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Recent developments in Vlasov-Fokker-Planck transport simulations relevant to IFE capsule compression. R. J. Kingham, C. Ridgers Plasma Physics Group, Imperial College London 9 th Fast Ignition Workshop, Boston, 3 rd —5 th Nov 2006. Outline. - PowerPoint PPT Presentation
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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