1975a) where the particle momenta are evaluated in the local fluid frame. It has the advantage of considerably simplifying the collision term on the r.h.s. of Eq. (96), which is parametrized by a scattering frequency $v_s$. The spherical harmonic expansion is stopped at the first order and $f_0$ and $f_1$ are used to evaluate the CR charge density and current to be re-injected into the ideal MHD equations (see Sect. 5.1.1). The vector $f_1 = f_x e_x + f_y e_y + f_z e_z$, where $f_x = f_1^0$, and the perpendicular components are $f_y = -2Rf_1^1$ and $f_z = 2I f_1^1$. The CR charge and current densities can be defined as

$$ \rho_{CR} = 4\pi q \int f_0^0 p^2 dp \\ J_{CR} = \frac{4\pi}{3} q \int \left[ \frac{f_1^0}{-2Rf_1^1} \right] p^2 v dp. \quad (101) $$

The current drives a Lorentz force $-\mathbf{J}_{CR}/c \times \mathbf{B}$ and induces a plasma heating $-\mathbf{J}_{CR} \cdot \mathbf{E}$. (We will come back on CR-MHD coupling in Sect. 5.5.)

The numerical scheme relies on the following approximations:

1. The particle distribution follows a power-law.

2. A ratio $v_s/\Omega_s < 1$ ensures the closing of the spherical harmonics expansion.

3. Another component $f_{LS}$ is added to $f_x$ in order to mimic the effect of a large scale CR component slowly varying over the simulation domain.

Spatially, the simulations are 3D and use periodic boundary conditions. The CFL condition requires some sub-cycling of the MHD step to account for the CR evolution.

In Fig. 38 we show the solution of the 3D VFP-MHD system at a particular time. The simulation has the following setup: a background gas density $n = 0.1 \text{ cm}^{-3}$, a background parallel magnetic field $B = 47 \mu\text{G}$, a shock speed $U_{sh} = 60,000 \text{ km s}^{-1}$, the CR are injected at 100 TeV. The background field is oriented in the z direction. The simulation box has 5676 cells in this direction and 32 in the x and y directions. The CR current is initialized along z. Figure 38 clearly shows two CR populations (see panel c). A first one is escaping ahead of the shock and generating a current (see panel e). The return current in the plasma then triggers non-resonant streaming modes and magnetic field fluctuations which induce a confinement of a second population of CRs at the shock front. This population is accelerated via DSA. Inoue (2019) has adapted this procedure to the case of a SNR propagating in a molecular cloud. There TeV CRs are released first and produce through their current enough magnetic perturbations to confine GeV CRs around the shock front.

## 5.5 Particle-in-cell-magnetohydrodynamics

The approach presented in this section combines a MHD approximation necessary to derive the evolution of the background thermal plasma coupled to a PIC module necessary to calculate the trajectories of supra-thermal particles imposed by the Lorentz force. Contrary to standard PIC simulations, now the Lorentz force is calculated using the electro-magnetic field derived from the system of MHD equations rather than
<figure><img src="image_65.png" /><figcaption>Fig. 38 2D slices of 3D simulation of the VFP-MHD system. From top to bottom: gas density, gas pressure, CR pressure, magnetic field component perpendicular to the shock normal, CR current. All quantities are obtained at a time of 1.3 year. The scales in the z direction are compressed by a factor 24. Image reproduced with permission from Bell et al. (2013), copyright by the authors</figcaption></figure>

Maxwell equations (Lucek and Bell 2000; Reville and Bell 2012). From the distribution of macro particles obtained by the PIC module it is now possible to derive the electric charge and current distribution associated with the supra-thermal particles and to re-inject these two quantities into the MHD equations. This modifies the dynamics of the background plasma. We also need to update the Ohm's law to account for the electric field associated with the energetic particles. Bai et al. (2015); van Marle et al. (2018); Mignone et al. (2018) are three recent works which propose the PIC-MHD approach adapted to the problem of CR in astrophysical flows including such a modified Ohm's law. They are based on three different MHD codes: respectively ATHENA, MPI-AMRVAC, and PLUTO.

The PIC-MHD model assumes global plasma electro-neutrality, i.e., $\sum_{\alpha} n_{\alpha} q_{\alpha} + e(n_{th,i} - n_{th,e}) = 0$, where the index $\alpha$ runs over the different supra-thermal population species: electrons, positrons, ions and $n_{th,p/e}$ stands for the thermal background electron and ion densities (here we have assumed the plasma to be composed of protons only for simplicity). Non-thermal particles carry a current density $\mathbf{J}_{EP} = \sum_{\alpha} n_{\alpha} q_{\alpha} \mathbf{u}_{\alpha}$ to be inserted into the Ampère's law. In the general Ohm's law Hall and electron pressure gradient terms can be safely neglected because they become important only at scales smaller than the ion inertial scale $c/\omega_{pi}$, see Bai et al. (2015). The modified Ohm's law can be written as

$$c\mathbf{E} = [(1 - R)\mathbf{u} + R \mathbf{u}_{EP}] \times \mathbf{B}, \qquad (102)$$

where

$$R = \sum_{\alpha} \frac{n_{\alpha} q_{\alpha}}{n_{th,e} e} \qquad (103)$$
is a measure of the relative density of energetic particles (the ideal Ohm's law is recovered by setting $R = 0$), and

$$ \mathbf{u}_{EP} = \frac{\sum_{\alpha} n_{\alpha} q_{\alpha} \mathbf{u}_{\alpha}}{\sum_{\alpha} n_{\alpha} q_{\alpha}} = \frac{\mathbf{J}_{EP}}{n_{EP} e} \qquad (104) $$

is the average velocity of the energetic particle population defined in terms of the energetic particles current $\mathbf{J}_{EP}$ and density $n_{EP}$. Energetic particles (or CRs) then induce a specific Hall effect. Even if $n_{EP}/n_{th,e} \ll 1$ as it is the case for CRs, the average speed of the non-thermal particles may not be small compare to c, so it has to be retained in the modified Ohm's law. The Lorentz force to be inserted as a source term in the Euler and energy conservation equations is now:

$$ \mathbf{F}_{EP} = (1 - R) \left( n_{EP} e \mathbf{u} \times \mathbf{B} + \frac{\mathbf{J}_{EP}}{c} \times \mathbf{B} \right). \qquad (105) $$

The Lorentz equation which controls supra-thermal particle trajectories can be written as (for particle j of type $\alpha$):

$$ \frac{\partial \mathbf{p}_{\alpha,j}}{\partial t} = \frac{q_{\alpha}}{c} (\mathbf{u}_{\alpha,j} - R \mathbf{u}_{EP} - (1-R)\mathbf{u}) \times \mathbf{B}. \qquad (106) $$

Again setting $R = 0$ gives the standard form of the Lorentz equation. Notice now that the ensemble of energetic particles produce an electric field which modifies the trajectory of each energetic particle.

Energy conservation can be obtained by imposing within each cell the condition:

$$ \sum_{\alpha} \sum_{j} n_{\alpha} \mathbf{u}_{\alpha,j} \cdot \frac{\partial \mathbf{p}_{\alpha,j}}{\partial t} = \mathbf{F}_{EP} \cdot \mathbf{u}_{EP}. \qquad (107) $$

### 5.5.1 Numerical schemes

All works cited above use a Boris pusher to integrate energetic particle tracks with time (Birdsall and Langdon 1985). In order to reach second-order accuracy of the PIC-MHD scheme, Bai et al. (2015) (see also Mignone et al. 2018) introduce a predictor-corrector scheme to calculate the background plasma evolution under the effects of energetic particles. In the predictor (resp. corrector) part the source terms in the MHD equations are deduced from the location of individual particles to neighboring grid cell centers at the initial time (resp. at half the MHD timestep). The momentum and energy feedback ensures second order accuracy. Aside the corner transport upwind (CTU) scheme (Colella 1990), Mignone et al. (2018) also introduce a second order Runge-Kutta time stepping method in the predictor-corrector scheme. van Marle et al. (2018) use a different strategy. Each component, i.e., the MHD fluid and the energetic particles, evolves over their own grid: the MHD grid is used as a base and the PIC grid is offset so that MHD cell centers stand as PIC cell corners.
### 5.5.2 Coupling MHD and PIC time steps

In order to properly resolve each particle trajectory the MHD timestep deduced from the CFL condition has to verify $\Delta t r_g \le \zeta$, with $\zeta$ a constant to be adjusted and $r_g$ the maximum gyroradius of the energetic particles. For instance in the case of a fixed grid, Bai et al. (2015) use $\zeta = 0.3$ for shock problem studies. In the case of non-relativistic MHD flows, the kinetic part evolves more rapidly than the MHD part. To treat this issue it is possible to impose some sub-cycling of the PIC step. Bai et al. (2015) typically use 10 sub-cycles. Mignone et al. (2018) improve this strategy and propose two sub-cycling schemes which recalculate the energetic particle induced Lorentz force at every sub-steps or at even sub-steps.

