VFP simulations of transport in Polar D-D

&

Effect of IB heating on B-field phenomena

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

A. Thomas P.W. McKentyUniversity of Michigan LLE, University of Rochester

DD+FIW, Prague, 3rd — 6th May 2009

Polar Direct Drive – the concept

• Method for doing direct-drive ICF on NIF - beam repointing (LLE)

• Greater heating on equator

Temperature gradients in

Density gradients in

[ Skupsky et al., Phys. Plasmas 11, 2763 (2005) ]

‘main’ gradient in r+

1 B ⎭⎬⎫

⎩⎨⎧ ∇×∇−=

∂∂ Tn

nt [ Stamper et. al., PRL 1012 (1971) ]

Shock Ignition – Has related transport issues

• Non-uniform irradiation – on LMJ, use 33o beam cone for driving shock (?)

I ~ 2.5 x 1015 W/cm2 I ~ 0.3—1.2 x 1015 W/cm2

• 2x higher peak intensity than in PDD

[ Skupsky et al., PoP 11, 2763 (2005) ]

PDDShock ignition

[ Ribeyre et al., PPCF 51, 015013 (2009) ]

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

Energy dep. rate300

14000

8 x 10-4

x / mfp

y /

mfp

• Took a ‘snapshot’ of ne(r,) , Te(r,), dU(r,)/dt from DRACO (2D-ALE code)

• Used as initial conditions & heating rate in VFP transport sim

- “IMPACT” B-fields, 2D-Cartesian, static density

ncr= 1022 cm-3

(Radius = 1.08mm)

770 m

Peak heating rate:

~ 1.5 keV / ns at ncr

I ~ 3 x 1014 W/cm2

~ 8 x 10-4 (neTeo / ei)cr

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

ncr= 1022 cm-3

Teo= 2.9 keV

ei = 5.5 m

ei = 0.17 ps

(Ln)r = 65 ei

(Ln) ~ 400 ei

Density scale length

Z = 3.5

300

1400 x / mfp

y /

mfp

ncr2 ncr 0.5 ncr

0 4000x / m

24

22

20

log 1

0(

n e /c

m3

)

Simulation set up – region from 1.9 < Te < 3.7 keV

ei = 5.5 m

ei = 0.17 ps

(LT)r = 120 ei

(LT) ~ 1000 ei

Temp. scale length

Simulation details

x = 2.5 ei

y = 7.5 ei

t = 0.5 ei

Te in keV

x / mfp

y /

mfp

0 4000x / m

4

2

Te

/ ke

V

X-BC fixed

fo = fm(neb,Teb)

Y boundaries

reflective

See B-field grow to ~ 0.03 by 500ps

at 8.5ps

• Initial B-field growth consistent with (n)r (T)

• Simulation t=0 ; only radial ne & Te gradients

at 500ps

• See Nernst advection

at 85ps

x / mfp

y /

mfp

€

vNernst = − β∧' ∂

xT

e

t = 85ps

with B-field

no B-field

B-field does affect lateral Te profile

t = 500ps

with B-field

no B-field

t = 1ns

with B-field

no B-field

y / mfp

Te

/ eV

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

• B-field modifies Te() via Righi-Leduc heat flow

€

q∧ = − κ∧ ˆ b ×∇Te

• Similar effect in “corona”

- 10eV change in Te

• “Modification” of similar size to

intrinsic non-uniformity due to

heater beams at ne = 2ncr

Flux limiter for q and qr not generally the same

• Implied flux limiter for q larger than that for qr where heating occurs €

fr =qr( )Brag rms−θ

qr( )VFP rms−θ

• Flux limiter measure: RMS ave. in

t = 85ps fr

f

0 140r / ei

10

1

6

• Er & E also show departures from locality

“Flux” limiter for Er and Eq3

1

2

• q “diffuses” toward ablation surface

- Braginskii underestimates q here!

- Analogous effect to Nernst (?)

c.f. [ Rickard, Epperlein & Bell., PRL 62, (1989) ]

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. [ 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

f1 min / max = -2.1e-3 / 1.7e-3

Anisotropic pressure --> makes a difference to B

IMPACT

IMPACTA

f1 min / max = -4.3e-3 / 1.5e-3

f1+ f2 min / max = -6.4e-3 / 4.6e-3

t = 85ps

( Thomas et al., NJP 11, 033001 (Mar. 2009) )

Bz

IB - heating

f1+ f2 min / max = -2.1 / +1.5 kG

f1 min / max = -1.4 / +0.49 kG

Preliminary !

