Proceedings of ITC/ISHW2007

ICNTS - Benchmarking of Momentum Correction Techniques








Craig D. BEIDLER , Maxim Yu.
ISAEV  , Sergei V. KASILOV  , Winfried KERNBICHLER  ,Georg O. LEITOLD Henning
MAASSBERG,
Sadayoshi
MURAKAMI ,
Victor V. NEMOV  , Donald A. SPONG , Victor TRIBALDOS
Max-Planck
Institut für Plasmaphysik, Association EURATOM, Greifswald, Germany
RussianResearch Center “Kurchatov Institute”, Moscow, Russia
Insitute
of Plasma Physics, NSC-KhIPT, Kharkov, UkraineInsitutfür Theoretische Physik, Association EURATOM-ÖAW, TU-Graz, Austria

Departmentof Nuclear Engineering, Kyoto University, Kyoto, Japan

Oak
Ridge
National Laboratory, Oak Ridge, Tennesee, USA
Laboratorio Nacional de Fusión, CIEMAT, Association EURATOM, Madrid, Spain
October 11, 2007
In the traditional neoclassical ordering, mono-energetic transport coefficients are evaluated using thesimplified Lorentz form of the pitch-angle collision operator which violates momentum conservation. In thispaper, the parallel momentum balance with radial parallel momentum transport and viscosity terms is analysed,
in particular with respect to the radial electric field. Next, the impact of momentum conservation in the stellarator
lmfp-regime is estimated for the radial transport and the parallel electric conductivity. Finally, momentum
correction techniques are described based on mono-energetic transport coefficients calulated e.g. by the DKES
code, and preliminary results for the parallel electric conductivity and the bootstrap current are presented.
Keywords: ICNTS, neoclassical transport, stellarators, parallel momentum conservation
1. Introduction
The International Collaboration on NeoclassicalTransport in Stellarators (ICNTS) was initiated in 2000.
The starting point is the drift-kinetic equation (DKE)
which is linearised with respect to the 1st-order distribution function defined as the (small) deviation from the 0thorder (unshifted) Maxwellian,  , with the density, electrostatic potential and temperature assumed to be constanton flux-surfaces. The 1st-order DKE becomes inhomogeneous with a radial driving force,     (  being the radial component of the   -drift velocity and   the radialderivative of the Maxwellian with total energy conserved),
and with a parallel driving force, !#"%$&'( . Splitting
this DKE with respect to the driving forces leads to two
1st-order distribution functions, ) and * , where ) is related to +  ,  (symmetric in "&$ ) and * to "-$&'  . The
Vlasov operator couples symmetric and asymmetric terms;
consequently, ) (* ) has also asymmetric (symmetric) contributions. With the linearised collision operator, .0/ 1 2 , theparallel momentum balances are given by
5
3
4




5 

4

)

6 6 7 "-$L I
6 6;:GK FHM
>J


<>

*98 8

6 6;:=<?>



N"-$

)
*98 8






@

" $ 

3


)
A " BDC6 6 "-$. / 1 2POS

author’s e-mail: Henning.Maassberg@ipp.mpg.de

 TK
 " UWV *E8 8)*&QR8 8 L

(1)where X XZYZY[Y \ \ means flux-surface averaging and velocity
space integration and the electrostatic potential. Here,I
the incompressible form of the ]drift is used for conF:
sistency with e.g. DKES (Drift-Kinetic Equation Solver)[1, 2] and both momentum correction techniques [3, 4] described later. The important advantage of this approximaC
 term intion is the disappearance of the
>_I
<`>
@ :^
a
a F_
S
both acceleration terms,  and "  ("&$;bc" ), in the conservative formulation; see [5]. Then, a  !
 , i.e.
:d<&>the mirror term, and "  S T allowing for a mono-energetic
treatment of the 1st-order DKE if the collision operator
is replaced by the simple Lorentz form of the pitch-angle
collision term which, however, violates momentum conservation. As shown in [5], the incompressibility approximation in the 1st-order DKE is justified for radial electric fields, egf , nearly up to the toroidal resonance value,oqpe,f hjilk Snm "r .
The 1st term in the parallel momentum balances (1)describes the radial transport of parallel momentum which,in general, is ignored in the traditional neoclassical theory(see e.g. the review [6]). In an axisymmetric configuration with esf S T , the tMuwvyx part of ) (symmetric in a )leads to the particle transport whereas the z|{}tMx part (asymmetric in a ) finally leads to the bootstrap current coefficient. This separation is broken at larger e f : the tMuwvyx
component of ) has also asymmetric contributions leading to radial transport of parallel momentum. The parallel
momentum transport coefficients in mono-energetic form
equivalent to eq.(1) are calculated with a new DKES ver-

