Real time detection of multiple stable MHD eigenmode growth rates towards kink/tearing modes avoidance in DIII-D tokamak plasmas

Real time detection of time evolving growth rates of multiple stable magnetohydrodynamic (MHD) eigenmodes has been achieved in DIII-D tokamak experiments via multi-mode three-dimensional (3D) active MHD spectroscopy. The measured evolution of the multi-modes’ growth rates is in good accordance with the variation of the plasma β N . Using experimental equilibria, resistive MARS-F simulations found the two least stable modes to have comparable growth rates to those experimentally measured. Real time and offline calculations of the modes’ growth rates show comparable results and indicate that cleaner system input and output signals will improve the accuracy of the real time stability detection. Moreover, the shortest real time updating time window of multi-mode eigenvalues can be about 2 ms in DIII-D experiments. This real time monitoring of stable, macroscopic kink and tearing modes thus provides an effective tool for avoidance of the most common causes of tokamak disruption.


Introduction
Steady-state operation for a long period of time is a crucial working foundation for advanced tokamak devices, such as 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.
ITER and future power plants, to sustainably output energy [1].During steady tokamak discharges, nonuniformity of plasmas means the storage of large amounts of free energy.The release of free energy can drive deleterious magnetohydrodynamic (MHD) instabilities [2], which can be terribly harmful to the steady-state operation of advanced tokamak devices.Global MHD instabilities, especially tearing modes (TMs) [3][4][5][6][7][8] often induce major disruptions of plasmas, which will lead to great damage of future large-scale tokamak devices and thus large economic loss.Although various control techniques, such as electron cyclotron current drive [9,10], have proven to be feasible for stabilizing unstable MHD modes, it is still necessary to develop tools oriented towards real time monitoring of stable MHD modes and predicting when they might become unstable.
In general, unlike unstable MHD modes, stable MHD modes cannot be easily detected via different kinds of diagnostics.Fortunately, the sensitivity of tokamak plasmas to an externally applied nonaxisymmetric magnetic perturbations [11,12] offers an opportunity to detect stable MHD modes through active detection.By applying nonaxisymmetric magnetic perturbations with given waveform via external threedimensional (3D) coils during experimental discharges, the corresponding plasma response can be measured through the multiple arrays of 3D magnetic sensors.Accordingly, the stability information of multiple MHD modes can be extracted by means of response analysis.Due to the fact that the intensity of magnetic perturbation (δB) is usually several orders lower than the equilibrium magnetic field (B), corresponding responses can be described via linear MHD equations [13][14][15][16][17].This type of active detection is so-called active MHD spectroscopy [18].At first, the single mode MHD spectroscopy has been developed to detect the stable resistive wall mode, identifying the mode growth rate and frequency [18].Similar to previous works [19][20][21][22][23][24], the single dominant mode is assumed in the MHD spectroscopy.However, theoretical studies [25][26][27][28][29][30] point out that multiple considerable stable eigenmodes can exist in the plasma responses at the same time, which has been observed in both EAST [31] and DIII-D experiments [32][33][34][35] recently.Motivated by the experimental observations, Wang et al developed a multi-mode 3D active MHD spectroscopy (M3DS) in the form of the multi-pole transfer function, with the intention to improve the reliability of plasma-state monitoring [36].The M3DS has been successfully applied in experiments to extract the eigenvalues of multiple MHD modes directly from stable DIII-D and EAST plasmas.However, the low efficiency of frequency analysis largely impedes the real time application of M3DS.A new technique is needed for real time application.
In this paper, the M3DS has been upgraded to work directly in the time domain [37], leveraging experience from previous control efforts [38,39].In this new method, the complicated frequency analysis is not necessary and the linear least square fitting guarantees the a high efficiency of the calculation.Furthermore, no requirement of initial guess leads to better numerical convergence than the frequency domain method.
The previous work [37] focused on verifying the feasibility of the method.The equilibria were kept stable with little change and the eigenmode extractions were all preformed via post-analysis.In this work, the M3DS applied in the DIII-D experiments to detect the growth rates of multiple stable modes simultaneously and in real time.The quick refreshing of the extracted eigenvalues, as short as about 2 ms, allows tracking a changing stability so that the temporal evolution of changing eigenvalues are obtained in real time.This allows the plasma control system to monitor and predict the low frequency MHD instabilities including kink mode and TM.Other MHD modes are actively excited by external antennas, such as Alfven eigenmodes in JET [40].The equilibrium plasma β N has been changed in several ways during the experimental discharges to vary the plasma stability, and the extracted temporal evolution trend of the eigenvalues in accordance with the corresponding equilibrium stability variation in all cases.
The rest of this paper is organized as follows.The model of this new method is presented in section 2. The experimental application and validation are reported and analyzed in section 3. The conclusion is drawn in section 4.

