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Energetic particle-driven instabilities in general toroidal configurations
D. A. Spong1, B. N. Breizman2, D.L. Brower3, E. D’Azevedo1, C. B. Deng3, A. Konies4,
Y. Todo5, K. Toi5
1Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A.
2Institute for Fusion Studies, The University of Texas, Austin, Texas, USA
3Department of Physics, University of California, Los Angeles, CA, USA
4Max-Planck Institut für Plasmaphysik, EURATOM-Association, Greifswald, Germany
5National Institute for Fusion Science, Japan
e-mail address: email@example.com
Key words: energetic particles, Alfvén eigenmodes, stellarator, toroidal plasmas, three-
PACS: 52.55.Pi, 52.55.Tn, 52.35.Bj, 52.55.Hc
Energetic-particle driven instabilities have been extensively observed in both tokamaks and
stellarators. In order for such devices to ultimately succeed as D-T fusion reactors, the super-
Alfvénic 3.5 Mev fusion-produced alpha particles must be sufficiently well confined. This
requires the evaluation of losses from classical collisional transport processes as well as from
energetic particle-driven instabilities. An important group of instabilities in this context are
the discrete shear Alfvén modes, which can readily be destabilized by energetic particles
(with velocities of the order of vAlfvén) through wave-particle resonances. While these modes
in three-dimensional systems have many similarities to those in tokamaks, the detailed
implementation of modeling tools has required development of new methods. Recent efforts
in this direction will be described here, with an emphasis on reduced models.
An improved understanding of energetic particle driven instabilities in three-dimensional
systems is motivated both by the difficulties they may cause as well as opportunities that may
be associated with these instabilities. In the former are concerns about first wall damage, the
negative impact of enhanced fast ion losses on heating efficiency  and thermal plasma
confinement. Under the topic of opportunities are the potential for new ways of inferring
diagnostic information via MHD spectroscopy  (e.g., i profile, fast ion density profile),
Alfvén mode driven transport barriers , bootstrap current profile shaping, possibilities for
stellarator shape optimization  of Alfvén instabilities, selective excitation of modes for ash
removal, and alpha channeling for direct ion heating . All of these areas will require
development and experimental validation of improved modeling capabilities, ranging from
rapidly applied reduced models to more comprehensive first principles calculations. This
paper will describe the physics components that are necessary to understand the frequency
spectrum, mode structure and stability of energetic particle-driven Alfvén instabilities in
three-dimensional equilibria. These tools will be illustrated through applications to several
2. Alfvén-Sound continua
The existence of Alfvén modes possessing a global mode structure in toroidal systems can be
attributed to the poloidal (tokamak) and toroidal (stellarator + rippled tokamak) spatial
variations in the equilibrium magnetic field. In the case of a straight cylindrical plasma with
uniform magnetic field, the Alfvén spectrum is degenerate, as a result of the crossing of the
adjacent continuum ( ω2= k
remove this degeneracy and create frequency gaps at locations where the continua would
have intersected in the cylindrical limit. The first step in identifying experimentally observed
Alfvén modes is the correlation of measured frequencies with these gaps. A number of
models have been developed for the calculation of such continua in stellarators [6,7,8] and
generally provide reasonable agreement with frequency lines that are measured in
experiments, to within uncertainties related to imprecise knowledge of the i and density
profiles, average ion mass and Doppler shifts related to plasma flows.
2) lines. Variations of the magnetic field within flux surfaces
A topic of recent interest both for tokamaks [9,10] and stellarators [4,6,8,11] has been the
inclusion of sound wave couplings to the shear Alfvén continua. In a three-dimensional
system these couplings might be expected to become of importance when
vA= B2/ µ0nionmion
density, mion ion mass, Telectron,ion = electron/ion temperatures, Γ = adiabatic index, and Z = ion
charge state. Normally, the small size of Cs
minimize such couplings. However, due to differences in magnitude between the coupled
k's (especially in the case of stellarators where Δn = jNfp; j =1,2,...; Nfp= field period)
the Alfvén and sound continua can become coupled even at moderate values of βelectron. An
example of the overlapping of the sound continua with the Alfvén continua as βelectron is
2 where Δm, Δn are the coupling intervals in the poloidal and toroidal
Cs= Γ ZTelectron+Tion
1/2is Alfvén velocity and
1/2is the sound speed. Here µ0 = 4π × 10-7 H/m, nion = ion
2= 0.5 ΓZβelectronmight be expected to
increased for an LHD discharge is illustrated in Fig. 1 based on using the STELLGAP model
 with acoustic couplings included.
