AVALANCHE GROWTH OF THE SECONDARY RUNAWAY ELECTRON GENERATIONChang Liu, Dylan Brennan, Amitava BhattacharjeePrinceton University, PPPL
Allen BoozerColumbia University
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July 14th 2015Theory and Simulation of Disruptions Workshop
• In recent dedicated runaway electron experiments on DIII-D with gas puffing during flat-top, a turning point of the runaway electron HXR signal was observed.
• Critical electric field found to be several times larger than Connor-Hastie Ec.
• Mysterious energy loss mechanisms?
Motivation
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R.S. Granetz et al., Phys. Plasmas 21, 072506 (2014).C. Paz-Soldan et al., Physics of Plasmas 21, 022514 (2014).
Caveats of Rosenbluth-Putvinski• Rosenbluth-Putvinski’s theory predicts Ec (Connor-Hastie critical
field) is threshold of secondary generation, and avalanche growth rate (almost) proportional to E/Ec-1.
• Issues with the theory• Calculation of secondary generation is based on simplified source
term that ignores energy and pitch angle distribution of seed electrons.
• Radiation effects (synchrotron, bremsstrahlung) ignored in kinetic model.
• Other kinetic effects (whistler wave, magnetic fluctuation) are also missing.
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S = nr4π lnΛ
δ (ξ −ξ2 )1p2
∂∂p
11− 1+ p2
⎛
⎝⎜
⎞
⎠⎟ ξ2 =
1+ p2 −1p
Outline• Kinetic model of runaway electrons
• Synchrotron radiation reaction force• Deriving source term for secondary RE generation
• Calculate runaway probability function• PDE solving method• Critical electric field for growth
• Avalanche growth simulation• Growth rate calculation• Simulation of gas-puffing case
• Conclusions
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Outline• Kinetic model of runaway electrons
• Synchrotron radiation reaction force• Deriving source term for secondary RE generation
• Calculate runaway probability function• PDE solving method• Critical electric field for growth
• Avalanche growth simulation• Growth rate calculation• Simulation of gas-puffing case
• Conclusions
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Kinetic model of runaway electrons• Collisions, radiation effects, and secondary RE generation
included in the kinetic equation.
• Collision operator gives correct limits for thermal electrons and relativistic electrons.
• Numerical scheme similar to code CODE.
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∂ f∂t
+ E{ f }+C{ f }+ R{ f } = S
E: Parallel electric field driveC: Collision operatorR: Synchrotron radiation reaction forceS: Source term for secondary RE generation
M. Landreman, A. Stahl, and T. Fülöp, Comp. Phys. Comm. 185, 847 (2014).
Synchrotron radiation reaction force
• Synchrotron radiation force is important for high energy electrons (comparable to E field and collisional drag)
• For electrons with γ<100 (most), contribution from the magnetic field curvature is negligible compared to Larmor motion (rg << R0).
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FS =
23remec
2β 2γ sin2θrg2 (1+ p⊥
2 )p⊥ + p⊥2 p!b⎡⎣ ⎤⎦ +
βγ 3
R02 b
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
B. Bernstein and D. C. Baxter, Phys. Fluids 24, 108 1981. A. Stahl, M. Landreman, G. Papp, E. Hollmann, and T. Fülöp, Phys. Plasmas 20, 093302 (2013).
R{ f } = ∇⋅ FS f( )
Deriving source term for secondary generation• We use Møller scattering cross section to get large angle
collision scattering probability for relativistic electrons.• Scattering angle derived from energy and momentum
conservation.
• Source term is integrated from scattering probability and electron distribution function
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S[ f ]= 12π p2
2π pe2∫ dpedξeS(p,ξ; pe,ξe ) f (pe,ξe ).
C. Møller, Ann. Phys. (Berlin), 406, 531 (1932)A.H. Boozer, Phys. Plasmas 22, 032504 (2015).
cosθδ =γ e +1γ e −1
γ −1γ +1
.
