[1,2]Patrick Reichherzer

1]Department of Physics, University of Oxford, Oxford OX1 3PU, UK 2]Exeter
College, Oxford OX1 3DP, UK 3]Trinity College, Oxford OX1 3BH, UK 4]Balliol
College, Oxford OX1 3BJ, UK 5]Lady Margaret Hall, Oxford OX2 6QA, UK
6]Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton NJ 08544, USA 7]Princeton Plasma Physics Laboratory, Princeton
University, PO Box 451, Princeton NJ 08543, USA 8]Merton College, Oxford OX1
4JD, UK

# Efficient micromirror confinement of sub-TeV cosmic rays in galaxy clusters

<EMAIL_ADDRESS>Archie F. A. Bott  Robert J. Ewart
Gianluca Gregori  Philipp Kempski  Matthew W. Kunz  and Alexander A.
Schekochihin  [  [  [  [  [  [  [  [

###### Abstract

Recent observations suggest a stronger confinement of cosmic rays (CRs) in
certain astrophysical systems than predicted by current CR-transport theories.
We posit that the incorporation of microscale physics into CR-transport models
can account for this enhanced CR confinement. We develop a theoretical
description of the effect of magnetic microscale fluctuations originating from
the mirror instability on macroscopic CR diffusion. We confirm our theory with
large-dynamical-range simulations of CR transport in the intracluster medium
(ICM) of galaxy clusters and kinetic simulations of CR transport in
micromirror fields. We conclude that sub-TeV CR confinement in the ICM is far
more effective than previously anticipated on the basis of Galactic-transport
extrapolations.

###### keywords:

Cosmic Ray, Diffusion, Mirror Instability, Magnetic Mirror, Resonant
Scattering, Galaxy Cluster

A good theory of cosmic-ray (CR) transport is crucial for advancing our
understanding of phenomena in the Universe, including the formation and
evolution of galaxies and galaxy clusters [43, 81]. CRs, through their
transport characteristics, not only influence their environment but also
modulate their own (re)acceleration efficiency [17, 18, 97, 20, 3, 11],
confinement/escape [82, 54, 100], spectra [74, 34], (an)isotropy [4], and
observable photon [19, 14, 45] and neutrino [46] emission.

The transport characteristics of CRs in magnetic-field structures depend on
the scattering efficiency and mechanism, both of which are influenced by the
properties of the ambient plasma. Specifically, within a weakly collisional,
high-$\beta$ plasma, i.e., one in which the thermal pressure much exceeds the
magnetic pressure, deviations from local thermodynamic equilibrium provide
free energy for fast-growing Larmor-scale instabilities, leading to
distortions in magnetic fields on thermal-ion kinetic scales. In such a
high-$\beta$ plasma, two characteristic scales are relevant for describing the
global transport of CRs: the macroscale correlation length $l_{\mathrm{c}}$ of
the magnetic turbulence, and the microscale $l_{\mathrm{mm}}$ of the
micromirrors111The prefix “micro” refers to scales much smaller than the
${\gtrsim}\mu\mathrm{pc}$ gyroradii of ${\gtrsim}100\,\mathrm{MeV}$ CRs, and
serves to distinguish the plasma-kinetic-scale “micromirrors” (mm) from the
large-scale magnetic mirrors that also influence CR transport [23, 76, 67].
created by the mirror instability [84, 66]. While there are other micro-
instabilities, the magnetic fluctuations created by the mirror instability are
stronger and thus more influential for CR transport [64, 16]. The physics
associated with these micro- and macroscales introduces three distinct
transport regimes, which depend on the CR energy.

First, in the high-energy (he) limit, CRs with gyroradii $r_{\mathrm{g}}\gg
l_{\mathrm{c}}$ (energies $E\gtrsim 100\,$EeV for typical turbulence-driving
scales in galaxy clusters) undergo scattering by small angles of the order
$\delta\Theta\sim l_{\mathrm{c}}/r_{\mathrm{g}}$ at characteristic times
$\delta t\sim l_{\mathrm{c}}/c$, leading to a scattering rate
$\nu_{\mathrm{he}}\sim\delta\Theta^{2}/\delta t\sim
cl_{\mathrm{c}}/r_{\mathrm{g}}^{2}$ and, therefore, to a CR diffusion
coefficient $\kappa_{\mathrm{he}}\sim c^{2}/\nu_{\mathrm{he}}\sim
cr_{\mathrm{g}}^{2}/l_{\mathrm{c}}\propto E^{2}/l_{\mathrm{c}}$ [e.g., 5].
This scaling is indeed observed both in numerical simulations [e.g., 58, 94,
77] and (scaled) laboratory experiments [26] and serves as input for
propagation models of ultra-high-energy CRs in galaxy clusters [30].

As the high-energy regime is, thus, believed to be understood, recent studies
of CR transport have predominantly focused on the second, mesoscale regime,
$r_{\mathrm{g}}\lesssim l_{\mathrm{c}}$, in which CRs scatter resonantly off
inertial-range turbulent structures [50, 37, 68, 55]. This second regime is
likely relevant for explaining CR spectra [7], e.g., the steepening of the CR
spectrum in the Galactic Center from GeV to PeV energies [77, 33, 22, 101].

