Proceedings of ITC/ISHW2007

Stability analysis of Wendelstein 7-X configurations
with increased mirror ratio
Joachim Geiger
Max-Planck-Institut für Plasmaphysik, IPP-EURATOM Association, Greifswald, Germany
Wendelstein 7-X has been optimized with respect to equilibrium, stability and neoclassical transport properties.
The resulting coil system offers a large variety of configurations. The paper summarizes the interchange stability
characteristics of the reference configurations and extends them towards configurations with very large toroidal
mirror fields. Two classes of configurations are found. In the first one stabilizing properties of shear and magnetic
well of the vacuum configurations are decreased, with the effect that extreme cases are found to be Mercierunstable for all investigated -values less than 6% with no indication of stabilization by the finite- magnetic wellformation. The second class exhibits a strong additional suppression of the parallel current densities, a factor 3larger than the reference configurations.
Keywords: stellarator, equilibrium, stability, Wendelstein 7-X, configuration variation

1. Introduction
The magnetic coil system of Wendelstein 7-X is theresult of an optimization procedure concerned with thephysics properties of a magnetic configuration important
for a fusion reactor, such as equilibrium, stability,
neoclassical transport and -particle confinement [1]. The
optimization led to the HELIcal-axis-Advanced-Stellarator
configuration (HELIAS) with 5 field periods with reduced
equilibrium and bootstrap currents. The good -particle
confinement requested a toroidal field mirror of 10% to bepresent in the configuration which was also beneficial for afurther reduction of the bootstrap current. The mirror ratio
is defined as (B( =0o)-B( =36o))/(B( =0o)-B( =36o)),where =0o is the bean-shaped plane in the corners where
the toroidal curvature is largest. To avoid major resonances
a rather flat profile of the rotational transform with a central value above 5/6 and below 1 at the boundary waschosen. A vacuum magnetic well of 1% together with thefinite shear in the outer half of the plasma provides
stability up to < >-values of 5%. For experimental realization, the configuration was cast in a set of 10 modular coils
per period for the optimized configuration. To increase theexperimental flexibility and to explore the configuration
space around the optimization point, the coil currents in themodular coil system can be adjusted independently. Thus,
the toroidal field mirror can be varied to influence the size,
location and behavior of the trapped particle population.
An additional set of 4 planar coils per period, which areslightly inclined, provide the possibility to adjust therotational transform and the plasma position. A number ofreference cases have been proposed already at the beginning of the project [2], with slight modifications later [3],E-mail: joachim.geiger@ipp.mpg.deto show the fundamental flexibility in the value of therotational transform (low- (5/6), standard (5/5), high(5/4)) and its shear (low-shear), the toroidal mirror term(high- (10%) and low-mirror (0%) compared to standard(5%)), the plasma position (in- and outward shifted) andthe plasma volume (limiter). Note, that the optimization
point in this set is the high-mirror configuration. The
standard configuration got its name because it is created by
the modular coil system only, with all coils carrying thesame current. However, up to now there was no systematicinvestigation of the large configuration space offered by
the W7-X coil system. Only limited work has been donefor an -variation of the high-mirror case with respect tostability [4], or for neoclassical transport in a more general
way for optimized HELIAS-type configurations [5].
This paper starts a broader investigation of the configuration space by investigating the stability behavior of configurations with very large toroidal mirror field. In the beginning the important vacuum field properties of the reference cases are summarized and the implications for interchange stability are shown. Based on these results, the region of the configuration space defined by very large mirror ratios (ca. 20%) is explored with respect to ideal andresistive interchange stability. If not otherwise stated, thepressure profiles used are linear in the normalized toroidal
flux s (p ~ (1-s)). The free-boundary 3D-equilibria werecalculated with VMEC2000 [6]. For the analysis of thelocal interchange stability, the JMC-code was used [7].

