Controlling Runaway Vortex Via Externally Injected High … Session 2... · 2017. 7. 18. ·...
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Controlling Runaway Vortex Via Externally Injected High-Frequency Electromagnetic Waves Zehua Guo, Chris McDevitt, Xianzhu Tang Theoretical Division, Los Alamos National Laboratory Work supported by DOE Office of FES and ASCR
Transcript of Controlling Runaway Vortex Via Externally Injected High … Session 2... · 2017. 7. 18. ·...
Controlling Runaway Vortex Via Externally
Injected High-Frequency Electromagnetic Waves
Zehua Guo, Chris McDevitt, Xianzhu Tang
Theoretical Division, Los Alamos National Laboratory
Work supported by DOE Office of FES and ASCR
Motivation
Damage of runaways to plasma-facing components is determined by runawaycurrent (jRE ≈ enREc) and runaway energy E = γmc2.
For given jRE (essentially nRE), higher runaway energy (larger γ) leads to◮ higher heat load (unit area) → melting, ablation, etc◮ deeper heat load deposition (longer stopping range) → unexpected subsurface
damage.
(a) Be melting by RE in JET (G. Matthews 2016 Phys. Scr.)
runaway electrons 2/23
Mitigation strategy
Can we limit RE energy under a few MeV to control the extent of damage?◮ Physics basis: RE energy is determined by runaway vortex in momentum space
∼ E2B−2n−1. [Guo, McDevitt, Tang, PPCF (2017)]◮ Approach: Phase-space engineering of runaway electrons by manipulating the
runaway vortex.⋆ This talk focuses on externally injected high-frequency electromagnetic waves
runaway electrons 3/23
Runaway vortex is a robust feature in momentumspace
The Fokker-Planck equation can be written in a conservative form
∂f
∂(t/τc)+
1
p2∂
∂pp2Γp +
1
p
∂
∂ξΓξ = 0 ,
p normalized to mec, define α ≡ τc/τs = (2/3)ǫ0B2/neme lnΛ
Γp =
[
(−ξ)E−1 + p2
p2− αpγ(1− ξ2)
]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf+
1+ Z
2
γ
p2∂f
∂ξ
)
6 7 8 9 10 11 12 13 14 15p
−1.00
−0.98
−0.96
−0.94
−0.92
−0.90
ξ O point
Γp =0
Γξ =0
A runaway vortex forms around the O point where Γp = Γξ = 0.
runaway electrons 4/23
Runaway vortex → runaway energy distribution
5 10 15 20 25 30 35 40p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20p/mc
−0.9
−0.8
−0.7
−0.6
ξ
phase space flux(E=3.0)
E = 3.0, α = 0.2, Z = 1. The red curve corresponds to zero energy flux. Resultscomputed with LAPS-RFP code (LANL).
The spread of RE distribution in both energy and pitch-angle is associatedwith the phase-space vortex;
A separatrix (X point) in phase-space separates RE into local and globalcirculating populations;
Topological change (runaway vortex is present or not) is crucial for REavalanche growth [Talk by C. Mcdevitt, this morning];
runaway electrons 5/23
O/X merging → avalanche threshold
At pO ≫√
E/α, an expression is derived for O point’s energy [Guo-PPCF17]
pO =√2(E + α)(E − 1)
(1 + Z)α,
The energy of X point can also be modeled by
pX =
√
1 + (1 + Z)/2√2
E.
1.5 2.0 2.5 3.0 3.5 4.0 4.5E/Ec
0
10
20
30
40
50
60p/m
cO point
X point
O model-full
O model-simp
X model-full
X model-simp
Avalanche threshold electric field is determined by the merging of O and Xpoints. (McDevitt talk this morning)
runaway electrons 6/23
Z increases → pO decreases & pX increases
At pO ≫√
E/α, an expression is derived for O point’s energy [Guo-PPCF17]
pO =√2(E + α)(E − 1)
(1 + Z)α,
The energy of X point can also be modeled by
pX =
√
1 + (1 + Z)/2√2
E.
Increasing Z can effectivelydecrease the O point energy andthe volume of the runaway vortex;
◮ bound electrons → Ec andsecondary source.
◮ partial screening → highereffective Z of impurities →enhanced scattering
Talk by Fulop this morning0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1/(1+Z)
0
5
10
15
20
25
30
p/m
c
O point
X point
peak
spread
runaway electrons 6/23
Reduce pO by enhancing pitch-angle diffusionRunaway vortex forms around an O point where Γp = Γξ = 0.
