Influence of the Weibel instability on the expansion of a plasma slab into a vacuum

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plasma slab into a vacuum

Cédric Thaury, P Mora, A Héron, Jean-Claude Adam, Thomas Antonsen

To cite this version:

Cédric Thaury, P Mora, A Héron, Jean-Claude Adam, Thomas Antonsen. Influence of the Weibel instability on the expansion of a plasma slab into a vacuum. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2010, pp.026408. �10.1103/PhysRevE.82.026408�.

�hal-01166851�

Influence of the Weibel instability on the expansion of a plasma slab into a vacuum

C. Thaury, P. Mora, A. Héron, and J. C. Adam

Centre de Physique Théorique, Ecole Polytechnique–CNRS, 91128 Palaiseau, France

T. M. Antonsen

Department of Electrical Engineering, University of Maryland, College Park, Maryland 20742, USA 共

Received 18 June 2010; published 25 August 2010

兲

The development of the Weibel instability during the expansion of a thin plasma foil heated by an intense laser pulse is investigated, using both analytical models and relativistic particle-in-cell simulations. When the plasma has initially an anisotropic electron distribution, this electromagnetic instability develops from the beginning of the expansion. Then it contributes to suppress the anisotropy and eventually saturates. After the saturation, the strength of the magnetic field decreases because of the plasma expansion until it becomes too weak to maintain the distribution isotropic. For this time, the anisotropy rises as electrons give progressively their longitudinal energy to ions, so that a new instability can develop.

DOI: 10.1103/PhysRevE.82.026408 PACS number

共

s

兲

: 52.38.Fz, 52.35.

⫺

g, 52.65.Rr, 52.38.Kd

I. INTRODUCTION

When an intense laser pulse impinges on a thin solid tar- get, it ionizes it within a few optical cycles and forms a dense plasma with an electron temperature in the 0.1–1 keV range. The laser pulse then interacts with the plasma around the critical density surface, where it is reflected, and heats a population of electrons, up to MeV energies for laser inten- sities I ␭

²

⬎ 10

¹⁹

W cm

⁻²

␮ ^m

² 关

1

兴

. These hot electrons flow through the target, and a part of them forms electrostatic sheaths, on both sides of the plasma, which pull ions into the vacuum and accelerates them, potentially up to several tens of MeV

关

2–10

兴

.

This acceleration process is commonly described using one-dimensional

共1D兲

models of plasma expansion into a vacuum

关11–14兴. The assumption of a dependence on a

single spatial dimension is justified as long as the laser focal spot is sufficiently large. However, it is incorrect to consider only one component of the electron velocities. Indeed, sev- eral mechanisms such as elastic Coulomb collisions and electromagnetic instabilities can induce a coupling between the transverse and longitudinal components of velocity.

These can strongly affect both the expansion dynamics

关15,16兴

and the properties of the accelerated ion beams

关17,18兴.

In this paper, we assume that the plasma is too hot for collisions to be efficient

关

15

兴

and consider the influence of the Weibel instability on the expansion. The principle of this electromagnetic instability, which occurs in plasmas with an- isotropic velocity distributions, is schematized in Fig. 1.

We assume that

共ex

, e

y

,e

z兲

is a standard basis, and that the initial electron velocity distribution function is f

⁰共

v

兲

⬀ exp关−v

x

2

/ v

_x0²

− v

_y²

/ v

_y0²

− v

_z²

/ v

_z0²兴

with v =

共vx

, v

y

, v

z兲

and v

_x0

⬎ v

_y0

= v

_z0

. We also suppose that a perturbation magnetic field B= B

⁰

cos kxe

z

arises from the noise. In these condi- tions, the magnetic Lorentz force bends the electron trajec- tories in the

共ex

,e

y兲

plane

共see Fig.

1兲 and forms current sheets j = je

x

which enhance the magnetic field

关19兴. As

B does not perturb the isotropic part of f

⁰

the currents build always along a hot axis and the instability develops only in anisotropic plasmas.

Such plasmas are naturally obtained when thin foils are irradiated by intense laser pulses. Indeed hot electrons gain most of their energy through collisionless mechanisms

共

e.g., resonant absorption

关

20

兴

, Brunel absorption

关

21

兴

, and j ⫻ B heating

关

22

兴兲

which depend on the polarization of the laser and heat the plasma in a preferred direction. Because of this anisotropy which can be very strong, the Weibel instability is triggered from the beginning of the expansion

共see the

growth of the mean magnetic energy by hot electron in Fig.

2兲. As electrons give progressively their longitudinal mo- menta to ions through the ambipolar field, and transfer a part of it to the transverse directions through the magnetic Lor- entz force, the anisotropy is eventually suppressed

共at

t

⬇

6 fs in Fig. 2兲. However, the mean-square longitudinal momentum p

_储²

is still decreasing because electrons continue to lose energy to ions. As the amplitude of the magnetic field is too low to keep the plasma isotropic, p

_储²

becomes lower than the mean-square transverse momentum p

_⬜²

. In other words the plasma becomes again anisotropic, so that a new instability can develop

共for

t ⲏ 9 fs in Fig. 2兲 关16兴. Note that during this second growth the former hot axis e

x

is a cold axis.

In this paper, these two stages are analyzed in detail. First, the relevant dispersion relations are derived and discussed in Sec. II. Then the expansion is analyzed using two- dimensional particle-in-cell

共PIC兲

simulations in Sec. III.

B y

x

j<0 vx

vy

j>0

FIG. 1.

共

Color online

兲

Principle of the Weibel instability. A per- turbation magnetic field

B

drives a current

j, which enhances the

original field. The arrows schematize the trajectories of the elec- trons which form the current sheets.

