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Published by Oak Ridge National Laboratory
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Editor: James A. Rome Issue 142 December 2013
E-Mail: [email protected] Phone (865) 482-5643
Enhanced microstability in
quasi-isodynamic stellarators
Optimized stellarators promise to show reduced neoclassical
transport thanks to the improved confinement of
trapped particles. Anomalous transport is therefore
expected to play an important role, and one must study
how the neoclassical optimization affects microinstabilities
and the turbulence they cause. We focus on quasi-isodynamic
stellarators, in which the contours of constant
magnetic field are poloidally closed (see Fig. 1) and, more
importantly, in which the bounce-averaged radial drift
vanishes,
Here b denotes the bounce time. This implies that the parallel
adiabatic invariant
where the integral is taken between two bounce points, is a
flux function, . In a maximum-J configuration [1]
where J has its maximum on the magnetic axis, so that
, the diamagnetic frequency of species a
and the bounce-averaged magnetic drift frequency
are in opposite directions (see Fig. 1),
(1)
There is thus no resonance between these two frequencies,
so that all orbits have favorable bounce-averaged curvature.
Instabilities that rely on a resonance between these
two frequencies should therefore be stabilized.
In order to analyze stability in the collisionless and electrostatic
limit we define the rate of gyrokinetic energy
transfer from the field to species a as
J    0 *a
da
Fig. 1. Magnetic field strength |B| of a six-period quasi-isodynamic
stellarator QIPC [2], with the directions of the diamagnetic
drift frequency and the bounce-averaged
magnetic drift frequency indicated by the arrows.
*a
da
In this issue . . .
Enhanced microstability in quasi-isodynamic
stellarators
Quasi-isodynamic stellarators with the maximum-J
property are optimized to have reduced neoclassical
transport. It can be shown analytically that these configurations
are stable towards trapped-particle instabilities
in large parts of parameter space, so that
turbulent transport might also be reduced. This
enhanced resilience to these instabilities is also seen
in numerical simulations of Wendelstein 7-X, which is
only approximately quasi-isodynamic. .................... 1
Coordinated Working Group Meeting
(CWGM12) for Stellarator-Heliotron Research
The 12th Coordinated Working Group Meeting
(CWGM11) was held on 20 September 2013 following
the Joint 19th International Stellarator-Heliotron Workshop/
16th IEA-RFP Workshop in Padova, Italy. ..... 4
Stellarator News -2- February 2012
(2)
where J0 is a zeroth-order Bessel function and is the
electrostatic potential perturbation. The perturbed distribution
function is given by
with fa0 being a Maxwellian at rest. Here we use the notation
for integration over velocity space and across the entire
flux surface.
The two governing frequencies of the system are defined
as with as the
perpendicular wave vector and
with the density n appearing in
Here denotes the normalized velocity,
with the temperature and particle mass . The ratio
between temperature gradient and density gradient is
given by
Note that, for a destabilizing species, the energy transfer
rate must be negative, . If marginal stability is
approached, so that the growth rate  of the mode is going
to zero, the contributions from all particle species should
cancel,
For modes whose frequency  lies well below the bounce
frequency ba of the particle species a, , we find
the following expression near marginal stability:
(3)
which is always positive in maximum-J devices if the temperature
gradient is not too large, .
If we consider low-frequency modes so that
holds for all species, then there cannot be a point of marginal
stability and thus no unstable mode exists. This
applies, for example, to the collisionless trapped-particle
mode, which should therefore be absent in quasi-isodynamic
stellarators. If only the electrons are bouncing faster
than the mode in question, so that Eq. (3) applies only to
the electrons, then the electrons are not contributing to
destabilization and it must be the ions that drive the instability.
Moreover, it can be shown that any possible mode
must be travelling in the ion diamagnetic direction.
Also, classical electron-driven trapped-electron modes
(TEMs) should therefore be absent in quasi-isodynamic
stellarators with the maximum-J property [3, 4].
While these findings are very promising, it has been
shown that perfectly quasi-isodynamic configurations cannot
exist [5]. This raises the question of whether enhanced
stability also prevails in only approximately quasi-isodynamic
configurations such as Wendelstein 7-X (W7-X).
