Geometric Gyrokinetic Theory and its Applications to Large-Scale Simulations of Magnetized Plasmas Hong Qin Princeton Plasma Physics Laboratory, Princeton University CEA-EDF-INRIA School -- Numerical models for controlled fusion Univ. of Nice, France Sept 11, 2008 www.pppl.gov/~hongqin/
Complementary to the lectures by Eric Sonnendrücker: Gyrokinetic Vlasov simulation method Laurent Villard: Noise reduction, gyrokinetic simulation of turbulent transport Jukka Heikkinen: Gyrokinetic particle simulation method
Gyrokinetic approximation and algorithm for fusion plasmas Magnetized plasmas fast gyromotion e v B + v + + f (,, t ) = 0 j E j t c x p x p Average out fast gyromotion Multiscale Gyrokinetic system $$ of DOE OFES Fast algorithm for fusion simulations
However, three problems e v B + v + + f (, v, t ) = 0 j E j t c x x v θ dependent 2 : v,? 2π dθ [Valsov Eq.] π x c v 0 1 Problem 1 f v B f Gyrokinetic system Problem 2: energy conservation? Fusion plasma simulations Problem 3: conservative algorithms?
Problem #3, example Gyrocenter dynamics and standard non-conservative algorithms dx μ B b E b u b ( b b) = ub + + + dt B B B du μb B = + b E dt B 2 Carl Runge (1856-1927) Martin Kutta (1867-1944) nth nth (4 (4 th th )) order Runge-Kutta methods Long time time non-conservation. Errors add add up up coherently.
Problem #3 banana orbit goes bananas 5 5 10 turns Exact banana orbit Numerical result by RK4 6 burn T banana ITER: T ~ 3 10 6 run T banana Tore Supra: T ~ 10 10 H. Qin and X. Guan, PRL 100, 035006 (2008).
60 million years ago
Outline Solution to Problem 1 decouple, not not average Solution to Problem 2 field field theory γ Poincare-Cartan- Einstein 1-form Solution to Problem 3 symplectic algorithm Gyrocenter gauge theory & algorithm for for RF RF physics
( xv) 1-form γ in the, coordinates 1 2 L = A v + v φ 2 1 2 = ( A+ v) v v + φ 2 dt base vectors Elie Cartan (1869-1951) 2 v γ = A+ p = ( A+ v) dx + φ dt 2 A ( φ, A), 4-vector potential 2 v p (, p), 2 4-momentum
Where does γ live? -- 7D phase space Phase space is a fiber bundle π : P M. Cotagent bundle 1 2 2 P = {( x, p) x M, p T M, g ( p, p) = m c } x 7D Spacetime Inverse of metric
Solution to Problem 1: Decouple not average out Find the gyrocenter coordinates ( X, u, μθ, ): B = const. d du dθ dμ,, = 0, = 0 θ dt dt dt dt X 2 b v v X = x +, u = v B, μ = B 2B f dx du f + f + X t dt dt u f dx du f = + f + = X t dt dt u θ 0 B const. How to find gyrocenter coordinates? Go geometric Gyrocenter is is about gyro-symmetry.
Poincare-Cartan-Einstein 1-form γ particle dynamics Hamilton s Eq. Exterior derivative t 1 t 1 I = Ldt = γ 0 0 Eular-Lagrangian of I Inner product id τ γ = 0 Worldline
Gyro-symmetry in γ Coordinate: L d L = 0, = 0. θ dt θ Geometric: general, stronger, enables techniques to find θ. S. Lie (1890s) Noether s Theorem (1918) Lie derivative L γ = η ds Symmetry vector field d( γ η) τ = ds τ γ η S is conversed. ( ) L γ = d i γ + i dγ η η η Cartan s formula
Gyrophase is the orbit of gyrosymmetry B = const. Kruskal (1958) η 1 1 = v x v + + y = B x v B y v θ y x Noether s Theorem μ 2 2 x + vy v = = 2B const. Amatucci, Pop 11, 2097 (2004). B const. but weakly inhomogeneous Coordinate perturbation to find symmetry η =. θ The dynamics transform as an 1-form!
Perturbation techniques quest of good coordinates Peanuts by Charles Schulz. Reprint permitted by UFS, Inc.
