Mildly Relativistic Shearing Flows in Astrophysics

• Mildly relativistic shearing flows are astrophysical systems with velocity gradients reaching 0.1c–0.9c, enabling efficient energy dissipation and particle acceleration.
• They exhibit complex hydrodynamic interactions, where instabilities like Kelvin–Helmholtz and mushroom instabilities generate turbulence and amplify magnetic fields.
• These flows produce distinct synchrotron spectra in phenomena such as supernovae, AGN jets, and GRB afterglows, offering valuable insights into high-energy processes.

Mildly relativistic shearing flows are astrophysical fluid or plasma systems exhibiting sustained velocity gradients between adjacent regions, where the bulk flow speeds reach substantial fractions of the speed of light (typically $0.1c \lesssim v \lesssim 0.9c$) but remain below the ultra-relativistic limit. Such shearing flows play a crucial role in the dynamics, particle acceleration processes, radiative signatures, and overall energetics of diverse high-energy astrophysical environments, including core-collapse supernovae, active galactic nucleus (AGN) jets, gamma-ray bursts (GRBs), transient outflows from compact mergers, and more. The micro- and macrophysics of mildly relativistic shear fundamentally regulate energy dissipation, the formation of non-thermal particle populations, and the production of electromagnetic and cosmic-ray emission in these systems.

1. Astrophysical Context and Observational Signatures

Mildly relativistic shearing flows have been robustly identified across a variety of explosive and jet-driven phenomena:

• Stripped-envelope supernovae: VLBI imaging of SN 2007gr revealed expansion corresponding to a minimum apparent velocity $v_{\rm app} \gtrsim 0.61\, c$, much faster than the $\sim6000$ km/s photospheric velocities derived from optical spectra, implying that a small fraction of the ejecta forms a mildly relativistic, bipolar jet (1001.5060). The inferred minimum kinetic energy in the radio-synchrotron emitting region is $E_{\rm radio, min} \approx (0.7$–$1.3) \times 10^{46}\ \mathrm{erg}$, only $\sim 0.1\%$ of the bulk explosion energy.
• Central engine-driven supernovae (e.g., SN 2009bb): Radio afterglow observations and analytic modeling demonstrate a massive, baryon-loaded relativistic shell with initial Lorentz factors $\gamma_0\sim1.2$–$1.4$ expanding nearly freely on timescales of months (1103.4109).
• Relativistic jets in AGN: Observations of X-ray and radio emission over kiloparsec scales, including limb-brightening and ridge-line features, are interpreted as signatures of transverse velocity shear and associated distributed particle acceleration (Wang et al., 2022, Rieger et al., 2016, 2206.13098).
• Fast blue optical transients (FBOTs) and merger afterglows: Outflows in FBOTs and kilonova remnants (e.g., GW170817) involve shock velocities $v \sim 0.1$–$0.7 c$, with synchrotron break frequencies (peak, cooling, self-absorption) providing diagnostics of mildly relativistic expansion, energy partitioning, and composition (Bykov et al., 2022, Rassel et al., 2023, Sadeh, 1 Oct 2024).

The transition from purely non-relativistic to mildly relativistic, shearing outflows is manifest in the spectral properties, angular structure, and temporal evolution of observed emission in both resolved imaging and broadband (synchrotron/inverse-Compton) spectra.

2. Hydrodynamics and Self-Similar Structure

The global dynamics of mildly relativistic shearing flows, especially in explosive events, often require analysis beyond the classic non-relativistic (Sedov) and ultra-relativistic (Blandford-McKee) regimes. Extensions of the Blandford-McKee solutions describe the internal regions behind the shock front where the flow decelerates to mildly relativistic velocities. In these regions, the velocity profile—expressed in terms of a similarity variable $\xi = t/r$—becomes independent of the shock Lorentz factor $\Gamma$:

$$\beta = \bar{b}(\xi), \quad p(r,t) = \bar{P}(t)\, \bar{f}(\xi), \quad n'(r,t) = \bar{N}'(t)\, \bar{h}(\xi)$$

The solution is accurate from $r = 0$ out to $R/\Gamma^2$ behind the shock. For ambient density profiles $\rho \propto r^{-k}$, singular points (sonic lines) may arise; for $k < 0.92$, a secondary shock forms within the decelerating flow (Faran et al., 2020). These features influence the thermalization, energy transport, and evolution of shearing flows in supernova remnants and GRB afterglows.

