Proceedings of ITC/ISHW2007

ICNTS - Benchmarking of Bootstrap Current Coefficients






 

Klaus ALLMAIER , Craig D. BEIDLER  , Maxim Yu. ISAEV
 , Sergei V. KASILOV
Winfried KERNBICHLER  ,  Henning MAASSBERG
 , Sadayoshi MURAKAMI ,
Victor V. NEMOV , Donald A. SPONG , Victor TRIBALDOS

,





Insitut für Theoretische Physik, Association EURATOM-ÖAW, TU-Graz, Austria
Max-Planck
Institut für Plasmaphysik, Association EURATOM, Greifswald, Germany
RussianResearch Center “Kurchatov Institute”, Moscow, Russia
Insitute of Plasma Physics, NSC-KhIPT, Kharkov, Ukraine
Departmentof Nuclear Engineering, Kyoto University, Kyoto, Japan

Oak
Ridge
National
Laboratory, U.S. Department of Energy, USA
Laboratorio Nacional de Fusión, CIEMAT, Association EURATOM, Madrid, Spain
October 11, 2007

The benchmarking of the mono-energetic particle transport coefficients for various stellarator configurations
is completed, very good agreement was obtained by quite different codes (several Monte Carlo techniques,
DKES, GSRAKE and NEO). Such an extensive benchmarking for the bootstrap current coefficient is still lacking.
The bootstrap current is an off-diagonal term in the neoclassical transport matrix and standard  -Monte Carlotechniques are well suited only for the diagonal transport terms in stellarator configurations. This benchmarking
is based on four different codes.
Keywords: ICNTS, neoclassical transport, stellarators, bootstrap current coefficient
1. Introduction
The International Collaboration on NeoclassicalTransport in Stellarators (ICNTS) was initiated in 2000 for
solving the linearised mono-energetic drift-kinetic equations




)+*

 

"!

,-"!

+*( $#%'&$#-.0/2143



Here,
is the Vlasov term acting only in the flux surface(the radius, ( , and the absolute velocity, / , are only parameters),  ! is the pitch-angle collision term (Lorentz form),.5#6/87:9;/ , ( & the radial component of the <>= -drift velocity,# = 9 =A@ the normalised magnetic field strength. and 1?
*
andare the 1st-order distributions functions, i.e. the(small) deviation from the 0th-order Maxwellian. The
mono-energetic transport coefficients are defined by thefollowing moments: B  #DC:( &  E is the particle diffusion
B  #F/GC .  :E the boostrap current coefficient,coefficient,**(
E the Ware pinch coefficent, and B K#F/LC .
E
B HI#JC:&
C
M
O
#
Q
N
R
P
A
M
U
S
V
T
.
E
the electric conductivity coefficient. Here,
where P3W3X3 S denotes flux-surface averaging. Onsager symmetry leads to the relation B Q#Y B  . The Z,[QZ thermaltransport matrix is obtained by energy convolution of themono-energetic coefficients for different weights with respect to / .
The early ICNTS phase concentrated on the benchmarking of the B  coefficient for the main stellarator configurations; see [1]. DKES (Drift-Kinetic Equation Solver)[2, 3] is based on a Fourier-Legendre expansion of both
author’s e-mail: Henning.Maassberg@ipp.mpg.de*andin the 3D phase space and evaluates all monoenergetic diffusion coefficients depending on collisionality
and radial electric field. MOCA [4] and DCOM [5] arefull-  Monte Carlo codes evaluating B  from the radialbroadening of an ensemle of test particles in time following [6]. A  MC technique [7] allows for a strictly local
estimation of B  , i.e. the radial drift velocity determines
only the weight of the markers \"] # N ( & T_^ for ( #a`b4cedV^ .
This B  benchmarking (also supported by other codes) islargely completed with very good agreement; the impactof the approximation of incompressible fD[-= flows forrather large radial electric fields, f"g 94/ =ihYklj ( 9m , is analysed in [8].


