Investigation of alpha-particle transport by Alfvén eigenmodes in CFETR using a simplified diffusion model

The aim of this work is to analyze the alpha particle transport induced by Alfvén eigenmodes (AEs) based on the latest design of China Fusion Engineering Test Reactor (CFETR). Firstly, the stability of AEs is analyzed in order to understand the physical characteristics of AEs in CFETR with latest design parameters. AEs driven by alpha particles are analyzed using the gyrokinetic ion/fluid electron hybrid code GEM. The transport of alpha particles is investigated using a reduced energetic particle transport model based on the resonance broadened quasi-linear model. GEM results show that AEs with many toroidal mode numbers can be destabilized simultaneously by alpha particles in CFETR, and the toroidal mode number of the most unstable mode is n=8 . It is found that the excitation threshold of the alpha particle central beta for the n=8 mode is substantially below the expected alpha particle beta value of CFETR. All these results indicate that AEs can be driven strongly in CFETR plasmas. Furthermore, simulation results using the reduced transport model show that, due to multiple unstable AEs, the alpha particle density decreases in the core and alpha particles are transported from the core region ( r/a<0.35 ) to the outer region ( 0.35$?> r/a>0.35 ). In addition, we find that the minimum value of the safety factor qmin and the central beta value of alpha particle have significant effects on the alpha particle transport, and the AE-induced alpha particle transport level becomes lower with smaller linear growth rates. Finally, it is found that the radial range of the redistributed alpha particle density profile depend on the qmin radial location.


Introduction
China Fusion Engineering Test Reactor (CFETR) is the next-generation fusion facility in the roadmap of Chinese fusion energy development [1][2][3], which aims to bridge the gap between International Thermonuclear Experimental Reactor (ITER) [4] and the magnetic fusion energy demonstration reactor (DEMO) [5].At the moment, key physics and engineering design challenges of CFETR are being investigated [6][7][8][9].One challenge is the confinement of energetic particles (EPs), especially the alpha particles produced by the deuterium-tritium fusion reaction.It has been demonstrated both theoretically and experimentally that Alfvén eigenmodes (AEs) can be destabilized by EPs through waveparticle interaction [10][11][12][13][14][15].In present tokamak devices, it is found that AE activity can flatten the EP profile radially, which may degrade fusion performance in a reactor.In addition, strong AE activity may cause losses of EPs and even damage the first wall.Therefore, it is crucial to investigate the physical mechanisms of AEs driven by EPs and EP transport.
EPs are mainly produced by various auxiliary heating methods in present tokamaks.However, the deuterium-tritium fusion reaction can also produce energetic alpha particles in future reactors, such as ITER or CFETR.It has been shown recently that AEs can be driven strongly in CFETR [16].Ren et al [16] numerically investigated AEs driven by alpha particles using the gyrokinetic code GEM.It was found that the most unstable toroidal mode number is n = 10, which is larger than those of beam-driven AEs in present tokamaks.It was demonstrated that these AEs are toroidal Alfvén eigenmodes (TAEs) [16].Recently, the key parameters and equilibrium are updated for the latest CFETR design, and this is expected to affect the AE instability significantly.Therefore, it is necessary to re-evaluate the alpha-driven AEs based on the latest design of CFETR.Yang et al investigated the linear stability of the AEs in CFETR with the updated parameters using the linear eigenvalue code NOVA/NOVA-K, and it is found that the n = 8 TAE is the most unstable mode [17].Furthermore, unstable AEs can induce significant alpha particle transport and degrade alpha particle heating [18][19][20].Bass et al investigated the EP transport induced by AE instabilities in ITER using a reduced 1D critical gradient model (CGM).In that model, the critical EP density gradient is determined by the rule of γ AE = γ ITG/TEM based on the gyrokinetic code GYRO [21][22][23][24].Their results show that the EPs are redistributed by AEs from inner core with insignificant alpha particle losses to the edge [23].Gorelenkov et al have recently developed the resonance broadened quasi-linear (RBQ) model for EP transport [25].RBQ is a quasi-linear EP transport model which evolves the EP distribution in both real space and velocity space during the EP relaxation.The RBQ model is designed not only for the isolated modes but also for multiple overlapping modes, whereas the conventional quasi-linear theory applies only to cases of multiple overlapping modes.In addition, it has been demonstrated that the RBQ model can predict EP transport in present tokamaks [25][26][27].
In this work, we investigate the AE instability using the hybrid gyrokinetic ion/massless fluid electron model in GEM code with the latest design parameters of CFETR.In addition, a simplified EP transport model based on the RBQ model is applied to investigate the transport of alpha particles in CFETR.Firstly, the stability of alpha-driven AE is analyzed.The results show that the toroidal mode number of the most unstable mode is n = 8 and the n = 7, 8, 9 modes are of TAE type.Secondly, we find that alpha particles are re-distributed significantly due to AEs from the inner core to the outer region.The value of linear growth rate of AEs has a decisive effect on the level of alpha particle transport.
The article is organized as follows: section 2 briefly describes the model of GEM code and presents the main parameters and profiles of CFETR.In section 3, we show the linear simulation results of alpha-driven AEs, including the excitation threshold and damping rate of the most unstable mode.In section 4, we describe the simplified alpha particle transport model developed based on the RBQ model, which is then used to investigate alpha particle transport induced by AEs.The effects of key parameters on alpha particle transport are analyzed, including alpha particle beta, minimum value of the safety factor q min , and the radial location of q min .In section 5, discussions and conclusions are given.

