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

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

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

The filamentation instability Nf, Vf Np, Vp Ni Wave vector is here normal to the beam flow. Produces current filaments and B fields. Perturbation k Tatarakis, PRL 90, , (2003) Why filaments, and not stripes? Two-stream “lost the race” because of system parameters (relativistic) MODES COMPETITION B. Fried, Phys. Fluids 2, 337 (1959).

Filamentation vs. Weibel Temperature anisotropy k k Weibel instability: instability of an anisotropic distribution – plasma alone Weibel, Phys. Rev. Lett. 2, 83 (1959) Kalman, Montes & Quemada, Phys. Fluids 11, 1797 (1968). Fastest growing mode

Strong interaction kFkF kWkW Filamentation vs. Weibel What if a beam enters the plasma ? Lazar, Phys. Plasmas 13, (2006) & 15, (2006) Stockem, Phys. Plasmas 15, (2008) - Bret, Phys. Rev. E 72, (2005) v // Beam: two-stream, filamentation… Plasma: stable Beam: two-stream, filamentation… Plasma: Weibel unstable (fastest k // v // ) v // Beam: filamentation STABLE (with enough T b  ) Plasma: Weibel unstable (fastest k // v // ) v // Beam: two-stream, filamentation… v // Plasma: Weibel unstable with fastest k  v //

More instabilities: Full unstable spectrum A real world perturbation does not consist in one single k perfectly aligned along the velocity (or perp.) Nf, Vf Np, Vp Ni Filamentation k Two-stream k What about these ones? Are they faster than F or TS ? GROWTH RATE ?

Full unstable spectrum: Growth rate – No thermal spreads Diluted beam Nb/Np=0.1,  b =1.01 Z=kVb/  p Beam Two-stream Filamentation

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

Full unstable spectrum: Growth rate – No thermal spreads Max two-stream Max Filamentation Max Oblique  b = 5 Zz Zx Growth rate/  p  =Nb/Np<<1  =Vb/c Y. B. Fainberg, Soviet Phys. JETP 30, 528 (1970). Oblique modes are linear. Not some mode-mode interaction.

Which mode grows faster? Which is the fastest growing mode = “First move” of the system Cold fluid answer in terms of (N b /N p,  b ): Bret, PoP 12, , (2005). Ultra- relativistic regime is oblique

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? Beam Z=kVb/  p 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) A. Bret, Phys. Rev. E 72, (2005). A. Bret, PRL 94, (2005) N b /N p =0.1  b =5

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

Magnetized case (a glimpse) Consider a B 0 aligned with the beam. Measure its strength through  B =  c /  p Godfrey, Phys. Fluids 18, 346 (1975)  c = NR Electron cyclotron frequency N b /N p =0.1  b =5 Cold

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 N b =N p ). Good agreement with PIC simulations (Dieckmann, PoP 13, , Gremillet, PoP 14, , 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, , 2006) Non-linear regime Two-stream driven: particle trapping (Luque, Phys. Rep. 415, ) Filamentation driven: filaments merging (Medvedev, ApJ 618, L ) Oblique driven: Massive 3D PIC showed: oblique -> Two-stream -> Filamentation (Bret, PRL 2008). Typical pattern, or there is more? Thanks for your attention