Post on 11-Mar-2020
Using the SOL: LSP simulations of fast ions slowing
down in cool magnetized plasma
E. S. Evans1, M. Chu-Cheong1, A. Creely1, E. J. Kolmes1, E. Paul1, S. A. Cohen1; T.Rognlien2, B. Cohen2; E. Meier3; D. R. Welch4
1Princeton Plasma Physics Laboratory, 2Lawrence Livermore National Laboratory, 3College of William andMary, 4Voss Scientific
February 23, 2016
DOE contract: DE-AC02-09CH11466
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 1/26
Outline
I SOL functions and transport processes
I The small, clean FRC
I Classical slowing down theory (transitional regime)
I Fast ion slowing down in strong magnetic fields
I Simulation results
I Conclusions
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 2/26
Functions of the scrape-off layer (SOL)
I Power exhaust (fusion and heating power)
I Normal operation
I Off-normal events
I Ash exhaust (also impurity shielding)
I Helium: for D-T reactors, to prevent fuel dilution
I Tritium: for D-3He reactors, to reduce neutron production
I Convert fusion products (power and ash) to desired form
I Thrust and specific impulse
I Electricity generation
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 3/26
Transport into tokamak divertorsI Diffusive B⊥ transport across separatrix
I Convective flow along B at ∼ cs
t|| ∼ πRqcs
, λ2⊥ ∼ Dt||
1λP
= 1λn
+ 32λT
, λP ∼ 0.1 cm
τHe ∼ τsd + a2
2.4D− a
vc(∼ seconds)
M. H. Redi, S. A. Cohen, E. J. Synakowski (1991)
τfusion ∼ 20 s
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 4/26
Non-local particle transport via divertors of small FRCs
I SOL density width set bydivertor orifice: r ∼ 7 cm
I SOL density set by gas flow
I Te ∼(
Pne
)2/3
I SOL power width set by densitywidth (slide after next)
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 5/26
Gas box model: a cooling, recombining plasma column
B. Cohen, T. Rognlien (LLNL)
G.S. Chu and S.A. Cohen, PRL 76, 1248 (1995)
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 6/26
Non-local energy deposition in the FRC SOLI Small D-3He FRC reactor: Fast ions drag in cold SOL
I Reactor parameters: rs = 25 cm, Be = 6 T, Ti = 100 keV, ne = 5× 1014/cm3
FRC Core: S∗/E ∼ 3, τE ,classical ∼ 9 s
s = 0.3rs/ρi , sfuel ∼ 9
fusion product s(D-D) 3He 5.6
(D-D) T 2.5(D-D), (D-3He) p 2.5, 1.15
(D-3He) 4He 2.3
Te,SOL ∼ 50 eV, ne,SOL ∼ 5× 1013/ccI τSD,classical ∼ 5 ms
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 7/26
Non-local energy deposition in the FRC SOLI Small D-3He FRC reactor: Fast ions drag in cold SOL
I Reactor parameters: rs = 25 cm, Be = 6 T, Ti = 100 keV, ne = 5× 1014/cm3
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 7/26
Classical slowing down (summarized by Stix)
I Frictional drag
I Small-angle coulomb collisions withplasma electrons
I Scattering by plasma ions
I vth,i vb vth,e
I nb ne
Wcrit ≡(αβ
)2/3
= 14.8kTe
[A3/2
ne
∑ nj Z2j
Aj
]2/3
ts = 6.27× 108 A(kTe )3/2
Z 2ne ln Λsec
τ = −∫W
0dW
dW /dt= ts
3ln
[1 +
(W
Wcrit
)3/2]
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 8/26
Other energy losses in the SOL
Buneman Instability
requirement:
I vb >(
memi
)1/3
vth,b =
7× 106 cm/s
I vb > vthe = 4.2× 108 cm/2
most unstable mode:I growth rate:
0.7(
memi
)1/3
ωpe =
1.6× 108 Hz
I ω = 0.4(
memi
)1/3
ωpe =
9.1× 107 HzI vg = 2
3vb
Beam-Plasma Instability
requirement:
I vb >(
np
nb
)1/3
vth,b
most unstable mode:I growth rate:
0.7(
nbnp
)1/3
ωpe =
8× 108 Hz
I ω = ωpe−0.4(
nbnp
)1/3
ωpe =
2.4× 109 HzI vg = 2
3vb
α density in SOL
I 0.4 MW in 3.6 MeV α’s⇒ Γα ∼ 1018α/s
I 2x SOL area ⊥ B⇒ASOL ∼ 103 cm2
I vα∣∣500keV
∼ 5× 108 cm/s
I Mirror ratio R ∼ 10I Γα = nαvαASOL/R ⇒
nα ∼ 2× 107 α/cm3
I βα ∼ 10−7,Pα/Pthermal ∼ 0.04
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 9/26
Classical slowing down: correct/adequate?
