Charged particle transport in turbulent media F. Spanier A. Ivascenko S. Lange C. Schreiner Center for Space Research, North-West University Astronum 2013, Biarritz
Motivation Particle transport in heliosphere and ISM What is the microphysics of transport? Turbulent magnetic fields ⇒ charged particle scattering Felix Spanier (NWU) Astronum 2013, Biarritz 2 / 26
Motivation Particle transport in heliosphere and ISM What is the microphysics of transport? Turbulent magnetic fields ⇒ charged particle scattering Felix Spanier (NWU) Astronum 2013, Biarritz 2 / 26
Fokker-Planck Equation Particle transport is described by the Fokker-Planck equation Vlasov equation in gyrocenter coordinates Fokker-Planck-Equation ∂ F T + v µ∂ F T ∂ Z − ǫ Ω ∂ F T ∂φ = S T ( X σ , t ) + 1 � ∂ F T � ∂ p 2 D X σ X ν p 2 ∂ t ∂ X σ ∂ X ν Diffusion-convection equation Pitch angle diffusion coefficient D µµ particularly important Mean free path λ � derived from that Felix Spanier (NWU) Astronum 2013, Biarritz 3 / 26
MHD Simulations Felix Spanier (NWU) Astronum 2013, Biarritz 4 / 26
MHD Simulations 1e+12 1e+10 E(k) [numerisch] 1e+08 t = 17 s t = 34 s t = 51 s t = 68 s 1e+06 t = 85 s Kolmogorov-Spektrum 10000 100 1 1 10 k L / 2 π Felix Spanier (NWU) Astronum 2013, Biarritz 4 / 26
Test particle Simulations Felix Spanier (NWU) Astronum 2013, Biarritz 5 / 26
Test particle Simulations 1500 initial distribution distribution after >20 gyrations 1000 particle number 500 0 -1 -0.5 0 0.5 1 pitch angle Felix Spanier (NWU) Astronum 2013, Biarritz 5 / 26
Wave-particle resonance Testing the interaction of particles with a single wave Inject isotropic, monoenergetic particle distribution Assume background plasma with one Alfvén wave Plot ∆ µ ( t ) vs. µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 6 / 26
Simple wave-particle resonance 2 gyrations, wave amplitude δ B / B 0 = 0 . 01, QLT prediction Felix Spanier (NWU) Astronum 2013, Biarritz 7 / 26
Simple wave-particle resonance 2 gyrations, wave amplitude δ B / B 0 = 0 . 01 Felix Spanier (NWU) Astronum 2013, Biarritz 7 / 26
Simple wave-particle resonance 10 gyrations, wave amplitude δ B / B 0 = 0 . 001 Felix Spanier (NWU) Astronum 2013, Biarritz 7 / 26
Simple wave-particle resonance 2 gyrations, wave amplitude δ B / B 0 = 0 . 1 Felix Spanier (NWU) Astronum 2013, Biarritz 7 / 26
Simple wave-particle resonance 50 gyrations, wave amplitude δ B / B 0 = 0 . 001 Felix Spanier (NWU) Astronum 2013, Biarritz 7 / 26
Turbulent transport Testing the interaction of particles with turbulence Undisturbed turbulence Excited turbulence S I E mag (t 1 ) 1e+14 S II E mag (t 2 ) 1e+15 S III E mag (t 3 ) S IV Gaussian fit Kolomogorov spectrum Gaussian fit 1e+12 Gaussian fit 1e+10 E(k || ) [numerical units] E(k) [numerical units] 1e+10 1e+08 100000 1e+06 1 10000 1e-05 100 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 1 10 k || L / 2 π k L / 2 π Not only µ 0 − ∆ µ plot, but additional � α 2 � D αα = lim 2 ∆ t t →∞ scattering angle diffusion coefficient Felix Spanier (NWU) Astronum 2013, Biarritz 8 / 26
Scattering in MHD turbulence 0.14 T = 1 gyr T = 5 gyr T = 10 gyr 0.12 T = 30 gyr 0.1 0.08 D αα [s -1 ] 0.06 0.04 0.02 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 9 / 26
Scattering in MHD turbulence Felix Spanier (NWU) Astronum 2013, Biarritz 9 / 26
MHD excitation Felix Spanier (NWU) Astronum 2013, Biarritz 10 / 26
Scattering with excited modes I k � 0.3 T = 1 gyr T = 5 gyr T = 10 gyr 0.25 n = -1 n = 0 n = 1 0.2 D αα [s -1 ] 0.15 0.1 0.05 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 11 / 26
Scattering with excited modes I k � Felix Spanier (NWU) Astronum 2013, Biarritz 11 / 26
Scattering with excited modes II k � , k ⊥ 1.4 T = 1 gyr T = 5 gyr T = 10 gyr 1.2 1 0.8 D αα [s -1 ] 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 12 / 26
Scattering with excited modes II k � , k ⊥ Felix Spanier (NWU) Astronum 2013, Biarritz 12 / 26
Quasilinear comparison 1 0.01 0.0001 D αα [s -1 ] 1e-06 1e-08 SQLT background SQLT driven stage SQLT decay stage Particle 30gyr background 1e-10 Particle 30gyr driven stage Particle 30gyr decay stage 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 13 / 26
