ICRF coupling in ASDEX upgrade magnetically perturbed 3D plasmas

The RF properties of the four ion cyclotron range of frequencies (ICRF) antennas in the ASDEX Upgrade tokamak are characterized in H-mode magnetically perturbed 3D discharges. An n = 2 magnetic perturbation (MP) field is applied and rigidly rotated, which allows diagnosing the separatrix displacement and consequent coupling change. We find the antenna loading resistance to be coherently modified by the resulting non-axisymmetric plasma equilibria, thus becoming a function of the applied MP field poloidal mode spectra. We perform a detailed statistical analysis, which correlates the change in loading resistance to the fast wave R-cutoff layer movements. From it, a 1D scaling is derived that differs from previous studies evaluated in pure axisymmetric plasma conditions. This experimentally derived scaling is used to predict the average loading resistance change of the ITER ICRF antenna under applied MPs. ICRF coupling simulations using measured 1D density profiles are performed with the RAPLICASOL code, in order to investigate the predictive capabilities of numerical state of the art tools. We find that both 1D conventional scaling laws and 1D numerical simulations fail to capture the 3D physics, and can substantially overestimate the measured loading resistance change up to a factor of ∼3.


Introduction
Ion cyclotron range of frequencies (ICRF) systems are a widespread method for plasma heating and current drive used in experimental magnetic confinement fusion reactors. The common working principle relies on the excitation of the fast oscillatory mode of a magnetized plasma at a frequency matching the cyclotron frequency of a given ionic species in the plasma core, such that resonant damping can occur. The excitation is produced by an embedded antenna in the reactor wall, which contains a finite number of radiating straps, powered by a generator through a transmission line and matching system. The fast wave, however, displays evanescence from where it is excited due to the low-density conditions at the plasma edge, and must reach the cutoff region before it starts propagating. This coupling process is equivalent to the optics phenomenon of frustrated total internal reflection, where the core plasma acts as the frustrating medium [1]. Therefore, the amount of ICRF power that can be coupled through the evanescent volume (rarer medium) at a given current amplitude in the antenna conductors is a function of the optical properties of this medium, determined by plasma density, composition, magnetic field, and the antenna parallel wavenumber (k P ) spectrum. These variables thus determine the antenna radiation impedance Z rad [2].
The dispersion relation that governs this phenomenon can be obtained from the cold plasma dielectric tensor formulation. For  w w p e , , the electron plasma frequency, and in the limit of negligible electron mass  m 0 e , we can express the dispersion relation in a simplified form that describes fairly well the fast wave evanescence and propagation properties [3,4]: where   w = n ck is the parallel refractive index, c is the speed of light, ω is the angular frequency of the antenna, n ⊥ is the perpendicular refractive index, R L , and S are the Stix cold plasma dielectric tensor elements, which are functions of the plasma density, composition and magnetic field: ). The edge evanescent region is comprised by the volume between the antenna, where < n 0 2 , and the R-cutoff, where the wave becomes propagative towards the plasma core.
Maximization of the radiated power requires increasing either ( ) Z Re rad or the current amplitude in the straps. The latter is limited by engineering constraints due to voltage stand-off [5]. The former depends on the evanescence conditions in front of the ICRF antenna, approximated by equation (1). Changes in the plasma kinetic profiles or magnetic field within the coupling region translate into changes of Z rad , and thus, changes of the radiated power for a given current amplitude in the antenna conductors. The dependence of Z rad on plasma parameters has been extensively studied assuming the evanescent medium to be one-dimensional, with gradients in the kinetic profiles and magnetic field only in the normal direction to the ICRF antenna [2,6]. The additional inclusion of poloidal asymmetries proved to be important in understanding the changes in coupling performance when the plasma triangularity is experimentally scanned [7]. Recently, special attention has been dedicated to the understanding of the effect of 3D density configurations due to the interest in improved antenna coupling capabilities with gas puffing techniques [8,9]. Integration of ICRF systems in stellarators [10], and the understanding of ICRF coupling physics in arbitrary 3D geometries, extends this interest to 3D kinetic profiles and 3D magnetic fields. In tokamaks, this latter case is present when plasma instabilities arise, such as MHD modes, or when magnetic perturbation (MP) in-vessel coils are applied for edge-localized mode (ELM) mitigation and suppression. The response of the ICRF system to MPs is also of special interest to ITER, which counts on an MP coil set for ELM control [11] and an ICRF system expected to deliver 20 MW of heating power [12,13]. So far, however, the effect of non-axisymmetric plasma configurations has been neglected in the design of the ITER antenna, and their impact is mostly unknown. In ASDEX Upgrade, dedicated experiments to understand the interplay between MPs and ICRF heating had been carried out so far with static MP fields, which can only account for the effect of the toroidally localized plasma asymmetries on the ICRF antennas [14,15]. No effect of the applied poloidal phasing, defined as the differential phase between the current waveforms of the upper and lower toroidal rows of MP coils in ASDEX Upgrade, j D UL , on the ICRF coupling was found. A more recent study using rotating MP fields, however, revealed that ICRF coupling can be coherently affected by the induced non-axisymmetric plasma structure produced by different MP configurations [16]. This observation, conjugated with the reduced availability of data for ICRF heated MP discharges, motivated a dedicated set of experiments. Therefore, in this paper, we have systematically studied for the first time the coupling properties of the fast wave when MP fields are applied and rigidly rotated. This allows us to assess the impact of plasma non-axisymmetric profiles, regardless of where the ICRF antennas are placed inside the vessel. For this purpose, we have utilized the MP and ICRF systems in the ASDEX Upgrade tokamak. In addition, the poloidal phasing of the MP coils is scanned, and the effect of the plasma response to the MPs on the ICRF system is discussed. It is shown that for the studied plasma conditions, j D UL plays a significant role on the response of the ICRF system. The paper is organized as follows: in section 2 the experimental setup is introduced. Plasma discharges with ICRF heating and MPs are described along with measurements of key RF quantities. The effect of different MP phasings on ICRF coupling is studied. In section 3, we compare the experimental results against 1D analytical formulas and experimentallyderived scalings found in the literature, in order to assess their validity under our experimental conditions. A statistical analysis of the data allows deriving a 1D scaling for the average loading resistance change under applied MPs. In section 4, numerical simulations are presented and compared to the experiments. In section 5, the experimentally-derived 1D scaling of section 3 is utilized to predict the average loading resistance changes in the ITER ICRF antenna due to MP plasma displacements. In section 6, we present the conclusions of this study.

