Visualization of fast ion phase-space flow in plasmas well-below, near and well-above Alfvén eigenmode stability threshold in tokamak

Transport of fast ions along certain local phase space paths, referred to as fast ion phase-space flow, has been systematically measured by an imaging neutral particle analyzer (INPA) in three plasma regimes, which are well-below, near, and well-above the Alfvén eigenmode stability threshold (Du et al 2021 Phys. Rev. Lett. 127 235002). (1) In plasmas well-below the Alfvén eignenmodes (AE) stability threshold, fast ions are well-confined on passing particle orbits without noticeable transport over the phase space. The observed INPA images agree well with the synthetic INPA images, using the fast ion distribution predicted by neoclassical theory. (2) In plasmas near the AE stability threshold, INPA images in the presence of AE activity moderately deviate from those without AE activity. The image difference can be well interpreted by AE-driven, phase-space fast ion flow. Paths of this flow over the velocity space (or streamlines) are reconstructed by the intersection lines of curved Eʹ and µ surfaces, referred to as Eʹ and µ line (where E′≡E−(ω/n)Pζ ; E, P ζ and µ are the energy, canonical toroidal momentum and magnetic moment of ions; ω and n are the angular frequency and toroidal mode number of AEs, respectively). Resonant fast ions move radially inward by gaining energy and move radially outward by losing energy and the trajectory well aligns with the Eʹ and µ lines that pass through the mode resonances near the injection energy of neutral beams. (3) In plasmas well-above the AE stability threshold, fast ion phase-space dynamics shows additional features. Fast ions are transported out of the birth positions so promptly along the streamlines that the slowing-down process from the injection energy is not observable, exhibiting strong critical gradient behavior at local phase space. As a result, the increase of electron temperature is very small, in spite of an increase of beam power by ∼45% . It should be emphasized that the directions of phase-space transport, induced by AEs with different frequencies, structures and mode numbers, do not largely differ.

Transport of fast ions along certain local phase space paths, referred to as fast ion phase-space flow, has been systematically measured by an imaging neutral particle analyzer (INPA) in three plasma regimes, which are well-below, near, and well-above the Alfvén eigenmode stability threshold (Du et al 2021 Phys. Rev. Lett. 127 235002). (1) In plasmas well-below the Alfvén eignenmodes (AE) stability threshold, fast ions are well-confined on passing particle orbits without noticeable transport over the phase space. The observed INPA images agree well with the synthetic INPA images, using the fast ion distribution predicted by neoclassical theory. (2) In plasmas near the AE stability threshold, INPA images in the presence of AE activity moderately deviate from those without AE activity. The image difference can be well interpreted by AE-driven, phase-space fast ion flow. Paths of this flow over the velocity space (or streamlines) are reconstructed by the intersection lines of curved E ′ and µ surfaces, referred to as E ′ and µ line (where E ′ ≡ E − (ω/n)P ζ ; E, P ζ and µ are the energy, canonical toroidal momentum and magnetic moment of ions; ω and n are the angular frequency and toroidal mode number of AEs, respectively). Resonant fast ions move radially inward by gaining energy and move radially outward by losing energy and the trajectory well aligns with the E ′ and µ lines that pass through the mode resonances near the injection energy of neutral beams. (3) In plasmas well-above the AE stability threshold, fast ion phase-space dynamics shows additional features. Fast ions are transported out of the birth positions so promptly along the streamlines that the slowing-down process from the injection energy is not observable, exhibiting strong critical gradient behavior at local phase space. As a result, the increase of electron temperature is very small, in spite of an increase of beam power by ∼45%. It should be emphasized that the directions of phase-space transport, induced by AEs with different frequencies, structures and mode numbers, do not largely differ. * Author to whom any correspondence should be addressed.
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Introduction
Future fusion reactors rely on good confinement of alpha particles to achieve self-sustained hot plasmas. However, wave-particle resonant interactions jeopardize good confinement. The resonant interactions redistribute alpha particles out of the hot plasma core and sometimes, induce loss to the plasma facing materials [1][2][3][4][5][6][7]. In rare circumstances, additional heat load from fast ion losses may damage plasma facing material and devices [8,9].
In current devices, ions of high energy (fast ions) are produced from neutral beam injection or ion cyclotron heating. Unlike fusion-born alpha particles, distributions of these fast ions are largely anisotropic [10]. Significant flattening of fast ion density profiles has been reported, which is caused by multiple, small-amplitude Alfvén eignenmodes (AE) [11]. Here, radial profiles of fast ions are measured by fast ion deuteriumalpha light (FIDA) [12], integrating over a broad phase space of energy and pitch [13]. Moreover, significant losses of fast ions, induced by large-amplitude bursting instabilities, are also observed in both tokamaks and stellarators. For example, it is well documented the severe loss of energetic ions, induced by fast ion driven resistive interchange mode [14][15][16], is able to decrease the neutron rate by ∼50% [17] and limits the achievable bulk ion temperature in the large helical device (LHD) [18].
To achieve a full understanding of how fast ions are transported and are finally removed from systems by these instabilities, it is important to trace fast ion migration paths across the radius-energy-pitch-resolved phase space. Great effort has been devoted to the measurement of phase-space transport paths. In LHD, a formation of clump and hole structures in real space is resolved by a passive neutral particle analyzer (NPA) during bursts of toroidicity-induced AE (TAE) [19,20]. The radial positions of clumps and holes are inferred from the slowing-down time of the phase-space structure. More recently, fast ion phase space transport above a critical gradient threshold has also been studied using neutron, FIDA and NPA diagnostics [21,22]. The systems probe fast ion transport at different phase-space volumes and find different fast ion transport threshold well above the linear stability threshold of AEs [21]. Nevertheless, a complete picture of continuous phase-space flow cannot be resolved, due to the limitation of the measurement capabilities.
To bridge the gap, an imaging neutral particle analyzer (INPA) is developed in the DIII-D tokamak [23,24]. Quoting from [23], 'The primary goal of the INPA in the DIII-D tokamak is to study the phase space flow of energetic particles generated by multiple small amplitude Alfvén eigenmodes.' The diagnostic is designed to resolve the phase space flow of fast ions on passing orbits. This is a class of orbits with excellent confinement in tokamak devices, at least when wave-particle resonant interactions are absent.
It should be pointed out that understanding the phase space dynamics of fast ion transport is not only of great interest from the physics point of view, but has practical impact on the design and operation of fusion devices. Resolving fast ion transport routes largely contributes to model validation and development of reduced, predictive modeling capabilities. The knowledge is also important to explore control actuators [25][26][27] that mitigate fast ion loss through phase space engineering [28].
Some of the results in this paper were concisely reported in the letter [29]. This paper documents details of the experiment and expands the discussion. A companion paper describes simulations of these results [30]. The paper is organized as follows. Section 2 discusses the apparatus used in this experiment. Section 3 details the experimental strategies and the results of the observed phase space flow in three plasma regimes: well below, near, and well-above the AE stability threshold. Section 4 reconstructs the path of phase space flow for a single frequency mode and discusses observed flattening of fast ion density along the E ′ and µ line. The impact of the frequency sweeping of AEs on phase space flow is also included. Summary and discussion appear in section 5.

