Stability of shock waves in high temperature plasmas

Abstract

The Dyakov-Kontorovich criteria for spontaneous emission of acoustic waves behind shock fronts are investigated for high temperature aluminum and beryllium plasmas. To this end, the Dyakov and critical stability parameters are calculated from Rankine-Hugoniot curves using a more realistic equation of state (EOS). The cold and ionic contributions to the EOS are obtained via scaled binding energy and mean field theory, respectively. A screened hydrogenic model, including l-splitting, is used to calculate the bound electron contribution to the electronic EOS. The free electron EOS is obtained from Fermi-Dirac statistics. Predictions of the model for ionization curves and shock Hugoniot are found to be in excellent agreement with available experimental and theoretical data. It is observed that the electronic EOS has significant effect on the stability of the planar shock front. While the shock is stable for low temperatures and pressures, instability sets in as temperature rises. The basic reason is ionization of electronic shells and consequent increase in electronic specific heat. The temperatures and densities of the unstable region correspond to those where electronic shells get ionized. With the correct modeling of bound electrons, we find that shock instability for Al occurs at a compression ratio ~5.4, contrary to the value ~3 reported in the literature. Free electrons generated in the ionization process carry energy from the shock front, thereby giving rise to spontaneously emitted waves, which decay the shock front.

D'yakov-Kontorovitch instability of shock waves in hot plasmas

“…The pressure is determined using the relativistic stress-tensor formula. We agree with the assertion of Das et al [37] that the instability is related to thermal as well as pressure ionization (i.e., that the shock waves become unstable for temperatures and pressures where sudden ionization of electronic shells occurs), but our conclusions are different as concerns the conditions in which the instability occurs. Our model predicts that the shock becomes unstable for higher temperatures, when the Hugoniot curve departs from the non-relativistic asymptote ρ/ρ 0 =4 towards the limit ρ/ρ 0 = 7, i.e., almost beyond the successive ionization of the electronic shells.…”

“…In the interesting study carried out by Das et al [37], the authors find, for aluminum, two instability regions: the first one exists for a much lower temperature than we find (around T ≈ 150 eV, i.e., about 1.7×10 6 K), after which D'yakov's parameter falls below the critical value h c , and a second instability region starts around T ≈ 550 eV, i.e., 6.4×10 6 K. Since this occurs in the conditions where ionization of K and L shells is important, we agree with the statement that D'yakov's instability is strongly connected to quantum electronic properties, but we find that the shock becomes unstable for much higher temperatures. This is probably due to the differences between our models (screened hydrogenic vs quantum bound states and non-relativistic vs relativistic).…”

“…However, it is not clear to us whether the plasma radiation should be accounted for or not in the Hugoniot energy balance. For instance, Das et al [37] did not include it in their study of D'yakov-Kontorovich instability. This is justified because their temperatures are lower than 1.1 keV≈ 1.28×10 7 K, which corresponds to the end of the signature of the shell structure, before the relativistic effects start to play a role.…”

“…In Ref. [37], Das et al consider an equation of state model relying on scaled binding energy for the cold contribution, mean-field theory for the ionic part, and a screened hydrogenic model with ℓ−splitting for the contribution of bound electrons. In their modeling, the free electrons are treated within the non-relativistic semi-classical Thomas-Fermi model.…”

Stability of expanding accretion shocks for an arbitrary equation of state

“…Distinguished regimes for isolated planar shocks ((a) known results) and expanding accretion shocks ((b) new findings) along the variable h. (Bates & Montgomery 2000), and magnesium (Lomonosov et al 2000;Konyukhov et al 2009); for ionizing shock waves in inert gases (Mond & Rutkevich 1994;Mond, Rutkevich & Toffin 1997); for shock waves dissociating hydrogen molecules (Bates & Montgomery 1999); for Gbar-and Tbar-pressure range shocks in solid metals, where the shell ionization affects the shapes of Hugoniot curves (Rutkevich, Zaretsky & Mond 1997;Das, Bhattacharya & Menon 2011;Wetta, Pain & Heuzé 2018); for shock fronts producing exothermic reactions, such as detonation (Huete & Vera 2019;Huete et al 2020). Other examples include EoS constructed ad-hoc specifically for analytical and numerical studies of shock instabilities: (Ni, Sugak & Fortov 1986;Konyukhov, Levashov & Likhachev 2020;Kulikovskii et al 2020).…”