We first study the nonlinear evolution of the TAE mode with single-n (n_single = 1 or 2) as a reference case. Figure 1 shows the time evolution of the TAE mode kinetic energy (a) and frequency [(b), (c)]. The mode frequency chirping is due to the evolution of the phase space structure. The injection speed range of beam ions υ_h = [ 2.19, 3.29]υ_A is used in the n_single = 1 case. Figure 2 shows the locations of the single-n TAE modes.

Fig. 1.

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	Fig. 1. Nonlinear evolution of TAE mode: (a) time evolution of single-n (nsingle = 1, 2) TAE mode kinetic energy, (b) and (c) fourier spectrogram of single-n (nsingle = 1, 2) TAE mode.

Fig. 2.

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	Fig. 2. The safety factor equilibrium profile and the locations of the single-n (nsingle = 1, 2, 3) TAE modes.

Figure 3 shows the single-n (n_single = 1, 2) TAE mode structure at three different times: [(a), (d)] in the linear phase, [(b), (e)] in the initial saturation phase, and [(c), (f)] in the late saturation phase. Here U is the velocity stream function of the incompressible part of the perturbed plasma velocity. TAE is characterized with two dominant poloidal harmonics (m and m + 1) localized between two mode rational surfaces. It is observed that for the n_single = 1 mode, the structure with the dominant m/n = 1/1 component in the initial saturation phase is similar to that in the linear phase.

Fig. 3.

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	Fig. 3. The single-n (nsingle = 1, 2) TAE mode structure of velocity stream function (U) at three time slices: [(a), (d)] in the linear phase, [(b), (e)] in near initial saturation, [(c), (f)] in late nonlinear phase.

In the late saturation phase of the evolution, however, the mode structure changes significantly with the dominant m/n = 2/1 component, as shown in Figs. 3(a)–3(c). It coincides with the fact that the n_single = 1 mode we investigate here produces the pitch-fork phenomenon and has two branches: down-chirping branch and up-chirping branch, as shown in Fig. 1(b). It is known that frequency chirping is always accompanied by the redistribution of energetic ions. The gradient of energetic ions near the resonant surface q = 1.5 may change. Thus the dominate mode is changed from m = 1 to m = 2. However, for the n_single = 2 case, the m/n = 3/2 mode is always dominant. This is due to the fact that the n_single = 2 mode has only one branch, i.e., down-chirping branch, as shown in Fig. 1(c).

It is known that the fast ions can destabilize Alfvén waves by passing and/or trapped particle resonance via tapping the free energy associated with fast ion pressure nonuniformity. The TAE modes can be excited by the free energy associated with fast ion pressure gradient through wave particle resonant interaction due to the fact that they are discrete in nature and have weak continuum damping. The wave particle resonant interaction mathematically can be described as the resonant condition nω_t + pω_p – ω = 0, where p is an integer, ω is the mode frequency, ω_t = Δϕ/Δt, ω_p = 2π/Δt are both bounce-average frequency in the calculation and Δt is the time for each particle to complete one round poloidally. For passing particles, ω_t is toroidal transit frequency and ω_p is poloidal transit frequency. For trapped particles, ω_t ≡ ω_d is toroidal precession drift frequency and ω_p is poloidal bounce frequency. If the right wave-particle phase is given, the linear momentum exchange may cause convective loss of fast ions. However, when the trajectories of resonant fast ions overlap with mode spatial structure, the fast ion stochastic diffusion is induced, thus a global redistribution of fast ions may occur. In Fig. 4, the beam ion perturbation distribution function δf for n_single = 1 is plotted, and the resonant particle locations in P_ϕ–E phase space around μ = 5.225 (normalized by E₀/B₀) are traced. The selected value of magnetic moment μ is a typical one, which contains the most regions of both passing and trapped resonant particles, around which the perturbation distribution function δf is relatively larger. Multiple kinds of fast ions and resonances exist in the system since the TAE instability in this work is studied with a beam ion distribution function including the beam ion pitch angle scattering effect. It is shown in Fig. 4 that the primary resonances are p = 1 and p = 2 for trapped particles, p = 1 for co-passing particles, and p = 2 for counter-passing particles.

