[PPT] - Non-linear Simulations of Edge Localized Modes in ASDEX Upgrade PowerPoint Presentation

SLIDE 1

Non-linear Simulations of Edge Localized Modes in ASDEX Upgrade

Matthias H¨

lzl

(Postdoc at IPP Garching)

SLIDE 2

1 Introduction

2 Model

3 Results

4 Outlook

5 Summary

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 2

SLIDE 3

1 Introduction

2 Model

3 Results

4 Outlook

5 Summary

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 3

SLIDE 4

Introduction Edge Localized Modes

Electron temperature measured with ECE-Imaging at an ELM onset in ASDEX Upgrade: Dominant toroidal Fourier harmonic n ≈ 11

[J. E. Boom, et al. 37th EPS, P2.119 (2010)] Te [eV]

q=4 1.0 1.5 2.0

1.0
0.5

0.0 0.5 1.0

R (m) Z (m)

200 400 600 800 Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 4

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Introduction Localization

⊲ ASDEX Upgrade: Expanded and localized ELMs observed (distribution)

#25764@1.7574s 6 7 5 t-tELM [ms]

0.2
0.1

0.1 0.2 dB/dt [a.u.] + ΦMAP [rad]

Signature of a Solitary Magnetic Perturbation in ASDEX Upgrade

[R. P . Wenninger, et al. Nucl.Fusion, 42, 114025 (2012)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 5

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Introduction Low-n Harmonics

toroidal mode number n Fourier harmonics δBav [mT] π/2 π 3π/2 2π

0.4

0.4 φ [rad] 0.4 0.2 4 2 6 8 15 10 5 2

TCV #42062

# dominant toroidal harmonic 1 3

Example for ELM signature with strong low-n component Histogram of dominant components in a TCV discharge (23 ELMs)

[R. P . Wenninger, et al. Nucl.Fusion (to be submitted)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 6

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1 Introduction

2 Model

3 Results

4 Outlook

5 Summary

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 7

SLIDE 8

Model JOREK

⊲ Originally developed at CEA Cadarache [G. Huysmans and O. Czarny. Nucl.Fusion, 47, 659 (2007)] ⊲ Non-linear reduced MHD in toroidal geometry (next slide) ⊲ Two-fluid extensions ⊲ Full MHD in development ⊲ Bezier finite elements + Toroidal Fourier decomposition ⊲ Fully implicit time evolution ⊲ GMRES with physics-based preconditioning

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 8

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Model Reduced MHD Equations

∂Ψ ∂t = ηj − R [u, Ψ] − F0 ∂u ∂φ ∂ρ ∂t = −∇ · (ρv) + ∇ · (D⊥∇⊥ ρ) + Sρ ∂(ρT) ∂t = −v · ∇(ρT) − γρT∇ · v + ∇ ·

K⊥∇⊥ T + K||∇||T
+ ST

eφ · ∇ ×

ρ∂v

∂t = −ρ(v · ∇)v − ∇p + j × B + µ∆v

B ·
ρ∂v

∂t = −ρ(v · ∇)v − ∇p + j × B + µ∆v

j ≡ −jφ = ∆∗Ψ

ω ≡ −ωφ = ∇2

pol u

Variables: Ψ, u, j, ω, ρ, T, v|| Definitions: B ≡ F0

R eφ + 1 R∇Ψ × eφ

and v ≡ −R∇u × eφ + v||B

[H. R. Strauss. Phys.Fluids, 19, 134 (1976)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 9

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Model Typical code run

⊲ Initial grid (Grids shown with reduced resolution) ⊲ Flux aligned grid including X-point(s) ⊲ Equilibrium flows ⊲ Time-integration

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 10

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Model Typical code run

⊲ Initial grid (Grids shown with reduced resolution) ⊲ Flux aligned grid including X-point(s) ⊲ Equilibrium flows ⊲ Time-integration

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 10

SLIDE 12

Model Typical code run

⊲ Initial grid (Grids shown with reduced resolution) ⊲ Flux aligned grid including X-point(s) ⊲ Equilibrium flows ⊲ Time-integration