We argue, and confirm numerically, that in high-$\beta$ plasmas, the presence
of microstructures caused by plasma instabilities introduces a third regime,
whose physics is similar to that of the first, but with the micromirror scale
$l_{\mathrm{mm}}$ playing the role of $l_{\mathrm{c}}$ and the requirement
that $r_{\mathrm{g}}\gg l_{\mathrm{mm}}$ ($E\gg 100\,$MeV). We show that this
microscale physics largely overrides the mesoscale resonant scattering and
streaming. We apply our theory to the intracluster medium (ICM), a
representative high-$\beta$ plasma, and determine the transition between
micro- and macrophysics-dominated transport to be at TeV energies, only weakly
influenced by mesoscale physics. We confirm this theory with a novel method
(described in Section 3) that incorporates the microscales
($l_{\mathrm{mm}}\sim 100\,$npc), the macroscales ($l_{\rm c}\sim 100\,$kpc),
and the vast range in between.

##  1 Results

###  1.1 Effect of micromirrors on large-scale CR transport

It has long been realized that plasma instabilities may dominate the transport
of low-energy CRs. However, the instabilities most often highlighted in the
literature arise from the CRs themselves, rather than from the thermal plasma.
One prominent example, especially for Galactic transport of CRs below
$100\,$GeV, is the streaming instability [63]. This instability generates
fluctuations in the magnetic field that in turn scatter the CRs and thereby
reduce their streaming velocity to be comparable to the Alfvén speed in the
plasma [63, 88, 12, 86]. By contrast, in high-$\beta$, weakly collisional
plasmas, there exists a variety of instabilities that are driven by pressure
anisotropies and generate magnetic fluctuations on ion-Larmor scales (see [66]
and references therein). The pressure anisotropies arise from the
(approximate) conservation of particle adiabatic invariants during the local
stretching and compression of magnetic fields [27]. In the present context,
the mirror instability [9, 42] is of particular interest because its saturated
amplitude $\delta B_{\mathrm{mm}}\sim B/3$ is of the same order of magnitude
as the ambient magnetic field $B$ [64, 79, 69].222 By contrast, the other
well-known instability arising in such plasmas, the firehose instability, is
unlikely to affect CR transport because its expected saturation amplitude is
small under ICM conditions: $\delta B_{\rm f}\sim(\tau\Omega_{i})^{-1/4}B$
[64, 16], where $\tau$ is the timescale over which a firehose-susceptible
plasma evolves macroscopically, and $\Omega_{i}\sim
0.01\,(B/3\,\mu\mathrm{G})\,\mathrm{s}^{-1}$ is the non-relativistic thermal-
ion gyrofrequency. In the ICM, $\tau\Omega_{i}\sim 10^{11}$ [80], so $\delta
B_{\rm f}\sim 10^{-3}B$. By analogy to (3), it follows that the CR scattering
rate $\nu_{\rm f}$ of CRs off firehoses in the ICM is much smaller than that
off the mirrors: $\nu_{\rm f}/\nu_{\rm mm}\sim 10^{-7}$ (with the firehose
scale taken to be comparable to the thermal-ion gyroradius [64, 16]).

Figure 1: The micromirror field generated by the PIC simulation discussed in
the text: color shows fluctuations along ($\delta B_{\parallel}$) the field
$B$, which is aligned with the $x$-axis. We show two snapshots of the 3D field
during its secular evolution, characterized by the different $\delta
B_{\mathrm{rms}}/B$ (the right snapshot is later in the evolution).

To determine the impact of these micromirrors on CR transport, we begin by
working out the relevant theoretical predictions for diffusion coefficients of
CRs scattering at such strong fluctuations.333A previous assumption of weaker
micromirror fluctuations led to a different, much larger diffusion coefficient
based on calculations using quasi-linear theory [41]. The velocity change
$\delta\boldsymbol{v}$ of a relativistic CR with gamma factor
$\gamma=(1-v^{2}/c^{2})^{-1/2}$, charge $q=Ze$, and mass $m$ in a magnetic-
field structure of scale $l_{\mathrm{mm}}\ll r_{\mathrm{g}}$ and vector
amplitude $\delta\boldsymbol{B}_{\mathrm{mm}}$ is given in the small-angle
limit $|\delta\boldsymbol{v}|\ll v$ by integrating the equation of motion
$\gamma
m\,\mathrm{d}\boldsymbol{v}/\mathrm{d}t=q\,(\boldsymbol{v}/c)\boldsymbol{\times}\delta\boldsymbol{B}_{\mathrm{mm}}$
along the CR path:

$\delta\boldsymbol{v}\sim\frac{q}{\gamma
mcv}\int_{0}^{l_{\mathrm{mm}}}\mathrm{d}l\,\boldsymbol{v}\boldsymbol{\times}\delta\boldsymbol{B}_{\mathrm{mm}}\,.$
(1)

Assuming relativistic CRs with $v\approx c$, energy $E=\gamma mc^{2}$, and
gyroradius $r_{\mathrm{g}}=\gamma mc^{2}/qB$ determined by the ambient
magnetic field $B\gtrsim\delta B_{\mathrm{mm}}$, the scattering angle at
characteristic times $\delta t~{}\sim~{}l_{\mathrm{mm}}/c$ is

$\delta\Theta\sim\frac{|\delta\boldsymbol{v}|}{c}\sim\frac{l_{\mathrm{mm}}}{r_{\mathrm{g}}}\frac{\delta
B_{\mathrm{mm}}}{B}\,.$ (2)

