Proceedings of ITC/ISHW2007

Simulation study of energetic ion transport due to Alfvén
eigenmodes in an LHD plasma
Y. Todo, N. Nakajima, M. Osakabe, S. Yamamoto a, and D. A. Spong ba

National Institute for Fusion Science,322-6 Oroshi-cho, Toki 509-5292, Japan
Kyoto University Pioneering Research Unit for Next Generation, Gokasho, Uji 611-0011, Japanb

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USAThe creation of hole and clump in energetic ion energy spectrum associated with the Alfvén eigenmodes wasobserved with the neutral particle analyzer (NPA) on LHD shot #47645. The difference in slowing-down time
between the hole and clump suggests that the energetic ions were transported for 10% of the plasma minor radius.
The spatial profile and frequency of the Alfvén eigenmodes was analyzed with the AE3D code. The phase spacestructures of the energetic ions on the NPA line-of-sight were investigated in an axisymmetric equilibriumcomparable to the LHD equilibrium with the Poincaré plots where one oscillating Alfvén eigenmode for each plotis employed. The phase space regions trapped by the Alfvén eigenmodes appear as islands in the Poincaré plots.
The radial width of the islands corresponds to the transport distance of energetic ions. The island width depends on
the Alfvén eigenmode amplitude. It was found that the Alfvén eigenmodes with amplitude "Br / B ~ 10#3 transportenergetic ions for 10% of the minor radius.
Keywords: energetic ion transport, Alfvén eigenmode, Poincaré plot, LHD, neutral particle analyzer!

1. Inroduction(a)

The creation of hole and clump in energetic ion
energy spectrum associated with the Alfvén eigenmodes
was observed with the neutral particle analyzer (NPA) on
LHD shot #47645 [1]. The frequencies of the Alfvén
eigenmodes are roughly 55 kHz and 68 kHz, respectively.
Both the Alfvén eigenmodes have toroidal mode number
n=1. The hole and clump are created around energy 150
keV. The difference in slowing-down time between thehole and clump suggests that the energetic ions weretransported for 10% of the plasma minor radius.
In this paper, we try to find the candidates of theAlfvén eigenmodes using the AE3D code [2]. We also
investigate the Alfvén eigenmode amplitude consistent
with the energetic-ion transport suggested by the NPA data.
The Poincaré plots were made to investigate the phasespace structures of the energetic ions on the NPA line-ofsight. The phase space regions trapped by the Alfvén
eigenmodes appear as the islands in the Poincaré plots.
The radial width of the islands corresponds to thetransport distance of energetic ions. As island width
depends on Alfvén eigenmode amplitude, we can discuss
the amplitude consistent with the NPA observation.

2. Analysis of Alfvén eigenmodes
The Alfvén eigenmodes with toroidal monde number
n=1 in the LHD shot #47645 were analyzed with theauthor’s e-mail: todo@nifs.ac.jp

(b)

Fig.1

Spatial profiles of the n=1 toroidal Alfvén
eigenmodes analyzed with the AE3D code forthe LHD shot #47645. The frequencies are (a)42.7 kHz and (b) 79.1 kHz, respectively.

Proceedings of ITC/ISHW2007
AE3D code where the Galerkin method is used in theBoozer coordinates. An MHD equilibrium wasconstructed in the Boozer coordinates. Two toroidal
Alfvén eigenmodes (TAE modes) were found with The
eigen-frequencies 42.7 kHz and 79.1 kHz. The
eigen-frequencies are comparable to the Alfvén
eigenmode frequencies observed at the experiment. The
spatial profiles of electrostatic potential are shown in Fig. 1.
The primary poloidal harmonics of both the two TAE
modes are m=1 and m=2. The primary poloidal
harmonics have the same sign for the TAE mode withfrequency 42.7 kHz, while they have the opposite sign toeach other for the mode with frequency 79.1 kHz.

3. Analysis of energetic-ion orbit!

The energetic-ion orbits were calculated withdifferent starting points on the NPA line-of sight. Theorbits were calculated in an MHD equilibrium
constructed in the HINT coordinates [3]. Theenergetic-ion energy is 150 keV with which the hole andclump creation in energy spectrum was observed with theNPA. The pitch angle is determined by the direction ofthe NPA line-of-sight. The poloidal orbit-frequency isdefined by "# = 2 $ / T# where T" is the poloidal
circulation time. The toroidal orbit-frequency is definedby "# = $# / T% where "# is the toroidal angle which
the energetic ion proceeds in T" .
!
!
The orbit frequency of the energetic ions is defined by! f m / n = n"# $ m"% / 2 & ,where n and m are toroidal
and poloidal mode nubers. We!
take n=1 and m=1 which are identical to the primarypoloidal harmonic of the TAE modes shown in Fig. 1. In!
Fig. 2, the orbit frequency is shown versus the major radiusof the starting point on the NPA line-of-sight. We see that
there exist energetic ions with the same orbit frequency as
the Alfvén eigenmode frequencies observed at the LHD
experiment. These energetic ions resonate with the Alfvén
eigenmodes.

