Non-linear MHD Simulations of Edge Localized Modes in ASDEX Upgrade Matthias Hölzl, Isabel Krebs, Karl Lackner, Sibylle Günter
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 2 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 3 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary
Introduction H-Mode Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 4 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)]
Introduction ELMs Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 5 0.8 Te [kev] 0.6 0.4 2.8 2.9 3.0 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
Introduction ELMs (2) Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 6 Z (m) 1.0 Te [ev] 800 0.5 600 Electron temperature at ELM onset in ASDEX Upgrade: Dominant toroidal Fourier harmonic n 11 [J. E. Boom, et al. 37th EPS, P2.119 (2010)] 0.0-0.5 400 200 q=4-1.0 1.0 1.5 2.0 R (m) 0
Introduction Localization Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 7 ASDEX Upgrade: Expanded and localized ELMs observed db/dt [a.u.] + Φ MAP [rad] 7 6 5 #25764@1.7574s -0.2-0.1 0 0.1 0.2 t-t ELM [ms] Signature of a Solitary Magnetic Perturbation in ASDEX Upgrade [R. P. Wenninger, et al. Nucl.Fusion, 42, 114025 (2012)]
Introduction Low-n Harmonics Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 8 0.4 Fourier harmonics δb av [mt] 0-0.4 0 π/2 π 3π/2 2π φ [rad] 0.4 0.2 0 0 2 4 6 8 toroidal mode number n Example for ELM signature with strong low-n component 15 10 # 5 0 TCV #42062 1 2 3 dominant toroidal harmonic Histogram of dominant components in a TCV discharge (23 ELMs) [R. P. Wenninger, et al. to be published (2013)]
Introduction Theory Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 9 Poloidal flux perturbation caused by a ballooning instability (linear MHD calculation) Non-linear simulations Low mode numbers Localization ELM sizes Mitigation...
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 10 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary
Model JOREK Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 11 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)]
Model Reduced MHD Equations Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 12 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)]
Model Typical code run Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 13 Initial grid (Grids shown with reduced resolution) Flux aligned grid including X-point(s) Radial and poloidal grid meshing Equilibrium flows Time-integration Postprocessing
Model Typical code run Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 13 Initial grid (Grids shown with reduced resolution) Flux aligned grid including X-point(s) Radial and poloidal grid meshing Equilibrium flows Time-integration Postprocessing
Model Typical code run Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 13 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 Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 14 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary
Results Overview Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 15 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 7 6 5 4 3 2 1 0 1 0.8 0.6 0.4 0.2 ρ T 0 0 0.2 0.4 0.6 0.8 1 Ψ N q = toroidal turns poloidal turns
Results Poloidal Flux Perturbation Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 16 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, 082505 (2012b)]
Results Poloidal Flux Perturbation Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 16 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, 082505 (2012b)]
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17 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-22 220 230 240 250 260 270 280 290 300 time [µs]
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17 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-22 220 230 240 250 260 270 280 290 300 time [µs] Simulation including n = 0, 1,..., 15, 16 n = 9 and 10 are linearly most unstable
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17 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= 1 220 230 240 250 260 270 280 290 300 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?
Results Mode Interaction Model Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 18 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-22 420 440 460 480 500 520 540 560 580 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
Results Mode Interaction Model (2) Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 19 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)]
Results Mode Interaction Model (2) Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 19 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)]
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 20 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-22 420 440 460 480 500 520 540 560 580 time [µs] Non-linear drive recovered Saturation not recovered (of course) Explains low-n features in experimental observations Poster: Isabel Krebs, P19.15 (Thursday)
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 21 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-20 300 400 500 600 700 800 time [µs] Energy time traces during the fully non-linear phase.
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 21 Results Non-linear phase 1e-05 E mag,08 E kin,00 E kin,08 energies [a.u.] 1e-06 1e-07 300 400 500 600 700 800 time [µs] Energy time traces during the fully non-linear phase.
Results Filament Formation Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 22 Detached filaments quickly lose their pressure due to fast parallel heat conduction. Substructures appear in divertor heat flux patterns.
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 23 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary
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 0.4 0.3 [kev] 0.8 0.6 Electron temperature (pedestal top) 0.4 16 perturbation coils are currently installed in ASDEX Upgrade [ka] 12 Outer divertor current 6 [W. Suttrop, et al. 24th IAEA, EX/3-4 (2012)] 0 2.8 2.9 3.0 3.1 3.2 3.3 3.4 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, 02/2013 24
Outlook Continue Investigations Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 25 Heat flux pattern Full ELM crash ELM types Two fluid Rotation...
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, 012010 (2012a)] Matthias Ho lzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 26
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 27 Outlook Resistive Walls (2) 10 4 CEDRES++ JOREK+STARWALL growth rate [s -1 ] 10 3 10 2 ITER wall 10 1 10-5 10-4 10-3 10-2 wall resistivity [Ω] Vertical Displacement Event in ITER-like limiter plasma Good agreement with CEDRES++ code Next Steps: X-point cases, 3D wall
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 28 1 Introduction 2 Model 3 Results 4 Outlook 5 Summary
Summary Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 29 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] 0.8 0.6 0.4 2.8 2.9 3.0 time [s] ELM mitigation with magnetic perturbations ELM types, heat flux patterns,... Resistive Walls Z (m) 1.0 0.5 0.0 Te [ev] 800 600 400-0.5 200 q=4-1.0 1.0 1.5 2.0 R (m) 0
Summary Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 29 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-22 420 440 460 480 500 520 540 560 580 time [µs] n=16 n=12 n= 8 n= 4 ELM mitigation with magnetic perturbations ELM types, heat flux patterns,... Resistive Walls
Summary ASDEX Upgrade #27585, n =2 [ka] 1 upper row Saddle coil currents 0 lower row -1 [MJ] 0.5 MHD stored energy 0.4 0.3 Edge Localized Modes in H-Mode plasmas. Mitigation important for ITER 0.6 Electron temperature (pedestal top) 0.4 12 [ka]. [kev] 0.8 Outer divertor current 6 0 2.8 2.9 3.0 large type-i ELMs. Non-linear simulations in realistic geometry. Localization. Low-n features. Filament formation 3.1 3.2 3.3 3.4 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, 02/2013 29
Matthias Hölzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 30 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, 012010 (2012a). M. Hölzl, 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. Günter Acknowledgements G. Huysmans, E. Strumberger, P. Merkel, R. Wenninger