The collective energy loss of the relativistic electron beam propagating through background plasma

O. Polomarov*, I. Kaganovich**, and Gennady Shvets*

*The University of Texas at Austin, Austin, TX, USA**Princeton Plasma Physics Laboratory, NJ, USA

#2

Motivation

Will the e-beam penetrate through the corona without falling apart due to filamentation?

Will filamentation expedite beam stopping?

– What is the final nonlinear outcome of the Weibel instability?

– PIC simulations indicate that the magnetic energy eventually decays. Why?

#3

Three Stages of Beam Filamentation

linear stage

trapping

small filaments coalescence

large filaments coalescence

• Linear growth and saturation via magnetic particle trapping

• small current filaments (c/p), small energy extraction.

• Nonlinear coalescence of current filaments

• each filament carries up to 17kA of current; significant energy conversion into magnetic fields.

• Coalescence of super-Alfvenic current filaments

• beam current reduction, formation of “hollow” current filaments, decrease of the B-field energy. time (1/p)

#4

Super-Alfvenic filaments, I>IA=mc3/e

Beam density Plasma density Current density

Beam density is equal to the back-ground ion density in the filament and sharply decreases at the periphery of the filament.

Ambient plasma is fully expelled from the filament.

Beam current is absent in the center of filament and localized at the edges of the filament.

return current

Beam current

Schematic of the electron velocity.

#5

Analytic solution for filament structure

Quasineutrality:

z B e

Conservation of canonical momentum for beam and plasma electrons:

22

0

44i b b

en en

mc

Vector potential, magnetic flux

0b bz b bz

em v m v

c

0be

emv

c

i b en n n

Amphere’s law:

The beam part of the solution has the form of the Hammer-Rostoker beam equilibrium.

Theory versus PIC Beam and plasma current

0 3 6

0.0

0.2

veloci

ty,

rp/c

Vb

Vp

#6

Magnetic energy decrease as a result of merger of large filaments, I>IA

500 1000 15000.0

0.5

1.0Particle energy

Current

Magnetic energy

Electric energy

En

erg

ies

and

Cu

rren

t

Time (1/p0

) Current I: 2.4IA, Current II: 2.7 IA,

Resulting current IV: 4.5IA

II

II

III

III

IV

+ =I0 I0 I1=2I0In small filaments, the current flows throughout the entire beam cross section the current doubles the magnetic energy doubles.

I0 + I0 = I1=21/2 I0 In large filaments, the current flows only at the periphery of the beam the magnetic energy decreases.

#7

Conclusions

Developed fast code for modeling of the Weibel filamentation instability (Darwin scheme + quasineutrality).

Analytical model of the filaments structure is developed.

Showed that super-Alfvenic filaments with I>IA=17kA can form and have hollow-current density structure, where plasma electrons are completely expelled from the beam filament.

Analytical model predicts, and PIC simulations confirm, that mergers of super-Alfvenic filaments are responsible for the magnetic energy decay during the late stages of the Weibel instability.

#8

Movie of the filamentation

np =8 109cm-3

nb =2 109cm-3

b=3.3