### 5.5.3 Adaptive mesh refinement for the PIC module

van Marle et al. (2018) have adopted an adaptive mesh refinement procedure for the PIC part, conversely Bai et al. (2015) use a fixed grid. The octree system of adaptive mesh refinement of the MPI-AMRVAC code is used but it involves an additional refinement condition on the energetic particles: if the number of energetic particles within a grid reaches 25% of a pre-set maximum the grid is no longer allowed to coarsen. If the number of particles reaches 80% of the maximum, the grid is refined (assuming it has not yet reached the maximum allowed refinement level). A supplementary condition is applied over the average gyroradius of the particles: if within the grid it becomes smaller than a pre-set number of times the size of the individual grid cells the grid is allowed to be refined. Hence, the particle gyro-radius is always resolved which is a necessary condition for a correct calculation of the source terms in the MHD equations.

A tricky aspect of mesh refinement appears when a particle is moving from one cell of a given level to another cell of a finer level. In that case it is essential to conserve charge and current. Then, the effective weight of the particle has to be increased by a factor equivalent to the reduction in effective volume. This effective weight has to be carefully calculated on the finer grid if a particle stands at the boundary of two grids at different levels. The physical extent where the weight is calculated is fixed by the coarser mesh, so the calculation of the particle weight may cover several rows (in a 2D view) of the finer grid (see Fig. 1 in van Marle et al. 2018).

### 5.5.4 Numerical tests

Bai et al. (2015) propose two tests of the PIC-MHD method. The first one follows the evolution in time of a test-particle in an uniform magnetic field, so it tests particle gyromotion. Different setups have been tested: non-relativistic and relativistic particles, background plasma moving perpendicularly to the background field. Figure 39 shows the time evolution of the particle energy in the co-moving plasma frame and the evolution of the particle position in the co-moving frame in the relativistic case. It can be seen that the Boris pusher conserves the energy perfectly in the case of no drift, and to better than 0.1% level in the case of a small drift (a bit better than in the non-relativistic case as the energy conservation degrades with respect to the relative speed of the drift with respect to the particle speed). The particle trajectory fits with the
<figure><img src="image_66.png" /><figcaption>**Fig. 39** Gyromotion test for a relativistic particle (with a Lorentz factor $\gamma = 10$). The normalized variable time step is $\Delta t \Omega_L = 0.5 \pm 0.1$. Dashed blue lines: numerical solutions obtained in the case of a null drift velocity. Red dashed lines: numerical solutions obtained in the case of a drift velocity with strength $u_d = U_A$. Left panel: time evolution of the particle energy in the co-moving frame (indicated by a prime). Right panel: particle motion in the co-moving frame. The black curve is the analytical solution. Image reproduced with permission from Bai et al. (2015), copyright by AAS</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>**Fig. 40** Real and imaginary part of the non-resonant instability relation dispersion as a function of $\epsilon = U_A/U_{CR}$. Solid lines: analytical solution. Diamond and circle: numerical solutions. Image reproduced with permission from Mignone et al. (2018), copyright by AAS</figcaption></figure>

analytical solution well for long time evolution. Mignone et al. (2018) obtain similar results for this test. The authors propose another setup which tests the motion of a particle in a non-orthogonal electric and magnetic field. Energy conservation to 0.1% level is found for a mildly relativistic particle with respect to the analytical solution (see their Fig. 3).

A second setup concerns the test of the CR feedback. It captures the linear growth rate of the non-resonant (or Bell) instability (Bai et al. 2015; Mignone et al. 2018). In this setup, the background plasma is uniform and at rest, it is pervaded by a uniform magnetic field and a CR current is propagating along the background magnetic field with a speed $U_{CR}$. The results are displayed in Fig. 40. The plots show the real and imaginary parts of the dispersion relation as a function of the ratio of the background Alfvén speed and the CR drift speed. A very good agreement between analytical (solid lines) and numerical (symbols) solutions is obtained.

Finally, Mignone et al. (2018) propose a setup to test the relative drift of the thermal gas and CRs. In this test, the two components drift in opposite direction perpendicularly to a background uniform magnetic field. This test is used to evaluate the performances of the predictor-corrector scheme and the impact of the number of CR sub-cyclings.
The results show that the predictor step is mandatory in order to reproduce the analyt-
ical expected solution and to reach second order accuracy in time.

5.5.5 Results on cosmic ray acceleration at non-relativistic shocks

CR acceleration and CR driven instabilities at SNR shocks can be investigated with the help of the PIC-MHD method, as, in contrast to the case of hybrid techniques, now the PIC part only has to handle supra-thermal particles. This improves a lot particle statistics, which is necessary to properly reconstruct the CR charge and current.

**Simulation setups** Reville and Bell (2012) investigate the development of CR driven instabilities at shocks in the PIC-MHD framework generalizing the work of Lucek and Bell (2000) restricted to the resonant streaming instability. In this work the background plasma is at rest and pervaded by a uniform magnetic field augmented with a turbulent spectrum composed of modes in the perpendicular plane. CR are initialized with a mono-energetic distribution, drifting with respect to the background plasma. A total of 1024 particles per grid cell is used to ensure gyrotropy. The drift speed of CRs corresponds to a shock Alfvénic Mach number of initially $M_A = 10^3$. CRs are injected at an energy that allows the gyromotion to be resolved. The fraction of incoming kinetic energy imparted into CRs is $10^{-4}$.

Bai et al. (2015) also report on fast super-Alfvénic shock simulations. Their setup is as follows: a plasma flow is launched against a static, conductive wall (Mignone et al. (2018) proceed similarly). Quasi instantaneously after the collision a shock forms which propagates in the upstream medium with a speed $u_{sh} > U_{A,u}$, where $U_{A,u} = B_u/\sqrt{4\pi\rho_u}$ is the upstream Alfvén speed. The shock position is identified by evaluating the transversely-averaged fluid speed along the shock propagation direction (let say $x$) $u_x$. This speed is further smoothed by a Gaussian kernel of typical size of 4 grid cells. Then the shock position is fixed at the location where $u_x$ has decreased by 40%. Supra-thermal particles are injected as CRs at a kinetic energy of $E_{inj} = 10E_{sh}$ at the shock position. The CR population injected is normalized to a small fraction of the incoming flux. Another important aspect is that the authors use an artificial light speed $C$ different from $c$ for CRs but still much larger than any fluid speed in the system. The Lorentz factor of species $\alpha$ of an individual CR is: $\gamma_j = C/\sqrt{C^2 - u_{ji}^2}$. This speed is inserted into the Lorentz equation. It has the advantage to be adapted to the investigation of different particle energy regimes (a large $C$ is adapted to follow non- or mildly-relativistic CRs and a small one is adapted to the relativistic regime). Bai et al. (2015) report on the production of non-thermal particle distribution consistent with the Fermi first order process, which is a strong indication that PIC-MHD methods can handle CR scattering off self-generated waves. van Marle et al. (2018) use a different shock generation procedure. The shock is set up from a configuration following the Rankine-Hugoniot conditions. Particles are then injected following the same procedure as in the Bai et al work. Table 4 summarizes the setups of these different works.