Model of real time multi-mode 3D MHD spectroscopy
Real time M3DS is established via the combination of the ideas from the frequency domain version of M3DS and subspace system identification (SSI) theory.Frequency domain version of M3DS serves as a foundation of the active detection via the analysis of 3D plasma responses described using linear MHD theory.SSI augments the part of data processing through the great enhancement of the numerical efficiency and convergence, as well as the noise-proof feature.Linear plasma response can be modeled by linear MHD equations including externally applied low level of magnetic perturbations.After certain reductions, the response model can be expressed as the following linear state-space model (1), which can be easily and efficiently solved using SSI by fitting this model through sequenced matrix operations.Accordingly, multiple eigenmodes can be extracted directly from the corresponding experimental signals.
Equation (1a) describes the coupling between the external current field perturbations and system eigenmodes, and equation (1b) describes the process of measuring the magnetic response to the external perturbations.Here, x k is system state vector.δB k , serving as the system output, is referred to as the total magnetic response of the whole system contributed by plasma, vacuum vessel and other peripheral structures.During the experimental discharges, δB k is measured using toroidal arrays of magnetic sensors located at multiple poloidal locations.The vector δJ k represents the system input that is the externally applied 3D current field perturbations.Matrices A, B, C, and D are system matrices required to be determined via SSI.More details can be found in [37,41].
In this work, the input and output data (externally applied current fields and the measurements of system magnetic responses, respectively) can be obtained via performing a group of purpose-designed system identification experiments.Instead of fitting multi-mode transfer function as in the frequency domain version [36], the time domain version estimates the sequence of system state x k through a series of matrix operations on the basis of input and output data.After the matrix operations, the model can be transformed into a simplified form so that one can obtain the system matrices A, B, C, and D via linear least square fitting.Eventually, the eigenvalues of the eigenmodes, serving as the quantified stability indices, can be obtained through the eigenvalue decomposition of the matrix A. Here, a finite amount of x k are evaluated for obtaining matrix A, which evolves on a longer time scale than the k step.The signals in a period of time adopted for the first extraction of the eigenvalues is defined as the sliding fitting window ∆t.After that, the fitting signals will only be partly updated with cutting out the beginning t u time data and adding up the following t u time data.Thus, the actual updating time window is defined as t u , which is much less than the sliding fitting window ∆t.