(Fig.2 Blue curve represents energetic-ion orbit
frequency with m/n=1/1 as a function of thestarting point major radius on the NPA
line-of-sight. Black lines denote Alfvén
eigenmode frequencies observed in the LHD
experiment.

(a))

(b)

4. Phase space structure of energetic ions
In axisymmetric systems, E" = E # $P% / n!

isconserved in the wave-particle interaction. Here, E andP" are respectively the total energy and the canonicaltoroidal momentum of the particle. This enables to utilize!
Poincaré plots where the particles with !the same E" areemployed. The phase space structure associated withfinite-frequency waves can be analyzed with the Poincaréplots. Furthermore, as the harmonics of!the two TAE
modes other than n=1 are negligibly small, we expect somesimilarities in phase space structure between the LHD
equilibrium and axisymmetric systems.
Then, we investigate an axisymmetric equilibrium

Fig.3 Poincaré plots for the lower frequency TAEwith amplitude (a) "Br / B = 10#3 and(b) "Br / B = 2 # 10$3 .

!
!

Proceedings of ITC/ISHW2007
minor radius.
(a)4. Summary
The spatial profile and frequency of the Alfvén
eigenmodes in the LHD shot #47645, where the creationof hole and clump in energetic ion energy spectrum
associated with the Alfvén eigenmodes was observed,
was analyzed with the AE3D code. The difference inslowing-down time between the hole and clump observedat the experiment suggests that the energetic ions weretransported for 10% of the plasma minor radius. Thephase space structures of the energetic ions on the NPA
line-of-sight were investigated investigated in an
axisymmetric equilibrium comparable to the LHD
equilibrium with the Poincaré plots where an oscillating
Alfvén eigenmode for each plot is employed. The phasespace regions trapped by the Alfvén eigenmodes appear
as the islands in the Poincaré plots. The radial width of
the islands corresponds to the transport distance ofenergetic ions. The island width depends on the Alfvén
eigenmode amplitude. It was found that the Alfvén
eigenmodes with amplitude "Br / B ~ 10#3 transportenergetic ions for 10% of the minor radius.

(b)

Fig.4 Poincaré plots for the higher frequency TAEwith amplitude (a) "Br / B = 10#3 and(b) "Br / B = 2 # 10$3 .

!
References
[1]
[2]!

!
!

!

!
comparable
to the LHD plasma #47645. The parameters of
the axisymmetric equilibrium are major radius 3.67m,
minor radius 0.54m, toroidal magnetic field 0.5T, andsafety factor profile q(r) = 2.11" 1.23(r / a)2 . The valueof E"  is that of a particle with energy 150keV atr = 0.7a . The two TAE modes are mapped to theaxisymmetric! equilibrium. The frequencies arerenormalized to the experimental values 55kHz for thelower frequency TAE shown in Fig. 1(a) and 68kHz for thehigher frequency TAE shown in Fig. 1(b).
The Poincaré plots for the lower frequency TAE are
shown in Fig. 3. The phase space regions trapped by theAlfvén eigenmodes appear as the islands in the Poincaréplots. The radial width of the islands corresponds to thetransport distance of energetic ions. The island width
depends on the Alfvén eigenmode amplitude. The lower
frequency TAE with amplitude "Br / B = 2 # 10$3
transports energetic ions for 10% of the minor radius. The
phase space structures with the higher frequency TAE isanalyzed in Fig. 4. Comparing Fig. 4(a) and (b), we see!
that the higher frequency TAE with amplitude"Br / B = 10#3 transports energetic ions for 10% of the[3]M. Osakabe et al., Nucl. Fusion 46, S911 (2006).
D. A. Spong and Y. Todo, “Shear Alfvén spectrum andmode structures for 3D configurations”, 49th Annual
Meeting of the Division of Plasma Physics (12-16
November 2007, Orlando), PP8.00098.
K. Harafuji, T. Hayashi, and T. Sato, J. Comput. Phys. 81,
169 (1989).