1539-3755/2010/82

共

2

兲

/026408

共

12

兲

026408-1 ©2010 The American Physical Society

II. DISPERSION RELATIONS

In this section we establish dispersion relations for the Weibel instability in different conditions. We first assume in Sec. II A that all electrons have been accelerated to multi- keV energies, so that the hot directions can be described by a single temperature, and consider successively the cases p

_储²

⬎ p

_⬜²

and p

_储²

⬍ p

_⬜²

. Then we discuss in Sec. II B the more realistic situation where only a part of the plasma electrons has been heated to high energies.

A. Single population plasma

1. Cold transverse temperature

We assume here that the electron velocities are nonrela- tivistic and that the distribution function is of the form

f

^共0兲共

v

兲

⬀ exp

关

−

兩

v

_⬜兩²

/ v

_⬜²₀

− v

_储²

/ v

_储²₀兴

with v

_⬜0

⬍ v

_储0

= v

_h0

. We look for transverse magnetic modes with wave vector k parallel to e

y

and nonzero field components

共Ex

,E

y

,B

z兲. We

express f

^共0兲

in the Cartesian basis

共ex

, e

y

,e

z兲, where

e

z

is chosen to be a cold direction, and where the hot direction lies in the

共

e

_x

,e

_y兲

plane such that the second cold direction makes an angle ␪ with respect to k

关Fig.

3共a兲兴:

f

^共0兲共

v

兲

⬀ exp

关

−

共

v

_x

+ ␮ ^v

y兲²

/ v

₀²

− v

_y²

/ v

_e²

− v

_z²

/ v

_⬜²₀兴

,

共

1

兲

with

v

_e²

= v

_h0² 共A

sin

²

␪ + 1兲/共A + 1兲,

v

₀²

= v

_h0²

/共A sin

²

␪ + 1兲,

␮ ^{= A cos} ␪ ^sin ␪ /共A sin

²

␪ + 1兲 ,

A =

共v储0

/v

_⬜0兲²

− 1.

In this basis, the current j lies in the

共

e

_x

,e

_y兲

plane

共

it always builds in a hot direction兲 and B is along the e

z

axis

共it is

parallel to k ⫻ j兲.

To derive a dispersion relation for the unstable modes, we linearize the Maxwell and Vlasov equations for perturbations

共

f

^共1兲

,E ,B

兲⬃

exp

关

i

共

ky − ␻ ^t

兲兴

as follows:

B

_z

= −

共

k / ␻

兲

E

_x

,

共

2

兲

ic

²

kB

z

= j

x

/ ⑀

0

− i ␻ ^E

x

,

共3兲

0 = j

_y

/ ⑀

0

− i ␻ ^E

y

,

共

4

兲

− i ␻ 冉 ^{1 − v} ␻

^y

^k 冊 ^f

^共1兲

^{= m e}

e

冋

^共Ex

^{+ v}

^y

^B

^z兲

^{⳵ ⳵ f v}

^共0x兲

+

共Ey

− v

x

B

z兲

⳵ ^f

^共0兲

⳵ ^v

y

册 ^,

^共5兲

where ⑀

0

is the vacuum permittivity, e is the electron charge, and m

_e

is the electron mass. Combining Eqs.

共

2

兲

and

共

3

兲

, and averaging −ev

x

over f

^共1兲

, gives

i ␻ ^j

x

⑀

0

=

共c²

k

²

− ␻

²兲Ex

= ␻

p

2

冕 ^d

³

^vv

^x

冋 ^E

^x

^{⳵ ⳵ f v}

^共0兲x

+ ␻ ^E

y

+ v

_x

kE

_x

␻ ^{− v}

y

k

⳵ ^f

^共0兲

⳵ ^v

y

册 ^,

^共

⁶

^兲

where ␻

p

= n

e

e

²

/ m

e

⑀

0

is the electron plasma frequency. Simi- larly, combining Eqs.

共2兲

and

共4兲

and averaging −ev

y

over f

^共1兲

leads to

i ␻ ^j

y

⑀

0

= − ␻

²

^E

y

= ␻

2p

冕 ^d

³

^v ␻ ^{− v}

^y

^v

y

k

关

␻ ^E

y

+ v

x

kE

x兴

⳵ ^f

^共0兲

⳵ ^v

y

.

共

7

兲

Then, using integration by parts, Eqs.

共6兲

and

共7兲

become

− 冓

^共

^{␻ v −}

^x

^{k v ␻}

^y

^k兲

²

冔 ^E

^y

^{= 冋 c}

²

^k

²

^{␻ −}

^p2

^␻

²

^{+ 1 +} 冓

^共

^{␻ v −}

^x2

^{v k}

^2y

^k兲

²

冔 ^{册 E}

^x

^,

共8兲

冋 ^{␻ ␻}

^22p

⁻ 冓

^共

^{␻ − ␻ v}

^2y

^k兲

²

冔 ^{册 E}

^y

⁼ 冓

^共

^{␻ v −}

^x

^{k v ␻}

^y

^k兲

²

冔 ^E

^x

^,

^共

⁹

^兲

0.01 0.1 1 10 100

0.1 1 10 100 1000

p2 /2me(keV)

Time (fs)

10^-3 10^-2 10^-1 1 10

First stage

Magneticenergy(keV)

p²_{⎢ ⎢}

p²_⊥

E_m

Second stage

FIG. 2.

共

Color online

兲

Energy transfers during the expansion of a plasma slab into a vacuum, in log-log scale. The initial density and thickness of the slab are, respectively,

n_e

= 5

⫻

10

²³

cm

⁻³

and

L

= 150 nm. At

t= 0 the plasma is nonmagnetized 关E_m共

0

兲

= 0

兴

, and the mean-square longitudinal and transverse momenta are, respec- tively,

p_储²/

2m

_e

= 1.5 MeV and

p_⬜²/

2m

_e

= 1 keV. These results origi- nate from a two-dimensional particle-in-cell simulation with peri- odic conditions in the transverse direction, and a transverse width of 180 nm.

