Stochastic fluctuation and transport in the edge tokamak plasmas with the resonant magnetic perturbation field
Minjun J. Choi, Jae-Min Kwon, Juhyung Kim, T. Rhee, J.-G. Bak, Giwook Shin, H.-S. Kim, B.-H. Park, Hogun Jhang, Gunsu S. Yun, M. Kim, J.-K. Park, S.K. Kim, H.H. Lee, Y. In, J. Lee, M.H. Kim, Hyeon K. Park
SStochastic fluctuation and transport in the edge tokamak plasmas withthe resonant magnetic perturbation field
Minjun J. Choi ∗ , Jae-Min Kwon , Juhyung Kim , T. Rhee , J.-G. Bak , Giwook Shin , H.-S.Kim , B.-H. Park , Hogun Jhang , Gunsu S. Yun , M. Kim , J.-K. Park , H.H. Lee , Y. In , J.Lee , M.H. Kim , and Hyeon K. Park Korea Institute of Fusion Energy, Daejeon 34133, Republic of Korea Pohang University of Science and Technology, Pohang, Gyungbuk 37673, Republic of Korea Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of KoreaMarch 4, 2021
Abstract
The stochastic layer formation by the penetration of the resonant magnetic perturbation (RMP)field has been considered as a key mechanism in the RMP control of the edge localized mode (ELM) intokamak plasmas. Here, we provide experimental observations that the fluctuation and transport in theedge plasmas become more stochastic with the more penetration of the RMP field into the plasma. Theresults support the importance of the stochastic layer formation in the RMP ELM control experiments.
Introduction
The edge localized mode (ELM) is a peeling-ballooning instability driven by the large current densityand the steep pressure gradient in the high confinement mode (H-mode) pedestal in tokamak plasmas.Periodic collapses of the pedestal due to the explosive growth of the ELM have been a serious issue forsteady-state operation of the H-mode plasmas. Applying the resonant magnetic perturbation (RMP) fieldis the most developed method to suppress the ELM and its pedestal collapse [1, 2, 3, 4, 5, 6]. Sincethe first demonstration of the RMP ELM suppression [1], many researches have followed for in-depthunderstanding of its mechanism. The initial idea of the stochastic edge transport [7] was suggested [1], andlater a narrow stochastic layer was considered promising to explain the experimental edge transport withthe RMP field [8, 9]. It was more elaborated in the recent two-fluid MHD simulation that explained theELM control process as sequential formation of narrow separated stochastic layers at the pedestal foot andtop [10]. Resonant penetration of the perturbation field and consequent temperature flattening, supportingthe stochastic layer formation inside the plasma, were observed in the experiment [11], but the lack of aproper measure of the stochasticity and the uncertainty in the profile diagnostics prohibit the improveddiagnosis which would be required for demonstration of the simulation results [10]. In this work, we suggestthe rescaled Jensen Shannon complexity [12] for an effective measure of the stochasticity to identify thestochastic layer. We found that the stochasticity in electron temperature fluctuations is enhanced nearthe pedestal top in the RMP ELM suppression plasmas. The stochastic layer full width is estimated as2 . ± . ∗ Corresponding Author: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] M a r igure 1: (Color online) (a) The D alpha emission (black) and the in-vessel coil current for the resonantmagnetic perturbation (RMP) field (red). The edge localized mode (ELM) is suppressed with the increasedRMP field. (b) The line averaged plasma density ( n e ) and the edge electron temperature ( T e ). (c)Spatiotemporal evolution of the rescaled Jensen Shannon complexity obtained by multiple T e measurementsin the pedestal region around the ELM location ( R ELM = 215 . Results
An example of the RMP ELM suppression in the KSTAR [15] H-mode plasma n = 1 RMP field is applied from t = 3 . n e ) drops about 17 % with applying the RMP field while the edge electron temperature( T e ) drops about 9 %. This is the so-called density pump-out process. n e and T e are obtained by two colorinterferometry diagnostics [16] and the electron cyclotron emission diagnostics [17], respectively. The ELMcrash is mitigated with the RMP field and it is completely suppressed after t ∼ . D alpha signal in Fig. 1(a) whose sharp rise indicates the ELM crash event. Modest decrease of n e and the edge T e during a transition from the ELM mitigation phase to the ELM suppression phase is observed, whichindicates that the pedestal pressure height is reduced to make the ELM stable.Formation of the stochastic layer in the pedestal region has been considered as a key mechanism of theRMP ELM suppression [1, 10]. Although a direct measurement of fine structures of the magnetic field isnot possible yet in tokamaks, local measurements of other plasma parameters in the pedestal region couldbe