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

The D'yakov-Kontorovich stability criterion for spontaneous emission of acoustic waves behind shock fronts is investigated for high-temperature carbon, aluminum, silicon, and niobium plasmas. The D'yakov and critical stability parameters are calculated along the principal Rankine-Hugoniot curve with an equation-of-state model in which the contribution of bound and free electrons is calculated through a relativistic quantum average-atom model, solving the Dirac equation. The pressure is determined using the stress-tensor formula in the relativistic framework. We find that the instability occurs at the end of the ionization of electronic shells, when the Hugoniot curve departs from the $\rho /{\rho }_0=4$ asymptote to tend to the $\rho /{\rho }_0=7$ limit. In such conditions, if the plasma is optically thick, the contribution of blackbody radiation to the EOS is dominant, and the system becomes always stable. Our results indicate that the conditions in which the instability takes place are different from previously published estimates, due to assumptions made in the corresponding equation-of-state models, especially as concerns the relativistic effects, and depend on the radiative opacity of the material.

Figure 1

Pressure along the principal Hugoniot of carbon. Comparison between our results (relativistic quantum average-atom model) and the Hugoniot obtained from density functional theory-molecular dynamics (for $T<{10}^6\mathrm{K}$) simulations and path integral monte carlo (for $T>{10}^6\mathrm{K}$) by Driver et al. [58].

Figure 2

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of carbon, compared to the critical value $h_c$ as a function of temperature $T$.

Figure 3

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of carbon, compared to the critical value $h_c$ as a function of compression $\rho /{\rho }_0$.

Figure 4

Temperature and compression along the principal Hugoniot of carbon. The instability region ($h>h_c$) is indicated in red.

Figure 5

Principal Hugoniot of aluminum.

Figure 6

Sound speed along the principal Hugoniot of aluminum.

Figure 7

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of aluminum, compared to the critical value $h_c$ as a function of temperature $T$.

Figure 8

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of aluminum, compared to the critical value $h_c$ as a function of compression $\rho /{\rho }_0$.

Figure 9

Temperature and compression along the principal Hugoniot of aluminum. The instability region ($h>h_c$) is indicated in red.

Figure 10

Principal Hugoniot of silicon. Comparison between our results (realtivistic quantum average-atom model) and the Hugoniot obtained from density functional theory-molecular dynamics (for $T<{10}^6\mathrm{K}$) simulations and path integral Monte Carlo (for $T>{10}^6\mathrm{K}$) by Driver et al. [58].

Figure 11

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of silicon, compared to the critical value $h_c$ as a function of temperature $T$.

Figure 12

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of silicon, compared to the critical value $h_c$ as a function of compression $\rho /{\rho }_0$.

Figure 13

Temperature and compression along the principal Hugoniot of silicon. The instability region ($h>h_c$) is indicated in red.

Figure 14

Principal Hugoniot of niobium.

Figure 15

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of niobium, compared to the critical value $h_c$ as a function of compression $\rho /{\rho }_0$.

Figure 16

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of niobium, compared to the critical value $h_c$ as a function of temperature $T$.

Figure 17

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of niobium, compared to the critical value $h_c$ as a function of compression $\rho /{\rho }_0$.

Figure 18

Principal Hugoniot of carbon with and without blackbody radiation.

Figure 19

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of carbon, compared to the critical value $h_c$ as a function of temperature $T$. The EOS model includes blackbody radiation.

Figure 20

Principal Hugoniot of aluminum with and without blackbody radiation.

Figure 21

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of aluminum, compared to the critical value $h_c$ as a function of temperature $T$. The EOS model includes blackbody radiation.

Figure 22

Principal Hugoniot of niobium with and without blackbody radiation.

Figure 23

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of silicon, compared to the critical value $h_c$ as a function of temperature $T$. The EOS model includes blackbody radiation.

Figure 24

Principal Hugoniot of niobium with and without blackbody radiation.

Figure 25

Value of D'yakov's instability parameter $h$ [17] along the principal Hugoniot of niobium, compared to the critical value $h_c$ as a function of temperature $T$. The EOS model includes blackbody radiation.

Figure 26

Value of ${\zeta }_t$ as a function of parameter ${\chi }_0$.

Figure 27

Variation of parameters $h$ and $h_c$ versus compression $\zeta =\rho /{\rho }_0$ for three different values of parameter ${\chi }_0$: 1.5, 2.5, and 3. The conditions in which the photon gas is the dominant contribution to the EOS in the present work starts at $\zeta \approx 5$.