FIG. 3.

共

Color online

兲

Standard basis used

共

a

兲

when

p_储²⬎p_⬜² 共

case of Sec. II A 1

兲

and

共

b

兲

when

p_储²⬍p_⬜² 共

case of Sec. II A 2

兲

.

where the angular brackets denote the averaging over f

^共0兲

. The dispersion relation is finally obtained by combining Eqs.

共

8

兲

and

共

9

兲

,

c

²

k

²

= ␻

²

⁻ ␻

p

2

冋 ^{1 +} 冓

^共

^{␻ v −}

^x2

^{v k}

^2y

^k兲

²

冔

− 冓

^共

^{␻ v −}

^x

^{k v ␻}

^y

^k兲

²

冔

²

冒 ^冉 冓

^共

^{␻ − ␻ v}

^2y

^k兲

²

冔 ^{− ␻ ␻}

^22p

^{冊 册 .}

共10兲

Note that this equation can also be derived using the general expression of the dielectric tensor established in Ref.

关23兴.

Purely growing solutions of Eq.

共10兲,

⌫

z共k兲

= −i ␻

共k兲, are

plotted in Fig. 4共a兲 for different values of ␪ ^{, A = 10, and v}

h0

= 0.2c. This figure shows that the most unstable modes are obtained for ␪ = 0, that is, when the wave vector lies in the cold plane. The maximum growth rate is divided by 2 when

␪ is increased from 0° to 20°, and is reduced by almost two orders of magnitude when ␪ ^{= 37.5°.}

The largest unstable

共

or critical

兲

wave number k

_c

is ob- tained by taking the limit ␻ → 0 in Eq.

共10兲. This leads to

k

_c²

c

²

/ ␻

p 2

= v

₀²

/v

e

2

− 1 + ␮

²关

␲ /2 − 1兴

= A

cos

²

␪ ^{− sin}

²

␪ 冋 ^{1 + A 冉 1 − ␲ 2 cos}

²

^{␪ 冊} 册

共A

sin

²

␪ + 1兲

²

.

共11兲

For A→ 0, this equation reduces to k

_c²

c

²

/ ␻

p2

⬇

A

共

1 – 2 sin

²

␪

兲

, which shows that for small A, purely grow- ing modes are obtained only for ␪ ⬍ 45°. Similarly, taking the limit A → ⬁ for ␪ ⫽ 0 in Eq.

共11兲

leads to k

_c²

c

²

/ ␻

p2

⬇关共

␲ /2兲cos

²

␪ − 1兴/ sin

²

␪ . In this case, k

c

does not depend on A and is much smaller than for ␪ = 0 where k

_c²

c

²

/ ␻

p

2

= A.

All these findings are confirmed in Fig. 5共a兲 which displays k

c共

␪

兲

for five different values of A.

Figure 5共b兲 shows that the maximum growth rate ⌫

z m

also decreases more steeply with ␪ ^{for large} A. As a result of the behavior of both k

c

and ⌫

z

m

, unstable modes for large A grow mainly with wave vector in the cold plane. In contrast, for small A and ␪ ⱗ 10°, k

c

and ⌫

z

m

depend weakly on ␪ , so modes can grow efficiently with 0 ⬍ ␪ ⱗ 10°.

2. Hot transverse temperature

We turn now to the case where the electron distribution function is of the form

f

^共0兲共v兲

⬀ exp关−

兩v_⬜兩²

/v

_⬜²₀

− v

_储²

/v

储20兴

with v

_⬜0

= v

_h0

⬎ v

_储0

.

This corresponds to the second stage of the plasma expan- sion in Fig. 2.

We first express f

^共0兲

in the standard basis

共ex

⬘ ^,e

y

⬘ ^,e

z

⬘

兲

in which e

_z

⬘ is a hot axis and k = ke

_x

⬘ ^:

f

^共0兲共v兲

⬀ exp关−

共vy

+ ␮ v

x兲²

/v

₀²

− v

_x²

/v

e

2

− v

_z²

/v

_⬜0² 兴

,

共12兲

where v

e

, v

₀

, and ␮ have the same definitions as in Eq.

共1兲,

except that A =

共v_⬜0

/ v

_储0兲²

− 1, and ␪ is the angle between the wave vector k and the cold axis or, equivalently, the angle between e

_y

⬘ and the hot plane

关see Fig.

3共b兲兴. In this situation, B must lie in the

共

e

_y

⬘ ^,e

z

⬘

兲

plane. One possible combination of nonzero fields is

共Ex

,E

y

,B

z兲. This case is identical to the one

studied in Sec. II A 1, just exchanging the roles of x and y in the calculations. A second combination is

共Ez

, B

y兲. We note

that in this case, where the cold direction and k are both

0 1 2 3

0.0 0.2 0.4

(a)

B

_y

θ=60°

θ=45°

θ=30°

θ=15°

Γ y

/ (

ω p

v

h0

/c )

kc /

_ωp

θ=0°

(b)

B

_z

θ=30°

θ=15°

θ=0°

0 1 2 3

0.0 0.2 0.4

Γ z

/ (

ω p

v

h0

/c )

kc /

ω_p

FIG. 4.

共

Color online

兲

Growth rates of the Weibel instability as a function of the wave number, for different angles

␪

between

k

and the cold plane

共

or axis

兲

,

v_h0

= 0.2c, and

A= 10. The angle␪

is incre- mented by 7.5° steps. In

共

a

兲

the current lies in the anisotropic plane

共e_x

,e

_y兲

, while in

共

b

兲

it is along the

e_z

axis.

