# An Introduction to Particle Acceleration in Shearing Flows

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Supra-Thermal Particle Acceleration in Microscopic Shear Flows

_{s}≲ 10

^{11}cm, which would be minute compared to the jet width. Increasing the shear gradient length in the noted PIC simulations, however, decreases the growth rate of the kKHI instability (e.g., [62]), so that it appears uncertain to which extent the noted effects should be expected. Nevertheless, the simulations mentioned above clearly demonstrate that given suitable conditions microscopic instabilities in collisionless relativistic shear flows can efficiently generate electron-scale electromagnetic turbulence, allowing for the dissipation of kinetic energy of the flow and the production of supra-thermal particles.

## 3. Fermi-Type Particle Acceleration in Macroscopic Shear Flows

#### 3.1. Gradual Shear Flows

#### 3.1.1. A Microscopic Approach—Momentum Space Diffusion

#### 3.1.2. Propagation and Acceleration in Non-Relativistic Shear Flows

#### 3.1.3. Generalization of the Particle Transport to Relativistic Shear Flows

#### 3.1.4. Recent Applications of Gradual Shear Acceleration

- (i) Shear Particle Acceleration in Expanding Relativistic Outflows:The jetted outflows from AGN and GRBs can exhibit highly relativistic speeds, regions of (quasi-conical) expansion and flow Lorentz factors varying with polar angle (e.g., [83,84,85,86]). This makes them possible sites where gradual shear particle acceleration could occur [28,59]. An application to AGN-type outflows has been presented recently, considering the case of a radial velocity shear profile ${u}^{\alpha}={\gamma}_{b}\left(\right)open="("\; close=")">\theta )(1,{v}_{r}(\theta )/c,0,0$, where $\theta $ denotes the polar angle, r the radial coordinate, and ${\gamma}_{b}(\theta )$ the bulk flow Lorentz factor [28]. When the impact of different functional dependencies for ${\gamma}_{v}(\theta )$ such as a power-law-, Gaussian- or Fermi-Dirac-type profile is explored (see Figure 6), the characteristic (co-moving) acceleration timescale is found to be a strong function of $\theta $. This could facilitate the generation of some prominent, non-axis (e.g., ’ridge line’) emission features in AGN jets [28].In order to overcome adiabatic losses ($\propto {\gamma}_{b}{v}_{r}/r$) and allow for efficient acceleration, relativistic outflow speeds and sufficient energetic seed particles (${\lambda}^{\prime}/r>{10}^{-3}$ for the example shown in Figure 6) would be needed. When put in GRB context, particle acceleration in expanding shear flows might result in a weak and long-duration leptonic emission component in GRBs, as well as be conducive to UHE cosmic-ray production [59].
- (ii) Multi-Component Particle Distributions and Extended Emission:Since ${t}_{\mathrm{acc}}\propto 1/\lambda $ (Equation (13)), gradual shear particle acceleration will begin to dominate over conventional first- and second-order Fermi acceleration (${t}_{\mathrm{acc}}\propto \lambda $) above a certain energy threshold. This could naturally result in the formation of multi-component particle distributions. A basic example assuming radiative-loss-limited acceleration in a cylindrical, mildly relativistic shearing flow is shown in Figure 7 [31]. The figure is based on a time-dependent solution of the Fokker-Planck equation for $f(p,t)$, or equivalently $f(\gamma ,t)$, including the effects of classical second-order Fermi and gradual shear particle acceleration as well as synchrotron losses. Employing a Kolmogorov-type ($q=5/3$) scaling for the particle mean free path, $\lambda (p)\propto {p}^{2-q}$, and using parameters applicable to mildly relativistic large-scale jets in AGN, electron acceleration up to Lorentz factors of $\gamma \sim {10}^{9}$ seems feasible (cf. Figure 7 (left)). In the example given, stochastic second-order Fermi acceleration dominates particle energization up to $\gamma \sim {10}^{4}$, while above this threshold shear acceleration becomes operative leading to a somewhat flatter spectral slope (with a change by $2/3$ in the example shown). Synchrotron radiation eventually introduces a spectral cut-off at high