Development of Moment Approach forNeoclassical Transport in Stellarators

Presented by

S.Nishimura

National Institute for Fusion Science

Graduate University for Advanced Studies

Acknowledgements:

D.R.Mikkelsen2), D.A.Spong3), H.Sugama1), Y.Nakamura4), N.Nakajima1),

CHS group1), LHD experimental group1), Heliotron-J group5)

1) National Institute for Fusion Science

2) Princeton Plasma Physics Laboratory

3) Oak Ridge National Laboratory

4) Graduate School of Energy Science, Kyoto University

5) Institute of Advanced Energy, Kyoto University

stellarator theory teleconferenceMarch 22, 2007, PPPL

OutlineIntroduction Developments of the stellarator moment equation approach. From Hirshman and Sigmar (1981), Shaing and Callen (1983) to Sugama and Nishimura (2002)

Three mono-energetic viscosity coefficients M* (parallel viscosity against flow), N*(driving force for the flows), L*(radial diffusion)

Although we do not show today any calculation examples on actual existing or

planned helical/stellarator devices, many application examples (QPS, HSX,

NCSX, W7X, LHD) by Dr.D.A.Spong were already reported by the QPS group. Fusion Sci.Technol 45, 15(2004), 46, 215(2004), in APS 2004,

Phys.Plasmas 12, 056114(2004), Nucl.Fusion 45, 918 (2005),

in 15th ISW 2005, Sellarator News 11 (Nov. 2005), in past teleconferences

But the DKES code were used in these examples to obtain M*, N*, L*

Analytical methods to obtain M*, N*,L* for applications in an integrated simulation system

please see, S.Nishimura, et al., Fusion Sci.Technol. 51, 61(Jan. 2007)

(1) Physical meaning of a constant H2 introduced by Shaing, et al. in their

bootstrap (BS) current theories.

(2) A role of numerical solvers for the DKE with the pitch-angle-scattering (PAS)

collision operator as benchmark tools to test the analytical formulas for

the neoclassical viscosities.

(3) A role of bounce- or ripple-averaging codes in the integrated simulation.

Outline (2)To include the Pfirsch-Schlüter transport in general 3-D configurations

Extension of tokamak P-S transport theory based on the moment approach. Impurity transport studies.

Some suggestions given by CHS/LHD experimental results for this extension. (1) spontaneous parallel flows of collisional impurity (2) poloidal variation of the plasma density electrostatic potential being a flux surface quantity. This kind of measurements have to be done also in future advanced stellarators (NCSX, QPS)

Steps toward this development discussions on the collaboration plan for the benchmarking using configuration

datum of NCSX, QPS, and other devices in the U.S. (1) mono-energetic viscosity coefficients M*, N*, L* (DKES, MonteCarlo) (e.g., 1/ 1/2 diffusion in CHS-qa) (2) total neoclassical fluxes a, qa, JBS for arbitrary multi-ion species plasmas (other kinetic codes in the U.S.)

Summary, Concluding Remarks

History of the “neoclassical” theory

Linked mirror

Without Ip,Without

Tokamaks Helical/Stellarators Bumpy torus

With magnetic flux surfaces with finite

Mirror

A.A.Galeev, R.Z.Segdeev, H.P.Furth, and M.N.Rosenbluth, Sov.Phys.JETP 26(1968)

PRL 22, 511 (1969)

S.P.Hirshman and D.J.Sigmar,

Nucl.Fusion 21, 1079 (1981)

B• a naea BE// = BF//a1

B• a = BF//a2

NCLASS code

W.A.Houlberg, K.C.Shaing,

S.P.Hirshman, and M.Zarnstorff,

Phys.Plasmas 4, 3230 (1997)

C.L.Hedrick, D.A.Spong,

D.E.Hastings, J.S.Tolliver, et al. in 1980’s

The bounce-averaged DKEs for the bounce-

averaged motion of ripple-trapped particles

Dama

B0 μ

μ Jμ

GXa(1/ ) =

mac

ea '

J

b• GXa (1/ )=0

Shaing-Hokin, FPSTEL, GSRAKE,

GIOTA, NEO, etc.,etc.,..