Assuming that these small-angle deflections add up as a correlated random
walk, the scattering rate is

$\nu_{\mathrm{mm}}\sim\frac{\delta\Theta^{2}}{\delta
t}\sim\frac{c\,l_{\mathrm{mm}}}{r_{\mathrm{g}}^{2}}\left(\frac{\delta
B_{\mathrm{mm}}}{B}\right)^{2}\,,$ (3)

which implies a spatial diffusion coefficient of

$\kappa_{\mathrm{mm}}\sim\frac{c^{2}}{\nu_{\mathrm{mm}}}\sim\frac{c\,r_{\mathrm{g}}^{2}}{l_{\mathrm{mm}}}\left(\frac{\delta
B_{\mathrm{mm}}}{B}\right)^{-2}\propto E^{2}l_{\mathrm{mm}}^{-1}\,.$ (4)

As usual, more energetic CRs diffuse much faster.

Figure 2: Diffusion coefficients of CRs in the micromirror fields presented in
Figure 1. The diffusion coefficient $\kappa$ is dominated by the parallel
diffusion, the perpendicular diffusion coefficient $\kappa_{\perp}$ is
negligible. Our predictions (5) (gray dashed line) and (6) (gray dash-dotted
line), valid for $l_{\mathrm{mm}}\ll r_{\mathrm{g}}$, agree well with the
computed diffusion coefficients. Note that $\kappa$ decreases with $\delta
B_{\mathrm{mm}}/B$ (triangles vs. stars). We include results for CRs with
gyroradii smaller than the micromirror scale $l_{\mathrm{mm}}$ to highlight
the change in transport regimes, which follows from our theory. The gray
markers show the transport of CRs through the residual field that results
after filtering out the wavenumbers associated with the micromirrors, leaving
only numerical noise. The purple vertical (dash-)dotted lines show grid
resolution and box sizes along the three axes. $Z=1$ is used for the energy
scale.

In arriving at (4), we effectively assumed that micromirrors are described by
only one characteristic scale, $l_{\mathrm{mm}}$. In reality, micromirrors are
anisotropic (see ellipsoid-like shapes in Figure 1) with scales perpendicular
($\perp$) and parallel ($\parallel$) to the ambient magnetic field that
satisfy $l_{\perp,\mathrm{mm}}\ll l_{\parallel,\mathrm{mm}}$. While gyrating
through this field, CRs with $r_{\mathrm{g}}\gg l_{\mathrm{mm}}$ mostly
traverse micromirrors perpendicularly444Only low-energy CRs satisfying
$r_{\mathrm{g}}\,v_{\perp}/c\lesssim l_{\perp,\mathrm{mm}}\ll
l_{\parallel,\mathrm{mm}}$ are an exception and should be treated analogously
to thermal electrons with negligible $r_{\mathrm{g}}$ getting scattered [79,
102] and trapped [57] in the micromirrors. This subpopulation makes a
negligible contribution to the overall transport of CRs with
$r_{\mathrm{g}}\gg l_{\perp,\mathrm{mm}}$ considered here.. For
$r_{\mathrm{g}}\gg l_{\perp,\mathrm{mm}}$, CRs will sample many different
micromirrors, with deflections adding up as a correlated random walk555During
one gyro-orbit, CRs will travel $\Delta l_{\parallel}\sim 2\pi
r_{\mathrm{g}}\,v_{\parallel}/c$ in the field-parallel direction. CRs with
large pitch angles satisfying $\Delta l_{\parallel}\lesssim
l_{\mathrm{\parallel,mm}}$, i.e., $v_{\perp}/v_{\parallel}\sim
c/v_{\parallel}\gtrsim 2\pi r_{\mathrm{g}}/l_{\mathrm{\parallel,mm}}$, that
sample the same micromirror repeatedly, may become relevant only at low
energies $r_{\mathrm{g}}\lesssim l_{\mathrm{\parallel,mm}}/2\pi$, not
considered in this study.. The scattering rate associated with the parallel
micromirror perturbation $\delta B_{\parallel}\sim B_{\mathrm{mm}}$ decreases
with decreasing pitch angle, but this is overcome by scattering at the
perpendicular micromirror component $\delta B_{\perp}\sim\delta
B_{\parallel}l_{\perp,\mathrm{mm}}/l_{\parallel,\mathrm{mm}}$ for
$v_{\perp}/v_{\parallel}\lesssim
l_{\perp,\mathrm{mm}}/l_{\parallel,\mathrm{mm}}\ll 1$. Except for this cone
containing CRs with small pitch angles, from which they escape quickly on the
time scale
$t_{\mathrm{esc}}\sim\nu_{\mathrm{mm}}^{-1}\,l_{\perp,\mathrm{mm}}/l_{\parallel,\mathrm{mm}}$,
gyrating CRs cross micromirrors perpendicularly faster than they traverse them
in the parallel direction, implying $l_{\mathrm{mm}}\sim
l_{\perp,\mathrm{mm}}$ to be the relevant scale.