Proceedings of ITC/ISHW2007Êåä® ägæçè , and with the trapped particle fraction

cÜ

£ Ç



 Ô

½ §



é

½ §

Ö× 
³

ØsÙrØ

ÉêÊ Ì ÍÉ ÚÛî ï ð

Øìë

½ §

µ
Ø

Û

ÊÍ


for trapped particles.
ÏZÏ is the 1st
Legendre component of, ion-electron collisions
can be neglected. Impurities can act as an additional
momentum sink (but are ignored here for simplicity).
In this collisionless picture, the viscous damping of theparallel ion flow is equivalent to the friction of the passing
with the trapped ions. In an equivalent approach [9], thecollisionless electric conductivity is obtained.
Fig. 1 Mono-energetic parallel momentum transport coefficients, ~9~l (open circle) and ~g|M~0W (fullsquare), vs. the normalised radial electric field,  forthe LHD-360 configuration at  DN} E;}- .

sion. In Fig. 1, the mono-energetic  M; % (normalised to the particle transport coefficient    q % )and  ¡ ¢;; -£¤ (normalised to the Ware pinch coefficient  ¥¡ ¦c £` ¨§s ¡ ¦§0 % ) are shown forrather large ©sª (normalised to the toroidal resonance value,©¬ª « ­© ª® ©,ª ¯j°l± , for the LHD vacuum configuration with²
´³&µ ¶`· m at half the plasma radius. (Here, ZµZµ[µ means
integration over and flux-surface averaging.) The value²
of the collisionality, ¸%«R­¸ ®rºW¹ » , places these results inthe lmfp-regime. For an axisymmetric configuration where
both   and   ¡ are independent of ©gª (for ©¬ªJ
« ¼¾½ ),
 ® ¿ slightly exceeds ¬|¡ ® ¿¥¡ , and both terms scalelinearly in © ª « . These results lead to a strong restriction for
the radial electric field: only for © ª « ¼¦½ , does the radialtransport of parallel momentum become negligible. This
restriction is stronger than the one related to validity of theincompressible À^ÁNÂ approximation; see [5]. Furthermore, also the 3rd term in eq.(1), the viscosity related tothe ÀÃÁÄÂ flow, can be ignored for ©Åª « ¼Æ½ .
The 2nd term in the parallel momentum balances (1)is the parallel viscosity describing the damping of parallel
flows due to the magnetic field inhomogenity. In an asymptotic collisionless limit, the 1st-order DKE for £ (parallel driving force) can be directly integrated for sufficiently
small © ª following Refs. [7, 8]. By flux-surface averaging, the 1st Legendre component of the DKE reduces to a
1D equation for the velocity dependence of the asymmetricpart of £ (here only for ions)
£-Ç

and

È§

Î ËÉ Ê;ÌÍ¸}ÏZÏÐjÑÓÒ

½
 ÔDÕ
}

Ö× 

ØsÙrØ

ÐjÑÒ
ÉwÚ

½ §

Ø
Ê|Í

×
Û

Ï[Ï Ð ÕcÜÒ(§}Ô ¸ Ï[Ï ÐWÑÓÒ ÕÝ¸ Ï Ñ%Þàßâá
×with the thermal ion collision frequency, ¸`Ï , Ñ'