GEM model
In this work, we use a hybrid model of GEM to investigate AE instability in CFETR and develop a simplified transport model to calculate the diffusion coefficient of EP, which is applied to study alpha particle transport in section 4. In the hybrid model of GEM, the electrons are treated as massless fluid, whereas the ions are described using the gyrokinetic equation with the δf particle-in-cell method [28].The GEM code has been extensively applied to investigate MHD instabilities and AEs in tokamaks, including alpha-driven AEs driven in ITER and CFETR [16,[29][30][31].More details of the GEM model are described in [32,33].

Equilibrium profiles and parameters
In this work, the parameters and equilibrium profiles according to the latest design of CFETR (steady state scenario) are used.The parameters are listed as follows: major radius R = 7.2m, minor radius a = 2.2m, toroidal magnetic field B t = 6.5T and plasma current up to 14MA.In the previous work, the AE stability in CFETR has been studied based on the former design parameters with a smaller device size and a lower magnetic field [16].The comparison of toroidal magnetic field and main geometry parameters used in the previous and present works are shown in table 1.The equilibrium profiles with the normalized minor radius r/a are shown in figure 1, where the minor radial coordinate r is defined as r/a = √ ψ t with ψ t being the normalized toroidal flux.Figure 1(a) shows the safety factor q profile.Figures 1(b)-(f ) show the radial profiles of the alpha particle density, and the densities and temperatures of thermal electrons and ions, separately.In figure 1, the green lines indicate equilibrium profiles used in this work, and the purple lines indicate equilibrium profiles used in former work.From this figure, it is found that the shape and the minimum value of q profile used in present work are all different from those in the former work.The central values of alpha particle density, as well as the densities and temperatures of thermal electrons and ions are all larger in comparison with those in the former work.These parameters and profiles play a significant role on the AE stability as shown in former work.Therefore, the AE instabilities are re-analyzed in this work with the latest parameters and profiles [34].
In addition, several other key parameters are also different.These parameters are as follows: central total beta β total = 6.626%, central alpha particle beta β α = 1.156%, the ratio of alpha particle birth speed to Alfvén velocity ν h /ν A = 1.42 and the Larmor radius of alpha particle normalized by the minor radius ρ h /a = 0.019.The simulation domain in radius is chosen to be 0.1 < r/a < 0.6 with fixed boundary condition.Furthermore, the beam ion beta value is much smaller than the beta value of alpha particles as discussed in the previous work.Therefore, only the contribution of alpha particles to the AE destabilization is considered in this work.An isotropic slowing down distribution in velocity space is applied for alpha particles, which is given by where C v is a normalization constant defined as C ν = ´νb 0 (4π ν 2 /(ν 3 + ν 3 I ))/dν.n h is alpha particle' density, ν b is the birth speed of alpha particle, and the cube of the critical velocity is given by where Z 1 is effective charge number and is given by