0 10 20 30 40

0 1

0 10 20 30 40

10^-3 10^-2 10^-1 1

(b)

A=10⁵

A_{→ 0}

A=10³A=100

A=1

k

c

c/A

1/2 ωp

θ

(in degree)

A=10

(a)

A=1

Γm z/(ω p

v

h0

/c )

θ

(in degree)

A=10⁵

FIG. 5.

共

Color online

兲

Influence of

␪

on the largest unstable

wave vector and on the maximum growth rate for

B_z

.

共

a

兲

Largest

unstable wave number

k_c

normalized to

冑A␻p/c, as a function of␪

,

for different anisotropy parameters

A. The dotted line corresponds

to the limit

A→

0.

共

b

兲

Maximum growth rate as a function of

␪

for

A

= 1 and

A= 10⁵

, in a logarithmic scale.

perpendicular to e

_z

⬘ , an electric field in the e

_z

⬘ direction pro- duces no current density in the

共ex

⬘ ^,e

y

⬘

兲

plane, due to the reflection symmetry of the distribution function in e

_z

⬘ ^{. Con-} sequently, E

x

= E

y

= 0.

As in Sec. II A 1 we begin by writing the linearized Max- well and Vlasov equations,

B

y

= −

共k/

␻

兲Ez

,

共13兲

ic

²

kB

y

= j

z

/ ⑀

0

− i ␻ ^E

z

,

共14兲

− i ␻ 冉 ^{1 − v} ␻

^x

^k 冊 ^f

^共1兲

^{= m e}

e

冋

^共Ex

^{− v}

^z

^B

^y兲

^{⳵ ⳵ f v}

^共0兲x

+

共Ez

+ v

x

B

y兲

⳵ ^f

^共0兲

⳵ ^v

z

册 ^.

^共15兲

Then we combine Eqs.

共

13

兲

and

共

14

兲

, and average −ev

_z

over f

^共1兲

, to get

i ␻ ^j

z

⑀

0

=

共c²

k

²

− ␻

²兲Ez

= ␻

p

2

冕 ^d

³

^v 冋 ^{␻ kv − v}

^z2x

^{k E}

^z

^{⳵ ⳵ f v}

^共0兲x

^{+ v}

^z

^E

^z

^{⳵ ⳵ f v}

^共0兲z

册 ^.

^共16兲

We finally apply integration by parts to obtain the following dispersion relation:

c

²

k

²

= ␻

²

⁻ ␻

p

2

冋 ^{1 + k 2}

²

冓

^共

^{␻ − v v}

^zx

^k兲

²

冔 ^册

^共17兲

= ␻

²

⁻ ␻

p2

冋 ^{1 − A sin A}

²

^{+ 1 ␪ + 1}

^{关1 +}

^{␰ Z共 ␰}

^兲兴

册 ^,

共18兲

where ␰ ⁼ ␻ / kv

e

and Z共 ␰

兲

= ␲

^−1/2兰dt

exp关t

⁻²兴

/

共t

− ␰

兲

is the plasma dispersion function

关24兴.

The growth rate ⌫

y共

k

兲

obtained from Eq.

共

17

兲

is plotted in Fig. 4共b兲. For ␪ = 0, the growth rates are the same for B

y

and B

z 关Fig.

4共a兲兴. In contrast, for ␪ ⬎ 0, ⌫

y共k兲

is larger than

⌫

z共

k

兲

. The reason for that is that when B grows along the e

_y

⬘ axis the currents are purely along a hot direction, which is not the case when it grows along the e

_z

⬘ ^axis

共except for

␪

= 0兲; currents are thus larger in the first case.

The critical wave number,

k

_c²

c

²

/ ␻

p2

= A cos

²

␪

A sin

²

␪ ^{+ 1 ,}

^共19兲

is also larger for B

y

than for B

z关compare Eqs.共11兲

and

共19兲

and the two panels of Fig. 4兴. Another difference is that purely growing modes are obtained for all ␪ ⬍ 90° when B is along the e

_y

⬘ axis. In contrast the influence of A on k

c

is similar in both cases. Indeed, Eq.

共19兲

shows that k

_c²

c

²

␻

p2

⬇

A cos

²

␪ when A → 0, and k

_c²

c

²

␻

p

2⬇

cot

²

␪ when A → ⬁ and

␪ ⫽ 0.

3. Discussion for␪= 0

Sections II A 1 and II A 2 show that, in all cases, the most unstable mode grows with a wave vector in a cold direction.

It is therefore interesting to study in detail the dispersion relation for ␪ = 0. Note that even though the most unstable modes are always obtained for ␪ = 0, for small anisotropy, k

c

and ⌫

^m

decrease slowly when ␪ is increased

共see Fig.

5兲.

Modes with ␪ ⬎ 0 can therefore develop during the second stage of the expansion during which the anisotropy is quite small.

In this section we assume that ␪ = 0 and we derive a simple expression for ⌫

^m

when the anisotropy is large. Then we discuss the case of relativistic velocities. The starting point for this study is the dispersion relation obtained by taking ␪ ^{= 0 in Eq.}

共10兲

or Eq.

共17兲:

c

²

k

²

= ␻

²

⁺ ␻

p

2关A

+

共A

+ 1兲 ␰ Z共 ␰

兲兴, 共20兲

with ␰ ⁼

冑

A + 1 ␻ / kv

_h0

. As mentioned before, the largest un- stable wave vector in this case is k

c

=

冑

A ␻

p

/ c.

