Transcript Slides for seminar at GSI on 2006.11.07

Theoretical modeling of radiation-dominated
plasmas
M. M. Basko
in collaboration with
J. Maruhn, An. Tauschwitz, P.V.Sasorov,V.G. Novikov, A.S. Grushin
EMMI Workshop, Speyer, September 26-29, 2010
Philosophical proposition
With this my talk I will try to illustrate the following proposition:
Numerical simulations of complex phenomena may add
a new quality to our understanding of these phenomena
Equations of hydrodynamics
The newly developed RALEF-2D code is based on a one-fluid, one-temperature
hydrodynamic model in two spatial dimensions (either x,y, or r,z):

    u  0,
t
 
  
 u
t
  u  u   p  0,
E
     E  p  u      T   Qr  Qdep ,
t
u2
E  e  , e  e(  , T )
2
  T  – energy deposition by thermal conduction (local), Qr – energy deposition by
radiation (non-local),
Qdep
– eventual external heat sources.
2D and 3D ideal hydrodynamics is already a very complex system!
Radiation transport
Transfer equation for radiation intensity
1 I
  I  k  B  I  ,
c t
I
in the quasi-static approximation:


I  I t , x , ,  , B  B  , T 
Quasi-static approximation: radiation transports energy infinitely fast (compared to the fluid
motion)  the energy residing in radiation field at any given time is infinitely small !
In the present version, the absorption coefficient
are calculated in the LTE approximation.
k
and the source function
B = B(T)
Coupling with the fluid energy equation:
Qr    
  d   I d  

 d   k  I  B  d
4
Radiation transport adds 3 extra dimensions (two angles and the photon frequency) 
the 2D hydrodynamics becomes a 5D radiation hydrodynamics !
New quality due to radiation transport
Pure hydrodynamics (with or without thermal conductivity) is local.
Radiation hydrodynamics is non-local !
 poses serious difficulties for the development of adequate numerical
algorithms in 2 and 3 dimensions;
 RALEF-2D is based on a newly developed original algorithm for radiation
transport (not published yet)
Main constituents of the RALEF-2D code
1.
Hydrodynamics: 2D, Godunov, 2nd order in space (CAVEAT)
2.
Thermal conduction: 2D, SSI, 2nd order in space
3.
Radiation transport: 2D, short characteristics, 1st order
4.
EOS and opacities: LTE, Hartree-Fock-Slater (KIAM, Moscow)
5.
Laser absorption: no refraction, rad.transport, inv.bremsstrahlung
Problem 1: Rayleigh-Taylor instability of thin
foils accelerated by direct laser
irradiation
(closely related to the ongoing experiments with the NHELIX/PHELIX
laser beams at GSI)
Foil and laser beam parameters
y
Carbon foil:
2g/cc  0.5μm = 1g/cc  1μm = 100 μg/cm2
Laser beam:
50J/1mm2/10ns = 51011 W/cm2,
2 Nd-glass, λ = 532 nm
Spatial pulse profile
x
1
Temporal pulse profile
Flas(y)
0.1
51011 W/cm2
1.0
FWHM
0.01
Flas(t)
0.5
1E-3
0.0
0
5
t (ns)
10
1E-4
0.0
0.2
0.4
0.6
y (mm)
The laser focal spot is assumed to be perfectly uniform at r < rfoc = 200μm !
Initial density perturbations
The initial density profile of the foil is perturbed along the y-axis (perpendicular to the
laser beam) by superimposing 2 cosine waves with wavelengths of 10 μm and 7.07 μm;
the r.m.s. deviation from ρ0 is set equal to 0.7%.

