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Plasma flow characteristics vary significantly across stellarator configurations - opens opportunities for a range of physics issues to be explored: Relevance to turbulence suppression/enhanced confinement regimes Experiments/simulations show importance of E x B shearing Reduced neoclassical viscosity lowers zonal flow electric field damping rate K. C. Shaing, Phys. Plasmas 12, (2005) Magnetic perturbations shielding - island suppression Toroidal flows shield resonant magnetic perturbations at rational surfaces A. Reiman, M. Zarnstorff, et al., Nuc. Fusion 45, (2005) 360 Possible source of “self-healing” mechanism in tokamaks Impurity accumulation/shielding analysis S. Nishimura, H. Sugama, Fusion Science and Tech. 46 (2004) 77 New hidden variable in the international stellarator transport scaling database? Productive area for experiment/theory comparison stellarator analog of NCLASS code Improved bootstrap current/ambipolar electric field analysis
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Stellarator Theory TeleconferenceJanuary 26, 2006
Don SpongOak Ridge National Laboratory
Acknowledgements: Hideo Sugama, Shin Nishimura, Jeff Harris, Andrew Ware, Steve Hirshman, Wayne Houlberg, Jim Lyon, Lee
Berry, Mike Zarnstorff, Dave Mikkelson, Joe Talmadge
• Recent stellarator optimizations make cross-field neoclassical losses << anomalous losses
• Ripple reduction, quasi-symmetry, isodynamicity, omnigeneity
• However, within these devices a variety of parallel momentum transport characteristics are present
• poloidal/toroidal velocity shearing • two-dimensional flow structure within magnetic surfaces
Plasma flow properties provide an additional dimension to the stellarator
transport optimization problem
Plasma flow characteristics vary significantly across stellarator configurations - opens opportunities for a
range of physics issues to be explored:• Relevance to turbulence suppression/enhanced confinement regimes
• Experiments/simulations show importance of E x B shearing
• Reduced neoclassical viscosity lowers zonal flow electric field damping rateK. C. Shaing, Phys. Plasmas 12, (2005) 082508
• Magnetic perturbations shielding - island suppression• Toroidal flows shield resonant magnetic perturbations at rational surfaces
A. Reiman, M. Zarnstorff, et al., Nuc. Fusion 45, (2005) 360
• Possible source of “self-healing” mechanism in tokamaks• Impurity accumulation/shielding analysis
S. Nishimura, H. Sugama, Fusion Science and Tech. 46 (2004) 77
• New hidden variable in the international stellarator transport scaling database?• Productive area for experiment/theory comparison
• stellarator analog of NCLASS code• Improved bootstrap current/ambipolar electric field analysis
decorrelation time ∝1
1+C ′ωE ×B( )2
Reference: K. C. Shaing, J. D. Callen, Phys.Fluids 26, (1983) 3315.
Stellarator flow damping:• weights viscous stress-tensor components differently than
cross-field transport• removes linear dependencies (from symmetry-breaking effects)
Advances in stellarator optimization have allowed the design of 3D configurations with magnetic structures that
approximate: straight helix/tokamak/connected mirrors:
U
Ufinal
U||
U
Ufinal
U||
U||
UfinalU
HSX<a>=0.15 m<R> = 1.2 m
NCSX<a> = 0.32 m<R> = 1.4 m
QPS<a> = 0.34 m
<R> = 1 m
Quasi-helical symmetry|B| ~ |B|(m - n)
Quasi-toroidal symmetry|B| ~ |B|()
Quasi-poloidal symmetry|B| ~ |B|()
In addition, LHD and W7-X achieve closed drift surfaces by inward shifts (LHD) and finite plasma effects (W7-
X)
U
Ufinal
U||
U
Ufinal
U||
LHD<a> = 0.6 m<R> = 3.6 m
W7-X<a> = 0.55 m<R> = 5.5 m
• Based on anisotropic pressure moment of distribution function rather than momentum or particle moments– Less affected by neglect of field particle collisions on test particles