For application to the ICM, we estimate $l_{\mathrm{mm}}$ using an asymptotic
theory of the mirror instability’s nonlinear evolution [78] supported by
previous numerical studies [64, 71]:
$l_{\mathrm{mm}}\sim(\tau\Omega_{i})^{1/8}r_{\mathrm{g},i}$, where $\tau$ is
the timescale over which a micromirror-susceptible plasma evolves
macroscopically. Applying the theory to an ICM [40, 66] with $B\sim 3\,\mu$G,
the thermal-ion gyroradius
$r_{\mathrm{g},i}\sim(2T/m_{i})^{1/2}/\Omega_{i}\sim 1\,$npc, the mean galaxy
cluster temperature $T\sim 5\,$keV, the thermal-ion mass $m_{i}$, and
$\tau\sim 10^{12}\,\mathrm{s}$ [80, 78] yields $l_{\mathrm{mm}}\sim
l_{\perp,\mathrm{mm}}\sim 100\,r_{\mathrm{g},i}\sim 100\,$npc [71], only a
factor of few smaller than the gyroradius of a GeV CR. In combination with
(4), this gives us the estimate

$\begin{split}\kappa_{\mathrm{mm}}&\sim
10^{30}\,Z^{-2}\,\left(\frac{l_{\mathrm{mm}}}{100\,\mathrm{npc}}\right)^{-1}\left(\frac{B}{3\,\mu\mathrm{G}}\right)^{-2}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\frac{\delta
B_{\mathrm{mm}}/B}{1/3}\right)^{-2}\left(\frac{E}{\mathrm{TeV}}\right)^{2}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1},\\\
&\sim
10^{30}\,Z^{-2}\,\left(\frac{T}{5\,\mathrm{keV}}\right)^{-1/2}\left(\frac{B}{3\,\mu\mathrm{G}}\right)^{-1}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\frac{\delta
B_{\mathrm{mm}}/B}{1/3}\right)^{-2}\left(\frac{E}{\mathrm{TeV}}\right)^{2}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}\,.\end{split}$
(5)

This estimate is valid provided $l_{\mathrm{mm}}\ll r_{\mathrm{g}}$ and
$\delta t\,\nu_{\mathrm{mm}}\sim(l_{\mathrm{mm}}/r_{\mathrm{g}})^{2}(\delta
B_{\mathrm{mm}}/B)^{2}\ll 1$ ($E\gg 100\,$MeV).

We now show that the diffusion coefficient (5) is associated with parallel
transport along field lines by demonstrating that the perpendicular diffusion
coefficient is negligible. Each scattering at characteristic times $\delta
t~{}\sim~{}l_{\mathrm{mm}}/c$ moves the gyrocenter by a distance $\Delta
r_{\perp}\sim r_{g}\delta\Theta$ in the plane perpendicular to the local
magnetic field line. Using the estimate (2) for the scattering angle leads to
the perpendicular diffusion coefficient

$\begin{split}\kappa_{\perp,\mathrm{mm}}&\sim\frac{\Delta
r_{\perp}^{2}}{\delta
t}\sim\frac{r_{g}^{2}\delta\Theta^{2}}{l_{\mathrm{mm}}/c}\sim
cl_{\mathrm{mm}}\left(\frac{\delta B_{\mathrm{mm}}}{B}\right)^{2}\\\
&\sim\frac{l_{\mathrm{mm}}^{2}}{r_{\mathrm{g}}^{2}}\left(\frac{\delta
B_{\mathrm{mm}}}{B}\right)^{4}\kappa_{\mathrm{mm}}\,.\end{split}$ (6)

Since $\kappa_{\perp,\mathrm{mm}}\ll\kappa_{\mathrm{mm}}$ for
$r_{\mathrm{g}}\gg l_{\mathrm{mm}}$ and $\delta B_{\mathrm{mm}}\lesssim B$, it
is the parallel diffusion
$\kappa_{\parallel,\mathrm{mm}}\sim\kappa_{\mathrm{mm}}$ along field lines
that dominates.

To validate our theoretical prediction for the diffusion coefficients (5) and
(6), we performed a numerical experiment in which a spectrum of CRs was
integrated in a magnetic field containing micromirrors generated self-
consistently via a particle-in-cell (PIC) simulation (Section 3.6). For our
numerical experiment, we selected two representative realizations of the 3D
field during its secular evolution, visualized in Figure 1. We then determined
the diffusion coefficients of the CRs in both fields by integrating the CR
equation of motion. The results are shown in Figure 2. The diffusion
coefficients in the micromirror fields show good agreement with (5) and (6).

###  1.2 The micro-macrophysics transition is at TeV CR energies

In Section 1.1, we derived the diffusion coefficient
$\kappa_{\mathrm{mm}}\propto l_{\mathrm{mm}}^{-1}E^{2}$ associated with CR
scattering at micromirrors of scale $l_{\mathrm{mm}}$. To determine the upper
bound for the energies at which the scattering off micromirrors dominates CR
transport, one needs a model of the competing contribution to it from the
resonant scattering off mesoscale magnetic turbulence. In Section 3.4, we
present models of CR diffusion based on resonant scattering in the (mesoscale)
inertial range of magnetic turbulence stirred at the macroscale
$l_{\mathrm{c}}$, leading to the diffusion coefficient