2. Reference cases
The reference cases can be viewed as basic variationsof the standard configuration in three directions of theProceedings of ITC/ISHW2007

configuration space: , mirror-ratio and plasma position.
The low-shear and the limiter case can be interpreted asconfigurations resulting from combinations of these 3 basicvariations. Except for the high- and low- cases, all reference cases are located at a boundary- value just below the5/5-resonance. Figs 1 and 2 show shear and vacuum magnetic well depth for the different configurations since these
are important quantities in the local interchange stabilitycriteria. It can be seen that the shear increases and the mag-Fig.4: Parallel current density suppression for reference cases. A classical stellarator has the value 2.

Fig. 1: Shear variation of reference cases.

Fig.2: Vacuum magnetic well for reference cases.

Fig.3: Summary of vacuum configuration changes
on shear and magnetic well (black arrows), and on
parallel current densities suppression (red arrows).
Interesting regions are marked with symbols(stars/crosses).
E-mail: joachim.geiger@ipp.mpg.denetic well deepens with , with inward-shifting and with alower mirror-ratio. Note, that with respect toinward/outward shifting the effect on the magnetic well is
opposite to the situation in LHD where outward shifting
deepens the magnetic well. For the reference cases, themirror ratio affects the shear only slightly, but it affects thevacuum magnetic well depth in a way similar toinward/outward shifting. The joint effect can be seen whenlooking at the profiles of the low-shear configuration
which is an outward-shifted configuration with anincreased mirror field. This is summarized in Fig.3, whereby following the arrows one moves towards configurations
less stabilized by shear or vacuum magnetic well.
An analysis with respect to the ideal and resistive
interchange criteria show that all configurations except thelow-shear configuration are Mercier stable at all
investigated < -values (up to 5-6%). The finite shear atouter radii stabilizes the ideal interchange modes in thecases where the derivative of V is marginal or positive atthese radii, e.g. low- and high-mirror cases. These show
the development of a resistive interchange unstable region
at the boundary for already low < >-values (below 1%).
Another figure of merit to assess the configuration
properties is the ratio of the average of the square ofparallel to perpendicular current density (see [7] for its
definition). Since the parallel current density is inversely
proportional to , the dependency can be removed bymultiplying the ratio by 2. For a classical stellarator thisfigure of merit can be estimated to have the value 2. The
suppression of the parallel current density by optimizationis then obvious by a reduction of the value. Fig. 4 showsthat the values for the reference cases are reduced byroughly a factor of 4. Note, that within the reference cases,
the ratio falls with increasing and mirror-ratio and alsowith inward-shifting. The latter gives rise to a different
behavior than the one observed for shear and vacuum
magnetic well.

Proceedings of ITC/ISHW2007

Fig.5: Shear for mirror ratio scan. Reference cases
are indicated with names.

3. Cases with very large toroidal mirror
In a first step the mirror ratio was scanned fromslightly inverted (ca -3%) to values twice as large as thatof the high-mirror reference case (ca 23%) without using
the planar coils, thus keeping the boundary value of therotational transform roughly at 5/5. To perform the scan,
the two coil currents around the triangular plane ( =36o)Fig.6: Vacuum magnetic well in mirror scan.
Reference cases are indicated with names.

Fig.7: Parallel current density suppression with mirror ratio. Reference cases are indicated with names.
E-mail: joachim.geiger@ipp.mpg.de

were varied simultaneously while the others were kept
constant. The reference cases which vary the toroidalmirror are included in the scan for comparison. Thetendency seen with mirror-ratio in the reference cases isobvious: with increasing mirror-ratio, the shear is slightly
reduced (Fig.5) and the vacuum magnetic well becomesless stabilizing, so that for the largest mirror ratio
considered here, a significant vacuum magnetic hillregion exists in the outer half of the configuration (Fig.6).
The parallel current densities show a significant suppression with increasing mirror ratio (Fig. 7). For the highest
mirror-case (ca 23%), the suppression is almost a factorof 2 larger compared to the high-mirror reference case.
The interchange stability behavior is dominated by the
reduction of the stabilizing shear and vacuum magnetic
well. With increasing mirror ratio the configurations
become susceptible to interchange modes in the outer halfwhere, as noted, for the highest mirror ratio a vacuum
magnetic hill situation develops. For the latter case, ideal
stability is still maintained at very low -values due to thefinite shear, but the outer half becomes unstable to idealinterchange already at moderate < >-values of 1%.
In a next step a variation around the configuration

Fig.8: Shear variation with planar coil currents.