Apparatus
The essential diagnostic to resolve fast ion phase-space flow is INPA, which is developed for the first time in DIII-D tokamak [23,24]. The principle of this diagnostic is briefly summarized as follows. As seen from figures 1(a) and (b), INPA collects charge-exchanged energetic neutrals by viewing an active neutral beam through a collimating slit. The collimation slit defines the sightline that the neutrals could escape out of the plasma and strike the ultra thin carbon stripping foils of 10 nm thickness at the rear side of the collimation box. Once the neutral influx hits the carbon stripping foils, ionization may happen. The local tokamak magnetic field acts as a magnetic spectrometer to disperse ionized neutrals onto the scintillator. A fast camera of 160 frames per second monitors the visible light from the phosphor, which is proportional to the incident neutral flux.
The phase-space weights of the INPA are computed using the FIDASIM [31] and INPASIM code [32]. The computation process is documented in great details in the [32]. Weight functions of a few percent of the available INPA pixels in phase space are given by the circles in figure 1(d). The circles are defined as the contour lines of 1/e of the maximum weights of each camera pixel in energy and radial space. It is seen that the system covers a broad radial range from the highfield side to the low-field side on the plasma midplane with a radial resolution of ∼6 cm. The INPA resolves the neutral energy from ∼30 keV to ∼100 keV with a resolution of ∼7 keV. The Monte Carlo particles at a nearly fixed pitch χ of χ ≡ v ∥ /v ∼ 0.78 can be captured by the INPA, given by the pink dots in figure 1(d) (v ∥ refers to the fast ion velocity parallel to the magnetic field line, with a positive value being in the direction of the plasma current). In this experiment, toroidal magnetic field is in the clockwise direction and plasma current is in the counter-clockwise direction from the top view of machine. The pitch resolution is about ∼0.05 at a fixed radial location.
As seen from figure 1(c), the phase space interrogated by the INPA system is dominated by the well-confined fast ions on passing orbits that travel in the same direction as the plasma current. Near the plasma axis of R axis ∼ 1.75 m, the INPA measures fast ions on stagnation orbits, i.e. a class of orbits that are confined on the low-field side of the magnetic axis near the device midplane. The INPA view does not cover the phase space near the fast ion confined-loss boundary and does not cover fast ions on trapped orbits.
Other key diagnostics, which are used in this experiment, are listed, as follows. As seen from figure 2, three vertical chords and one horizontal chord of CO 2 interferomenter [33] measure the line-integrated density and AE-induced density fluctuation from the plasma core to the edge. Electron cyclotron emission (ECE) [34] measures the local temperature and AE-induced electron temperature fluctuation near the midplane from the high-field side to the plasma edge in the low-field side. The motional Stark effect diagnostic measures the Balmer alpha line emitted from the deuterium beam neutrals. The pitch angle of the magnetic field line or safety factor is deduced by measuring the polarization angle of the light emission [35].

