# Analysis of the HAE activity in the TJ-II stellarator using a Landau closure model

J. Ortiz, J. Varela, L. Herrera Quesada, P. Pons-Villalonga, A. Cappa, U.

Losada, F. Papoušek, B. Ph. van Milligen, L. Garcia, D. Spong, et al.

► To cite this version:

J. Ortiz, J. Varela, L. Herrera Quesada, P. Pons-Villalonga, A. Cappa, et al. Analysis of the HAE activity in the TJ-II stellarator using a Landau closure model. Physics of Plasmas, 2025, 32 (7), pp.072506. 10.1063/5.0263300. hal-05468250

HAL Id: hal-05468250

https://hal.science/hal-05468250v1

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# Analysis of the HAE activity in the TJ-II stellarator using a Landau closure model

J. Ortiz

E-mail: juortiz1@fis.uc3m.es

Aix Marseille Univ, CNRS, Centrale Med, M2P2, Marseille, France

Universidad Carlos III de Madrid, 28911 Leganes, Madrid, Spain

Institute for Fusion Studies, University of Texas, Austin, TX 78712, United States

J. Varela

Institute for Fusion Studies, University of Texas, Austin, TX 78712, United States

Universidad Carlos III de Madrid, 28911 Leganes, Madrid, Spain

L. Herrera Quesada

Universidad Carlos III de Madrid, 28911 Leganes, Madrid, Spain

P. Pons-Villalonga

Laboratorio Nacional de Fusion, Av. Complutense 40, 28040 Madrid, Spain

A. Cappa

Laboratorio Nacional de Fusion, Av. Complutense 40, 28040 Madrid, Spain

U. Losada

Auburn University, 182 S College Street, Auburn, AL 36849, USA

F. Papoušek

Laboratorio Nacional de Fusion, Av. Complutense 40, 28040 Madrid, Spain

B. Ph. van Milligen

Laboratorio Nacional de Fusion, Av. Complutense 40, 28040 Madrid, Spain

L. Garcia

Universidad Carlos III de Madrid, 28911 Leganes, Madrid, Spain

D. A. Spong

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA
Y. Ghai

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8071, USA

M. A. Ochando

Laboratorio Nacional de Fusion, Av. Complutense 40, 28040 Madrid, Spain

C. Hidalgo

Laboratorio Nacional de Fusion, Av. Complutense 40, 28040 Madrid, Spain

Abstract.

The aim of this study is to analyze the stability of helical Alfven Eigenmodes (HAE) in TJ-II discharges and the stabilizing effect of the energetic particles generated by the neutral beam injector (NBI) on pressure gradient driven modes (PGDM). HAE and PGDM stability is studied using the linear version of the gyro-fluid code FAR3d and the continuous structure by the STELLGAP code. First, AE and PGDM activity observed in the experiments is reproduced by the simulations, identifying unstable $m/n = 4/7-2/3$ and $7/12-5/8$ HAEs triggered around $\rho = 0.66$ showing a frequency of 209 and 204 kHz, respectively, as well as 5/3 PGDM. Next, a parametric study is performed with respect to the thermal ion density and iota profile at the middle-outer plasma region to verify the robustness of the simulation results with respect to the uncertainty of the experimental profiles. The analysis confirms the experimental uncertainty does not cause large deviations in the simulation results, showing the destabilization of the same HAEs for all the configurations tested. The simulations also indicate the decay of the 5/3 PGDM growth rate as the EP population in the plasma increases, consistent with the experiment. The stability analysis of the $n = 3, 7, 11, n = 5, 9, 13, n = 6, 10, 14$ and $n = 8, 12$ helical families is performed with respect to the NBI operational regime for different EP energies, $\beta$ and deposition profiles. The most unstable configuration is the radially localized on-axis NBI operation (stiff EP density profile gradients nearby the magnetic axis). Using the simulation model that reproduces the observed Alfvén activity, we extend the study to analyze NBI performance within a theoretical framework. It shows that increasing NBI voltage (which raises EP energy) leads to a degradation in NBI performance for a given power (related to EP $\beta$ and their density). To achieve better NBI operation, higher voltage must be balanced with lower injection power, ensuring stable AEs while keeping the same EP $\beta$.

PACS numbers: 52.35.Py, 52.55.Hc, 52.55.Tn, 52.65.Kj

Keywords: Stellarator, TJ-II, MHD, AE, energetic particles
# 1. Introduction

Energetic particles (EPs) generate instabilities that can negatively impact the performance of fusion devices. Therefore, it is crucial to understand their effects reflected in experiments and theoretical predictions. To achieve this, tools and analysis methods must be developed and tested, providing an explanation for these phenomena through theoretical models based on the current understanding of plasma physics.

EPs generated by neutral beam injection (NBI) can destabilize Alfvén Eigenmodes (AE) driven in the gaps of the continuum [1, 2]. The presence of these modes leads to increased EP losses before thermalization [3] reducing the NBI heating efficiency [4, 5]. These Alfvén Eigenmodes are classified as: Toroidicity-induced Alfvén Eigenmodes (TAE) coupling m with m+1 modes [6, 7, 8] (m is the poloidal mode number of the shear wave), Ellipticity-induced Alfvén Eigenmodes (EAE) coupling m with m+2 modes [9, 10], Non-circularity-induced Alfvén Eigenmodes (NAE) coupling m with m+3 or higher modes [9, 11, 12], Beta-induced Alfvén Eigenmodes (BAE) driven by compressibility effects [13, 14] and combinations of Alfvén Eigenmodes coupling modes n and n+N (n is the toroidal mode number of the shear wave and N is the number of device periods) induced by helicity (HAE) [15, 16]. HAEs are of special relevance to the instabilities detected in TJ-II [17, 18, 19, 20] and LHD [21]. HAE also have been predicted for other configurations as QPS [22], CFQS [23] and could also been destabilized in W7-X if the NBI power is increased [24].

Pressure gradient driven modes (PGDM) are instabilities that deteriorate the plasma confinement, triggered if the thermal plasma $\beta$ exceed a given threshold as the pressure gradient in the plasma increases [25]. Interchange and ballooning modes are examples of PGDM. They become destabilized when the averaged (interchange) or local (ballooning) magnetic field curvature of the device is unfavorable. [26, 27].

TJ-II device is a heliac type stellarator with major radius 1.5 m, minor radius 0.22 m, four periods, average magnetic field on axis 0.95 T, and plasma volume $\le 1 \text{ m}^3$). Plasma discharges of 150 – 250 ms are sustained using Electron Cyclotron Resonance Heating (ECRH) (two gyrotrons, 53.2 GHz, $P \le$

300 kW each) and Neutral Beam Injection (NBI) heating (two H0 injectors, $E \le 32$ kV, and $P \le 600$ kW each). This article is dedicated to analyze the instabilities detected in TJ-II using electrostatic probes and magnetic coils at low (0 – 10 kHz), medium (10 – 100 kHz), and high (100 – 250 kHz) frequency ranges. Previous studies identified low frequency instabilities as PGDM and high frequency instabilities as global Alfvén eigenmodes (GAE) and discrete shear-Alfvén eigenmodes induced by a non-rotating magnetic island (MIAE) [28]. Nevertheless, the analysis did not include the effect of non inductive currents and the study was performed using the vacuum iota profile. The modification of the magnetic field topology due to non inductive currents, particularly the NBI current drive, is included in the present study.

