Richtmyer-Meshkov instability of an imploding flow with a two-fluid plasma model

The two-fluid (ions and electrons) plasma Richtmyer-Meshkov instability of a cylindrical light-heavy density interface is numerically investigated without an initial magnetic field. Varying the Debye length scale, we examine the effects of the coupling between the electron and ion fluids. When the coupling becomes strong, the electrons are restricted to co-move with the ions and the resulting evolution is similar to the hydrodynamic neutral fluid case. The charge separation that occurs between the electrons and ions is responsible for the self-generated electromagnetic fields. We show that the Biermann battery effect dominates the generation of magnetic field when the coupling between the electrons and ions is weak. In addition to the Rayleigh-Taylor stabilization effect during flow deceleration, the interfaces are accelerated by the induced spatiotemporally varying Lorentz force. As a consequence, the perturbations develop into the Rayleigh-Taylor instability, leading to an enhancement of the perturbation amplitude compared with the hydrodynamic case.

Figure 1

Initial conditions of ions and electrons in the first quadrant of the domain: (a) number density, (b) pressure. Reflected conditions: inner boundaries ($x=0$, $y=0$); outflow conditions: outer edges ($x=3$, $y=3$).

Figure 2

Space-time diagrams along the line $\theta =\pi /4$ with $d_{D,0}=0.1$. (a) Ion number density, (b) electron number density, (c) ion entropy $S_i=ln(p_i/{\rho }_i^{{\gamma }_i})$, (d) electron entropy $S_i=ln(p_e/{\rho }_e^{{\gamma }_e})$, (e) charge density, (f) Lorentz acceleration of electrons in $r$ direction. “IS” is incident shock, “CD” is contact discontinuity, “RW” is rarefaction wave.

Figure 3

Evolution of number densities with $d_{D,0}=0.1$. Each time plot is arranged in double rows with ions (top) and electrons (bottom). (a) $t=0.09$, (b) $t=0.18$, (c) $t=0.27$, (d) $t=0.39$, (e) $t=0.42$, (f) $t=0.45$, (g) $t=0.57$, (h) $t=0.72$, and (i) $t=0.87$.

Figure 4

Radial electron Lorentz acceleration ${[{\mathcal{L}}_e/{\rho }_e]}_r$ for the case with $d_{D,0}=0.1$ at various time overlaid with density interface of ions (black) and electrons (green). (a) $t=0.075$, (b) $t=0.09$, (c) $t=0.105$, (d) $t=0.12$, (e) $t=0.345$, (f) $t=0.36$.

Figure 5

Ion Lorentz acceleration $[{\mathcal{L}}_i/{\rho }_i]$ along radial (top) and azimuthal (bottom) direction for the case with $d_{D,0}=0.1$ at various time overlaid with ion interface (black line). (a) $t=0.45$, (b) $t=0.6$, (c) $t=0.75$

Figure 6

Generated field and velocities overlaid with ion DI (black) and electron DI (green) at $t=0.075$. (a) $E_r$, (b) $E_{\theta }$, (c) $B_z$, (d) $u_{e,r}$, (e) $u_{e,\theta }$, (f) $u_{i,\theta }$.

Figure 7

Evolution of charge density and electromagnetic field overlaid with ion interface (black line) and electron interface (green line). In each subplot, from top to bottom: charge ${\sum }_{\alpha }{n}_{\alpha }{q}_{\alpha }$, radial electric field $E_r$, azimuthal electric field $E_{\theta }$, and magnetic field $B_z$. (a) $t=0.09$, (b) $t=0.15$, (c) $t=0.21$, (d) $t=0.27$, (e) $t=0.42$, (f) $t=0.48$, (g) $t=0.54$, (h) $t=06$.

Figure 8

Local strength of the magnetic field, $1/{\beta }_{\alpha }=B_z^2/2p_{\alpha }$, at (a) $t=0.18$ and (b) $t=0.57$.

Figure 9

Comparison between magnetic part ${\mathcal{L}}_{\alpha }^M$ and electric part ${\mathcal{L}}_{\alpha }^E$ of Lorentz force in the region of $-0.9<{\varphi }_{\alpha }<0.9$ at (a) $t=0.57$ and (b) $t=0.75$, $f_{\alpha }=|{\mathcal{L}}_{\alpha }^M|/(|{\mathcal{L}}_{\alpha }^M|+|{\mathcal{L}}_{\alpha }^E\left|\right)$.

Figure 10

Space-time diagrams along the line $\theta =\pi /4$ with $d_{D,0}=0.01$: (a) ion number density, (b) electron number density, (c) ion entropy $S_i=ln(p_i/{\rho }_i^{\gamma })$, (d) electron entropy $S_i=ln(p_e/{\rho }_e^{\gamma })$, (e) charge density, (f) Lorentz acceleration of electrons in $r$.

Figure 11

Number density of ions (top) and electrons (bottom) at various time with $d_D=0.01$. (a) $t=0.36$, (b) $t=0.39$, (c) $t=0.42$, (d) $t=0.57$, (e) $t=0.72$, (f) $t=0.87$.

Figure 12

Evolution of charge density and electromagnetic field overlaid with ion interface (black line) and electron interface (green line). In each subplot, from top to bottom: charge ${\sum }_{\alpha }{n}_{\alpha }{q}_{\alpha }$, radial electric field $E_r$, azimuthal electric field $E_{\theta }$, and magnetic field ($B_z$). (a) $t=0.42$, (b) $t=0.48$, (c) $t=0.54$, (d) $t=0.6$.

Figure 13

Evolution of self-generated magnetic field ${\overline{B}}_z$, $K_{\alpha }{\overline{\omega }}_{z,e}+{\overline{B}}_z$ and the cumulative time integral of Biermann battery term ${\overline{\mathcal{B}}}_e$. All quantities are averaged over the region $S$: $(r,\theta )\in (0,1.1)\times (3\pi /4,\pi /2)$. (a) $d_{D,0}=0.1$, (b) $d_{D,0}=0.01$.

Figure 14

Evolution of (a) relative perturbation amplitudes, $\eta /{\eta }_0$ and (b) averaged interface positions, $\overline{r}$. “i” denotes ions, “e” denotes electrons, and “HD” denotes the corresponding hydro-case.

Figure 15

Evolution of (a) ion and (b) electron circulations with various $d_{D,0}$. “HD” is the corresponding hydro-case. “- -” is the predicted circulation of the HD case with the SZ model.

Figure 16

Relative difference between perturbation amplitudes in the two sectors. (a) $d_{D,0}=0.1$, (b) $d_{D,0}=0.01$.

Figure 17

Number densities with $d_{D,0}=0.1$. Each time plot is arranged in double rows with ions (top) and electrons (bottom). (a) $t=0.57$, (b) $t=0.72$, (c) $t=0.87$.

Figure 18

Number densities with $d_{D,0}=0.01$. Each time plot is arranged in double rows with ions (top) and electrons (bottom). (a) $t=0.57$, (b) $t=0.72$, (c) $t=0.87$.