Simulation of the Interaction Between Alfvén Waves and Fast Particles In

Home , Nova (laser)

Simulation of the interaction between Alfven´ waves and fast particles in stellarators

T. Feher,´ A. Konies¨ , R. Kleiber, M. Borchardt, R. Hatzky

Max-Planck-Institut f¨ur Plasmaphysik, Greifswald, Germany

IAEA TM on Energetic Particles in Magnetic Conﬁnement Systems Austin, Texas, 09.09.2011

Tamas´ Feher´ CKA-EUTERPEbenchmarks 0/21 Outline

Introduction The hybrid model

CKA–EUTERPE Hybrid code MHD description Kinetic description Interface between the codes

Benchmarks Benchmark 1 Benchmark with NOVA GYGLES benchmark

Stellarator results

Tamas´ Feher´ CKA-EUTERPEbenchmarks 1/21 Outline

Introduction The hybrid model

CKA–EUTERPE Hybrid code MHD description Kinetic description Interface between the codes

Benchmarks Benchmark 1 Benchmark with NOVA GYGLES benchmark

Stellarator results

Tamas´ Feher´ CKA-EUTERPEbenchmarks 1/21 Introduction

Alfven´ waves ◮ Alfven´ waves are present in magnetised plasmas ◮ There are global eigenmodes in the gaps of the Alfven´ wave spectrum ◮ These modes could be driven unstable by energetic particles

CKA-EUTERPE Hybrid Code ◮ To describe wave particle interaction ◮ In real 3D geometry of stellarators ◮ Tool for perturbative stability analysis

Tamas´ Feher´ CKA-EUTERPEbenchmarks 2/21 The hybrid approach

◮ The particles are described by kinetic equations

◮ The Alfven´ waves are calculated from MHD

◮ The particles move in the calculated waveﬁeld

◮ Nonlinear: the energetic particles modify the waveﬁeld

Why do we need a hybrid model? ◮ Kinetic simulation of Alfven´ waves is complicated

◮ We need a numerical tool to determine stability

◮ Better physical model than present hybrid codes (CAS3D-K)

Tamas´ Feher´ CKA-EUTERPEbenchmarks 3/21 Outline

Introduction The hybrid model

CKA–EUTERPE Hybrid code MHD description Kinetic description Interface between the codes

Benchmarks Benchmark 1 Benchmark with NOVA GYGLES benchmark

Stellarator results

Tamas´ Feher´ CKA-EUTERPEbenchmarks 3/21 CKA–EUTERPE Hybrid code

Φ δ , δA code: CKA EUTERPE δp describes: Alfven´ waves Energetic particles solves: MHD EQs Gyrokinetic EQs method: Splinedecomposition PIC

◮ Real 3D magnetic geometry ◮ No assumptions on particle orbits

Tamas´ Feher´ CKA-EUTERPEbenchmarks 4/21 MHD description

◮ Linearised reduced MHD equations ◮ Transformed to an eigenvalue problem:

2 1 3 2 1 2 ω ∇ ∇⊥Φ + ∇∇⊥ ρ ∇∇⊥Φ =−∇ b∇ (b∇)Φ v2 4 i v2 [ ] A A

◮ The CKA code is used to solve the MHD equations in 3D real magnetic geometry ◮ Determines the mode frequency ω and the mode structure Φ(r), A(r)

∂A E = −∇Φ − = 0 ∂t

Tamas´ Feher´ CKA-EUTERPEbenchmarks 5/21 MHD description

◮ Linearised reduced MHD equations ◮ Transformed to an eigenvalue problem:

2 1 3 2 1 2 ω ∇ ∇⊥Φ + ∇∇⊥ ρ ∇∇⊥Φ =−∇ b∇ (b∇)Φ v2 4 i v2 [ ] A A j µ δp µ0δp⊥ 0 −∇ b∇ µ0 B b×∇Φ −∇ 2 b×∇B −∇ 2 b×κ h B i B

◮ The CKA code is used to solve the MHD equations in 3D real magnetic geometry ◮ Determines the mode frequency ω and the mode structure Φ(r), A(r)

∂A E = −∇Φ − = 0 ∂t

Tamas´ Feher´ CKA-EUTERPEbenchmarks 5/21 Kinetic description

∂δF ˙ (0) ∂δF (0) ∂δF ˙ (1) ∂F0 (1) ∂δF0 (1) ∂δF0 +R +v˙ = −R −v˙ −v˙⊥ ∂t ∂R ∂v ∂R ∂v ∂v⊥

dR B∗ q 1 = v − A + b × (µ∇B + q∇ Φ − v A ) dt B∗ m qB∗ ∗ dv 1 B = − (µ∇B + q∇ Φ − v A ), dt m B∗

◮ The EUTERPE code is used to integrate the equations ◮ PIC code, δf method ◮ Full particle orbits ◮ Currently the linearised version is used ◮ Possible to use multiple species

