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Tomo-Hiko Watanabe Department of Physics, Nagoya University [email protected] www.p.phys.nagoya-u.ac.jp 2016/7/13 EASW2016@Tsukuba 1 Outline Introduction Research target, motivations for gyrokinetics Fluid and kinetic equations Gyrokinetic equations and simulation models Gyrokinetic ordering and df and full-f GK equations Basic properties of the gyrokinetic equations Applications to plasma turbulence Solar wind turbulence and cascades on the phase space Zonal flow and turbulence controle in fusion plasmas Multi-scale turbulence simulation on HPC 2016/7/13 EASW2016@Tsukuba 2 2016/7/13 EASW2016@Tsukuba 3 Strongturbulenceandtransportin magnetizedplasma Strong turbulence drives the particle and heat transport, if mean gradients of n and/or T exist in magnetic fusion plasma. TFTR experiments [Fonck+ PRL (1993)] Turbulence with ion gyro-radius scales Turbulent transport coefficients Turbulent density fluctuations observed in a whole torus Normalized minor radius Poloidal wavenumber Normalized minor radius 2016/7/13 EASW2016@Tsukuba 4 Researchtargetsofgyrokinetics are… Low frequency (w << Wi) waves and instabilities in magnetized plasma Alfven waves, drift waves, MHD and drift wave instabilities, micro-tearing mode … Turbulent transport of particle, heat, and momentum driven by the low-frequency waves Anomalous transport in fusion and space plasmas Energy conversion through plasma kinetic processes Particle acceleration, heating, and dissipation Magnetic reconnection, … 2016/7/13 EASW2016@Tsukuba 5 Motivations- Whydoweneed kineticapproaches? A set of fluid equations, such as MHD equations, represent conservation of mass, momentum, and energy: 𝑑𝜌 𝑑𝒗 𝑑𝑝 = −𝜌𝛻 ' 𝒗, = −𝛻𝑝 + 𝒋×𝑩, = −𝛾𝛻 ' 𝒗 𝑑𝑡 𝑑𝑡 𝑑𝑡 Only fluid quantities, i.e., the charge and current densities, are used in the Maxwell equations … Insufficient to describe the collisionless plasma? Where is a flaw? 2016/7/13 EASW2016@Tsukuba 6 Fluidapproximationmaybreak downinhightemperatureplasmas Fluid approximation can be valid for L >> lii : mean-free-path ( // to B ) >> ri : gyro-radius (perpendicular to B) In fusion plasmas of Ti~10 keV, n~1014/cc, nii ~ 102 s-1, lii ~ 104 m, a~1m, qR0~10m Thu, the Knudsen number lii / qR0 ~ 103 !! How large is lii in the Earth’s magnetosphere? lii ~ O(108 km) !! (for Ti~10 eV, n~5/cc) 2016/7/13 EASW2016@Tsukuba 7 StartfromtheVlasov equations Advection of f along particle trajectories in the phase space (Hamiltonian flow), 𝜕𝑓 𝑞 𝜕𝑓 + 𝒗 ' 𝛻𝑓 + 𝑬 + 𝒗×𝑩 ' =0 𝜕𝑡 𝑚 𝜕𝒗 or 𝜕𝑓 + 𝐻, 𝑓 = 0 𝜕𝑡 includes a variety of kinetic effects, i.e., Landau damping, particle trapping, finite gyroradius effects, … Coupling to higher-order moments is generated by the advection term, 𝒗 ' 𝛻𝑓. 2016/7/13 EASW2016@Tsukuba 8 MomentsofVlasov equations Definethe𝑁th-order moment of 𝑓 as D 𝑀@A…C ≡ F 𝑓𝑣@ 𝑣A … 𝑣C 𝑑 H 𝑣 Taking the 𝑁th moment of the Vlasov equation, one finds D 𝜕𝑀A…C 𝜕𝑡 DJK + where 𝐴@ 𝐵AN…C 2016/7/13 𝜕𝑀@A…C 𝜕𝑥@ @ 𝑞 DOK − 𝐸A 𝑀N…C 𝑚 A −⋯=0 = 𝐴A 𝐵@N…C + 𝐴N 𝐵A@…C + ⋯ + 𝐴C 𝐵AN…@ EASW2016@Tsukuba 9 Weneedaclosuremodelto completeasetoffluidequations D Equation for 𝑀@A…S involves the higher-order moment, DJK 𝑀@A…C . DJK To truncate the moment hierarchy at 𝑁, 𝑀@A…C should be modeled by means of lower-order moments, 𝑀 K 𝑀@ , T , D …, 𝑀@A…S . c.f. Ideal fluid models often neglect the heat flux, i.e., the 3rd-order moment. How is it justified ? 