Approximate pitch-angle diffusion: ∂ ln f∂ξ
∼ (√2∆ξp)
−1 with Γp(∆ξp) = 0
Γp =
[
(−ξ)E − 1 + p2
p2− αpγ(1− ξ2)
]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf+
1 + Z
2
γ
p2∂f
∂ξ
)
5 10 15 20 25 30p/mec
−1.00
−0.95
−0.90
−0.85
−0.80
ξ
Γp =0
Γξ =0(Z=1)
Γξ =0(Z=3)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01/(1+Z)
0
5
10
15
20
25
30
p/m
c
O point
X point
peak
spread
runaway electrons 7/23
Enhance scattering via wave-particle interactionRunaway vortex forms around an O point where Γp = Γξ = 0.
Approximate pitch-angle diffusion: ∂ ln f∂ξ
∼ (√2∆ξp)
−1 with Γp(∆ξp) = 0
Γp =
[
(−ξ)E − 1 + p2
p2− αpγ(1− ξ2)
]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf+
1 + Z
2
γ
p2∂f
∂ξ+Dξξ
∂f
∂ξ
)
5 10 15 20 25 30p/mec
−1.00
−0.95
−0.90
−0.85
−0.80
ξ
Γp =0
Γξ =0(Z=1)
Γξ =0(Z=3)Whistler wave is a good candidatefor pitch-angle scattering ofrelativistic electrons [Lyon JPP1971;Guo PRL2012]
We can use it to control runawayenergy!?
runaway electrons 7/23
Capture wave-particle interaction with quasilineardiffusion
In general, the relativistic quasilinear diffusion of electromagnetic waves canbe written as [Lerche POF68, Lyons JPP71]
C(f) =1
p2∂
∂pp2
[
Dpp∂f
∂p+
√
1− ξ2
pDpξ
∂f
∂ξ
]
+1
p
∂
∂ξ
[
√
1− ξ2Dξp
∂f
∂p+
1− ξ2
pDξξ
∂f
∂ξ
]
The diffusion coefficients (Dpξ = Dξp) are
Dpp =2ω2
peτc
ωce
(1− ξ2)n=∞∑
n=−∞
∫
d3kEn,kδ(ωkr − k‖v‖ +n
γ)
Dpξ = −2ω2
peτc
ωce
√
1− ξ2n=∞∑
n=−∞
∫
d3kEn,kδ(ωkr − k‖v‖ +n
γ)(ξ −
k‖v
ωkr
)
Dξξ =2ω2
peτc
ωce
n=∞∑
n=−∞
∫
d3kEn,kδ(ωkr − k‖v‖ +n
γ)(ξ −
k‖v
ωkr
)2
◮ ω normalized to ωce, k to ωce/c, t to τc;
◮ For ITER-like plasma, ωpeτc ∼ 1010 and ωpe ∼ ωce
◮ The perturbation energy density normalized to electron inertial energy density
En,k =1
V (2π)3|Ψn,k|2
8πn0mec2ω3ce
c3.