The linear stability of AEs in CFETR
In this section, the linear stability of intermediate n AEs in CFETR is analyzed with the parameters and profiles described in section 2.2. Figure 2 shows the linear growth rates of AEs with toroidal mode number 4 ⩽ n ⩽ 12.It is found that the toroidal mode number of the most unstable AE is n = 8, which is larger than those in present tokamaks.This result is consistent with the theoretical prediction as the toroidal magnetic field and minor radius are larger in CFETR than those of present tokamaks.Fu and Cheng showed that the linear growth rate is maximized at k θ ρ α on the order of unity for the TAE instability [14].In this work, the alpha particles' Larmor radius normalized by the minor radius is ρ h /a = 0.019, the largest mode amplitude is located around r/a ∼ 0.32, the effective radius is √ κr with elongation κ = 2.0, and the q value is around 3.95 at r/a ∼ 0.32.Thus the toroidal mode number n for the most unstable TAEs is about n ∼ √ k(r/a)/q/(ρ h /a) ∼ 6 according to the formula k θ ρ α = (nq/ √ κr)(mv/ZeB) ∼ 1, which is similar to the n = 8 from the simulation.In addition, Yang et al also showed that the most unstable toroidal mode number is around n = 8 [17].
In addition, figure 3 shows the 1D radial structure of n = 6 ∼ 9 modes with different poloidal harmonics.Compared with the previous work on CFETR, the AE mode structures obtained based on the latest parameters and profiles are  more global in radial direction with more poloidal harmonics, which is favorable for inducing larger alpha particle transport.Furthermore, figure 4 shows Alfvén continuous spectra of n = 7, 8, 9 modes calculated by NOVA using the slow-sound approximation.The radial locations of the modes combined with the mode frequencies relative to Alfvén spectra indicate that these modes are inside or a little lower than the TAE gaps, which indicate that these modes are TAEs.It should be noted that the n = 7 mode frequency is a little lower than the TAE gap, which is probably because NOVA code uses MHD model to calculate the continuous spectra, and the simulation is carried out using a gyrokinetic model for ions.
In addition, the excitation threshold of alpha particle for the n = 8 mode is analyzed.Figure 5 shows the linear growth rate and mode frequency of the n = 8 mode as a function of alpha particle' central beta value β α .The relationship between the linear growth rate and EP beta value is expected to be linear approximately for a TAE mode when the mode frequency and mode structure remain nearly the same for a range of alpha beta value.Therefore, we can obtain the excitation threshold of alpha particle and the total damping from the calculated linear growth rate with different β α .As shown in figure 5, it is found that the excitation threshold of alpha particle beta for the n = 8 TAE instability is around β α = 0.206% and the linear extrapolation to β α = 0 indicates that the damping rate of the mode is γ d /ω A = 0.983%.This excitation threshold of alpha particle central beta is much less than the projected central alpha particle beta value of β α = 1.156%, which indicates that the multiple AEs can be excited strongly in CFETR.

