Confinement of Fast Ions in the Presence of the Radial Electric Field inWendelstein-line Stellarators
Ya.I. Kolesnichenko¤, V.V. Lutsenko¤, A.V. Tykhyy¤, A. Weller†, A. Werner†,
H. Wobig† and J. Geiger†
¤Institute for Nuclear Research, Prospekt Nauky 47, 03680 Kyiv, Ukraine
†Max-Planck-Institut für Plasmaphysik, IPP-EURATOM Association, D-17489 Greifswald, Germany
Abstract. Effects of the radial electric field, Er, on the confinement of locally trapped and transitioning fast ions in the
Wendelstein-line stellarators are investigated. In particular, it is shown that when the electric-field-induced drift compensates
the particle precessional motion, the particles escape from the plasma, which can take place for Er > 0; this may facilitate
the ash removal in reactor plasmas. On the other hand, the negative electric field, even being localized in a narrow region,
improves the fast-ion confinement. This effect can be especially important for the injected ions in Wendelstein 7-X.
Keywords: Stellarator, fast ions, particle confinement, radial electric field
PACS: 52.50.Gj, 52.55.Hc, 52.65.Cc, 02.60.Cb
Fast ions of various energies and pitch-angles will be present in plasmas of theWendelstein 7-X (W7-X) stellarator 
and a Helias reactor . The sources of fast ions in W7-X are the Neutral Beam Injection (NBI), Ion Cyclotron
Resonance Heating (ICRH), and Negative-ion-based Neutral Beam Injection (NNBI) at the later stage of operation,
which will provide the plasma heating. In addition, it is planned to install an NBI injector for the diagnostics. Some
energetic ions will be produced due to fusion reactions, mainly due to beam-plasma interaction. On the other hand, an
important source of fast ions in reactor plasmas, in particular, in a Helias reactor will be a thermonuclear reaction.
The basic idea to provide good confinement of the trapped fast ions in the Wendelstein-line stellarators is to
achieve a sufficiently high b (the ratio of the plasma pressure to the magnetic field pressure), b » 5%. At high b the
plasma diamagnetism “kills” unclosed superbanana orbits by making contours of the longitudinal adiabatic invariant
vkdl) closed and weakly deflecting from the magnetic flux surfaces. On the other hand, the radial electric field,
Er, also affects the particle confinement. However, in contrast to the plasma diamagnetism, the electric field effects
are important only for particles with the energy less than a certain magnitude determined from the condition that the
electric drift velocity is about or more than the magnetic drift velocity. In addition, electric field effects depend on the
sign of the electric field.
The purpose of this work is to study effects of the electric field on the confinement of trapped fast ions in the
RESONANCE AND NON-RESONANCE EFFECTS OF THE ELECTRIC FIELD
We start from a qualitative analysis based on a simple approximation of the magnetic field, B, in W7-X and Helias
reactor. We write A number of simplifying assumptions were used in the analysis above. Nevertheless, direct numerical calculations
of the particle orbits in realistic equilibria of W7-X and Helias reactor confirmed our conclusions, see Figs. 1-3. These
calculations made by the code ORBIS (Orbits in Stellarators) solving guiding center equations (non-averaged) of the
The obtained results are relevant to Wendelstein-7X and a Helias reactor. They can be summarized as follows:
(i) The presence of the negative electric field tends to improve the trapped ion confinement. The particles are
confined by the electric field when their energy does not exceed a certain magnitude. An electric field localized in
a ring region can play the role of a transport barrier for the energetic ions.
(ii) The positive electric field deteriorates the confinement of trapped ions, unless the magnitude of the electric field
is very large. A detrimental influence of the positive electric field is especially strong when it leads to a rigid plasma
rotation with the frequency that satisfies a certain condition, which we refer to as a resonance condition. The resonance
rotation frequency, Wres
E , is a function of the particle energy, E . When WE = Wres
E , well-trapped particles with a certain
energy escape from the plasma. The resonance can be rather wide, i.e., the range of rotation frequencies, DWE around
E for which particles with the given energy are not confined can be large. When this is the case, the effect of the
E-field induced loss of the fast ions is robust, i.e., the effect exists for Wres
E which arbitrary varies with radius around
E in a certain range DWres
(iii) The confinement of the transitioning particles is also affected by the radial electric field (mainly at the stage
when these particles are in the locally trapped state).
(iv) A positive electric field satisfying the resonance condition for ions with the energy in the range T ¿E ¿ 3.5
MeV will remove partly thermalized trapped a-particles (ash) from the plasma in the Helias reactor. This helps to
solve the problem of ash removal in the Helias reactor.
1. G. Grieger and the W7-X Team, J. Plasma Fusion Res. SERIES 1, 53 (1998).
2. J. Kisslinger et al., 1998 Fusion energy Proc. 17th Int. Conf. (Yokohama, 1998) vol 4, (Vienna: IAEA) 1239 (1999).
3. K. Itoh K, S-I. Itoh, Plasma Phys. Contr. Fusion 38, 1 (1996).
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