J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)69–73
000–000

Energetic Ion Transport due to Alfvén Eigenmode Bursts
TODO Yasushi, BERK Herbert L.1 and BREIZMAN Boris1
1National Institute for Fusion Science, Toki 509-5292, Japan
Institute for Fusion Studies, University of Texas at Austin, Austin, Texas 78712, USA(Received: 10 December 2003 / Accepted: 6 July 2004)

Abstract
Recurrent bursts of toroidicity-induced Alfvén eigenmodes (TAE) are studied using a self-consistent simulation
model. Bursts of beam ion losses observed in the neutral beam injection experiment at the Tokamak Fusion Test
Reactor are reproduced using experimental parameters. Only co-injected beam ions build up to a significant stored
energy even though their distribution is flattened in the plasma center. They are not directly lost as their orbits extend
beyond the outer plasma edge when the core plasma leans on a high field side limiter. The time evolution of the beam
ion density during a burst and the particle loss mechanism are presented. In the simulation, the distance to the limiteris decreased after the stored beam energy saturates. We show that the co-passing beam particles may not readily reachthe limiter even if the orbit width of edge-located particles is larger than outer edge spacing between the limiter andplasma. For the parameters of this run the stored energy of the co-injected beam density only drops rapidly, after thegap width from the plasma edge to the limiter becomes less than 0.3 of the minor radius. The existence of KAM(Kolmogorov-Arnold-Moser) surfaces of the edge mode even at large field amplitudes apparently inhibits energeticco-passing particles from being lost to the limiter at larger gap widths.
Keywords:
Alfvén eigenmode burst, energetic particles, transport, limiter, computer simulation

1. Introduction

results of the simulation reproduced quite closely thefollowing aspects of the experimental parameters; a)synchronized bursts of multiple TAEs taking place at regulartime intervals close to the experimental value; b) a modulation
depth in the stored beam energy that is close to the one
inferred in experiment; c) stored beam energy that is about
one-third of the classical slowing-down distribution. Only coinjected beam ions build up to a significant stored energy eventhough their distribution is flattened in the plasma center.
They are not directly lost as their orbits extend beyond theouter plasma edge when the core plasma leans on a high fieldside limiter.
The effects of the distance from the plasma edge to thelimiter are important for energetic ion confinement. To
demonstrate it, we carried a run where the distance to thelimiter in the low field side is decreased after the stored beamenergy reaches roughly a constant level. In this paper, we
report on the simulation results and also present the time
evolution of the beam ion density during a burst and theparticle loss mechanism.

The toroidicity-induced Alfvén eigenmode (TAE) [1] canbe destabilized by fast ions which have velocities comparable
to the Alfvén velocity. A decade ago recurrent bursts of TAEs
were observed with neutral beam injection (NBI) in theTokamak Fusion Test Reactor (TFTR) [2] and DIII-D [3]experiments. Nearly synchronous with these TAE excitations,
there were observed drops in neutron emission. Hence it wasinferred that the TAE excitations caused a direct loss of theinjected beam ions. In the experiments cited multiple TAE
mode bursting at regular time intervals were observed. The
modulation depth of the drop in neutron emission in the TFTR
plasma was typically ~10 % (Fig. 4 of ref. [2]) and the beam
confinement time is about one-half to one-third of thecollisional slowing-down time [4]. This means that the TAE
activity in these experiments substantially reduced the beam
ion energy confinement time because TAE activity expels asubstantial fraction of the energetic beam ions before thisenergy is absorbed by the core plasma through drag that iscaused by classical collisions.
Recently, simulations, based on a reduced magnetohydrodynamic (MHD) method for a configuration typical of
the TFTR experiment which had balanced beam injection [2],were carried out and the results were reported in ref. [5]. The