Fig. 4.

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	Fig. 4. For nsingle = 1 the perturbed distribution function δf around the magnetic moment μ = 5.225 in phase space of canonical momentum and energy, i.e., Pϕ–E. For a fixed E, small Pϕ corresponds to the plasma core and large Pϕ corresponds to the plasma edge.

For the single-n case, the moment of a particle in phase space can be described mathematically as

To analyze the loss mechanism in the n_single = 1 case, a typical lost particle is taken to represent the lost particle resonant with the n_single = 1 mode. Figures 5(a) and 5(b) show the time evolution of P_ϕ and energy E, respectively, and Figure 5(c) shows the position of the lost particle in the P_ϕ–E phase space. P_ϕ can be regarded as a radial variable, and the change of P_ϕ can represent the size and motion of saturated island, since P_ϕ conservation is broken in the presence of the perturbation. The mode’s growth and chirping induce island broadening and drift. When the saturated island moves outwards and intersects with the loss boundary, the fast ion may pass the last closed flux surface. It is found from Fig. 5 that with the TAE mode growth (see Fig. 1(a)), the saturated island becomes broad, first moves inwards radially and then outwards.

Fig. 5.

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	Fig. 5. The evolution of Pϕ (a), E (b) and the trajectory (c) of a typical initially resonant particle in the nsingle = 1 simulation. Note that the red dot “lost” in the figure denotes the position at the lost moment and that the black triangle “initial” in subplot (c) denotes the position at initial moment.

To study the effect of the beam ion injection speed on the beam ion loss, the time evolution of the beam ion loss induced by the n_single = 1 TAE mode is given in Fig. 6. Here f_loss, representing the beam ion loss level, is defined as

Fig. 6.

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	Fig. 6. (a) The time evolution of the loss level floss and (b) the time evolution of the mode energy in the nsingle = 1 case for υh = [2.19, 3.29]υA with the fast ion beta βh = 1.81%.

In this section, the results of simulations simultaneously including n (or n_multiple) = 1 – 4 modes (multiple-n case) are presented. The parameters used in the multiple-n simulation are the same as those in the n_single = 1 simulation (see Section 2). Comparisons are also made with the results of simulations including only one individual mode (single-n cases). The evolutions of the n = 1, n = 2, n = 3, and n = 4 mode amplitudes in multiple-n and single-n simulations are compared in Fig. 7. In the multiple-n simulation, the n > 4 fluctuations are filtered out, so only n = 1, n = 2, n = 3, n = 4 modes are kept. It is clearly observed that due to the nonlinear interaction between different n modes, the evolution of n = 1 mode is significantly different after t/τ_A ⩾ 300. The n = 1 mode amplitude in the multiple-n simulation is clearly reduced by such nonlinear interactions after t/τ_A ⩾ 300. Moreover, the n = 4 mode in the single-n case is stable, while it is unstable in the multiple-n case. The mode frequency evolution of n = 1, n = 2, n = 3, n = 4 modes are shown in Fig. 8. Since two possible free energy channels, wave-wave coupling and wave-particle interaction, exist for Alfvén wave instabilities, there are two possible explanations for the n = 4 mode excitation. However, we have examined the frequency matching condition between n = 4 and n = 2, between n = 4 and n = 1 3, and it is found that the frequency matching condition is not satisfied very well. One possible mechanism for n = 4 mode excitation is due to the wave-particle interaction. Since the redistribution and loss of energetic ions induced by n = 1, n = 2, n = 3 modes lead to the modification of energetic ion density gradient, the n = 4 mode can be excited.

Fig. 7.

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Fig. 7. Nonlinear evolution of kinetic energy Ek for multiple-n and single-n cases with βh = 1.81% and beam ion injection speed υh = 2.55υA. Multiple-n (nmultiple = 1, 2, 3, 4) stands for n = 1, 2, 3, 4 in the multiple-n cases, respectively. Single-n (nsingle = 1, 2, 3, 4) stands for n = 1, 2, 3, 4 in the single-n cases, respectively. The growth rates of nmultiple = 1, 2, 3, 4 modes are γn = 1 = 0.0167ωA, γn = 2 = 0.0525ωA, γn = 3 = 0.0656ωA, γn = 4 = 0.0937ωA, respectively.