energies.As shearing conditions are likely to prevail along astrophysical jets, stochastic-shear particle acceleration is expected to be of relevance for understanding the extended X-ray emission in the large-scale jets of AGN (cf. Section 1) [31]. In reality, the anticipated change in spectral slope will also depend on the spatial transport and escape properties (see below). As a consequence, higher speeds would be needed to achieve comparable, moderate breaks. When put in UHE cosmic-ray context, gradual shear acceleration of protons up to $\sim {10}^{19}$ eV seems feasible in the large-scale jets of AGN [31,33,82], cf. also Figure 7 (right). Higher energies might be achieved for faster flows and for heavier particles.
- (iii) Incorporating Spatial Transport and Diffusive Escape:In the previous Fokker-Planck approach details of the spatial transport, and possible modifications introduced by the diffusive escape of particles from the system, have not been incorporated. Implications of the spatial transport could in principle be studied by using the full relativistic particle transport Equation (23). Analytical examples in this regard have been recently presented by Webb et al. [33,82]. Focusing on steady-state solutions ${f}_{0}(r,{p}^{\prime})$ for a cylindrical jet with longitudinal shear ${u}_{z}(r)$ and allowing for a specific radial dependence $g(r)$ of the scattering time, $\tau (r,p)={\tau}_{0}\phantom{\rule{0.166667em}{0ex}}g(r)\phantom{\rule{0.166667em}{0ex}}{(p/{p}_{0})}^{\alpha}$, they showed that diffusive escape can counter-act efficient acceleration. In particular, while the local particle distribution still follows a power law $f({p}^{\prime})\propto {p}^{\prime -\mu}$, its momentum index $\mu $ becomes dependent on the maximum flow speed ${\beta}_{0}$ on the jet axis, and significantly steepens with decreasing ${\beta}_{0}$ (approaching $\mu \to \infty $ for ${\beta}_{0}\to 0$) [33,82]. Though possible limitations due to the chosen $\tau $-dependence may deserve some further studies, these results imply that efficient gradual shear particle acceleration requires relativistic flow speeds. The analytical solutions [33] can be used to explore the full radial evolution of the particle transport. Figure 8 represents an example for a hyperbolic, relativistic shear flow profile ${\beta}_{z}(r)={\beta}_{0}[1-tanh{(r)}^{2}]$ with a maximum Lorentz factor ${\gamma}_{b}=20$ on the jet axis [72].As can be seen, away from injection at ${r}_{1}$ the known power-law momentum dependence, Equation (18), is approximately recovered at high flow speeds (${\beta}_{0}\to 1$). Clearly, advancing our understanding of the (radial) diffusion properties in astrophysical jets will be important to further improve our understanding of the particle acceleration in gradual shear flows.

#### 3.2. Non-Gradual Shear Flows

^{17}eV in blazar-type AGN to ultra-high energies > 10

^{18}e [89,90]

_{g}becomes larger than the width of the jet r

_{j}. Note that for non-gradual shear one has t

_{acc}∝ λ, while for gradual shear t

_{acc}∝ 1/λ (Equation (13)).

## 4. Particle Acceleration by Large-Scale Velocity Turbulence

## 5. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Growth rate $Im(\omega )$ of unstable kinetic Kelvin-Helmholtz modes in an electron-proton plasma following linear theory for a microscopic tangential shear flow with step-functional velocity jump across the $x=0$ plane, corresponding to two counter-propagating flows with speed ${v}_{0}$ and different densities ${n}_{+}$ (for the flow in the upper region $x>0$) and ${n}_{-}$ (for the flow in the lower region $x<0$). The curves shown are for a density contrast ${n}_{+}/{n}_{-}=1,10,100$, respectively. The growth rates display a cut-off at $k\le 1$ (in units of ${\omega}_{p+}/[{v}_{0}{\gamma}_{0}]$, where ${\omega}_{p+}={[4\pi {n}_{+}{e}^{2}/{\gamma}_{0}{m}_{e}]}^{1/2}$), and take on a maximum value somewhat below. The three curves qualitatively resemble each other, but with a growth rate that is lowered for ${n}_{+}/{n}_{-}>1$. Note, however, that the results of linear fluid theory do not predict a growth of a DC ($k=0$) mode as found in PIC simulations (see below). From Ref. [62].

