing for the formation of steep tem-perature and density gradients that would otherwise beeliminated by diffusion . Second, various metastable mi-croinstabilities induce transport across the pedestal de-spite the high levels of turbulent shear, controlling theevolution of the pedestal structure . If left unmitigated,non-linear interactions between these microinstabilitiesperiodically spark global explosive events called edge-localized modes (ELMs) which can melt and erode themachine wall . As such, understanding the details ofthese microinstabilities is not only for crucial for the op-timization of fusion parameters but also for successfulplasma control .Over the past few decades, largely theoretical andcomputational work has uncovered five plasma instabili-ties that may contribute to inter-ELM transport throughthe H-mode pedestal. These include three electrostaticmodes: the trapped electron mode (TEM), the electrontemperature gradient (ETG) mode, and the ion tem-perature gradient (ITG) mode; and two electromagneticmodes: the kinetic ballooning mode (KBM) and the mi-crotearing mode (MTM). Extensive modeling has shownthat each of these modes could become unstable in thetokamak edge under certain conditions, but an experi-mental validation of which modes are actually active in the pedestal remains elusive due to the nebulous natureof the turbulence. Without an empirical determinationof individual modes, it is difficult to improve the physicsbasis of leading turbulent models.Notably, recent theoretical work suggests that theMTM , a small-scale resistive magnetohydrodynamic(MHD) mode not yet included in leading predictivemodels , might play a critical role in limiting electronthermal transport through the pedestal . The pres-ence of pedestal MTMs has been suggested through anal-ysis of so-called “transport fingerprints” and throughcomparisons of measured magnetic fluctuations with sen-sitive theory-based (gyrokinetic) simulations . How-ever, a conclusive experimental identification of thesemodes has not yet been presented and is needed to vali-date the theoretical results.In this article, we introduce novel experimental ev-idence to unambiguously demonstrate the existence ofMTMs in the tokamak pedestal. MTMs are theorizedto destabilize at particular resonant locations within theplasma and to oscillate at the electron diamagnetic fre-quency, which is a function of plasma radius. We uti-lize an innovative experimental technique in the form oflarge vertical plasma displacements to dissociate thesetwo phenomena by dynamically shifting magnetic sur-faces in the edge region. Experimental observations ofMTM evolution in a series of plasma discharges on theDIII-D tokamak are found to be in overall agreement withtheoretical expectations, providing a compelling valida-tion of the model. The presented work describes a clear-cut experimental identification of MTMs, focusing atten-tion on the need to include MTM physics into predictivetokamak models. TIME-DEPENDENT MTM IDENTIFICATION
Microtearing modes are finite-collisionality electro-magnetic modes destabilized by the electron temperaturegradient ∇ T e13,14 . In tokamaks, magnetic surfaces have a r X i v : . [ phy s i c s . p l a s m - ph ] F e b FIG. 1: Physics of the time-dependent thermal forcedrive for MTMs. (a) A temperature gradient ∇ T e projected onto a q -resonant magnetic perturbationcreates spatial variation in the thermal drag force ( R T (cid:107) )between electrons and ions. (b) Due to parallel motionat v e ∗ , a time lag is introduced to the thermal force,creating a de-phased electric field through chargeseparation. (c) The resulting inductive field dB r /dt isin-phase with the initial perturbation, causing theinstability to grow.a helical structure defined by the ratio q = m/n , whichdescribes the winding of a field line in the poloidal ( m )and toroidal ( n ) directions. At rational values of m and n , a radial perturbation B r can be driven unstable if thepresence of ∇ T e creates an instability drive stronger thanthe stabilizing influence of magnetic curvature . In fig-ure 1, the destabilizing effect of ∇ T e is illustrated usingthe thermal drag force R T (cid:107) ∝ ( ν +e − ν − e ), where the dif-ference in collision frequency ν e along a field line is dueto changes in the electron temperature since ν e ∝ T − / .Importantly, plasma motion at the electron diamagneticvelocity v e ∗ introduces a time-lag to R T (cid:107) . As a result,the emergent parallel electric field E (cid:107) creates an induc-tive field that adds in-phase to the initial perturbation,leading to growth of the instability.This description brings to light two important facetsof MTM instability drive: (1) MTMs should be localizedaround rational magnetic surfaces and (2) MTMs shouldoscillate at the electron diamagnetic frequency ω e ∗ ( ψ n ),where ψ n is a radial unit given by the normalized poloidalflux,. Here ω e ∗ is given by ω e ∗ = k y ρ s c s (cid:18) L n e + 1 L T e (cid:19) , (1)which depends explicitly on the density and temperaturegradient length scales L n e and L T e . More details are given in the methods section. Since ω e ∗ is inversely re-lated to L T e , a peak in the ω e ∗ ( ψ n ) profile correspondsto a peak in the MTM instability drive from ∇ T e ( ψ n ).Therefore MTMs are most likely to occur when a ratio-nal q surface aligns with the peak of the ω e ∗ ( ψ n ) profile .This formulation has been used to explain steady-statefrequency bands observed in magnetic fluctuation dataon the JET tokamak, which were identified as MTMsthrough comparisons with gyrokinetic simulations , andit forms the theoretical foundation of the dynamical ex-perimental analysis presented here.In plasma experiments, magnetic fluctuations mea-sured in the lab frame will have an additional frequencycomponent given by the Doppler shift ω dop ( ψ n ). By ex-ploiting high spatial and temporal resolution diagnosticson the DIII-D tokamak , we can track the structureof both ω e ∗ and ω dop through time, enabling an investi-gation of the dynamical evolution of plasma microinsta-bilities in tokamaks.In figure 2, we demonstrate this process for a single n = 3 MTM, providing unambiguous evidence for MTMactivity in the H-mode pedestal. Figure 2(a) shows theedge f e ∗ , n=3 and f dop , n=3 profiles for a single represen-tative timeslice. As a result of the steep temperaturegradients in the pedestal, a large peak in the n = 3MTM destabilization potential occurs near the plasmaedge. Also shown are the locations of four possible ra-tional q surfaces in the pedestal, with m varying from15 −
18 throughout the steep gradient region.In figure 2(b), the evolution of the projected MTMfrequency ( f MTM = f e ∗ + f dop ) at these four radial lo-cations is tracked through time between explosive ELMevents. The rational surface at q = 16 / as follows: The recovery of densityand temperature gradients after an ELM introducesperiodic growth into the ω e ∗ profile described by equa-tion 1. MTMs, being locked at a particular rational q surface, will simultaneously experience a local increasein ∇ T e and ω e ∗ . Therefore, once these modes turn on ata critical ∇ T e31 , their frequency will continue to increaseuntil saturation of the pedestal gradients is achieved. EXPERIMENTAL MTM FREQUENCY MODIFICATION
With the dynamics of MTM evolution established, wenow introduce a novel perturbation scheme to explore theeffect of rational surface displacement on the modes. Pre-viously, small vertical oscillations of the plasma volume(“jogs”) have been used to perturb the edge current in or-der to destabilise peeling modes and trigger ELMs . ψ n F r e q u e n c y ( k H z ) m = m = m = m = DIII-D f e * f dop f MTM (a)
Time since last ELM (ms) P o t e n t i a l M T M f r e q u e n c y ( k H z ) DIII-D m/n=15/3m/n=16/3m/n=17/3m/n=18/3m/n=15/3m/n=16/3m/n=17/3m/n=18/3 (b)
Time since last ELM (ms) F r e q u e n c y ( k H z ) DIII-D m/n=16/3 magnetics ( f B)(c)
FIG. 2: Time-dependent MTM frequencies computed from experimental profiles match well with those observed inmagnetic fluctuations. (a) The MTM frequency f MTM is shown as the sum of f dop and f e ∗ for a representativetimeslice, along with the possible rational q surface locations for an n = 3 mode. (b) At each rational q surface, theexpected MTM frequency is plotted as a function of time since the last ELM. (c) Profile-based predictions for the m/n = 16 / n = 3 chirped mode observed in magnetic fluctuations. −1 0 1 2 R (m) −0.80.00.8 Z ( m ) pre-jogpost-jogDIII-D ψ n q q=16/3(b) R (m) −0.6−0.5−0.4−0.3 Z ( m ) w r w r s e p a r a t r i x s e p a r a t r i x (c) 0.00.10.20.30.40.50.60.70.80.91.0 J t ( M A / m ) FIG. 3: (a) With vertical control algorithms, the plasma is rapidly dropped ∼
