Alfvénic fluctuations measured by in-vessel Mirnov coils at the Wendelstein 7-X stellarator

125 individual Mirnov coils are installed in four out of five toroidal modules (red squares in figure 1(a)). They are located at the inner wall of the plasma vessel between CuCrZr heat shield elements (figure 1(c)). During plasma operation the coils are covered by graphite wall protection panels (not shown here). The harsh environment of the fusion experiment W7-X in terms of space, geometry and material constraints leads to demanding requirements for the technical realization of individual coils, signal cable shielding and routing. Figure 1(b) depicts a single coil with two layers (9 windings each) of an insulating, high temperature resistant Constantan^®wire wound on a ceramic coil former (height 15 mm, width 26 mm, length 43 mm) resulting in a low self-inductance of ∼3.7 µH. The ceramic body has a high heat conductivity to avoid critical temperature gradients within the material and to provide sufficient cooling by attaching the coils to the water-cooled inner wall components. Between the two winding layers of the coil a center tap is directly connected to the plasma vessel ground to prevent potential DC charging during plasma operation. The twisted-pair signal cables are guided within a copper pipe toward the closest distribution box inside the plasma vessel behind the plasma facing components. Here the signal cables of nearby coils are bundled and routed within copper pipes to a vacuum port with suitable electrical feedthroughs. More details and specifications, describing design and construction of the coils as well as used materials, can be found in [43]. The design covers a wide dynamic range with respect to amplitude levels and relevant frequencies (table 2) as well as complex mode structure detection capability due to the arrangement of the individual coils (figure 1(a)).

The DAQ of the Mirnov diagnostic is build upon ATCA (advanced telecommunications computing architecture) ADC (analog to digital converter module) boards [44]. The technology is specifically capable of recording with high data rate in the quasi-steady state long-pulse operation at W7-X (planned up to 1800 s). It is similar compared to the DAQ systems of the magnetic equilibrium diagnostics [45] without the countermeasures against signal drifts, which become relevant for long-time integrated signals. The Mirnov coils are constructed to measure AC fluctuations above 1 kHz, therefore the compensation of DC-drifts can be neglected. In total four ATCA boards are distributed on two electrical racks, each board involving 32 measurement channels. A single signal, which is recorded in-vessel by an individual coil, is amplified by a pre-amplifier close to the feedthrough directly at the vacuum port to minimize loss of information over the 90–120 m long signal cables toward the diagnostic racks. The Mirnov DAQ is fully integrated into the general W7-X operational control system in a way that centrally pre-programmed experiments automatically start and stop the measurement and a timing module ensures synchronization with the general W7-X time. A specifically designed software, which is operated on a measurement PC, combines in total up to 64 individual channels of measured data with synchronized time stamps and transfers them to the W7-X archive data base. The individual components of the signal chain are depicted in figure 2.

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An essential part of the diagnostic commissioning represents the calibration of the Mirnov diagnostic system. It has been performed step-wise individually for all relevant components (depicted in figure 2) determining the signal path of the coils measurements. Starting at the pick-up coil (figure 2, (1)), the signal cables are routed within copper pipes toward the closest vacuum feedthrough (figure 2, (2)). The low amplitude measurement signals are pre-amplified (figure 2, (3)) before they enter the DAQ in the electronic rack via 90–120 m long cables (figure 2, (4)). The individual DAQ components (figure 2, (5) signal receiver and power supply for pre-amplifier, (6) converter panel, (7) ATCA board including individual measurement ADCs), which are directly involved in the signal path, were also calibrated separately. Finally the timing module and the PC are labeled in figure 2 as (8) and (9), respectively.

