Recent developments in Vlasov-Fokker-Planck

transport simulations relevant to IFE capsule

compression

R. J. Kingham, C. Ridgers Plasma Physics Group, Imperial College London

9th Fast Ignition Workshop, Boston, 3rd—5th Nov 2006

Outline

• We are coupling our electron transport code, IMPACT, to an MHD code

(previously, IMPACT used static density)

• Example of enhanced code in use Froula (LLNL) & Tynan’s (USD) expt.

effect of B-fields on non-local transport in hohlraum gas-fill context

• We are starting to investigate transport & B-field generation on outside

wall of cone, during implosion

• Preliminary results B-field of > 1 in 0.5ns

affects lateral Te profile next to cone (beneficial?)

lateral heat flow non-local

Interested in departures from Braginskii transport……even in classical transport, B-fields add complexity

Braginskii’s transport relations (stationary plasma)

€

q = −κ ⋅∇Te − Teβ ⋅j

€

κ||∇

||Te + κ⊥∇⊥Te + κ∧

ˆ b ×∇Te

€

eneE = −∇⋅Pe + j×B + eneα ⋅j − neβ ⋅∇Te

€

L + β∧ˆ b ×∇Te

Nernst effect

Convection of B-field with heat flow

Righi-Leducheat flow

T

qRL

Implicit finite-differencing very robust + large t (e.g. ~ps for x~1m vs 3fs)

Solves Vlasov-FP + Maxwell’s equations for fo, f1, E & Bz

IMPACT – Parallel Implicit VFP code

First 2-D FP code for LPI with self consistent B-fields

IMPLICT LAGGED EXPLICIT

Kingham & Bell , J. Comput. Phys. 194, 1 (2004)

fo can be non-Maxwellian

get non-local effects

VFP equation for isotropic component f0

Moving with ion fluid -

include bulk convection.

Compressional heating -

from bulk plasma compression/rarefaction

Fictitious forces - we are no longer in an inertial frame

€

∂∂t

+ C ⋅∇ ⎛

⎝ ⎜

⎞

⎠ ⎟f0 − ∇ ⋅C( )

w

3

∂f0

∂w+

w

3∇ ⋅f1 +

1

3w2

∂

∂w−w2 eE

me

+ C ×eB

me

+∂C

∂t+ C ⋅∇ r( )C

⎡

⎣ ⎢

⎤

⎦ ⎥⋅f1

⎧ ⎨ ⎩

⎫ ⎬ ⎭

€

= ν 'ee

w2

∂

∂wC f0( ) + D f0( )

∂f0

∂w

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

v → w + C(z, t)€

∂f0

∂t+

v

3∇ ⋅f1 −

e

3mev2

∂

∂v(v 2E ⋅f1) =

ν ee

v 2

∂

∂vC( f0) + D( f0)

∂f0

∂v

⎡ ⎣ ⎢

⎤ ⎦ ⎥

c

f(v)f(w)

VFP equation for “flux” component f1

€

∂f1 j

∂t+ Ck

∂f1 j

∂rk

⎛

⎝ ⎜

⎞

⎠ ⎟+ w

∂f0

∂rj

−eB

me

× f1

⎡

⎣ ⎢

⎤

⎦ ⎥j

−∂f0

∂w

eE j

me

+ C ×eB

me

⎡

⎣ ⎢

⎤

⎦ ⎥j

−∂C j

∂t− Ck

∂Ck

∂rj

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

−f1k

∂Ck

∂rj

−w2

3

∂Ck

∂rj

∂

∂w

f1k

w

⎛

⎝ ⎜

⎞

⎠ ⎟+

∂C j

∂rk

∂

∂w

f1k

w

⎛

⎝ ⎜

⎞

⎠ ⎟+

∂Ck

∂rk

∂

∂w

f1 j

w

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥= −ν ei f1 j

Moving with ion fluidBulk flow terms -

Bulk momentum flowFictitious Forces

[ Chris Ridgers’ PhD project ]

€

∂ ρe ve + C( )[ ]

∂t + ∇ ⋅ ve + C( ) ve + C( )[ ] + ∇ ⋅Pe = −

ρ ee

me

E + C × B( ) + α ⋅ j − β ⋅∇Te

Ist velocity moment of this yields

Using IMPACT with MHD to model magnetized transport experiments

N2 gas jet1, 1J, 1ns laser beam

2, 1J, 200ps probe beam - Thompson scattering

Te

(eV

)

Radius (m)

B=0 nonlocal heat wave

B=12T local heat wave

10

100

1000

-600 -400 -200 0 200 400 600

LA

SE

R

• Experiment of D. Froula (LLNL), G. Tynan (UCSD) and co.

