A linear parameters study of ion cyclotron emission using drift ring beam distribution

Ion cyclotron emission (ICE) holds great potential as a diagnostic tool for fast ions in fusion devices. The theory of magnetoacoustic cyclotron instability (MCI), as an emission mechanism for ICE, states that MCI is driven by a velocity distribution of fast ions that approximates a drift ring beam. The influence of key parameters on the linear MCI is systematically investigated using the linear kinetic dispersion relation solver BO (Xie H. 2019 Comput. Phys. Comm. 244 343). The computational spectra region considered extends up to 40 times the ion cyclotron frequency. By examining the influence of these key parameters on MCI, several novel results have been obtained. In the case of MCI excited by super-Alfv\'enic fast ions, the parallel velocity spread significantly affects the bandwidth of harmonics and the continuous spectrum, while the perpendicular velocity spread has a decisive effect on the MCI growth rate. As the velocity spread increases, the linear relationship between the MCI growth rate and the square root of the number density ratio transitions to a linear relationship between the MCI growth rate and the number density ratio. This finding provides a linear perspective explanation for the observed linear relation between fast ion number density and ICE intensity in JET. Furthermore, high harmonics are more sensitive to changes in propagation angle than low harmonics because a decrease in the propagation angle alters the dispersion relation of the fast Alfv\'en wave. In the case of MCI excited by sub-Alfv\'enic fast ions, a significant growth rate increase occurs at high harmonics due to the transition of sub-Alfv\'enic fast ions to super-Alfv\'enic fast ions.

A successful theoretical explanation for the excitation mechanism of ICE is magnetoacoustic cyclotron instability (MCI) in the locally uniform approximation [53][54][55][56][57][69][70][71][72][73].The MCI theory was first developed by Belikov and Kolesnichenko [71], in which the fast Alfvé n and ion Bernstein branches at frequencies close to the cyclotron harmonics Ω F (where  is the harmonic number) of a fast ion species F , are excited and propagate strictly perpendicular to the magnetic field in frequencies  ≫ Ω F (where Ω F is the ion cyclotron frequency).Dendy et al. later extended this analysis to the frequency range ~Ω F , providing an explanation for ICE excitation [69].The inclusion of finite parallel wave number  ∥ allowed MCI to being further successful in explaining the ICE excited by super-Alfvé nic fast ions in JET [53].Subsequently, MCI was used to account for ICE excited by sub-Alfvé nic fast ions generated by fusion reactions and NBI [55].Additionally, a variant of MCI successfully accounted for ICE excited by greatly sub-Alfvé nic fast ions from NBI [54].Analysis further considering magnetic field gradient and curvature drift effects predicted higher MCI growth rates [56,57].
However, the aforementioned MCI theory is valid when the instability growth rate  is larger than the inverse bounce/transit period of the fast ions   −1 ( >   −1 ).When the instability growth rate  is lower than   −1 ( <   −1 ), the analysis must consider toroidal effects and eigenmode structures [73][74][75][76][77][78][79][80][81][82][83][84].In this work, our focus mainly lies on the locally uniform approximation MCI theory, where  >   −1 .A large number of linear and nonlinear simulations have confirmed the validity of MCI in the locally uniform approximation [14,38,44,85], and these simulations capture most of the key observed features of the ICE measurements in JET and TFTR experiments, including the simultaneous excitation of all cyclotron harmonics, the splitting of spectral peaks, ICE intensity scaling approximately linearly with the fast ion number density, the strong growth rate for nearly perpendicular wave propagation, and the congruence between the linear theory and the observed signal intensities.The experimental phenomenon of the un-captured continuous spectrum for  > 8 in JET has also gained further understanding in this work.What needs to be emphasized is the congruence between linear theory and the observed signal intensities, including the linear simulation results agreeing surprisingly well with both the peaks in ICE intensity at ion cyclotron harmonics and the trend of increasing intensity with harmonic number [58,59], and a striking correlation between the time evolution of the maximum linear growth rate and the observed time evolution of the ICE amplitude [55,66].Nonlinear simulations [58,59] suggest that MCI is intrinsically self-limiting on very fast timescales, providing an explanation for the observed correlation between linear theory and ICE intensity.Therefore, the self-limitation of MCI suggests that linear simulation is still a very important way to study ICE.
The key observed features of the ICE measurements captured by the simulations are closely related to the velocity spread of the fast ions, the number density ratio  F (the ratio of the fast ion number density to the background ion number density), and the instability propagation angle  (the angle between the wave propagation and the ambient magnetic field), and these three parameters have been the focus of MCI research.However, previous linear and nonlinear simulations have not systematically studied the influence of these key parameters on MCI for the drift ring beam distribution in three cases: super-Alfvé nic, sub-Alfvé nic, and greatly sub-Alfvé nic fast ions.This is a limitation for using ICE diagnosis to obtain fast ion information in future experiments.Therefore, this work provides a systematic investigation of the influence of these key parameters on linear MCI.The computational spectra region considered in this work is up to 40 times of the ion cyclotron frequency, which is rarely explored in other simulations.In addition, a more realistic experimental condition is considered in the simulation of ICE excited by super-Alfvé nic fast ions.The present work is carried out by using the fully kinetic dispersion relation program BO (further details provided in Appendix A) [86][87][88].The BO program has the advantage of providing all solutions of a linear kinetic plasma system without requiring an initial guess for root finding, which distinguishes it from previous simulation programs.This allows for parameter scanning across a wide range.Furthermore, the BO program model includes the wave electric field parallel to the ambient magnetic field.Therefore, MCI simulation carried out by using the BO program can be extended to arbitrary angle.
In section 2, the simulation results of BO program are successfully compared with the previous simulation results.In sections 3, 4 and 5, detailed simulation results are presented for MCI excited by super-Alfvé nic, sub-Alfvé nic, and greatly sub-Alfvé nic fast ions, respectively.Conclusions are presented in section 6.

