Radiation effects of the runaway electrons · Runaway electron avalanche and distribution in...

Radiation effects of the runaway electronsChang Liu, Lei Shi, Dylan Brennan, Eero Hirvijoki, Amitava BhattacharjeePrinceton University, PPPLAllen BoozerColumbia UniversityCarlos Paz-SoldanGeneral Atomics

58th Annual Meeting of the APS Division of Plasma PhysicsYI2.00003

San Jose, CA Nov 4, 2016

1

Outline•  Overview of runaway electron in plasma

•  Radiation in plasma and feedback to resonant particles

•  Radiation reaction effect on runaway electrons•  Radiation loss from Cherenkov radiation and synchrotron radiation•  Radiation scattering from fluctuation electric field

•  Radiation scattering from normal modes

•  ECE synthetic diagnostic of runaway electrons

•  Summary

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•  Summary

3

Overview of runaway electrons in tokamak•  In tokamak disruptions, runaway electrons (RE) can be generated in the thermal quench.

•  Plasma current transfer to RE current.•  Given the high energy (~25MeV) and significant number (1015 m-3), RE can cause severe damage to the

tokamak wall.

•  To study physics of RE, quiescent runaway electron (QRE) experiments have been conducted.

•  Critical electric field is several times larger than theory predict•  RE distribution in momentum space is non-monotonic.•  Strong radiation effect associated with RE (HXR, ECE, Synchrotron).

R.S. Granetz et al., Phys. Plasmas 21, 072506 (2014).C. Paz-Soldan et al., Physics of Plasmas 21, 022514 (2014).E.M. Hollmann, et al., Phys. Plasmas 22, 56108 (2015).R.J. Zhou, et al, Plasma Phys. Control. Fusion 55, 55006 (2013).

4

Prompt ECE signal growth observed in RE experiments•  REs give remarkable contribution to ECE, given their small population (≲10-4 ne).

•  The contribution to high harmonics is more prompt.•  The growth rate of ECE signal is much higher than HXR growth rate.•  This effect is a result of the radiation scattering of RE.

C. Paz-Soldan, et al., Nucl. Fusion 56, 56010 (2016).

Shot 157209

5

0 10 20 30 40−15

−10

−5

0

p (mc)

log 10

f

Electron distribution function at zero pitch angle

Runaway electron avalanche and distribution in momentum space•  Two major runaway electron generation mechanism

•  Dreicer growth: Slide-away of electron high energy tail•  Secondary generation: Knock-on collision of high energy electrons with thermal electrons (avalanche growth)

•  The RE tail is flat in p, but strongly anisotropic in pitch angle especially for highly relativistic electrons.

Dreicer

Secondary generation

M. Landreman, A. Stahl, and T. Fülöp, Comp. Phys. Comm. 185, 847 (2014).A.H. Boozer, Phys. Plasmas 22, 032504 (2015).

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•  Summary

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Radiation in plasma: interactions of particles and E&M fields

Moving particles generate E&M fields•  Straight line motion: Cherenkov radiation•  Circular motion: Cyclotron (synchrotron) radiation

Moving Particles

E&M waves feedback to moving particles (radiation reaction)•  For single particle, polarization E&M fields give back reaction force (radiation loss)•  Particles can be diffused by E&M fields generated by other resonant particles (radiation scattering)

E&M Fields

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•  Summary

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Radiation reaction electric field on a test electron•  The excited electric field from a single moving test electron in plasma.

•  j is the current formed by the single test electron.•  For electron doing straight line motion

•  The radiation reaction electric field

Ampere's lawFaraday's law

⎫⎬⎪

⎭⎪→ k × k ×E+ ω

c⎛⎝⎜

⎞⎠⎟

2

ε ⋅E = N ⋅E = − 4πiωc2 j

j(x,t) = −evδ (x − x0 − vt)→ j(k,ω ) = −evδ (ω − k ⋅v)exp(ik ⋅x0 )