The calibration measurement of the DAQ electronics was achieved by sending a differential sine signal of constant frequency, supplied by a function generator, to each of the relevant components via a transmitter (figure 3(a)) and detecting the response signal with a corresponding receiver (figure 3(b)). Electrical connectors and adapters (BNC, RJ45, LEMO—not shown here) of each individual DAQ component, the Mirnov coil signal cables and an oscilloscope were chosen to match both the electrical output of the transmitter (E-out) and the electrical input of the receiver (E-in). A PC, connected to the oscilloscope, was then used to automatically compare the received signal to the transmitted signal regarding amplitude and phase shift. The frequency of the supplied signal was increased step-wise (one hundred steps from 1 kHz to 1 MHz) and the amplitude was kept constant at 500 mVpp. The gain and phase response of the calibration setup itself has been determined (figure 3(c)) and corrected for in the final calibration curves for each component. The calibration procedure has been repeated at least ten times for an individual coil, which results in an accuracy of $\lt$1% and ≈1° regarding the amplitude and phase shift, respectively.

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Figures 4(a) and (b) show frequency dependent calibration curves of an example Mirnov coil QXM10CE040, which is located in HM10 at the outboard side. An almost constant frequency response for the gain and phase shift of the combined power supply and converter panel (i.e. the rack electronics, blue dashed curves in figure 4(a) and (b)) has been obtained. The phase shift slightly rises ($\lt10$ deg) for frequencies $\gt$500 kHz. The characteristics of the pre-amplifier (solid red curves) and the long signal cables from the torus hall to the electronic rack (magenta dotted curves), however, are significant. They clearly dominate the resulting sensitivity and phase shift curves shown in figures 4(c) and (d), where the individual component calibration curves have been combined numerically. It is noted that each individual Mirnov coil was calibrated in a Helmholtz test-stand in a laboratory before installation [43]. It was found that the frequency response of the relevant poloidal pick-up direction shows an almost constant amplitude gain and phase shift in the frequency range from 1 to 1000 kHz (green dash-dotted curves in figure 4(a) and (b)).

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Comparing the calibration curves of different coils, we find a variation in sensitivity and phase shift, which is indicated by the gray shaded area in figures 4(c) and (d). For the sensitivity a change of up to 10% above 200 kHz between different coils is derived (The gray shaded area in figure 4(c) is barely visible in this representation.). The phase shift variation is up to $5^{\circ}\!-\!35^{\circ}$ for frequencies $\gt$100 kHz (gray shaded area in figure 4(d)). The effect of the 90–120 m long signal cables from pre-amplifier at the vacuum feedthrough to the electronic rack is mainly responsible for the differences between individual coils.

Advanced spectral as well as mode analysis techniques are applied to measured data and are accompanied by numerical calculations, which model the structure of possible stable and unstable modes considering the complex three-dimensional geometry of W7-X. A detailed description of the theoretical approach and used simulation codes to identify the nature of the observed MHD fluctuations is given in [46, 47]. The following paragraph summarizes the tools, which are relevant in the context of this work and emphasizes newly implemented improvements.

A standard non-parametric FFT algorithm is used to calculate spectrograms of measured time traces by single Mirnov coils. Considering the variety of events observed on very short time scales (e.g. pellet [48] and Tespel injection [49] or sawtooth-like plasma crashes due to externally driven plasma currents [40]) and in case of partly weak Alfvénic broadband fluctuation activity, the simple FFT analysis often remains insufficient in terms of frequency resolution versus time range length [50]. Therefore in addition the advanced parametric DMusic algorithm (Damped Multiple signal classification [51–53]) is used for calculating spectrograms. Features of interest, which are barely visible in standard FFT spectrograms, can become much more clearly pronounced, whereas in the corresponding context other less important features may fade or even vanish. That allows a significantly more profound follow-up analysis, e.g. for identifying complex mode structures related to certain frequency regimes.

For the latter the SSI method (Stochastic System Identification, [53–55]) is adapted to calculate mode number spectra. The implementation of the basic SSI algorithm has been complemented by an interface, which is specifically tailored to process measurements of poloidally and toroidally arranged Mirnov coils.

