Modeling Of Anomalous Transport in ECRH
Plasmas At HSX

Walter Guttenfelder
HSX Plasma Laboratory
Electrical & Computer Engineering, UW-Madison

Acknowledgements: Travel Support from NIFS &ITC17/ISHW2007 - Thank You!!
ITC17/ISHW2007 Ceratopia Toki, Gifu, Japan October 17, 2007

Outline• Transport in HSX plasma– Mostly anomalous• Modeling stellarator turbulent transport – stealing fromtokamaks (Weiland ITG/TEM model)– Using 3D linear gyrokinetics (GS2) to justify approximations usingtokamak model• 1-D predictive transport modeling of HSX profiles andconfinement• Conclusions

HSX is Helically Symmetric in |B|•

Quasihelical symmetry (QHS) reduces direct loss orbits (IAEA 2002), flow
damping (PRL, 2005), and neoclassical transport (PRL, 2007)
QHSMirror
<R>

1.2 m

<a>

0.12 m

ι

1.05
→1.12

B00.5 -1.0 TECRH
28 GHz

100 kW

N=4, m=1ι≈1ιeff = |N-mι| ≈ 3

B = B0 [1 − εh cos(N − mι)φ]B = B0 [1 − εh cos(N − mι)φ + εM cos(Nφ)]

Typical plasma parameters•<ne> ≤ 6 × 1012 cm-3•

Te ~ 0.5 - 2.5 keV >> Ti ~ 20-100 eV→ Opportunity to study electron heat transport in LMFPν ∗e =

ν e / εH
≤ 0 .1v
Te1/ 2εHqeff R

Reduction of Core Neoclassical Transport isObserved with Quasihelical Symmetry• Similar Te achieved in QHS with half the power of Mirror→ Neoclassical χe reduced in core via quasihelical symmetry (reduced 1/ν rippletransport)•Transport is anomalous over most of minor radius

Modeling Anomalous Transport•“State of the art” drift wave turbulent transport models (ITG/TEM/ETG) exist fortokamaks (MMM / Weiland, IFS-PPPL, GLF23, TGLF)•

These are quasi-linear transport models (γlin/k⊥2) that have been tweaked to best
match non-linear simulations

Modeling Anomalous Transport•“State of the art” drift wave turbulent transport models (ITG/TEM/ETG) exist fortokamaks (MMM / Weiland, IFS-PPPL, GLF23, TGLF)•

These are quasi-linear transport models (γlin/k⊥2) that have been tweaked to best
match non-linear simulations•

A number of 3D linear & non-linear stability calculations (ITG/TEM/ETG) nowexist (Rewoldt et al.; Kendl & Wobig; Jost et al.; Belli et al.; Jenko, Kendl, Merz; Rafiq, Nadeem, et al.; Kuroda et al.;Sugama et al.; Yamagishi et al.; + others)•

However, no anomalous (ITG/TEM/ETG) transport models (usable for predictivesimulations) have been formally developed for generic 3D stellarator
configurations→ No non-linear simulations for HSX (previous linear calculations by Jost; Rafiq;Rewoldt)

Modeling Anomalous Transport•“State of the art” drift wave turbulent transport models (ITG/TEM/ETG) exist fortokamaks (MMM / Weiland, IFS-PPPL, GLF23, TGLF)•These are quasi-linear transport models (γlin/k⊥2) that have been tweaked to best
match non-linear simulations•

A number of 3D linear & non-linear stability calculations (ITG/TEM/ETG) nowexist (Rewoldt et al.; Kendl & Wobig; Jost et al.; Belli et al.; Jenko, Kendl, Merz; Rafiq, Nadeem, et al.; Kuroda et al.;Sugama et al.; Yamagishi et al.; + others)•However, no anomalous (ITG/TEM/ETG) transport models (usable for predictivesimulations) have been formally developed for generic 3D stellarator
configurations→ No non-linear simulations for HSX (previous linear calculations by Jost; Rafiq;Rewoldt)
REMAINDER OF THIS TALK•

