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