Gyrokinetic flux-tube simulations were performed with
the GENE code [6]. Collisions and electromagnetic effects
were neglected. To highlight the effect of geometry on the
stability of trapped-particle modes, a realistic tokamak
equilibrium for DIII-D and a vacuum equilibrium for the
high-mirror configuration of W7-X were compared.
Tokamaks are non-maximum-J devices, since Eq. (1) is
violated for deeply trapped particles, so that significant
TEM growth rates can be expected. W7-X is not perfectly
quasi-isodynamic either; that is, the stability criterion is
not fulfilled for all orbits, but the fraction of trapped particles
experiencing an unfavorable curvature is much
smaller than in DIII-D. As a result, TEM activity should
therefore be reduced.
In simulations with adiabatic electrons, in which only the
ion temperature gradient scale length and the density
gradient scale length Ln were varied and thus only ion
temperature gradient (ITG) modes are expected, comparable
growth rates are found in both devices (Fig. 2). In both
machines a strong destabilization is seen with increasing
temperature gradient a/ , where a is the minor radius of
the configuration. Reduced growth rates were not
expected in W7-X, because the predicted stabilization
through the electrons is an effect of the nonadiabatic electrons.

LTi
LTi
Stellarator News -3- February 2012
If kinetic electrons are considered in the simulations, the
growth rates differ greatly, see Fig. 3. While a clear destabilization
by the electrons can be observed in DIII-D, the
growth rates in W7-X are reduced in large parts of parameter
space. Only modes existing at low temperature gradient
are somewhat destabilized. Analyzing the energy
transfer in the simulations near marginal stability shows
that the electrons draw energy from the mode and therefore
stabilize it, just as predicted from analytical theory.
The same is true for simulations in which the ion temperature
gradient was set to zero and only density and electron
temperature gradients were varied. This combination of
gradients should lead to classical TEMs. However, in W7-
X the modes are clearly driven by the ions and not the
electrons, so that the modes, even though they are located
in the magnetic wells and therefore count as trapped-particle
modes, cannot be called classical TEMs. Again, the
growth rates are significantly lower in W7-X than in DIIID.
Therefore, it can be concluded that the enhanced stability
against TEMs and other trapped-particle modes also holds
in approximately quasi-isodynamic configurations. Some
of the next steps will be to investigate whether this resilience
still prevails in nonlinear simulations and in simulations
covering the entire flux surface rather than just flux
tubes.
References
[1] M.N. Rosenbluth, Phys. Fluids 11, 869 (1968).
[2] A.A. Subbotin, M.I. Mikhailov, V.D. Shafranov, M.Yu.
Isaev, J. Nührenberg, C. Nührenberg, R. Zille, V.V.
Nemov, S.V. Kasilov, V.M. Kalyuzhnyj and W.A.
Cooper, Nucl. Fusion 46, 921 (2006).
[3] J.H.E. Proll, P. Helander, J.W. Connor and G.G. Plunk,
Phys. Rev. Lett. 108, 245002 (2012).
[4] P. Helander, C.D. Beidler, T.M. Bird, M. Drevlak, Y.
Feng, R. Hatzky, F. Jenko, R. Kleiber, J.H.E. Proll, Yu.
Turkin and P. Xanthopoulos, Plasma Phys. Control. Fusion
54, 124009 (2012).
[5] J. Cary and S. Shasharina, Phys. Plasmas 4, 3323
(1997).
[6] F. Jenko, W. Dorland, M. Kotschenreuther and B.N.
Rogers, Phys. Plasmas 7, 1904 (2000).
J.H.E. Proll,1 P. Helander,1 P. Xanthopoulos,1G.G. Plunk,1,2 and
J.W. Connor3,4
1Max Planck Institute for Plasma Physics, EURATOM Association,
Wendelsteinstr.1, 17491 Greifswald, Germany
2Max-Planck-Princeton Research Center for Plasma Physics
3Culham Centre for Fusion Energy, Abingdon OX143DB, United
Kingdom
4Imperial College of Science, Technology and Medicine, London
SW72BZ, United Kingdom
Fig. 2. Growth rates for ITG modes with adiabatic electrons
in DIII-D and W7-X.