A set of linear simulations are performed reproducing the AE and (PGDM) activity observed in the experiment, as well as identifying the optimal NBI operational regime to improve plasma heating by minimizing the AE activity [18]. In addition, the stabilizing effect of the EPs on the stability of PGDM is analyzed.

The analysis is conducted using the gyro-fluid FAR3d code [18, 29, 30, 31, 32, 33]. FAR3d solves the reduced linear resistive MHD equations to describe the stability of the thermal plasma coupled with moments of the gyro-kinetic equations that describe the destabilizing effect of the EP. The quilibrium is calculated using the VMEC code [34].

The FAR3d code has been verified through participation in bench-marking studies of gyro-kinetic and hybrid codes [35] and shows a reasonable agreement with the experimental data. FAR3d has been validated against linear AE stability in LHD [36, 37, 38, 39], TJ-II [18, 40, 41] and DIII-D [42, 43]. The nonlinear version of the code has analyzed successfully sawtooth-like and internal collapse events as well as EIC and TAE burst observed in LHD plasma [44, 45, 46, 47, 48, 49] and AE saturation in DIII-D plasma [50].

This study is part of a global project dedicated to analyze the AE stability in devices with different magnetic traps, including stellarators as LHD [51, 52], TJ-II [18, 20], Heliotron J [53, 54], W7-AS / X [55], CFQS [23] and QPS [22] as well as Tokamaks as DIII-D [56, 57, 43, 58, 42], JT60SA [59, 60], ITER [61] and CFETR [49], identifying the operational scenarios that show an optimal plasma heating with respect to the NBI operational regime, thermal plasma properties
and magnetic configuration.

This paper is organized as follows. The experiments are described in section 2. The numerical scheme and equilibrium properties are presented in section 3. The simulations dedicated to reproduce the instabilities observed in the discharge, the effect of the profiles uncertainty in the analysis results, the stabilizing effect of EPs on PGDM and AE stability optimization trends with respect to the NBI operational regime are discussed in section 4. Next, the conclusions of this paper are presented in section 5.

## 2. Experiment

A set of experiments in TJ-II, discharges 43702 to 43709 [62, 63], were performed to study the AE activity in plasma heated by co-injected neutral beam (NBI1) [52, 20]. The beam is injected parallel to the toroidal field, resulting in an increase in rotational transform at the core through NBI-driven currents.

Instabilities are measured through Langmuir probes [64] and Mirnov coils [65], as shown in Figure 1. Langmuir probes are capable of determining electron temperature, electron density, and electric potential of a plasma in the periphery while Mirnov coils allow to estimate the amplitude and frequency of the perturbations of the magnetic field outside the plasma.

The experiment consists of progressively inserting a Langmuir probe 2 mm into each shot for plasmas with identical parameters, in order to have more precise information about the Alfvénic activity generated at the periphery.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Scheme of the TJ-II device divided by sectors indicating the position of the Mirnov coils (normal font) and the heating systems NBI and ECRH (bold font).</figcaption></figure>

To provide an overview of the main plasma

parameters we aim to replicate in each shot, Figure 2 shows data from shot 43702. Electron cyclotron emission diagnostic (ECE) [66] is used for measuring electron temperature. The ionic temperature is measured with a Neutral Particle Analyzer (NPA) [67], which measures the distribution of neutrals. Based on the distribution of neutrals and the charge exchange at low energies, an ionic temperature can be extracted. Plasma current and line density come from standard Rogowski coil and microwaves interferometer.

To provide an overview of the main plasma parameters we aim to replicate in each shot, Figure 2 shows data from shot 43702. Electron cyclotron emission diagnostic (ECE) [66] is used for measuring electron temperature. The ionic temperature is measured with a Neutral Particle Analyzer (NPA) [67], which measures the distribution of neutrals. Based on the distribution of neutrals and the charge exchange at low energies, an ionic temperature can be extracted. Plasma current and line density come from standard Rogowski coil and microwaves interferometer. The sign reversal observed in the plasma current during NBI injection is due to the inductive coupling between the plasma and the ripple of the external magnetic coils. This effect induces screening currents that can momentarily lead to negative plasma current values, especially in the low-current regime of TJ-II. [68].

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Discharge 43702. (a) Injection power of ECH 1 (red line, $P_{ECRH1}$), ECH 2 (blue line $P_{ECRH2}$) and NBI (black line $P_{NBI1}$) Red and blue are almost identical and superimposed. (b) Line integrated plasma density ($\langle n_e \rangle$ blue line), thermal electron temperature at the magnetic axis ($T_e^0$, red line) and thermal ion temperature ($T_i^0$, green line). (c) toroidal plasma current ($I_p$, black line) and diamagnetic energy ($W_p$, green line).</figcaption></figure>

The red and blue lines of panel a indicate the plasma is firstly heated using ECRH, and from $t = 1140$ ms, the NBI heating phase begins (black line).
Panel b shows the thermal plasma density (blue line) and temperature (red and green lines) are almost stable during the NBI phase. The black line in panel c shows the toroidal current has rather large excursions during the NBI phase, thus the iota profile evolves along the discharge. The large oscillations observed in the plasma current are generated by ripple in the currents that flow through the main coils of the device [65]. These are very close to the plasma column and couple inductively.

Figure 3 shows spectrograms of magnetic fluctuations measured by Mirnov coils data for the discharges 43702, 43704, 43706 and 43708. The characteristic frequency of the observed instabilities is ~200 kHz showing up/down frequency sweeping between 150 and 230 kHz. It should be noted that the intensity of low-frequency instabilities, in the following identified as PGDM, decays once the ~200 kHz instability arises, pointing out correlations between EP population and PGDM activity. The evolution of the modes frequency and the plasma toroidal current (red lines) are also correlated because the iota profile change during the discharge, affecting the AE activity and frequency range [19, 69]. Among the modes appearing at frequencies close to 200 kHz, there are distinct higher-intensity modes at slightly below 200 kHz and other modes that follow the same trends but at frequencies approximately 30 kHz higher.

It is important to note that the oscillations observed in the mode frequencies are characteristic of this type of edge localized mode in this device. As mentioned above, the fluctuations in plasma current are attributed to the proximity of the plasma to the central coil, that influence the mode frequency through variations in rotational transform. The presence of these oscillations reflects the specific dynamics of the modes being analyzed in this study.

Focusing on defining the most intense mode and its position within the plasma, Figure 4 presents the Langmuir probe measurements, indicating plasma potential and filtered data across various frequency ranges. These include low-frequency modes such as PGDM or zonal structures (1–10 kHz), low-frequency AEs and PGDM between 10 and 50 kHz, and high-frequency AEs in the range of 100 to 250 kHz. Additionally, spectroscopic graphs for the same shots as in Figure 3 are shown for the frequency range of 100 to 300 kHz. The mode detected for this periphery measurements matches with the modes observed in Figure 3 slightly below 200 kHz and the perturbation analysis indicates that it is extending inward from the plasma periphery. Although it is not possible to precisely determine the mode's position, since it is unfeasible to insert the Langmuir probe into the plasma core, the maximum of the perturbation has to

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Magnetic fluctuations spectrogram of the discharges (a) 43702, (b) 43704, (c) 43706 and (d) 43708 during the NBI heating phase. Plasma current is shown as the $I_p$ red line.</figcaption></figure>

be located below $\rho = 0.8$ due to no maximum is reached in the data and tendency keeps growing towards the inner plasma.