10 cm during a jogging event. (b) Dueto the jog, the resonant q surface moves substantially through the pedestal. (c) The effects of the jog are confined toa small edge region with width w r that contains a thin, strong current layer. Image Credit: General Atomics.Analytical studies have shown that, during a joggingevent, toroidal current is induced in the pedestal due pri-marily to the compression of the plasma cross section asit travels through an inhomogeneous magnetic field , asdescribed further in the methods section. Changes in the edge current impact the poloidal magnetic field throughAmpere’s law, which in turn impacts the winding ratioof magnetic field lines q = m/n and modifies the locationof rational magnetic surfaces in the pedestal.Here we apply this same principle in a more intense F r e q u e n c y ( k H z ) magnetics ( f B)(a) DIII-D F r e q u e n c y ( k H z ) n=5n=4 n=3n=5n=3n=2 (b)−9−6−303 z p ( c m ) (c)0.940.971.00 I p ( M A ) (d)3500 3550 3600 3650 3700 3750 3800 3850Time (ms)4.675.005.33 q (e) FIG. 4: (a) Magnetic fluctuations show an invertingchirping behavior after a jogging event compared toafter a normal ELM. ELM times are indicated withvertical dashed lines. (b) The chirping modes are n = 3 , n = 5 modes, shown in orange, magentaand red, respectively. Additional core (black) andpedestal (blue) modes are also shown. The evolution ofthe (c) magnetic axis Z p , (d) plasma current I p and (e)edge safety factor q through a jogging event show thedirect effects of the jog.manner with large ( ∼
10 cm) and fast ( <
10 ms) down-ward jogs designed to produce the largest possible per-turbations in the edge current. Figure 3(a) comparestwo equilibria before (red) and after (blue) a large jog-ging event. The effects of the jog on the plasma areprimarily constrained to an edge region w r , which can beapproximated as the MHD skin depth w r ∝ √ η , where η is the plasma resistivity . As a result of the jog, thereconstructed q profile presented in figure 3(b) dramat-ically changes. As expected, the radial location of the q = 16 / J t for the pre-jog equilibrium, highlighting that the jogs are large enough to influence the edge peak in J t butsmall enough not to significantly disturb the core plasma.This perturbation scheme is uniquely capable of investi-gating the behavior of microinstabilities in the edge bydecoupling the q and ω e ∗ profiles.When applied in experiment, the jogs successfully pro-duce clear and quantifiable differences in microinstabil-ity signatures distinct from observations during naturalinter-ELM periods. In figure 4(a), a magnetic spec-trogram from high-frequency Mirnov coils is shown fora time period including two natural ELM periods fol-lowed by a large jogging perturbation. Multiple insta-bilities are evident in the inter-ELM periods, but thehigher frequency modes at ∼
60 kHz and ∼
110 kHz showan inverted chirping behavior after the jogging eventat 3700 ms. Again, local density fluctuation measure-ments place these modes in the plasma edge. UsingFourier analysis techniques on a set of fast magneticdiagnostics , the chirped modes are identified in fig-ure 4(b) as n = 3 , z p ), thetotal plasma current I p and the edge q profile magnitude( q ) are shown during a jogging event in figure 4(c)-(e).Notably, the robust analysis developed above can beapplied after a jogging event. Figure 5(a) shows the edge f MTM , / profile (solid curves) and the q = 16 / (cid:46)
100 ms), the location ofthe rational q = 16 / q = 16 / f MTM profile andthen moves inwards over the course of ∼
80 ms, fallingslightly off the peak destabilizing frequency. In contrastto the growth of f MTM after a natural ELM, however,the expected MTM frequency falls after a large joggingevent due to the inwards motion of the magnetic sur-face. In figure 5(b), the computed decrease of f MTM , / is overlayed on the n = 3 mode extracted from magneticsmeasurements in figure 4(b). The time-dependent profileanalysis matches the experimentally-observed fluctuationdynamics, showing strong agreement between theory andexperiment. Moreover, it is noted that the n = 3 modeamplitude (see figure 4(a)) is strongest when the 16 / ω e ∗ profile, inagreement with the expectation that the electron ther-mal gradient, which peaks with ω e ∗ , acts as main MTMdrive .In figure 5(c), the analysis is augmented by match-ing the decreasing n = 4 mode observed early in theELM cycle after jogs. In this case, the rational surface m/n = 21 /
4, which lies just inside of the m/n = 16 / ω e ∗ profile. Directly after a jog, the MTM drive on thissurface is high and the mode appears in magnetic fluc-tuation measurements. However, as the q surface movesinwards, the drive, amplitude and frequency drop untilthe n = 3 mode dominates. Power is transferred to more n F r e q u e n c y ( k H z ) q=16/3 location f MTM (a) n=3 Time since jog F r e q u e n c y ( k H z ) (b) m/n=16/3 F r e q u e n c y ( k H z ) (c) m/n=21/4 DIII-D
FIG. 5: The automated analysis presented in figure 2 is repeated for data averaged over several jogging cycles. (a)Due to the current recovery after a jog, the q = 16 / f MTM predictions and the (b) orange n = 3 and (c) magenta n = 4 modes extracted from magnetic fluctuation measurements.unstable rational surfaces during this transition throughnon-linear coupling between various pedestal modes .This manifests in the magnetics measurements as a dis-appearance of the n = 4 signature coincident with a peakin the n = 3 amplitude around ∼
