J. Plasma Fusion Res. SERIES, Vol.4 (2001) 146-152

Electric Potential Cells at the Diverted Tokamak Separatrix
SCHAFFER Michael J.*, PORTER Gary D.1, BOEDO Jose A.2, BRAY Bruce D.,
HSIEH Chung-Lih, MOYER Richard A.2, ROGNLIEN Thomas D.1,

STANGEBY Peter C.3 and WATKINS Jonathan G.a
General Atomics, P.O. Box 85608, San Diego, Califurnia 92186-5608, USAlLawrence Livermore National Laboratory, Livermore, Califurnia 94551, USA2(Jniversity of Calfornia, San Diego, Calfornia 92093-0417, USA
3University of Toronto, Toronto, ON M3H 576, Canada
aSandia National Laboratories, Albuquerque, New Mexico, USA(Received: 5 December 2000 / Accepted: 18 August 2001)

Abstract
Two-dimensional measurements by probes and Thomson scattering reveal unanticipated electric
potential and electron pressure (p") maxima near the divertor X-point in L-mode plasmas in the DIII-D
tokamak. The potential hill (-100 V) drives ExB circulation ("potential cell") of particles, energy andtoroidal momentum around the X-point and in and out across the magnetic separatrix. Modeling by the
UEDGE two-dimensional edge transport code with plasma drifts shows similar X-point potential andpressure hills. The code predicts additional drift-driven nonuniformity poloidally around the separatrix.
Potential cells in UEDGE arise from parallel (to B) viscous stress acting on the Pfirsch-Schliiter ion
return flow of the VB drift. These experimental and theoretical results demonstrate that the boundary
layer just inside the separatrix of low power tokamak plasmas can be far from poloidal uniformity. We
speculate that separatrix potential cells might be a major feature of L-mode edge transport and theirsuppression an important feature of H-mode.

Keywords:
electric potential, electric field, ExB drift, plasma flow, plasma viscosity, X-point, divertor, divertor X-

point

region, because its magnetic lines do not encircle theconfined plasma. It is beneath the X-point in Fig. l. The
region roughly from the X-point to the targets is usuallycalled "the divertor."
The X-point is the apex of four quadrant regions:privateregion, the inner and outer SOLs (at major
the

1. lntroduction
Many tokamaks divert exhaust plasma along magnetic lines to target surfaces somewhat removed fromthe main plasma. The magnetic separatrix, indicated inFig. 1, separates the interior region of plasma, confinedon closed toroidal magnetic surfaces, from the open line

radius R smaller and larger than the X-point, res-scrape-off layer (SOL), where plasma flows almost
parallel to the magnetic field B to targets, where it
deposits its energy and recombines. The X-point, apoloidal magnetic field null, defines the separatrix inconventionally diverted tokamaks. The region between
the separatrix-target strike points is called the private

pectively) and closed confinement surfaces. The X-point
region is complex, because the four plasma quadrants,

each with distinct temperature, density and electric
potential, meet there. Theoretical analysis predicts the*Corresponding author's e-mail: schaffer@fusion.gat.com

t46@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchSchaffer M.J. et al., Electic Potential Cells at rhe Diverted Tokamak Separatrix

2. Experimental Results
The experiments were performed in the DIII-D

Vacuumvesseltokamak [8]. The plasma was diverted by a single mag-outlinenetic null to the lower target. Figure I shows a typicalgeometry. The divertor plasma was attached to the target
at the outer strike point and detached at the inner, which
is typical in DIII-D.
The plasma electric potential @ was measured by apair of fast-stroking Langmuir probe ,urays, one moving
vertically through the divertor region [1 1] and the other

CER

Upstreamprobe

horizontally through the upstream SOL just below the

Separatrix

torus equatorial plane [12]. The probe stroke paths are

Target.\ Thomson
viewpoints andPrivate region
0.0

0.5

1.0

indicated in Fig.l. Two Thomson scattering systems
provided the primary measurements of Z" and electron
density n..The divertor Thomson scattering system [13]measured at eight vertically separated locations at thedivertor probe

1,5 2.0 m

same radius as the divertor probe, as shown in Fig. 1.
The upstream Thomson scattering system [14] measured

Fig. 1 Single-null diverted plasma geometry inside theDlll-D vacuum vessel outline, with the disposition

at many closely spaced points (= 13 mm separation)
vertically across the edge and SOL, as shown in Fig. l.
Plasma ion temperature I was measured near theof probe. Thomson scattering and CER

diagnostics. The 98% and 99% 14, surfaces are
drawn inside the separatrix and the 101o/o andequator by charge exchange recombination spectroscopy

102% surfaces outside.

