Effect of electric field on filamentation in counterstreaming beams

22
Effect of electric field on filamentation in counterstreaming beams Gareth Murphy 1 , Mark Dieckmann 2 , Luke Drury 1 1. Dublin Inst for Advanced Studies, 2. Univ. of Linkoping Friday, 16 September 2011

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Transcript of Effect of electric field on filamentation in counterstreaming beams

Page 1: Effect of electric field on filamentation in counterstreaming beams

Effect of electric field on filamentation in

counterstreaming beamsGareth Murphy1, Mark Dieckmann2, Luke Drury1

1. Dublin Inst for Advanced Studies, 2. Univ. of Linkoping

-20 0 20X, !s

0

20

40

60

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

Friday, 16 September 2011

Page 2: Effect of electric field on filamentation in counterstreaming beams

Shock waves at all scales in astrophysics

2

•Many astrophysical objects exhibit collisionless shock waves.•Some of them feature synchrotron emission - implying highly relativistic electrons, as well as a magnetic field far in excess of the compressed ISM or IGM field.•We seek an instability which can self-generate magnetic fields, from zero initial field.

Friday, 16 September 2011

Page 3: Effect of electric field on filamentation in counterstreaming beams

Motivation

• Explore the filamentation instability at mildly relativistic speeds with PIC simulations

• How does the plasma composition affect filament formation?

• Electron-positron vs Electron-only (fixed ion background)

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Page 4: Effect of electric field on filamentation in counterstreaming beams

JplasmaJbeam

The filamentation instability:

COUNTERSTREAMING

CURRENTS

SIMULATION

PLANE

•Parallel current elements mutually attract each other•Ant i -para l le l cur rent elements repel each other•R e s u l t s i n c u r r e n t elements bunching and increase in size•Process runs away and forms filaments in the plasma

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Page 5: Effect of electric field on filamentation in counterstreaming beams

Filamentation Instability

• Linear theory: Exponential growth in magnetic field strength (Weibel 1959, Fried 1959)

• Nonlinear theory: Saturation due to magnetic trapping (Davidson, Hammer, Haber, Wagner 1973)

• Electrostatic field growth rate predicted (Califano et al., Phys Rev E.,1998)

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Page 6: Effect of electric field on filamentation in counterstreaming beams

Basic Setup

•2D simulation strictly orthogonal to flow vector

•Two simulations: fixed immobile ions, mobile electrons, and mobile electrons and positrons

•Choose flow velocity such that growth rate of the filamentation i n s t a b i l i t y i s t h e o re t i c a l maximum. (Bret et al , Phys Plasma 2010)

COUNTERSTREAMING

CURRENTS

SIMULATION

PLANE

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Page 7: Effect of electric field on filamentation in counterstreaming beams

Initial Conditions

✤ Physical

• Beam momentum +/-1.487 c

• 0.13 c electron thermal speed

• Simulation electrically quasi-neutral initially

• Maxwell-Juttner distribution

• Magnetic & electric fields zero initially

• Periodic boundary conditions

✤ Computational

• 144 particles per cell, 4,000 x4,000 cells in 2D resolving 133 x133 skin

depths

• Simulation runtime of 952 inverse plasma

frequencies

• Runtime = 36 hours on 4096 Bluegene cores

COUNTERSTREAMING

CURRENTS

SIMULATION

PLANE

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Page 8: Effect of electric field on filamentation in counterstreaming beams

Numerical Method

• Particle In Cell (PIC) Simulations

• Plasma Simulation Code (PSC; Ruhl et al 2003)

• MPI-Parallel

• Scaling to 32,000 cores on PRACE Tier-0 BlueGene

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Electron Density 9

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 10.2"p

0.0000 0.0208 0.0415 0.0623 0.0831 0.1038 0.1246

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 16.3"p

0.0000 0.0148 0.0297 0.0445 0.0593 0.0741 0.0889

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 20.4"p

0.0002 0.0205 0.0408 0.0612 0.0815 0.1018 0.1221

FIG. 16: (Colour online) The EP simulation at times T1,2,3.Electron density shows the formation, growth and merging offilaments.

