Plasma effect on error fields correction at high βN in ASDEX Upgrade

Tokamak plasmas can amplify very small resonant components of error fields (EFs) when operating close to the ideal magneto-hydrodynamic (MHD) limits. Such EFs are well diagnosed in ASDEX Upgrade tokamak (Igochine V et al 2017 Nucl. Fusion 57 116027, Maraschek M et al 40th EPS Conf. on Plasma Physics 2013 P4.127), which allows to model EF as well as the correction required for the optimal compensation. Experiments on ASDEX Upgrade show that EF correction considering the plasma effect, as it is foreseen for ITER, is necessary even in the case of small resonant EF. Such correction improves the achievable βN by 10% and makes discharges more stable with respect to ideal modes.


Introduction
Correction of the error field (EF) in fusion devices has a long history of research motivated by potential detrimental effects of the EFs on the plasma confinement (see for example [1][2][3][4][5][6][7][8][9]). These effects are resonant and non-resonant braking of the plasma rotation [10][11][12], triggering of ideal and 5  * Author to whom any correspondence should be addressed.
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. tearing modes [13,14], locking of these modes [15][16][17], and modification of fast particle confinement [18,19]. This subject is equally important for all main concepts of the toroidal plasma confinement concepts: tokamak, stellarator [20], and reversed field pinch [21]. In a tokamak, the long-wavelength instabilities with toroidal mode number n = 1 are most sensitive to the EFs. Application of the external fields could avoid these problems or control these modes like it is done for resistive wall modes [22,23] or used for controllable rotation of n = 1 modes [24,25] for further stabilization with electron cyclotron current drive [26].
The required EF compensation for a particular tokamak depends on the amplitude of intrinsic EFs. The n = 1 EFs in ASDEX upgrade are very small: B EF /B tor < 10 −5 . This allows tokamak operations in most regimes in L and H-modes without any correction of the EFs. Moreover, even the maximal currents in the external B-coils [27] could only slightly change the magneto-hydrodynamic (MHD) mode behavior in most of the operation space and have a small influence on the stability of n = 1 modes [28]. In these plasmas, the toroidal rotation is non-uniform and sheared which effectively screens external perturbations and improves the plasma confinement. The situation changes in particular areas of the operational space, where the ideal modes or tearing modes are marginally stable. One of these areas is the high beta plasmas with the safety factor profile slightly above one. This is typical for improved H-mode scenarios (or high β hybrid operations) in ASDEX Upgrade. Such scenarios are one of the main candidates for high fusion performance tokamak operation, which offers potentially steady-state tokamak operations. Thus, the investigation of these scenarios is of particular importance. In addition to this, even a small increase in the achievable beta in these regimes will be translated into at least twice the relative increase of the fusion power (P fus ∼ β 2 N ). The normalized beta is defined as β N = β (aB t /I p ) , β = 2µ 0 ⟨p⟩/B 2 t ; ⟨p⟩ is the volume average pressure, B t is the vacuum toroidal magnetic field at the axis, a is the minor radius and I p is the plasma current. In this situation, even a small correction becomes important if it helps to improve the overall performance as shown in the next sections.