Experimental method
To directly visualize the AE-driven fast ion phase-space flow, the experiment is conducted in the plasma near the AE marginal stability boundary. A neutral beam modulation technique is developed to simultaneously obtain the INPA images with and without the presence of AEs for nearly the same bulk plasma parameters at each beam modulation period.
Taking one pixel of the INPA camera as an example, a neutral beam is selected to directly populate its interrogated local phase space volume. As illustrated in figure 3(a), when  the plasma is well below the AE stability boundary, the finite power of the neutral beam may be insufficient to drive AEs. In this case, switch-on of the beam leads to a gradual increase of the signal, due to birth of fast ions through atomic physics of beam deposition in the relevant phase space. Switch-off of the neutral beam leads to a gradual decay of INPA signal to nearly zero, because of thermalization of fast ions through Coulomb collisions. The time evolution of the INPA signal depends on the injected neutral beam power and the whole process can be accurately simulated using neoclassical theory.
Adding one steady beam, which populates the same phase space as the modulated beam does, may or may not drive AEs, depending on the plasma conditions: (1) If the increased total beam power is insufficient to drive AEs, the INPA signal would increase in an amount that approximately depends on the added power, on the assumption that the impact on the bulk plasma parameters is negligibly small, as illustrated in figure 3(b). That is, the neoclassical theory would expect the INPA signal approximately increases from Γ m to Γ m + Γ s , where Γ m and Γ s are the neutral flux from only the modulated beam and only the steady beam, respectively. It should be noted that Γ s can be obtained directly at the end of the modulation period, when Γ m decays to zero (see the blue arrows in figure 3(b)). (2) If the increased total beam power is sufficient to drive AEs, the INPA signal may suffer from AE-induced resonant transport, as illustrated in figure 3(c). That is, the INPA signal Γ ob during the two-beam phase would not equal Γ m + Γ s . The deficit, i.e. ∆Γ ≡ Γ ob − (Γ m + Γ s ), quantitatively reflects the AE-induced transport at the local phase space. It should be noted that Γ s is obtainable near the end of the modulation period, when Γ m decays to zero. This is because AEs will be stabilized after switching off the modulated beam, due to the lack of the beam power. Γ m can be obtained from a reference shot with nearly matched plasma conditions, but without the steady beam.

Experimental setup
The controlled experiment is conducted in the plasma current ramp-up phase, at a toroidal magnetic field of ∼2.1 T and in an upper single null magnetic configuration. A neutral beam, nearly-perpendicular to magnetic axis (see green band in figure 4(a)), is injected constantly at beam ion energy of 55 keV and a power of 1 MW. It is used as an active neutral source to promote charge-exchange reactions in the plasma core for the INPA system. The low beam energy of 55 keV (compared to the standard beam voltage of 80 keV in DIII-D) is selected for the following reasons. The diagnostic beam does not populate the phase space of interest, and it populates the phase space occupied by fast ions on trapped orbits. The diagnostic beam of low energy at the power of 1 MW does not largely drive the AE modes [36]. This is because v fast /v Alfvén is reduced from 0.38 for the 81 keV beam to 0.31 for the 55 keV beam [37] and the beam power is lowered to ∼1 MW, where v fast is the velocity of the beam ions and v Alfvén is the Alfvén velocity.
Besides the diagnostic beam, all other beams, which are employed in this experiment, are injected nearly-tangential to the magnetic field line in the direction of the plasma current. These beams directly populate INPA-interrogated phase space, i.e. the phase space occupied by fast ions on passing orbits. The arrangements are as follows: 1. In the low-power shot of #179416, a deuterium neutral beam of 81 kV, modulated at a cycle time of 70 ms and a duty cycle of 50%, is injected at a power of 2.5 MW (see the red band of figure 4(a)). 2. In the high-power shot of #179415, the other 82 kV deuterium neutral beam of 1.7 MW (see the blue band in figure 4(a)) steadily populates the same velocity space that the modulated beam does.
Note that, compared to the low-power shot, the steady neutral beam in the high-power shot introduces an additional particle source in the plasma. In order to match the density of the low-power shot, the target plasma density in the highpower shot is carefully tuned. It is seen from figure 5(a) that the line-integrated electron density, measured by CO 2 interferometer from plasma core to the edge at R = 1.48 m, 1.94 m and 2.1 m, agrees well between the two shots. This assures the same birth profiles of fast ions during the beam deposition in the low-power and high-power shots.
The electron temperature (T e ), measured by ECE, does not match as well as the density between the low-power and highpower shots. Figure 5(c) shows that, before 1.0 s, adding the neutral beam power of 1.7 MW causes a modest increase of T e at R ∼ 2.03 m. On the other hand, the increase of T e is barely noticeable in the plasma core and edge. After ∼1.0 s, T e in the high-power shot start to largely exceed that in the lowpower shot in all radii by ∼20%. Time evolution of plasma profiles, including electron temperature and density, carbon temperature and rotation of the high-power shot, are given in figure 6.   The cross power spectra of the electron temperature fluctuation measured by ECE in high-power shot (a1)-(a6) and in the low-power shot (b1)-(b6). The measured radial locations of ECE channels are labeled.

