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Hydrodynamic Simulations of HEDPlasmas
P. B. Radha Laboratory for Laser EnergeticsUniversity of Rochester
Fusion Science CenterSummer SchoolSan Diego, 2011
Geometry effects
Nonuniformity growth
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Hydrodynamic codes are used to design, interpret orunderstand the physics of HEDP plasmas
• The complexity and non-linearity of HED plasmas require simulations.
• Fluid codes include a large range of physics, adequate for many applications.
• Many different numerical techniques are used to model the necessary physics.
• Some examples of hydrodynamic simulations of laser generated plasmas including planar foil acceleration, implosions with nonuniformities are shown.
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Outline
• Why simulations? • What physics do we need to model? • How good is the fluid approximation? • How are hydrodynamics codes typically constructed?• Applications - spherically symmetric implosion - Multi-dimensional simulations - Laser driven planar foil - Laser driven thin shell implosions - x-ray images from laser driven thin shell implosions - movie of shock transit through nonuniform shells
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Hydrodynamic codes are used to design, interpret andunderstand the physics of HEDP plasmas
• Simulations play a key role because of the complexity and non-linearityof several concurrently occurring physical processes.
• Length and time scales of these processes vary over several orders of magnitudes.
• Need simulations to design experiments and quantitatively interpret experiments. - Hydrodynamic simulation codes typically have a suite of postprocessors to calculate observables.
• Simulations are extremely critical for insight into the physics of HEDP plasmas and assessing analytical models.
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D istance (mic)
electrondensity(/cc)
Complex physical processes need to be included inHEDP codes
Laser
Heat conductionRadiation transport
Shell velocity
Laserirradiation
Target
Time (ns)
Power(TW)
0 1 2 30
2
4
6
8
10
12
Critical surface
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Several complex processes in the low density corona ofthe target can, in principle, influence dynamics
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It is impractical to include kinetic effects inhydrodynamic codes
HEDP plasmas
Particle kineticsSolve Boltzman equation(Particle-in-Cell or Monte Carlo)Solution: Velocity distribution of electrons and ionsIntractable on the length and time-scales required for the problem
Fluid approximationApproximate kinetic effects - ignore laser plasma interactions - multi-group flux limited diffusive heat and radiation transport
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The fluid approximation works reasonably well for alarge class of laser generated plasmas
• Distribution of electrons in the corona is close to Maxwellian for 2 14 2 210I Wcm m! µ"<
• Ratio of velocity of electrons, , accelerated by laser field, , to thermal velocity,
22 2 2
14 2
1 [ ( )( ) ( ) 3.6 10
10 ( )
o o
e e o e e
eE I m
m Wcm T keV
! " µ
! # !
$
$% = &
Kruer, “Radiation in plasmas”, World Scientific, Singapore 1984Atzeni, Plasma Phys. and Cont. Fus. 29, 1535 (1987)
• Typical values 15 210 , 351 , 3
eI Wcm nm T keV!"= = =
2( ) ~ 0.02 1o
e
!
!<<
oE
o!
e!
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Fluid approximation works reasonably well for a largeclass of problems
• Electron mean free paths are small compared to temperature scale lengths
• Exception: near critical density and ablation surface (come back to this later)
Assume same temperature for electrons and ions
~e TL!
Braginskii, Reviews of Plasma Physics, Consultants Bureau, New York 1965Atzeni, Plasma Phys. and Cont. Fus. 29, 1535 (1987)
2
2 2
( )2 0.064 [ ( ) ]
4 ( ln ln ) ( 1) ln
e
e
i ei ee ei
kT A T kev
e n Z Z Z Z!
" #=
$ + $ + $
• Typical values 33.5, 6.5, 3 , 6 10 / , ln ~ 8eiZ A T keV g cc! "
= = = = # $
~ 4 ( )e Tm L quartercritical! µ <<
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Fluid equations describe conservation laws
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( )0
( ) ( )0
( ) ([ ] )0
( , )
u
t x
u P u
t x
P u
t x
P P
! !
! !
!" !"
! "
# #+ =
# #
# # ++ =
# #
# # ++ =
# #
=
In the frame of the box Eulerian form of fluid equations
Mass
Momentum
Energy
Equation-of-State (EOS)
1 1i iu!
+ +
2
1 1i iu!
+ +
1 1i i! "
+ +
i iu!
2
i iu!
i i! "
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Fluid equations describe conservation laws
dF F Fu
dt t x
! != +! !
In the fluid frame for any function F
Eulerian derivative
Replace to get flow equations in the fluid frame (Lagrangian equations)
2
0d u
dt x
du P
dt x
d P d
dt dt
!!
!
" !
!
#+ =
#
#= $
#
=
( , )P P ! "= ( )V t
( )V t t+ !
iu 1i
u+
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Numerical approaches to solving the fluid equationsinvolve discretization in space and time
space
time
• Variables can be associated with zone centers, zone vertices or zone corners.• Finite difference constructions to fluid equations must be reasonably accurate, numerically stable and reduce to the correct equation when and
1/ 2nt+
!