2. Simulation model
The simulation uses a perturbative approach where theTAE spatial profile is assumed fixed, while amplitudes and©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: todo@nifs.ac.jp
169Todo Y. et al., Energetic Ion Transport due to Alfvén Eigenmode Bursts

phases of the eigenmodes and the fast-ion nonlinear dynamics
is followed self-consistently. The algorithm to advance theamplitude and phase of TAE modes is similar to the one
developed in refs. [6] and [7], and is described in detail inref. [5]. For the TAE burst simulation the safety factor (q)profile is taken to vary quadratically with minor radius froma central value of 1.2 to an edge value of 3.0, q(r) = 1.2 +1.8(r/a)2, where a is the minor radius and r is the minor radiuscoordinate. For simplicity we consider concentric circularmagnetic surfaces to describe the equilibrium magnetic field.
In the “vacuum” region the q-profile is modeled with asimplified form of q(r) = 3(r/a)2. The major and minor radii
are R0 = 2.4 m and a = 0.75 m. The magnetic field is 1.0 T
on axis. The spatial structure and the real frequency of theeigenmodes are obtained from a Fokker-Planck-MHD
simulation [8]. The plasma density in the simulation is chosen
for simplicity to be uniform 2.2 × 1019 m-3. Both the coreplasma ions and the beam ions are deuterium. Fiveeigenmodes are taken into account. Their toroidal mode
number and real frequency are, respectively, a) n = 1, ω =0.283ωA (mode 1), b) n = 2, ω = 0.404ωA (mode 2), c) n = 2,
ω = 0.278ωA (mode 3), d) n = 2, ω = 0.257ωA (mode 4), ande) n = 3, ω = 0.330ωA (mode 5), where ωA ≡ VA /R0 = 1.35 ×
106 s-1 and is the Alfvén velocity. The spatial profile of theeigenmodes is shown in Fig. 1. The linear damping rate ofeach mode is assumed to be constant at 4 × 103 s-1. The
Fokker-Planck-MHD simulation does not give the part of themode damping rate which depends on the kinetic propertiesof the bulk plasma. This leads to an arbitrariness in the choiceof the damping rate and the eigenmodes in the presentsimulation. We have chosen the aforementioned set of
eigenmodes and damping rates that roughly reproduces theexperimental results.
Beam ions have balanced injection with a constantheating power of 10 MW and with a spatial Gaussian profile
whose radial scale length is 0.3 m. The injection energy is110 keV which roughly corresponds to the Alfvén velocity
parallel to the magnetic field. The injected particle speed is V
= V 0 . The injected beam ion has a uniform pitch angle
distribution in the range of 0.7 ≤ ⏐λ⏐ ≤ 1, where λ ≡ V|| /Vand V|| is the velocity parallel to the magnetic field. In theTFTR experiment two types of limiters, toroidal belt limiterand three poloidal limiters, were used. In the poloidal crosssection the limiters roughly defined a circle of radius 1 m.
We model these limiters by removing particles if they reacha torus with axis at R/a = 3.53 (R = 2.65 m) on the midplaneand minor radius 1.33a (1 m). Figure 2 shows the configuration of the plasma and the limiter where particles areremoved. Thus the plasma is leaning on the limiter at the highfield side, while at the low field side there is a space fromthe plasma edge to the limiter whose width is ∆ = 0.67a (0.5m). In addition to the plasma and the limiter, examples ofcounter-injected beam ion orbit and co-injected beam ion orbit
are shown in Fig. 2. By convention the velocity of a coinjected ion is parallel to the plasma current and in our casethe velocity is parallel to the toroidal magnetic field as well.
270Fig. 1

Major four harmonics of the electric potential of Alfvén
eigenmodes with the toroidal mode number of a) n = 1,
ω = 0.283ωA (mode 1), b) n = 2, ω = 0.404ωA (mode 2), c)n = 2, ω = 0.278ωA (mode 3), d) n = 2, ω = 0.257ωA(mode 4), and e) n = 3, ω = 0.330ωA (mode 5), where ωA≡ VA /R0 = 1.35 × 106 s-1 [5].

Fig. 2

Configuration of the plasma and the limiter, andexamples of counter-injected beam ion orbit and coinjected beam ion orbit. The velocity of the co-injected
ion is parallel to the plasma current. The orbits of coinjected (counter-injected) beam ions are displaced
from magnetic surfaces towards the low (high) fieldside. The distance from the plasma edge to the limiterin the low field side is denoted as ∆.