utilized to investigate the pedestal state for identification of the stochastic layer. In this work, localmeasurements of T e fluctuations from the electron cyclotron emission imaging (ECEI) diagnostics [13] areused. In order to assess the degree of the stochasticity in the T e fluctuations, the rescaled Jensen Shannoncomplexity is calculated.The Jensen Shannon complexity ( C JS = QH ) is defined as the product of the normalized Shannonentropy H = S/S max , where S = S ( P ) = − (cid:80) i p i log( p i ) is the Shannon entropy of the given probabilitydistribution function P = { p i } and S max is the maximum entropy, and the Jensen Shannon divergence Q ∝ S (cid:0) P + P e (cid:1) − S ( P )2 − S ( P e )2 where P e is the uniform probability distribution function. It measures kind ofthe distance between the given probability distribution function and the uniform probability distributionfunction which would correspond to the most stochastic case. The C JS - H plane analysis based on theBandt-Pompe (BP) probability distribution function [18] is known as a useful method to quantify thestochastic or chaotic nature of the time series data [12, 19, 20]. The signal can be called stochastic if its BPprobability results in C JS close to or smaller than C JS of fractional Brownian motion (fBm) and fractionalGaussian noise (fGn) signals having the same H [12]. The BP probability represents the occurrence2robability of an amplitude ordering of finite size ( d ) consecutive values with the same time step (∆ t ) [18].For d consecutive values, there are d ! possible ways of the amplitude ordering, and the BP probability p i means how frequently the i -th order-permutation appears in a time series data with total N points, i.e. p i = the number of d consecutive values whose amplitude order is represented by the i th permutationtotal number of d consecutive values in available data N should be large enough than d ! ( N (cid:29) d !) for the reliable calculation of the BP probability [18].The BP probability calculation parameters such as d , ∆ t , and N should be carefully determined formeaningful results. In this work, d = 5 and ∆ t = 1 us (for the Langmuir probe data, see below) or∆ t = 2 us (for the ECEI data) are used so that d ∆ t can be close to the time of interest. The timeof interest in the ECEI data analysis is the transit time of pedestal structures over the ECEI channelmeasurement area where its size would be about 2 cm and the laboratory frame transit velocity is around1–10 km/s. For the Langmuir probe data analysis, the time of interest is determined by considering theshort pulses in the data [20]. N is chosen as 5000 (for ∆ t = 1 us) or 2500 (for ∆ t = 2 us) which satisfies N (cid:29) d ! = 120 and provides the 5 ms time resolution of the calculation. A fast temporal resolution of theBP probability or C JS calculation is required to study the prompt plasma response to the RMP field.For the efficient comparison of C JS to that of fBm or fGn signals, it is rescaled asˆ C = C JS − C | C bdry − C | (1)where C ( H ) is the Jensen Shannon complexity of fBm or fGn signals and C bdry ( H ) is the maximum (if C JS > C ) or minimum (if C JS < C ) Jensen Shannon complexity at the given H . The rescaled complexity( ˆ C ) ranges from -1 ( C JS = C min ) to 1 ( C JS = C max ), and the less ˆ C means the more stochastic.The analysis result of the ECEI data shows that measured T e fluctuations over the pedestal top regionbecome more stochastic in the RMP ELM suppression phase. Spatiotemporal evolution of ˆ C measurementsusing multiple ECEI channels in R, z ) space are mapped into the radial coordinate on the z line of the vertical plasma center to enhance theradial resolution of the measurement (see Fig. 2(a)). Note that some channels whose noise contributionsare exceptional and the data during the ELM crash period are excluded in the measurements. It dropssignificantly near the pedestal top ( R ELM = 215 . T e fluctuations) witha transition to the ELM suppression phase, and remains at a lower level for the entire ELM suppressionphase. The formation of the stochastic layer by the RMP field penetration would result in a complexmagnetic geometry. The T e fluctuations measured by the laboratory frame channel going through thestochastic layer are expected to be more stochastic than the fluctuations over regular pedestal geometry.Therefore, the lowered ˆ C (enhanced stochasticity) of T e fluctuations near the pedestal top suggests that thestochastic layer forms at the pedestal top and it plays an important role in the transition and maintenanceof the ELM suppression.The time averaged profile of ˆ C shown in Fig. 2(b) provides an estimate of the stochastic layer fullwidth. By the stochastic layer, we mean the region of the reduced ˆ C compared