Here, the scaling of the bootstrap current coefficient in
the different  collisionality regimes is briefly summarised:
B KnJo49)prq in the Pfirsch-schlüter regime, B ?nso49ptq`b4cedV^3 at very-low
in the plateau regime, and B vu
k
L
p
w
q
x
#
8
p
I
m
;
9
/
collisionalities, where
j . This scaling holds
both for stellarators and tokamaks.
2. The Numerical Tools
The bootstrap current coefficient evaluated by DKES
could only be benchmarked with standard  MC techniques either for an axisymmetric configuration or only formoderate collisionalities, so far. For estimating B  and(B  , the marker equations \"] # N & T_^ and \2y #6/ N .AT)^
increase the weights of those markers which dominate thecontribution to the corresponding coefficients at low collisionalities, i.e. ripple-trapped particles with large ( & toB  and passing particles with large . to B  . The ripple-

Proceedings of ITC/ISHW2007
trapped particle distribution function is nearly symmetricin z (allowing for a bounce-averaged treatment), but {}|~is determined by the asymmetry in the passing particle distribution with rather small weights, " . Consequently, thelarge   of ripple-trapped particles leads to huge statistical problems for evaluating the {|~ coefficient in stellarators at very-low collisionalities. This problem does notappear in axisymmetric configurations since   is a periodic
function along the banana orbit, and the tokamak { |~ canbe evaluated directly with  -MC techniques; see e.g. [9].
The situation is equivalent for evaluating the Ware pinch
coefficient: {~H| is mainly determined by the trapped particles with small z whereas the far passing particles dominate in the marker weights Q . This equivalence was foundin direct  -MC simulations of { |~ and { ~| : rather similar convergence problems appeared at the same (intermediate) collisionalities.
One possibility to overcome this artificial noise problem in  -MC simulations of the bootstrap current coefficient is a filtering technique used in the VENUS- ;
code [10, 11]. The strongly weighted ripple-trapped particles, i.e. the markers with large  "L , are omitted for evaluating {|~ , and only barely trapped particles and passing
particles contribute. This filtering technique based on 
bounds is equivalent to a forced localisation in  : if theradial drift of ripple-trapped particles exceeds a maximum
radial displacement, their contribution to { |~ is simply ignored. The main disadvantage of this approach, however,is the somewhat artificial definition of the  bounds: thesensitivity of {|~ on these bounds must be controlled foreach magnetic configuration depending on the degree ofdrift-optimisation and fraction of ripple-trapped particles.
At very-low collisionalities, VENUS-  calculations arevery time expensive.
Another approach used in the NEO-MC code, is basedon the iterative evaluation of the distribution function in
those phase space regions (e.g. close to the trapped-passing
boundary) mainly contributing to the bootstrap current coefficient. The standard  -MC methods are based on the
0th-order Maxwellian, and  must describe the full deviation without any iteration. Iterative re-discretisationsof the distribution function in the relevant phase-space region for evaluating {|~ by using this information for the
next step reduce significantly the artificial noise introducedby the ripple-trapped particles [12]. The convergence of
this method is equivalent to  within the iteration.
Furthermore, a constant marker weight scheme with decreasing number of test particles is used for trapped particles leading to an additional statistical improvement for a
fixed number of starting test particles. The variance (noise)in this method scales with the mean free path (instead ofmean free path squared as in the standard method) and finally the computing time needed in NEO-MC for the same
statistical error decreases by a factor proportional to themean free path to the power  .
In addition, NEO-2 calculations of the bootstrap

current coefficient are included in the benchmarking.
NEO-2 [13] integrates the linearised drift-kinetic equations along field lines for arbitray collisionalities, however,the poloidal  drift cannot be included. In NEO2, the Green’s function technique in combination with anadaptive (3rd-order) conservative finite-difference discretisation scheme with respect to the normalised magnetic moment is applied which allows for high accuracy at low collisionalities (where DKES and both MC techniques become
very time consuming due to the high localisation of the 1storder distribution functions in the 3D-phase space). Furthermore, NEO-2 is extended to the full linearised collison
operator [14] (instead of the simplified Lorentz form of thepitch-angle collision term).
Finally, the collisionless asymptotic value of { |~ isused for the benchmarking. In particular the axisymmetricvalue in the large aspect ratio limit, Q  ;Yx , is used
for normalisation. In this limit, the collisionless asymptote
is easily obtained from the mono-energetic flux-friction relation for an axisymmetric configuration:for this case, therelation s_'iK ¢¡R£¤'¦¥§;¨© holds, where ¨U©
is the radial unit vector, and all 3 mono-energetic transportcoefficients are linked for small radial electric fields, A© ;see [15]. The parallel electric conductivity is given by{|| ¡Rªr«A¬_¥,a{|| ¡Rªr«A®­¯¥,