In jet systems, spine–sheath structures—characterized by a relativistic spine and a sheath exhibiting smooth, sometimes nearly linear, velocity gradients—arise due to the growth and saturation of Kelvin–Helmholtz instabilities (KHI) at the interface with the ambient medium (Wang et al., 2022, Rieger et al., 2021). The resultant large-scale flow profiles critically shape conditions for distributed particle acceleration.

3. Microphysical Instabilities, Turbulence, and Energy Dissipation

The microphysics of mildly relativistic shear layers is governed by a hierarchy of plasma instabilities:

• Mushroom instability (MI): On electron scales, MI in sheared, collisionless $e^+e^-$ beams leads to rapid conversion of bulk kinetic energy into electromagnetic field energy, providing a non-contact mechanism for flow deceleration, even in the absence of direct plasma–plasma overlap (Alves et al., 2015).
• Kelvin–Helmholtz instability (KHI) and kinetic KHI (kKHI): KHI operates from MHD to kinetic scales, producing turbulence and mixing at shear interfaces (Shukla et al., 2016, Wang et al., 2022). PIC simulations show that KHI in mildly relativistic shear can produce electromagnetic vortices, turbulence, and, in strongly relativistic scenarios, intense upper hybrid electrostatic oscillations via compressibility effects.
• Electromagnetic field generation and beaming: At boundary layers between electron–ion and $e^\pm$ plasmas, PIC simulations find that lepton acceleration is highly anisotropic, with the highest-energy leptons beamed near $1/\Gamma$ in the observer frame, but broader energy and angle distributions at lower energies in the hybrid spine-sheath configurations (Liang et al., 2016).

Emergent turbulence (with spectra consistent with Kolmogorov scaling) serves as the source of scattering centers for stochastic Fermi-type acceleration (Wang et al., 2022). The scale, amplitude, and spatial distribution of turbulence are governed by the shear gradient and the development of instabilities, which in turn determine both the efficiency and energy dependence of particle diffusion.

4. Particle Acceleration Processes and Spectral Properties

Mildly relativistic shearing flows act as sites for efficient particle acceleration through a combination of Fermi-type systematic and stochastic processes:

• Shear (systematic) acceleration: Charged particles gain energy by repeatedly crossing regions of differential velocity. The momentum diffusion coefficient is

$D_p = \Gamma_s\, p^2\, \tau \propto p^{2+\alpha},$

where $\Gamma_s \propto \gamma_b^4 (d\beta/dr)^2$ and $\tau \propto p^{\alpha}$ defines the momentum dependence of the mean free path.

The associated acceleration timescale is inversely proportional to the particle mean free path ($t_{\rm acc} \propto 1/\lambda$), enhancing the energization of high-energy particles.

• Stochastic (second-order Fermi) acceleration: Turbulence-driven random scattering further contributes a momentum-diffusion term, typically dominating at lower energies and acting as a pre-acceleration channel.
• Transition between mechanisms and multicomponent distributions: Numerical solutions of the Fokker–Planck equation incorporating both shear and stochastic terms yield multibranch particle spectra: at low energies, stochastic acceleration dominates ($n(\gamma) \propto \gamma^{1-q}$), at intermediate energies shear acceleration establishes a harder power law ($n(\gamma) \propto \gamma^{q-3}$), and at the highest energies synchrotron or escape losses produce an exponential-like cutoff (Liu et al., 2017).
• Spectral indices and energy cutoffs: In the absence of losses, the asymptotic spectrum for a steady-state, highly relativistic shear reduces to $N(E)\propto E^{-(1+\alpha)}$ (2206.13098, Rieger et al., 24 Jul 2025). At mildly relativistic flow speeds (e.g., $\Gamma_0\simeq 3$), the spectrum steepens and becomes sensitive to the flow velocity profile; “power-law–type” shear profiles yield harder spectra than Gaussian or linear profiles (2206.13098).
• Synchrotron-limited maximum energies: The balance between acceleration and synchrotron losses imposes a cutoff Lorentz factor,

$$\gamma_{\rm max} = \left[\frac{(4+\alpha)\,D_0}{\tilde{\beta}_s}\right]^{1/(1-\alpha)},$$

with $D_0$ the normalization of the diffusion coefficient and $\tilde{\beta}_s$ the synchrotron loss term. Numerical studies find that electrons can be accelerated to $\gamma_e \sim 10^8$ and above in suitable mildly relativistic jets (Rieger et al., 24 Jul 2025).