for δBn,k ∼ 10−5B0, En,k ∼ 10−10
runaway electrons 8/23
Whistler wave dispersion and resonance conditionWhistler waves are effective in inducing pitch-angle scatterings of electrons inspace [Kennel& Engelmann PF66] and laboratory [Guo PRL12] plasmas;
For ωci ≪ ω ≪ ωce, ωce|k‖/k| ≫ ω, a simple dispersion for whistler wave
ωk = k|k‖|v2Ac2
ωce
ωci
◮v2
A
c2ωce
ωci∼ 1 (ratio between magnetic and electron inertial energy densities) for
ITER plasma;
Two primary cyclotron resonances:
◮ n = −1, normal Doppler resonance [Stix WIP83, Davidson BPP92]
ωkr − k‖v‖ − γ−1 = 0
◮ n = 1, anomalous Doppler resonance [Fulop POP06, Aleynikov NF15]
ωkr − k‖v‖ + γ−1 = 0
Assume a spectrum with a fixed k⊥ and a narrow band of k‖, so that
E±1,k = E0
δ(k⊥ − k⊥0)
(√2π)3k⊥∆k
exp
[
− (k‖ − k‖0)2
2∆k2
]
where∫
d3kE±,k = E0, and only consider the primary harmonics n = ±1;
runaway electrons 9/23
Chopping the runaway vortex
No wave, E = 3.0Ec, α = 0.2, vt = 0.1c, a long high-energy tail forms;
5 10 15 20 25 30 35 40p(mc)
−1.0
−0.5
0.0
0.5
ξdistribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20 25 30 35 40p/mc
−20
−15
−10
−5
0 fv
ξ= − 1average
5 10 15 20 25p/mc
−0.9
−0.8
−0.7
−0.6
−0.5
ξ
phase space flux(E=3.0)
runaway electrons 10/23
Chopping the runaway vortex
No wave, E = 3.0Ec, α = 0.2, vt = 0.1c, a long high-energy tail forms;
5 10 15 20 25 30 35 40p(mc)
−1.0
−0.5
0.0
0.5
ξdistribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20 25 30 35 40p/mc
−20
−15
−10
−5
0 fv
ξ= − 1average
5 10 15 20 25p/mc
−0.9
−0.8
−0.7
−0.6
−0.5
ξ
phase space flux(E=3.0)
Idea: locate the resonance to theleft of O point & at low enoughenergy → chopping the vortex tokill high-energy runaways
runaway electrons 10/23
Normal Doppler resonance is preferredNormal Doppler resonance, k‖0 = 0.2, k⊥0 = 0.1, E0 = 10−9,∆k = 0.05
0 2 4 6 8 10p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
log10Dpp(p, ξ)
0.00
0.38
0.75
1.12
1.50
1.88
2.25
2.62
3.00
0 2 4 6 8 10p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
log10Dpξ(p, ξ)
0.00
0.38
0.75
1.12
1.50
1.88
2.25
2.62
3.00
0 2 4 6 8 10p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
log10Dξξ(p, ξ)
0.00
0.38
0.75
1.12
1.50
1.88
2.25
2.62
3.00
Anomalous Doppler resonance, k‖0 = −0.2, k⊥0 = 0.1, E0 = 10−9,∆k = 0.05
5 10 15 20 25p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
log10Dpp(p, ξ)
0.00
0.38
0.75
1.12
1.50
1.88
2.25
2.62
3.00
5 10 15 20 25p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
log10Dpξ(p, ξ)
0.00
0.38
0.75
1.12
1.50
1.88
2.25
2.62
3.00
5 10 15 20 25p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
log10Dξξ(p, ξ)
0.00
0.38
0.75
1.12
1.50
1.88
2.25
2.62
3.00
The normal Doppler resonance tends to peak at lower energy comparing tothe anomalous Doppler resonance, so the former is preferred for our purpose;
runaway electrons 11/23
Whistler waves can limit the RE energy!
With wave, k‖0 = 0.2, k⊥0 = 0.1,∆k = 0.05, E0 = 5× 10−10;
5 10 15 20
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20
p/mc
−20
−15
−10
−5
0fv
ξ= − 1average
2 4 6 8 10 12 14 16p/mc
−0.9
−0.8
−0.7
−0.6
−0.5
ξ
phase space flux(E=3.0)
The long high-energy runawaytail above ∼ 2MeV is removedby the resonance;
runaway electrons 12/23
Whistler waves can limit the RE energy!
With wave, k‖0 = 0.2, k⊥0 = 0.1,∆k = 0.05, E0 = 5× 10−10;
5 10 15 20
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20
p/mc
−20
−15
−10
−5
0fv
ξ= − 1average
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
The long high-energy runawaytail above ∼ 2MeV is removedby the resonance;
A new vortex is created at lowerenergy below the center ofresonance due to whistler waves;
runaway electrons 12/23
Reduce resonance energy → the new vortex shrinks
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
Pushing the resonance to lower energy (k‖0 = 0.3, 0.4, 0.5, 0.6) → lowerenergy/higher pitch O point & vortex volume decreases;The X point moves to higher pitch quickly once the resonance is close to it;The new vortex eventually disappears when O/X points merge;
runaway electrons 13/23
Reduce resonance energy → the new vortex shrinks
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
Pushing the resonance to lower energy (k‖0 = 0.3, 0.4, 0.5, 0.6) → lowerenergy/higher pitch O point & vortex volume decreases;The X point moves to higher pitch quickly once the resonance is close to it;The new vortex eventually disappears when O/X points merge;
runaway electrons 13/23
Resonance energy controls the RE distribution
5 10 15 20 25 30 35
p(mc)
−1.0
−0.5
0.0
0.5ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20 25 30 35
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20 25 30 35
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20 25 30 35
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
Runway electrons start to accumulate in the original vortex again as the newvortex’s volume decreases; Don’t overdo it!