ICRF system and measurements
Four ICRF antennas are in current use on ASDEX Upgrade: a pair of 2-strap antennas using boron-coated limiters (ICRH1 and ICRH3) and a pair of 3-strap antennas using tungsten coated limiters (ICRH2 and ICRH4, see figure 1(a)). Both 2-strap antennas are fed jointly from two RF generators connected together by a combiner-splitter system [7]. Each antenna is then individually matched by two stub tuners and a T-Junction that splits the power to each strap. The 3-strap antennas are also fed by two RF generators, such that one generator feeds the central strap of both antennas and the other one feeds the side straps. This is done to allow for different power balances and phasing between the outer straps and the inner strap of each antenna, in order to reduce the net RF image currents on the antenna limiters [19]. Each transmission line feeding a central strap as well as the ones that feed the side straps are matched with two stub tuners. In order to diagnose the change in ICRF coupling, measurements of the loading resistance, R L , are taken. The loading resistance gives a good estimate of the degree of matching between the transmission line-strap system and the plasma. Its measurement is performed by a directional coupler placed behind the generator-side stub tuner and a voltage probe located at the voltage antinode in the unmatched antenna-side transmission line, such that: where P V , coupled max and = W Z 25 0 are the power coupled to, the maximum voltage in, and the characteristic impedance of the unmatched transmission line. Due to the feeding arrangement in ASDEX Upgrade, measurements of the loading resistance are available for each individual transmission line-strap system, except in the case of the 3-strap antennas, for which a single measurement is present for both side straps together and another one for each central strap. The reader is referred to [20] for a detailed description of the ASDEX Upgrade ICRF antenna feeding arrangement.

Discharges description
ELM control requires the MP field to be applied with a specific poloidal mode spectrum which is amplified the most by ideal MHD modes excited in the plasma edge. These  modes create a field-aligned plasma displacement which has been well predicted and characterized in many tokamaks, such as ASDEX Upgrade [21,22], DIII-D [23] and MAST [24]. The induced 3D plasma displacement creates a nonaxisymmetric density profile that extends into the ICRF fast wave coupling region in the scrape-off layer (SOL). The degree of displacement and its topology depend upon the toroidal and poloidal mode number spectrum of the MP field, the edge plasma current and the edge pressure gradient. On ASDEX Upgrade, two toroidally distributed, poloidally spaced rings of 8 MP coils each, are installed as depicted in figure 2. It is then possible to modify the MP applied poloidal mode spectrum by varying the differential phase of the coil current waveforms in the two rings, termed Δj UL here.
A set of five H-mode discharges was performed in order to study the effect of MPs on ICRF coupling. In order to explore a reactor-relevant scenario, two main parameters were optimized: high plasma β N ∼2 and low electron and ion pedestal collisionalities n ñ -~-0.25 0.65, 0.15 0.25 e i * * . Time traces for the edge safety factor q 95 , plasma current I p , line-averaged electron density á ñ n e , deuterium puff D puff tot and applied total power P tot can be seen in 3 . A total heating power of P tot ∼11.5 MW was applied, out of which P ICRF ∼2.2 MW was delivered simultaneously by the four ICRF antennas in dipole phasing. This value is the generator power after subtracting the dissipated power in the 3 dB hybrid coupler loads and the Ohmic losses in the transmission lines, accounted for by vacuum measurements on the 2-strap antennas. The ICRF heating scenario was minority (H)-D, second harmonic D (ω ICRF =2ω c,D ), with central deposition at a frequency ν ICRF =36.5 MHz. The rest of the heating power was supplied by NBI (∼6.8 MW) and ECRH (∼2.5 MW). The discharges were fueled with deuterium gas by a single lower divertor valve located in sector 13 ( figure 1(a)). One of the discharges was performed as a reference (#34632), without MPs. For the rest of discharges the MPs were switched on at 2 s, during the flat top, and were supplied with I coil =5 kA× turns. Two differential phasings, Δj UL , were applied per discharge in n = 2 toroidally symmetric configuration and rigidly rotated with a frequency of ν 0 = 3 Hz for diagnostic purposes. The last two discharges terminated prematurely due to disruptions, and only one Δj UL phase was acquired for them, resulting in a set with  discharge with UL a sputtered tungsten flake produced a density increase from 1.5 to 3 s, hence this time slice is rejected in the analysis. The resulting average plasma beta poloidal, b á ñ t pol , electron and ion collisionalities, n n á ñ á ñ , e t i t * * , are presented in table 1 for reference. The collisionalities were computed according to the formulas included in [25,26] using the electron density and temperature profiles obtained from the integrated data analysis [27] and the ion temperature profiles obtained from charge exchange recombination spectroscopy (CXRS). All the profiles are evaluated at ρ pol =0.9 , i.e. at the pedestal top. The Z eff value was obtained from Bremsstrahlung emission. For the ion density, a fractional boron impurity percentage of n B /n e ∼0.005 was consistently computed by the CHICA code [28]. In order to represent all the impurity species, this value was scaled to match the measured Z eff , such that ( )~- , results in the ELM behavior to be unchanged by the MPs, that is, the discharges behaved as regular ELMy H-modes. This behavior is consistent with previous studies of ELM mitigation and suppression accessibility in ASDEX Upgrade [29].
