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 Diagnostic Solve 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)/B 2 ~X 2 General magnetic coordinates: J~X Low aspect-ratio, high-  equilibrium [W. X. Wang]

Global Field-aligned Mesh in GTC Discretization in (  )), rectangular mesh in (  ),  =  -  /q # of computation ~ (a/  ) 2, reduce computation by n~10 3 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 Diagnostic Solve 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=f 0 +  f, L=L 0 +  L, L 0 f 0 =0, then L  f=-  Lf 0 F 0 : arbitrary function of constants of motion in collisionless limit. Canonical Maxwellian [Idomura, PoP2003] Neoclassical  f simulation [Lin et al, PoP1995] f 0 =f M +f 02, L 0 =L 01 +L 02, L 01 f M =0, L 0 f 02 =-L 02 f M Coupling neoclassical physics with turbulence? Long time simulation with profile evolution? Full-f?

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?

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]

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]

Global Gyrokinetic Toroidal Code (GTC) Coordinate and mesh Particle dynamics Field solver Poisson solver Numerical methods Parallelization Turbulence Spreading Integrate orbit Diagnostic Solve 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]

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 f k =cos 2 (  k/2k max ) for (0.25,0.5,0.25)

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: 3D Solver via PETSc Preconditioning HPRE Initial 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”

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” k r =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

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 (e  T)  spectrum (k  e ) vs. time (L T /v e ) ~200 eigenmodes k  e ~0.03 exited first before saturation Successive cascade from k  e >0.2 to k  e <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

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.