runaway electrons 14/23
Dependence on magnetic field strengthThe synchrotron damping parameter α ∝ B2n−1
Scan α = 0.2, 0.1, 0.05, k‖0 = 0.2, k⊥0 = 0.1, E0 = 5× 10−10,∆k = 0.05
5 10 15 20
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
5 10 15 20
p(mc)
−1.0
−0.5
0.0
0.5
ξ
distribution function
−20.0
−18.5
−17.0
−15.5
−14.0
−12.5
−11.0
−9.5
−8.0
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5 6 7 8
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
2 4 6 8 10
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
Since the synchrotron damping is proportional to αpγ(1− ξ2), the O pointenergy only increases weakly with
√α;
runaway electrons 15/23
Chopping the vortex during avalanche growthUsing chiu’s avalanche source [Chiu NF98], snapshot at Jre ≃ 2.5MA;Unmitigated avalanche shows a long high-energy tail;
5 10 15 20 25 30 35 40
p/mc
−1.0
−0.5
0.0
0.5
ξ
distribution function
−18.00
−16.38
−14.75
−13.12
−11.50
−9.88
−8.25
−6.62
−5.00
0 50 100 150 200t/τc
10-1210-1110-1010-910-810-710-610-510-410-310-210-1100101 ne(t)−ne0
2 4 6 8 10 12 14p/mc
−1.0
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
ξ
phase space flux (E=3.0)
5 10 15 20 25 30 35 40
p/mc
−20
−15
−10
−5
0fv
runaway electrons 16/23
The mechanism works during avalanche growth!Whistler wave, k‖0 = 0.2, k⊥0 = 0.1,∆k = 0.05, E0 = 5× 10−10;
5 10 15 20
p/mc
−1.0
−0.5
0.0
0.5
ξ
distribution function
−18.00
−16.38
−14.75
−13.12
−11.50
−9.88
−8.25
−6.62
−5.00
5 10 15 20
p/mc
−20
−15
−10
−5
0fv
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux (E=3.0)
A new vortex is created at lowerenergy below the center ofresonance due to whistler waves;
Avalanche electrons can belimited to lower energy by waveinjections;
runaway electrons 17/23
Reduce resonance energy → the new vortex shrinks
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux (E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux (E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux (E=3.0)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux (E=3.0)
Pushing the resonance to lower energy (k‖0 = 0.3, 0.4, 0.5, 0.6) → lowerenergy/higher pitch O point, vortex volume decreases;Runway electrons start to accumulate in the original vortex again as the newvortex’s volume decreases; Don’t overdo it!
runaway electrons 18/23
Resonance energy controls the avalanche distribution
5 10 15 20
p/mc
−1.0
−0.5
0.0
0.5
ξ
distribution function
−18.00
−16.38
−14.75
−13.12
−11.50
−9.88
−8.25
−6.62
−5.00
5 10 15 20 25 30 35
p/mc
−1.0
−0.5
0.0
0.5
ξ
distribution function
−18.00
−16.38
−14.75
−13.12
−11.50
−9.88
−8.25
−6.62
−5.00
5 10 15 20 25 30 35
p/mc
−1.0
−0.5
0.0
0.5
ξ
distribution function
−18.00
−16.38
−14.75
−13.12
−11.50
−9.88
−8.25
−6.62
−5.00
5 10 15 20 25 30 35
p/mc
−1.0
−0.5
0.0
0.5
ξ
distribution function
−18.00
−16.38
−14.75
−13.12
−11.50
−9.88
−8.25
−6.62
−5.00
Pushing the resonance to lower energy (k‖0 = 0.3, 0.4, 0.5, 0.6) → lowerenergy/higher pitch O point, vortex volume decreases;Runway electrons start to accumulate in the original vortex again as the newvortex’s volume decreases; Do not overdo it!