**Figure 2.**Results of two-dimensional PIC simulations of counter-propagating flows of equal density (${n}_{+}={n}_{-}$) and with $|{v}_{0}|=0.2$ c, showing the emergence of a dominant DC (${k}_{1}=0$) magnetic field component along the shear on top of the harmonic structure inferred from the linear fluid regime at times ${\omega}_{pe}t=35$ (a1), 45 (b1) and 55 (c1). From Ref. [62].

**Figure 3.**

**Left:**Resultant electron energy distribution for a relativistic, cold electron-proton shear flow with ${\gamma}_{0}=3$ at time $t={10}^{3}/{\omega}_{pe}$. At low energies ($1<{\gamma}_{e}<5$) the distributions resembles a thermal one, while in the intermediate range ${\gamma}_{e}=5-25$ it exhibits some smooth (power-law-type $\sim {\gamma}_{e}^{-5}$) evolution. After some hardening, a tail is seen extending up to ${\gamma}_{e}\sim {\gamma}_{0}^{4}\simeq 80$. From Ref. [62].

**Right:**Electron energy distribution in the case of a relativistic shear flow with ${\gamma}_{0}=5$ at late times $t={10}^{4}/{\omega}_{pe}$. The distribution peaks around ${\gamma}_{e}\sim 3000$, but shows little evidence for a conventional power-law behaviour. Both distributions refer to the center-of-momentum frame. From Ref. [65].

**Figure 4.**Schematic illustration of a simple two-dimensional velocity shear profile, in which a flow, directed along the z-axis, is characterized by a velocity whose magnitude smoothly varies with the x-coordinate. $\Delta x$ denotes the shear width.

**Figure 5.**Time-dependent solution $f(p,t)$ of the Fokker-Planck diffusion equation for non-relativistic gradual shear acceleration assuming an impulsive, mono-energetic injection with ${p}_{0}$ at ${t}_{0}=0$. A linear momentum-dependence $\tau \propto p$ ($\alpha =1$) has been used for the scattering time The distribution broadens with time due to momentum dispersion. The (double logarithmic) inlet illustrates the formation of a power law like distribution $n(p)\propto {p}^{2}f(p)\propto {p}^{-2}$ above ${p}_{0}$ for ${t}^{\prime}\ge 0.3$. From Ref. [60].

**Figure 6.**

**Left:**Illustration of a simple conical flow whose radial outflow speeds varies with polar angle $\theta $.

**Right:**Ratio of viscous shear gain versus adiabatic losses multiplied by ($r/{\lambda}^{\prime}$), illustrated assuming a core angle ${\theta}_{c}=0.03$ rad and an on-axis flow Lorentz factor ${\gamma}_{b}=30$. A non-axis preference becomes particularly evident for a Gaussian or Fermi-Dirac shaped flow profile. From Ref. [28].

**Figure 7.**

**Left:**Time-evolution of the electron spectrum, ${\gamma}^{2}\phantom{\rule{0.166667em}{0ex}}n(\gamma )$, in the presence of stochastic-shear particle acceleration, where $n(\gamma )\propto {\gamma}^{2}f(\gamma )$ represents a solution of the corresponding Fokker-Planck equation for a linearly decreasing (trans-relativistic) velocity shear of width $\Delta l\sim {r}_{j}/10$, and an Alfven speed ${\beta}_{A}\sim 0.007$. Above particle Lorentz factors of a few times ${10}^{4}$ the spectrum is shaped by shear acceleration, with a high-energy spectral cut-off around $\gamma \sim {10}^{9}$ being introduced by synchrotron losses. The successive operation of different acceleration processes here naturally results in a broken-power law distribution.

