status report on modeling the isr spectrum perpendicular...

Status report on modeling the ISR spectrum perpendicular to

the magnetic field

Marco Milla & Erhan KudekiElectrical and Computer Engineering

University of Illinois at Urbana-Champaign

June 27, 2007

Jicamarca ISR measurements perp. to B

• Phasing Jicamarca antenna perp. to B, we can measure:

- Drifts:

- Old days, using the phase of the pulse-to-pulse correlation (Woodman & Hagfors, 1969).

- Modern times, using Kudeki et al (1999) spectral technique (Doppler shift of ISR signal).

- Densities: depolarization effects in the ISR signal as it propagates in the ionosphere (Kudeki et al, 2003).

PropagationEffect

Incoherentscatter process

MagneticField lines

What happened with temperatures?

Jicamarca ISR spectrum perp. to B

!150 !100 !50 0 50 100 15021

22

23

24

25

26

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

Doppler velocity (m/s)

@oAe

r Bpe

ctrCm

(dB)

280F00 km300F00 km320F00 km340F00 km

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

Doppler velocity (m/s)

Ran

ge (k

m)

!150 !100 !50 0 50 100 150

200

300

400

500

600

700

800

dB15

20

25

30

The doppler shift of the spectrum is a direct measurement of the drifts. The width should give us

information of the temperatures.

The actual spectrum was narrower than what the non-collisional theory predicted. Thus, something was missing.

Kudeki et al (1999) analyzed the spectrum using non-collisional IS

theory. But, the temperatures that he obtained were about half of

what is expected.

The effect of electron Coulomb collisions

0 500 10000

1

2

3

4

5

6

7

8 x 10!4

Frequency (Hz)

Spe

ctra

l Den

sity

! = 1o

w colw/o col

0 100 200 3000

1

2

3

4

5

6 x 10!3

Frequency (Hz)

! = 0.1o

w colw/o col

0 10 20 300

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency (Hz)

! = 0.01o

w colw/o col! = 0o • Sulzer & Gonzalez (1999)

showed that Coulomb collisions are responsible for the narrowing of the ISR spectrum as the radar beam approaches perp. to B.

• However, their model was not able to consider exact perpendicularity (α > 0.125o).

Here our work started. We have extended the approach of S&G and considered the effect of collisions at all aspect angles including exact

perpendicularity (α = 0.0o).

How do we include the effect of collisions?

• The standard theory of incoherent scatter formulates the spectrum of the ISR signal in terms of the so-called Gordeyev integrals

• The Gordeyev integrals can be interpreted as the one sided Fourier transform of the correlation of the signal scattered by a single particle in a plasma where collective interactions have been neglected.

• Thus, if we know the trajectories of the particles in this fictitious plasma, we should be able to calculate the single particle ACF and also their corresponding Gordeyev integrals. The effect of collisions is considered in the modeling of the particle trajectories.

!|ne(!,"k)|2"Ne

=|j(k2h2

e + µ) + µ#iJ(#i)|2

|j(k2h2e + 1 + µ) + #eJ(#e) + µ#iJ(#i)|2

2ReJ(#e)#2kCe

+|j + #eJ(#e)|2

|j(k2h2e + 1 + µ) + #eJ(#e) + µ#iJ(#i)|2

2ReJ(#i)#2kCi

• Generalized Langevin equation:

• Link with plasma kinetic theory:

• Either of these equations describe the same process. We prefer Langevin approach because it gives us more insight into the physics of the problem.

Modeling of electron and ion trajectories

!fs

!t+ "v ·!rfs +

qs

ms("v " "B) ·!vfs =

!#fs

#t

"

coll!

!fs

!t

"

coll

= !"v#!"v

!t$sfs +

12"v"v :

##!"v!"v

!t$sfs

$

d!v(t)dt

! q

m!v(t)" !B = !"(v)!v(t) +

!D!(v)

2!1(t)v!1(t) +

!D!(v)

2!2(t)v!2(t) +

"D"(v)!3(t)v"(t)

Friction and diffusion coefficients

• These friction and diffusion functions were first calculated by Chandrasekhar (40's).

• Next, Rosenbluth, Spitzer and others (60's) re-solve the problem for plasma physics,

• S&G (1999) used the same coefficients in their simulations, however, they neglected the effect of diffusion across B.

!!v!

!t" = #ADl2s

!

1 +me

ms

"

G(lsv)

!(!v!)

2

!t" =

AD

vG(lsv)

!(!v")2

!t" =

AD

v(!(lsv) # G(lsv))

0 1 2 3 4 5 6 7 8!1

0

1

2

3

4

5

6x 10!3

Velocity (v/Ce)

Veloc

ity in

creme

nt (!v

/Ce)

Diffusion coefficients ! Electron!electron collisions!T = 1e!08 s Ce = 1.23e+05

"!vpar#

"|!vpar2 |#1/2

"|!vper2 |#1/2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1!0.4

!0.35

!0.3

!0.25

!0.2

!0.15

!0.1

!0.05

0

0.05

0.1

Velocity (v/Ce)

Veloc

ity in

creme

nt (!v

/Ce)

Diffusion coefficients ! Electron!ion collisions!T = 1e!08 s Ce = 1.23e+05

"!vpar#

"|!vpar2 |#1/2

"|!vper2 |#1/2

Computer simulations

• Assumption: the magnetic field is weak enough such that within a Debye cube the trajectories of electrons and ions have a very small curvature (caused by the magnetic field). Under this condition, the Rosenbluth/Spitzer velocity dependent coefficients are valid.

• Then, we simulate the motion of the particle in the plasma using

for the velocities, and the trajectories are computed by numerical integration of the velocity series.

