Hydrodynamics Driven by High-Energy

Introduction to
High Energy Density Physics
R. Paul Drake
University of Michigan
High-Energy-Density Physics
The study of systems in which the pressure exceeds 1 Mbar (= 0.1
Tpascal = 1012 dynes/cm2), and of the methods by which such
systems are produced.
In today’s introduction to this field, we will cover
– Part 1: An overview of the physics
– Part 2: The toys (hardware and code)
– Part 3: The applications
My task is to give you a perspective and some context, within
which you can better appreciate the lectures from experts you will
hear this week.
2003 HEDP Class
Inroductory Lecture
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How is HEDP connected to other areas?
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The equilibrium regimes of HEDP
Adapted from:
National Research
Council Report, 2002
“Frontiers in High Energy
Density Physics: The X Games
of Contemporary Science”
2003 HEDP Class
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What is Equation of State or an EOS?
Simple example: p = RT
In general an equation of state relates one of the four
thermodynamic variables (, T, p, ) to two others.
Codes for HEDP often work with density and temperature(s), and
thus need p(, T) and (, T). This may come in formulae or tables.
An equation of state is needed to close the fluid equations, as we
will see later.
Another important example is the adiabatic EOS: p = C 
  = 5/3 for an ideal gas or a Fermi-degenerate gas
  = 4/3 for a radiation-dominated plasma
  ~ 4/3 for an ionizing plasma
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The EOS Landscape for HEDP
Rip Collins will
discuss EOS at much
more length on
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
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EOS results are often shown as the pressure
and density produced by a shock wave
This sort of curve is
called a Shock
Hugoniot (or RankineHugoniot) relation.
The other two
variables (,T) can be
inferred from the
properties of shocks
Compression (density ratio)
Credit: Keith Matzen, Marcus Knudson, SNLA
2003 HEDP Class
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Why do we care about EOS?
Whether we want to
– make inertial fusion work,
– model a gas giant planet, or
– understand the structure of a
white dwarf star,
we need to know how the
density of a material varies
with pressure
Here is one theoretical model
of the structure of hydrogen
Saumon et al., 2000
2003 HEDP Class
Inroductory Lecture
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What is Opacity?
The spatial rate of attenuation of radiation
For radiation intensity (power per unit area per steradian) I:
  I  m I
The opacity has units of 1/cm or cm2/g
Opacity matters because the interaction of matter and radiation is
for much of the HEDP regime
The opacity has contributions from absorption and scattering. In
HEDP absorption typically dominates. The absorption opacity is
often labeled .
2003 HEDP Class
Inroductory Lecture
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Examples of opacity
Opacity of Aluminum
Can see regimes
affected by atomic
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
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One application: Cepheid variable stars
These stars have regions
on uphill slopes of an
opacity “mountain”
As the star contracts,
increases, holding in
more heat and producing a
greater increase in
As the star expands, 
decreases, letting more
radiation escape and
increasing the pressure
Iron transmission based on
Da Silva 1992
- d
Both HEDP experiments and
sophisticated computer
calculations were essential to
quantitative understanding
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Inroductory Lecture
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X-ray absorption and emission has major
implications for the universe
X-ray opacity measurements have other important applications
– Understanding the universe: light curves from Type Ia supernovae
Credit: Joe Bergeron
Jha et al.,
Harvard cfa
Studies of photoionized plasmas are required
– To resolve discrepancies among existing models
– To interpret emission near black holes regarding whether Einstein had
the last word on gravity
– To interpret emission near neutron stars to assess states of matter in
huge magnetic fields
2003 HEDP Class
Inroductory Lecture
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Many exciting phenomena in HEDP
come from the dynamics
Shock waves and other hydrodynamic effects
Hydrodynamic Instabilities
Dynamics involving radiation (radiation hydrodynamics)
– Radiative heat waves
– Collapsing shock waves
Relativistic dynamics
2003 HEDP Class
Inroductory Lecture
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So how does one start HEDP dynamics?
Shoot it, cook it, or zap it
Shoot a target with a “bullet”
– Pressure from stagnation against a very dense bullet ~ target (vbullet)2/2
– 20 km/s (2 x 106 cm/s) bullet at 2 g/cc stuff gives ~ 4 Mbar
Cook it with thermal x-rays
– Irradiance T4 = 1013 (T/100 eV)4 W/cm2 is balanced by outflow of
solid-density matter at temperature T and at the sound speed
T 4   T / Mi  p T / Mi /( 1)
– From which
2003 HEDP Class
T /Mi
p   1 Mi T 3.5
Inroductory Lecture
 T  3.5
~ 20
 Mbars
100 eV
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… or zap it with a laser
The laser is absorbed at less than 1% of solid density
Bill Kruer will
explain laserplasma
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
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We can estimate the laser ablation pressure
from momentum balance
Temperature from energy balance
– Irradiance IL = 1014 I14 W/cm2 is carried away by flowing electrons
– Energy balance is
– One finds
IL ~ f T /me
T ~ 2I14  
with f ~ 0.1 and
 ~ 1.5ncrit kBT ~ 2.6 105
TkeV J
2 cm3
Pressure from momentum balance
 (p = momentum flux)
p  Mi
k BT
 ncrit
 ncrit kBT  3.5 2 / 3 Mbars
– This is a bit low; the flow is actually faster (3.5 -> 8.6)
2003 HEDP Class
Inroductory Lecture
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Most HEDP dynamics
begins with a shock wave
If I push a plasma boundary forward at a speed below cs, sound
waves move out and tell the whole plasma about it.
