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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!