global electromagnetic gyrokinetic and hybrid …global electromagnetic gyrokinetic and hybrid...
TRANSCRIPT
Global electromagnetic gyrokinetic and hybridsimulations of Alfven eigenmodes
Michael Cole,
A. Mishchenko, A. Biancalani, A. Bottino, R. Kleiber and A.Konies
Max Planck Institut fur Plasmaphysik, Germany
in the framework of the:“Nonlinear Energetic-Particle Dynamics” European Enabling Research Project
Theory Seminar, Princeton Plasma Physics Laboratory
December 9, 2016
1/32
1. Introduction→ 1.1 Motivation
→ 1.2 Models
2. Nonlinear Toroidal Alfven Eigenmodes→ 2.1 ITPA benchmark
→ 2.2 Wave-wave interaction
→ 2.3 Continuum interaction
3. Realistic geometry→ 3.1 ASDEX Upgrade
→ 3.2 Wendelstein 7-X
4. Outlook and Summary
2/32
1. Introduction→ 1.1 Motivation
→ 1.2 Models
2. Nonlinear Toroidal Alfven Eigenmodes→ 2.1 ITPA benchmark
→ 2.2 Wave-wave interaction
→ 2.3 Continuum interaction
3. Realistic geometry→ 3.1 ASDEX Upgrade
→ 3.2 Wendelstein 7-X
4. Outlook and Summary
3/32
[1.1] Introduction, Motivation
Fusion devices requireexternal heating, produceenergetic Alpha particles.
These fast particles caninteract with Alfven modes byinverse Landau damping.
Interaction redistributes fastparticles→ reduction in confinement,damage to the device.
→ could be used to control
plasma profiles.
0 0,2 0,4 0,6 0,8
sqrt norm. toroidal flux
0,0
2,0×105
4,0×105
6,0×105
8,0×105
1,0×106
1,2×106
ω (r
ad s
-1)
m= 9, n= -6m= 10, n= -6m= 11, n= -6m= 12, n= -6
|φ|
4/32
[1.2] Introduction, Models
EUTERPE code package:
Particle-in-Cell approach (δf ).
Global to last closed flux surface, with arbitrary 3D toroidalgeometry (e.g. stellarator, tokamak with RMPs, etc.).
Electromagnetic (currently δB⊥ only).
Standard model: Gyrokinetic ion species, drift kinetic electrons.
Reduced models:
gyrokinetic ion species, fluid electrons.
gyrokinetic fast ions, MHD bulk plasma (self-consistent).
gyrokinetic fast ions, prescribed background field.
Pitch-angle-scattering collision operator (not used here).
5/32
[1.2] Introduction, Models
Gyrokinetics - solve Vlasov equation for the perturbed distributionfunction, f1s, for each species,
∂f1s∂t
+ ~R · ∂f1s∂ ~R
+ v‖∂f1s∂v‖
= −~R(1) · ∂F0s
∂ ~R− v
(1)‖∂F0s
∂v‖,
with equations of motion - p‖-formulation,
~R =(v‖ −
q
m
⟨A‖⟩)~b∗ +
1
qB∗‖
~b ×[µ∇B + q∇
⟨φ− v‖A‖
⟩]v‖ = − 1
m
[µ∇B + q∇
⟨φ− v‖A‖
⟩]· ~b∗
~B∗ = ~B +m
qv‖
(∇× ~b
)quasi-neutrality equation and Ampere’s law,
−∇·n0ms
B2∇⊥φ =
∑s=i,e,f
qs n1s ,
∑s=i,e,f
βsρ2s
−∇2⊥
A‖ = µ0
∑s=i,e,f
j‖1s .
6/32
[1.2] Introduction, Models
Problem: for electromagnetic simulations, numerical cancellation oftwo large terms leads to inaccuracy. ∑
s=i,e,f
βsρ2s
−∇2⊥
A‖ = µ0
∑s=i,e,f
j‖1s
Unphysical ‘skin terms’ are the result of the formulation, balancedby ‘adiabatic current’:
βeρ2e
A‖ =µ0n0e
2
meA‖ = µ0j
(ad)1e ≈ µ0qs
∫v‖
v‖qeF0e
TeA‖d
3v
Cancellation problem most severe with kinetic electrons, high βe ,
low k‖ - electromagnetic global modes.
