Proceedings of ITC/ISHW2007

Transport Modeling for W7-X on the Basis of W7-AS
Experimental Results
C.D. Beidler, J. Geiger, H. Maaßberg, N.B. Marushchenko, Yu. Turkin, W7-AS Team
Max-Planck Institut für Plasmaphysik
IPP-EURATOM Association, Greifswald Germany
October 12, 2007
Exploratory simulations of plasma confinement in the Wendelstein 7-X (W7-X) stellarator are presented,
concentrating on scenarios which simultaneously achieve high temperature and high β (normalized plasmapressure) at the full magnetic field of B = 2.5 T. Efficient 1-D transport and ray-tracing codes are combined toprovide an approximately self-consistent description of the heating and current drive (ECCD) to be expectedfrom up to 10 MW of electron cyclotron resonance heating (ECRH) in W7-X. Best performance is exhibited inhigh density (n = 1.8 × 1020 m−3 ) simulations heated at the second harmonic of the ordinary mode, althoughcontrol of the magnetic topology at the plasma edge — needed to insure functioning of the island divertor —becomes problematic due to the imbalance of the bootstrap current and ECCD.
Keywords: Wendelstein 7-X, neoclassical transport, bootstrap current, electron cyclotron current drive.
The design specifications of the Wendelstein 7-X(W7-X) stellarator were chosen so as to enable this device to demonstrate the reactor potential of the advanced
stellarator concept. The ultimate goal of the experimental
program is thus to heat, confine and exhaust plasmas withreactor-relevant β and collisionality values under steadystate conditions. This goal must be viewed in its entirety,
and it is therefore mandatory to avoid the common experimental expedient of breaking it up into a number of selfexclusive portions (e.g. performing “high-β” experiments
at small values of the magnetic field or “low-collisionality”
experiments by reducing the density). Currently, there exists no single numerical tool capable of simulating all aspects of such a multifaceted problem but exploratory investigations have been carried out to determine the expected
range of W7-X plasma parameters assuming electron cyclotron resonance heating (ECRH) and the prospects formagnetic configuration control using the accompanying
current drive (ECCD). The results of these investigations
are the subject of this paper.
The basis for these investigations is provided by thetheoretical interpretation of experimental results from theW7-AS device, which operated in Garching, Germany,from 1988 until 2002. Three observations are of principal
importance for the simulations carried out here:(1) With the exception of the low-temperature edge
region, high-performance W7-AS discharges (with central
densities n(0) > 5 × 1019 m−3 and central ion and electron temperatures T i,e (0) > 1 keV) were described wellby the predictions of stellarator neoclassical theory. This
was true for all quantities of interest including the radial
particle fluxes, energy fluxes and ambipolar electric fieldauthor’s e-mail: crb@ipp.mpg.de[1] as well as the bootstrap current [2, 3]. For the radiallosses, this may be attributed to the strong temperature dependence of the neoclassical transport coefficients in theso-called 1/ν regime, which scale
∝

7/2
e3/2
ff T

nB20 R20

where e f f is the effective helical ripple (a measure of how
well a stellarator’s magnetic field confines trapped particles), B0 is the magnitude of the magnetic field and R0 is
the major radius of the torus. For W7-AS, a device withmoderate values of e f f , this led to neoclassical energy diffusivities χ > 1 m2 /s when T > 1.5 keV at n = 1020 m−3 .
Although W7-X was optimized to have much smaller values of e f f , the temperature dependence of this scaling remains unchanged and justifies the continued relevance ofneoclassical theory for describing high-temperature bulk
plasmas in W7-X.
(2) The low-temperature periphery of W7-AS discharges exhibited anomalously large radial transport, considerably in excess of neoclassical expectations. Assuming
this anomalous transport to be purely diffusive, a data setof experimental profiles for ECRH discharges was used tosolve the local energy balance in the gradient region and to
determine the dependence of the energy diffusivity on local
plasma parameters using linear regression; results foundthat χ was inversely proportional to density and independent of temperature [4]. This result is noteworthy as thedensity scaling is identical to that of 1/ν transport so thata simple “combination” of these anomalous and neoclassical results would predict a global energy confinement timeobeying τE ∝ n (assuming n is varied while holding its
profile shape constant). Experimental results for such adensity scan are shown in Fig. 1 for two different W7-AS