Fig. 8.

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	Fig. 8. The mode frequency evolutions of n = 1, n = 2, n = 3, n = 4 modes with the observation of mode frequencies chirping down or up in the multiple-n simulation.

In general, multiple modes can interact with each other indirectly via mutual scattering effect on their individual resonant particles. Such indirect interaction may occur in two ways. On the one hand, the modes have to share the energy delivered by the particles. On the other hand, the energy can also be transferred when a particle is scattered from one mode into the other mode instead of escaping from the wave. Such a particle recycling, as well as multiple resonances covering a large region of phase space, can also increase the total energy transferred to the fluctuating fields in the multiple-n simulation. This can be easily seen in Fig. 7 by comparing the mode energies between the single-n and multiple-n cases.

Moreover, the wave-particle resonance and particle trajectory in the phase space are analyzed. The orbits of beam ions are usually defined by three invariants of the motions: the energy E, the magnetic moment μ, and the toroidal canonical momentum P_ϕ. When a resonant interaction between a mode and a beam ion takes place, the magnetic moment μ of the ion is conserved, whereas both the energy E and the toroidal canonical momentum P_ϕ change during the process. In Fig. 9, the particle trajectories in P_ϕ–E space are followed in the presence of time-dependent multiple-n TAE perturbation, and the position in P_ϕ–E phase space is recorded when each particle passes through the mid-plane. The different components of the multiple-n mode are represented by different colors: n = 1 (red), n = 2 (green), n = 3 (black), n = 4 (blue). The initially resonant particles are with a fixed value of magnetic moment μ = 9.933 (normalized by E₀/B₀). Of particular concern in the multiple-n case is whether the mode amplitudes become large enough to cause resonances overlap, which can determine the global fast ion loss. It is found that the resonance overlap is greatly facilitated by the simultaneous excitation of multiple modes, as studied previously.^[33] Moreover, when the closely spaced resonances overlap, there is an enhanced release of mode energy, which in turn can cause nearby but more widely spaced resonances to overlap. This thereby rapidly expands the phase space region over which particles are redistributed to the plasma edge or lost. Figure 10 indicates that the radial mode structures overlap in the radial direction, in particular for the n_multiple = 1, 2 modes in the multiple-n case. As a result, the resonances in the P_ϕ–E phase space are neighboring and thus easily overlap for the multiple-n case, especially at higher amplitudes. This resonance overlap in the phase space is evidently observed for the multiple-n case in Fig. 9(a). Although the modes discussed here come from the components of multiple-n TAE, one can clearly see the overlap, especially in the region of the low energetic particles.

Fig. 9.

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Fig. 9. The weak resonance-overlapping multiple-n TAE (βh = 1.81%) (a) and the relatively strong resonance-overlapping multiple-n TAE (γh = 6%) (b). Fast ion trajectories in the phase space of canonical momentum Pϕ and energy E in the presence of multiple-n TAE modes at t = 0–2000τA. Only the initially resonant particles are plotted in the figure. The particles which initially resonate with the different components of the multiple-n mode are represented by colors: n = 1 (red), n = 2 (green), n = 3 (black), n = 4 (blue). Since the particles are trapped by the n = 1, 2, 3, 4 modes, the positions of resonant particles in phase space can be regarded as the eigenmodes. Namely, the eigenmodes are represented by colors: n = 1 (red), n = 2 (green), n = 3 (black), n = 4 (blue). The frequencies of ω1, ω2, ω3, ω4 stand for the mode frequencies of n = 1, 2, 3, 4, respectively, in the multiple-n case.

Fig. 10.

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	Fig. 10. Comparison of velocity stream function at the midplane with t = 251τA in the multiple-n (nmultiple = 1, 2, 3, 4) and single-n (nsingle = 1, 2, 3, 4) simulations.

