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Verification of Gyrokinetic Codes: Theoretical Background & Numerical Implementations

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Verification of Gyrokinetic Codes: Theoretical Background & Numerical Implementations N.Tronko 1, T.Goerler 2 , A.Bottino 2 , B.D.Scott 2, E.Sonnendrücker 1 Verification of Gyrokinetic codes: Theoretical background & Numerical implementations NumKin 2016, IRMA, Strasbourg, France 1 NMPP, Max Planck Institute für Plasmaphysik 2 TOK, Max Planck Institute für Plasmaphysik VeriGyro Project Participants Enabling Research Project on Verification of Gyrokinetic codes • Germany! Max Planck Insitute for Plasma Physics (Garching and Greifswald) • Switzerland SPC-EPFL, Lausanne • Finland! Aalto Univeristy • Great Britain University of Warwick • France CEA Cadarache, Université Paris IV, Rennes, Toulouse, Lorraine, Bretagne IRMA, Maison de la Simulation (Saclay) • USA Saint Michel’s College, VT NumKin 2016 VeriGyro Project @ IPP Garching Interdisciplinary Project in IPP Max Planck (Garching) NMPP Division (Numerical Methods for Plasma Physics) • Development and implementation of new algorithms for fusion and astrophysical plasma modeling • Development of libraries (i.e. Selalib) • Verification of existing codes • Participants: N. Tronko, E. Sonnendrücker Enabling Research Project VeriGyro TOK (Tokamak Theory) • Development of major codes • Plasma modeling and comparison with experimental results (ASDEX Upgrade) • Participants: T. Görler, A. Bottino, B. D. Scott NumKin 2016 VeriGyro Project: Motivation • Verification of Global (Electromagnetic) Gyrokinetic codes: Why? • Most popular tools for magnetised plasmas simulations: • Significant Development since last 10 years • Electrostatic gyrokinetic implementations : well established ! [Dimits Phys. Pl. 2000 , Bottino&Sonnendrücker J.Pl.Phys. 2015] • Global Electromagnetic gyrokinetic implementations: • Another level of complexity with respect to electrostatic codes • A lot of freedom for approximations (Poisson and Ampère equations) • Different codes use different version of GK equations) • What was missing in the beginning of VeriGyro (2014) ? • General hierarchy of the GK equations implemented numerically • Global Electromagnetic Intercode benchmark NumKin 2016 VeriGyro: Strategy & Accomplishments • Theory: Establishing Hierarchy of models for existing codes! (Task leader Natalia Tronko) • Systematic derivation from the Variational GK framework ! [Tronko, Bottino, Sonnendrücker, Phys.Plasmas 2016] • Verification of approximations consistency and regimes of applicability • Intercode Benchmark: challenging electromagnetic problems! (Task leader Tobias Görler) • Implicit numerical schemes verification • Microinstability characterization and exploration of required grid • Microturbulence: one of the next steps Linear intercode benchmark [Görler, Tronko, Hornsby et al Phys.Pl. 2016] NumKin 2016 VeriGyro: Intercode Benchmark Participating codes • PIC codes: • ORB5 [Jolliet Comp. Phys. Comm. 2007; Bottino PPCF, 2011; Tronko Phys. Pl., 2016] N. Tronko (IPP Garching) • EUTERPE [Kornilov Phys. Pl. 2004] R. Kleiber (IPP Greifswald) • Eulerian codes: • GENE [Jenko Phys. Pl.,2000; Görler, J.Comp. Phys. 2011] ! T. Görler (IPP Garching) • GKW [Peeters Comp. Phys. Comm. 2009; Buchholz Phys. Pl. 2014] ! W. Hornsby (IPP Garching) • Semi-Lagrangian: • GYSELA[Grandgirard J. Comput. Phys. 2006; Grandgirard Comp. Phys. Comm. 2016] V. Grandgirard, C. Norcini (CEA Cadarache) NumKin 2016 VeriGyro: Numerical resources • Supercomputer Helios (Japan) Hydra (IPP Garching)! • Since Oktober 2017 Marconi Fusion (Italy) • Using between 512 and 2048 processors (for ORB5) • CPU time pro simulation: 2h (adiab) towards several days (electromagnetic linear) NumKin 2016 PART I: Theoretical foundations • Gyrokinetic theory for numerical implementations: • Systematic derivation from the first principle of dynamics! • Second order Gyrokinetic Maxwell-Vlasov Parent model • Focus on the ORB5 code model • Continuous descriptions • Gyrokinetic dynamical reduction on single particle phase space • Systematic coupling of reduced particle dynamics with fields dynamics • Conservation laws and code diagnostics (ORB5) • Discretized description (ORB5) • Finite Element Monte-Carlo Particle-In-Cell Lagrangian • Discrete equations of motion NumKin 2016 PART II : Intercode Benchmark • Establishing common test case: • Adiabatic electrons simulations • Electrostatic (2 species) • Electromagnetic linear Benchmark • Identifying differences in simulations:! in function of the implemented model and numerical scheme! NumKin 2016 GK Orderings ⇢ ln B ✏ • Guiding- center: background quantities | ir | ⇠ B δf c δE δB • Gyrocenter: fluctuating fields | ?| | | ✏δ F ⇠ Bv ⇠ B ⇠ th e δφ ✏δ =(k ⇢i) • Gyrokinetics ? Ti • Drift-kinetics k ⇢i 1 k ⇢i 1 ? ⇠ ? ⌧ • Maximal ordering • Code ordering ✏ ✏ ✏B ✏δ B ⇠ δ ⌧ [Brizard, Hahm Rev.Mod.Phys., 2007] NumKin 2016 Local Particle coordinates • Non-canonical local particle coordinates 6D: Z↵ =(x, v) x,v ,µ,✓ ! k ? ⇢ • Rotating particle basis: b ✓ = b sin ✓ b cos ✓ b ? − 1 − 2 ⇢ = b cos ✓ b sin ✓ b b1 − b 2 • Magnetic • Velocity vector bdecompositionb b Momentum: 2 mv 1/2 adiabatic invariant µ = ? v = v b = v = v b +(2µmB) 2B k ? k ? b b b • Goal of the dynamical reduction: build up a change of variables such that µ˙ =0 fast gyromotion is uncoupled • Relevant dynamics on the reduced (4+1)D phase space: X,v ; µ k NumKin 2016 Two step dynamical reduction • Two polarization displacements x X + ⇢ (X,µ,✓)+⇢ (X,µ,✓) ⌘ 0 1 • Guiding-center displacement • Gyrocenter displacement 2 mc 2µ mc pz ⇢0 ⇢ ⇢0⇢ ⇢1 = φ1(X) A1 (X) ⌘ e mB ⌘ B2 r? − mc k r 0 ⇣ ⌘ (✏B) (✏δ) b⇠ O b ⇠ O • First order electromagnetic potential e 1 = φ1 pzA1 − mc k • Canonical gyrocenter momentum: e pz = mv + A1 (X + ⇢0 + ⇢1) k c k NumKin 2016 Second order Gyrocenter Lagrangian • Particle Lagrangian on the reduced phase space (X,pz,µ,✓) e mc GK theory does L = A + p b X˙ + µ ✓˙ H p c z · e − not stop here!!! ⇣ ⌘ b p2 • Non-perturbed dynamics (guiding-center) H = z + µB 0 2m ep • First order dynamics H = e gc (φ (X + ⇢ )) z gc (A (X + ⇢ )) 1 J0 1 0 − mc J0 1 0 The gyroaveraging operator: with respect to the guiding-center displacement 1 2⇡ gc [ (X + ⇢ (✓))] = d✓ (X + ⇢ (✓)) J0 1 0 2⇡ 1 0 Z0 NumKin 2016 Second order Hamiltonian hierarchy • Full FLR 2nd order electromagnetic Hamiltonian equivalent to Hahm 1988 electrostatic model 1 e 2 e2 @ Hfull = gc ( (X + ⇢ )) gc (X + ⇢ ) (X + ⇢ ) 2 2m c J0 1 0 −2mcJ0 @µ 1 0 1 0 ⇣ ⌘ ✓ ◆ Electromagnetice couplinge between GK Poisson and • Truncated: All 2nd FLR terms Ampère equations 2 FLR e 2 µ 2 1 µ 2 H2 = A1 (X) + A1 (X) + A1 (X) A1 (X) 2mc2 k 2B r? k 2 B k r? k 2 2 mc pz φ1(X) A1 (X) −2B2 r? − mc k nd • ORB5 2 Order : Second order FLR magnetic part and first order FLR in Electrostatic Polarization part 2 2 ORB5 e 2 1 µ 2 mc 2 H2 = A1 (X) + A1 (X) A1 (X) φ1(X) 2mc2 k 2 B k r? k − 2B2 |r? | Uncoupled GK Poisson and Ampère equations NumKin 2016 Gyrokinetic field theory Which components we do really need for building up a self- consistent model for a code? [Sugama Phys. Pl. 2000,! Brizard PRL 2000] 2 2 E1 B1 L = dV dW F (Z ,t ) L Z[Z ,t ; t], Z˙ [Z ,t ; t]; t + dV | | − | ?