Even though all discharges were performed with the same input waveforms on the actuators, some differences are noticeable. Discharges #34622 and #34632 have higher electron collisionalities than the others, which arise from lower pedestal electron temperatures. They were performed on the same experimental session and thus with very similar machine conditioning. On the other hand, the rest of discharges were distributed into two different experimental sessions, and the difference in plasma parameters is attributed to different machine conditioning and core MHD activity. Furthermore, since no feedback control was used on the density profile, the average value is seen to increase during the discharge time for all the performed experiments, due to NBI fueling and the forward gas puff. While electron density changes have an impact both on the plasma response to the MPs and on the ICRF coupling, such changes are taken into account for all the studied quantities, as discussed in section 2.5, through baseline correction and sinusoidal fits representing the average value over the considered MP periods. The plasma edge 3D displacement was evaluated with toroidally localized diagnostics in the same fashion as in [21,22]. The MP field is rigidly rotated and the diagnostics register variations in the plasma kinetic profiles locked to the external field as a time-dependent oscillation of their measurement quantity. In this study, we have used the lithium beam diagnostic [30], two chords of the CXRS diagnostic (termed CMZ: edge toroidal CXRS and CPZ: edge poloidal CXRS) [31], three X-mode reflectometry channels embedded in the 3-strap antenna ICRH4 [32], labeled as Ref. X{1,4,8} in figure 1, and the midplane O-mode reflectometer [33]. X8 below the midplane. The CXRS diagnostics track the + B 5 impurity line intensity corresponding to λ≈494 nm, which is taken as a proxy for the + B 5 impurity density, while in the lithium beam and reflectometers case we use the reconstructed electron density profiles. The plasma displacements are evaluated in time-domain by tracing the position (d LOS iso ) of constant intensity or density layers over the line of sight (LOS) of each diagnostic during the MP rotation. The data is ELM-filtered, and an ordinary least square (OLS) fit is performed to the position of these iso-layers. The OLS fit computes the estimators for the amplitudes of sine functions (| ( )| x n n j ) with up to 4 harmonics of the fundamental frequency, ν 0 , i.e.
, and the coefficients of a cubic polynomial (α k ) for baseline subtraction, such that: where ω j = 2πν j and f j is the phase of each sine function. The resulting displacement profiles of the density layers for | ( )| x n n 0 can be seen in figure 4 for the Δj UL = 0°and reference cases. During the MP phases, the plasma displacements decay monotonically in the SOL. This is wellregistered by both the lithium beam and the reflectometers, with the exception of a 'bump' observed in the lithium beam at around~-´n 7 8 10 m e 18 3 , which matches the position of the H-mode density shoulder [34]. Very good agreement is found among all the diagnostics in the region Table 1. Discharge numbers with MP configuration and average parameters of interest: number of used MP periods in the analysis, timeaveraged poloidal plasma beta b á ñ t pol , time-averaged effective charge number á ñ Z t eff , time-averaged electron/ion collisionality at the pedestal top n 3 , which includes the Rcutoff for all the ICRF antennas. However, in the far SOL no absolute agreement is found when MPs are active, despite the fact that the fluctuation level for ν 0 is very similar when no MPs are applied. The ICRF limiter position in densityspace is included along the lithium beam LOS, to give a reference of the measurements alignment and discard its effect on the displacements. 3 region. Measurements of the density profile in reflectometry depend on different parameters: X-mode reflectometry requires an accurate magnetic equilibrium reconstruction in order to correctly place the position of the first fringe, or first plasma echo [35]. The precision of the magnetic reconstruction is hampered when MPs are applied, since the used tool (the CLISTE code [36]), is a Grad-Shafranov solver that can only handle axisymmetric plasmas. Furthermore, there are built-in assumptions when computing the density profiles, such as a constant residual density for the wave upper cut-off determination, that can affect the density profile initialization [37]. Similarly, in O-mode reflectometry while the magnetic field plays no role, the assumption of a constant value for the ( ) On the other hand, the profiles should be less sensitive to these effects the farther they are from their initialization point, which can explain the good agreement with the lithium beam at the~n 10 m e 18 3 density layers. A detailed study of the differences between the diagnostics and possible improvements to X-mode reflectometry when a proper 3D magnetic reconstruction is used, goes however, beyond the scope of this paper. It should also be kept in mind that the lithium beam profiles become more inaccurate inside the pedestal region, 1. Reflectometry profiles are always used in relative terms, that is, to compare the changes between the minimum and maximum plasma-antenna gap distances. In this manner, systematic errors play a smaller role. 3 density layers further motivates their validity in this region. For the ICRF cutoff oscillation, either using the lithium beam or reflectometry density profiles does not change the conclusions of our study. The R-cutoff displacements are nearly equivalent for both diagnostics (we refer to figure 4(a)). 3. The ICRH4 embedded X-mode reflectometers offer a description of the poloidal distribution of the density profile. This information is very valuable to the study at hand, and unavailable otherwise.