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Proceedings of ITC/ISHW2007

Fig. 1 Energy confinement times for 1.2 MW ECRH discharges
in two W7-AS configurations for different central densities compared to the expectations of ISS04. A linear
scaling of τE with n is shown by the red line.

magnetic configurations heated by 1.2 MW ECRH. A linear scaling of τE with density is indicated by the red line
and describes the results considerably better than the n0.54
dependence of the International Stellarator Scaling ISS04[5] which is shown by the dashed line for comparison.
(3) A linear description of ECCD using the adjoint
approach with a Spitzer function modified to account fortrapped-particle effects provided an excellent descriptionof electron cyclotron current drive experiments in W7-AS
at densities of 6 × 1019 m−3 and larger [3]. As a consequence, high-density ECRH simulations in W7-X can accurately model both power deposition and ECCD using the
ray-tracing technique without the need to resort to lengthy
solutions of the Fokker-Planck equation.
At the moment, simulations are restricted to a handful of fixed W7-X equilibria which are calculated using
VMEC [6] for specified pressure profiles. Modificationsof the equilibrium caused by internal current densities (dueto ECCD and the bootstrap current) or changes in the pressure profile are not accounted for. For each equilibrium themono-energetic neoclassical transport coefficients are determined using the Drift Kinetic Equation Solver (DKES)[7] for the relevant range of electric field, E r , and collisionality values on several magnetic flux surfaces (typicallyentailing more than 3000 individual DKES calculations);routines are also provided for interpolation of intermediate
values in this three-dimensional space and to perform theenergy convolutions necessary to calculate the neoclassical transport matrix. With the specification of a model forthe anomalous energy diffusivity (based on W7-AS results,
χan = 1019 n−1 m2 /s is taken here) one has sufficient information to solve a system of four coupled diffusion equations for the radial profiles of T e , T i , Er and the toroidal
plasma current using a 1-D transport code [8]; the density
profile is held fixed for these simulations so that a modelfor the anomalous particle diffusion coefficient is not re-quired. Heating of the plasma by ECRH and calculationsof the associated ECCD are simulated with the ray-tracing
code TRAVIS [9]. Through direct coupling with the transport code, temperature effects on the ECRH deposition andthe current-drive efficiency are fully accounted for. Radiation losses have been ignored for these simulations andimpurities are only accounted for insofar as an effective
charge number Ze f f = 1.5 is assumed to exist throughout
the plasma.
The ECRH system of W7-X will consist of ten 1 MW
gyrotrons, designed for thirty minutes of continuous wave
operation at a frequency of 140 GHz (resonant at B =2.5 T) [10]. Operation at the second harmonic of the extraordinary mode (X2) offers an extremely flexible meansof heating the plasma due to its excellent first-pass absorption, allowing a wide choice of launch angles and thus
the ability to vary the ECCD over a broad range of values in both the co- and counter-directions (by convention,
current in the co-direction increases the rotational transform, ι , and decreases it when flowing in the counterdirection). A limitation is the X2 density cut-off which
occurs at ne = 1.2 × 1020 m−3 , roughly a factor of twoless than possible if operation is switched to the second
harmonic of the ordinary mode (O2). Unfortunately, firstpass absorption of O2 ECRH is relatively poor for the projected plasma temperatures in W7-X, forcing the adoptionof a multi-pass absorption scheme with a fixed launchinggeometry to conform to the placement of mirrors and reflecting surfaces within the vacuum vessel. This launching geometry has been chosen to provide equal ECCD in
the co- and counter-directions so that full-power operation
implies doing without any net driven current. This drawback is not as serious as it might seem, however, since thecurrent-drive efficiency falls rapidly with increasing density, regardless.
High-density O2 operation in W7-X is potentially ofcritical importance for optimum divertor performance inthe device. Preliminary simulations with the 3-D edge
package EMC3/EIRENE [11] have shown that pumping of
neutral gas in the W7-X divertor geometry becomes efficient at edge densities of at least 3×1019 m−3 [12]; a similarvalue of the edge density was found to provide the impurity
screening effect observed experimentally in W7-AS HDHdischarges. Ratios of central to edge density in W7-AS
were seldom less than five making it appear questionable
whether such high edge densities can be reached in W7X with X2 ECRH. Heating with O2 was not attempted in
W7-AS but is the most attractive option for high-density
operation in the early W7-X campaigns as neutral-beaminjection heating will be limited to ten-second pulses during this time.
The requirement of sufficient ECCD to balance thebootstrap current is also a consequence of the divertor concept chosen for W7-X. This makes use of the naturally occurring magnetic islands with ι = 5/m at the plasma edge
(configurations with m = 4, 5 or 6 may be realized with

