#### 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 Page 2 How is HEDP connected to other areas? 2003 HEDP Class Inroductory Lecture Page 3 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 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 Page 5 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 Page 7 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 Page 8 What is Opacity? • The spatial rate of attenuation of radiation • For radiation intensity (power per unit area per steradian) I: 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 Page 9 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 radiation escape and 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 Page 11 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 Page 12 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 Page 13 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 ~ 2I14 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 Page 16 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 Page 17 Much of the excitement in HEDP comes from the dynamics Shock waves establish the HEDP regime of an experiment 2003 HEDP Class Inroductory Lecture Page 18 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 2003 HEDP Class Inroductory Lecture Page 19 Most phenomena can be seen with a singlefluid approach • Continuity Equation • Momentum Equation • • • • u t u uu 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 Page 20 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 PeRad rad m hydro Smaller or Hydro-like Inroductory Lecture m Pe m Re Page 21 So let’s discuss dynamic phenomena We start with hydrodynamics • 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) Irradiance ≥ 1013 W/cm2 (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 Page 23 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 1M 2 u 1M 2 2 2 p d 2M 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 • Rarefactions can be steady – Steady (more or less) – 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 Page 26 Many HEDP experiments have both shocks and rarefactions SN 1987A Sketch of Experiment Radiographic 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 Page 29 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 pressure gradient 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 nearly steady shocks – 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 ux s us t • • 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 kxU 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) 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 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