Charged particle transport in turbulent media F. Spanier A. - - PowerPoint PPT Presentation

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Charged particle transport in turbulent media F. Spanier A. - - PowerPoint PPT Presentation

May 24, 2023 316 likes •704 views

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

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Charged particle transport in turbulent media

  • F. Spanier
  • A. Ivascenko
  • S. Lange
  • C. Schreiner

Center for Space Research, North-West University

Astronum 2013, Biarritz

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Motivation

Particle transport in heliosphere and ISM What is the microphysics

  • f transport?

Turbulent magnetic fields ⇒ charged particle scattering

Felix Spanier (NWU) Astronum 2013, Biarritz 2 / 26

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Motivation

Particle transport in heliosphere and ISM What is the microphysics

  • f transport?

Turbulent magnetic fields ⇒ charged particle scattering

Felix Spanier (NWU) Astronum 2013, Biarritz 2 / 26

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Fokker-Planck Equation

Particle transport is described by the Fokker-Planck equation Vlasov equation in gyrocenter coordinates Fokker-Planck-Equation ∂FT ∂t + vµ∂FT ∂Z − ǫΩ∂FT ∂φ = ST(Xσ, t) + 1 p2 ∂ ∂Xσ

  • p2DXσXν

∂FT ∂Xν

  • Diffusion-convection equation

Pitch angle diffusion coefficient Dµµ particularly important Mean free path λ derived from that

Felix Spanier (NWU) Astronum 2013, Biarritz 3 / 26

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MHD Simulations

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MHD Simulations

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1 100 10000 1e+06 1e+08 1e+10 1e+12 1 10 E(k) [numerisch] k L / 2 π t = 17 s t = 34 s t = 51 s t = 68 s t = 85 s Kolmogorov-Spektrum

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Test particle Simulations

Felix Spanier (NWU) Astronum 2013, Biarritz 5 / 26

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Test particle Simulations

Felix Spanier (NWU) Astronum 2013, Biarritz 5 / 26

  • 1
  • 0.5

0.5 1 pitch angle 500 1000 1500 particle number initial distribution distribution after >20 gyrations

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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. µ0

Felix Spanier (NWU) Astronum 2013, Biarritz 6 / 26

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Simple wave-particle resonance

2 gyrations, wave amplitude δB/B0 = 0.01, QLT prediction

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Simple wave-particle resonance

2 gyrations, wave amplitude δB/B0 = 0.01

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Simple wave-particle resonance

10 gyrations, wave amplitude δB/B0 = 0.001

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Simple wave-particle resonance

2 gyrations, wave amplitude δB/B0 = 0.1

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Simple wave-particle resonance

50 gyrations, wave amplitude δB/B0 = 0.001

Felix Spanier (NWU) Astronum 2013, Biarritz 7 / 26

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SLIDE 15

Turbulent transport

Testing the interaction of particles with turbulence

Undisturbed turbulence

100 10000 1e+06 1e+08 1e+10 1e+12 1e+14 1 10 E(k) [numerical units] k L / 2 π S I S II S III S IV Kolomogorov spectrum

Excited turbulence

1e-05 1 100000 1e+10 1e+15 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 E(k||) [numerical units] k|| L / 2 π Emag(t1) Emag(t2) Emag(t3) Gaussian fit Gaussian fit Gaussian fit

Not only µ0 − ∆µ plot, but additional Dαα = lim

t→∞

α2 2∆t scattering angle diffusion coefficient

Felix Spanier (NWU) Astronum 2013, Biarritz 8 / 26

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Scattering in MHD turbulence

0.02 0.04 0.06 0.08 0.1 0.12 0.14

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dαα [s-1] µ0 T = 1 gyr T = 5 gyr T = 10 gyr T = 30 gyr

Felix Spanier (NWU) Astronum 2013, Biarritz 9 / 26

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Scattering in MHD turbulence

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MHD excitation

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Scattering with excited modes I k

0.05 0.1 0.15 0.2 0.25 0.3

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dαα [s-1] µ0 T = 1 gyr T = 5 gyr T = 10 gyr n = -1 n = 0 n = 1

Felix Spanier (NWU) Astronum 2013, Biarritz 11 / 26

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Scattering with excited modes I k

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Scattering with excited modes II k, k⊥

0.2 0.4 0.6 0.8 1 1.2 1.4

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dαα [s-1] µ0 T = 1 gyr T = 5 gyr T = 10 gyr

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Scattering with excited modes II k, k⊥

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Quasilinear comparison

1e-10 1e-08 1e-06 0.0001 0.01 1 0.2 0.4 0.6 0.8 1 Dαα [s-1] µ0 SQLT background SQLT driven stage SQLT decay stage Particle 30gyr background Particle 30gyr driven stage Particle 30gyr decay stage

Felix Spanier (NWU) Astronum 2013, Biarritz 13 / 26

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Turbulent transport

Good agreement of testparticle and QLT results in Dαα 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

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Derivation of coefficients

µ − ∆µ plots show the physics of scattering For further use Dµµ or Dαα is needed! Determination is - as seen - flawed, especially for strong turbulence.