2016/7/13 EASW2016@Tsukuba 10 Closuremodelneedstobe validatedincomparisonwithf Simple closure models: adiabatic, CGL model The heat flux vanishes for 𝑓 = 𝐹V , which is the simplest closure for ideal fluids. Hammett-Perkins (Landau fluid) closure is designed to mimic the Landau damping. Taking the fluid moments of 𝑓 is equivalent to projection of 𝑓 onto a polynomial basis of the velocity space, e.g. Hermitian polynomial expansion. Higher-order moments are related to finer-scale structures of 𝑓. Closure models introduce coarse-graining of fine structures of 𝑓. 2016/7/13 EASW2016@Tsukuba 11 Howdoesthedistributionfunction developinthephasespace? Nonlinear Landau damping in 1-D Vlasov-Poisson system, where 𝑓 𝑥, 𝑣, 𝑡 = 0 = 𝐹V 1 − 𝐴 cos 𝑘𝑥 Fine structures of 𝑓 continuously develop Ballistic modes with scale-lengths of 1/𝑘𝑡 in 𝑣-space Stretching of 𝑓due to shear of the Hamiltonian flow t=0 wpe-1 t=10 wpe-1 t=40 wpe-1 v 2016/7/13 x EASW2016@Tsukuba 12 Thus,weneedkineticdescriptions In high-temperature plasma, Collisionality is quite low. Mean-free-path >> system size Distribution function can be far from FM. Construction of closure models for a collisionless regime is still in progress, e.g., FLR closure We need to deal with the kinetic equation of f on multi-dimensional phase space… 2016/7/13 EASW2016@Tsukuba 13 2016/7/13 EASW2016@Tsukuba 14 Kineticmodelshouldbesimplified forlow-frequencyphenomena Although the Vlasov equation is “the first principle” for describing collisionless plasma behaviors, it OK OK involves short time scale of ΩOK , Ω , 𝜔 _ a ... ^ In a magnetic fusion plasma with 𝐵 = 1T, 𝑒𝐵 Ω^ = ~1×10e[rad ' sec OK ] 𝑚^ We need reduced kinetic equations to eliminate the fast gyro-motion as well as 𝜔a, while keeping finite gyro-radius and other kinetic effects. => Gyrokinetic equations 2016/7/13 EASW2016@Tsukuba 15 FromVlasov togyrokinetic eqs. To deal with fluctuations slower than the gyro-motion, reduce the Vlasov equation to a gyro-averaged form: Gyrokinetic ordering and perturbation expansion 𝜔 𝜌 𝑘∥ 𝛿𝑓 𝑒𝜙 𝛿𝐵 𝜀~ ~ ~ ~ ~ ~ , 𝑓 = 𝑓T + 𝛿𝑓 Ω 𝐿 𝑘n 𝑓T 𝑇 𝐵T Recursive formulation of linear gyrokinetic equations [Rutherford & Frieman (1968); Antonsen & Lane (1980)] 2016/7/13 EASW2016@Tsukuba 16 Perturbedgyrokinetic equation Gyrocenter coordinates 𝑋 (t), 𝑣∥, 𝜇, 𝜉 µ: magnetic moment, x: gyrophase (a) Particle and gyrocenter distributions 𝛿𝑓x,𝒌 δf ( p) k⊥ =δ f (g) k⊥ rs X(g) eφk⊥ X(p) and (t) 𝛿𝑓x,𝒌 exp (−ik ⊥ ⋅ ρ ) − FM ⎡⎣1− J 0 ( k⊥ ρ ) exp (−ik ⊥ ⋅ ρ )⎤⎦ !