Todo Y. et al., Energetic Ion Transport due to Alfvén Eigenmode Bursts

Thus negative values of λ correspond to the counter-injectedbeam ions and positive values of λ correspond to co-injectedbeam ions. The orbits of co-injected (counter-injected) beamions first encounter the plasma edge on the low (high) fieldside. Note that the co-injected particles can stick out of theplasma on the low field side, whereas the counter-injectedparticles are immediately removed by the limiter when theyreach the edge of the high field side.
Particle collisions, slowing down and pitch angle
scattering, are taken into account in the simulation. The
slowing-down time is assumed to be 100 ms. For an
experimental electron temperature of 2 keV the criticalenergy, above which the collisions with electrons dominate
the slowing down process, is 37 keV. The pitch angle
scattering rate is given by vd = vVc3/2V3, where v is the rate of
the slowing down and Vc is the critical velocity corresponding
to 37 keV. Because the pitch angle scattering rate diverges as
the particle speed reaches zero, we remove particles when
they reach V = 0.1V0 . The number of particles used in thesimulation runs is 2 × 106. Convergence with particle number
was investigated in ref. [5]. It was concluded that we have agood numerical convergence so that the number of particles
used is sufficient.

Fig. 3 Amplitude evolution of all the eigenmodes.

3. TAE bursts with variable distance tothe limiter3.1 Simulation results
We start the simulation at an initial time taken as t = 0
when the beam ions are first injected. As time passes,
energetic ions gradually accumulate. After t = 66.8 ms when
the stored beam energy have reached roughly a constant level,
the distance from the plasma edge to the limiter ∆ isdecreased linearly in time with a time scale 30 ms. The time
evolution of the amplitude of each mode is shown in Fig. 3.
We see that before t = 66.8 ms synchronized bursts take placerecurrently at a burst interval that is roughly 2.9 ms which isreasonably close to that of the experimental value 2.2 ms in
the TFTR experiment that we are comparing with. Figure 4
shows the time evolution of the distance from the plasma edge
to the limiter in the low field side (denoted as ∆) and of thestored energy of co and counter injected beams. Before we
discuss the effects of the change in ∆, we summarize theresults with the steady limiter configuration before t = 66.8
ms. The modulation depth of the drop in the stored beamenergy is 10 % which is close to the inferred experimental
value of 7 %. In the relative units of Fig. 4, the sum of thestored beam energy of the co and counter injected beams withTAE bursts saturates at a relative level of 0.31 which is 40 %of that of the classical distribution which is established with
only a particle source and particle collisions. We thereby findgood agreement between the simulation and the experiment
where the stored beam energy is about one-half to one-thirdof the classical distribution [4]. A basic feature of thesimulation that is apparent in Fig. 4 is the dramatic difference
between the stored beam energy of co- and counter-injectedbeams whose velocity is parallel and anti-parallel to theFig. 4Time evolution of the distance from the plasma edge tothe limiter in the low field side (denoted as ∆) and of
the stored energy of co and counter injected beams.

plasma current, respectively. The loss in counter-injectedbeam energy induced by the TAEs’ activity is 88 %, whilethat in co-injected beam energy is 37 %. Figure 5 shows thetime evolution of the dominant two modes 2 and 5 and thedensity of the co-injected beam ions [5]. We can see that themode 2, which is located at the plasma center, has precursory
growth before the modes grow together during each burst.
Because the beam injection profile peaks at the plasma center,mode 2 is destabilized before mode 5. We can see a complete
flattening of the density at the plasma core (r/a < 0.72) whilesmall increase in the density at the plasma edge (r/a > 0.72).
The beam ions stored at the plasma core during the quiescentphases are transported to the plasma edge and lost primarily
during the bursts.