AE activity in the low-power and high-power shots
The AE activity is drastically different in the low-power and high-power shots. Figure 7 shows electron temperature fluctuation, measured by ECE. In the high power shot (see figures 7(a1) and (a2)), edge ECE channels detect the steadyfrequency electron temperature fluctuation and the mode amplitude is modulated, associated with the injected neutral beam power. These modes are identified as TAE in [38]. The ECE channels near the radial locations of minimum safety factor (q min ) in the plasma core observes the modes with frequency sweep-ups (see the figures 7(a4)-(a6)), which are known as reversed-shear AE modes (RSAE) [38]. The time The time evolution of q profiles is given in figure 8. The TAE and RSAE modes are simultaneously observed from 2.01 m to 2.04 m (see the figure 7(a3)), indicating the radial mode structure overlaps in the radial region.
In comparison, in the low-power shot (see figures 7(b1)-(b6)), similar AE modes with much weaker amplitude appear only at the very beginning of the shot. The TAEs are not observed in ECE after ∼0.6 s and the RSAEs are not observed after ∼0.8 s. Moreover, TAE and RSAE are gradually stabilized in both low-power and high-power shots, as plasma An estimate of the uncertainty in the q-profiles is obtained using 50 independent equilibrium reconstructions, for which the constraining measurements (magnetic probes, MSE, pressure profiles, etc) were perturbed using normal distributions defined by their reported uncertainties. current ramps up and electron density increases, as shown in figures 5 and 7. This is because the critical fast ion density gradient for AE drive scales as ∼q −2 [39] and the classical fast ion pressure, predicted by TRANSP code, stays nearly constant after ∼0.4 s, as shown in figure 9. The increasing plasma density causes a large reduction in fast ion fraction at later times.
A chart of AE stability in the low-power shot and highpower shot is summarized in figure 10 and the approximate stability boundary is indicated by the solid line. Figure 10. The density fluctuation induced by AE activity as the function of minimum safety factor (q min ) and classical beam pressure in the plasma core. The density fluctuation is integrated over the range between the GAM and EAE frequencies from the cross power spectra of horizontal and vertical CO 2 chords. The classical beam pressure is averaged over ρ ∼ 0.1 to ∼0.2, estimated by the NUBEAM module of the TRANSP code using experimental data. The grey dots represent the noise level in the spectra and the solid line indicates the approximate stability boundary of AE activity.
1. At high q min , for example from 0.3 s to 0.6 s, AEs are destabilized in both low-power and high-power shots, the plasma rests well beyond the AE stability threshold. 2. At a reduced q min , from 0.7 s to 1.3 s, AEs are not present in the low-power shot, but are unstable in the high-power shot for well-matched bulk plasma parameters. That is, the plasma is near the AE stability threshold. 3. At low q min , from 1.3 s to 1.7 s since AEs are not present in the low-power shot and the AE amplitudes are negligibly small or below the measurable level in the high power shot, the plasma is considered to be below the AE stability boundary in the neoclassical transport regime.
Overall, the time evolution of AE activity in the low-power and high-power shots provides the opportunity to study the fast ion phase space flow in three different AE stability regimes, as discussed above.
Besides AEs, beam emission spectroscopy also reveals the perturbation at frequency from 50 kHz to 150 kHz after ∼1.3 s near q min , as seen from figure 11(a). This instability shows 'Christmas light' pattern [40]. This low-frequency mode is theoretically identified as a reactive-type kinetic ballooning mode with a dominant Alfvénic polarization [41,42], which is at lower frequency than beta-induced Alfvén-acoustic modes [43]. The previous study [40] finds that the destabilization of instability is correlated to the increased electron temperature and the mode can be destabilized without fast ions. In this study as shown later, this mode does not induce noticeable transport of fast ions. It is interesting to point out that the electron temperature fluctuation induced by the mode is significantly weaker than the RSAE modes, i.e. below the detectable level of the ECE. Nevertheless, the density fluctuation caused by this low-frequency mode is comparable to the RSAE modes. The toroidal mode number of magnetohydradynamic (MHD) activities, measured by a set of magnetic loops, is given in figure 11(b). It should be noted that the modes at frequencies above 150 kHz, which are measured by magnetic loops, are ellipticity-induced AE (EAE) modes.

