stellarator theory teleconference march 22, 2007, pppl ... · development of moment approach for...
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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*.