gyrokinetic particle simulation of plasma turbulence zhihong lin department of physics &...
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Gyrokinetic Particle Simulation of Plasma Turbulence
Zhihong Lin
Department of Physics & Astronomy University of California, Irvine
Workshop on ITER Simulation Beijing, May 15-19, 2006
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Turbulence in Fusion Plasmas
• Pressure gradients drive Rayleigh-Taylor type microscopic instability: “drift wave instability”
• Turbulence as a paradigm for cross-field transport
Size (and cost) of a future fusion reactor determined by: turbulent transport = self-heating
• Turbulence as a complex, nonlinear, dynamical system
Wave-wave coupling, wave-particle interaction
• Turbulence measurements hindered by high temperature
• Nonlinear analytic theory often intractable
ITER
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Gyrokinetic Particle Simulation of Plasma Turbulence
• Linear micro-instabilities theory well understood & computationally “solved”
• Various nonlinear theories: applicable in limiting regimes
Wave-wave interactions: energy transfer to damped modes
Wave-particle interactions: Compton scattering, resonance broadening
• Particle simulations: treat all nonlinearities on same footing
Nonlinear wave-particle interactions
Complex geometry
• Gyrokinetic particle simulations of ion temperature gradient (ITG) turbulence
Paradigm of 3-mode coupling [Lee & Tang, PF1988]
Realistic toroidal spectra [Parker et al, PRL1993] (1GF)
Device size dependence of transport (Bohm scaling) [Sydora et al, PPCF1996]
Turbulence self-regulation via zonal flow [Lin et al, Science1998; PRL1999] (100GF)
Nonlinear up-shift of threshold [Dimits et al, PoP2000]
Transition of transport scaling from Bohm to gyroBohm via turbulence spreading [Lin et al, PRL2002; PoP2004] (1TF)
• Impacts on theory and experiment: zonal flow, turbulence spreading (24TF)
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Global Gyrokinetic Toroidal Code (GTC)
• Coordinate and mesh
Toroidal geometry
Magnetic coordinates
Global field-aligned mesh
• Particle dynamics
• Field solver
• Parallelization
• Turbulence Spreading
Integrate
orbit
DiagnosticSolve
field
Particle Simulation
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Toroidal Geometry
• Magnetic field lines form nested flux surfaces
• Radial poloidal toroidal
• Safety factor q, magnetic shear s
• Major radius R, minor radius a ITER
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Magnetic Coordinates
• Magnetic coordinate ()
• Flux surface:
• Straight field line:
Efficient for integrating particle orbits & discretizing field-aligned mode
• Boozer coordinates [Boozer, PF1981]: J=(gq+I)/B2~X2
• General magnetic coordinates: J~X
Low aspect-ratio, high- equilibrium [W. X. Wang]
ζBθBψB
ψζq
θψ
B
B1
0 ψB
q
BB
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Global Field-aligned Mesh in GTC
• Discretization in ()), rectangular mesh in (), =-/q
# of computation ~ (a/)2, reduce computation by n~103
No approximation in geometry, loss of ignorable coordinate
Twisted in toroidal direction: enforce periodicity
Magnetic shear: radial derivative, unstructured mesh, complicating FEM solver & parallelization
• Flux-tube approximation [Dimits, PF1993; Beer et al, PF1995; Scott, PoP2001]
• Decomposition in toroidal mode? ~ (a/)3
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Global Gyrokinetic Toroidal Code (GTC)
• Coordinate and mesh
• Particle dynamics
Toroidal perturbative method
Guiding center motion
Collision
• Field solver
• Parallelization
• Turbulence Spreading
Integrate
orbit
DiagnosticSolve
field
Particle Simulation
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Toroidal Perturbative Method
• Perturbative method: discrete particle noise reduced by (f/f)2 [Dimits & Lee, PF1993; Parker & Lee, PF1993; Hu & Krommes, PoP1994]
• ES GK equation: Lf(R,v||,)=0
• Define f=f0+f, L=L0+L, L0f0=0, then Lf=-Lf0
• F0: arbitrary function of constants of motion in collisionless limit.
Canonical Maxwellian [Idomura, PoP2003]
• Neoclassical f simulation [Lin et al, PoP1995]
f0=fM+f02, L0=L01+L02, L01fM=0, L0f02=-L02fM
• Coupling neoclassical physics with turbulence?