Turbulent transport Good agreement of testparticle and QLT results in D αα SQLT misses Cherenkov resonance ( n = 0) Limited spectrum yields resonance gap Finite simulation time results in broadened resonances Felix Spanier (NWU) Astronum 2013, Biarritz 14 / 26
Derivation of coefficients µ − ∆ µ plots show the physics of scattering For further use D µµ or D α α is needed! Determination is - as seen - flawed, especially for strong turbulence. Felix Spanier (NWU) Astronum 2013, Biarritz 15 / 26
Running coefficient Derivation of D µµ via its definition: (∆ µ ) 2 ≈ (∆ µ ) 2 t ≫ t 0 D µµ = lim 2 ∆ t 2 ∆ t t →∞ 0.14 T = 1 gyr T = 5 gyr T = 10 gyr 0.12 T = 30 gyr 0.1 0.08 D µµ [s -1 ] 0.06 0.04 0.02 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 16 / 26
Kubo-Formalismus Integration along trajectories: t 1 � � D µµ = ∆ t ˙ µ ( t 0 ) ˙ µ ( t ) N T T t 0 = 0 0.4 T = 1 gyr T = 5 gyr 0.3 T = 10 gyr T = 30 gyr 0.2 0.1 D µµ [s -1 ] 0 -0.1 -0.2 -0.3 -0.4 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 17 / 26
Matrix inversion Discretisation of the diffusion equation for each µ n : ∂ t f = D n + 1 µµ − D n − 1 µµ ∂ µ f + D n µµ ∂ µµ f 2 · ∆ µ Tridiagonal matrix equation: ∂ µ f 0 ∂ µµ f 0 0 0 D 0 ∂ t f 0 2 ∆ µ µµ ... − ∂ µ f 1 D 1 ∂ t f 1 ∂ µµ f 1 0 µµ 2 ∆ µ · = . . . . ... ... ∂ µ f n − 1 . . 0 2 ∆ µ D n ∂ t f n − ∂ µ f n µµ ∂ µµ f n 0 0 2 ∆ µ Matrix inversion with standard methods! Felix Spanier (NWU) Astronum 2013, Biarritz 18 / 26
Integration method Ensemble averaging over several simulations would be statistically correct, but expensive Fitting or smoothing is usually required Integration of the diffusion equation over µ smoothes time derivatives � µ ∂ f T ( µ, t ) d µ = D µµ ( µ ) ∂ f T ( µ, t ) = − j µ ( µ ) ∂ t ∂µ − 1 Diffusion coefficients are calculated via the integration of µ -stream Felix Spanier (NWU) Astronum 2013, Biarritz 19 / 26
Results 0.15 0.1 D µµ [s -1 ] 0.05 0 MIV background turb polynomial fit MIV background turb integration -0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 20 / 26
Results 0.25 0.2 0.15 D µµ [s -1 ] 0.1 0.05 0 MIV peaked turb polynomial fit MIV peaked turb integration -0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 µ 0 Felix Spanier (NWU) Astronum 2013, Biarritz 20 / 26
Wave-Particle Interaction with PiC Physical Background: magnetized plasma in heliosphere / solar wind thermal background plasma non thermal component of energetic particles resonant scattering of particles on plasma waves Fermi-II-acceleration Felix Spanier (NWU) Astronum 2013, Biarritz 21 / 26
Wave-Particle Interaction with PiC Physical Background: magnetized plasma in heliosphere / solar wind thermal background plasma non thermal component of energetic particles resonant scattering of particles on plasma waves Fermi-II-acceleration Numerical Setting: magnetized thermal background plasma one excited wave mode population of relativistic test particles Felix Spanier (NWU) Astronum 2013, Biarritz 21 / 26
PiC simulations E y in x direction (transverse) Simulation Setup: 0.1 L-Mode R-Mode 10 1 excitation of low frequency 0.08 10 0 wave (ideal case: Alfvén 0.06 ω ( ω pe ) wave ) 10 -1 0.04 → huge number of cells and 10 -2 0.02 timesteps required 0 10 -3 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 k (1/cm) use resonance condition to determine the parallel component of the test particles’ velocities → parameters k w , ω w and Ω i give constraints → resonant pitch angle µ res is free initialize monoenergetic test particles ( | v | = v � ) with isotropic angular distribution → resonant scattering only for particles with µ = µ res Felix Spanier (NWU) Astronum 2013, Biarritz 22 / 26
Pitch Angle Diffusion Scatter Plots: peaks at ± µ res left peak: lefthanded wave right peak: righthanded wave ballistic transport (smaller peaks) QLT approximation Felix Spanier (NWU) Astronum 2013, Biarritz 23 / 26
Pitch Angle Diffusion with PiC lefthanded wave mode Felix Spanier (NWU) Astronum 2013, Biarritz 24 / 26
Pitch Angle Diffusion with PiC righthanded wave mode Felix Spanier (NWU) Astronum 2013, Biarritz 25 / 26
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