Ions lost on their first orbit can impact Alfv (cid:2) en eigenmode stability

Some neutral-beam ions are deﬂected onto loss orbits by Alfv (cid:2) en eigenmodes on their ﬁrst bounce orbit. The resonance condition for these ions differs from the usual resonance condition for a conﬁned fast ion. Estimates indicate that particles on single-pass loss orbits transfer enough energy to the wave to alter mode stability. V C 2015 AIP Publishing LLC .


I. INTRODUCTION
Alfv en eigenmode stability is a competition between energy extracted from the fast-ion population and energy lost to the background plasma. 1 The energy that a particle exchanges in a single orbit is Þ eẼ Áṽ dt, where e andṽ are the charge and velocity of the particle,Ẽ is the wave's electric field, and the integral is over the orbit. Customary analysis assumes that fast ions interact with the mode for many wave cycles. In this case, a particle may exchange energy on a single orbit but, after many cycles, only orbits that preserve the wave-particle phase over multiple orbits exchange net energy. These particles are the resonant particles. If the wave frequency x is much smaller than the cyclotron frequency, the resonance condition is (1) where x / and x h are the toroidal and poloidal orbital frequencies, n and m are the toroidal and poloidal mode numbers, and l is an integer. Depending on the initial phase, some resonant particles gain energy from the wave and others lose energy. In linear theory, the net exchange of energy between the waves and the particles depends upon the slope of the distribution function at the resonant frequency, as in Landau damping. In a DIII-D experiment, Chen et al. discovered that toroidal Alfv en eigenmodes (TAE) 2 and reversed shear Alfv en eigenmodes (RSAE) 3,4 can deflect newly injected beam ions $10 cm on their first bounce orbit through the mode. 5,6 In the experiment, the loss detectors are spatially localized, so coherent losses are only measured when the unperturbed orbit passes close to a fastion loss detector. [5][6][7] Presumably, however, the same process occurs for all ions on similar orbits, irrespective of toroidal angle. These lost particles do not satisfy the resonant condition, Eq. (1). Nevertheless, the particles do maintain approximately constant phase on a single pass through the mode, so the net energy exchange Þ eẼ Áṽ dt is nonzero. Recently, Zhang et al. 8 analyzed this situation for a model RSAE and equilibrium. They found that particles that satisfy a single-pass resonant condition deflect $5 cm as they orbit through the mode for conditions similar to the experiment. Here, h _ /i and h _ hi are the average toroidal and poloidal angular velocities during the portion of the orbit that traverses the mode. Because the orbit is large and the Alfv en eigenmode is of limited spatial extent, h _ /i and h _ hi differ considerably from the customary (orbit-averaged) precession and bounce frequencies x / and x h .
In both theory and experiment, the magnitude of the deflection depends linearly on the mode amplitude. The direction of the radial "kick" depends on the phase of the wave during the orbit's transit through the mode. Hamiltonian theory implies that, in an interaction with a single mode, the change in particle energy DW is related to the change in toroidal canonical angular momentum DP / by the relation Ions that are deflected inward to smaller major radius gain energy from the wave; ions that are deflected outward, lose energy. In a normal interaction with a confined non-resonant particle, a particle that gains energy from the wave eventually gives it back; similarly, a particle that initially loses energy eventually regains it. Irrespective of the initial wave-particle phase, the net energy transfer for non-resonant particles averages to zero. Only resonant particles produce secular terms that survive after averaging over many wave cycles.
A loss boundary destroys this reversibility. If the initial wave-particle phase pushes the ion outward so that it collides with the wall, it loses energy. Because the ion is lost, the return cycle is never completed. Ions that initially move inward gain energy from the wave. Since they move inward, they remain confined until the phase changes sign. They then move outward, collide with the wall, and have lost energy. Irrespective of the initial wave-particle phase, all particles that are near the loss boundary deliver energy to the wave.
The goal of this paper is to assess the impact of this effect on mode stability. The rough estimates presented here show that the power transfer is potentially significant.

