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

Instabilities of a relativistic electron beam in a plasma A Review Talk. Antoine Bret Universidad Castilla la Mancha – Ciudad Real – Spain. KINETIC MODELING OF ASTROPHYSICAL PLASMAS Krakow, Poland, October 5-9, 2008. Outline of the talk. The system considered The two-stream instability

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

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  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. Ni Np, Vp Nb, Vb The system • Beam + plasma with return current • Fixed ions • Linear collisionless theory (Vlasov + Maxwell)

  4. Perturbation k // E 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. v// Beam: two-stream, filamentation… v// Plasma: Weibel unstable (fastest k // v//) Beam: filamentation STABLE (with enough Tb) v// Plasma: Weibel unstable (fastest k // v//) Strong interaction kW kF Beam: two-stream, filamentation… v// Plasma: Weibel unstable with fastest kv// Filamentation vs. Weibel Beam: two-stream, filamentation… What if a beam enters the plasma ? 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. What about these ones? Are they faster than F or TS ? GROWTH RATE ? More instabilities:Full 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. Beam Full unstable spectrum:Growth rate – No thermal spreads • Diluted beam Nb/Np=0.1, gb=1.01 Filamentation Two-stream Z=kVb/wp

  10. In real systems, thermal effects tend to stabilize this part: Non-relativistic diluted systems governed by Two-stream Beam Full unstable spectrum:Growth rate – No thermal spreads • Diluted beam Nb/Np=0.1, gb=1.01 Z=kVb/wp Y. B. Fainberg, Soviet Phys. JETP 30, 528 (1970) F. Califano, Phys. Rev. E 58, 7837 (1998).

  11. gb= 5 Zx Zz Full unstable spectrum:Growth rate – No thermal spreads a=Nb/Np<<1 b=Vb/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. Ultra-relativistic regime is oblique 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. Beam Full unstable spectrum: Transverse beam temperature (waterbag) • 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. • Transverse beam temperature reduces filamentation (Silva, PoP, 2002). • Weak effect on two-stream • Where is the border of the zone of influence? (waterbag kinetic calculation) Nb/Np=0.1 gb=5 Z=kVb/wp A. Bret, Phys. Rev. E 72, 016403 (2005). A. Bret, PRL 94, 115002 (2005)

  14. Tb = 2 MeV Nb/Np = 1 gb = 1.5 Tb = 100 keV Nb/Np = 1 gb = 1.5 Tb = 500 keV Nb/Np = 0.1 gb = 1.5 Which mode grows faster?Relativistic Maxwellians T_plasma: 5 keV A. Bret, PRL 100, 205008 (2008).

  15. Magnetized case (a glimpse) wc= NR Electron cyclotron frequency • Consider a B0 aligned with the beam. • Measure its strength through WB=wc/wp Nb/Np=0.1 gb=5 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 Nb=Np). • 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 -> Two-stream -> Filamentation (Bret, PRL 2008). Typical pattern, or there is more? Thanks for your attention

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