**Non-resonant CR driven instability studies and shock acceleration** The main objective of these first studies is the onset of non-resonant (NR) streaming modes in the

<figure><img src="image_68.png" /></figure>
<table><caption>**Table 4** Setups for shock acceleration studies</caption><thead><tr><th>Authors</th><th>Resolution<br>(r<sub>g</sub>)</th><th>t<sub>max</sub><br>(ω<sub>c</sub><sup>-1</sup>)</th><th>M<sub>A</sub></th><th>θ<sub>B</sub><br>(°)</th></tr></thead><tbody><tr><td rowspan="2">Bai et al. (2015)</td><td>1.2 10<sup>5</sup> × 3000</td><td>3 10<sup>3</sup></td><td>30</td><td>0</td></tr><tr><td>3.89 10<sup>5</sup> × 4800</td><td>1.2 10<sup>4</sup></td><td>30</td><td>0</td></tr><tr><td>van Marle et al. (2018)</td><td>240 × 30</td><td>2 10<sup>3</sup></td><td>3–30–300</td><td>0–70–90</td></tr><tr><td rowspan="2">Mignone et al. (2018)</td><td>1.2 10<sup>5</sup> × 3000</td><td>3 10<sup>3</sup></td><td>30</td><td>0</td></tr><tr><td>3.84 10<sup>5</sup> × 4800</td><td>1.1 10<sup>4</sup></td><td>30</td><td>0</td></tr></tbody></table>

All simulations have been done in 2D space configuration. The spatial resolution for van Marle et al. (2018) is the basic one, the simulations also include four levels of refinement. The maximum time $t_{max}$ is in units of $\omega_c^{-1}$. The last column displays the shock magnetic obliquity. The second row for Bai et al. (2015) and Mignone et al. (2018) shows the setup for relativistic runs with a reduced light speed $C$.

context of strong shocks.<fnref n="50" /> All these studies have considered the case of parallel super-Alfvénic shocks and conclude that indeed CRs trigger a streaming instability in the non-resonant regime (see Fig. 41). The theoretical linear growth timescale is recovered by the simulations. A Fourier analysis shows the destabilization of the main NR modes in the upstream medium at a scale fixed by the CR current (see Fig. 42). The interest of the PIC-MHD method with respect to previous MHD simulations (e.g., Bell 2004; Zirakashvili and Ptuskin 2008) is that it accounts for the time evolution of the CR-MHD including CR back reaction. A clear transfer to smaller wave numbers (hence large scales) can be noticed on Fig. 42. The back-reaction of CR also induces a strong corrugation of the shock front as can be seen on Fig. 41. The corrugation is so strong that it becomes difficult to identify the position of the shock front. Longer term simulations in 3D are required to investigate the transition from a non-relativistic to relativistic regime and to capture the propagation of the CRs in the self-generated turbulence (van Marle et al. 2019).

van Marle et al. (2018) have also considered an oblique shock configuration with a magnetic obliquity angle of 70° with respect to the shock normal. The authors find that particle acceleration is delayed but still present. Particle acceleration proceeds in two steps in this configuration. First SDA occurs that injects a population of charged CRs upstream, which trigger non-resonant streaming modes parallel to the background magnetic field, which in turn perturb the magnetic field downstream. This perturbation leads to a corrugation of the shock front, which changes the orientation of the magnetic field and ultimately allows particles to be injected in a parallel configuration. Here again long-term 3D evolution are necessary to investigate the dynamics of particle acceleration and transport. At perpendicular (super-luminal) shocks the electromotive electric field cannot be compensated upstream and the authors do not observe any particle acceleration. The results on the oblique shock configuration have triggered some discussions. The main argument against is (Caprioli and Spitkovsky 2014a; Caprioli et al. 2018) that at high obliquity CR injection is not efficient because incoming

<footnote n="50">⁵⁰ Reille and Bell (2012) discuss a filamentation instability generated by the CRs drifting ahead the shock front. This instability, contrary to the non-resonant streaming instability, is able to generate large scale perturbations which helps to confine high energy CRs.</footnote>
<figure><img src="image_69.png" /><figcaption>**Fig. 41** Super-Alfvénic parallel shock at time $225\omega_c^{-1}$, $450\omega_c^{-1}$ and $600\omega_c^{-1}$. The Alfvénic Mach number is $M_A = 300$. The upper row shows the magnetic field strength relative to the original background magnetic field. The middle row shows the non-thermal particle charge density relative to the thermal gas density. The lower row shows the thermal gas mass density relative to the upstream density at the start of the simulation, combined with the magnetic field stream lines. Filamentary structures characteristic of the non-resonant streaming instability develop in the upstream medium. The shock front gets strongly corrugated with time. Image reproduced with permission from van Marle et al. (2018), copyright by the authors</figcaption></figure>

<figure><img src="image_70.png" /><figcaption>**Fig. 42** Fourier spectra of the turbulence in the up- and downstream media at times $225\omega_c^{-1}$ and $600\omega_c^{-1}$. At early times the non-resonant Bell-like mode develops at the maximum wavelength predicted by the linear theory. The turbulent scale is smaller in the downstream medium due to shock compression of the transverse magnetic field component. At later times non-linear effects start to inject turbulent motions at larger scales and modify the upstream CR current. Image reproduced with permission from van Marle et al. (2018), copyright by the authors</figcaption></figure>
ions are reflected at the shock overshoot, a structure which can not be captured by
a MHD code. The main argument in favour is that hybrid simulations have a small
simulation box and cannot be performed over duration much exceeding a few hundreds
of $c/\omega_{c,i}$ so they are not able to capture instabilities that develop at larger scales
in the downstream medium once the non-resonant streaming instability is triggered.
Moreover, hybrid simulations treat the entire ion population (both thermal and supra-
thermal) as kinetic and hence have much lower statistics to reconstruct the CR current
at the origin of the streaming instability. Both techniques however agree at early
timescales for high Alfvénic Mach number shocks and both show the development of
a supra-thermal tail associated with the SDA mechanism.

5.5.6 Other astrophysical applications

We discuss here recent setups designed to investigate CR or energetic particles trans-
port in specific astrophysical contexts. Two subjects are considered: other type of CR
driven instabilities and energetic particles acceleration near X points in relativistic
magnetic reconnection. The number of scientific cases will likely rapidly increase
with the availability of numerical resources.

**CR driven instabilities** In Lucek and Bell (2000) perturbations are produced by the drifting of a mono-energetic or Gaussian distribution of CRs in a background medium at rest. The background magnetic field is composed of a uniform component of strength B and Alfvén waves of amplitude $\delta B = 0.1B$. CR drift with a speed ten times the local Alfvén speed. The authors consider the growth of the resonant streaming instability and the energy/impulsion transfer to the background plasma using 1-, 2- and 3-D simulations. Rapid magnetic field generation is obtained. The linear growth rate can be sustained even in the non-linear regime. Magnetic field generation saturates when the level of perturbations are $\sim B$.

Lebiga et al. (2018) use a similar numerical approach as the one developed in Bai
et al. (2015) (also using the MHD code ATHENA) to investigate the growth of the gyro-
resonant instability produced by an anisotropic pressure of the CR gas with respect to
the background magnetic field. If the strength of the magnetic field changes over scales
larger than particle Larmor radius then the first adiabatic invariant $p_{\perp}^2/B$ is conserved
along the particle trajectory. In this instability the driving term is given by the relative
perpendicular to parallel CR pressure $A = P_{\perp}/P_{\|} - 1$. The simulation setup involves
a background uniform magnetic field on which is superimposed a flat spectrum of cir-
cularly polarized MHD waves, and an anisotropic CR energy distribution scaling as
$E^{-2.8}$ consistent with the Galactic CR spectrum. The main parameters in the simula-
tions are the ratio of CR density to gas density $n_{CR}/n_g \ll 1$, and the initial anisotropy
parameter $|A| < 1$. The main findings are: the linear growth phase well reproduces
the quasi-linear growth rate deduced from Bykov et al. (2013), the non-linear satura-
tion of the instability occurs due to particle isotropization, a process faster at low CR
energies which contain most of the CR pressure. Bai et al. (2019) perform PIC-MHD
simulations to study the resonant streaming instability. The authors use the $\delta f$ method
(see Kunz et al. 2014 and references therein) which consists in affecting a weight $w_i$ to
each particle i as: $w_i = 1 - f(\mathbf{x}_i(t), \mathbf{p}_i(t))/f$. This method permits a drastic reduction
<table><caption>**Table 5** A list of the main fluid codes used in astrophysics</caption><thead><tr><th>Code name</th><th>physics modules</th><th>AMR</th><th>geometry</th><th>parallelization</th></tr></thead><tbody><tr><td>ATHENA</td><td>CR, PIC-MHD</td><td>y</td><td>Ca, Cy, Sp, Po</td><td>OpenMP/MPI</td></tr><tr><td>BATS-R-US</td><td>PIC-MHD</td><td>y</td><td>Ca</td><td>OpenMP/MPI</td></tr><tr><td>ENZO</td><td>CR</td><td>y</td><td>Ca, Sp</td><td>MPI</td></tr><tr><td>MPI-AMRVAC</td><td>CR, PIC-MHD, PIC-RMHD</td><td>y</td><td>Ca, Cy, Sp, Po</td><td>OpenMP/MPI</td></tr><tr><td>PIERNIK</td><td>CR</td><td>y</td><td>Cy</td><td>MPI</td></tr><tr><td>PLUTO</td><td>CR, PIC-MHD</td><td>y</td><td>Ca, Cy, Sp, Po</td><td>OpenMP/MPI</td></tr><tr><td>RAMSES</td><td>CR</td><td>y</td><td>Ca</td><td>OpenMP/MPI</td></tr></tbody></table>