Test of the axisymmetric Weiland ITG/TEM anomalous transport model fordominant electron heated HSX stellarator plasmas•

Beginning tests of the validity of the Weiland ITG/TEM model against 3D
gyrokinetic linear stability calculations (GS2)GS2 Is Used For Stellarator Microstability•

Flux tube gyrokinetic code in ballooning
coordinates (Kotchenreuther et al., 1995; Dorland

NCSX linear stability(Belli et al., APS 2001)

et al., 2000)– Initial value (can do non-linear)– 3D equilibrium input (shaped tokamak orstellarator, Belli et al., 2001)– Electromagnetic (β)– Momentum conserving collision operator– No assumptions on k⊥ (ITG/TEM/ETG)•Used for 12+ years by 30+ users foranalysis on numerous tokamaks•

Benchmarked in numerous scenarios,
including stellarator configurations(NCSX, HSX)

•We are using GS2 electrostatically forlinear calculations → compare toWeiland model

QHS linear stability
ηe=ηi=2.66
Rewoldt et al.
(2005)

What is the Weiland Model?
•A linear fluid model for toroidal ITG and TEM instabilities, including:
••••
•

•

Multiple ion ITG
Collisionless TEM
Collisional stabilization of TEM
Electromagnetic (finite β)Parallel ion dynamics (momentum transport)

Heat and particle transport predictions come from quasi-linear mixing length
estimates which compare well to limited non-linear simulations (Nordman et al., 1990;
Dimits et al., 2000)•Easy to solve (quick) – useful for predictive transport modelingWhat is the Weiland Model?
•A linear fluid model for toroidal ITG and TEM instabilities, including:
••
•
••

•

Multiple ion ITG
Collisionless TEM
Collisional stabilization of TEM
Electromagnetic (finite β)Parallel ion dynamics (momentum transport)

Heat and particle transport predictions come from quasi-linear mixing length
estimates which compare well to limited non-linear simulations (Nordman et al., 1990;
Dimits et al., 2000)•Easy to solve (quick) – useful for predictive transport modeling•

For HSX, using collisionless electrostatic form including particle, and ion andelectron heat transport
Geometry parameters required are trapped electron fraction (ft) and“appropriate” toroidal drift scale length (LB = R for a tokamak)•⎞⎡ ωr , γ ⎤ ρs2 cs ⎛ aa a a Te⎜
F,,,=⋅
ρ,,f,k⎢χ , χ , D ⎥ L⎜ L L L L T t ⊥ s ⎟⎟⎣ e i ⎦n
⎝ Te Ti n B i⎠•

What are the “appropriate” geometry parameters for HSX?

Microstability Estimates Can Be Made Using Axisymmetric
Models With “Quasisymmetric” Approximation•

3D stability calculations find most unstableeigenmodes (ITG/TEM) ballooning in the low field,
bad curvature region in HSX (also seen in Rewoldt 2005;
Rafiq 2006)•Dominant particle trapping comes from helicalripple, εH (0.14⋅r/a = 1.4⋅r/R)•

Reduced connection length, Lc = qeffR = R/|N-mι|≈ R/3, leads to very low collisionality electrons
across the minor radius → CTEM (Te >> Ti)ν e / εHν ∗e =≤ 0 .1v Te1/ 2
εHqeff R•

Normal curvature rotates helically, with bad
curvature following the location of low fieldstrength
κN,max ~ 1/45 cm-1 ≠ 1/R (R=120 cm)
To account for toroidal drifts in drift wave models,R → R/3••

Comparison Of Linear Growth Rates Between
Weiland Model And 3D Gyrokinetics•Weiland ITG/TEM model is used with approximations:ft ≈ 2ε T → 2ε H , κN & |∇B|/B ~ 1/R → 3/R•

Linear growth rates from Weiland and 3D GS2 are in agreement near
experimental gradients (a/Ln, a/LTe = 2 → 5, largest difference ~30%)
Larger deviations exist near marginal stability
Weiland growth rates 2× smaller without “quasisymmetric” approximation••

