Formation of small-scale modes via ECCD injection into KSTAR plasma core

In KSTAR experiments exhibiting sawtooth instability, the formation of multiple flux tubes (MFTs) has been frequently observed when electron cyclotron resonance heating or a current drive is applied near the inversion radius of the sawtooth. On the global scale, these MFTs evolve into a single flux tube mode or dual modes. The modes are observed as multiple Fourier harmonics in the spectrogram. A comprehensive correlation analysis of 2D imaging diagnostic data reveals a notable energy transfer within structures of varying sizes during the global mode transition. Broadband fluctuations are enhanced, and energy transfer between Fourier harmonics occurs in the presence of MFTs. Cross-power spectrum in the presence of multiple Fourier harmonics aligns with power law of inverse cascade. This suggests that energy inverse cascade process can contribute to formation of MFTs.


Introduction
Sawtooth instability is characterized by a periodic sequence involving a gradual accumulation of core heat and density followed by their rapid relaxation [1][2][3].This occurs in plasma where the central safety factor drops below unity.In particular, due to accumulation of current in the core region, precursor oscillation attributed to the m/n = 1/1 kink instability Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.[1,2,4] is often observed.The sawtooth instability can interfere with core temperature or density increase, possibly leading to degradation of plasma performance.As sawtooth are considered beneficial because it can spread out the heat and current and prevent the accumulation of impurity ash in the core region [5,6], strategies to mitigate the rapid nature of sawtooth relaxation must be developed.
Decades of physical understanding and experimental observations have shed light on sawtooth behavior [7][8][9].These insights underscore the potential of auxiliary heating systems as an advantageous tool for sawtooth control, and research on their incorporation into ITER [10] as a control actuator is ongoing.In particular, localized heating or current driven through an electron cyclotron (EC) wave affects the sawtooth dynamics by changing the magnetic shear [10][11][12][13].Pioneering investigations into the consequences of injected electron cyclotron heating or current drive (ECH/CD) have employed Porcelli's model [9] to successfully analyze the effect of electron cyclotron current drive (ECCD) on the sawtooth period.
Moreover, previous experiments conducted at KSTAR have delved into the effects of ECCD on sawtooth phenomena.Electron cyclotron emission imaging (ECEI) [14,15] has shown through 2D images that applying ECCD can produce multiple flux tubes (MFTs) [16,17].Specifically, the MFTs appear when an EC wave is injected within or near the q = 1 surface with a finite toroidal angle in the co-current direction, and their dynamics vary depending on the ECCD deposition relative to the q = 1 surface.
Using a simulation based on the resistive magnetohydrodynamic (MHD) model, Bierwage et al [18] investigated the conditions required for generating MFTs.They compared various mode dynamics or patterns that appear depending on the ECCD deposition relative to the sawtooth inversion radius, demonstrating that the development and sustainment of MFTs require flat q profile with a value approximating unity.Nam et al [19] suggested that for the growth of MFTs, the central safety factor immediately after the crash must be greater than unity, and a possible sawtooth control scheme involving slow crashes [20] was also investigated.
This study introduces the observation of an energy inverse cascade phenomenon triggered by ECCD injection near the q = 1 region, where the ECCD acts as an external energy forcing, leading to energy transfer from smaller to larger mode structures.Our study focuses on the dynamics of MFTs that commonly appear in the core of the KSTAR plasma during ECCD deposition near q = 1, a method used for sawtooth control.While the previous works [16][17][18] also investigated the dynamics of MFTs and suggested that they can be interpreted as large filamentary flux tubes, the purpose of this study is to propose that MFTs emerge from small-scale structures via the process of inverse cascade.
Additionally, this study was conducted with a correlation analysis of the ECEI data.The implementation of channel correlation analysis for ECEI measurements [21,22] has enabled research on the interaction between turbulent fluctuation and magnetic islands [23,24], flow changes, coupling of turbulent eddies due to resonant magnetic perturbation [25], and whistler-frequency emission measurement based on the estimation of the vertical wavenumber [26].Similarly, [27] investigated the change in fluctuations during sequences of sawtooth oscillations and suggested that the growth of fluctuations results from an increase in the T e gradient owing to the heat pulse stemming from the sawtooth crash.However, this study differs in the following aspects: we have measured T ECE fluctuations with ECEI channels close to the inversion radius (or q = 1 surface), where the ECCD pulse strongly affects the sawtooth dynamics and growth of fluctuations.
In section 2.1, we review a previous KSTAR experiment [20] to investigate MHD patterns appearing when ECCD is injected near the q = 1 surface.In section 2.2, we introduce the implementation of a signal correlation technique for our phenomena of interest, and the results reveal an increase in broadband fluctuation.The methodology used to implement the correlation analysis for assessing the sawtooth dynamics is similar to that used in [27]; however, we have utilized an automatic postprocessing code introduced in section 2.2 and appendix to classify distinctive features during the sawtooth sequence.Section 3 presents an analysis of the multiple modes identified in another experiment, which further suggests the development of an energy inverse cascade via a localized ECCD pulse near the q = 1 surface.Section 4 presents concluding remarks regarding the analysis and interpretation of our findings.Finally, this study and its findings are summarized in section 5.