Benchmark
In this section, the simulation results of the BO program are successfully compared with the linear theory for super-Alfvé nic and greatly sub-Alfvé nic fast ions, respectively.In all our simulations, the velocity space distribution of the fast ions follows a drift ring beam distribution [86]: Here  ∥ and  ⊥ denote velocity components parallel and perpendicular to the magnetic field, and  d and  ⊥ are constants that define the average parallel drift speed and the unique perpendicular speed, respectively.  and   are parameters that define the parallel and perpendicular velocity spread of the fast ions, respectively.
In figure 1, we compare the output of BO program with both linear analytical theory and the linear stages of first principles fully kinetic nonlinear simulations for MCI excited by super-Alfvé nic fast ions (alpha-particle).The simulation parameters of the BO program are the same as those in [59], where bulk plasma parameters approximate the outer midplane edge conditions of the JET Preliminary Tritium Experiment (PTE) pulse 26148.The magnetic field is 2.1T, electrons and bulk deuterons are thermalized at From figure 1, it can be seen that the simulation results obtained using the BO program are basically consistent with those from the nonlinear simulations and linear analytical theory.
Figure 2 shows the comparison between the BO program and the variant of the linear MCI theory regarding the growth rate of instability excited by greatly sub-Alfvé nic fast ions (deuteron).The simulation parameters used in the BO program are, following Ref.[54], magnetic field  0 = 5T , bulk deuteron temperature  D = 3keV , electron temperature  e = 1.5keV, bulk deuteron number density   Through the successful verification of MCI excited by super-Alfvé nic and greatly sub-Alfvé nic fast ions, the maturity and reliability of the BO program for MCI simulations have been demonstrated.Next, we will conduct detailed simulation on the velocity spread of the fast ions, the number density ratio, and the instability propagation angle for the cases of the MCI excited by super-Alfvé nic, sub-Alfvé nic, and greatly sub-Alfvé nic fast ions, respectively.It is important to note the classification of fast ions into these categories [4].On the one hand, the physical mechanism of ICE excited by greatly sub-Alfvé nic fast ions differs from that of ICE excited by super-Alfvé nic and sub-Alfvé nic fast ions.On the other hand, the excitation conditions of MCI are distinctly different for the above three fast ions.In general, the super-Alfvé nic fast ions can drive the MCI even if they are isotropic or have a relatively broad distribution of speeds [4].The sub-Alfvé nic fast ions that are isotropic or have undergone a certain degree of thermalization cannot drive the MCI [4].For the greatly sub-Alfvé nic fast ions with a very narrow spread of velocities in the parallel direction the instability can occur [4].In addition, the above three ions show more differences in this work.

Super-Alfvénic fast ions
In the JET Preliminary Tritium Experiment, ICE excited by super-Alfvé nic fast ions (alpha-particle) exhibits numerous features, and linear and nonlinear simulations have captured and explained most of the essential features [4,53,56,[58][59][60].However, the high cyclotron harmonic range has been seldom considered in the previous simulations.Based on this, the present work studies that the computational spectra region is up to 40 times of the ion cyclotron frequency while a more comprehensive MCI simulation results on the velocity spread of the fast ions, the number density ratio, and the instability propagation angle are presented.Additionally, we consider a more realistic experimental condition by including a certain percentage of tritium in the background plasma for deuterium-tritium plasma simulations.The simulated parameters are consistent with those used in section 2 regarding MCI excited by super-Alfvé nic fast ions.