7/32
[1.2] Introduction, Models
Unphysically large terms not present in v‖ − formulation:−∇2
⊥A‖ =∑
s=i,e,f
j‖1s .
But equations of motion are hard to solve(
∂A‖∂t
):
~R = v‖~b∗ +
1
qB∗‖
~b ×
[µ∇B + q
(∇〈φ〉+
∂⟨A‖⟩
∂t~b
)]
v‖ = − 1
m~b∗ · µ∇B − q
m
(~b∗ · ∇ 〈φ〉+
∂⟨A‖⟩
∂t
)~B∗ = ~B +
m
qv‖
(∇× ~b
)+ ~b · ∇ × A‖~b
One solution: truncate physics with Ohm’s law closure, e.g.∂A‖∂t +∇‖φ = 0.
8/32
[1.2] Introduction, Models
Fluid model: derive electron continuity equation from 0th momentof the Vlasov equation:
∂n1e
∂t= f (u‖1e ,P1e , φ,A‖)
Fluid moments connected to gyrokinetic quantities byquasineutrality equation and Ampere’s law:
−∇⊥min0
eB2∇⊥φ = n1i − n1e , j‖1i = en0u‖1e −
1
µ0∇2⊥A‖
Closures needed for pressure and E‖ (Ohm’s law):
E‖ = −∇‖φ−∂A‖∂t
= −η∇2⊥A‖
Eliminates time derivative in the v‖ equations of motion, andtruncates physics.
9/32
[1.2] Introduction, Models
Fully gyrokinetic simulations still desirable. New approach: mixedvariables formulation.
Magnetic potential is split into symplectic (s, v‖-like) and
hamiltonian (h, p‖-like) components, A‖ = A(s)‖ + A
(h)‖
New perturbed guiding-centre phase-space Lagrangian:
γ = q ~A∗·d~R+m
qµdθ+qA
(s)‖~b·d~x+qA
(h)‖~b·d~x−
[mv2‖
2+ µB + qφ
]dt
Appropriate Lie transformation separates A(h)‖ and A
(s)‖ into the
Hamiltonian and symplectic structure respectively:
Γ = q ~A∗·d~R+m
qµdθ+q
⟨A
(s)‖
⟩~b·d~R−
[mv2‖
2+ µB + q
⟨φ− v‖A
(h)‖
⟩]dt
A. Mishchenko, M. Cole, R. Kleiber and A. Konies, Phys. Plasmas, 21 052113 (2014)
10/32
[1.2] Introduction, Models
Deriving equations of motion by a variational method gives rise to‘mixed’ form:
~R(1) =~b
B∗‖×∇
⟨φ− v‖A
(s)‖ − v‖A
(h)‖
⟩− q
m
⟨A
(h)‖
⟩~b∗
v‖(1) = − q
m
[~b∗ · ∇
⟨φ− v‖A
(h)‖
⟩+
∂
∂t
⟨A
(s)‖
⟩]− µ
m
~b ×∇BB∗‖
· ∇⟨A
(s)‖
⟩∂/∂t term still present, but can be eliminated with new degree of
freedom, e.g.∂A
(s)
‖∂t +∇‖φ = 0.
Ampere’s law becomes: ∑s=i,e,f
βsρ2s
−∇2⊥
A(h)‖ −∇
2⊥A
(s)‖ = µ0
∑s=i,e,f
j‖1s
No truncation of the physics, but ‘guess’ for A‖ evolution can
remove the cancellation problem from much of the solution.A. Mishchenko, M. Cole, R. Kleiber and A. Konies, Phys. Plasmas, 21 052113 (2014)
11/32
[1.2] Introduction, Models
Particles are pushed and fieldequations solved in ‘mixed
space’, where∂A‖∂t problem
not present.
After each time-step,transform to symplectic (v‖)coordinates:
f(s)
1 (Zs ,A(s)‖ ) = f
(m)1 (Zm,A
(s)‖ ,A
(h)‖ )
v(s)‖ (Zs ,A
(s)‖ ) = v
(m)‖ (Zm,A
(s)‖ ,A
(h)‖ )
In v‖ coordinates, set
A(s)‖(new) = A‖ = A
(h)‖(old)+A
(s)‖(old)
→ set problematic A(h)‖ to 0
at each time-step.
