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NLTE effects in the coupled CRASH-RADIOM code.!
Igor Sokolov, Bart van der Holst , R. Paul Drake (UofM)!Marcel Klapisch and Michel Busquet (ARTEP) !
The NLTE study is relevant to the main objectives!
� Key objectives:!o Predictive Science!o Uncertainty Quantification!
� CRASH target, key features:!o “Solid” (=high-density) wall!o Axial symmetry (not for year 5)!o Wall shock!
LTE Simulation Result !
Governing Equations!
€
Tg (Eg ) : Eg = B(Tg )dεε g
∫ , Prad =13
Egg∑ ,
Cg (T) =dB(T)dT
dεε g
∫ , Eg = Eεdεε g
∫
€
∂ρ∂t
+∇ ⋅ (ρu) = 0,
∂ρu∂t
+∇ ⋅ ρu⊗ u+ (P + Prad )I[ ] = 0,
∂(ρ u2
2+ E)
∂t+∇ ⋅ u(ρ u
2
2+ E + P + Prad )
'
( )
*
+ , = Prad∇ ⋅u,
∂Eg
∂t+∇ ⋅ (uEg ) −
13∇ ⋅u
∂(εEg )∂ε
dεε g
∫ = −13Eg∇ ⋅u+
+∇ ⋅ (cCg (Tg )3κRoss
∇Tg ) + cκPlanckCg (T)(T −Tg )
€
P = PEOS (E,ρ), T = TEOS (E,ρ),
What do we need?!� For xenon, beryllium and plastic we need both
the direct EOS (the electron temperature is an input) and the invesrse EOS (the internal energy density is an input) !
� For high-resolution schemes we need the sound speed, that is:!
!
� The implicit solver for the radiation energy diffusion uses a linearized preconditioner, so that CV is needed. !
€
Cs =∂P∂ρ
$
% &
'
( ) S
=γPρ, γ =
ρP
∂P∂ρ
$
% &
'
( ) T
+TCVP
∂P∂T$
% &
'
( ) ρ
2
Reaction rates in xenon are a possible issue for CRASH !
� LTE happens when!o radiative
recombination (red curve) << 3-body collisional recombination (green curve)!
o Then collisions ensure that Z=Z(Te) only. !
� Non-LTE happens when!o radiation
recombination dominates over 3-body recombination !
o For optically thin media, of course!
!
1012
1014
1016
1018
1020
1022
10−10
10−5
100
105
1010
1015
Plasma Density (cm−3
)
Reactio
n R
ate (sec−
1)
Xenon at 50 eV
Coll. Ion.
Rad. Recom.
Diel. Recom.
3Body Recom.
Atomic density [per cm3]
1022 1020 1018 1016 1014 1012
CRASH density
LTE
Reac
tion rate Radia
tive recom
bination
3-‐body Dielectronic
50 eV Xe
Principle of the RADIOM model !
� Non-LTE of charge state distribution (and excited states to some extent) is mimicked by an "ionization temperature" Tz!o See backup slides for more detail!
� We are able to derive numerically Tz from Ne, Te,!
� Non-LTE total energy is a function of the internal energy density and ionization degree (both functions of Tz ), Te ρ:!
ENLTEtot � 3
2kB�Z
� (Tz)⇥ (Te � Tz) = ELTE (Tz)
hνg ,Eg
Bg (Te )
"#$
%$
&'$
($
0
10
20
30
40
50
Z
-2 -1 0 1 2 3LOGTE
24
25
26
27
28
29
LO
GN
A
nx= 201, 201, it= 0, time= 0.0000
0
10
20
30
40
50
Z
-2 -1 0 1 2 3LOGTE
24
25
26
27
28
29
LO
GN
A
nx= 201, 201, it= 0, time= 0.0000
Ionization degree in Xe
NonLTE with Erad=0 LTE
CRASH experiment
Steps to merge NonLTE to the “fancy” radhydro code !
� Recalculate all thermodynamic derivatives, accounting for derivatives of and over Tz !
� Address the issue of Erad/B(Te) (ratio of actual-to-LTE energy density of radiation, for each energy group)!o Let them vary continuously !
§ Would require 30 more thermodynamic variables, computationally not feasible and inconsistent with assumptions in RADIOM, or!
o Fix them within a time step, probably necessary in codes that split radiation and hydrodynamics !§ Implies a time-step-dependent inconsistency between Te found from the
internal energy, density and fixed Erad/B(Te) and the value of Te used to determine B(Te). !
� A “Fake LTE” test worked:!o Set Erad/B(Te) = 1 in data passed to RADIOM!o Results of simulation were identical to those obtained from LTE
CRASH when they should have been (see quarterly report) !
NLTE movie (3d_laser_rz test)!
NLTE Vs LTE comparison (the same test, time 0.9 ns)!
LTE!
!
NLTE!
NLTE effects shorten the length of the radiative precursor!
1.4 ns high-resolution run (NLTE)!
1.4 ns high-resolution run (LTE)!
Results!
� NonLTE version of the code is slower by a factor ≥ 5. !o More computationally intense EOS function !o Degrade in the preconditioner efficiency!
� More work would have been needed on the implementation and verification to achieve definitive results. !
� Results to date !o Differences in coronal temperatures are seen, as would be
expected, and differences in shock location at early times. !o Any impact of these on our present predictions is probably now
absorbed within the Laser Energy Scale Factor.!o NLTE effects appear to produce modest changes in the radiative
precursor.!§ This is consistent with previous conclusions involving opacity
variations. !
Conclusion!
� It proved far more challenging than anticipated to incorporate NLTE effects in to CRASH using the RADIOM approach !o RADIOM is not just a subroutine, alas !o The interplay of NLTE and thermodynamic quantities is complex!
� We produced a functioning code but have yet to complete long runs with it !o Still trying !
� NLTE remains an area where the fidelity of the CRASH simulation could be improved in future work !
Backup slides#
NLTE model accounts for the deviation from the detailed equilibrium!
� We use the simplified scheme as described in Busquet, PoP,1993 as we illustrate for two-level atom !
� The Maxwellian distribution of free electrons assures the Boltzmann distribution of bound electrons: The Planckian distribution of photons with the same temperature balances the photo-transitions with the same distribution of bound electrons.!
€
∝Ne+↓−−−−−−−−−−−−−− E + = E − + ΔE
∝Nγ (Eγ = ΔE)↑ ↓∝[1+ Nγ (Eγ = ΔE)]
∝Ne− ↑− −−−−−−−−−−−−− E −
0 =dN +
dt= Ccoll (Ne
−N− − Ne+N +) + Crad N−Nγ (Eγ = ΔE) − N +[1+ Nγ (Eγ = ΔE)]{ }
€
N + = N− exp(−ΔE /kBTe )
NLTE model accounts for the deviation from the detailed equilibrium!
!
� Once the radiation is out of equilibrium with the matter, the actual number of photons differs from that for the Planckian spectrum. The terms in braces do not cancel each other, therefore, where the “effective” temperature, Tz , tends to that of electrons in two limiting cases: the electron density is very high or the photon spectrum is Planckian.!
€
∝Ne+↓−−−−−−−−−−−−−− E + = E − + ΔE
∝Nγ (Eγ = ΔE)↑ ↓∝[1+ Nγ (Eγ = ΔE)]
∝Ne− ↑− −−−−−−−−−−−−− E −
0 =dN +
dt= Ccoll (Ne
−N− − Ne+N +) + Crad N−Nγ (Eγ = ΔE) − N +[1+ Nγ (Eγ = ΔE)]{ }
€
N + = N− exp(−ΔE /kBTz )