Ambipolar condition ea a(Es)=0

(1/ )

K.C.Shaing and J.D.Callen,

Phys.Fluids 26, 3315 (1983)

N.Nakajima and M.Okamoto,

J.Phys.Soc.Jpn. 61, 833 (1992)

H.Sugama and W.Horton,

Phys.Plasmas 3, 304 (1996)

Unified theory!

H.Sugama and S.Nishimura, Phys.Plasmas 9, 4637 (2002)

“How to calculate the neoclassical viscosity, diffusion, and current coefficients in general toroidal plasmas”

“with magnetic flux surfaces with finite rotational transforms 0, and with sub-sonic flows u• u 0.”

Fortunately, it seems that researchers of the bumpy tori are not included in

the participants of this conference !

Therefore, I present here only treatments of configurations with magnetic

flux surfaces with finite rotational transforms. (i.e., helical/stellarators)

When the magnetic flux surface is formed….(1) We have to calculate all of the contributions of circulating, toroidally-trapped, ripple-trapped

particles.Non-bounce-averaged motion of these particles make parallel particle and heat flow fluxes

such as the bootstrap(BS) currents and the Pfirsch-Schlüter(P-S) currents. And therefore the particle,

momentum, and energy balances in these flows have to be taken into account.

(2) Because of a characteristic of the perturbations corresponding to the ripple diffusions b• GXa(1/ )=0,

the theories and/or codes for the ripple diffusions and those for the flows had been constructed

independently at least until the former half of 1990’s.

But an integration and/or unification of these two types of theories is required now.

(For e.g., BS current calculation under the self-consistent ambipolar radial electric fields)

(3) By the way, when we consider the 0 ( bumpy torus) limit in present B-S currents and the P-S currents formulas, the current , since these formulas generally include B10/ . This singularity with respect to is caused by a fact that circulating,

toroidally-trapped particles correspond to the “loss cone” in the bumpy torus.

The unification including the bumpy torus is difficult in resent status.

Why do we discuss now on the moment method by Hirshman and Sigmar?

The generalized “correspondence principle”in the “neoclassical” theory

“correspondence principle” in our 21 century :c1 B/ +c2 B/ 0 intrinsic ambipolar

rotations minimizing the viscosities

axisymmetry (c1=0),

helical symmetry (c1c2 0),

poloidal symmetry (c2=0).

Quantum Mechanics (h 0)

Theory of Relativity (c )Newtonian Mechanics

Non-symmetric :

Viscous damping of the flows

non-ambipolar

Symmetric (axisymmetric, straight stellarators) :

Rigid rotation in the direction of the symmetry

intrinsic ambipolar

In 1960’s and 1970’s, this characteristic of the

neoclassical diffusions in the symmetric systems

was often explained as a conservation of

the canonical angular momentum of the systems.

DKES requires Bmn, but does not require

Rmn, Zmn, mn.

Local structure of the flow pattern

before the flux surface averaging has a winding determined by

·(nau//a) = ·(nau a)

(Later,we will consider this structure determining the P-S transport)

Even though it is well known that the radial diffusions are dominated by the turbulent

transport, plasma flows along the flux surfaces will be determined by the neoclassical

processes. The momentum balance including friction forces for the flows determines

impurity accumulation and/or shielding.

In contrast to toroidally rotating tokamaks, however, this winding structure will not be

simply determined by the incompressible condition ·ua=0, ·qa=0.

Integrated Simulation System(Y.Nakamura, et al., in 15th ISW 2005, LHD coordinated research)

Moment equations(to the MHD equilibrium p=J B and a higher order

corresponding to the Grad’s 13M approximation)

taking d3v v v n and d3v v n moments of distribution function and this equation itself

na fa d3v

naua v fa d3v

pa ma

3v ua

2fa d3v

Ta pa/na

Qa vv 2fa d3v

Fa1 ma v Ca(fa) d3v , Fa1

a

= 0

(momentum conservation)

