Collisionless Shock Driven by a Supersonic Velocity Shear

The gist: The long-term evolution of a relativistic collisionless velocity shear in an unmagnetized electron-positron plasma leads to the formation of collisionless shocks and magnetic field turbulence, which are essential for particle acceleration.

Introduction and Background

Velocity shears are ubiquitous in the universe and on the Earth, from the flow of a river to black hole accretion disks <ref:2608.16656#pg2> How how particles are accelerated are also fundamental issues to understand the nonthermal emission from high-energy astrophysical objects and the origin of cosmic rays <ref:2608.16656#pg2>. In a velocity shear, particles can be accelerated if they can move back and forth across the shear layer <ref:2608.16656#pg2> The shear acceleration mechanism requires magnetic field turbulence to scatter the particle <ref:2608.16656#pg2>. Moreover, the particle energy must be sufficiently large so that the mean free path is larger than the thickness of the shear layer <ref:2608.16656#pg2> Otherwise, the particle cannot be accelerated efficiently (injection problem) <ref:2608.16656#pg2>. In this study, we investigate the long-term evolution of a relativistic supersonic velocity shear, where the free energy is much larger than the initial thermal energy <ref:2608.16656#pg2>. We first show that collisionless shocks are produced by dissipating the supersonic velocity shear <ref:2608.16656#pg2>.

Simulation Setup

The simulation was performed using a two-dimensional particle-in-cell simulation in the x-y plane using the open code, Wuming (Matsumoto et al. 2025) <ref:2608.16656#pg3>. Since the AI is the most unstable mode in a relativistic collisionless velocity shear (Alves et al. 2015), we chose the simulation plane perpendicular to the shear velocity (U0 = U0ez, where U0 is the four-velocity of the shear flow) <ref:2608.16656#pg3>. We set the initial velocity profile as U0/c = +10 in Lx/4 3Lx/4, where Lx and c are the x-directional size of the simulation box and the light speed, respectively <ref:2608.16656#pg3>. We set 30 simulation particles per cell for electrons and positrons <ref:2608.16656#pg3>. The time step is (∆t)ωpe = 0.1

Results of Shock Formation

After the AI grows, the kinetic energy of the shear flow around the shear layer (x/(c/ωpe) = 0) is converted to the electromagnetic and particle thermal energies, so that the pressure around x = 0 becomes much larger than the initial value because the shear velocity is supersonic As a result, plasmas on both sides of the shear layer are strongly pushed out perpendicular to the shear layer, leading to formation of collisionless shocks on both sides as shown in the top panel of Fig.1. The speed of the shocks is vsh/c ∼ 0.25 in the simulation frame at tωpe = 600. Three regions with different characteristics are formed as shown in Fig.1: low-density dissipated region with uz/c ∼ 0 (1), high-density and high-pressure shocked regions with uz/c = ±10 (2), and shock upstream regions with uz/c = ±10 (3). These regions are clearly separated by shock fronts (SF) and contact discontinuities (CD). This structure is not observed in the long-term evolution of the hydrodynamic KHI because the hydrodynamic KHI occurs only in a subsonic velocity shear.

Magnetic Field Turbulence and Particle Acceleration

Fig.3 shows the y-directional power spectrum of Bx at x/(c/ωpe) = 0 shown for each time. In the early phase (tωpe = 5 − 10), the power at the inertial scale increases exponentially due to the linear phase of the AI (Alves et al. 2015). In the later phase (tωpe = 40−600), the peak wavelength becomes larger, which corresponds to the large scale structure as shown in Fig.2. Although this nonlinear evolution of the AI has been observed in previous work (Kawashima et al. 2022), the small scale magnetic field fluctuation in the upstream region is newly observed in this work, which provides the scattering bodies required by the shear acceleration. To understand the origin of the strong B′x and B′y in our simulation, we plot the phase space distribution in the upstream rest frame in Fig.5. The leaking plasma with a large negative u′z excites the Weibel instability in the shock transition region, leading to the generation of the x′ and y′ components of the magnetic field.