Velocity spread
The velocity spread of fast ions generated by D-T fusion is influenced by the rise time of neutron emission and the fast ion slowing-down time.The research indicates that if the rise time of neutron emission exceeds the fast ion slowing-down time, collisions will cause the energy distribution of fast ions to broaden before new fast ions are added [89][90][91].Therefore, it is necessary to consider the impact of energy distribution on ICE.Studies on the effects of velocity spread on ICE have explored both the impact of considering only parallel velocity spread [50,57] and the combined effects of parallel and perpendicular velocity spread [2,55,70] , which corresponds to the case of isotropic temperature.These studies have all shown a suppressive effect on ICE.In our simulation, we consider isotropic temperature, i.e.,   =   , with the simulation range from 0 to 0.4 ⊥ .Figures 3 and 4 show that the growth rates of cyclotron harmonics up to  = 40 are plotted as a function of   for velocity spread ranging from 0 to 0.4 ⊥ .The results show that the growth rates of MCI gradually decrease with increasing   and   when the cyclotron harmonics are relatively low, i.e., less than 17, which is consistent with the previous simulation results (c.f., [55]).However, when considering higher harmonics, the simulations on MCI reveal more unique phenomena.Firstly, when   =   < 0.1 ⊥ , the harmonics greater than 18 are divided into four intervals, with the centers of these intervals located at  = 25, 32, 36, and 39, respectively.Secondly, the harmonics, specifically  = 19, 20, 21, and 29, are obviously suppressed in   =   = 0. Thirdly, in terms of the high harmonics, there is a significant suppression as   and   increase when   =   < 0.1 ⊥ .Overall, the growth rate of most harmonics decreases with increasing   and   .However, for boundary harmonics such as  = 22, in the four intervals, the growth rate decreases sharply first, then increases slightly, and finally decreases gradually with increasing   and   .Hence, the previous rule does not apply to such harmonics.
A particularly interesting phenomenon arises when   and   exceed 0.22 ⊥ : a continuous spectrum forms at the high harmonics, representing the first instance in which a simulation captures a continuous spectrum resembling experimental results.To understand the factors contributing to this phenomenon, we conduct a separate investigation into the influence of   and   on MCI. Figure 5 shows that the growth rates of cyclotron harmonics are plotted as a function of   for (a)  = 0.4 ⊥ ,   = 0, and (b)  = 0,   = 0.4 ⊥ .By comparing Figure 5 with Figure 3(a), we can clearly see that   exerts a minor suppressive effect on the growth rate of MCI but plays a decisive role in determining the bandwidth of harmonics and the presence of the continuous spectrum.Conversely,   significantly influences the growth rate of MCI.It should be noted that it is understandable that the previous simulations did not capture this key feature.In previous linear simulations,   was consistently small, and only a single low harmonic was considered.In previous nonlinear simulations, where the excitation energy of MCI stems from perpendicular speed,   remained small during the nonlinear evolution.Thus, in future nonlinear simulations containing a greater   , a continuum spectrum is foreseen.

Number density ratio
In JET [1], a linear relation was found between ICE intensity and neutron rate over six orders of magnitude.This observation not only suggests that ICE is excited by fast ions but also indicates a strong connection between fast ion number density and MCI.The simulations have successfully reproduced this experimental phenomenon and revealed a linear relationship between √ F and the MCI growth rate, with  F spanning 2 to 3 orders of magnitude in the linear phase [58,60].In our simulations, our study explores a wider range of  F spanning 5 orders of magnitude:  F = 5 × 10 −7 , 10 −6 , 10 −5 , 10 −4 , and 10 −3 , while keeping other simulation parameters constant.Figure 6 shows the ratio of the MCI growth rate  to √ F versus  F , where a straight horizontal trend implies a relationship ~√ F .It should be noted that the properties of the four high harmonic intervals regarding the relation between  and √ F is similar.Therefore, figure 6 only displays the relationship between  and √ F for each harmonic within one harmonic range (22 ≤  ≤ 28).Comparing figure 6 and figure 7(a), we observe that these high harmonics conforming to the linear relation appear only in the centers of the four intervals.In addition, when  < 22, the growth rate of all harmonics except  = 17,  = 18, and  < 6 varies linearly with √ F , which is consistent with the previous simulation results [60]. Figure 7 shows the growth rates of cyclotron harmonics up to  = 40 are plotted as a function of   for  F ranging from 10 −6 to 10 −3 .It is evident that as  F decreases, the growth rate of each harmonic gradually decreases.Similar to   ,  F also exerts a significant influence on the bandwidth of harmonics.
Lastly, considering the self-limitation of MCI, it is reasonable to infer that the linear relationship between ICE intensity and neutron rate during the nonlinear stage is not unrelated to the linear stage.Therefore, it is natural to speculate that the velocity spread plays a crucial role in the relationship between  and  F . Figure 8