-60 -40 -20 0 20 40 60X, !s

0

20

40

60

80

100

120

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

FIG. 17: Magnetic energy density for E beam

sponding growth in planar electrical current, for the elec-510

tron beam-Weibel instability. We have increased the size511

of the simulation box to 4000 square cells wihch allows us512

to track the growth of filaments at a good resolution. We513

confirm the previous results of [13, 14], who highlighted514

the importance of electric field growth on the filament515

-60 -40 -20 0 20 40 60X, !s

0

20

40

60

80

100

120

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

FIG. 18: Magnetic energy density for EP beam

size and statistics. We find remarkable di!erences in the516

growth of the electric field between the E and EP beam.517

The electric field grows at approximately twice the speed518

of the magnetic field in the E beam. This growth is519

quenched in the EP beam, as positrons neutralise the520

electric fields.521

For many years, magnetic trapping was assumed to522

dominate the filamentation instability, so much so that523

only recently [6, 18] the role of the electric field was thor-524

oughly examined.525

Previous authors have found that varying the526

plasma composition can reduce the net current527

carried, decrease the size of individual filaments.528

We summarise the physical mechanism is as fol-529

lows. The Jz current filaments induce an in-plane530

Bx,y in a closed loop around each filament, due to531

Ampere’s law. This Bx,y exerts a magnetic pres-532

sure gradient force MPGF, in the plane, which533

drives in-plane currents, Jx,y. In the case of534

electron-only, these currents are large and cor-535

related with an in-plane electric field Ex,y. In536

the case of electron-positron plasma, the MPGF537

drives both positive and negative charge carri-538

ers in the same direction, which partially cancels539

the Jx,y, reducing it to negligibly small values, so540

a large Ex,y is not excited. In our results, the541

Jx,y is about 2 orders of magnitude larger in the542

electron-only case than in the electron-positron543

case. So the process of growth of electric fields is544

choked o! by the presence of positrons.545

The loss of quasi-neutrality and the generation546

of strong electrostatic fields in the plasma is im-547

portant as it may limit the lifetime of current fil-548

aments. These strong magnetic fields associated549

with the current filaments are expected to play in550

important role in accelerating electrons in plasma551

shocks, for example. Such fields are expected to552

decay rapidly, due to phase space mixing [21].553

The toy model first proposed by Medvedev554

et al. [19] suggested that the field cannot dissi-555

pate e"ciently due to di!usion as the field corre-556

lation lengthscale grows as the light crossing time.557

However this toy model does not include the elec-558

trostatic fields which we have shown in this paper559

Friday, 16 September 2011

Page 10: Effect of electric field on filamentation in counterstreaming beams

9

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 10.2"p

0.0000 0.0208 0.0415 0.0623 0.0831 0.1038 0.1246

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 16.3"p

0.0000 0.0148 0.0297 0.0445 0.0593 0.0741 0.0889

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 20.4"p

0.0002 0.0205 0.0408 0.0612 0.0815 0.1018 0.1221

FIG. 16: (Colour online) The EP simulation at times T1,2,3.Electron density shows the formation, growth and merging offilaments.