Source of n = 1 EF in ASDEX upgrade and correction variants
The main source of small EFs in ASDEX upgrade are the feedthroughs for the poloidal field coils in ASDEX Upgrade, which was identified in previous experiments [17,29,30]. These feedthroughs are located on the upper and lower part of the tokamak as shown in figure 1(a). In each location, the currents in and out are close to each other, but they still produce a small magnetic field which is the source of the small global n = 1 EF. The current amplitude in the feedthroughs depends on the plasma shape and the resulting EFs are identical only for discharges with identical currents in the poloidal field coils. Thus, the discharges with different plasma shapes require different correction currents which can be calculated assuming the realistic geometry of the system and applied currents. The EFs were calculated with the electromagnetic code CAFÉ [31] using realistic geometries of the currents and all relevant internal components. The same code is also used to model AUG magnetic actuators (i.e. the B-coils) and thus to calculate the required EF correction through an optimization procedure.
As was mentioned before, these fields do not set any limit in most cases, except the one where these fields are amplified by the plasma. The plasma is a good amplifier for the resonant component when the correspondent mode is marginally stable, which is typical for improved H-mode scenarios. Experiments on ASDEX Upgrade show that the non-resonant components are sufficiently small to avoid direct effects like neoclassical toroidal viscosity. Thus, the main aim of the EF correction in ASDEX Upgrade was to avoid amplification of the n = 1 component. Also, the different poloidal mode numbers are coupled for n = 1 mode, the strongest and broadest components have poloidal mode numbers m = 1 and m = 2. For our discharges, m = 2 is the main source of the problems, and m = 1 and m = 3 are mainly the results of the toroidal coupling [17]. In practice, one can either nullify this particular (2, 1) component using the B-coils or make this in combination with the optimization of (3, 1) and other components. The result in magnetic field has local amplitudes comparable to the EF without correction, but the most important resonant components can be canceled very effectively. The case in figure 1(b) has a negligible (m = 2, n = 1) component after correction in the case of a vacuum. This is called ' (2,1) vacuum' correction in the following.
Two other correction recipes have been developed, taking the influence of plasma on the applied perturbations into account. The results of the CAFÉ code are coupled with the output of the linear MHD code MARS-Q at the plasma boundary. One of the metrics applied in this work has also been used for ITER, following calculations with the generalized perturbed equilibrium code (GPEC) code [32], and is presently the reference EF correction criterion for ITER operations. In the present implementation, plasma response is defined as the real penetrated perturbed field (δB m,n ) calculated over the whole plasma radius and including non-ideal effects. The MARS-Q code calculates the plasma response taking into account the plasma rotation profile measured in preliminary ASDEX Upgrade experiments with charge exchange diagnostics. The effect of rotation, which is important due to the high neutral beam injection (NBI) heating, is assumed constant in the plasma response calculations, thus neglecting the non-linear evolution induced by the external magnetic perturbations. Variation of the magnetic perturbations at the boundary allows us to find the optimal reduction of the relevant components in the plasma. To be precise, two corrections with plasma in the following are: a) Correction of (2,1) component only, which is called '(2,1) plasma' correction in the following. In this case, only the (2,1) component in the plasma was canceled. b) 'Overlap' correction, which includes also other m's as described in [32,33]. In this case, the dominant mode of the perturbed field δB m,n is calculated with singular value decomposition (SVD) and the result is used to optimize the correction of the so-called 'overlap' external field [34]. The resulting vector is an optimized projection for EF compensation (δB m,1 ) and contains weight for different mcomponents as a column in SVD decomposition. This is the correction which is used for ITER.
The resulting magnetic field produced by the B-coils at the q = 2 surface and the corresponding currents in the coils are shown in figure 2. One can see that the required current amplitudes are comparable for all three cases. At the same time, the phase shift around 90 degrees is present due to the plasma effect. In our case, the main contribution in the case of the plasma is still (2,1) which one can see from similar current values for '(2,1) plasma' and 'overlap'. One has to note, that the required maximum correction currents are only about 10%  of the maximal allowable current for B-coils. This show again how tiny the corrections are. In the next section, the application of these corrections in experiments is discussed. The correspondent amplitudes of different poloidal components and their phases are shown in figure 3 for the q = 2 surface and the control surface, where the MARS-Q and the CAFÉ code are coupled. The coupling surface is very close to the plasma boundary.

