Parametric study of helicon wave current drive in CFETR

This paper evaluates the feasibility of helicon current drive (HCD) in a hybrid scenario for the China Fusion Engineering Test Reactor (CFETR). Utilizing the GENRAY/CQL3D package, a large number of simulations (over 5 000) were conducted, with parametric scans in the antenna's poloidal position, launched parallel refractive index, and wave frequency. The analysis reveals that helicon has excellent accessibility under reactor-level conditions, and smaller n|| and higher wave frequency result in enhanced wave absorption. The simulations demonstrate an optimal launched parallel refractive index of approximately 1.6 for the CFETR hybrid scenario. The best launch position is found to be within a poloidal angle range of 25 degrees to 65 degrees. Additionally, it is preferable to have a narrow parallel refractive index spectrum for wave absorption when operating below the threshold value of {\Delta}n|| (~0.6), beyond which the effect of {\Delta}n|| on wave absorption is negligible. This study provides valuable insights into the potential application of HCD in CFETR.


Introduction
Fast waves in magnetized plasmas can be broadly classified into two types. The first type, commonly referred to as 'fast waves' in tokamaks, has a frequency in the ion cyclotron frequency range. The second type is known as the 'helicon' wave and has a much higher frequency, close to the lower hybrid frequency. This second type of fast wave is also referred to as a 'whistler wave' or 'fast wave in the lower hybrid frequency * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. range'. The propagation of the first type of fast wave is predominantly influenced by ion absorption in the ion cyclotron resonance layer. In contrast, the second type, the 'helicon' wave, mainly experiences absorption by electrons through the process of Landau damping [1].
Helicon current drive (HCD), also known as fast wave current drive (CD) in the lower hybrid range of frequencies, is regarded as a promising tool for driving off-axis currents in reactor-grade plasmas. In future steady-state fusion devices, it could provide a solution to the drawbacks associated with electron cyclotron waves and lower hybrid CD (LHCD). Since the lower hybrid wave (LHW) cannot access to the higher density region because of the lower hybrid resonance. However, the helicon wave will not be affected by it if the wave frequency is less than the lower hybrid resonance frequency while a particular value of n ϕ leads the depart of the two branches. This excellent accessibility from the edge of the plasma to the desired damping location makes HCD become an attractive choice [1,2]. The helicon wave system was first installed and tested on the JFT-2M tokamak using a 12-element combline antenna [2]. The traveling wave antennas (TWA) has valuable features such as load resiliency, a narrow n || spectrum, and a simple radio frequency (RF) circuit that does not require external matching systems. The combline TWA antenna exhibited the expected characteristics during the JFT-2M experiments. The input impedance matched well with the transmission line impedance under various plasma loads, including H-mode, Lmode, ohmic, and vacuum (for pre-ionization) conditions. The antenna operated reliably within the available power range of 400 kW and showed no signs of power saturation. The measured wave electric fields at the plasma center were consistent with the full-wave simulations, indicating successful excitation of the fast wave [3].
Currently, mid-sized to large tokamaks such as KSTAR and DIII-D are equipped with helicon systems. In KSTAR, a low power (mW) mock-up combline-type TWA was tested during discharges, and various plasma conditions such as L/H modes were studied to measure changes in impedance matching, coupling, and dominant n || [4]. To distinguish between slow and fast wave coupling, the magnetic field tilt angle was scanned. The experimental and analytical results showed that the fast wave dominated in relatively high coupling, and the use of the TWA antenna was found to provide load flexibility. To achieve practical off-axis CD applications, KSTAR is preparing to install a 1 MW helical wave CD system [5]. A 30-module traveling wave antenna has been installed and optimized in-vessel at DIII-D in early 2020, achieving good performance in the 10 MHz band around 476 MHz, with approximately 2% reflected power and approximately 1.5% dissipated power per module. The main elements of the design, construction, installation, and commissioning of the 1.2 MW helical wave system have been successfully completed. At present, this RF system has low reflectivity, low losses, and good antenna directionality. The physical basis for helical wave CD may soon be validated through this system [6,7].