EEyy

f1 min / max = -0.69 / +0.56 kG

2--3x larger

Anisotropic pressure --> makes a difference to B

f1 min / max = -9.0e-3 / 4.3e-3

f1 min / max ~ -5e-3 / 2e-3f1+ f2 min / max = -2e-2 / 1.7e-2

Preliminary !

t = 340 ps

( Thomas et al., NJP 11, 033001 (Mar. 2009) )

IMPACT

IMPACTA

Bz

IB - heating

f1 min / max = -3.0 / +1.4 kG

f1 min / max ~ -1.5 / 0.7 kGf1+ f2 min / max = -6.6 / +5.6 kG

~4x larger

Anisotropic pressure – Suppresses B-field advection?

y / mfp y / mfp

IMPACT

IMPACTA

IMPACTA + f2

Bz(y) at ncr

IMPACT

IMPACTA

IMPACTA + f2

Bz(y) at 2 ncr

t = 340 ps

Units: “0.002” --> 0.7 kG

PART 2 — Effect of Inverse-Bremsstrahlung heating on transport & B-field phenomena

• Theoretical: Better understanding of B-fields and transport

• Practical: Inertial Confinement Fusion and other experiments

D.H. Froula et al., PRL 98, 085001 (2007)

1m, 100J, 1ns laser

20 < Te < 800eV

ne ~ 1.5 x 1019 cm-3

Bapplied up to 120 kG

(12 T)

I~ 4x1014 W/cm2 ~ 150 m

Super-Gaussian electron distribution functionBreakdown of Maxwellian Assumption

• A. B. Langdon, PRL 44, 9 (1980): EDF f0(r,v,t) tends to Super-Gaussian due to I.B.

Super-Gaussian fit (m=3.3)

Langdon (m=5)

Maxwellian (m=2)

€

f0(r,v, t)∝ exp(−V m )

General m

€

f ≈ f0 + δf

• Involved in transport

Where IB heating distortion is important

• Langdon parameter ‘’ & Matte’s fit for ‘m’

€

= Z (vosc /vTh )2

€

m = 2 + 3 1+( 53)α −0.724

[ ]−1

• PDD

~ 0.02 , m ~ 2.1

Te = 3keV , I~ 3x1014 , 0.33 m , Z=3.5

• Froula’s N2 gas jet expt

Te = 200 eV , I~ 4x1014 , 1 m , Z = (4) — 7

~ 6 , m ~ 4

shock ign.

I~ 3x1015 , 1 m

~ 0.15 — 1.5m ~ 2.4 — 3.3

€

Kn = λ mfp

LT

= ν c

vT4

Z nine

1

LT

• Non-local transport

Transport RelationsExtension to Super-Gaussian EDF

€

E = −∇Pe

ene

− α ⋅ j − 1

eβ ⋅∇Te +

1

ene

j × B

• Dum (1978) & Ridgers (2008): transport theory for 2 m 5

€

E = − 1

ene

γ ⋅∇Pe − α ⋅ j − 1

eβ ⋅∇Te +

1

ene

j × B

€

q = − κ ⋅∇Te − ψ ⋅ jTe

e − φ ⋅∇Pe

€

q = − κ ⋅∇Te − β ⋅ jTe

e

• Braginskii: valid m=2 ( f0 = fM )

• New coefficients , old ones changed , Onsager symmetry broken

Components of transport coefficients (tensors)

€

−E = −1

ene

j × B −α ⋅ j +1

eβ ⋅∇Te +

1

ene

∇Pe

€

⋅∇Te = β ||∇ ||Te + β∧ b ×∇Te( ) + β⊥∇⊥Te

€

b

€

y€

x€

∇Te

€

b ×∇Te

€

∇⊥Te

€

∇||Te

Extended Transport Theory:Ridgers’ Ohm’s Law

€

E = − 1

ene

γ ⋅∇Pe − α ⋅ j − 1

eβ ⋅∇Te +

1

ene

j × B

• Functions of Hall parameter and m:

C. P. Ridgers, PoP 15, 092311 (2008)

Extended Transport Theory:Ridgers’ Ohm’s Law

€

E = − 1

ene

γ ⋅∇Pe − α ⋅ j − 1

eβ ⋅∇Te +

1

ene

j × B

• Functions of Hall parameter and m:

C. P. Ridgers, PoP 15, 092311 (2008)

B-field Evolution with Super Gaussian EffectsInduction equation

Fourth termThird term

€

∂B

∂t=∇ × −

1

ene

j × B ⎡

⎣ ⎢

⎤

⎦ ⎥+∇ × −α ⋅ j[ ]

€

+∇×1

e(β + γ ) ⋅∇Te

⎡ ⎣ ⎢

⎤ ⎦ ⎥+∇ ×

Te

ene

γ ⋅∇ne

⎡

⎣ ⎢

⎤

⎦ ⎥

B-field Evolution with Super Gaussian EffectsModifies Nersnt advection

€

∂B

∂t=∇ × β∧ + γ∧( ) b ×∇Te( )[ ]

€

=∇× −∧+γ∧( )∇Te

B× B

⎡

⎣ ⎢

⎤

⎦ ⎥

€

+γ( )⋅∇Te = β∧ + γ∧( ) b ×∇Te( ) + other terms

€

∂B

∂t=∇ × vN × B( ) Nernst Velocity

Nersnt effect

Advection of B-fieldby heat flow

J. Bissell — PhD research

B-field Evolution with Super Gaussian EffectsSuppression of Nernst advection

80% Suppression

J. Bissell — PhD research

€

vBrag. =−β∧(m = 2)

eB∇Te

Classical transport

€

vN =−(β∧(m) + γ∧(m))

eB∇Te

Extended transport

Extended / Classical

€

VN

B-field Evolution with Super Gaussian EffectsDensity gradient effects

Fourth term

€

∂B

∂t=∇ × −

1

ene

j × B ⎡

⎣ ⎢

⎤

⎦ ⎥+∇ × −α ⋅ j[ ]