3.2 Particle loss mechanism
We now consider how the energetic particle loss
mechanism is to be understood. To study this, we studied
surface of section plots where only one eigenmode is taken
into account and the amplitude of the eigenmode is at aconstant value [5]. In the surface of section plot we print out
the major radius of a counter-passing (co-passing) particle371

Todo Y. et al., Energetic Ion Transport due to Alfvén Eigenmode Bursts

Fig. 5each time the poloidal angle of the particle reaches ϑ = 180°(0°). We show in Fig. 6 surface of section plots for the (a)counter-passing and (b) co-passing particles, respectively,
where the field amplitude of mode 5 is fixed in time at δB/B= 6 × 10-3. At this amplitude the stored beam energy takes on
relative maximum values during the simulation run. Thisamplitude is higher than the ambient amplitudes between
bursts, but considerably lower than the peak amplitudes thebursts reach. We see in Fig. 6(a) and (b) that the KAM
surfaces (see e.g. [9]) are destroyed for mode 5 near theplasma edge R/a < 2.6 and R/a > 4.6, respectively, which thenleads particle loss even before the modes reach their peak
amplitude. The destruction of KAM surfaces takes place due
to overlap of higher-order islands [5,10]. We should noticethat in Fig. 6(b) the KAM surfaces exist at 4.4 < R/a < 4.6
for co-injected beam ions, which do not allow the particle
diffusion from the plasma center to the edge at that field
amplitude and lead to substantial delay in particle loss
compared with the counter-injected beam ions. We havestudied surface of section plots for all the eigenmodes bothfor counter-passing and co-passing particles at the field
amplitude that induces enough loss to prevent the increase in
the stored beam energy. We have found that no eigenmodehas a global stochastic region which extends from the plasma
center to the limiter not only for co-passing particles, but also
for counter-passing particles which are lost at this amplitude.
This and the synchronization of all the eigenmodes suggeststhat the resonance overlap of different eigenmodes [11] is also
important as the particle loss mechanism.

The time evolution of the dominant two modes 2 and 5
and the density of the co-injected beam ions at various
minor radius [5].

3.3 Effects of the distance to the limiter
Next, we turn to the results after t = 66.8 ms. The
distance from the plasma edge to the limiter in the low fieldside ∆ is decreased in 30 ms. We can see in Fig. 4 that thestored energy of co-injected beam drops rapidly after ∆/a isreduced to about 0.3. In Fig. 6(b) the KAM surfaces exist at
4.4 < R/a < 4.6 for co-injected beam ions, which do not allow
the particle diffusion from the plasma center to the edge andlead to substantial delay in particle loss compared with thecounter-injected beam ions. After ∆/a becomes less than about
0.3, the limiter comes inside the KAM surfaces leading todisappearance of this effective barrier. After ∆/a reaches 0,
the time evolution of the stored energy of the co-injectedbeam is roughly the same as that of the counter-injected beam.
Thus, it is clear that the difference in the stored energy of theco and counter beams before t = 66.8 ms was created due tothe limiter configuration which allows only the co-injectedbeam ions to stick out of the plasma. We see in Fig. 3 that as∆ decreases in time after t = 66.8 ms, the burst intervalbecomes longer. After ∆ reaches 0 at t = 96.8 ms, the burst
intervals are about 5 ms.

4. SummaryFig. 6

Surface of section plots for a) counter-injected and b)
co-injected beam ions where the amplitude of mode 5
is fixed in time at δB/B = 6 × 10-3 and ϕ denotes thetoroidal angle.

Recurrent bursts of toroidicity-induced Alfvén
eigenmodes (TAE) have been studied using a self-consistent
simulation model. Bursts of beam ion losses observed in the472

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neutral beam injection experiment at the TFTR werereproduced using experimental parameters. Only co-injectedbeam ions build up to a significant stored energy even though
their distribution is flattened in the plasma center. They are
not directly lost as their orbits extend beyond the outer plasmaedge when the core plasma leans on a high field side limiter.
To demonstrate the effects of the distance from the plasmaedge to the limiter in thí low field side, the distance to thelimiter is decreased after the stored beam energy saturates inthe simulation. The stored energy of co-injected beam drops
rapidly, after the distance from the plasma edge to the limiter
becomes roughly less than 0.3 of the minor radius. Surfaceof section plots have demonstrated that both the resonanceoverlap of different eigenmodes and the disappearance of
KAM surfaces in phase space due to overlap of higher-orderislands created by a single eigenmode lead to particle loss.

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