Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade. Matthias Hölzl, Isabel Krebs, Karl Lackner, Sibylle Günter

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1 Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade Matthias Hölzl, Isabel Krebs, Karl Lackner, Sibylle Günter

2 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Introduction 2 Model 3 Results 4 Outlook 5 Summary

3 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Introduction 2 Model 3 Results 4 Outlook 5 Summary

4 Introduction H-Mode Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ 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)]

5 Introduction ELMs Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Te [kev] 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

6 Introduction ELMs (2) Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Z (m) 1.0 Te [ev] Electron temperature at ELM onset in ASDEX Upgrade: Dominant toroidal Fourier harmonic n 11 [J. E. Boom, et al. 37th EPS, P2.119 (2010)] q= R (m) 0

7 Introduction Localization Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ ASDEX Upgrade: Expanded and localized ELMs observed db/dt [a.u.] + Φ MAP [rad] t-t ELM [ms] Signature of a Solitary Magnetic Perturbation in ASDEX Upgrade [R. P. Wenninger, et al. Nucl.Fusion, 42, (2012)]

8 Introduction Low-n Harmonics Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Fourier harmonics δb av [mt] π/2 π 3π/2 2π φ [rad] toroidal mode number n Example for ELM signature with strong low-n component # 5 0 TCV # dominant toroidal harmonic Histogram of dominant components in a TCV discharge (23 ELMs) [R. P. Wenninger, et al. to be published (2013)]

9 Introduction Theory Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Poloidal flux perturbation caused by a ballooning instability (linear MHD calculation) Non-linear simulations Low mode numbers Localization ELM sizes Mitigation...

10 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Introduction 2 Model 3 Results 4 Outlook 5 Summary

11 Model JOREK Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ 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, (2011)] RMP field penetration [M. Becoulet, et al. 24th IAEA, TH/2-1 (2012)]

12 Model Reduced MHD Equations Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Ψ t = ηj R [u, Ψ] F u 0 φ ρ t = (ρv) + (D ρ) + S ρ (ρt) = v (ρt) γρt v + (K T + K T ) + S T { t e φ ρ v } = ρ(v )v p + j B + µ v t { B ρ v } = ρ(v )v p + j B + µ v t j j φ = Ψ ω ω φ = 2 pol u Variables: Ψ, u, j, ω, ρ, T, v Definitions: B F 0 R e φ + 1 R Ψ e φ and v R u e φ + v B [H. R. Strauss. Phys.Fluids, 19, 134 (1976)]

13 Model Typical code run Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Initial grid (Grids shown with reduced resolution) Flux aligned grid including X-point(s) Radial and poloidal grid meshing Equilibrium flows Time-integration Postprocessing

14 Model Typical code run Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Initial grid (Grids shown with reduced resolution) Flux aligned grid including X-point(s) Radial and poloidal grid meshing Equilibrium flows Time-integration Postprocessing

15 Model Typical code run Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Initial grid (Grids shown with reduced resolution) Flux aligned grid including X-point(s) Radial and poloidal grid meshing Equilibrium flows Time-integration Postprocessing

16 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Introduction 2 Model 3 Results 4 Outlook 5 Summary

17 Results Overview Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ ELMs in typical ASDEX Upgrade H-mode equilibrium Many toroidal harmonics Resistivity too large by factor 10 due to numerical constraints (improving) q-profile normalized quantities ρ T Ψ N q = toroidal turns poloidal turns

18 Results Poloidal Flux Perturbation Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ n = 0, 8, 16 Red/blue surfaces correspond to 70 percent of maximum/minimum values [M. Hölzl, et al. 38th EPS, P2.078 (2011); M. Hölzl, et al. Phys.Plasmas, 19, (2012b)]

19 Results Poloidal Flux Perturbation Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ 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. Hölzl, et al. 38th EPS, P2.078 (2011); M. Hölzl, et al. Phys.Plasmas, 19, (2012b)]

20 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Results 1e-06 1e-08 Energy Timetraces n=10 magnetic energies [a.u.] 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e time [µs]

21 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Results 1e-06 1e-08 Energy Timetraces other n=10 n= 9 magnetic energies [a.u.] 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e time [µs] Simulation including n = 0, 1,..., 15, 16 n = 9 and 10 are linearly most unstable

22 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Results magnetic energies [a.u.] 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e-22 Energy Timetraces other n=10 n= 9 n= 2 n= time [µs] 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?

