Gyrokinetic Particle Simulation of Plasma Turbulence

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Gyrokinetic Particle Simulation of Plasma
Turbulence

• Zhihong Lin
• Department of Physics Astronomy University of
  California, Irvine
• Ack.
• GTC Team, US DOE Fusion SciDAC GPS Center

4th Workshop on Nonlinear Plasma Sciences
International School on Plasma Turbulence and
Transport Zhejiang University, Hangzhou, China
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General Remarks

• Turbulence simulation is not simply a matter of
  computing
• Theoretical understanding critical for designing
  simulation models and interpreting simulation
  results
• Simulation complement theory and experiment as a
  tool for scientific discovery
• Turbulence simulation is a team work
• Computational plasma physicist, applied
  mathematician, computational scientist
• One size does not fits all
• PIC discrete particle noise
• Vlasov (continuum) simulation velocity space
  resolution
• Fluid approximate kinetic effects
• PIC high-dimensionality, complex geometry
• PIC method for toroidal plasmas
• No physics/movie

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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
• Device size dependence of transport (Bohm
  scaling) Sydora et al, PPCF1996
• Turbulence self-regulation via zonal flow Lin et
  al, Science1998 PRL1999
• Nonlinear up-shift of threshold Dimits et al,
  PoP2000
• Transition of transport scaling via turbulence
  spreading Lin et al, PRL2002
• Impacts on theory and experiment zonal flow
  physics

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Global Gyrokinetic Toroidal Code (GTC)

• Coordinate and mesh
• Toroidal geometry
• Toroidal eigenmode
• Magnetic coordinates
• Global field-aligned mesh
• Particle dynamics
• Field solver
• Parallel computing
• Discrete particle noise

Integrate orbit
Solve field
Diagnostic
Particle Simulation
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Toroidal Geometry

• Magnetic field lines form nested flux surfaces
• Radial y, poloidal q, toroidal z
• Safety factor q, magnetic shear s
• Major radius R, minor radius a

ITER
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Toroidal Eigenmode

• Ballooning mode peak near q0
• Parallel k 1/qR
• Perpendicular
• Radial streamers

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Toroidal Eigenmode

• Linear toroidal coupling of an eigenmode n
• Poloidal wavevector kqqn/r
• Parallel structure radial width of m-harmonics
• Radial structure envelope of m-harmonics
• Hidden krs(q-q0)kq
• Spatial resolution in simulation
• Parallel R
• Radial poloidal r

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Magnetic Coordinates

• Magnetic coordinate (y,q,z)
• Flux surface
• Straight field line
• Efficient for integrating particle orbits
  discretizing field-aligned mode
• Boozer coordinates Boozer, PF1981
  J(gqI)/B2X2
• General magnetic coordinates JX
• Low aspect-ratio, high-b equilibrium W. X. Wang

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Global Field-aligned Mesh in GTC

• Discretization in (y,a,z)), rectangular mesh in
  (a,z), aq-z/q
• of computation (a/r)2, reduce computation by
  n103
• 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/r)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

Integrate orbit
Solve field
Diagnostic
Particle Simulation
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Toroidal Perturbative Method

• Perturbative method discrete particle noise
  reduced by (df/f)2 Dimits Lee, PF1993
  Parker Lee, PF1993 Hu Krommes, PoP1994
• ES GK equation Lf(R,v,m)0
• Define ff0df, LL0dL, L0f00,
  
  then Ldf-dLf0
• F0 arbitrary function of constants of motion in
  collisionless limit.
• Canonical Maxwellian Idomura, PoP2003
• Neoclassical df simulation Lin et al, PoP1995
  
  f0fMf02, L0L01L02, L01fM0,
  L0f02-L02fM
• Coupling neoclassical physics with turbulence?
• Long time simulation with profile evolution?
  Full-f?

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Electron Models

• For low frequency mode w/kltltv, electron
  response mostly adiabatic
• Dynamically evolve non-adiabatic part
• Perturbed potential fdfF(k0)
• Split-weigh scheme Mamuilskiy Lee, PoP2000
• Fluid-kinetic hybrid model Lin Chen, PoP2001
• Lowest order fluid, adiabatic response
  non-resonance current
• Higher order kinetic, resonant contribution
• Extension of hybrid model for tearing mode?
• Implicit method?

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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

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Collisions Monte-Carlo Method

• Electron-ion pitch angle xv/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
• Full-f simulation discrete particle collides
  with background plasma fluid

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Global Gyrokinetic Toroidal Code (GTC)

• Coordinate and mesh
• Particle dynamics
• Field solver
• Poisson solver
• Numerical methods
• Parallelization

Integrate orbit
Solve field
Diagnostic
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 Nishimura Adams

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Numerical Methods

• Gyroaveraging performed on poloidal plane
  (zconstant)
• Assuming
• Gyro-orbit is elliptic
• Linearized
• Field gathering charge scattering
• Linear interpolation in (y, q, z)
• Radial derivative finite difference in real
  space
• Numerical filter
• fkcos2(pk/2kmax) for (0.25,0.5,0.25)
• Gather-scatter operations via finite element
  method?

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Concluding Remarks

• New GK frontiers
• Electron dynamics electromagnetic turbulence
• Multi-scale, multi-process
• Space astrophysical plasmas
• Simulation is interdisciplinary SciDAC, FSP

RF-heating of fusion plasmas nonlinear effects
in coupling and mode-conversion regions
Turbulence heating of solar wind
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• Physics Today, November 2004, Kadanoff
• Computational Scenarios
• There are two ways in which a simulation may be
  useful. First, the simulation may prove
  something. A second kind of simulation is the
  exploratory one that suggests new mechanisms for
  complex physical processes.
• Physics Today, February 2005, Batchelor
• Integrated Simulation of Fusion Plasmas
• Although physicists cannot self-consistently
  incorporate the short-time and short-wavelength
  phenomena, they can use reduced models to
  simulate evolution occurring over long time
  scales.
• Physics Today, January 2005, Post Votta
• Computational Science Demands a New Paradigm
• Preliminary computational predictions in 1996 of
  inadequate performance by ITER were wrongly
  characterized as definitive. Those prediction
  contributed to the 1998 US withdrawal from
  ITER.
• Recent PR on ITER
• Computer simulation shows ITER would work