Outline• Kinetic model of runaway electrons
• Synchrotron radiation reaction force• Deriving source term for secondary RE generation
• Calculate runaway probability function• PDE solving method• Critical electric field for growth
• Avalanche growth simulation• Growth rate calculation• Simulation of gas-puffing case
• Conclusions
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Calculate runaway probability function• When E/Ec>1, the electron phase space is separated into the
runaway region (electron will run away) and lost region (electron will fall back to the thermal population).
• Two methods to study this phase space structure• Test particle method - truncate the kinetic equation to make it
deterministic, and locate the singular point in phase space.
• Monte-Carlo simulation – Random sampling to obtain runaway probability
• We develop a new method to get runaway probability by solving PDE.
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J.R. Martín-Solís, J.D. Alvarez, R. Sánchez, and B. Esposito, Phys. Plasmas 5, 2370 (1998).I. Fernández-Gómez, J.R. Martín-Solís, and R. Sánchez, Phys. Plasmas 19, 102504 (2012).
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∂∂ξ(1−ξ 2 ) ∂ f
∂ξ= ∂∂ξ
ξ f( ) + ∂2
∂ξ 21−ξ 2
2f⎛
⎝⎜⎞⎠⎟
PDE solving method• Introduce function P representing the runaway probability
• P is found as a solution to a PDE derived from the kinetic equation. Derivation is similar to first passage problem.
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P = 1 at high energy boundaryP = 0 at low energy boundary
v(x) dP(x)dx
+ D(x) d2P(x)dx2
= 0
∂ f∂t
= − ∂∂x
v(x) f[ ]+ ∂2
∂x2D(x) f[ ]
Adjoint equation of Fokker-Planck equation
0.4 0.5 0.6 0.7 0.8 0.9 1 1.10
0.2
0.4
0.6
0.8
1
Run
away
Pro
babi
lity
Test particle methodPDE solved ProbabilityMonte−Carlo Simulation
Results of runaway probability function
• New method gives smooth probability function rather than separatrix.• Overcomes caveats of test particle method (truncation & coordinates
dependence).• Agrees well with Monte-Carlo simulation. (Efficiency is better.)
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Runaway
Lost
Separatrix(Test-particle method)
p (θ chosen as the Rosenbluth-Putvinski source term)
0 5 10 15 201.5
2
2.5
3
3.5
4
4.5
5
5.5
τr/τ
E r/Ec
Z=10Z=5Z=1
Critical electric field for growth• In presence of synchrotron radiation force, if E is below a
threshold Er, transition solution is missing, with only a (almost) uniform solution left.
• Er is the critical electric field for runaway electron growth.
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2
2.5
3
3.54
4.5 5
Z
τ r/τ
2 4 6 8 102
4
6
8
10
12
14
16
18
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P. Aleynikov and B.N. Breizman, Phys. Rev. Lett. 114, 155001 (2015).
Contour of Er
Outline• Kinetic model of runaway electrons
• Synchrotron radiation reaction force• Deriving source term for secondary RE generation
• Calculate runaway probability function• PDE solving method• Critical electric field for growth
• Avalanche growth simulation• Growth rate calculation• Simulation of gas-puffing case
• Conclusions
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0 10 20 30 40 50 60−53
−52
−51
−50
−49
−48
−47
−46
−45
−44
p
log
f
t=0τt=2τt=10τt=20τt=30τt=40τ
(b)
Simulation Result – Avalanche growth
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• With strong radiation, distribution function is non-monotonic.0 10 20 30 40 50 60
−53
−52
−51
−50
−49
−48
−47
−46
−45
−44
p
log
f
t=0τt=2τt=10τt=20τt=30τt=40τ
(a)
• Time-dependent kinetic equation solved using backward Euler.