-60 -40 -20 0 20 40 60X, !s

0

20

40

60

80

100

120

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

FIG. 17: Magnetic energy density for E beam

sponding growth in planar electrical current, for the elec-510

tron beam-Weibel instability. We have increased the size511

of the simulation box to 4000 square cells wihch allows us512

to track the growth of filaments at a good resolution. We513

confirm the previous results of [13, 14], who highlighted514

the importance of electric field growth on the filament515

-60 -40 -20 0 20 40 60X, !s

0

20

40

60

80

100

120

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

FIG. 18: Magnetic energy density for EP beam

size and statistics. We find remarkable di!erences in the516

growth of the electric field between the E and EP beam.517

The electric field grows at approximately twice the speed518

of the magnetic field in the E beam. This growth is519

quenched in the EP beam, as positrons neutralise the520

electric fields.521

For many years, magnetic trapping was assumed to522

dominate the filamentation instability, so much so that523

only recently [6, 18] the role of the electric field was thor-524

oughly examined.525

Previous authors have found that varying the526

plasma composition can reduce the net current527

carried, decrease the size of individual filaments.528

We summarise the physical mechanism is as fol-529

lows. The Jz current filaments induce an in-plane530

Bx,y in a closed loop around each filament, due to531

Ampere’s law. This Bx,y exerts a magnetic pres-532

sure gradient force MPGF, in the plane, which533

drives in-plane currents, Jx,y. In the case of534

electron-only, these currents are large and cor-535

related with an in-plane electric field Ex,y. In536

the case of electron-positron plasma, the MPGF537

drives both positive and negative charge carri-538

ers in the same direction, which partially cancels539

the Jx,y, reducing it to negligibly small values, so540

a large Ex,y is not excited. In our results, the541

Jx,y is about 2 orders of magnitude larger in the542

electron-only case than in the electron-positron543

case. So the process of growth of electric fields is544

choked o! by the presence of positrons.545

The loss of quasi-neutrality and the generation546

of strong electrostatic fields in the plasma is im-547

portant as it may limit the lifetime of current fil-548

aments. These strong magnetic fields associated549

with the current filaments are expected to play in550

important role in accelerating electrons in plasma551

shocks, for example. Such fields are expected to552

decay rapidly, due to phase space mixing [21].553

The toy model first proposed by Medvedev554

et al. [19] suggested that the field cannot dissi-555

pate e"ciently due to di!usion as the field corre-556

lation lengthscale grows as the light crossing time.557

However this toy model does not include the elec-558

trostatic fields which we have shown in this paper559

Friday, 16 September 2011

Page 11: Effect of electric field on filamentation in counterstreaming beams

9

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 10.2"p

0.0000 0.0208 0.0415 0.0623 0.0831 0.1038 0.1246

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 16.3"p

0.0000 0.0148 0.0297 0.0445 0.0593 0.0741 0.0889

-4 -2 0 2 4X, !s

2468

Y, !

s

Electron Density t= 20.4"p

0.0002 0.0205 0.0408 0.0612 0.0815 0.1018 0.1221

FIG. 16: (Colour online) The EP simulation at times T1,2,3.Electron density shows the formation, growth and merging offilaments.

-60 -40 -20 0 20 40 60X, !s

0

20

40

60

80

100

120

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

FIG. 17: Magnetic energy density for E beam

sponding growth in planar electrical current, for the elec-510

tron beam-Weibel instability. We have increased the size511

of the simulation box to 4000 square cells wihch allows us512

to track the growth of filaments at a good resolution. We513

confirm the previous results of [13, 14], who highlighted514

the importance of electric field growth on the filament515

-60 -40 -20 0 20 40 60X, !s

0

20

40

60

80

100

120

Y, !