ECCD modulation experiment
In KSTAR experiment #13504, which is dedicated to studying the modification of a sawtooth pattern with a high toroidal guide field (B T = 2.9 T) [20], a power-modulated ECCD (5 Hz) was injected close to the sawtooth inversion radius (ρ dep ≃ ρ inv , where ρ is the normalized radius).For the ECCD, a 170 GHz gyrotron was operated, and the toroidal injection angle was fixed (ϕ tor = 10 • ) for the entire discharge.Moreover, on-axis neutral-beam injection has been applied at a constant power of 1 MW to stabilize the kink mode.The plasma is sustained in the L-mode until the termination of the discharge.
The 2D distribution of the electron temperature fluctuation was measured with ECEI diagnostics installed on KSTAR [14,15] to detect millimeter waves of electron cyclotron emission (ECE).Equipped with 24 vertical arrays and 8 frequency channels per array, the ECEI system can digitize the 2D distribution of local electron temperature fluctuations (δT ECE / ⟨T ECE ⟩) at a sampling rate of 500 kHz and a high spatial resolution (1 ∼ 2 cm).A heterodyne array enables the detection of the W-band ECE signal from the plasma by downconverting the signal into a detectable range of several GHz (2-9 GHz).For this experiment, three ECEI systems were operated to detect the ordinary fundamental (O1) mode of EC emission because the frequency range of the extraordinary second (X2) mode was higher than that of the W-band under high toroidal field conditions [15].Note that two ECEI systems are located at the same toroidal position, whereas the other system is installed at a different toroidal location (∆ϕ = 22.5 • ), enabling the estimation of toroidal mode number and mode pitch angle [28][29][30].
As the current accumulates in the central region of the plasma, the increased current triggers a kink mode, commonly known as a sawtooth crash precursor [4,9].Figure 1 illustrates the evolution of δT ECE / ⟨T ECE ⟩ distribution, depicting spectral features and snapshots during the sawtooth oscillation when the ECCD is switched off (interpulse period).The sawtooth crash dynamics during the ECCD interpulse are similar to those of a typical sawtooth, involving the formation of a kinked core owing to the accumulation of heat in the central region before relaxation [31] but exhibit a slow crash with a duration of 0.5 ∼ 1 ms [20].When the ECCD is injected near the inversion radius, filamentary flux tube structures (or MFTs) characterized by a spectrum of Fourier harmonics with m/n > 2/2 can be identified, as previously reported in [17,20].The change in δT ECE / ⟨T ECE ⟩ distribution during the evolution of the sawtooth with the ECCD injection is shown in figure 2. The ECCD injection increases the sawtooth crash duration up to ∼1 ms [20].The flux tubes appear as hot regions in the ECE snapshot, capable of carrying additional heat and current as the electrons gain additional energy and momentum from the injected EC wave [16,18], thereby forming a filament structure along the resonant field line in the q = 1 region.Note that time periods without any detectable Fourier mode exists before the generation of flux tubes, as indicated in figure 2(b), which will be discussed in detail in next section.

