#### Transcript 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?
2003 HEDP Class
Inroductory Lecture
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The equilibrium regimes of HEDP
National Research
Council Report, 2002
“Frontiers in High Energy
Density Physics: The X Games
of Contemporary Science”
2003 HEDP Class
Inroductory Lecture
Page 4
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
2003 HEDP Class
Inroductory Lecture
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The EOS Landscape for HEDP
•
Rip Collins will
discuss EOS at much
more length on
Thursday
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
Page 6
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
thermodynamic
variables (,T) can be
inferred from the
properties of shocks
Pressure
(GPa)
Compression (density ratio)
Credit: Keith Matzen, Marcus Knudson, SNLA
2003 HEDP Class
Inroductory Lecture
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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
•
dI
  I  m I
dx
•
The opacity has units of 1/cm or cm2/g
•
Opacity matters because the interaction of matter and radiation is
important
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
•
From LANL “SESAME”
tables
•
Can see regimes
affected by atomic
structure
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
Page 10
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
pressure
•
As the star expands, 
decreases, letting more
increasing the pressure
decrease
Iron transmission based on
Da Silva 1992
Transmission
e
- d
Both HEDP experiments and
sophisticated computer
calculations were essential to
quantitative understanding

2003 HEDP Class
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
Credit:
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
•
– 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
so
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
Page 14
… or zap it with a laser
•
The laser is absorbed at less than 1% of solid density
Bill Kruer will
explain laserplasma
interactions
tomorrow
morning
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
Page 15
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  
2
2/3
with f ~ 0.1 and
 ~ 1.5ncrit kBT ~ 2.6 105
keV
TkeV J
2 cm3
Pressure from momentum balance
 (p = momentum flux)

p  Mi
2/3
k BT
kBT
I14
 ncrit
 ncrit kBT  3.5 2 / 3 Mbars
Mi
Mi

– 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
plasma
Denser,
Hotter
downstream
csd > vs
here
Shock velocity, vs
csu < vs
here
Initial plasma
2003 HEDP Class
Inroductory Lecture
upstream
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
2003 HEDP Class
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
– 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”
density
•
A single particle or a PIC (particle-in-cell) approach is needed for
the relativistic regime and may help when there are strong waves
2003 HEDP Class
Inroductory Lecture
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
Most phenomena can be seen with a singlefluid approach
•
Continuity Equation
•
Momentum Equation
•
•
•
•

  u
t
u
  uu  p  pR     Fother
t
, velocity u , pressure p , radiation pressure pR
Density 
Viscosity tensor

, other force densities
Fother
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
Equation:
Material Energy
Flux m
  u 2 


 
u 2
 E R    u 
 
 pu
t 
2
2 

 

J  E  Fother  u    FR  pR  E R u  Q   v  u
 ei
~1
 pe
Or
Ideal
MHD
2003 HEDP Class
Typ.
small
m

m
 hydro
Smaller
or
Hydro-like

Inroductory Lecture

m
Pe
m
Re
Page 21
So let’s discuss dynamic phenomena
• Sound waves  = cs k
or
f (Hz) = cs / 
• Shock waves
• Rarefactions
• Instabilities
2003 HEDP Class
Inroductory Lecture
Page 22
It’s easy to make a shock wave with a laser
Laser beam
Any material
Thicker layer for
Laser: 1 ns pulse (easy)
diagnostic
≥ 1 Joule (easy)
(implies spot size of 100 µm at 1 J,
1 cm at 10 kJ)
Emission
From rear
Time
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
Page 24
We analyze shocks in a frame of reference
where the shock is at rest
Matter leaves at
slower velocity, vd
Denser,
Hotter
Density u here
Density d here
From continuity
equation:
From momentum
and energy
equations:
Matter comes in at velocity
of shock in lab frame, vs

vd  vs u
d
d
  1M 2

u   1M 2  2
2
p d 2M   1

pu
  1
For strong shocks

  1
  1

2
M2
  1
Marcus Knudson will tell you much more about shocks
2003 HEDP Class
Inroductory Lecture
Page 25
Where the density drops,
plasmas undergo rarefactions
• The outward flow of matter
with a density decrease is a
rarefaction
– The Sun emits the solar
wind
• Rarefactions can be abrupt
– When shock waves or blast
waves emerge from stars or Density
dense plasma, a rarefaction
occurs
Position
2003 HEDP Class
Inroductory Lecture
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Many HEDP experiments have both shocks and
rarefactions
SN 1987A
Sketch of
Experiment
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
rarefactions
Inroductory Lecture
Page 27
When rarefactions overtake shocks, “blast
waves” form
•
Planar blast
wave
produced
by a 1 ns
laser pulse
on plastic
From Drake, High-Energy-Density Physics,
Springer (2006)
2003 HEDP Class
Inroductory Lecture
Page 28
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
determines
http://www.chaseday.com/PHO
TOSHP/2JUL76/01-cbnw.JPG
Local density determines
local gravitational force
Net upward force = (<> - )g
2003 HEDP Class
Inroductory Lecture
Page 30
Two mechanisms reduce
Rayleigh-Taylor in HEDP experiments
kg
 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
n

•
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
2003 HEDP Class
Inroductory Lecture
Page 31
Rayleigh-Taylor also occurs in flow-driven
systems
• Ejecta-driven systems
– Rarefactions drive
– Supernova remnants
– Experiments
– Rarefactions often
evolve into blast waves
2003 HEDP Class
A rarefaction can produce
flowing plasma that can drive
instabilities
Inroductory Lecture
Page 32
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.
2003 HEDP Class
Inroductory Lecture
Page 33
.. in observation, and in lab experiment
Remnant
E0102
Blast-wave
driven lab
result
Dmitri Ryutov
will tell you
more….
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
2003 HEDP Class
Inroductory Lecture
Page 34
Here’s how we do such experiments
•
Precision structure inside a
shock tube
2003 HEDP Class
•
Interface with 3D
modulations
Inroductory Lecture
From Drake et al.
Phys. Plas. 2003
Page 35
The second major instability driver is lift
U
Flow
Rippled
interface
Flow
U
Airplane wing
2003 HEDP Class
Kelvin-Helmholtz Instability
Inroductory Lecture
Page 36
For simple abrupt velocity shear the theory is
simple
x s
 ux s  us
t
•
•
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
2 ( a  b )
Wave
propagates
If A ≠ 0
•
Wave
Grows for all kx

However,
velocity gradients with scale length Lu stabilze modes
with k > ~ 2/ Lu
2003 HEDP Class
Inroductory Lecture
Page 37
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.
Page 38
Instead, “vortex shedding” is important in
clump destruction
Clump destruction by blast wave (Robey et al. PRL)
flow (Kang et al. PRE)
Simulation of 1987A ejecta-ring collision
This process is also driven by lift
2003 HEDP Class
Inroductory Lecture
Page 39
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
t
•
Vortex motion can produce large structures in systems that are
not technically “unstable” (as they have no feedback loop).

2003 HEDP Class
Inroductory Lecture
Page 40
A major vorticity effect in astro & ICF is
the Richtmyer-Meshkov “instability”
•
•
Richtmyer Meshkov occurs when a shock crosses a rippled
interface.
Related processes happen with a rippled shock reaches any
interface.
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
Page 41
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
Page 42
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