Fa2 ma v mav 2

2Ta 52

Ca(fa) d3v

Here we use the particle conservation of Ca(fa)

a ma (v ua)(v ua) v ua2I/3 fa d3v

b• pa + b• • a naeaE/ / = F//a1

parallel force balanc neglecting only d/dt

Fa1a

= 0, naeaa

= 0

•J/ / = •J = c •

paa

B

B 2

(well-known Pfirsh-Schlueter(P-S) current)

radial balance and particle and energy conservation

neglecting only higher order with respect to B/B in • a and d/dt

fa

t + v •

fa

x +

ea

ma E +

v Bc

• fa

v = Ca(fa) Cab(fa1, fbM)

b

+ Cab(faM, fb1)b

na

t = •(naua)

manaua

t + ua• ua = naea E +

ua Bc

+ Fa1 pa • a

p aa

= J B

c, J (naua)

a

Moment equations for non-symmetric plasmas

(E B• na and E B• Ta are retained)

•(naua//) = •(naua ) c • na B

B 2 +

cea

•

pa B

B 2

neglecting B• =(4 /c)J• , B • pa=(4 /c)J• pa by following usual transport ordering

•(naua//) cea

pa + na ea B•1

B 2 +

c B

B 2 • na

Here,we used B/B<<1 approximation in parts including • .This relation still retains the P-S current as a consequence of the equilibrium condition.

1st term corresponds to the flow divergence given by vda• faM in DKE as shown later, and 2nd term is a part of E B• fa1

When neglecting c B

B 2 • na, this equation corresponds to •ua=0 (2002)

But this assumption was not esssential.

with a local transport anzatz of fa1

s 0 and resulting pa

pa

ss,

ss

naua// = U cea

pa

s + na ea

s + B dl

c B

B 2 B • na

l

+integration constant

Even in this approximation, the Pfirsch-Schlüter current eanaua is still unchanged.

However, flow velocities ua of individual species a is now not generally incompressible

( ·ua 0, na const). Although we previously neglected this effect in our explanation(2002)

for simplicity, it is not negligible in impure plasmas with high collision frequencies.

I skip today the derivations of moment equations in higher orders and drift kinetic equation,

although they are important and essential. (I show only a final result here.)

Linearized approximations for the Vlasov

and the collision operators(Here is an essential reason to use viscosity and friction coefficients)

d3v Cab(fa1, fbM) + Cab(faM, fb1) =0

d3v mav Cab(fa1, fbM) + Cab(faM, fb1)a,b

= 0

d3v mav 2 Cab(faM, fbM) + Cab(fa1, fbM) + Cab(faM, fb1)a,b

= 0

Cab fa1 Y lm( , ), fbM Y lm( , ), Cab faM, fb1 Y lm( , ) Y lm( , )

Y lm( , ) Plm ( ) exp(im )

collision operator in (v , , ) or (v , v , v ) coordinates in the velocity space

Cab(fa1, fbM) = Dab

(v )2

(12) +

1

12

2

2fa1 + v 2

v v 3 ma

ma+mb s

ab(v )fa1 + / /

ab(v )2

v fa1

vpitch-angle-scattering (PAS) and energy scattering (ES)

Cab(faM, fb1) = 2 ea

2eb2ln

ma

v d3v '

faM(v )mb

fb1(v ')

v '

faM(v )

v fb1(v ')

ma U (v v ')

= 4 ea

2eb2ln

ma2

v

faMhab(fb1)

v

12

2

v v faM

2gab(fb1)

v v

U (x) x 3(x2 x x ), hab(fb1) (1+ma/mb) d3v ' fb1(v ')

v v '

, gab(fb1) d3v ' fb1(v ')v v '

Linearized approximations for the Vlasov

and the collision operatorsLegendre expansion of the gyro-phase averaged distribution functions

F(x,v , ) = F(l)(x ,v , )l=0

F(l)(x ,v , ) Pl( ) 2l+1

2d Pl( ) F(x,v , )

1

1

P0( ) =1, P1( ) = , P2( ) = 32

2 12

, P3( ) = 52

3 32

, ...

(l,j)=(0,0) : na1( , )

(l,j)=(0,1) : Ta1( , )

(l,j)=(1,0) : u//a( , )

(l,j)=(1,1) : q//a( , )

F(l)(x ,v , ) is expressed by a Laguerre series of F (l)(x ,v , ) = Pl( )v lfaM Fl,jLj(l+1/2)(xa2)

j especially for l=0,1

xa2 mav 2

2Ta

L0(1/2)(xa2)=1, L1

(1/2)(xa2) = 32

xa2, L2(1/2)(xa2) = xa4

2 5

2xa2 + 15

8, ....