(a) Limit for large anisotropy. When A is large, Z共 ␰

兲

can be substituted by its asymptotic expansion for

兩

␰

兩

Ⰷ 1, Z

共

␰

兲

⬇

− ␰

⁻¹共1 + 1/

2 ␰

²

+ 3/ 4 ␰

⁴兲 关25兴, so that Eq.共20兲

becomes

c

²

k

²⬇

− ⌫

²

+ ␻

p2

冋 ^{− 1 + k}

²

^{v 2}

^h02

^⌫

⁻²

^{− 4 3k}

^共

^A

⁴

^{v + 1}

^{h04 兲}

^⌫

⁻⁴

册 ^.

共

21

兲

Differentiating Eq.

共

21

兲

with respect to k, we find that for A Ⰷ 1 the maximum growth rate is

⌫

^m⬇

冋 ^{1 − 冑 6共2 + A v}

^h0

^/c兲 册

^1/2

冑 ^v 2c

^h0

␻

p

⬇

v

_h0

冑 2c ␻

p

when A → ⬁ .

共

22

兲

Then, combining Eqs.

共21兲

and

共22兲, we find the wave num-

ber of the most unstable mode,

k

^m⬇

␻

p

c 冋 ^{2 + v 6}

^h0

^{/c A} 册

^1/4⬇

^{␻ c}

^p

冋 ^{A 3} 册

^1/4

^{for v}

^h0

^{Ⰶ c.}

共23兲

Equation

共22兲

shows that, for large A, ⌫

^m

is almost a linear function of the hot velocity. It indicates also that ⌫

^m

saturates when A→ ⬁. Further Eq.

共23兲

states that the wave number of the most unstable mode k

^m

increases much more slowly with A than the critical wave number k

c

=

冑

A ␻

p

/ c.

This means that the instability develops mainly with rela- tively small wave numbers

共

for A = 10 000, k

^m⬇

7.6 ␻

p

/ c, while k

c

= 100 ␻

p

/c兲. Note that this is not the case for A Ⰶ 1, where k

^m

= k

c

/

冑

3. In summary for large anisotropy, the growth rate saturates and the most unstable wave number increases slowly with A

共more slowly that

k

c兲.

(b) Relativistic velocities. For now this study is quite aca- demic. In an experiment the velocity is generally relativistic when the anisotropy is very large. It is thus necessary to analyze how the previous results are modified when v

_h0

→ c.

As it is much more complicated to derive a dispersion rela-

tion in this case

关26–28兴, we limit the present discussion to

an analysis of one spatial and three velocity dimensions

共

1D3V

兲

relativistic PIC simulations. In these simulations, the

initial distribution function is f

^共0兲

⬀ exp关−共 ␥

x

− 1兲/

共

␥

储0

− 1兲

−

共

␥

y

+ ␥

z

− 2兲/

共

␥

_⬜0

− 1兲兴 with ␥

x

=

共1 −

v

_x²

/ c

²兲^−1/2

and ␥

0储

=

共1

− v

_储²₀

/ c

²兲^−1/2共and so on for

␥

y

, ␥

z

, and ␥

0⬜兲. The wave vector

is necessarily along e

y

, which is the only spatial dimension in these 1D simulations, so ␪ is forced to zero. Ions are fixed and the width of the plasma along the e

y

axis is 42c/ ␻

p共see

Sec. III A 1 for supplementary information on the simula- tions兲.

Figure 6 displays the results of simulations performed for different values of E

_储0

=

共

␥

储0

− 1

兲

mc

²

, and a mean-square transverse momentum p

_⬜0²

/ 2m

e

= 500 eV. For E

储0

ⱗ 50 keV, the maximum growth rate ⌫

^m

is almost equal to the one given by Eq.

共20兲. For larger longitudinal energies, the simu-

lation results start to differ significantly from the classical model. The growth rate saturates for E

储0⬇

1 MeV

共v储0

⬇

0.94c兲 as in the classical limit, but the value reached by ⌫

^m

is lower in the relativistic regime

共0.43

␻

p

, instead of 0.71 ␻

p兲

. Note that this value is not modified when p

_⬜0²

is reduced. The most important difference is however observed for E

储0

⬎ 5 MeV, where ⌫

^m

exhibits a slow decay.

To explain this behavior, we derive a dispersion relation for the simplest relativistic distribution function f ⬀ ␦

共

␥

x

− ␥

储0兲

␦

共

v

_y兲

␦

共

v

_z兲, where

␦

共x兲

is the Dirac delta function. Us- ing the general dispersion relation established in Ref.

关23兴,

we obtain after integration,

c

²

k

²

+ ⌫

²

+ ␻

p 2

␥

储0

−3关1 −

c

²

k

²共

␥

储0

2

− 1兲⌫

⁻²兴

= 0.

共24兲

Then, differentiating Eq.

共24兲

we find that the maximum growth rate, for ␥

储0

Ⰷ 1, is ⌫

r

m

= ␻

p

/

冑

␥

储0

. In agreement with Fig. 6 this growth rate decreases when the longitudinal en- ergy is raised. Physically, this originates from the relativistic increase of the mass, which makes electrons harder to move in the e

y

direction when ␥

储0

increases. Note that this effect is also observed analytically when more realistic distribution functions are used

关27兴.

To sum up this section, the growth rate of the most un- stable mode ⌫

^m

increases with A before saturating for very large anisotropy. This property which has been demonstrated for nonrelativistic thermal velocities is also verified when v

_h0

is relativistic. Additional PIC simulations actually show that for E

_储0

= 50 MeV, ⌫

^m

varies by less than 1% when 10

³

ⱕ A ⱕ 10

⁵

. In the classical regime

共E储0

ⱗ 50 keV兲, the maximum growth rate rises almost linearly with the thermal velocity in

the hot direction. In contrast, in the relativistic regime, ⌫

^m

is observed to decrease when the longitudinal energy is in- creased above approximately 5 MeV. As a result, the optimal growth rate is obtained for E

_储0⬇

1 – 5 MeV. This growth rate can however be much smaller than predicted in Fig. 6 if the plasma contains a large population of cold electrons.

