Gyrokinetic simulations with GYSELA: Main current issues in physics & numerics

Transcription

1 Gyrokinetic simulations with GYSELA: Main current issues in physics & numerics Y. Sarazin, Y. Asahi 2, N. Bouzat, G. Dif-Pradalier, P. Donnel, C. Ehrlacher, C. Emeriau 3, X. Garbet, Ph. Ghendrih, V. Grandgirard, G. Latu, C. Passeron CEA, IRFM, Saint Paul-lez-Durance, France. 2 also JAEA, Japan. 3 also CEA, IRFU, Saclay, France.

2 Basics on Gyrokinetics Fusion plasmas weakly collisional (ITER: n*~n/t 2 ~10-3 ~ qr/ lpm ) F can depart from a Maxwellian kinetic description mandatory Gyro-freq. dj c /dt~10 8 >> Turb. freq.~10 5 j c can be safely "averaged out" phase-space reduction: 6D F(x,v //,m,j c,t) 4D+1D F G (x G,v //,m,t) particle gyrocenter Gyro-average operator J 1) Magnetic momentum m 2) Polarization Adiabatic invariant m=mv 2 /2B [Frieman-Chen, 1982, Littlejohn, 1983; Brizard-Hahm, 2007] Maxwell's eqs. on F requires relation F F G polarization density: n = n G + n pol Page 2

3 The semi-lagrangian GYSELA code Self-consistently coupled Gyrokinetic & Quasi-Neutral eqs: df Gs /dt = S + C(F Gs ) + D BC 4D advection Source Collisions Boundary Conditions L(f) = S s dv J s.f Gs [Grandgirard, CPC 2016] Peculiarities: global full-f boundary conditions multi-scale physics flux-driven steady-state on t E GYSELA Backward semi-lagrangian scheme: Trajectories (F G =Cst) followed backwards on fixed grid (weak noise, moderate dissip.) t F G =Cst r, v Page 3

4 Outline Physics upgrades numerical challenges present solutions & issues Boundary conditions: core (r=0) & scrape-off layer Kinetic electrons: Field aligned method & open issues Small scale physics: gyro-average operator & boundary issue Neoclassical transport: collision operator & parallelization issue Page 4

5 Removing Inner Boundary Condition: r = 0 Issue at r=0: divergence of metric (1/r) + too many q points Previously: r min >0 Dirichlet for f mn Neumann for f 00 Upgrade: Poisson (trick): r min =Dr/2 no BC required in r Vlasov: bilinear interpolation in 0<r<r min [Latu-Mehrenberger, 2016] Page 5

6 Towards Scrape-Off Layer physics r>a Coupling core (r/a<1) SOL (r/a>1) is important: H-mode, impurities & neutrals Critical challenges: close/open magnetic surfaces (periodicity; plasmasurface interaction) relative fluctuation levels particle sources/sinks Forced relaxation towards SOL-like profiles: df Gs /dt = S + C(F Gs ) - n (F Gs -F SOL ) [Dif-Pradalier, 2016] Smooth transition towards vanishing fluctuations Some evidence of SOL core interplay Possible alternatives: penalization and/or transition towards fluid description? Page 6

7 FT(f) Kinetic electrons & spurious w H modes Kinetic electrons mandatory: particle transport + trapped electron modes Electrostatic limit spurious "w H " modes: w H /w ci = (k // / k ) (m i /m e ) 1/2 [Lee 1987] Correspond to hydro-dynamical limit (w >> k // v th ) of ITG disp. rel. Also: electrostatic limit (b=0) of kinetic Alfvén wave [Scott 1997] Should disappear in electromagnetics (for b > (k r i ) 2 m e /m i ~ ) Trick: disappear when filtering out (m 0,n=0) modes in QN eq. [Idomura 2016] [Ehrlacher 2016] m i /m e = 1 m i /m e = 25 Filtering out m i /m e = 1 w H w H (m 0,n=0) modes in Quasi-Neutrality Frequency w/w ci Frequency w/w ci Frequency w/w ci Page 7

8 Kinetic electrons & Field-aligned method Kinetic electrons mandatory: particle transport + trapped electron modes Numerical issues v the ~ (m i /m e ) 1/2 v thi ~ 10 8 m.s -1 time step / (m i /m e ) 1/2 r e ~ r i /(m i /m e ) 1/2 ~ r i /60 ~ 50mm nb grid points (m i /m e ) 3/2 Reducing numerical cost: filtering passing electrons (adiabatic) artificially large (m e /m i ) ~OK Using field-aligned method nb grid points (m i /m e ) only [Bottino 2016] [Ottaviani 2011; Hariri 2013] Foot point of trajectory (q*, j*) Points used for interpolation Field Line Optimized parallelization: memory OK, time+30% q j j j*-1 j j* j j*+1 [Latu-Mehrenberger, 2016] Page 8

