study of the effects of coulomb collisions on h , he and o ... · study of the effects of coulomb...

Study of the effects of Coulomb collisions on H+, He+ and O+ plasmas for

ISR applications at Jicamarca

Marco Milla1, Erhan Kudeki2, and Jorge Chau1

1 Radio Observatorio de Jicamarca, Instituto Geofísico del Perú, Lima, Perú.2 Department of Electrical and Computer Engineering, University of Illinois at Urbana-

Champaign, Urbana, IL.

June 24, 2013

2013 CEDAR WorkshopMillennium Hotel - Boulder, Colorado

Monday, June 24, 13

Radio Observatorio de Jicamarca - Instituto Geofísico del Perú

Project OutlineCoulomb collision effects on H+, He+, and O+ plasmas

• Massive simulation of particle trajectories in H+, He+ and O+

plasmas (Langevin equation and Fokker-Planck collision model).

• Statistical analysis of the simulated trajectories and construction of a numerical library of single-particle ACF's.

• Comparison of the collisional model with standard incoherent scatter theories.

• Application of the model to ISR experiments at Jicamarca.

�0.10

0.1�0.1

00.1�5

0

5

10

15

x-axis [m]

Time Interval: 0-400 µs

y -axis [m]

z-a

xis

[m]

�0.1 �0.05 0 0.05 0.1

�0.1

�0.05

0

0.05

0.1

x-axis [m]

y-a

xis

[m]

Displacement ! to B

0 50 100 150 200 250 300 350 400�5

0

5

10

15

Time [µs]z-a

xis

[m]

Displacement " to B

Monday, June 24, 13


Motivation: ISR spectrum perp. to BChannel 1 ! 2004!06!08 11:55:00

Doppler velocity (m/s)

Range (

km

)

!150 !100 !50 0 50 100 150

200

300

400

500

600

700

800

dB

15

20

25

30

To develop a model for the IS spectra measured with antenna beams pointed perpendicular-to-B at Jicamarca with the goal of estimating ionospheric physical parameters (e.g. densities, temperatures).

Monday, June 24, 13


Motivation: ISR spectrum perp. to B

Monday, June 24, 13



!150 !100 !50 0 50 100 15021

22

23

24

25

26

27Channel 1 ! 2004!06!08 11:55:00


Po

we

r S

pe

ctru

m (

dB

)

280.00 km

300.00 km

320.00 km

340.00 km

Doppler shift of the spectrum is a direct measurement of the drift.

The width of the spectrum is related to the plasma temperature.

Monday, June 24, 13



!150 !100 !50 0 50 100 15021

22

23

24

25

26

27Channel 1 ! 2004!06!08 11:55:00


Po

we

r S

pe

ctru

m (

dB

)

280.00 km

300.00 km

320.00 km

340.00 kmKudeki et al (1999) fitted the

measurements using a simplified spectral model. This model was developed based on the collisionless IS theory.But, the temperatures they obtained were about half of

what is expected.

Monday, June 24, 13



Sulzer and Gonzalez (1999) showed that, due to Coulomb collision effects, the IS spectrum becomes narrower than what

the collisionless theory predicts at small magnetic aspect angles.

!150 !100 !50 0 50 100 15021

22

23

24

25

26

27Channel 1 ! 2004!06!08 11:55:00


Po

we

r S

pe

ctru

m (

dB

)

280.00 km

300.00 km

320.00 km

340.00 kmKudeki et al (1999) fitted the

measurements using a simplified spectral model. This model was developed based on the collisionless IS theory.But, the temperatures they obtained were about half of

what is expected.

Monday, June 24, 13


Simulation of particle trajectories based on Langevin equation

Monday, June 24, 13


IS spectrum and Gordeyev integrals

• The power spectrum of the incoherent scatter signals is proportional to the spectrum of electron density fluctuations in a plasma (e.g., Kudeki & Milla, 2011)

• Thanks to the fluctuation-dissipation (or Nyquist) theorem, the self-spectra of thermal density fluctuations and species conductivities are link to each other. Moreover they can be written in the following forms

where denotes the so-called Gordeyev integral for each particle species.

⇥s(⇤, k)j⇤�o

=1� j⇤Js(⇤)

k2h2s

�|ne(⌅k,⇤)|2⇥ =|j⇤�o + ⇥i(⌅k, ⇤)|2�|nte(⌅k,⇤)|2⇥ + |⇥e(⌅k, ⇤)|2�|nti(⌅k, ⇤)|2⇥

|j⇤�o + ⇥e(⌅k,⇤) + ⇥i(⌅k, ⇤)|2

⇤|nts(⌥k, �)|2⌅Ns

= 2Re{Js(�)}

Js(!)