• For a given plasma configuration, the simulations run for several hours to to obtain good statistics of the particle trajectories.

• Using these trajectories, we have computed the pdf (histograms) of the displacements and also the Gordeyev integrals.

!vi+1 = !vi +q

m!vi × !B!t− "(vi)!vi!t +

!!tD!(vi)

2n1vi,!1 +

!!tD!(vi)

2n2vi,!2 +

"!tD"(vi)n3vi,"

Statistics of ion trajectories

• Velocity probability distributions have a gaussian shape.

• PDF of the displacement in the direction perpendicular to B is gaussian as function of delay τ.

• In the parallel direction, the PDF also looks gaussian.

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

0 2 4 6 8 10

!20

2

0

0.1

0.2

0.3

0.4

Time (ms)

Perpendicular displacement distribution

!r"(t) / #

"(t)

0 2 4 6 8 10

!20

2

0

0.1

0.2

0.3

0.4

Time (ms)

Parallel displacement distribution

!r||(t) / #||(t)

Statistics of electron trajectories

• Velocity probability distributions have also a gaussian shape.

• PDF of the displacement in the direction perp. to B is almost gaussian as function of delay τ.

• In the parallel direction, the PDF looks gaussian at short time, but rapidly becomes narrow.

• The Brownian motion approach that assumes constant friction and diffusion coefficients is not good enough.

0 2 4 6 8 10

!20

2

0

0.1

0.2

0.3

0.4

Time (ms)

Perpendicular displacement distribution

!r"(t) / #

"(t)

0 2 4 6 8 10

!20

2

0

0.1

0.2

0.3

0.4

Time (ms)

Parallel displacement distribution

!r||(t) / #||(t)

Database of electron Gordeyev integrals

Frequency (Hz)

Aspect angle

(de

g)

Electron Gordeyev integral (Ne=1E12m!3, Te=1000K, !B=3m)

!1500 !1000 !500 0 500 1000 15000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ReJ

e(f

) (

dB

)

!20

!10

0

10

20

30

40 • We have built a library for O+ ions that considers

- 600K < Te, Ti < 2000K (each 200K)

- |B| = 20, 25, 30 μT

- Ne = 1E11, 1E12, 1E13 m3

- Large set of aspect angles from 0o to 90o.

• A web-page with the results is under construction.

• DVD experiment: north and south quadrants of Jicamarca antenna are phased to point perp to B.

• Beam-width: ~1 deg.

• The measured spectrum is the sum of signal coming from different magnetic aspect angles.

Main-lobe

Locus ofPerpendicularity

X-Pol (S-E)

Perpendicular to B

Parallel to B

0 50 100 150 200

10!4

10!3

10!2

10!1

're*uen-. (12)

4opp

ler Sp

e-tru

m [lo

g!s-

ale]

ISR spe-tra ! Be = 1000 Bi = 1000

Radarequation:

Application: Beam-weighted ISR spectrum

S!(ω, τ) ∝! !

drdΩ|G(r)|2

r2

1T

"

n

S(−2kr,ω − 2πn

T,$r)

####χ(τ − 2r

c,−ω − 2πn

T)####2

Magnetic field is obtained from the IGRF 2005 model.

!200 !150 !100 !50 0 50 100 150 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity (m/s)

Dopp

ler S

pectr

um

ISR beam!weighted spectra ! Te = 1800 Ti = 1000

Aliasing & windowingNo aliasiang

!200 !150 !100 !50 0 50 100 150 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity (m/s)

Dopp

ler S

pectr

um

ISR beam!weighted spectra ! Te = 1000 Ti = 1000

Aliasing & windowingNo aliasiang

Some preliminary results

• The beam-weighted spectrum is aliased.

• The spectral width is roughly proportional to thermal velocity squared and collision frequency

• As Te/Ti increases the aliasing also increases for the same density.

k2BC2

e!e

!2e + !2

e

! Ne"Te

Aliasing

AliasingWidth

narrows

More preliminary results

• Does the model fit the data?

• Using reasonable temperatures (from IRI model). We have computed our modeled spectrum and plotted it against the data.

• The agreement is good! Although it is not perfect yet.

• The beam-weighted spectrum need to be calculated more precisely (this will sharp the peak).

• In addition, magneto-ionic effects probably are needed to account for the reduction in the floor of the spectrum.

!200 !150 !100 !50 0 50 100 150 2000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity (m/s)

Dopp

ler S

pectr

um

ISR Spectra ! 2005!09!17 12:00:00 ! Height = 360km

North ! Z!polNorth ! M!polSouth ! Z!polSouth ! M!polTe = 900K Ti = 800K

!200 !150 !100 !50 0 50 100 150 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity (m/s)

Dopp

ler S

pectr

um

ISR Spectra ! 2005!09!17 08:00:00 ! Height = 360km

North ! Z!polNorth ! M!polSouth ! Z!polSouth ! M!polTe = 1300K Ti = 850 K

Current and Future work

• The aliasing is less strong in the north-south cross-spectrum data.

• Thus, its shape is dominated by the signal coming at exact perp to B.

• Using this data together with the spectrum, we expect to be able to obtain Te and Ti.

• Since densities are needed, we will combine this fitting together with the differential phase technique.

• DP will provide densities and the spectral fitting the temperatures that will be used again by DP to correct for Te/Ti and so on.

!200 !150 !100 !50 0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity (m/s)

Cro

ss S

pect

rum

ISR Cross!Spectra ! 2005!09!17 12:00:00 ! Height = 360km

North x South (Z!pol)North x South (M!pol)

status report on modeling the isr spectrum perpendicular...

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