It is also worth noting that, although the magnetic probes detect two peaks in the frequency spectrum, the Langmuir probe only detects one. This is because
the Langmuir probe provides a local measurement near the plasma edge, while the magnetic coils integrate the effect of perturbations across the entire plasma. Therefore, modes located further inside may not be captured by the Langmuir probe, but can still generate a measurable magnetic signature.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Plasma potential (a) and perturbation amplitude in the frequency range of 1-10 kHz (b), 10-50 kHz (c) and 100-250 kHz (d) obtained with the floating potential oscillations data using the Langmuir probe for the discharges 43702 to 43709 averaged over the interval 1200-1220 ms. Spectrograms obtained using the Langmuir probe for the shots 43702, 43704, 43706, and 43708 (e).</figcaption></figure>

Figure 5, panel a, shows the mode number of the low frequency instability measured in the discharge 43707. For all the discharges analyzed, the poloidal mode number identified is $-4 \pm 1$. An example of the high frequency modes, calculated during the local maxima of the instability frequency in the discharge 43704 of 216 kHz, indicates instabilities with poloidal mode numbers $4 \pm 1$ and $9 \pm 1$. Since poloidal array of Mirnov coils is not located in a constant toroidal angle in magnetic coordinates the result of the mode number analysis depends on the actual toroidal mode number of the perturbation, which is unknown.

Summarizing the experimental analysis, the most intense detected mode oscillates around 200 kHz and is located at the plasma periphery below $\rho = 0.8$, with possible poloidal mode number candidates of

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Lomb periodogram [70] of (a) discharge 43707 at $t = 1164$ ms for the low frequency mode (22 kHz) and (b) discharge 43704 at $t = 1190$ ms for the high frequency mode (216 kHz). White text indicate the poloidal modes numbers and uncertainty.</figcaption></figure>

$m = 9 \pm 1$ and $m = 4 \pm 1$. As we observe in figures 3 and 4, an increase in current leads to an increase in this mode's intensity and a decrease in its frequency. Low-frequency PGDM modes are also detected, which disappear as the peripheral mode emerges, with potential poloidal mode number candidates of $m = 4 \pm 1$. Additionally, other modes of lower intensity appear with frequencies approximately 30 kHz higher than the peripheral mode, showing a similar behavior, as well as a mode around 100 kHz for which localization cannot be determined.

### 3. Numerical model

The FAR3d numerical model for the thermal plasma uses a set of reduced MHD equations retaining the toroidal angle dependency in a three-dimensional equilibrium [34, 29]. The EPs perturbation is introduced by moments of the gyro-kinetic equation truncated with a closure relation. The closure coefficients are optimized to bring in the effects of the parallel Landau resonance (Landau coefficients) [71] in the first (EP density $n_f$) and the second (EP velocity parallel to the magnetic field lines $v_{\|,f}$) moment equations. Landau closure coefficients used for the model calibration are calculated by matching the analytic TAE growth rate of the two-pole approximation of the fluid plasma response function with a kinetic response function for energetic particles obeying a Lorentzian distribution. The lowest order Lorentzian can be matched to a Maxwellian or to a slowing-down distribution by choosing an equivalent average energy (a Maxwellian for the FAR3d model). The FAR3d includes a velocity averaged grad-B orbit drift propagator in each of the fast ion moment equations. This comprises radial, poloidal, and toroidal grad-B drift components. Thus it effectively incorporates the average orbit deviation from magnetic surfaces by factoring these drifts into the fast ion
moment dynamic evolution.

The effect of the electron and ion Landau damping is not added in the simulations because the effect is negligible due to the low temperature and density of the plasma. The EP FLR stabilization effects are taken into consideration although the impact in the simulations is rather small as is indicated in the Appendix, reason for this damping is no further discussed in the text. It should be noted that the simulations only reproduce the destabilizing effect of passing EPs injected by the tangential NBIs (zero pitch angle), thus the destabilizing effect of the EP trapping is neglected in the study assumed small compared to the passing EP. References [72, 73] include further details about the model equations and numerical scheme.

### 3.1. Equilibrium properties

A fixed boundary VMEC equilibria is calculated for the TJ-II configuration with a Hydrogen plasma based on the discharge 43702. The initial iota profile (vacuum) corresponds to the vacuum configuration, prior to the onset of beam-driven current (the thermal plasma $\beta$ of the discharges is low). The iota profile of the equilibrium is calculated including the current drive generated by NBI1 injection [68], shown in Figure 6. The methodology is inspired by previous work on current penetration and ripple effects in TJ-II [20]. It is considered as the reference case of the study, indexed by the letter O in the following, used to reproduce the AE and PGDM activity in the discharge 43702.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Vacuum iota profile (solid line) and iota profile used as reference case in the simulations (dotted line) calculated for $t = 1200$ ms.</figcaption></figure>

Due to the lack of precise information on the evolution of the plasma current and iota profile, especially at the edge linked to the ripple effects, an exact time-resolved reconstruction is not feasible. Instead, a parametric scan of modified iota profiles

is performed to emulate the effect of increasing or decreasing the plasma current during the NBI heating process in Figure 9. It should be noted that the effect of the NBI current drive is included in the iota profile for the reference case. In addition, the variation of the iota profile in the plasma region where the mode is destabilized is mainly caused by the effect of the ripple, while the NBI current drive is approximately constant from the discharge time studied.

### 3.2. Simulation parameters

Figure 7, panels a and b, show the thermal plasma density and temperature profiles measured by Thomson scattering [74, 75] for discharges 43702 to 43709 (solid lines). The model profile used in the simulations as the reference case employs simplified profiles based on the experimental data (dashed black lines). The experimental uncertainty in both cases is shown as gray error bars. For the density profiles, there are no experimental data beyond $\rho = 0.65$, and what is shown is a reconstruction of the profiles, although the uncertainty approaching the plasma periphery is substantial. The EP density profile in the simulations is also indicated in the panel a (dotted red line). The EP energy profile is assumed constant with $T_f = 10$ keV and the EP distribution function is a Maxwellian distribution fitted to the average beam slowing-down temperature calculated with ASCOT [76, 20]. The uncertainty of the experimental temperature profiles is represented as gray error bars, using shot 43704 as reference example. The experimental density profiles are indicative due to the inability to achieve high precision with the available tools, especially in the plasma periphery. The choice of the reference case profile (O) was made considering this uncertainty and will be discussed in section 4.2. The choice for the temperature profile was made to simplify the modeling process while still capturing the key physical behaviors relevant to the study due the simulations are far more sensitive to other dominant factors such as the iota or the density.

The analysis of the AE activity will be focused on the plasma periphery, where the mode is destabilized. This study emphasizes the variation of profiles that are most relevant to the stability, particularly the iota profile and the thermal ion density profile. In contrast, the variation in thermal electron profiles is not considered, as these variations are negligible in the present analysis.

The dynamic and equilibrium toroidal and poloidal modes included in the simulations are listed in table 1. The dynamic modes evolve in time representing the plasma instabilities while the equilibrium modes are time independent depicting the device magnetic field configuration. The notation used
<figure><img src="image_8.png" /><figcaption>**Figure 7.** (a) Experimental thermal density profile (n) with colored continuous lines with uncertainty margin for 43704 shot as grey bars until $\rho = 0.65$. The reference case (O index) is indicated by the black dashed line. Energetic particle density ($n_{EP}$) shown as the red dashed line. (b) Experimental thermal electron temperature ($T_e$) profiles with colored continuous lines with uncertainty margin for 43704 shot as grey bars. Data obtained by Thomson scattering at $t = 1200$ ms Reference case (O index) indicated by the black dashed line.</figcaption></figure>

to identify the modes is $m/n$. The linear simulations are limited to $n = 3$ to 14 toroidal mode families. The mode selection includes all the resonant modes between the magnetic axis and the plasma periphery. The number of points of the radial grid is 1000 and the magnetic Lundquist number is $S = 5 \times 10^6$.

