high-frequency kinetic instabilities driven by anisotropic electron beams 20 march 2013 anna kómár...

HIGH-FREQUENCY KINETIC INSTABILITIES DRIVEN BY ANISOTROPIC ELECTRON BEAMS

20 March 2013

Anna Kómár1, Gergő Pokol1, Tünde Fülöp2

1) Department of Nuclear Techniques, Budapest University of Technology and Economics, Association EURATOM

2) Department of Applied Physics, Chalmers University of Technology and Euratom-VR Association

I. Chalmers Meeting on Runaway Electron Modelling

A. Kómár - Kinetic instabilities driven by anisotropic electron beams Chalmers Meeting on Runaway Electron Modelling

Description of the instability Runaway electron distributions Wave dispersions Growth rate of the waves Critical runaway densities Plans, current problems

Outline

20 March 2013

2/29


Runaway distribution

20 March 2013

T. Fülöp, PoP 13(062506), 2006

p = γve/cnormalized momentum

3/29

Anisotropic → Instabilities


20 March 2013

0

ie 1

1212

112

22||

22222

22

11

2

r

rr ckck

2122

22

222

22||

11

ckck

response Imaginar

y part

Growth rate

Particle-wave interaction

Imi

Dispersion of the plasma waves:(homogeneous plasma) Dielectric tensor

Perturbative

analysis

k: wave numberω: wave frequencyc: speed of light

4/29

0

0

Re

Im


Generalresonance condition

Ultrarelativistic resonance condition

Generalizations

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Magnetosonic-whistler

wave

Electron-whistler,

EXEL wave

High electric field

distribution function

Near-critical field

distribution function

5/29


Near-critical distribution

Avalanche distribution

Distribution function

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T. Fülöp, PoP 13(062506), 2006

P. Sandquist, PoP 13(072108), 2006

6/29

p = γve/cnormalized momentum


Distribution function

Qualitatively similar Lower electric field →

less anisotropy

Growth rate of the waves: Ensured by the

anisotropy Not affected by the

details20 March 2013

7/29


Electron plasma waves

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Relaxing the electromagnetic approximation:

Approximations for the dielectric tensor:ciiece mm /

8/29

0

0

0det

2

22

332

2||

2

22

2212

2

2||

122

22||

11

ckckk

ck

ckkck T.H. Stix, Waves in Plasmas

(AIP, 1992)



20 March 2013

Dielectric tensor:

Wave dispersion:

9/29

22

2

11 1ce

pe

22

2

22 1ce

pe

2

2

33 1

pe

22

2

12

ce

cepei

2

122

22

222

22||

11

ckck

2

22

224

422||

2

22

33

ckckkck

0



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10/29

Qualitatively different for B > 2.6 T



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11/29


20 March 2013A. Kómár - Kinetic instabilities driven by anisotropic

electron beams Chalmers Meeting on Runaway Electron Modelling

12/29

Cold plasma approximation is used(th. motion << gyro-motion)

Validity: 20 keV, 10 keV, 1 keV

0,,, 12332211 F

Calculating the growth rate of the waves

20 March 2013A. Kómár - Kinetic instabilities driven by anisotropic

electron beams Chalmers Meeting on Runaway Electron Modelling

13/29

0 Imi

Unperturbed dispersion:Perturbed dispersion:

runawaythGF ,,,, 12332211

The wave frequency changes: →0

Calculating the runaway susceptibilities:

resn

rce

rr ppcpkp

fn

p

fnLpd

||||||

3~Im

1 ,0 ,1n


General resonance condition

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Resonance condition: implicit

0

,

||

0

ck

ppnp perpparce

par

Ultrarelativistic

General caseparp

0||

ck

np ce

res

221 parperp pp

14/29

2

022

||

22220

22||0|| 1

ck

npcknckp

cece

res

T.H. Stix, Waves in Plasmas

(AIP, 1992)