L0(3/2)(xa2)=1, L1

(3/2)(xa2) = 52

xa2, L2(3/2)(xa2) = xa4

2 7

2xa2 + 35

8, ....

How to treat the Legendre orders of

l=0 ( na, Ta, pa, r ' a )

l=1 ( ua, qa Qa52

paua )

l=2 ( a, a maTa

(r' a r ' aI) 52

a ),

l 3,...

For l 2, test/field ratio l2. Furthermore, PAS/ES ratio l2 (l ) in transport applications.For l=0,1, full parts of the collision operator are comparable and indispensable in viewpoint of the cons ervation laws

Linearized approximations for the Vlasov

and the collision operators

friction-flow relation

Fa1Fa2Fa3

=

l11ab l12

abl13ab

l21ab

l22ab l23

ab

l31ab l32

abl33ab

1nb

(nbub)

25 pb

qb

ub2

b

Energy exchange or scattering coeeficients for l=0 components

•(naua)

•qa

•u2a (collision)

=

0 0 0

0 0 Pab12

0 Pab21+Pab

11 Pab22+Pab

12b

na1Ta1na2

+

0 0 0

0Ta

Tb Qab

11 Qab12

0 Pab21+Pab

11 Qab22+Qab

12

nb1Tb1nb2b

Fa1 ma v Ca(fa) d3v , Fa2 ma v xa2

52

Ca(fa) d3v , Fa3 ma v xa4

2 72

xa2 +

358

Ca(fa) d3v

Friction forces as the Legendre (order l=1) and Lagurre moments of the collision operator

In the flux surface averaged part of the momentum balance using the 13M approximation,

the energy scattering are neglected, and the 3rd row and the 3rd column (Laguerre order of

j=2) in the friction-flow relation are truncated.

But we retain them when calculating the Pfirsch-Schlüter transport as show later.

S.P.Hirshman and D.J.Sigmar, Nucl.Fusion 21, 1079 (1981)

Full neoclassical matrix for general toroidal plasmas(except the bumpy torus)

collisionless regime in non-symmetric configurations (so-called ripple diffusion)

abn

qabn

= Lbn

11aa

Lbn12aa

Lbn21aa

Lbn22aa

X a1X a2

(GIOTA, NEO, GSRAKE, FPSTEL, etc.)

Pfirsch-Schlueter

aPS

qaPS

= L PS

11ab

L PS12ab

L PS21ab

L PS22ab

X b1X b2

b

ea aPS

a = 0

(intrinsic ambipolar condition due to the momentum conservation)

banana-plateau + ripple

abn

qabn

= Lbn

11ab

Lbn12ab

Lbn21ab

Lbn22ab

X b1X b2

b

+ Lbn

E1a

LbnE2a

BE/ /

JBS = LbnE1b

L bnE2b

X b1X b2

+ LbnEE BE/ /

b

a,b = electron, ion, impurities

X a1 1na

pa

s ea

s, X a2

Ta

s

symmetric configurations

ea abn

a = 0

(intrinsic ambipolar cpndition due to the momentum conservation)

M,N,L matrix and

flux surface averaged parts of the parallel momentum balance

determining nau//aB , q//aB as the integration constants of

•(nau//a), •q//a (H.Sugama and S.Nishimura, Phys.Plasmas 9, 4637(2002))

a, b = e, D+, T+, He+, He2+, Li+, Li2+, Li3+, …

A non-diagonal coupling between particle species is introduced in this step.

Given by an approximated DKE

(numerically and/or analytically)

(with energy integrations)

combined with

the friction-flow relation

B• a

B• a

abn

qabn

=

Ma1Ma2Na1Na2

Ma2Ma3Na2Na3

Na1Na2La1La2

Na2Na3La2La3

1na

nau//aB / B 2

25 pa

q//aB / B 2

1na

pa

s ea

s

Ta

sIn symmetric cases

Laj Naj Maj

ab

B 2

Ma1 Ma2Ma2 Ma3

l11ab l12

ab

l21ab l22

1nb

nbu//bB

25 pb

q//bBb

= Na1 Na2Na2 Na3

1na

pa

s ea

s

Ta

s

+ naea BE//a0

DKES as a benchmark tool

10-7

10-6

10-5

10-4

10-3

10-2

10-6 10-5 10-4 10-3 10-2 10-1 100 101

N* [m

-1]

