Theory and simulations of hydrodynamic instabilities in inertial fusion R. Betti Fusion Science Center, Laboratory for Laser Energetics, University of Rochester IPAM/UCLA Long Program PL2012 - March 12 - June 15, 2012 "Computational Methods in High Energy Density Plasmas"
Acknowledgments Special thanks to: Radha Bahukutumbi (LLE) Ryan Nora (UR-Physics Dept)
Outline Theory of the Rayleigh-Taylor instability in inertial fusion Ablation fronts in laser-driven targets Classical linear Rayleigh-Taylor instability Ablative linear Rayleigh-Taylor instability Single-Mode Nonlinear Theory Multimode ablative RT Hydrodynamic simulations Eulerian and Lagrangian hydrodynamics Two-fluids, nonlocal heat conduction, radiation transport, laser absorption Single mode and multimode simulations of hydrodynamic instabilities
Theory of the Rayleigh-Taylor instability in inertial fusion
Corona Isothermal expansion Time-dependent supersonic flow M>1 The laser energy deposition generates a thermal conduction zone and an ablative flow Sonic point M=1 U b conduction zone steady state subsonic flow M<1 T D c U b =blow-off velocity u M=Mach = V/C s U a =Ablation velocity= Heat fron velocity in dense cold target Laser energy deposited near critical surface Heat flows by conduction Ablated plasma Light and Hot U a Target heavy cold X Critical surface N e 10 22 electrons/cc a=acceleration in the lab frame
The outer surface of an imploding capsule separates a dense fluid supported by a lighter one a=acceleration in the lab frame g= -a=acceleration in the shell frame In the shell frame Denseheavy Imploding dense shell Low density ablated expanding plasma Outer shell surface g Low densitylight
density The outer surface of an imploding capsule is unstable to the Rayleigh-Taylor instability EQUILIBRIUM CONDITIONS Pressure gradient is opposite to density gradient h h dp dx g h dp dx g heavy light g x g
Derive the classical R-T from Newton s law g x HEAVY LIGHT P h P ~ F S( P P ) ma S ~ h dp g heavy 0 h 0 g dx g light dp0 P ~ ~ P0 P0 g dx h k 2 / P h dp0 P ~ ~ 0 P0 hg dx h F ma S h g ~ h S ~ ~ kg ~ ~ ~ t e kg Rayleigh, Proc. London Math. Society, 1883 Taylor, Proc. Royal Soc. of London, 1950 Classical growth rate
The ABLATIVE R-T is just Newton s law at work again but with a restoring force: the dynamic pressure. Target (heavy) x Heat flux P P h U b ~ Laser absorbed isot Ablated plasma (light) Critical surface Newton s law S[ P h ( P u 2 b Dynamic pressure )] Perturbed dynamic pressure 2 b ~ u U u~ h S ~ Ablation rate = m a u U u b b b b b
The perturbed dynamic pressure is stabilizing Target (heavy) x Heat flux P P h U b ~ Laser absorbed isot Ablated plasma (light) Critical surface ~ ~ 2 k( g m u ) ~ Energy flow balance (see Appendix) Ablation introduces a cutoff (wave number) in the unstable spectrum S. Bodner, Phys. Rev. Lett. 33, 761 (1974) H. Takabe et al, Phys. Fluids 28, 3676 (1985) J. Sanz, Phys. Rev. Lett. 73, 2700 (1994) V. Goncharov PhD Thesis, U. Rochester (1996) 5 2 h pu b q kg heat a b u ~ 2 b k U U h ku ~ R. Betti, et al, Phys. Plasmas 3, 2122 (1996) R. Piriz. J. Sanz, L. Ibanez, Phys Plasmas 4, 1118, (1996) R. Betti et al, Phys. Plasmas 5., 1446 (1998) a b b
Appendix: Perturbed blow-off velocity
The ablative growth rate is significantly less than the classical value. Modes with k> k c are stable u a =3.5m/ns g=100 m/ns 2 Cutoff wave number
Nonlinear classical RT: the bubble velocity saturates when the bubble amplitude is ~0.1. The bubble amplitude does not saturate drag buoyancy bubble heavy Buoyancy ~ Drag ~ Sg U 2 h Saturation buoyancy=drag h S U sat bubble ~ g 1 h U sat(2d) bubble g 3k 1 h U sat(3d) bubble g 1 k h ight Transition to saturation: linear bubble velocity = saturated velocity spikes D. Layzer, Astrophys. Journal 122, 1 (1955) ~ ~ U bubble ~ ~ (0) e t sat ~2D 0. 1 sat
What can we infer about the nonlinear ablative RT by simply looking at the linear spectrum?
