Proceedings of ITC/ISHW2007

Searching for a flux-expansion divertor in TJ-II
Francisco Castejón1,2) , Antonio López-Fraguas1) , Alfonso Tarancón2,3) and José L. Velasco2,3)1)

2)

Laboratorio Nacional de Fusión-Asociación Euratom/Ciemat, Madrid 28040, Spain
Instituto de Biocomputación y Fı́sica de Sistemas Complejos, Universidad de Zaragoza, Zaragoza 50009, Spain
3)
Departamento de Fı́sica Teórica, Universidad de Zaragoza, Zaragoza 50009, Spain
October 11, 2007

In this work, we explore the possibility of having a flux-expansion divertor in TJ-II. A 3D map of theparticle flux has been obtained for two different plasma regimes using the code ISDEP, which follows the ion
guiding-centre trajectories. In TJ-II, one must consider the particle trajectories rather than the field lines due tothe fact that common ion orbits can separate from the field lines, and moreover the plasma electric field and thecollisionality must be considered. We have chosen a configuration that presents flux expansion at given toroidal
positions. We have estimated the heat and particle fluxes and checked that it is possible to reduce them strongly
by intersecting the trajectories at a given zone of the space. Future studies, maybe including the creation of anergodic zone, will determine the strategy for intercepting such trajectories.
Keywords: flux expansion, divertor, ion kinetic transport, Montecarlo code.

1

Introduction and motivation.

The quest for the stellarator reactor needs a robust divertor concept to guarantee low plasma-wall interaction andpower exhaust [1]. A good divertor concept should reduce the particle and heat fluxes on its divertor. Additionally, the path of the recycled neutrals that will enter
the plasma must be large in the real space, and the plasma
profiles should ideally present a steep pressure gradient inthe edge. These facts ensure that they cannot go beyond
the plasma edge. The tokamak divertor, based on locating one or two X points inside the vacuum chamber, hasbeen demonstrated as a good solution in such devices. It
has been recently suggested that this kind of effects canalso be reached in tokamaks locating the X point outside
the vacuum chamber (see [2] and references therein). Thistransforms the divertor configuration into a limiter configuration with flux expansion.
The divertor programme in stellarators needs to consider awider range of concepts due to the diverse possible configurations. To start with, LHD presents the helical divertor
concept. It is based in a natural ergodic zone of its magnetic configuration that rotates with the same law as thehelical coils of the device [3].
The island-based divertor is a promising concept, as has
been demonstrated in W7-AS [4], where excellent results
have been obtained. This concept is suitable for devices
like W7-X, which has a fixed robust magnetic configuration. For this concept to work, it is necessary that the island positions and widths do not change substantially during plasma operation. The same can be said about the helical divertor: its topology must remain unchanged during
plasma operation. This fact makes island and helical diauthor’s e-mail: francisco.castejon@ciemat.esvertors not appropriated for devices that rely their configuration on the bootstrap current, like QPS or NCSX, or for
devices that present high flexibility in their rotational transform values, like TJ-II. For these cases, the flux expansion
concept [5] could be a good candidate for the divertor. This
concept is based on intercepting the particle and energy
fluxes with plates in an ergodic area of the plasma where
the magnetic lines are well separated, so that the power flux
onto the plates is small enough and the resulting neutrals
and impurities can be pumped. The large flux expansion
should also guarantee that the neutrals entering the plasma
have to perform a large path, thus diminishing the probability that they go deep inside the device core.
TJ-II presents specific plasma-wall interaction issues. Dueto its magnetic configuration, the groove is the preferredzone for the escaping particles to strike. Since the grooveis physically close to the centre of the device, one should
try to diminish such flux by intersecting the particle trajectories far from that position.
We have found several magnetic configurations that aresuitable for such a divertor concept, since they have plasma
zones where the density of magnetic surfaces is especiallylow. The point is to look for a position in which the efficiency of the divertor is maximum (i.e. intersects a large
fraction of the flux) and to try to make compatible this requirement with a low enough heat flux on the plates.
No natural ergodic zones appear outside the last closed flux
surface in TJ-II. Therefore, a second phase of this work
may imply the creation of such ergodic zone by introducing extra coils that create a resonant magnetic field.

Proceedings of ITC/ISHW2007

2.2

ECHNBIn [ 1019-3m ]

1.8
1.4
1
0.6
0.2
1Fig. 2 Plasma toroidal sections. The surfaces in the ends lay in
φ = 0 (left) and φ = π2 (right).

Te [ keV ]

0.8
0.6

tions are taken from experimental measurements: the density and electron temperature are obtained from ThomsonScattering measurements [9], the ion temperature from theCX-NPA diagnostic [10], and the electrostatic potential
comes from HIBP measurements [11].
Obviously, the quantitative results will depend on plasma
characteristics, namely the collisionality and the electrostatic potential. Nevertheless, we expect our results on thedivertor effect to hold in a range of plasma parameters. In
order to check this, two characteristic (and very different)
plasma regimes have been chosen in this work. The plasma
parameters are shown in Fig. 1 as a function of the effective
radius ρ: a low density, low collisionality ECRH plasma,
which presents a positive electric field in the core, according to the electron root [12], and an NBI plasma, with highdensity and low electron temperature. The electrostatic potential is negative in this case, which will have strong consequences on the ion confinement.