_ª
°

;±

where ;± is the trapped particle fraction± %²Q
³ ´Rµ·,}¶ ¸¹)º

»¼ ~

½0¾½
´À¿

²Q

½

µ ·,¸¹ ºwith µ ·,¸¹ 6> ·,¸¹ . In the large aspect ratio limit, ±ÂÁ³_Ä²)Ã¿  for a tokamak with circular cross section, and{|:~ ¡Rª « ¬_¥,




4±
 

Á

8Ã ÅÇÆ)
 ¿

ÉÈ

(1)

equivalent to the value given e.g. in [16] (here, the energy convolution with the Maxwellian leads to the factorÇ¥ is used for the normalisation
3/2). This { |~ ¡Rª « of all bootstrap current coefficients in this benchmarking.
DKES and NEO-2 calculations for an axisymmetric configuration have shown, that the convergence to this collisionless asymptote is rather slow, Ê ¿ ªrË , confirming thescaling of the size of the relevant boundary layer between
trapped and passing particles; see e.g. [17]. Finally, { |~is independent of  © in tokamaks as long as  ©ÌÍ :Î°holds.
In the stellarator lmfp-regime, {|~ depends on  © andª « in a rather complex manner. Nevertheless, for ª « x ,a purely geometrical factor, the Shaing-Callen limit[18, 19], is equivalent to the collisionless asymptote foraxisymmetric configurations. However, the collisionalities
in this benchmarking for stellarator configurations are in
general too high to check the convergence of {}|~ to thecollisionless Shaing-Callen limit. The convergence ofProceedings of ITC/ISHW2007

Fig. 1 Mono-energetic bootstrap current coefficients, Ï¦ÑÐ Ò (normalised to the collisionless asymptote, eq.(1)), for NCSX
vs. collisionality, Ó Ð . DKES (open squares) forÔÖÕ×ØÙÚ Û
ÛÞUß
Û)Þß
(red), Ü¢ÝÜ
(green), à,ÝÜ(dark blue),
Û Þ Ñ
Û Þ Ñ
(light blue), àeÝHÜ
(magenta), and VENUS-á4âÜLÝHÜÔ Õ ×ØÙaÚãÛ
(full triangles only for). The Shaing-Callenvalue is given for reference (dot-dashed line).

äå:æcalculations at extremely low collisionalities (downto çrè5éëêíìUîGï is accessible for the tokamak case) is very
poor and requires huge computing times. Consequntly,
the Shaing-Callen limit is not seriously included in theä å:æ
-benchmarking.
3. Benchmarking Results
Here, the benchmarking of the bootstrap current coefficient is discussed for 3 magnetic (vacuum) configurations at half the plasma radius: NCSX (ncsx-2), the “standard” LHD with ðñóòrôöõ)÷ m (lhd-375), and the W7X “standard” (w7x-sc1) configuration. NCSX is a quasiaxisymmetric configuration with high elongation [22], andthe bootstrap current is expected to be rather similar toto the equivalent tokamak. This LHD configuration represents a classical stellarator configuration [20]. Finally,
the W7-X configuration is strongly drift-optimised, and thebootstrap current is minimised [21].
Fig. 1 shows the benchmarking results for DKES and
VENUS- øù for NCSX. The impact of úAû is rather small,i.e. this quasi-axisymmetric configuration is similar to a
tokamak with respect to the bootstrap current coefficientäææ(but the radial transport coefficient,, is dominated byripple-trapped particles on which ú û has a large effect [1]).
ä åæThe fairly high NCSX elongation leads to the reduced èat the low collisionalities (for the normalisation in eq. (1),an axisymmetric configuration without elongation was assumed).
For the LHD configuration, all 4 codes contribute to
the benchmarking shown in Fig. 2 (for a 1st benchmarking of VENUS-øù and DKES, see Ref. [23]). As in the
NCSX case, the agreement of the quite different codes isä å:æ
at very low ç è are decreased
very good. With ú"û , theFig. 2 Normalised mono-energetic bootstrap current coeffiÚ
cients, Ï ÑÐ Ò , for LHD with üàý þíÿ m vs. Ó Ð .
DKES (open squares), VENUS-áíâ (full triangles), NEOMC (open diamonds), and NEO-2 (full circles only forÛ ÞÔ Õ ×ØÙëÚÉÛÔ Õ ×ØÙ£Ú Û) for(red), àÎÝÇÜ
(green)Ñ
Û Þß
Û Þß
Û Þ
(dark blue), à"ÝÜ(light blue), and Ü§ÝÜÜ§Ý;Ü
(magenta); the Shaing-Callen value (dot-dashed line).