(CER) t151. No suitable Z, diagnostic covers the X-point

in DIII-D. To obtain data in two dimensions, the Xpoint was magnetically scanned radially past theformation of a boundary layer across the separatrix toaccommodate the disparate regions [1]. Numerical

divertor diagnostics, between the extremes as shown in
Fig. 2. The X-point was positioned about 0.13 m above

solutions of the edge plasma by the UEDGE computercode [2,3] with electric potentials and magnetic gradient

the target, closer than usual, to provide good diagnostic

drifts self consistently included [4-6] indeed exhibit acoverage above the X-point. Magnetic surfaces were
calculated by the equilibrium fitting code EFIT [16].
They are labeled by their normalized poloidal magnetic
flax, tyn. W, = I is the separatrix, ry,> | is the SOL withtg, increasing outward from the separatrix, and 14" < 1 isin either the closed confinement or open private region
with rg, decreasing inward from the separatrix. Additional information about the measurements is containedin Refs. [9,10].
We present data from low power L-mode plasmas,
which display the most interesting X-point behavior.
The applied toroidal magnetic field was Br = 2.0-2.1 T
at radius R = 1.7 m; plasma current was 1o - 1.0 MA;
line average electron density was 2.5x10re m'3; andheating was 0.6 MW Ohmic and 0.3 MW neutral beam
power. Both 87 directions were employed. The plotted
data are from discharges with -B7 directed out of thepage in Fig. I ("standard" 87, ion VB drift toward the
X-point). The X-region profiles for "reversed" 87 (ionVB drift away from the X-point) are not markedly

boundary layer just inside the separatrix. These solutions
also demonstrated the existence and importance of alarge ExB plasma convection in the private region [4].
Recent measurements of the electric potential [7] in thedivertor of the DIII-D tokamak [8] confirmed the private
ExB convection in quantitative agreement with the Ref.
[4] prediction. Further measurements revealed unexpected local electric potential and electron pressure (p")

maxima near the X-point in DIII-D L-mode (lowconfinement) plasmas [9,10]. The potential hill (-100V) drives ExB circulation ("potential cell") ofparticles,
energy and toroidal momentum around the X-point andin and out across the magnetic separatrix. New UEDGE
modeling at lower power and confinement than in Refs.

[4-6] develops similar X-point potential cells and further
predicts additional drift-driven poloidal nonuniformity
around the separatrix [0]. The present paper reports theexperimental and numerical potential cells near the
separatrix and explains them as a consequence ofparallel ion viscous stress acting on the Pfirsch-Schltiterdifferent.
Figure 3 shows the plasma potential distribution in

ion return flow of the VB drift.
t47

Schaffer M.J. et aI., Electic Potential Cells at the Diverted Tokamak Separatrix

VM{QXk$(

o
o
N
@
o

-1.10

z (m)

Fig.2 Range of divertor geometry used to measure-1.20

around the X-point.

-1,30

-1.10
N

z (m)

@

o

1.70

1

a

.1.20

R (m)

Fig. 4 Electron pressure for ion VB drift toward the X-

point. Small squares indicate measurements.
Upstream p" are 60 Pa, 40 Pa, and 30 Pa at y, =0.995. 1.000 and 1.005, respectively.

1.30

1.40

1.50

1.60

fore, there is also a p" maximum or hill near the X-point.
Thep" hill is associated with a corresponding greater nenear the X-point relative to upstream. The p" hill is

1.70
R (m)

Fig. 3 Plasma potential for ion VB drift toward the X-

observed in L-mode with both -B1 directions and in the
few Ohmically heated plasmas for which we have data.
The X-point non-uniformity extends inward for about

point. Small squares indicate measurements.
Outer midplane potential is = 50 V with almost noradial variation across the magnetic surfaces
shown.

l7o of poloidal flux from the separatrix. In contrast, Pe
in H-mode appears to be uniform on the closed surfaces

the divertor and X-region. The potential in the three
quadrants with data is positive (100-250 V) and muchlarger than the upstream potential (-50 V) on the same

tl0l.