Felix Spanier (NWU) Astronum 2013, Biarritz 15 / 26

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Running coefficient

Derivation of Dµµ via its definition: Dµµ = lim

t→∞

(∆µ)2 2 ∆t

t≫t0

≈ (∆µ)2 2 ∆t

0.02 0.04 0.06 0.08 0.1 0.12 0.14

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dµµ [s-1] µ0 T = 1 gyr T = 5 gyr T = 10 gyr T = 30 gyr

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Kubo-Formalismus

Integration along trajectories: Dµµ =

  • T

1 NT

t

  • t0=0

∆t ˙ µ(t0) ˙ µ(t)

  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 0.4

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dµµ [s-1] µ0 T = 1 gyr T = 5 gyr T = 10 gyr T = 30 gyr

Felix Spanier (NWU) Astronum 2013, Biarritz 17 / 26

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Matrix inversion

Discretisation of the diffusion equation for each µn: ∂tf = Dn+1

µµ − Dn−1 µµ

2 · ∆µ ∂µf + Dn

µµ∂µµf

Tridiagonal matrix equation:         ∂µµf 0

∂µf 0 2∆µ

− ∂µf 1

2∆µ

∂µµf 1 ... ... ...

∂µf n−1 2∆µ

− ∂µf n

2∆µ

∂µµf n         ·      D0

µµ

D1

µµ

. . . Dn

µµ

     =      ∂tf 0 ∂tf 1 . . . ∂tf n      Matrix inversion with standard methods!

Felix Spanier (NWU) Astronum 2013, Biarritz 18 / 26

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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

−1

∂fT(µ, t) ∂t dµ = Dµµ(µ)∂fT(µ, t) ∂µ = −jµ(µ) Diffusion coefficients are calculated via the integration of µ-stream

Felix Spanier (NWU) Astronum 2013, Biarritz 19 / 26

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Results

  • 0.05

0.05 0.1 0.15

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dµµ [s-1] µ0 MIV background turb polynomial fit MIV background turb integration

Felix Spanier (NWU) Astronum 2013, Biarritz 20 / 26

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Results

  • 0.05

0.05 0.1 0.15 0.2 0.25

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Dµµ [s-1] µ0 MIV peaked turb polynomial fit MIV peaked turb integration

Felix Spanier (NWU) Astronum 2013, Biarritz 20 / 26

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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

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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

  • ne excited wave mode

population of relativistic test particles

Felix Spanier (NWU) Astronum 2013, Biarritz 21 / 26

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PiC simulations

Simulation Setup: excitation of low frequency wave (ideal case: Alfvén wave) → huge number of cells and timesteps required

ω (ωpe) k (1/cm) Ey in x direction (transverse) L-Mode R-Mode 0.02 0.04 0.06 0.08 0.1

  • 0.02
  • 0.015
  • 0.01
  • 0.005

0.005 0.01 0.015 0.02 10-3 10-2 10-1 100 101

use resonance condition to determine the parallel component of the test particles’ velocities → parameters kw, ωw and Ωi give constraints → resonant pitch angle µres is free initialize monoenergetic test particles (|v| = v) with isotropic angular distribution → resonant scattering only for particles with µ = µres

Felix Spanier (NWU) Astronum 2013, Biarritz 22 / 26

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Pitch Angle Diffusion

Scatter Plots: peaks at ±µres left peak: lefthanded wave right peak: righthanded wave ballistic transport (smaller peaks) QLT approximation

Felix Spanier (NWU) Astronum 2013, Biarritz 23 / 26

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Pitch Angle Diffusion with PiC

lefthanded wave mode

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Pitch Angle Diffusion with PiC

righthanded wave mode

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Some Comments

basic characteristics of resonant wave-particle interaction are found results are comparable to MHD simulations deviations from QLT due to → dispersive wave modes? → thermal broadening of test particle population? → 1D and 2D simulation effects? full 3D simulations are yet to come → waiting for computing time...

Felix Spanier (NWU) Astronum 2013, Biarritz 26 / 26