Observation of broadband temperature fluctuation during sawtooth evolution
The dynamics of electron temperature fluctuations associated with ECCD pulse deposition were analyzed using a signal correlation technique aided by an automatic postprocessing code for ECEI data.Because correlation analysis requires a sufficient number of data points, the plasma must be maintained in a steady state for extended periods (up to ∼ 100 ms) [21].However, given the transient nature of sawtooth phenomena, data samples representing similar MHD phenomena across several sawtooth oscillations were manually classified and collected.As the categorized MHD activity varies on a timescale of less than a few milliseconds, data segments must be accurately selected with a precision of 10 µs.To ensure reliable categorization, we identified the dynamics of MHD using automatically generated ECEI snapshots.
The automatic post-processing code, which facilitates and accelerates the identification of dynamics during sawtooth oscillations, reduces white noise components with a power level below a threshold value.Owing to the inherent noise components in the ECE signal, proper postprocessing is crucial for obtaining a clear snapshot of δT ECE / ⟨T ECE ⟩ distribution.In this regard, retaining only the MHD-relevant frequencies using the bandpass filtering method serves as an effective strategy; however, this generally entails a manual load owing to the dynamic nature of the MHD phenomena [16,20].Based on the statistical characteristics of white noise in the ECE signal, the automatic post-processing code eliminates the power components below the threshold level in the frequency domain.This method is called threshold filtering [32,33]; the mathematical details regarding the determination of threshold level are reported in appendix.
To detect small-scale fluctuations in the ECE signal, the sensitivity determined from the module bandwidth parameters of ECEI electronics and number of data points are essential considerations [34,35].By suppressing the thermal and intrinsic noise using correlation analysis, two frequencydomain parameters are obtained: coherence and cross-phase.The coherence is analogous to the convolution of two signals and measures the extent to which a channel signal shares a common waveform with adjacent channels [35,36].The cross-phase evaluates the phase difference between signals from adjacent channels.In particular, given that the relation between the cross-phase and frequency is linear, the fluctuations exhibit an averaged phase velocity equal to group velocity.The phase velocity of the fluctuations can be calculated based on the cross-phase relation [21,23,25,36], as follows: v p = ∆ω/k = 2π∆x/ (∆φ/∆f ), where ∆x is the distance between the channels and ∆φ = k∆x is the cross phase.The phase relations of incoherent noise components are random, and their coherence has a value comparable to the noise level.
To resolve small-scale T ECE fluctuations, an upper limit for the detectable wavenumber(k) has been proposed [25], which is determined by the channel separation.For the analysis of experiment #13504, radial channel pairs were used as shown in figure 3(a), each with a separation of ∆x ∼ 1.95 cm.This corresponded to the upper limit of the detectable wavenumber k θ ⩽ 1.0 cm −1 .Additionally, the sampling extent of each channel determined by the IF bandwidth was greater than that of the channel separation, satisfying the decorrelation scheme for thermal noise [35,36].
To understand the behavior of small-scale fluctuations using correlation analysis, the phenomena during sawtooth oscillation were classified into several categories according to the MHD mode features as following: (1) MFTs characterized by a spectrum of high Fourier harmonics, (2) kinklike mode during the sawtooth crash, (3) periods without a dominant mode.Note that small-scale filamentary structures may be present in the category (3).Using the categorized ensembles of data, each containing distinctive features during evolution, the dynamics of the temperature fluctuation associated with the deposition of ECCD pulse were subsequently analyzed.Note that the automatic postprocessing code was used only for data classification, and a correlation analysis was conducted using the T ECE data before denoising.illustrates the result of correlation analysis with an ensemble of data containing MFTs, performed with the channels marked on the equiflux surfaces constructed with EFIT.Channel pair near the radius where ECCD is deposited(ρ ≃ ρ dep ) exhibited higher coherence (figure 3(c)) and a broader linear region in the cross-phase relation (figure 3

