Introduction to High Energy Density Physics R. Paul Drake University of Michigan

Introduction to High Energy Density Physics

R. Paul Drake University of Michigan

2003 HEDP Class Inroductory Lecture

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.


How is HEDP connected to other areas?


The equilibrium regimes of HEDPAdapted from:

National Research Council Report, 2002

“Frontiers in High Energy Density Physics: The X Games

of Contemporary Science”


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


The EOS Landscape for HEDP

• Rip Collins will discuss EOS at much more length on Thursday

From Drake, High-Energy-Density Physics, Springer (2006)


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 Rankine-Hugoniot) relation.

• The other two thermodynamic variables (,T) can be inferred from the properties of shocks

Credit: Keith Matzen, Marcus Knudson, SNLA

Compression (density ratio)

Pressure (GPa)


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


What is Opacity?

• The spatial rate of attenuation of radiation

• For radiation intensity (power per unit area per steradian) I:

• 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 .

€

dI

dx= −χI = −ρχ mI


Examples of opacity

• Opacity of Aluminum

• From LANL “SESAME” tables

• Can see regimes affected by atomic structure



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 radiation escape and increasing the pressure decrease

Iron transmission based onDa Silva 1992

Transmission

Both HEDP experiments and sophisticated computer calculations were essential to quantitative understanding

€

∝ e-χd


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

• 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

Credit: Joe Bergeron

Credit: Jha et al., Harvard cfa


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


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

– From which €

T /M i

€

T 4 = ε T /M i = p T /M i /(γ −1)

€

p = γ −1( ) M iσT3.5 ~ 20

T

100 eV

⎛

⎝ ⎜

⎞

⎠ ⎟3.5

Mbars


… or zap it with a laser

• The laser is absorbed at less than 1% of solid density

Bill Kruer will explain laser-plasma interactions tomorrow morning



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 with f ~ 0.1 and

– One finds

• Pressure from momentum balance (p = momentum flux)

– This is a bit low; the flow is actually faster (3.5 -> 8.6)

€

~ 1.5ncritkBT ~ 2.6 ×105 TkeV

λ μ2

J

cm3

€

IL ~ fε T /me

€

T ~ 2 I14λ μ2

( )2 / 3

keV

€

p = M i

kBT

M i

×ncrit

kBT

M i

= ncritkBT = 3.5I14

2 / 3

λ μ2 / 3

Mbars


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

Initial plasma

Shock velocity, vs

csd > vs here

csu < vs here

upstreamdownstream

Mach number M = vs / csu


Much of the excitement in HEDP comes from the dynamics

Shock waves establish the HEDP regime of an experiment


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” 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


Most phenomena can be seen with a single-fluid approach

• Continuity Equation

• Momentum Equation

• Density , velocity , pressure , radiation pressure

•

• Viscosity tensor , other force densities

• Hydrodynamics is complicated because the nonlinear terms in these equations matter essentially

€

∂∂t

= −∇ • ρu( )

€

∂u∂t

+ ρ u • ∇( )u = −∇p−∇pR −∇ • σ +Fother

€

€

u

€

p

€

pR

€

€

Fother


The energy equation has a number of terms that often don’t matter

• General Fluid Energy Equation:

€

∂∂t

ρε +ρu2

2+ ER

⎛

⎝ ⎜

⎞

⎠ ⎟+∇ ⋅ ρu ε +

u2

2

⎛

⎝ ⎜

⎞

⎠ ⎟+ pu

⎡

⎣ ⎢

⎤

⎦ ⎥=

−J • E+ Fother • u−∇ • FR + pR + ER( )u+Q − σ v • u( )[ ]

Material Energy Flux m

€

m

Pe

€

m

Re

€

m

PeRad

=

τ rad

τ hydro

Γm

Smalleror Hydro-like

€

ν ei

ωpe

~ 1 Typ. small

Or Ideal MHD


So let’s discuss dynamic phenomena We start with hydrodynamics

• Sound waves = cs k or f (Hz) = cs /

• Shock waves

• Rarefactions

• Instabilities


It’s easy to make a shock wave with a laser

Any material

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)

This produces a pressure ≥ 1 Mbar (1012 dynes/cm2, .1 TP).

This easily launches a shock.

Sustaining the shock takes more laser energy.