Experimental setups
In order to validate the feasibility of real time M3DS, the corresponding algorithm has been integrated into the DIII-D plasma control system.A series of designed system identification experiments were carried out in this work.Both the upper and the lower rows of internal 3D coils [42] are employed in the experiments to generate n = 1 3D fields, which makes the coil current δJ k , a vector with a length of 2. It is noted that the two elements of δJ k are complex-valued representations of the n = 1 component of the upper and lower coil currents.The waveform of the current was chosen as square waves because it is the best for the time domain analysis.The time intervals between each wave crest were set as periodic.It is noted that, in the previous work [37], random time intervals were adopted.Offline analysis indicated that there was no obvious difference between random and periodic time intervals.As shown in figure 1, a time segment of signals of discharge 178583 was chosen to do the test.The offline M3DS extraction was performed using signals in the vicinity of the turning point at about t = 1.94 s.The growth rates of multiple modes obtained using signals within t 1 − t 4 , t 1 − t 5 , t 1 − t 6 , t 2 − t 6 , t 3 − t 6 time segments are shown in figure 1(d).It is found that prolonging flattop of the square wave will not evidently influence the extracted multi-modes' growth rates.The effective response information is mainly included in the vicinity of the turning point.In this work, we use a shorter fitting window (50 ms) to do the real time detection.In order to include more response information for each time of fitting, time intervals between each crest is shortened.More detailed differences of the M3DS extraction between random and period time intervals will be further tested in future experiments.To get more information in the system response measurement δB k , the phases of current fields between the upper coil and the lower coil are changed with time to excite more variation.Here, the coil phasing ∆ϕ = ϕ up − ϕ low is the difference between the upper coil current phase, ϕ up , and the lower coil current phase, ϕ low .In this model, the magnetic response corresponds to the total (plasma and vacuum) magnetic perturbations measured by the 3D magnetic sensor arrays located at both the low field side (LFS) and high field side (HFS).Generally, including more than one array of signals into the analysis will improve the accuracy and robustness of the extraction.Figure 2 shows the corresponding schematic diagram of DIII-D cross section including the positions of employed sensor arrays and current coils.Moreover, more information about the magnetic sensor configuration in DIII-D can be found in [43,44].In this work, the plasma shape and q 95 were kept constant while scanning the neutral beam injected power to adjust the plasma pressure.Two different ways to vary β N have been used.One consists in increasing β N by steps, the other is increasing β N continuously.The results obtained via offline analysis and real time detection will be shown in the following section.

Real time detection results
Discharge 186292 exemplifies the real time M3DS behavior during steps in the β N .Plasma current (I p ), safety factor at the edge (q 95 ) and β N time traces are displayed in figure 3(a).and (c) display the temporal evolution of externally applied upper current field and the phasing between upper and lower row of I-coils respectively.Figure 3(d) shows the amplitude of the measured n = 1 magnetic field in the low LFS.In the discharge 186292, the plasma is stable in the beginning.The n = 1 external perturbed fields are applied and the system response is continuously detected.Then, the plasma β N is stepped up to a higher value in a short time and kept steady for about 0.5 s.The step-up operation is repeated for several times until the plasma becomes unstable.Time evolution of multiple eigenmodes' growth rates via real time detection of discharge 186292 are shown in figure 4. In the DIII-D plasma control system, amplitude and phase of the n = 1 component of the signals from the magnetic sensors are extracted in real time.The sliding fitting time window in this case is set as ∆t = 50 ms, and the actual updating time window is t u = 2 ms.A 50ms averaging window means 1/50ms = 20 Hz, so the Nyquist frequency is 10 Hz.Finally, the time evolution of eigenvalues of multiple modes are obtained via M3DS.Note that the real part of the eigenvalue Re(γ) is identified as the growth rate of mode.The negative level of the growth rate indicates the mode is stable.The imaginary part of eigenvalue represents the natural mode rotating frequency.As shown in 4(a) the plasma β N increases in a clear staircase pattern.Every time the β N is stepped up, the growth rate of the least stable mode does not instantly have a large leap as expected.However, the overall trend of the least stable mode's growth rate is approaching marginal stability, in accordance with the variation of the plasma β N .The growth rate is increasing at the first (at t = 3.05 s) and the third (at t = 4.05 s) stepups of the plasma β N .At the second step-up (at t = 3.55 s), the growth rate suddenly drops instead.Clearer variation of the  least stable modes' growth rate can be seen in the following offline analysis, in which the system input and output signals are clearer after being optimized in offline.The growth rate of the mode exceeding zero means that the mode is becoming unstable.The growth rate of the least stable mode is beyond zero when the plasma disrupts, which is in agreement with the amplitude of the fast rotating (a few kHz) unstable n = 1 mode diagnostic signals (n1rms) in figure 4(a).Moreover, the M3DS is capable of extracting the growth rate of the secondary mode.As shown in figure 4(c), the secondary mode stays well below the marginal stability boundary throughout the discharge.It is noted that large fluctuations appear in the real time calculated evolution of growth rates and there are several times when the growth rate of the least stable mode is beyond zero before the plasma disrupts.The quality of the extracted curves can be improved using cleaner signals, which can be proved via offline analysis in the following section.
Figure 5 shows two cases where the β N is varied continuously and the M3DS is applied to track the plasma stability.In the discharge 186286, the β N is increased continuously until the plasma disrupts.Before the plasmas crash, the MHD becomes unstable, as indicated by the n1rms signal.The evolution of the least stable mode's growth rate follows an increasing trend approaching marginal stability.The secondary mode is stable all the time.In the discharge 186288, the β N first continuously increases and then continuously decreases without a plasma crash.The n1rms signal indicates the MHD modes are always stable.The evolution of the least stable mode's growth rate follows the change of β N to some extent.Considering the β N shows more fluctuations than that in the discharge 186286, the secondary mode behaves like continuously switching between two states.

