J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)534–537
000–000

Quasi-isodynamic Configuration with Small Number of Periods
SAMITOV Marat, COOPER Wilfred1, ISAEV Maxim, MIKHAILOV Mikhail, NÜHRENBERG Jürgen2,
SHAFRANOV Vitaliy, SUBBOTIN Alexey, YAMAZAKI Kozo3 and ZILLE Regine2
Russian Research Centre “Kurchatov Institute”, 123182, Moscow, Russia
1
Centre de Recherches en Physique des Plasmas, Association
Euratom-Confédération Suisse, Ecole Polytechnique Fédérale de Lausanne, PPB, 1015 Lausanne, Switzerland
2Max-Planck-Institut für Plasmaphysik, IPP-EURATOM Association, Germany3
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 13 March 2004)

Abstract
The possibility to fulfill the quasi-isodynamicity condition [1] for all reflected particles for moderate (N=6) [2]and high (N=9) [3] numbers of periods was shown earlier by numerical optimization. It was demonstrated that thefulfillment of the quasi-isodynamicity condition leads to significant improvement of long-time collisionless fast particle
confinement. In the present report, the same task is considered for a small number of periods, N=2. It is shown thatfor sufficiently high three-dimensionality of the magnetic axis (deviation of the magnetic axis from a plane), it ispossible to satisfy the quasi-isodynamicity condition along with the Mercier- and resistive-mode stability criteria.
Keywords:
stellarator, numerical optimization, poloidal pseudo-symmetry, quasi-isodynamicity,
collisionless fast particle confinement

1. Introduction

calculations, the JMC code [6] for the transition to magnetic(Boozer) coordinates and the MCT code [7] for direct
calculation of particle drift orbits following the sameprocedure as that considered early for the N=6 and N=9
configurations.
The optimization was performed within the class ofconfigurations with so-called “stellarator symmetry”, forwhich, in Fourier space, the coordinates R and Z of cylindricalcoordinate system R,Z,φ can be represented as sums of cosine(for R) and sine (for Z ) terms. Also, it was assumed that thelongitudinal magnetic field on the magnetic axis has only one
minimum and one maximum within one period. In this caseit follows from the condition of quasi-isodynamicity that themagnetic axis is a spatial (non-planar) curve. The initialgeometry of the boundary magnetic surface was prescribedin the same manner as for the configurations consideredearlier: the cross-sections were chosen to be triangular in theregion with minimal B on the magnetic axis and bean-shapedin the region with maximal B.
Compact configurations have some peculiarities incomparison with systems with larger number of periods.
Firstly, because of the small number of periods the gaps in
the rotational transform between low-order resonances arenarrower. Here, from the possible gaps (4/10 – 4/9), (4/9 – 4/8) and (4/8 – 4/7) the last one was chosen, so that theAs has been shown earlier (see, e.g., Refs. [1-3]),stellarator configurations with poloidal direction of thecontours of the magnetic field strength B on magnetic surfaces
can be optimized with respect to poloidal pseudosymmetry[4] and quasi-isodynamicity [1], so that for all reflectedparticles, the second adiabatic invariant J = ∫ V|| dl becomes afunction of the flux label s of the magnetic surface to a very
good approximation. Approaching these conditions guarantees
good long-time collisionless confinement of fast particles. It
was shown also that for moderate (N=6) and high (N=9)
numbers of periods, the requirement of good confinement offast particles is well compatible with the stability conditions.
It is interesting to study the possibility to realize such kind of
configurations for compact systems with smaller number ofperiods. Here some preliminary results for a two-period
configuration are presented. The configuration considered has<β > = 2< p>/< B2 > ≈ 0.03(< ... > means the averaging overplasma volume, p is the plasma pressure) and is assumed tohave zero net toroidal current.

