Theoretical modeling of radiation-dominated plasmas

1. Theoretical modeling of radiation-dominated plasmas M. M. Basko in collaboration with J. Maruhn, An. Tauschwitz, P.V.Sasorov,V.G. Novikov, A.S. Grushin EMMI Workshop, Speyer, September 26-29, 2010

2. Philosophical proposition With this my talk I will try to illustrate the following proposition: Numerical simulations of complex phenomena may add a new quality to our understanding of these phenomena

3. – energy deposition by thermal conduction (local), – energy deposition by radiation (non-local), – eventual external heat sources. Equations of hydrodynamics The newly developed RALEF-2D code is based on a one-fluid, one-temperature hydrodynamic model in two spatial dimensions (either x,y, or r,z): 2D and 3D ideal hydrodynamics is already a very complex system!

4. Radiation transport Transfer equation for radiation intensity I in the quasi-static approximation: Quasi-static approximation: radiation transports energy infinitely fast (compared to the fluid motion)  the energy residing in radiation field at any given time is infinitely small ! In the present version, the absorption coefficient k and the source function B= B(T) are calculated in the LTE approximation. Coupling with the fluid energy equation: Radiation transport adds 3 extra dimensions (two angles and the photon frequency) the 2D hydrodynamics becomes a 5D radiation hydrodynamics !

5. New quality due to radiation transport Pure hydrodynamics (with or without thermal conductivity) is local. Radiation hydrodynamics is non-local ! • poses serious difficulties for the development of adequate numerical algorithms in 2 and 3 dimensions; • RALEF-2D is based on a newly developed original algorithm for radiation transport (not published yet)

6. Main constituents of the RALEF-2D code Hydrodynamics: 2D, Godunov, 2nd order in space (CAVEAT) Thermal conduction: 2D, SSI, 2nd order in space Radiation transport: 2D, short characteristics, 1st order EOS and opacities: LTE, Hartree-Fock-Slater (KIAM, Moscow) Laser absorption: no refraction, rad.transport, inv.bremsstrahlung

7. Problem 1: Rayleigh-Taylor instability of thin foils accelerated by direct laser irradiation (closely related to the ongoing experiments with the NHELIX/PHELIX laser beams at GSI)

8. Foil and laser beam parameters y Carbon foil: 2g/cc  0.5μm = 1g/cc  1μm = 100 μg/cm2 Laser beam: 50J/1mm2/10ns = 51011 W/cm2, 2 Nd-glass, λ = 532 nm Spatial pulse profile x Temporal pulse profile 51011 W/cm2 The laser focal spot is assumed to be perfectly uniform at r < rfoc = 200μm !

9. Initial density perturbations The initial density profile of the foil is perturbed along the y-axis (perpendicular to the laser beam) by superimposing 2 cosine waves with wavelengths of 10 μm and 7.07 μm; the r.m.s. deviation from ρ0 is set equal to 0.7%.

10. Simulation without radiation transport: (thermal conduction only) temperature plots

18. Simulation with radiation transport: temperature plots

28. Comparison of density distributions at t= 3.5 ns

29. Comparison of temperature distributions at t= 3.5 ns

30. Column density variation along the foil surface Radiation smoothing reduces variations in dx from about 5:1 down to ±13%.

31. What do we learn from the simulations • Thermal conduction has little effect (under the conditions studied) on the development of the Rayleigh-Taylor instability: the turbulent-mixing zone near the accelerated interface expands as (0.05–0.07) gt2 . • Smoothing due to thermal radiation transport invalidates the classical picture of the Rayleigh-Taylor instability. • If the focal spot is sufficiently uniform, a directly irradiated foil may be accelerated in essentially 1D manner– but with a dramatic increase in the foil thickness. • Transition from the 1 to the 2 (or to the 3) laser light is quite helpful.

32. Problem 2: A strongly radiating central Z-pinch in tungsten multi-wire arrays (experiments at Sandia, USA and in Troitsk, Russia)

33. Multi-wire Z-pinches (Sandia, Angara-5) 40-mm diameter array of 240, 7.5-μm-diam. wires. Z-machine at Sandia (USA): • 11.5 MJ stored energy • 19 MA peak load current • 40 TW electrical power to load • 100-250 TW x-ray power • 1-1.8 MJ x-ray energy

34. X-ray pulses at Z (W.A.Stygar et al., PRE 2004)

35. Initial MHD phase of wire implosion (J.P.Chittenden et al.)

36. v0 Problem statement for an RH simulation Cylindrical implosion of an initially cold tungsten plasma cloud Initial shell parameters: • radial thickness: 2 mm; • implosion velocity: v0 = 400 km/s; • uniform temperature: T0 = 20 eV; • mass: m0 = 0.3 mg/cm (A); 6.0 mg/cm (Z); • kinetic energy: 24 kJ/cm (A); 480 kJ/cm (Z); • far from the axis, mass is uniformly distributed over the radius; • possible influence of the magnetic field is ignored. In its present formulation, the problem is one-dimensional.

37. The goal of the simulation Our primary goal: to calculate and compare with the experiments the X-ray spectrum and spectral images of the imploding tungsten plasma.

38. Tungsten EOS and opacities The equation of state and opacities (LTE) of tungsten have been provided by the V.G.Novikov et al. from KIAM (Moscow) ; calculated with the THERMOS code based on the Hartree-Fock-Slater atomic model. • Hydrodynamics was simulated with either 8 or 32 spectral groups. • The output spectra were calculated in the post-processor mode by solving the radiation transfer equation with 200 spectral groups.

39. Case A: the Angara-5 (RUS) parameters m0 = 0.3 mg/cm