s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

FIG. 18: Magnetic energy density for EP beam

size and statistics. We find remarkable di!erences in the516

growth of the electric field between the E and EP beam.517

The electric field grows at approximately twice the speed518

of the magnetic field in the E beam. This growth is519

quenched in the EP beam, as positrons neutralise the520

electric fields.521

For many years, magnetic trapping was assumed to522

dominate the filamentation instability, so much so that523

only recently [6, 18] the role of the electric field was thor-524

oughly examined.525

Previous authors have found that varying the526

plasma composition can reduce the net current527

carried, decrease the size of individual filaments.528

We summarise the physical mechanism is as fol-529

lows. The Jz current filaments induce an in-plane530

Bx,y in a closed loop around each filament, due to531

Ampere’s law. This Bx,y exerts a magnetic pres-532

sure gradient force MPGF, in the plane, which533

drives in-plane currents, Jx,y. In the case of534

electron-only, these currents are large and cor-535

related with an in-plane electric field Ex,y. In536

the case of electron-positron plasma, the MPGF537

drives both positive and negative charge carri-538

ers in the same direction, which partially cancels539

the Jx,y, reducing it to negligibly small values, so540

a large Ex,y is not excited. In our results, the541

Jx,y is about 2 orders of magnitude larger in the542

electron-only case than in the electron-positron543

case. So the process of growth of electric fields is544

choked o! by the presence of positrons.545

The loss of quasi-neutrality and the generation546

of strong electrostatic fields in the plasma is im-547

portant as it may limit the lifetime of current fil-548

aments. These strong magnetic fields associated549

with the current filaments are expected to play in550

important role in accelerating electrons in plasma551

shocks, for example. Such fields are expected to552

decay rapidly, due to phase space mixing [21].553

The toy model first proposed by Medvedev554

et al. [19] suggested that the field cannot dissi-555

pate e"ciently due to di!usion as the field corre-556

lation lengthscale grows as the light crossing time.557

However this toy model does not include the elec-558

trostatic fields which we have shown in this paper559

Friday, 16 September 2011

Page 12: Effect of electric field on filamentation in counterstreaming beams

Friday, 16 September 2011

Page 13: Effect of electric field on filamentation in counterstreaming beams

Magnetic field Energy

•Agreement between 2 simulations•Exponential growth: growth rate matches theoretical value•Saturates when magnetic trapping occurs electron cyclotron

frequency is comparable to growth rate

Fixed Ions

Electron Positron

0 200 400 600 80010-610-510-410-310-210-1

(a) L

og E

mag

netic

/EK

IN

0 200 400 600 80010-810-710-610-510-410-310-2

(b) L

og E

XY

,ele

ctric

/EK

IN

0 200 400 600 80010-810-710-610-510-410-310-2

(c) L

og E

Z,el

ectri

c/EK

IN

0 200 400 600 80010

100

1000

10000

(d) L

og J Z

,ele

ctric

Friday, 16 September 2011

Page 14: Effect of electric field on filamentation in counterstreaming beams

Electrostatic field growth 0 200 400 600 800

10-610-5

10-4

10-3

10-210-1

(a)

Lo

g E

mag

net

ic /

EK

IN

0 200 400 600 800

10-810-710-610-510-410-310-2

(b)

Lo

g E

XY

,ele

ctri

c /E

KIN

0 200 400 600 800

10-810-710-610-510-410-310-2

(c)

Lo

g E

Z,e

lect

ric/

EK

IN

0 200 400 600 800

10

100

1000

10000

(d)

Lo

g J

Z,e

lect

ric

•Qualitative differences•Growth rate is twice that of magnetic field

Fixed Ions

Electron Positron

Friday, 16 September 2011

Page 15: Effect of electric field on filamentation in counterstreaming beams

Jz

0 200 400 600 800

10-610-5

10-4

10-3

10-210-1

(a)

Lo

g E

mag

net

ic /

EK

IN

0 200 400 600 800

10-810-710-610-510-410-310-2

(b)

Lo

g E

XY

,ele

ctri

c /E

KIN

0 200 400 600 800

10-810-710-610-510-410-310-2

(c)

Lo

g E

Z,e

lect

ric/

EK

IN

0 200 400 600 800

10

100

1000

10000

(d)

Lo

g J

Z,e

lect

ric

Electric Current, Jz

•Electric current increases with time•peaks when magnetic trapping sets in•Decreases due to redirection of energy

Fixed Ions

Electron Positron

Friday, 16 September 2011

Page 16: Effect of electric field on filamentation in counterstreaming beams

Time-Wavenumber Plot of Ex

•Wavenumbers decrease exponentially with time•Coherent structures increase exponentially with time•Similar behaviour for electric field as magnetic field

7

FIG. 11: (Colour online) Spatial power spectrum of theelectron-positron beam complex planar electric field Exy in-tegrated over the azimuth in (kx, ky) as a function of time.

they are averaged in x-space over larger intervals, de-421

creasing the 1/sqrt(N) error.422

FIG. 12: (Colour online) Spatial power spectrum of the elec-tron beam complex planar field Exy integrated over the az-imuth in (kx, ky) as a function of time.