Figure 1 – Coupled Alfvén-sound continua with increasing β for n = 1 modes in LHD. Pink curves
are dominated by acoustic (sound) couplings while blue curves are Alfvén dominated. Far right-hand
figure is a magnification of the β = 0.002 case showing small Alfvén-sound induced gaps.
For simplicity, the plots in this figure show only the n = 1 mode continua. Inclusion of
Alfvén-sound coupling effects in these continua introduces several new effects. One is the
existence of new gaps where the Alfvén and sound continua intersect (these are present on a
fine-scale in the case of Fig. 1). A second is the presence of a finite frequency minimum in
the coupled continua. The latter effect is illustrated in Fig. 2 for several time slices in an LHD
discharge  where a local minimum was present in the i profile. In this plot, couplings to
the acoustic mode have been included, but the sound continua have been removed for clarity.
The minima in the coupled Alfvén-sound continua of Fig. 2(b) generally agree with the
frequency minima measured experimentally. Alfvén-sound couplings can introduce new
eigenmodes near the minima of these continua, such as the BAAE [10,11] (beta-induced
Alfvén-acoustic eigenmode) and EGAM  (energetic-particle-induced geodesic acoustic
Figure 2 – (a) Reversed shear rotational transform profiles for two times in LHD RSAE discharge
(inset shows measured frequency spectrogram); (b) associated n = 1 shear Alfvén only continua
(lighter lines) and coupled Alfvén-sound continua (bolder lines).
Sound wave coupling effects have also been studied in the HSX stellarator experiment .
In this device coupled sound-Alfvén modes were initially suspected. However, systematic
experimental variation of the rotational transform profile indicated that the observed
frequencies were insensitive to changes in i. This led to the conclusion that the ECH-
produced electron tail predominantly excited sound waves with only weak Alfvénic
couplings . In Fig. 3 a sequence of n = 1 coupled Alfvén-sound continua are shown for
HSX equilibria with increasing i. The i variation causes frequency up-shifts in the central
m = 1, n = 1 Alfvén-sound gap to a greater degree than was observed in the experimental
frequency lines, suggesting that the observed modes were at least not associated with this
particular gap. As reported in ref. , sound waves existing in the m,n = 5,1 and 7,1
continua (large dots in Fig. 3) were consistent with the measured frequencies (dashed lines in
Fig. 3) and their scalings. The question remains as to why these specific modes, out of the
available multitude of sound and Alfvén waves were destabilized. In order to understand this
both the drive (from the energetic electron tail) and damping mechanisms (continuum, e,i
Landau, etc.) will need to be evaluated for the different modes in this frequency range; also,
other mechanisms, such as non-ideal effects (resistivity, viscosity) and sheared plasma flows
could influence the structure of these continua. These remain topics for future research.
Figure 3 – Variation of n = 1 coupled Alfvén-sound continua in HSX with central value of rotational
transform. Color-coding is used to indicate the strength of the sound vs. Alfvén components. Magenta
dashed lines indicate the experimental frequencies, which were not observed to change significantly
with variation in i.
3. Alfvén eigenmodes in stellarators
The next step in the analysis of Alfvén modes is the calculation of the eigenmode structures.
These are the eigenmodes of the ideal MHD operator. Such an approximation is appropriate
for modes that are weakly driven and near marginal stability. For this paper results from the
AE3D code  are used, which is based on a low β, reduced MHD model, and currently
does not include acoustic couplings. An efficient, solution technique has recently been
developed  that provides eigenmodes clustered about a user specified frequency; this
simplifies the process of sorting out modes that are likely to be more readily de-stabilized
(i.e., less affected by continuum damping). The targeted frequencies are likely to be readily
known either from experimental measurements or from the location of open gap regions in
the calculated continua. An example of such an eigenmode calculation for the NCSX
compact stellarator  is shown in Fig. 4. This is an HAE (helical Alfvén eigenmode) with
strong poloidal and toroidal couplings. The continuum plot given on the left also shows
evidence of continuum-crossing gap structures , which are often present in the continua
of three-dimensional systems.
