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

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

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Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade Matthias H olzl, Isabel Krebs, Karl Lackner, Sibylle G unter 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H olzl Nonlinear ELM Simulations

slide-1
SLIDE 1

Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade

Matthias H¨

  • lzl, Isabel Krebs, Karl Lackner, Sibylle G¨

unter

slide-2
SLIDE 2

1

Introduction

2

Model

3

Results

4

Outlook

5

Summary

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 2

slide-3
SLIDE 3

1

Introduction

2

Model

3

Results

4

Outlook

5

Summary

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 3

slide-4
SLIDE 4

Introduction H-Mode

⊲ High Confinement Mode first observed in ASDEX [F. Wagner, et al. PRL, 49, 1408 (1982)] ⊲ Sudden rise of edge gradients and confinement time ⊲ Extremely beneficial for fusion

Formation of density pedestal during L-H transition [M. E. Manso. PPCF, 35, B141 (1993)]

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 4

slide-5
SLIDE 5

Introduction ELMs

0.4 0.6 0.8 Te [keV] 2.8 3.0 2.9 time [s]

⊲ Edge Localized Modes (ELMs) appear in H-Mode ⊲ Periodic collapse of pedestal ⊲ Up to 10% of stored energy lost ⊲ Critical for ITER → mitigation

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 5

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SLIDE 6

Introduction ELMs (2)

Electron temperature at 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

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 6

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SLIDE 7

Introduction Localization

⊲ ASDEX Upgrade: Expanded and localized ELMs observed

#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

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 7

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SLIDE 8

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. to be published (2013)] Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 8

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SLIDE 9

Introduction Theory

Poloidal flux perturbation caused by a ballooning instability (linear MHD calculation)

Non-linear simulations

⊲ Low mode numbers ⊲ Localization ⊲ ELM sizes ⊲ Mitigation

. . .

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 9

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SLIDE 10

1

Introduction

2

Model

3

Results

4

Outlook

5

Summary

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 10

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SLIDE 11

Model JOREK

⊲ Originally developed at CEA Cadarache

[G. Huysmans and O. Czarny. Nucl.Fusion, 47, 659 (2007);

  • O. Czarny and G. Huysmans. J.Comput.Phys, 227, 7423 (2008)]

⊲ Non-linear reduced MHD in toroidal geometry (next slide) ⊲ Full MHD in development ⊲ Toroidal Fourier decomposition ⊲ Bezier finite elements ⊲ Fully implicit time evolution ⊲ Selected results:

  • Pellet ELM triggering [G. Huysmans, et al. 23rd IAEA, THS/7-1 (2010)]
  • ELMs in JET [S. J. P

. Pamela, et al. PPCF, 53, 054014 (2011)]

  • RMP field penetration [M. Becoulet, et al. 24th IAEA, TH/2-1 (2012)]

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 11

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SLIDE 12

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

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 12

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SLIDE 13

Model Typical code run

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

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 13

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SLIDE 14

Model Typical code run

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

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 13

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SLIDE 15

Model Typical code run

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

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 13

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SLIDE 16

1

Introduction

2

Model

3

Results

4

Outlook

5

Summary

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 14

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SLIDE 17

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

q = toroidal turns

poloidal turns

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 15

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SLIDE 18

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

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 16

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SLIDE 19

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

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 16

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SLIDE 20

Results Energy Timetraces

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 220 230 240 250 260 270 280 290 300 magnetic energies [a.u.] time [µs] n=10

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17

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SLIDE 21

Results Energy Timetraces

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 220 230 240 250 260 270 280 290 300 magnetic energies [a.u.] time [µs]

  • ther

n=10 n= 9 ⊲ Simulation including n = 0, 1, . . . , 15, 16 ⊲ n = 9 and 10 are linearly most unstable

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17

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SLIDE 22

Results Energy Timetraces

1e-22 1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 220 230 240 250 260 270 280 290 300 magnetic energies [a.u.] time [µs]

  • ther

n=10 n= 9 n= 2 n= 1 ⊲ Simulation including n = 0, 1, . . . , 15, 16 ⊲ n = 9 and 10 are linearly most unstable ⊲ low-n modes driven non-linearly to large amplitudes ⊲ Can we understand this with a simple model?

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17

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SLIDE 23

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 ⊲ Simplified case with n = 0, 4, 8, 12, 16 ⊲ Quadratic terms lead to mode coupling (n1, n2) ↔ n1 ± n2 ⊲ For instance: (16, 12) ↔ 4

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 18

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SLIDE 24

Results Mode Interaction Model (2)

⊲ 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

[I. Krebs, et al. to be published (2013)] Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 19

slide-25
SLIDE 25

Results Mode Interaction Model (2)

⊲ 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, et al. to be published (2013)] Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 19

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SLIDE 26

Results Mode Interaction Model (2)

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) ⊲ Explains low-n features in experimental observations

→ Poster: Isabel Krebs, P19.15 (Thursday)

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 20

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SLIDE 27

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 the fully non-linear phase.

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 21

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SLIDE 28

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 the fully non-linear phase.

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 21

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SLIDE 29

Results Filament Formation

Detached filaments quickly lose their pressure due to fast parallel heat conduction. Substructures appear in divertor heat flux patterns.

