instabilities of a relativistic electron beam in a plasma a review talk
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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 - PowerPoint PPT PresentationTRANSCRIPT
Instabilities of a relativisticelectron beam in a plasma
A Review Talk
Antoine BretUniversidad Castilla la Mancha – Ciudad Real – Spain
KINETIC MODELING OF ASTROPHYSICAL PLASMASKrakow, 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, VpNi
Beam + plasma with return current Fixed ions Linear collisionless theory (Vlasov + Maxwell)
The two-stream instability
Nb, Vb Np, VpNi
The system is in « static » equilibrium. No net current, no net charge. But unstable
Perturbationk // E
Flow
Bohm & Gross, Phys. Rev. 75, 1851 & 1864 (1949)Bludman, Watson & Rosenbluth, Phys. Fluids 3, 747 (1960)
The filamentation instability
Nf, Vf Np, VpNi
•Wave vector is here normal to the beam flow.•Produces current filaments and B fields.
Pert
urb
ati
on
k
Tatarakis, PRL 90, 175001, (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
Temperatureanisotropyk
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 interactionkFkW
Filamentation vs. WeibelWhat if a beam enters the plasma ?
Lazar, Phys. Plasmas 13, 102107 (2006) & 15, 042103 (2006)Stockem, Phys. Plasmas 15, 014501 (2008) - Bret, Phys. Rev. E 72, 016403 (2005)
v//Beam: two-stream, filamentation…
Plasma: stable
Beam: two-stream, filamentation…
Plasma: Weibel unstable (fastest k // v//)v//
Beam: filamentation STABLE (with enough Tb)
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
Filam
en
tati
on k
Two-streamk
What about theseones? 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
Tw
o-s
tream
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
Gro
wth
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 (Nb/Np, b):
Bret, PoP 12, 082704, (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, 016403 (2005).A. Bret, PRL 94, 115002 (2005)
Nb/Np=0.1
b=5
Which mode grows faster?Relativistic Maxwellians
Tb = 500 keVNb/Np = 0.1b = 1.5
T_plasma: 5 keV
Tb = 2 MeVNb/Np = 1
b = 1.5
Tb = 100 keVNb/Np = 1
b = 1.5
A. Bret, PRL 100, 205008 (2008).
Magnetized case (a glimpse)
Consider a B0 aligned with the beam. Measure its strength through B=c/p
Godfrey, Phys. Fluids 18, 346 (1975)
c= NR Electron cyclotron frequency
Nb/Np=0.1
b=5Cold
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