II. COMPARISON OF RESONANT AND SINGLE-PASS ENERGY TRANSFER
For fast-ion driven instabilities, the contribution to the growth rate c drive is proportional to where N res is the number of particles that interact with the wave, DW is the amount of energy exchanged in an interaction, and f net depends upon the fraction of the particles that lose or gain energy. This section compares resonant interactions with single-pass loss interactions for these three factors qualitatively and quantitatively.
For the quantitative comparison, we use an actual case (discharge #146096 at 365 ms) that is shown in many figures in the experimental papers. 5,6 At 365 ms, this discharge has a circular shape (elongation j ' 1.2), a line-averaged density of 1.4 Â 10 19 m À3 , a plasma current of 0.6 MA, and a toroidal field of 2.1 T. The minimum and edge safety factor are q min ¼ 3.5 and q 95 ¼ 6.0, respectively. The neutral beams inject 4.6 MW of deuterium neutrals at 74 and 80 keV in the co-current direction, with approximately equal numbers injected by near-perpendicular and near-tangential sources (tangency radii of 74 and 115 cm, respectively). The Larmor radius (banana width) of a typical trapped particle is approximately 5 (33) cm. The beams drive many Alfv en modes unstable, including an n ¼ 2, m ¼ 7, 117 kHz RSAE with a peak fluctuation amplitude of dT e /T e ' 1.4% and a radial eigenfunction that extends between R ¼ 195-215 cm. (T e is the electron temperature and R is the major radius.) The calculations in Ref. 8 are based on a model RSAE that resembles this mode.
The first factor that determines the growth rate is the number of resonant fast ions N res . To interact with the wave effectively, an ion must be on an orbit that satisfies either the standard resonance condition (Eq. (1)) or the single-pass resonance condition (Eq. (2)). Figure 1 shows the orbit topology in this equilibrium for ions at the injection energy. The orbits are classified according to their angular momentum P / and magnetic moment l. For these large-orbit fast ions, in addition to the usual trapped ions and co-passing and counterpassing circulating ions, there are lost orbits and non-axis encircling orbits. 9 Orbits that satisfy one of the resonance conditions and transit through the measured RSAE eigenfunction are marked on the orbit topology map. For the single-pass resonance, the guiding-center orbit must also pass within 10 cm of the wall. Standard resonances (Eq. (1)) with this mode exist for trapped, co-passing, and counterpassing orbits. Single-pass resonances (Eq. (2)) exist with trapped orbits and occupy a different region of phase space than the standard resonance. The single-pass resonance occurs on the inner (counter-going) leg of the banana orbit, where h _ /i has opposite sign from the precession frequency x / .
In addition to the resonance condition, another factor that influences the number of ions that interact with the wave is the width of the resonance Dx. The standard resonance is much narrower than the single-pass resonance. The calculations in Ref. 8 indicate that Dx is an order of magnitude wider for the single-pass resonance. For the single-pass resonance, effective wave-particle interaction occurs as long as the phase variation across the interaction region is Շp, so Dx is large. 8 In addition to the resonance conditions and resonance widths, the number of ions that interact with the mode depends upon the distribution function. The initial orbits are determined by the beam deposition; subsequently, Coulomb scattering populates other orbits. In Fig. 1, the orbital parameters that are initially populated by the neutral beams are overlaid on the resonance map. The calculation 10 uses measured density and temperature profiles and includes orbits that ionize in the scrape-off layer. The figure shows that the beams populate the co-passing and trapped portions of phase space. As a rough estimate of the number of fast ions that interact with the mode, we count the fraction of deposited orbits that satisfy a resonance condition (within the resonance width). Approximately 2% of the recently ionized fast ions interact with the RSAE through the single-pass resonance condition. Approximately 6% of recently ionized fast ions satisfy a standard resonance condition. The figure shows orbits at the time of fast-ion birth; however, additional ions come into resonance with the mode as they thermalize. Since thermalization causes the banana width to shrink, moving ions away from loss boundaries, additional single-pass resonances from thermalizing ions are rare. In contrast, many confined ions do eventually satisfy a standard resonance condition. To estimate this fraction, we track the deposited ions through phase space as they thermalize, modifying ðW; l; P / Þ and hence the orbital frequencies x / and x h ; at each energy step, we check to see if the tracked ion encounters a resonance with the observed RSAE.
FIG. 1. Orbit topology map for DIII-D discharge #146096 at 365 ms for W ¼ 75 keV. The abscissa is the toroidal canonical angular momentum normalized to the poloidal flux at the last-closed flux surface. The ordinate is the magnetic moment normalized by the particle energy and magnetic field at the magnetic axis. Orbits that satisfy the standard resonance condition (Eq. (1)) for l ¼ À3 to 3 and that pass through the RSAE eigenfunction are indicated by squares; orbits that satisfy the single-pass resonance condition (Eq. (2)), which pass through the RSAE eigenfunction, and pass within 10 cm of the wall are indicated by diamonds. The small triangles represent orbits that are populated by the injected neutral beams. The various orbit types listed in the legend are lost orbits (including orbits on open field lines), co-passing orbits that do not encircle the magnetic axis, trapped orbits, axisencircling co-passing orbits, and counter-passing orbits.
Ultimately, $20% of the injected ions eventually encounter a resonance with the RSAE. (For this estimate, resonances between l ¼ À3 and 3 in Eq. (1) are considered.) The second factor is the amount of energy exchanged in the interaction DW. This depends on the time-averaged value of Þ eẼ Áṽ dt, where the average is over many cycles for the standard resonance and over one cycle for the single-pass resonance. Calculations with a model RSAE (Figs. 2 and 3 of Ref. 8) find that the most effective standard resonances exchange about twice as much energy as the single-pass resonance. In the calculation, single-pass resonant particles exchange $3.5% of their energy. An independent estimate of the energy exchange can be found from the measured displacement. Experimentally, the single-pass resonant particles are deflected $10 cm. This deflection implies a change in average flux surface Dw/w, where w is the poloidal flux. With the assumption that DP / =P / ' Dw=w, Eq.
The third factor is the fraction of resonant particles that gain or lose energy from the wave f net . As argued in Sec. I, virtually all single-pass resonant particles ultimately deliver energy to the wave, so f net ' 1 for the single-pass resonance. In contrast, for the standard resonance, the fraction of particles that deliver energy to the wave depends upon the slope of the distribution function. Since Alfv en waves are driven by the spatial gradient of the distribution, the relevant slope is @F=@P / . In the nonlinear theory of Landau damping, 11 particles that are trapped by the wave exchange net energy. The fast-ion spatial gradient is steep for some of the resonances shown in Fig. 1 and is shallow for others. For the resonance with co-passing particles at P / =w wall ¼ 0:25; lB 0 =W ¼ 0:3, a NUBEAM calculation 12 indicates that the slope of the distribution function near the injection energy is ð@F=@P /;norm Þ=F ' 10. Assuming that the wave flattens the distribution over a range of DP /;norm ¼ 60:25, which corresponds to a spatial step >10 cm, this implies that twice as many ions lose energy as gain energy, so the effective energy transfer of a typical resonant particle is approximately ðf lose À f gain Þ=ðf lose þ f gain Þ ' 1 3 . Comparing these factors for the specific case in Fig. 1, the fast-ion drive from the standard resonance is larger than the fast-ion drive from the single-pass resonance by a factor of For this case, the estimated contribution to the fast-ion drive by the single-pass resonances is sub-dominant but not negligible. It is evident that, for different conditions (particularly different beam deposition), single-pass resonances could provide the dominant drive term. The estimates given above are for a finite amplitude wave at the nominal experimental amplitude of dT e /T e ' 1%. Since the energy transferred to the wave is proportional to (dB) 2 for both the standard resonance and the single-pass resonance, the ratio of drive terms is independent of mode amplitude. (Here, dB is the mode amplitude.) For the standard resonance, the width of the trapping region scales as ffiffiffiffiffi ffi dB p ; assuming that all trapped particles acquire the wave speed yields an energy transfer that scales as (dB) 2 @F/@v. 11 For the single-pass resonance, the displacement scales linearly with mode amplitude, which implies that the number of ions that reach the loss boundary N res scales linearly with dB. Through Eq. (3), the energy lost by an ion DW also scales linearly with dB. Since the energy transfer is proportional to N res dW, the transferred energy scales with (dB) 2 .

