[PPT] - Scattering length of relativistic particles in aperiodic PowerPoint Presentation

SLIDE 1

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Scattering length of relativistic particles in aperiodic fluctuations

Anne Stockem

Ruhr-University Bochum, Theoretical Space and Astrophysics

October 2008

RUB, TPIV Anne Stockem Scattering length

SLIDE 2

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation:

RUB, TPIV Anne Stockem Scattering length

SLIDE 3

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation

RUB, TPIV Anne Stockem Scattering length

SLIDE 4

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields

RUB, TPIV Anne Stockem Scattering length

SLIDE 5

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields Many applications in Space and Astrophysics Solar eruption:

RUB, TPIV Anne Stockem Scattering length

SLIDE 6

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields Many applications in Space and Astrophysics Reduction of instabilities Solar eruption:

RUB, TPIV Anne Stockem Scattering length

SLIDE 7

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields Many applications in Space and Astrophysics Reduction of instabilities Solar eruption: Scattering length: Condition: Instability

RUB, TPIV Anne Stockem Scattering length

SLIDE 8

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields Many applications in Space and Astrophysics Reduction of instabilities Solar eruption: Scattering length: Condition: Instability Interaction: Particles/Field

RUB, TPIV Anne Stockem Scattering length

SLIDE 9

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields Many applications in Space and Astrophysics Reduction of instabilities Solar eruption: Scattering length: Condition: Instability Interaction: Particles/Field Inside the system?!

RUB, TPIV Anne Stockem Scattering length

SLIDE 10

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Outline

Motivation: AF: Magnetic field generation Origin of magnetic fields Many applications in Space and Astrophysics Reduction of instabilities Solar eruption: Scattering length: Condition: Instability Interaction: Particles/Field Inside the system?! Scattering length

RUB, TPIV Anne Stockem Scattering length

SLIDE 11

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles

RUB, TPIV Anne Stockem Scattering length

SLIDE 12

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles Magnetic field fluctuation δBy(x, t) = By(t)eıkx

RUB, TPIV Anne Stockem Scattering length

SLIDE 13

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles Magnetic field fluctuation δBy(x, t) = By(t)eıkx Lorentz force FL = qv × B

RUB, TPIV Anne Stockem Scattering length

SLIDE 14

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles Magnetic field fluctuation δBy(x, t) = By(t)eıkx Lorentz force FL = qv × B Charge separation

RUB, TPIV Anne Stockem Scattering length

SLIDE 15

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles Magnetic field fluctuation δBy(x, t) = By(t)eıkx Lorentz force FL = qv × B Charge separation Right hand rule: amplification of By

RUB, TPIV Anne Stockem Scattering length

SLIDE 16

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles Magnetic field fluctuation δBy(x, t) = By(t)eıkx Lorentz force FL = qv × B Charge separation Right hand rule: amplification of By Non-linear phase: Biot-Savart interaction

RUB, TPIV Anne Stockem Scattering length

SLIDE 17

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Physical Principle

Linear phase: Counterstreaming particles Magnetic field fluctuation δBy(x, t) = By(t)eıkx Lorentz force FL = qv × B Charge separation Right hand rule: amplification of By Non-linear phase: Biot-Savart interaction Merging of the current filaments

RUB, TPIV Anne Stockem Scattering length

SLIDE 18

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

RUB, TPIV Anne Stockem Scattering length

SLIDE 19

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L No interaction:

RUB, TPIV Anne Stockem Scattering length

SLIDE 20

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No interaction:

RUB, TPIV Anne Stockem Scattering length

SLIDE 21

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

RUB, TPIV Anne Stockem Scattering length

SLIDE 22

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

RUB, TPIV Anne Stockem Scattering length

SLIDE 23

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

λ ≪ L Interaction:

RUB, TPIV Anne Stockem Scattering length

SLIDE 24

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

λ ≪ L Interaction inside the system Interaction:

RUB, TPIV Anne Stockem Scattering length

SLIDE 25

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

λ ≪ L Interaction inside the system Instability possible Interaction:

RUB, TPIV Anne Stockem Scattering length

SLIDE 26

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

λ ≪ L Interaction inside the system Instability possible Interaction: Consequences:

RUB, TPIV Anne Stockem Scattering length

SLIDE 27

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

λ ≪ L Interaction inside the system Instability possible Interaction: Consequences: Calculation of λ

RUB, TPIV Anne Stockem Scattering length

SLIDE 28

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Interaction between Particles and Field

1. Case:

λ ≪ L Particle leaves the system before being scattered No instability No interaction:

2. Case:

λ ≪ L Interaction inside the system Instability possible Interaction: Consequences: Calculation of λ Appropriate model is necessary!