runaway electrons 19/23
The avalanche growth rate is affected
The low energy vortex is still present during avalanche, and keeps a largepopulation of runaways at relatively low energy ( MeV );
The growth rate reaches a minimum near a particular k‖0 = 0.3 with givenparameters;
The whistler waves can be applied to lift the avalanche threshold electric field;
The vortex appears at larger pitch which may be favorable in tokamakgeometry where trap electrons can not runaway;
0 20 40 60 80 100 120 140 160t/τc
−7
−6
−5
−4
−3
−2
−1log10|n−n0|
w0 =0
k ∥ 0 =0. 2
k ∥ 0 =0. 3
k ∥ 0 =0. 4
k ∥ 0 =0. 5
0.100.150.200.250.300.350.400.450.50k ∥ 0
0.5
0.6
0.7
0.8
0.9
1.0γ(k)/γ0
runaway electrons 20/23
Conclusion
The vortex is a local circulation of runaway electrons in momentum-space,which results from the competition between the parallel electric fieldacceleration, the Coulomb collision and the radiational damping force.
The vortex determines the energy distribution of runaway electrons, andgoverns the threshold of avalanche growth;
Pitch-angle diffusion due to Coulomb collisions alone becomes ineffective atrelativistic energy > MeV , and thus the vortex extends to tens of MeVeven with moderate electric fields;
A narrow spectrum of small amplitude whistler waves (δB < 10−4B0) caneffectively enhance the momentum scatterings in the MeV energy rangethrough the normal Doppler cyclotron resonance. A new vortex forms belowthe peak of resonant energy ∼ MeV ;
By properly choosing the wave (thus the resonance), the runaway electronscan be limited to below a few MeV in the presence of avalanche growth;
Future work: the new vortex also appears at higher pitch, so the toroidaleffect (finite trap-region) becomes very important;
runaway electrons 21/23
Enhanced diffusion ⇒ local “increase” of ZFor simplicity, we can consider the pitch-angle diffusion alone;
The O point is where Γp = 0 and Γξ = 0 [Guo PPCF17]
Γp =[
(−ξ)E −CF − αpγ(1− ξ2)]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf +
1 + Z
2
γ
p2∂f
∂ξ+Dξξ
∂f
∂ξ
)
1 2 3 4 5p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
The X point is not affected as long asthe resonance is away from it;
The wave induced pitch-anglediffusion corresponds to a localincrease of Z near the resonance(1 →∼ 6.5);
runaway electrons 22/23
Energy/cross diffusions modify the O point location
Energy and pitch-angle fluxes [Guo PPCF17]
Γp =[
(−ξ)E − CF − αpγ(1− ξ2)]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf+
1 + Z
2
γ
p2∂f
∂ξ+Dξξ
∂f
∂ξ
)
From left to right: only Dξξ, Dξξ + Dpp, Dξξ + Dpp + Dpξ
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
Both energy and cross diffusions introduce additional energy flux, so the Opoint moves slightly to larger energy and higher pitch;
runaway electrons 23/23
Energy/cross diffusions modify the O point location
Energy and pitch-angle fluxes [Guo PPCF17]
Γp =
[
E(−ξ)− CF − αp√
1 + p2(1− ξ2)−Dpp∂ ln f
∂p
]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf+
1 + Z
2
γ
p2∂f
∂ξ+Dξξ
∂f
∂ξ
)
From left to right: only Dξξ, Dξξ + Dpp, Dξξ + Dpp + Dpξ
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
Both energy and cross diffusions introduce additional energy flux, so the Opoint moves slightly to larger energy and higher pitch;
runaway electrons 23/23
Energy/cross diffusions modify the O point location
Energy and pitch-angle fluxes [Guo PPCF17]
Γp =
[
E(−ξ)−CF − αp√
1 + p2(1− ξ2)−Dpp∂ ln f
∂p−Dpξ
∂ ln f
∂ξ
]
f
Γξ = −(1− ξ2)
(
Ef − αp
γξf+
1 + Z
2
γ
p2∂f
∂ξ+ Dξξ
∂f
∂ξ+Dξp
∂f
∂p
)
From left to right: only Dξξ, Dξξ + Dpp, Dξξ + Dpp + Dpξ
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
1 2 3 4 5
p/mc
−1.0
−0.8
−0.6
−0.4
−0.2
ξ
phase space flux(E=3.0)
Both energy and cross diffusions introduce additional energy flux, so the Opoint moves slightly to larger energy and higher pitch;
runaway electrons 23/23