The dynamic variables must include both mode parities because the moments of the gyro-kinetic equation break the MHD symmetry. Eigenfunctions in FAR3d code are represented in terms of sine and cosine components, using real variables:

$$f(\rho, \theta, \zeta, t) = \sum_{m,n} f_{mn}^{s}(\rho, t) \sin(m\theta + n\zeta) + \sum_{m,n} f_{mn}^{c}(\rho, t) \cos(m\theta + n\zeta)$$

Where $\rho$ is the normalized radius, $\theta$ is poloidal angle, $\zeta$

<table><caption>**Table 1.** Dynamic and equilibrium toroidal and poloidal modes used in the simulations. The modes highlighted in gray indicate that, for the simulations of the dynamic modes associated with $n = 8$ and $n = 12$, only the equilibrium modes with $n = 0$ and $n = 4$ are used.</caption><thead><tr><th></th><th>n</th><th>m</th></tr></thead><tbody><tr><td rowspan="4">equilibrium</td><td>0</td><td>0,1,2,3,4,5</td></tr><tr><td>4</td><td>0,1,2,3,4,5</td></tr><tr><td>8</td><td>0,1,2,3,4,5</td></tr><tr><td>12</td><td>0,1,2,3,4,5</td></tr><tr><td rowspan="12">dynamic</td><td>3</td><td>1,2,3</td></tr><tr><td>5</td><td>2,3,4</td></tr><tr><td>6</td><td>3,4,5</td></tr><tr><td>7</td><td>3,4,5,6</td></tr><tr><td>8</td><td>4,5,6,7</td></tr><tr><td>9</td><td>4,5,6,7,8</td></tr><tr><td>10</td><td>5,6,7,8</td></tr><tr><td>11</td><td>5,6,7,8,9</td></tr><tr><td>12</td><td>6,7,8,9,10</td></tr><tr><td>13</td><td>6,7,8,9,10,11</td></tr><tr><td>14</td><td>7,8,9,10,11,12</td></tr></tbody></table>

is toroidal angle, $t$ is the time, $m$ is the poloidal mode number and $n$ is the toroidal mode number, $f_{mn}^s$ is the sine component and $f_{mn}^c$ is the cosine component. The eigenfunctions will be shown as $|f_{mn}|$:

$$|f_{mn}| = \sqrt{(f_{mn}^s)^2 + (f_{mn}^c)^2}$$

# 4. Simulation results

This section is dedicated to study and identify the AE activity observed in the experiments. In addition, the effect of the thermal ion density and iota profiles uncertainty on the simulations results is evaluated.

## 4.1. Mode identification

Figure 8, panel a, shows the continuum gaps in the reference case calculated using the code Stellgap [77] including the effect of the sound wave and helical couplings. The analysis indicates the presence of helical gaps around 200 kHz for the helical families 3, 7, 11 and 8, 12 in the middle-outer plasma region around $\rho = 0.66$, indicated by a dashed black oval. Panels b and c show the results of the simulations performed for the helical families 3, 7, 11 and 8, 12 indicating the destabilization of $m/n = 4/7 - 2/3$ and $m/n = 7/12 - 5/8$ HAEs at the middle-outer plasma region around $\rho = 0.7$ with frequencies 209 and 204 kHz. The eigenfunctions show oscillation linked to the interaction of the mode with the continuum, that is to say, the mode width is broader than the radial gap extension. The results of the simulations and the
estimations of the radial location and frequency range of the helical gaps are consistent with the experimental data for the frequencies associated with the instabilities shown in Figure 3, as well as with the relative position at $\rho < 0.8$, as indicated in Figure 4.

<figure><img src="image_9.png" /><figcaption>**Figure 8.** (a) Alfvén continuum of the reference case for the $n = 1$ to 15 toroidal families calculated for $t = 1200$ ms. Eigenfunctions of (b) $n = 3, 7, 11$ (red) and (c) $n = 8, 12$ (blue) AEs if $\beta_{EP} = 0.06$</figcaption></figure>

## 4.2. Effect of the plasma equilibrium uncertainties on the simulation results.

The rather large uncertainty of the thermal plasma profiles may affect the final conclusions of the analysis; for this reason the simulation results must be confirmed by modifying the iota profile and thermal plasma density. It should be noted that the profiles of the

EP density and temperature also have an essential impact in the AE activity. Also, by analyzing different iota profiles that represent changes in plasma current at the edge, we investigate the correlation between plasma current and the mode frequency and evolution, as observed in Figure 3. This is why a study about this issue is also performed in the following section of this article.

Figure 9, panel a, shows 7 different iota profiles included in the study, emphasizing the iota profile uncertainty in the middle-outer plasma region where the calculated HAEs are located. It should be noted that the excursion of the iota profile analyzed in the most extreme cases is larger than the expected effect of the toroidal current variation along the discharge. Since plasma current variations are the result of inductive coupling with the main field conductors [69] the inductive current contribution must penetrate from the plasma edge and thus it is reasonable to assume iota variations in this zone. The reference case is identified by the index (O). Panels b and c show the variation of the radial location and frequency range of the HAE gaps for the equilibrium representing the extreme cases C (strongest negative magnetic shear, gray line) and F (strongest positive magnetic shear, green line) at the middle-outer plasma region. The helical gap shows a shift with respect to the reference case (O) around in the radial direction, from $\rho = 0.66$ to 0.62 (6%)(case C) and to 0.65 (1.5%)(case F) and in the frequency, from $f = 195$ kHz to 200 kHz (2.5%)(case C) and 206 kHz (5.6%)(case F). The radial width and frequency range of the helical gaps are reduced in the configuration C compared to the reference case because the magnetic shear is stronger in the region where the gap is observed. The opposite happens for the case F due to a reduction of the magnetic shear. The decrease of the gap radial width and frequency range as the magnetic shear increases is also observed in LHD and Heliotron J plasma, leading to a stabilizing effect on the AEs [53].

Figure 10, panels a and b, indicate the variation of the growth rate and frequency of the AEs triggered by 3, 7, 11 and 8, 12 helical families with respect to the iota profile. The cases with negative magnetic shear at the middle-outer plasma region (A, B, and C) result in a slight decrease in growth rate for both helical families, approximately 5%, consistent with an enhancement of the magnetic shear stabilizing effect in the radial location where the HAEs are destabilized. In addition, $n = 3, 7, 11$ AE frequency increases by 5% and $n = 8, 12$ AE decreases by 10%. Panels c and d show the eigenfunction obtained in the simulations for the extreme case C, identifying a similar HAE structure compared with the reference case. It should be mentioned that $n = 8, 12$ AE shows a much larger
<figure><img src="image_10.png" /><figcaption>**Figure 9.** (a) Iota profiles ($\iota$) with colored continuous lines (dashed black line for the reference case O). Alfvén continuum in the extreme cases (b) C (gray line) and (c) F (green line).</figcaption></figure>

5/8 mode amplitude compared to 7/12 mode, pointing out the helical coupling is weaker compared to the reference case and the HAE may switch to a 5/8 GAE as the negative magnetic shear is reinforced. Regarding the configuration with positive magnetic shear at the middle-outer plasma region (D, E, F) there is an increase of the growth rate for both helical families by 6%, consistent with the reduction of the magnetic shear

in the plasma region where the modes are triggered. Likewise, $n = 3, 7, 11$ AE frequency decrease by 2% and $n = 8, 12$ AE increases by 10%. Panels e and f show the eigenfunction obtained in the simulations for the extreme case F, identifying again the same HAE structure compared with the reference case. It is also important to mention that although the aim of this part of the analysis was to address experimental uncertainty, the information obtained from examining the different iota profiles reveals that the 4/7-2/3 HAE mode candidate reproduces the experimental behavior more accurately showing a clear correlation where an increase in current (case F) is accompanied by a reduction in frequency and an increase in growth rate, as was also observed in Figure 3.