)1( resp

Restrictions on the wave dispersion

20 March 2013

Resonant momentum is physical if:

(1)

(2)

)2( 0resp

0,0|| kck and

0n

Whistler and high-k region of EXEL

15/29

01 22220

22|| cenpck

0,0|| kck and

0n


Growth rate in near-critical field (Whistler)

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Maximum out of the EW region of validity

16/29

E/Ec = 1.3

B = 2 T

ne = 5 ∙ 1019 m-3

nr = 3 ∙ 1017 m-3

0 ,1n

Order of resonance:


Growth rate in near-critical field (EXEL)

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No growth rate for k < 1300 m-1

17/29

E/Ec = 1.3

B = 2 T

ne = 5 ∙ 1019 m-3

nr = 3 ∙ 1017 m-3

0 ,1n

Order of resonance:

k||c = ω0


Growth rate in near-critical field

20 March 2013

Electron-whistler wave Extraordinary electron wave

Near-critical case → max. energy (2.6 MeV, p = 5)18/29

most unstable

wave

E/Ec = 1.3, B = 2 T, ne = 5 ∙ 1019 m-3, nr = 3 ∙ 1017 m-3

102 γ / ωce 102 γ / ωce


Damping rates of the wave, stability

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Collisional damping:

Convective damping: The runaway beam has a finite radius, Lr

19/29

20

320

2/3

42

3

ln5.1

Tee

id

vm

eZn

rv L

k

4

/

M. Brambilla, PoP 2(1094), 1995

G. Pokol, PPCF 50(045003), 2008


Critical density (Whistler)

20 March 2013

Stability:Finding (Growth rate – Damping rates) =

0 (for the most unstable wave)

Critical runaway density

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T = 20 eVT = 1000 eV

Unstable

StableE/Ec = 1.3

ne = 5 ∙ 1019 m-3


Critical density (EXEL)

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Orders of magnitude lower critical density Break at B ~ 2.6 T (for high temperature)

T = 20 eVT = 1000 eV

E/Ec = 1.3

ne = 5 ∙ 1019 m-3


What is different with EXEL?

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22/29

Slightly higher growth rate Orders of magnitude lower convective

damping Collisional damping smoothes this effect

for low T

E/Ec = 1.3, ne = 5 ∙ 1019 m-3, nr = 3 ∙ 1017 m-3



20 March 2013

23/29

E/Ec = 1.3, ne = 5 ∙ 1019 m-3, nr = 3 ∙ 1017 m-3

Parameters of the most unstable wave change from θ ~ 0



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For the parameters of the most unstable wave:

T 5T 2

kk


Wave instability in near-critical electric field Generalization

Relaxing the electromagnetic approximation General resonance condition

Linear stability The most unstable wave is dependent on the

maximum runaway energy Stability threshold is significantly lower for the

EXEL

20 March 2013

Conclusions25/29


Plans, current problems

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26/29

Growth rates for high electric field Avalanche runaway distribution No (or much higher) maximum energy

Whistler waveElectron-whistler

Magnetosonic-whistler

T. Fülöp, PoP 13(062506), 2006

E/Ec = 865

B = 2 T

ne = 5 ∙ 1019 m-3

nr = 3 ∙ 1017 m-3


Extraordinary electron wave Most unstable wave would be in the region of

no-growth rate (if there is a maximum runaway energy)


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27/29

Maximum runaway energy

k||c = ω0This is not

a problem!



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28/29

For k||c > ω0: n ≤ 0

For k||c < ω0: n > 0

2

022

||

22220

22||0|| 1

ck

npcknckp

cece

res

22||

20

2221

ck

np ce

2

022

||

22220

22||0|| 1

ck

npcknckp

cece

res



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29/29

k||c > ω0 k||c < ω0

n = 0, -1

n = 1

Maximum at k||c = ω0: with n ≠ 0

high-frequency kinetic instabilities driven by anisotropic electron beams 20 march 2013 anna kómár...

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