Computationally efficient description of relativistic electron beam transport in dense plasma

  Oleg Polomarov*, Adam Sefkov**, Igor Kaganovich**and Gennady Shvets* 

   *IFS, The University of Texas at Austin, TX, 78712;

**PPPL, Princeton, NJ 08543 

Abstract A reduced model of the Weibel instability and electron beam

transport in dense plasma is developed. Beam electrons are modeled by macro-particles and the background plasma is represented by electron fluid. Conservation of generalized vorticity and quasineutrality of the plasma-beam system are used to simplify the governing equations. Our approach is motivated by the conditions of the FI scenario, where the beam density is likely to be much smaller than the plasma density and the beam energy is likely to be very high. For this case the growth rate of the Weibel instability is small, making the modeling of it by conventional PICs exceedingly time consuming. The present approach does not require resolving the plasma period and only resolves a plasma collisionless skin depth and is suitable for modeling a long-time behavior of beam-plasma interaction. An efficient code based on this reduced description is developed and benchmarked against the LSP PIC code. The dynamics of low and high current electron beams in dense plasma is simulated. Special emphasis is on peculiarities of its non-linear stages, such as filament formation and merger, saturation and post-saturation field and energy oscillations.

Supported by DOE Fusion Science through grant DE-FG02-05ER54840.

Motivation

• To design an computationally efficient approach for modeling electromagnetic Weibel instability

for the case of propagation of relativistic electron (ion) beams into dense background plasma

• Interesting features: pinching, filamentation, generation of strong magnetic field and its saturation

• Applications: Fusion - fast ignition, Accelerator physics - beam transport, Astrophysics - origin of cosmic strong electro-magnetic

field.

Problem Setup

Beam current vz >> vx, vy

Plasma return current

Bx,y

x

z

y

Ez

In the quasi-neutral plasma the rising beam current produces encircling transverse magnetic field Bx,y, which leads to the beam pinching and generates Ez by the induction law, which produces a return plasma current and tries to stop and screen the beam current.

Instability loop: Increase of the beam (filament) density -> larger beam (filament) current -> larger magnetic field -> larger pinching -> larger beam (filament) density

The instability is collective, not resonant

Relativistic beam with vz >> vx, vy

propagates into dense background plasma nb < np.The beam-plasma system tends to be quasi-neutral.

Existing computational approaches

• PIC L. O. Silva et al., Phys Plasmas 10, 1979 (2003), M. Honda et al., Phys Plasmas 7, 1302, (2000). Advances all plasma and beam particles

• Hybrid modeling. LSP (hybrid mode), T. Taguchi et al., Phys. Rew. Lett., 86, 5055 (2001). Solves hydrodynamic equation for background plasma

Both computationally expensive

Main assumptions

• Plasma electrons are warm non-relativistic fluid

• Beam electrons are relativistic particles streaming along the z-direction

• Plasma ions are fixed neutralizing background, with ni=const

• “Plasma + beam” is quasi-neutral: nb+np=ni

L >> D and t >> pe

• All quantities of interest are z independent

Conservation of generalized vorticity

• Assuming

• Combining the Faraday’s law and the equation of motion of fluid collisionless background plasma yields the conservation of generalized vorticity

• For initially quiescent plasma giving

),(),( yxeeyxBB zzz

SdSdorvt

iniinie

zzeez eBmc

evand

mc

ev

Bmc

ev e

Bmc

ev eini 0

Consequences of the vorticity conservation

• No need to solve the equation of motion for background plasma. Plasma velocity is related to magnetic field from conservation of vorticity and plasma density is from quasi-neutrality

• The only equations to be solved are equations for (x,y) and Bz(x,y) and equations of motions of the beam particles

Field equations Magnetic field:• The axial projection of Ampere’s law produces the equation for axial vector potential

• Curl of the transverse part of Ampere’s law yields the equation for Bz

Note: Coefficients in the both equations depend only on the beam density and current. No time dependence - the equations are solved at each time

step as the beam particles are evolved. For underdense beams the equations for Bz can be neglected.