B. Two population plasma

In this section, we analyze the influence on the Weibel instability of a cold isotropic population in the classical limit.

This case is of particular interest in the context of laser- plasma interaction, where the laser heats only a part of the plasma electrons to very high energies. To investigate the impact of a cold population, we consider distribution func- tions of the form

f

_2p^共0兲共

v

兲

= ␣ ^f

h共0兲共

v

兲

+

共

1 − ␣

兲

f

_c^共0兲共

v

兲

,

共

25

兲

with ␣ = n

_h0

/

共nh0

+ n

_c0兲

as the fraction of hot electrons

共nh0

and n

_c0

are, respectively, the hot and cold electron densities兲, f

_h^共0兲

= f

^共0兲

, and f

_c^共0兲共v兲

⬀exp关−兩v兩

²

/ v

_c0² 兴.

For simplicity we assume that ␪ = 0, so that we can easily derive from Eq.

共17兲

the dispersion relation for the two popu- lation plasma,

c

²

k

²

= ␻

²

⁺ ␻

ph2 关A

+

共A

+ 1兲 ␰ Z共 ␰

兲兴

+ 1 − ␣

␣ ␻

ph2

␰

c

Z共 ␰

c兲, 共26兲

with ␻

ph2

= ␣␻

p2

and ␰

c

= ␻ / kv

_c0

. Taking the limit ␻ ^{= 0, we} find that the critical wave number is k

c

=

冑

A ␻

ph

/ c. This sug- gests that the presence of a cold population does not perturb much the growth of the instability. But Fig. 7共a兲, which dis- plays the growth rate of purely growing modes for different hot electron fractions, demonstrates that this conclusion is not correct since ⌫/ ␻

ph

is observed to decrease with ␣ . Fur- ther, Fig. 7共b兲 shows that ⌫ / ␻

ph

is also reduced when the

10^-3 10^-2 10^-1 1 10 10² 10³ 0.0

0.2 0.4

0.6 Rel. PIC code Class. model

Γm /ωpe

E_{⎜ ⎜0}= (γ_{⎜ ⎜0}-1)mc²(MeV)

FIG. 6. Growth rates for relativistic thermal velocities. The tri- angles are the results of 1D3V relativistic PIC simulations per- formed for a mean-square transverse momentum

p_⬜²₀/

2m

_e

= 500 eV. The dashed line is the maximum growth rate provided by Eq.

共

20

兲

.

0 1 2 3

0.0 0.2 0.4

α=0.1

kc /ω_ph

α=0.1 κ=10³

(b)

α=0.25 α=0.5

Γ/(ωphvh0/c)

α=1

(a) _κ=10³

0 1 2 3

0.00 0.02 0.04

κ=10⁵ κ=∞

κ=10⁴ Γ/(ωphvh0/c)

kc /ω_ph

FIG. 7.

共

Color online

兲

Dispersion relations in the presence of cold electrons for

A

= 10 and

v_h0

= 0.2c.

共

a

兲

Growth rate

⌫/共␻phv_h0/c兲

as a function of

kc/␻ph

for

␬

=

共v_h0/v_c0兲²

= 10

³

and

different fractions of hot electrons

␣

.

共

b

兲

Influence of

␬

on the

dispersion curve for

␣

= 0.1. The dashed line corresponds to the

limit

␬→⬁

.

ratio ␬ =

共vh0

/ v

_c0兲²

is increased, i.e., when the cold electrons become colder or, equivalently, when the hot electrons be- come hotter

共while

A is kept constant兲. Finally, we remark that the wave number of the most unstable mode shifts to lower value when ␬ ^rises.

To understand these different effects, we take the limit v

_c0

→0 in Eq.

共26兲

to obtain another dispersion relation,

c

²

k

²

= ␻

²

⁺ ␻

ph2 关1 − 1/

␣ ^{+ A +}

共A

+ 1兲 ␰ Z共 ␰

兲兴. 共27兲

The purely growing solution of Eq.

共27兲

is plotted in Fig. 7

共dashed line兲. We see that the growth rate

⌫

^m

is 200 lower in this case than in the single population case. Besides, the critical wave vector,

k

c

= ␻

ph

c 冑 ^{A + 1 −} ␣ ^{1 ,}

^共28兲

is smaller than in the finite- ␬ case. This means that for very low cold thermal velocity, the growth rate for

冑

A+ 1 − 1 / ␣

⬍ k

^m

c/ ␻

ph

⬍

冑

A tends to zero. Thus, the instability is almost suppressed when ␣ ⬍ 1 /

共

A + 1

兲

. In other words, cold elec- trons can stabilize the plasma if their fraction is sufficiently high.

This is illustrated in Fig. 8共a兲 which shows that the maxi- mum growth rate ⌫

^m

decreases strongly with ␣ ^when ␣ ⱗ 1 /

共

A+ 1

兲

, even when ␬ is relatively small. It also reveals a very good agreement between Eq.

共26兲

and 1D PIC simula- tions with fixed ions, thus validating the theoretical predic- tions. Further, Fig. 8共a兲 shows that for ␣ Ⰷ 1 /

共A

+ 1兲 the so- lution obtained for ␬ →⬁ is very close to the one given by Eq.

共26兲. As Fig.

7 proves, in addition, that Eq.

共27兲

provides a good estimate of ⌫

^m

when ␬ ⬎ 10

⁵

, even for ␣

⬇

1 / A, we can use it—as a first approximation—to study the influence of the cold population.

To explain the “stabilization” by the cold population, we rewrite Eq.