Ep = d 3k dω E(k,ω )δ (ω − k ⋅v − nω ce /γ )∫∫

N = kk − k2 + ω

c⎛⎝⎜

⎞⎠⎟2

ε

10

•  Electrons moving in magnetized plasma can interact with E&M fields through•  Cherenkov resonance ω-k∥v∥=0 ➛ Cherenkov radiation•  Doppler resonance ω-k∥v∥=nΩe (n>0, Ωe=ωce/γ) ➛ Cyclotron (Synchrotron) radiation•  Anomalous Doppler resonance ω-k∥v∥=-nΩe

•  Cherenkov radiation energy loss gives a correction to lnΛ for relativistic electron

Radiation loss of Cherenkov radiation and synchrotron radiation

Resonance condition

ω − k!v! −nω ce

γ= 0

Unmagnetized: lnΛ→ ln λD

bmin

⎛⎝⎜

⎞⎠⎟+ ln v

vth

⎛⎝⎜

⎞⎠⎟

,

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•  Electrons moving in magnetized plasma can interact with E&M fields through•  Cherenkov resonance ω-k∥v∥=0 ➛ Cherenkov radiation•  Doppler resonance ω-k∥v∥=nΩe (n>0, Ωe=ωce/γ) ➛ Cyclotron (Synchrotron) radiation•  Anomalous Doppler resonance ω-k∥v∥=-nΩe

•  Cherenkov radiation energy loss gives a correction to lnΛ for relativistic electron

•  In DIII-D QRE experiments, for highly relativistic runaway electrons, this correction is about 20%.•  Synchrotron radiation gives strong radiation-reaction force for high energy RE.

•  It can form a bump-on-tail distribution of RE tail (For details, see [C. Paz-Soldan, CO4.00010])

Radiation loss of Cherenkov radiation and synchrotron radiation

D. Pines and D. Bohm, Phys. Rev. 85, 338 (1952).A.A. Solodov and R. Betti, Phys. Plasmas 15, 042707 (2008).D.K. Geller and J.C. Weisheit, Physics of Plasmas 4, 4258 (1997).E. Hirvijoki, I. Pusztai, J. Decker, O. Embréus, A. Stahl, and T. Fülöp, J. Plasma Phys. 81, 475810502 (2015).

Resonance condition

ω − k!v! −nω ce

γ= 0

Magnetized: lnΛ→ ln ρL

bmin

⎛⎝⎜

⎞⎠⎟+ ln v

vth

⎛⎝⎜

⎞⎠⎟+ Δ2, Δ2 = ln

ω pe2 +ω ce

2

ω pe

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Radiation scattering electric field in plasma•  The scattering of RE is determined by fluctuation electric field ⟨ẼẼ⟩ in plasma.•  Similarly to Ep, ⟨ẼẼ⟩ (electric field correlation tensor) can be calculated from test particle current and

Maxwell equations

•  K is the resonant current correlation tensor, calculated from electrons satisfying resonance condition ω-k∥v∥=nΩe .•  ϵH is the Hermitian part of the dielectric tensor. In this work we use cold plasma approximation.•  ϵA is the anti-Hermitian part of the dielectric tensor, including both the resonant particle contribution

and the collisional effect.

⇒ E!E! (k,ω ) = 4πω

c2⎛⎝⎜

⎞⎠⎟2

N −1(k,ω )K(k,ω )(N −1(k,ω ))† k × k ×E+ ω

c⎛⎝⎜

⎞⎠⎟2

ε ⋅E = N ⋅E = − 4πiωc2

j

N = kk − k2 + ω

c⎛⎝⎜

⎞⎠⎟2

ε(k,ω ), ε = ε H + ε A , K(k,ω ) = j! j!

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Radiation scattering from normal modes

•  For normal modes (det N ➛ 0), including both whistler and EXEL waves, electric fluctuation is significant

•  Collision damping gives positive C.•  Additional damping can be caused by spatial diffusion of the waves (not included in the current model).