If I push a plasma boundary forward at a speed above cs, a shock
wave is driven into the plasma.
In front of the shock wave, the plasma gets no advance warning.
The shock wave heats the plasma it moves through, increasing cs
behind the shock.
Behind the shock, the faster sound waves connect the entire
csd > vs
Shock velocity, vs
csu < vs
Initial plasma
2003 HEDP Class
Inroductory Lecture
Mach number
M = vs / csu
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Much of the excitement in HEDP comes from
the dynamics
Shock waves establish the HEDP regime of an experiment
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Inroductory Lecture
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HEDP theory: a fluid approach often works, but
not always
Most phenomena can be grasped using a single fluid
– with radiation,
– perhaps multiple temperatures
– perhaps heat transport, viscosity, other forces, and
A multiple fluid (electron, ion, perhaps radiation or other ion)
approach is needed at “low” density
Magnetic fields sometimes matter
Working with particle distributions (Boltzmann equation and
variants) is important when strong waves are present at “low”
A single particle or a PIC (particle-in-cell) approach is needed for
the relativistic regime and may help when there are strong waves
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Most phenomena can be seen with a singlefluid approach
Continuity Equation
Momentum Equation
  u
  uu  p  pR     Fother
, velocity u , pressure p , radiation pressure pR
Density 
Viscosity tensor
, other force densities
Hydrodynamics is complicated because the nonlinear terms in
these equations matter essentially
2003 HEDP Class
Inroductory Lecture
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The energy equation has a number of terms that
often don’t matter
• General Fluid Energy
Material Energy
Flux m
  u 2 
 
u 2
 E R    u 
 
 pu
t 
2 
 
J  E  Fother  u    FR  pR  E R u  Q   v  u
 ei
 pe
2003 HEDP Class
 rad
 hydro
Inroductory Lecture
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So let’s discuss dynamic phenomena
We start with hydrodynamics
• Sound waves  = cs k
f (Hz) = cs / 
• Shock waves
• Rarefactions
• Instabilities
2003 HEDP Class
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It’s easy to make a shock wave with a laser
Laser beam
Any material
Thicker layer for
Laser: 1 ns pulse (easy)
≥ 1 Joule (easy)
Irradiance ≥ 1013 W/cm2
(implies spot size of 100 µm at 1 J,
1 cm at 10 kJ)
From rear
This produces a pressure ≥ 1 Mbar (1012 dynes/cm2, .1 TP).
This easily launches a shock.
Sustaining the shock takes more laser energy.
2003 HEDP Class
Inroductory Lecture
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Astrophysical jets and supernovae make
shocks too
Supernova Remnant
Astrophysical Jet
J. Hester
2003 HEDP Class
Burrows et al.
Inroductory Lecture
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We analyze shocks in a frame of reference
where the shock is at rest
Matter leaves at
slower velocity, vd
Density u here
Density d here
From continuity
From momentum
and energy
Matter comes in at velocity
of shock in lab frame, vs
vd  vs u
  1M 2
u   1M 2  2
p d 2M   1
  1
For strong shocks
  1
  1
  1
Marcus Knudson will tell you much more about shocks
2003 HEDP Class
Inroductory Lecture
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Where the density drops,
plasmas undergo rarefactions
• The outward flow of matter
with a density decrease is a
• Rarefactions can be steady
– Steady (more or less)
– The Sun emits the solar
• Rarefactions can be abrupt
– When shock waves or blast
waves emerge from stars or Density
dense plasma, a rarefaction
2003 HEDP Class
Inroductory Lecture
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Many HEDP experiments have both shocks and
SN 1987A
Sketch of
data at 8 ns
R.P. Drake, et al.
ApJ 500, L161 (1998)
Phys. Rev. Lett. 81, 2068 (1998)
Phys. Plasmas 7, 2142 (2000)
2003 HEDP Class
This experiment to reproduce the
hydrodynamics of supernova
remnants has both shocks and
Inroductory Lecture
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When rarefactions overtake shocks, “blast
waves” form
Planar blast
by a 1 ns
laser pulse
on plastic
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
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Hydrodynamic instabilities are common
Chevalier, et al.