The three dimensional ideal MHD continuum code CONTI [56] is used to identify the type of the observed AE. Based on a numerical equilibrium provided by the VMEC code [57, 58], considering pressure profiles, which are e.g. measured by the Thomson scattering diagnostic [59], CONTI calculates the continuous frequency spectrum of Alfvén waves, which depends on the normalized toroidal flux label and poloidal and toroidal mode numbers. Empty frequency bands are revealed (so-called gaps within the continuum), where unstable modes could be excited [19, 60]. Considering experimentally estimated mode numbers for certain frequencies, specific AEs can be assigned to the observations.

In addition CONTI optionally delivers the full sound wave spectrum. Usually for observations at medium and lower frequencies (here $\lt$150 kHz) the coupling of sound waves to AEs deforms the gap structure of the continuum mainly close to the plasma core, which has to be considered for a more precise identification of a possible mode type. Another aspect, which also affects the location and structure of continuum gaps, is the quality of the measured plasma profiles. In the context of this work, the achieved accuracy of the profile fits is sufficient to discuss the relevant observations. Additionally the radial electric field could lead to a Doppler shift within the continuum of the order of up to a few 10 kHz. The gap structure can be distorted substantially, especially for low frequencies. Nevertheless the influence on the typical Alfvénic broadband activity around 200 kHz, which is discussed in this work, can be neglected, since the associated gap itself is rather wide and remains open, when taking the corresponding radial electric field into account (similar as in [46]).

Relevant gap modes are further analyzed using the ideal MHD-code CKA [61] to find the radial structure of corresponding AEs. The perturbative gyrokinetic, three-dimensional, electromagnetic particle-in-cell code EUTERPE (CKA-EUTERPE [62]) can be applied to study the power transfer between relevant modes and kinetic particles. The latter lays beyond the scope of this paper and is demonstrated in [46].

To illustrate typical measurement data of a Mirnov coil (here: QXM10CE040, installed in HM10) during plasma operation, the W7-X experiment ID 20180918.045 has been chosen (figure 5). Standard plasma parameters, depicted in figure 5(a), are the line integrated electron density [63] (blue line), a central ECE channel indicative for the electron temperature [36] (red line), diamagnetic energy [45] (magenta line) and toroidal plasma current [64] (black line). The plasma is heated via ECRH (electron cyclotron resonance heating [65]) with a constant power of 3 MW from 0–3 s and 4.5 MW from 3 to 6 s. At about 1.9 s frozen hydrogen pellets have been injected into the plasma [48] (green shaded region in figure 5(a)), which leads to an increase of density and energy and decrease of temperature and current. The energy reaches about 1 MJ, which marks a high performance phase of the plasma (red shaded region in figure 5(a)). The high energy plateau holds for approximately one confinement time (≈200 ms), followed by an energy and density decrease to a level, which is slightly higher compared to the pre-pellet phase at the beginning of the discharge. The underlying mechanisms of the observed dynamic of high performance plasmas at W7-X, also with respect to the previously published record triple product [66], are discussed in detail in [35, 67, 68].

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Several observations in the measured Mirnov coil raw data are noticeable (figure 5(b)). During the plasma pulse a reasonable signal-to-noise ratio S ≥ 20 is found, where the noise level is measured before the onset of the ECR heating. The magnetic fluctuation amplitude slightly decreases during the pellet injection phase (marked green in figure 5(b)). The individually injected pellets provoke short and strong magnetic bursts with a length of about 1 ms. During the high energy phase (marked red in figure 5(b)) the amplitude increases again to a similar level compared to the pre-pellet phase at the beginning of the discharge. Finally the magnetic fluctuation amplitude remains at a slightly higher level compared to the pre-pellet phase after the high energy phase abruptly ends and the plasma energy and density drop down by a factor of about two.

A spectral analysis of the measured Mirnov coil time trace shown in figure 5(b) is depicted in figure 6. Both plots were generated based on the same data, but different analysis methods have been used—a simple short-time FFT algorithm in figure 6(a) and the advanced spectral analysis method DMusic in figure 6(b). The previously mentioned short magnetic bursts due to injected pellets are clearly spread out over a wide range of frequencies (figure 6(a)). During the following high energy phase a well pronounced, narrow frequency band occurs at about $t = 3{-}3.5$ s around 30 kHz within the overall enhanced fluctuation activity for $f\lt50$ kHz. We note that it is also detected by the PCI and XMCTS system, but not shown here. Recent investigations, which are out of the scope of this work, show evidence, that these observations might be related to kinetic ballooning modes.