GS2 - HSX

Weiland - HSX

Weiland - TOKPredictive 1D Transport Modeling Is Performed
Using “Multi-Mode” Approach•(χ,D) = (χ,D)neoclassical + (χ,D)anomalous– Neoclassical from DKES (Hirshman et al., 1986)– Anomalous from Weiland ITG/TEM and RBM (MMM, Bateman et al., Phys. Plasmas, 1998)•Electron energy source from ECRH
– Profile from ray tracing– Total absorbed power from measurement
Particle source adjusted to minimize difference in ne (within factor of ~2 of DEGAS
calculations)•••Radiation, electron-ion coupling negligibly small
Boundary conditions from experiment•1D flux-surface averaged transport equations integrated
⎞∂n
∂
1 ∂ ⎛
2
∇ρ + V ( n ) n ∇ρ ⎟⎟ = ∑ S(ρ)
n+V' ⎜⎜ − D∂tV' ∂ρ ⎝∂ρ
⎠⎞∂T
11 ∂ ⎛3 ∂2∇ρ + V ( nT ) nT ∇ρ ⎟⎟ = ∑ P(ρ)
V' ⎜⎜ − nχn T+∂ρ
eV ' ∂ρ ⎝2 ∂t⎠

Density Profiles In Good Agreement•

Core Te underpredicted in QHS

QHS
ne (1012 cm-3)

Simulation

Experiment
PECRH = 44 kW

Te (keV)ρDensity Profiles In Good Agreement••

Core Te underpredicted in QHS
Global confinement predicted within 10%QHS

ITER Physics Basis Figures of Merit (1999)

ne (1012 cm-3)(STD (n e ) =( )21n eexp∑,jN j

Simulation(ExperimentPECRH = 44 kW)

21sim
n e (ρ j ) − n exp
∑e, j
N j

STD (Te ) == 9.1%)

2
1
sim
Te (ρ j ) − Teexp∑,jN j= 40%( )2
1Teexp∑,jN jTe (keV)ρPECRH (kW)τexp (ms)τsim (ms)Δτ/τexp

26

3.0

3.3

10%

44

2.6

2.4-8%

70

1.9

1.90%

100

1.6

1.5-6%

Predicted Confinement Time Scaling Close To
Experiment•

Experimental power scaling slightly weaker than simulation
τEexp,diamagnetic ~ P-0.36
τEsim~ P-0.57

QHS
B=1T
<n> ~ 4 × 1012 cm-3

3D Linear Stability Similar Between QHS &Mirror
ρ = 0.86••|B| no longer symmetric
Local geometry similar in ballooning
region→ Results in similar growth rates• Similar to results with approximate
DTEM response in HSX (Rafiq &Hegna, Phys. Plasmas 2006)

GS2 (3D)

Mirror

Profiles In Reasonable Agreement for Mirror•Overall confinement overpredicted at low power
Mirrorne (1012 cm-3)

STD (ne) = 11%

Simulation

STD (Te) = 30%
Experiment
PECRH = 44 kW

Te (keV)ρPECRH (kW)τexp (ms)τsim (ms)Δτ/τexp

26

1.9

3.1

63%

44

1.9

2.4

26%

70

1.6

1.8

13%

100

1.6

1.60%

Summary•

Anomalous transport is significant in both quasisymmetric(QHS) and non-symmetric (Mirror) configurations– With low collisionality electrons, CTEM expected to be dominant
instability•First test of Weiland ITG/TEM model for dominant electron
heated stellarator plasmas– With “quasisymmetric” approximations, linear growth rates from
Weiland model agree within ~30% of 3D gyrokinetic (GS2) linear
stability calculations (near experimental gradients)– Density profiles and QHS energy confinement times predictedwithin ~10%
– Electron temperature profiles further off (~40%)Δγlin < 30%
σne,στ < 10%
σTe < 40%