Thicker layer for diagnostic

Laser beam

Emission From rear

Time


Astrophysical jets and supernovae make shocks too

Supernova RemnantAstrophysical Jet

Burrows et al. J. Hester


We analyze shocks in a frame of reference where the shock is at rest

Denser,Hotter

Matter comes in at velocity of shock in lab frame, vs

Density d here Density u here

Matter leaves at slower velocity, vd

From continuity equation:

From momentum and energy equations:

vd =vs

ρuρd For strong shocks

ρdρu

=γ+1( )M 2

γ−1( )M 2 +2 ≈

γ+1( )γ−1( )

pdpu

=2γM2 − γ−1( )

γ+1( ) ≈

2γγ+1( )

M 2

Marcus Knudson will tell you much more about shocks


Where the density drops, plasmas undergo rarefactions

• The outward flow of matter with a density decrease is a rarefaction

• Rarefactions can be steady– Steady (more or less) – The Sun emits the solar

wind

Density

Position

• Rarefactions can be abrupt – When shock waves or blast

waves emerge from stars or dense plasma, a rarefaction occurs


Many HEDP experiments have both shocks and rarefactions

SN 1987ASketch of

ExperimentRadiographic data at 8 ns

R.P. Drake, et al.ApJ 500, L161 (1998)Phys. Rev. Lett. 81, 2068 (1998)Phys. Plasmas 7, 2142 (2000)

This experiment to reproduce the hydrodynamics of supernova remnants has both shocks and rarefactions


When rarefactions overtake shocks, “blast waves” form

• Planar blast wave produced by a 1 ns laser pulse on plastic



Hydrodynamic instabilities are common

• Three sources of structure

– Buoyancy-driven instabilities (e.g. Rayleigh-Taylor)

– Lift-driven instabilities (e.g. Kelvin-Helmholtz)

– Vorticity effects (e.g. Richtmyer-Meshkov)

Instability in a simulation of supernova remnant

Chevalier, et al. ApJ 392, 118 (1992)


Buoyancy-driven instabilities are very important

• The most important are bouyancy-driven– Rayleigh Taylor

– “Entropy mode” or “Convective mode”

• Examples of this:

http://www.chaseday.com/PHOTOSHP/2JUL76/01-cbnw.JPG

Convective cloud formation

Rayleigh Taylor

Average density determines pressure gradient

Local density determines local gravitational force Net upward force = (<> - )g


Two mechanisms reduce Rayleigh-Taylor in HEDP experiments

• Approximate exponential growth rate

• Gradient scale length (L) reduces growth rate

• Ablation removes material at a speed vAblation, stabilizing Rayleigh-Taylor 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

€

n ≈kg

1+ kL−βkvAblation

Remington et al. Phys. Fl. B 1993


Rayleigh-Taylor also occurs in flow-driven systems

• Ejecta-driven systems – Rarefactions drive

nearly steady shocks– Supernova remnants – Experiments– Rarefactions often

evolve into blast waves

A rarefaction can produce flowing plasma that can drive instabilities


Supernova remnants produce the Rayleigh-Taylor driven by plasma flow in simulation, …

• 1D profile and 2D simulations

Chevalier, et al. ApJ 392, 118 (1992)

In supernova remnants

and supernovae

Kifonidis, et al.


.. in observation, and in lab experiment

Supernova Remnant E0102- 72 from Radio to X- Ray Credit: X- ray (NASA/C XC/ SAO); optical (NASA/HST); radio: (ATNF/ ATCA) http://antwrp.gsfc.nasa.gov/apod/ap00 0414.html

Blast-wave driven labresult

RemnantE0102

Dmitri Ryutov will tell you more….


Here’s how we do such experiments

• Precision structure inside a shock tube • Interface with 3D

modulations

From Drake et al. Phys. Plas. 2003


The second major instability driver is lift

Airplane wing

Rippled interface

Flow

Flow

Kelvin-Helmholtz Instability

U

U


For simple abrupt velocity shear the theory is simple

• 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

• However, velocity gradients with scale length Lu stabilze modes with k > ~ 2/ Lu

€

n = −ikx

A

2ΔU +

kxΔU

2

2 ρ aρ b

(ρ a + ρ b )Wave

propagates If A ≠ 0

Wave Grows for all kx

€

∂δx s

∂t+ u • ∇δx s = us


Kelvin-Helmholtz makes mushrooms on Rayleigh-Taylor spike tips

Supernova simulation by Kifonidis et al. Lab simulation: Miles et al.

Data in Robey et al.

But not so much along the stems.

A big difference among codes is how much “hair” they grow on the stems.


Instead, “vortex shedding” is important in clump destruction

Simulation of 1987A ejecta-ring collision

QuickTime™ and aGIF decompressorare needed to see this picture.

Clump destruction by blast wave (Robey et al. PRL)

Clump destruction by steady flow (Kang et al. PRE)

This process is also driven by lift


This is a natural entry to the third category:vorticity effects

• Vorticity is defined as

• Volumetric vorticity corresponds to swirling motions

• Shear flows generate surface vorticity

• Volumetric vorticity is transported like magnetic fields in plasmas

• Vortex motion can produce large structures in systems that are not technically “unstable” (as they have no feedback loop).

€

=∇×u

€

∂∂t

=∇ × (u×ω) + ν∇ 2ω


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


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

Introduction to High Energy Density Physics R. Paul Drake University of Michigan

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