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 22

slide-30
SLIDE 30

1

Introduction

2

Model

3

Results

4

Outlook

5

Summary

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 23

slide-31
SLIDE 31

Outlook ELM Mitigation

0.3 0.4 0.5 [MJ]

MHD stored energy

upper row lower row

  • 1

1 [kA]

Saddle coil currents

ASDEX Upgrade #27585, |n|=2

Outer divertor current

0.4 0.6 0.8 [keV]

Electron temperature (pedestal top)

2.8 3.1 3.2 3.3 3.4 3.0 2.9 6 12 [kA] time [s]

large type-I ELMs small ELMs

16 perturbation coils are currently installed in ASDEX Upgrade

[W. Suttrop, et al. 24th IAEA, EX/3-4 (2012)]

⊲ ELM mitigation with magnetic perturbations ⊲ Important option for ITER

→ Simulate penetration and interaction with ELMs (with M. Becoulet and F. Orain)

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 24

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SLIDE 32

Outlook Continue Investigations

⊲ Heat flux pattern ⊲ Full ELM crash ⊲ ELM types ⊲ Two fluid ⊲ Rotation ⊲ . . .

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 25

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SLIDE 33

Outlook Resistive Walls

Discretization of first ITER wall in the STARWALL code which describes vacuum region and wall currents

[P . Merkel and M. Sempf. 21st IAEA, TH/P3-8 (2006);

  • E. Strumberger, et al. 38th EPS, P5.082 (2011)]

⊲ Interaction of instabilities with conducting structures ⊲ Coupling via natural boundary condition [M. H¨

  • lzl, et al. JPCS, 401, 012010 (2012a)]

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 26

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SLIDE 34

Outlook Resistive Walls (2)

10

  • 5

10

  • 4

10

  • 3

10

  • 2

wall resistivity [Ω]

10

1

10

2

10

3

10

4

growth rate [s

  • 1]

CEDRES++ JOREK+STARWALL

ITER wall

⊲ Vertical Displacement Event in ITER-like limiter plasma ⊲ Good agreement with CEDRES++ code ⊲ Next Steps: X-point cases, 3D wall

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 27

slide-35
SLIDE 35

1

Introduction

2

Model

3

Results

4

Outlook

5

Summary

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 28

slide-36
SLIDE 36

Summary

⊲ Edge Localized Modes in H-Mode plasmas ⊲ Mitigation important for ITER ⊲ Non-linear simulations in realistic geometry ⊲ Localization ⊲ Low-n features ⊲ Filament formation

→ ELM mitigation with magnetic perturbations → ELM types, heat flux patterns, . . . → Resistive Walls

0.4 0.6 0.8 Te [keV] 2.8 3.0 2.9 time [s]

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

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 29

slide-37
SLIDE 37

Summary

⊲ Edge Localized Modes in H-Mode plasmas ⊲ Mitigation important for ITER ⊲ Non-linear simulations in realistic geometry ⊲ Localization ⊲ Low-n features ⊲ Filament formation

→ ELM mitigation with magnetic perturbations → ELM types, heat flux patterns, . . . → Resistive Walls

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

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 29

slide-38
SLIDE 38

Summary

⊲ Edge Localized Modes in H-Mode plasmas ⊲ Mitigation important for ITER ⊲ Non-linear simulations in realistic geometry ⊲ Localization ⊲ Low-n features ⊲ Filament formation

→ ELM mitigation with magnetic perturbations → ELM types, heat flux patterns, . . . → Resistive Walls

0.3 0.4 0.5 [MJ]

MHD stored energy

upper row lower row

  • 1

1 [kA]

Saddle coil currents

ASDEX Upgrade #27585, |n|=2

Outer divertor current

0.4 0.6 0.8 [keV]

Electron temperature (pedestal top)

2.8 3.1 3.2 3.3 3.4 3.0 2.9 6 12 [kA] time [s]

large type-I ELMs small ELMs

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 29

slide-39
SLIDE 39

References

  • M. Becoulet, et al. 24th IAEA, TH/2-1 (2012).
  • J. E. Boom, et al. 37th EPS, P2.119 (2010).
  • O. Czarny and G. Huysmans. J.Comput.Phys, 227, 7423 (2008).
  • M. H¨
  • lzl, et al. 38th EPS, P2.078 (2011).
  • M. H¨
  • lzl, et al. JPCS, 401, 012010 (2012a).
  • M. H¨
  • lzl, et al. Phys.Plasmas, 19, 082505 (2012b).
  • G. Huysmans and O. Czarny. Nucl.Fusion, 47, 659 (2007).
  • G. Huysmans, et al. 23rd IAEA, THS/7-1 (2010).
  • I. Krebs, et al. to be published (2013).
  • M. E. Manso. PPCF, 35, B141 (1993).

P . Merkel and M. Sempf. 21st IAEA, TH/P3-8 (2006).

  • S. J. P

. Pamela, et al. PPCF, 53, 054014 (2011).

  • H. R. Strauss. Phys.Fluids, 19, 134 (1976).
  • E. Strumberger, et al. 38th EPS, P5.082 (2011).
  • W. Suttrop, et al. 24th IAEA, EX/3-4 (2012).
  • F. Wagner, et al. PRL, 49, 1408 (1982).
  • R. P

. Wenninger, et al. Nucl.Fusion, 42, 114025 (2012).

  • R. P

. Wenninger, et al. to be published (2013).

Slides and Publications

http://me.steindaube.de

Co-Authors

  • I. Krebs, K. Lackner, S. G¨

unter

Acknowledgements

  • G. Huysmans, E. Strumberger, P

. Merkel,

  • R. Wenninger

Matthias H¨

  • lzl

Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 30