1n !
n
1n +
2n +
1i ! i 1i + 2i +
1/ 2ix +!
0t! "
0x! "
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Simulations are typically advanced using explicit time-stepping
1/ 2nt+
!
1n !
n
1n +
2n +
1i ! i 1i + 2i +
1/ 2ix +!
Example: Typical Lagrangian formulation
Update acceleration from pressure gradientUpdate time step (for stability of numerical scheme ) Update velocities Determine new positions Get new densitiesDetermine new pressures from EOS
, , ,n n n nx P! "
Known
/s
t x c! < !
1/ 2nu
!
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Additional physics is important for HED plasmas
• Shocks launched by the laser
• Different temperatures for electrons and ions - Electrons collisionally heated by the laser - ions heated by the shock
• Source terms to the energy equation - laser energy deposition - radiation energy - energy deposited by charged particles (alpha particles and ignition)
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Shocks are treated by adding an artificial viscous termto the ion pressure
• Dissipative mechanisms like viscosity and heat conduction introduce a thin transition layer instead of a sharp discontinuity.
• Use an “artificial viscosity”
• in fluid equations
• Originally proposed by Richtmyer and von Neumann
• No special internal boundary conditions required• Shocks as approximate discontinuities in • Obeys the basic conservation laws
2 2( )o
Q a u! "=
0=
0u! <
0u! "
P P Q! +
, ,P! "
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The position of the shock is captured adequately in ashock tube calculation
Analytic First orderSecond order
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The energy equation is simply extended to ions andelectrons
Lagrangian form of energy equation written in terms of two temperatures
Specific heatconductivity Electron-ion
relaxation timeSource terms Ions – shock, alphasElectrons – radiation, laser energy, alphas
Electrons
Ions
( ) ( )
( ) ( )
e e
ve e e ei e i e
i i
vi i i ei e i i
dT T uC P T T S
dt x x x
dT T uC P T T S
dt x x x
! " #
! " #
$$ $= % % % +$ $ $
$$ $= % + % +$ $ $
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The approximation of operator splitting substantiallysimplifies the solution
• N-coupled non-linear equations with N unknowns
• Brute force – matrix inversion of complete set of equations intractable
• Instead, separately solve each piece of physics in a certain order
• example: where are some function of
( , , ,..) ( , , ,..) ...F
A F x t B F x tt
!= + +
!
FA
t
FB
t
!=
!
!=
!
,A B
, ,F x t
Solve sequentiallyUpdate each timeF
• Accuracy depends on the time step
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Deposition of laser energy into a low density plasma istypically modeled using a ray trace algorithm
• Geometrical optics approximation is used with Inverse Bremsstrahlung as the mechanism for energy deposited in cell
• A 2D/3D raytrace is necessary to account for refraction in the low-densitycorona.
Ray propagation equationDispersion relation
22 21/ 2
2 3/ 2 2
ln~ (1 )
( )
peIB
o e e o
n ZK
m kT Z
!
! !
"< > #"
< >
j
Laser ray
1
1(1 )
r
IB
ro
K ds
j jW E e
!
+
"= !
0r
1r
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Geometrical optics ignores various coronal laser-plasma interactions
• Typically thresholds for evaluated to identify if they are exceeded• Ultimately empirical evidence is the best indicator of the adequacy of the raytrace approximation
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Most fluid codes approximate radiative transfer usingmulti-group diffusion
•The frequency spectrum is divided into bins.
• For each group solve the diffusion equation.
, 1, 2,...g g gv v g G! !< < +" =
G
( ) ( ) v
v v v v v
qu uu U c B U P
t x x x x!"
## # # #+ + = $ $ $
# # # # #
Radiationenergy density opacity
Planck function
Radiativeflux
Radiative pressure
emission absorption
• The energy removed or added to each zone is summed over all energy groups and added as a source term to the heat equation.
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Most fluid codes approximate radiation transport withmultigroup diffusion
mfp
v L! <<• Good approximation when
• Assumes that radiation and electrons are in equilibrium so a single temperature describes both.
• In highly ionized plasmas, inverse Bremsstrahlung dominates.
• At higher densities photo-ionization and line absorption are important.
• Radiation transport should accurately take this into account.
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Empirical studies indicate that the heat flux inhibition isrequired in laser driven plasmas
• Refer to equations on VG 17
• An implicit procedure is typically used to solve the diffusive heat-conduction equation.
• Near the critical surface . Diffusion breaks down in this limit.
• The heat flux may exceed the flux of freely streaming electrons,which is unphysical.
• This makes it necessary to limit the flux in an ad-hoc manner.
• Flux inhibition may be due to the breakdown of the diffusion approximation, megaGauss magnetic fields, laser-plasma instabilities.
~e TL!
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Empirical studies indicate that heat flux inhibition isrequired in laser driven plasmas
•
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Equations-of-State close the fluid equations
( , )P P T!=
Implosions can reach extreme states of matter
Density – 10-3-1500 g/ccTemperature 1-100 eV
Matter can be degenerate,strongly coupled
EOS must capture these physics
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Inline or tabulated equations of state are used in fluidcodes
• The implementation of an EOS is relatively straightforward.