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Focussing on the broad band fluctuation around 200 kHz we notice that the dominating frequency follows n^−0.5( n is the line integrated electron density) very well (black line in figure 6(a)) during the whole plasma pulse, which implies an Alfvénic character. In contrast to the simple FFT algorithm, which has been used to generate the spectrogram in figure 6(a), the more advanced method DMusic is able to provide a much clearer picture of the frequency band of interest (figure 6(b)). The overall signal amplitude decreases during the pellet injection and high energy phase, but recovers afterwards until the end of the discharge. The following analysis will refer to the highlighted time interval from 4 to 5 s.

The measured signals of in total 41 poloidally arranged Mirnov coils in HM11 are used to perform a poloidal mode number analysis using the SSI-Method. Within the time interval 4–5 s in the W7-X experiment 20180918.045 some slightly pronounced poloidal mode numbers m = 3, 11 for a corresponding frequency of 180 kHz (figure 7) are found. It is noted that for the calculation of this mode spectrum no experimentally derived information about the radial localization of the mode was available. Hence a mapping of the Mirnov coil positions to magnetic coordinates of a specific flux surface has not been performed, which neglects a possible distortion of the mode spectrum. However, since the almost equidistant coil positions at the given toroidal location in HM11 (cf figure 8(a)) reflect the triangular plasma shape well and the observed mode likely exists at a radially outer flux surface of the plasma (derived from a comparison to theoretical predictions, cf next paragraph and figure 8(b)), the distortion is assumed to be small. Improved mode spectra calculations taking into account an exact mapping of the poloidal coil positions to a given magnetic flux surface and considering an experimentally determined radial localization of the observed mode (e.g. by XMCTS or ECE diagnostic measurements) are being prepared and will be subject of a future publication.

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Although the experimentally deduced mode number spectrum is not very conclusive, a comparison with theoretical calculations delivers evidence for the type and location of the AE based on the observed mode number and frequency. Considering the relevant VMEC equilibrium (depicted in figure 8(a), left hand side: flux surfaces at the toroidal location φ = 33° of the Mirnov coil arrangement in HM11, Mirnov coils indicated as red squares; right hand side: corresponding pressure and iota profiles), the Alfvén continuum has been calculated with CONTI at 5.02 s (figure 8(b), gray lines). The full sound wave spectrum has been considered to account for the coupling effects to the Alfvén wave continuum, but for better readability the branches related to sound mode dispersion relations are not depicted in the continuum. Associated AEs are stated within the gaps as TAE—toroidally-induced AE, HAE—helicity-induced AE, EAE—ellipticity-induced AE and NAE—noncircularity-induced AE (Details and continuative literature for the different AEs are given in [19]). A selection of branches are colored corresponding to their poloidal mode numbers, which were chosen considering the findings of the experimental mode analysis (figure 7) and exist within the depicted frequency range. Several regions can be identified, where such a colored mode crosses a continuum gap. However, only within the gray shaded area in figure 8(b) frequency, mode number and strength (according to figure 7) match the observations well. Close to the observed frequency of 180 kHz (marked by the black dashed line within the continuum) the mode with poloidal mode number m = 11 is found (colored in red) at approximately s ≈ 0.65, where s is the normalized toroidal flux label. The dedicated gap is associated with EAE. We note that based on the same approximation, which has been used in [46], the expected Doppler shift at the identified location due to the radial electric field (measured by the Doppler reflectometry system) is of the order of 10 kHz and does not affect the findings so far.