III. POWER ESTIMATES
This section presents a back-of-the-envelope calculation that demonstrates the plausibility of appreciable power transfer from single-pass resonances.
The measured losses occur during neutral-beam injection. The beams that populate these lost orbits inject a power P inj . The fraction of these injected particles that are lost through single-pass resonance and give energy to the wave is f depo . When a particle is lost, a fraction of its energy f E ¼ DW/W inj is delivered to the wave. Here, W inj is the injection energy. Putting these factors together, the power delivered to the wave is P drive ¼ f depo f E P inj ¼ ð0:02Þð0:13Þð4:6e6Þ ¼ 12 kW; (6) for the example of Fig. 1.
To estimate the power required to sustain the mode, assume a saturated state, where c drive ' c damp . For an Alfv en wave, the energy density u is approximately twice the magnetic energy density (the other half is kinetic energy density), so u ¼ 2(dB) 2 /2l 0 . The volume of the mode is roughly V ¼ (2pR)(2pr)DR so the power required to sustain the mode is approximately For dB/B ¼ 10 À3 , r ¼ 0.2 m, DR ¼ 0.1 m, and c damp /x ¼ 10 À2 , P damp ' 30 kW.
These estimates indicate that it is plausible that singlepass losses help sustain the mode. In comparison, a NOVA-K calculation of the fast-ion drive due to standard resonances 13 finds a stronger contribution to mode stability, c drive ' 6c damp , for an n ¼ 2 DIII-D RSAE. Thus, for the case considered here, we again conclude that single-pass losses make a significant but sub-dominant contribution to mode stability.