ä

å:æ

to the Shaing-Callen value. This reduction of thewithincreasing ú û starts at the same çtè as the reduction of theäææ
coefficient (from the ê ç -regime). In the analytic theory [24], the coupling of the ripple-trapped particle distribution function and the passing particle distribution func
tion changes in the transition from the ê ç - to the  ç regime.
For this “standard” LHD configuration (without elonäåæ
gation), thefor ú û ñ ì (except the very low çtè )is rather similar to the equivalent tokamak (with circular cross section). An inward-shift of the magnetic axis
leads to a drift-optimised stellarator configuration for ð%ñ
ò8ô )ì m [20] in the sense of  -optimisation [25], and also
äåæ
the
for ú û ñ ì at low çrè are reduced by nearly a factor of 2, but the Shaing-Callen value is less affected. A
stronger inward-shift (e.g. ðJñ ò8ôö÷ò m) further improves
the lmfp-confinement (but violates the  -optimisation), thebootstrap current coefficients are also reduced (found by
DKES computations). Finally, an outward-shift of theä}å:æ
magnetic axis results in negative(reducing  ) in theplateau regime [23], the effect of the helical curvatureterms in the -Fourier spectrum being increased.
The benchmarking for the W7-X “standard” configuration is shown in Fig. 3 with all 4 codes included. Inä åæthis case, the è for only 3 ú û values are shown (moreare available). Very time-expensive VENUS-øù compuä åætations at very low çtè indicate the “convergence” ofto the Shaing-Callen value in the limit çLè ì even forú0ûñ ì (here, the statistical error of VENUS- øù is about25%). At these low collisionalities, in particular DKES
results have rather large errors (see Fig. 3), up to 2000
Fourier modes in the angle coordinates and 250 Legendre
polynomials are used for the representation of the distri-Proceedings of ITC/ISHW2007
calculations. His work was supported by the CRPP-KIAM
contract, by the MPG-grant, by Max-Planck Institut für
Plasmaphysik, by NIFS/NINS under the Project of Formation of International Network of Scientific Collaboration(Japan), by the Fond National Suisse pour la Recherche
Scientifique, and by the Departement of Atomic Scienceand Technology, RosAtom (Russia).

Fig. 3 Normalised mono-energetic bootstrap current coefficients, for the W7-X “standard” configuration vs.
. DKES (open squares), VENUS- (full triangles),NEO-MC (open diamonds), and NEO-2 (full circles) for(red),(green), and(darkblue); the Shaing-Callen value (dot-dashed line).

    "!$#%'& % #)(+*

%'& % #)(

bution function), however, the ,./1- 0 are fairly small (e.g.
the relative errors are much smaller for the LHD configuration with large ,2/3- 0 ). This W7-X example is similar to
the LHD case where , /30 is reduced with 465 (at larger 465 ,however, , /30 increases), however, the , /30 are nearly one
order of magnitude smaller than in the LHD case. These
results confirm the W7-X optimisation criterium of minimised bootstrap current.
The variation of the toroidal mirror term in W7-X has
a fairly strong impact on the bootstrap current coefficient;see [26]. The low-mirror configuration represents aclassical highly-elongated stellarator configuration, andthe ,7/3- 0 at low 89- are increased by more than a factor of
2. On the other hand, the high-mirror configuration (with:<; 0= :<;>;@?BADCFE ) has the lowest ,7/1- 0 , but only at verysmall 4G5 . However, , /30 increases with 465 to equivalentvalues as in the W7-X “standard” configuration, i.e. thehigh degree of bootstrap current minimisation in thehigh-mirror configuration is lost even for intermediate 4<5 .
4. Conclusions
For 3 quite different stellarator configurations, thebenchmarking of the mono-energetic bootstrap current
coefficient by 4 codes, i.e. DKES, VENUS-H I , NEO-MC
and NEO-2, is performed and very good agreement
was found. As a next step in the ICNTS activity, theimpact of the violation of the momentum conservation in
the simplified Lorentz form of the pitch-angle collision
operator on the boostrap current will be analysed in detail;see [27].
Acknowledgements
One of the authors, M. Isaev, thanks W.A.Cooper and
T.M.Tran for the help and the support of the VENUS-H I[1] C.D. Beidler et al., 30th EPS Conf. Plasma Phys. Contr.
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