Electric field: is expressed through vector potential and plasma fluid velocity which in turn is

expressed through Bz and beam density and current.

bze J

cc

42

22

e

bzez

ee n

Je

c

nB

cn

4)ln(

2

22

Beam equations

2

22

2

2ψ

cm

e

mc

ev

dt

vd jzjj

0

mc

ev

dt

djzj

Newton equations for “particles”:

jj vrdt

d

Note: Although the beam electrons phase space is {Vx,Vy,Vz,X,Y}, actual integration is done in {Vx,Vy,X,Y}.

2

2

eejz

z v

t

vve

mc

eB

Block-scheme of simulation

• Initialization: Given initial density and velocities of the beam, the Eq. for axial vector potential is solved.

• Cycles of time integration: a - > b - > a a) the Eqs. for beam particles are solved by PIC technique. The

beam electrons momentae and positions are advanced in time dt using fields on a grid. The beam density and current are recalculated on the grid.

b) the Eqs. for axial vec. potential and Bz are solved by the MUDPACK multigrid solver for non-separable 2D elliptic equations.

See: J. Adams, "MUDPACK: Multigrid Fortran Software for the Efficient Solution of Linear Elliptic Partial Differential Equations,“ Applied Math. and Comput. vol.34, Nov 1989, pp.113-146.

Time evolution of the underdence electron beam in ambient plasma.a) initial exponentially growing stage:

• Weibel (filamentation) instability of relativistic electron beam with diameter Kp D=20.

• Simulation box is 256x256 (or 32 Kp x 32 Kp), 2x106 particles. Peak

beam density compression ~ 100 times.

bn

X Y

emcII A /3

b) Final stage of instability. SaturationMaximal pinching,Saturation.

Saturation of the beam-plasma energies

pe

bcbz

b

b

cR

vR

vR

~and~

~

The instability saturates when

In agreement with

of R. C. Davidson et al., Phys. Fluids 15, 317 (1972).

bounce ~

Initial rate of instability growth for warm beam

)()(),,( 22

20 zzyxb

zyx PpppPP

npppf

,ˆ),,( )(

11

txkxki yxeftpxf

),,()(),,(where0 10011 tpxfpftpxfp

fBep

mc

e

x

f

m

p

t

fzz

Linearized Vlasov for the perturbed part of the distribution function under assumptions:

,neglecting,,dependenceno E,pppz yxz leads to:

Assuming “waterbag-like” distribution function

Calculating z-Fourier component of the beam current

note: no singularities in denominator as is imaginary,

,ˆˆ1fm

ppdj z

bz

substituting it into the linearized Fourier expanded Eq. for axial vector potential

bz

pe jcc

k ˆ4ˆˆ

2

2

2

gives the dispersion relation

)(

1

vk

The dispersion relation

2222222

22222222

222

22

/4)/(

/2)/(

/

1

cvVck

cvVck

ck

k

bbzpe

bbzpe

pe

2

2

222

222

/ c

v

ck

k bz

pe

b

22

2

2

222

222

4

3

/

kVc

v

ck

k bz

pe

b

Waterbag:

Cold beam:

/V kFor warm beam with

0V Possible unstable solutions are only with pure imaginary frequency:

Im

0Imand0Re

Typical dispersion behavior

14

/ 2

2

2

2

max e

bbz

pe V

v

c

k

e

b

bz n

n

v

V

2

Maximal possible wave vector: Maximal possible transverse thermal velocity:

Cold

Waterbag

Warm

009.0

89.0

001.0/

V

v

nn

bz

eb

Extracted magnetic energy

Abeam

magn

I

I

TCE

E

)1(2)(

1

(?)1)(

1~)(

/3

TCIIfor

TCIIfor

emcI

A

A

A

Abeam II 33.1

0.885,z

0 1000 2000 3000 4000pet

1. 106

0.00001

0.0001

0.001

0.01

0.1

EnergyFieldEnergyBeam

nbeam/nplasma=0.007

0.05 0.025

0.01

Plasma density

19 57

L ωpe /c 9.5 27.5

L ωpe /c

initiallynbeam/nplasma=0.007

Initially nbeam/nplasma=0.025

ωpe t = 3000

Abeam II 33.10.885,z

0.5

0.01

2

0.01

nbeam/nplasma nbeam/nplasma

Conclusion and further work

• Robust hybrid code for modeling Weibel instabilities is developed.

• Detailed studies of filamentation, tearing, instability saturation and non-linear stage of the beam dynamics will be continued.