Appearance of multiple modes and their interaction
An additional experiment with ECCD deposition near the inversion radius demonstrated the interaction and energy exchange of the Fourier harmonics with m/n ⩾ 2/2.In experiment #32309, a 105 GHz gyrotron was used for ECCD with a modulation of 2 Hz, power of 450 kW, and a fixed toroidal angle of ϕ tor = 10 • .A smaller guide field was applied (B T = 1.8 T) to satisfy the operating conditions and ECEI detection range, and the plasma was maintained in the L-mode before the undesirable L-H transition.
As shown in figure 5, several sawtooth cycles exhibit the coexistence of multiple Fourier harmonics during the injection of ECCD pulse.Each cycle contains the following sequence: (1) phase A, T ECE ramp-up, (2) phase B, growth of multiple harmonics, and (3) phase C, transition to single or dual harmonics.Each phase is shaded with different colors; magenta, gray, and green in figure 5(b).Figures 5(c)-( (e), it can be identified that the flux tube structures evolve to a larger size during the progression of a sawtooth cycle.Figure 5(f ) is the beginning of the relaxation phase: here the flux tube structure is pushed outwards toward the mixing radius, similar to the phenomenon observed in figure 2(d).
In phases B and C, the Fourier harmonics with a lower toroidal mode number(n) survived for a longer period compared to those with a higher mode number.The toroidal mode numbers of the spectral components constituting the Fourier harmonics can be estimated by measuring the time delay between the temperature fluctuation of the ECEI and the ECE radiometer [17].This estimation is subsequently validated by measuring the distance between the periodic mode structures inside a single toroidal view of the ECEI (as shown in equation ( 3) of [29]).During phase B, the toroidal mode number (n) constituting the Fourier harmonics ranges from n = 2 to 5, and n = 3 (occasionally coexisting with n = 2) dominates after the transition to phase C. Because the mode structures corresponding to Fourier harmonics appear near the sawtooth inversion radius, the safety factor should approach unity; this observation has been validated via MSE diagnostics.Therefore, the poloidal mode number (m) is considered to be equal to the toroidal mode number (m = n).
During ECCD inter-pulse (figure 6), the Fourier harmonics are only observed during the relaxation process of the sawtooth.Contrary to the development of Fourier harmonics in the interim of core temperature ramp-up, figure 6(a) shows only occasional appearance of fundamental 1/1 Fourier mode near f mode ∼ 10 kHz.Also, MFTs that are observed in In this experiment, sawtooth cycles exhibiting the aforementioned sequence were commonly observed during ECCD pulse, accounting for approximately 30 ∼ 40 % of all sawtooth cycles.Therefore, sufficient data points to perform a signal correlation analysis can be gathered using the procedure described in section 2.2.In addition, the bicoherence given hereafter measures the degree of phase coupling between the fluctuating components, which also implies a nonlinear interaction between the fluctuating components [21,23,25,26].
Herein, F(f i ) and F * (f i ) are the components f i of the Fourier transform and its conjugate, respectively, and the bracket ⟨•⟩  denotes ensemble averaging.The LHS of the equation is the calculated bicoherence from signals f 1 and f 2 .
For the analysis, a vertical channel pair was used with separation ∆z ≃ 1.7 cm, which limited the range of detectable wavenumbers to k θ ⩽ 1.2 cm −1 .Note that the expression k θ is used in place of k z as the vertical position of the channel pair lies near the plasma midplane (z = −1.0 and − 2.7 cm, respectively).In addition, the field line pitch is minimal near the channel position (∼2 − 3 • ), and assuming that the mode or eddy structures are elongated along the field line, the contribution of k ∥ to the measured k θ is minimal.Therefore, k θ ≃ k ⊥ is a reasonable assumption for the analysis described hereafter.
Figure 7 presents the coherence spectrum and bicoherence image obtained with adjacent ECEI channels near the outermost region wherein modes develop.Each subplot in figure 7 represents the distinctive phases(A-C) shown in figure 5(b), compared with the analysis for the period when the m/n = 1/1 mode and its harmonics appeared in the sawtooth cycle .The shaded region marks the range wherein the cross-power aligns with the power law; the lower limit corresponds to the mode frequency of m/n = 3/3 mode, while the upper limit is determined from the extent to which the relation between frequency and cross-phase is linear, allowing for conversion of the frequency axis to wavenumber.without injection of ECCD (figure 6).In phases B and C, an enhancement in broadband fluctuations is observed in the coherence spectrum (figures 7(b) and (c)).In phase A, where the core heat gradually increases, a corresponding enhancement in fluctuations is not observed.
Moreover, the bicoherence image (figures 7(e)-(h)) reveals that the interaction of fluctuations varies depending on the prescribed phases and the application of ECCD.In particular, figure 7(f ) demonstrates that the interaction of high Fourier harmonics in the frequency range of 30-100 kHz with those at lower frequencies (30-60 kHz) is dominant during phase B. It can be inferred that frequency components with a summed frequency range of f 1 + f 2 = 25 − 55 kHz (shaded region in figure 7(f )) exhibit significant coherence levels.This frequency range is where Fourier harmonics are observed in figure 5(b).Furthermore, in phase C (figure 7(g)), the interaction of the Fourier harmonics over 50 kHz is diminished, and the frequency range of the interaction is downshifted to f 1 + f 2 = 8 − 45 kHz, as indicated by the shaded region.In contrast, during Phase A (figure 7(e)), significant interactions between the fluctuations could not be identified.
During the sawtooth cycles in the ECCD inter-pulse, only the interacting components close to f 1 = 20 kHz and f 2 = −10 kHz are identified.Figures 7(d) and (h) present the results of correlation and bicoherence analysis, with the data containing MHD modes during ECCD inter-pulse period.The peaks in figure 7(d) indicate m/n = 1/1-mode with f mode ∼ 10 kHz, which is typically known as the crash precursor oscillation of the sawtooth and its harmonics.
An energy inverse cascade [37,38] is observed in the cross-channel spectrum obtained from phase B, representing a sequence with multiple Fourier harmonics when the ECCD pulse is switched on.Figure 8(a) presents the crossphase relation between neighboring channels.The relationship between the frequency and cross-phase is linear up to 75 kHz.Moreover, with the distance between the channels (∆z = 1.7 cm) utilized for cross-phase analysis, the characteristic wavenumber of the mode can be calculated as: k = θ (f ) /∆x.Thus, in the range wherein the relation between the frequency and cross-phase is linear, the frequency axis can be converted into a wavenumber as shown in figure 8(b).This figure presents the square root of the cross-power S(k ⊥ ) 1/2 versus wavenumber and frequency.Note that the cross-power is presented in this figure prior to being normalized to coherence.The cross-power (S(k ⊥ )) has a dimension of the cross product of two signals per unit frequency.Also, in the core of KSTAR plasma, the optical thickness (τ ) is τ ≫ 1, so it is assumed that the effect of density fluctuation on ECE intensity is much smaller than temperature fluctuation.Therefore, assuming that ECEI signal level is proportional to the electron temperature fluctuation, S(k ⊥ ) is analogous to the square of the thermal energy stored per unit wavenumber.In phase B (blue dashed line in figure 8(b)), the square root of the crosspower aligns with the red dotted line, representing the powerlaw relation of the energy inverse cascade in anisotropic, weak MHD turbulence with a strong guide field: E ∼ k −5/3 ⊥ [39].The enhanced power value near f ∼ 30 kHz in the cross-power spectrum of phase C (black dotted line in figure 8(b)) indicates that the energy is more concentrated in the lower wavenumber components compared to phase B. Note that the peaks near k ⊥ ∼ 0.9 cm −1 , both observed in phases B and C, are instrumental noise.
The energy injection wavenumber scale (k in ) is estimated to be bigger than the detectable range of ECEI, k θ ⩽ 1.2 cm −1 < k in , for the size of the filamentary flux tube for seed perturbation generated by the injection of ECCD would be smaller than the beam size (w) determined from the FWHM of the EC absorption profile calculated using TORAY: The measurement of the mode number, discussed earlier in this section, further validates the energy transfer from a smaller to a larger scale.After the transition from phase B to C, the high Fourier harmonics with m/n > 3/3 disappears, but m/n = 3/3 component with a characteristic frequency of f mode ∼ 30 kHz sustains until the crash.The shaded region in figure 8(b) indicates the range in which the cross-power aligns with the power law.In particular, the lower limit of the shaded region is the frequency of the m/n = 3/3 mode, and the upper limit is the extent to which the conversion from frequency to wavenumber is valid.Therefore, based on the comparison of the power law and estimation of the mode number, it can be concluded that there is an inverse cascade of energy from a smaller scale with a higher mode number (m/n > 3/3) to a larger scale with m/n = 3/3 and 2/2.