60 ms after the joggingevent (see figure 4(a)). Again, figure 5(c) shows excellentagreement between the profile and fluctuations measure-ments, verifying the dynamic behavior of edge-localizedMTMs. We note further that the very low MTM fre-quency predicted directly after the jog ( t jog = 20 ms) infigures 5(b) and 5(c) occurs at a time before the MTMonset and thus is not expected to produce magnetic fluc-tuations. This analysis can be reproduced to the sameeffect for the n = 5 modes shown in figure 4(b) and isrobust to changes in the background plasma across sev-eral DIII-D discharges, showing that minor backgroundplasma changes do not modify the fundamental MTMbehavior. DISCUSSION AND OUTLOOK
To support the above dynamic identification of MTMsin the H-mode pedestal, we note here that additionalmeasurements were taken to rule out the possibility ofthe above dynamics being caused by other instabilities. • Large amplitudes of these modes in magnetic fluc-tuation measurements suggest that the modes areelectromagnetic in nature, eliminating considera-tion of ETGs, ITGs and TEMs. • The propagation of the chirped modes is deter-mined to be strongly in the electron diamagneticdirection, as is expected for MTMs, both through E × B profile calculations and detailed analysis of local density fluctuation data . Conversely, KBMs(the other primary electromagnetic edge turbulencecandidate) rotate in the ion diamagnetic direction,which is inconsistent with the measured data. • Experimental transport studies show that D e /χ e ∼ /
10 in the vicinity of the n = 3 and n =5 chirped modes. This is consistent with gy-rokinetic predictions stating that pedestal MTMsshould contribute predominately to electron ther-mal transport, whereas KBMs contribute approxi-mately equally to both electron thermal and parti-cle transport . • Finally, the modes turn on simultaneously with thesaturation of ∇ T e , consistent with models tyingdestabilization to the growth of pedestal gradientsbeyond a critical threshold .In sum, these compounding observations combined withthe novel dynamic frequency evolution described abovepaint an explicit experimental picture of the existence ofMTMs in the DIII-D H-mode pedestal. Further, mea-surements showing that the associated transport is pre-dominantly in the electron heat channel and is closelylinked to the saturation of ∇ T e reinforce the establishedtendency of MTMs to contribute significantly to electronheat flux through the plasma edge.By regulating electron thermal transport, MTMs areexpected to establish limits on the maximum electrontemperature gradients within the H-mode pedestal .Since the pedestal contributes substantially to totalplasma pressure and fusion performance, identifying andunderstanding the full impact of MTMs could have sig-nificant implications on the design of future pilot planscenarios. While numerous theoretical studies have pre-dicted the presence of these instabilities under commonpedestal conditions, an experimental validation is re-quired to build confidence in the MTM physics basis ifpredictive pedestal models are to be expanded to includethe relevant effects. In this article, we experimentallydemonstrate the existence of edge-localized MTMs byexploiting the time response of the plasma to verticaljogs. These results validate leading analytic and numeri-cal theories and motivate the future incorporation ofMTM effects into advanced predictions of the full plasmaperformance.The success of the perturbative approach applied herealso implies a possibility for the expansion of dynamicturbulence identification to other unexplained tokamakregimes. Similar instability markers have been reportedon a wide variety of machines and scenarios , but theunderlying physics remains largely undetermined. Theanalysis presented here offers a new mechanism to un-cover explanations for these observations, potentially en-abling a comprehensive perturbative study of experimen-tal transport signatures in tokamak devices. METHODS
This investigation is predicated on the accurate si-multaneous measurement of many different plasma pa-rameters, including electron and ion densities and tem-peratures and the equilibrium magnetic structure. TheDoppler-shift ω dop and electron diamagnetic frequency ω ∗ e are calculated from Carbon impurity measurementsfrom charge-exchange recombination and electron pro-file measurements from Thomson scattering , respec-tively. Localized density fluctuations measurements aremade with beam emissions spectroscopy in order to lo-calize the instabilities in the plasma edge . To acquirerobust statistics, measurements are taken every ∼