Transport model
A simplified alpha particle transport model based on the RBQ model is proposed and applied in this work.We now describe the model in some details below.In the RBQ model, the diffusion coefficient is given by [25] where J = J 1 , J 2 , J 3 is the vector of actions of EP unperturbed motion, C k is the amplitude of the kth AE driven by EPs, m is the resonant particle's mass.ω k − I • ω AI (J) = 0 is the resonance condition, and where ω k is the mode frequency, n k is the toroidal mode number of mode k, p is an integer, ω ϕ is the toroidal transit frequency and ω θ is the poloidal transit frequency.F I (I • ω AI (J) − ω k ) is the window function with a finite resonance width, and , where e k is the electric field of the kth mode, and j k I (x|J) is the resonant particle's current.More detailed descriptions of equation ( 3) are given in [25].Considering the resonant particles, the variation in the direction of toroidal angular momentum can be estimated as δP ϕ ≈ Ze ∂ψ ∂r δr, which means that the resonant particle movement in the P ϕ direction can be regarded as in the radial direction.In addition, if the mode frequency is small or the toroidal mode number is large, the diffusion of resonant EPs can be assumed just in the radial direction, which is justified by the estimate When the dependence on the velocity space is ignored, the diffusion coefficient can be simplified as follows.Considering that the nonlinear trapping frequency of a resonant particle is proportional to the square root of the mode amplitude C 2 k ∝ ω 4 b and that |ω b,sat | = 8 1/4 [γ/(γ + γ d )] 1/4 ν k at saturation for weakly unstable AEs where γ d is the damping rate, γ is the linear growth rate, and ν k is the collision frequency, we have 2 is proportional to the square of mode amplitude, we have D ∝ γA 2 (r), where A(r) represents the effective mode structure in the radial direction.
The information of linear growth rate and mode structure is calculated by GEM code in this work.The mode amplitudes of different toroidal mode numbers are normalized by the maximum value, and then the actual diffusion coefficient used in this simplified model is given by n , where n is the toroidal mode number, D c is a constant coefficient.
It should be noted a number of simplifications have been made in this simplified model of the radial diffusion coefficient.First, because evaluating the damping rates of AEs for all the toroidal mode numbers is very time-consuming, we have assumed that the damping rate of each mode is nearly the same and it does not affect the diffusion coefficient.Moreover, as shown in figure 2(b), the differences in the frequencies of different toroidal mode number AEs are very small, we assume the mode frequencies are the same and do not include them in the expression of C 2 k .Second, regarding the coefficient α k I (J) in equation ( 3), it is actually a weighted orbit integral of eigenmode structure and it is dependent on particle's energy, toroidal angular momentum and pitch angle.Therefore, the radial structure of α k I (J) is not simply given by A n (r) of each eigenmode.However, as the green line shown in figure 7, the effective mode width is around 0.15a, and the ratio of finite orbit width of the resonant alpha particles to effective mode width is around qρ α /0.15a = 0.36.As a result, we assume the orbit width of the resonant alpha particle is much smaller than the effective mode width in the simplified model, and we ignore the finite orbit width effect and assume that α k I (J) is proportional to the square of mode amplitude.Finally, D(J; t) defined in equation ( 3) is actually a diffusion matrix in phase space and it is a strong function of particle energy and pitch through the resonance window function F I .Therefore, strictly speaking, the simple formula of n is a phase-spaceaveraged radial diffusion coefficient for resonant particles.Our treatment is similar to the CGM, in which the diffusion coefficient is given by the deviation from the critical gradient of AE instability and regardless of whether these particles are resonant or not.Nonetheless, our model is a significant improvement over the CGM.The CGM model is local and its diffusion coefficient is given locally through the deviation from the local critical gradient of AE instability.In contrast, our model is global and the radial profile of diffusion coefficient is more self-consistently determined by radial mode structures of unstable AEs.Furthermore, the unstable eigenmodes evolve self-consistently with the evolving EP density profile.
Similar to the treatment in the CGM, the transported alpha particle density profile is converged when the value of D c is chosen to be sufficient large.Here the value of D c is chosen to be 19 m s −2 .As shown in figure 6, the transported alpha particle density profile is converged when D c ⩾ 10m s −2 for our case, which indicates that the chosen value of D c = 19 m s −2 is appropriate.The radial distribution of the initial  diffusion coefficient of alpha particle in CFETR is shown in figure 7.
The 1D diffusion equation of EP is given by [23] ∂n where n j (r) is the density profile of alpha particles, r is minor radius, D j is the diffusion coefficient for alpha particles, V ′ = (2π R)(2π rκ 1/2 ) is the radial derivative of volume enclosed by the local flux surface V at major radius R, minor radius r, and elongation κ.In addition, S 0,j = n iD n iT ⟨σ v⟩ DT is the alpha particle source rate from DT fusion, and n SD j is the initial alpha particle density profile.
The RBQ model allows the relaxation of the EP distribution function in the presence of multiple AEs.Compared with the RBQ model, our simplified model does not include the dependence on the velocity space, which is similar to that of CGM [21][22][23].In comparison with CGM, the diffusion coefficient defined globally in the simplified model including the dependence on AE mode structure.Furthermore, the linear growth rates of AEs are calculated globally by GEM code in this work, which is also different from that of CGM where the linear growth rates of AEs are calculated locally by TGLF [35].