 1.414 y 
  y 

0.77
cos



5μm
5μm




 ( y) / 0  1  0.01 0.63cos 

1.015
1.010
(y)/0
1.005
1.000
0.995
0.990
0.985
-0.10
-0.05
0.00
y (mm)
0.05
0.10
Simulation without radiation transport:
(thermal conduction only)
temperature plots
Simulation with radiation transport:
temperature plots
Comparison of density distributions at t= 3.5 ns
Comparison of temperature distributions at t= 3.5 ns
Column density variation along the foil surface
t = 3.5 ns
no radiation
rad. on, 1 laser
rad. on, 2 laser
3.0
2
<x> (g/mm )
2.5
2.0
1.5
1.0
0.5
0.0
0.00
0.02
0.04
0.06
0.08
0.10
y (mm)
Radiation smoothing reduces variations in  dx from about 5:1 down to ±13%.
What do we learn from the simulations
 Thermal conduction has little effect (under the conditions studied) on the
development of the Rayleigh-Taylor instability: the turbulent-mixing zone near
the accelerated interface expands as (0.05–0.07)
gt2 .
 Smoothing due to thermal radiation transport invalidates the classical picture
of the Rayleigh-Taylor instability.
 If the focal spot is sufficiently uniform, a directly irradiated foil may be
accelerated in essentially 1D manner – but with a dramatic increase in the foil
thickness.
 Transition from the 1 to the 2 (or to the 3) laser light is quite helpful.
Problem 2: A strongly radiating central Z-pinch
in tungsten multi-wire arrays
(experiments at Sandia, USA and in Troitsk, Russia)
Multi-wire Z-pinches (Sandia, Angara-5)
40-mm diameter array of 240, 7.5-μm-diam. wires.
Z-machine at Sandia (USA):
• 11.5 MJ stored energy
• 19 MA peak load current
• 40 TW electrical power to
load
• 100-250 TW x-ray power
• 1-1.8 MJ x-ray energy
X-ray pulses at Z (W.A.Stygar et al., PRE 2004)
Initial MHD phase of wire implosion (J.P.Chittenden et al.)
Problem statement for an RH simulation
Cylindrical implosion of an initially
cold tungsten plasma cloud
Initial shell parameters:
• radial thickness: 2 mm;
• implosion velocity: v0 = 400 km/s;
• uniform temperature: T0 = 20 eV;
v0
• mass: m0 = 0.3 mg/cm (A); 6.0 mg/cm (Z);
• kinetic energy: 24 kJ/cm (A); 480 kJ/cm (Z);
• far from the axis, mass is uniformly distributed over
the radius;
• possible influence of the magnetic field is ignored.
In its present formulation, the problem is one-dimensional.
The goal of the simulation
Our primary goal: to calculate and compare with the experiments the X-ray spectrum
and spectral images of the imploding tungsten plasma.
Tungsten EOS and opacities
The equation of state and opacities (LTE) of tungsten have been provided by the
V.G.Novikov et al. from KIAM (Moscow) ; calculated with the THERMOS code based on
the Hartree-Fock-Slater atomic model.
-1
Absorption coefficient k (cm )
W: T=0.25 keV, =0.01 g/cc
100
 Hydrodynamics was simulated
with either 8 or 32 spectral
groups.
THERMOS data
8 -groups
32 -groups
 The output spectra were
calculated in the post-processor
mode by solving the radiation
transfer equation with 200
spectral groups.
10
1
0.1
1
Photon energy h (keV)
10
Case A: the Angara-5 (RUS) parameters
m0 = 0.3 mg/cm
t = 3 ns
Density
t = 3 ns
Total X-ray emission power: case A
X-ray emission power PX (TW/cm)
6
Case A
5
The imploding tungsten plasma
radiates away about 90% of its
initial kinetic energy.
4
3
The nominal power of implosion is
2
1
MU im2 / t pulse 
2
 24 kJ cm 1 / 5 ns  4.8 TW cm 1
Wnom 
nominal power
DEIRA
RALEF, 8 -groups
RALEF, 32 -groups
1
0
0
1
2
3
Time (ns)
4
5
6
Stagnation shock: case A
Case A: t = 3 ns
Density  (g/cc), temperature T (keV)
4
2

0.35
0.3
1
0.8
0.6
 T14 
0.25
0.2
0.4
0.041
0.042
0.043
0.1
0.08
0.06
The shocked material radiates
away about 90% of its initial kinetic
energy.

The shock front is marked by an
extremely narrow peak of matter
temperature, which defines the
hard tail of the emitted spectrum.

The density jumps by a very large
factor of ~60.
0.044
density
matter temperature
radiation temperature
0.04
0.05
0.10
0.15
Radius (mm)
0.20
0.25
1
U 03
2

0.2
0.02
0.00
Here we deal with a supercritical
RD shock front (see Zel’dovich and
Raizer, chapter VII):
Comparison with the 1D 3T DEIRA code
20
Te, RALEF
Tr, RALEF
Te, DEIRA
Tr, DEIRA
Case A: t = 3ns
0.35
10
8
6
4