• Long time simulation with profile evolution? Full-f?
|||| )(*)(
vBv
tL BEd
bR
vvb
||||0 )(*)(
vBv
tL d
bR
vb
Cv
Bvt
L
||
||01 )(* bR
b
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Electron Models
• For low frequency mode /k||<<v||, electron response mostly adiabatic
• Dynamically evolve non-adiabatic part
• Perturbed potential =+k||=0
• Split-weigh scheme [Mamuilskiy & Lee, PoP2000; J. Lewandowski; Y. Chen]
• Fluid-kinetic hybrid model [Lin & Chen, PoP2001; Y. Nishimura]
Lowest order: fluid, adiabatic response & non-resonance current
Higher order: kinetic, resonant contribution
• Implicit method?
])([/
dee
MTe
T
e
tT
efegL e vvv
geff eTeMe /
)1(||||
vk
fT
ef M
ee
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Guiding Center Equation of Motion
• Gyrocenter Hamiltonian [White & Chance, PF1984]
• Canonical variables in Boozer coordinates
• Equation of motion
• Only scalar quantities needed conserve phase space volume
• Canonical variables in general magnetic coordinates [White & Zakharov, PoP2003]
BBH 22||2
1
pgP
IP
||
||
P
H
dt
dH
dt
dP
P
H
dt
dH
dt
dP
,
,
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Collisions: Monte-Carlo Method
• Electron-ion pitch angle =v||/v scattering in ion frame: Lorentz operator
• Linear like-species guiding center collision operator [Xu & Rosenbluth, PFB1991]
• Conserve momentum and energy, preserve Shifted Maxwellian [Dimits & Cohen, PRE1994; Lin et al, PoP1995]
• Evolve marker density [Chen et al, PoP1997; Wang et al, PPCF1999]
• Evolve background [Brunner et al, PoP1999]
eeei ffC
)1(2
1)( 2
02/12
00 ])1(12)[5.0()1( trt
)()(2
1)(
)(2
1)(
)()(),(),(),()(
22
2
||2||
2
||2||
2
2||||
000
fv
fv
fv
fv
fv
fFPfFPFfCfC ss
Ev
v
dx
xdxPv
v
vxw thth ]
)()([
23))((
23 ||
3 2
2
||2
3
2
2
jj
j
th
jj
th
vwnv
E
vwnv
P
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Global Gyrokinetic Toroidal Code (GTC)
• Coordinate and mesh
• Particle dynamics
• Field solver
Poisson solver
Numerical methods
• Parallelization
• Turbulence Spreading
Integrate
orbit
DiagnosticSolve
field
Particle Simulation
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Poisson Solver
• Gyrokinetic Poisson equation [Lee, JCP1987]
• Polarization density
• Solve in k-space: Pade approximation
• Solve in real space [Lin & Lee, PRE1995]
• Need to invert extremely large matrix
• Iterative method: good for adiabatic electron
• Electromagnetic: FEM via PETSc [Y. Nishimura; M. Adams]
)(4)~
(2 ei
D
nne
kk
F
0
~)()(
~
MRx
j
jji
vkJcdF
vkJk )()()()( 2
0200 M
])(1/[1 20 ik
ijeinm
mnmnij nnc )(,
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Numerical Methods
• Gyroaveraging: performed on poloidal plane (=constant)
Assuming
Gyro-orbit elliptic
Linearized
• Field gathering & charge scattering
Linear interpolation in (
• Radial derivative: finite difference in real space
• Numerical filter
fk=cos2(k/2kmax) for (0.25,0.5,0.25)
ththv
vρρ
ρxR
)()( kk||
nnk
nn
kkncf
xnxcx
)/cos(
)()(
max
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Global Gyrokinetic Toroidal Code (GTC)
• Coordinate and mesh
• Particle dynamics
• Field solver
• Parallelization
Domain-decomposition
Mixed-Mode decomposition
• Turbulence Spreading
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Domain Decomposition
• Massively parallel computer: tightly-coupled nodes• Domain-decomposition for particle-field interactions
Dynamic objects: particle points
Static objects: field grids
DD: particle-grid interactions on-node
• Communication across nodes: MPI On-node shared memory parallelization: OpenMP
• Computational bottleneck: gather-scatter
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Mixed-Mode Domain Decomposition• Particle-field DD: existence of simple surfaces enclosing sub-domains• Field-aligned mesh distorted when rotates in toroidal direction
Not accurate or efficient for FEM solver
• Re-arrangement of connectivity: no simple surfaces• Particle DD: toroidal & radial [S. Ethier]
• Field DD: 3DSolver via PETScPreconditioning HPREInitial guess value from previous time step
• Field repartitioning: CPU overhead minimal
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Physics of Turbulence Spreading
• Coordinate and mesh
• Particle dynamics
• Field solver
• Parallelization
• Turbulence Spreading due to nonlinear mode coupling
Role of zonal flow?