RUB, TPIV Anne Stockem Scattering length

SLIDE 29

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Scattering Length

Parallel mean free path length: λ = 3v 8 1

−1

dµ (1 − µ2)2 Dµµ(µ)

Jokipii (1966); Hasselmann and Wibberenz (1968); Earl (1974)

RUB, TPIV Anne Stockem Scattering length

SLIDE 30

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Scattering Length

Parallel mean free path length: λ = 3v 8 1

−1

dµ (1 − µ2)2 Dµµ(µ)

Jokipii (1966); Hasselmann and Wibberenz (1968); Earl (1974)

Diffusion coefficient: Dµµ(µ) = const. ∞

kmin

dk⊥ ∞ ds eΓ(k⊥)s f(k⊥, µ, s)

RUB, TPIV Anne Stockem Scattering length

SLIDE 31

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Scattering Length

Parallel mean free path length: λ = 3v 8 1

−1

dµ (1 − µ2)2 Dµµ(µ)

Jokipii (1966); Hasselmann and Wibberenz (1968); Earl (1974)

Diffusion coefficient: Dµµ(µ) = const. ∞

kmin

dk⊥ ∞ ds eΓ(k⊥)s f(k⊥, µ, s) Damping rate: Assumption: Γ(k⊥) = c lr−1

e

kr

⊥

Chang et al. (2008)

RUB, TPIV Anne Stockem Scattering length

SLIDE 32

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Model Dependency

Thermal particles:

r = 1:

λ L ≈ 8.18·10−20 n−3L2

1

m

me

3

2 θ− 1 2

r = 3:

λ L ≈ 3.84·10−3 n1/6

−3 L1/3 1

m

me

7

12 θ 5 12 RUB, TPIV Anne Stockem Scattering length

SLIDE 33

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Model Dependency

Thermal particles:

r = 1:

λ L ≈ 8.18·10−20 n−3L2

1

m

me

3

2 θ− 1 2

r = 3:

λ L ≈ 3.84·10−3 n1/6

−3 L1/3 1

m

me

7

12 θ 5 12

Scattering length:

RUB, TPIV Anne Stockem Scattering length

SLIDE 34

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Model Dependency

Thermal particles:

r = 1:

λ L ≈ 8.18·10−20 n−3L2

1

m

me

3

2 θ− 1 2

r = 3:

λ L ≈ 3.84·10−3 n1/6

−3 L1/3 1

m

me

7

12 θ 5 12

Scattering length: Highly relativistic particles:

r = 1:

λ L ≈ 8.65·10−23 n8L2

15

m me γ

r = 3:

λ L ≈ 3.08·10−3 n1/6

8

L1/3

15

m me γ

RUB, TPIV Anne Stockem Scattering length

SLIDE 35

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Model Dependency

Thermal particles:

r = 1:

λ L ≈ 8.18·10−20 n−3L2

1

m

me

3

2 θ− 1 2

r = 3:

λ L ≈ 3.84·10−3 n1/6

−3 L1/3 1

m

me

7

12 θ 5 12

Scattering length: Highly relativistic particles:

r = 1:

λ L ≈ 8.65·10−23 n8L2

15

m me γ

r = 3:

λ L ≈ 3.08·10−3 n1/6

8

L1/3

15

m me γ

Scattering length:

RUB, TPIV Anne Stockem Scattering length

SLIDE 36

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry:

RUB, TPIV Anne Stockem Scattering length

SLIDE 37

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry: Interpretation: Magnetic field fluctuations δB ⊥ ez

RUB, TPIV Anne Stockem Scattering length

SLIDE 38

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry: Interpretation: Magnetic field fluctuations δB ⊥ ez Ambient field B0 ez

RUB, TPIV Anne Stockem Scattering length

SLIDE 39

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry: Interpretation: Magnetic field fluctuations δB ⊥ ez Ambient field B0 ez Charge separation slows down

RUB, TPIV Anne Stockem Scattering length

SLIDE 40

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry: Interpretation: Magnetic field fluctuations δB ⊥ ez Ambient field B0 ez Charge separation slows down Particle confinement for B0 ≥ Bc

RUB, TPIV Anne Stockem Scattering length

SLIDE 41

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry: Interpretation: Magnetic field fluctuations δB ⊥ ez Ambient field B0 ez Charge separation slows down Particle confinement for B0 ≥ Bc Consequences: Growth rate is reduced

RUB, TPIV Anne Stockem Scattering length

SLIDE 42

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Influence of an Ambient Magnetic Field

Geometry: Interpretation: Magnetic field fluctuations δB ⊥ ez Ambient field B0 ez Charge separation slows down Particle confinement for B0 ≥ Bc Consequences: Growth rate is reduced No amplification for B0 ≥ Bc

RUB, TPIV Anne Stockem Scattering length

SLIDE 43

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Analytical Results of the FI

Additional assumption: Cold plasma approach: T = 0

RUB, TPIV Anne Stockem Scattering length

SLIDE 44

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Analytical Results of the FI

Additional assumption: Cold plasma approach: T = 0 Analytics: Aperiodic fluctuations: δB⊥ ∝ eık·x+σt Magnetic field:

RUB, TPIV Anne Stockem Scattering length

SLIDE 45

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Analytical Results of the FI

Additional assumption: Cold plasma approach: T = 0 Analytics: Aperiodic fluctuations: δB⊥ ∝ eık·x+σt Maximum growth rate:

σmax =

Ω2

max − Ω2

Linear growth rate:

5 10 15 20 0.2 0.4 0.6 0.8 1 k c / ωp σ (k) / σM B0 = 0 B0=Bc/2 B0=3Bc/4 B0=7Bc/8

RUB, TPIV Anne Stockem Scattering length

SLIDE 46

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Analytical Results of the FI

Additional assumption: Cold plasma approach: T = 0 Analytics: Aperiodic fluctuations: δB⊥ ∝ eık·x+σt Maximum growth rate:

σmax =

Ω2

max − Ω2

Ωmax = ωpU/√γc Ω = eB0/m

Linear growth rate:

5 10 15 20 0.2 0.4 0.6 0.8 1 k c / ωp σ (k) / σM B0 = 0 B0=Bc/2 B0=3Bc/4 B0=7Bc/8

RUB, TPIV Anne Stockem Scattering length

SLIDE 47

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Analytical Results of the FI

Additional assumption: Cold plasma approach: T = 0 Analytics: Aperiodic fluctuations: δB⊥ ∝ eık·x+σt Maximum growth rate:

σmax =

Ω2

max − Ω2

No growth for B0 ≥ Bc

Ωmax = ωpU/√γc Ω = eB0/m

Linear growth rate:

5 10 15 20 0.2 0.4 0.6 0.8 1 k c / ωp σ (k) / σM B0 = 0 B0=Bc/2 B0=3Bc/4 B0=7Bc/8

RUB, TPIV Anne Stockem Scattering length

SLIDE 48

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density:

RUB, TPIV Anne Stockem Scattering length

SLIDE 49

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density: Electric energy density:

RUB, TPIV Anne Stockem Scattering length

SLIDE 50

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density: Electric energy density: Results:

RUB, TPIV Anne Stockem Scattering length

SLIDE 51

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density: Electric energy density: Results: Same amplification level independent of B0

RUB, TPIV Anne Stockem Scattering length

SLIDE 52

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density: Electric energy density: Results: Same amplification level independent of B0 ǫB: Growth for B0 = Bc

RUB, TPIV Anne Stockem Scattering length

SLIDE 53

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density: Electric energy density: Results: Same amplification level independent of B0 ǫB: Growth for B0 = Bc Non-linear effect

RUB, TPIV Anne Stockem Scattering length

SLIDE 54

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

The Energy Densities

Magnetic energy density: Electric energy density: Results: Same amplification level independent of B0 ǫB: Growth for B0 = Bc Non-linear effect Amplification of ǫE starts later

RUB, TPIV Anne Stockem Scattering length

SLIDE 55

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Magnetic Field Evolution

Perpendicular component:

RUB, TPIV Anne Stockem Scattering length

SLIDE 56

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Magnetic Field Evolution

Perpendicular component:

RUB, TPIV Anne Stockem Scattering length

SLIDE 57

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Magnetic Field Evolution

Perpendicular component: Parallel component:

RUB, TPIV Anne Stockem Scattering length

SLIDE 58

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Magnetic Field Evolution

Perpendicular component: Parallel component:

RUB, TPIV Anne Stockem Scattering length

SLIDE 59

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Magnetic Field Evolution

Perpendicular component: Parallel component: Comparison: Strong perpendicular component has developed

RUB, TPIV Anne Stockem Scattering length

SLIDE 60

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Magnetic Field Evolution

Perpendicular component: Parallel component: Comparison: Strong perpendicular component has developed Amplification of B: temperature effect

RUB, TPIV Anne Stockem Scattering length

SLIDE 61

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations

RUB, TPIV Anne Stockem Scattering length

SLIDE 62

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field

RUB, TPIV Anne Stockem Scattering length

SLIDE 63

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field Interaction between particles and field

RUB, TPIV Anne Stockem Scattering length

SLIDE 64

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field Interaction between particles and field Scattering length

RUB, TPIV Anne Stockem Scattering length

SLIDE 65

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field Interaction between particles and field Scattering length Appropriate model

RUB, TPIV Anne Stockem Scattering length

SLIDE 66

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field Interaction between particles and field Scattering length Appropriate model Physical principle of the B-generation process

RUB, TPIV Anne Stockem Scattering length

SLIDE 67

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field Interaction between particles and field Scattering length Appropriate model Physical principle of the B-generation process Analytics – PIC simulations

RUB, TPIV Anne Stockem Scattering length

SLIDE 68

Scattering length Motivation Particle/Field The Filamentation Instability Conclusions

Conclusions

Aperiodic fluctuations Instability generating a magnetic field Interaction between particles and field Scattering length Appropriate model Physical principle of the B-generation process Analytics – PIC simulations Thanks for your attention!

RUB, TPIV Anne Stockem Scattering length