»
®

» ÜWã

,

Ð

½ Òí¨·

2. Simple Pictures of Momentum Corrections
The computations of even mono-energetic transportcoefficients (e.g. by DKES) are rather time expensive inthe lmfp-regime. The treatment of the linearised collision
operator with momentum conservation would require thesolution of the DKE in the 4D-phase space instead of the
3D mono-energetic solution. Consequently, momentum
correction techniques [3, 4] based on the mono-energetictransport coefficients become attractive. A rough estimation of the correction, however, can be obtained from eq.(1)with incompressible ©nÁÅä flow for the different transportcoefficients. The 1st-order distribution functions are split:




£
ñ
 and  areand £  £ ñ £ whereÛÛî ï ð the solutions of theÌ mono-energetic Ì DKE with the Lorentz form ofthe pitch angle collision term,, instead of. Then,£the corrections  Ûandaredefined
by
îïð
Û îïð
Û
Ì

ô
òNó 
£ Ì

Ì ô
ó 
£ Ì

§

Ì

ô
ó  
£



 ô
 ó 
£

§

Ì

Neglecting parallel momentum transport and the À´Á_Â viscosity term for sufficiently small ©õª leads to the balance
equations for the 1st and 2nd Legendre components of thecorrections,  and £ ,öÌÛ î ï ð(ûÂ=÷cø+äNù

Ì

Ì

Ð




Ò £

ñ
Ì

³

ö
ä
Î

Ì;ú
¸



¡;


¡¡

§






Û î£ ï ð Ì
Ì;ú}ü
ó 

¡


¡¡

ô

(2)

where ý¡}Ð » ÒEG    and Å¡¡`Ð » ÒEG &£   are the monoenergetic transport coefficients, calculated e.g. by DKES,
and ù Ð  Ò is the 2nd Legendre polynomial.
InÌ the lmfp-regime, the parallel viscosity evaluated for (by DKES) is proportional to ¸&þ , and only weakly dependent on © ª . Although this term is also determined by
the symmetric component of   , the large ÿ' ;}   
has a quite different dependence on ¸Ó« and © ª , e.g. in
the ½ ® ¸ - and the ¸ -regimes. If any distribution function is split into aÚ slow and a fast scale with respect tobounce-averaging, the slow part determines   [10], butdoes not contribute to the parallel viscosity. This canProceedings of ITC/ISHW2007
be shown by formulating the flux-surface averaging as
bounce-averaging for the ripple-trapped particles, and theterm (differslow contribution vanishes due to theent toin ). Consequently, the impact of momentum conservation on the radial transport in the stellarator
lmfp-regime is negligible. For tokamaks, however, the situation is different. Here, the radial transport coefficientsin the lmfp-regime (banana regime) are proportional toand corrections from the momentum conservation are notnegligible. NEO-2 [11], based on the field-line integration
technique, is generalised for a full linearised collision operator with momentum conservation. As shown in Ref. [12],the radial transport coefficients are reduced whereas thebootstrap current coefficient are increased with the full linearised collision operator compared to the Lorentz model.
The impact of momentum conservation for the parallel
electric conductivity is of the order of 100%. The classical “Spitzer problem” in the collisional limit [13] can beanalytically generalised to the collisionless limit [9]. The
collisionless Spitzer function,, is defined by the 1D
integro-differential equation in  


