Proceedings of ITC/ISHW2007

Effect of rotational transform and magnetic shear on
confinement of stellarators
E. Ascasíbar(1), D. López-Bruna(1), F. Castejón(1), V.I. Vargas(1), V. Tribaldos(1), H. Maassberg(2), C.D.
Beidler(2) R. Brakel(2), A. Dinklage(2) J. Geiger(2), J.H. Harris(3,4), A. Kus(2) T. Mizuuchi(5), S.
Murakami(5), S. Okamura(6), R. Preuss(2), F. Sano(5), U. Stroth(7), Y. Suzuki(6), J. Talmadge(8), K.Y.
Watanabe(6), H. Yamada(6), M. Yokoyama(6)(1) CIEMAT, Madrid, Spain(2) Max-Planck-Institut für Plasmaphysik, EURATOM Association, Greifswald, Germany(3) Oak Ridge National Laboratory, USA(4) Australian National University, Canberra, Australia(5) Kyoto University, Kyoto, Japan(6) National Institute for Fusion Science,322-6 Oroshi-cho, Toki 509-5292, Japan(7) Institut für Plasmaforschung, Universität Stuttgart, Germany(8) University of Wisconsin, USA
ABSTRACT: This papers studies the effect of rotational transform and magnetic shear on the confinement of low
shear stellarators.
Keywords: confinement, rotational transform, magnetic shear

1. Introduction
The fact that the confining field in stellarators is largely
independent of the plasma itself provides anextraordinary freedom of design and is the reason behindthe variety of configurations existing or underconstruction [ 1]. A significant degree of freedom is thecapability of choosing different values and profiles ofrotational transform, " . Historically, two approaches
have been followed to prevent the expected confinement
deterioration due to the presence of low orderresonances. The first one is to design machines with# profiles (extremely low magnetic shear, sˆ ),very flat " !
trying to avoid resonances (for example, theWendelstein family [ 2, 3], TJ-II [4], HSX [ 5], Heliotron-J[6]). W7-X, the large stellarator appearing in the horizon! follows the low shear approach and its design! has been
optimized to have a weak dependence of " on theplasma pressure. The second approach is to look forstrongly varying " # profiles that do not avoid theresonances but force the islands to shrink and prevent!
them to overlap (for example, Heliotron-E[7], L2 [ 8],91011ATF [ ], CHS [ ], LHD [ ]).
After !
those experiments rotational transform andmagnetic shear remain as two important issues whoserole must be considered in future designs of candidates
for a stellarator reactor.
The extensive inter-machine 0-D global scaling studies
performed to date do not yield a clear picture about therole of " on confinement, as will be discussed later.
Additional 1-D information contained in the temperatureand density profiles should be considered as well. Local!

transport analysis can provide valuable information on
the role played by the resonances and help inunderstanding the influence of rotational transform on
confinement. This work points in this direction, within
the framework of the International Stellarator/Heliotron
Confinement DataBase (ISHCDB) under auspices of theIEA Implementing Agreement for Cooperation in theDevelopment of the Stellarator Concept [ 12]. As a firststep, it has been found convenient starting with a survey
based only on low shear machines, which should help in
deciding further steps where large shear devices must be
included.
A summary of the results obtained so far on theinfluence of rotational transform value on global
confinement is presented in section 2. The effect of loworder resonances on confinement is discussed in section
3. The role of magnetic shear is presented in section 4.

2. Dependence of global confinement on therotational transform value
Early tokamak studies predicted that high rotationaltransform values are favourable for confinement, withexponent 0.4 [13]. In stellarators, the W7-AS group was
the first to publish experimental results on theconfinement dependence on " owing to the capability of
their device to perform " # scans. They reported ageneral improvement (energy confinement time andelectron heat diffusivity) at higher rotational transform
values provided!that distinct optimum confinement
windows close !
to low order resonances (1/3 and 1/2)
were chosen [ 14, 15].