Figure 4 – (a) Strongly coupled n = 1 mode family continua (left figure) for NCSX compact
stellarator; (b) example of an HAE eigenmode with global radial structure (upper right figure); (c)
three dimensional mode structure in potential function (lower right figure).
4. Stability analysis using a wave-particle energy transfer method
The linear stability of Alfvén modes in stellarators has been addressed by both continuum
kinetic calculations  and more recently  using particle based methods. Due to the
large deviations of trapped and transitional particle populations in some stellarators, finite
orbit width effects (FOW) are expected to be of significance since they allow energetic
particles to sample the Alfvén mode structure over a larger radial width than if they were
confined close to fixed flux surfaces. These effects have been analyzed previously using
kinetic  methods. In order to address these issues using particles, a technique  has
been developed that evaluates the stability of a fixed, oscillating Alfvén mode by tracking the
accumulated wave-particle energy transfers between the wave and a large collection of test
particles. This method has been successfully benchmarked against a tokamak case for which
results are available from other codes. Several example applications to stellarators have been
made; Fig. 4 shows results from one such application to the LHD configuration. This was for
a case (not an experimental discharge) where the ion density was chosen to align the n = -1
gaps in radius, resulting in a rather global m, n = (2, -1) mode at 74.1 kHz with coupling to
(1, -1) and (3, -1) sidebands. In the left-hand plot of Fig. 5 the time variation of the growth
rate is plotted as the ratio of the average fast ion velocity to the Alfvén velocity is increased.
A distribution function that was Maxwellian in energy and beam-like in pitch angle [i.e.,
f ∝δ(µ /ε)] was used here. As the plot shows, the growth rate function is initially
oscillatory for several wave periods as the particle markers acquire and average over resonant
perturbations from the wave. For later times the growth rate settles into a quasi-stationary
state with small oscillations that are presumably related to the various particle
bounce/precession/transit motions. A dual weight δf scheme has been developed that
removes the fixed driving frequency of the waves. The averaged growth becomes positive
(unstable) at around < vf> /vA ≈ 0.3 and increases with increasing < vf> /vA. In the right-
hand figure, the time-averaged growth rates are plotted vs. < vf> /vAfor both a beam and an
isotropic fast ion distribution, showing that they reach a maximum and then start to decrease.
This drop-off is caused both by the fact that for 〈vf〉 > vA the fast ion population is moving out
of Landau resonance (v|| = vA) with the wave and by the increasing FOW effects at the fast
ion energies associated with these parameters. The decreased growth rate for the isotropic
distribution vs. the beam distribution is also related to FOW effects and the greater dispersion
of particle velocities in the case of the isotropic distribution – resulting in a smaller fraction
of the distribution being in resonance with the wave.??
Figure 5 – (a) Time evolution of linear TAE growth rate for n = -1 mode family in LHD as a function
of <vf>/vA; (b) Dependence of time-averaged growth rates on <vf>/vA for beam-like and isotropic
Development of the theoretical/computational techniques needed to understand energetic
particle-driven instabilities in three-dimensional systems is an important component of
stellarator physics and will be essential for the projection to future stellarator fusion reactors.
The basic steps in the analysis of such instabilities have been outlined here and consist of
calculation of the coupled Alfvén-sound continuum (identifies frequency windows where
such modes are likely), computation of eigenmode structures (useful for comparison with
experiment and perturbative stability analysis), and assessment of stability, including finite
orbit effects (allows mapping out unstable parameter regimes). In addition, the nonlinear
evolution of such instabilities and their impact on confinement must be considered and is a
topic of current research. The techniques developed for energetic particle physics in
stellarators are also applicable to other three-dimensional systems, such as tokamaks with
symmetry-breaking effects  and self-organized helical states in reversed field pinches .
Acknowledgements?? –?? Research?? sponsored?? by?? the?? U.S.?? Department?? of?? Energy?? under??
Contract?? DE-‐AC05-‐00OR22725?? with?? UT-‐Battelle,?? LLC.??
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