Instabilities%20of%20a%20relativistic%20electron%20beam%20in%20a%20plasma%20A%20Review%20Talk

1
Instabilities of a relativisticelectron beam in
a plasmaA Review Talk
Antoine Bret Universidad Castilla la Mancha
Ciudad Real Spain
KINETIC MODELING OF ASTROPHYSICAL PLASMAS Krakow,
Poland, October 5-9, 2008
2
Outline of the talk

• The system considered
• The two-stream instability
• The filamentation instability
• Filamentation vs. Weibel
• More instabilities the full unstable spectrum
• Kinetic effects
• Modes hierarchy
• Magnetized system (a glimpse)
• Conclusion

3
The system

• Beam plasma with return current
• Fixed ions
• Linear collisionless theory (Vlasov Maxwell)

4
The two-stream instability
Ni
Np, Vp
Nb, Vb
The system is in  static  equilibrium. No net
current, no net charge. But unstable
Flow
Bohm Gross, Phys. Rev. 75, 1851 1864
(1949) Bludman, Watson Rosenbluth, Phys. Fluids
3, 747 (1960)
5
The filamentation instability
Ni
Np, Vp
Nf, Vf
k
Perturbation

• Wave vector is here normal to the beam flow.
• Produces current filaments and B fields.

Why filaments, and not stripes? Two-stream lost
the race because of system parameters
(relativistic) MODES COMPETITION
Tatarakis, PRL 90, 175001, (2003)
B. Fried, Phys. Fluids 2, 337 (1959).
6
Filamentation vs. Weibel

• Weibel instability instability of an anisotropic
  distribution plasma alone

Fastest growing mode
Temperature anisotropy
k
k
Weibel, Phys. Rev. Lett. 2, 83 (1959) Kalman,
Montes Quemada, Phys. Fluids 11, 1797 (1968).
7
Filamentation vs. Weibel

• What if a beam enters the plasma ?

Beam two-stream, filamentation
Plasma stable
Lazar, Phys. Plasmas 13, 102107 (2006) 15,
042103 (2006) Stockem, Phys. Plasmas 15, 014501
(2008) - Bret, Phys. Rev. E 72, 016403 (2005)
8
More instabilitiesFull unstable spectrum

• A real world perturbation does not consist in one
  single k perfectly aligned along the velocity (or
  perp.)

Ni
k
Filamentation
Nf, Vf
k
Two-stream
Np, Vp
9
Full unstable spectrumGrowth rate No thermal
spreads

• Diluted beam Nb/Np0.1, gb1.01

Filamentation
Two-stream
ZkVb/wp
10
Full unstable spectrumGrowth rate No thermal
spreads

• Diluted beam Nb/Np0.1, gb1.01

ZkVb/wp
Y. B. Fainberg, Soviet Phys. JETP 30, 528
(1970) F. Califano, Phys. Rev. E 58, 7837 (1998).
11
Full unstable spectrumGrowth rate No thermal
spreads
aNb/Npltlt1 bVb/c

• Max two-stream
• Max Filamentation
• Max Oblique

Growth rate/wp
Oblique modes are linear. Not some mode-mode
interaction.
Y. B. Fainberg, Soviet Phys. JETP 30, 528 (1970).
12
Which mode grows faster?

• Which is the fastest growing mode First move
  of the system
• Cold fluid answer in terms of (Nb/Np, gb)

Bret, PoP 12, 082704, (2005).
13
Full unstable spectrum Transverse beam
temperature (waterbag)

• Transverse beam temperature reduces filamentation
  (Silva, PoP, 2002).
• Weak effect on two-stream
• Where is the border of the zone of influence?

• Transverse beam temperature kills
  filamentation, and everything beyond a given
  critical angle.
• There is now ONE most unstable mode.
• Temp effects are NOT homogenous
• The max growth rate is still 65 of the cold
  value.

(waterbag kinetic calculation)
Nb/Np0.1 gb5
ZkVb/wp
A. Bret, Phys. Rev. E 72, 016403 (2005). A. Bret,
PRL 94, 115002 (2005)
14
Which mode grows faster?Relativistic Maxwellians
T_plasma 5 keV
A. Bret, PRL 100, 205008 (2008).
15
Magnetized case (a glimpse)

• Consider a B0 aligned with the beam.
• Measure its strength through WBwc/wp

wc NR Electron cyclotron frequency
Nb/Np0.1 gb5 Cold
Godfrey, Phys. Fluids 18, 346 (1975)
16
Conclusions

• Old and (still) interesting problem.
• The relativistic regime demands the investigation
  of the full 2D k spectrum.
• Linear kinetic theory with relativistic
  Maxwellians gives access to the hierarchy of the
  3 competing kind of modes.
• Highly relativistic regime governed by oblique
  modes (unless NbNp).
• Good agreement with PIC simulations (Dieckmann,
  PoP 13, 112110, 2006 - Gremillet, PoP 14, 040704,
  2007).
• Need to provide an easier access to oblique
  modes.
• Electrostatic approximation
• Fluid model (Silva, Bull. Am. Phys. Soc. 46, 205,
  2001 Bret, PoP 13, 042106, 2006)
• Non-linear regime
• Two-stream driven particle trapping (Luque,
  Phys. Rep. 415, 261 2005.)
• Filamentation driven filaments merging
  (Medvedev, ApJ 618, L75 02005)
• Oblique driven Massive 3D PIC showed oblique -gt
  Two-stream -gt Filamentation (Bret, PRL 2008).
  Typical pattern, or there is more?

Thanks for your attention