23 Results Mode Interaction Model Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ magnetic energies [a.u.] 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18 n=16 n=12 n= 8 n= 4 1e-20 1e time [µs] Simplified case with n = 0, 4, 8, 12, 16 Quadratic terms lead to mode coupling (n 1, n 2 ) n 1 ± n 2 For instance: (16, 12) 4

24 Results Mode Interaction Model (2) Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Model assuming mode rigidity and fixed background: linear non-linear interaction {}}{{}}{ Ȧ 4 = γ 4 A 4 + γ 8, 4 A 8 A 4 + γ 12, 8 A 12 A 8 + γ 16, 12 A 16 A 12 [I. Krebs, et al. to be published (2013)]

25 Results Mode Interaction Model (2) Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Model assuming mode rigidity and fixed background: linear non-linear interaction {}}{{}}{ Ȧ 4 = γ 4 A 4 + γ 8, 4 A 8 A 4 + γ 12, 8 A 12 A 8 + γ 16, 12 A 16 A 12 Ȧ 8 = γ 8 A 8 + γ 4,4 A 4 A 4 + γ 12, 4 A 12 A 4 + γ 16, 8 A 16 A 8 Ȧ 12 = γ 12 A 12 + γ 4,8 A 4 A 8 + γ 16, 4 A 16 A 4 Ȧ 16 = γ 16 A 16 + γ 8,8 A 8 A 8 + γ 4,12 A 4 A 12 Linear growth rates from JOREK simulation + Energy conservation Determine few free parameters by minimizing quadratic differences [I. Krebs, et al. to be published (2013)]

26 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Results Mode Interaction Model (2) magnetic energies [a.u.] 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18 n=16 n=12 n= 8 n= 4 1e-20 1e time [µs] Non-linear drive recovered Saturation not recovered (of course) Explains low-n features in experimental observations Poster: Isabel Krebs, P19.15 (Thursday)

27 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Results Non-linear phase 1 1e-05 E mag,00 E mag,08 E kin,00 E kin,08 energies [a.u.] 1e-10 1e-15 1e time [µs] Energy time traces during the fully non-linear phase.

28 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Results Non-linear phase 1e-05 E mag,08 E kin,00 E kin,08 energies [a.u.] 1e-06 1e time [µs] Energy time traces during the fully non-linear phase.

29 Results Filament Formation Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Detached filaments quickly lose their pressure due to fast parallel heat conduction. Substructures appear in divertor heat flux patterns.

30 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Introduction 2 Model 3 Results 4 Outlook 5 Summary

31 Outlook ELM Mitigation ASDEX Upgrade #27585, n =2 [ka] 1 upper row Saddle coil currents 0 lower row -1 [MJ] 0.5 MHD stored energy [kev] Electron temperature (pedestal top) perturbation coils are currently installed in ASDEX Upgrade [ka] 12 Outer divertor current 6 [W. Suttrop, et al. 24th IAEA, EX/3-4 (2012)] time [s] large type-i ELMs small ELMs. ELM mitigation with magnetic perturbations. Important option for ITER Simulate penetration and interaction with ELMs (with M. Becoulet and F. Orain) Matthias Ho lzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/

32 Outlook Continue Investigations Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Heat flux pattern Full ELM crash ELM types Two fluid Rotation...

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. Ho lzl, et al. JPCS, 401, (2012a)] Matthias Ho lzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/

34 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Outlook Resistive Walls (2) 10 4 CEDRES++ JOREK+STARWALL growth rate [s -1 ] ITER wall wall resistivity [Ω] Vertical Displacement Event in ITER-like limiter plasma Good agreement with CEDRES++ code Next Steps: X-point cases, 3D wall

35 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Introduction 2 Model 3 Results 4 Outlook 5 Summary

36 Summary Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Edge Localized Modes in H-Mode plasmas Mitigation important for ITER Non-linear simulations in realistic geometry Localization Low-n features Filament formation Te [kev] time [s] ELM mitigation with magnetic perturbations ELM types, heat flux patterns,... Resistive Walls Z (m) Te [ev] q= R (m) 0

37 Summary Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ Edge Localized Modes in H-Mode plasmas Mitigation important for ITER Non-linear simulations in realistic geometry Localization Low-n features Filament formation magnetic energies [a.u.] 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e time [µs] n=16 n=12 n= 8 n= 4 ELM mitigation with magnetic perturbations ELM types, heat flux patterns,... Resistive Walls

38 Summary ASDEX Upgrade #27585, n =2 [ka] 1 upper row Saddle coil currents 0 lower row -1 [MJ] 0.5 MHD stored energy Edge Localized Modes in H-Mode plasmas. Mitigation important for ITER 0.6 Electron temperature (pedestal top) [ka]. [kev] 0.8 Outer divertor current large type-i ELMs. Non-linear simulations in realistic geometry. Localization. Low-n features. Filament formation time [s] small ELMs ELM mitigation with magnetic perturbations ELM types, heat flux patterns,... Resistive Walls Matthias Ho lzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/

39 Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/ 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. Hölzl, et al. 38th EPS, P2.078 (2011). M. Hölzl, et al. JPCS, 401, (2012a). M. Hölzl, et al. Phys.Plasmas, 19, (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, (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, (2012). R. P. Wenninger, et al. to be published (2013). Slides and Publications Co-Authors I. Krebs, K. Lackner, S. Günter Acknowledgements G. Huysmans, E. Strumberger, P. Merkel, R. Wenninger

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