**Right:**Required (blue-hatched) range of parameters (magnetic field strength B, shear layer width $\Delta l$) to allow shear acceleration of protons to $\sim {10}^{18}$ eV given confinement and loss constraints for the noted conditions. The required conditions might be met in large-scale AGN jets. From Ref. [31].

**Figure 8.**Normalized steady-state particle distribution function $f(r,{p}^{\prime})$ in the presence of a gradual, hyperbolic relativistic shear flow as a function of momentum $({p}^{\prime}/{p}_{0}^{\prime})$, shown for three different spatial locations, $r=0.06$ (blue), $r=0.50$ (orange), $r=1.20$ (green). Mono-energetic particle injection with ${p}_{0}^{\prime}$ at ${r}_{1}=0.02$ and an outer (escape) boundary at ${r}_{2}=2$ have been assumed. A linear momentum-dependence $\alpha =1$ has been used for the scattering time. The red curve (top) shows the expected power-law dependence $f({p}^{\prime})\propto {p}^{\prime -4}$ above ${p}_{0}^{\prime}$, as inferred from the Fokker-Planck approach in Equation (18).

**Figure 9.**

**Left:**Cartoon of the considered scenario assuming a recycling of galactic cosmic rays by non-gradual shear acceleration in a jet - (turbulent) cocoon system. Some fraction of galactic cosmic rays are considered to be swept up by the kiloparsec-scale jet and reaccelerated to high energies. The return probability of a particle here is dominated by the scattering (turbulence) properties (i.e., particle mean free path) in the cocoon and not in the jet.

**Right:**Reconstruction of the observed UHECR spectrum assuming mildly relativistic (${\Gamma}_{j}\simeq 1.4$), non-gradual shear acceleration in an extragalactic jet-cocoon system with a thin transition layer $\Delta r=5$ pc (${B}_{j}=0.3$ mG, ${r}_{j}=0.5$ kpc). The composition at the highest end is dominated by intermediate and heavy nuclei. From Ref. [32].

**Figure 10.**Particle distribution function for turbulent shear acceleration at two different times assuming static incompressible velocity turbulence with mean amplitude $\langle \delta {u}^{2}\rangle ={(0.05c)}^{2}$. Red histograms show results of Monte Carlo simulations for a mono-chromatic wave spectrum $S(k)\propto \delta (k-{k}_{0})$ with ${\tau}_{0}c{k}_{0}=0.01$, where $\tau (p)={\tau}_{0}\phantom{\rule{0.166667em}{0ex}}(p/{p}_{0})$ (i.e., $\alpha =1$) has been employed. Thin and thick lines show analytical solutions at times $t/{\tau}_{0}=5\times {10}^{6}$ and ${10}^{7}$, respectively. The formation of a power-law tail $f(p)\propto {p}^{-4}$ above injection ${p}_{0}$ and below $p/{p}_{0}\sim {10}^{2}$ becomes apparent with time. Deviations with regard to the analytical solutions are seen towards higher momenta ($p/{p}_{0}>{10}^{2}$) where the particle mean free path ${\lambda}_{res}$ starts exceeding the turbulence scale ${L}_{0}=2\pi /{k}_{0}$. From Ref. [50].

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Rieger, F.M.
An Introduction to Particle Acceleration in Shearing Flows. *Galaxies* **2019**, *7*, 78.
https://doi.org/10.3390/galaxies7030078

**AMA Style**

Rieger FM.
An Introduction to Particle Acceleration in Shearing Flows. *Galaxies*. 2019; 7(3):78.
https://doi.org/10.3390/galaxies7030078

**Chicago/Turabian Style**

Rieger, Frank M.
2019. "An Introduction to Particle Acceleration in Shearing Flows" *Galaxies* 7, no. 3: 78.
https://doi.org/10.3390/galaxies7030078