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Proceedings of ITC/ISHW2007

Fig. 2 Radial profiles of the toroidal mirror harmonic (left), the fraction of trapped particles (center) and the effective helical ripple (right)
for the W7-X high-mirror (blue), standard (red) and low-mirror (green) configurations.

the W7-X coil system) to guide the outflowing plasma into
prepared divertor regions where recycling and exhaust is
to take place. For this “island divertor” concept to be effective the magnetic topology of the edge region must be
maintained throughout the discharge, which is equivalent
to preserving the original value of the rotational transformat the edge. This is possible to sufficient accuracy for net
plasma currents of the order ±10 kA.

Simulation Results
The simulations presented here were performed for threedifferent W7-X configurations with the currents in themodular coils chosen to either increase or decrease themagnitude of the toroidal mirror term in B relative to a
standard case. (Having the designation b01 in the poloidal-,toroidal-angle Fourier decomposition of B, the toroidal
mirror is also commonly referred to as the “bumpiness”
of the magnetic field strength.) Radial profiles of relevant quantities for the vacuum fields are plotted in Fig. 2.
The high-mirror configuration (blue curves) has the largest
fraction of trapped particles ( ft ) at all radii which generally helps to reduce the bootstrap current while the lowmirror configuration (green curves) provides the opposite
extreme. With regard to confinement, however, the standard configuration (red curves) has the smallest valuesof e f f over most of the plasma radius and these values
are further reduced by finite-β effects, especially near theplasma axis. With its much larger e f f , the high-mirror
configuration can be expected to have the poorest confinement of these three. The fraction of trapped particles isalso of relevance to the current-drive efficiency with smallvalues of ft most conducive for maximizing ECCD. It will
be noted that the simultaneous goals of minimum bootstrap
current, maximum confinement and maximum ECCD arecontradictory and that the simulations are therefore also ameans to search for the best compromise.
A first set of simulations was carried out to determine
whether compensation of the bootstrap current by ECCD is

possible in all W7-X configurations assuming X2 heating
at a density value close to cut-off. The bootstrap current
flows in the co-direction in these simulations so that thelaunch angles of the ECRH system were chosen to maximize counter-ECCD. For significant counter-current near
the axis the rotational transform goes to zero in the vicinity,
implying a loss of confinement in this region. Indeed, forsimilar discharges in W7-AS, this loss of confinement was
clearly indicated by flat temperature profiles throughout
the ι ≈ 0 portion of the plasma [3]. Such an effect is notaccounted for in the present simulations as self-consistentalterations to the equilibrium are not attempted (the original equilibrium with ι (0) > 5/6 is maintained during
the simulation). To avoid this difficulty, off-axis heating
was also considered; an example of the results for the< β >= 2% equilibrium of the W7-X standard configuration
with 10 MW ECRH is presented in Fig. 3. The shift of the
ECCD current density away from the plasma axis (green
curve labeled Jcd in the lower center) conforms closely tothe deposition profile (lower left). The total driven current
in this simulation, IECCD = −88 kA, is more than sufficient to compensate the bootstrap current of Ibs = 75 kA as
evinced by the slight reduction of the edge rotational transform relative to the current-free VMEC equilibrium value(compare black and green ι curves on the lower right).
The significant reduction of the rotational transform overa large portion of the plasma is the price one pays for using ECCD to compensate the bootstrap current (due to theinherent mismatch of the current-density profiles); nevertheless, ι remains well above zero for off-axis deposition.
The simulation results yield < β >= 2.70%, in reasonably
good agreement with the assumed equilibrium.
The impact which the magnetic configuration has on
the results can be illustrated by considering a similar offaxis X2 simulation for the high-mirror W7-X. As shownin Fig. 4, the bootstrap current density assumes small values over the entire plasma radius due to the approximate
cancellation of the ion and electron contributions. The negeis due to the lower temperatures reached in this
ative Jbs

Proceedings of ITC/ISHW2007
C.D. Beidler, et al.