Ti
Te
Tr
2
1
0.8
0.6
0.4
0.2
0.30
0.25
0.20
0.15
0.1
0.08
0.06
0.0365
Temperature (keV)
Density (g/cc), temperature (keV)
0.40
 rs
0.0370
0.0375
0.0380
Radius (mm)
0.10
0.0385
0.0390
0.041
0.042
0.043
Radius (mm)
0.044
Spatially integrated emission spectra: case A
High-resolution emission spectra can be obtained in the post-processor regime even when
hydrodynamics is simulated with a relatively small number (8, 32) of spectral groups.
The hard component contains ~16% of the total X-ray pulse energy.
1
-1
-1
RALEF 8 -groups
RALEF 32 -groups
Planckian,
T = 0.11 keV
0.5
0.0
0.01
Case A: t = 3 ns
RALEF, 8 -groups
RALEF, 32 -groups
Planckian,
T = 0.34 keV
0.1
-1
-1
-1
1.0
Spectral flux F (TW cm sr keV )
-1
Spectral flux F (TW cm sr keV )
Case A: t = 3 ns
0.01
1E-3
1E-4
1E-5
0.1
Photon energy h (keV)
1
0
1
2
3
4
5
Photon energy h (keV)
6
7
Optical thickness of the imploding plasma: case A
Radial optical depth
20
Case A: t = 3 ns
h = 40 eV
h = 210 eV
h = 330 eV
h = 2.0 keV
h = 5.1 keV
10
8
6
4
Depending on a selected
frequency, the shock front lies at
an optical depth of  = 0.5–10.0
2
1
0.8
0.6
0.4
 a large portion of the X-rays
emitted by the shock front is
reprocessed in the tenuous
infalling plasma.
0.2
0.1
0.08
0.06
0.0
0.1
0.2
0.3
0.4
0.5
Radius (mm)
0.6
0.7
0.8
X-ray images of the imploding pinch: case A
-1
Radiation intensity I (TW cm ster keV )
Spectral X-ray images provide information on the internal structure of the imploding
pinch.
Case A: t=3 ns
h = 0.205 keV
h = 0.27 keV
h = 1.81 keV
-1
30
-2
25
20
15
10
5
0
-1.0
-0.5
0.0
0.5
1.0
Distance along observation slit (mm)
Case Z: the Z-accelerator (USA) parameters
m0 = 6 mg/cm
Total X-ray emission power: case Z
120
Case Z
X-ray emission power PX (TW/cm)
100
The imploding tungsten plasma
radiates away about 80% of its
initial kinetic energy.
80
60
The nominal power of implosion is
1
MU im2 / t pulse 
2
 480 kJ cm 1 / 5 ns  96 TW cm 1
Wnom 
40
nominal power
DEIRA
RALEF 8 -groups
RALEF 32 -groups
20
0
0
1
2
3
Time (ns)
4
5
6
Density  (g/cc), temperature T (keV)
Stagnation shock: case Z
10
8
6
Case Z: t = 3 ns
4
0.55
0.5
2
Here the shock front is
almost a factor 2 more
narrow than in case A –
which is counter-intuitive!
0.45
1
0.8
0.6
0.4
0.130
0.131
0.132
0.4
0.2
0.1
0.08
0.0
density
T matter
T radiation
0.1
0.2
0.3
0.4
Radius (mm)
0.5
0.6
Spatially integrated emission spectra: case Z
Here both Planckian components have significantly higher temperatures than in case A.
The hard component contains ~7% of the total X-ray pulse energy.
RALEF, 8 -gr.
RALEF, 32 -gr.
Planckian,
T=0.21 keV
Case Z: t = 3 ns
-1
-1
10
RALEF, 8 -groups
RALEF, 32 -groups
Planckian, T=0.53 keV
1
-1
10
Case Z: t = 3 ns
Spectral flux F (TW cm sr keV )
-1
-1
-1
Spectral flux F (TW cm sr keV )
15
5
0.1
0.01
1E-3
1E-4
1E-5
0
0.1
1
Photon energy h (keV)
0
1
2
3
4
5
6
7
Photon energy h (keV)
8
9
10
The observed spectrum (Sandia)
Optical thickness of the imploding plasma: case Z
600
400
Case Z: t = 3 ns
h = 40 eV
h = 210 eV
h = 260 eV
h = 2.15 keV
h = 5.1 keV
200
Radial optical depth
100
80
60
40
20
10
8
6
4
 practically all the X-rays
emitted by the shock front
are reprocessed by the
infalling plasma.
The “photosphere” radius
is r ~ 0.7 mm.
2
1
0.0
Here the shock front lies at an
optical depth of  = 4–100
0.1
0.2
0.3
0.4
0.5
Radius (mm)
0.6
0.7
0.8
X-ray images of the imploding pinch: case Z
-2
-1
-1
Radiation intensity I (TW cm ster keV )
Here the hot shock front, buried in the infalling material, can only be observed at
significantly higher frequencies yhan in case A.
50
Case Z: t=3 ns
h = 0.205 keV
h = 1.81 keV
h = 8.0 keV
40
30
I x 3000
20
10
0
-1.0
-0.5
0.0
0.5
1.0
Distance along observation slit (mm)
What do we learn from the simulations
 The X-ray pulse is generated within a hair-thin supercritical RD shock front,
buried deep in the imploding plasma cloud.
 The radiated spectrum has two major components: the hard component
emerges from the RD shock front, the soft one is emitted from a broad halo in
the colder infalling plasma.
Final remarks
 Unlike in the ideal hydrodynamics, in radiation hydrodynamics there are
virtually no analytical or self-similar solutions to rely upon; hence
 Without numerical simulations, it is virtually impossible to achieve adequate
qualitative understanding of dynamic plasma processes, where energy
transport by thermal radiation is important.