# #"## $ !T######"###### $ differenceof particle position and gyrocenter polarization (t) Nonlinear gyrokinetic equation for 𝛿𝑓x ⎡∂ µs ∂ ⎤ (g) c ⎧ v|| esϕ ⎫ (g) ⎬ ⎢ + v||b ⋅ ∇ + v ds ⋅ ∇ − b ⋅ ∇B ⎥δ fs + ⎨Φ − Ψ, δ fs + m ∂v|| ⎦ B0 ⎩ c Ts ⎭ ⎣ ∂t Magnetic s drift Mirror force ExB drift and advection along 𝐵z ⎤ ⎛ es ⎛ 1 ∂Ψ ⎞ es ⎡ v|| ⎞ = −v||F0s ⎜ b ⋅ ∇Φ + ⎟ + F0s ⎢v∗s ⋅ ∇ ⎜ Φ − Ψ ⎟ − v ds ⋅ ∇Φ⎥ ⎝ Ts ⎝ c ∂t ⎠ Ts ⎣ c ⎠ ⎦ [Friemann & Chen, ’82] Parallel electric field 2016/7/13 EASW2016@Tsukuba Diamagnetic drift 17 Gyrokinetic Poissonequationand Ampere’slaw Fluctuations of the electrostatic potential are given by the quasi-neutrality condition, 2 e s φk % −k ρ (g) 3 2 2 ' e J k ρ δ f d v − n 1− e I k ∑ s ∫ 0 ( ⊥ s ) sk 0∑ 0 ( ⊥ ρ ts )( = 0 & 2 2 ⊥ ts ⊥ ⊥ s s Difference of X(g) and X(p) Ts Polarization of backgrounds Fluctuations of the flux function are calculated from the Ampere’s law, 4π 2 (g) 3 k⊥ψ = e v δ f ∑ s ∫ || sk d v c ⊥ s The finite gyroradius effect is also taken into account, (Φk = J 0 (k⊥v⊥ /Ωs )φ k Effective potential ) acting on gyrocenters Ψ = J (k v /Ω ) ψ 0 ⊥⊥ s k * k 2016/7/13 ⊥ ⊥ ⊥ ⊥ EASW2016@Tsukuba 18 AdifferentformofGKequation preservingthephase-spacevolume Fast gyromotion is eliminated by the Lie transform to gyrocenter coordinates and the gyrophase average [Littlejohn, 1979, ’81, ’82, ’83] fs (R, u, µ ) Gyrocenter Hamiltonian: Independent of the gyrophase angle x Adiabatic invariant: µ [Hahm, 1988; Brizard, 1989] Poisson brackets 2016/7/13 EASW2016@Tsukuba 19 Basicpropertiesofgyrokinetics Suitable to describe time-varying fluctuations slower than the cyclotron period, ΩOK x Introduction of finite gyroradius effects Gyrocenter effectively feels the potential averaged over the gyro-orbit 2016/7/13 EASW2016@Tsukuba Electric field Particle density at x is given by sum of particles with different gyrocenter positions 𝑋 (t) , of which orbits are polarized by the electric field 20 Moreadvantagesofgyrokinetics Parallel electric field involved in GK can describe Landau damping, drift instabilities, particle acceleration, and magnetic reconnection Strong anisotropic fluctuations of 𝑘∥ ≪ 𝑘n are resolved. Consistent to the flute ordering in MHD Small amplitude fluctuations of 𝑒𝜙 ≪ 𝑇 can be considered. Magnetic and diamagnetic drift and mirror motions are included. Polarization term modifies the plasma dielectricity. 2016/7/13 EASW2016@Tsukuba 21 ThreeapproachesinGKsimulation PIC model combined with df-method Compute motion of charged rings [Lee, 1987] Time-varying weight function describing df [Dimtis+, 1993] Vlasov approach Solve the GK equations on grids or spectral methods df and full-f methods; local flux tube and global models Semi-Lagrangian method Compute mappings of f with interpolations 2016/7/13 EASW2016@Tsukuba 22 Plasmatheorycoversmultiplespaceandtime-scales Fluid Macroscopic MHD Eqs. Extended MHD or Tow-Fluid Eqs. Velocity-Space Moment & Closure Model Gyrokinetic Eqs. Kinetic Drift Kinetic Eqs. Ordering & Gyrophase Average Boltzmann (Vlasov) Eq. of One-Body Distribution Function f Microscopic Reduced Models Kinetic (gyro) Fluid Eqs. BBGKY Hierarchy Newton (Klimontovich) Eq. for N-Particles + Maxwell Eqs. 2016/7/13 EASW2016@Tsukuba 23 2016/7/13 EASW2016@Tsukuba 24 Howdoesthedissipationworkin thesolarwindturbulence? Mean flow energy => Turbulence energy Turbulence spectrum from the MHD scale to the ion or electron gyroscales (assuming Taylor’s frozen hypothesis). Spectrum in the inertial subrange can be given by reduced MHD or EMHD Dissipation process? 