+N*(sym)

N*(asym)

+N*

(asym)

/v [m-1]

+N*(total)

N*(total)

t=0.01

h=0.05

-10

-5

0

5

10

15

20

25

30

10-6 10-5 10-4 10-3 10-2 10-1 100 101

G(B

S)

axisymmetric

t=0.1

h=0.01

t=0.1

h=0.05

t=0.01 h

=0.05

/v [m-1]

helical symmetricE

r/v = 1.0e-3, 3.0e-3 [T]

B=B0[1 t cos B + h cos(l B n B )], l=2, n=10, B0=1T,

'=0.15T•m, '=0.4T•m, B =0, B

=4T•m are assumed.

Procedure to obtainoff-diagonal coefficients N*(driving force for BS currents)

Parallel viscosityagainst flows M*

Geometrical factorG(BS) <B2>N*/M*

N*(sym) = M*{( 'B

'B )/ B2 +H2V'/4 2}/(2 ' ')

(generating the “rigid rotation”)

10-7

10-6

10-5

10-4

10-3

10-2

10-6 10-5 10-4 10-3 10-2 10-1 100 101

M*[

m-1

T2 ]

/v [m-1]

analytical

h=0.05

h=0.01

t=0.1

These coefficients are the function of B expressed in the flux surface coordinates as

M*=M*( ’, ’,B ,B ,Bmn, /v ,Es/v), N*=N*( ’, ’,B ,B ,Bmn, /v,Es/v),

L*=L*( ’, ’,B ,B ,Bmn, /v,Es/v). The formulas are summarized in FS&T 51, 61(Jan. 2007)

Diagonal radial diffusion coefficients L*

and the boundary layer correction to the parallel viscosity force N*

(Role of existing bounce- or ripple-averaging codes in the integrated simulation

and the analytical calculation of N*)

Procedure to obtain

diagonal coefficients L* Obtained L*N* including the boundary layer

correction in the 1/ regime

10-4

10-3

10-2

10-1

100

101

102

103

10-6 10-5 10-4 10-3 10-2 10-1 100 101

t=0.1

h=0.05

Er/v = (1.0e-6) - (3.0e-3) [T]

( Xa

(asym), GXa

(asym))

M*

ripplediffusion

N*(asym)com

pone

nts

of L

*[m

1 T2 ]

/v [m 1]

10-3

10-2

10-1

100

101

102

103

10-6 10-5 10-4 10-3 10-2 10-1 100 101

L*[

m1 T

2 ]

analytical

t=0.1

h=0.05

Es/v = (1.0e-6) - (3.0e-3) [T]

/v [m 1]

-10

-5

0

5

10

15

20

25

30

10-6 10-5 10-4 10-3 10-2 10-1 100 101

/v [m-1]

t=0.1

h=0.05 DKES

analytical

Es/v = (1.0e-6) - (3.0e-3) [T]

G(B

S)

mono-energetic coefficients L* in the LMFP regime

(Importance of 1/ 1/2 diffusion in quasi-axisymmetric systems)

1/ 1/2 diffusion coefficient: extending the theory for rippled tokamaks (K.C.Shaing and J.D.Callen,Phys.Fluids 25,1012(1982))

to multi-helicity stellarators

1/ and collisionless detrapping regimes diffusion coefficients: Here we show an example using theory by Shaing and Hokin, and a scaling law by E.C.Crume,Jr. Other advanced bounce averaging codes (FPSTEL, GSRAKE, GIOTA, NEO, etc.) for 1/ , and scaling recently obtained by using DCOM, MOCA, GSRAKE for regime are also applicable for this part.

L*( ) = 6

t3/2 D

a/v

Es/v 2

We used these formulas also in the boundary layer correction term in N*,to determine the boundary collision frequency between 1/ and regimes.

