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 )

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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. !

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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.!

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∝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 )