Note that although a particle can be lost in the n_single = 1 or 2 mode, or other TAE mode, it is only recorded as one lost particle in the sum loss of all individual single-n modes. That is, this sum loss is not the simple addition of all individual single-n losses. Therefore, it is shown in Fig. 11 and Table 1 that in the single-n case, the sum loss level is certainly higher than the loss level from any single-n mode, though less than the loss level from the simple addition of all individual single-n losses. Even more important, it is interestingly found that the loss level in the multiple-n case is rather higher than the sum loss level in the single-n case because of the resonance overlap. Such a loss level increase in the multiple-n case is enhanced by 16% for β_h = 1.81%. Further simulations indicate that increasing β_h can significantly rise the loss level in the multiple-n case.

Fig. 11.

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	Fig. 11. Evolutions of the level of beam ion losses floss in nsingle = 1, nsingle = 2, nsingle = 3, nsingle = 4 simulations and multiple-n simulation with the same parameter βh = 1.81%, υh = 2.55υA. Note that sum denotes the sum of loss over nsingle = 1–4 modes, where every lost particle is calculated for one time.

Table 1.

Table 1.

Table 1. Comparison of the linear growth rate and fast ion losses in the single-n and multiple-n cases. Note that every lost particle is calculated for one time. . Simulation type Growth rate Loss fraction multiple-n TAE for nmultiple = 1, 2, 3, 4 2.45 0.0167, 0.0525, 0.0656, 0.0937 nsingle = 1 TAE 0.0167 0.98 nsingle = 2 TAE 0.0511 1.93 nsingle = 3 TAE 0.0178 0.84 nsingle = 4 TAE stable 0.09 Sum 2.12

Table 1. Comparison of the linear growth rate and fast ion losses in the single-n and multiple-n cases. Note that every lost particle is calculated for one time. .

Figure 12 shows that for β_h = 6.0%, the loss level is greatly enhanced and even reaches ∼ 13% of beam ions, approximately 5 times higher than that for β_h = 1.81%. This significant increment may result from the strong resonance overlap, as shown in Fig. 9(b). Once the overlap of closely spaced resonances occurs in the multiple-n case, the release of mode energy is increased so that the widely spaced resonances can also take place. As a result, such a process can effectively enlarge the phase space region and then particles are redistributed or lost. It is also demonstrated that the trajectory of the same lost ions in the presence of the multiple-n TAE is significantly different from that in the presence of the single-n TAE. Additionally, the loss level increment between the multiple-n and single-n cases is enhanced to 44% for β_h = 6.0%. This high loss level is indeed of critical importance for the fast ion confinement in high parameter plasmas like in ITER and CFETR.

Fig. 12.

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	Fig. 12. Evolutions of the level of beam ion losses floss in nsingle = 1, nsingle = 2, nsingle = 3, nsingle = 4 simulations and multiple-n simulation with high fast ion beat βh = 6%, υh = 2.55υA. Every lost particle is calculated for one time.

The evolution of P_ϕ is given in Fig. 13(a), which can be regarded as a radial variable used to describe the radial position of a particle. The particle we plot here is a typical resonant trapped particle which is initially in resonance with the multiple-n TAE mode. The oscillation of P_ϕ shown by the green triangles indicates that the particle is trapped in the multiple-n TAE mode. In comparison with the single-n cases, the amplitude of oscillation of P_ϕ in the multiple-n simulation is much larger, especially in the later nonlinear phase. Figure 13(b) indicates that the energy change in the multiple-n simulation (shown by green triangles) is significantly larger than that in the single-n simulation. This is due to the fact that the particles can have multiple resonances and the overlap even occurs in the multiple-n simulation, increasing the total energy transferred to the background plasmas. For the n_single = 1 case, the oscillation of P_ϕ indicates that the particle is also trapped by the mode. At each resonance, the wave-particle momentum exchange, proportional to the mode perturbation amplitude, corresponds to a radial drift. It is found by comparison that in spite of the same particles in the simulations, the trajectories and the energy changes in the presence of different n modes are quite different from each other. For example, the trajectories of the particle studied here in the multiple-n and n_single = 1 simulations all exhibit diffusive characteristic, since such random walk (oscillation of P_ϕ) is the typical character of diffusive fast ion transport. However, in the n_single = 2 and 3 simulations, a convective characteristic is exhibited since direct walk is the typical character of convective fast ion transport. Finally, according to Eq. (3) and the change of P_ϕ, it is revealed that the particles in the n_single = 1 and 3 simulations gain energy, while those in the multiple-n and n_single = 2 simulations lose energy.