L * ( 1/2) = 2.92 22 v

Da

1/2 23/4

( ') 2V '

4 2 B0

'

1/2 d2

eff3/4

2sin 1 *

N'' L

1/2T

2

1 *2 T H + 29

(1 *2) H2

L*(1/ ) = 1

2 2 2 ' 2 v

Da(K)

d B

0

2

eff3/2 16

9T

B

2

3215

1 *2 T

B

H

B

+ 0.684(1 *2) H

B

2

A Numerical Example in CHS-qa

major radius : R =1.5m

minor radius : a =0.47m

magnetic field : B 1.5T

toroidal period : N=2

rotational transform: (r=a)/2 =0.4

(a) mono-energetic radial diffusion coefficients L*,

parallel viscosity coefficients M*

(b) mono-energetic geometrical factor associated with the bootstrap currents G(BS).

S.Okamura, K.Matsuoka, S.Nishimura, et al., in 19th IAEA (Lyon, 14-19,Oct. 2002)

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-6 10-5 10-4 10-3 10-2 10-1 100 101

/v [m-1]

L* [m-1T-2]

M* [m-1T2]L

* , M*

axisymmetric (pure QA)

(a) 2b32(r/a=0.5, B=1.5T, =1%)

0

5

10

15

20

25

10-6 10-5 10-4 10-3 10-2 10-1 100 101

0.03.0E-051.0E-043.0E-041.0E-033.0E-03pure QA

G(B

S)

/v [m-1]

axisymmetric (pure QA)

Er/v[T]

2b32(r/a=0.5, B=1.5T, =1%)

(b)

On poloidally and toroidally varying parts offorce balances and flows

In this theory separating the flows and the force balances into two parts, B· · a naea BE// = BF//a1 determining nau//aB , q//aB

bi pa naeaE // = F //a1 determining

nau//a1 ,q//a1

where •(naua)=0, •qa=energy exchange

later part determining the Pfirsh-Schlüter diffusions also must be solved.

(But we didn’t discussed about it in 2002.)

Although one simplification of the formulations which often used (Hirshman-

Sigmar, 1981, Shaing-Callen, 1983, Sugama-Nishimura, 2002) may be

•ua=0, ua• na =0, it is not generally valid in non-symmetric configurations.

Only in the rigid rotation of the symmetric plasmas in the direction of the

symmetry, ua• may be ua• =0.

The procedure to solve the momentum balance must…

(1) It must automatically include the previous tokamak theory in axisymmetric

limits. Particle and energy conservation laws must include not only E B•

term but also uE B//• . uE B

// is obvious when considering the symmetric

plasmas with the rigid rotation.

(2) How is it in general non-symmetric configurations ?

(E// : by the quasi-neutrality)

(V// CaPAS)GXa = Xa = Xa

(sym)+ Xa(asym) + Xa

(avg)

When we divide the radial drift term Xa following 3 parts,the solution is given by the linear combination of those for (V// Ca

PAS)GXa(sym) = Xa

(sym)

(V// CaPAS)GXa

(asym) = Xa(asym)

(V// CaPAS)GXa

(avg) = Xa(avg)

These parts are solved by different asymptotic expansions(1/ , banana, plateau, Pfirsch-Schlüter)

0.8

0.9

1

1.1

1.2

1.3

-4 -3 -2 -1 0 1 2 3 4

B o

n a

fiel

d lin

e

The flux surface averaged parallel flows nau//aB , q//aB are

composed of two components with contrastive characteristics.

This concept of superposed components is developed in

our derivations of analytical formulas for N* and L* in 2003-2005.

/ /V v b •

v2(b • lnB)(1 2 ) a

PASC D

a

2(1 2 )

GXa=GXa(sym) + GXa

(asym) + GXa(avg), N*=N* (sym) + N* (asym) + N* (boundary)

The Er driven parallel flows also include parts corresponding to them.

Characteristics of (sym)/(asym) separation

• (sym) rigid rotation

Remaining non-bounce-averaged guiding center motion is separated into 2 parts.extension of the theory for the banana regime to collisional regimes.