Discussion and remarks
The analysis of figures 3 and 4 indicates that the localized ECCD deposited on the q = 1 surface was responsible for the increase in the electron temperature fluctuation.In addition, in a recent experiment, the distinct presence of modes with m/n ⩾ 3/3 were identified and their interactions were investigated through bicoherence analysis.
The findings in section 3 can be understood using the approach used to simulate the generation of MFTs [18].When the ECCD is deposited near the rational q surface, numerous electrons passing through the resonance point return to the poloidal spot after toroidal rotation(s), thereby carrying the additional energy and momentum gained from the injected EC wave [40].In the simulation of [18], due to the unavailability of a self-consistent model that includes kinetic effect of ECH and resulting distributions of pressure and current, the current source is modeled as a repeating pulse.This source model is a realization of the assumption that after a flux tube has grown to a certain size, the source will disconnect from that flux tube as the plasma continues to rotate, and produce another flux tube as the source latches onto a new seed.This cycle repeats until the first flux tube returns to the ECCD deposition and gets reinforced.Based on this mechanism, the simulation predicted that the few hot spots in figure 2(c) will grow and merge to yield a mode with a larger scale, especially when q approaches unity.Therefore, this approach can explain the generation of multiple Fourier harmonics and the transition to single/dual harmonics in experiment #32309 by merging small-scale filaments into larger flux tubes.The observations and analysis presented in section 3 provide further insight on the formation of flux tubes in the view of energy inverse cascade.
Moreover, previous studies have reported the merging of modes or flux tubes [16] and the formation of filaments [41].Our observations further suggest that the interaction of flux tubes at smaller scales contributes to the electromagnetic perturbation on the rational q surface.As discussed in [18], the appearance of small-scale modes and their merging can be beneficial for the sustainment of plasma by avoiding largeamplitude sawtooth crashes owing to the enhancement of core transport [36,42].Although the interaction of turbulence emerging as a mode has been partly demonstrated using an inverse cascade, the analysis to demonstrate the transport by turbulent fluctuations is beyond the scope of this study.Further investigations into the direct effects of heating on turbulence generation [43,44] may help us understand this phenomenon.
By comparing the spectrograms representing each experiment (figures 2(a) and 5(a)), we recognize that the dynamic characteristics of the spectrum were not entirely identical.The main difference manifested in the dynamics before the appearance of MFTs.In #13504 (figure 2(a)), a dominant mode is not detected before the growth of MFTs, whereas in #32309 (figure 5(a)), the interaction of high Fourier harmonics results in the growth of either m/n = 3/3 and 2/2, or both, as described in section 3.As described earlier in this section, the simulation in [18] assumed that the sole action of ECCD generates large flux tubes due to a cyclic saturation and detachment process.This was replicated by applying a repeated source pulse despite the continuous injection of ECCD in the experiment.While the simulation's hypothesis of saturation and detachment during plasma rotation appears plausible, it is not the sole possibility.An alternative scenario is that continuous ECCD injection strengthens numerous small filamentary flux tubes in the region where q is close to unity, along the annulus that contains the resonance position of the EC wave.As these filaments rotate at the same speed as the plasma, they can be identified with ECEI through enhanced broadband coherence and uniform crossphase, as presented in section 2.2.Subsequently, these small flux tubes may merge into larger ones through a process reminiscent of an inverse cascade, potentially involving long-range interactions.The findings presented in this paper support this perspective.
A slow crash is another feature identified in the sawtooth cycles of interest, as analyzed in section 3.As previously discussed in [20], the time required for current dissipation to exceed the ECCD modulation period can explain the appearance of a slow crash during the interpulse period.Moreover, in another experiment with similar plasma conditions but without the injection of ECCD, the interaction between the fluctuating components shown in figure 6(h) is not observed.Therefore, the ECCD may have affected the interaction between the fundamental mode and harmonics in the sawtooth cycles during ECCD interpulse by modifying the current profile near q = 1 surface.
According to Kolmogorov's energy cascade theory, in a 3D isotropic and homogeneous fluid, energy is transferred from a larger scale to a smaller scale.However, the fluctuations parallel to the field are suppressed in the presence of a strong external magnetic field.Therefore, the turbulent fluctuations perpendicular to the magnetic field (k ⊥ < k in ) exhibit an inverse energy cascade from larger to smaller wavenumbers [38,45] in the wavenumber range below the energy-injection scale.
The linear relationship between frequency and cross-phase (below 75 kHz) has been utilized to match the frequency with wavenumber to demonstrate that the cross power spectrum aligns with the power law of energy inverse cascade in figure 8(b).But, the linear relationship becomes ambiguous above 50 kHz, meaning that the spectrum can deviate from the power law in the actual wavenumber space.There are several possibilities that can cause such deviation.First, the ideal inverse cascade occurs in the form of self-organization of incoherent turbulence, while the merging of filamentary flux tubes is an interaction between coherent Fourier harmonics.Also, the bicoherence shows the interaction between components that are distant (or non-local) in frequency space, but the inverse cascade accounts for the local interactions.These two kinds of interaction can coexist, causing the spectrum to deviate from the power law.In the present study, because of the limitations of our diagnostics, we could not compare the power law at larger wavenumbers above the energy-injection wavenumber scale, where the spectrum is expected to follow a different power law corresponding to a direct cascade of enstrophy.We expect that measurements with a higher sampling rate [26] in the future will help resolve smaller turbulent modes beyond the energy injection scale and answer the above question.

