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

Non-linear Simulations of Edge Localized Modes in ASDEX Upgrade Matthias H¨ olzl (Postdoc at IPP Garching)

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 2

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 3

Introduction Edge Localized Modes Z (m) Te [eV] 1.0 800 Electron temperature measured with ECE-Imaging at an ELM onset in 0.5 600 ASDEX Upgrade: Dominant toroidal Fourier harmonic n ≈ 11 [J. E. Boom, et al. 37th EPS, P2.119 (2010)] 400 0.0 -0.5 200 q=4 -1.0 R (m) 0 1.0 1.5 2.0 Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 4

Introduction Localization ⊲ ASDEX Upgrade: Expanded and localized ELMs observed (distribution) #25764@1.7574s 7 dB/dt [a.u.] + Φ MAP [rad] Signature of a Solitary Magnetic 6 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, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 5

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. Nucl.Fusion (to be submitted)] TCV #42062 0 1 2 3 dominant toroidal harmonic Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 6

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 7

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¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 8

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 e φ · ∇ × ∂t = − ρ ( v · ∇ ) v − ∇ p + j × B + µ∆ v � � ρ∂ v B · ∂t = − ρ ( v · ∇ ) v − ∇ p + j × B + µ∆ v 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, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 9

Model Typical code run ⊲ Initial grid (Grids shown with reduced resolution) ⊲ Flux aligned grid including X-point(s) ⊲ Equilibrium flows ⊲ Time-integration Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 10

Model Typical code run ⊲ Initial grid (Grids shown with reduced resolution) ⊲ Flux aligned grid including X-point(s) ⊲ Equilibrium flows ⊲ Time-integration Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 10

Model Typical code run ⊲ Initial grid (Grids shown with reduced resolution) ⊲ Flux aligned grid including X-point(s) ⊲ Equilibrium flows ⊲ Time-integration Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 10

1 Introduction 2 Model 3 Results 4 Outlook 5 Summary Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 11

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 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, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 12

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, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 13

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, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 13

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] ⊲ Non-linear drive of low-n modes ⊲ Start with simplified case including n = 0, 4, 8, 12, 16 (periodicity 4) Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 14

Results Mode Interaction Model ⊲ Quadratic terms lead to mode coupling ( n 1 , n 2 ) ↔ n 1 ± n 2 ⊲ For instance: ( 16, 12 ) ↔ 4 ⊲ 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. Master’s thesis, LMU, Munich (2012)] Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 15

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] ⊲ Non-linear drive recovered ⊲ Saturation not recovered (of course) Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 16

Results Mode Interaction Model n=10 1e-06 n=9 n=2 1e-08 n=1 magnetic energies [a.u.] 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e-22 500 520 540 560 580 time [ µ s] ⊲ 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¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 17

Results Mode Interaction Model n=10 1e-06 n=9 n=2 1e-08 n=1 magnetic energies [a.u.] 1e-10 1e-12 1e-14 1e-16 1e-18 1e-20 1e-22 500 520 540 560 580 time [ µ s] ⊲ 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¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 17

Results Non-linear phase 1 E mag,00 E mag,08 E kin,00 E kin,08 1e-05 energies [a.u.] 1e-10 1e-15 1e-20 300 400 500 600 700 800 time [ µ s] ⊲ Energy time traces during an ELM crash ⊲ Simulation with n = 0, 8 ⊲ Several bursts Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 18

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 [ µ s] ⊲ Energy time traces during an ELM crash ⊲ Simulation with n = 0, 8 ⊲ Several bursts Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 18

Results Filament Formation Detached filaments quickly lose their pressure due to fast parallel heat conduction. Substructures appear in divertor heat flux patterns. Matthias H¨ olzl, I. Krebs, K. Lackner, S. G¨ unter Nonlinear ELM Simulations ITPA PEP Meeting, Garching, 04/2013 19