$\kappa_{\mathrm{res}}\sim
c\,l_{\mathrm{c}}\left(\frac{r_{\mathrm{g}}}{l_{\mathrm{c}}}\right)^{\delta}\propto
E^{\delta}l_{\mathrm{c}}^{-\delta+1}\,,$ (7)

where the model-dependent exponent is $0\leq\delta\leq 1/2$ (Section 3.4). The
mechanism with the smallest diffusion coefficient dominates CR transport. The
transition between the micromirror and resonant-scattering transport regimes
occurs when $\kappa_{\mathrm{mm}}\sim\kappa_{\mathrm{res}}$. Equating (4) and
(7) determines the gyroradius corresponding to this transition:

$r_{\mathrm{g}}\sim l_{\mathrm{c}}\left(\frac{\delta
B_{\mathrm{mm}}}{B}\right)^{2/(2-\delta)}\left(\frac{l_{\mathrm{mm}}}{l_{\mathrm{c}}}\right)^{1/(2-\delta)}\,.$
(8)

Figure 3: CR diffusion coefficients (Section 3.5) of cosmic rays (CRs) with
$Z=1$ in the ICM displayed as functions of CR energy for a number of model
turbulence spectra. These diffusion coefficients are calculated from simulated
CR trajectories, taking into account their equation of motion and scattering
by micromirrors (mm) (Section 3). The black stars show the diffusion
coefficients in the micromirror field generated by a PIC simulation (Section
3.6). The olive empty crosses show the diffusion coefficients of CRs in MHD
turbulence without a guide field (Section 3.7). The other empty markers show
the diffusion coefficients computed in isotropic synthetic turbulence with a
large inertial range (Section 3.1), together with our stochastic micromirror-
scattering model (Section 3.3). The grey dash-dotted lines represent theories
for CR transport depending on the macroscale $l_{\mathrm{c}}$ according to
(7), including the most efficient (Bohm) and the least efficient (energy-
independent) diffusion scenarios. The grey dotted line represents the
diffusion due to streaming instability, according to (14). The gray dashed
line represents our prediction of the diffusion due to micromirrors according
to (5). The vertical light-blue bar indicates our estimate (9) for the micro-
macro transition for the most likely range of $\delta$ values between $1/3$
and $1/2$. Simulation results from Subedi et al. [94] and Kuhlen et al. [62]
illustrate the best resolution towards the limit $r_{\mathrm{g}}\ll
l_{\mathrm{c}}$ achieved prior to the present results with synthetic
turbulence on a grid ($2048^{3}$ grid points) and nested grids, respectively.

This translates into a $\delta$-dependent estimate for the transition energy:

$\displaystyle E\sim
Z\,\left(\frac{B}{3\,\mu\mathrm{G}}\right)\left(\frac{l_{\mathrm{c}}}{100\,\mathrm{kpc}}\right)\times\begin{cases}\displaystyle
300\,\left(\frac{\delta
B_{\mathrm{mm}}}{B}\right)^{2/(2-\delta)}\left(\frac{l_{\mathrm{mm}}}{l_{\mathrm{c}}}\right)^{1/(2-\delta)}\,\mathrm{EeV}\,,&\text{general
$\delta$}\,,\\\ \displaystyle 5\,\left(\frac{\delta
B_{\mathrm{mm}}/B}{1/3}\right)^{6/5}\left(\frac{l_{\mathrm{mm}}/l_{\mathrm{c}}}{10^{-12}}\right)^{3/5}\,\mathrm{TeV}\,,&\delta=1/3\,,\\\
\displaystyle 600\,\left(\frac{\delta
B_{\mathrm{mm}}/B}{1/3}\right)^{4/3}\left(\frac{l_{\mathrm{mm}}/l_{\mathrm{c}}}{10^{-12}}\right)^{2/3}\,\mathrm{GeV}\,,&\delta=1/2\,.\end{cases}$
(9)

Below this energy, magnetic micromirrors dominate CR diffusion. The factor
involving the ratio $l_{\mathrm{mm}}/l_{\mathrm{c}}$ accounts for the scale
separation between micro- and macrophysics, which we estimate to be
${\sim}10^{-12}$ in the ICM.

To test this prediction, we performed numerical simulations of CR transport in
the ICM (detailed in Section 3), modeling the effects of both the micromirrors
(Section 3.3) and of the turbulent cascade up to $l_{\rm c}\sim 100\,$kpc
(Section 3.4). Figure 3 summarizes our results by presenting the CR diffusion
coefficient as a function of energy. The vertical light-blue bar indicates our
estimate (9) for the micro-macro transition assuming the most likely range of
$\delta\in[1/3,1/2]$. While this estimate of the micro-macrophysics transition
at TeV CR energies is numerically confirmed using synthetic turbulence, we
also used magnetic fields from direct PIC and MHD simulations to validate the
consistency of our numerical approach at micro- and macroscales, respectively,
and capture all relevant diffusion coefficients discussed in the literature,
and detailed as points (i) to (iii) in Section 3.4.