Figure 12 shows a plot of the spatial power spectrum423

of the E beam complex planar field Exy integrated over424

the azimuth in (kx, ky) as a function of time. Comparing425

with the EP beam (Figure 11) , the peak value of Ex,y426

remains lower for the EP beam at any given time.427

C. Phase 3: Saturation428

At later times, electric and magnetic fields are satu-429

rated. The linear description is no longer valid. The430

magnetic bounce frequency becomes comparable to the431

linear growth rate immediately prior to saturation [20].432

! !=!

!

ekm

VBc

!

!

433

The approximation of magnetic trapping ap-434

plies well to the EP simulation, since the elec-435

trostatic field can be neglected. The electric en-436

40 42 44 46 48 500.000.02

0.04

0.060.08

Ne

40 42 44 46 48 50-0.04-0.02

0.00

0.020.04

Ex

40 42 44 46 48 50-0.0002-0.00010.00000.00010.00020.0003

grad

Bz2

40 42 44 46 48 50x [!d]

-0.0002-0.0001

0.0000

0.00010.0002

Bz

FIG. 13: For E beam, profiles of Ne,Bx,Ex,!Bz.

ergy density is high in the E simulation and the437

electric forces will play a role in the saturation438

of the instability. However, we demonstrate here439

that for the case studies we consider, the electric440

field neither modifies the time-evolution of the fil-441

ament size, nor the magnetic energy density. It442

does clearly a!ect the filament shape and the cur-443

rent distribution as we see in the later plots.444

By looking at profiles of individual filaments the role445

of the electric and magnetic fields in defining the struc-446

ture can be clarified. For the E beam, electrostatic forces447

are stronger than in the EP beam. The ions are not448

free to move, and provide a flat uniform distri-449

bution of positive charge everywhere in the pe-450

riodic simulation box. The electrostatic in-plane451

electric fields will eventually set ions in motion in452

a real plasma. However, since the leptonic fila-453

ments move and merge, this ion motion may not454

be significant because the electric fields are strong455

only in limited spatial intervals and act only for456

short times. This means a larger electronic cur-457

rent Jx,y,! can be driven by the MPGF, which458

would otherwise be reduced and partial cancelled459

by the mobile positronic current Jx,y,+, which is460

driven in the same direction by the MPGF. The461

Jx,y are associated with electrostatic forces, which462

act to increase the radial extent of the filament,463

while decreasing its density ( Figure 13). Over464

7

FIG. 11: (Colour online) Spatial power spectrum of theelectron-positron beam complex planar electric field Exy in-tegrated over the azimuth in (kx, ky) as a function of time.

they are averaged in x-space over larger intervals, de-421

creasing the 1/sqrt(N) error.422

FIG. 12: (Colour online) Spatial power spectrum of the elec-tron beam complex planar field Exy integrated over the az-imuth in (kx, ky) as a function of time.

Figure 12 shows a plot of the spatial power spectrum423

of the E beam complex planar field Exy integrated over424

the azimuth in (kx, ky) as a function of time. Comparing425

with the EP beam (Figure 11) , the peak value of Ex,y426

remains lower for the EP beam at any given time.427

C. Phase 3: Saturation428

At later times, electric and magnetic fields are satu-429

rated. The linear description is no longer valid. The430

magnetic bounce frequency becomes comparable to the431

linear growth rate immediately prior to saturation [20].432

! !=!

!

ekm

VBc

!

!

