using the sol: lsp simulations of fast ions slowing down ... · outline i sol functions and...

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


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


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


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)


Gas box model: a cooling, recombining plasma column

B. Cohen, T. Rognlien (LLNL)

G.S. Chu and S.A. Cohen, PRL 76, 1248 (1995)


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


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


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]


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


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


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


Background plasma only, Bz = 0 kGI Baseline for particle behavior in absense of fast ionsI Background particles exchange energy with field; ∆Ef = −∆Ee


Simulation (100 keV αs): Zα = 200, Bz = 0 kGI Fast ions slow down on electrons; ∆Ee = −∆Eα


Trend with EαI Slowing down time increases with EαI Magnetic field effect?


Trend with Zα (subthermal)I Simulations validate the Z 2

α of the modelI No apparent effect from magnetic field


Trend with ζα (subthermal)I Simulations validate the ζα of the modelI No apparent effect from magnetic field


Combined trend with ζα, Zα (subthermal)I Simulations validate the combined ζα ∗ Z 2

α of the modelI No apparent effect from magnetic field


10 MeV αs: Superthermal isotropic; Bz = 0 kG, Zα = 200

I Superthermal α’s heat electrons and increase field energy


10 MeV αs: Oscillation appears; Bz = 2.5 kG, Zα = 200


10 MeV αs: Bz = 5 kG, Zα = 200


10 MeV αs: Bz = 5 kG, Zα = 20

I large proton energy oscillation scales with Bz and ZαI related to Ωα?


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


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


References[1] Redmond Plasma Physics Laboratory, University of Washington, http://depts.washington.edu/rppl/images/frcintropict.gif

[2] M. Chu-Cheong, S. A. Cohen, et al., Reducing Neutron Emission from Small Fusion Rocket Engines, (2015), IAC, IAC-15,C4.7-C3.5,9,x28852

[3] M. Chu-Cheong, Energetic Particle Slowing in FRC Edge, http://w3.pppl.gov/ppst/docs/cheong_pres12a.pdf

[4] J. D. Huba, NRL Plasma Formulary, (Naval Research Laboratory, Washington, DC 2011), pp. 28–29, 44–45. NRL/PU/6790–11-551

[5] E. J. Kolmes, Applications of Particle-in-Cell Simulations to Fast Ion Slowing-down (2014) http://w3.pppl.gov/ppst/docs/kolmes.pdf

[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

[7] A. Creely, Particle-in-Cell Simulations of the Slowing Down of Energetic Charged Particles in a Background Plasma (2012), http://w3.pppl.gov/ppst/docs/creely.pdf

[8] M. N. Rosenbluth et al, Fokker-Planck Equation for an Inverse-Square Force (1957), Phys. Rev. 107, 1

[9] P. M. Bellan, Fundamentals of Plasma Physics, (Cambridge University Press, Cambridge, United Kingdom, 2006), pp. 384–393

[10] P. E. Grabowski, et al., Molecular Dynamics Simulations of Classical Stopping Power (2013), Phys. Rev. Lett. 111, 215002

[11] W. L. Hsu, M. Yamada, et al., Experimental Simulation of the Gaseous Tokamak Divertor (1982), Phys. Rev. Lett. 49:14

[12] G. D. Porter, et al., Simulation of experimentally achieved DIIID detached plasmas using the UEDGE code (1996), Phys. Plasmas 3

[13] M. E. Fenstermacher, et al., UEDGE and DEGAS modeling of the DIII-D scrape-off layer plasma (1995), Journal of Nuclear Materials 220-222, 330-335

[14] N. S. Wolf, et al., Effect of divertor geometry on plasma detachment in DIII-D (1999), Journal of Nuclear Materials 266-269, 739-741

[15] G. S. Chiu and S. A. Cohen, Experimental Observations of Steep Temperature Steps in Dense Magnetized Plasmas (1995), Phys. Rev. Lett. 76:8

[16] S. A. Cohen and Omer Artun, A scaling relation for the ITER scrape-off layer thermal diffusivity (1992), Journal of Nuclear Materials 196-198, 888-893

[17] G. Belyaev, et al., Measurement of the Coulomb energy loss by fast protons in a plasma target (1995), Phys. Rev. E 53:3

[18] J. A. Frenje, P. E. Grabowski, et al., Measurements of Ion Stopping Around the Bragg Peak in High-Energy-Density Plasmas (2015), Phys. Rev. Lett. 115, 205001

[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

[20] J. Y. Park, Studies on a transition to strongly recombining plasmas (1998), PhD Thesis, Princeton University

[21] G. Chiu, Studies of Magnetized Plasmas Interacting with Neutral Gas (1994), PhD Thesis, Princeton University

[22] M. H. Redi, S. A. Cohen, E. J. Synakowski, Transport simulations of helium exhaust in ITER using recent data from TFTR, TEXTOR and JT-60 (1991), Nucl. Fusion 31, 9


http://depts.washington.edu/rppl/images/frcintropict.gif

http://w3.pppl.gov/ppst/docs/cheong_pres12a.pdf

http://w3.pppl.gov/ppst/docs/kolmes.pdf

http://w3.pppl.gov/ppst/docs/paul_pres.pdf

http://w3.pppl.gov/ppst/docs/creely.pdf

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),


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