to that of the inter ELMperiods. Since each channel has different noise contribution, the relative change of ˆ C profile among differentaveraging periods should be more meaningful than the absolute value of ˆ C . The averaged ˆ C profile is nearlysame in the inter ELM periods of the H-mode w/o the RMP phase (black crosses) and the ELM mitigationw/ the RMP phase (blue circles). A significant drop of ˆ C compared to other periods is observed locallynear the pedestal top ( R = R ELM ) in the ELM suppression period w/ the RMP phase (red squares). Thefull width of the enhanced stochasticity region can be estimated as the range of the ˆ C reduction more thanthe standard deviation of ˆ C measurements, which is about 2 . ± . ∼ C (enhanced stochasticity) of the pedestal top T e fluctuations in the KSTAR experiment can beconsistent with the simulation result, but no evidence for multiple separated layers (reduced ˆ C regions) is3igure 2: (Color online) (a) The channel positions (squares) of the electron cyclotron emission imagingdiagnostics used for calculation of the rescaled Jensen Shannon complexity. They are mapped into the radialcoordinate (black circles) following the flux surfaces (red thin lines) on the black vertical line crossing theplasma center. (b) The radial profile of the time averaged rescaled complexity of electron temperaturefluctuations in different phases. The pedestal top ( R = R ELM , the location of the edge localized mode) isindicated by black dashed line. Error bars indicate the standard deviation during the time average period.observed. However, the ECE measurement in the pedestal foot region is uncertain due to the limited diag-nostics capability with the low density and temperature of that region. For the quantitative comparison ofthe stochastic layer width, a sophisticated cross validation analysis matching all the important parameterssuch as the plasma resistivity would be necessary. A relatively large uncertainty in the stochastic layerwidth measurement due to the finite channel size suggests that the growing tendency of the width withthe resistivity rather than the absolute value would be more feasible to be confirmed in the experiment,which is left for the future work.Although a reliable analysis based on local measurements of the pedestal foot region was not available,the Langmuir probe data which measures the ion saturation current ( I sat ∝ the particle flux) could be usedto investigate any change of stochasticity in the particle transport in the far edge region with the RMP field.Measurements of the particle flux around the striking point are obtained by the divertor Langmuir probewhen the striking point drifts across the probe location from three different plasmas for different phases, i.e. q
95 and RMP configurations, we shouldassume that the stochasticity of particle flux measured by the Langmuir probe can reflect the stochasticityof the particle transport in the pedestal region regardless of local pedestal parameters. Nonetheless, all thedata could be obtained by one probe in the near term so that the absolute value of the rescaled complexityfrom different plasmas can be directly compared assuming the similar noise contribution. In fact, the baselevel of ˆ C measurements in different plasmas has shown a similar value. ˆ C measurements in time as thestriking point moves across the probe location are transformed into ˆ C measurements along the distance( D ). This distance is defined as a distance between two flux surfaces containing the striking point and theprobe as illustrated in Fig. 3(a). Equilibrium flux surfaces are reconstructed using EFIT code [21] withthe magnetic data, and D measurements can involve about 1 cm absolute error especially when the RMPfield is applied. By the definition, D = 0 would correspond to the striking point, D <
D > C profiles over the distance in Fig. 3(b) show that the edge particle transport across the separatrixbecomes more stochastic with the RMP field. The ˆ C profiles from different plasmas have a similar shape,i.e. it drops fast from D (cid:28) D = 0 and increases slowly over the D > C values of all plasmas are around thesimilar noise level) to the striking point. Interestingly, the minimum of ˆ C which is expected to reside closeto the striking point is found to be significantly different in different plasma conditions. It is lower in theELM mitigation phase than in the H-mode phase, and the lowest value was found in the ELM suppression4igure 3: (Color online) (a) The illustration for the distance ( D ) measurement between the flux surfacesof the striking point (black dot) and the Langmuir probe (red dot). D can have about 1 cm absolute error.The particle flux measurements along D are obtained as the striking point moves across the Langmuirprobe