Collision of two radial rarefaction waves in unmagnetized ambient plasma: effects of the ambient plasma density

Abstract

The expansion of two circular rarefaction waves in vacuum or in a thin ambient plasma is examined with particle-in-cell simulations that resolve two spatial dimensions. In the simulation with no ambient plasma, the rarefaction waves interpenetrate near the symmetry line between both rarefaction wave centers. The exponential density decrease of rarefaction waves with distance implies that the sum of their density does not lead to a density maximum near the symmetry line. The absence of a density maximum, which would yield a repelling electric potential for the inflowing rarefaction wave ions near the symmetry line, and the high interpenetration speed of the ion beams lead to ion-ion instabilities rather than shocks in the overlap layer. The simulations with ambient plasma show that the rarefaction waves pile up the ions of the ambient plasma near the symmetry line. A localized piston of hot ambient ions forms. If its density is large enough, its thermoelectric field allows reverse shocks to grow in the rarefaction waves. These reverse shocks move slowly in the simulation frame and enclose a slab of downstream plasma. A decrease in the speed of the rarefaction wave ions upstream of the shocks with time leads to their collapse.

Figures (8)

• Figure 1: A circular dense plasma with radius $r_D$ is placed in a 2D simulation box, which resolves $- (L_x / 2 + r_D) \le x \le L_x/2 -r_D$ and $-L_y/2 \le y \le L_y/2$. The circular dense plasma is centered at $(x_c, y_c) =(-L_x/2,0)$. The separation vector starts at $(x_c,y_c)$ and returns to this point after crossing the right boundary. The vector $\mathbf{d}$ starts at $(x_c,y_c)$ and ends at the point on the symmetry axis $x=0$ that is defined by its angle $\alpha$ relative to the separation vector. The space outside the circle is either vacuum or ambient plasma with the same fully ionized nitrogen ions as the dense plasma. The hashed zone corresponds to the displayed zone in the following figures.
• Figure 2: Square root of the ion phase space densities $f_i(x, v_x)$, normalized to the peak value in the dense plasma at time t = 0, $f_{i0}$, and ion number densities of Simulations 1. Panels (a) and (b) show $f_i(x, v_x)^{1/2}$ and $n_i(x)$ along the separation vector at times $1800\omega_{pe}^{-1}$ and $7000 \omega_{pe}^{-1}$, respectively. The dashed curves are exponential fits. Panel (c) shows ${f_i(y, v_x)}^{1/2}$ and $n_i(y)$ along the symmetry axis x = 0, at $7000 \omega_{pe}^{-1}$ (red curve) and $1800\omega_{pe}^{-1}$ (brown curve). The dashed curve is a Maxwellian fit. In all panels, the color scale and the velocity scale to the left apply to the phase space densities while the density scale to the right applies to the number densities.
• Figure 3: Panel (a) shows the ratio $log10(n_{i,b1}/n_{i,b2})$ of the density $n_{i,b1}$ of beam 1 over $n_{i,b2}$ of beam 2. Panel (b) shows the difference $\Delta_{vx}=(\langle v_x \rangle_{b1}+\langle v_x \rangle_{b2})/c_s$ of the mean velocities along $x$ of both beams. Both panels show the values at time $7000\omega_{pe}^{-1}$.
• Figure 4: Ion density $n_i/n_{i0}$ of Simulation 1 at times (a) $7000\omega_{pe}^{-1}$, (b) $8600 \omega_{pe}^{-1}$, and (c) $10200\omega_{pe}^{-1}$. All panels share the same color scale.
• Figure 5: Square root of the dense ion phase space densities $f_i(x, v_x)$, normalized to the peak value in the dense plasma at time t=0, $f_{i0}$, and ion densities of the dense plasma, the ambient plasma, and their sum of Simulation 2. Panel (a) shows $f_i(x,v_x)^{1/2}$ and the ion densities at time $1800\omega_{pe}^{-1}$ and along the separation vector. Panels (b) and (c) show the same quantities at the times $5200\omega_{pe}^{-1}$ and $8600\omega_{pe}^{-1}$, respectively. The velocity scale to the left and the color map apply to $f_i(x,v_x)$. The density scale to the right and the inset apply to the ion densities. The grey dashed line highlight the density of the hybrid structure in (b) and the reverse shock density in (c).