Monday, June 24, 13


Gordeyev integral, Fokker-Planck collision model and Langevin equation

• The Gordeyev integrals are effectively Fourier transforms of the electron or ion particle ACFs (Hagfors & Brockelman, 1971)

• Instead of computing the integrals solving a kinetic equation with the Fokker-Planck collision operator, we decided to compute them from simulated electron and ion trajectories.

• The trajectories are simulated using a Generalized version of the Langevin equation in which Coulomb collisions are modeled by a deterministic friction force and random diffusion forces acting on a test particle.

�ej⇥k·�⇥rs⇥ = �ej⇥k·(⇥rs(t+�)�⇥rs(t))⇥Js(⇥) =� ⇥

0d�e�j⇥� �ej⇤k·�⇤rs⇥

d v(t)dt

=q

m v(t)⇥ B � �(v) v(t) +

�D⇥(v)W1(t) v̂⇥(t)

⇥D�(v)

2W2(t) v̂�1(t) +

⇥D�(v)

2W3(t) v̂�2(t)

Monday, June 24, 13


3D particle trajectory sample

Ion moving in an O+ plasma experiencing Coulomb collisions

104 sequences of 217 samples are generated (~30 GB), however, only the statistics (pdfs and ACFs) are stored (~60 MB).

Ne = 1011 m-3

Te = 2000 KTi = 2000 KB = 20000 nT

Monday, June 24, 13


Simulation of multiple trajectories using CUDA

Significant saving in simulation and processing time.

Monday, June 24, 13


Statistics of test-ion displacementsin H+, He+, and O+ plasmas and

comparison to the Brownian model

Monday, June 24, 13


Ion displacement distributions• H+, He+, and O+ ion displacement

distributions are Gaussian in the direction perpendicular to B as function of delay τ.

• In the parallel direction, the distributions also look gaussian.

• A Brownian motion model with Gaussian trajectories is a good representation of the ion process (Woodman, 1967).

• The single-ion ACF can be approximated by

0 2 4 6 8 10−20

2

0

0.1

0.2

0.3

0.4

Time delay [ms]

Ion displacement distributions ! to B

!r!(! )/"!(! )

!r !

pdf

0 2 4 6 8 10−20

2

0

0.1

0.2

0.3

0.4

Time delay [ms]

Ion displacement distributions " to B

!r"(! )/""(! )

!r "

pdf

�ej⇧k·�⇧r

⇥= e�

12 k2 sin2 �⇤�r2

⇥⌅ � e�12 k2 cos2 �⇤�r2

�⌅

Monday, June 24, 13


H+ single-ion ACFs

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ion

AC

Ffu

ncti

on

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ion

ACF

func

tion

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

1

kBCi

Correlation time proportional to

At perpendicular to B, periodicity of the H+ ion ACF

is damped by ion-ion collisions. At larger angles is a mixed effect of collisional and

non-collisional damping.

Brownian-motion model captures well all the details of

the ion ACFs.

Simulation results

Monday, June 24, 13


He+ and O+ single-ion ACFs

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ion

AC

Ffu

ncti

on

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ion

ACF

func

tion

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ion

ACF

func

tion

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ion

AC

Ffu

ncti

on

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

In the case of He+, ion resonance is weak, but

there is still dependence on aspect angle.

1

kBCi

Correlation timesare proportional to

In the case of O+, ion resonance is completely destroyed and there is

no dependence onaspect angle.

In both cases, the Brownian motion model captures all the details of the

ion ACFs.

He+ O+

Monday, June 24, 13


Statistics of test-electron displacements in H+, He+, and O+ plasmas and

comparison to the Brownian model

Monday, June 24, 13


Electron displacement distributions

0 2 4 6 8 10−20

2

0

0.1

0.2

0.3

0.4

Time delay [ms]

Electron displacement distributions ! to B

!r!(! )/"!(! )

!r !

pdf

0 2 4 6 8 10−20

2

0

0.1

0.2

0.3

0.4

Time delay [ms]

Electron displacement distributions " to B

!r"(! )/""(! )

!r "

pdf

• Electron distributions look the same for H+, He+, and O+ plasmas.

• In the direction perp. to B, the distribution is approximately Gaussian as function of time delay.

• In the parallel direction, the distribution looks Gaussian at short delays, but becomes narrower within a “collision time”.

• Brownian motion (gaussian displacements) is not a good model for the electron motion.

�ej⇧k·�⇧r

⇥= e�

12 k2 sin2 �⇤�r2

⇥⌅ � e�12 k2 cos2 �⇤�r2

�⌅

Monday, June 24, 13


More on electron displacement distributions• The distribution in the direction

perpendicular to B remains Gaussian for long time delays.