Alpha particle transport prediction in the CFETR baseline case
In this section, all the AE instabilities are simulated using GEM [28,29], including the information of linear growth rates and mode structures used in the transport model.In addition, alpha particle transport due to AE instabilities is investigated using the simplified transport model described in the above section.
Multiple iteration steps are applied to calculate the alpha particle transport in the presence of multiple unstable AEs according to the transport model, and the iteration process is shown as follows.Firstly, the diffusion coefficient is calculated based on the linear growth rates and mode structures of AEs, which are simulated by GEM with the initial alpha particle density profile, and alpha particle density profile is evolved by solving the diffusion equation.As shown in figure 8, the evolving alpha particle density profile would reach a steady state after many time steps, and the red line is the updated alpha particle density profile.At the next iteration step, we recalculate the linear growth rates and mode structures of AEs with the updated alpha particle density profile from the previous iteration step, and evolve alpha density profile again until reaching a steady state using the diffusion coefficient corresponding to the updated mode structures and growth rates.This iteration process is repeated until the alpha particle density profile is almost the same with that of the previous iteration step.
In figure 9, the black line shows the initial alpha profile of CFETR.In figure 10, the black line shows the corresponding linear growth of AEs driven by alpha particles in the range of 4 ⩽ n ⩽ 12.In these two figures, the iteration 1 denotes the alpha particle density profile after the first iteration step and the corresponding linear growth rates of AEs with the updated alpha particle density profile.It is found that AEs can induce significant alpha particle redistribution for the first iteration step.Afterwards, alpha particle density profile and AE instabilities are updated repeatedly for each subsequent step by the same method.The results of iteration 3 to iteration 6 are not shown in figures 9 and 10 for clarity.Finally, when alpha density profile remains almost unchanged between two iteration steps, the corresponding alpha particle density profile is judged to be the final redistribution profile due to AEs, which is shown in figure 9.Moreover, AEs are marginally unstable with the alpha particle density profile at iteration 9, as shown in figure 10.From figure 9, it is found that the alpha particles are  substantially redistributed radially due to AEs and are transported from the core region (r/a < 0.35) to the outer region (r/a > 0.35).As shown in figure 7, in comparison with the diffusion coefficient calculated by the initial alpha particle density profile, the diffusion coefficient of the final redistributed alpha particle density profile is much smaller as expected.
In order to ascertain whether the redistributed alpha particle density profile due to the AEs with toroidal mode numbers 4 ⩽ n ⩽ 12 is reasonable, AEs are analyzed with toroidal mode numbers from n = 13 to n = 20 added.Low-n modes with n < 3 are excluded because of a high-n approximation the field solvers of GEM.The linear growth rates from n = 4 to n = 20 with the initial alpha profile of CFETR and the final transported alpha particle density profile at iteration 9 in figure 9 are shown in figure 11.It is that the linear growth rates of AEs with n > 12 decreases when n increases with the initial  alpha profile.More importantly, the AEs with n > 12 are still marginally unstable with the final alpha particle density profile at iteration 9, which means that these AEs with higher n would lead to little change in alpha particle transport with the final alpha particle density profile.