SYMPLECTIC-VARIABLE SPACE
MIXED-VARIABLE SPACE
mix
ed-v
aria
ble ∆t
mix
ed-v
aria
ble ∆t
mix
ed-v
aria
ble ∆t
pullb
ack
pullb
ack
...
A. Mishchenko, A. Konies, R. Kleiber and M. Cole, Phys. Plasmas, 21 092110 (2014)
12/32
1. Introduction→ 1.1 Motivation
→ 1.2 Models
2. Nonlinear Toroidal Alfven Eigenmodes→ 2.1 ITPA benchmark
→ 2.2 Wave-wave interaction
→ 2.3 Continuum interaction
3. Realistic geometry→ 3.1 ASDEX Upgrade
→ 3.2 Wendelstein 7-X
4. Outlook and Summary
14/32
[2.1] Nonlinear TAEs, ITPA benchmark
ITPA TAE benchmark: worldwidestandard test case for Alfven wavephysics.
B0 = 3.0 T.
Circular tokamak,R0/a0 = 10.0.
Flat bulk plasma profiles(βbulk ≈ 0.18%).
Drive from energetic particledensity profile, gradientpeaking at r/ra = 0.5(βfast ≈ 0.1%).
0 0,2 0,4 0,6 0,8
sqrt norm. toroidal flux
0,0
2,0×105
4,0×105
6,0×105
8,0×105
1,0×106
1,2×106
ω (r
ad s
-1)
m= 9, n= -6m= 10, n= -6m= 11, n= -6m= 12, n= -6
|φ|
A. Mishchenko, A. Konies and R. Hatzky, Phys. Plasmas, 16 082105 (2009) 15/32
[2.1] Nonlinear TAEs, ITPA benchmark
ITPA TAE benchmark: worldwidestandard test case for Alfven wavephysics.
B0 = 3.0 T.
Circular tokamak,R0/a0 = 10.0.
Flat bulk plasma profiles(βbulk ≈ 0.18%).
Drive from energetic particledensity profile, gradientpeaking at r/ra = 0.5(βfast ≈ 0.1%).
1×104
1×105
1×106
1×107
1×108
1×109
N (markers)
380
400
420
440
460
480
ω (
10
3 r
ad
s-1
)
gyrokinetic, mixed variablesfluid-electron hybrid
CKA-EUTERPE γd= 1.33.10
4 s
-1CKA-EUTERPE γd= 2.5.10
3 s
-1CKA-EUTERPE γd= 2.5.10
3 s
-1
0 200 400 600 800T/ keV
0
10
20
30
γ/1
03 s
-1
CAS3D-K (ZOW)
FLU-EUTERPE (FLR)
GYGLES (FLR)
CKA-EUTERPE (FLR)
MEGA (FLR)
NOVA-K (FLR)
LIGKA (FLR)
EUTERPE (FLR)
A. Konies et al., Proc. 24th IAEA Fusion Energy Conf., ITR/P1 (2012), M. Cole et al., PPCF, 57 054013 (2015) 16/32
[2.1] Nonlinear TAEs, ITPA benchmark
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 50000 100000 150000 200000 250000
δ B
/B0
t (Ω-1
)
EUTERPE, mixed variables, γd=0FLU-EUTERPE, ideal, γd=0
FLU-EUTERPE, resistive, γd=3x103 rad s
-1
Linear exponential growth phase gives way to clean saturationonly with damping.
Fluid-electron, kinetic ion model requires ad-hoc (here,resistive) damping.
M. Cole et al. Phys. Plasmas, submitted 17/32
[2.1] Nonlinear TAEs, ITPA benchmark
Fast particle density profiledrives the mode; non-lineareffects result in fastparticle density profileflattening (right).
Modification of modestructure (below).
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
nf/n
0r/a
Initial
EUTERPE saturated
ORB5 saturated
linear, nf (0.5)/n0 ≈ 0.0035 NL, nf (0.5)/n0 ≈ 0.0035 NL, nf (0.5)/n0 ≈ 0.007
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12
M. Cole et al. Phys. Plasmas, submitted, A. Biancalani et al. PPCF, submitted 18/32
[2.1] Nonlinear TAEs, ITPA benchmark
1×104
1×105
γ/rad s-1
0.1
1
10
δB
/B0 /
10
-3
EUTERPE, ZLR, γd=1.2.10
3rad s
-1
ORB5 γd=1.7.10
3 rad s
-1
FLU-EUTERPE, FLR, γd=5.5.10
3rad s
-1
CKA-EUTERPE γd=5.6.10
3rad s
-1
δΒ/Β0 ∼ γ
2
δΒ/Β0 ∼ γ
Analytical theory predicts transition from δB/B0 ∼ c · γ2 (resonancedetuning) to δB/B0 ∼ c · γ (radial decoupling) regime.