Propagation angle
In the development of linear MCI theory, the fast Alfvé n and ion Bernstein branches is extended from perpendicular to oblique propagation.While Landau damping and ion cyclotron damping have been introduced into MCI, it has been observed through both linear and nonlinear simulations [53,56,57,62] that all ion cyclotron harmonics are excited simultaneously, exhibiting a strong growth rate for nearly perpendicular wave propagation.However, previous linear simulations neglected the wave electric field parallel to the ambient magnetic field, limiting the strict self-consistency of the results to large instability propagation angles.In nonlinear simulations, the instability propagation angle is also limited to a small range close to perpendicular direction.Therefore, we use the BO program, which includes the wave electric field parallel to the ambient magnetic field, to simulate the effect of a large angle deviating from the perpendicular direction on MCI excitation.We consider a large angle of deviation ( 15°≤  ≤ 90 °) and account for the parallel drift speed v d , which cannot be neglected in such cases.Based on [1,53], we set   =   = 0.04 ⊥ and v d = 0.25 ⊥ .Figure 9 shows that the growth rates of cyclotron harmonics up to  = 40 are plotted as a function of   for  ranging from 50 ° to 88 °.We do not present simulations with propagation angles greater than 90 ° since the forward and reverse propagating waves of angles greater than 90 ° are equivalent to the reverse and forward propagating waves of angles less than 90 °, respectively.Overall, as the propagation angle decreases, almost all harmonics are basically suppressed at propagation angles  < 15 °.At about 85 °, the harmonics || ≥ 15 are almost suppressed, indicating that the high harmonics are more sensitive to the propagation angle than low harmonics.
The rapid suppression of MCI with increasing the angle of deviation from the perpendicular direction is related to the dispersion relation of the fast Alfvé n wave.Figure 10 shows the dispersion relation of the fast Alfvé n wave at different angles, where the straight lines and curves correspond to ion Bernstein and fast Alfvé n waves, respectively.The dispersion relation changes when the propagation angle changes from 88 ° to 80 °, which causes  ⊥ to become smaller than the perpendicular phase velocity of the fast Alfvé n wave  A⊥ at harmonics || > 10 , invalidating the super-Alfvé nic condition and making the sub-Alfvé nic condition valid.This eventually leads to the suppression of the harmonics.It needs to be noted that previous references focused on the case of low harmonic numbers and large propagation angles, where the dispersion relationship of linear Alfvé n waves  =  ⊥  A can describe the dispersion relationship of Alfvé n waves well.Therefore, comparing  ⊥ with the Alfvé n speed  A is meaningful.However, when the harmonic number is large, the dispersion relationship of Alfvé n waves changes, and it makes sense to compare the  ⊥ with the perpendicular phase velocity of fast Alfvé n wave  A⊥ , as defined in this work.
One important observation from figure 9 is that each harmonic consists of both forward and reverse propagating waves, with the forward wave having a higher frequency than the reverse wave.The line splitting, which is evident in figure 9 due to  and  d , provides a simple explanation [4,53] for the spectral peak splitting observed in JET experiments.Furthermore, recent computational results reported in Ref. [85] indicate that the origin of spectral peak splitting is Doppler-shifted resonances and the intricate landscape of the MCI growth rate on the dispersion surface in ( ⊥ ,  ∥ ) space.

Containing tritium
In previous simulations of the JET Preliminary Tritium Experiment [50,53,[55][56][57][58][59][60], the background plasma was typically assumed to be deuterium plasma, which is a reasonable approximation for low tritium number density.However, for the higher tritium-to-deuterium number density ratios expected in the future ITER, it is necessary to contain a corresponding percentage of tritium in the background plasma.Specifically, we set the maximum ratio of tritium to total ion number density to 0.3, while keeping other parameters unchanged.Figure 11 shows that the growth rates of cyclotron harmonics up to  = 40 plotted as a function of   for tritium number density ratio ranging from 0 to 0.3.It is clear from the figure that, as the tritium number density ratio increases, the four high harmonic intervals move toward the low harmonics.The harmonics that are significantly suppressed not only exhibit the same trend, but also experience a slight increase in their number.

Velocity spread
As mentioned in section 2, ICEs excited by sub-Alfvé nic fast ions depend more on the velocity spread of the fast ions than those excited by super-Alfvé nic fast ions.However, this conclusion is obtained based on low harmonics, and new results have emerged with the inclusion of high harmonics in the present study.Figure 12, where the blue lines represent  ≤ 28 and the red lines represent  ≥ 29, show that the growth rates of cyclotron harmonics up to  = 40 are plotted as a function of   for velocity spread ranging from 0 to 0.35 ⊥ .The figure shows that the growth rate of harmonics less than 29 is one to two orders of magnitude smaller than those of the harmonics greater than 29, indicating that high harmonics are more likely to be excited.As the velocity spread increases, the harmonics less than 29 are rapidly suppressed, consistent with the previous simulation results that only consider the velocity spread in the perpendicular direction [64] or the combined effects of velocity spread in parallel and perpendicular [55], i.e., the isotropic temperature case, which all show a suppressive effect on ICE.The growth rate of harmonics greater than 28 gradually decreases, while continuous spectrum features appear, similar to the MCI excited by super-Alfvé nic fast ions.Further analysis reveals that the characteristics observed in high harmonics are related to the dispersion relation of the fast Alfvé n wave.Figure 13 shows that  A⊥ decreases gradually with the increase of harmonics, eventually resulting in  ⊥ >  A⊥ .Consequently, the sub-Alfvé nic condition becomes invalid at harmonics above 28 (the black circle mark in the figure), while the super-Alfvé nic condition becomes valid, resulting in a drastic increase in the growth rate at harmonics greater than 28.Finally, it should be noted that from the figure 12(a) the simulation results of the BO differ significantly from those in [62] at  = 11.This is because the wave-wave coupling, which leads to an increase in the growth rate [59], occurs in the late linear phase of the simulations in [62] while the results of the BO here regarding the MCI excited by sub-Alfvé nic fast ions in the linear phase.