3. X-point Potential Cell
The experimental near-separatrix plasmas are
sufficiently collisional to justify use the Braginskii

magnetic surface. Therefore, @ is high (potential hill) by
as much as 200 V near the X-point. The potential was
roughly similarly distributed in shots with reversed 87,

MHD plasma equations [17]. The potential gradient on amagnetic surface is governed by a balance of the par-but it was lower, -100 V in the X-region and -25 V
upstream. We conclude that a positive potential hill
exists near the X-point on SOL, closed and private

allel forces acting on electrons, which in the low
resistivity collisional limit issurfaces, in L-mode plasmas for both 87 directions. The

eYu(D=-eE11=Y1p"ln"+0'7IV,kT..

large potential difference between the X and upstream(1)

The last term is the parallel thermal force. On closed
magnetic surfaces just inside the separatrix and on open
SOL surfaces at or upstream of the X-point, T" has very
little parallel gradient, so eV11@ = kT"Ynln(2"), and the
locally high potential is associated with locally highlocations on closed surfaces is remarkable, because
neoclassical plasma theory predicts only a weak poloidal

potential variation, eL@ << kT.. ln the SOL, the nonmonotonic poloidal potential variation, from target (@ =0) to potential hill in the X-region to a lower value
upstream, is also noteworthy, because it is commonly
expected that @ should increase monotonically with Z.
(2" -5 eV at target, -50 eV upstream).
Figure 4 shows electron pressure, P" = k.7., from
Thomson scattering from a shot in the same series as
Fig. 3. The data show thatp" near the X-point (100-150

electron density. However, T" decreases toward thetargets, and the full equation must be used downstream.
In general, a p. hill at the X-point implies an associatedpotential hill and can be used as a proxy for it. This isconvenient, because direct potential measurement by
Langmuir probes is difficult as plasma heating power isincreased. In fact, an X-point p" hill is measured byThomson scattering in high-power L-mode plasmas, butPa) is about 2-3 times greater than upstream on
magnetic surfaces near the separatrix (= 50 Pa). Theret48

Schaffer M.J. et al., Electdc Potential cells at the Diverted rokamak separatrix

not in H-mode [10]. If Eq. (l) is evaluated just inside
the separatrix, where T" = 50 eV, and the X-point andupstream densities are 1.5xlOre m-3 and 0.6xlOre m-3,
respectively, it yields an X-point potential only 46 V
more positive than upstream. This is much less than the-200 V measured. The potential data are very noisy andperhaps are not accurate. The agreement between Eq.
(1) and data for reversed BT is better.

I
I

The origin of the near-X p" hill relative to upstream

I

I HiS.h p"

.

I ano
I potential

pe must be explained. We proposed a mechanism based

circulationon constancy of total pressure, p = pe t p1, between theTargetX-region and upstream, combined with strong ion
cooling in the divertor [9,10]. This model is plausible

Fig. 5 Oualitative E<Ecirculation around the X-ooint.

for low power L-mode plasmas like the ones in Figs. 3
and 4, but not for high-power L-mode. New modelingby the UEDGE edge transport code, presented in the

charges in the same experimental series [18]. X-region

charge exchange has been advanced as significant

next Section, suggests that parallel ion viscosity plays amajor and more general role in generating potential cells

mechanism to remove plasma angular momentum andperhaps influence the tokamak H-mode 119,201.
However, the X-region ExB circulation is much grearer.
For f"* = 0.3 ms, the rate of ExB angular momentum

near the separatrix.