The effects of key parameters on alpha particle transport
The diffusion coefficient decreases when the linear growth rate decreases according to the formula D ∝ γA 2 (r).In this section, we first analyzed the alpha particle transport due to AEs with lower linear growth rates using the same method used in sections 4.1 and 4.2.We mainly consider two cases: (A) lower alpha particle beta (β α = 0.46%), (B) different q min value.
In figure 5, the red square represents the linear growth rate of the n = 8 mode with alpha particle central beta β α = 0.46%, and the linear growth rate of the n = 8 mode is lower than that with the original alpha particle beta value.Figure 12 shows the linear growth rate of AEs with different n, in which the green line is obtained with β α = 1.156% and the red line is obtained with β α = 0.46%.It is found that the linear growth rates of all modes for the case of β α = 0.46% are lower than the corresponding modes with the original alpha particle beta β α = 1.156%.
In figure 13(b), the black line shows the linear growth of AEs driven by alpha particle in the range of 4 ⩽ n ⩽ 12 with the initial alpha particle density profile.In figure 13(a), the black line shows the initial alpha particle density profile in CFETR and the red line indicates the final redistributed profile due to AEs.It is found that the density profile of alpha particle remains almost unchanged at the last two iterations.It is worth noting that the transport of alpha particles is also significant with alpha particle beta β α = 0.46%, although it is substantially smaller than that of β α = 1.156% as expected.
In addition, the alpha particle transport with q min = 2.2 is analyzed.Figure 14 shows the linear growth rates of AEs with different values of q min , which shows that the linear growth increasing with q min increasing, and the linear growth rates of AEs with q min = 2.2 are significantly lower than those with the original safety factor profile (q min = 2.73).The results of alpha particle transport with q min = 2.2 are shown in figures 15(a) and (b), in which the black lines represent the initial alpha particle density profile and the corresponding linear growth rates of AEs in the range of 4 ⩽ n ⩽ 12, respectively.In addition, the red line indicates the final redistributed profile in figure 15(a), and the blue line shows the linear growth rates of AEs with the alpha particle density profile at iteration 7 in figure 15(b).In figures 15(a) and (b), the results of iteration 3 to iteration 5 are not shown for clarity.We observe that the alpha particle profile at iteration 8 is almost the same as that of iteration 7 and is judged to be the converged profile, and the corresponding linear growth rates of AEs at iteration 7 are very small.
Figure 16 shows the comparison between the final transported alpha particle density profiles with different parameters and the initial profile.The black line indicates the initial alpha particle density profile, and the red line indicates the final redistributed density profile for the baseline case.The purple line and green line indicate the final redistributed alpha particle density profile with q min = 2.2 and a lower alpha particle beta of β α = 0.46%, respectively.These results show that the transport level is substantially lower for the cases with q min = 2.2 or β α = 0.46% in comparison with the original case.As we expect, the alpha particle radial transport is lower with smaller linear growth rates of AEs.As a result, we can reduce the transport level by decreasing alpha particle beta and the value of q min .
In order to verify the effect of the q min value on the stability of AEs, the excitation threshold of alpha particle  with q min = 2.2 for the n = 8 mode is analyzed, as shown in figure 17.It is found that the excitation threshold is around β α = 0.515% and the damping rate of the mode is γ d /ω A = Figure 16.The final transported alpha particle density profiles for different cases.The red line indicates the final redistributed alpha particle density profile for the baseline case, the purple line indicates the final redistributed alpha particle density profile with lower q min value q min = 2.2, and the green line indicates the final redistributed alpha particle density profile with lower alpha particle beta βα = 0.46%.The black line indicates the initial alpha particle density profile.1.42%.The excitation threshold and damping rate with q min = 2.2 are both larger than the original case with q min = 2.73, which indicates that the damping is changed for different safety factor profiles.
Finally, we analyzed the effect of q profile on alpha particle transport by changing the radial location of the minimum value of safety factor q min .The q profile with r q min /a = 0.492 is chosen to be analyzed in this section, as the green line shown in figure 18.The safety factor profile with different q min radial locations.The purple line indicates the baseline q profile with q min located at rq min /a = 0.565, and the green line indicates the q profile with q min located at rq min /a = 0.492.density profile, which is larger than the baseline case.A possible explanation is that the q value is smaller at the radial location of mode peak.The radial mode structures with toroidal mode number n = 11, 12 are shown in figure 20.The range of 7 ⩽ n ⩽ 16 are included in the calculation of the diffusion coefficient because the toroidal mode number of the most unstable mode is larger than the baseline case.The red line indicates the final redistributed profile in figure 19(a), and the blue line shows the linear growth rates of AEs with the alpha particle density profile at iteration 7 in figure 19(b).The corresponding linear growth rates of AEs at iteration 7 are very small.
The comparison between the final transported alpha particle density profile in this section and that using the baseline profile is shown in figure 21.The black line indicates the initial alpha particle density profile, and the red line indicates the final redistributed density profile of the baseline case.The green line indicates the final redistributed alpha particle density profile with r q min /a = 0.492.We define the ratio of transported EPs to total EPs as ´|f − f 0 | rdr/2 ´f0 rdr, and the transported EP ratio is found to decrease slightly from 6.72% of the baseline case to 6.05% with r q min /a = 0.492.In addition, the radial transport range of the final transported alpha particle density profile moves outwards compared to the baseline case, which is consistent with the radial location change of maximum mode amplitudes.As a result, we find that the change of the q min radial location would mainly affect the radial range of the redistributed alpha particle density profile obviously.