Proceedings of ITC/ISHW2007

ISS95, the first extensive stellarator study based oninternational collaboration compiled a database from themain devices at that time and deduced an intermachine
scaling law, which predicted an improvement of global
confinement with increasing " [16]. However, the factthat an offset between the shear-less machines and theheliotron/torsatrons appeared in the unified scaling
required the use of an "ad-hoc" parameter to arrive at a
! This so-called "parameter s"
unified expression.
accounted for the difference in confinement between
stellarators with and without shear.
The individual results from the TJ-II flexible heliac
showed again the beneficial effect of " on the global
energy confinement and supported the W7-AS finding ina wider " range (1.2 < " 2 / 3 < 2.2) [ 17].

2

!

!

! NF42 (2002)] (f=0.60)
W7-AS !-scan [Brakel,
IotaScan
00:34:40
9/10/07
TJ-II (ISS04
data)(f=0.30)
Heliotron J (ISS04 data) (f=0.60)
ISS04 without !-dependence"ISS04 ! !

0.41

a2.28

" exp/f * " ISS04

1.5

1

0.5

0

configuration subgroup, was used [ 19, 20]. This factor was
related to specific properties of the helical field structureof each device; it appears to be correlated, for instance,with the corresponding effective helical ripple, plateau
factor and elongation. Taking into account thecorresponding configuration factors, the " -dependenceof different machines can be compared.
Fig. 1 shows, for example, the renormalized
confinement times of data subsets from W7-AS, TJ-II
and Heliotron-J [19, 21]. It !can be seen that theintermachine comparison is consistent as regards the " dependence (the data clouds are positioned over theISS04 horizontal line) but there is a saw-like finestructure within each data subset, which is a typical!to the effectfeature of low shear stellarators. It is relatedof low order rationals and influenced also by the
reduction of plasma radius due to the appearance of
natural islands at the boundary [16, 22]
The conclusion of this summary is that the search forphysical mechanisms behind the configuration factors
suggested by the ISS04 study requires a step beyond,considering detailed profile information as well as
neoclassical and turbulent transport effects. Localtransport analysis appears as an essential tool in thisprocess to understand which are the plasma regions
where the confinement is improved and to go deeper inthe physics involved in the transport reduction.

3. Effect of low order resonances on confinement
0

0.5

1

1.5

2

2.5

!/2#

Fig. 1: Comparison of the dependence ofrenormalised confinement times on " , for datasubsets of W7-AS, TJ-II and Heliotron J.
For the ISS04 revision of ISS95, new devices entered!
the confinement database: LHD, TJ-II, Heliotron-J and
HSX. Particularly important was the contribution from
LHD, the largest device, which extended the parameter
regime to substantially lower values of normalized
collisionality and ion gyro radius, much closer to reactor
regimes than those of the ISS95 devices [18]. Besides,
W7-AS had discovered significant high confinement
regimes with divertor operation. Thus, a new larger
database was compiled including these new results.
ISS04, the new scaling expression was derived from a
restricted set from the full database. The dependence of
global confinement on " deduced from ISS04 is again
positive, with exponent 0.4, in line with ISS95, but again
a caveat questions this result. In ISS04, the use of the old"parameter s" is not sufficient to arrive at a unified
scaling and !a new empirical configuration-descriptive
renormalization factor, fren, derived for each

Both low and high shear approaches face challenges
regarding the effect of low order resonances. Examples
are i) keeping the configuration free of resonances athigh β in low shear devices or ii) maintaining thedivertor capabilties in high shear machines. In any case,there are some open questions that need to be addressed:-Is there a threshold for the magnetic shear over which
low order resonances lose their detrimental character?
Does this threshold depend on the value of the rotationaltransform itself?
-What is the effect of resonances when the shear isabove this threshold?
-Is the sign of the shear important, or is only its
magnitude what matters?
The logic behind this type of questions is clear: Provided
that low order resonances do not deteriorate confinement
anymore in the presence of enough (perhaps very low)magnetic shear then the design constraints of future
stellarators might be relieved: a strict control of themagnetic shear (i.e. internal currents) would no longer
be necessary and the available configuration space tooptimize other physics aspects would expand.
In the process of answering the questions posed above(goal beyond this article) a first step would be tosummarize some well-established results from low shear

