Hybrid MHD-Gyrokinetic Simulations of Global Alfvn Modes in Fusion - PowerPoint PPT Presentation

Hybrid MHD-Gyrokinetic Simulations of Global Alfvén Modes in Fusion Plasmas HYMHDGK: Marconi-fusion 1 st cycle project (Oct. 2016-Dec. 2017) NLED: WP-ENR EUROfusion project (2014-2017) G. Vlad 1 (PI), S. Briguglio 1 , G. Fogaccia 1 , V. Fusco 1 , C. Di Troia 1 , E. Giovannozzi 1 , X. Wang 2 , F. Zonca 1,3 1 ENEA, Dipartimento FSN, C. R. Frascati, Via E. Fermi 45, 00044 Frascati (Roma), Italy 2 Max-Planck-Institut für Plasmaphysik, Boltzmannstr. 2, 85748 Garching, Germany 3 Institute for Fusion Theory and Simulation and Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China G. Vlad - EUROfusion Science meeting - 16 May 2018 1

Outline of the presentation • Introduction and Motivation • Numerical Model • Numerical Simulations - Single- n Simulations - Multiple- n Simulations - Role of MHD and Energetic Particle Non-linearities • Conclusions G. Vlad - EUROfusion Science meeting - 16 May 2018 2

Introduction and Motivation • The study of the effects of energetic particles (EPs), characterized by supra- thermal velocity, on magnetically confined plasmas approaching ignited conditions is a very relevant field of investigation in the magnetic confinement plasma community since several tens of years. • The main concern is that the mutual interaction of globally extended Alfvén modes and EPs (as, e.g., the fusion generated alpha particles and/or the energetic ions accelerated by auxiliary heating systems) could drive shear Alfvén modes unstable and, eventually, enhance the radial transport of the EPs themselves. • This can results, in turn, in increased difficulties in approaching and/or maintaining the ignited conditions (the EPs being displaced from the inner, hot core of the plasma discharge toward the edge, colder region before slowing down and heating the bulk species), or even damaging the vacuum vessel, if the EPs escape the plasma and hit the reaction chamber. • Single- n dynamics quite extensively studied in the past; Multiple- n effects on radial transport only recently addressed ( n : toroidal mode number). G. Vlad - EUROfusion Science meeting - 16 May 2018 3

Numerical Model Key ingredients of our model are: • Shear Alfvén waves (to be studied in toroidal geometry => magnetically confined fusion in tokamak devices); • Effect of Energetic Particles (EPs) on Alfvénic modes; • The mutual interaction of globally extended Alfvén modes and EPs; • Their effect on the energetic particle radial density profile (radial transport). Hybrid MHD-Gyrokinetic model: • Magnetohydrodynamics is used to describe the bulk plasma and Alfvén waves (Alfvén continua, various Alfvén modes, e.g., TAEs, RSAEs, …); • Gyrokinetics is used to describe the dynamics of the EPs, in order to keep the detail of the resonant interaction between EPs and MHD waves; • The two plasma components (thermal and EPs) being coupled (W. Park et al., Phys. Fluids 1992) via the divergence of the pressure tensor term of the EPs entering in the extended momentum equation of the bulk plasma. G. Vlad - EUROfusion Science meeting - 16 May 2018 4

HMGC code • Thermal (core) plasma: - described by reduced O( ε 0 3 ) visco-resistive MHD equations in the limit of β =0 ( ε 0 ≡ a / R 0 ) ➡ equilibria with shifted circular magnetic surfaces can be investigated - MHD fields: ψ , φ (poloidal magnetic flux function and electrostatic potential) • Energetic Particle population: - described by the non-linear gyrokinetic Vlasov equation, expanded up to order O( ε ) and O( ε B ), with ε ~ ρ H / L n the gyrokinetic ordering parameter and ε B ~ ρ H / L B < ε , in the k ⊥ ρ H <<1 limit (guiding-center approximation); - coupling term between MHD and GK is the energetic particle pressure: Π ⊥ , Π || ; - fully retaining magnetic drift orbit widths; - solved by particle-in-cell (PIC) techniques. k ⊥ : perpendicular component of the wave vector; ρ H : energetic ion Larmor radius; L n , L B : the equilibrium density and magnetic field scale lengths. • Toroidal coordinates system ( r , θ , φ ) G. Vlad - EUROfusion Science meeting - 16 May 2018 5

Numerical Simulations Equilibrium: • ε 0 ≡ a / R 0 =0.1; T H / T H0 =1, ρ H0 / a =0.01, v H0 / v A0 =1, m H / m i =2; n H0 / n i0 =1.75 × 10 -3 • q ( r )= q 0 +( q a - q 0 )( r / a ) 2 with q 0 =1.1 and q a =1.9 • n i ∝ 1/ q 2 => toroidal Alfvén gaps for different n aligned • EP equilibrium distribution function F H;eq is isotropic Maxwellian • n H = n H0 exp(-19.53 (1- ψ / ψ 0 ) 2 ) 1.9 Fourier space for perturbed quantities: ( m , n ) and (- m ,- n ) 1 1.8 modes included in the simulations; 1 ≤ n ≤ 15, n q 0 ≲ m ≲ n q a ; 0.8 n i /n i0 (r/a) 1.7 1.6 0.6 15 Grid and particle per cell for GK: 1.5 n=m/1.1 n n 0.4 1.4 N ppc =8, N r,GK =256, n H /n H0 (s) 1.3 N θ ,GK =160, N φ ,GK =80, 10 q(r/a) 0.2 n=m/1.9 1.2 N p = N ppc N r,GK N θ ,GK N φ ,GK ≈ 26.2 × 10 6 1.1 0 0 0.2 0.4 0.6 0.8 1 5 r/a, s With s defined as: = | | | | coordinate s ψ ψ ψ ψ 0 / eq 0 edge − − ψ eq the equilibrium magnetic poloidal flux poloidal fm ux function, and and 0 function, and ψ 0 and ψ edge its values, at the 0 5 10 15 20 25 m 30 m magnetic axis and at the edge, respectively. G. Vlad - EUROfusion Science meeting - 16 May 2018 6