2. Choice of the initial configuration andthe target functions
The optimization for N=2 case was performed on theNEC SX-5 super-computers himiko (Germany) and prometeo(Switzerland) with the VMEC code [5] for equilibrium©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: samitov@nfi.kiae.ru1534

Samitov M. et al., Quasi-isodynamic Configuration with Small Number of Periods

rotational transform was required to be in the region 0.5 < ι< 0.57 for all plasma radii. In this case there are no resonant
magnetic surfaces inside the plasma column with poloidal
numbers m ≤ 10. Secondly, the transition to a small numberof periods and small aspect ratio leads to an increase of thecurvature of the magnetic axis. That means that the poloidal
variation of the magnetic field strength increases, so that tofulfill the pseudo-symmetry condition the mirror component
in the field strength must be increased, too. This increase
leads to a widening of the Fourier spectrum of B, that makes
the VMEC calculations and the optimization procedure itself
more difficult.
The value of < β > was chosen as a compromise: low< β > does not create the radial dependence of < J >θ necessary
for improvement of fast particle confinement, while a large
value of < β > leads to problems with convergence andstability. In the present report a configuration with < β > ≈0.03 is considered.
In addition to the requirement mentioned above on therotational transform, further requirements such as pseudosymmetry, Mercier- and resistive-mode stability andrestrictions on the magnetic axis shift were imposed asnonlinear conditions. As a target function the measure ofquasi-isodynamicity was used. This measure was constructedin such a way that optimization leads to an increase of thearea inside the closed contours of the second adiabatic
invariant for a sufficiently large set of pitch angles.

profiles of the rotational transform, magnetic well and plasmapressure. The high elongation of the cross-section at half aperiod is partly due to the high slope of the magnetic axis
(the cross-sections are shown for φ =constant).
Fig. 3 shows the contours of B on the magnetic surface
that corresponds to 1/2 of the small plasma radius (1/4 of thetoroidal flux). As in previously found quasi-isodynamic
configurations [2,3], the curvature of the magnetic axis in theregions of the extrema of B on the magnetic axis is nearlyzero. Fig. 4 shows the contours of the second adiabatic
invariant for a set of different values of Bref , the value of B
from which the trapped particle is reflected, or for different
pitch angles, starting from deeply trapped particles (top left,Bref = Bmin + ΔB/5) to barely reflected particles (bottom right,
Bref = Bmin + 4ΔB/5). Here ΔB = Bmax – Bmin with Bmax , Bmin
the maximal and minimal values of B on 1/2 of the minor
plasma radius. It is seen that for the inner part of the plasma
column the contours of J are closed inside the plasma for all
values of pitch angles. The quality of the reflected particle
confinement in the optimized configuration was verified by

3. Results of the numerical optimization
The optimization of the N=2 configuration wasperformed for < β > = 2.7%. A 3D view of the boundary
magnetic surface of the optimized configuration is shown inFig. 1. The configuration obtained is similar to a figure-eightstellarator. The ratio Bmax /Bmin is quite large here, Bmax /Bmin ≈3.5. Another characteristic feature of the configuration foundis a large vertical excursion of the magnetic axis, so that inspite of a small toroidal aspect ratio, R/a0 ≈ 3.6 (here R, a0
are averaged large and small plasma radii), the ratio L /2πa0
is about 6 with L the magnetic axis length. The boundary
magnetic surface cross-sections at the beginning, one quarterof and half a period are shown in Fig. 2, as well as radial

Fig. 1

Boundary magnetic surface of the optimized
configuration.

2535

Fig. 2Magnetic surface cross-sections at the beginning,
quarter of and half a period and radial profiles of therotational transform, magnetic well and plasmapressure.

Fig. 3

Contours of B on the magnetic surface with s = 0.25 forthe optimized configuration.

Samitov M. et al., Quasi-isodynamic Configuration with Small Number of Periods

Fig. 4J contours for increasing values of B ref in polar—
coordinate representation √s , θ with s the flux label.

