Kinetic Physics 2016 WorkshopLawrence Livermore National Laboratory, CA, USA

April 5-7, 2016

Vlasov-Fokker-Planck simulation of ion-kinetic effectsin ICF implosions

O. Larrochea

B. Canauda, A. Ingleberta, B.-E. Peigneya, R. D. Petrassob, H. G. Rinderknechtc,M. J. Rosenbergd, V. T. Tikhonchuke

aCEA DIF, 91297 ArpajonCEDEX, FrancebPlasma Science and Fusion Center, MIT, Cambridge, MA 02139,USAcLawrence Livermore National Laboratory, Livermore, CA 94550, USA

dLaboratory for Laser Energetics, University of Rochester,Rochester, NY 14623, USAeUniversity of Bordeaux, CNRS, CEA, CELIA, 33405 Talence, France

Summary

Are time and length scales in ICF implosions sufficiently larger than collisional scales ?• Hydro simulation post-processing for collisional times, lengths, diffusion coefficients

Kinetic (Vlasov) formalism with Fokker-Planck ion collision modeling : VFP• FPION : hybrid formalism valid on the hydro time scale : kinetic ions, thermal electrons• Faster particles are less collisional than thermal bulk : shock fronts• Dedicated experiments deviate from hydro modeling, betteragreement with VFP• Full-scale ICF kinetic and hydro simulations of hot-spot formation show discrepancies• Temperature decoupling between ion species in shock-ignition targets.

Kinetic α-particle transport in igniting ICF targets : Knudsen number ∼ 1• FUSE : a two-scale ion-kinetic code for ignition• First results ofFUSE simulation of baseline target : indications of reduced gain

Work to do :• Stratification and/or diffusion or mix at the fuel-pusher interface• Ability to treat ion species with large differences in charge and mass• Including pusher dynamics in ignition simulations to investigate ignition threshold• Need for more quantitative results in longer ignition target calculations

3

Collisional metrics from hydro simulations

λcD (µm)

0.4 0.6 0.8 1time (ns)

0

100

200

300

400

radi

us (

µm)

10-2 1 102 104

0.4 0.6 0.8 1time (ns)

DD (cm2/s)10-2 1 102 104 106 108

Implosion of D3He gas atρ = 0.4 mg/cm3 in a 430µm-radius SiO2 capsule with a 14.6kJ, 0.6 ns flat-top laser pulse : hydro simulation post-processing.Knudsen numberλc/L and diffusion coefficientD = λcvth for a thermal D ion highcompared with relevant valuesDref = L2/T whereL andT are typical length and time :L ∼ 100 µm andT ∼ 200 ps⇒ Dref ∼ 5× 105 cm/s [Larrocheet al, PoP (2016)].

4

The ion Vlasov-Fokker-Planck codeFPION

Ion Vlasov-Fokker-Planck equation in spherical geometry (in reduced units) :

∂fi∂t

+ vr∂fi∂r

+v⊥r

(

v⊥∂fi∂vr

− vr∂fi∂v⊥

)

+EiAi

∂fi∂vr

=

n∑

j=1

(

∂fi∂t

)

ij

+1

2τei

∂

∂vα

[

(vα − uiα)fi(v) +TeAi

∂fi∂vα

(v)

]

Coulomb collisions⇒ advection-diffusion in velocity space :(

∂fi∂t

)

ij

=4πZ2i Z

2

j LogΛij

A2i

∂

∂vα

[

AiAj

∂Sj∂vα

fi −∂2Tj

∂vα∂vβ

∂fi∂vβ

]

with “Rosenbluth potentials” : ∆vSj = fj , ∆vTj = SjCoulomb collisions⇒ τc ∼ v3/n, several scales, localized metastable states inv spaceFluid electrons (τ ≫ τee), quasineutrality (L ≫ λD), ion/hydro timescale (τ ≫ ω−1pe )⇒ energy equation left to solveNumerical implementation : seeLarroche, PFB (1993) – EPJ-D (2003)Other kinetic simulation tools : LSP (hybrid PIC/fluid) usedby Bellei et al, PoP (2013,2014), VFP code under development :Taitanoet al, JCP (2015)

5

Shock simulations show intrinsically kinetic features

0

200

400

-10 0 10 20

0 0.05 0.1 0.15 0.2 0.25

dist

ance

(µm

)

velocity vr (107 cm/s)

D

F//(r,vr) (reduced units)

-10 0 10 20

0 0.1 0.2 0.3F//(r,vr) (reduced units)

3He

velocity vr (107 cm/s)

Asymptotic strong (M = 5) shock front in an equimolar D-3He mixture.Maps in phase space(r, vr) of the longitudinal velocity distribution functions for both ionspecies [Larroche, PoP (2012)] : notice upstream kinetic features.