• In the parallel direction, the distribution becomes Gaussian again after a few “collision times”.

• The electron distribution is independent of the ion type.

• For ISR applications around perpendicular to B, the correlation times are of the order of the electron “collision time”, therefore the choice of the collision model does matter to define the shape of the spectrum.

02000

40006000−2

02

0

0.1

0.2

0.3

0.4

Time delay [ms]

Electron displacement distributions ! to B

!r!(! )/"!(! )

!r !

pdf

02000

40006000−2

02

0

0.1

0.2

0.3

0.4

Time delay [ms]

Electron displacement distributions " to B

!r"(! )/""(! )

!r "

pdf

Test electron in a H+ plasma

Monday, June 24, 13


Simulated single-electron ACFs

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ele

ctro

nA

CF

funct

ion

! = 0.0!! = 0.2!! = 0.4!! = 0.6!! = 0.8!! = 1.0!

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ele

ctro

nA

CF

funct

ion

! = 0.00!! = 0.02!! = 0.04!! = 0.06!! = 0.08!! = 0.10!

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [ms]

Ele

ctro

nA

CF

funct

ion

! = 0.000!! = 0.002!! = 0.004!! = 0.006!! = 0.008!! = 0.010!

The simulated electron ACFs are effectively the same despite the plasma configuration (i.e., despite the type of ions to which the electrons collide).Very close to perp. to B (α<0.01o) the electron ACFs have long correlation

times (of the order of a “collision time”), while as the angle increases, correlation times decrease very rapidly (100 times within 1 deg).

0o<α<0.01o 0o<α<0.1o 0o<α<1o

Monday, June 24, 13


Comparison of single-electron ACFs

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time [ms]

(f ) Electron ACF (! = 1 !)

SimulationBrownianNo-collisions

0 100 200 3000

0.2

0.4

0.6

0.8

1

Time [ms]

(a) Electron ACF (! = 0 !)

SimulationBrownian

0 100 200 3000

0.2

0.4

0.6

0.8

1

Time [ms]

(b) Electron ACF (! = 0.01 !)


0 20 40 600

0.2

0.4

0.6

0.8

1

Time [ms]

(c) Electron ACF (! = 0.05 !)


0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Time [ms]

(d) Electron ACF (! = 0.1 !)


0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Time [ms]

(e) Electron ACF (! = 0.5 !)


Simulated electron ACF's for λB=3m at different magnetic aspect angles: (a) α=0°, (b) α=0.01°, (c) α=0.05°, (d) α=0.1°, (e) α=0.5°, and (f) α=1°.

The Brownian motion model

does not capture all the details of the

electron ACFs.

Monday, June 24, 13


Database of single-electron ACFs and Gordeyev integrals

• Before, we built a library of single-electron ACFs (and corresponding Gordeyev integrals).

• The library spans different values of Ne, Te, B, and α.

• As the electron ACFs are independent of the plasma configuration, there is no need to develop another library but to parametrize it in order to use it for ISR applications.

Frequency [kHz]

Asp

ect

angl

e[d

eg]

Electron Gordeyev integral (!B = 3 m)

!1 !0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

Re{

Je}

[dB

]

!60

!50

!40

!30

!20

!10

0

Monday, June 24, 13


Collisional IS Spectrum

Milla & Kudeki [2011]

Monday, June 24, 13


Collisional IS Spectrum

The spectrum becomes super narrow at

perpendicular to B

Jicamarca antenna beam illuminates this range of magnetic aspect angles

Milla & Kudeki [2011]

Monday, June 24, 13


Conclusions

• Coulomb collision effects (as modeled by the Fokker-Planck equation with Spitzer coefficients) on the ion motion can be approximated as a Brownian motion process for H+, He+, and O+ (ionospheric) plasmas.

• In the case of the electrons, Brownian motion does not capture all the details of the electron ACFs because the electron displacement distributions are not Gaussian. The approximation is not appropriate in this case.

• Electron displacement statistics are independent of the plasma configuration, therefore, electron ACFs are the same for H+, He+, and O+ plasmas. We expect the same to happen in multiple-ion component plasmas.

Monday, June 24, 13


Current WorkISR radar experiments and data analysis

• Validation of the collisional ISR spectrum model with standard experiments at Jicamarca.

• Multi-beam radar experiments to measure perpendicular-to-B and off perpendicular ISR data from the topside ionosphere.

• Analysis of radar data and inversion of ionospheric parameters (Densities, temperatures, and drifts).

Monday, June 24, 13

study of the effects of coulomb collisions on h , he and o ... · study of the effects of coulomb...

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