•  For Cherenkov and Doppler resonance, R[f] is mostly positive.•  For anomalous Doppler resonance (n<0), anisotropicity gives negative R[f]

•  Counter the collision damping and gives abrupt growth of ⟨EE⟩

•  Continuous emission of waves with very little damping•  Exponential growth of normal mode due to fan instabilities

E!E! (k,ω ) = 4πωc2

⎛⎝⎜

⎞⎠⎟2

N −1KN −1† ~K

detN2

E!E! (k,ω ) ~K

ε A2 ~

K[ f R ]C + R[ f R ]

R[ f R ] ~ − d 3pδ (ω − k"v" −nω ce

γ) 1v∂ fe∂p

−ωξ − k"vω pv

∂ fe∂ξ

⎡

⎣⎢

⎤

⎦⎥∫

n=−∞

∞

∑ , ξ = cosθ

13

P. Aleynikov and B. Breizman, Nucl. Fusion 55, 43014 (2015).

Quasi-linear analysis of pitch angle scattering•  In quasi-linear theory, we assume particles’ trajectories are little affected by the perturbation (test-

particle method), and calculate the perturbed 𝛿f.•  Quasi-linear operator (including both electric and Lorentz force)

•  In this work, we only take into account n=0, ±1, and only use the pitch angle scattering part of the operator.

∂ f0∂t

= 12e2 d 3k L̂p2 (1−ξ 2 )δ (ω − k!v! − nΩ)ψ (n,k,ω )

2 L̂f0∫n=−∞

∞

∑

L̂ = 1p

∂∂p

− 1p2(ξ −

k!vω) ∂∂ξ

ψ (n,k,ω ) = 12(Ex + iEy )Jn−1(k⊥ρL )+

12(Ex − iEy )Jn+1(k⊥ρL )+

p!p⊥EzJn (k⊥ρL )

V. L. Yakimenko 1963, C. F. Kennel & F. Engelmann 1966T. H. Stix, Waves in PlasmasV.V. Parail and O.P. Pogutse, Nucl. Fusion 18, 303 (1978).

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RE can be strongly scattered by normal modes•  Scattering from normal modes (plasma waves) can be

300× electron-electron pitch angle scattering.

•  When RE density reaches a threshold, anomalous Doppler resonance leads to abrupt wave scattering at certain p.

•  Wave scattering stops RE going to higher energy regime.

0 10 20 30 400

50

100

150

200

250

300

350

400

p (mc)

Wav

e sc

atte

ring

(nor

mal

ized

to e−e

sca

tterin

g)

RE tail distribution at t=0.5s

E/ECH=9, Te=1keV, ne=1019 m-3, B=1.5T

log10f

15





E/ECH=9, Te=1keV, ne=1019 m-3, B=1.5T


0 10 20 30 400

50

100

150

200

250

300

350

400

p (mc)

Wav

e sc

atte

ring

(nor

mal

ized

to e−e

sca

tterin

g)

log10f

15





E/ECH=9, Te=1keV, ne=1019 m-3, B=1.5T


0 10 20 30 400

50

100

150

200

250

300

350

400

p (mc)

Wav

e sc

atte

ring

(nor

mal

ized

to e−e

sca

tterin

g)

log10f

15



•  Radiation reaction effect on resonant electrons•  Radiation loss from Cherenkov radiation and synchrotron radiation•  Radiation scattering from fluctuation electric field



•  Summary

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ECE synthetic diagnostic of RE•  Reciprocity method: Instead of solving the radiated power P by

collecting waves from all the sources, we solve the reciprocal problem, calculate P using reciprocity theorem.•  Result of the original problem (power received) can be easily

obtained through the reciprocal problem•  Much faster: calculate one wave propagation instead of many

•  Simplifications•  ne and Te are constant, B~1/R.•  ky=kz=0, k is perpendicular to B.•  Only include X-mode, ignoring mode conversion

•  For details, see [Lei Shi et al. YP10.00059] xy

za

𝐵 =𝐵𝑧 

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Abrupt growth of ECE signal is linked to normal mode scattering of RE

3

0 2 4 6 80

1

2

3

4

5

6

t (s)

ECE

T rad (k

eV)

ω=2ωce0

ω=3ωce0

ECE synthetic diagnostic withno normal mode scattering

Shot 157209

18

Abrupt growth of ECE signal is linked to normal mode scattering of RE

•  ECE signals shows abrupt growth, as the normal mode scattering becomes important.•  Abrupt growth mainly comes from normal mode

scattering at intermediate energy regime (1.5<γ<3).

