1

Comparison ofMicroinstability Propertiesfor Stellarator Magnetic

Geometries*

G. Rewoldt, L.-P. Ku,and W.M. Tang

Princeton UniversityPlasma Physics Laboratory

With thanks to W. Guttenfelder,V. Kornilov, N. Nakajima,

J. Nuehrenberg, D. Spong,J.N. Talmadge, H. Yamada,

and M. Zarnstorff,for providing input data and for

discussions

*Work supported by U.S.D.O.E. Contract No.

DE-AC02-76-CHO-3073

2

Introduction• Differing regions of localization of

trapped electrons, and of “good”(stabilizing) and “bad”(destabilizing) magneticcurvature, along magnetic fieldlines, can affect linear growthrates and real frequencies of IonTemperature Gradient (ITG)modes and Trapped-ElectronModes (TEMs) in tokamaks andstellarators.

• Here, we compare ITG-TEMmode properties for stellaratorcases corresponding to differentpresent and planned stellarators,and to correspondingaxisymmetric cases.

3

Introduction-2• Use stellarator (non-

axisymmetric) version of FULLcode, in electrostatic,collisionless limit, on single,chosen magnetic surface.Includes trapped particles,complete FLR (Bessel functions),transit frequency, bouncefrequency, and magnetic driftfrequency resonances, for allincluded species, in high-n,radially-local limit (ballooningrepresentation).

4

Cases• LHD• NCSX (non-zero total current)• NCSX-J=0 (zero total current)• Wendelstein 7-X (W7-X)• HSX• QPS• NCSX-SYM (like NCSX, but keep

n=0 components only in MHDequilibrium)

• NCSX-TOK (like NCSX-SYM, butchange to tokamak-like q or ιprofile)

• NCSX-BETA (like NCSX, but with4% volume-average β)

5

Cases-2• All cases have same geometric-

center major radius• All cases have same B0 at

geometric center• All cases have same density and

temperature profile shapes, andthus pressure profile shape(taken from LHD inward-shiftedexperimental shot), but with verylow volume-average β = 0.1%(except for NCSX-BETA case)

6

Cases-3• Shapes of last closed flux

surfaces shown in Fig. 1,with magnetic field strengthindicated (red=strongest,blue=weakest)

• Rotational transform ιprofiles shown in Fig. 2

• MHD equilibrium fromVMEC, with resultsprocessed throughTERPSICHORE and VVBAL

NCSX & NCSX-J=0& NCSX-BETA

LHD

W7-X

HSX

QPS

NCSX-SYM& NCSX-TOK

Fig. 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.0 0.2 0.4 0.6 0.8 1.0

Iota

S

LHD

HSX

W7-X

NCSX-TOK NCSX

NCSX-BETA

NCSX-J=0

QPSNCSX-SYM

Fig. 2

7

Results• All calculations for magnetic

surface with s = 0.74 ∝ (r/a)2, withni = ne and Ti=Te (includeelectrons & deuterium only)

• Start with α = ζ-qθ = 0, θ0 (=θk) =0,η = ηi = ηe = 2.66 (LHDexperimental value), and k⊥(θ = 0)ρi = 0.30, with ηj = d ln Tj / d ln nj

• ITG mode and TEM mode“hybridize” to form single ITG-TEM root!

• First, maximize linear growth rateγ over α (gives αMax for eachcase)

• Second, maximize linear growthrate γ over θ0 (gives θ0

Max for eachcase) for α = αMax

8

Results-2• Third, maximize linear growth

rate γ over k⊥(θ=0)ρi (gives k⊥Max(θ=0)ρi for each case) for α

= αMax and θ0 = θ0Max

• Finally, vary η = ηi = ηe for α =αMax and θ0 = θ0Max and k⊥(θ=0)ρi= k⊥Max(θ=0)ρi

• Magnetic field strength variationalong field line (in ballooningrepresentation) for α = αMax andθ0 = θ0Max shown for nine cases inFig. 3, with illustrative trapped-particle orbit extents

• Corresponding variation ofk⊥2(θ)/n2 shown in Fig. 4

• Corresponding variation ofcurvature function k⊥•{b×[(b•∇)b]}/n (where b=B/B) shown inFig. 5

W7-X|B(θ)| (T)

|B(θ)| (T)

|B(θ)| (T)

|B(θ)| (T)

|B(θ)| (T)

|B(θ)| (T)

|B(θ)| (T)LHD

NCSX

NCSX-J=0

HSX

QPS

NCSX-SYM

0 3-3θ (radians)

21-1-2

1.2

1.7

1.5

1.0

1.5

2.01.3

1.5

1.7

1.5

1.7

1.3

1.5

1.3

1.5

1.71.2

1.6

2.0

NCSX-TOK

NCSX-BETA

|B(θ)| (T)

1.3

1.8

1.5

|B(θ)| (T)

1.3

1.7

1.5

Fig. 3

0

400

800

k (θ

) / n

(m

)⊥2

2-2

LHD

80

40

0

NCSX

k (θ

) / n

(m

)⊥2

2-2

40

20

0

k (θ

) / n

(m

)⊥2

2-2 NCSX-J=0

0

50

100

150W7-X

k (θ

) / n

(m

)⊥2

2-2

HSX60

40

20

0

k (θ

) / n

(m

)⊥2

2-2

QPS180

120

60

0

k (θ

) / n

(m

)⊥2

2-2

NCSX-SYM16001200800400

0

k (θ

) / n

(m

)⊥2

2-2

0 3-3θ (radians)

