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

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 DPG Spring Meeting, Jena, 02/2013 2

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 3

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

Introduction ELMs 0.8 T e [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 Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 5

Introduction ELMs (2) Z (m) Te [eV] 1.0 800 0.5 600 0.0 400 Electron temperature at ELM onset in ASDEX Upgrade: Dominant toroidal Fourier harmonic n ≈ 11 -0.5 200 [J. E. Boom, et al. 37th EPS, P2.119 (2010)] q=4 -1.0 R (m) 0 1.0 1.5 2.0 Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 6

Introduction Localization ⊲ ASDEX Upgrade: Expanded and localized ELMs observed #25764@1.7574s 7 dB/dt [a.u.] + Φ MAP [rad] 6 Signature of a Solitary Magnetic Perturbation in ASDEX Upgrade [R. P . Wenninger, et al. Nucl.Fusion, 42, 114025 (2012)] 5 -0.2 -0.1 0 0.1 0.2 t-t ELM [ms] Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 7

Introduction Low-n Harmonics 0.4 δB av [mT] 0 -0.4 0 π/2 π 3π/2 2π φ [rad] Example for ELM signature with 0.4 strong low-n component harmonics 0.2 Fourier 0 0 2 4 6 8 toroidal mode number n 15 10 Histogram of dominant components # in a TCV discharge (23 ELMs) 5 [R. P . Wenninger, et al. to be published (2013)] TCV #42062 0 1 2 3 dominant toroidal harmonic Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 8

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

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 10

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

Model Reduced MHD Equations ∂Ψ ∂u ∂t = ηj − R [ u , Ψ ] − F 0 ∂φ ∂ρ ∂t = − ∇ · ( ρ v ) + ∇ · ( D ⊥ ∇ ⊥ ρ ) + S ρ ∂ ( ρT ) � � = − v · ∇ ( ρT ) − γρT ∇ · v + ∇ · K ⊥ ∇ ⊥ T + K || ∇ || T + S T ∂t � � ρ∂ v ∂t = − ρ ( v · ∇ ) v − ∇ p + j × B + µ∆ v e φ · ∇ × � � ρ∂ v ∂t = − ρ ( v · ∇ ) v − ∇ p + j × B + µ∆ v B · 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)] Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 12

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

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 14

Results Overview ⊲ ELMs in typical ASDEX Upgrade H-mode equilibrium ⊲ Many toroidal harmonics ⊲ Resistivity too large by factor 10 due to numerical constraints (improving) 7 6 5 q-profile 4 3 2 1 q = toroidal turns poloidal turns 0 1 normalized quantities 0.8 0.6 0.4 ρ 0.2 T 0 0 0.2 0.4 0.6 0.8 1 Ψ N Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 15

Results Poloidal Flux Perturbation n = 0, 8, 16 ⊲ Red/blue surfaces correspond to 70 percent of maximum/minimum values [M. H¨ olzl, et al. 38th EPS , P2.078 (2011); M. H¨ olzl, et al. Phys.Plasmas , 19, 082505 (2012b)] Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 16

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¨ olzl, et al. 38th EPS , P2.078 (2011); M. H¨ olzl, et al. Phys.Plasmas , 19, 082505 (2012b)] Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 16

Results Energy Timetraces n=10 1e-06 1e-08 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 [ µ s] Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17

Results Energy Timetraces other 1e-06 n=10 n= 9 1e-08 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 [ µ s] ⊲ Simulation including n = 0, 1, . . . , 15, 16 ⊲ n = 9 and 10 are linearly most unstable Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17

Results Energy Timetraces other 1e-06 n=10 n= 9 1e-08 n= 2 n= 1 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 [ µ s] ⊲ 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¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 17

Results Mode Interaction Model 1e-06 n=16 n=12 n= 8 1e-08 n= 4 1e-10 magnetic energies [a.u.] 1e-12 1e-14 1e-16 1e-18 1e-20 1e-22 420 440 460 480 500 520 540 560 580 time [ µ s] ⊲ 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 Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 18

Results Mode Interaction Model (2) ⊲ Model assuming mode rigidity and fixed background: linear non-linear interaction � �� � � �� � ˙ A 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)] Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 19

Results Mode Interaction Model (2) ⊲ Model assuming mode rigidity and fixed background: linear non-linear interaction � �� � � �� � ˙ A 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 ˙ A 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 ˙ A 12 = γ 12 A 12 + γ 4,8 A 4 A 8 + γ 16, − 4 A 16 A 4 ˙ A 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 H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 19

Results Mode Interaction Model (2) 1e-06 n=16 n=12 n= 8 1e-08 n= 4 1e-10 magnetic energies [a.u.] 1e-12 1e-14 1e-16 1e-18 1e-20 1e-22 420 440 460 480 500 520 540 560 580 time [ µ s] ⊲ Non-linear drive recovered ⊲ Saturation not recovered (of course) ⊲ Explains low-n features in experimental observations → Poster: Isabel Krebs, P19.15 (Thursday) Matthias H¨ olzl Nonlinear ELM Simulations DPG Spring Meeting, Jena, 02/2013 20