###  1.3 Case of spatially intermittent micromirrors

Thus far, we have effectively assumed that the micromirrors permeate the
plasma uniformly. In reality, the situation is more complicated: micromirrors
will most likely appear in spatially intermittent and temporally transient
patches wherever turbulence leads to local amplification of the magnetic field
at a rate that is sufficiently large to engender positive pressure anisotropy
exceeding the mirror-instability threshold (${\sim}1/\beta$; see, e.g., [71,
91] and references therein). This gives rise to an effectively two-phase
plasma, with two different effective scattering rates: $\nu_{\mathrm{mm}}\sim
c^{2}/\kappa_{\mathrm{mm}}$ in micromirror patches and $\nu_{\mathrm{res}}\sim
c^{2}/\kappa_{\mathrm{res}}$ elsewhere (instead of $\kappa_{\mathrm{res}}$,
one could also use $\kappa_{\mathrm{st}}$ associated with the streaming
instability – see Section 2). For the purpose of modeling CR scattering in
such a plasma, we introduce the effective micromirror fraction
$f_{\mathrm{mm}}$ to quantify the probability of CR being scattered by the
micromirrors. By definition, the effective scattering rate in a two-phase
medium is [52, 59]:

$\nu_{\mathrm{eff}}=f_{\mathrm{mm}}\,\nu_{\mathrm{mm}}+(1-f_{\mathrm{mm}})\,\nu_{\mathrm{res}}\,.$
(10)

The effective diffusion coefficient is then

$\kappa_{\mathrm{eff}}\sim\frac{\kappa_{\mathrm{mm}}}{f_{\mathrm{mm}}+(1-f_{\mathrm{mm}})\kappa_{\mathrm{mm}}/\kappa_{\mathrm{res}}}\,.$
(11)

The transition at which micromirror transport takes over from resonant
scattering is entirely independent of $f_{\mathrm{mm}}$:
$\kappa_{\mathrm{eff}}\sim\kappa_{\mathrm{res}}$ when
$\kappa_{\mathrm{mm}}\sim\kappa_{\mathrm{res}}$. However, the asymptotic
scaling $\kappa_{\mathrm{eff}}\sim\kappa_{\mathrm{mm}}/f_{\mathrm{mm}}$ is
only reached at CR energies for which
$\kappa_{\mathrm{mm}}/\kappa_{\mathrm{res}}\lesssim
f_{\mathrm{mm}}/(1-f_{\mathrm{mm}})$, pulling the transition energy down by a
factor of $f_{\mathrm{mm}}^{1/(2-\delta)}$. This is not a very strong
modification of our cruder ($f_{\mathrm{mm}}=1$) estimate (9) unless
$f_{\mathrm{mm}}$ is extremely small.

The most intuitive interpretation of $f_{\mathrm{mm}}$ is that it is the
fraction of the plasma volume occupied by the micromirrors. This, however,
requires at least two caveats: (i) the lifetime of micromirror patches,
determined by the turbulent dynamics, can be shorter than the time for a CR to
diffuse through the patch; (ii) if the micromirror patches form solid
macroscopically extended three-dimensional blobs, it is possible to show that
CRs typically do not penetrate much farther than $\lambda_{\mathrm{mm}}$ into
the patches. This leaves the patch volume largely uncharted (see Appendix 3 in
[59]). The second concern obviates the first (see Section 3.8). Note that
under such a scenario, the effective CR diffusion in the ICM might be
determined primarily by such factors as the typical size of the patches and
the distance between them [21]. However, the scenario of micromirror-dominated
transport is made more plausible as the diffusion is mostly along the field
line, which will guide CRs naturally into and out of the micromirror patches,
especially if the two phases of the scattering medium intertwine narrowly. For
example, with micromirror patches forming sheet-like structures – in a
turbulent environment, where those patches should coincide with regions of
greatest amplification of the magnetic field, such a scenario appears most
likely.

Figure 4: Effective diffusion coefficient of charged particles in a two-phase
medium vs. the effective micromirror fraction $f_{\mathrm{mm}}$ modeled by the
volume-filling fraction of magnetic fields whose strength exceeds a certain
threshold (see Section 3.8). We model the CR transport through a two-phase
medium in 3D and 1D (to reduce simulation costs), as explained in Section 3.8.
The 3D case (red diamonds) shows the diffusion coefficient of $300\,$GeV CRs
computed at their trajectory lengths of ${\simeq}10l_{\mathbf{c}}$. The blue
and black lines with corresponding error bands show the 1D results on 400 test
trajectories using our recurrent neural network (RNN) and the classical
analysis method. The dash-dotted black line represents expected values for
averaged scattering frequencies of CRs in the two-phase plasma, formally
expressed in (11). The diffusion coefficients $\kappa_{\mathrm{mm}}$ and
$\kappa_{\mathrm{res}}$ are recovered for $f_{\mathrm{mm}}=1$ and
$f_{\mathrm{mm}}=0$, respectively.

Indeed, our model formula (11) proves to be a good prediction even in a simple
modification of our numerical experiment with synthetic fields, designed to
model micromirror patches (see Section 3.8). Its results are shown in Figure
4. It is a matter of future work to determine the precise dependence of
$f_{\mathrm{mm}}$ on the morphology and dynamics of magnetic fields and
micromirror-unstable patches in high-$\beta$ turbulence – itself a system that
has only recently become amenable to numerical modeling [8, 91]. Here, we
proceed to discuss the implications of dominant micromirror transport,
assuming that $f_{\mathrm{mm}}$ is not tiny.