The good agreement among all the diagnostics in thẽ
From now on, we will focus exclusively on three positions of interest for this study: the separatrix and the two main R-cutoffs for the 2-strap and 3-strap antennas. The iso-density and isointensity values for the separatrix are taken to be a time-average of all MP periods in a given Δj UL of the ρ pol =1 value, as determined by CLISTE, and were found to be »ń 1. . The lithium beam measurements are corrected for the geometrical angle between the diagnostic LOS and the normal vector to the axisymmetric CLISTE flux surface, such that a correct normal displacement,  x n , can be obtained. This results in displacements about ∼6% smaller than those measured by the diagnostic. The R-cutoff positions are obtained from equation (1) with the assumption of a H-D mixture of 5%-95% in the plasma, and using the CLISTE axisymmetric magnetic induction field calculation. This way, the total tokamak field is included in the analysis, although, partially inaccurate due to the assumption of pure axisymmetry. For the k P values, we will take those for which the predicted Fourier spectra of the field-aligned for the used ν ICRF ) is usually representative for dipole phasing, as the power balance between the central and outer straps was kept at 1.5:1 [20]. The resulting R-cutoff density is of the order of´n 7.5 10 m e cutoff 18 3 , which is positioned beyond the X-mode reflectometry density measurements, hence, only the 3 , that is well-resolved by X-mode reflectometry measurements, and thus are also included in the analysis. The | ( )| x n n 0 fit amplitudes as a function of the MP phasing, and the time traces for Δj UL =0°can be seen in figure 5. It is to be noted that the abscissa values in (a)-(c) are slightly different from the ones presented in table 1. The PSL model from [39] has been incorporated into the calculations, which results in slightly different Δj UL values from the ones applied to the coil current waveforms of the order of ∼3°.

From the measurements, it is clear that | ( )|
x n n 0 depends on the applied Δj UL , not only for the separatrix, but also for the ICRF R-cutoffs. The maximum separatrix displacements are seen around −90°<Δj UL <0°while the minimum displacements lie somewhere between 90°Δj UL 180°. In general, very good agreement is found at the separatrix between the CXRS diagnostics and the lithium beam, even though they sit at different poloidal positions. The biggest scatter in the data is seen for the Δj UL =90°case, which has the least number of MP periods available, and thus the poorest statistics, also resulting in larger measurement error bars. On the other hand, the O-mode reflectometer tends to overestimate the displacements in this region. In the SOL, the reflectometers and the lithium beam measure the same dependence on Δj UL differs slightly from that of the R-cutoffs. The density profile in the SOL can be locally affected by fueling and recycling processes due to plasma-wall interactions, as well as by ICRF induced density perturbations. In the case of discharge #34634, the early detachment of a tungsten flake is seen to increase the SOL density throughout the discharge. Therefore, these differences suggest that SOL physics and machine conditioning also play a role in the amplitude of the edge plasma displacements. Knowledge of the plasma response to the MPs becomes in this case not enough to predict which Δj UL phasings will have the biggest impact on ICRF coupling. Finally, while the complete non-axisymmetric structure will dictate the behavior of the ICRF antenna, the SOL trend already hints to which phasings might have the biggest impact on the ICRF coupling. A detailed study of the measured loading resistance is presented in the next sections.
2.4. Impact of perturbed SOL density profiles on ICRF coupling performance X-mode reflectometry density profiles and loading resistance measurements are presented in figure 6 for the Δj UL = {0°, 180°} cases and the reference discharge. All considered time traces were analyzed with ELM-filtered data. The density profiles are plotted along the LOS coordinate of the X-mode reflectometer channels d LOS , where d LOS =0 represents the reflectometer horn antenna position. Since the reflectometer uses X-mode, the maximum resolvable density differs among channels depending on their radial location inside the vessel. A clear density oscillation is observed in front of ICRH4 when the MPs are rotated. The R-cutoff position (included in the figure is the one for the 2-strap antennas) oscillates with the density profile, and the amplitude of the oscillation directly depends on the k P value, as it was observed in figure 4. Coherent loading resistance oscillations are registered, as measured in the two feeding lines of the same antenna. It is worth noting that when the MPs change from in-phase (Δj UL =0°) to out of phase (Δj UL =180°), the plasma response changes, decreasing the displacements in front of the antenna and thus the coupling oscillations. It is also seen that the loading resistance oscillations follow closely the density perturbation on Ref. X4, which is the reflectometer channel best aligned with the geometrical center of ICRH4. A more comprehensive way of plotting the density perturbation versus the loading change can be found in figure 7. There, we plot the density structure arising from the intersection of a poloidal plane with the three X-mode reflectometer channels. Since the gap between the ICRF antenna and the plasma changes poloidally, the distance between the ICRF antenna and the plane is also made to change accordingly. . A mapping function for the density has been used such that the oscillation can be commonly observed using the same normalized scale:  arrows. The antenna loading resistances have been fitted with an OLS fit, in the same fashion as indicated before, in order to obtain the times when the local maxima happen (dashed red and blue lines in figure 7). These lines for ICRH4 have been extended upwards to the density plot for comparison. The maxima are well correlated with the density perturbation when it is closer to the antenna center. We hypothesize that this is due to the ballooned structure of the perturbation on the LFS of the tokamak, thus covering a larger portion of the ICRF antenna the closer it is to its center. On the right, one-dimensional density profiles are plotted for the three reflectometer channels, representative of local maximum and minimum approach to the ICRF antenna position when the MP phasing was set to Δj UL =0°. The profiles were calculated by performing local averages centered at each extrema over 50 ms windows. Each local averaged profile is then averaged again together for all the maxima and minima separately. The density profiles for the reference discharge with no MPs are evaluated in the same way, and included as dashed lines. Error bars are added for some points as one standard deviation of the profiles. Significant displacements of the density profiles are seen close to the ICRF antenna in the MP case, while for the reference case the profiles have a steady position. In the next section, we will investigate how the different MP phasings affect the ICRF antennas loading resistance.