The cross-B potential gradient drives plasma alongequipotentials around the X-point at the electric drift
velocity, vn = ExB/82 = -Y1D x Br/827. We use thetokamak approximation, B = 87. Although the potential
distribution cannot be measured over the full edge
plasma volume with the available DIII-D diagnostics,

circulation across the separatrix is = 0.14 N.m, which iscomparable with = 0.16 N.m injected by the neutral
beam. Similarly, the ExB circulation of ions out acrossthe separatrix is large, = 3.102t s-r, more than half of thethe X-point potential hill must contribute an ExB

the X-point potential cell plays a significant andcirculation qualitatively like the one sketched in Fig. 5.

previously unappreciated role in transport across thetotal cross-separatrix transport rate - 5.102r s-I. Clearly,

It convects particles, energy and momentum out and in

separatrix.

across the separatrix. The divertor drift, especially thedrift across the private region, has been discussed

4. Numerical Modelingtheoretically [4"5] and observed experimentally [7]. The
private drift was shown to be the main factor causing

The two-dimensional multi-species edge transport
code UEDGE [2,3] was previously used to model ageneric, single-null-diverted, DIII-D H-mode discharge
with self-consistent inclusion of ExB and VB drifts [46]. Recent numerical improvements to the code havefacilitated its use to model other plasmas with the drifts.
Here we show new results from modeling a low-power(0.7 MW radial power flow), single-null-diverted DIII-D
L-mode plasma. A constant impurity (carbon) fraction
was used here. UEDGE obtains T" = 50 eV, just inside
the separatrix, like the experiment, and it is nearly
uniform poloidally. The model plasma differs from theexperimental ones reported above mainly in its divertor
geometry, which has about twice the divertor leg lengthof the experiments. The UEDGE inner divertor plasma
is detached from its target, as in the experiment.
Figures 6 and 7 show, respectively, the UEDGEcalculated potential and p" distributions in the divertor
and X-region for standard 87 direction. Tne p"
distribution is close to the experimental one in Fig. 4.

the long-observed sensitivity of the inner-outer divertor
target plasma differences to the B. direction [4,5]. The

potential hill observed on closed surfaces near the Xpoint, reported above, extends that circulation into theconfinement volume.
The ExB exchange time, defined as the time for a
fluid element traveling at yE to go from an entry to anexit point on an equipotential surface, is ?"* = lB dAl
d@1. Here dA is the area in the R-Z plane enclosed
between equipotential surfaces aD and, @ + d@ in thevolume of interest. For the potential hill region above
the X-point, roughly the area bounded by the separatrix,

the ryn - 0.99 surface and extending part way upstream,A - 0.01 m2. ApproximatingdA/d@by Al@z- @11, withl@z- @rl = 65 V, yields z"* - 0.3 ms. It is noteworthy
that cex is much shorter than the ion-neutral charge
exchange time, which is ) 3 ms for neutral density
measured at S 1.1016 m-3 in the X-region of similar dist49

Schaffer M.J. et al., Electic Potential Cells at the Diverted Tokamak Separatrix

Potential

-0.9

z(m)

-1.1
Fig. 8 Potential @ versus poloidal angle, just inside theseparatrix, for both directions of 8r' Dashedseparatrix shape is added for reference.

-1.3

0

20

60 80v40

Fig. 6 UEDGE-calculated-0.9@ distribution.

Electron Pressure
Fig. 9 Pressures p" and p. as in Fig. 8.

z(m)Figure 8 shows the UEDGE-calculated potential @just inside the separatrix (V, = 0.9975) for both 87-1.1

directions as a function of polar angle. The separatrix
outline is shown to help the reader relate poloidal
nonuniformities with geometric features. The UEDGE
peak-to-peak variation, = 30 V, is much smaller than
measured difference between X-point and upstream. The
UEDGE potential for standard Br peaks at the X-point'
but with reversed BTthere is only a small X-point peak'
and the main peak is somewhat upstream. The
corresponding calculated p" on the same magnetic
surface, Fig. 9, follow the @ nonuniformity closely'
because T" is nearly uniform. Figure 9 also shows ionpressures, pi, which are relatively lower at the X-point

-1.3

0

1.4

R(m)

1.7

50

100

150Pa

Fig. 7 UEDGE-calculated p" distribution.

than p", because T1 decreases to about 50 eV near the X-

point, from about 90 eV upstream on this surfaceCalculated total pressures, p, arc shown in Fig. 10.
The poloidal gradients of total pressure are surprising.