Linear toroidal driftwave eigenmode
Spreading in ITG turbulence (with zonal flow)
Spreading in ETG turbulence (without zonal flow)
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Toroidal Driftwave Eigenmode• Ballooning: mode peak near =0
• Parallel k||~ 1/qR
• Perpendicular
• Radial “streamers”
/1~k
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Toroidal Driftwave Eigenmode
• Linear toroidal coupling of an eigenmode n
Poloidal wavevector k=qn/r
Parallel structure: radial width of m-harmonics
Radial structure: envelope of m-harmonics
“Hidden” kr=s(k
• Spatial resolution in simulation
Parallel ~ R
Radial ~ poloidal ~
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ITG Turbulence Self-Regulation by Zonal Flows
• Nonlinear ITG simulation: turbulence saturated by zonal flows [Lin et al, Science1998; PRL1999]
Zonal flows spontaneously generated by secondary instability
Sheared rotations twist and break up ITG eigenmode: saturation
Coupling of flow poloidal shearing and turbulence radial scattering leads to enhanced decorrelation and suppression of turbulence
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Device size dependence of ITG eddy & transport
• ITG turbulence: eddy size does not increase when device size increase
• Transport size scaling: extrapolation of transport property from existing devices to future larger reactors
• Mixing length rule: r
Large eddy size: Bohm ~ CT/eB
Microscopic fluctuation: gyro-Bohm GB ~ /a
• Experimental evidence of microscopic fluctuation, while transport scaling includes gyro-Bohm, Bohm, …
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ITG: Gradual Transition from Bohm to Gyro-Bohm
• Gradual transition from Bohm to gyro-Bohm [Lin et al, PRL2002]
• Intensity key for resolving the contradiction [Lin & Hahm, PoP2004]
Transport driven by local intensity
Intensity driven nonlocally
JET
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Turbulence Spreading Breaks Gyro-Bohm: nonlocality
• Radial spreading of fluctuation into stable region
• Nonlinearity of ExB drift: local turbulence damping and radial diffusion [Hahm et al, PPCF2004; Hahm et al, PoP2005; Gurcan et al, PoP2005]
• Radial propagation of toroidal drift wave [Chen, White & Zonca, PRL2004; PoP2004, PoP2005]
• Role of zonal flow in turbulence spreading?
• Spatial scale separation important
• Spreading common in fluid turbulence
• Fluid vs. plasma turbulence
Wave-dominated turbulence
Wave-particle interaction
)()( 02 I
rI
rIrI
t
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ETG Ballistic Spreading at Saturation (No Zonal Flow)
• Envelope: t/100=1, 2, 3, 4, 5, 6, 7, 8
• History: r/10=2, 3, 4, 5, 8, 16
• Overlap: t=500, r=40
• Propagation speed:
v=1.8v_drift, 0.3v_dia
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• Turbulence spreading at saturation & steady state
• Spectral inverse cascade
• Eddy rotation
• Particles do not rotate with eddies
• Need feature tracking?
• 5D phase space structure for particle dynamics?
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ETG Energy Inverse Cascade
• (eT)spectrum
• (ke) vs. time (LT/ve)
• ~200 eigenmodes
• ke ~0.03 exited first before saturation
• Successive cascade from ke>0.2 to ke<0.2
• Reach spectral steady state t~1000
• Low-k modes stronger than high-k modes
• Zonal flows significant t>1000
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ETG Saturates via Nonlinear Toroidal Coupling
• Generation of low-n quasi-mode
• Energy transfer to nonlinear mode
Streamers nonlinearly generated
• Cascade facilitated by low-n quasi-mode
Nonlocal in k-space, “Compton Scattering”
• Saturation via nonlinear toroidal coupling before onset of Kelvin-Helmholtz instability
• Consistent with nonlinear gyrokinetic theory
Lin, Chen, & Zonca, PoP2005
Chen, Zonca, & Lin, PPCF2005
),(),(),(),( 12122211 mmnnmnmnmn
),(),(),( 1111 mmnnmnmn
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Turbuelnce Spreading: Channels of Mode Coupling
• Slab: three-mode resonant interaction
• Toroidal ITG: modulational instability via zonal flow
• Toroidal ETG: nonlinear toroidal coupling
• Zonal flow: regulate both intensity and mode coupling
• Nonlocal interaction in k-space
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From Fusion to Space Plasma Physics
• Physics insights
• Theoretical and computational tools
• Alfven turbulence
spectral cascade
plasma heating
scattering of cosmic ray
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Discussions
• There are interesting physics in plasma turbulence
• Physics simulation is NOT about codes; it is about physics understanding
• Improvement of plasma confinement in fusion experiment comes from better physics understanding
• Fusion may be here (and ITER gone) in 35 years, but plasma physics will carry on.