€

+∇×1

e(β + γ ) ⋅∇Te

⎡ ⎣ ⎢

⎤ ⎦ ⎥+∇ ×

Te

ene

γ ⋅∇ne

⎡

⎣ ⎢

⎤

⎦ ⎥

B-field Evolution with Super Gaussian EffectsSuppression of

30% Suppression

€

∇Te ×∇ne

J. Bissell — PhD research

€

∂B

∂t=∇ ×

Te

ene

γ ⋅∇ne

⎡

⎣ ⎢

⎤

⎦ ⎥

€

∂B

∂t = γ⊥

∇Te ×∇ne

ene

+ K

B-field Evolution with Super Gaussian EffectsA New Effect — Advection up density gradients

€

∇× vNew × B( )

€

−γ∧• positive advection of B-field up density gradients

• As with Nernst write

€

∂B

∂t= γ⊥

∇Te ×∇ne

ene

+∇ ×Te

ene

γ∧ b ×∇ne( ) ⎛

⎝ ⎜

⎞

⎠ ⎟ + other terms

€

∇× −γ∧Te∇ne

eneB× B

⎛

⎝ ⎜

⎞

⎠ ⎟

€

VNernst ≈ b∧ λ ei

LT

⎛

⎝ ⎜

⎞

⎠ ⎟ vT

€

VNew ≈ c∧ λ ei

Ln

⎛

⎝ ⎜

⎞

⎠ ⎟ vT

• Low magnetization limit ( << 1) ….

m = 2.5 m = 5

€

VNew

VN

≈ 0.03 0.2 LT

Ln

⎛

⎝ ⎜

⎞

⎠ ⎟

Conclusions

• Polar D-D ; get ~ 0.03 after 500ps of heating

• B-field strong enough to modify Te()

At 2 ncr similar in size to intrinsic variation due to laser non-uniformity

• Implied flux limiter varies in space (and time) + limiter for q > qr

• Inclusion of anistropic pressure (f2) stronger B-field

• Calc. prone to instabilities + need moving plasma + radial BCs tricky !

• IB heating new & mod. B-field dynamics via new tranport coeffs

Suppression of Nernst & n x T + advection up density gradient

• PDD relevant to shock ignition IB hohlraum walls ( + shock ignition ? )

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

Higher resolution run with shell – 0.1 ncr < ne < 84 ncr

0 4000x / m

4

2

Te

/ ke

V

4000x / m

24

22

20

log 1

0(

n e /c

m3

)

0

Low res

x = 2.5 ei

y = 7.5 ei

t = 0.5 ei

Hi. res

x = 1.9 ei

y = 1.1 ei

t = 0.2 ei

nx = 160

ny = 280

nv = 80 v = 0.1 vt0

min / max = -1.1e-2 / 5.4e-3min / max = -3.6 / +1.8 kG

Bz t = 85ps

min / max = -0.97e-2 / 3.4e-3

Low resHi. res

Higher resolution agrees well early on

min / max = -3.2 / +1.1 kG

x / mfp

y /

mfp

x / mfp

y /

mfp

min / max = -0.13 / 0.4 min / max = -1.8e-2 / 5.4e-3

Low resHi. res

Bz t = 255ps

… but an instability grows later on

min / max = -5.9 / +1.8 kGmin / max = -33 / +33 kG

x / mfp x / mfp

y /

mfp

€

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

€

q = −κ ⋅∇Te − Teβ ⋅j

Braginskii’s transport relations

VFP heat flow profile more diffuse than Braginskii

t = 85ps

Braginskiiheat flow

VFPheat flow

x / mfp

y /

mfp

qqxxqqyy

Larger when initial ne & Te not -averaged

• substantially larger; ~5x at 2ncr

PLANAR ne & Te(t=0)

y /

mfp

x / mfp

EXACT ne & Te(t=0)

t = 85ps

y / mfp0 300

0

0.001

-0.002

-0.001

() at ne ~ 2 ncr

Exact

Planar

• Some problems with simulation - Numerical instability happen early on - Need better spatial resolution

• Due mainly to (n) (T)r

Heating mechanism (Maxwellian or IB) -- Same to 10%

min / max = -10e-3 / 6.6e-3 min / max = -9.4e-3 / 5.5e-3

Maxwellian HeatingInverse Bremsstrahlung

Bz t = 500ps

Extended Transport Theory:Modified heat-flux

Extended theory has been shown to predict Heat-flux betterthan classical transport theory (magnetized plasma)

C. P. Ridgers, PoP 15, 092311 (2008)

m = 3.3