共27兲

to make appear the cold plasma frequency

␻

pc2

=

共1 −

␣

兲

␻

p2

,

c

²

k

²

= ␻

²

⁻ ␻

pc 2

+ ␻

ph

2 关A

+

共A

+ 1兲 ␰ Z共 ␰

兲兴. 共29兲

We see that the plasma is stabilized if ␻

pc

2

Ⰷ ␻

ph 2 关A

+

共A

+ 1兲 ␰ Z共 ␰

兲兴, that is, if the cold population response to the per-

turbation is much stronger than the one of the hot population.

To elucidate the influence of v

_c0

, we take one more term in the Taylor expansion of Z共 ␰

c兲

in Eq.

共26兲, replacing

␻

pc2

with

␻

pc

2共

1 − k

²

v

_c0²

/ 2 ⌫

²兲

in Eq.

共

29

兲

. Thus we see that a finite ve- locity v

_c0

tends to reduce the influence of cold electrons. In fact the cold return current associated with the ␻

pc2

term is the same whether v

_c0

= 0 or v

_c0

⫽ 0, but in the latter case electrons are also subjected to the magnetic Lorentz force which drives currents in the opposite direction, and hence reduces the influence of cold electrons. The impact of this second term increases with v

_c0

.

We study now the influence of the anisotropy parameter.

For ␣ ^A Ⰷ 1, we find, in the same way that we obtained Eq.

共22兲, that the maximum growth rate is

⌫

^m⬇

冋 ^{1 − 冑 6}

^共

^{2 + ␣ ␣ A v}

^h0

^{/ c}

^兲

册

^1/2

^{冑 v 2c}

^h0

^␻

^p⬇

^{冑 v 2c}

^h0

^␻

^p

^.

共

30

兲

The maximum growth rate is thus almost unchanged by the presence of cold electrons if ␣ ^A is sufficiently large. In con- trast, when ␣ A

⬇

1, ␰ Z共 ␰

兲⬇

−

冑

_␲

_共A

_{+ 1兲⌫ /} kv

_h0

and Eq.

共27兲

leads to

⌫

^m⬇

1

冑 ␲ 冉 ^{2 3} 冊

^3/2

冋 ^{1 − ␣}

^共

^{A 1 + 1}

^兲

册

^3/2

冑 ^v 2c

^h0

␻

ph

,

共31兲

which confirms that ⌫

^m

strongly decreases with ␣ when ␣

→

共A

+ 1兲

⁻¹

, whatever is the anisotropy. To illustrate these effects, we plotted in Fig. 8共b兲 ⌫

^m

as a function of ␣ ^{for three} different anisotropy parameters

共solid lines兲. We observe that

Eqs.

共

30

兲

and

共

31

兲

provide a good approximation of ⌫

^m

for, respectively, large and small ␣ A. For large A, the influence of cold electrons is very weak if ␣ Ⰷ 1 /A, while the instability can be suppressed, even for very high anisotropy, if the plasma is mainly made of cold electrons.

To conclude this section, we address the specific case of a cold population with a temperature equal to the cold axis temperature

关v_c0²

= v

_h0²

/

共A

+ 1兲兴. This case may be of impor- tance in the context of laser heated plasma foils, where such a distribution is obtained when only a part of the plasma electrons gains energy from the laser and is accelerated in a preferential direction. In this situation, ␰

c

= ␰ ^{and Eq.}

共26兲

becomes

c

²

k

²

= ␻

²

⁻ ␻

pc 2

+ ␻

ph

2 关

A ⬘ ⁺

共

A ⬘ ^{+ 1}

兲

␰ ^Z

共

␰

兲兴

with A ⬘ ^{= A +}

共1 −

␣

兲/

␣ .

共32兲

Comparing Eqs.

共29兲

and

共32兲

reveals that this dispersion relation is the same as the one obtained for an infinitely cold population substituting A by A ⬘ . Thus, the previous analysis, and in particular Eqs.

共30兲

and

共31兲, can be used in this case,

1E-3 0.01 0.1 1

10^-4 10^-3 10^-2 10^-1 1

A=1000 A=100 Γm /(ωphvh0/c)

Hot electron fraction_α A=10

0.0 0.2 0.4 0.6 0.8 1.0

10^-3 10^-2 10^-1 1

Theory,κ= 100 Theory,κ=∞ PIC,κ= 100 Γm /(ωphvh0/c)

Hot electron fractionα (b)

(a)

FIG. 8.

共

Color online

兲

Influence of the hot electron fraction

␣

on the maximum growth rate

⌫^m

. In

共

a

兲

, the model for

A

= 10 and

␬

= 100 is compared to 1D3V PIC simulations with fixed ions. The

theoretical growth rate obtained for

␬→⬁

is also plotted in dashed

line. In

共

b

兲⌫^m

is plotted as a function of

␣

for three different values

of

A. The dotted line corresponds to the limitA→⬁

and the dashed

lines correspond to the limits obtained when

␣A⬇

1.

without approximation, by just replacing A with A ⬘ ^{. For in-} stance, we get for small ␣ A

⌫

^m⬇

1

冑 _␲ 冉 ² 3 冊

^3/2

冋 ^{1 − ␣ A 1 + 1} 册

^3/2

^{冑 v 2c}

^h0

^␻

^ph

^.

^共33兲

Note that, in contrast with Eq.

共

31

兲

, ⌫

^m

⬎ 0 for all ␣ ^A ⬎ 0.

III. SIMULATIONS OF THE PLASMA EXPANSION

Up to now, we have studied the development of the Wei- bel instability in a plasma with fixed ions. To take into ac- count the influence of the expansion, we perform in this sec- tion two spatial and three velocity dimensions

共

2D3V

兲

PIC simulations with mobile ions, and use the results of Sec. II to analyze the numerical findings. First we present in Sec. III A simulations with an initial anisotropy and without laser pulse, and then we discuss in Sec. III B a simulation without initial anisotropy but with a laser pulse focused on the plasma slab.