•  Onset of abrupt growth is correctly predicted.

•  The catch-up of higher harmonic signal is reproduced.

•  The growth rate is still smaller compared to experiments.

3

Shot 157209

0 2 4 6 80

1

2

3

4

5

6

t (s)

ECE

T rad (k

eV)

ω=2ωce0

ω=3ωce0

ECE synthetic diagnostic withnormal mode scattering

18

2 2.5 3 3.5 4 4.5 5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

ω (ωce0)

log 10

EC

E T ra

d (keV

)

t=7st=4st=0s

Spectrum of ECE signal becomes flat when RE grows •  For thermal electrons, the radiation power spectrum is like

step-function•  Flat density and temperature profile.

•  Growth of RE makes the spectrum flatter, which is consistent with the experiments.•  RE can have larger values of k⟂ρL, which is important for

high harmonics.

−3 −2 −1 0

−4−2

t−tLM (s)log H

XR (a

.u.)

100 200 300−3

−2

−1

0

1

f (GHz)

T rad (

log10

keV)

Michelson 157209Shot 157209

19



•  Radiation reaction effect on resonant electrons•  Radiation loss from Cherenkov radiation and synchrotron radiation•  Radiation scattering from fluctuation electric field



•  Summary

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Open questions•  What is the best way to stop RE generation in ITER disruption?

•  RE mitigation through MGI and SPI•  Diffuse RE by creating stochastic magnetic field through MHD instability•  Generate negative E field to stop runaway production•  Use fast pitch angle scattering to lower RE growth rate

•  Understand the current results of RE experiments•  Critical electric field and avalanche growth rate•  Synchrotron radiation spectrum and spatial distribution•  Criterion for significant RE generation in disruption•  Current spike at the beginning of thermal quench

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•  Physics of RE in tokamak•  Generation of seed RE in disruption

for avalanche•  Spatial distribution of RE current•  RE interaction with MHD

instabilities and magnetic fluctuations

Summary•  Radiation electric field in plasma can give feedback to resonant particles, through radiation loss effect

and radiation scattering.

•  Radiation electric field can give strong contribution to runaway electron pitch angle scattering.•  Most of the radiation scattering comes from normal modes in plasma.•  The normal mode scattering can be 300 times e-e collisional scattering.

•  Synthetic diagnostic shows that normal modes scattering results in the abrupt ECE signal growth found in experiments.

•  Future work•  Use a more consistent dielectric tensor including thermal correction•  Study the full quasi-linear operator including both p∥ diffusion and pitch angle scattering•  Apply this model to JET/ITER to study RE generation criterion

21

Thanks!

Characteristic time to reach thermal equilibrium

•  The amplitude of electric field fluctuation in thermal equilibrium state with electrons

•  The characteristic time τE to reach a thermal equilibrium is 1/ν, where ν is the damping rate associated with C+R[f]

•  For a Maxwellian distribution, this time τE0 is the maximum of thermal electron collision time (C) and the Landau damping time (R[f]).

•  For runaway electron tail with negative R[f], τE is larger and can be 300 τE0.

•  This time is still smaller than the characteristic time of RE tail growth, which is the collision time of relativistic electrons (~3000 τE0).

E!E! (k,ω ) ~K

ε A2 ~

K[ f R ]C + R[ f R ]

R[ f R ] ~ −δ (ω − k"v" −nω ce

γ) 1v∂ fe∂p

−ωξ − k"vω pv

∂ fe∂ξ

⎡

⎣⎢

⎤

⎦⎥, ξ = cosθ

Inverted-sawtooth of RE ECE signal

E.D. Fredrickson, M.G. Bell, G. Taylor, and S.S. Medley, Nucl. Fusion 55, 13006 (2015).R.J. Zhou, et al., Plasma Phys. Control. Fusion 55, 55006 (2013).

•  Inverted sawtooth of ECE signal in RE experiments has been observed in both TFTR and EAST experiments.

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