Z=1τ/τr=0.76 (B=3T, ne=1019m3
∂ f∂t
+ E{ f }+C{ f }+ R{ f } = S{ f }
Distribution function with no radiation Distribution function with Radiation added
1 2 3 4 5 60
0.05
0.1
0.15
0.2
E/Ec
Gro
wth
Rat
e
Rosenbluth−PutvinskiSimulation resultCalculation with Runaway probability
1 2 3 4 5 60
0.05
0.1
0.15
0.2
E/Ec
Gro
wth
Rat
e
Rosenbluth−PutvinskiSimulation ResultCalculation with Runaway probability
Avalanche growth rate
• With synchrotron radiation force added, a new threshold Er>Ec is observed, below which there is no avalanche growth.
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Growth rate with no radiation Growth rate with radiation added
Z=1τ/τr=0.76 (B=3T, ne=1019m3
Simulation of gas-puffing case
0 1 2 3 4 51
2
3
4
t (s)
n e
0 1 2 3 4 50
5
10
E/E c
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0 1 2 3 4 50
0.1
0.2
0.3
n RE
t (s)0 1 2 3 4 5
0
5
10
15
HXR
RE
0 20 40 60 80−8
−7
−6
−5
−4
−3
−2
−1
0
p
log
f
t=0st=0.5st=1st=2st=3st=5s
• Three effects after gas puffing: Dreicer loss (loss of low energy electrons), Radiation loss (loss of high energy electrons) and secondary generation.
• HXR signal turning point reflects redistribution of RE energy• Qualitative agreement with experiment observed. Other loss mechanism not necessary
C. Paz-Soldan et al., Physics of Plasmas 21, 022514 (2014).
Outline• Kinetic model of runaway electrons
• Synchrotron radiation reaction force• Deriving source term for secondary RE generation
• Calculate runaway probability function• PDE solving method• Critical electric field for growth
• Avalanche growth simulation• Growth rate calculation• Simulation of gas-puffing case
• Conclusions
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Conclusions• A PDE solving method is developed to calculated the runaway
probability function.• The method can also identify the critical electric field for
runaway electron growth.• In presence of synchrotron radiation reaction and the pitch angle
scattering, the threshold electric field for avalanche growth increases from Ec to Er, which depends on B and Z.
• Simulation of gas-puffing experiment shows qualitatively agreement with the experimental result.• Synchrotron radiation• Pitch angle scattering Zeff
• Redistribution of the runaway electron energy
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Thank you!
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τ r =
6πε0me3c3
e4B2
τ = 4πε0
2me2c3
e4ne lnΛ
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f (x,t)P(x)dx = const∫0 = ∂ f
∂tP(x)dx∫
= − ∂∂x
v(x) f[ ]+ ∂2
∂x2 D(x) f[ ]⎧⎨⎩
⎫⎬⎭P(x)dx∫
= v(x) dP(x)dx
+ D(x) d2P(x)dx2
⎧⎨⎩
⎫⎬⎭f (x,t)dx∫ + Surface term
v(x) dP(x)dx
+ D(x) d2P(x)dx2 = 0
Theoretical estimation of the growth rate
• If a distribution function is given, the growth rate can be calculated using the runaway probability function.
• If a growth rate is given, an approximate distribution can be obtained from the kinetic equation.
• The solution is thus the stationary point.
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γ = 1nre
S∫ (p,ξ )Q(p,ξ )2π p2dξdp
∂ f∂t
+ E{ f }+C{ f }+ R{ f } = SΓ(p) f + E{ f }+C{ f }+ R{ f }− S{ f } = 0
Next steps• Study other loss mechanisms, including the bremsstrahlung
radiation loss, the magnetic field fluctuation and the whistler wave scattering.
• Study the RE generation and decay for sudden cooling of plasma.
Future work• Couple the kinetic simulation to MHD code.• Collaborate on the future DIII-D experiments to study the critical
electric field for runaway electron growth and the runaway electron energy distribution.
• Develop more complicated synthetic diagnostics simulations and compare the results with the experiments.
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