0.40.20.12

Ti [ keV ]0.11

0.1

0.09
0.10Φ [kV]

-0.1
-0.2
-0.3
-0.4
00.2

0.4

0.6

0.8

1

ρ3

The chosen configuration.

Fig. 1 Radial plasma profiles in the two regimes.

2ISDEP code and the chosen plasma
conditions.

Previous calculations performed to explore the flux expansion concept divertor in NCSX [5] followed magnetic field
line trajectories, including a diffusion coefficient of about
1 ms−2 , of the order of the one experimentally measured.
This approach happens to be valid for devices where theparticle trajectories do not separate very much from thefield lines. This is not fulfilled in TJ-II [6], and the particle
trajectories must be followed to estimate the fluxes.
ISDEP (Integrator of Stochastic Differential Equations
in Plasmas) [7], is a Montecarlo code that follows ion
guiding-centre trajectories considering a given electrostatic potential profile and ion-ion as well as ion-electron
Coulomb collisions [8]. This code has been used in thepresent work to study the ion collisional flux properties
in the chosen magnetic configuration and various plasma
regimes. The plasma parameters used in our simula-

One of the main properties of TJ-II heliac is its flexibility.
By changing the currents that circulate by the two coils of
the central conductor, it is possible to change the plasma
size and shape as well as the rotational transform (TJ-II isan almost shearless device).
After studying several magnetic configurations, we havechosen the 100 68 91 (the numbers stand for the currents
that circulate by the coils), which presents a large flux expansion in given toroidal positions. The average minor radius is a = 0.2 m and the rotational transform in the edge is1.825. In this magnetic configuration, similarly to the majority of TJ-II ones, a large fraction of the particle fluxes
strike the groove, as will be shown in Section 4. Therefore,
the neutrals coming from the wall appear very close to theplasma bulk. Hence, the main goal of this divertor should
be to diminish as much as possible the fluxes that are directed to the groove. Fig. 2 shows several Poincaré mapsof the field lines of the chosen configuration (the sections
have been rotated for a better comparison). It is possible toappreciate that the maximum flux expansion happens fora toroidal angle around φ = π4 (and of course also aroundProceedings of ITC/ISHW2007
0.40 < θ < π/810.30.2

fraction intercepted ECH

0.8

0.1

0-0.1

-0.2

π/8 < θ < π/4π/4 < θ < 3π/83π/8 < θ < π/2

0.5 1 1.5 2
ϕ/π

0.5 1 1.5 2ϕ/π

0.5 1 1.5 2
ϕ/π

ρ>0.8
ρ>0.9
ρ>1.0
ρ>1.1

0.6

0.4

0.2

-0.3

0-0.3

-0.2

-0.1

00.1

0.2

0.3

0.5 1 1.5 2
ϕ/π

0.4Fig. 3 Plasma coronas at φ ≈ π6 and sketch of the wall of thevacuum chamber.

7π
φ = 3π4 , 5π/4 and 4 ). Fortunately, the zone with large
flux expansion lays on a wide range around these angles.
This gives us quite a lot freedom in our optimizing task.
Fig. 3 shows a particular toroidal section, together with asketch of the wall of the vacuum chamber. The distancefrom the magnetic axis to the groove is about 12 cm, andonly a thin sheet separates the edge and the plasma bulk.
If the plasma-wall interaction can be concentrated at thezone where the flux expansion is maximum, the particle
flux onto the groove will be strongly reduced. The amountof neutrals that enter the plasma bulk will also decrease.

Fig. 4 Proportion of ions intersected as a function of the angular
position of the plate for the ECH plasma.

0 < θ < π/8

0.8

0.6

Design and results.

π/4 < θ < 3π/8

3π/8 < θ < π/2

0.5 1 1.5 2

0.5 1 1.5 2

0.5 1 1.5 2

ρ>0.8
ρ>0.9
ρ>1.0
ρ>1.1

0.4

0.2

04

π/8 < θ < π/41fraction intercepted NBI-0.4-0.4

0.5 1 1.5 2ϕ/πϕ/πϕ/πϕ/πFig. 5 Same as Fig. 4 for the NBI plasma.