• Effect of B-fields on non-local transport in hohlraum gas-fill context

[ Tynan et al. submitted to PRL ]

[ Divol et al. APS2006 Z01.0014 ]

No B-field: k mfp > 0.03 non-local

Strong B-field expected to “localize”

D ~ mfp / D ~ r2

ge /

means krge << kmfp

mfp rge

“Bottling up” of Te for >1 seen in VFP simulation too

• Simulations start at Te=100eV + heating via inverse bremsstrahlung

No B-field 12T B-field

• 1D problem with cylindrical symmetry code 2D Cartesian so do 2D calc

• See “bottling up” of temperature in VFP sims with B-field

200 m

VFP suggests heat flow is marginally non-local at 12T

Radial heat flow

€

qr = −κ⊥∂rTe − Teβ∧j

θ

VFP code successfully moving plasma & B-field

• Magnetic Reynold’s # large resistive diffusion small

• … Nernst covection responsible for majority of central B-field reduction

• B-field convecting with plasma…

electron pressure blowing out plasma

Allowing for plasma motion affects evolution

Te(r) Heat flow - |q| (r)

e Bz(r) / me eio

with hydro

w/o hydro

• Simulations starts at Te= 20eV

• B = 12T

“What does the gold cone do to thermal transport in the vicinity of ncr in the adjacent shell?”

• Focusing on critical surface 0.25 ncr < ne <4 ncr

r

n , T

rcrit

• Could be susceptible to n x T B-fields?

Radial Te & Lateral ne gradients ?

r

r

r T

qRL

n

B (T)r (n)

Lateral Te & Radial ne gradients ?

r

r

r n

qRL

T

B (T) (n) r

Simulation set up – region from 0.25 ncr < ne < 4 ncr

0 4000r / m

24

22

20

log 1

0(

n e /c

m3

)

0 4000r / m

4

2Te

/ ke

VRadial densprofile

Radial Te profile

• DRACO ‘snapshot’ of ne(r,) , Te(r,), dU(r,)/dt used as init. cond. for IMPACT

[ … as used in APS talk on PDD. DRACO data courtesy Radha & McKenty ]

Heating RateHeating Rate

y /

mfp

x / mfp

Peak heating:

~8 keV / ns

I ~1.5 x 1014 W/cm2

~ 4 x10-4 (neTeo/ ei)cr

ne

niZ

y / m

Gold cone:

Lni ~ 80m

Z ~ 50

Te ~ 3 keV !!!

log10( n/ncr , Z)

ei = 5.5 m

ei = 0.17 ps

B-fields strong enough to magnetize plasma develops via n x T

t = 85ps

~ 1.3 t = 500ps

x / mfp

y /

mfp

log10(ne)

(n) (T)r (n)r (T)

Simulation details

x = 2.5 ei (nx = 56) fixed x-bc

y = 7.5 ei (nx = 40) refl. y-bc t = 0.5 ei ei = 5.5 m ei = 0.17 ps

B-field does affect lateral Te profile

Te = Te(y) - Tey at ne = 2 ncr

with B-field

no B-field

t = 8.5pst = 85ps

with B-field

no B-field

t = 1ns

with B-field

no B-field

Te

/ eV

(n) (T)r

(n)r (T)

Lowering due to Righi-Leduc

heat flow from

B-field (?)

€

q∧ = − κ∧ ˆ b ×∇Te

Flattening due to Righi-Leduc

heat flow from

B-field (?)

y / mfp

• Virtually no change in Te in cone Low thermal cond.

T5/2

c κ ~

(Z ln

Large heat capacity

Classical heat flow into cone up to 4x too large

qqxx qqyy

t = 0.5ns

VFPheat flow

Braginskiiheat flow

x / mfp

y /

mfp

Units qfso= neo mevTo3

B-field alters lateral heat flow in VFP sims

qy – B=0

t = 500psqy – with B-field

Conclusions

• IMPACT (VFP code) + MHD moving plasma + B-field in 2D

• Fielded on Froula & Tynan’s experiment; B-field suppr. of non-local effects

still some non-locality at 12 Tesla

B-field cavity, primarily due to Nernst advection

• Transport & B-field generation on outside wall of cone during CGFI implosion

• Preliminary results B-field of > 1 in 0.5ns flattens lateral Te profile next to cone (beneficial?)

lateral heat flow non-local

• Future: use enhanced code + working on adding f2 + f3

no radiation transp., ionization (yet) + Au too hotLn to large?

Simulation: Teo = 100eV (Au), 500eV (shell)

Lni ~ 20m Radial dens. gradient ~ 3x shorter than before

T = 17ps

VFP predicts 5x larger B-field than with Classical sim

Bz

t = 510ps

• Used an equivalent non-kinetic transport simulation

• Solves 1) Elec. energy equation 2) Ohm’s law 3) heat-flow eqn 4) Ampere-Maxwell 5) Faraday’s law

• Transport coeffs. κ [ Epperlein & Haines, Phys. Fluids 29, 1029 (1986) ]

• No flux limiter used in classical simulation --> Te(y) smaller --> less B-field

• Collapse of Te(y) outweighs tendancy for Braginskii to overestimate E ?

VFP Classical