5. Radiative Hydrodynamics and Observable Synchrotron Signatures

The coupling of radiation and matter in relativistic shear flows leads to distinctive anisotropic and transient properties:

• Radiation-matter interaction: Self-similar analyses demonstrate that the radiation energy-momentum tensor can be interpolated between viscous (small shear) and streaming (large shear) limits, offering a covariant prescription for radiative transfer in arbitrary relativistic shearing flows (Coughlin et al., 2016). Shear can substantially “heat” the radiation field, modifying both the energy density and the angular dependence of emission.
• Spectral break frequencies: For collisionless shocks in mildly relativistic ($\gamma \lesssim 10$) flows, analytic formulae for the observed synchrotron spectrum break frequencies (e.g., $\nu_m$, $\nu_c^{\rm syn}$, $\nu_a$) have been developed for arbitrary observer geometry (Sadeh, 1 Oct 2024). These formulae enable constraints on the Lorentz factor, ambient density, and post-shock energy partition when applied to multi-band observations (e.g., GW170817 yielding $2.6 < \gamma < 12$ at $t\sim$10–$16$ days and $$n\varepsilon_B\lesssim3\times10^{-7}\,\mathrm{cm}^{-3}$$).
• Shear-generated synchrotron emission in jets: Extended, distributed acceleration produces high-energy electrons ($\gamma_e\gtrsim10^8$) that generate optical–X-ray synchrotron emission on kiloparsec scales, matching observed profiles of AGN jets (Wang et al., 2022, Rieger et al., 2021, Liu et al., 2017).

6. Efficiency Limits, Confinement, and Implications for Cosmic Rays

The efficiency of shear acceleration and the maximum attainable particle energies are controlled by both microphysical and macroscopic parameters:

• Efficiency dependence: The shear acceleration process is most efficient at relativistic speeds owing to the $\gamma_b^4$ dependence of the diffusion coefficient (Rieger et al., 2019). In mildly relativistic flows (e.g., $\gamma_b\sim3$), electrons achieve Lorentz factors $\sim$~$10^8$ and protons are limited to $\sim$~EeV energies, subject to the Hillas criterion $\lambda(E) \lesssim \Delta r$ for confinement.
• Constraints from turbulence and stability: To avoid disruptive developments of KHI, AGN jets must have relatively thick shear layers $(\Delta R_{\rm jet}/R_{\rm jet})\gtrsim0.1$ (Rieger et al., 2021). The acceleration of electrons to multi-TeV energies typically requires significant pre-acceleration or the existence of strong turbulence. For protons and heavier nuclei, larger mean free paths and weaker losses allow acceleration to highest observed cosmic-ray energies (Rieger et al., 2019, Bykov et al., 2022).
• Application to transients: Simulations of mildly and moderately relativistic outflows in FBOTs, kilonovae, and other transients confirm the formation of power-law electron tails that produce the observed radio–X-ray signatures and enable the acceleration of protons and nuclei to $\sim$~PeV energies (Bykov et al., 2022, Rassel et al., 2023).
• Spectral break estimation and diagnostics: Analytical predictions for the growth, saturation, and observational position of spectral breaks in off-axis merger events improve parameter estimation and resolve ambiguities in the modeling of galactic and extragalactic transients (Sadeh, 1 Oct 2024).

7. Future Directions and Multi-Messenger Relevance

Mildly relativistic shearing flows remain a key focus for understanding energy dissipation, particle acceleration, and radiation in many astrophysical contexts. Advances in analytic theory and numerical simulation (PIC, MHD, Monte Carlo transport) have enabled direct connections between flow microphysics, acceleration processes, and observables. Particularly in the era of multi-messenger astrophysics—with routine gravitational wave detections of compact mergers and coordinated radio–X-ray follow-up—the accurate modeling of synchrotron spectra, spectral breaks, and energy budgets in mildly relativistic shearing flows is central for physical inference (Sadeh, 1 Oct 2024, Bykov et al., 2022, Rassel et al., 2023). Upcoming facilities and systematic observations of transients, AGN jets, and supernova remnants are expected to provide constraints on the spatial structure, turbulence properties, and maximum energies reached in these flows, thereby refining our understanding of high-energy cosmic accelerators and the environments surrounding explosive astrophysical phenomena.