In order to search for the optimal position of the plates, wehave performed a map of the ion flux on several magnetic
surfaces and at different toroidal and poloidal angles. Since
TJ-II is a four-field-period device, we will center our discussion in one of the periods, but will consider all of themin calculation. We have accomplished this task defining
our plates as the locus of points such that:ρ > ρ0 ,2π2π2π2πi < φ < (i + 1),j < θ < ( j + 1).(1)NφNφNθNθ

By setting Nφ = 4 (four φ intervals in each period) andNθ = 32, we have 128 plates defined. An sketch of one
ensemble of them will be shown in Fig. 6. We follow a
number of trajectories and study the individual effect ofeach plate on the particle flux. More precisely, in Fig. 4we show, for the ECH plasma, the fraction of the trajectories that would be intercepted by each plate in the casewhere this plate were the only one in our device. Note thatthe contributions of two plates cannot be directly added,
since they may partly shadow each other. The structuresin Fig. 4 suggest the positions where a plate can be more
effective. We will be interested in plates in the outer regionof the plasma, where ρ > 1.0, for an acceptable interaction
with the hot plasma. In such radial positions, θ ≈ 3π
2 forπ3π3ππ4 < φ < 8 and 8 < φ < 2 looks promising, since 10% ofparticles would be intercepted by each of them. Considering the mirror images of these plates in the other three periods (altough, as we know, their contributions do not simplyadd up), one could expect to concentrate a great proportionof the plasma-wall interaction in these plates. Note that ourchoice is not the optimal intercepting particles, but it is thebest that makes it far from the groove.
Fig. 5 shows the same quantity for the case of the NBI
plasma. Here, the radial electric field clearly improves theconfinement. One of the consequences is that each ion has
more probability of being intercepted by each plate, since it
performs more toroidal turns around TJ-II. Our former proposal of divertor configuration still seems one of the best
possible. These are good news, since one would desire adivertor design valid for a wide range of plasma parameters. Looking at both figures, our first tentative design will
23π
be plates located at ρ > 1.0, 11π
8 < θ < 16 along all thetoroidal angle. This configuration is sketched in Fig. 6.
In the ECH case, our plates intercept about half the particles in the plasma. This includes ions that end their trajectories in the groove of and ions that do not. The originalProceedings of ITC/ISHW2007
theless, from about 50% of ions colliding with the groove,
one reduces it to about 35%.

5Fig. 6 Tentative design of plates. The 0.9 < ρ < 1.0 surfaces areplotted in black, and the plates in red.

0 < θ < π/8angular collision distribution ECH

0.3

π/8 < θ < π/4π/4 < θ < 3π/83π/8 < θ < π/2

0.5 1 1.5 2

0.5 1 1.5 2

0.5 1 1.5 2

orig.
div.

0.25
0.2
0.15
0.1
0.05
00.5 1 1.5 2ϕ/πϕ/πϕ/πϕ/πFig. 7 Angular distribution of the collisions with the vacuum
chamber and the plates for the ECH plasma with andwithout divertor.

and the modified angular distribution of trajectory ends areshown in Fig. 7. The high original peaks correspond tocollisions with the groove, which in usual operation represent around 60% of the collisions with the vaccum chamber. The effect of our plates is to diminish this quantity
to be around 35%, about half the proportion existing before. New peaks appear, corresponding to the location ofour plates. This means that we have concentrated a greatpart of the plasma-wall interaction on the divertor plates.
In the case of the NBI plasma, see Fig. 8, the same effectstill exists, although a bit weaker. The proportion of thetotal trajectories that are intercepted is about 63%. Never-0 < θ < π/8angular collision distribution NBI

0.3π/8 < θ < π/4π/4 < θ < 3π/8

3π/8 < θ < π/2

0.5 1 1.5 20.5 1 1.5 20.5 1 1.5 2

orig.
div.

0.25
0.2
0.15
0.1
0.05
00.5 1 1.5 2ϕ/π

ϕ/π

ϕ/π

Fig. 8 Same as Fig. 7 for the NBI plasma.

ϕ/πConclusions and future work.

We have found a promising configuration for having a flux
expansion divertor, since the particle collisional flux maps
are characterised by presenting strong poloidal asymmetries, showing a high value in the poloidal position corresponding to one of the extremes of the ”bean”.
This magnetic configuration has the property that the flux
expansion is maximum in a region where the main part ofthe particle flux that goes onto the groove passes through.
Therefore, one may minimize at the same time the flux
onto the groove and onto the plates. Due to the TJ-II configuration characteristics, this effect is especially beneficial because we move the main plasma wall interaction toa zone much farther from the plasma bulk.
The beneficial effect of the divertor is larger in the NBI
regime despite of the fact that this case presents a smaller
mean free path, because the structure of electrostatic potential ensure that a large fraction of particles is intercepted.
The next step of this work is to optimize the design of thedivertor plates in order that the interrupted particle flux on
the groove is maximum. It may be also necessary to create
an ergodic zone in order to minimize the particle and heat
flux on the plates. Before assessing the feasibility of thisconstruction, new flux calculations with the ergodic zone
are mandatory. For this last phase of calculation, the effect of turbulence on particle trajectories must be included
in order to have more refined calculations, especially inhigh-beta plasmas. Once the flux on the divertor plate isestimated, including the field ergodization effect, and before starting the engineering assessment of the coils andthe divertor plates, it will be necessary to calculate the outgasing coming from the plate. A 2D neutral transport codeis available and will be applied to this case. Experiments
are also foreseen in TJ-II to benchmark all these theoreticalresults.
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