In summary, the effect of the iota profile uncertainty in the simulations affects the modes growth rate and frequency, consistent with the evolution of the AE frequency range and intensity along the discharge as the plasma current changes. Nevertheless, the simulations indicate the type of mode destabilized is the same, HAEs, and points to the 4/7-2/3 candidate as the one that best fits the experimental data. It should be recalled that the evolution of the HAE frequency and intensity along the discharge is not the topic of the present study, only a consequence of the analysis performed. To reproduce correctly the variation of the HAE frequency and intensity along the discharge it is required a correct identification of the different current sources in the plasma, an analysis out of the scope of the present study.

Figure 11, panel a, shows 7 thermal ion density profiles modifying the gradient at the middle-outer plasma region. The reference case is identified by the index (O). Panels b and c indicate the variation of the radial location and frequency range of the HAE gaps in the extreme cases C (lowest density and largest gradient, gray line) and F (largest density and lowest gradient, green line). Both cases show an outward displacement of the gap compared to the reference configuration, located at $\rho = 0.65$ for the case (C) and 0.63 for the case (F). The frequency profile of the gap in both cases is similar with respect to the reference although it is up shifted to 225 kHz in case C and down shifted to 175 kHz in case F, effect linked to the variation of the Alfvén velocity of the plasma.

Figure 12, panels a and b, show the variation of the growth rate and frequency of the AEs triggered by 3, 7, 11 and 8, 12 helical families with respect to the thermal ion density profile. The decrease of the thermal ion density (cases A, B, and C) results in an increase in growth rate and frequency, 5% higher growth rate and 20% frequency for $n = 3, 7, 11$ family, 20% increase in growth rate and frequency for $n = 8, 12$ family comparing case C and reference case. This
<figure><img src="image_11.png" /><figcaption>**Figure 10.** (a) Growth rate (b) frequency of the AEs driven by the helical families *n* = 3, 7, 11 (red line) and *n* = 8, 12 (blue line) with respect to the iota profile. Eigenfunction of (c) *n* = 3, 7, 11 AE and (d) *n* = 8, 12 AE in the configuration C. Eigenfunction of (e) *n* = 3, 7, 11 AE and (f) *n* = 8, 12 AE in the configuration F.</figcaption></figure>

is explained by the increment of the plasma Alfvén velocity as the thermal ion density decreases, leading to an enhancement of the EP resonance. Panels c and d indicate the eigenfunction obtained in the simulations for case C, identifying a similar HAE structure compared with the reference case. Again, 5/8 mode amplitude is much larger compared to 7/12 mode for the *n* = 8, 12 AE, thus the HAE may switch to a 5/8 GAE as the thermal ion density decreases. Conversely, increasing the thermal ion density (cases D, E, and F) leads to a decrease of the growth rate for both helical families, around 5% comparing case F and reference case. Likewise, the frequency decreases

<figure><img src="image_12.png" /><figcaption>**Figure 11.** (a) Thermal ion density profiles with colored continuous lines (dashed black line for the reference case O). Alfvén continuum for the equilibria with the extreme cases (b) C (gray line) and (c) F (green line) modifying the thermal ion density profile.</figcaption></figure>

around 10%. This is explained by the EP resonance weakening as the plasma Alfvén velocity decreases. Panels e and f indicate similar HAE eigenfunctions comparing case F and the reference. Consequently, the uncertainty of the thermal ion density profile may not modify the identification of the instabilities as 4/7–2/3
and 7/12–5/8 HAEs, although it affects the predicted growth rate and frequency range. It should be recalled that the line averaged thermal plasma density is almost constant along the discharge phase heated by the NBI, thus the frequency oscillations observed in the AE activity should by mainly caused by the evolution of the toroidal current in the plasma (see figure 3).

<figure><img src="image_13.png" /><figcaption>**Figure 12.** (a) Growth rate and (b) frequency of the AEs driven by the helical families $n = 3, 7, 11$ (red line) and $n = 8, 12$ (blue line) with respect to the thermal ion density profile. Eigenfunction of (c) $n = 3, 7, 11$ AE and (d) $n = 8, 12$ AE in the configuration C. Eigenfunction of (e) $n = 3, 7, 11$ AE and (f) $n = 8, 12$ AE in the configuration F.</figcaption></figure>

In conclusion, the uncertainty in the iota and thermal ion density profiles may not affect the instabilities identification as 4/7 – 2/3 and 7/12 – 5/8 HAEs, although the growth rate and frequency may vary up to 20% comparing the most extreme configurations analyzed with respect to the reference

case. That means the mode identification by the simulations is robust. In addition, the variation of the HAE frequency with respect to the iota profile may explain the frequency sweeping of the instabilities observed along the discharge, induced by the evolution of the toroidal plasma current while the thermal plasma density remains almost constant.

### 4.3. Stabilizing effect of the EP on low-frequency modes

The discharges analyzed indicate a anti-correlation between local maxima activity of AEs with local minima of low frequency modes, pointing out cross-stability effects between the EP population and pressure gradient driven modes. In figure 13, panels a and b, indicate opposite activity phases of AEs and low frequency modes around 20 kHz along the discharge 43702 in magnetic fluctuation spectrogram and radiation fluctuation spectrogram. This effect is observed in all the discharges analyzed. This section is dedicated to study the effect of the EP population in the stability of pressure gradient driven modes.

The frequency of EP-driven modes depends on the drive induced by the gradients of the EP distribution function, particularly the density gradient; the dispersion relation of EP-driven instabilities can be expressed by an implicit equation for $\omega + i\gamma$ [78], with $\omega$ the frequency and $\gamma$ the growth rate of the mode. The resonance between EPs and plasma waves can exist even if the EP density is low; that is, the EPs can affect the stability of the plasma even though the amount of EPs in the system is not large enough to overcome the threshold to trigger an EP-driven mode [79, 80]. Indeed, there can be an EP stabilizing effect on thermal plasma instabilities caused by the non-reactive part of the resonance term in the dispersion relation at low EP density. On the other hand, as the EP density increases, the reactive and non-reactive elements compete until the EP-driven branch is destabilized [81]. There is experimental evidence that indicates stabilizing or destabilizing effects of the EP particles on the stability of thermal plasma instabilities, for example by increasing or reducing the waiting time between saw-tooth linked to the destabilization of kinks or tearing modes in JET, ASDEX, EAST and DIII-D plasma [82, 83, 84] or by damping the growth rate of interchange and ballooning modes in LHD plasma [85, 86, 87]. This is explained by the resonance between EPs and plasma waves, active even if the EP population is below the EP beta threshold required to destabilize EP-driven modes.