20 msthroughout several dedicated discharges (each of whichlasts ∼ . Essential to these re-constructions are accurate calculations of the bootstrapcurrent , requiring the consideration of constraints fromboth magnetic and internal profile data. Senstive equilib-rium reconstructions are necessary to calculate both theedge q profile, for which no direct measurement currentlyexists on DIII-D, and the radial alignment between the ω dop and ω ∗ e profiles. To facilitate robust analysis, multi-ple reconstructions are made for each measurement timeto generate effective uncertainties in the plasma magneticstructure, which are propagated through the final MTMfrequency calculation.Throughout this work, we define ω e ∗ as in equation 1, reproduced here for convenience: ω e ∗ = k y ρ s c s (cid:18) L n e + 1 L T e (cid:19) . (2)Here k y = nq/aρ tor is the binormal wavenumber, ρ tor = √ Φ n is the square root of the normalized toroidal mag-netic flux, ρ s = c s / Ω i is the sound gyroradius, c s = (cid:112) ZT e /m i is the sound speed, Ω i is the ion gyrofrequency,and the electron density and temperature gradient scalelengths are defined as a/L n e = (1 /n e )( dn e /dρ tor ) and a/L T e = (1 /T e )( dT e /dρ tor ), respectively . The Dopplershift ω dop is correspondingly given by: ω dop = nE r RB p , (3)where n is the toroidal mode number, E r is the radialelectric field, R is the major radius and B p is the poloidalmagnetic field.During a jogging event, the induction of current dueto the motion of the plasma through an inhomogeneousmagnetic field can be by is described by δI w r φ = 4 πµ R (cid:20) δψ ext ( a ) − B θ ( r ) R δw r − ηJ φ δt (cid:21) , (4)as first reported by Artola et. al. . Here the change intotal toroidal edge current ( δI w r φ ) is given as a function ofthe local change in external magnetic flux ( δψ ext ), the in-homogeneous poloidal magnetic field ( B θ ), plasma com-pression ( δw r ) and a small resistive decay term ( ηJ φ δt ) .The width w r of the edge region is generally smallcompared to the plasma minor radius ( r ) such that w r /r << w r ∼ (cid:112) η/ ( πµ f ), where f is the oscillation frequencyand η is the plasma resistivity.During and after a jogging event, the correspondingchanges in the current profile are taken from kinetic equi-librium reconstructions based on fast magnetic measure-ments and internal plasma profiles. In tokamaks, thesafety factor q = m/n can be defined as q = rB φ RB θ , (5)where r is the minor radius, R is the major radius, and B φ and B θ are the toroidal and poloidal magnetic fields,respectively. Since B θ is directly related to the toroidalplasma current through Ampere’s Law, the q profile inthe plasma edge is significantly modified during a jog.This is the necessary perturbation for the study of edgemicroinstabilities, as is discussed in the main text.Throughout this work, 1D and 2D transport sim-ulations are conducted with the TRANSP andautoUEDGE codes in order to verify the trans-port fingerprints of pedestal MTMs. Assessment ofthe toroidal mode numbers was completed with theMODESPEC code; magnetic diagnostic resolution is nothigh enough to directly determine the poloidal modenumbers , suggesting m >
12 as found through theprofile analysis. Part of data analysis for this workwas performed using the OMFIT integrated modelingframework . DATA AVAILABILITY
The data discussed and used for all figures in this ar-ticle are available from the corresponding author uponreasonable request.
ACKNOWLEDGMENTS
The authors would like to especially thank D.R. Hatchand M. Curie for several helpful discussions relating tothe frequency identification of MTMs, as well as W.Guttenfelder and R. Nazikian for valuable advice dur-ing preparation of the manuscript. This material wassupported by the U.S. Department of Energy, Office ofScience, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Scienceuser facility, under Awards DC-AC02-09CH11466, DE-SC0015480, DE-SC0015878 and DE-FC02-04ER54698.This report is prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof,nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibil-ity for the accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercial prod-uct, process, or service by trade name, trademark, man-ufacturer, or otherwise, does not necessarily constituteor imply its endorsement, recommendation, or favoringby the United States Government or any agency thereof.The views and opinions of authors expressed herein donot necessarily state or reflect those of the United StatesGovernment or any agency thereof. J. Wesson,
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