with the plasma movement. (b) The radial profile of the rescaled complexity of the particle fluxaround the striking point in different phases. The same Langmuir probe is used for direct comparison ofthe profiles from different plasmas. Error bars indicate the standard deviation of the measurements.phase. This means that the particle flux becomes more and more stochastic as the plasma state changesfrom the H-mode to the ELM mitigation and from the ELM mitigation to the ELM suppression, whichcan result from the more RMP field penetrations as observed in the previous numerical simulation [10].Next, another interesting observation has been made using the pedestal top T e fluctuations in the RMPELM suppression experiment T e measurements from one ECEI channel close to the pedestal top ( R = R ELM ).The bicoherence calculations with the fast temporal resolution (milliseconds scale) could be obtained uti-lizing the Morlet wavelet transformation [22] instead of the conventional Fourier transformation, and it issummed over 10–100 kHz range to identify the nonlinear three-wave coupling among the fluctuations.A strong anti-correlation between ˆ C and the total bicoherence was observed during the RMP ELMsuppression phase. The total bicoherence at the pedestal top before the ELM suppression is around thenoise level, and it rapidly increases [23] with the reduction of ˆ C in a transition to the ELM suppressionphase as shown in Fig. 4. They are pulled back a little in about 200 ms but kept at the higher (the totalbicoherence) and lower (the rescaled complexity) level than the levels before the suppression.It should be noted that the insignificant bicoherence before the ELM suppression does not mean theabsence of the broadband fluctuations or turbulence. The turbulence almost always exist in the pedestalregion when the pedestal grows sufficiently regardless of the RMP field. The significant bicoherence meansthat the T e turbulence have strong three-wave coupling which could be a passage for the fluctuation energytransfer. The synchronized enhancement of the bicoherence with the RMP field penetration in the ELMsuppression phase and its strong anti-correlation with the rescaled complexity imply that the couplingamong the fluctuations is via a resonant island at the pedestal top. Indeed, the three-wave coupling amongthe turbulence via an island was already observed in the initial phase of the neoclassical tearing mode [24].Recently, the interaction between an island and turbulence has been extensively studied [25, 26]. Theirinteractions are known to result in coupled evolution between an island and turbulence [27, 28, 29, 30, 24].Turbulence can be modified by the island associated change of the background pressure [31, 32] andflow [33, 34, 35, 36, 37, 38] profiles [39], which in turn affects the evolution of the island [27, 28, 29, 30, 24].Understanding the formation and evolution of the stochastic layer in the pedestal region should also requireconsidering intricate turbulence effects. 5igure 4: (Color online) (a) The D alpha emission (black) and the in-vessel coil current for the resonantmagnetic perturbation (RMP) field (red). (b) The rescaled complexity and (c) the total bicoherence ofthe electron temperature fluctuation at the pedestal top (the edge localized mode location). Error barsindicate the standard deviation of the measurements. Summary
In summary, we report on the experimental observation of the enhanced stochasticity in the pedestal regionwith the RMP field in the ELM control experiment. Both the electron temperature fluctuations and theparticle flux become more stochastic as the RMP field penetrates into the edge plasma. This observationis not inconsistent with the expected formation of the stochastic layer through which the fluctuation andtransport would have the stochastic behavior. The formation and evolution of the stochastic layer caninvolve the complex dynamics such as the nonlinear interaction among small scale modes which wouldaffect the transport in the pedestal region.
Acknowledgements
This research was supported by R&D Programs of “KSTAR Experimental Collaboration and Fusion PlasmaResearch(EN2101-12)” and “High Performance Fusion Simulation R&D(EN2141-7)” through Korea Insti-tute of Fusion Energy (KFE) funded by the Government funds and by National Research Foundation ofKorea under NRF-2019M1A7A1A03088462.
Competing interests
The authors declare no competing interests.
Data availability
Raw data were generated at the KSTAR facility. Derived data are available from the corresponding authorupon request. 6 ode availability
The data analysis codes used for figures of this article are available via the GitHub repository https://github.com/minjunJchoi/fluctana [40].
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