Transport Modeling for W7-X

Off-Axis X2 for1% W7-X Standard Configurationne [10 20 /m3]ne

T e, T i [keV]= 0.721vol0.75

ne 4.5= 0.834

-5

0.5

3
W= 2.726MJ
1.5
tau E= 0.273s
0
0
0.15

-10

line

0.25
0
0

2.4

0.15

0.3
r [m]

0.45

PECRH [MW/m3]

0.3
r [m]

-2 .10 5-4 .10 5

0.6-6 .10 5
0.15

0.3
r [m]

0.45

0

0.15

0.3
r [m]

-20
0

0.45

Pecrh =09.99MW

1.2

0
0

-15

jcd jebs jibs jbs [A/m2]

1.8

Er [kV/m]

0

0.45

0.15

0.3
r [m]

0.45

ι, ιcf, ιbse

I bs = 66.36kA
0.9
Iibs = 8.99kA
0.75
Ibs = 75.34kA
0.6
Icd= -87.63kA
0.45
0
0.15

0.3
r [m]

0.45

Fig. 3 Simulation results for 10 MW off-axis X2 ECRH in the < β >= 2% standard configuration. Density, temperature and radial electricfield profiles are plotted in the upper row; power deposition, current densities and rotational transform are given in the lower row.
ei
In the temperature plot, T e is shown in red, T i in blue. In the current density plot, Jcd is given in green, Jbs
in red, Jbs
in blue andthe total bootstrap current density in black. In the rotational transform plot, the current-free (VMEC equilibrium) ι is shown ingreen, the alteration of this profile due to the bootstrap current appear in red and the rotational transform profile accounting for all
currents is given in black.
C.D. Beidler, et al.

Transport Modeling for W7-X

Off-Axis X2 for1% W7-X Standard Configurationne [10 20 /m3]ne

T e, T i [keV]= 0.721

vol4

0.75

ne

3= 0.834

-4

0.5

2
W= 2.726MJ
1
tau E= 0.273s
0
0
0.15

-8

0.25
0
0

3.2

0.15

0.3
r [m]

0.45

-12
0.3
r [m]

.

0

0.8

-10
0.15

0.3
r [m]

0.45

4

0

0.15

0.15

0.3
r [m]

0.45

ιcf,ιbse

I bs = -4.17kA
0.96
Iibs = 9.75kA
0.93
Ibs = 5.58kA
0.9

4

2 10= 9.99MW
Pecrh
10 4

1.6

-16
0

0.45

jcd jebs jibs jbs [A/m2]

2.4

0
0

line

PECRH [MW/m3]

Er [kV/m]

0

0.3
r [m]

0.45

0.87
0

0.15

0.3
r [m]

Fig. 4 As in Fig. 3 for 10 MW off-axis X2 ECRH in the vacuum high-mirror configuration.

0.45

Proceedings of ITC/ISHW2007

simulation, shifting the collisionality to higher values
where the helical components of the magnetic field are predominant and can reverse the sign of the bootstrap current
coefficients. IECCD as large as −48 kA can be achievedin the high-mirror configuration but is not required forthis simulation as the residual bootstrap current is only
Ibs = 6 kA, which has little effect on the rotational transform profile (red curve on the lower right). The poorer
confinement also reduces the normalized plasma pressure
to < β >= 2.40%.
A summary of the X2 results is provided in Fig. 5
which depicts the values of Ibs , IECCD and the plasma
stored energy, W, for on-axis and off-axis heating simulations as a function of the axis value of b01 for the equilibrium. From largest to smallest value of b01 , these equilibria are for the high-mirror vacuum, standard vacuum,
standard < β >= 2%, standard < β >= 4% and low-mirror
vacuum configurations. The toroidal mirror is reduced byincreasing plasma pressure; as already noted, this is expected to improve confinement in the standard configuration which is indeed verified by the increase in W. At thesame time, however, both Ibs and IECCD are significantly
increased. Also as expected, confinement is poorest in thehigh-mirror configuration. The stored energy is nearly independent of the deposition profile so that off-axis heatingmust be favored due to its more benign effects on the rotational transform profile.
For O2 operation current drive can only be realized bydoing without a portion of the ECRH power. To maximize
the counter-ECCD, the five gyrotrons which would provide
co-ECCD are therefore not utilized, limiting the heating
power to 5 MW. Additionally, O2 absorption is a strong
function of the electron temperature so that O2 ECRH is
only possible when a high-T e “target” plasma is already inexistence before O2 heating is begun. This is the basis forthe series of simulations which are summarized in Fig. 6,
which were performed in the < β >= 2% standard configuration. Here X2 heating is assumed at low to moderate densities but with the identical launching geometry necessary
for O2 operation; as in the previous X2 simulations, I ECCD
exceeds Ibs at all densities with their sum ≈ −12 kA at
n = 1020 m−3 (note that −IECCD has been plotted in the figure). For O2 operation at this density, however, the ECCD
is considerably reduced with its magnitude falling belowthat of the bootstrap current which is only slightly affected
through the differing ECRH deposition profiles of X2 andO2. At higher densities, the “gap” between the two currents closes but is probably initially too great to expect anadequate degree of configuration control during a simple
density ramp.
For scenarios where ECCD compensation of the bootstrap current is not possible, there remains the option offactoring the resulting change in the edge value of the rotational transform into the experimental plans. For example,
a vacuum magnetic configuration could be chosen with an