Summary
Based on the observation of MFTs in the KSTAR experiment with sawtooth control via ECCD, we identified an increase in electron temperature fluctuations before the appearance of MFTs.In our additional experiment to investigate the dynamics of MFTs, we observed the generation of multiple Fourier harmonics with m/n ⩾ 3/3 and the transition to structures with smaller mode numbers before the crash.Moreover, we classified the phenomenological features within the sawtooth period involving multiple and compared crosscoherence and bicoherence.These results suggested that the high harmonics interacted to form structures with larger scales or smaller characteristic frequencies.A comparison with the power law and an analysis of the mode number supported the possibility of energy transfer from a smaller scale to a larger scale.Therefore, the fluctuations exhibit an energy inverse cascade, which is predicted to occur in MHD turbulence with a strong guide field.
Furthermore, based on the occurrence of slow crashes following the merging of multiple harmonics, we suggest that the injection of ECCD near a rational q-surface can be beneficial for the operation of future fusion machines to prevent rapid core collapse.Moreover, we used the signal correlation technique to extract the fluctuating components and facilitate analysis using an automatic post-processing code.Although post-processed ECEI data are yet to be routinely available, we expect this to enable physical studies based on the detection of phenomenological features.before and after removal, as the MHD should be retained.By examining up to ∼100 discharges, we determined α for white noise elimination to be 0.95.Because the noise characteristics and response amplitude differ by channel, the threshold value (θ) for each channel is calculated, and the components below θ are eliminated automatically.Moreover, instrumental noise components from nonplasma sources (i.e.internal circuits of the ECEI system and plasma heating/control units) typically have higher spectral power than white noise.These components can also be eliminated by treating the threshold as the higher limit for the signal components that should be retained.The removal of instrumental and white noise is performed in two steps: (1) elimination of system noise with a ratio α sys = 0.9999), followed by (2) elimination of white noise with α sig = 0.95.We apply a weighting function to the signal components comparable to the threshold value for both steps, as shown in figure A1.
Subfigures (b)-(d) of figure A1 illustrate the noise elimination process for ECEI data representing various MHD condi-tions of KSTAR plasma experiments.The filtering method is adaptive to various test signals representing different plasma conditions, as summarized in table A1.The three examples are temporally separated; therefore, the average white noise level and system noise characteristics differ.Figure A2 is an example comparing ECEI single channel spectrogram and 2D snapshots before and after the noise elimination.This figure demonstrates that the threshold-filtering method is advantageous compared to the conventional bandpass filtering method for removing dynamic signal components.To maximize the advantage of the threshold filtering method which does not require manual selection of filtering parameters, we have developed an automated post-processing code for ECEI data.The code was written in a MATLAB script to utilize the proficiency of matrix calculation, with a Python wrapper for implementing a semi-automatic process.Through parallel computing, the code could generate post-processed ECEI snapshots with a virtual frame rate of 100 Hz and spectrograms for selected channels within 5 min per experiment(discharge).