 "!
#%&$% "')(+*-,   %&% "0/  %21 : %21 <;7= >35476289,.
with the thermal electron-ion collision frequency,  %21 ,and the (energy-dependent) electron-electron collision fre# $quency,  %&% " . %&% is the 1st Legendre moment of thelinearised collision operator. The classical “Spitzer problem” corresponds to  @? . In the mono-energetic ap,
proach, the normalised A 3B3  CDFE in the collisional
limit and is reduced to the passing (circulating) particle
fraction, A 3G3 HC, . , i.e. the trapped particles definedby  do not contribute. With momentum conservation,,a stronger weighting of the trapped particles appears (defined by   ) which reflects the passing-trapped electron, ,.
fricton adding to the friction with ions and impurities (defined by the
4I628 term). Rather accurate approximationsof both the collisional and collisionless Spitzer function,
J" , have been developped; see [14] and [9], respec-tively. Momentum correction is important for the parallel
electric conductivity for all collisionalities.
Fig. 2 shows the benchmarking for the parallel electric conductivity, , normalised to the collisional SpitzerHärm value [6] for the W7-X high-mirror (w7x-hm1) andlow-mirror (w7x-lm1) vacuum configurations atwithand, respectively. Instead of
the Maxwellian, the collisional Spitzer function is used in
the energy convolution of the mono-energetic DKES coefficients,(solid lines). This approach was already used
for the evaluation of the experimental current balance in
W7-AS; see e.g. [15]. The first preliminary results (fullcircles) of the momentum correction technique developed
by Taguchi [3] agree quite well with the simplified (collisional) correction technique implemented in the DKES energy convolution. (These data are normalised to the DKES
data at the highest , so far). Furthermore, the asymptotic

K0, 7?SRUTVWTV, I?CR XYT X[Z

A 3G3



MOL NQP E\^]

Fig. 2 Parallel electric conductivity, , normalised to the collisional Spitzer-Härm value vs. the electron collisionality
for the high- and low-mirror W7-X configurations (lower
and upper curves, respectively) at half the plasma radius
for.
based on the collisionless Spitzer function is given for reference (dashed lines).

_a`bdce \ ] gf ?

limit
based on the collisionless Spitzer function
with all the trapped-particle effects included (completely
kinetic modelling) is given for reference (dashed lines).
3. Momentum Correction Techniques
The momentum correction techniques [3, 4] are basedon moment methods, i.e. a low-order expansion of the kinetic equation with the linearised collision operator bothin Legendre (with respect to ) and in Sonine (with respectto ) polynomials. With flux-surface averaging, a linear
system of equations is obtained which can be closed by using the 3 mono-energetic transport coefficients calculated
numerically for different collisionalities and radial electricfields. In particular, the parallel particle and heat viscosities defined by the 2nd Legendere coefficients ofandin eq.(2) are obtained. The expansion of the linearised
collision operator couples the radial transport and parallel
flows of all species, and, consequently, the thermodynamic
fluxes are corrected and not the transport coefficients ofeach species. An exception is the parallel electric conductivity where ion effects can be neglected. As dicussed in
the previous section, the radial transport in the stellarator
lmfp-regime is nearly unaffected by the momentum correction.
Both momentum correction techniques have been implemented in the energy convolution based on databases of
the 3 mono-energetic DKES transport coefficients. Results
by using the [4] approach are given in [16, 17]. Recently,
benchmarking was initiated, however, this activity is onlyin the preliminary phase (i.e. benchmarking the interfaces
to the DKES data and the energy convolution algorithms).
Rather preliminary results for the impact of the momentum conservation technique [3] for the bootstrap current
density are shown in Fig. 3 both for the W7-X low- andhigh-mirror configuration (equivalent to Fig. 2). In these

j



i

h

,[i

Proceedings of ITC/ISHW2007
sion operator. The distribution function is evaluated by
the field-line integration technique which leads to the. In the strict sence, NEO-2 is not
restriction
a momentum correction technique since it is not basedon the 3 mono-energetic transport coefficients. With thehigher accuracy of the expansion, NEO-2 is an attractive
tool for benchmarking with both momentum correction
techniques described above.