433

The approximation of magnetic trapping ap-434

plies well to the EP simulation, since the elec-435

trostatic field can be neglected. The electric en-436

40 42 44 46 48 500.000.02

0.04

0.060.08

Ne

40 42 44 46 48 50-0.04-0.02

0.00

0.020.04

Ex

40 42 44 46 48 50-0.0002-0.00010.00000.00010.00020.0003

grad

Bz2

40 42 44 46 48 50x [!d]

-0.0002-0.0001

0.0000

0.00010.0002

Bz

FIG. 13: For E beam, profiles of Ne,Bx,Ex,!Bz.

ergy density is high in the E simulation and the437

electric forces will play a role in the saturation438

of the instability. However, we demonstrate here439

that for the case studies we consider, the electric440

field neither modifies the time-evolution of the fil-441

ament size, nor the magnetic energy density. It442

does clearly a!ect the filament shape and the cur-443

rent distribution as we see in the later plots.444

By looking at profiles of individual filaments the role445

of the electric and magnetic fields in defining the struc-446

ture can be clarified. For the E beam, electrostatic forces447

are stronger than in the EP beam. The ions are not448

free to move, and provide a flat uniform distri-449

bution of positive charge everywhere in the pe-450

riodic simulation box. The electrostatic in-plane451

electric fields will eventually set ions in motion in452

a real plasma. However, since the leptonic fila-453

ments move and merge, this ion motion may not454

be significant because the electric fields are strong455

only in limited spatial intervals and act only for456

short times. This means a larger electronic cur-457

rent Jx,y,! can be driven by the MPGF, which458

would otherwise be reduced and partial cancelled459

by the mobile positronic current Jx,y,+, which is460

driven in the same direction by the MPGF. The461

Jx,y are associated with electrostatic forces, which462

act to increase the radial extent of the filament,463

while decreasing its density ( Figure 13). Over464

Fixed Ions Electron-Positron

Friday, 16 September 2011

Page 17: Effect of electric field on filamentation in counterstreaming beams

Azimuthally Averaged E (k)

t=100: Electric fields qualitatively similar but FI 10^4 times larger

t=600 Decreases to 10^2 times larger

1 10 100Normalised Wavenumber, k s

10-11

10-10

10-9

10-8

10-7

10-6

E x,y(k

)

EEP

1 10 100Normalised Wavenumber, k s

10-10

10-9

10-8

10-7

10-6

E x,y(k

)

EEP

Friday, 16 September 2011

Page 18: Effect of electric field on filamentation in counterstreaming beams

Magnetic Pressure Gradient Force

• JZ induces, by Ampere’s law, magnetic field BX,Y in plane

• The magnetic pressure gradient force, grad( B2X,Y)drives JX,Y

• For fixed ions, JX,Y currents are large and correlated with EX,Y

• In electron-positron plasma, JX,Y current is partially cancelled as MPGF drives both charged species in same direction.

Friday, 16 September 2011

Page 19: Effect of electric field on filamentation in counterstreaming beams

Plot of EX,Y for Fixed-ion

5

di!erences building up. The electric field seems331

to counteract the formation of small-scale struc-332

tures.333

1 10 100Normalised Wavenumber, k !s

10-9

10-8

10-7

10-6

10-5

J z(k

)

EEP

FIG. 3: Spatial power spectra of the currents in the 2d boxat time t=T2, integrated over the azimuth. E corresponds tothe electron beam and EP to the electron-positron beam.

1 10 100Normalised Wavenumber, k !s

10-11

10-10

10-9

10-8

10-7

10-6

E x,y(k

)

EEP

FIG. 4: Spatial power spectra of the electric field Exy in the2d box at time t=T2, integrated over the azimuth in (kx, ky).E corresponds to the electron beam and EP to the electron-positron beam.