6

FPION modeling of dedicated gas-filled target implosions

Simulations of high- and low-density cases (ρ = 3.3 and 0.4 mg/cm3).

Initial condition : Maxwellian distributions for D and3He att = 350 ps.

Boundary condition on the gas/pusher interface as given by the hydro calculation.

In the low-density case, the ion temperatureTi for D and3He is taken from slightly insidethe pusher to take into account the (possibly unphysical) large Ti gradient across theinterface found in the hydro simulation.

Numerical grid : 200 cells inr, the spatial grid shrinks or expands to follow the fuel/pusherboundary.

128 × 64 cells in (vr, v⊥) space respectively,δvr = δv⊥ adjusts as a function of thethermal velocity found in each spatial cell.

Flux-limited electron thermal conduction withf = 0.07 .

Distribution moments, nuclear reactivities and DD-n and D3He-p reactivity-averaged iontemperatures are computed as a post-processing treatment,using cross-section data fromBosch & Hale, Nucl. Fusion (1992).

7

A summary of experimental data and simulation results

××××

××××

××××

××××

FromRosenberget al (PRL 2014). ×× D3He –FPION ×× DD – FPIONOther observables better rendered as well : reactivity profiles [Larrocheet al (PoP 2016)]

8

Hot spot formation in full-scale ICF targets

FromLarroche, EPJ-D (2003)Baseline ICF target near stagnation (17 ns< t < 17.9 ns), single speciesFPION simu-lation of DT (usingA = 2.5) :• Timing modification• LowerTi but≈ sameρr profile• No direct effect on reactivity, but lower

Ti → 30% lower reaction rate, althoughφ = ρr/(ρr + 6) unchanged ?

Two-species results similar in this stageLonger simulations ?

0

0.5

1

1.5

2

0 20 40radius (µm)

ρr (

g/cm

2 )

0 20 40radius (µm)

fluid kinetic

0.5

1

1.5

2

2.5

dens

ity (

1026

cm

-3)

-3

-2

-1

0

velo

city

(10

7 cm

/s)

Ti

Te

10 20 30 40

0

radius (µm)

int.

reac

tivity

(10

26 s

-1)

0

2

4

6

Te

Ti

tem

pera

ture

(ke

V)

0

2

4

6

0 10 20 30 40radius (µm)

reac

tivity

(10

33 c

m-3

s-1 )

0

0

0.5

1

1.5int.

int.

n

u

n

u

fluid kinetic

9

Temperature decoupling between ion species in ignition-scale targets

From Inglebertet al, EPL (2014):very CPU-consuming calculation !

Shock-ignition target, kinetic ef-fects show up near stagnation.

Tritium is appreciably hotterthan Deuterium in hot spot atstagnation.Interpretation :miv

2

impl → 3kB∆Ti beforeinter-species temperature relaxa-tion.

Average Ti at stagnation inmulti-species calculation lar-ger than Ti in single-speciesaverage-ion calculation→ re-visit previous results (Larroche,EPJ-D (2003)).

10

Kinetic modeling of ignition with fast α-particle treatment

A new hybrid Fokker-Planck treatment of fast particles has been developed : distributionsin the form of the sum of a “cold” component and a “hot” component : fa = fa,c + fa,hBy construction, the hot componentfa,h varies slowly everywhere in velocity space ; itcan thus be discretized on acoarse grid. From the diffusion (≈ transverse) point of view,fa,h is close to equilibrium, because the source term from nuclear reactions is isotropic.

fα(v)

vfα,h

fα,cα-

parti

cle

inje

ctio

n

from

fusi

on re

actio

ns

collisionalconvectionat constant

rate

ther

mal

isat

ion

into

cold

com

pone

nt

v⊥

vr

hot grid

cold grid

Hot → hot collisions can be neglected with respect to hot→ cold ones. Collisions withelectrons are treated as usual.Implementation : code «FUSE » (Peigneyet al, JCP (2014)).

11

Reduced gain in kinetic simulation of baseline target

First simulations :Peigneyet al, PoP (2014)NIF/LMJ-like target withcodeFUSE.

Pusher modeled by a mo-ving piston with adjus-table mass. Interface tra-jectory :

: optimal kinetic: mass× 0.5: mass× 2: fluid calculation

α-particle heating morespread out in space andtime, total energy yield∼2× lower than in hydro +multigroup diffusion cal-culation.