• Finite difference numerical stability is determined by sound speed
( )s s
Pc
!
"="
- Sound speed is obtained from the EOS
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The range of physics used in these codes is broad
Particletransport
Control
Setup
Hydrodynamics
Radiation transport
Laser deposition
Heat conduction
Equation-of-State
Rezoning (ALE)
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Several different codes exist for modeling lasergenerated plasmas
LILAC 1D ALE University of RochesterDRACO 2D ALE University of Rochester 2D Sliding mesh EulerianLASNEX Lawrence Livermore National LabsHYDRA 3D Lawrence Livermore National LabsFAST 2D / 3D Sliding mesh Eulerian Naval Research LaboratoryRAGE 2D / 3D Eulerian AMR Los Alamos National Lab.Allegra 2D / 3D ALE MHD Sandia National LaboratoryFLASH 1D/2D/3D AMR University of ChicagoCRASH 2D/3D AMR University of MichiganFCI 2 2D ALE CEA, FranceHyades 1D Cascade Applied Sciences
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Application 1: Spherically symmetric implosion (LILAC)
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Application 1: Spherically symmetric implosion
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1 F. Seguin et al., Phys. Plasmas 9, 2725 (2002)
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Application 1: Spherically symmetric implosion
SimulatedMeasured
Neutron production history27µm-thick-CH shell, 1ns-square pulse
Nonuniformities are the likely reason that measurements deviate from simulation
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Implosions are Rayleigh-Taylor unstable duringacceleration and deceleration
Distance (mic)
Density(g/cc)
Pr essure(GBar)
0 10 20 30 40 50 600
30
60
90
120
0
10
20
30
40
Distance (mic)
Density(g/cc)
Pressure( GBar)
200 225 250 275 3000
1
2
3
4
5
0
20
40
60
80
Classical Rayleigh-Taylor instabilityAcceleration
RT unstable
Deceleration
RT unstable
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Multidimensional fluid codes can be written byextending the 1D models
Replace
u u!r P
Px
!"#
!
uu
x
!"#
!
r
z
r
( , , )T P!
( , )u i jr
( 1, )u i j+r
( 1, 1)u i j+ +r
( , 1)u i j +r
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Typically a combination of two different parallelizationschemes are used in HEPD codes
Domain decomposition Group parallel
P1 P2 P3 P1(G1-GN), P2(GN+1-GN+M), P3(GM+1-GO)
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Typically a combination of two different parallelizationschemes are used in HEPD codes
Hydro stepThermal TransportEOSCalculate Opacities
Master MasterSlaves by domain
Slaves by group
Laser DepositionMGD Radiation transportMonte Carlo Rad. Transp. Charged Particle Transp.
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Parallelization speeds up computation considerably butonly up to a point
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Application 2: Ablatively driven thin shells
1 J. D. Lindl, Inertial Confinement Fusion, Springer Verlag New York 19982 C. D. Zhou and R. Betti, Phys. Plasmas 14 (2007)3 P. B. Radha et al., Phys. Plasmas 18 (2011)
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Application 2: Ablatively driven thin shells
2D DRACO1 simulations of implosions
Nonuniformity seeds: single beam speckle Beam-to-beam variationsAt peak neutron production
1 P. B. Radha et al., Phys. Plasmas 12, 056301 (2005)
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Application 2: Ablatively driven thin shells
20 30 40 50 60 70 IFAR
Simulated areal densities Measured areal densities
Laser speckle sources
Imbalances between beams
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Codes are needed for complex geometries which canbechallenging for analytic approaches
Fast Ignition1Astrophysical Jets2
Planar foil acceleration3
1 R. Kodama et al., Nature 418, 933 (2002)2 S. Sublett, Division of Plasma Physics Meeting, 2004. 3 S. X. Hu Phys. Rev. Lett., 101, 055002 (2008)
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Application 3: Ablatively driven thin foils
S. X. Hu et al., Phys. Rev. Lett. 101, 055002 (2008)
• Test of heat conduction models
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Application 3: Ablatively driven thin foils
S. X. Hu et al., Phys. Rev. Lett. 101, 055002 (2008)
Sidelit image Postprocessed image from DRACO
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Application 4: Ablatively driven thin shells withbacklighting
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Application 4: Ablatively driven thin shells withbacklighting
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Application 5: Movie of shock transit through clumpyfoam
http://www.pas.rochester.edu/~tim/introframe/150.2.lr.p2_4.mov
Available at:
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References
1. “Numerical Modeling in Applied Physics and Astrophysics” - Bowers and Wilson
2. Computational Methods for Astrophysical Fluid Flow, Vol 199 - LeVeque et al.
3.David J. Benson, “Computational Methods in Lagrangian and Eulerian Hydrocodes”, Computer Methods in Applied Mechanics and Engineering 99, 235 (1992).
4. Hirt, Amsden, and Cook, “An Arbitrary Lagrangian-Eulerian Computing Method for all Flow Speeds”, Journal of Computational Physics 14 (3):227-253 (1974).