AE-induced fast ion phase-space transport
Considering that T e is slightly higher in the high-power shot than that in the low-power shot for a same density after ∼1.3 s, neoclassical theory expects reduced collision between fast ions and electrons and a longer slowing-down time of fast ions in the high-power shot, leading to more confined fast ion content and thus a higher energetic neutral flux toward the INPA. This is demonstrated in figure 4(b) by calculating INPA synthetic signals at the interrogated phase space of (80 keV, 2.0 m). The synthetic signal is calculated by FIDASIM and INPASIM codes, using the fast ion distribution from NUBEAM module of TRANSP code [44]. The synthetic signal associated with the modulated beam (Γ m ) in the low-power shot is given by the solid curve. The synthetic signal of the high-power shot is Γ m + Γ s , given by the dashed curve. The Γ s derived at the timings when Γ m decays to nearly zero (see the blue arrow in figure 4(b)) is comparable or slightly lower than the up-shift of the signal in the high power shot (see the red dashed arrow). This is due to the slightly lower T e when the modulated beam is off.
When the plasma is near and well beyond the marginal AE stability threshold, the observed INPA signal (Γ ob ) in the high power shot evolves differently than the signal predicted by neoclassical theory. For example, INPA signal of the modulated beam Γ m is obtained from the low-power shot, as seen from the black line in figure 12(c). The observed INPA signal in the high-power shot is given by the red dashed line. Γ s is estimated near the end of each modulation period from the high-power shot (see the black arrow). It is found that, during the on-period of two-tangential beams, the INPA signal (Γ ob ) at the phase space of (81 keV and 2.0 m) is much lower than the expected Γ m + Γ s (see the dashed arrow). The signal deficit, i.e. ∆Γ = Γ ob − Γ m − Γ s , is depicted as the vertical width of the colored band in figure 12(c). The signal deficit gradually diminishes, as the plasma enters the neoclassical transport regime after ∼1.3 s. The result is consistent with the expectation that the low-frequency mode does not transport fast ions out of the interrogated phase space volume at the injection energy.
As shown in figure 12(e), the observed INPA signal at this phase space is consistent with the measured neutron rate. The measured neutron rate in the high power shot (red solid line) is comparable to that in the low-power shot during the onperiod of the modulated beam, before 0.7 s, although the beam power of 1.7 MW is superimposed, i.e. an increase of power by ∼45%. The predicted neutron rates by NUBEAM module of the TRANSP code (green dashed line) is also significantly higher than the measured neutron rates, showing a large neutron deficit to the neoclassical-theory-predicted neutron rates. The observation suggests significant fast ion transport occurs, correlated to the destabilization of AEs. Consistently, after ∼1.3 s, when AEs activity is very weak or absent, the neutron rates in the high power shot is nearly doubled compared to that in the low-power shot, and the predicted neutron rates start to match the measured neutron rates.
However, the evolution of the INPA signal is different in other interrogated phase space volumes. Figure 12(d) shows that, at a neighboring phase space (70 keV, 2.2 m), Γ ob is similar with Γ s + Γ m before 1 s, but starts to exceed Γ s + Γ m near ∼1 s. Moreover, the large positive ∆Γ is rapidly reduced after ∼1.2 s, coinciding with the stabilization of edge TAE modes in the high-power shot, as seen in figure 12(b).
The dependence of ∆Γ on the AE amplitudes is statistically analyzed for two INPA-interrogated phase-space volumes in figure 13 for different beam modulation periods of this shot. The result shows that the signal deficit, i.e. negative value of ∆Γ, strongly depends on the amplitude of local AE activities, but the positive ∆Γ does not depend on the local AE amplitude. For example, at the phase space of E inj = 80 keV and  figure 13). The absolute calibration of the INPA system is not currently available and a direct quantitative estimation of the fast ion transport flux through tomography inversion is left for a future work. Nevertheless, the order of magnitude of the fast ion transport flux can be crudely inferred from the measured fractional loss of the INPA signal induced by AE activity, using the equation f loss n f L redis /T int . Here, the maximum f loss is 80%, as shown in figure 13; n f ∼ 1.25 × 10 10 cm −3 is the fast ion density at the local phase space estimated by NUBEAM module of TRANSP without considering AE modes; L redis ∼ 0.15 m is the observed traveling distance in the radial direction during the redistribution; T int ∼ 0.012 s is the time scale of the redistribution. Therefore, the inferred, time-averaged, net fast ion transport flux out of the local phase space at the injection energy of 80 keV and R ∼ 2.05 m is O (10 17 The results suggest the existence of complicated phasespace transport, associated with multiple AE activities. The remainder of this paper shows complete pictures of the phase space transport process by expanding the analysis to all INPA pixels in three different plasma regimes, i.e. well-below, near and well-above the linear AE stability threshold.

Phase-space flow well-below the AE stability threshold.
Firstly, INPA images in nearly AE-free periods in both low-power and high-power shots are analyzed. As seen from figure 12, at ∼1.5 s, MHD instabilities are absent in  images consistently show the ionization and slowing-down process of fast ions, after switching on the modulated beam. More importantly, it is found that these derived images are similar with the measured, modulated-beam-only images from the low-power shot at the corresponding timings (compare row a and b of figure 15). This is not surprising, since the plasma equilibrium, electron density and temperature are well matched between low-power and high-power shots.
The image series also agrees with synthetic images of the modulated beam at the corresponding time slices, as seen from figures 15(c1)-(c3). The synthetic images are obtained by a convolution integral of the diagnostic phase-space weights and fast ion distributions, computed by NUBEAM module of TRANSP code [44]. In the NUBEAM model, the ionized neutrals are computed using realistic neutral beam geometry and plasma profiles. The drift orbits of the fast ions are followed and Coulomb collisions are taken into account. The consistency suggests that the fast ion confinement without AEs is near the neoclassical-theory predicted level. It should be mentioned that the validation of synthetic modeling over a broad range of the plasma parameters in the MHD-quiescent plasma is systematically studied and documented in [45].

3.4.2.
Phase-space flow near the AE stability threshold. As seen from figure 12, at ∼1.25 s, RSAE activity dominates the plasmas in the high-power shot, but AEs are stable in the low-power shot. This can also be seen in figure 16(i), in which the density fluctuation of RSAE activity is only detected in the high-power shot (see the red crosses) during the on-period of the modulated beam, but not in the low-power shot (see the black crosses). Time series of the modulated-beam-only INPA images in the low-power shot is given in the first row of figure 16(A, 1-6) and the corresponding time slices are marked by vertical green lines A1-A6 in figure 16(ii). Similarly with those in MHD-quiescent plasma above, the image series exhibits the birth and thermalization process of fast ions.
Unlike the previous case, the image series of the high power-shot, as seen from the second row of figure 16 The distortion of the INPA image in the high-power shot cannot be attributed to the change of plasma profiles. Since the density from the core to the edge matches well, the atomic physics of beam deposition is similar, so similar birth profiles are expected. Changes in deposition cannot explain the observation of reduced neutrals near the injection energy in the plasma core. Moreover, higher T e in the high-power shot causes fast ions to take longer to reach 70 keV. On contrary, in experiment, a prompt increase of the fast ion density toward the low energy of ∼70 keV and the plasma edge of R ∼ 2.2 m is observed in the high power shot, contradicting with the expectation from neoclassical theory. The process happens in a time scale significantly shorter than the slowing-down time. Note that the generation of the phase-space flow, induced by AE activity, is in a time scale of ∼0.2 ms, as predicted by the MEGA code [30]. Therefore, the uncertainty of measured image differences near the injection energy is quite small.
It should be emphasized that the observed image distortion in the high-power shot is quite robust for all modulation periods, as demonstrated in the image groups B1-B6 and C1-C6 in figure 16. The corresponding time slices of each image are labeled in figure 16(ii). Among these three modulation periods, the AE activity in the high-power shot is the most intense for the modulation period near ∼0.95 s, corresponding to image group C1-C6. It is seen that more severe image distortion is observed in this image group C1-C6, further demonstrating the causality of INPA image distortion and AE activity.  modulation periods A, B and C. The first row of six images is directly measured in the low-power shot; the second row of six images is derived from the high-power shot by removing the steady beam images; the third row of six images is the differential images between the low-power and high-power shot, emphasizing the image differences.

