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Synthesis of full MHD simulation results of neoclassical tearing modes in ITER geometry. H.Lütjens, J.F.Luciani CPHT-Ecole polytechnique UMR-7644 du CNRS Palaiseau, France. Outline. XTOR and theory NTM: nonlinear thresholds NTM: saturation NTM: toroïdal interaction. XTOR equations:. - PowerPoint PPT Presentation
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Synthesis of full MHD simulation results of neoclassical tearing modes in ITER geometry
H.Lütjens, J.F.Luciani
CPHT-Ecole polytechnique
UMR-7644 du CNRS
Palaiseau, France
Outline
• XTOR and theory• NTM: nonlinear thresholds• NTM: saturation• NTM: toroïdal interaction
XTOR equations:
Full toroïdal geometry.
Mapping:
Bootstrap:
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ρ Dr v
Dt=
r J ×
r B −∇p +∇ν∇
r v
∂t
r B =∇ × (
r v ×
r B ) −∇ ×η (
r J −
r J boot )
∂tT = −r v .∇T − (Γ −1)T∇
r v +∇χ ⊥∇T +
r B .∇χ //
r B .∇T
B2+ H
∂tρ = −r v .∇ρ − ρ∇
r v +∇D⊥∇ρ + Q
H = −∇χ ⊥∇Tequil; η equil (Jφ − Jφ,boot )equil = const.
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ηequil (r); Tequil (r)Spitzer,p _ edge
⏐ → ⏐ ⏐ η equil (Tequil ) t≠0 ⏐ → ⏐ η (T(t))
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rJ boot (t) = fbs
r J boot,equil .∇ r p(t) / p'equil
r B (t) /
r B (t)
Nonlinear theory
• Generalized Rutherford equation
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τ r
1.22
dw
dt= Δ'(w) + Δ'GGJ (w) + Δ'boot (w) (+ non MHD)
(Rutherford (1973),White(1977),Thyagaraja (1981) Militello et al., Escande et al., Hastie et al. (2004),
with Kotschenreuter (1985), Lütjens & al.(2001), Fitzpatrick (1995))
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Δ'GGJ = 6.35DR
w2 + 0.65wc2
Δ'boot = 6.35Roq
Boss
Jboot,o
w
w2 + 1.8wc( )2and
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wc = 2 2χ ⊥
χ //
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 4rsR
nss
; ss =rsq'
q
(curvature)
(bootstrap)
Equilibrium (CHEASE):
ITER:A=3; =1.75; =0.4
NTM: linear stability thresholds
•S=107
•Open:•Closed:•ITER:m/n=4/3 (circles) m/n=3/2 (squares) m/n=2/1 (triangles) TS: m/n=2/1 (diamonds)
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χ // /χ ⊥ =108
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χ // /χ ⊥ = 6.25.106
•Threshold with given geometry and depends on S.
•For ITER, S>1010----> threshold at fbs >> 2
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χ // /χ ⊥
NTM: nonlinear stability thresholds
• NTM dynamics (m=4/n=3) about its nonlinear threshold (ITER)
•Thresholds: numerics (XTOR) vs. Theory•Closed symbols: with linear correction i.e
.
•Opens symbols: without linear corrections
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τ r
1.22
dw
dt= Δ'eff
w
w + wlin
+ Δ'boot ; Δ'eff = Δ'+ 2π3
2DR
Wc
NTM: saturation
• Comparison of NTM saturation levels in ITER geometry with leading edge theory:
XTOR gives much smaller saturation sizes than predicted with Rutherford
Validity field of Rutherford vs. Numerical XTOR results:
•Rutherford ---> Boundary layer approximation ---> w and Δ’ are small
•XTOR saturation:
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m
rs
w ≈1;ψ '
ψw ≈1
•Theory derived with constant approx. Shape of (r)•XTOR does not satisfy these assumption.
NTM: toroïdal interactions
Equilibrium bootstrap:(~20%)
Example:Growth of 2 NTM’sm/n=4/3 et 3/2
•NTM’s with m/n=2/1,3/2,4/3•Single, double or triple mode simulations•Initial perturbation W_ or Wsat.•S=107 and •Iter geometry
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χ // /χ ⊥ =108
Observations:
•Within the framework of the XTOR model, and theSimulations times (about 60000 τa), no toroïdal coupling was observed. No interaction as measured in experiments•In multiple mode simulations, island overlap cause large stochastics zones, which empty the central pressure.
Conclusions•Full numerical simulations show a reasonable agreement with generalized Rutherford’s equation in the small island regime. Acceptable results are obtained for nonlinear NTM thresholds.
•In the NTM saturation regime, simulation results and theory disagree. XTOR results give much smaller saturation sizes than theory.
•We have not observed toroïdal mode coupling effects in multiple NTM runs.
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