Criteria for ion acceleration in laboratory magnetized quasi-perpendicular collisionless shocks: when are 2D simulations enough?

Technical Summary: Criteria for Ion Acceleration in Laboratory Magnetized Quasi-Perpendicular Collisionless Shocks

Problem Statement
The study of collisionless shocks is central to understanding cosmic ray acceleration in astrophysical environments such as supernova remnants and galaxy clusters. While numerical simulations have advanced the field, a critical discrepancy exists regarding dimensionality: previous 1D and 2D particle-in-cell (PIC) and hybrid simulations of quasi-perpendicular shocks have struggled to reproduce efficient ion acceleration and non-thermal tails, whereas recent 3D simulations by Orusa & Caprioli (2023) demonstrated that significant ion energization emerges only when full 3D magnetic field structures are included. In 2D, magnetic field lines act as walls, preventing particles from returning to the upstream region to undergo multiple shock crossings (shock drift acceleration). In 3D, particles can leak back upstream through out-of-plane trajectories.

The central question addressed in this work is whether existing 1D and 2D simulations are sufficient to model ion acceleration in current and near-future laboratory experiments using high-power lasers, or if 3D effects are necessary. The authors aim to establish scaling criteria to determine when 3D modeling is required and to identify experimental conditions where efficient 3D ion acceleration can be achieved.

Methodology
The authors employed hybrid particle-in-cell simulations (kinetic ions and fluid electrons) using the non-relativistic dHybridR code. The simulations modeled a supersonic flow propagating toward a reflecting wall in a static, homogeneous magnetic field with a perpendicular orientation ($\vartheta_{Bn} = 90^\circ$). This setup mimics the post-decoupling phase of laboratory shocks where a piston drives a shock into an upstream medium, though the simulations do not explicitly model the laser-piston interaction or the initial shock formation.

Key parameters varied included:

• Alfvénic Mach number ($M_A$): Ranging from 19 to 28.
• Sonic Mach number ($M_s$): Ranging from 7 to 19.
• Plasma beta ($\beta$): Ranging from 2 to 18.
• Dimensionality: Comparisons were made between 2D and 3D configurations.

The study focused on the first tens of ion cyclotron times ($t \approx 10 \omega_c^{-1}$), analyzing shock structure, ion energy spectra, maximum particle energy ($E_{max}$), and acceleration efficiency ($\epsilon$), defined as the fraction of post-shock energy density in ions with energies $\ge 10 E_{sh}$.

Key Results

1. Dimensionality Dependence: The simulations confirmed that efficient ion acceleration is a strictly 3D phenomenon in the quasi-perpendicular regime. In 2D simulations, no non-thermal tails developed regardless of parameters; particles were advected downstream after at most one gyration. In 3D, significant non-thermal populations emerged, allowing particles to cross the shock multiple times and gain energy.
2. Acceleration Thresholds: The authors identified specific thresholds for "strong" ion acceleration:
   • $M_A \gtrsim 25$
   • $M_s \gtrsim 13$
   • $\beta \lesssim 5$
     Under these conditions, acceleration efficiency reached $\sim 0.2\%$, with maximum energies reaching $E_{max} \approx 40 E_{sh}$.
3. Parameter Regimes:
   • Strong Acceleration: $M_A \gtrsim 25, M_s \gtrsim 13$. Requires 3D modeling.
   • Weak Acceleration: $19 \lesssim M_A < 25, M_s \gtrsim 7$. Differences between 2D and 3D exist but are less pronounced; high-energy tails are difficult to detect.
   • No Acceleration: $M_A < 19, M_s < 7$. 2D and 3D results are indistinguishable; particles gain at most $\sim 10 E_{sh}$.
4. Scaling Criteria: The authors derived scaling relations linking laboratory parameters (magnetic field $B_0$, density $n_e$, temperature $T$, shock velocity $v_{sh}$, and ion species $A/Z$) to the critical thresholds. They found that to achieve the necessary number of ion gyrations ($N \gtrsim 10$) within a typical experimental timeframe ($\tau_{exp} \approx 10$ ns), specific combinations of $B_0$ and $n_e$ are required. For instance, a shock velocity of $v_{sh} = 1000$ km/s requires $B_0 \approx 10$ T and $n_e \approx 3.5 \times 10^{19}$ cm$^{-3}$ for hydrogen.

Significance and Claims
The paper claims to provide the first comprehensive scaling criteria for determining when 3D effects become crucial for modeling ion acceleration in laboratory magnetized shocks.

• Current Experiments: The authors surveyed existing laboratory experiments (Schaeffer et al. at OMEGA, Yamazaki et al. at Gekko-XII, Yao et al. at LULI2000) and concluded that their parameters ($M_A < 25$, $M_s < 13$, or short durations) fall within the "weak" or "no acceleration" regimes. Consequently, the authors assert that 1D and 2D simulations are currently sufficient to model the ion spectra observed in these experiments.
• Future Potential: The study identifies that current facilities are realistically close to accessing the 3D acceleration regime. Specifically, experiments driving shocks at velocities exceeding 1000 km/s and sustaining them for $\ge 10$ ns could access the strong acceleration regime.
• Proposed Configurations: The authors propose specific experimental configurations (e.g., using hydrogen or helium plasmas with $B_0 \approx 10-20$ T and $v_{sh} \approx 1000-2000$ km/s) that could maximize ion acceleration. These setups are suggested as benchmarks for validating plasma astrophysics kinetic codes and as controllable sources of energetic particles.

The work emphasizes that while 3D simulations are computationally expensive, they are essential for accurately predicting ion spectra in future high-velocity laboratory experiments, whereas 2D models remain adequate for current lower-velocity regimes.