J. Sanz, J. Ramirez, R. Ramis, R. Betti, RPJ Town, Phys. Rev. Lett. 89, 195002 (2002) J. Sanz, R. Betti, R. Ramis, R. Ramirez 12B, B368-B370, Plasma Phys. Cont. Fus. (2004)
R. Betti, J. Sanz, APS-DPP Bulletin (2004)
In the deeply nonlinear phase, the vorticity accumulates inside the bubble raising the bubble terminal velocity R. Betti and J. Sanz, Phys. Rev. Lett. 97, 205002 (2006)
Single-mode simulation of the deeply nonlinear ablative Rayleigh-Taylor instability
R. Betti and J. Sanz, Phys. Rev. Lett. 97, 205002 (2006)
R. Betti and J. Sanz, Phys. Rev. Lett. 97, 205002 (2006)
Multimode nonlinear interaction leads to an envelop growth of the bubble front h=gt 2 with dependent on the box size h classical bubbles 2 cl gt cl ~ 0.05 ablative h gt abl bubbles 2 abl cl 1 k k box nonlcutoff 2 This conclusion does not include the effect of bubble acceleration J. Sanz, R. Betti, R. Ramis, R. Ramirez 12B, B368- B370, Plasma Phys. Cont. Fus. (2004)
Hydrodynamic Simulations Excellent tutorial at http://hedpschool.lle.rochester.edu/1000_proc2011.php by Radha Bahukutumbi
Hydrodynamic codes use a combination of Eulerian and Lagrangian grids (ALE = arbitrary Lagrangian-Eulerian)
-1-1
Laser Radius (mm)
density (g/cc) Electron temperature (ev) L T = T/T = temperature gradient scale length e = electron mean free path 5 e << L T diffusive heat transport SH e L T nonlocal heat transport FS 4.5 4 3.5 3 2.5 T e 10 3 10 2 2 T e L T ~ e 1.5 10 1 1 0.5 0 10 0 1000 1100 1200 1300 1400 Radius (microns)
Single mode simulations are carried out to assess the suitability of the ALE (arbitrary Lagrangian-Eulerian) grid
Multimode laser-imprinting simulations of CH+DT targets show Rayleigh-Taylor instability growth at the ablation front Simulations by R. Nora (LLE)
density (g/cc) Electron temperature (ev) density (g/cc) Electron temperature (ev) Imploding shells can develop multiple unstable interfaces depending on the material opacities 6 5.5 5 4.5 4 3.5 3 2.5 T e DT ice + SiO 2 Double ablation front DT rad T e rad-front 10 3 10 2 DT ice + CH+CHSi(5%)+Si Double ablation front + Classical interface 5 4.5 4 3.5 3 2.5 2 T e DT Classical interface rad-front 10 3 10 2 2 1.5 1 0.5 SiO 2 e-front 0 10 0 400 450 500 550 600 Radius (microns) 10 1 1.5 1 0.5 CH CHSi e-front 0 10 0 400 450 500 550 600 Radius (microns) 10 1
Multimode laser-imprinting simulations of SiO 2 +DT targets show Rayleigh-Taylor instability growth at the radiative/classical front E-ablation front radiative front Simulations by R. Nora (LLE)
conclusions Hydrodynamic instabilities in ICF are generally understood but challenges remain in developing an accurate predictive capability for 2D-3D multimode interactions The linear theory of the ablative RT is fully developed and the physics is well understood. Ablation is stabilizing in the linear phase. The nonlinear single-mode evolution is well understood and ablation is destabilizing in the deeply nonlinear phase (bubble acceleration) The effect of the initial conditions on the nonlinear multimode ablation front dynamics is not well understood An accurate evaluation of the instability seeds from laser non-uniformities (imprinting) is difficult Three dimensional simulations are computational expensive and great difficulties remain in developing accurate nonlinear multimode simulations (this also applies to 2D simulations).
BACK UP SLIDES ON NONLINEAR ARTI
Mitigation techniques for the Rayleigh-Taylor instability in laser accelerated targets: 1. Reduce the seeds 2. Reduce the growth rates
P. McKenty et al, Phys. Plasmas 2000 Reducing the seeds for the RT (by making uniform laser beams) improve the integrity of the imploding shell No SSD 1THz SSD
The growth rates can be reduced by shaping the entropy of the imploding shell Stabilize the RT by increasing the ablation velocity 0.94 kg 2. 7ku a P app 3/5 ~ α measure of entropy u a m a m ~ P a 3/5 app 3/5 Increase the entropy (α) at the ablation front while keeping α low in the unablated material
Adiabat shaping reduces the RT growth without degrading the final compression V. N. Goncharov et al, Phys. Plasmas 10, 1906 (2003) K. Anderson and R. Betti, Phys. Plasmas 10, 4448 (2003) K. Anderson and R. Betti, Phys. Plasmas 11, 5 (2004) R. Betti et al, Phys. Plasmas 12, 042703 (2005)