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

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.

💡 Research Summary

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The paper investigates the interaction of two circular rarefaction waves (RWs) generated by the expansion of dense, fully ionized nitrogen plasma clouds, using two‑dimensional particle‑in‑cell (PIC) simulations performed with the Smilei code. Three simulation scenarios are examined: (1) expansion into vacuum, (2) expansion into an ambient plasma whose electron and ion densities are 1/50 of the dense plasma, and (3) expansion into a more tenuous ambient plasma with densities 1/75 of the dense plasma. The dense plasma clouds have a radius of 1.1 mm, electron temperature 1500 eV, ion temperature 100 eV, and a peak electron density of 10¹⁶ cm⁻³. The simulation box is 2.95 mm × 8.85 mm with periodic boundaries, and the ion‑to‑electron mass ratio is set to the physical value (≈2.6 × 10⁴).

Vacuum case (Simulation 1).
In the absence of ambient plasma, each RW propagates outward from the initial circular boundary at roughly the ion acoustic speed (c_s≈3.5×10^5) m s⁻¹, driven by the thermoelectric field that arises from electron thermal diffusion across the steep density gradient. As the RW expands, its density falls off exponentially with distance from the front. When the two RWs meet at the symmetry axis (x = 0), the sum of the two exponentially decreasing profiles yields a density minimum rather than a maximum. Consequently, no repulsive electric potential is generated to decelerate the incoming ion streams. The ion beams from opposite sides interpenetrate with relative speeds up to ~5 (c_s). Because the relative beam speed is modest compared to the electron thermal speed (≈46 (c_s)), the dominant instability is a resonant ion‑ion mode rather than a current‑driven ion‑acoustic instability. The ion‑ion waves are aperiodic, remain near the overlap region, and do not evolve into a shock. Magnetic fields generated by electron temperature anisotropy (Weibel‑type) stay at the level (\omega_{ce}/\omega_{pe}≈0.01) and are neglected. Thus, in vacuum the collision produces a turbulent overlap layer dominated by ion‑ion instability, with no electrostatic shock formation.

Ambient plasma case with density 1/50 (Simulation 2).
When an ambient plasma of the same ion species (N⁷⁺) is present at 2 % of the dense plasma density, each RW sweeps up ambient ions, forming a hybrid structure: an electrostatic shock for the ambient ions and a double‑layer for the RW ions. The swept‑up ambient ions accumulate at the mid‑plane, creating a thin, hot ion slab (“piston”) that is essentially stationary. The two hybrid fronts collide at the piston, and the strong thermoelectric field associated with the high‑density slab drives reverse shocks that propagate back into the RW plasma. These reverse shocks travel slower than the incoming RW ions, thereby compressing a downstream slab of plasma between them. The piston’s electric field exceeds the ion acoustic speed, providing the necessary deceleration for shock formation. Over time, the RW ion front slows down, the piston’s electric field weakens, and the reverse shocks lose strength and eventually collapse, allowing the ambient ions to disperse. This sequence demonstrates that a sufficiently dense ambient plasma can convert the interpenetrating ion beams into a shock‑mediated structure.

Ambient plasma case with density 1/75 (Simulation 3).
Reducing the ambient density to 1.33 % of the dense plasma still produces a piston, but its charge density is too low to generate a thermoelectric field strong enough to launch reverse shocks. Consequently, the RW ions pass through the piston with little deceleration, and the system remains dominated by interpenetrating ion beams and ion‑ion instability, similar to the vacuum case. This outcome identifies a critical ambient density (approximately between 1/70 and 1/75 of the dense plasma density) below which shock formation is suppressed.

Key physical insights.

1. The exponential density profile of a freely expanding RW prevents the formation of a density peak at the collision point; therefore, a repulsive electrostatic potential does not arise spontaneously.
2. Ambient plasma acts as a “piston” that can accumulate charge and generate a strong thermoelectric field, enabling the transition from ion‑ion interpenetration to shock formation.
3. There exists an ambient‑density threshold: above it, reverse shocks appear; below it, the system remains shock‑free.
4. The dominant instability switches from ion‑ion resonant modes (vacuum or low‑density ambient) to electrostatic shock formation when the piston field exceeds the ion acoustic speed.
5. The dynamics are strongly time‑dependent: early‑time acceleration of RW ions is rapid, later stages are governed by the slowing of the ion front and the eventual decay of the reverse shocks.

Implications for laboratory experiments.
The results provide quantitative guidance for designing laser‑plasma experiments that aim to study colliding plasma flows or generate electrostatic shocks. By adjusting the residual gas pressure (which sets the ambient plasma density), researchers can control whether the interaction yields a turbulent ion‑beam overlap or a well‑defined shock structure. The identified density threshold (~1/70 of the target plasma density) offers a practical target for vacuum‑chamber conditioning and gas‑fill selection. Moreover, the study clarifies that simply increasing the separation between the two dense targets (thereby increasing the collision speed of the RWs) is insufficient to guarantee shock formation; the ambient plasma must be dense enough to provide the necessary electrostatic piston.

Overall conclusion.
Two radial rarefaction waves colliding in vacuum interpenetrate and generate ion‑ion instabilities without forming shocks, because their exponential density tails never produce a central density maximum. Introducing an ambient plasma changes the picture dramatically: the ambient ions are swept up, forming a high‑density slab that can act as an electrostatic piston. If the ambient density exceeds a critical value, the piston’s thermoelectric field drives reverse electrostatic shocks that propagate back into the rarefaction flows, temporarily trapping a downstream plasma slab. Below this critical density, the piston field is too weak, and the system behaves like the vacuum case. These findings elucidate the conditions under which shocks can be generated in laser‑driven plasma expansions and provide a framework for interpreting and designing future experiments involving colliding plasma flows.