Number density ratio
Similar to the MCI excited by super-Alfvé nic fast ions, detailed simulations about the number density ratio  F is presented for the MCI excited by sub-Alfvé nic fast ions.The simulation sets  F = 10 −6 , 10 −5 , 10 −4 , and 10 −3 , while keeping other parameters unchanged.Figure 14 illustrates that as the number density ratio  F decreases, the growth rate of each harmonic also decreases.Moreover, for harmonics above 29, they are divided into three intervals centered around  = 33, 37, and 39, respectively.We find that these high harmonics, which exhibit a linear relationship between √ F and , appear only at the centers of the three intervals, as shown in figure 15(b), where the interval encompassing  = 36 , 37 and 38 is depicted.Therefore, the high harmonics with the typical characteristics resembling MCI excited by super-Alfvé nic fast ions further supports the conclusions presented in section 4.1.Lastly, for harmonics less than 29, such as  = 15 in figure 15(a), the growth rate  is close to linear relation with √ F , which is consistent with the previous results [62].

Propagation angle
In our research on the relationship between the MCI excited by super-Alfvé nic fast ions and the propagation angle, simulation results indicate that the larger the angle of deviation from the perpendicular direction, the stronger the suppression of high harmonics.We have explored the relationship between MCI excited by sub-Alfvé nic fast ions and the propagation angle, and the conclusions at the high harmonics are consistent with those of the MCI excited super-Alfvé nic fast ions.From figure 16, which shows the relation between MCI growth rate and propagation angle, we can see that the high harmonics are first strongly suppressed as the angle of deviation from the perpendicular direction increases.Near  = 85 °, the growth rate of high harmonics is in the same order of magnitude with that of low harmonics, indicating that the super-Alfvé nic condition becomes invalid at harmonics above 28, while the sub-Alfvé nic condition becomes valid.As the propagation angle decreases further, the growth rate of high harmonics first gradually decreases.However, the growth rate of low harmonics does not decrease monotonically with decreasing propagation angle, but increase first and then decrease, which is consistent with the previous results [66].When the propagation angle is less than 10 °, all harmonics are basically suppressed.

Greatly sub-Alfvénic fast ions
Different from the electromagnetic instability excited by super-Alfvé nic and sub-Alfvé nic fast ions, the instability excited by greatly sub-Alfvé nic fast ions, which is a variant of MCI [54], is mainly electrostatic.There have been relatively few simulation studies on the ICE excited by greatly sub-Alfvé nic fast ions.Here we conducted a comprehensive simulation study on MCI, taking into account the velocity spread of the fast ions (deuteron), the number density ratio, and the instability propagation angle.The simulation parameters are, following Ref.[54,92], bulk deuteron temperature  D = 4keV, and electron temperature  e = 3keV.Other parameters remain the same as those used for greatly sub-Alfvé nic fast ions discussed in section 2. In our subsequent simulations, we focus solely on the forward propagating waves, as the forward and reverse propagating waves exhibit similar properties, and this choice enhances the clarity of the presented results.In addition, we compare the electromagnetic and electrostatic results carried out by using the program BO. Figure 17 illustrates that the results are largely consistent, supporting the analytical conclusion [54].

Velocity spread
As mentioned earlier, the excitation of ICE by greatly sub-Alfvé nic fast ions requires a very narrow spread of velocities in the parallel direction [4,54], on the order of 10 −2  ⊥ .The simulation results of the BO are consistent with the previous simulation results at low harmonics.However, when considering high harmonics, the excitation condition for MCI by greatly sub-Alfvé nic fast ions becomes relatively relaxed.Figure 18 shows that the growth rates of cyclotron harmonics up to  = 40 are plotted as a function of   for velocity spread ranging from 0 to 0.1 ⊥ .The figure shows that MCI excited by greatly sub-Alfvé nic fast ions exhibits a similar characteristic to that excited by sub-Alfvé nic fast ions at high harmonics, where the growth rate of high harmonics is one to two orders of magnitude larger than that of low harmonics.Different from the MCI excited by super-Alfvé nic and sub-Alfvé nic fast ions, the MCI excited by the greatly sub-Alfvé nic fast ions shows more line splitting (the line splitting of high harmonics in figure 18(a) is similar to that of low harmonics, but the details are not shown due to the relatively small values of the finite structure compared to its maximum value), which can be well understood through equation (11) in Ref. [54].Another important characteristic is that with increasing velocity spread, low harmonics are rapidly suppressed, while a few high harmonics still persist when   =   = 0.1 ⊥ .This expands the parameter range for studying ICE excited by greatly sub-Alfvé nic fast ions and is significant for experimental investigations of ICE.