• Viscosities incorporate all needed kinetic information– Momentum balance invoked at macroscopic level rather than kinetic level– Multiple species can be more readily decoupled
• Recent work has related viscosities to Drift Kinetic Equation Solver (DKES) transport coefficients– Moments method, viscosities related to D11 D31, D33 M. Taguchi, Phys. Fluids B4 (1992) 3638 H. Sugama, S. Nishimura, Phys. of Plasmas 9 (2002) 4637
– DKES: D11 (diffusion of n,T), D13 (bootstrap current), D33 (resistivity enhancement)
W. I.Van Rij and S. P. Hirshman, Phys. Fluids B, 1, 563 (1989)
• Implemented into a suite of codes that generate the transport coefficient database, perform velocity convolutions, find ambipolar roots, and calculate flow components
– D. A. Spong, Phys. Plasmas 12 (2005) 056114– D. A. Spong, S.P. Hirshman, et al., Nuclear Fusion 45 (2005) 918
Development of stellarator moments methods
rB⋅
r∇⋅
tΠa( ) −naea BE|| = BF||a1
rB⋅
r∇⋅
tΘa( ) = BF||a2
Moments Method Closures for Stellarators
€
BF||a1
BF||a2
⎡ ⎣ ⎢
⎤ ⎦ ⎥=
l11ab −l12
ab
−l12ab l22
ab
⎡ ⎣ ⎢
⎤ ⎦ ⎥
b∑
Bu||b
25pb
Bq||b
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
The parallel viscous stresses, particle and heat flows are treated as fluxes conjugate to the forces of parallel momentum, parallel heat flow, and gradients of density, temperature and potential:
rB⋅
r∇⋅
tΠa( )
rB⋅
r∇⋅
tΘa( )
Γa
Θa / Ta
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
=
Ma1 Ma2 Na1 Na2
Ma2 Ma3 Na2 Na3
Na1 Na2 La1 La2Na2 Na3 La2 La3
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
u||aB / B2
25pa
||aB / B2
−1na
∂pa∂s
−ea∂F∂s
−∂Ta∂s
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
• Analysis of Sugama and Nishimura related monoenergetic forms of the M, N, L viscosity coefficients to DKES transport coefficients• Combining the above relation with the parallel momentum balances and friction-flow relations
Leads to coupled equations that can be solved for <u||aB>, <q||aB>, Γa, Qa
Γeqe /Te
Γ i
qi /TiJBSE
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
=
L11ee L12
ee L11ei L12
ei L1Ee
L21ee L22
ee L21ei L22
ei L2Ee
L11ie L12
ie L11ii L12
ii L1Ei
L21ie L22
ie L21ii L22
ii L2Ei
LE1e LE2
e LE1i LE2
i LEE
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
Xe1Xe2Xi1Xi2XE
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
where Xa1 ≡ −1na
∂ pa∂s
− ea∂Φ∂s
, Xa2 ≡ −∂Ta∂s
, XE = BE|| / B2 1/2
Bu||i
25pi
Bq||i
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥=
1B2
Mi1 Mi2
Mi2 Mi3
⎡⎣⎢
⎤⎦⎥−
l11ii −l12
ii
−l12ii l22
ii
⎡⎣⎢
⎤⎦⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
−1Ni1 Ni2
Ni2 Ni3
⎡⎣⎢
⎤⎦⎥Xi1
Xi2
⎡⎣⎢
⎤⎦⎥
Using solutions for an electron/ion plasma, the self-consistent electric fields, bootstrap currents and parallel flows can be obtained
(Appendix C - H. Sugama, S. Nishimura, Phys. Plasmas 9 (2002) 4637)
Radial particle flows required for ambipolar condition self-consistentenergy fluxes and bootstrap currents
Parallel mass and energy flows:
fluxsurface 1
fluxsurface 2
fluxsurface 3
fluxsurface 4
fluxsurface n. . .
processor 1 processor 2 processor 3 processor 4 processor n
DKES Transport coefficientCode: D11, D13, D33
(functions of n/v, Er/v)
results vs. n/v, Er/v concatenated together
DKES results supplementedat low/high collisionalitiesusing asymptotic forms
Energy integrations, parallel force balancerelations, ambipolarity condition solved, profilesobtained for: Er, ΓiΓe, qi, qe, <u
i>, <ui>, JE
BS
Parallel Environment for Neoclassical Transport Analysis (PENTA)
• delta-f Monte Carlo• superbanana effects (i.e.,
limits on 1/n regime)• better connection formulas• DKES extensions• convergence studies• E effects• Magnetic islands
Work in progress:
10
100
1000
107 108 109
(# of Legendre modes)*(# of Fourier modes)**2
POWER3(Seaborg)
POWER4Cheetah
POWER5(Bassi)
The neoclassical theory provides <u||B> and the ambipolar electric field Es (from solving Γi=Γe). The
final term needed is U, the Pfirsch-Schlüter flow.