Loading resistance change as a function of applied MP phasing
At a constant safety factor q and pressure profile, the plasma kink displacement varies as a function of the applied poloidal mode spectra from the MPs, which we tune through different Δj UL . In the same fashion as observed in figure 6, the loading resistance variation of each feeder will depend on the plasma displacements. We perform an OLS fit to the loading resistances of each antenna feeder for every one of the applied Δj UL phasings, such that the amplitude of the coupling variation can be obtained. The procedure is displayed in figure 8, where the loading resistances are fitted for the case of Δj UL =0°. The loading resistances for the discharge with no MPs during the same time period are added for comparison. A sliding median filter with a window size of 51 points is performed to the no-MP time traces in order to better visualize the baseline level of the loading. Aside from the harmonic oscillation when the MPs are present, it is also observed that, on average, the loading resistance is higher with MPs than without them, as already observed in previous studies [14,15]. This observation is consistent with the plasma density increase in the SOL due to the pump-out effect, which is visible in the line-integrated density measurements in figure 3. In the lower plot, we present the loading variation in percentage as a function of the applied Δj UL . Here, ΔR L is the amplitude of the fundamental ν 0 component of each feeder and á ñ R L is the loading resistance time average of the considered window. Thus, the minimum loading resistance for a given Δj UL is taken to be á ñ -D R R L L and the maximum á ñ + D R R L L . The error bars representing the uncertainty of the OLS fit alone are comparable to the marker size, and thus have not been included. The noise level, marked as a horizontal dashed line on the lower plot, is calculated by performing the same OLS fit to the loading resistances for the discharge with no MPs, and then taking the average over the antenna pairs.
An interesting observation is that the loading resistance baseline increases during the discharge, especially for the 2-strap antennas. This is due to the general electron density increase in the plasma over time, also seen in figure 3. Since the baseline variation is effectively accounted for and removed by the polynomial fit, it is assumed throughout the analysis that the amplitudes of the OLS fit represent an average value of the loading resistance amplitude over the considered time trace. From the performed analysis, a clear dependence of the loading resistance change on the applied Δj UL is observed well above the noise level. This corroborates the picture of the coupling change being directly correlated to the plasma kink displacements. A maximum loading resistance change of ∼25% is reached, with a similar Δj UL dependence between the 2-strap antennas and the 3-strap antennas. The 3-strap antennas experience larger coupling changes, as it would be expected from their higher ~> -k 11 m 7.7 m 1 1 combined with the fact that the displacements decay radially in the SOL. Even larger coupling changes would be expected, well correlated with larger plasma displacements, if the MP field attenuation by the PSL becomes smaller, e.g. with slower MP rotation frequencies, or if larger currents are applied to the MP coils.

Experimental scaling of antenna loading resistance with cutoff distance. Comparison to 1D formulas
It is instructive to compare the experimental data obtained in these non-axisymmetric scenarios against one-dimensional analytical and experimentally-derived formulas, often used to describe the scaling behavior of the loading resistance with plasma parameters. In particular, we would like to know if the three-dimensionally perturbed cases behave similarly to radial scans of the (axisymmetric) plasma position in the limit of very small perturbations, and how large does the non-axisymmetry need to become before large deviations occur. For the comparison, we build data sets based on the displacements measured by the different diagnostics. We, therefore, use the density data measured by Ref.  maximum loading resistance for a given MP phasing. We will also evaluate the data sets for the different ICRF antennas separately and grouped by antenna type. Median values are computed over the feeders of the antennas for each MP phasing and then fitted to an exponential function of the shape ( ) = a -D f d e d fit cutoff from which the α value is obtained. The motivation behind this fitting function stems from the derived expressions in [6,40]. The radiation impedance of an ICRF antenna depends on the optical properties of the medium in front of it. If an evanescent region exists, we obtain from the cold plasma theory an scaling for the coupling resistance such that: k dl , with 6 c e the optical thickness of the medium, Ω e the domain that bounds the evanescence region, β a constant that depends on the assumptions made in the derivation and R c the coupling resistance, equal to the real part of the radiation impedance. In order to evaluate this integral, knowledge of the RF near fields is necessary. Some common approximations rely on stating k 0 , with k 0 the free-space wavenumber. Here, we acknowledge the dependence of η on the distance between the strap and the cutoff (size of the coupling region) and retain the functional form µ a R e L d cutoff for convenience. The fitting is performed with two statistical tools: OLS and orthogonal distance regression (ODR). OLS minimizes the squared vertical distance between the fit and the loading resistance measurements in order to determine α. ODR, on the other hand, finds the estimators which minimize the squared errors of the observed points for both the dependent and independent variables { 1 }. This approach should be more robust than OLS for obtaining the α estimator. The results for both approaches can be visualized in table 2. The covariance matrix is computed both for OLS and ODR, and a 99% confidence interval is constructed assuming the samples to be normally distributed.
Among the mentioned formulas found in the literature, we first turn our attention to the analytically derived one proposed in [6]. This formula is based on the cold plasma dielectric tensor formulation. The plasma is assumed homogeneous in toroidal and poloidal directions and the antenna is assumed perfect, producing a single k P value. It is further assumed that the only excited plasma mode is the fast wave, and thatn 2 increases linearly from the antenna to the R-cutoff within the linear model derivation. In order to allow flow of energy from the antenna to the cutoff, the poloidal electric field of the fast wave is decomposed in a sum of two opposite-'propagation' evanescent waves. The coupled power can then be expressed in terms of the plasma admittance, and in the limit of low optical thickness (0<η<3 ) it reduces to:  (7) can be used to give an estimate of the loading resistance change due to a radial movement of the cutoff position, Δd cutoff , such that: The results of the comparison between the experimental scaling and this analytical formula can be seen in figure 9. It is observed that the exponent correctly reproduces the experimental data within the given uncertainties for ICRH1, while it overestimates the changes of the other data sets. We now further compare the data to numerically and experimentally-derived scalings found in the literature. A statistical analysis of the ASDEX Upgrade 2-strap antennas coupling properties, combining L-mode and H-mode discharges, was performed in [15]. The numerically computed coupling resistance (R c ) from the Rsolver code was related to specific features of the experimentally measured density profiles. A scaling law was derived of the shape: where d min is the distance between the main R-cutoff and the minimum density gradient, ( ) n min e , used to characterize the effect of the density shoulder that often appears in H-mode density profiles.