A. Two temperature expanding plasmas 1. Numerical conditions

The PIC code used in this study is relativistic and noncol- lisional. It involves two spatial

共

e

_x

and e

_y兲

and three velocity dimensions. The simulation box size is 1000c/ ␻

ph

⫻ 24c / ␻

ph

. The plasma is a foil with an initial thickness along the e

x共longitudinal兲

axis of L = 20c/ ␻

ph

. The boundary conditions in the transverse direction are periodic for both fields and particles. In the longitudinal direction, open boundaries are used for the fields while electrons are re- flected. The time step is ⌬t = 0.1 ␻

p−1

and the spatial grid sizes are ⌬x= ⌬y = 0.16c / ␻

p

. The ion to electron mass ratio is m

_i

/ m

_e

= 1836, and the ion charge is Z = 1.

In this section we consider plasmas with an initial mean- square longitudinal momentum p

_储²₀

larger than the transverse one p

_⬜0²

, and with or without a cold isotropic electron popu- lation. The transverse and the cold population temperatures are both of 1 keV, while the ion temperature is 30 eV. To limit both the computation time and the noise level, we take in our simulations ␣ ⱖ 0.1, which is sufficient to analyze the influence of cold electrons. There are initially

⬇

9 ⫻ 10

⁶

hot macroparticles when ␣ = 1, and

⬇3

⫻ 10

⁶

when ␣ = 0.1.

2. Simulation results

Figures 9 and 10 are the results of 2D3V PIC simulations performed for different initial hot longitudinal energies E

_h0

=

共

␥

h0

− 1兲m

e

c

²

and different fractions of hot electrons ␣ ^. They display, respectively, the mean-square longitudinal and transverse momenta and the mean magnetic energy as func- tions of time

共Fig.

9兲, and the magnetic field as a function of space for different times

共Fig.

10兲.

(a) Nonrelativistic case. In Figs. 9

共

a

兲

and 10

共

a

兲

, ␣ ^{= 1 and} E

_h0

= 10 keV

共A

= 9兲. The magnetic energy is observed to increase with a growth rate 2⌫

PIC⬇

2 ⫻ 0.055 ␻

ph

, before saturating at t⬇ 100 ␻

ph−1 关Fig.

9共a兲兴. According to the linear theory of Sec. II, the maximum growth rate in this nonrela- tivistic case should be ⌫

^m

= 0.053 ␻

ph

. The theoretical and nu-

merical values are very close, which indicates that the linear model describes correctly the development of the Weibel in- stability, even for a two-dimensional expanding plasma. Fig- ure 10共a兲 shows the spatial variations of the magnetic field, when it reaches its maximum at t

⬇

100 ␻

ph−1

. In agreement with Sec. II A 1, the wave vector lies in the transverse plane

共

␪ = 0兲. Further, the value of the most unstable wave number k

^m_PIC⬇

1.3 ␻

ph

/ c is also close to the value predicted by the linear model

共

k

^m

= 1.2

兲

, which confirms its accuracy. Note that at earlier times there are several other unstable modes which eventually coalesce

关29,30兴.

Figure 9共a兲 shows that, due to the magnetic field, elec- trons transfer a part of their longitudinal momentum to the transverse direction up to t

⬇

1000 ␻

ph

−1

, where p

_储²

= p

_⬜² 共A

= 0兲. At this time, the mean-square transverse momentum stops rising, while the longitudinal one continues decreasing because electrons lose their energy to ions. As the magnetic field at t

⬇

1000 ␻

ph

−1

is not intense enough to isotropize “in- stantaneously” the electron distribution, the anisotropy A

⬇

p

_⬜²

/ p

_储²

− 1 grows. The maximum anisotropy reached during this second stage is A

⬇

0.12, which is sufficient to drive a new Weibel instability. Indeed, for A

⬇

0.12 we have k

c

⬇

0.35 ␻

ph

/ c, while the smallest unstable wave number which can be sustained by the plasma is k= 2 ␲ / L

⬇

0.31 ␻

ph

/ c ⬍ k

_c

. Note that for simplicity the plasma expan- sion is neglected in this analysis. A more comprehensive dis- cussion can be found in Ref.

关16兴.

As a consequence of this second instability, the magnetic energy restarts to rise and it reaches a second maximum at t

⬇

2700 ␻

ph

−1

. The magnetic field at this time is plotted in Fig.

10共a兲

共right column兲. The wave vector of the most unstable

1 100

0.1 10 1000

1 10 100 1000

0.1 10 1000 1 100

(d) (c) p²_⊥

10^-3 10^-1

E_m p²_{⎢ ⎢}

Meanmagneticenergybyhotelectron,Em(keV) 10^-3 10^-1 10 E_m

ω_pht

10^-3 10^-1 10 E_m

Meansquaremomentum,p2 /2me(keV) p²_{⎢ ⎢}

p²_⊥ ¹⁰

-3

10^-1 (b) (a)

α=0.1,E_h0=1 MeV α=1,E_h0=1 MeV α=0.1,E_h0=10 KeV α=1,E_h0=10 KeV Em

p²_{⎢ ⎢}

p²_⊥

p²_{⎢ ⎢}

p²_⊥

FIG. 9.

共

Color online

兲

Mean magnetic energy by hot electron,

and mean-square transverse and longitudinal momenta as functions

of time, in log-log scale, for different hot fractions

␣

and hot lon-

gitudinal energies

E_h0

=

共␥h0

− 1

兲m_ec²

.