The table displays: the code name, the CR treatment module CR = fluid or PIC-MHD, if or not a relativistic option exists (SR = special relativistic), if or not an AMR version does exist, the geometrical options (Ca= Cartesian, Cy= Cylindrical, Sp= Spherical, Po= Polar coordinates), the parallelization model. Code presentation can be found in Stone et al. (2008) for ATHENA, Powell et al. (1999) for BAT-R-US, Bryan et al. (2014) for ENZO, Keppens et al. (2012) for MPI-AMRVAC, Hanasz et al. (2010), for PIERNIK, Mignone et al. (2007) for PLUTO, Teyssier (2002) for RAMSES

of the noise inherently associated with particle simulations. In their fiducial setup the authors inject 2048 particles per cell. The results show that the quasi-linear theory of wave growth is well reproduced for both right- and left-handed mode polarizations. The technique permits to investigate the problem of 90° pitch-angle scattering which involves non-linear wave particle interactions.

**CR acceleration near an X-point** Mignone et al. (2018) propose a study of particle acceleration near a 2D X-point in relativistic flows. The simulations are performed in the test-particle limit. The out-of plane electric $E_z$ and guiding magnetic field $B_z$ strength are varied. Initially particles are distributed over a Maxwellian with a thermal speed of 0.1 $U_A$. Particles are accelerated near the null point where the electric field intensity is the highest. An energy distribution with a power-law approximately $E^{-2}$ is obtained. The particle energy shifts to high energies as the guiding magnetic field strength increases because the acceleration zone where $E \cdot B \neq 0$ extends with respect to the case when $B_z$ is smaller than the in-plane magnetic field strength.

## 5.6 A list of HD and MHD codes with CR physics

In Table 5, we present the MHD codes in use in various astrophysical applications that include a module (fluid or kinetic) to treat CR physics, a more complete list can be found in the appendix of Martí and Müller (2015).

## 5.7 Semi-analytical approaches for cosmic ray acceleration

Because direct simulations of DSA are numerically expensive, simpler approaches are desirable for the modelling of large-scale and multi-physics problems. Most of the methods described above deal with the microphysics of collisionless shocks and particle acceleration in some way, and are not readily applicable to simulate a
macroscopic object like, say, a supernova remnant. The most precise methods, PIC simulations, are restricted to tiny scales and narrow dynamical ranges. For instance, in the most advanced numerical simulations of DSA relevant to SNRs to date, by Caprioli and Spitkovsky (2014b), lengths are normalized by $l_p = c/\omega_p$ where $\omega_p$ is the ion plasma frequency, which for a typical density $n_p = 1 \text{ cm}^{-3}$ evaluates to $l_p \simeq 2 \times 10^7 \text{ cm} = 2 \times 10^{-12} \text{ pc}$. So in their Figs. 6 and 7, the “far upstream” reaches about $3 \times 10^{-7} \text{ pc}$, whereas in state of the art simulations of the 3D evolution of a young SNR, by Ferrand et al. (2010), the smallest length resolved is $\simeq 5 \text{ pc}/1024 \simeq 5 \times 10^{-3} \text{ pc}$, nearly 5 orders of magnitude larger. Similarly in the same figures the range of magnetic turbulence scales probed is $\sim 10^{-11}-10^{-8} \text{ pc}$, whereas in one of the most recent study of the effect of turbulence on the SNR emission, by West et al. (2017), the smallest resolved scale is $4 \times 10^{-2} \text{ pc}$, 6 to 9 orders of magnitude larger. It is therefore necessary to make use of sub-grid models in simulations of astrophysical objects. In hydro-kinetic approaches of Sect. 5.4.1, the complex interaction between the particles and the magnetic turbulence is encapsulated in the diffusion coefficient $D$. With such approaches it is possible to operate on space- and time-scales that are relevant to an object like a SNR (e.g. Kang 2015), although simulations still have a high computational cost when using a realistic dependence of $D$ on $p$, and to our knowledge their use has been restricted to 1-dimensional problems (in either slab or spherically symmetric geometries). When the focus is on properly describing the geometry of the SNR, and a 3-dimensional modeling is required, the treatment of DSA needs to be simplified even further.

### 5.7.1 General considerations

To fill this need, Berezhko and Ellison (1999) proposed a simple analytical model of NLDSA, where the spectrum of the particles is assumed to be a three-parts power-law, with slopes linked to the shock properties. This model was used to make the first studies of the effect of NLDSA on young SNRs, by coupling it to 1D self-similar solutions (Decouchelle et al. 2000) then to 1D hydrodynamic simulations (Ellison et al. 2004). A more physical, semi-analytical model was proposed by Blasi (2002), in the framework of the hydro-kinetic treatment. The key idea (developed in the next section), which allows to greatly simplify the mathematics, is that the energy-dependent diffusion of the particles allows to establish a one-to-one correspondence between the particle energy $E$ (or equivalently momentum $p$), the position variable $x$, and the fluid velocity $u$. This trick was already used in Eichler (1979) and developed by Eichler (1984), Ellison and Eichler (1984), Berezhko (1996), but somehow its usefulness was not fully realized until Blasi (2004; 2005) published their NLDSA model.

Being physically motivated while computationally extremely fast, Blasi’s model was quickly adopted by the community doing SNR simulations: it replaced the Berezhko and Ellison model in the 1D hydrodynamic simulations of Ellison et al. (2007) and following works, and it was used for the 3D hydrodynamical simulations of Ferrand et al. (2010) and following works. It allowed Kosenko et al. (2014) to perform parametric studies of the efficiency of CR acceleration in young SNRs. Note that an important limitation of the model is that it is looking only for stationary solutions,
and so needs to be re-run at each time step in order to compute the time evolution of the coupled shock-particles system, assuming quasi-stationarity is reached at each step. This is justified at most energies, but will break down close to the highest energies when the acceleration time becomes of the same order as the age of the simulated shock. The model jointly solves the particle spectrum and the fluid velocity profile as functions of the momentum of particles. As inputs, it requires basic information on the shock (speed $V_{sh}$ and Mach number $M_{sh}$), which can be determined from a hydrodynamic simulation, as well as an injection recipe at some $p_{inj}$ and a cutoff recipe to set $p_{max}$. Amongst the outputs, it provides the total shock compression ratio $r_{tot}$, that can be used to determine an effective adiabatic index $\gamma_{eff}$ for the fluid+particles system. The back-reaction of the particles on the flow can then be imposed by tweaking the value of $\gamma$ in the hydro model according to the prediction of the NLDSA model. Ellison et al. (2004) showed good agreement between this pseudo-fluid approach and two-fluid calculations in 1D. It is worth mentioning that in the original (and most popular) version of Blasi’s model one does not deal explicitly with the diffusion coefficient $D$ (although often an assumption on $D(p)$ is made in order to estimate $p_{max}(t)$). Accordingly, hydrodynamic simulations typically do not need to explicitly resolve the shock precursor generated by the particles, although for a given $D(p)$ law the velocity profile in the precursor may be reconstructed if desired (the position $x_p$ where the fluid velocity is $u_p$ is given by $x_p = D(p)/u_p$). In a subsequent model, Amato and Blasi (2005) introduced the explicit spatial dependence of the distribution $f$ and the diffusion coefficient $D$. This generalization is more complex to derive and significantly longer to compute, and Amato et al. (2008) showed that the two models provide similar results.