Theoretical studies have shown that a HCD is highly promising for off-axis CD in DIII-D, ITER, FNST, and DEMOgrade plasmas [8][9][10][11][12], as well as some spherical tokamaks such as EXL-50 [13]. The primary methods used include linear [14], quasi-linear [8], and full-wave approaches [12]. The full-wave and linear calculations have been well-verified with respect to wave trajectories. However, the extent to which quasi-linear effects impact the results at current device scale remains unclear [8,15]. It is certain that non-linear effects due to parameter decay cannot be ignored when the wave input power is above 1 MW from the current DIII-D experiments. In nonlinear simulations, the global gyrokinetic code (GTC) has been extensively validated for both linear and nonlinear electromagnetic modeling of the LHWs. The propagation, mode conversion, and absorption of LHW have been investigated, employing a combination of fluid ion and drift kinetic electron treatments in a toroidal geometry [16,17]. Jenkins et al delved into the nonlinear effects of wave behavior in a slab configuration, notably illustrating similarities between scenarios of parameter decay instability observed in Alcator Cmod experiments and particle-in-cell (PIC) simulations [18]. Kuley et al explored the quasi-linear physics of LHW and Ion Bernstein Waves in both cylindrical and toroidal plasma geometries, with parametric decay instability being observed through simulations [19,20].
Additionally, recent research has focused on the impact of density turbulence on wave absorption [12], as well as the effects of the scrape-off layer region [14,21]. Another significant challenge associated with wave propagation is the nonlinear damping effect in the boundary region and density threshold. Wallace et al reported a lack of penetration of LHCD in divertor discharges in high-density Alcator C-Mod experiments. Addressing this phenomenon, Horton et al developed a theory of LHCD considering drift wave turbulence. The results indicate that electron temperature gradient turbulence leads to diffusion, influencing the description of lower hybrid plasma wave phase front propagation as described by rays. Lower hybrid rays might not penetrate the high-beta core region of fusion plasmas [22]. Additionally, Kuley et al developed a magnetohydrodynamic formulation to study the stabilization of LHWs by ion temperature gradient driven modes. The results revealed that the parametric coupling between lower hybrid and drift waves produces lower hybrid sideband waves, influencing the frequency and growth rate of drift waves. The required lower hybrid power is approximately ∼900 kW at 4.6 GHz [23,24]. Therefore, although there has been numerous theoretical and simulation studies on helicon and LHW heating and drive, several facets of helicon physics and its comparative analysis with LHW remain unclear and require further investigation.
This paper presents a scoping study on the potential use of HCD for the China Fusion Engineering Test Reactor (CFETR). One of the potential operating scenarios for CFETR, namely the 'hybrid' scenario in which some of the plasma current is sustained by the ohmic transformer, is chosen [25]. The equilibrium reconstruction code EFIT [26] provides the necessary GEQDSK file, while GENRAY [27] and CQL3D [28] solve the ray tracing and Fokker-Planck equations, respectively. The CQL3D calculations include quasi-linear effects. In order to perform a parametric study, over 5000 cases have been carried out. For our studies, a fixed operating frequency of 1.4 GHz is assumed, unless otherwise stated.
The structure of this paper is as follows. In section 2, we briefly describe the helicon physics in toroidal plasma. A broad parametric scan of antenna poloidal location and launched n || for CFETR are presented in section 3. In section 4, we summarize the results.

Wave accessibility and absorption
To study the characteristics of the propagation of the slow and fast branches, we are considering the linear (small-amplitude) propagation of waves in steady-state conditions, where the parameters of the medium vary only in one direction, which we refer to as the x-direction in the Cartesian coordinate system. This direction corresponds to the minor radius direction in circular magnetic confinement devices (such as tokamaks, stellarators, or RFPs). According to the [1], the critical n ϕ to depart the two branches can be estimated by the below formula, where the S, D and T is the elements of the dielectric tensor in Stix's style. However, we need to consider the behavior of n 2 ⊥ from the edge of the plasma, where the density is zero, up to the maximum density. A more accurate method to study the accessibility of waves can be carried out through solving the dispersion relation directly, which is shown in figure 1. This figure shows the density range of slow and fast wave propagation effects resulting from solving the cold plasma dispersion relation at fixed parameters. The parameters are B 0 = 7.2 T, n e0 = 1.2 × 10 20 m −3 , f = 1.4 GHz, and the ion species is deuterium. To enhance the presentation effect, we use the inverse hyperbolic sine function (sinh −1 ) to the quantity n 2 ⊥ for the ordinate. As can be seen, the slow wave branch cannot access the higher densities because of the lower hybrid resonance, while the fast wave branch is unaffected by it. At the parameters that n ∥ (n z ) = 2 or 3, there is no indistinguishable part to make the fast branch inaccessible for the higher densities. In addition, the cut-off density for fast wave is about the order of 10 19 m −3 , which is satisfied with our expectation. Thus, as demonstrated by the above analysis, the helicon branch (or fast wave branch) exhibits excellent accessibility to the mid-radius region from the low-field side.