Number density ratio
In the TFTR experiment, the number density ratio  F is on the order of 10 −2 , and relevant simulations have shown that exciting MCI for low harmonics is difficult at lower number density ratios [54].Here we further study the influence of number density ratio on each harmonic, especially the high harmonics.Figure 19 plots the growth rates of cyclotron harmonics up to  = 40 as a function of   for the number density ratio  F ranging from 10 −4 to 10 −1 .From the figure, we can see that the low harmonics are suppressed at a lower number density ratio.However, even when  F is reduced to 10 −4 , a high harmonic persists.In addition, at  F = 10 −1 , a continuous spectrum is formed at the high harmonics.Overall, as  F decreases, the growth rates of both low and high harmonics decrease rapidly, accompanied by narrower bandwidths.This behavior is similar to that of super-Alfvé nic and sub-Alfvé nic fast ions.

Propagation angle
Here we study in detail the relationship between the instability excited by greatly sub-Alfvé nic fast ions and the propagation angle.Figures 20 and 21 show that the growth rates of cyclotron harmonics up to  = 40 are plotted as a function of   for propagation angle ranging from 87 ° to 93 °.Notably, the instability excited by greatly sub-Alfvé nic fast ions is more sensitive to the propagation angle than the instability excited by super-Alfvé nic and sub-Alfvé nic fast ions, and is basically suppressed when the propagation angle deviates from the perpendicular direction by about 4 °.Overall, the growth rate of each harmonic is strong for nearly perpendicular propagation.As the angle of deviation from the perpendicular direction increases, the most unstable harmonic remains in the high harmonic range, while the harmonics at the middle harmonic range are the first to be suppressed.This is different from the instability excited by super-Alfvé nic and sub-Alfvé nic fast ions, where the high harmonics are typically suppressed first.

Summary and Conclusions
In this work, we systematically investigate the effects of key parameters on linear MCI in three cases: super-Alfvé nic, sub-Alfvé nic, and greatly sub-Alfvé nic fast ions with drift ring beam distribution.Our simulation results are consistent with the previous ones, which are summarized in table 1, in the low harmonic range.However, when expanding the computational spectra region up to 40 times of the ion cyclotron frequency, many new features appear.For MCI excited by super-Alfvé nic fast ions, harmonics greater than 18 are divided into four intervals.For MCI excited by sub-Alfvé nic and greatly sub-Alfvé nic fast ions, the growth rate of high harmonics is one to two orders of magnitude larger than that of low harmonics.Additionally, our simulations on velocity spread, number density ratio, and instability propagation angle yield interesting results, summarized in table 2.
The first one is about the simulations of velocity spread.For MCI excited by super-Alfvé nic fast ions,   has a small suppressive effect on the growth rate but has a decisive effect on the bandwidth of harmonics and the continuous spectrum, while   has a decisive effect on the growth rate.For MCI excited by sub-Alfvé nic fast ions, the high harmonics form a continuous spectrum at a greater velocity spread.This indicates that the sub-Alfvé nic condition becomes invalid above harmonics 28, while the super-Alfvé nic condition becomes valid.For MCI excited by greatly sub-Alfvé nic fast ions, the parameter range of the velocity spread, where fast ions can excite MCI, is expanded.
The second one is about the simulations of number density ratio.For MCI excited by the super-Alfvé nic fast ions, an important conclusion is that the relationship ~√ F transitions to ~ F with increasing the velocity spread.This promotes the understanding of the linear relation of the fast ion number density with ICE intensity in the JET.For MCI excited by sub-Alfvé nic fast ions, high harmonics conforming to the linear relation between √ F and  appear only in the centers of the three intervals, which shows the typical characteristics of MCI excited by super-Alfvé nic fast ions.For MCI excited by greatly sub-Alfvé nic fast ions, the parameter range of the number density ratio allowing fast ions to excite MCI is expanded.
The last one is about the simulations of propagation angle.For MCI excited by the super-Alfvé nic and sub-Alfvé nic fast ions, high harmonics are highly sensitive to the propagation angle compared with low harmonics.This is because the change in the dispersion relation of the fast Alfvé n wave with propagation angle results in the transition of super-Alfvé nic fast ions to sub-Alfvé nic fast ions.In addition, low harmonics still persist at large angles of deviation from the perpendicular direction.For MCI excited by greatly sub-Alfvé nic fast ions, the most unstable harmonic is still at the high harmonic range when the angle of deviation increases.The instability excited by the greatly sub-Alfvé nic fast ions is highly sensitive to the propagation angle and is basically suppressed when the propagation angle deviates from the perpendicular direction by about 4 °.
Lastly, we consider a more realistic experimental condition for MCI excited by super-Alfvé nic fast ions, that is, the background plasma contains a certain percentage of tritium.The simulation results show that as the tritium number density ratio increases, the four high harmonic intervals move toward the low harmonics, and the harmonics that are significantly suppressed not only exhibit the same moving trend but also have a slight increase in their numbers.
In our current work, we have simulated the key parameters in detail on different devices such as JET (super-Alfvé nic fast ions), LHD (sub-Alfvé nic fast ions), TFTR (greatly sub-Alfvé nic fast ions), and roughly summarized the rules.However, for different parameters of some different devices such as LHD (super-Alfvé nic fast ions) [62], JET (sub-Alfvé nic fast ions) [38], and TFTR (sub-Alfvé nic fast ions) [55], we have also made corresponding simulations.The specific results may change for super-Alfvé nic, sub-Alfvé nic, and greatly sub-Alfvé nic fast ions, but the rules are similar to those of the present work.For more wide parameters and more detailed results, the MCI simulation carried out by using the BO program is still anticipated.Finally, nonlinear simulations about the ICE continuous spectrum as well as wave-wave coupling in the linear phase would be interesting future works.Table 1.Summary of the previous linear simulation results on MCI excited by super-Alfvé nic fast ions, sub-Alfvé nic fast ions, and greatly sub-Alfvé nic fast ions, respectively, in low harmonic range.