Fig.9: Variation of vacuum magnetic well withplanar coil currents.

Proceedings of ITC/ISHW2007

Fig. 10: Parallel current density suppression withplanar coil current variation.
with the largest mirror (23%) was performed using theadditional planar coils, to investigate the effect of varying
and the plasma position. The effects seen in Figs 8-10
can be explained from the tendencies discussed
previously. The names used for descriptive purposes referto the center point for the variation, i.e. the case with amirror ratio of 23%. From the variation two classes ofinteresting configurations emerge. The most unstableconfigurations result from going to low(lo-iota isaround 5/6 and lo-iota-2 is below 5/6) possibly in
combination with outward-shifting. In this case, a vacuum magnetic hill region with low shear extends over theentire plasma volume. These configurations are Mercierunstable from the very beginning with no indication ofstabilization with higher -values. The other class showsa strong suppression of the parallel current density andarises when shifting inward or when going to high- . In
these configurations, the shear increases and the vacuum
magnetic well deepens. The parallel current density issuppressed by an additional factor of 3 compared to thestandard configuration of the reference cases. Althoughthese configurations are stable for very low -values, they
develop a Mercier-unstable region at the outer half of theplasma volume which grows asincreases. Not
surprisingly, the strong reduction of the parallel current
densities leads to very stiff configurations, i.e. very smallchanges in the -profile and the plasma position(Shafranov-shift) with increasing .

4. Summary and discussion
The configuration space of W7-X is explored forconfigurations with very large toroidal mirror ratios. The
properties of these configurations are the extensions of
the properties already found in the reference cases. The
exploration led to two configuration classes with large
mirror field displaying interesting equilibrium andstability properties.
E-mail: joachim.geiger@ipp.mpg.deFirst, combining low- and outward-shifting withlarge mirror creates configurations with very low shear
and a vacuum magnetic hill over the entire plasma crosssection. These configurations are unstable to interchange
modes for all -values. This offers the possibility toinvestigate the importance of interchange stability in a
low-shear device which is expected to behave differently
than in large shear devices like LHD. Second, the high- ,inward-shifted branch of the very large mirror cases
reveals configurations with a strong suppression of theparallel current densities. These configurations are lessunstable than their low- , outward-shifted counterparts.
However, they are also unstable from medium < >values onward. Nevertheless, the strong reduction of theparallel current densities leads to rather stiffer configurations with respect to finite- induced equilibrium changes
like the Shafranov-shift.
In a next step, it is planned to extend the analysis toneoclassical transport properties. Due to the large toroidal
mirror, particle and energy transport will not be improved
with respect to the standard case. However, it is known
that the bootstrap current reduces with increasing mirror
and may change sign [5]. With respect to neoclassicaltransport it might be interesting to explore the low oreven inverted mirror region of the configuration space insearch of configurations being closer to quasi-helicalsymmetry than the low-mirror configuration.

References
[1]

[2]

[3]
[4]
[5]
[6]
[7]

W. Lotz, J. Nuehrenberg and C. Schwab, 13th Int. Conf.
Plasma Physics Control. Nucl. Fus. Res., Washington, 1990
(Int. Atomic Energy Agency, vienna, 1991), Vol. 2, p.603
J. Kisslinger et al., Proc. 16th Symp. On Fusion Technology,London 1990 (North Holland Publ., Amsterdam, 1991),
Vol.2, 1520-1524
T. Andreeva, IPP-Report, IPP III/270 (2002)
C. Nuehrenberg, Phys. Plasmas 3, 2401 (1996)
H. Maassberg, W. Lotz and J. Nuehrenberg, Phys. Fluids B
5, 3728-3736 (1993)
S. Hirshman, W. van Rij and P. Merkel, Comput. Phys.
Comm. 43, 143-155 (1986)
J. Nuehrenberg and R. Zille, in Theory of Fusion Plasmas
Varenna 1987 (Editrice Compositori, Bologna, 1988), Vol.
EUR 11336 EN, p.3