Tamas´ Feher´ CKA-EUTERPEbenchmarks 6/21 Wave particle interaction

◮ The ﬁeld energy

2 1 |∇⊥Φ| 1 2 Eﬁeld = mn0 2 dV + |∇⊥A| dV 2 Z B 2µ0 Z

◮ The growth rate of the mode is given by:

1 ∂E 1 ∂E 1 γ = field = − kin = − j Ed3r 2Efield ∂t 2Efield ∂t 2Efield Z

◮ Later the pressure perturbation will be calculated

δp = dv mv 2δF, δp = dv µBδF Z ⊥ Z

Tamas´ Feher´ CKA-EUTERPEbenchmarks 7/21 Interface between the codes

◮ The magnetic equilibrium is calculated by VMEC ◮ The PEST coordinates are implemented in CKA ◮ Data transfer routines from CKA to EUTERPE are written ◮ The feedback from EUTERPE will be implemented later

VMEC

Boozer,PESTBoozer PEST Φ CKA δ , δA EUTERPE

Tamas´ Feher´ CKA-EUTERPEbenchmarks 8/21 Outline

Introduction The hybrid model

CKA–EUTERPE Hybrid code MHD description Kinetic description Interface between the codes

Benchmarks Benchmark 1 Benchmark with NOVA GYGLES benchmark

Stellarator results

Tamas´ Feher´ CKA-EUTERPEbenchmarks 8/21 Benchmark 1

q profile 2 Plasma parameters

◮ Circular tokamak, R0 = 4m, a=1 m 1.5 ◮ B0 =5T ◮ Deuterium plasma 1 ◮ 19 −3 ne = 5 10 m , constant n proﬁle 0 0.5 1 s pol

Energetic particles 19 −3 x 10 n (m ) 6 ◮ Hydrogen ◮ T = 6keV - 1.4 MeV, constant T proﬁle 4 ◮ β kept constant 2 ◮ density gradient ◮ Maxwellian velocity distribution 0 0 0.5 1 s Tamas´ Feher´ CKA-EUTERPEbenchmarkstor 9/21 Mode structure

6 x 10 1.8 1.6 ◮ Alfven´ spectrum calculated by CKA 1.4 ◮ toroidal mode number n=2 1.2 (rad/sec) ◮

ω 1 global TAE at the crossing of (2,2) 0.8 and (2,3) modes 0.2 0.4 0.6 0.8 s tor

Tamas´ Feher´ CKA-EUTERPEbenchmarks 10/21 Simulation with EUTERPE

The data is transferred to EUTERPE Fast particles started

Tamas´ Feher´ CKA-EUTERPEbenchmarks 11/21 Simulation with EUTERPE

The data is transferred to EUTERPE Fast particles started

1 γ = − j Ed3r 2Eﬁeld Z

Tamas´ Feher´ CKA-EUTERPEbenchmarks 11/21 Simulation with EUTERPE

The data is transferred to EUTERPE Fast particles started

20000

1) 10000 −

1 3 (s γ = − j Ed r γ 2E Z ﬁeld 0

0 0.05 0.1 0.15 time (ms)

Tamas´ Feher´ CKA-EUTERPEbenchmarks 11/21 Simulation with EUTERPE

The data is transferred to EUTERPE Fast particles started

20000

1) 10000 −

1 3 (s γ = − j Ed r γ 2E Z ﬁeld 0

0 0.05 0.1 0.15 time (ms)

Tamas´ Feher´ CKA-EUTERPEbenchmarks 11/21 Stability of the mode

The energetic particle velocity was changed 4 x 10 5 CKA−Euterpe 4 NOVA−K KIN−2DEM 3 LIGKA fow LIGKA zow ) 2

−1 CAS3D−K (s

γ 1

−1

−2 0 0.5 1 1.5 v/v A

Tamas´ Feher´ CKA-EUTERPEbenchmarks 12/21 NOVA benchmark

q profile Plasma parameters 2.5

◮ Circular tokamak, R0 = 3m, a=1 m 2 ◮ B = 1T,q = 1.1+ Ψ 0 pol 1.5 ◮ Deuterium plasma ◮ n = 4.142 1019m−3, constant n proﬁle 1 e 0 0.5 1 ◮ T 3 14 keV s e = . pol

Energetic particles 19 −3 x 10 n (m ) 6 ◮ Deuterium ◮ EEP = 173 keV, constant T proﬁle 4 ◮ density proﬁle ∼ exp(−Ψ /0.37) pol 2 ◮ Maxwellian velocity distribution ◮ 1 0 or slowing down f (v) ∼ 3 3 v +vcrit 0 0.5 1 s Tamas´ Feher´ CKA-EUTERPEbenchmarkstor 13/21 NOVA benchmark, linear n=1 TAE

100

60 The CKA code is used to

f (kHz) 40 calculate the Alfven´ mode

20 spectrum

0 0 0.5 1 s The mode structure from CKA, CAS3D and NOVA:

ω ω EV = 1.028471e-02 +ω i 9.235399e-16=0.6544 v /(q R ) ω =0.63978 v /(q R ) =0.6544 v /(q R ) CKA=0.6470 vA/(q1 R0) NOVA A 1 0 CAS3D A 1 0 NOVA A 1 0 1 m= 1, n= -1 CKA m=2 CAS3D m=1 CAS3D 0.6 m= 2, n= -1 CKA 0.8 m=3 CAS3D m= 0, n= -1 CKA m=4 CAS3D 0.5 m= 3, n= -1 CKA m=0 CAS3D m= 4, n= -1 CKA m=0 NOVA m= 2, n= -1 CAS3D 0.6 m=1 NOVA 0.4 m= 1, n= -1 CAS3D m=2 NOVA n

φ m=3 NOVA m= 3, n= -1 CAS3D ξ r m=4 NOVA 0.3 m= 4, n= -1 CAS3D 0.4 m= 0, n= -1 CAS3D 0.2 0.2 0.1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Ψ Ψ 1/2 (Ψ / Ψ (a))1/2 ( tor/ tor(a) ) pol pol

Tamas´ Feher´ CKA-EUTERPEbenchmarks 14/21 Stability of the mode

The charge of the fast particles was changed.

Maxwellian Slowing down 5 2.5 4 2 3 f 1.5 f β 2 β

ω ω

/ 1 1 / γ

CKA−Euterpe γ CKA−E ZLR 0 CKA−Euterpe 0.5 Theory Theory NOVA NOVA −1 Nova, passing only 0 CAS3D−K −2 CAS3D−K

0 0.5 1 −1 1.5 2 2.5 0 1 2 3 4 5 z z−1 f f

Tamas´ Feher´ CKA-EUTERPEbenchmarks 15/21 GYGLES benchmark

q profile 1.9 Plasma parameters

◮ Circular tokamak, R0 = 10m, a=1 m 1.8 ◮ B0 = 3T ◮ Hydrogen plasma 1.7 ◮ 19 −3 ne = 2 10 m , constant n proﬁle 0 0.5 1 s ◮ Te = 1 keV pol 16 −3 x 10 n (m ) Energetic particles 15 ◮ Deuterium 10 ◮ T = 100 keV-800 keV constant T proﬁle 5 ◮ 17 −3 n0hot = 1.44 10 m , density gradient ◮ Maxwellian velocity distribution 0 0 0.5 1 s Tamas´ Feher´ CKA-EUTERPEbenchmarkstor 16/21 Mode structure, GYGLES benchmark

Mode structure calculated by CKA

GYGLES mode structure (T = 400keV):

Tamas´ Feher´ CKA-EUTERPEbenchmarks 17/21 Stability of the mode

The temperature of the fast particles was changed 50

30 ) −1 s 3 20 (10 γ GYGLES 10 CKA−EUTERPE GYGLES ZLR 0 CKA−E ZLR CAS3D−K

100 200 300 400 500 600 700 800 T (keV)

Tamas´ Feher´ CKA-EUTERPEbenchmarks 18/21 Outline

Introduction The hybrid model

CKA–EUTERPE Hybrid code MHD description Kinetic description Interface between the codes

Benchmarks Benchmark 1 Benchmark with NOVA GYGLES benchmark

Stellarator results

Tamas´ Feher´ CKA-EUTERPEbenchmarks 18/21 W7-AS #39042

m=6, n=-2 ◮ Alfven´ spectrum 0.6 m=5, n=-2 determined by CKA 0.4 ◮ Stability of a global TAE is

calculated Φ(a.u.) 0.2 0 ◮ Slowing down distribution function 0.2 0.4 0.6 0.8 Stor

Tamas´ Feher´ CKA-EUTERPEbenchmarks 19/21 W7-AS #39024 Energy transfer

Fast ions 25 CKA−EUTERPE CAS3D−K 20

) 15 −1 s 3 10 (10 γ

0 0.2 0.4 0.6 0.8 1 v /v s A

Tamas´ Feher´ CKA-EUTERPEbenchmarks 20/21 W7-AS #39024 Energy transfer

Fast ions Background ions 25 CKA−EUTERPE CAS3D−K CAS3D−K 4 20 CKA−EUTERPE local theory

) 2 15 ) −1 −1 s s 3 3 10 0 (10 (10 γ γ

5 −2

0 0.2 0.4 0.6 0.8 1 −4 v /v 0 200 400 600 800 1000 s A Ti

Tamas´ Feher´ CKA-EUTERPEbenchmarks 20/21 Summary

CKA-EUTERPE ◮ Codes were coupled to create a hybrid code ◮ Perturbatively study the stability of Alfven´ modes

Linear benchmarks ◮ Benchmark 1: successful ◮ NOVA benchmark: differences maybe due to the distribution function ◮ GYGLES benchmark: successful

Stellarator results ◮ W7-AS case calulated ◮ FLR and FOW effects are strongly stabilizing

Tamas´ Feher´ CKA-EUTERPEbenchmarks 21/21