Parameter
1keV.The bulk deuteron number density  D is 10 19 m −3 , and the number density ratio  F =  F  D ⁄ = 10 −3 , where  F represents fast ion number density.The rest of the simulation parameters are that  ⊥ = 1.294 × 10 7 m s ⁄ ,  d = 0 m s ⁄ ,   =   = 0 m s ⁄ , the Alfvé n velocity  A = 1.02 × 10 7 m s ⁄ , and the propagation angle  = 88 °.

Figure 1
Figure 1 Growth rate of analytical linear theory from figure 1(b) of [59], nonlinear simulation from figure 1(b) of [59], and BO simulation for MCI excited by super-Alfvé nic fast ions in cyclotron harmonics up to 12. Ω F is the cyclotron frequency of fast ion.

Figure 2
Figure2Analytical linear growth rate from figure3of[54] along with corresponding results from BO simulations for instability excited by greatly sub-Alfvé nic fast ions.Through the successful verification of MCI excited by super-Alfvé nic and greatly sub-Alfvé nic fast ions, the maturity and reliability of the BO program for MCI simulations have been demonstrated.Next, we will conduct detailed simulation on the velocity spread of the fast ions, the number density ratio, and the instability propagation angle for the cases of the MCI excited by super-Alfvé nic, sub-Alfvé nic, and greatly sub-Alfvé nic fast ions, respectively.It is important to note the classification of fast ions into these categories[4].On the one hand, the physical mechanism of ICE excited by greatly sub-Alfvé nic fast ions differs from that of ICE excited by super-Alfvé nic and sub-Alfvé nic fast ions.On the other hand, the excitation conditions of MCI are distinctly different for the above three fast ions.In general, the super-Alfvé nic fast ions can drive the MCI even if they are isotropic or have a relatively broad distribution of speeds[4].The sub-Alfvé nic fast ions that are isotropic or have undergone a certain degree of thermalization cannot drive the MCI[4].For the greatly sub-Alfvé nic fast ions with a very narrow spread of velocities in the parallel direction the instability can occur[4].In addition, the above three ions show more differences in this work.
shows  Ω F ⁄ versus  F at different velocity spreads.The figure illustrates that the relationship  Ω F ⁄ ~√ F (figure 8 (a)) transitions to  Ω F ⁄ ~F (figure 8 (d)) with increasing the velocity spread.Hence, this work confirms that the linear relationship between ICE intensity and neutron rate may be determined by a linear mechanism.

Figure 6
Figure 6 Ratio of growth rate to √ F as a function of  F for the harmonic number  = 22, 23, 24, 25, 26, 27, and 28.

Figure 8
Figure 8 For super-Alfvé nic fast ions instability growth rate  Ω F ⁄ as a function of the number density ratio  F , calculated for the harmonic number  = 8 at different velocity spreads.The red dashed and solid red lines correspond to  Ω F ⁄ ~√ F and  Ω F ⁄ ~F , respectively.The solid blue line shows the result of a numerical calculation.

Figure 10
Figure 10 Dispersion relation of fast Alfvé n wave for (a)  = 88 °, (b)  = 80 °.The straight lines and curves correspond to ion Bernstein and fast Alfvé n wave, respectively.Here,  F = Ω F  A ⁄ .