Offline post-analysis
To provide a clear trend of the eigenvalue time evolution, the offline analysis is performed here.The method of offline analysis is similar to the work in [37].Amplitude and phase of the n = 1 component of the signals from the magnetic sensors are extracted offline to lower the potential uncertainty as far as possible.Then, according to the real time detection experiments, the sliding fitting time window is set as ∆t = 50 ms, and the actual updating time window is set as t u = 2 ms.Finally, the temporal evolution of eigenvalues of multiple modes can be obtained via M3DS post-analysis. Figure 6(a) shows a comparison of the growth rate evolution of the least stable mode of shot 186292 computed offline and in real time.The blue curve represents the result of real time M3DS detection using real time signals, while the red curve represents the offline calculated growth rate.Although a similar trend is observed, especially when the mode is closer to marginal stability, larger fluctuations are shown for the real time calculated Re(γ).The two main differences between the real time and the offline  calculations are the code itself and the amplitude and phase of the n = 1 plasma response.The yellow curve shows the growth rate calculated using the real time n = 1 magnetic response calculation and the offline code.The results are in good agreement with the real time calculated Re(γ), showing that the real time version of the code can reproduce the offline results.This also suggests that cleaner n = 1 system input and output signals are necessary for improved accuracy.After a simple smoothing operation, the tendency of the growth rate evolution can be clearer as shown in 6(b).It is noted that, in real time smoothing, we can only utilize the data before the time of interest.Although only simple smoothing operation has been made here, it is expected that smoothing operation can be favorable for future control purposes.The errors of the eigenvalues are propagated from the uncertainties of offline fitting of n = 1 magnetic response signals [44,45].Here, best fitting, error upper limit and error lower limit of magnetic response signals are adopted for each fitting time window.The standard deviation of the three eigenvalues is used for error bar plotting.It is found that the error is relatively large while the mode is far below the marginal stability.Approaching marginal stability, the error could be neglected.Once a clear trend is obtained, an extrapolation can be made to predict the unstable moment on some level, and corresponding control techniques could be turned on in advance to prevent the plasma crash.Offline analysis results of shots 186292, 186286, and 186288 are shown in figure 7. The behaviors of the growth rates' evolution are all clearer than that of the real time detection.It is found in 7(a) that, for the case with staircase pattern plasma β N evolution, every time the β N is stepped up, the growth rate of the least stable mode also has a large leap towards the marginal stability.Just before the moment that the plasma disrupts, the growth rate of the least stable mode extracted via M3DS is almost equal to zero, meaning the least stable mode is becoming unstable, which is in good agreement with the unstable n = 1 mode diagnostic signals (n1rms).The secondary mode stays far below the marginal stability edge all the time.
With the aim to identify the type of the two extracted modes, numerical simulation using MARS-F code [46][47][48] is carried out here.Two moments of equilibrium, indicated by the short red lines in figure 7(a), are chosen as the simulation input parameters when the plasma β N is relatively flat, one for the smallest and one for the largest values.By solving the eigenvalue problem, two stable eigenmodes are found by MARS-F code.The comparison of the eigenvalue between the M3DS extractions and MARS-F solutions is listed in table 1.It is noted that the eigenvalue listed in table 1 via M3DS is extracted through fitting a longer period of time than the real time detection case.The fitting window is indicated by the red bars in figure 7(a).It is found that the growth rates of the two eigenmodes obtained via MARS-F code are very close to those extracted directly from experimental data via M3DS.Moreover, as plasma β N increases, both eigenmodes become less stable.MARS-F simulations also identify the eigenmodes 2D structure.Contour plots of these two modes are displayed in figure 8.The least stable mode shows a m = 2 harmonic dominant mode with odd parity in perturbed displacement, typical of a TM, indicating that the M3DS is capable of detecting stable TMs.This also suggests it is a TM causing the plasma crash.The secondary mode is a m = 1 harmonic dominant mode showing a kink structure.
The offline analysis of shots 186286 and 186288 also show better results than that in real time detection.The results are shown in figures 7(b) and (c).In the discharge 186286, the β N is increased continuously until the plasma disrupts.The evolution of the least stable mode's growth rate follows a clearer increasing trend than that of real time detection.Different from the real time detection, the behavior of the least stable mode is always below marginal stability edge before the plasmas crash, in better accordance with the n1rms signal.The growth rate is close to zero before the plasmas crash.The two large dips shown at about 2.8 s and 3.2 s, and a large leap at about 3.15 s may be due to the variation of the equilibrium such as profile effect.It is found that near the dips, the growth rates of the secondary mode are close to the least stable mode.This indicates that, during the discharge, the capability of simultaneously monitoring multiple modes is important.Because the secondary mode has the potential to become the least stable mode as the equilibrium varies.After the extrapolation of the growth rate evolution curve as aforementioned in section 2.2, the secondary mode could be considered as the one that becomes unstable first.In order to better predict and prevent the plasma crash, it is essential to simultaneously trace all the dominant stable modes from the beginning of the discharge.In this work, two modes are enough.Previous results show that three dominant stable modes can be extracted during one discharge [36].More than three modes are difficult to find so far.In the discharge 186288, the β N first continuously increases and then continuously decreases without a plasma crash.The n1rms signal indicates the MHD modes are always stable.The evolution of the least stable mode's growth rate is always under zero and follows the change of β N in a good manner.Moreover, it is interestingly found that the growth rate extracted via M3DS can follow the fluctuation of plasma β N on some level.Since the eigenvalue calculation uses 50 ms of data, high frequency fluctuations will be naturally excluded due to the average effect so that only large fluctuations of the β N behavior are reflected in the eigenvalue curve and are somewhat delayed.