Figure 1 .
Figure 1.Evolution of electron temperature fluctuation (δT ECE / ⟨T ECE ⟩, where δT ECE = T ECE − ⟨T ECE ⟩) measured with ECEI during ECCD inter-pulse in KSTAR experiment #13 504.(a) The top panel indicates electron temperature fluctuation inside (gray) and near (pink) the inversion radius (offsets are added to distinguish the time traces).The bottom panel shows the spectrogram measured near the inversion radius.The cyan dotted line indicates the time at which snapshots (b)-(d) are plotted.(b)-(d) are snapshots showing the distribution of δT ECE / ⟨T ECE ⟩ (b) before the crash, (c) during the crash, and (d) after the crash.Gray and pink rectangles in snapshots (b)-(d) denote the channel position, whose signal is shown in the top panel of (a).The white dotted line in snapshots specifies the sawtooth inversion radius.The arrow in (c) indicates the rotation direction of the m/n = 1/1 kink, and that in (d) indicates the relaxation of core heat during the sawtooth crash.

Figure 2 .
Figure 2. Evolution of electron temperature fluctuation measured with ECEI when ECCD pulse is applied in KSTAR experiment #13 504.The top and bottom panels for (a) represent the same quantity as in figure 1. (b)-(d) are the snapshots showing the distribution of δT ECE / ⟨T ECE ⟩, (b) in the period where dominant Fourier mode is undetected, (c) in the presence of MFTs (here multiple Fourier harmonics are observed in spectrogram), and (d) during the final stage of the crash.Note that the two large hot spots indicated by yellow dotted circles in (c) are interpreted as large filamentary flux tubes.These hot spots merge to become the single hot spot, also indicated by the yellow dotted circle in (d).As the crash proceeds, the single hot spot annihilates at the inversion radius.

Figure 3 .
Figure 3. (a) ECCD deposition (orange), ECEI channels used for correlation analysis marked as colored squares, q = 1 surface (dashed violet), and region affected by ECCD (yellow).The dashed, thin black lines indicate equiflux surfaces calculated via EFIT.(b)-(g) present examples of correlation analysis, where the plot color corresponds to the colors of the squares marked in (a); blue denotes channels outside the ECCD deposition, red denotes channels nearer to the deposition, and black denotes channels within the deposition.(b)-(d) illustrate examples of coherence, and (e)-(g) depict cross-phase plots of the signals from adjacent channels.Note that time window around 5 − 10 ms is used for the calculation of coherence and cross-phase presented in this figure and hereafter. Figure

Figure 4 .
Figure 4. (a) Heating power(top) and summed coherence(bottom).The black, red, and blue lines in the bottom panel indicate the summed coherence calculated with channel pairs inside, proximate to, and outside the inversion radius, respectively.(b) and (c) are 2D plots illustrating the radial(ρ) dependency of the coherence spectrum(γxy( f, ρ)), (b) during the ECCD pulse and (c) ECCD inter-pulse, respectively.The shaded region in (b) marks the radial range within which the ECCD pulse is deposited.
(f )) compared to those farther away.The results in figure 3 were obtained using an ensemble of data containing MFTs (figure 2(c)).Narrowband filters have been applied before the calculation of coherence in figures 3(b)-(d), to eliminate peaks corresponding to the Fourier harmonics constituting MFTs.Note that a similar spectrum can be obtained with ensembles containing periods without a dominant MHD mode.In other words, broadband fluctuations are observed to increase before the MFTs become clearly visible in figure 2(c), suggesting that the small-scale structures in figure 2(b) evolve to become large-scale MHD modes.A clear relation exists between the injection of ECCD and the increase in the broadband fluctuation of T ECE .The time trace of the summed coherence plotted against the heating power is plotted in figure4(a).Here the summed coherence is the coherence above noise integrated by the frequency range of interest, which in this case is set to 5 − 70 kHz.The summed coherence of the channel pair close to the inversion radius shows a higher correlation compared to those outside or inside the radius.Figures4(b) and (c) illustrates the radial dependency of the coherence spectrum obtained with channel pairs at ρ ⩽ 0.3, each representing the coherence during ECCD pulse and ECCD inter-pulse, respectively.It must be noted that figures 4(b) and (c) have been obtained with an ensemble of data containing MFT phase (figure 2(c)) and have been interpolated for both axes for clarity.
Figure 4(b) clearly indicates that during the ECCD pulse, an increase in broadband fluctuations is apparent when compared with the narrow-band spectrum in figure 4(c) obtained during ECCD inter-pulse.The following conclusion can be drawn from the observations in figures 3 and 4: (1) during the injection of ECCD pulse, broadband fluctuations distinct from discrete narrow-band modes can be identified from the cross-channel coherence spectrum, and (2) In-phase broadband fluctuations increase before the development of clearly visible MFTs.
e) are the ECEI snapshots showing the evolution of flux tubes corresponding to the multiple Fourier harmonics.As indicated by red dotted lines in figure 5(b), subplots figures 5(c)-(e) show the snapshot at the beginning of phase B, end of phase B, and end of phase C respectively.Comparing the snapshots in figures 5(c)-

