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

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

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)

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

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

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

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

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

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

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

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

,

,

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

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

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 )(,

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

Global Gyrokinetic Toroidal Code (GTC)

• Coordinate and mesh

• Particle dynamics

• Field solver

• Parallelization

Domain-decomposition

Mixed-Mode decomposition

• Turbulence Spreading

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

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

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)

Toroidal Driftwave Eigenmode• Ballooning: mode peak near =0

• Parallel k||~ 1/qR

• Perpendicular

• Radial “streamers”

/1~k

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 ~

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

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, …

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

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

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

• 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?

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

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

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

From Fusion to Space Plasma Physics

• Physics insights

• Theoretical and computational tools

• Alfven turbulence

spectral cascade

plasma heating

scattering of cosmic ray

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.