Phase-space flow well-above the AE stability threshold.
In DIII-D, the critical fast ion density gradient for AE drive scales as ∼q −2 [39]. TAE and RSAE become marginally unstable even in the low-power shot at the beginning of the plasma current ramp-up, as seen from figure 12(a).
INPA images of the modulated beam in the low-power and high-power shots are given in figure 17 for three beam modulation periods i.e. A1-A6, B1-B6 and C1-C6. The corresponding time slices of each frame are labeled in figure 17(ii). In the low power shot, all of the image series still show typical beam deposition and thermalization process. It should be pointed out that, since AEs are destabilized in the low-power shot, the INPA images in the low-power shot are 'contaminated' by the perturbed fast ion distributions by AEs. Not surprisingly, key features of the image distortion, discussed in the previous section, appear in the INPA images of the low-power shot.
Added upon the image distortion, INPA images of the modulated beam in the high power shot show interesting new features in plasmas well-above the AE stability threshold: 1. The INPA signal near the injection energy is not observed in the high-power shot and the thermalization of fast ions from the birth locations is also not observable. It suggests that fast ions are rapidly transported toward the plasma edge and low-energy phase space, in a time scale much shorter than the sampling rate of the camera, i.e. ∼6 ms. This is consistent with the experimental observation that the increase of T e is small, even though the energy transfer is dominated by the electron channel for the injected energy of beam ions and bulk electron temperature (the critical beam energy is ∼38 keV). Moreover, since the AE stability threshold is met in the low power shot, adding more beam power in the high-power shot only enhances AE modes to drive more fast ions out of unstable phase space, i.e. an open system with a fast relaxation mediation by the threshold [46]. 2. In contrast to the previous case, the redistributed fast ions toward the plasma edge and lower energy disappear from the INPA view. Redistributed fast ions are close to or at the confined-loss boundary, as seen from figure 1(c). This coincides with the strong destabilization of TAE modes in the plasma edge, which are not seen in previous case, as seen in figures 12(a) and (b). At the end of the modulation periods, the INPA interrogated phase space is dominantly occupied by the deficit region, i.e. ∆Γ < 0. 3. The acceleration of the fast ions towards the plasma core and above the injection energy of 81 keV is also observed (see the red bands in figure 17). As with the radially outward transport, it occurs in a time scale much faster than the time resolution of the INPA camera. The tomography inversion [32] suggests the fast ions, which travel into the plasma core, gain the energy of ∼5 keV.
It should be pointed out that the beam modulation technique used in this experiment is developed from Heidbrink's method [47], which is a 'fast ion variant' of perturbative experiment for thermal transport studies [48,49]. Heidbrink's method uses the fluctuation components of the signal to obtain the divergence of flux (∇ · Γ) from diagnostic-weighted phase-space volume. That is, The signalñ is measured in experiment. Diagnosticweighted fast ion source termS and thermalization sink term time scale τ can be obtained by the best-fitting of the experimental data in a reference MHD-quiescent shot.
The analysis result for the modulated component of the INPA signal, using Heidbrink method, is shown in figure 18. A brief discussion is given, as follows: (1) in the plasma wellbelow AE stability threshold, the modulated components of the INPA signal in the high-power and low-power shots are similar, as seen in figure 18(a1). The modulation component of the INPA data (crosses) can be well interpreted by the transport equation above (see the solid lines); (2) in the plasma near AE stability threshold, the modulation component of INPA signal in the high-power shot (red crosses in figure 18(b1)) is largely distorted, compared to that in the low-power shot (black crosses). The distortion comes from the increase of INPA signal in the high-power shot much less than that in the low-power shot during the on-period of the modulated beam. The saturation of the INPA signal is also faster in the highpower shot, due to the intense AE activities. (3) In the plasma well above the AE stability threshold, besides the large distortion of the modulated waveform of the INPA signal in the high-power shot, the decrease of the INPA signal occurs even before the neutral beam switch-off, accompanied by the rapid decay of the AE amplitude. The exact reason of the early signal decay is not clear. It is speculated that it relates to the strong fast ion transport due to the rapid growth of AE, leading to a time-dependent fast ion diffusivity in local phase space. That is, the transport is so fierce that the collapse of the fast ion pressure gradient cannot sustain the saturated mode amplitude. This is especially possible when the drift orbit size is large at the early of the discharge, as the case presented here. In this case, equation (1) with a constant fast ion diffusivity cannot accurately interpret the modulation component of the INPA signal. It should be pointed out that the early decay of NPA signal before the beam switch-off, which interrogates the phase space of fast ions on banana orbits, was previously found in the simulation using the TRANSP kick model (see figure 14 of [22]). The model uses slowly evolving AE activity without bursting modes.
The advantage of Heidbrink's method is that it is suitable for a beam power scan experiment without the limitation on plasma parameters and plasma regimes. In comparison, the beam modulation method in this experiment does not rely on the distortion of the fluctuation waveform. It relies on quantitative difference of the images in low-power and high-power shots with well-matched plasma parameters, but distinct AE activities. In other words, the plasma must rest near AE stability threshold and the AE activities can be modified without a dramatic change of plasma equilibrium and bulk plasma parameters. The advantage of this method is the simplicity to handle large amount of experimental data in a computationally-efficient way, such as INPA with pixels of  modulation periods A, B and C. The first row of six images is directly measured in the low-power shot; the second row of six images is derived from the high-power shot by removing the steady beam images; the third row of six images is the differential images between the low-power and high-power shot, emphasizing the image differences.
>30 000, and the high accuracy to interpret the result without relying on the accuracy of the model or the input parameters to the model, which avoids the propagation of error. The analysis can be conducted in an empirical way.