Fig. 3. Growth rates for ITG modes with kinetic electrons in
DIII-D and W7-X.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.5 1 1.5 2 2.5 3
γ[cs/a]
a/Ln
DIII-D, kyρ=0.6
Wendelstein 7-X, β= 0%, kyρ=1.3
a/LT=0.0
a/LT=1.0
a/LT=2.0
a/LT=3.0
a/LT=4.0
a/LT=5.0
a/LT=6.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.5 1 1.5 2 2.5 3
γ[cs/a]
a/Ln
DIII-D, kyρ=0.5
Wendelstein 7-X, β= 0%, kyρ=0.8
a/LT=0.0
a/LT=1.0
a/LT=2.0
a/LT=3.0
a/LT=4.0
a/LT=5.0
a/LT=6.0
Fig. 4. Growth rates for TEMs in DIII-D and W7-X.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.5 1 1.5 2 2.5 3
γ[cs/a]
a/Ln
DIII-D, kyρ=1.5
Wendelstein 7-X, β= 0%, kyρ=1.6
a/LT=0.0
a/LT=1.0
a/LT=2.0
a/LT=3.0
Stellarator News -4- February 2012
The 12th Coordinated Working
Group Meeting
The 12th Coordinated Working Group Meeting
(CWGM12) was held on 20 September 2013 following the
Joint 19th International Stellarator-Heliotron Workshop/
16th IEA-RFP workshop in Padova, Italy. This was a very
brief meeting (1.5 hour) after the adjournment of the
workshops. Nevertheless, more than 25 participants from
6 nations attended. In this very short meeting, we focused
on progress since CWGM11 (March 2013 at CIEMAT)
and future joint activities on some topics. A report of
CWGM11 is available in Stellarator News, June 2013.
The materials presented at the 12th CWGM are available
at http://ishcdb.nifs.ac.jp/ and http://fusionwiki.ciemat.es/
wiki/Coordinated_Working_Group (search for CWGM12)
for those of you having further interest. Below, you will
find a brief summary of the meeting.
The Inter-Machine Validation Study on transport models
progressed enough for the contents of the joint presentation
(IAEA-FEC 2012 by A. Dinklage, IPP) published in
Nuclear Fusion (2013). Joint activities have been performed
on LHD, TJ-II, and W7-AS (using materials from
published literature). Discharges with comparable ion and
electron temperature in medium to high density (say, ~ 5 ×
1019 m3) have been gathered to form “step-ladder” data
sets approaching the reactor-relevant collisionality regime.
Complete sets of equilibrium (VMEC) and measured profiles
(density, temperatures, and radial electric field) have
been prepared to calculate, as the first phase, neoclassical
energy diffusion properties using benchmarked (verified)
numerical codes. This has been summarized in the
Nuclear Fusion paper. As an extension of this joint activity,
a systematic study has been performed to investigate
nonlocal features of neoclassical transport using codes
such as FORTEC-3D (see the ISHW2013 poster by S.
Satake, NIFS) to analyze this data set (in relation to flows
and viscosity). Gyrokinetic simulations are also anticipated
by systematically utilizing this data set to simulate
validation activity. This data set will be registered in the
International Stellarator-Heliotron Profile Database to
facilitate common use. Further joint experiments in LHD
using increased electron cyclotron heating (ECH) power
are being planned. It was pointed out by D. Lopez Bruna
(CIEMAT) that we should facilitate study of particle transport
issues in addition to ongoing energy transport issues.
Collaborations on flows and viscosity have been promoted
in terms of neoclassical viscosity analysis using
numerical codes (mainly FORTEC-3D). The biasing
experiment in LHD has been numerically investigated and
has successfully predicted a relevant biasing voltage for
transition. In such a way, experimental validation of the
numerical codes has progressed. FORTEC-3D is now in
preparation to be transitioned to “open source.” It has
already been transferred to CIEMAT (V.L.Velasco) and
used for direct comparison with DKES results (nonlocal–
local). It will be also transferred to HSX (October 2013) to
investigate neoclassical viscosity in plasmas with a high
poloidal Mach number. Other possible collaborations in
progress or under consideration are:
 IPP (potential asymmetry on a flux surface, high-Z
impurity transport by EUTERP, J. M. Garcia-
Regana),
 PPPL (J.K. Park), Heliotron J (H-J)/TU-Heliac (biasing
experiment), and
 JAEA (RMP effects on JT-60/JT-60SA, M. Honda).