IV. DISCUSSION
We have assumed that virtually all single-pass resonant particles ultimately lose energy to the mode. This assumption is reasonable for the example considered here but could be invalid elsewhere. If a particle that gains energy is displaced so that it no longer interacts with the wave (for example, by crossing a topological boundary), it is unable to return the gained energy to the wave when the phase changes. In our case, the loss boundary is nearer than other topological boundaries for most resonant ions but this need not be the case in general.
Nonlinear effects are another neglected complication. Experimentally, when multiple Alfv en eigenmodes produce coherent losses on the fast-ion loss detector (FILD) signals, peaks at sum and difference frequencies often appear in the FILD spectra. 14 Presumably, these nonlinear interactions alter the energy delivered to the fundamental modes; however, since the nonlinear spectral peaks are typically an order of magnitude smaller than the fundamental peaks, the impact on the estimates given here is small.

V. CONCLUSION
In conclusion, a comparison of the factors that influence energy transfer to the wave (Eq. (4)) shows that, under certain conditions, single-pass loss orbits can play an important role in mode destabilization. Rough estimates for an actual DIII-D case suggest a 10%-30% contribution to the fast-ion drive. These estimates motivate more accurate calculations of the actual energy transfer using realistic mode structures, beam deposition, distribution functions, and loss boundaries.
Although loss boundaries have been considered in the previous work, the importance of single-pass resonances that expel particles is new. For example, in their analysis of the energetic particle driven geodesic acoustic mode (EGAM), Berk and Zhou 15 find that the loss boundary for counterpassing particles plays an important role in mode excitation. In a stability calculation using the standard resonance condition, a loss boundary enhances the drive because the slope of the distribution function is infinite at the boundary between confined and lost particles; this effectively increases our f net factor to unity. What distinguishes our work from the previous studies is the recognition that, for a single-pass loss orbit, the resonance condition occupies a different region of phase space. The contribution of single-pass loss orbits to EGAM stability should be considered in future work.
The effect discussed here is unimportant for the main fusion-product population in burning plasmas. In a successful ignited reactor, few fusion products will populate regions near loss boundaries. On the other hand, in present-day magnetic-fusion experiments, neutral beams often deposit appreciable power on orbits that are near loss boundaries. In these plasmas, single-pass resonances can impact mode stability.

ACKNOWLEDGMENTS
The assistance of the DIII-D team and helpful discussions with Ruibin Zhang are gratefully acknowledged. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award Nos. SC-G903402, DE-FC02-04ER54698, and DE-AC0209CH11466. DIII-D data shown in this paper can be obtained in digital format by following the links at https:// fusion.gat.com/global/D3D_DMP.