effect of electric field on filamentation in counterstreaming beams
DESCRIPTION
TRANSCRIPT
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
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
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)
Friday, 16 September 2011
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
Friday, 16 September 2011
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)
Friday, 16 September 2011
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
Friday, 16 September 2011
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
Friday, 16 September 2011
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
Friday, 16 September 2011
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
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
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
Friday, 16 September 2011
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
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
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
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
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
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
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
-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
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
Perspectives
• Plasma composition may affect the lifetime of current filaments generated by the filamentation instability
Friday, 16 September 2011