Figure 5 .
Figure 5. (a) Several sawtooth cycles are observed during the injection of ECCD pulse in experiment #32309.The top panel indicates the time trace of temperature fluctuation, measured via ECEI channels at the midplane.Four plots at the top panel represent ECEI channels 8, 7, 5, and 3, respectively, indicating the temperature change from the inner (ch.8) to the outer region (ch.3).The bottom panel presents the spectrogram of channel 7. The normalized radius of each channel is: ρ ≃ 0.06, 0.08, 0.14 and 0.20 respectively.(b) Magnified view of (a), with shaded regions indicating distinctive phases (A)-(C) during the evolution of sawtooth.(c)-(f ) are the ECEI snapshots, taken at the time marked as the red dotted lines in (b).(c) and (d) are snapshots in the early and final stage of phase B, (e) is the snapshot in the final stage of phase C, and (f ) is the snapshot in the interim of temperature relaxation (sawtooth crash).The orange solid line in (c) is the ECCD deposition calculated using TORAY.The white dotted circle in the snapshots (c)-(f ) marks the sawtooth inversion radius confirmed with the signals of the ECE radiometer.Crossmarks in the snapshots (c)-(f ) indicate the channels whose time-trace signals are shown in (a): channels 8, 7, 5, and 3, arranged radially outwards.

figures 5 (
figures 5(d)-(f ) are not identified during ECCD inter-pulse (figures 6(b)-(c)).In this experiment, sawtooth cycles exhibiting the aforementioned sequence were commonly observed during ECCD pulse, accounting for approximately 30 ∼ 40 % of all sawtooth cycles.Therefore, sufficient data points to perform a signal correlation analysis can be gathered using the procedure described in section 2.2.In addition, the bicoherence given hereafter measures the degree of phase coupling between the fluctuating components, which also implies a nonlinear interaction between the fluctuating components[21,23,25,26].

Figure 6 .
Figure 6.(a) Top panel is the ECEI signal, and the bottom is the spectrogram during ECCD inter-pulse.(b)-(d) are ECEI snapshots taken at the time marked as red dotted lines in (a).(b) and (c) are snapshots during the ramp-up of the ECEI signal, and (d) is the snapshot in the interim of temperature relaxation (sawtooth crash).The crossmark in snapshots (b)-(d) indicates the channel position of the signal shown in (a).

Figure 7 .
Figure 7. Result of bi-channel analysis with vertical channel pair near midplane (z = −1.0 and − 2.7 cm) and close to inversion radius(ρ ≃ 0.11 and ρ inv ≃ 0.12 ± 0.02), obtained via experiment #32 309.(a)-(d) compare the coherence spectrum, and (e)-(h) compare the results of bicoherence analysis.The same pair of ECEI channels are used for the analysis presented throughout this figure.The orange plots ((a)-(c) and (e)-(g)) represent different phases of sawtooth evolution described in figure 5 when the ECCD pulse is triggered.The dark blue plots, (d) and (h), represent the analyses of modes appearing during the sawtooth evolution in ECCD inter-pulse, described in figure 6.Note that the gray horizontal line in (a)-(d) marks the minimum sensitivity level of ECEI.The violet-shaded area in (f ) and (g) indicates the range of constant f 1 + f 2 , where interactions among fluctuations are most active, f 1 + f 2 = 25 − 55 kHz for (f ) and f 1 + f 2 = 8 − 45 kHz for (g).

Figure 8 .
Figure 8.(a) Cross-phase between channels used in the analysis presented in figure 7, measured with data segments representing phase B during ECCD pulse.(b) Square root of the cross-power spectrum (S(k ⊥ ) 1/2 ) measured with same channel pair; the blue and black lines represent measurements during phases B and C, respectively.Both axes are plotted on a logarithmic scale for comparison with the power law of inverse cascade (red dotted line); E(k ⊥ ) ∼ k −5/3 ⊥

Figure A2 .
Figure A2.Comparison of ECEI single channel spectrogram(top) and 2D snapshots(bottom) at the time points indicated by the dashed black lines on the spectrogram, produced with (a) raw data and (b) post-processed data.As a result of threshold filtering, the background white noise is reduced up to ∼35 dB.