Path of phase-space flow
For the above observations, key questions include: why does the observed image distortion follow a certain pattern in phase  space? How can we reconstruct the phase-space transport path to interpret the experimental data? These issues are considered in this section.

Construction of the phase-space path for a single frequency mode
It is known that magnetic moment is an adiabatic invariant during interactions of energetic particles and low-frequency AE waves. 'Low-frequency' here is relative to the ion cyclotron frequency, i.e. ω AE ≪ (qB)/m D .
The other adiabatic invariant during the wave-particle resonant interaction with a single wave of frequency ω and toroidal mode number n is E ′ ≡ E − (ω/n)P ζ , where E and P ζ are the energy and canonical toroidal angular momentum of ions [7,50]. This can be understood as follows, since, for a wave whose Hamiltonian varies as Using these two adiabatic invariants, the path of fast ion phase-space flow can be well defined. Figure 19(a) shows the reconstructed constant µ = 18 keV,T −1 surface (see the purple colored, curved surface). The surface is nearly tangential to the INPA observed pitch from ∼0.75 to ∼0.82. The constant E ′ surface is also overlaid, using one of the dominant RSAE modes, having f = 86 kHz and n = 2 (see the 'mountain'-shaped surface). Since fast ions, which resonate with AE waves, have to rest on both the constant µ surface and constant E ′ surface during the wave-particle interaction, they must travel along the intersection line of these two curved surfaces, as indicated by the black arrows in figure 19, referred to as E ′ and µ line. That is, fast ions born near q min of R = 2.0 m at the injection energy of 81 keV can travel along the path either radially outward to a lower energy or inward to a higher energy. Fast ions that migrate radially outward along this path lose energy to the waves. Fast ions that migrate radially inward gain energy and damp the waves. It should be noted that finite pitch resolution of the INPA is crucial to view the continuous phase space flow that travels across the phase space.
As seen from figure 19(b), projecting the reconstructed E ′ and µ line on the observed image pattern shows that the E ′ and µ line passes through the radial positions of the minimum safety factor at R ∼ 2.0 m near the beam injection energy, and connects the depleted phase-space region (∆Γ < 0) to two redistributed, pile-up regions (∆Γ > 0).

Flattening of fast ion density along the E ′ and µ path
The drive and damping of the mode by the fast ion distributions in the INPA interrogated phase space is linked to the fast ion migration direction along the E ′ and µ line across the phase space. Figure 20 shows fast ion distributions as a function of energy, pitch and major radius during the on-period of the neutral beam from 369 ms to 419 ms. The distributions are calculated by the NUBEAM module of the TRANSP code, based on neoclassical theory. The profile of the fast ion density at the INPA interrogated pitch of ∼0.78 is hollow, (see, for example, three-dimensional plots at 376 ms and 382 ms). It should be emphasized that the amplitude of RSAEs promptly increases after switching on the neutral beams, indicating that fast ions near the injection energy play an essential role on RSAE drive, as seen from figures 17(i) and (ii).
The fast ion density along the E ′ and µ line in the INPA view is also hollow, as seen from the bottom figure of figures 20 and 21(a). Neoclasscial theory predicts that the hollowed profile persists during the entire beam modulation period. This means that collisions between fast ions and bulk plasmas cannot flatten the hollowed fast ion density profile along the E ′ and µ line. On the other hand, the growth rate of the AE mode is proportional to ∂f ∂E | E ′ , i.e. fast ion density gradient along the E ′ and µ line. The hollowed profile suggests that whether fast ions interrogated by INPA drive or damp the mode depends upon resonance locations in phase space.
On the experimental side, the INPA signal consistently shows a hollow profile along the E ′ and µ line in the lowpower shot, as seen from the black curve in figure 21(b). The width of the peak is broader than predicted, dominantly due to the instrumental broadening effect from the finite pinhole size and finite size of the carbon stripping foil in the INPA system (in other words, the finite spatial resolution of the diagnostic system). However, in the high power shot, the INPA signal of only the modulated beam shows a flattened profile along the E ′ and µ line. The profile flattening only occurs during strong AE activity, when plasmas are well above the AE stability threshold. That is, fast ions travel both radially inward by gaining energy of ∼5 keV from the waves and also migrate radially outward by losing energy to the waves. The observed flattening of the fast ion density along the E ′ and µ line shows fast ions in the INPA interrogated phase space not only drive the AE waves, but also damp the AE waves. The flattening is likely caused by the shift of resonances in phase space during the frequency sweeping of RSAE or the excitation of multiple AEs at different frequencies. The result emphasizes that realistic beam distributions have a significant impact on the estimation of the linear growth rate of AE modes. It also hints at a possibility to mitigate the AE activities by controlling beam deposition through phase-space engineering.