##  2 Discussion

We have argued that CR diffusion in the ICM is determined by microscale
($l_{\mathrm{mm}}$) mirrors at energies $E\lesssim{\rm TeV}$ and by macroscale
($l_{\mathrm{c}}$) turbulence at $E\gtrsim{\rm TeV}$. Although the micro-
macrophysics transition mainly depends on these two scales, there is a degree
of fine-tuning at mesoscales. This refers to the role played in (9) by the
exponent $\delta$, which depends on the details of the scattering mechanism
and of the turbulent cascade (see Section 3.4). While our study is tailored to
the ICM, it can be adapted to other nearly collisionless high-$\beta$ plasmas,
such as the hot interstellar medium and the Milky Way halo.

Let us now discuss the self-confinement of CRs due to the streaming
instability, believed to play an important role in the Galaxy [12] and in
galaxy clusters [48]. In this picture, the streaming instability generates
fluctuations of the magnetic field, which in turn can scatter CRs. It is
believed that this mechanism may take over at lower CR energies, with details
depending on the instability’s growth rate at wavenumber $k$ [63]:

$\begin{split}\gamma_{\mathrm{SI}}&\sim\Omega_{i}\frac{n_{\mathrm{CR}}({>}E)}{n_{i}}\left(\frac{v_{\mathrm{st}}}{v_{\mathrm{A}}}-1\right)\\\
&\sim
10^{-14}\,\left(\frac{B}{3\,\mu\mathrm{G}}\right)\left(\frac{E}{\mathrm{TeV}}\right)^{-1.6}\,\mathrm{s}^{-1}\,,\end{split}$
(12)

where ${n_{\mathrm{CR}}({>}E)}$ is the density of CRs with energies above
energy $E$ corresponding to the resonance condition that can interact
resonantly with waves with wavenumber $k$, $n_{i}$ is the ambient ion density,
$v_{\mathrm{A}}$ is the Alfvén speed, and $v_{\mathrm{st}}$ is the streaming
speed, believed to be on the order of $v_{\mathrm{A}}$ in saturation for the
$\sim\,$GeV CRs [48, 61]. In the second estimate in (12), we employed the
common assumptions (see, e.g., [103, 54] and references therein) that
$v_{\mathrm{st}}/v_{\mathrm{A}}\sim 1$ and $n_{\mathrm{CR}}({>}E)/n_{i}\sim
10^{-7}(E/\mathrm{GeV})^{1-\alpha}$ in galaxy clusters, with $\alpha\approx
2.6$.

Scattering of sub-TeV CRs at micromirrors increases the effective CR
collisionality in high-$\beta$ environments. This could potentially undermine
the CR streaming instability. A comparison of the growth rate (12) with the
scattering rate at micromirrors (3) gives

$\begin{split}\frac{\gamma_{\mathrm{SI}}}{\nu_{\mathrm{mm}}}&\sim
10^{-5}\,Z^{-2}\left(\frac{T}{5\,\mathrm{keV}}\right)^{-1/2}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\frac{\delta
B_{\mathrm{mm}}/B}{1/3}\right)^{-2}\left(\frac{E}{\mathrm{TeV}}\right)^{0.4}\,.\end{split}$
(13)

With such a large effective collisionality isotropizing and homogenizing CRs,
it is doubtful that a collisionless, resonant instability can operate.

Another way to gauge the importance of the streaming instability is to compare
the putative CR diffusion coefficient $\kappa_{\mathrm{st}}$ due to it with
$\kappa_{\mathrm{mm}}$ at the same energies. Let us pretend that the
instability is not suppressed, despite (13), estimating $\kappa_{\mathrm{st}}$
in an environment with magnetic-field lines stochastically tangled on the
scale $l_{\mathrm{c}}$ [94, 61]:

$\begin{split}\kappa_{\mathrm{st}}&\sim l_{\mathrm{c}}v_{\mathrm{st}}\gtrsim
l_{\mathrm{c}}v_{\mathrm{A}}\\\ &\sim
10^{30}\,\left(\frac{l_{\mathrm{c}}}{100\,\mathrm{kpc}}\right)\left(\frac{v_{\mathrm{A}}}{100\,\mathrm{km/s}}\right)\,\mathrm{cm}^{2}~{}\mathrm{s}^{-1}\,,\end{split}$
(14)

where we have used the Alfvén speed $v_{\mathrm{A}}\sim 100\,\mathrm{km/s}$ as
the lower limit of the streaming speed. Comparing this CR diffusivity with the
one caused by micromirrors (5) gives

$\begin{split}\frac{\kappa_{\mathrm{mm}}}{\kappa_{\mathrm{st}}}&\sim
Z^{-2}\left(\frac{E}{\mathrm{TeV}}\right)^{2}\left(\frac{l_{\mathrm{c}}}{100\,\mathrm{kpc}}\right)^{-1}\left(\frac{v_{\mathrm{A}}}{100\,\mathrm{km/s}}\right)^{-1}\\\
&~{}~{}\times\left(\frac{B}{3\,\mu\mathrm{G}}\right)^{-1}\left(\frac{\delta
B_{\mathrm{mm}}/B}{1/3}\right)^{-2}\left(\frac{T}{5\,\mathrm{keV}}\right)^{-1/2},\end{split}$
(15)

demonstrating that $\kappa_{\mathrm{mm}}$ dominates CR transport in the ICM up
to TeV energies. Note that the effective Alfvén speed is increased due to
positive pressure anisotropy in micromirror patches [65, 104, 15], which
should reduce the growth rate (12) of the streaming instability and further
increase the resulting CR diffusion coefficient (14). While the suppressed
streaming instability within micromirror regions of the ICM challenges the
universal applicability of the prevailing models that seek to explain the
thermal balance between heating and cooling of galaxy clusters based on
collisionless, resonant mechanisms [48, 49], a patchy distribution of these
micromirrors may allow for the existence of regions where the streaming
instability remains active – indeed possibly more so than usually expected, as
those micromirror-free regions are likely to feature negative pressure
anisotropies and, therefore, reduced effective Alfvén speeds.

