Post on 29-Dec-2015
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