Artificial damping, γd , no longer needed in fully gyrokinetic
treatment → accurate determination of level.M. Cole et al. Phys. Plasmas, submitted 19/32
[2.1] Nonlinear TAEs, ITPA benchmark
Numerical model comparison - computational requirements forlinear ITPA TAE benchmark:
Model Markers Timestep CPU-hrs (to 105Ω−1c )
EUTERPE (p‖) 3×107 0.75 80000EUTERPE (mixed) 3×106 10.0 256FLU-EUTERPE 2×105 10.0 18CKA-EUTERPE 8×105 20.0 6
Two order of magnitude improvement for full gyrokineticsimulations - moving from p‖ formulation to mixed variablesformulation with pullback.
Fluid-electron, gyrokinetic ion hybrid model an order ofmagnitude faster, where applicable.
M. Cole et al. Phys. Plasmas, submitted20/32
[2.2] Nonlinear TAEs, Wave-wave interaction
Simplest case: two (n = −2,n = −6) TAEs interactingnonlinearly.
Through wave-particlenonlinearity: large δB/B0
(black).
Through bulk ion nonlinearity:order of magnitude lowersaturation amplitude (red).
Significant mode structuremodification via bulk ionnonlinearity.
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 50000 100000 150000δ B
/B0
t (Ω-1
)
Wave-particle nonlinearity onlyWave-wave nonlinearity included
wave-particle only (black) with wave-wave (red)
21/32
[2.3] Nonlinear TAEs, Continuum interaction
1.4
1.6
1.8
2
2.2
2.4
2.6
0 0.2 0.4 0.6 0.8 1
q(r
)
r/ra
ITPA caseHigh shear case
0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux
0
2×105
4×105
6×105
8×105
1×106
1×106
ω (
rad
s-1
)
m= 7m= 8m= 9m= 10m= 11m= 12m= 13m= 14m= 15m= 16
Steeper q profile → more complex continuum.
Higher bulk ion temperature moves mode frequency upwards→ interaction possible between the mode and the continuum.
Otherwise, parameters are the same.
A. Mishchenko et al. Phys. Plasmas 21 052114 (2014), M. Cole et al. PPCF 57, 054013 (2015)
22/32
[2.3] Nonlinear TAEs, Continuum interaction
Apply hierarchy of models tomore realistic continuum.
Continuum interaction
creates global structure
(middle) modified by finite E‖(right); growth rate increases
at each stage. 0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux
0
2×105
4×105
6×105
8×105
1×106
1×106
ω (
rad s
-1)
m= 7m= 8m= 9m= 10m= 11m= 12m= 13m= 14m= 15m= 16
MHD bulk plasma, kinetic fast ions fluid electrons, kinetic bulk ions kinetic bulk plasma
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12m=13m=14
0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux
0
0,2
0,4
0,6
0,8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12m=13m=14
0 0,2 0,4 0,6 0,8 1sqrt norm. toroidal flux
0
0,2
0,4
0,6
0,8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12m=13m=14
A. Mishchenko et al. Phys. Plasmas 21 052114 (2014), M. Cole et al. PPCF 57, 054013 (2015)23/32
[2.3] Nonlinear TAEs, Continuum interaction
MHD-kinetic fluid-electron
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12m=13m=14
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12m=13
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12m=13m=14
0 0.2 0.4 0.6 0.8 1sqrt norm. toroidal flux
0
0.2
0.4
0.6
0.8
1
|φ| (
arb
. u
nit
s)
m=9m=10m=11m=12
1e-06
1e-05
1e-04
1e-03
1e-02
0 0.0002 0.0004 0.0006 0.0008 0.001
δB
/B0
t (s)
fluid electron, kinetic ionMHD bulk plasma
Global structure grows more quickly and saturates at a levelone order magnitude higher.