Discussion and conclusion
In this article, the AE instability driven by alpha particles in CFETR with the latest design profiles and parameters has been investigated using the gyrokinetic code GEM.The corresponding alpha particle transport induced by multiple AEs has also been investigated using a simplified quasi-linear model.The simulation results show that multiple AEs can be driven unstable by alpha particles and the toroidal mode number of the most unstable mode is n = 8.In addition, the shear Alfvén continuous spectra calculated by NOVA show that these modes are TAEs.The effect of central alpha particle beta value on the linear growth rate of n = 8 mode is analyzed.It is found that the excitation threshold of the most unstable n = 8 mode is β α = 0.206%, which is substantially below the alpha particle central beta value β α = 1.156% in CFETR.These results indicate that alpha particle-driven AEs are strongly unstable in CFETR with the latest parameters.
This work mainly aims at analyzing the transport of alpha particle driven by AEs in CFETR.Multiple AEs with toroidal mode numbers 4 ⩽ n ⩽ 12 are included in the transport model.These results indicate that the AEs can induce significant radial transport of alpha particles from core region (r/a < 0.35) to the outer region (r/a > 0.35).We have also studied the effect of the minimum value of the safety factor q min and alpha particle central beta value α on the alpha particle transport.We find that alpha particle transport is strongly dependent on the strength of AE instabilities in CFETR, and the transport levels are lower for the cases with lower linear growth rates.Therefore, the transport of alpha particle can be reduced by controlling the AE instability.In addition, we find that the change of the q min radial location would affect the radial range of the redistributed alpha particle density profile.It should be noted that a simplified EP transport model based on the RBQ model is used in this work.Specifically, global mode structures of multiple AEs are included in this model.Future work will also include alpha particle redistribution in velocity space.An eigenvalue code is under development and will be applied to calculate the mode structures and linear growth rates of AEs for numerical efficiency.

Figure 1 .
Figure 1.Equilibrium profiles of (a) safety factor, (b) alpha particle density, (c) electron density, (d) electron temperature, (e) ion density and (f ) ion temperature.The green lines indicate equilibrium profiles used in this work, and the purple lines indicate equilibrium profiles in previous work [16].

Figure 2 .
Figure 2. Linear growth rate and mode frequency of AE versus toroidal mode number n in the range of 4 ⩽ n ⩽ 12.