The general appearance of the identified fluctuation band in W7-X plasmas is investigated. Therefore the spectrograms of about 490 discharges were automatically analyzed with respect to dominant frequency components. For each discharge a time interval with a length of 1 s has been determined, where plasma temperature, density and energy are constant or show only minor variations. Within this time period and in the frequency range from 100 to 400 kHz the amplitude maximum in most spectrograms is found at around 170–200 kHz (figure 9(a), underlaying histograms in gray). The individual W7-X experiments have been sorted by different main magnetic field configurations [69, 70] or applied plasma heating method. For each case the frequency distribution was calculated and drawn as an extra line (figure 9(a), left hand side configuration standard: blue, low iota: black, high iota: green, high mirror: red and right hand side heating method only ECRH: dashed line, additional or only NBI: dash-dotted line). They show very similar dominant frequencies around 170–200 kHz. Hence a significant shift of the dominant frequency regime with global parameters, like heating scenario and magnetic configuration is not found and the observed fluctuation band can be considered as a common W7-X plasma phenomenon.

The mentioned Alfvénic nature of the fluctuation activity between 150 and 250 kHz is reflected in figure 9(b). The colors have been chosen according to figure 9(a) (left hand side), whereas circles indicate plasma heating only via ECRH and crosses via additional or only NBI. In the same time period, where previously the dominant frequency component has been identified, the corresponding measured line integrated electron density is determined and plotted against the dominant frequency. Generally the resulting distribution is well approximated by the expected n^−0.5 trend (black dashed line). The indicated trend fits best the measurements in discharges, which were operated in standard magnetic configuration. It slightly overestimates the observations in case of high iota and high mirror configurations and underestimates in case of the low iota configuration.

Various aspects have been studied regarding potential benefits of an extra on- or off-axis plasma current drive [71], the so-called ECCD, which is driven by the ECR heating system. However within a number of W7-X experiments involving ECCD a disturbing observation was the appearance of fast, sawtooth-like plasma crashes. In some cases they even led to a total plasma collapse. On a time scale of up to 100 times faster than expected, compared to the plasma confinement time, the toroidal and poloidal plasma currents collapsed, causing high induced currents in the adjacent in-vessel components. Resulting mechanical forces due to the interaction of these induced currents and the main magnetic field could lead to severe damage of vital machine parts and relevant diagnostics. Fortunately this was not the case during past operational phases of W7-X and also has to be ruled out in future high-beta plasma experiments foreseen for later campaigns. Hence, understanding the nature of the observed crashes and related plasma collapses is important and detailed investigations concerning the underlying instability and trigger mechanism are underway [40, 72, 73].

In the context of this work a typical crash event is selected from the W7-X experiment ID 20180918.022. Figure 10(a)) shows the global plasma parameters, i.e. line integrated electron density (blue line), a central ECE channel indicative for the electron temperature (red line), diamagnetic energy (magenta line) and toroidal plasma current (black line). The plasma is heated via ECRH with a constant power of 2 MW (ECCD included). Abrupt crashes are clearly seen in the electron temperature (measured by the ECE diagnostic located in HM41) and the diamagnetic energy (measured by the compensated diamagnetic loop located in HM31). The magnetic fluctuations (measured by the Mirnov coil QXM11CE120 located in HM11, figure 10(b)) show related distinct, short-time peaks. In figure 11(a) a zoom into one selected crash event at 3.8146 s (marked with a light blue bar in figure 10(b)) is shown, which is complemented by the 46 kHz component of a calculated real-valued Morlet wavelet spectrum (red curve). The frequency spectrogram of the same crash, which has been calculated using DMusic (figure 11(b)), confirms the obviously good agreement of the fit and the measured data. During the crash a clear dominant fluctuation activity slightly below 50 kHz is found accompanied by some weaker signatures around 150 and 250 kHz. Considering only the time interval for the selected crash (3.8146–3.815 s) the SSI mode analysis of the measured data of 41 poloidally arranged Mirnov coils in HM11 indicates a slightly pronounced mode with a poloidal mode number of m = 7 for a corresponding frequency of 46 kHz (figure 11(c)).

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