ApJ 392, 118 (1992)
Instability in a simulation of supernova remnant
• Three sources of structure
– Buoyancy-driven instabilities (e.g. Rayleigh-Taylor)
– Lift-driven instabilities (e.g. Kelvin-Helmholtz)
– Vorticity effects (e.g. Richtmyer-Meshkov)
2003 HEDP Class
Inroductory Lecture
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Buoyancy-driven instabilities
are very important
• The most important are bouyancy-driven
Convective cloud formation
– Rayleigh Taylor
– “Entropy mode” or “Convective mode”
• Examples of this:
Rayleigh Taylor
Average density
pressure gradient
Local density determines
local gravitational force
Net upward force = (<> - )g
2003 HEDP Class
Inroductory Lecture
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Two mechanisms reduce
Rayleigh-Taylor in HEDP experiments
 kv Ablation
1 kL
Approximate exponential growth rate
Gradient scale length (L) reduces growth rate
Ablation removes material at a speed vAblation, stabilizing RayleighTaylor at large k
There is an interplay of initial conditions and allowable growth
Riccardo Betti will discuss the ICF case Thursday
Experiments have gone beyond ICF-compatible growth
Remington et al.
Phys. Fl. B 1993
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Rayleigh-Taylor also occurs in flow-driven
• Ejecta-driven systems
– Rarefactions drive
nearly steady shocks
– Supernova remnants
– Experiments
– Rarefactions often
evolve into blast waves
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A rarefaction can produce
flowing plasma that can drive
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Supernova remnants produce the RayleighTaylor driven by plasma flow in simulation, …
1D profile and 2D simulations
In supernova remnants
Chevalier, et al. ApJ 392, 118 (1992)
and supernovae
Kifonidis, et al.
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.. in observation, and in lab experiment
driven lab
Dmitri Ryutov
will tell you
Supernova Remnant E
0 102 - 72 fr o m Radio to X- Ray
Credit: X - ra y (NASA/C XC/ SAO); optical (NASA/HST ); radio : (ATNF/
ATCA) htt p ://antwrp.gsfc.nasa.gov/apod/ap00
0 414.h t ml
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Here’s how we do such experiments
Precision structure inside a
shock tube
2003 HEDP Class
Interface with 3D
Inroductory Lecture
From Drake et al.
Phys. Plas. 2003
Page 35
The second major instability driver is lift
Airplane wing
2003 HEDP Class
Kelvin-Helmholtz Instability
Inroductory Lecture
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For simple abrupt velocity shear the theory is
x s
 ux s  us
Start with Euler equations
Plus continuity of the interface:
For abrupt shear flow (i.e., velocity difference) at an interface, find
Kelvin Helmholtz instability growth rate
 A
kxU 2 a b
n  ikx U 
2 ( a  b )
If A ≠ 0
Grows for all kx
velocity gradients with scale length Lu stabilze modes
with k > ~ 2/ Lu
2003 HEDP Class
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Kelvin-Helmholtz makes mushrooms on
Rayleigh-Taylor spike tips
Supernova simulation by Kifonidis et al.
Lab simulation: Miles et al.
But not so much along the stems.
A big difference among codes is how
much “hair” they grow on the stems.
2003 HEDP Class
Inroductory Lecture
Data in Robey et al.
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Instead, “vortex shedding” is important in
clump destruction
Clump destruction by blast wave (Robey et al. PRL)
Clump destruction by steady
flow (Kang et al. PRE)
Simulation of 1987A ejecta-ring collision
This process is also driven by lift
2003 HEDP Class
Inroductory Lecture
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This is a natural entry to the third category:
vorticity effects
  u
Vorticity is defined as
Volumetric vorticity corresponds to swirling motions
Shear flows generate surface vorticity
Volumetric vorticity is transported like magnetic fields in plasmas
   (u  )   2
Vortex motion can produce large structures in systems that are
not technically “unstable” (as they have no feedback loop).
2003 HEDP Class
Inroductory Lecture
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A major vorticity effect in astro & ICF is
the Richtmyer-Meshkov “instability”
Richtmyer Meshkov occurs when a shock crosses a rippled
Related processes happen with a rippled shock reaches any
The shear flow across
the interface drives it
to curl up.
The ripple may or may
not invert in phase,
depending on details.
The modulations grow
at most linearly in time
2003 HEDP Class
Inroductory Lecture
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Richtmyer Meshkov can produce spikes and
bubbles like those from Rayleigh-Taylor
Strong-shock case
The vorticity deposited by a
shock on a rippled interface
causes the denser material
to penetrate to the shock
From Glendinning et al., Phys.
Plas. 2003
2003 HEDP Class
Inroductory Lecture
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