( )( )  = -n n r r eff edge co edge co with n edge defined at the pedestal top following the convention described in [15] and r edge its position, =´n 3.5 10 m co 18 3 the main R-cutoff density for the 2-strap antennas and r co its position. ∇ max is defined as the maximum density gradient of the profile. A linear relationship was also found between the actual loading resistance as computed by the TOPICA code [42]  The evaluation of equation (10) is only possible with a diagnostic that covers the region from the far SOL to the pedestal top. We use lithium beam profiles to give an estimate of the importance of each term in the exponential factor under our experimental conditions. All discharges in table 1 are investigated. We again produce representative profiles analogously to previously performed with the X-mode reflectometers. First, the time traces are ELM-filtered and local maxima and minima density profiles are computed for each MP cycle as the median of the extrema neighboring profiles over ∼40 ms. Next, the median is computed again over all the local maxima and minima in order to yield representative profiles for a given Δj UL . The result of this procedure is seen in figure 10 for the Δj UL =0°c ase. Points for , min e e edge are added as per definition in [15]. It is found that in all studied cases 11.5 3.9 0.064 edge cutoff min eff ( )| D  0.004 max , and hence can be safely neglected. Among the rest of parameters | | -Dd 11.5 cutoff is usually much larger than 0.064 0.004 min eff max , except in the cases where Δd cutoff is very small, like in the Δj UL =180°c ase, or the density shoulder is not clearly defined and thus cannot be captured by the min(∇n e ) condition. The evaluation of all the MP phases yielded: 24.5 18.9, 11 cutoff where the uncertainty represents s 1 . We hence estimate the main parameter to be Δd cutoff and further reduce DR L approx -D e d 11.5 cutoff . This scaling, which approximates in the lower limit the expected loading resistance change computed by TOPICA for the 2-strap antennas, overestimates the measurements here presented, only agreeing occasionally with some of the feeders as seen in figure 9. Naturally, this scaling can only be applied to the 2-strap antennas, since it was performed prior to the installation of the 3-strap antennas in ASDEX Upgrade and the statistical analysis was never repeated for these.
A purely experimental scaling was also derived for the ASDEX Upgrade 2-strap antennas when studying the influence of different gas puffing locations on the loading resistance [41]. The O-mode reflectometer was utilized to find the distance from the antenna to the R-cutoff, and this was correlated to the loading resistance measured by a set of voltage and current probes in the unmatched transmission lines (not used in this paper). The scaling was derived by uniquely  [6] is added as a blue dashed curve. The scaling predicted in [15] is added as a dashed green curve. The scaling predicted in [40] is added as a yellow dashed curve. The scaling measured in [41] is added as a magenta solid curve. The marker size is representative of the vertical OLS 1σ uncertainty.
using H-mode discharges with divertor gas puff. The reported exponent factor was α∼18 , i.e. D µ R e L d 18 cutoff . This factor greatly overestimates the measured loading resistance for all the antenna feeders. Despite this fact, it will be seen in section 4 that this scaling is in very good agreement with 1D ICRF coupling simulations.
In Tore Supra, the measured loading resistance of a database of ICRF heated discharges could be fit with the exponent α ∼ 17.3 [40]. This scaling compares favorably to the one found in ASDEX Upgrade (α ∼ 18), which is further accentuated by the fact that both scalings were obtained with 2-strap antennas, despite the slightly different k P , i.e.

~>
-k 9.1 m 7.7 m Tore Supra 1 1 . A generalization of the experimental scaling was proposed from purely theoretical grounds stemming from the cold plasma approximation, such that: Here, β=2 and h~á ñ k d x cutoff in equation (6). The value for á ñ k x was taken to be approximately the maximum k P of the radiated power spectrum in the plasma, as computed by the ICANT code. The change in loading resistance would be: We utilize the RAPLICASOL code as described in the next section in order to better estimate the radiated power max(k P ) for the ASDEX Upgrade 2-strap antenna coupling to an axisymmetric density plasma. We assume the fast wave to carry most of the radiated power, and therefore, we compute the power spectral density (PSD) of the field-aligned RF magnetic field component, which can be seen in figure 13. The vacuum profile is evaluated in the private region in front of the ICRF antenna, close to the FS. The PML profile is evaluated at the end of the simulation domain, where outward radiation boundary conditions exist, and therefore, where we consider the power to be coupled to the core. We have obtained (| |) ~-k max 7.45 m FS 1 , in good agreement with the value used in the experimental determination of the R-cutoff. We therefore approximate k 0 , and included this value in the scaling displayed in figure 9. It can be seen that this theory-based scaling also overestimates the measured change in loading resistance.
With the presented comparison, we find that the diverse scalings overestimate the measured loading resistance change in the investigated non-axisymmetric configurations. The best agreement is found with Bilato's scaling for the 2-strap antennas. Nevertheless, this scaling clearly departs from the other experimental and simulation observations reported in the literature. Particularly, it strongly diverges from the experimental 1D scaling in axisymmetric conditions obtained by Jacquet. A marginal improvement can be found if the 2-strap antenna current spectrum ~-k 9.25 m current 1 is used instead (as suggested in [6]), leading to µ R e L d 10 cutoff . This departure could be attributed to the assumption thatn 2 increases linearly from the strap to the cutoff. In reality, it is likely that a sharp decrease in plasma density occurs inside the ICRF limiter. Under these circumstances, the step model also proposed in [6] could be a better representation of reality, but has not been here investigated.