12

Conclusion and work to do

Dedicated gas-filled target implosions with moderate to large Knudsen number• ModerateKN : satisfactory agreement between experiments and VFP simulations• High KN : encouraging results, but possible problems at the gas/pusher interface• To do :kinetic treatment of diffusion across fuel/pusher boundary

collisional interaction with heavier elements : Si, O

Hot spot formation in ignition-scale ICF targets• Kinetic simulations shows effects on hot-spotTi and some species decoupling, possibly

detrimental to ignition• To do :longer simulations including main shock propagation in cold gas

more satisfactory treatment of other physics is needed

Kinetic effects on ignition and burn• New kinetic code treatsα-particle transport and thermalization• First proof-of-principle simulation results show lower gain• To do :extensive quantitative simulations of ignition (threshold ?) and burn

same improvements as above are needed

13

References

C. Belleiet al, Phys. Plasmas20, 012701 (2013).C. Belleiet al, Phys. Plasmas21, 056310 (2014).H.-S. Bosch, G. M. Hale, Nucl. Fusion32, 611 (1992).A. Inglebert, B. Canaud, O. Larroche, Europhys. Lett.107, 65003 (2014).O. Larroche, Phys. Fluids B5, 2816 (1993).O. Larroche, Eur. Phys. J. D27, 131 (2003).O. Larroche, Phys. Plasmas19, 122706 (2012).O. Larrocheet al, Phys. Plasmas23, 012701 (2016).B. Peigney, O. Larroche, V. T. Tikhonchuk, J. Comput. Phys.278, 416 (2014).B. Peigney, O. Larroche, V. T. Tikhonchuk, Phys. Plasmas21, 122709 (2014).M. J. Rosenberget al, Phys. Rev. Lett.112, 185001 (2014).M. J. Rosenberget al, Phys. Plasmas22, 062702 (2015).W. T. Taitanoet al, J. Comput. Phys.297, 357 (2015).

14

Backup : Ion temperature in hydro simulations of gas-filled targets

0.4 0.6 0.8 1

0 5 10 15 20

time (ns)

0

100

200

300

400ra

dius

(µm

)ρ = 3.3 mg/cm3

ion temperature (keV)

0.4 0.6 0.8 1

0 10 20 30 40 50ion temperature (keV)

time (ns)

ρ = 0.4 mg/cm3

LargeTi jump across the pusher/fuel interface in the low-density case [Rosenberget al,PoP (2015)- Larrocheet al, PoP (2016)]

15

Backup : The ion Vlasov-Fokker-Planck codeFPION (1)

VFP equation in spherical geometry, to lowest order inme/mi, for speciesi :

∂fi∂t

+ vr∂fi∂r

+v2⊥

r

∂fi∂vr

− v⊥vrr

∂fi∂v⊥

+EiAi

∂fi∂vr

=

n∑

j=1

(

∂fi∂t

)

ij

+

(

∂fi∂t

)

ie

Ai mass number of species-i ions ; electron-ion collision term :(

∂fi∂t

)

ie

=1

2τei

∂

∂vα

[

(vα − uiα)fi(~v) +TeAi

∂fi∂vα

(~v)

]

with electron-ion collision time

τei =3√πAiT

3/2e

2ǫ√2Z2i neLogΛie

ǫ =√

me/mp (me, mp : electron and proton masses),Zi charge number of species-i ions,na, ~ua, Ta : density, velocity and temperature of speciesa, LogΛab Coulomb logarithm forcollisions between species-a and -b particles, effective electric field :

Ei = −Zine

∂Pe∂r

16

Backup : The ion Vlasov-Fokker-Planck codeFPION (2)

Pe : electron pressure. Units used are :

time : τ0 =(kBT0)

3/2m1/2p

4πe4n0, space : λ0 =

(

kBT0mp

) 1

2

τ0

kBT0 andn0 are given reference values of the thermal energy per particle and number den-sity, e is the elementary charge.

ne =

n∑

j=1

Zjnj , ue =1

ne

n∑

j=1

Zjnjuj , Z̃ =

∑nj=1 Z

2

j njLogΛejneLogΛee

LogΛab = LogλD

max(λbar, ρ⊥)where λD =

λD0λ0

neTe

+

n∑

j=1

njZ2

j

Tj

−1/2

λD0 =

(

kBT04πn0e2

)1/2

= reference Debye length ;ρ⊥ =

(

λD0λ0

)2 ZaZbAabu2

λbar =~

τ0kBT0

1

Aabuwhere Aab =

AaAbAa +Ab

and u =√3

(

TaAa

+TbAb

)1/2

17

Backup : The ion Vlasov-Fokker-Planck codeFPION (3)