As you can see, various viscosity verification and validation
activities are proceeding.
Collaborations have been successfully developed among
TJ-II, LHD, and H-J on Alfvèn eigen (AE) modes/energetic
particles. Contents of the joint presentation (AE
modes in low-shear helical plasmas, IAEA-FEC 2012, S.
Yamamoto, Kyoto U.) are anticipated to be published.
Recent highlight topics are the observed effect of ECH/
ECCD on AE control in TJ-II (Nuclear Fusion paper, K.
Nagaoka, NIFS) and H-J. Mechanisms have not yet been
clarified, and theoretical explanations will be explored by
analyzing information from experiments (D.A. Spong,
ORNL). An experiment in this regard will be also performed
in the coming LHD experimental campaign with
participation of E. Ascasibar (CIEMAT). There were also
discussions on the strong interaction of this topic with the
ITPA Energetic Particle (EP) Topical Group (especially on
EP-7: ECH effect on AEs). The next ITPA-EP meeting
will be held at CIEMAT in April, and contributions from
CWGM are anticipated. Anomalous transport of energetic
particles by MHD instabilities is also of common concern
among LHD, H-J and TJ-II.
Linking to ITPA is also a major area of CWGM outreach
to the wider fusion community. As introduced at
CWGM11, the organizational process of the Steady State
Operations Coordination Group (SSOCG), co-chaired by
T. Mutoh (NIFS) and G. Sips (chair of ITPA Integrated
Operation Scenario Topical Group), has progressed by
creation of related International Energy Agency (IEA)
Implementing Agreements with national laboratories. It
has formulated seven work packages for coordinated
actions; one of which (#7) is “A draft roadmap for developing
steady state operation,” to which stellarator-heliotrons
certainly should contribute. Proposals for this issue
are anticipated from the S-H community. It was also
pointed out that it is odd not to have topics such as divertor
operation included in the seven packages. This comment
was to be transmitted to the next SSOCG meeting (to be
held in Fukuoka, October, 2013).
Stellarator News -5- February 2012
Miscellaneous
It was pointed out by K. Ida (NIFS) that, during the joint
ISH-RFP workshop, it was recognized that magnetic
topology (e.g., stochasticity, magnetic islands) affects
impurity transport, and systematic understanding of this
should be an urgent issue. M. Kobayashi (NIFS) proposed
to lead an international collaboration on this issue via the
EMC3/EIRENE code, based on discussions with groups
such as DIII-D (E.A. Unterberg) and TEXTOR (O.
Schmitz) during the week. S. Satake (NIFS) also proposed
to facilitate studies of the impurity transport issue in core
plasmas with the FORTEC-3D code, by utilizing collaboration
with EUTERP. Progress on impurity transport
issues resulting from these joint activities is foreseen in
coming CWGMs.
Finally, a data viewer named “Myview” is under development
at NIFS. It was introduced and demonstrated by K.
Ida as a tool to facilitate joint experiments on LHD.
Details will be available soon. Visitors coming for LHD
experiments are cordially invited to test Myview and submit
comments for improvement. Also, Sam Lazerson
(PPPL) mentioned that he has developed a utility to convert
VMEC output files to the old v.6.90 format. Once
compiled, it will accept any VMEC output supported by
the LIBSTELL package it was compiled with. These kinds
of data handling/numerical tools should facilitate our joint
activities.
Next Meeting
The 13th CWGM will be held at the Uji Campus (Heliotron
J site) of Kyoto University on 26 (Wed)–28 (Fri),
February 2014. Those who are interested in participating,
please contact K. Nagasaki: nagasaki@ iae.kyoto-u.ac.jp
or M. Yokoyama: yokoyama@ LHD.nifs.ac.jp.
Acknowledgements
We are deeply indebted to Dr. D. Terranova (Consorzio
RFX, Italy) and local organizing committee members for
the use of the auditorium after adjournment of the joint
workshop. The 12th CWGM is partly supported by NIFS
(National Institute for Fusion Science)/NINS (National
Institutes of Natural Sciences) under the project, “Promotion
of the International Collaborative Research Network
Formation,” and a grant-in-aid from Future Energy Association
(Kyoto).
M. Yokoyama (NIFS) on behalf of all participants in the 12th
CWGM