The simple model was gradually improved to incorporate other physical processes. Of particular importance is the improvement of the treatment of the magnetic turbulence (Caprioli et al. 2008, 2009b). A recipe for magnetic field amplification (MFA) was included, and the fate of magnetic waves generated by the particles was considered: they may either be damped in the plasma upstream of the shock or be carried through the shock, which leads to very different magnetic fields in the downstream region, although similar overall levels of back-reaction on the shock. These effects were included in SNR simulations as well (by Lee et al. 2012 in 1D and Ferrand et al. 2014b in 3D). Another point of interest, regarding SNRs as sources of cosmic rays, is the role of particles escaping the accelerator. In the base model only the spectrum at the shock is computed and escape is treated implicitly, but if desired the escape of particles can be treated explicitly, in two ways (Caprioli et al. 2009a, 2010a). Other recent developments include the role of ionization (Morbino 2011) and the role of neutrals (Morbino et al. 2013) in the DSA process.

A comparison between Blasi’s semi-analytical model and two popular numerical approaches, hydro-kinetic simulations à la Kang & Jones (Sect. 5.4.1) and Monte Carlo simulations à la Ellison & Eichler (Sect. 3.6.3), can be found in Caprioli et al. (2010b). Typical results are shown in Fig. 43.

In the following last section, we outline the inner workings of Blasi’s NLDSA model.
<figure><img src="image_71.png" /><figcaption>**Fig. 43** Comparison of different methods for non-linear diffusive shock acceleration. The plot at the top shows the distribution function $f(p)$ of particles, the Maxwellian component and a non-thermal tail are apparent. The two plots at the bottom show the hydro profiles of the shock: velocity $u(x)$ and density $\rho(x)$. The models assume efficient acceleration and include backreaction effects: the reduced sub-shock and extended precursor are apparent, and accordingly the non-thermal spectrum is concave. The three different sets of curves show the results of three different methods to solve the coupled fluid and particles system: a semi-analytical model (solid lines, labelled CBA for Caprioli–Blasi–Amato, described in Sect. 5.7.2), a numerical model coupling hydro equations and a diffusion-convection equation (dashed lines, labelled KJ for Kang and Jones, described in Sect. 5.4.1), and another numerical model relying on a Monte Carlo approach (dot-dashed lines, labelled EV for Ellison & Vladimirov, described in Sect. 3.6.3). Images reproduced with permission from Caprioli et al. 2010b, copyright by the authors</figcaption></figure>

## 5.7.2 Construction of a NLDSA model

We restrict ourselves to 1D slab geometry along direction $x$, with a velocity discontinuity (sub-shock) located at $x = 0$, and a velocity ramp (precursor) extending for $x < 0$ over a distance $x_{max}$. Subscripts distinguish between three distinct media, with usual notations: 0 denotes the far upstream (unperturbed) medium ($x < -x_{max}$), 1 denotes the region immediately upstream of the sub-shock ($x = 0^-$), and 2 the region downstream of the sub-shock ($x = 0^+$).

**Distribution of accelerated particles at the shock** The evolution of the distribution function $f$ is described by a convection-diffusion equation which, under the assumption of stationarity $\partial f/\partial t = 0$ reads
$$ -u \frac{\partial f}{\partial x} + \frac{\partial}{\partial x} \left( D \frac{\partial f}{\partial x} \right) + \frac{1}{3} \frac{du}{dx} p \frac{\partial f}{\partial p} + Q = 0, \quad (108) $$

where D is the diffusion coefficient, and Q represents injection of particles, assumed to occur only at the shock front: $Q(x, p) = Q_1(p) \delta(x)$. By integrating Eq. (108) across the shock (from $x = 0^-$ to $x = 0^+$), using the continuity of the distribution function $f_2 = f_1$, then integrating it from far upstream ($x < -x_{max}$) to just ahead of the sub-shock ($x = 0^-$), and assuming homogeneity downstream and of the shock: $(\partial f / \partial x)_2 = 0$, the equation can be recast as

$$ \frac{1}{3} (u_p - u_2) p \frac{df_1}{dp} - \left( u_p + \frac{1}{3} p \frac{du_p}{dp} \right) f_1 + (u_0 f_0 + Q_1) = 0, \quad (109) $$

where we have introduced the key function

$$ u_p = u_1 - \frac{1}{f_1} \int_{-x_{\text{max}}}^{0^{-}} dx \frac{du}{dx} f(x, p), \quad (110) $$

which is the average fluid velocity experienced by particles with momentum p while diffusing upstream of the shock front. Assuming that D is a growing function of p, particles of a given momentum p explore a region of a certain extent $x_p$ upstream of the shock, and thus sample only a part of the precursor in the velocity profile. Hence, $u_p$ can be thought of as being the typical velocity of the fluid at the point $x_p$ that particles of momentum p can reach.

The solution of Eq. (109) can be written in implicit form

$$ f_1(p) = \frac{3}{U_p - R_{\text{tot}}^{-1}} \int_{p_{\text{min}}}^{p} \frac{dp'}{p'} \left( f_0(p') + \frac{Q_1(p')}{u_0} \right) \exp \left( - \int_{p'}^{p} \frac{dp''}{p''} \frac{3U_{p''}}{U_{p''} - R_{\text{tot}}^{-1}} \right), \quad (111) $$

where we have noted $p_{\text{min}}$ the minimum momentum of particles and we have introduced the total compression of the shock $R_{\text{tot}} = u_0/u_2$ and, in a similar way, the normalized velocity everywhere in the precursor: $U_p = u_p/u_0$.

If we further assume that there are no pre-existing particles: $f_0 = 0$, and that a fraction $\eta$ of the particles crossing the sub-shock are “injected” in the acceleration process at a single momentum $p_{\text{inj}}$:

$$ Q_1(p) = \frac{\eta n_1 u_1}{4\pi p_{\text{inj}}^2} \delta(p - p_{\text{inj}}), \quad (112) $$

then Eq. (111) simplifies to

$$ f_1(p) = \frac{3}{U_p - R_{\text{tot}}^{-1}} \frac{\eta n_0}{4\pi p_{\text{inj}}^3} \exp \left( - \int_{p_{\text{inj}}}^{p} \frac{dp'}{p'} \frac{3U_{p'}}{U_{p'} - R_{\text{tot}}^{-1}} \right). \quad (113) $$
The injection momentum $p_{\text{inj}}$ can be parametrized as

$$p_{\text{inj}} = \xi p_{\text{th},2}. \qquad (114)$$

where $p_{\text{th},2} = \sqrt{2m_p k T_2}$ is the mean downstream thermal momentum. Continuity of the thermal and non-thermal distributions at $p_{\text{inj}}$ imposes that

$$\eta = \frac{4}{3\sqrt{\pi}} (R_{\text{sub}} - 1) \xi^3 \exp(-\xi^2), \qquad (115)$$

where we have introduced the compression of the sub-shock $R_{\text{sub}} = u_1/u_2$. The factor $R_{\text{sub}} - 1$ acts as a regulator: injection is switched off when the sub-shock gets smoothed. We have a single parameter $\xi$ to describe injection, but note that the value of $\eta$ is extremely sensitive to the value of $\xi$.

**Velocity profile of the fluid in the precursor** In the previous paragraph we have expressed the distribution of accelerated particles $f$ as a function of the velocity profile of the thermal fluid $U$. As particles back-react on the shock dynamics, $U$ is itself a function of $f$. To find this second relation, we make use of conservation of momentum, which involves 4 terms: dynamical pressure $\rho u^2$, thermal pressure $P_{\text{th}}$, non-thermal pressure $P_{\text{CR}}$, and waves pressure $P_{\text{w}}$. We write it from a point far upstream ($x < -x_{\text{max}}$), where the fluid velocity is $u_0$, to the point $x_p$ (reached by particles of momentum $p$), where the fluid velocity is $u_p$:

$$\rho_p u_p^2 + P_{\text{th},p} + P_{\text{cr},p} + P_{\text{w},p} = \rho_0 u_0^2 + P_{\text{th},0} + P_{\text{cr},0} + P_{\text{w},0}. \qquad (116)$$

The upstream fluid pressure $P_{\text{th},0}$ can be expressed as

$$P_{\text{th},0} = \frac{\rho_0 u_0^2}{\gamma_{\text{th}} M_{S,0}^2}. \qquad (117)$$