21-1-2

NCSX-TOK

NCSX-BETA70

40

0

k (θ

) / n

(m

)⊥2

2-2

k (θ

) / n

(m

)⊥2

2-2

400

200

0

Fig. 4

-606

1218

“bad curvature”

“good curvature”

LHD

“good curvature”

“bad curvature”0.5

0

-0.5

NCSX

“bad curvature”

“good curvature”-1.2

0

1.2 NCSX-J=0

-2

0

“bad curvature”

“good curvature”

W7-X

“bad curvature”

“good curvature”

HSX

-2

-1

0

1

QPS

-2.4

-1.2

0

“bad curvature”

“good curvature”

-6

-4

-2

0“bad curvature”

“good curvature”

NCSX-SYM

0 3-3θ (radians)

21-1-2

“bad curvature”

“good curvature”“bad curvature”

“good curvature”

NCSX-TOK

NCSX-BETA

-1.2

-0.8

-0.4

0

-0.6-0.3

0

0.30.6

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

k ⋅

{b×[

(b⋅∇

)b]}/

n⊥

Fig. 5

9

Results-3• Variation of growth rate γ and

real frequency ωr with Mα (whereM = number of periods) forstarting values for nine casesshown in Fig. 6

• Corresponding variation of γ andωr with θ0 (for α = αMax and otherstarting values) shown in Fig. 7

• Corresponding variation of γ andωr with k⊥(θ=0)ρi (for α = αMax andθ0 = θ0

Max and other startingvalues) shown in Fig. 8

• Linear eigenfunctions along fieldline (in ballooning representation)(for α = αMax and θ0 = θ0

Max andk⊥(θ=0)ρi = k⊥Max(θ=0)ρi) shownin Fig. 9

0

1

2

0 1 2 3

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

γ (10

5 sec

-1)

M α = M (ζ - q θ) (radians)

(a)

-2

-1

0

1

0 1 2 3

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

ωr (1

05 sec

-1)

M α = M (ζ - q θ) (radians)

(b)

Fig. 6

0

1

2

0 1 2 3

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

γ (10

5 sec

-1)

θo = θk (radians)

(a)

-2

-1

0

1

0 1 2 3

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

ωr (1

05 sec

-1)

θo = θk (radians)

electron diamagnetic direction

ion diamagnetic direction

(b)

Fig. 7

0

2

4

0 0.4 0.8

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

γ (10

5 sec

-1)

k⊥(θ = 0) ρi

(a)

-6

-4

-2

0

0 0.4 0.8

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

ωr (1

05 sec

-1)

k⊥(θ = 0) ρi

electron diamagnetic direction

ion diamagnetic direction

(b)

Fig. 8

0(arb

itrar

y un

its)

Re φ

Im φ

LHD

Re φ

Im φ0

(arb

itrar

y un

its) NCSX

Re φ

Im φ(arb

itrar

y un

its)

0

NCSX-J=0

Re φ

Im φ

W7-X

0(arb

itrar

y un

its)

Re φ

Im φ

HSX

0(arb

itrar

y un

its)

Re φ

Im φ

|φ|

0

(arb

itrar

y un

its) QPS

NCSX-SYM

0 4π2π-2π−4πθ (radians)

Im φRe φ

|φ|

0

(arb

itrar

y un

its)

NCSX-BETA

NCSX-TOK

0

0Re φ

Re φ

Im φ

Im φ

|φ|

|φ|

(arb

itrar

y un

its)

(arb

itrar

y un

its)

Fig. 9

10

Results-4• Variation of γ and ωr with η = ηi =ηe (for α = αMax and θ0 = θ0

Max

and k⊥(θ=0)ρi = k⊥Max(θ=0)ρi)shown in Fig. 10

• ITG destabilization mechanism(non-resonant) dominant at largeηi - strongly unstable for all cases

• Collisionless TEM destabilizationmechanism (resonant with orbit-average magnetic drift frequency)dominant for small ηi - stronglyunstable for some cases andweakly unstable or stable forother cases

• Small-ηi result depends ondetails of localization of trappedelectrons and of localization ofgood and bad curvature!

0

2

4

0 1 2 3 4

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

γ (10

5 sec

-1)

η = ηi = ηe

(a)

-4

0

4

8

0 1 2 3 4

LHDNCSXNCSX-J=0W7-XHSXQPSNCSX-SYMNCSX-TOKNCSX-BETA

ωr (1

05 sec

-1)

η = ηi = ηe

electron diamagnetic direction

ion diamagnetic direction

(b)

Fig. 10

11

Conclusions• For NCSX, NCSX-J=0, and LHD

cases, γ rises as η approacheszero, indicating rather strongdestabilization from trapped-electron magnetic drifts (badcurvature)

• For W7-X, NCSX-SYM, andNCSX-BETA cases, γ falls as ηapproaches zero, indicating weaktrapped-electron destabilization.For HSX and NCSX-TOK cases,γ also falls as η approaches zero,but overall destabilization islarger

• For QPS case, γ also falls as ηapproaches zero, and ITG-TEMmode is completely stable for η <0.5!

12

Conclusions-2• NCSX-SYM case, with negative

magnetic shear (in tokamaksense) less unstable than NCSX-TOK case, with positive shear, asexpected

• NCSX-BETA case (with largevolume-average β) less unstablethan NCSX case, (with almostzero β), showing Shafranov-shift-like reduction in trapped-electronbad curvature!

• For all of these magneticgeometries, this ITG-TEM modeis strongly unstable linearly if thetemperature gradient is large!