The absorption of high harmonic fast waves can be calculated from the imaginary part of the wave number Here ω cs = q s B/m s is the cyclotron frequency of the The expressions for the coefficients A1 and A2 can be find in [14,29]. Z is the dispersion function. When considering only real numbers, its expression is as follows Figure 2 shows the contour lines of 0.2k ⊥i a, where k ⊥i is from equation (2) and a is the minor radius, for parameters characteristic of a hybrid scenario in CFETR that will be described later. The parameters used in the calculations correspond to the location (ρ = 0.6) where the helical wave is mostly absorbed in the CFETR configuration: B = B 0 = 7.2 T, n e = 9.2 × 10 19 cm −3 , kT e = 12 keV. In figures 2(a) and (c), f = 1.4 GHz, while in figures 2(b) and (d), β e = 0.02. The parameter 0.2k ⊥i a characterizes the wave absorption, so it can be used to represent the absorption of the wave within the small radius. When 0.2k ⊥i a is greater than 1 (indicated by bold lines), the wave is absorbed within a small range of the small radius. Within the parameter range shown in the figure, figures 1(a) and (c) indicates that absorption becomes stronger as β e increases. Figures 2(b) and (d) shows that for low wave frequencies, absorption becomes stronger as the frequency increases. When the wave frequency is approximately greater than 1.1 GHz, the absorption rate does not change significantly with frequency. In addition, absorption increases as ξ e decreases. From (a)-(c) or (b)-(d), it can be observed that when n || increases from 2 to 3, the contour lines 'shrink' towards the lower right corner, indicating a decrease in absorption.

Helicon wave heating and CD in CFETR
In this section, we investigate the capability of high harmonic fast wave heating and CD using the GENRAY and CQL3D packages. GENRAY is a linear ray-tracing code that solves the wave trajectory equation, while CQL3D incorporates the quasi-linear effect and collisionality. In our simulations, we use the CFETR hybrid scenario, which has a plasma temperature and density as shown in figure 3, with a core temperature of 30 KeV and core density of 1.2 × 10 20 m −3 . The input power for all cases is 10 MW, and the cold plasma dispersion relationship is adopted in the ray-tracing calculations.
A typical run of helicon heating and CD with the GENRAY/CQL3D code is depicted in figure 4, with a poloidal injection angle of 5 • . Figures 4(a) and (b) represent the wave trajectories in both real and phase space, while figures 4(c) and (d) demonstrate the power absorption density and driven current density. We observe that the waves slowly spiral radially inwards, gradually depositing their energy. The peak of the deposition profile is located at ρ = 0.51. The total current driven by helicons is 360 KA, which corresponds to a CD efficiency of η helicon = n e IR/P helicon = 2.4 × 10 19 A·W −1 ·m −2 . Figures 5 and 6 summarize the simulation results at the optimal launched parallel refractive index (n || = −2) for emission positions near the plasma top (95 • ) and the high-field side (185 • ). It can be observed that the emission on the highfield side yields weaker CD results compared to the top emission and even lower than that of the midplane emission (as also evident from figure 7). This is in contrast to the conclusion that emission on the high-field side with LHW can achieve higher drive efficiency. For LHW, emission on the high-field side should be optimized at lower n|| to better utilize wave accessibility. Lower n || allows the wave to penetrate deeper into the higher temperature plasma core (as shown in equation (1)), which is inaccessible from the low-field side due to strong Landau damping of high n || waves at the edge. However, for the high harmonic fast wave, the strong damping of the wave also depends on ξ e , and considering the electron temperature, it is important to note that figure 2 shows that local ξ e ≈ 0.8 is the condition for the strongest absorption.
