Understanding+ITG+turbulence+in+stellarator+ geometries+...Conclusion •...
33
Stellarator in a Box: Understanding ITG turbulence in stellarator geometries G. G. Plunk, IPP Greifswald Collaborators: T. Bird, J. Connor, P. Helander, P. Xanthopoulos, (Yuri Turkin)
Transcript of Understanding+ITG+turbulence+in+stellarator+ geometries+...Conclusion •...
Stellarator in a Box: Understanding ITG turbulence in stellarator
geometries
G. G. Plunk, IPP Greifswald Collaborators: T. Bird, J. Connor, P.
Helander, P. Xanthopoulos, (Yuri Turkin)
W7X vs. a Tokamak
W7X vs. a Tokamak (r.m.s. ϕ)
Turbulence in stellarators is localized along the magneKc field (W7X)
Black contours show local magne2c shear
Credit: P. Xanthopoulos; see [P Helander et al 2012 Plasma Phys. Control. Fusion 54 124009]
Turbulence in stellarators is localized along the magneKc field (W7X)
Credit: P. Xanthopoulos; see [P Helander et al 2012 Plasma Phys. Control. Fusion 54 124009]
Black contours show normal curvature
Overview
• LocalizaKon of Turbulence • Part I: Local MagneKc Shear* – Model integral equaKon
– Numerical experiments with Gene
• Part II: “Bad curvature” wells – “Boxing” or “Ballooning?” – SA-‐W7X: looks like a tokamak, feels like W7X
• Part III: Nonlinear theory (TBC)
*[R. E. Waltz and A. H. Boozer, Phys. Fluids B 5, 2201 (1993)]
“Slab branch” ordering • • most unstable mode is at finite parallel wavenumber
“Toroidal branch” ordering • • fininte parallel wavenumber is generally stabilizing
Overture: The “Local” ITG Mode Wave soluKon with dispersion relaKon:
[Kadomtsev & Pogutse, Rev. Plasma Phys., Vol. 5 (1970)]
Stabilizing Effect of Parallel “LocalizaKon” on Slab Mode
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
20
12
8
5
3.5
2
Conventional Slab ITG Stability Boundaries
Tokamak esKmate:
Part I: Local MagneKc Shear.
W7X
TheoreKcal Model* * [J W Connor et al, 1980 Plasma Phys. 22 757]
Eikonal representa4on for non-‐adiabaKc part of δf1:
Linear GK EquaKon:
SoluKon:
“Ballooning”
[Source: hPp:GYRO]
Eikonal representa4on for non-‐adiabaKc part of δf1:
Linear GK EquaKon:
SoluKon:
TheoreKcal Model* * [J W Connor et al, 1980 Plasma Phys. 22 757]
Theory… Outgoing boundary condi4ons
ApproximaKon:
SoluKon becomes:
SubsKtute soluKon into quasineutrality equaKon
Integral EquaKon
W7-‐X Parameters
Also let
Aspect RaKo
So
Growth rate curves (pure slab)
0.2 0.4 0.6 0.8 1.0 1.2
0.1
0.2
0.3
0.4
0.5
0.6
second harmonicfundamental harmonic
-L L
Slab mode: half wavelength fits in the box
Growth rate curves (pure slab)
0.0 0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
2.0
2.5
NCSX Linear SimulaKons
-20
-15
-10
-5
0
5
10
15
- - /2 0 /2
Loca
l she
ar
NCSXNCSX_SYM
NCSX minus curvature
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
[cs/a
]
ky s
SLAB mode for NCSX ( =3/0)
w shearw/o shear
Eigenmode Structure for kρ ~ 1
-15
-10
-5
0
5
10
- - /2 0 /2
Re(
)SLAB mode for NCSX (ky s=1)
w shearw/o shear
Linear SimulaKons using GENE W7-‐X (including curvature)
• center of flux tube: on the helical ridge, at the “triangular plane”
Eigenmode Structure for kρ ~ 1 W7-‐X No Shear
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
- - /2 0 /2
Re(
)W7-X triangular plane UN-SHEARED
Eigenmode Structure for kρ ~ 1 W7-‐X with Shear
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
- - /2 0 /2
Re(
)
W7-X triangular plane WITH SHEAR
How much does shear “boxing” actually mamer?
0.5 1.0 1.5 2.0
0.05
0.10
0.15
0.20
0.25
W7X α = 0 (GENE)W7X α = 0 NO SHEAR (GENE)
(a/L_n = 1.0, a/L_T = 3.0)
Part II: “Bad curvature”
• “Ballooning” = mode localizaKon via global magneKc shear
• “Boxing” = mode localizaKon via sudden shear spike
• Op4on 3: Curvature wells set the parallel extent (i.e. the usual rule-‐of-‐thumb kıı = 1/qR)
Curvature Landscape of W7X
“S-‐Alpha W7X”
(A) No magneKc shear (B) a/R = 1/11 to mimic W7X (C) (un)safety factor q = 0.6 = (L/πR)W7X
(D) Periodic “Bloch wave” soluKons
“S-‐Alpha W7X”
Jukes, Rohlena, Phys. Fluids 11, 891 (1968)
(A) No magneKc shear (B) a/R = 1/11 to mimic W7X (C) (un)safety factor q = 0.6 = (L/πR)W7X
(D) Periodic “Bloch wave” soluKons
Growth Rate Comparison (a/Ln = 1.0, a/LT = 3.0)
0.5 1.0 1.50.00
0.05
0.10
0.15
0.20
“BOXED” MODE (GP)W7X α = 0 (GENE)SA-W7X (GS2)Local Disp. Reln. (kıı = 0)
Growth Rate Comparison (a/Ln = 1.0, a/LT = 3.0)
0.5 1.0 1.50.00
0.05
0.10
0.15
0.20
“BOXED” MODE (GP)W7X α = 0 (GENE)SA-W7X (GS2)Local Disp. Reln. (kıı = 0)
“Boxing”
Growth Rate Comparison (a/Ln = 1.0, a/LT = 3.0)
0.5 1.0 1.50.00
0.05
0.10
0.15
0.20
“BOXED” MODE (GP)W7X α = 0 (GENE)SA-W7X (GS2)Local Disp. Reln. (kıı = 0)
“Boxing”
“Bloching”
Growth Rate Comparison (a/Ln = 1.0, a/LT = 3.0)
0.5 1.0 1.50.00
0.05
0.10
0.15
0.20
“BOXED” MODE (GP)W7X α = 0 (GENE)SA-W7X (GS2)Local Disp. Reln. (kıı = 0)SA-W7X + SHEAR (GS2)
“Ballooning”
Part III: Nonlinear Theory (a preview)
Conclusion • Generally, magneKc geometry determines parallel variaKon of ITG
turbulence in two ways – curvature and (local or global) magneKc shear – TheoreKcal model of shear “boxed” ITG mode demonstrates the
potenKal for strong stabilizaKon. – At large k, numerical simulaKons confirm significant stabilizing
influence of local magneKc shear. • W7X ITG mode behaves as convenKonal curvature (“toroidal
branch”) mode at low k, matching closely to a ficKKous tokamak (s-‐alpha-‐W7X) ITG mode at the same parameter point.
• Summary: Simple “boxed” models capture the ITG mode in a flux tube. The modes match expectaKons from Tokamak calculaKon at a similar parameter point. – ExoKc stellarator geometry effects do not seem to strongly effect the
ITG mode. – We can understand the turbulence with exisKng theoreKcal machinery
(+ epsilon)
