Between Filamentation and Between Filamentation and Two-Stream instabilities in e-Two-Stream instabilities in e-beam/plasma interactionbeam/plasma interaction

A. Bret, M.-C. Firpo and C. Deutsch

Laboratoire de Physique des Gaz etdes Plasmas – Paris Orsay Université

The problem of The problem of kk-orientation-orientation

Two-Streamk

Filamentation, Weibel

?

Beams

FormalismFormalism

• Linear Maxwell-Relativistic Vlasov system in Linear Maxwell-Relativistic Vlasov system in 3D3D

• Collisionless – Fixed plasma ionsCollisionless – Fixed plasma ions• AnalyticalAnalytical

• Dielectic tensor Dielectic tensor kk non-diagonal non-diagonal

• Det T(Det T(kk) = 0 : proper frequencies) = 0 : proper frequencies• E along eigenvectors : gives angle (k,E)E along eigenvectors : gives angle (k,E)

The (k,E) angleThe (k,E) angle

Filamentation

?k

Two-StreamEk

Dispersion RelationDispersion Relation

Det T(Det T(kk) = 0) = 0

(2yy - k2c2)×

- (2xz + kxkzc2)2] = 0

[(2xx - kz2c2)(2zz - kx

2c2)

Two BranchesTwo Branches

(2yy - k2c2) = 0

- (2xz + kxkzc2)2] = 0

[(2xx - kz2c2)(2zz - kx

2c2)

Branch 1

Branch 2

Model 1: T = 0 (Fluid)Model 1: T = 0 (Fluid)

Plasma : npvxvyvz+Vp

Beam : nbvxvyvz-Vb

nbVb = npVp

Model 2: Hot PlasmaModel 2: Hot Plasma

Plasma : npvz+Vp×

Beam : nbvxvyvz-Vb

nbVb = npVp

vx+Vt)+ (vx-Vt)] ×

vy+Vt)+ (vy-Vt)]

2Vt

Model 3: Hot PlasmaModel 3: Hot Plasma + Relativistic Beam + Relativistic Beam

Beam : Relativistic energyE ~ 2 MeV (b ~ 4)

Plasma : Non relativisticT ~ 2 keV

Branch 1 behaviorBranch 1 behavior

(2yy - k2c2) = 0•Stable at T = 0•Unstable T ≠ 0 for small angles•No relativistic effects (model 3 = model 2)•Purely transverse mode in ANY model•Original « Weibel » instability

kE // y Beam

UNSTABLE

Branch 1 behaviorBranch 1 behavior

Beam

kk

Weibel result

Z = kVb /p

103

= Vtp/Vb, =Vb/c

Rm Vtp/Vb

Branch 2 behaviorBranch 2 behavior

k

EBeam

[(2xx - kz2c2)(

•Yields unstable modes for all models and all angles•Bridge between Two-Stream and Filamentation

x

z

Between two-streamBetween two-streamand filamentation, T=0and filamentation, T=0

Z = kVb /p

p

Between two-streamBetween two-streamand filamentation, T=0and filamentation, T=0

Z = kVb /p

p

Between Two-StreamBetween Two-Streamand filamentation, T≠0and filamentation, T≠0

p

Between Two-StreamBetween Two-Streamand filamentation, T≠0and filamentation, T≠0

p

Critical angleCritical angle

Tan =1+nb /np

Vth /Vb

/2Vth

Angle comes when 2 dispersion function singularities crossHigh Z instability just is just shifted in another direction

Between two-stream and Between two-stream and filamentation, T≠0filamentation, T≠0Relativistic effectsRelativistic effects

Between two-stream and Between two-stream and filamentation, T≠0filamentation, T≠0Relativistic effectsRelativistic effects

Same Angle

Highest Growth RateHighest Growth Rate

BEAM

1

Zxm

Zxm =

Vb / c

(Vth /Vb)1/2

(k,E) angle(k,E) angle

BEAM

Longitudinal approximation OK below

Growth Rates ScalingGrowth Rates Scaling

Vb

/cnb/np

=

0

1/3 /b

/2 (/b)1/2

(/b)1/3

m

b

Transverse Beam Temp. Transverse Beam Temp. EffectsEffects

Beam

No effectk

k

Important effect

Vtb

Which effect ?

Beam Temp. EffectsBeam Temp. Effects

Tan =1+nb /np

(Vthb/b+Vth) /Vb

Beam Temp. damps growth rate beyond

Non relativistic beamNon relativistic beam

Vtb=0 Vtb=Vb/30 Vtb=Vb/30+

k // E

1. Beam Temp. damps instabilities beyond .2. Longitudinal approximation fails beyond .3. Longitudinal approximation even better with hot

beam.

ConclusionConclusion• Electromagnetic formalism• Exhaustive instabilities search

– Weibel Branch– TSF Branch

• TSF Branch: Two k-oblique effects– Critical angle – Max. Growth Rate for oblique k (Vb~c)

• Longi. Approx. Fails beyond • Beam Temp. damps instabilities beyond ,

not bellow• Maxwelian distribution, Collisions, Density

gradient