At a fixed frequency of 1.4 GHz, we further scanned two key parameters: the parallel refractive index and the launch angle. Figure 7 presents the scanning results of the peak position of the driven current and the current driven by helicons in the two-dimensional polar coordinate system. The polar angle determines the position where the wave is launched, thus well representing the poloidal injection angle. On one hand, we can clearly see that there exists an optimal parallel refractive index (n || ) of about 1.6 for the best wave absorption, which is reasonable. Theoretically, according to equation (2), for wave propagation, absorption, and CD efficiency, the key parameter is ξ e , or n || , for a target plasma with a specified electron temperature. There exists an optimal ξ e (n || ) value due to the competition between two effects: higher ξ e leads to higher CD efficiency per absorbed power, but on the other hand, power absorption decreases exponentially with e −ξ 2 e . Therefore, both too high or too low n || are not favorable for wave absorption. From a physical point of view, absorption weakens as n || parallel to the wave increases, and the wave deposits its energy and momentum in fast electrons near the wave phase velocity v || = c/n || . Electrons with higher velocity are less likely to undergo collisions, thus lower n || should lead to higher CD efficiency. However, as we analyzed previously, too low n || will cause bad accessibility of the wave. In addition, we find that the optimal launch position has a poloidal angle range from 25 degrees to 65 degrees, which is very similar to the cases in the DIII-D tokamak where a poloidal angle of 45 degrees for the antenna location is a good choice [8]. Furthermore, when the driven current is at its peak, the current density profile peaks at around ρ = 0.5-0.6 as shown in figure 7(a). The key to off-axis CD is to have plasma conditions that allow the wave to travel toward the plasma center at a not too fast speed, while also having strong enough absorption to sufficiently damp the wave before it gets too close to the plasma axis. From our simulations, we find that too close to the center or the edge of the plasma, the absorption efficiency is greatly reduced.
We conducted further scoping studies on helicon heating and CD for CFETR. The simulation results of a parametric scan in the launched parallel index of refraction  figure 8(a), we still observe an optimal n || (∼1.6), at which the driven current can reach 414 kA, corresponding to a CD efficiency of 2.8 × 10 19 A·W −1 ·m −2 . This implies that HCD is highly promising under reactor-relevant conditions, as the highest achievable drive efficiency for LHWs under the same conditions is only 4.0 × 10 19 A·W −1 ·m −2 [30]. Another interesting phenomenon is that a small n || results in wave deposition over a wider range of profile peak, from about ρ = 0.2 to ρ = 0.7. This is consistent with the fact that a lower n || can allow the wave to penetrate more deeply into the hotter core region of the plasma, resulting in a wider radial range of peak value. Regarding the launch position, upward launch is better than downward launch, and the main deposition range for upward launch is approximately at ρ = 0.4 to ρ = 0.65. In general, helicon waves achieve approximately 70% of the CD efficiency compared to LHWs. While LHWs still demonstrate their characteristic of high CD efficiency when compared to helicon waves, the advantage of helicon waves lies in their power deposition location. They exhibit a significant capability to drive substantial current across a broad range of offaxis positions, spanning from ρ = 0.2 to 0.7, which is not easily achievable by LHWs within the ρ < 0.6 range [30]. Consequently, considering the power deposition location, both helicon waves and LHWs offer complementary and promising CD schemes for future fusion reactors.
The previous calculations were based on a frequency of 1.4 GHz, which is currently designed with CFETR parameters. Nevertheless, it is noteworthy that there exist 500 kW continuous wave klystrons currently accessible at 3.7 GHz and 4.6 GHz. Consequently, an inquiry may arise regarding the potential CD capabilities of helicon waves at varying frequencies. We can perform a simple calculation to evaluate the absorption behavior of the helical wave at higher frequencies.
The impact of wave frequency on wave absorption and CD is presented in figure 9. The plot displays the power absorption and CD as a function of wave frequency for a given poloidal emission position and n || . It is observed that over a wide frequency range (from 1.4 GHz to 5.5 GHz), wave absorption exhibits a strong dependence on frequency, where higher frequencies result in increased absorption and CD. This finding is consistent with the analytical results presented in figure 2. Nevertheless, the frequency has little impact on the deposition location, with only a slight inward shift observed for frequencies above 2.45 GHz. In general, at higher frequencies, the  wave trajectory tends to spiral slightly upward and is absorbed relatively deeper inside the plasma, resulting in higher absorption efficiency. In contrast, at lower frequencies, the wave cannot spiral as deeply into the plasma and is absorbed with lower efficiency.