-66.5 -66.0 -65.5 -65.0 -64.5X, !s

0.5

1.0

1.5

2.0

2.5

Y, !

s

Electric Field Ex,y2, exy t= 59.4"p

0.0000 0.0003 0.0006 0.0009 0.0012 0.0015 0.0018

FIG. 5: Spatial zoom on the plot of electric field Exy in theE beam simulation at time t=T2

Comparing the electric field spectrum in the saturated334

stage we see that only the electron beam has a strong335

peak in the power spectrum of Exy. The EP beam has336

a peak at the same wave number but almost 4 orders of337

magnitude less (Figure 4). We can also say that the338

power spectra are qualitively but not quantitively339

similar and that, thus, the presence of positrons340

does not fully prevent the build-up of the elec-341

tric fields, although they are strongly reduced (by342

4 orders of magnitude). This is not unexpected343

since quasi-neutrality implies that the densities344

of oppositely charged species should be approxi-345

mately the same but do not have to locally cancel346

each other.347

In Figure 5 we plot a spatial zoom on the plot of348

electric field Exy in the E beam simulation at time349

t=T2. We see that the filaments at this time have350

a typical spatial size of x and a spatial separation351

of y.352

1 10 100Normalised Wavenumber, k !s

10-10

10-9

10-8

10-7

10-6

E x,y(k

)

EEP

FIG. 6: Spatial power spectra of the electric field Exy in the2d box at time t=T3, integrated over the azimuth in (kx, ky).E corresponds to the electron beam and EP the electron-positron beam.

Figure 6 shows the spatial power spectra of the353

electric field Exy in the 2d box at time t=T3.354

At this time at lower k-values the E beam domi-355

nates the power spectrum. The power spectrum356

is again qualitatively similar but di!ers by almost357

2 orders of magnitude.358

The spatial power spectrum of the electron beam or-359

thogonal current Jz shows that the region where most of360

the power is concentrated shifts in time as k ! t!1. This361

indicates that the current carrying filament characteris-362

tic length ! k!1 has a linear dependence on t (Figure 7).363

This confirms previous results [19].364

The time dependent power spectrum of the electron-365

positron beam orthogonal current Jz also shows a similar366

growth in filament correlation length (Figure 8). The t!1367

fit represents a good first order fit to the (k, t) spectrum.368

Current generated in the plane by the electron fila-369

ments can be seen in the time dependent power spectrum370

of the complex planar current Jxy = Jx+iJy for the elec-371

tron beam (Figure 9). They have coherent structures,372

which increase in amplitude and correlation length. The373

currents are driven by the planar electric field Ex + iEy,374

which increases similarly (see Figure 2).375

The presence of positrons in the beam reduces the co-376

herency of of the in-plane currents, however a definite377

signal is present, as may be seen from the time dependent378

power spectrum of Jxy (Figure 10). Because the mag-379

netic pressure gradient force accelerates electrons380

Friday, 16 September 2011

Page 20: Effect of electric field on filamentation in counterstreaming beams

-20 0 20X, !s

0

20

40

60

Y,

!s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

-20 0 20X, !s

0

20

40

60

Y,

!s

Magnetic Energy Density t= 680.0"p

-6.8000 -5.8833 -4.9667 -4.0500 -3.1333 -2.2167 -1.3000

Final stages show larger voids between filaments for e-e+ plasma

20

Comparison of magnetic energy densitieselectron-positron plasma vs fixed ion plasma

Fixed IonsElectron-Positron

Friday, 16 September 2011

Page 21: Effect of electric field on filamentation in counterstreaming beams

Conclusions• Plasma composition influences electrostatic

field growth and saturation.

• Magnetic energy saturation level is unchanged

• Magnetic pressure gradient force causes differences in size of filaments, their separation, filling factor

• Long timescale reduction in current

Friday, 16 September 2011

Page 22: Effect of electric field on filamentation in counterstreaming beams

Perspectives

• Plasma composition may affect the lifetime of current filaments generated by the filamentation instability

Friday, 16 September 2011