Proceedings of ITC/ISHW2007

evolution of the rotational transform can be estimatedconsidering the measured net current as due only to the
OH transformer and taking into account the evolving Te
profiles from the ECE diagnostic. As the plasma current
density diffuses inwards due to the effect of (Spitzer)
resistivity, it is found that the 3/2 " -value disappears
from the plasma at t ≈ 1170 ms with null shear. An
effective " e is obtained simultaneously from power
balance calculations. The results shown in Fig. 3 are aclear indication that the short! transient with large " e (orsmall "Te ) around 1170 ms seen in the experimental!data are a consequence of the " = 3/2 resonanceoccupying a large fraction of the plasma core, thus
causing a transient flattening of the Te !
profile [ 31].

devices. In the following subsections we try to survey
the major effects of the low order rationals on
confinement and transport. In section 4 we put theemphasis on the effect of magnetic shear through its
impact on the rationals.

3.1 Degraded confinement due to low orderresonances at low β
As is well known, in the absence of magnetic shear, loworder resonances placed in the confined region produce
large magnetic islands, which cause confinement
degradation. The stellarators of the Wendelstein line
have documented this result extensively [ 23, 24, 14, 25].
In Heliotron J, a clear transient degradation ofconfinement was observed around " a = 0.59-0.62 for
ECH+NBI plasmas (<β> < 0.5%) with accompanying
bursting coherent magnetic fluctuations with m~5/n=3.
On the other hand, no obvious degradation of!around " a = 0.49-0.51 despite
confinement was observed
the existence of the m~2/n=1 mode [ 26].
In conditions of high ECR heating power density andlow plasma density (≈ 0.5-07 x 1019 m-3) transientdegraded confinement!states -sometimes interspersedwith phases of improved confinement- are observed in
TJ-II in a variety of different experimental conditions,normally associated to the presence of low orderrationals. The experimentally measured characteristicsof the features associated to the observed transport
events depend on the position of the resonance [27, 28, 29,
30].

Fig. 2: Pressure profile (Thomson scattering) for three
reproducible discharges with induced OH current.
Black and blue profiles are measured at t=1125 ms. A
clear flattening is found at t=1170 ms (red) (see Fig. 3).
An example of flattening of the pressure profile in TJ-II
is shown in Fig. 2, where a Thomson Scattering profile
measured in an ECH discharge with induced OH current
to study magnetic shear effects is shown. The time!

!

Fig. 3: Time evolution of the effective " e # 1$Teprofile obtained from ECE data for a TJ-II ECRH
discharge with small induced OH current. The
evolving " # profile has its 3/2 value moving as
shown by the black/white line.
!
Note that the presence of the 3/2 resonance (white/black!line) does not really alter transport unless something in
the rotational transform profile favours a singular effectof transient nature. Despite the approximated characterof these calculations, the figure suggests that thetransient of large diffusivity at t ≈ 1170 ms, coincident
with the narrowed profiles shown in Fig. 2, is aconsequence of the " -profile having been forced by the
OH current to flatten and occupy a portion of the plasma
with very small shear. It is worthwhile noting that, after
the clearly degraded transport at t ≈ 1170 ms due to the!
3/2 resonancein no shear condition, the resonancedisappears completely from the plasma due to the larger
induced current. In spite of this, after the crash, thetransport coefficient remains larger than before it. Thisbehaviour is attributed to the fact that the shear
continues decreasing in the external part of the plasma[32].