Collision term between ions of speciesi andj :

(

∂fi∂t

)

ij

=4πZ2i Z

2

j

A2iLogΛij

∂

∂vα

[

AiAj

∂Sj∂vα

fi −∂2Tj

∂vα∂vβ

∂fi∂vβ

]

Rosenbluth potentialsSj andTj :

Sj = −1

4π

∫

fj(~v′)

|~v − ~v′|d3v′ and Tj = −

1

8π

∫

|~v − ~v′|fj(~v′)d3v′

→ Poisson equations in velocity space :∆vSj = fj and∆vTj = Sj with appropriateboundary conditions.Electron temperatureTe from conservation of electron thermal energy densityWe :

∂We∂t

+1

r2∂

∂r

(

r2ueWe)

+1

r2∂

∂r(r2ue)Pe −

1

r2∂

∂r

(

r2κe∂Te∂r

)

=

=

n∑

j=1

3nj2τej

(Tj − Te) +(

∂We∂t

)

rad

18

Backup : The ion Vlasov-Fokker-Planck codeFPION (4)

κe : Spitzer-Härm electron thermal conduction in multi-ion-species case :

κe ≈64√2δT (Z̃)√π

T5/2e

ǫZ̃LogΛee

We andPe from electron fluid equation of state, with low-density limit :

We(ne, Te) −−−→ne→0

3

2neTe ; Pe(ne, Te) −−−→

ne→0neTe

Electron EOS takes Fermi degeneracy into account through :

ne = 4π

(

2mekBTeh2

)3/2

I1/2(z) ; We =3

2Pe = 4π

(

2mekBTeh2

)3/2

kBTeI3/2(z)

“Fermi integrals” In/2(z) =∫

∞

0

yn/2

z−1ey + 1dy

Bremsstrahlung losses :(

∂We∂t

)

rad

= −Prad = −4.14× 10−4T0(keV )neT 1/2e∑

i

niZ2

i

19

Backup : Space-time domain for the kinetic simulation of ICFcapsules

Green: DT/pusherinterfaceRed: initial interfacesolid DT/DT gasBlue : kinetic treatmentregion

0 5 10 15

time (ns)

2

1.5

1

0.5

0

radi

us (

mm

)

20

Backup : Fusion emissivities in exploding-pusher gas-filled targets

Reaction rate :Rij(r, t)

Emissivity :Eij(r) =

∫

∞

−∞Rij(r, t)dt

Yield :Yij =

∫

∞

0Eij(r)4πr

2dr

Median Radius :0.50 × Yij =

∫ R500

Eij(r)4πr2dr

0 50 100 150

104

reac

./µm

3

radius (µm)

D3He-p

DD-p

ρ = 0.4 mg/cm3

0 50 100 150 200

radius (µm)

ρ = 3.3 mg/cm3

0 50 100 150

radius (µm)

ρ = 0.4 mg/cm3

0 50 100 150 200

radius (µm)

ρ = 3.3 mg/cm3

0

5

10

15

104

reac

./µm

3

0

0.2

0.4

0.6

0.8

104

reac

./µm

30

2

4

104

reac

./µm

3

0

0.5

1

1.5

2

R50R50

R50 R50

2.5

× 7

× 20

: FPION calculation of D3He gas-filled target implosions[Larrocheet al, PoP (2016)]

— — — : Measured profiles [Rosenberget al, PoP (2015)]

21

Backup : Fusion surface brightness in exploding-pusher targets

Surf. brightness :Bij(r) =

∫

∞

−∞Eij(

√r2 + s2)ds

r

s

0

5

10

15

0 50 100 150

106

reac

./µm

2

radius (µm)

D3He-p

DD-p

0

0.5

1

1.5

2

0 50 100 150 200

radius (µm)

106

reac

./µm

2

0

1

2

3

4

0 50 100 150

radius (µm)

106

reac

./µm

20

0.5

1

0 50 100 150 200

radius (µm)

106

reac

./µm

2

ρ = 3.3 mg/cm3

ρ = 3.3 mg/cm3

ρ = 0.4 mg/cm3

ρ = 0.4 mg/cm3

× 20

× 10

: FPION calculation of D3He gas-filled target implosions[Larrocheet al, PoP (2016)]

— — — : Measured profiles [Rosenberget al, PoP (2015)]