0.001 Electron / ion heating? Bale+ PRL (2005); Schekochihin+ ApJ (2009) 2016/7/13 EASW2016@Tsukuba <E2> <B2> 0.01 0.1 kri 1 10 25 DirectsimulationbyGK Forced GK turbulence with ion and electron fluctuations Energy spectra of E and B Roughly consistent with observations and dimensional analysis => MHD or EMHD Multi-scale turbulence including ions and electron gyroradii Mixing process in the phase space is important 2016/7/13 EASW2016@Tsukuba Collisional dissipation 1 kri 10 Told+ PRL (2015) 26 Driftwaveturbulenceinsimple slabgeometrywithuniformB0 A simple test problem with fixed gradients of density and temperature normal to the confinement field Drift wave turbulence and transport driven by ion temperature gradient z Snapshot of electrostatic potential (stream function) B0 ∇n, ∇T y y Transport flux € 2016/7/13 x EASW2016@Tsukuba x 27 Small scale structures of the perturbed distribution function df continuously develops in the velocity space Turbulence intensity spectrum on the phase space δf 2 Time Turbulencecascadesfrommacroto micro-scalevelocityspace 2FM Velocity 2016/7/13 EASW2016@Tsukuba Watanabe & Sugama, PoP 2002 & 2004 28 Velocity-spacespectralanalysisfor kineticplasmaturbulence ∞ Hermite polynomial expansion of δ fk (v) = ∑ fˆk,n H n (v)FM (v) gives the n=0 “entropy” transfer equation 2 δf Entropy variable: δ S ≡ ∫ dv 2FM Transfer function is defined by Watanabe & Sugama, PoP 2004 2016/7/13 EASW2016@Tsukuba 29 “Inertialsub-range”isdiscovered alsointhevelocity-space Constant profile of the transfer function Jn shows an intermediate scale free from “entropy” production and dissipation Collision frequency is changed from 2x10-3 to 2x10-6 Similarity to the turbulence cascade in neutral f luids 2016/7/13 EASW2016@Tsukuba Watanabe & Sugama, PoP 2004 30 2016/7/13 EASW2016@Tsukuba 31 ZonalFlowsinNature Zonal Flows in Jupiter Hubble/Cassini, NASA/ESA 2016/7/13 EASW2016@Tsukuba Differential Rotation in Sun SOHO/MDI, NASA/ESA 32 ZonalFlowsandTurbulencein FusionPlasmaSimulations Hasegawa & Wakatani, PRL 59, 1581 (1987). Zonal flows have been found in various types of plasma turbulence simulations. A pioneering work by Hasegawa & Wakatani for zonal f low generation in turbulence (upper) with zonal flows w/o zonal flows Zonal f lows distorting eddies lead to turbulence regulation and transport reduction (lower left) 2016/7/13 EASW2016@Tsukuba Lin et al., Science 281, 1835 (1998). 33 IdentificationofZonalFlowsin FusionPlasma Zonal flow in a torus: potential fluctuations with poloidal and toroidal symmetries but with radial variations Heavy ion beam probe system in CHS experiment 1 Zonal flow pattern (Electrostatic potential f ) r =12cm 90 degree apart 1 C (r1,r2) 0.5 0 -0.5 V=c R=1m, a=0.2m A. Fujisawa et al., PRL 93 165002 (2004) 2016/7/13 EASW2016@Tsukuba -1 10 B × ∇φ B2 11 12 radius r (cm) 2 13 14 34 Fluxtubesimulationmodelfor fusionplasmas Beer, Hammett, Cowley, PoP 1995 After 20 years of the BCH paper of the toroidal flux tube model, GK simulations have largely been advanced ... A variety of codes, GS2, GENE, GKW, GKV, ... Widely used for theoretical and experimental studies of turbulent transport and zonal flows in fusion plasmas 2016/7/13 EASW2016@Tsukuba 35 HelicalFieldEnhancingZonalFlow Generation Poloidal coil Helical plasma is characterized by non-symmetric confinement field Radial drift motion of helicalripple-trapped particles leads to shielding effect of zonal flow. Plasma Theory and simulation suggest increase of zonal flow response B (to a source) in optimized helical confinement field Helical coils passing trapped vdr b => Enhancement of zonal flow generation Sugama & Watanabe, PRL 2008 Ferrando, Sugama & Watanabe, PoP 2007 2016/7/13 EASW2016@Tsukuba Classification of particle orbits in helical system 36 Turbulencecontrolledbyconfinement fieldoptimizationviazonalflows Un-optimized case Ion Heat Transport Optimized helical field Un-optimized Optimized Strong zonal flows generated by optimizing particle orbits in the helical field lead to further reduction of ion heat transport Watanabe, Sugama, Ferrando-Margalet, PRL 2008 2016/7/13 EASW2016@Tsukuba 37 2016/7/13 EASW2016@Tsukuba 38 HighPerformanceComputingwithGKV Maeyama et al (2013) Multi-scale ITG/TEM/ETG turbulence simulations demand huge computational costs Grids~1011; time steps~105; parallelization ~100k Improvement of strong scaling is critically important 2016/7/13 EASW2016@Tsukuba Strong scalings of GKV Performance [TFLOPS] Optimization on K computer Five-dimensional domain decomposition Optimized MPI process mapping on 3D torus Computation-communication overlap Excellent strong scaling for ~600k cores keeping 99.99994% parallelization rate 103 Maeyama et al (2013) K (6x1011 grids) 102 10 1 103 K (3x1010 grids) BX900 (4x109 grids) 104 105 Number of cores 106 39 Gyrokinetic simulationresolving theITGandETGturbulence The flux tube code, GKV, has been applied to the direct numerical simulation of the ITG and ETG turbulence. The highly scalable code enables the peta-scale computing on ~100k cores of the K computer (~100hrs) Cyclone base case with β = 2% and 𝑚^ ⁄𝑚_ = 1836 Maeyama+ PRL (2015) 2016/7/13 EASW2016@Tsukuba 5 2.5 ITG/TEM 0 -2.5 -5 ETG 40 Transportinmulti-scale turbulence:“Moreisdifferent” Electron thermal diffusion coefficient cek/cgB (ce/ cgBi)/( Dkyrti ) Transport in the multi-scale turbulence is characterized neither of the ITG and ETG transport in a single-scale. 102 100 Electron-scale simulation 10-2 0.01 10-4 0.0001 Maeyama+ PRL (2015) 2016/7/13 ce=5.4cgB 1010 Ion-scale simulation ce=1.2cgB Multi-scale simulation ce=4.5cgB 0.1 0.1 11 10 10 Poloidal wave number kyρti EASW2016@Tsukuba Poloidal wave number ykrti 41 Summary Introduction Kinetic approach is essential in descriptions of collisionless plasma behaviours Gyrokinetic equations and simulation models Overviewed the gyrokinetic equations Several advanges brought by the gyrokinetic ordering Applications to plasma turbulence Solar wind turbulence and cascades on the phase space Zonal flow and turbulence controle in fusion plasmas Multi-scale turbulence simulation on HPC 2016/7/13 EASW2016@Tsukuba 42