3.2 Improved electron transport in the vicinityof low order resonances

Proceedings of ITC/ISHW2007

W7-AS results show that narrow optimum confinement
windows with smaller transport are located close to (butno at) the low order resonances. The explanation given
for this result is based on an empirical model, whichassumes that transport is always enhanced at resonantsurfaces and that this enhancement is reduced bymagnetic shear [22]. This model invokes the rarefactionof high order resonances in the immediate vicinity of
low order ones, which is expected to decrease theturbulent transport
There is abundant experimental evidence of the role of
low order resonances as triggers for different improved
transport events [28]. Transport barriers close to resonantsurfaces have been found in tokamaks [ 33, 34]. Several
stellarators like LHD and TJ-II have also found electrontransport barriers close to rational surfaces located in thecore region [ 35, 36] or in the plasma edge [ 37, 38]. In TJ-II
low collisionality plasmas, when the 3/2 resonance is in
the core, an increase of the positive radial electric field ismeasured, synchronized with the electron transportbarrier formation [35, 39]. The phenomenon is similar to
the improved heat confinement found in the neoclassicalelectron root feature in several stellarators, the so-called
CERC [ 40]. Recent experiments in TJ-II, in low
collisionality plasmas, have shown also for the first time
an increase in the central ion temperature, simultaneousto the increase of electron temperature and triggered by
the 4/2 resonant surface [ 41].
Heliotron J has reported experimental evidence ofrotational transform windows for the high quality Hmode (τexp/ f×τISS04 > 1.5) close to the low orderrationals of the vacuum rotational transform at the lastclosed flux surface (LCFS). In these windows, Langmuirprobe measurements show reduced fluctuation-induced
transport in the plasma edge region. Simultaneously, anegative radial electric field Er (or Er -shear) forms near
the LCFS at the transition. The power and density
thresholds of the H-mode are observed to depend on therational surface, but the systematic dependences between
them are not fully understood at present [20]. It might
have to do with the influence of the topology (“shape”)
of the magnetic surfaces on the poloidal viscousdamping rate [ 42, 21].

3.3 Tracking the local lowering of " e due to loworder rationals in TJ-II plasmas
In this section the results of local power balance analysis
of a series of TJ-II discharges obtained in several
magnetic configuration scans!in TJ-II are presented [ 43].
The low order rationals included in this scans havepoloidal mode number m in the range 2 < m ≤ 6.
In TJ-II, one possible way of scanning the rotationaltransform profile through the confined plasma consistsof varying in small steps (down to 0.1 kA) the current
through the helical coil, which is the one with strongest
effect over the " # value, and keeping the currents in therest of coils essentially unchanged. This procedure, in a
shot-to-shot basis, allows "sweeping" a certain low orderrational through the confined plasma region in a very!
controlledway, in a close-to-vacuum shear condition.

Fig. 4: Contour plot of the effective " e obtained from
TS data for a set of 13 magnetic configurations labelledby their helical coil current. Each profile is an averageover a few discharges with similar line density!
The results for " e are shown in Fig. 4, in the form of acontour plot. The experimental data have beeninterpolated linearly along a rotated mesh aligned withthe path of the " = 8/5 in the range 0.6 < ρ < 0.8 (see[46]!for details). The white solid and dashed lines
correspond to the path followed by the lowest orderrationals of the vacuum- " present in the scan (from left
! " = 8/5, 13/8, 18/11 and 5/3). The figure shows
to right,
a pattern of "ridges" and "grooves" whose direction
follows roughly the path of rationals along the minor
radius as Ihc! increases (i.e, as the rationals move! inwards). Therefore, the power balance analysis suggeststhat low order rationals retain heat fluxes at their radial
location. This is very likely for ρ > 0.6, where " should
be practically the vacuum one due to the small net
plasma currents typically found in these experiments. At
this respect, preliminary estimations of the bootstrap
current [ 44] indicate that there should!be a change of signnear half radius, according to which the net current
would approach zero as one moves inwards in radius up
to ρ ≈ 0.6, making the vacuum " to be even closer to theone with plasma. All these aspects require further study
and a careful estimation of internal currents. However,taking as a hypothesis that the grooves are coincident! rationals, the proximity of thewith the low ordergrooves and the location of vacuum values of the loworder rationals in Fig. 4 would be indicating that theProceedings of ITC/ISHW2007

bootstrap currents are indeed small in TJ-II ECH
discharges.