Experimental results
High beta discharges, discussed in this paper, were performed with dominant neutral beam (NBI) heating in ASDEX upgrade and additional electron cyclotron resonance heating (ECRH). This allows us to reach high β N values with flat q-profile in the core close to unity. The evolution of time traces is shown in figure 4. The shaded regions are the time when EF correction was applied. The discharges were executed during a single session in an attempt to make them as identical as possible. The corrections were the following: The safety factor and pressure profiles were obtained by integrated analysis of data from different diagnostics using the coupled Grad-Shafranov solver with the current diffusion   equation. This analysis includes also local internal measurements of the magnetic field with motional Stark effect diagnostics as described in [35].
The first discharge ends with fast disruptive ideal n = 1 mode, which is seen on the constant phase from ECE measurements (insert of figure 5(a)). The mode is too fast to be visible on the presented overview spectrogram ( figure 5(a)) and the magnetic signal of the mode is shown in the insert of figure 5(a). This signal allows us to estimate the growth rate: In experiments, beta increases during the mode growth which requires correction to get the real γ MHD growth rate. The correction details are given in [36], where γ h ≈ 0.85s −1 determines the increase of beta: β N = β 0 (1 + γ h t) at the mode onset. The result growth rate (γ MHD ≈ 1.06 · 10 6 [ s −1 ] ) gives the times ( τ MHD ≈ 1/γ MHD = 9 · 10 −7 s ) comparable to Alfven time (τ A ≈ 5.5 · 10 −7 s), which confirms the ideal character of the mode. The maximal achieved normalized beta (β N = 2.95) is measured just before the mode onset. The spectrogram of the discharge is shown in figure 5(a).
We compare first this ideal mode limiting discharge with the 'overlap' case. Although, discharge #41100 temporarily lost one of the NBI power sources around 4 s (see figure 4(b) and (c)), it is identical to the previously discussed #41097 in all other parts and during the ramp-up phase of the discharge. In this case, the maximum archivable beta is 10% higher (β N = 3.32) and the discharge is limited by the resistive n = 2 mode. The onset of the mode and the position of the maximal beta are shown in figure 5(b). The phase of the temperature perturbations from ECE confirms the resistive character of the mode (see insert in figure 5(b)).
Comparison with the last case '(2,1) plasma' #41098 is not so straightforward. This discharge has a bit higher core plasma rotation and consequently, a higher beta normalized already from the beginning, before the switch-on of the Bcoils at 1.0 s. Thus, the plasma conditions are similar but not identical. At the same time, the main features remain the same. The discharge reaches a higher beta compared to the pure '(2,1) vacuum' correction case ( figure 6(b)). It is limited by the increase of impurities in the core (figure 6(c)). The impurity accumulation rises even further after the onset of the ideal n = 1 mode which converts later into an island ( figure 6(a)). Thus, it is better from the correction point of view compared to the 'vacuum' EF correction. The comparison between '(2,1) plasma' and 'overlap' cases is more difficult and new experiments are required. At the same time, the difference between these two cases is much smaller compared to the vacuum case as shown in figures 2(a) and (b).

Conclusions and discussion
Experiments in ASDEX Upgrade show that the main problems related to the small EFs come from the resonant n = 1 component. The n = 1 EF is well diagnosed in the ASDEX Upgrade tokamak using the compass scan approach and the influence of EF on the plasma rotation [17,29]. This allows modeling both the EF fields and the required correction with the electromagnetic code CAFÉ. The plasma effect is modeled by coupling the CAFÉ code with linear MHD code MARS-Q at the plasma boundary. In the present experiments, the direct comparison of EF correction for n = 1 was done with and without the effect of the plasma amplification. The correction with plasma increases β N by about 10% in ASDEX Upgrade and the limiting events are not the onset of ideal n = 1 kink anymore. In one case this was impurity accumulation, in the other n = 2 tearing mode. The general importance of the plasma response for EF correction was already demonstrated in other tokamaks, see for example [34,37,38]. The current paper confirms these results and demonstrates the importance in the case of small resonant EF. Contrary to typical compass scan studies mentioned above, the direct comparison of EF correction on similar discharges was done in this work. This shows that the source of the EF is identified correctly and the procedure can be used routinely for all discharges where this correction can be important. The developed modeling framework can account for realistic conductors and toroidal calculations of plasma response via the full solution of the linear resistive MHD problem. Toroidal flow has been included in this work and other non-ideal effects can also be treated (such as drift-kinetic effects). This makes possible forward modeling of required EF for next-generation tokamaks like ITER and DEMO.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.

Acknowledgments
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200-EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.
Part of the data analysis was performed using the OMFIT integrated modeling framework.