Figure 11
Figure 11 Growth rate of MCI excited by super-Alfvé nic fast ions as a function of   for (a)T = 0, (b)T = 0.12, (c)T = 0.24, and (d)T = 0.30.
ICEs excited by sub-Alfvé nic fast ions are commonly observed and have been widely studied through simulations.Here, as with the simulation of the MCI excited by super-Alfvé nic fast ions, we conduct a comprehensive simulation of the key parameters for MCI excited the sub-Alfvé nic fast ions, with consideration for cyclotron harmonics up to 40.The simulation parameters obtained from LHD are, following Ref.[62], magnetic field  0 = 0.46T , bulk proton temperature  H = 150eV , electron temperature  e = 150eV , bulk proton number density  H = 10 19 m −3 , the ratio of the fast ion (proton) number density to the background proton number density  F =  F  H ⁄ = 5 × 10 −4 ,  ⊥ = 2.77 × 10 6 m s ⁄ ,  d = 0 m s ⁄ ,   =   = 0 m s ⁄ ,  A = 3.17 × 10 6 m s ⁄ , and  = 89 °.

Figure 12
Figure 12 Growth rate of MCI excited by sub-Alfvé nic fast ions as a function of   for (a)  =   = 0.0 ⊥ , (b)  =   = 0.02 ⊥ , (c)  =   = 0.06 ⊥ , (d)  =   = 0.35 ⊥ .The blue lines represent  ≤ 28 , and its ordinate is on the left side of the figure.The red lines represent  ≥ 29, and its ordinate is on the right side of the figure.

Figure 13
Figure 13 Dispersion relation of fast Alfvé n wave.The blue straight lines and blue curves correspond to ion Bernstein and fast Alfvé n wave, respectively.The cold plasma dispersion relation for the fast Alfvé n wave, is calculated by the fluid module of the BO program, is shown by the red dashed line.

Figure 14
Figure 14 Growth rate of MCI excited by sub-Alfvé nic fast ions as a function of   for (a) F = 10 −3 , (b) F = 10 −4 , (c) F = 10 −5 , and (d) F = 10 −6 .The blue lines represent  ≤ 28 , and its ordinate is on the left side of the figure.The red lines represent  ≥ 29, and its ordinate is on the right side of the figure.

Figure 15
Figure 15 For sub-Alfvé nic fast ions instability growth rate  Ω F ⁄ as a function of the number density ratio  F , calculated for (a)  = 15 and (b)  = 36, 37 and 38.The red dashed and solid red lines correspond to  Ω F ⁄ ~√ F and  Ω F ⁄ ~F , respectively.The solid blue line shows the result of a numerical calculation.

Figure 16
Figure 16 Growth rate of MCI excited by sub-Alfvé nic fast ions as a function of   for (a) = 89 °, where the blue lines, whose ordinate is on the left side of the figure, represent  ≤ 28, and the red lines, whose ordinate is on the right side of the figure, represent  ≥ 29, (b) = 85 °, (b) = 75 °, and (d) = 60 °.

Figure 17 (
Figure 17 (a)electrostatic results along with corresponding (b)electromagnetic results for instability excited by greatly sub-Alfvé nic fast ions.The blue lines represent  ≤ 26 , and its ordinate is on the left side of the figure.The red lines represent  ≥ 27, and its ordinate is on the right side of the figure.

Figure 18
Figure 18 Growth rate of MCI excited by greatly sub-Alfvé nic fast ions as a function of   for (a)  =   = 0, where the blue lines, whose ordinate is on the left side of the figure, represent  ≤ 26 , and the red lines, whose ordinate is on the right side of the figure, represent  ≥ 27, (b)  =   = 0.03 ⊥ , (c)  =   = 0.05 ⊥ , and (d)  =   = 0.1 ⊥ .

Figure 19
Figure 19 Growth rate of MCI excited by greatly sub-Alfvé nic fast ions as a function of   for (a) F = 10 −1 , where the blue lines, whose ordinate is on the left side of the figure, represent  ≤ 17, and the red lines, whose ordinate is on the right side of the figure, represent  ≥ 18, (b) F = 10 −2 , where the blue lines, whose ordinate is on the left side of the figure, represent  ≤ 26, and the red lines, whose ordinate is on the right side of the figure, represent  ≥ 27, (c) F = 10 −3 , and (d) F = 10 −4 .

Figure 20
Figure 20 Growth rate of MCI excited by greatly sub-Alfvé nic fast ions as a function of   for (a) = 93 °, (b) = 92.5 °, (c) = 91.5 °, where the blue lines, whose ordinate is on the left side of the figure, represent  ≤ 26, and the red lines, whose ordinate is on the right side of the figure, represent  ≥ 27, and (d) = 90 °.

Figure 21
Figure 21 Growth rate of MCI excited by greatly sub-Alfvé nic fast ions as a function of   for (a) = 88.5 °, (b) = 88 °, (c) = 87.5 °, and (d) = 87 °.The blue lines represent  ≤ 26 , and its ordinate is on the left side of the figure.The red lines represent  ≥ 27, and its ordinate is on the right side of the figure.

Table 2 .
Summary of the new simulation results on MCI excited by super-Alfvé nic fast ions, sub-Alfvé nic fast ions, and