Optimization of M3DS
For the purpose of making better use of the real time M3DS for plasma control in future experiments, the optimization of M3DS through offline analysis is attempted in this section.As already mentioned in the previous sections, two settings play a major role in determining the quality of the extracted growth rate time evolution: the update window and the fit window.How their variation impacts the results is shown in figures 9 and 10 respectively.In figure 9(a), offline M3DS analysis of offline fitted n = 1 is shown, while real time fitted n = 1 signals and offline M3DS analysis are employed in figure 9(b).It is found both in figures 9(a) and (b) that updating too frequently often leads to more fluctuations.Updating slowly can help automatically smooth the curves to some extent.Thus, it is better to find out an optimal updating frequency to make a balance between the efficiency and the accuracy.Figure 10 shows the results of changing the fitting time window.Here, the updating window, t u , is fixed as 5 ms.It is found that the relation between the fitting quality and the fitting time window is not in a linear regime.In the figure 10(a), all the curves show a increasing trend of growth rates versus time.However, the ∆t = 25 ms case has more fluctuations.In order to examine the degree of fluctuating, the fluctuation factor ) 2 is defined here, with y o being original offline M3DS extracting data and y f being quadratic fitting data.Due to the fact that the best fitting is unknown, a quadratic function is used for the fitting, capturing the main  trend of the growth rate evolution.Thus, the quadratic fitting data y f are different for each case with different ∆t.The results are shown in figure 10(b).It is found that the ∆t = 50 ms case is sufficient for a good fitting quality.By setting fitting window as 25 ms, the evolution curve of growth rate can be more fluctuating, which is not favorable for real time control.Moreover, increasing fitting window to 100 ms or more does not provide a evidently less fluctuating curve, indicating that it will not give any benefit for real time monitoring the plasma state and for the prediction of the MHD instability.