K.C.Shaing, E.C.Crume, Jr., J.S.Tolliver, et al., Phys.Fluids B1, 148 (1989) K.C.Shaing, B.A.Carreras, N.Dominguez, et al., Phys.Fluids B1, 1663 (1989)

Local(short Lc)

Global(long Lc)

Bounce averaged motion of trapped particles ( ( Xa(avg)/v//)dl 0)

Ripple trapped : 1/ diffusionRipple untrapped and boundary layer : (V// Ca

PAS)GXa(avg) = 0

1/ 1/2 diffusion, parallel viscosity

Xa(avg)

CaPASGXa

(1/ )

Xa(sym) mac

2ea ' '

'B 'B

B 2 + V'

4 2H2 V / /(v B)

X a(asym)

mac

2ea ' '

B

B 2 v 2P2( ) '(1 H2)

B

B

'(1+H2)B

B

+ macea

B

B 2 v 2P2( ) G

B

B

B

G

B

B

B

X a(avg)

(asym): Er driven BS current eanaua (N.Nakajima, et al. J.Plasma Fusion Res.68, (1992))

(sym): Er driven “rigid rotation” velocity ua without friction, viscosity, and heat flow

Resulting momentum balance equations for

the poloidally and toroidally varying part ( P-S diffusion)

These equations are linear and therefore can be converted to algebraic equations by Fourier

expansions u///B=(u///B)mnexp[i(m n )], n=nmnexp[i(m n )] and so on.

B· (u///B) (V’/4 2) 1( ’m ’n)(u///B)mn, b· n (BV’/4 2) 1 ( ’m ’n)nmn, …

The perturbation functions in the limit have to become a shifted Maxwellian. However,this characteristic of thedistribution function cannot be automatically obtained by the approximated mono-energetic kinetic equations.

We use moment equations to calculate separated perturbation component (l=0,1) expressed by truncated Laguerreseries (j=0,1,2) (corresponding to that in the tokamak P-S transport theory).

pa

1 0 0

052

0

0 0358

• (nau//a)/ na(2/5)q//a/ pa(nau//a2)/ na

+ na

1 0 0

132

0

1 32

158

c B

B 2 + u//a

(rigid)b •

na1(j=0) Ta / na

Ta1(j=1)

na1(j=2) Ta / na

0 0 0

0 eab11 0

0 eab11 eab

22

nb1(j=0) Tb / nb

Tb1(j=1)

nb1(j=2) Tb / nb

b

= cea

s B •1

B 2

Ta pa

s + na ea

s

5

2 pa

Ta

s

0

na

1 1 0

0 5

2

5

2

0 0 35

8

b•

na1(j=0) Ta / na

Ta1(j=1)

na1(j=2) Ta / na

naeaE/ /

00

= F//a1F//a2F//a3

=

l11ab l12

abl13ab

l21ab

l22ab l23

ab

l31ab l32

abl33ab

(nbu/ /b)/ nb

(2/5)q//b/ pb(nbu//b2)/ nbb

Force balance

Particle and

Energy

conservation

Suggestions and Supports from experimental results (1)

(spontaneous parallel flows of collisional impurity induced by

the positive Er in the “neoclassical-ITB” operation)

0.0

0.5

1.0

0

1

2

3

0 40 80 120 160 200

wih 2nd ECHno 2nd ECH

with 2nd ECHno 2nd ECH

Te(0

) (k

eV),

ne(0

) (1

019m

-3)

time (ms)

NBI (co-injection)ECH

2nd ECH

(a)

-30

-20

-10

0

10

20

30

0 40 80 120 160 200

no 2nd ECHwith 2nd ECH

V (

km/s

)

time (ms)

2nd ECH

= 0.38

NBI (co-injection)ECH

(b)

ion diamagneticdirection

electron diamagneticdirection

-60

-40

-20

0

20

40

60

0 40 80 120 160 200

no 2nd ECHwith 2nd ECH

V(0

) (k

m/s

)

time (ms)

2nd ECH

NBI (co-injection)ECH

(c)

co

counter

0

1

2

3

0.0 0.2 0.4 0.6 0.8 1.0

with 2nd ECHno 2nd ECH

Te(k

eV)

(d)

C6+

C6+

K.Ida, et al., Phys.Rev.Lett.86(2001)3040

Vperp

"tokamak like"

direction

"helical plateau" direction

B

Vperp

"tokamak like"

direction

"helical plateau" direction

B

=

h cos(L M )

+

t cos

An analogous phenomenon was recently observed

also in LHD (M.Yoshinuma, et al.)