ru = 1
eB−1n∂p∂s
+Es
⎛⎝⎜
⎞⎠⎟r∇s×
rBB+
u||B
B2+ 1eB
−1n∂p∂s
+Es
⎛⎝⎜
⎞⎠⎟U
⎡
⎣⎢⎢
⎤
⎦⎥⎥
rBB
<U2> can be obtained by:-solving this equation directly (with damping to resolve singularities at rational surfaces)
- matching to high collisionality DKES coefficient: <U2> = 1.5D11v/n (for large n)
rBg
r∇
UB
⎛⎝⎜
⎞⎠⎟=
rB ×
r∇g
r∇
1B2
⎛⎝⎜
⎞⎠⎟ (stellarator)
U =I ( )B2
B2−1
⎡
⎣⎢⎢
⎤
⎦⎥⎥ (tokam ak)
Integrating leads to an equation for U: r∇grv = 0
The flow model will be applied to two parameter ranges with radially continuous/stable electric field roots
• ECH regime:– n(0) = 2.5 1019 m-3, Te(0) = 1.5 keV,
Ti(0) = 0.2 keV
• ICH regime– n(0) = 8 1019 m-3, Te(0) = 0.5 keV, Ti(0) = 0.3 keV
Roots chosen to give stable restoring force for electric field perturbations through:
e⊥∂Er
∂t=Γe −Γi
Γion > Γelec
Γion < Γelec
Γion < Γelec
Γion > Γelec
ECH: electron root
ICH: ion rootE r/E
0E r/E
0
Γion=Γelec
Γion=Γelec
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
n/n(0)
Te,i
/Te,i
(0)
(χ/χedge
)1/2
Electric field profiles for ICH (ion root) and ECH (electron root) cases
-10
0
10
20
30
40
50
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ΘΠS
W7-X
NCSX
HSX
LHD
(r/redge
)1/2
-25
-20
-15
-10
-5
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(r/redge
)1/2
NCSX
HSX
W7-X
ΘΠS
LHD
ICHparameters
ECHparameters
ECH Regime
The neoclassical parallel flow velocity can vary significantly among configurations. The lowest levels are present in W7-X.
Higher u|| flows characterize HSX/NCSX
ICH regime
-1.5 104
-1 104
-5000
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
HSX
LHD
W7-XQPS NCSX
(r/redge
)1/2
-2 104
-1 104
0
1 104
2 104
3 104
4 104
5 104
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(r/redge
)1/2
HSX
LHD
NCSX
W7-X
ΘΠS
2D flow variation within a flux surface:
• Visualization of flows in real (Cartesian space)• Indicates flow shearing (geodesic) within a flux surface over shorter
connection lengths than for a tokamak– Could impact ballooning, interchange stability, microturbulence
• Implies need for multipoint experimental measurements and/or theoretical modeling support
rV =
X1eB2
r∇s×
rB +
u||BB2
rB +
X1e%UrB ωhereX1 ≡−
1n∂p∂s
−e∂F∂s
diamagneticand E x B
neoclassicalparallel flow
Pfirsch-Schlüter- for
r∇• r
rV( ) = 0
1/B B f()B (variation within a flux surface)
HSX 2D-flow streamlines
|B| variation
LHD 2D-flow streamlines
|B| variation
|B| variation
W7-X 2D-flow streamlines
QPS 2D-flow streamlines
|B| variation
|B| variation
NCSX 2D-flow streamlines
Plasma flow velocity - averages
• Components taken:
• Reduction to 1D - flux surface average:
• Due to 1/B2 variation of Jacobian, averages will be influenced by whether 1/B (EB, diamagnetic) or B (neo. Parallel, PfirshSchlüter) terms dominate
rV =
X1eB2
r∇s×
rB +
u||BB2
rB +
X1e%UrB ωhereX1 ≡−
1n∂p∂s
−e∂F∂s
diamagneticand E x B
neoclassicalparallel flow
Pfirsch-Schlüter- for
r∇• r
rV( ) = 0
e =
r∇ /
r∇ and e =
r∇ /
r∇
K =
1′Vdθdζ g∫∫ K( )
Comparison of flux-averaged toroidal flow components (contra-variant) among devices indicates NCSX and
HSX have the largest toroidal flows
ICH parametersECH parameters
-1.5 104
-1 104
-5000
0
5000
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(r/redge
)1/2
LHD
ΘΠS
HSX
NCSX W7-X
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
ICH parameters