Conclusion
In this work, the M3DS is applied in the DIII-D experiments to detect multiple MHD eigenmodes' growth rates with changing plasma equilibrium in real time.In all the cases, the measured evolution of the multi-modes' growth rates are in accord with the expected variation of the plasma stability correlated with β N .The new method has been validated in different kinds of cases with different variations (step-up increasing, continuously increasing, and first increasing and then decreasing) of plasma β N .Using the experimental equilibrium of the step-up increasing β N case, MARS-F simulations found two modes with close growth rates to the experimentally extracted ones via real time M3DS.The perturbed displacement structure of the least stable mode in the simulation shows an odd parity, indicating it is a stable m = 2 dominant tearing mode.The perturbed displacement structure of the secondary mode in the simulation shows a m = 1 dominant kink structure.
Real time detection shows similar results to the offline calculations, showing the capability to calculate the stable mode growth rate time evolution.It also shows more fluctuations in the growth rate evolution compared to the offline testing.Detailed comparisons of the real time versus offline data and fitting routines show cleaner system input and output signals (n = 1 fits at each of the poloidal locations) are the key to improving the accuracy and quality of the M3DS extraction.Improving the real time system input and output quality will be the future direction of efforts.Moreover, even though the real time updating window of the extracted eigenvalues can be as short as about 2 ms in DIII-D, it is not always good to update too frequently.A slower updating provides a smoother curve of the evolution of the growth rate.A balance should be made between the efficiency and the accuracy by exploring an optimal updating frequency.In addition, the fitting time window enough for good trend of mode growth rate is found to be as short as 50 ms.The efficiency and effectiveness of the new method show great capability to monitor and predict the low frequency MHD instabilities in real time, including deleterious TM.The new technique can be extremely essential to evaluate plasma stability in real time and thus to predict the MHD instability triggering severe plasma disruption in the present advanced large-scale tokamak devices and future fusion power plants.

Disclaimer
This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Figure 1 .
Figure 1.Temporal evolution of a time segment of discharge 178583.The waveform of applied current in one of the six upper coils is shown in (a).The n = 1 magnetic response is measured by multiple radial sensor arrays located at the mid-plane of the (b) LFS and (c) HFS, respectively.(d) Growth rates of multi-modes extracted via offline M3DS using different time segments of signals.