Slower slowing down expected from two sources:
I Magnetic field effects: λD > ρe
I Speed effects: vfi > vth,e
I vfi ∼ 5× 109 cm/sI vth,e ∼ 5× 108 cm/s
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 10/26
Analytical model (for PIC comparison)
Model assumptions:
I mono-energetic isotropic homogenous fast ion distribution
I Maxwellian background particles (Te = 100 eV, Ti = 1 eV)
I background proton contribution neglected
I modifications due to macroparticle clumping factor (ζ):
v → v , n→ n/ζ, m→ ζm, q → ζq, T → ζT
I velocity regime: vth,i vα < vth,e
Test analytic solution:
W (t) = W0e−t/τs , 1
τs= ζαZ
2α
(ne
(κTe)3/2
) (m
1/2e e4 ln Λ
12√
2π3/2ε20mp
)Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 11/26
Analytical model (for PIC comparison)
Model assumptions:
I mono-energetic isotropic homogenous fast ion distribution
I Maxwellian background particles (Te = 100 eV, Ti = 1 eV)
I background proton contribution neglected
I modifications due to macroparticle clumping factor (ζ):
v → v , n→ n/ζ, m→ ζm, q → ζq, T → ζT
I velocity regime: vth,i vα < vth,e
Test analytic solution:
W (t) = W0e−t/τs , 1
τs= ζαZ
2α
(ne
(κTe)3/2
) (m
1/2e e4 ln Λ
12√
2π3/2ε20mp
)Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 11/26
PIC simulation parameters
I extent: ∼ 5λDe box in 3D cartesian coordinates
I grid resolution: ∼ λDe/7
I densities: ne = 1012/cm3, nα = 108 − 109/cm3
I clumping factors: ζe = ζi = 15.6 (64 ppc), ζα = 6.25− 200
I α particle charge: Zα = 2− 200
I magnetic field: Bz = 0− 10 kG
I ρe/λDe ≈ 0.64 for ne = 1012/cc, Bz = 5 kG
ne = 1014/cc, Bz = 50 kG
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 12/26
PIC simulation parameters
I extent: ∼ 5λDe box in 3D cartesian coordinates
I grid resolution: ∼ λDe/7
I densities: ne = 1012/cm3, nα = 108 − 109/cm3
I clumping factors: ζe = ζi = 15.6 (64 ppc), ζα = 6.25− 200
I α particle charge: Zα = 2− 200
I magnetic field: Bz = 0− 10 kG
I ρe/λDe ≈ 0.64 for ne = 1012/cc, Bz = 5 kG
ne = 1014/cc, Bz = 50 kG
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 12/26
Background plasma only, Bz = 0 kGI Baseline for particle behavior in absense of fast ionsI Background particles exchange energy with field; ∆Ef = −∆Ee
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 13/26
Simulation (100 keV αs): Zα = 200, Bz = 0 kGI Fast ions slow down on electrons; ∆Ee = −∆Eα
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 14/26
Trend with EαI Slowing down time increases with EαI Magnetic field effect?
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 15/26
Trend with Zα (subthermal)I Simulations validate the Z 2
α of the modelI No apparent effect from magnetic field
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 16/26
Trend with ζα (subthermal)I Simulations validate the ζα of the modelI No apparent effect from magnetic field
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 17/26
Combined trend with ζα, Zα (subthermal)I Simulations validate the combined ζα ∗ Z 2
α of the modelI No apparent effect from magnetic field
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 18/26
10 MeV αs: Superthermal isotropic; Bz = 0 kG, Zα = 200
I Superthermal α’s heat electrons and increase field energy
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 19/26
10 MeV αs: Oscillation appears; Bz = 2.5 kG, Zα = 200
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 20/26
10 MeV αs: Bz = 5 kG, Zα = 200
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 21/26
10 MeV αs: Bz = 5 kG, Zα = 20
I large proton energy oscillation scales with Bz and ZαI related to Ωα?
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 22/26
Superthermal beam simulations
I Wake formation (simulated previously)
simulation parameters
I Te = Ti = 10 eVI ne = ni = 1010/ccI Ebeam = 14.7 MeVI λD = 0.0235 cmI ωp ∼ 109 rad/s, c/ωp = 30 cm
I vbeam = 5× 109 cm/s,vth,e ∼ 108 cm/s
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 23/26
Summary
I ζα ∗ Z 2α scaling confirmed, speed up possible for future simulations
I Slowing down time increases with increasing Eα
I Subthermal τSD ∼ 1.3τsim, even with Bz 6= 0
I Unclear if Bz will have substantial effect ⇒ more simulations needed
I In current parameter space, slowing down times are sufficient for
SOL to absorb energy and remove ash from an FRC reactor
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 24/26
References[1] Redmond Plasma Physics Laboratory, University of Washington, http://depts.washington.edu/rppl/images/frcintropict.gif
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[6] E. Paul, 2-d Particle-in-cell Simulations of the Energetic-ion Slowing Down in Cool Plasma (2013) http://w3.pppl.gov/ppst/docs/paul_pres.pdf
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[19] T. Eich, A. W. Leonard, R. A. Pitts, W. Fundamenski, R. J. Goldston, et al., Scaling of the tokamak near the scrape-off layer H-mode power width and implications for
ITER (2013), Nucl. Fusion 53,093031
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Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 25/26
Extra
useful frequencies (ne = ni = 1012/cm3)[4]:
I ωpe/2π = 8.98 GHz, ωpi/2π = 210 MHz
I Ωe/2π = 14 Ghz, Ωi/2π = 7.6 MHz (B=5 kG)
Ωe/2π = 7 GHz, Ωi/2π = 3.8 MHz (B=2.5 kG)
I ωLH ≈√
ΩeΩi = 326 MHz (B=5 kG)
ωLH ≈√
ΩeΩi = 163 MHz (B=2.5 kG)
I ωpα/2π = 105 MHz, Ωα/2π = 380 MHz (B=5 kG, Zα = 200,
nα = 108/cm3),
Eugene S. Evans Using the SOL: LSP simulations of fast ions slowing down in cool magnetized plasma — EPR 2016 February 23, 2016 26/26