Single- n simulations-1 Single- n simulations on A1-A3 partitions of Marconi-fusion: γ ∝ ω *H = k ･ v *H ∝ nq W tot,n W n=4 n=6 tot,n ( ω *H is the diamagnetic frequency 10 -6 n=5 n=7 of the “hot” particles; modes tap n=11 10 -8 n=12 energy from EP spatial gradients) n=2 n=13 n=10 n=8 10 -10 0.06 0.6 γ / ω n=9 ω / ω n=14 n=3 γ / ω A0 ω / ω A0 A0 A0 0.04 10 -12 0.4 n=15 0.02 10 -14 0.2 0 n=1 -0.02 0 10 -16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 n n 0 100 200 300 400 t ω A0 500 600 n n t ω A0 γ decreased by FOW effects ω / ω A0 ω / ω A0 (eigenfunction spatial width smaller than EP drifts) Saturation occurs because of axisymmetric modification of EP r/a r/a distribution (in configuration and/or velocity space) G. Vlad - EUROfusion Science meeting - 16 May 2018 7

Single- n simulations-2 Frequency vs. r spectra ( t ω A0 =120): 10 -6 10 -8 10 -10 10 -12 10 -14 10 -16 0 100 200 300 400 500 600 t ω A0 n =1 n =2 n =3 n =4 n =5 ω / ω A0 r/a n =6 n =7 n =8 n =9 n =10 n =11 n =12 n =13 n =14 n =15 G. Vlad - EUROfusion Science meeting - 16 May 2018 8

Single- n simulations-3 Frequency vs. r spectra ( t ω A0 =360): 10 -6 10 -8 10 -10 10 -12 10 -14 10 -16 0 100 200 300 400 500 600 t ω A0 n =1 n =2 n =3 n =4 n =5 ω / ω A0 r/a n =6 n =7 n =8 n =9 n =10 n =11 n =12 n =13 n =14 n =15 G. Vlad - EUROfusion Science meeting - 16 May 2018 9

Single- n simulations-4 Spectrograms ( ω vs. t ): 10 -6 10 -8 frequency chirping up and down 10 -10 after saturation 10 -12 10 -14 10 -16 0 100 200 300 400 500 600 t ω A0 n =2 n =3 n =4 n =5 n =1 ω / ω A0 t ω A0 n =6 n =7 n =8 n =9 n =10 n =11 n =12 n =13 n =14 n =15 G. Vlad - EUROfusion Science meeting - 16 May 2018 10

HPC is important… Typical Non-linear simulation on Marconi-fusion A1 partition: • N toroidal Fourier components =10; γ / ω A0 • M poloidal Fourier components =76; • t ω A0 =355.2, Δ t ω A0 =0.02 => n N steps, MHD =17760, N steps, GK =5920 (+sub-cycling when required) • 120 nodes, 36 cores/node: 4320 cores; elapsed time ≈ 24h Typical Non-linear simulation on Marconi-fusion A3 partition: • N toroidal Fourier components =15; 0.06 γ / ω γ / ω A0 n spectrum A0 0.04 • M poloidal Fourier components =142; extended 0.02 0 • t ω A0 =393.6, Δ t ω A0 =0.02 => -0.02 0 2 4 6 8 10 12 14 16 n n N steps, MHD =19680, N steps, GK =6560 (+sub-cycling when required) 120 nodes, 48 cores/node: 5760 cores; elapsed time ≈ 24h • G. Vlad - EUROfusion Science meeting - 16 May 2018 11

Multiple- n simulations-1 2 10 -5 Multiple- n simulation W n=4 n=5 W tot,n tot,n n=2 W W tot,n 10 -6 tot,n • n =1,…,15 n=3 • Both fluid (mode-mode) and EP non- 10 -8 n=10 linearities included 10 -10 1 10 -5 n=9 • n =0 not evolved (eventual formation n=4 10 -12 n=13 of zonal structure not considered) n=14 n=2 n=15 10 -14 • Non-linear coupling drives all the n=5 n=3 n=2 n=1 n=3 modes unstable and makes them 10 -16 0 t ω A0 t ω A0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 saturate almost simultaneously t ! t ! A0 A0 • No evidence of “domino effect” • Saturation amplitude of MHD fields 2 10 -5 is smaller than single- n simulations W n=4 n=6 W tot,n tot,n W W tot,n 10 -6 n=5 tot,n n=7 n=11 10 -8 n=12 n=4 n=2 n=13 n=10 Single- n simulations n=8 10 -10 1 10 -5 n=9 n=14 n=3 10 -12 n=15 n=7 10 -14 n=5 n=6 n=1 10 -16 0 t ω A0 t ω A0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 t ! t ! A0 A0 G. Vlad - EUROfusion Science meeting - 16 May 2018 12

Multiple- n simulations-2 Frequency vs. r spectra ( t ω A0 =150): n =1 n =2 n =3 n =4 n =5 ω / ω A0 r/a n =6 n =7 n =8 n =9 n =10 n =11 n =12 n =13 n =14 n =15 G. Vlad - EUROfusion Science meeting - 16 May 2018 13