Dynamics under Lie group of coordinate transformation Continuous Lie group g : z Z = g( z, ε) Vector field (Lie algbra) G = dg / d ε ε = 0 γ in Z Pullback Taylor expansion 1 1 2 Γ ( Z) = g γ( z) = γ g ( Z) = γ( Z) LGZ ( ) γ( Z) + O( ε ) 2 = γ( Z) i dγ( Z) d[ γ G( Z) ] + O( ε ). GZ ( ) Cartan s formula Insignificant
Gyrocenter dynamics 1 2 ( A b) X ( ) γ = + u d + μdθ u + μb + φ dt 2 id τ γ = 0 1 2 ( A b) X μθ ( μ φ ) L = + u + u + B + 2 R. Littlejohn (1983) E-L Eq. E B, B drift curvature drift dx B μ b E = ( u + b b) dt B 2 B du dt dθ dt B E = B dμ = B, = 0 dt B A, A A+ ub B, = B b E E μ B Banos drift (1967)
Solution to Problem 2: Field theory Vlsov-Maxwell field theory in ( xv, ) 1 Lf t d x E B 2 3 2 2 ( ) = ( ) 3 3 0 0 0 0 0 L ( t) = 4π d x d v f ( v, x ) p 2 v φ ( x, t) + x [ A( x, t) + v] 2 x x( x, v, t), v v( x, v, t) 0 0 0 0 I[ φ, Axv,, ] = dt[ L ( t) + L ( t)] f p δi δi, =0 Maxwell Eqs. δa δφ δi δi, =0 Vlasov Eq. δx δv I t Noether s Theorem =0 Energy conservation
Geometrically generalized Vlasov-Maxwell system --- A field theory Ω 1 6 dγ dγ dγ γ I = L, x 1 L( x) = da da + 4π f Ω 2 π 1 ( x ) γ L f L p Valid for any γ. Exact conservation propertie. Allow physics models, approximations build into γ.
Even for curved spacetime
Second order field theory 4-diamagnetic current δi δa = 0 response to potentials d da = 4π j α δγ( x ) j ( x) = f 1 x Ω π ( x ) δaα ( x) γ( x ) = δ ( x x ) f 1 x Ω π ( x ) Aα ( x ) γ( x ) δ( x x ) f ( α 0 12 3) β 1 x x Ω, =,,, π ( x ) Aαβ, ( x ) 4-diamagnetic current A x α β response to strength
Modern gyrokinetics = field theory + gyro-symmetry 1 2 ( A b) X ( ) γ = + u d + μdθ u + μb + φ dt 2 Field theory Gyro-symmetry Interaction between particle and fields Decouple gyromotion Gyrokinetic Maxwell s Eq. Eq. F dx du F + XF + = 0 t dt dt u F f θ
Electrostatic gyrokinetic system Exact energy conservation Inhomogeneous background γ H = A dx + μdθ Hdt 2 2 u + w = + ψ + 2 ψ 1 2 ψ = φ ( X + ρ), φ φ ( X + ρ) dθ ψ 1 1 1 1 1 1 = φ φ b B 2 1 2 1 B 2 B 0 0 φ μ 2 1 2 1 φ = 4πn ( x ) n polarization density δ ψ1 δ ψ2 = FΩ + x π 1 ( x ) δφ1( x) δφ1( x) F φ 1 φ F B φ = F+ + F b + μ B 0 B B u 0 B B0 3 δ x x ρ x Bdx dudμdθ ( x) δ ( x x ) [ ( )] 1 1
Physics of 2 nd order field theory polarization density x = X + ρ + G 1X Gyrocenter density e e p = G1 X Z g0( z) dθ = φ 2π 2 Ω G 1 = S. B 1X 1
Problem 3: existing numerical algorithms do not bound errors dx μ B b E b u b ( b b) = ub + + + dt B B B du μb B = + b E dt B 2 Carl Runge (1856-1927) Martin Kutta (1867-1944) nth nth (4 (4 th th )) order Runge-Kutta methods Long time time non-conservation. Errors add add up up coherently.
Problem #3 banana orbit goes bananas 5 5 10 turns Exact banana orbit Numerical result by RK4 6 burn T banana ITER: T ~ 3 10 6 run T banana EAST: T ~ 10 10 H. Qin and X. Guan, PRL 100, 035006 (2008).