Summary and Discussion

In this study, we have investigated the long-term evolution of a relativistic collisionless velocity shear in an unmagnetized electron-positron plasma using a particle-in-cell simulation. We found that collisionless shocks and small-scale magnetic field turbulence are generated around the shear layer. The dissipation of the free energy by the AI results in large thermal and magnetic field pressures. These strong pressures push out the plasma outside the shear layer, and Weibel-mediated collisionless shocks are generated. As a result, three characteristic regions are formed as shown in Fig.1: (1) a low-density region with large-scale magnetic field and uz ∼ 0, (2) high-density regions with small-scale magnetic field and uz = ±U0, and (3) initial upstream regions with small-scale or no magnetic field and uz = ±U0. It has been demonstrated that particles are accelerated in the Weibel-mediated shock (Spitkovsky 2008; Sironi et al. 2013), although it was not observed in this simulation owing to the limitation of the short simulation time. Therefore, the high energy particles accelerated by the Weibel-mediated shock could be injected to the shear acceleration, which could be one of the solution for the injection problem. In addition to the AI, the Weibel-mediated shock generates magnetic field turbulence around the shear layer, which is also required to accelerate particles by the shear acceleration. As shown in the third panel from the top of Fig.1, our simulation showed that sharp velocity shears are still present at contact discontinuities, while a smooth shear structure is found in the low-density region. This is also unexpected structure and would affect the energy spectrum of particles accelerated by the shear acceleration.

The composition of plasma, magnetization, and velocity of shear affect the long-term evolution of collisionless velocity shear and particle acceleration around the shear layer are also interesting questions. Large-scale kinetic simulations for the velocity shear will open a new window to understanding fundamental plasma physics, astrophysical plasmas, and the origin of cosmic rays. The study concludes by highlighting the importance of these findings for understanding particle acceleration mechanisms in collisionless systems.

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Draft version August 18, 2026

Typeset using LATEX twocolumn style in AASTeX631

Collisionless Shock Driven by a Supersonic Velocity Shear

Kazuki Kamiido1 and Yutaka Ohira1

1Department of Earth and Planetary Science, The University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

ABSTRACT

The long-term evolution of a relativistic collisionless velocity shear in an unmagnetized electronpositron plasma is investigated using a first-principle particle-in-cell simulation.

1. Bold Header: Shear-Driven Collisionless Shock Generation in Electron-Positron Plasmas

The improved AI system can accurately model and predict how a relativistic collisionless velocity shear dissipates its free energy, specifically by simulating the conversion of kinetic energy into thermal and magnetic field energy via the Alves instability, leading to the formation of collisionless shocks.

2. Bold Header: Non-Thermal Particle Acceleration Mechanism

The system can model the generation of magnetic field turbulence that is required for the shear acceleration to work, enabling it to predict how this turbulence, generated by collisionless shocks, facilitates particle acceleration by providing necessary scattering bodies.

3. Bold Header: Structure Prediction of Collisionless Shocks

The AI can distinguish between different plasma structures formed during shear evolution, specifically identifying three characteristic regions as shown in Figure 1: (1) a low-density region with large-scale magnetic field and uz ∼ 0, (2) high-density regions with small-scale magnetic field and uz = ±U0, and (3) initial upstream regions.

4. Bold Header: Shock Front Propagation Dynamics

The system can predict the propagation characteristics of Weibel-mediated collisionless shocks by analyzing phase space distribution in the upstream rest frame, specifically identifying particles with large negative u'z in the shock transition region which excites magnetic field components.

The long-term evolution of a relativistic collisionless velocity shear in an unmagnetized electron-positron plasma is investigated using a first-principle particle-in-cell simulation. The Alves instability converts the shear kinetic energy into thermal and magnetic field energy. The resulting pressures push the plasma, leading to the formation of collisionless shocks. The generated collisionless shocks would accelerate high energy particles, which is a possible solution to the injection problem of shear acceleration. In addition, the collisionless shocks generate a magnetic field turbulence that is required for the shear acceleration to work.