Phase-pace path during frequency sweeping/chirping
Rapid frequency sweeping and chirping are common signatures of fast ion driven instabilities, such as RSAE frequency sweeping in this experiment. It is important to study how the change of mode frequency alters the fast ion phase space flow along the E ′ and µ lines.  The impact of mode frequency sweeping-up on the fast ion phase space flow are: 1. As mode frequency sweeps up, the flow trajectories that pass the injection energy bend towards the lower energy. It means that the mode with higher frequency is less efficient to transport fast ions in radial direction, because fast ions need to transfer more energy for migrating the same radial distance. 2. Interacting with the high frequency modes, fast ions on passing orbits tends to cross the passing-trapped boundary before loss, if the energy exchange is large enough. In contrast, for the low-frequency AEs, fast ions are more likely to cross the passing-loss boundary. 3. As the frequency sweeps up, the resonance line moves radially outward and the resonance lines are more dense in the plasma edge. This increases the possibility for an overlap of phase-space islands in the plasma peripheral region [29].
As seen from figure 23, the E ′ and µ lines are more tangential to the contour lines of the pitch in the low-field side, but nearly perpendicular in the high field side. Since INPA observes a nearly constant pitch from 0.75 to 0.82, the continuous phase space flow can be captured by the INPA only from the low field side, especially for AE modes with lower ω/n. In the high field side, continuous fast ion phase space flow rapidly escape the INPA interrogated phase space. This is consistent with the observation that the redistribution of fast ions is clearly observed by INPA only in the low-field side.

Summary and discussion
The INPA interrogates the phase space volumes that are nearly tangential to the E ′ and µ lines in the low-field side of DIII-D tokamak, i.e. intersection lines of constant µ and constant E ′ surfaces (see figure 19(a)). This makes a direct measurement of the continuous, AE-driven phase space flow possible, for the first time.
In plasmas well-below the AE stability threshold, the time evolution of the observed INPA images is consistent with the atomic phyiscs of beam deposition and collisions in both lowpower and high-power shots.
In plasmas near the AE stability threshold, adding moderate neutral beam power leads to the excitation of AEs in the high power shot. For the well-matched electron density, fewer fast ions are observed near the plasma core and more fast ions promptly appear in the plasma edge at low energy. This type of distortion of the INPA images by AE modes are routinely observed in all modulation periods, and the level of distortion correlates strongly with AE amplitudes.
In plasmas well-above the AE stability threshold, adding moderate beam power significantly enhances the AE mode amplitude. Compared to the low-power shot, fast ions populated by the modulated beam are not observed near the injection energy of neutral beams. The thermalization process from the birth locations is also not observable, and as a result, the increase of electron temperature is barely noticeable. This is not caused by enhanced heat transport, but the limit to the local heating power due to prompt fast ion phase-space transport in a time scale much shorter than the slowing-down time.
It is also found that some fast ions gain energy and move radially inward, causing damping of the AE waves. Other fast ions lose energy to the waves and move outward along E ′ & µ lines. Fast ion transport flattens the hollowed fast ion density profile along E ′ & µ lines, exhibiting strong critical gradient behavior at the local phase space. At the INPA interrogated phase space, the beam deposition profile provides both drive and damping to the AE waves.
Extensive simulations of these data are reported elsewhere [30]. To summarize the modeling, ad-hoc fast ion diffusivity model in TRANSP code does not reproduce any key features of the observed fast ion flow. A reduced fast ion transport model, solving Hamiltonian guiding center equations in the presence of AEs using ASCOT5 code [51], successfully reproduces some key features of the observation. However, there is a significant discrepancy between the experimental data and the ASCOT5 simulation near the magnetic axis, where fast ions are on stagnation orbits. Self-consistent, first-principle, multiphase hybrid simulations (MEGA code) [52][53][54] that include realistic neutral beam injection and collisions are able to reproduce most features of the observed fast ion phase-space flow. It is found that nonperturbative effects [14,55,56] are required to reproduce the depletion of the fast ions on stagnation orbits at the beam injection energy. Detailed comparison of the simulation results with the data, using these three numerical models, is reported in [30].
One of the interesting findings from this study is that, even though multiple small-amplitude AEs with a broad range of frequencies and toroidal mode numbers are destabilized in plasmas with very different safety factors q and bulk plasma parameters, the direction of phase space flow does not largely alter for fast ions on passing orbits. This strongly indicates the possibility to develop a computationallyefficient fast ion phase-space transport model, which flattens local fast ion density profiles at resonances along E ′ and µ lines. The radial widths of flattening are determined by AE amplitudes, which could be obtained empirically from experiments.