A direct comparison of a helicon (or LH fast wave) system at 4.6 GHz compared with a traditional LH slow wave system also at 4.6 GHz (as in [30]) has also been carried out in addition to the above calculations. Figure 10 illustrates a comparison between trajectories, as well as wave absorption and CD capabilities of helicons and LHW at a frequency of 4.6 GHz. The GENRAY/CQL3D program package was employed with the following parameters: the wave frequency was set at 4.6 GHz, the poloidal injection angle was 0 • , n || = −1.6, ∆n || = 0.4, and the input power was 10 MW. A total of 18 rays were utilized. From figures 10(a) and (b), it is evident that there  are distinct differences in the trajectories of LHW and helicon waves. Specifically, helicon waves propagate along helical paths toward the plasma center, whereas LHW gradually spiral into the plasma from its edge. The wave at the plasma edge is reflected before reaching the plasma center, making it challenging to reach off-axis positions. Similar observations can be made from the profiles of power absorption and CD. The absorption location for helicon waves is notably closer to the center compared to LHW, with a deposition position at ρ = 0.53 for the former and ρ = 0.71 for the latter. In terms of power absorption and CD magnitude, LHW outperforms helicon waves. The driven current of LHW is approximately 705 kA, while that of helicon waves is around 424 kA, making the former 60% more efficient in terms of absorbed power. In the GENRAY calculations, we employed an n || spectrum with a full width (∆n || ). The results of a parametric scan in the n || spectrum full width (∆n || ) are depicted in figure 11, in which we fixed the frequency (1.4 GHz) and n || (1.6). From figure 11, it can be observed that within a certain range (∆n || < 0.5), a more localized n || spectrum can drive higher current. As ∆n || is further increased, there are no significant changes in power absorption and current profiles. Overall, a more localized n || spectrum is more favorable for wave absorption, and this is relatively easy to achieve for the traveling wave (combline) antenna of the helicon wave system.

Summary
In this work, we have demonstrated the results of helicon heating and CD in a hybrid scenario for CFETR. Utilizing the GENRAY/CQL3D package, a large number of simulations (over 5000) were conducted, with parametric scans in the antenna's poloidal position, launched parallel refractive index, and wave frequency. The analysis demonstrates that helicon waves exhibit excellent accessibility under conditions of a reactor size, and that higher wave frequency and smaller parallel refractive index lead to slightly improved wave absorption. Moreover, an optimal n || value of approximately 1.6 is identified for the hybrid scenario in CFETR, with helicon achieving a maximum drive efficiency of 2.8 × 10 19 A·W −1 ·m −2 . This means that it can achieve 70% of the optimal drive current for LHW in the same configuration, and obtain higher drive efficiency than electron cyclotron waves [8,31]. The best launch position is found to be within a poloidal angle range of 25 degrees to 65 degrees. In addition, it is desirable to have a narrow n || spectrum for wave absorption when operating below the threshold value of ∆n || (≈0.6), beyond which the impact of ∆n || on wave absorption becomes insignificant.
In contrast to previous studies [14], we have employed a quasi-linear approach to investigate HCD with CFETR parameters. This approach allows for a more accurate assessment, particularly at higher-power wave injection, compared to the utilization of a sole ray-tracing code. Moreover, our investigation has encompassed a comprehensive consideration of factors such as emission position, launch angle, and wave antenna parameters, within the context of the DEMO configuration, and their impact on wave power absorption. Besides, our work has introduced a novel comparison between helicon waves and LH slow waves. This comparative analysis spans aspects of their propagation, absorption, and CD mechanisms. As a result, these comprehensive computations offer valuable insights into the selection considerations for helicon and LH slow wave systems in the context of future DEMO scenarios.
Finally, it should be emphasized that the utilization of CQL3D in this context incorporates considerations of quasilinear effects, rendering this quasi-linear approach capable of addressing wave heating and CD under certain reactor-relevant conditions [30]. Nevertheless, as previously highlighted, as the wave power increases, the significance of nonlinear damping becomes more pronounced. In such scenarios, the extension of PIC simulation codes assumes importance, offering a foundation for investigating intricate aspects encompassing toroidal geometry, electromagnetic influences, nonlinearitydriven dynamics, nonlinear ion Landau damping, and the phenomenon of parameter decay instability [16][17][18][19][20]. These areas will serve as focal points for our forthcoming endeavors.