The effect of the EP population on PGDM stability is explored performing a set of simulations modifying the $\beta_{EP}$, that is to say, by changing the amount of EPs in the plasma. Figure 14, panels a and
<figure><img src="image_14.png" /><figcaption>**Figure 13.** (a) Magnetic fluctuation spectrogram and (b) radiation fluctuation spectrogram in the discharge 43702. Pink bold arrows indicate discharge periods with opposite AE / PGDM activity intensity.</figcaption></figure>

b, show the instability growth rate and frequency with
respect to the simulation $\beta_{EP}$. The threshold $\beta_{EP}$ for
destabilization of AE is between 0.00160 and 0.00385,
identified as a change in the trend of growth rate.
Below the $\beta_{EP}$ threshold only PGDM are unstable
showing frequencies below 25 kHz (no equilibrium
toroidal rotation is included in the simulation thus
the Doppler effect on the mode frequency is ignored).
The data from $\beta = 0.0001$ up to $\beta = 0.00160$ are
plotted in green reflecting PGDM activity, showing a
decreasing trend in growth rate and an increasing trend
in mode frequency as $\beta_{EP}$ rises. From $\beta = 0.00160$ to
$\beta = 0.00385$, there is a transition between PGDM and
Alfvénic regimes. At $\beta = 0.00385$, the points switch to
black, marking the onset of a new Alfvén regime with
a starting frequency of approximately 50 kHz in which
the growth rate and frequency rises again along with
$\beta_{EP}$.

Figure 15 indicates the *n* = 5 perturbation
eigenfunction if the simulation *β*<sub>EP</sub> is 0.0001, panel
a, or 0.1, panel b. The simulation with *β*<sub>EP</sub> = 0.0001
shows the destabilization of a 3/5 interchange mode
(single m mode) in the middle-outer plasma region
although the simulation with *β*<sub>EP</sub> = 0.1 show an
unstable 3/5 GAE with 140 kHz also at the middle-
outer plasma region.

The thermal plasma density and temperature
(thermal plasma β) remain approximately constant
throughout the discharge phase heated by the NBI,
suggesting that the reduction in interchange mode

activity can be attributed to a combination of factors:
the increase in the EP population and the evolution
of the iota profile. As shown in figures 2 and 13,
the significant rise in plasma current during the early
NBI phase, from *t* = 1160 to 1200 ms, indicates that
the variation in the iota profile plays a major role
in influencing the stability of the interchange mode.
However, at *t* = 1230 ms, despite a reduction in
plasma current to almost half of the local maximum at
*t* = 1200 ms, although the interchange mode activity
barely changes.

This observation suggests that the EPs provide
an additional stabilizing effect on the interchange
mode. It is important to highlight that the evolution
of the iota profile is not the only factor affecting
thermal plasma stability; the variation in the EP
population due to AE activity must also be considered,
as evidenced by experimental observations in other
devices, such as the stabilization of interchange modes
in LHD and internal kinks in DIII-D plasmas [88, 89].
Both the iota profile and the EP population play
important roles in the stabilization of thermal plasma
instabilities.

4.4. Optimization

The destabilization threshold and characteristics of the
AE activity in the experiment strongly depends on the
EP population generated in the plasma. The intensity
of the EPs resonance and the AEs destabilization
is given by the amount of EPs and the energy of
<figure><img src="image_15.png" /><figcaption>**Figure 14.** Growth Rate (a) and frequency (b) for $n = 5$ toroidal family in the reference model if the $\beta_{EP}$ is modified (X axis in logarithm scale).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 15.** Eigenfunctions for $n = 5$ mode if $T_F = 10$ keV for $\beta_{EP} = 0.0001$ (a) and $\beta_{EP} = 0.1$ (b)</figcaption></figure>

the EP populations along the thermalization process. Consequently, the NBI operational regime with respect to the power injection (linked to the amount of EPs in the plasma), voltage (related with the EP energy) and radial deposition region (EP density profile) may lead to scenarios with strong or marginal stable AE activity [90, 57, 91]. An efficient plasma heating by the NBI requires stable AEs to minimize EPs losses and an optimal transfer of the EP energy towards the

thermal plasma.

This section is dedicated to identify operational regimes that minimize the Alfvén activity in the system. On that aim, for the plasma equilibrium used in the previous section, a set of parametric studies are performed to calculate the stability of the different helical families with respect to the simulation $\beta_{EP}$, EP energy and EP radial density profile (intensity of the EP density gradient and radial location). In the following, modifications of the EP $\beta$ referrers to a variation of the EP density for a fixed EP energy. It should be noted that the analysis provides theoretical trends obtained by modifying individual parameters of the model, although in the real experiment, any change of the NBI operation regime has consequences on the thermal plasma and iota profiles. Consequently, this study must be considered as an initial step of the optimization process and further analysis is required to obtain a more realistic approximation including a self-consistent model. Nevertheless, such self-consistent models need for a rather large accuracy in the thermal plasma, iota and EP profiles to correctly reproduce the experiment and tendencies, thus the conclusion obtained could be rather unclear. Consequently, simplified optimization studies are useful to identify the isolated effect of each model variable, providing useful information that must be considered as a first order approximation of the AE stability dependency with respect to the parameters analyzed.

Figure 16 indicates the different EP density profiles tested. Panel a shows profiles with the same EP density gradient although displaced at different radial locations between the inner plasma to the plasma periphery, mimicking peaked and broad heating NBI operational regimes. Panel b shows EP density profiles with different gradient intensities located at $\rho = 0.7$, dedicated to study configuration with different EP radial spreading along the minor radius, linked to the beam penetration inside the plasma core (normally a strong function of the thermal plasma density).

Figure 17 shows the growth rate, panel sets a to c, and frequency, panel sets d to f, of the helical families $n = 3, 7, 11, n = 5, 9, 13, n = 6, 10, 14$, and $n = 8, 12$ calculated in simulations modifying $\beta_{EP}$, EP energy and EP density profile. The study of each configuration is conducted as follows: EP energy ($T_f$) ranges from 10 to 60 keV in steps of 10 keV, $\beta_{EP}$ ranges from 0.01 to 0.1 in steps of 0.01, the axis variation cases at $\rho = 0.2, 0.4, 0.7$, and $0.8$, as well as gradient variation cases with $\partial n_f / \partial \rho = -1.5, -2.5, -3.4$, to $-4.9$ at $\rho = 0.7$. Panels a and d are dedicated to the reference model, panels b and e to the effect of the EP density gradient radial location and panels c and f to the effect of the EP density gradient intensity. Peaked NBI configurations show the highest AE growth rates
<figure><img src="image_17.png" /><figcaption>**Figure 16.** Simulation EP density profiles if (a) the radial location of the gradient is modified (fixed the gradient intensity to $dn_f/d\rho = -3.4$) and (b) the gradient intensity is modified (fixed the gradient radial location at $\rho = 0.7$. The solid black line indicates the reference case.</figcaption></figure>

and the lowest frequencies because, as can be observed in figure 8a, there are wide continuum gaps (with respect to the radial and frequency range extension) in the inner plasma region, thus the continuum damping effect in small and broad AEs can be destabilized. In addition, radially localized NBI injection leading to intense EP density profile gradients leads to further AE destabilization, because a stronger gradient implies a larger amount of free energy available to generate instabilities. The $\beta_{EP}$ threshold for AE destabilization if $T_f = 10$ keV is $\beta_{EP} \sim 0.02$, except for the peaked configuration showing a threshold below 0.01. On the other hand, the configuration with a less intense EP density gradient shows an $\beta_{EP}$ threshold of 0.04. In addition, the $\beta_{EP}$ threshold increases with the EP energy although it does not follow a linear dependency, meaning that a smaller density of more energetic EP

are able to destabilize AEs in the plasma.