It should also be remarked that the experimental scalings here presented were obtained under different (and axisymmetric) plasma conditions, with unknown S-matrices, making a comparison with our measured data only qualitative, but still valuable. In addition, we have purposely not included error bars for the scaling laws, since even though they would help to improve the agreement, our objective is simply to compare the experimental data sets against the general trend. We can also attempt to give an explanation to the measured scaling deviation from the ones of the literature in the following manner: in opposition to axisymmetric plasma movements, where the whole plasma column is moved towards/away from the ICRF antenna, the poloidal and toroidal inhomogeneities brought by the MP plasma deformation can make part of the plasma to be closer to the antenna, and another part to be farther away simultaneously with respect to the axisymmetric case. It can be seen in figure 7 that when the plasma is the closest to the antenna center, it is farther away at the antenna top and bottom. An 'integration' of this 3D density structure by the relatively large (compared to the plasma size) ASDEX Upgrade antenna could, therefore, result in a smaller net effect on the coupling characteristics. Thus, one would expect the loading changes to scale slower with the measured 1D cutoff deformation amplitude than those produced by axisymmetric movements of the plasma volume. In other words, if the change in the 3D deformation could be represented by a single 'effective' integrated one-dimensional displacement áD ñ d cutoff eff , we would likely find áD ñ <D d d cutoff eff cutoff . A similar integration was already attempted in [7] for an axisymmetric plasma with changing triangularity, and when weighted with a filtering factor to account for the exponential evanescent behavior, resulted in a consistent scaling of the measured loading resistance. This approach could also shed some light on why the found radial decay parameters, α, are very similar between the 2-strap and the 3-strap antennas, which is, in principle, an unexpected result due to their different k P . Such study would, however, require precise knowledge of the 3D magnetic topology and needs to be supported by dedicated MHD and/or transport simulations.
It can also be noted that the loading resistance change measured in different feeders scatters more significantly the higher the cutoff oscillation amplitude becomes. Systematic errors in the V max determination between different feeders can become more apparent the larger the perturbation grows. Furthermore, cross-coupling terms in the unknown S-matrices play a role in these differences, which is difficult to asses since no direct measurements of the antennas S-matrices are currently possible. However, it is also plausible that the larger scatter between the feeders' experimental data can be partly explained by the poloidal and toroidal effects on coupling becoming of greater importance. In this situation, it is not possible to approximate the coupling change in each feeder accurately with a single 1D scaling anymore (i.e. a single value of Δd cutoff ), but rather, a functional description of the Smatrix dependence on the 3D density profile becomes mandatory.
In order to better elucidate the impact of arbitrary plasma asymmetries on ICRF coupling, and other important antenna performance quantities, further dedicated 3D experiments and modeling must be performed in the future.

ICRF simulations using the COMSOL based RAPLICASOL code
In the previous section, we investigated the scaling of the experimentally measured loading resistance change with respect to the 1D cutoff movements measured by different diagnostics. We found that the derived 1D experimental scaling deviated from previous experimentally-derived and analytical scalings found in the literature. In this section, we focus on the numerical change of the loading resistance when 1D density profiles are fed into an ICRF coupling code. By doing so, we want to test our expectation that considering a 1D density profile for the whole ICRF antenna array, even if representative of the OMP measurements, is not enough to replicate the experimental results, which are inherently of three-dimensional nature. Furthermore, we can test how large the deviation is between the experimental results and the numerical predictions. We will use for the simulations the COMSOL based RAPLICASOL code [43][44][45]. RAPLICA-SOL is able to handle the realistic geometry of ICRF antennas, and perform full-wave simulations on a finite element mesh. Maxwell equations are solved in vacuum or plasma conditions, imposed via a dielectric tensor. A 'perfectly matched layer' (PML) provides an absorbing boundary at the radial, toroidal and poloidal ends of the simulation domain, that serves to mimic the outward radiation boundary condition, i.e. no wave reflection. So far, only flat models of the ASDEX Upgrade antennas had been used in RAPLICASOL studies. In order to be as close as possible to the experimental conditions, and to avoid artificial geometric transformations of the density profile from the tokamak toroidal geometry to the usual simulation flat geometry, a new curved model of the ASDEX Upgrade 2-strap antenna has been developed and used, in addition to the usual flat model for comparison. Both models are presented in figure 11.
In order to provide representative density profiles for the given experimental conditions, we will use the lithium beam diagnostic, which most extensively covers the ICRF coupling region. We follow the same procedure previously introduced: first, an OLS fit was performed to the ELM-filtered lithium beam time traces for each Δj UL , and the time points of local density maxima and minima were found, corresponding to the minimum and maximum antenna-R-cutoff gap distance. Second, all density profiles in a given time-window centered at the local extrema are averaged. Third, all the maxima, and, all the minima profiles are averaged together. The time window length is adjusted in order to ensure that the final profiles preserve, as close as possible, the same R-cutoffs and separatrix displacements as the ones obtained via OLS in figure 5. A time window of ∼40 ms was found to yield the closest agreement between OLS and the profile reconstruction procedure. Since the angle between the lithium beam LOS and the normal vector to the axisymmetric separatrix is of the order of ( ) a ã 20 , cos 0.94 libe libe , this is regarded as a second-order contribution and has not been corrected for in the reconstruction. The resulting density profiles are displayed in figure 12.