Qualitative mode structure differences persist in nonlinearphase.
24/32
1. Introduction→ 1.1 Motivation
→ 1.2 Models
2. Nonlinear Toroidal Alfven Eigenmodes→ 2.1 ITPA benchmark
→ 2.2 Wave-wave interaction
→ 2.3 Continuum interaction
3. Realistic geometry→ 3.1 ASDEX Upgrade
→ 3.2 Wendelstein 7-X
4. Outlook and Summary
25/32
[3.1] Realistic geometry, ASDEX Upgrade
Baseline case suggested by Ph. Lauber(Lauber-ITPA-14) for ‘Non-Linear En-ergetic particle Dynamics’ project -“ASDEX-like” equilibrium:
B0 = 2.248 T
R0/a0 = 3.36
κ = 1.7
No triangularity, not numericalreconstruction of experimentaldata
26/32
[3.1] Realistic geometry, ASDEX Upgrade
Apply hierarchy of models toASDEX parameters.
Bulk plasma kinetic effectschange mode structure, increasegrowth rate.
Mode identification on-going:more investigation needed.
Complex geometry → stepping
stone to stellarator simulations.
1e-02
1e+00
1e+02
1e+04
1e+06
0 1x10-5
2x10-5
3x10-5
4x10-5
5x10-5
|φ|
t (s)
MHD-gyrokineticFluid electrons, gyrokinetic ions
MHD bulk plasma, kinetic fast ions
0 0.2 0.4 0.6 0.8 1r
|φ| (a
rb. unit
s)
m=-2m=-1m=0m=1m=2m=3m=4m=5
fluid electrons, kinetic bulk ions
0 0.2 0.4 0.6 0.8 1r
|φ| (a
rb. unit
s)
m=-1m=0m=1m=2m=3m=4m=5m=6m=7m=8
27/32
[3.1] Realistic geometry, ASDEX Upgrade
On-going work: takeself-consistent MHD-kineticcase, add wave-particlenonlinearity.
Nonlinear saturation, withsignificant mode structuremodification.
0 0.2 0.4 0.6 0.8 1r
|φ| (a
rb.
un
its)
m=-2m=-1m=0m=1m=2m=3m=4m=5
linear phase nonlinear phase|φ| |φ|
28/32
[3.2] Realistic geometry, Wendelstein 7-X
Global W7-Xgeometry with 3DEUTERPE code, highfast particleβf ≈ 2β0.
Cut-off in growth rateof numerical origin,related to strongcoupling of poloidaland toroidalharmonics.
Future work: reduceβf to experimentallyrelevant values.
|φ|
0 2,5 5 7,5 10
Fast fraction (%)
0
100
200
300
400
500
γ (k
Hz)
29/32
1. Introduction→ 1.1 Motivation
→ 1.2 Models
2. Circular tokamak→ 2.1 Nonlinear response
→ 2.2 Kinetic mode modification
3. ASDEX-like simulations→ 3.1 Case definition
→ 3.2 Model comparison
4. Outlook and Summary
30/32
[4.] Outlook
Interaction of multiple effects incompletelyunderstood:
Citrin ’13: Electromagnetic effects,fast particles suppressmicroturbulence locally in tokamaks.
Xanthopoulos ’16: Full flux surfacetreatment vital to model electrostaticmicroturbulence in the Wendelstein7-X stellarator without fast particles.
Dumont ’13: Fast particle-drivenglobal EGAMs temporarily suppresselectrostatic microturbulence.
EUTERPE code package could provide a
complete model of this physics.
(Ph. Lauber et al.)
Linear EMITG in W7-X with EUTERPE
(A. Mishchenko et al.)
31/32
[4.] Summary
Theory and numerics: novel formulations of the gyrokineticequations have led to order of magnitude improvements inperformance simulating electromagnetic global modes.
Physics results: theoretical predictions by Berk & Breizman,S. Briguglio et al. confirmed; predictions for fast particleredistribution now possible; mode structure modification andmode-mode interaction may be qualitatively crucial forpredicting fast particle transport.
On-going work: applying the tool with realistic tokamakprofiles; stellarator simulations in support of Wendelstein 7-X.
Prospects: self-consistent, global, electromagneticMHD-turbulence interaction with fast particles.
32/32