Problem #3: Can we do better than RK4? -- symplectic integrator Conserves symplectic structure; Bound energy error globally Feng (1983) Ruth (1984) Symplectic integrator? Requires canonical Hamiltonian structure q 0 1 H, q = p 1 0 H, p Application areas: Accelerator physics (everybody) Planetary dynamics (S. Tremaine) Nonlinear dynamics (everybody) Gyrocenter dynamics does not have a global canonical structure
Gyrocenter dynamics does not have a global canonical structure ω Only has a non-canonical symplectic structure Ω dω 1 2 ( A b) X ( ) γ = + u d u + μb + φ dt 2
Darboux Theorem Darboux Theorem (1882): Every symplectic structure is is locally canonical. Jean Gaston Darboux (1842-1917) Gyrocenter dynamics can be canonical locally. No symplectic integrator for gyrocenter?
Variational symplectic integrator id τ γ = 0 A= t 1 discretize on t = [ 0, h, 2h,...,( N 1) h] 0 Ldt 2 ( A b 1 ) X ( μ φ) L = + u u + B + 2 J. Marsden (2001) N 1 A A = hl ( k, k + 1) d k= 0 d L ( k, k + 1) L ( x, u, x, u ) d d k k k+ 1 k+ 1 ( x,u ) minimize w.r.t. k k ( xk 1 uk 1), ( xk uk ) ( x u ),,, k+ 1 k+ 1 x u j k k [ L ( k, 1k) + L ( k, k + 1) ] = 0, ( j =,, 12 3) d [ L ( k, 1k) + L ( k, k + 1) ] = 0 d d d discretized Euler- Lagrangian Eq.
Conserved symplectic structure θ θ + ( kk, + 1) L ( kk, + 1) dx + L( kk, + 1) du x u d k+ 1 d k+ 1 k+ 1 k+ 1 ( kk, + 1) L ( kk, + 1) dx L( kk, + 1) du x u k d k d k k dl ( kk, + 1) = θ ( kk, + 1) θ ( kk, + 1) d + ( x,u ) minimize w.r.t. k k + Ω d ( kk, + 1) dθ = dθ + da = θ ( 01, ) θ ( N 1, N ) d Ω d( 01, ) =Ωd( N 1, N )
Variational symplectic integrator, banana orbits Δ t = T / 50 RK4 RK4 Δ t = T / 17 Variational symplectic 5 5 10 turns Transport reduction 6 by integration errors ITER: T ~ 3 10 T burn banana
Variational symplectic integrator bounds global error Xiaoyin Guan (2008) Energy Parabolic growth of energy error 02. T burn H. Qin and X. Guan, PRL 100, 035006 (2008).
Bonus: Gyrocenter gauge theory and algorithm for RF physics 5 10 21 nt τ > m 3 s KeV heating confinement Lawson Gyrokinetics for for multiscale RF RF physics Resonance heating and and CD CD Mode conversion Parametric decay
Bonus: Gyrocenter gauge theory and algorithm for RF Physics Gyrokinetic eq. F dx du F + XF + = t dt dt u 0 Gyrocenter gauge S S S S + X + u + θ t X u θ = φ ( X + ρ, t) v A( X + ρ, t) 0 0 Resonance heating & CD CD Mode conversion Parametric decay DOE RF RF SciDAC ASCR MultiScale Gyrokinetics
Kruskal ring as simulation particle Kruskal (1958) Kruskal ring ring species X u μ S[ n] θ Gyrocenter gaugue replaces gyrophase gyrocenter Gyrokinetic is is about gyro-symmetry. Gyrophase is is decoupled, not not removed.
Pullback of distribution function Particle distribution Don't know f ( z). Gyrocenter gauge S pullback from FZ ( ) : fz () = g[ FZ ( )] = F( gz ()).
G-Gauge (Gyrocenter-Gauge) Code Zhi Yu (2008) Numerical moment integral of of gyrocenter gauge algorithm
Electron Bernstein Waves
X-Wave (cold)
X-Wave reflection, tunnelling, and mode conversion to Bernstein wave Cut-off resonance X-wave Bernstein wave
Larger conversion rate for hotter plasmas Cut-off resonance X-wave Bernstein wave
Conclusions Multiscale gyrokinetics requires good good theories and and algorithms. Good theories are are geometric. So So are are good good algorithms.