| 0 0 p 0 0 0 0 8⇡ sp X Z particles⇣ ⌘ Z fields 2⇡ 2 Z =(X,pz,µ,✓); dW = 2 B⇤dpzdµ; B1 = A1 m k ? r? k Goal: Coupling reduced particle dynamics with fields within the common mathematical structure Getting consistently reduced set of Maxwell-Vlasov equations F (Z0,t0) Distribution function of species “sp” at arbitrary initial time t0 Lp Gyrocenter Lagrangian: reduced motion of a single particle NumKin 2016 How to build up a gyrokinetic model consistently? • All approximations should be performed on the Lagrangian L before deriving the equations of motion • Approach guarantees energetic consistency of final equations • Symmetry properties of Lagrangian are automatically transferred to the equations • Set up polarization effects into the system (hierarchy of models): • Choice of the second order gyrocenter Hamiltonian defines reduced field- particles dynamics NumKin 2016 Quasi-neutrality approximation • Electrostatic contributions from fields and gyrocenter polarization 2 2 2 E1 mc 2 1 ⇢s 2 dV | | + dW dV F φ1 = dV 1+ φ1 8⇡ 2B2 |r? | 8⇡ λ2 |r? | Z Z Z ✓ d ◆ Polarization from the second order particle dynamics: ORB5 model 2 2 ORB5 e 2 1 µ 2 mc 2 H2 = 2 A1 (X)+ A1 (X) A1 (X) 2 φ1 (X) 2mc k 2 B k r? k − 2B |r? | k T k T mc2 λ2 B e ⇢2 B e d ⌘ 4⇡ne2 s ⌘ e2B2 Debye length Sound ion Larmor radius 2 2 2 ⇢s 4⇡nmc c 2 = 2 = 2 1 λd B vA NumKin 2016 Linearised polarisation approximation e mc B2 L = dW dV A + p b X˙ + µ✓˙ H F dV ? c z · e − − 8⇡ sp X Z ⇣⇣ ⌘ 2 ⌘ Z Up to (✏ ) b ⇠ O δ • Initially: Hamiltonian decomposition H = H0 + H1 + H2 • Dynamical part of distribution function (H0 + H1)F • Background part of distribution function H2F0 e mc L = dW dV A + p b X˙ + µ✓˙ H H F c z · e − 0 − 1 sp X Z ⇣⇣ ⌘ ⌘ b B2 dW dV H F dV ? − 2 0 − 8⇡ sp X Z Z This approximation leads to linearised field equations NumKin 2016 GK Field equations Phase space volume d⌦ dV dW ⌘ • Polarization equation in a weak form Test function δL φ1 =0 φ1 = φ1 δφ1 ◦ 2 mc 2 gc ✏δ d⌦ F0 φ1 = d⌦ b0 (φ1)(F0 + b✏δF1) B2 |r? | J sp sp X Z X Z δL • Ampere’s equation in a weak form A1 =0 Test function δA1 ◦ k k A1 = A1 b k k 2 1 2 e gc 2 pz gc dV A1 + ✏δ d⌦ 0 A1 F0 = d⌦ 0 A1 (F0 + ✏δF1) 4⇡ r? k mc2 J k mcJ bk Z sp Z sp Z X X Weak form of the equations of motion is suitable for the finite elements discretisation!!!! NumKin 2016 Vlasov equation Vlasov equation is reconstructed from the characteristics δL ˙ @ (H0 + H1) B⇤ c =0 X = b (H0 + H1) δZ @pz B⇤ − eB⇤ ⇥ r k k b B⇤ p˙z = (H0 + H1) Z =(X,pz,µ) −B⇤ · r k d @ dZ @ f(Z [Z ,t ,t];t)= f(Z,t)+ f(Z,t) dt 0 0 @t dt · @Z Linearized characteristics: only H0 contributions NumKin 2016 PIC code ORB5 “J.E” diagnostics PIC code: separation in calculation of fields and particles dynamics: Particles: characteristics Fields: Grid evaluation Controlling quality of the simulation: contributions from particles energy and fields energy should be calculated independently.
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