4. The role of magnetic shear
There is a robust experimental evidence showing that theconfinement degradation produced by the presence of
low order rationals in the confined plasma is restored if
enough magnetic shear is generated, no matter the originof the current (pressure driven, inductive, EC driven)[24].
An example of W7-AS is shown in Fig. 5, which showsthe effect of " a and plasma current (using the OH
transformer) on the plasma energy content, fordischarges with identical plasma radius and densities.
The strong dependence on rotational transform of the! content, at zero current, decreases as the current
energyis raised and disappears at the highest current value.

with similar density, in plasmas operated under ohmic
induction. Negative induction drives the " -values
towards more negative and conversely. Positive
induction cases do not reach high sˆ > 0 values because
the largest plasma current (≈ 10 kA) does not allow for!
further variation of the rotational transformprofile. The
results indicate that the largest " e is found around zero
shear. The confinement!is clearly improved for negative
shear. Positive shear seems also to reduce transport
although the result is less clear due to the smaller! TJ-II results are in line with the
explored range. These
W7-AS conclusion.

Fig. 5: Plasma energy content vs. " a for four valuesof induced OH current.
Local power balance analysis also provides clear!
evidence of restoration of degraded confinement via
magnetic shear in W7-AS, as illustrated in Fig. 6. It
shows degraded confinement in the no shear situation (0kA). Increasing the shear with inductively driven
current, reduces strongly the electron heat diffusivity inthe gradient region. The discharges with Ip = 10 kA and
25 kA have " = 1/2 in the plasma region without any
significant local degradation of confinement. It is clearthat confinement improves with shear independently of
the sign.
!
Dedicated
experiments in TJ-II have also allowed
studying the influence of shear on confinement [32].
Figure 7 shows the effective thermal diffusivity obtainedin ρ = 0.75 for a number of different ECRH discharges

Fig. 6: T e (upper box) and " e (middle box)
profiles for discharges with " a = 0.42 and different
values of plasma current. The corresponding " profiles are shown in the lower box.

!
!

!

negligible impact on the local " e . So, complete analysisof temperature and density profiles would be needed.

20

2

thermal diffusivity (!=0.75), m /s

Proceedings of ITC/ISHW2007

18

This
work
contributes
totheInternational! Confinement DataBase (ISHCDB)
Stellarator/Heliotron
under auspices of the IEA Implementing Agreement for
Cooperation in the Development of the Stellarator
Concept [12]

16
14
12
10
8
6
-0.1

-0.05
0
0.05
magnetic shear at !=0.75

0.1

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2

Fig.7: Thermal diffusivity at ρ = 0.75 vs. magnetic
shear for several discharges with induced OH current.

5. Summary!

- Keeping the configuration free of low order resonancesin real experimental conditions is a difficult task for
devices with low vacuum magnetic shear. The optimized
design of Wendelstein 7-X will allow it to fully explore
and characterize this scenario at high β. The fine-tuning
capability of the available modern ECRH systems
provides an additional tool, if needed, to compensate
undesired internal currents by means of localised
heating.
- Certain amount of shear allows the presence of eventhe lowest order rationals (1/2 in the W7-AS case)within the confinement region without degradation. Low
shear values, sˆ ≈ 0.1, are enough in TJ-II -which is thestellarator with highest " - to observe the beneficial
effect of the shear. Further comparisons between devices
with different " are needed in order to study whether
there! is a threshold value for the shear, which depends!
on the " # value.
-Narrow optimum confinement windows are found in! and Heliotron-J close to low order rational
W7-ASvalues. Provided a small amount of magnetic shear ispresent, the low order resonances are found to trigger avariety of improved transport events in TJ-II. Fineconfiguration scans in this machine have shown that low
order rationals retain heat fluxes at their radial location.
-Both W7-AS and TJ-II results show that the beneficial
effect of shear on confinement does not depend on thesign.
-Many of the results presented in this paper are based on
local heat balance analysis. A word of warning
concerning the ISHCDB must be considered seriously inthis respect: W7-AS has shown that also the particle
confinement can depend on " and shear [ 45]. This fact
could give rise to misleading conclusions if only thepower balance is considered because quite different Te
profiles, indicating a degraded global " E , could have a!

!