We now compute with the curved antenna model the PSD of the field-aligned RF magnetic field component, which can be seen in figure 13. The obtained ( ) ~-k max 7.45 m 1 in the vacuum region is in good agreement with previous HFSS models of the ASDEX Upgrade 2-strap antennas [20], giving confidence in its robustness. Furthermore, we have constructed 3 is set in the plasma domain in order to ensure no density gradients exist in the plasma-PML interface. The curved model is run with the direct solver, whereas the flat model is converged with the iterative solver, setting 1% relative tolerance. The mesh was set with hexahedral elements of a characteristic radial length of ∼1 cm for the curved model, whereas the flat model uses tetrahedral elements with a radial length of ∼2 cm. Both resolutions should be enough to resolve the perpendicular wavelength of the fast wave in the coupling region, which for our conditions is of the order of (| |) lmin 15 cm. Two sets of simulations are performed: using the profiles in figure 12 for comparison between the maximum and minimum approach to the ICRF antenna per MP phasing, and a radial scan using only the density profile for Δj UL =0°'maxima'. The radial scan was performed by outward shifting the density profile in steps of 4 mm and thickening the vacuum layer in each iteration by the same width. With these two sets, we are able to distinguish between loading resistance changes due to a pure radial shift of the density profile from coupling changes that include variation in the density gradients. The computed S-matrices are transformed into loading resistances by assuming a perfect coaxial line with Z 0 =25 Ω characteristic impedance, such that [46]: 13 We present the simulation results, alongside with the scaling laws introduced in the previous section in figure 14. , the predicted change exceeds that of experiments by a factor of ∼3 for both the curved and flat models. The radial scans display a similar scaling to that obtained when using the full density profiles. We also find that the experimental scaling proposed by Jacquet for the ASDEX Upgrade 2-strap antennas in axisymmetric plasma radial scans is in excellent agreement with the 1D simulations. This observation further validates the RAPLI-CASOL results, and supports the conclusion obtained in the previous section.

1D empirical extrapolation to the ITER ICRF system
It has been shown in table 2 that the experimentally-derived ODR exponents for the combined data sets of 2-strap and 3-strap antennas are very similar. This is despite the fact that the two types of ICRF antennas have different geometry, k P wavenumber spectra and thus R-cutoff positions. We can take advantage of this fact to forecast the average coupling change of an ASDEX-like ICRF antenna in an ITER-like plasma. This exercise is meant to provide an overview of the similarities and differences between both machines, and will highlight the main knowledge gaps that need to be bridged in the future.
We will use the same scaling function with the experimentally-derived radial decay parameter α. If each of the derived α from the combined 2-strap and 3-strap data sets are taken to be independent realizations, and assume α normally  monotonically in the SOL, registered by the lithium beam and reflectometry diagnostics. However, the SOL can be noticeably affected by local sources and sinks, meaning that the SOL displacement dependence on Δj UL might not correspond to that of the separatrix.
The loading resistance measured in each ICRF antenna feeder reacts accordingly to the SOL density modification, well diagnosed by embedded X-mode reflectometry in three different poloidal positions. Two overlapping effects participate in the loading resistance changes: the plasma pump-out effect, seen as an average increase of the loading resistance with respect to a reference discharge with no MPs, and the rotating kink displacement, seen as a coherent oscillation of the loading resistance locked to the MP rotation. The maximum change in loading resistance is experienced approximately when the density profile is shifted closer/farther to the antenna center. For the studied cases in ASDEX Upgrade, the net effect is an increase of the overall loading resistance at the expense of time variable coupling conditions. For the first time, it is reported that the loading resistance changes do depend on the poloidal phasing of the MPs. A 1D scaling is derived that relates the change in the distance between the ICRF antenna and fast wave R-cutoff, as measured by several diagnostics, to the average change in loading resistance. This scaling is seen to predict smaller average loading resistance changes with Δd cutoff than that of scalings previously reported in the literature. It is plausible that the underlying reason is the 'integration' by the ICRF antenna over the whole 3D perturbed density profile, which would result in a milder loading resistance change than that produced by axisymmetric plasma movements.
ICRF simulations are performed with the 3D full-wave solver RAPLICASOL code. The input density profiles have been reconstructed from the lithium beam diagnostic. The magnetic induction field is treated in 1D approximation, decaying with 1/R. The predicted loading resistance changes exceed that of experiments by a factor of ∼3 when using either the curved or flat ASDEX Upgrade 2-strap antenna models. This prediction is in very good agreement with previous coupling studies evaluated in axisymmetric plasma conditions and highlights the need to treat the problem in a full 3D fashion, as the reduction to 1D prevents predictive capabilities.
The experimentally-derived scaling exponent is utilized for a prediction of the change in loading resistance for the ITER ICRF antenna in MP active plasmas. This scaling predicts changes up to áD ñR 65% L when the R-cutoff displacement can be identified with the maximum predicted separatrix displacement by different MHD simulations. However, this scaling allows only to give an estimation on the average array behavior, and not on the individual straps, for which the S-matrices need to be known. Furthermore, it is expected that the lower k P value used in ITER and the relative size increase between the ITER plasma and ICRF antenna of l l2.2 r r ITER AUG will affect this prediction. In order to diagnose the full effect of MPs on the ITER ICRF system, where a complex interplay exists among the 8 strap triplets, and the ratio between ICRF antenna size to the plasma perturbation characteristic wavelength in the midplane is different from ASDEX Upgrade, the problem needs to be addressed by considering a 3D magnetic field, density and antenna model altogether. Lastly, we would like to point out that coupling oscillations such as the ones presented in this study can have implications for antennas working close to their operational limits. If the plasma cutoff displacement becomes large enough to revert the benefit of the induced pump-out, the loading resistance can decrease below the level with no MPs for some rotation phases. This would lead to higher voltages in the antenna structure. Nevertheless, such a situation has not been observed in the studied scenarios.
Overall, the data here presented should help clarify the possible impact of plasma asymmetries on the total delivered RF power to the core plasma, help improve the matching and controller strategy, as well as shed some light on the implications on RF image current compensation on ICRF limiters.