Assuming adiabatic compression, the fluid pressure $P_{\text{th},p}$ at any point $x_p$ in the precursor is given by

$$\frac{P_{\text{th},p}}{P_{\text{th},0}} = \left(\frac{\rho_p}{\rho_0}\right)^{\gamma_{\text{th}}} = U_p^{-\gamma_{\text{th}}} \qquad (118)$$

where in the second equality we have made use of the conservation of mass. Various processes may lead to non-adiabatic compression in the precursor. One of the most discussed is heating through the damping of Alfvén waves, for which Berezhko and Ellison (1999) propose the following recipe (obtained for large $M_{A,0}$):

$$\frac{P_{\text{th},p}}{P_{\text{th},0}} = U_p^{-\gamma_{\text{th}}} \left( 1 + \zeta (\gamma_{\text{th}} - 1) \frac{M_{S,0}^2}{M_{A,0}} (1 - U_p^{\gamma_{\text{th}}}) \right) \qquad (119)$$

where $M_{A,0}$ is the upstream Alfvénic Mach number, and $\zeta \in [0, 1]$ is a free parameter added by Caprioli et al. (2009b) and discussed more below.
The particle pressure $P_{cr,p} = P_{cr,p,0} + P_{cr,p,1}$ at point $x_p$ is the sum of two terms. The first term is the pressure contributed by the adiabatic compression of an upstream population $f_0$:

$$P_{cr,p,0} = U_p^{-\gamma_{cr}} \times P_{cr,0} \qquad (120)$$

where $\gamma_{cr} \simeq 4/3$ is the adiabatic index of the particles “fluid” and

$$P_{cr,0} = \int_{p_{min,0}}^{p_{max,0}} \frac{p'v(p')}{3} f_0(p') 4\beta p'^2 dp' = \frac{4\pi}{3} m_p c^2 \int_{p_{min,0}}^{p_{max,0}} \frac{p'^4 f_0(p')}{\sqrt{1+p'^2}} dp' \qquad (121)$$

(with momenta expressed in $m_p c^2$ units in the right expression). The second term is the pressure of the particles accelerated at the shock (with distribution $f_1(p)$ extending up to $p_{max,1}$) and able to reach the position $x_p$ (that is those of momenta $\ge p$):

$$P_{cr,p,1} = \int_{p}^{p_{max,1}} \frac{p'v(p')}{3} f_1(p') 4\beta p'^2 dp' = \frac{4\pi}{3} m_p c^2 \int_{p}^{p_{max,1}} \frac{p'^4 f_1(p')}{\sqrt{1+p'^2}} dp'. \qquad (122)$$

Finally we turn to the pressure in magnetic waves. Far upstream, we assume that the magnetic field is not turbulent: $P_{w,0} = 0$. In the precursor, particles are believed to generate themselves the turbulence required for their scattering, hence as a first approach we may parametrize the pressure of waves as being some fraction $\alpha < 1$ of the pressure of particles:

$$P_{w,p} = \alpha P_{cr,p}. \qquad (123)$$

According to quasi-linear theory, $\alpha \sim v_A/u_0$ for the resonant streaming instability, and $\alpha \sim u_0/c$ for the non-resonant modes. Caprioli et al. (2009b) propose the following recipe for the resonant instability (obtained for large $M_{S,0}$ and $M_{A,0}$):

$$\frac{P_{w,p}}{\rho_0 u_0^2} = \frac{1-\zeta}{4M_{A,0}} U_p^{-3/2} (1-U_p^2). \qquad (124)$$

The term $U_p^{-3/2}$ represents adiabatic compression. The factor $1-\zeta$ is introduced to balance the factor $\zeta$ in relation (119): the amount of wave damping has to remain reasonably small for the magnetic field to be substantially amplified. Different parametrizations of $\delta B$ in the precursor are possible, see Kang et al. (2013) for a comparison of four different models.

Using the above relations for $P_{th}$, $P_{cr}$, $P_{w}$, together with mass conservation, Eq. (116) can be written as
$$
\begin{aligned}
& U_p + \left( \begin{array}{l} \frac{U_p^{-\gamma_{\text{th}}}}{\gamma_{\text{th}} M_{S,0}^2} \left( 1 + \zeta (\gamma_{\text{th}} - 1) \frac{M_{S,0}^2}{M_{A,0}} (1 - U_p^{\gamma_{\text{th}}}) \right) \\ + \frac{U_p^{-\gamma_{\text{cr},0}}}{\gamma_{\text{th}} M_{S,0}^2} \frac{P_{\text{cr},0}}{P_{\text{th},0}} + \frac{4\pi m_p c^2}{3 \rho_0 u_0^2} \int_p^{p_{\text{max},1}} \frac{p'^4 f_1(p')}{\sqrt{1 + p'^2}} dp' \\ + \frac{1 - \zeta}{4M_{A,0}} \frac{1 - U_p^2}{U_p^{3/2}} \end{array} \right) \\
& = 1 + \frac{1}{\gamma_{\text{th}} M_{S,0}^2} \left( 1 + \frac{P_{\text{cr},0}}{P_{\text{th},0}} \right).
\end{aligned}
\qquad (125)
$$

Deriving relation (125) with respect to $p$, we finally obtain

$$
\left(1 - \frac{U_p^{-(\gamma_{\text{th}}+1)}}{M_{S,0}^2} \left(1 + \zeta (\gamma_{\text{th}} - 1) \frac{M_{S,0}^2}{M_{A,0}} + \frac{\gamma_{\text{cr},0}}{\gamma_{\text{th}}} \frac{P_{\text{cr},0}}{P_{\text{th},0}} \frac{U_p^{-\gamma_{\text{cr},0}}}{U_p^{-\gamma_{\text{th}}}}\right) - \frac{1-\zeta}{8M_{A,0}} \frac{U_p^2 + 3}{U_p^{5/2}}\right) \\
\times \frac{dU_p}{dp} = \frac{4\pi m_p c^2}{3 \rho_0 u_0^2} \frac{p^4 f_1(p)}{\sqrt{1+p^2}}. \qquad (126)
$$

The distribution of particles $f_1$ at the shock being known, the velocity profile $U_p$ of the fluid can be computed by integrating Eq. (126) from one of these two boundary conditions to the other:

$$ U_p (p=0) = U_p (x=0^-) = \frac{1}{R_{\text{prec}}}, \qquad (127) $$

$$ U_p (p = p_{\text{max}}) = U_p (x = -x_{\text{max}}) = 1, \qquad (128) $$

where we have introduced the compression factor of the whole precursor $R_{\text{prec}} = u_0/u_1$. In practice, we will be looking for a $R_{\text{prec}}$ such that, starting from condition (127), condition (128) is matched after integration of Eq. (126).

**Compression at the sub-shock** So far we have expressed $f_1$ as a function of $U_p$, $R_{\text{tot}}$ and $R_{\text{sub}}$ [Eq. (111) with injection recipe (112)–(115)], and $U_p$ as a function of $f_1$ and $R_{\text{prec}}$ [Eq. (126) with boundary conditions (127)–(128)], To solve the coupled system $f_1 - U_p$, we need another independent relation between any two of the three compression ratios $R_{\text{prec}}$, $R_{\text{sub}}$ and $R_{\text{tot}}$ (the third one being deduced from $R_{\text{tot}} = R_{\text{prec}} \times R_{\text{sub}}$). To obtain this relation, we once again use conservation of momentum, this time across the sub-shock (from $x = 0^-$ to $x = 0^+$):

$$ \rho_2 u_2^2 + P_{\text{th},2} + P_{\text{cr},2} + P_{\text{w},2} = \rho_1 u_1^2 + P_{\text{th},1} + P_{\text{cr},1} + P_{\text{w},1}. \qquad (129) $$

The pressure of accelerated particles is always continuous across the shock: $P_{\text{cr},2} = P_{\text{cr},1}$. For the magnetic waves, using a simplified treatment of the parallel shock and in the limit of large Alfvénic numbers, Caprioli et al. (2008, 2009b) estimate the jump in pressure $P_w$ and in energy flux $F_w$ to be
$$[P_w]_1^2 = R_{sub}^2 - 1, \qquad (130)$$

$$[F_w]_1^2 = 2 (R_{sub} - 1) P_{w,1} u_1. \qquad (131)$$

Then the jump in fluid pressure at the shock is

$$\frac{P_{\text{th},2}}{P_{\text{th},1}} = \frac{(\gamma_{\text{th}} + 1) R_{\text{sub}} - (\gamma_{\text{th}} - 1) \left(1 - (R_{\text{sub}} - 1)^3 \frac{P_{w,1}}{P_{\text{th},1}}\right)}{(\gamma_{\text{th}} + 1) - (\gamma_{\text{th}} - 1) R_{\text{sub}}}. \qquad (132)$$