Fig. 5Particle drift trajectory for the N = 2 configuration. Inthe additional frames single bananas are shown.

direct calculation of the drift particle orbits. Fig. 5 shows anexample of the projection of the reflected α-particle trajectory
for a fusion-plant-sized machine (B0 = 5T, V = 1000 m3). The
history of losses of 1000 α-particles born at a half of plasma
minor radius is shown in Fig. 6. In addition, the results of thesame calculations are shown for B0 = 3 T. It is seen thatdiminishing the magnetic field strength does not lead to astrong increase of losses.
Fig. 7 shows that the Mercier and the resistive modes
are stable for all plasma radii for < β > = 2.7%.
The configuration described above was examined withrespect to local ballooning-mode stability. It was found thatthese modes are unstable for a/a0 > 0.3. Some attempts weremade to stabilize the ballooning modes. It was found that toimprove the ballooning mode stability it is necessary toincrease the vertical excursion of the magnetic axis, i.e. tomake the configuration less compact. The modified

Fig. 6

Collisionless α-particle confinement in the optimized
configuration as a function of the time of flight.
Particles are started at 1/2 of the plasma radius. Normalizations: plasma volume 1000 m3, B0 = 5 T (circles)
and B0 = 3 T (squares).

Fig. 7

Mercier- and resistive-mode stability vs. normalized
flux label.

configuration obtained requires further optimization ofparticle confinement. The possibility to stabilize Mercier
modes for larger < β > was investigated too. It was shown thatfor < β > up to 5% Mercier and resistive modes can bestabilized, with retaining fast particle confinement. Someattempts were made to decrease the ratio R = Bmax /Bmin. It
was found that R can be diminished from 3.5 to 3.0 without
violation of confinement.
The configuration found is not fully optimized. However,it demonstrates that the method of the optimization consideredhitherto can be applied not only to configurations with highnumber of periods but to compact systems as well.

4. Conclusions
By computational optimization, it is shown that compactconfigurations with poloidal direction of the contours of B
exist in which the confinement of fast particles can be3536

Samitov M. et al., Quasi-isodynamic Configuration with Small Number of Periods

References

significantly improved by approaching pseudo-symmetry andquasi-isodynamicity. Also, it is shown that the condition ofimproved confinement is well compatible with magnetic wellcreation and Mercier- and resistive-mode stability requirements. The optimization towards the ballooning-mode
stability, diminishing of the neoclassical effective ripple andbootstrap current are still under investigation.

[1] S. Gori, W. Lotz and J. Nührenberg, Theory of Fusion
Plasmas (Intern. School of Plasma Physics) Bologna: SIF(1996) p. 335.
[2] M.I. Mikhailov, V.D. Shafranov, A.A. Subbotin et al.,Nucl. Fusion 42, L23 (2002).
[3] V.D. Shafranov, W.A. Cooper, M.Yu. Isaev et al., 30th
EPS Conference, St.-Petersburg, Russia, 2003.
[4] M.I. Mikhailov, W.A. Cooper, M.Yu. Isaev et al., Theoryof Fusion Plasmas (Intern. School of Plasma Physics)
Bologna: SIF (1998) p. 185.
[5] S.P. Hirshman and O. Betancourt, J. Comput. Physics 96,99 (1991).
[6] J. Nührenberg and R. Zille, Theory of Fusion Plasmas(Varenna 1987), Editrice Compositori, Bologna (1988)
p. 3.
[7] R.H. Fowler, J.A. Rome and J.F. Lyon, Phys. Fluids 28,
338 (1985).

Acknowledgments
This work was supported by INTAS Grant No 99-00592,
by the Russian-Germany agreement WTZ-RUS-01/581, byRussian Federation President program on support of leading
scientific schools, Grant No 2024.2003.2, by the Russian
Fund for Basic Research, Grant No 03-02-16768, by theFonds National Suisse de la Recherche Scientifique, Euratom.

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