Enabling Research Project Tünde Fülöp and Ola Embréus, Eero Hirvijoki and Adam Stahl
Chalmers University of Technology
March 6, 2015
1
Ion runaway: Motivation
Low-mode number TAEs observed?
f ≈ 60 -260 kHz Resonant velocity:
vA /3 < vT e Runaway of ions responsible?
[Fülöp & Newton, PoP 21, 080702 (2014)]
2
Ion runaway: Intro
Largely analogous to electron
1
Non-monotonic friction
2
Non-monotonic distribution!
CODION:
F//
runaway, but:
eE
an extension of CODE to ions vc1
vc2
3
Ion runaway: Results
10
Initial dist. E=200 V/m 220 V/m 240 V/m 260 V/m
-1
fD / n D
10-2
10
-3
10 -4
10
-5
0
2
4
6
8
10
v|| / vTD
Typical runaway ion distribution
Bump energy too low to drive
exhibiting a large high-energy
Alfvénic instabilities:
bump.
vA /3 ∼ 30 − 50 vD 4
Guiding-center radiation-reaction force Lorentz-Abraham-Dirac force
The Lorentz-Abraham-Dirac force 3γ 2 γ2 3γ 2 2 ¨ + 2 (v · v) ˙ v˙ + 2 v · v ¨ + 2 (v · v) ˙ v v , c c c
Force on a particle:
K=
e2 γ 2 6πε0 c3
insert the Lorentz force into K, (no E − f ield) p2⊥ e4 B 2 −1 −1 ˙ p − ν Ω B B × p, with ν = K = −νr p⊥ + r r B (mc)2 6πε0 γ(mc)3
Guiding-center transformation: Z α = (X, pk , µ), · is the gyroaverage n o −1 α −1 i α Tgc Ki , Kgc = Tgc x , Z Landau Approximation:
gc
5
Guiding-center radiation-reaction force Gyro-averages
˙ is the guiding-center velocity X νr 2µB ˆ ˙ + 3vk %k κ , = −B ? b × X Ωk mc2
Force on spatial position:
KX
Force on the parallel momentum:
Kpk = −νr pk
p⊥ γ 2 µB 2 + % τ − ν %⊥ τB , B k B B r mc2 2
Force on the magnetic moment:
2µB K = −νr µ 1 + mc2 µ
2 + B %k τB ,
De nitions:
%k =
pk p⊥ ˆ · ∇ × b, ˆ κ=b ˆ · ∇b ˆ , %⊥ = , Ω?k = Ω (1 + B %k τB ), τB = b eB eB
6
Critical electric eld for runaway electron generation: • • • •
Experiments seem to show E/Ec > 3 − 5 needed for RE generation We study the RE dynamics using CODE RE growth rate strongly temperature dependent at xed E/Ec Raises the eective critical eld 0
3000 1000
E/E
D =1
−8
300 100 30 10 1.2 2 3 5 10
−4
0.1
−12 0.0 2 0.0
−16
1
30
100
300
−1 log10 (n−1 e dnr /dt) [s ]
At Te = 24 keV, a eld of at least E/Ec = 35 is needed for signi cant growth
Te [eV]
10000
−20
E/Ec
A. Stahl, E. Hirvijoki, J. Decker, O. Embréus and T. Fülöp, Accepted for publication in PRL, http://arxiv.org/abs/1412.4608
7
Critical electric eld for runaway electron generation: Synchrotron back-reaction Plots show change in growth rate
• Classical critical eld derivation • Synchrotron radiation
back-reaction eectively introduces additional drag
• Leads to reduction in growth
(Γw. synch /Γno synch ) 10000
∝
3/2 B 2 Te /ne
• Important at high temperature
and low density
0.5 300
0.3
100
0.1 30
2
3
5
10
30
E/Ec 30
ne [1019 m−3]
• Strength of synchrotron eects
0.7
1000
rate which can be very strong for weak electric elds
• Raises the eective critical eld
0.9
a)
3000
Te [eV]
only considers Coulomb collisions
0.9
b)
10
0.7
3
0.5
1
0.3
0.3 0.1
0.1 2
3
4
5
6
7
8
9
10
E/Ec
8
Critical electric eld for runaway electron generation: Growth-to-decay transition
• Experiments also considered transition from RE growth to decay • Build up RE tail, then ramp down E/Ec (ramp up density) • Visual synchrotron and HXR signals shows transition at E/Ec = 35 •
Simulations show transition at only slightly above
Ec
( 1.1)
E/Ec
BUT Synchrotron emission agrees with experiments!
•
Emitted synchrotron power sensitive to
•
Runaways are still gaining energy when the emission declines
4.2
1.3
B = 2.5
T
a)
0 1 (a.u.)
not RE decay but
7.4
E RE
Observed reduction is
redistribution of REs in momentum space
B
10.6
.5 T =3
B = 1.5 T
particle energies and pitches
•
12
1
Psynch.
•
12
0
b) 0.5
1
1.5
2
Time (s)
9
Critical electric eld for runaway electron generation: Summary
Experiments showing increased threshold electric eld for runaway growth can be explained by:
1
Temperature dependence of RE growth rate at xed E/Ec
2
Dampening eect of synchrotron emission (to some extent)
Apparent elevated RE growth-to-decay threshold is likely an artifact due to:
• Redistribution of runaways in momentum space as the eld
strength decreases
=⇒
• Change in emitted synchrotron and bremsstrahlung power