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1

co-NBI

ctr-NBI

-20

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

co

ctr

co-NBI

ctr-NBI

We previously showed a 2-ion-species model calculation using measured

na(r), Ta(r) to reproduce these Er, Vt. (in 14th ISW, 2003)

Next themes:

(1) extensions to general multi-species cases

(2) self-consistent determination of na(r) by including the P-S diffusion.

(i.e., impurity accumulation/shielding studies)

Suggestions and Supports from experimental results (2)poloidal variation of the plasma density

under the electrostatic potential being a flux surface quantity.

-6

-4

-2

0

2

4

6

0.85 0.9 0.95 1 1.05 1.1

(/)

R(m)

c (k

m/s

)

Lower Array

Upper Array

R(m)

0.6 0.8 1.0 1.2 1.4-0.4

-0.2

0

0.2

0.4

R(m)

(a)

(b)

S.Nishimura, K.Ida, M.Okasabe, et al.,

in 12th ISW, 1999(Madison), Phys.Plasmas 7, 437 (2000)

-30

-20

-10

0

10

20

0.85 0.9 0.95 1 1.05 1.1

ER (

V/c

m)

R (m)

-VtB

p

VpB

t

ER

diamagnetic term

(c)

0

5

10

15

20

25

30

35

40

-1 -0.5 0 0.5 1

C6

+ D

ensi

ty (

arb

.un

its)

(a)When retaining uE B· na,

uE B· Ta in the particle and

energy balances,

·(naua)=0 but ·ua 0

C6+ density nI

-100

-80

-60

-40

-20

0

20

-1 -0.5 0 0.5 1

(V)

(a)

-250

-200

-150

-100

-50

0

50

-1 -0.5 0 0.5 1

inte

gra

ted

flu

x (

arb

.un

its)

(b)

In this case, ·(naE B/B2) 0

na B2

Electrostatic potential

nIVpRdR

-10

-5

0

5

10

15

-1 -0.5 0 0.5 1

Vt (

km

/s)

Vt(B

t:CCW)

Vt(B

t:CW)

VPS

(Bt:CW) V

PS(B

t:CCW)

Vt(fit)

(b)

DKES AnalyticalORNL, NIFS,

IPPNIFS PPPL bechmarking

B data of NCSX, QPS, HSX, W7X, LHD, CHS, H-J, CHS-qa....

', ', B (Boozer), B (Boozer), Bmn

Database for NCSX, QPS, HSX, W7X, and LHD

is already prepared by Dr.Spong (ORNL).

+bounce-

averaging

codes

mono-energetic viscosity and diffusion coefficients M*, N*, L*

NEO, GIOTA,

GSRAKE,

FPSTEL, etc.

The flux surface averaged part of the moment equations

B• a naea BE// = BF//a1

B• a = BF//a2

2-ion-species (NIFS, 2003)

PENTA (ORNL, 2005)

multi-ion-species (2007?)

The poloidally and toroidally

varying part part of the

moment equations

(NIFS-PPPL, 2007?)

abn, qa

bn, JBS aPS, qa

PS

The future integrated simulation system

bechmarkingother kinetic codes in the U.S.

SummaryDevelopment of the stellarator moment method :

(1) A difficulty to treat the field particle portion of the collision operator.

An algebraic treatment of them based on the Legendre(l)-Laguerre(j)

expansions.

(2) By a characteristic of the DKE, it is better to use only the Legendre order

of l=2 component given by the approximated DKE, while we forsake the

l=0 component.

(3) Remaining components of the distribution function (l=0,1) are determined

by combining the viscosity-flow relation and the friction-flow relation.

(4) In non-symmetric plasmas, 3 mono-energetic viscosity coefficients

(M*, N*, L*) are required for this procedure, while the theories for

symmetric plasmas use only one coefficient.

(5) A reduction of the computational efforts for these coefficients is required

for a planned integrated simulation system, and therefore derivations and

tests of analytical expressions for them are now in progress.

(6) Numerical solvers for the DKE in the 3 dimensional phase space(pitch-

angle, poloidal-angle, toroidal-angle) are useful as benchmark tools in this

study. Bounce- or ripple- averaging codes are also useful for N* and L*.