+

k ml

Fig. 3 Bootstrap current density,(normalised to the equivalent tokamak value without momentum correction for), vs.
without (open symbols) and with
momentum correction (full symbols) for the W7-X lowmirror (squares) and for the W7-X high-mirror configuration (circles) at half the plasma radius based on Taguchi’s
momentum correction technique [3].

nporqtsuavw

calculations, an additional impurity species (fully ionisedoncarbon) was introduced to describe the impact ofthe momentum balance. The radial electric field was calculated from the ambipolarity condition, and, additionally,
the impurity density gradient from the condition of vanishing radial impurity flux. The inverse gradient lengths,
and
(with
) are the same for all scenarios. The plasma parameters correspond to low collisionalities, and an “ion-root”
is established. Since the electron-regime are reduced withtransport coefficients in theincreasing(and also the bulk ion density), this complex-dependence is eliminated by the normalisation of
the bootstrap current density (sum over all species) to theequivalent tokamak value for the same
for
, but
without momentum conservation. With this normalisation,
the impact of the momentum conservation on the bootstrap
current is highlighted.

xzy2{

5}~
x7y2{

x y2{

|)}~|

- ^a

~H

xy2{

  

Momentum correction has only a moderate impact on
the bootstrap current for both W7-X configurations; see
Fig. 3. With this correction, the bootstrap current is reduced (a slight increase, however, was found in NEO-2
calculations for a tokamak case with). Momentum
correction is largest atand becomes less important at higher, and the optimisation criterion of minimised bootstrap current is not affected which is realisedin the W7x high-mirror configuration, at least for small. Furthermore, the radial particle and energy fluxes arenearly identical with and without momentum correction as
was analysed in Sec. 2.

xIy2{

x y2{ 



Also the NEO-2 version with the momentum conserving collision operator [11] will be included in thebenchmarking. NEO-2 expands the distribution function
to higher orders in the Sonine polynomials compared
to Refs. [3, 4] and solves for the full linearised colli-

4. Discussion/Conclusions
The benchmarking of momentum correction techniques has been initiated and preliminary results havebeen obtained. The particle and energy fluxes (and,consequently, the ambipolar radial electric field) in thestellarator lmpf -regime are only weakly affected by theviolation of momentum conservation in the simplified
pitch-angle collision operator used in evaluating themono-energetic transport coefficients in all codes includedin the ICNTS activity. The impact of momentum conservation on estimating the parallel electric conductivity
is very well known, and the correction is rather large.
Preliminary results for the bootstrap current indicate
a moderate reduction with momentum conservation
taken into account. Very good agreement is obtained in
the benchmarking of mono-energetic bootstrap current
coefficients [18]. Consequently, a detailed analysis of
momentum conservation effects is a logical next step inthe ICNTS.
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[3] M. Taguchi, Phys. Fluids B4, 3638 (1992).
[4] H. Sugama and S. Nishimura Phys. Plasmas 9, 4637 (2002).
[5] C.D. Beidler et al., ICNTS - Impact of Incompressible
Flow in Estimating Mono-Energetic Transport Coefficients,
this workshop, (2007).
[6] F.L. Hinton and R.D. Hazeltine, Rev. Mod. Physics 48, 239(1976).
[7] M. Taguchi, Plasma Phys. Control. Fusion 30, 1897 (1988).
[8] M. Taguchi, Plasma Phys. Control. Fusion 31, 241 (1989).
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[10] A.A. Galeev and R.Z. Sagdeev, Review of Plasma Physics
7, 307 (1977).
[11] W. Kernbichler et al., 32nd EPS Conf. Plasma Phys., Tarragona 2005,
http://epsppd.epfl.ch/Tarragona/pdf/P1 111.pdf (2005).
[12] W. Kernbichler et al., 33rd EPS Conf. Plasma Phys., Rome
2006, http://epsppd.epfl.ch/Roma/pdf/P2 189.pdf (2006).
[13] L. Spitzer and R. Härm, Phys. Rev. 89, 977 (1953).
[14] S.P. Hirshman, Phys. Fluids 23, 1238 (1980).
[15] H. Maaßberg et al., Plasma Phys. Control. Fusion 47, 1137
(2005).
[16] D.A. Spong, Phys. Plasmas 12, 056114 (2005).
[17] D.A. Spong et al., Nucl. Fusion 45, 918 (2005).
[18] K. Allmaier et al., ICNTS - Benchmarking of Bootstrap Current Coefficients, this workshop, (2007).

n