Figure 2 .
Figure 2. Schematic diagram of DIII-D cross section.(Blue and red bars represent Br sensors and Bp sensors, respectively.Here, only Br signals are used in this work.Grey bars represent the I-coils.Grey arrows are the auxiliary symbols without physical meanings.δB k represents the magnetic signal measured on magnetic sensors.δJ k represents externally applied 3D current field perturbation.) Figures 3(b)  and (c) display the temporal evolution of externally applied upper current field and the phasing between upper and lower row of I-coils respectively.Figure3(d)shows the amplitude of the measured n = 1 magnetic field in the low LFS.In the discharge 186292, the plasma is stable in the beginning.The n = 1 external perturbed fields are applied and the system response is continuously detected.Then, the plasma β N is stepped up to a higher value in a short time and kept steady for about 0.5 s.The step-up operation is repeated for several times until the plasma becomes unstable.Time evolution of multiple eigenmodes' growth rates via real time detection of discharge 186292 are shown in figure4.In the DIII-D plasma control system, amplitude and phase of the n = 1 component of the signals from the magnetic sensors are extracted in real time.The sliding fitting time window in this case is set as ∆t = 50 ms, and the actual updating time window is t u = 2 ms.A 50ms averaging window means 1/50ms = 20 Hz, so the Nyquist frequency is 10 Hz.Finally, the time evolution of eigenvalues of multiple modes are obtained via M3DS.Note that the real part of the eigenvalue Re(γ) is identified as the growth rate of mode.The negative level of the growth rate indicates the mode is stable.The imaginary part of eigenvalue represents the natural mode rotating frequency.As shown in 4(a) the plasma β N increases in a clear staircase pattern.Every time the β N is stepped up, the growth rate of the least stable mode does not instantly have a large leap as expected.However, the overall trend of the least stable mode's growth rate is approaching marginal stability, in accordance with the variation of the plasma β N .The growth rate is increasing at the first (at t = 3.05 s) and the third (at t = 4.05 s) stepups of the plasma β N .At the second step-up (at t = 3.55 s), the growth rate suddenly drops instead.Clearer variation of the

Figure 3 .
Figure 3. Temporal evolution of discharge 186292 in the DIII-D tokamak.(a) The steady flattop Ip, safety factor q 95 and step-up increasing β N .(b) The equal-interval square waveform of applied current field in one of the six upper coils.(c) The phasing (∆ϕ = ϕup − ϕ low ) for each interval.(d) The n = 1 magnetic response measured by LFS sensor arrays located at the mid-plane.

Figure 4 .
Figure 4. Temporal evolution of discharge 186292 in DIII-D tokamak.(a) The step-up increasing β N and the n1rms diagnostic signals.The growth rates of (b) the least stable mode and (c) the secondary mode obtained from real time detection experiments via M3DS.

Figure 5 .Figure 6 .
Figure 5. Temporal evolution of the plasma β N , the n1rms diagnostic signals and the growth rate of the least stable mode and the secondary mode extracted via M3DS of discharges (a) 186286 and (b) 186288 in DIII-D real time detection experiments.(The green dot-dashed lines are auxiliary lines drawn by hand to guide the eyes, showing the trend of the evolution.)

Figure 7 .
Figure 7. Temporal evolution of the plasma β N , the n1rms diagnostic signals and the growth rates of the least stable mode and the secondary mode extracted via offline M3DS of discharges (a) 186292, (b) 186286 and (c) 186288 in DIII-D tokamak.(The green dot-dashed lines are auxiliary lines, showing the trend of the evolution.)

Figure 8 .
Figure 8. Eigenmode structures of the least stable mode (a) and (b), and the secondary mode (c) and (d) calculated using MARS-F.(a) and (c) represent the poloidal Fourier harmonics of radial displacement ξ • ∇ψ 1/2 in PEST coordinates.(b) and (d) plot the 2D cross section of radial magnetic perturbation.(The toroidal mode number is n = 1.The labels R and Z are in (m)).

Figure 9 .
Figure 9. Temporal evolution of the growth rate of the least stable mode of discharges 186292 under different length of updating-window via (a) post-fitting signals and offline M3DS post-analysis, and (b) real time detected signals and offline M3DS post-analysis.Fitting time window is set as 50ms.

Figure 10 .
Figure 10.(a) Temporal evolution of the growth rate of the least stable mode of discharges 186292 under different length of fitting-window via post-fitting signals and offline M3DS post-analysis.(b) Fluctuation factor C f versus fitting time window ∆t.Updating time window is set as 2ms.

Table 1 .
Comparison of eigenvalues between M3DS and MARS-F (The fitting window of M3DS results are indicated by the red bars in figure 7(a)).