To study the effect of EP energy on the AEs, table 2 compares simulations with the same EP density although different EP energy and $\beta_{EP}$. Increasing the EP energy causes an increase in the AE growth rate and frequency. That means, increasing the EP energy at fixed EP density will cause an enhancement of the AE activity at least in the range of EP energies analyzed. Moreover, if the simulation $\beta_{EP}$ is fixed and the EP energy / EP density rate is modified by reducing proportionally the EP density as the EP energy increases, the analysis indicated in table 3, an enhancement of the AE growth rate and frequency is observed, thus a smaller amount of EPs is required to trigger AEs as the EP energy increases. On the other hand, if the EP density is small enough, the $\beta_{EP}$ threshold to destabilize AEs is not exceeded and the AEs are stable. Consequently, there is an optimization trend increasing the EP energy while reducing the EP density that may lead to operational regimes with stable AEs.

To study the effect of EP energy on the AEs, Table 2 compares simulations with the same EP density but different EP energy and $\beta_{EP}$ by following the relation $n_{EP} \propto \beta_{EP}/T_f$. Increasing the EP energy causes an increase in the AE growth rate and frequency. That means, increasing the EP energy at fixed EP density will cause an enhancement of the AE activity, at least in the range of EP energies analyzed.

Moreover, Table 3 analyzes the case where $\beta_{EP}$ is fixed and the EP energy/density ratio is modified by reducing the EP density proportionally as the EP energy increases. In this case, the AE growth rate and frequency initially increase with EP energy, but then suddenly drop. At fixed $\beta_{EP}$, raising the EP energy $T_f$ reduces the EP density while simultaneously changing the resonance condition between the EPs and the Alfvén eigenmode. Initially, increasing EP energy enhances the resonance, but beyond a threshold $T_f$, changes in the resonance condition dominate and lead to mode stabilization, even though $\beta_{EP}$ remains constant.

Therefore, while a smaller amount of EPs can destabilize AEs at higher energies, if the EP density is too low, the $\beta_{EP}$ threshold is not exceeded and the AEs remain stable. Consequently, there is an optimization trend in increasing the EP energy while reducing the EP density, which may lead to operational regimes with stable AEs.

Figure 18 indicates the radial width and frequency range of the Alfvén Eigenmodes triggered by each helical family with respect to the continuum. Panel (a) shows that for the helical family $n = 3, 7, 11$, an unstable $4/7-2/3$ HAE appears at $\sim 200$ kHz when $T_F = 10$ keV and $\beta_{EP} = 0.05$; for $T_F > 10$ keV,
<figure><img src="image_18.png" /><figcaption>**Figure 17.** Growth rate and frequency of the AE triggered by the helical families $n = 3, 7, 11$; $n = 5, 9, 13$; $n = 6, 10, 14$; $n = 8, 12$ for different $\beta_{EP}$ and energies in the reference case (a) and (d); Axis variation study (b) and (e); and gradient variation study (c) and (f).</figcaption></figure>

a higher-frequency 4/7 GAE emerges above 230 kHz, both localized around $\rho = 0.65–0.70$. In panel (b), corresponding to the $n = 6, 10, 14$ helical family, a 6/10 GAE is observed with $f < 200$ kHz for $T_F = 10–20$ keV, located in the mid–outer plasma region. This mode increases its frequency above 200 kHz as $T_F$ increases while maintaining its EP density, yet it remains a 6/10 GAE. Panel (c), for the $n = 5, 9, 13$ family, shows an unstable 8/13 GAE with frequencies above 200 kHz at $\rho = 0.70$. In panel (d), for the $n = 8, 12$ family, a 7/12–5/8 HAE at $\sim 200$ kHz becomes unstable when $\beta_{EP} = 0.06$ and $T_F = 10$ keV.

By reducing the EP density for $\beta_{EP} = 0.03$, a 5/8 GAE with $f < 200$ kHz dominates at $\rho = 0.75$, and if $T_F$ is increased while maintaining the EP density with $T_F = 20$ keV and $\beta_{EP} = 0.06$, a 7/12 GAE above 230 kHz is triggered, showing a change in the dominant mode.

In conclusion, a localized peaked NBI operation regime may lead to configurations with a rather low $\beta_{EP}$ threshold for AEs destabilization as well as strong AEs activity. It should be noted that, the AE growth rate increases between simulations with the same amount of EPs as the EP energy increases.
<table><caption>**Table 2.** Growth Rate and frequency of each helical family in simulations with fixed EP density and different EP $\beta$ and energy.</caption><thead><tr><th>Helical family</th><th>β<sub>EP</sub></th><th>T<sub>F</sub> [keV]</th><th>γτ<sub>A0</sub></th><th>f [kHz]</th></tr></thead><tbody><tr><td rowspan="2">n=3,7,11</td><td>0.05</td><td>10</td><td>0.15</td><td>209</td></tr><tr><td>0.10</td><td>20</td><td>0.35</td><td>247</td></tr><tr><td rowspan="3">n=5,9,13</td><td>0.03</td><td>10</td><td>0.17</td><td>190</td></tr><tr><td>0.06</td><td>20</td><td>0.32</td><td>261</td></tr><tr><td>0.09</td><td>30</td><td>0.45</td><td>318</td></tr><tr><td rowspan="5">n=6,10,14</td><td>0.02</td><td>10</td><td>0.17</td><td>121</td></tr><tr><td>0.04</td><td>20</td><td>0.26</td><td>183</td></tr><tr><td>0.06</td><td>30</td><td>0.34</td><td>233</td></tr><tr><td>0.08</td><td>40</td><td>0.41</td><td>275</td></tr><tr><td>0.10</td><td>50</td><td>0.48</td><td>310</td></tr><tr><td rowspan="3">n=8,12</td><td>0.03</td><td>10</td><td>0.14</td><td>186</td></tr><tr><td>0.06</td><td>20</td><td>0.29</td><td>280</td></tr><tr><td>0.09</td><td>30</td><td>0.42</td><td>321</td></tr></tbody></table>

<table><caption>**Table 3.** Growth Rate and frequency of helical family $n = 3, 7, 11$ and $n = 8, 12$ in simulations with fixed EP $\beta$ (0.1 and 0.05) and different EP density and energy.</caption><thead><tr><th>Helical family</th><th>β<sub>EP</sub></th><th>T<sub>F</sub> [keV]</th><th>γτ<sub>A0</sub></th><th>f [kHz]</th></tr></thead><tbody><tr><td rowspan="4">n=3,7,11</td><td>0.10</td><td>10</td><td>0.28</td><td>176</td></tr><tr><td>0.10</td><td>20</td><td>0.35</td><td>228</td></tr><tr><td>0.10</td><td>30</td><td>0.39</td><td>263</td></tr><tr><td>0.10</td><td>40</td><td>Stable</td><td>Stable</td></tr><tr><td rowspan="6">n=8,12</td><td>0.10</td><td>10</td><td>0.36</td><td>232</td></tr><tr><td>0.10</td><td>20</td><td>0.43</td><td>284</td></tr><tr><td>0.10</td><td>30</td><td>0.46</td><td>322</td></tr><tr><td>0.10</td><td>40</td><td>0.47</td><td>352</td></tr><tr><td>0.10</td><td>50</td><td>0.48</td><td>377</td></tr><tr><td>0.10</td><td>60</td><td>Stable</td><td>Stable</td></tr><tr><td rowspan="2">n=3,7,11</td><td>0.05</td><td>10</td><td>0.15</td><td>209</td></tr><tr><td>0.05</td><td>20</td><td>Stable</td><td>Stable</td></tr><tr><td rowspan="3">n=8,12</td><td>0.05</td><td>10</td><td>0.20</td><td>193</td></tr><tr><td>0.05</td><td>20</td><td>0.26</td><td>236</td></tr><tr><td>0.05</td><td>30</td><td>Stable</td><td>Stable</td></tr></tbody></table>