Substituting relations (132) and (130) in Eq. (129), and using mass conservation, we obtain

$$M_{S,1}^2 = \frac{2R_{\text{sub}}}{(\gamma_{\text{th}} + 1) - (\gamma_{\text{th}} - 1) R_{\text{sub}} - 2R_{\text{sub}}(\gamma_{\text{th}} - (\gamma_{\text{th}} - 2) R_{\text{sub}}) P_{w,1}^*}, \qquad (133)$$

where we have introduced the sonic Mach number $M_{S,1}$ of the sub-shock, and where we have noted

$$P_{w,1}^* = \frac{P_{w,1}}{\rho_1 u_1^2} = R_{\text{prec}} \frac{P_{w,1}}{\rho_0 u_0^2} = \frac{1 - \zeta}{4M_{A,0}} R_{\text{prec}}^{5/2} (1 - R_{\text{prec}}^{-2}), \qquad (134)$$

where we used recipe (124) for the last equality. In the case where the pressure of magnetic waves is negligible [$P_{w,1} \simeq 0$, that is $\zeta \simeq 1$ with recipe (124)], Eq. (133) reduces to the well-known hydrodynamics relation

$$M_{S,1}^2 = \frac{2R_{\text{sub}}}{(\gamma_{\text{th}} + 1) - (\gamma_{\text{th}} - 1) R_{\text{sub}}} \iff R_{\text{sub}} = \frac{(\gamma_{\text{th}} + 1) M_{S,1}^2}{(\gamma_{\text{th}} - 1) M_{S,1}^2 + 2}. \qquad (135)$$

In the case $P_{w,1} > 0$, Eq. (133) is a quadratic relation for $R_{sub}$ as a function of $M_{s,1}$ and $P_{w,1}^*$, and thus of $R_{prec}$. We can thus solve the sub-shock.

**Alfvénic drift** At this point, we should make the distinction between the velocity of the flow $u$, and the velocity of the scattering centers $\tilde{u}$. In the MFA picture Alfvén waves are generated by particles counter-streaming the flow, so that in the precursor $\tilde{u}_p = u_p - v_{A,p}$ where $v_{A,p}$ is the Alfvén speed at the location reached by particles of momentum $p$, while in the downstream region $\tilde{u}_2 = u_2 + v_{A,2}$. It is this velocity $\tilde{u}$ that should be used in the transport equation for the particles. The effective velocity jumps experienced by the particles are smaller than $R_{sub}$ and $R_{tot}$, which leads to steeper spectra. In the preceding derivation we have assumed $u \simeq \tilde{u}$ for simplicity, but if MFA is efficient this may not be true. Now a difficulty is that, when the magnetic field is strongly turbulent, it is not clear how the waves speed should be calculated. The common approach (used, e.g., by Lee et al. 2012; Kang 2012; Ferrand et al. 2014b) is to parametrize the Alfvénic drift in the form
$$v_{A,p} = \frac{B_0 + f_A \times (B_p - B_0)}{\sqrt{4\pi\rho}}, \qquad (136)$$

where $B_p = \sqrt{B_0^2 + \delta B_p^2}$ is the total magnetic field at point $x_p$ and we have introduced the free parameter $f_A \in [0, 1]$. In this model, MFA is thus described by two free parameters $\zeta$ [Eqs. (119) and (124)] and $f_A$. The Alfvénic drift is an important correction when $\zeta$ is close to 0 and $f_A$ is close to 1.

### Escaping flux
A steady-state solution to the problem can only exist if particles can escape above some maximum momentum $p_{max}$, or upstream of some maximum diffusion length $x_{max}$. In the model, the two approaches are equivalent, the two quantities being related through the relation $x_{max} = D(p_{max})/u_0$. However, the two approaches do not provide the same information on the escape of particles: imposing $f(x_{max}) = 0$ allows to compute the energy spectrum $\phi_0(p)$ of particles leaving the shock around $p_{max}$, whereas imposing $f(p_{max}) = 0$ only allows to compute the net (integrated) energy flux $F_{esc,0}$ at the boundary $x_{max}$.

If one integrates Eq. (108) from the point $x_{max}$ where particles are supposed to leave the system, defined so that $f(x_{max}) = 0$, then a new term $\phi_0$ appears on the l.h.s. of Eq. (109):

$$\phi_0 = -D \left( \frac{\partial f}{\partial x} \right)_0, \qquad (137)$$

which is the flux of particles leaving the system through the boundary $x_{max}$. Assuming as before that no seed particles are present upstream and that injection at the shock front is mono-energetic, the solution (113) becomes

$$f_1(p) = \frac{3}{U_p - R_{\text{tot}}^{-1}} \frac{\eta n_0}{4\pi p_{\text{inj}}^3} \exp \left( -\int_{p_{\text{inj}}}^{p} \frac{dp'}{p'} \frac{3(U_{p'} + \Phi_0(p'))}{U_{p'} - R_{\text{tot}}^{-1}} \right), \qquad (138)$$

where we have noted the normalized escape flux $\Phi_0(p) = \phi_0(p)/(u_0 f_1(p))$. According to Caprioli et al. (2010a), to a very good approximation we have

$$\frac{1}{\Phi_0(p)} = \int_0^{x_{\text{max}}} dx' \frac{u_0}{D(x', p)} \exp \left( -\int_0^{x'} dx'' \frac{u(x'')}{D(x', p)} \right). \qquad (139)$$

The net flux of energy through the upstream boundary is

$$\begin{aligned} F_{\text{esc},0} &= \int_{p_{\text{inj}}}^{p_{\text{max}}} K(p') \phi_0(p') 4\pi p'^2 dp' \\ &= 4\pi m_p c^2 \int_{p_{\text{min}}}^{p_{\text{max}}} \left( \sqrt{1 + p'^2} - 1 \right) p'^2 \phi_0(p') dp', \end{aligned} \qquad (140)$$

where $K(p)$ is the kinetic energy of a particle of momentum $p$ (expressed in $m_p c^2$ units in the right expression).
If one integrates Eq. (108) from sufficiently far upstream, one can assume that the upstream gradient of particles vanishes, so that $\phi_0 = 0$. One can still compute the flux of escaping energy, by requesting that the particle distribution vanishes at the maximum momentum $p_{max}$: $f(p_{max}) = 0$. From Caprioli et al. (2009a), this condition imposes that

$$F_{\text{esc},0} = \frac{4\pi}{3} (u_2 - u_0) p_{\text{max}}^3 K(p_{\text{max}}) f(p_{\text{max}}), \quad (141)$$

where $K(p)$ is the kinetic energy of a particle of momentum $p$.

Finally we note that, to obtain all the required relations between hydrodynamic and kinetic quantities, we have only made use of the conservation of mass and of the conservation of momentum. Once the particle distribution, shock velocity profile, and escape flux have been obtained, the third conservation law, namely conservation of energy, can be checked, which provides a way to assess the precision of the model. We write it between upstream ($x = -x_{max}$) and downstream of the shock ($x = 0^+$):

$$\begin{aligned} & \frac{1}{2} \rho_2 u_2^3 + \frac{\gamma_{\text{th}}}{\gamma_{\text{th}} - 1} P_{\text{th},2} u_2 + \frac{\gamma_{\text{cr}}}{\gamma_{\text{cr}} - 1} P_{\text{cr},2} u_2 + F_{\text{w},2} \\ & = \frac{1}{2} \rho_0 u_0^3 + \frac{\gamma_{\text{th}}}{\gamma_{\text{th}} - 1} P_{\text{th},0} u_0 + \frac{\gamma_{\text{cr}}}{\gamma_{\text{cr}} - 1} P_{\text{cr},0} u_0 + F_{\text{w},0} - F_{\text{esc},0}, \end{aligned} \quad (142)$$

where the different terms account for kinetic energy, thermal energy, CR pressure, magnetic waves pressure, and CR escape.