Comparison of flux-averaged poloidal flow components (contra-variant) among devices indicates
QPS has largest poloidal flows
ECH parameters
0
1 104
2 104
3 104
4 104
5 104
6 104
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
LHD
QPS
W7-XHSX
NCSX
(r/redge
)1/2
<u•e>(m/seχ)
-5000
-4000
-3000
-2000
-1000
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
<u•e>(m/seχ)
ΘΠS
LHD
HSX
W7-X
NCSX
(r/redge
)1/2
Comparison of shearing rates from ambient flows with ITG growth rates:
gITG = (CS/ LT)(LT/R)m
where 0 < m < 1, CS = sound speedfrom J. W. Connor, et al., Nuclear Fusion 44 (2004) R1
Recent stellarator DTEM-ITGgrowth rates
from G. Rewoldt, L.-P. Ku, W. M. Tang, PPPL-4082, June, 2005
1000
104
105
106
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
QPS
LHD
HSX
W7-X
NCSX
(ρ/ρedge)1/2
γITG
(μ = 1)
γITG
(μ = 0)
}QPS, NCSX, W7-X, LHDg=0.2 to 1.6 105 sec-1
HSXg=4 105 sec-1
Transport barrier condition: shearing rate > gITG
Even without externaltorque, QPS, NCSX,LHD, W7-X shearingrates can exceedgDTEM-ITG at edge region
|d<v
EB>/
dr|, g I
TG
Flow variations within flux surfaces can also impact MHD ballooning/interchange thresholds:
t A0
)b •
r∇( )
ru where τ A0 = R / vA0
QPS flow shearing along B• Maximum flow shearing rates are ~0.5 of Alfvén time
• Could influence MHD stability thresholds
inboard
outboard
outboard
10-5
0.0001
0.001
0.01
0.1
1
W7-X HSX NCSX LHD QPS
ECHICH
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QPS-ICH
Er is modified by parallel momentum source(40 keV H0 beam, F||
(i,e) = 0.1, 0.2, 0.5, 0.7 Nt/m3)
NCSX-ICH
rB⋅
r∇⋅
tΠa( )
rB⋅
r∇⋅
tΘa( )
Γa
Θa / Ta
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
=
Ma1 Ma2 Na1 Na2
Ma2 Ma3 Na2 Na3
Na1 Na2 La1 La2Na2 Na3 La2 La3
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
u||aB / B2
25pa
||aB / B2
−1na
∂pa∂s
−ea∂F∂s
−∂Ta∂s
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
Same viscosity coefficients thatrelate gradients in n, T, to u||
also imply that changes in u||
modify Γ
Conclusions• A self-consistent ambipolar model has been developed for the
calculation of flow profiles in stellarators– Ambipolar electric field solution with viscous effects
– Predicts profiles of Er, <u||>, <u>, and <u>; 2D flow variations
• Magnetic field structure influences flows in quasi–symmetric stellarators
– QPS: poloidal flows dominate over toroidal flows
– W7-X: poloidal flows dominate, but flows reduced from other systems
– NCSX: toroidal flows dominate except near the edge
– HSX: flows are dominantly helical/toroidal
– LHD: toroidal flows dominate for ICH case, poloidal flows for ECH case
Conclusions (contd.)• Ambient flow shearing rates approach levels that could suppress turbulence
– ITG, ballooning
– Further stellarator-specific turbulence work needed, but with 2D neoclassical flow fields
– Collaborations initiated with LHD, HSX to further apply model and look for correlations with confinement data
• The sensitivity of flow properties to magnetic structure is an opportunity for stellarators– Experimental tests of flow damping in different directions
– Possible new hidden variable in confinement scaling data
– Stellarator optimizations/flexibility studies -> should use E B shear (turbulence suppression to complement neoclassical transport reduction
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