However, the UEDGE potential is considerably smaller
in magnitude than the experimental one in Fig. 3- Thismight arise in part from the different divertor geometry'

Still more surprising, UEDGE-calculated parallel
velocities are small, in that the divergence of the inertial
tensor is much less than Vp. Instead, Fig. l0 shows thatmost of - V11p is balanced by the gradient of the parallel

but, as noted in Section 3, it is also likely that themeasured potentials have substantial errors. The
UEDGE potential has a shallow valley on the large-R
side of the X-point, for which there is weak evidence inthe experimental data (the 150 V contour at z = -l-18

viscous stress,

nr,=-(4/9)piz1-11dv11lds11:-{;;dvx/dstt Q)

m).
t50

schaffer M.J. et al., Electic Potential cells at the Diverted rokamak seoaratrix
Total Pressure,

ViscousStresstheir associated cross-separatrix transport might be acommon, perhaps ubiquitous, feature of low confine-

lon
Drift

\t\l

ment behavior.

Although not shown here, the UEDGE toroidal
rotation frequency, C), is very nonuniform near the separatrix, even changing sign around the poloidal circumference. We conjecture that the transition from L- to Hmode requires formation of a more poloidally-uniform
boundary layer, one that permits a neoclassical-like
uniform Q and does not have the large cross-separatrix

Fig. 10 Total pressure and parallel viscous stress as inFig. 8. An arbitrary constant is added to 2,,, forplotting convenience.

transport associated with electric potential cells. poloidal

homogenization was a feature of electrode-driven L- to
H-transitions in the CCT limiter tokamak [23].

Here r1-1 is the collisional relaxation time between ion
parallel and perpendicular pressures f2l,6f, and cLu =5. Conclusion

2.511, where 11 is the usual ion collision time [17]. The

An electric potential hill and an associated electronpressure hill are measured at the divertor X-point in L-

parallel viscous force, 1[01,"11 = -Y11ni11, opposes velocity
derivatives. Since 411 - Tf'', u flux limit factor was usedin the UEDGE calculations to limit lz;111 to < 0.5 p1, inmode plasmas in DIII-D. The potential hill drives an

ExB, circllation about the X-point (potential cell),thereby exchanging plasma between closed and open

order to avoid unphysically large viscosity, but the numerical solutions were insensitive to the choice of flux
limit (tested between 0.1 p1 and 1.0 p,). The boundary

magnetic surfaces at rates that can be comparable to thetotal cross-separatrix transport. Modeling by the 2D

layer is sufficiently collisional to justify this viscosity
According to the UEDGE modeling, the poloidal

edge transport code UEDGE with VB and ExB drifts
reproduces many of the observations. The code identifies parallel (to B) viscous stress acting on the pfirsch-pressure nonuniformities arise where the parallel viscous

Schltiter ion return flow of the VB drift as the origin ofmodel.

force sufficiently impedes the Pfirsch-Schliiter ion
return flow of the VB drift. For example, while ionsthe pressure nonuniformity that generates potential cells

in the boundary layer near the magnetic separatrix. The

drift vertically in the non-uniform 87 field toward the X-

code also shows examples where small perturbations

point, other ions retum along B to the top of the plasma,

generate additional potential cells.

and re-supply the VB drift. Due to the weak near-X
poloidal B, the return flow divides between the inner

Acknowledgement(small R) and outer (large R) return options. ThisThe authors acknowledge helpful discussion andestablishes dv;1/ds;1 in the X-region. If 4 is large, thecontributions from D.R. Baker, K.H. Burrell, C.J.
Lasnier, A.W. Leonard, R. Bulner, and T.S. Taylor. This
work was supported by the U.S. Department of Energy

viscous force exceeds the ion inertia, and the parallel
gradient of the total pressure grows against the viscous
force. Quasineutrality couples p. and pi through n", and,the electron nonuniformities then generate the potential

under Contracts DE-AC03-99ER54463, DE-ACO494AL85000, and Grant No. DE-FG03 -958R54294.

cells.

The potential peak at about 45o below the equator

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