Ibs [kA]

150
120
90
60

on-axis BS
on-axis CD
off-axis BS
off-axis CD

30
0

0

0.03

0.06
b01

0.09

W [MJ]
4
3
2
1
0

on-axisoff-axis
0

0.03

0.06
b01

0.09

Fig. 5 Values of the bootstrap current and ECRH driven current(top) and plasma stored energy (bottom) as functions of
the axis value of the toroidal mirror for X2 simulations
with on-axis and off-axis deposition.

Fig. 6 Values of the bootstrap current and ECRH driven current
for 5 MW ECRH simulations in the < β >= 2% standard
configuration. The X2 simulations were performed with
the O2 launching geometry.

C.D. Beidler, et al.

Transport Modeling for W7-X

Off-Axis X2 for
1.8%Proceedings
W7-X Standard
Configurationof ITC/ISHW2007

ne [10 20 /m3]

T e, T i [keV]

6

1.35

4.5

-4.5

0.9

3

-9

0.45

1.5

-13.5

0
0

0.15

0.3
r [m]

0
0

0.45

PECRH [MW/m3]
6

0.15

0.3
r [m]

0.45

jebs jibs jbs [A/m2].

1.3 10

1.5

6.5 .10 4

0
0

0
0

0.15

0.3
r [m]

0.45

0.15

0.3
r [m]

0.45

ιcf,ιbse

Iibs= 18.82kA
0.99

5

3

0

I1.045= 63.45kA
bs

1.95 .10 5

4.5

Er [kV/m]

0

Ibs= 82.26kA
0.935
0.88

0.15

0.3
r [m]

0.45

0

0.15

0.3
r [m]

0.45

Fig. 7 As in Figure 3 for 10 MW on-axis O2 ECRH in the < β >= 4% standard configuration.

edge ι reduced by an amount which is then to be provided
by the net current of the planned discharge. Such a scenario poses additional complications since one must provide co-ECCD at the initial stage to provide the “missing”
bootstrap current and be able to adjust the ECCD duringlater stages according to the changes in Ibs . This is plausible using X2 heating and offers the only opportunity for
subsequently continuing with the discharge, transferring
the full 10 MW of ECRH into the O2 mode. At the moment, such simulations are beyond the capabilities of theapproach used here, but it is possible to determine the magnitude of the bootstrap current in the steady-state portionof such a discharge. Results for 10 MW on-axis O2 ECRH
assuming a central density of 1.8 × 1020 m−3 are given in
Fig. 7 for the < β >= 4% standard configuration. The total
bootstrap current of 82 kA determined for this simulationand its rather peaked current density profile lead to the alteration of the ι profile shown in red on the lower right.
It is expected that a more monotonic profile of ι can berealized by shifting the deposition off axis, although thisis likely to result in a modest deterioration of the 97.5%
three-pass absorption for the O2 heating in this simulationand in the < β >= 4.13% value which on-axis deposition
makes possible.
From the preceding discussion it is clear that timedependent discharge scenarios will be high on the priority
list of future tasks. Near-term plans also include determining free-boundary VMEC equilibria with the pressure profiles and internal current densities calculated here so as toenable modeling of the magnetic field topology in the (assumed) vacuum region outside the last-closed-flux-surface.
This will enable one to reach a considerably more detailed
verdict concerning the compatibility of edge topology with
the island divertor than was possible in the present workwhere only the edge value of ι was considered.
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