**Procedure for solving the coupled problem** As a summary, we outline the practical way for the numerical resolution of the system. For a given compression $R_{\text{prec}}$ in the precursor, the non-linearly coupled system ($f, U$) can be solved iteratively as follows:

compute quantities upstream of the sub-shock
compute $R_{\text{sub}}$ and $R_{\text{tot}} = R_{\text{prec}} \times R_{\text{sub}}$
compute quantities downstream of the sub-shock
set injection: $p_{\text{inj}}$ and $\eta$, and $p_{\text{max}}$
set $f_1 = 0$
set $U_p = 1/R_{\text{prec}}$
(set $\phi_0 = 0$)
repeat until convergence of ($f_1, U_p$):
compute $f_1$ from $U_p$ (and $\phi_0$)
set $U_p(0) = 1/R_{\text{prec}}$
compute $U_p$ from $f_1$
(compute $\phi_0$ from $f_1$)

We have neglected the Alfvénic drift for simplicity, and the lines in parentheses apply only when computing particle escape by imposing a spatial boundary condition. Possible values for $R_{\text{prec}}$ range from 1 (no precursor, the shock is not modified, $M_{S,1} = M_{S,0}$ and $R_{\text{sub}} = R_{\text{tot}}$) to some value $R_{\text{prec,max}}$ obtained by requesting that $R_{\text{sub}} = 1$ (limit case of a totally smoothed shock, all the compression is done in the precursor). For a pure hydrodynamical shock, $R_{\text{sub}} \to 1$ is equivalent to $M_{S,1} \to 1$ [see Eq. (135)],
although this is no longer true for a magnetized shock, when taking into account the pressure $P_{w,1}$ of waves [see Eq. (133)], that reduces the compressibility of the medium. Note that when $P_{w,1}^*$ reaches 1/2, a shock can no longer form.

For each value of $R_{\text{prec}}$, the couple $(f_1, U_p)$ will be accepted as a solution of the model if and only if $U_p$ ($p_{\text{max}}$) = 1. Note that a solution may be found for more than one value of $R_{\text{prec}}$. For most of the parameter space, a single solution is found, but sometimes three solutions are found. One corresponds to a weakly modified shock, while the other two correspond to significantly modified shocks (see Blasi et al. 2005 and Amato et al. 2008). Multiple solutions for CR-modified shocks had already been observed before (using completely different methods, see Sect. 5.3) but their physical meaning is unclear. In reality only a single solution will be realized, it is commonly assumed that the others will be suppressed because they are not stable. At this point we should keep in mind that Blasi’s NLDSA model is not time-dependent, and thus cannot describe how the modified shock structure progressively takes shape. This requires numerical simulations of the kind presented in the previous Sects. 5.4 and 5.5.

# 6 Summary and conclusions

This review addresses the numerical techniques developed in the community of high-energy astrophysics and high-energy lasers to investigate non-thermal particle acceleration and transport in magnetized turbulent flows. We first review the main theoretical frameworks developed for the study of particle acceleration in astrophysical flows: diffusive shock acceleration, shock drift and shock surfing processes, stochastic acceleration, and also provide a short survey of recent developments in the field of laser plasmas. We do not cover the process of shear acceleration, the reader can advantageously consult the work of Rieger and Duffy (2006) for further reference. We then detail the technical numerical techniques necessary to investigate problems which appear in the kinetic treatment of particle acceleration. We only give at this stage a short introduction on hybrid methods, the reader is referred to Lipatov (2002) for further details. Kinetic problems in cosmic ray physics can also be treated in the framework of Fokker-Planck or diffusion-convection equations. The Fokker-Planck model finds many astrophysical applications, from the study of cosmic ray transport in the Galaxy to the study of hot plasmas around compact objects. We then focus on recent developments in the theory of particle acceleration at collisionless non-relativistic or relativistic shocks and in reconnection sites based on particle-in-cell and/or hybrid simulations. The final part of the review addresses large-scale particle transport and acceleration studies mostly in the magnetohydrodynamic approximation. We review the rapid developments of numerical techniques coupling MHD with the kinetic description of non-thermal components. We end with the developments made to find semi-analytical solutions of the diffusion-convection equation in the context of CR acceleration at shocks. For completeness, we recommend interested readers to consult some recent and complementary reports and monographs on particle-in-cell methods and Vlasov methods (Palmroth et al. 2018), hybrid methods (Lipatov 2002), and magnetohydrodynamics (Martí and Müller 2015; Leveque 1998).
The subject of energetic particle acceleration and transport in turbulent flows is rapidly growing thanks to the increase in computational power. This applies to standard techniques for catching the propagation of energetic particles like particle-in-cell simulations. Beside this, we have seen that the real challenge is to handle the dynamics over the space, time and energy scales of the high-energy phenomena in astrophysics. This is mandatory because as the highest energetic particles are accelerated they trigger magnetic perturbations necessary to the acceleration of lower energetic particles. This back-reaction requires numerical tools able to treat the inter-connections between large and micro scales. The recent effort in developing PIC modules in MHD codes goes in this direction. This is also true for magnetic reconnection. The microphysics of current sheets depends on the way the magnetic field lines are forced to reconnect by large scale motions. Here it seems important to have simulations which combine MHD and kinetic simulations. One major difficulty remains however to control the numerical noise inherently related to PIC simulations (either due to Cherenkov radiation, or as in the case of PIC-MHD method due to the perturbations generated by the energetic particles themselves). An alternative resides in using a Vlasov approach (Lautenbach and Grauer 2018), but this possibility remains limited by numerical resources to investigate multi-dimensional problems properly. This aspect is crucial to a proper description of particle acceleration and turbulence around shocks and in reconnection zones. One way to make progress beyond the increase of computational power is to combine different numerical techniques to investigate different regions, as it is the case with implicit PIC simulations coupled with MHD solvers (Makwana et al. 2017; Rieke et al. 2015). Another challenge is to adapt the simulations developed for Newtonian flows to special and now general relativistic cases. This aspect is of particular importance since the era of multi-messenger astrophysics is now a reality, in a near future we will obtain an unprecedentedly accurate description of high-energy particle sources with the advent of high precision/high sensitivity gravitational wave, neutrino and gamma-ray detectors.

A positive aspect we can see is that the different numerical tools discussed in this review are and will be further routinely used by very different communities, i.e., high-energy astrophysics, high-energy laboratory plasmas and space plasmas to study the energetic events from the Sun. These converging interests will undoubtedly contribute to the emergence of new fruitful interdisciplinary research subjects.

**Acknowledgements** The authors thank Fabien Casse and Laurent Gremillet for several fruitful discussions. RW thanks Mickaël Melzani who contributed a lot to his knowledge of magnetic reconnection and Doris Folini for profound discussions and proof-reading. AM and MG thank Anna Grassi for her important contributions to Sects. 2.6 and 3.1–3.4. MG also thank Simon Bolanos for his input on magnetic reconnection experiments (Sect. 2.6.4), Francesco Califano, Daniele Del Sarto and Alain Ghizzo for various discussions on Vlasov codes (Sect. 3.4) and the SMILEI dev-team for various discussions on the PIC method. This work is supported by the ANR-14-CE33-0019 MACH project. We acknowledge support by the Programme National de Haute Énergie (PNHE).

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## Affiliations

Alexandre Marcowith¹ · Gilles Ferrand² · Mickael Grech³ · Zakaria Meliani⁴ · Illya Plotnikov⁵ · Rolf Walder⁶

<figure><img src="image_72.png" /></figure>

Alexandre Marcowith
Alexandre.Marcowith@umontpellier.fr

1 Laboratoire Univers et Particules de Montpellier (LUPM), CNRS/IN2P3, CC72, Université Montpellier, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

2 Astrophysical Big Bang Laboratory (ABBL), RIKEN Cluster for Pioneering Research, and Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wakō, Saitama 351-0198, Japan

3 Laboratoire d'Utilisation des Lasers Intenses (LULI), CNRS, Ecole Polytechnique, Sorbonne Université, CEA, 91128 Palaiseau cedex, France

4 LUTH, Observatoire de Paris, PSL Research University, CNRS UMR 8102, Université Paris Diderot, Meudon, France

5 IRAP, Université de Toulouse, UPS-OMP, 9 av. Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France

6 École Normale Supérieure de Lyon, CNRS, Centre de Recherche d'Astrophysique de Lyon (CRAL), UMR 5574, Université de Lyon, Lyon, France