Fig. 13.

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	Fig. 13. Comparison of the evolution of Pϕ (a) and the trajectory (b) of a typical initially resonant particle in multiple-n simulation with that in the nsingle = 1, nsingle = 2 and nsingle = 3 simulations. The black circle denotes initial position of the particle in phase space.

Figure 14 gives the distributions of lost beam ions in the phase space of the beam ion energy E and the change of pitch angle ΔΛ in the single-n and multiple-n simulations. It should be noted that the loss discussed here is all associated with the presence of TAE. As observed in Fig. 14, a slight shift of the beam ion losses is towards the higher pitch angle Λ value in the single-n and multiple-n simulations. Aside from the dominant characteristic, there are some other lost beam ions for which the pitch angles become lower. Since the definition of pitch angle Λ in the M3D-K code depends on the energy (see Section 2), it is changed to

Fig. 14.

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Fig. 14. Comparison of distribution of lost beam ions in E–ΔΛ space for (a)–(c) multiple-n and (d)–(f)nsingle = 2 simulations with the same parameter βh = 1.81%, υh = 2.55υA, where E is referred to the energy of lost beam ion at the lost moment and ΔΛ is referred to the difference between the pitch angle Λl1 at the lost moment and the pitch angle Λl0 at the initial moment when the particle has not suffered from the perturbation. [(a), (d)] The lost passing particle distribution in each simulation. [(b), (e)] The lost trapped particle distribution in each simulation. [(c), (f)] The overall lost particle distribution in each simulation. The color coding is in arbitrary units.

Table 2 gives a summary of the loss fraction characterized by contributing energy to the wave and the loss fraction of trapped ions in the single-n and multiple-n simulations. It is found that in both the single-n and multiple-n simulations, the main loss mechanism is through contributing energy to the wave. This is due to the fact that when a beam ion drives the mode, its energy decreases (ΔE < 0) and then it moves radially outward in the minor radius (ΔP_ϕ > 0) and eventually may be easily lost. However, a small fraction (∼ 5%) of the fast ion losses characterized by gaining energy from wave is observed in both the single-n and multiple-n simulations. This may be due to the fact that the particle is accelerated and its orbit is influenced.^[34]

Table 2.

Table 2.

Table 2. A statistical analysis on the characteristics of fast ion losses performed. The contributing energy is a ratio of the amount of losses that contribute energy from wave to the total amount of losses. The trapped fraction is a ratio of the number of lost trapped particles to the total number of lost particles. . Simulation type Contributing energy/% Trapped fraction/% multiple-n TAE 93 72 nsingle = 1 TAE 95.48 78.93 nsingle = 2 TAE 95.16 76.82 nsingle = 3 TAE 96.90 81.60

Table 2. A statistical analysis on the characteristics of fast ion losses performed. The contributing energy is a ratio of the amount of losses that contribute energy from wave to the total amount of losses. The trapped fraction is a ratio of the number of lost trapped particles to the total number of lost particles. .

Another characteristic in common, as shown in Table 2, is that the lost fraction of trapped ions is two to four times larger than that of passing ions. Particles traveling on banana orbits are more easily lost, since the trapped orbits are wider. Especially for so-called ‘potato orbits’, they are more sensitive to the perturbation near their turning points. For passing particles, in addition, the transformation from passing particles to trapped particles during the nonlinear evolution of TAE mode is a well-known loss mechanism for beam ions. Also, we can find that the multiple-n simulation has a higher fraction of lost fast ions coming from passing population. It can be attributed to the fact that compared with the single-n case, more passing particles can gain energy from the multiple-n TAE modes and render their orbits to become unconfined.