9 Kinetic electrons & Field-aligned method Kinetic electrons mandatory: particle transport + trapped electron modes Numerical issues v the ~ (m i /m e ) 1/2 v thi ~ 10 8 m.s -1 time step / (m i /m e ) 1/2 Reducing numerical cost: Standard r e ~ r i /(m i /m e ) 1/2 ~ r i /60 ~ 50mm nb grid points (m i /m e ) 3/2 filtering passing electrons (adiabatic) artificially large (m e /m i ) ~OK Using field-aligned method Time evolution of most unstable (m,n) modes of f N j =32 N j =128 N j =256 nb grid points (m i /m e ) only [Bottino 2016] [Ottaviani 2011; Hariri 2013] Optimized parallelization: memory OK, time+30% Aligned N j =32 N j =128 Less toroidal points: ~ N j / 8 Page 9

10 Kinetic electrons & Field-aligned method Kinetic electrons mandatory: particle transport + trapped electron modes Numerical issues v the ~ (m i /m e ) 1/2 v thi ~ 10 8 m.s -1 time step / (m i /m e ) 1/2 r e ~ r i /(m i /m e ) 1/2 ~ r i /60 ~ 50mm nb grid points (m i /m e ) 3/2 Reducing numerical cost: filtering passing electrons (adiabatic) artificially large (m e /m i ) ~OK Using field-aligned method nb grid points (m i /m e ) only [Bottino 2016] n Log FT{f(q,j)} t=6000 t=7000 n [Ottaviani 2011; Hariri 2013] Optimized parallelization: memory OK, time+30% m Aliasing OK m Spurious origin? Less toroidal points: ~ N j / 8 Spurious modes: under investigation Page 10

11 Kinetic electrons & Field-aligned method Kinetic electrons mandatory: particle transport + trapped electron modes Numerical issues v the ~ (m i /m e ) 1/2 v thi ~ 10 8 m.s -1 time step / (m i /m e ) 1/2 r e ~ r i /(m i /m e ) 1/2 ~ r i /60 ~ 50mm nb grid points (m i /m e ) 3/2 Reducing numerical cost: filtering passing electrons (adiabatic) artificially large (m e /m i ) ~OK Using field-aligned method nb grid points (m i /m e ) only [Bottino 2016] n Log FT{f(q,j)} t=6000 t=7000 n [Ottaviani 2011; Hariri 2013] Optimized parallelization: memory OK, time+30% m Aliasing OK m Less toroidal points: ~ N j / 8 Spurious modes: under investigation OK when filtered out Page 11

12 Small scales & gyro-average operator Gyro-average Finite Larmor Radius effects From Padé to N-point average Padé: small scales filtered out Padé Bessel J 0 N-point average (Hermite): Convergence reached for N=8 points (ion turb.) 16-points Issue: boundary condition? [Steiner, 2015; Rozar, 2015; Bouzat, 2016] IPL FRATRES, Coord. [a.u.] Page 12

13 Collisions: physical & numerical issues Critical for: Constraints: [Hirshman-Sigmar 1977; Helander-Sigmar 2005] flow damping: friction on trapped particles, Zonal Flow damping impurity transport (synergy turbulent-neoclassical) momentum & energy exchanges between species [Estève 2016] Boltzmann H-theorem (entropy production, equil.=maxwellian) Neoclassical transport = collisions + trajectories Collisions break down m-invariance parallelization issue [Garbet 2009; Dif-Pradalier 2011; Estève 2015] Energy exchange Momentum exchange v-motion (radial & deflection) Conservation properties OK (on t coll. ): [Donnel 2016] Projection on Laguerre polynomials in Neoclassical results under investigation replace differential operators by integrals Page 13

14 Collisions: physical & numerical issues Critical for: Constraints: [Hirshman-Sigmar 1977; Helander-Sigmar 2005] flow damping: friction on trapped particles, Zonal Flow damping impurity transport (synergy turbulent-neoclassical) momentum & energy exchanges between species [Estève 2016] Boltzmann H-theorem (entropy production, equil.=maxwellian) Neoclassical transport = collisions + trajectories Collisions break down m-invariance parallelization issue [Garbet 2009; Dif-Pradalier 2011; Estève 2015] Enregy exchange Momentum exchange v-motion (radial & deflection) H-theorem OK (in v // & v ): Projection on Laguerre polynomials in replace differential operators by integrals Page 14

15 Summary - conclusions Intricate upgrades of physics & numerical methods / parallelization "Boundary Conditions": towards a model for the SOL gyro-fluid? Fully kinetic electrons (trapped & passing) still out of reach on present HPC Electromagnetics cure electrostatic artifacts Trapped kinetic + heavy electrons Multi-scale physics requires accurate gyro-average operator Collisions mandatory BUT: complex (linearized) operator + parallelization issue! Page 15

16 Back-up slides Page 16

17 Collisions [Donnel 2016] Energy exchange Momentum exchange v-motion (radial & deflection) These terms are treated by projection on Laguerre polynomials, using a Crank-Nicolson scheme All the other terms are treated via finite differences with an explicit scheme Page 17