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.