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