Consequently, the EP resonance is stronger as the EP energy increases. This implies, smaller populations of high energy EPs are required to trigger intense AEs, thus weakly thermalized EP are easily unconfined, potentially leading to the most deleterious effects on TJ-II integrity. Therefore, increasing the EP energy while keeping the same EP density may cause an enhancement of the EP losses. On the other hand, if the increment of the EP energy is combined with a reduction of the EP density, operational regimes with stable AEs are possible by keeping the same plasma heating intensity, leading to an improved performance of the device.

<figure><img src="image_19.png" /><figcaption>**Figure 18.** Alfvén continuum as well as the radial width and frequency range of the AEs triggered for the helical families $n = 3, 7, 11$ (a), $n = 6, 10, 14$ (b), $n = 5, 9, 13$ (c) and $n = 8, 12$ (d) in simulations with different EP energy and $\beta$.</figcaption></figure>

## 5. Conclusions

The present study is dedicated to analyze high-frequency (200 kHz) and low-frequency (20 kHz) instabilities generated in the outer plasma region of the TJ-II device, measured by Langmuir probes and Mirnov coils diagnostics.

The experiments are modelled using FAR3d and Stellgap codes, calculating the stability of AEs and PGDMs as well as the Alfvén continuum configuration. $n = 3, 7, 11$ and $n = 4, 8, 12$ helical gaps exist at the middle-outer plasma region around the frequency range of 200 kHz. The simulations indicate the high-frequency instabilities observed at the periphery could be 4/7 – 2/3 and 7/12 – 5/8 HAEs with frequencies of 209 and 204 kHz, respectively, destabilized at $\rho = 0.66$, with 4/7 – 2/3 as the candidate that best reproduces the behavior observed in the experiment.

The robustness of the simulation results have been tested considering the experimental uncertainty of iota and thermal ion density profiles at the middle-outer plasma region, the plasma region where instabilities are measured. The analysis indicates the profile uncertainties do not alter the modes identification as HAEs although it modifies the growth rate and frequency of the modes. Indeed, the instability frequency sweeping in the experiment can be explained by the iota profile changes caused by the toroidal plasma current evolution. These profiles reproduce the observed experimental trend: an inverse correlation between plasma current and mode frequency, in agreement with the measurements. It should be recalled that a detailed analysis of the AE frequency sweeping along the discharges requires an accurate description of the different current sources in the
plasma, a topic out of the scope of the present study.

The analysis of the interchange modes stability with respect to the density of EP in the plasma may indicate a stabilizing effect of the NBI on low frequency instabilities, consistent with the experimental observation showing a decrease of low frequency mode activity during local maxima of high frequency mode activity.

The AE stability is also analyzed for different NBI operational regimes with respect to the EP $\beta$ (related with the NBI power), EP energy (connected to the NBI voltage) and EP density profile (NBI deposition region and beam spreading), aiming for the optimization of the device's AE stability. Simulations reproducing radially localized peaked NBI operation show the largest AE growth rates and the lowest frequencies. The AE $\beta_{EP}$ threshold is 0.02 if the EP energy is 10 keV in the broad NBI configuration tested, although the peaked case shows a $\beta_{EP}$ threshold below 0.01. On the other hand, the $\beta_{EP}$ threshold increases to 0.04 in the peaked configuration if the EP density gradient is reduced, indicating that a non radially localized NBI injection is preferred to improve the AE stability. Optimized NBI operational regimes showing stable or reduced AE activity require balancing the EP density and energy. The simulations show that an increment of the EP energy keeping the same EP density leads to a further destabilization of the AE activity. On the other hand, if the EP energy is increased while reducing the EP density, operation scenarios showing stable AEs are identified. It must be recalled that the theoretical trends identified cannot be straight forward applied in the experiment because the simulations do not include self-consistent EP, thermal plasma and iota profiles. For example, simulations with higher EP density may show a larger distortion of the iota profile in the inner-middle plasma due to an enhancement of the NBI current drive. Nevertheless, the analysis provides useful information of the AE stability with respect to different model parameters linked to the NBI operation. Simulations including self consistent profiles will be the topic of a future analysis, although such studies demand a rather high reliability on the EP, thermal plasma and iota profiles used. In addition, identifying optimization trends in simulations including the variation of several model parameters at the same time could be challenging.

Optimized scenarios in TJ-II with respect to the AE stability are identified attending to the NBI operational regime. Increasing the EP energy (NBI voltage) and reducing the EP density (NBI power) leads to configurations with stable AEs and a potential improvement of the plasma heating intensity by reducing EP losses. The AE stability is further improved if the NBI is deposited broad avoiding a localized beam injection. These optimization trends

are consistent with the AE stability experimental and theoretical analysis performed in DIII-D device with respect to the NBI operational regime [91, 90, 57, 92].

## Acknowledgments

This research has been partially supported by the Spanish National Research Project No. PID2022-137869OB-I00, by project NIFS07KLPH004, and by project 2019-T1/AMB-13648 funded by the Comunidad de Madrid. Additional support was provided by the U.S. Department of Energy under grant DE-FG02-04ER54742, and by the AIM4EP Project (ANR-21-CE30-0018), funded by the French National Research Agency (ANR). Computational resources were provided by the Uranus supercomputing cluster at Universidad Carlos III de Madrid (Spain), funded jointly by EU FEDER funds and the Spanish Government through national projects UNC313-4E-23612, ENE2009-12213-C03-03, ENE2012-33219, ENE2012-31753, and ENE2015-68265. Data supporting the findings of this study are available from the authors upon request. We also thank David Zarzoso Fernández for valuable discussions regarding key aspects of the work.

## Appendix (FLR)

This section is dedicated to analyze the effect of the EP finite Larmor radius (FLR) damping on the simulation results. On that aim, a parametric study is performed with respect to the EP Larmor radius, calculating the resulting AE growth rate. FLR stabilization is stronger if the EP Larmor radius is closer to the AE width, leading to a stronger averaging out of the AE drive.

Figure 19 shows the growth rate and frequency of AEs destabilized by the $n = 3, 7, 11$ and $n = 8, 12$ helical families in simulations including the effect of the EP FLR damping for different EP Larmor radius and EP energies. The Larmor radius tested range from 0.003 to 0.015 m. Simulations with EP Larmor radius below 0.005 m show a rather weak effect on the FLR damping on the AEs growth rate for all the EP energies tested. The Larmor radius of the EPs injected by the NBIs in TJ-II plasma show a Larmor radius around 0.005 m, thus the effect of the FLR damping can be neglected in the first approximation.

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<figure><img src="image_20.png" /><figcaption>**Figure 19.** Growth Rate at different FLR values for helical families *n* = 3, 7, 11 and *n* = 8, 12 for energies of 10 keV (a), 30 keV (b) and 40 keV (c)</figcaption></figure>

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