Figure 4 .
Figure 4. Alfvén continuous spectrum with different toroidal mode number n calculated by NOVA.The red lines indicate the mode frequencies and radial locations of the AEs simulated by GEM.

Figure 5 .
Figure 5.The linear growth rate and mode frequency of the n = 8 mode versus βα.Based on the simulated growth rate data extrapolated to γ = 0 and βα = 0, the threshold βα = 0.206% and the damping rate γ d /ω A = 0.983% are obtained.

Figure 6 .
Figure 6.The black line indicates the initial alpha particle density profile, and the other lines indicate the transported alpha particle density profiles with different Dc.

Figure 7 .
Figure 7.The diffusion coefficient as a function of radius.The green line shows the diffusion coefficient with the initial alpha particle density profile, and the red line shows the diffusion coefficient with the final alpha particle density profile.

Figure 8 .
Figure 8. Evolution of alpha particle density profile at different time steps within one iteration.

Figure 9 .
Figure 9. Evolution of alpha particle density profile.The black line indicates the initial profile, and other lines indicate the density profiles at different iteration steps.The red line indicates the final redistributed profile in the presence of multiple AEs.

Figure 10 .
Figure 10.Linear growth rates of AEs versus toroidal mode number n in the range of 4 ⩽ n ⩽ 12 with different alpha particle density profiles.The black line indicates the initial growth rates, and other lines indicate the results at different iteration steps.The blue line indicates the results of iteration 9.

Figure 11 .
Figure 11.The red line shows the linear growth rate as a function of toroidal mode number with the initial alpha profile, and the green line shows the linear growth rate with the final transported alpha particle density profile at iteration 9 of figure 9.

Figure 13 .
Figure 13.(a) Alpha particle density profile evolution with a lower alpha particle beta (βα = 0.46%).The black line indicates the initial profile.Other lines indicate the transported density profiles at different iteration steps and the red line indicates the final profile.(b) The black line indicates the initial growth rates, and other lines indicate the results with transported alpha particle density profiles at different iteration steps and the blue line indicates the final results.

Figure 14 .
Figure 14.The green line indicates the linear growth rates of the 4 ⩽ n ⩽ 12 modes with q min = 2.73 and the red line indicates the linear growth rates of the 4 ⩽ n ⩽ 12 modes with q min = 2.2.

Figure 15 .
Figure 15.(a) Alpha particle density profile evolution with q min = 2.2.The black line indicates the initial profile.Other lines indicate the transported density profiles at different iteration steps and the red line indicates the final profile.(b) The black line indicates the initial result, and other lines indicate the results with transported density profiles at different iteration steps and the blue line indicates the final results.

Figure 17 .
Figure 17.The linear growth rate and mode frequency of the n = 8 mode with q min = 2.2 versus βα.Based on the simulated growth rate data extrapolated to γ = 0 and βα = 0, we obtain the threshold βα = 0.515% and the damping rate γ d /ω A = 1.42%.

Figure 18 .
Figure18.The safety factor profile with different q min radial locations.The purple line indicates the baseline q profile with q min located at rq min /a = 0.565, and the green line indicates the q profile with q min located at rq min /a = 0.492.

Figure 19 .
Figure 19.(a) Alpha particle density profile evolution with rq min /a = 0.492.The black line indicates the initial profile.Other lines indicate the transported density profiles at different iteration steps and the red line indicates the final profile.(b) The black line indicates the initial result, and other lines indicate the results with transported density profiles at different iteration steps and the blue line indicates the final results.The results of iteration 3 to iteration 5 are not shown for clarity.

Figure 20 .
Figure 20.The radial mode structures with different poloidal harmonics for n = 11 and n = 12 AEs with rq min /a = 0.492.

Figure 21 .
Figure 21.The red line indicates the final redistributed alpha particle density profile for the baseline case, the green line indicates the final redistributed alpha particle density profile with new q profile.The black line indicates the initial alpha particle density profile.

Table 1 .
Magnet field and main geometry parameters used in two works.