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 10

SLIDE 13

1 Introduction

2 Model

3 Results

4 Outlook

5 Summary

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 11

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Results Overview

⊲ ELMs in typical ASDEX Upgrade H-mode equilibrium ⊲ Many toroidal harmonics ⊲ Resistivity too large by factor 10 due to numerical constraints (improving)

0.2 0.4 0.6 0.8 1

ΨN

0.2 0.4 0.6 0.8 1

normalized quantities

ρ T 1 2 3 4 5 6 7

q-profile

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 12

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Results Poloidal Flux Perturbation

n = 0, 8, 16

⊲ Red/blue surfaces correspond to 70 percent of maximum/minimum values [M. H¨

lzl, et al. 38th EPS, P2.078 (2011);
M. H¨
lzl, et al. Phys.Plasmas, 19, 082505 (2012b)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 13

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Results Poloidal Flux Perturbation

n = 0, 1, 2, 3, 4, . . . , 16

⊲ Red/blue surfaces correspond to 70 percent of maximum/minimum values ⊲ Localized due to several strong harmonics with adjacent n

⇒ Similar to Solitary Magnetic Perturbations in ASDEX Upgrade

[M. H¨

lzl, et al. 38th EPS, P2.078 (2011);
M. H¨
lzl, et al. Phys.Plasmas, 19, 082505 (2012b)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 13

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Results Mode Interaction Model

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 420 440 460 480 500 520 540 560 580 magnetic energies [a.u.] time [µs] n=16 n=12 n= 8 n= 4 ⊲ Non-linear drive of low-n modes ⊲ Start with simplified case including n = 0, 4, 8, 12, 16 (periodicity 4)

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 14

SLIDE 18

Results Mode Interaction Model

⊲ Quadratic terms lead to mode coupling (n1, n2) ↔ n1 ± n2 ⊲ For instance: (16, 12) ↔ 4 ⊲ Model assuming mode rigidity and fixed background:

˙ A4 =

linear

γ4 A4 +

non-linear interaction

γ8,−4 A8 A4 + γ12,−8 A12 A8 + γ16,−12 A16 A12

˙ A8 = γ8 A8 + γ4,4A4 A4 + γ12,−4A12 A4 + γ16,−8 A16 A8 ˙ A12 = γ12 A12 + γ4,8 A4 A8 + γ16,−4 A16 A4 ˙ A16 = γ16 A16 + γ8,8 A8 A8 + γ4,12 A4 A12

⊲ Linear growth rates from JOREK simulation + Energy conservation ⊲ Determine few free parameters by minimizing quadratic differences [I. Krebs. Master’s thesis, LMU, Munich (2012)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 15

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Results Mode Interaction Model

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 420 440 460 480 500 520 540 560 580 magnetic energies [a.u.] time [µs] n=16 n=12 n= 8 n= 4 ⊲ Non-linear drive recovered ⊲ Saturation not recovered (of course)

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 16

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Results Mode Interaction Model

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 500 520 540 560 580 magnetic energies [a.u.] time [µs] n=10 n=9 n=2 n=1 ⊲ Applied to full simulation with n = 0 . . . 16 ⊲ Explains low-n features in experimental observations

[I. Krebs, et al. Phys.Plasmas (to be submitted)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 17

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Results Mode Interaction Model

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 500 520 540 560 580 magnetic energies [a.u.] time [µs] n=10 n=9 n=2 n=1 ⊲ Applied to full simulation with n = 0 . . . 16 ⊲ Explains low-n features in experimental observations

[I. Krebs, et al. Phys.Plasmas (to be submitted)]

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 17

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Results Non-linear phase

1e-20 1e-15 1e-10 1e-05 1 300 400 500 600 700 800 energies [a.u.] time [µs] Emag,00 Emag,08 Ekin,00 Ekin,08 ⊲ Energy time traces during an ELM crash ⊲ Simulation with n = 0, 8 ⊲ Several bursts

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 18

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Results Non-linear phase

1e-07 1e-06 1e-05 300 400 500 600 700 800 energies [a.u.] time [µs] Emag,08 Ekin,00 Ekin,08 ⊲ Energy time traces during an ELM crash ⊲ Simulation with n = 0, 8 ⊲ Several bursts

Matthias H¨

lzl, I. Krebs, K. Lackner, S. G¨

unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 18

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