With this revised picture of CR diffusion in the ICM, we can discuss the
emission signatures of diffusing CRs within galaxy clusters. Galaxy clusters
have been identified as potentially significant contributors to the observed
diffuse gamma-ray background through hadronic interactions of CR protons with
background gas and photons (see, e.g., references in [31]). However, direct
observations of diffuse gamma-ray emission from galaxy clusters remain elusive
[31]. Our model offers a viable explanation for this non-detection:
micromirrors in the ICM confine sub-TeV CR protons near their emission sources
within galaxy clusters. This should result in (extended) point-like gamma-ray
emission rather than widespread diffuse emission, as indeed observed in [75,
2]. While this suggests a similar confinement of CR electrons, the prevalent
synchrotron radiation in some galaxy clusters, evident in the form of large
radio halos, could be attributed, e.g., to electron reacceleration
processes666The reduction in diffusion coefficients due to micromirror
scattering does not necessarily promote the reacceleration of CRs. Because the
phase velocity of the micromirrors is zero in the rest frame of the plasma,
the motions of the effective scattering sites are tied to the bulk flow.
Nevertheless, an effective non-resonant second-order Fermi acceleration may
occur in compressible turbulence when the diffusion coefficient is comparable
to the product of the sound speed and the scale of the compressing wave, so
that CRs can random walk in energy with energy gains (losses) when confined
during compressions (rarefactions) with a net energy gain [20]. [31] or to
secondary electrons that might be produced by interactions between very-high-
energy gamma rays and diffuse background radiation [96].

Finally, the suppressed CR diffusion coefficient due to micromirrors offers a
compelling explanation for how CRs can be effectively “frozen” within the ICM
as key parameters such as gas density and magnetic-field intensity evolve, an
assumption fundamental to recent models of the dynamical evolution of galaxy
clusters and their surroundings [32, 11, 14]. This impacts the interplay
between CRs and the broader astrophysical mechanisms within these massive
cosmic structures.

##  3 Methods

Modeling CR transport across a wide energy spectrum from GeV to EeV energies
in a multiscale high-$\beta$ plasma presents methodological and computational
challenges. For our numerical simulations, we choose parameters from the ICM,
a high-$\beta$ plasma. Rotation-measure data indicate magnetic-field strengths
${\sim}0.1$–$1\,\mu$G averaged over a Mpc3 ICM volume [40, 93], with typical
field strengths of several $\mu$G in central regions [e.g., 13]. Numerical
simulations support these estimates [e.g., 92]. We choose $B\sim 3\,\mu$G,
$\delta B_{\mathrm{mm}}/B\sim 1/3$, the micromirror scale $l_{\mathrm{mm}}\sim
100\,$npc, and turbulence correlation length $l_{\rm c}\sim 100\,$kpc [66].
Simulations are performed using the publicly available tool CRPropa 3.2 [6]
with additional extensions and modeling choices described below.

###  3.1 Model of synthetic magnetic turbulence

Modeling the competing micro- and macrophysical transport effects requires
resolving turbulence over at least ten decades in scale. Current MHD and PIC
simulations are unsuitable for this as they only allow for limited scale
ranges [72]. Even the current best grid resolutions of more than $10^{12}$
cells resolve less than four decades in scale separation. Synthetic
turbulence, on the other hand, can be generated by summing over
$n_{\mathrm{m}}$ plane waves at an arbitrary particle position
$\boldsymbol{r}$ as follows [37, 95]

$\begin{split}\delta\boldsymbol{B}(\boldsymbol{r})&=\mathrm{Re}\left(\sum_{n=1}^{n_{m}}\delta\boldsymbol{B}_{n}^{*}\,\mathrm{e}^{{\rm
i}\boldsymbol{k}_{n}\boldsymbol{\cdot}\boldsymbol{r}}\right)\\\
&=\sqrt{2}\,\delta
B\sum_{n=1}^{n_{m}}\boldsymbol{\xi}_{n}A_{n}\,\cos\left(k_{n}\,\hat{\boldsymbol{k}}_{n}\boldsymbol{\cdot}\boldsymbol{r}+\phi_{n}\right)\,,\end{split}$
(16)

with normalized amplitudes $A_{n}$ determined by the assumed turbulent energy
spectrum, uniformly distributed phase factors $\phi_{n}~{}\in~{}[0,2\pi]$,
unit wavevectors $\hat{\boldsymbol{k}}_{n}$, and polarizations
$\boldsymbol{\xi}_{n}$, satisfying
$\hat{\boldsymbol{k}}_{n}\boldsymbol{\cdot}\boldsymbol{\xi}_{n}=0$. We employ
the performance-optimized method described in [85]. We investigated the number
of wavemodes $n_{\mathrm{m}}$ needed by analyzing turbulence characteristics
and diffusion coefficients of CRs and found that $n_{\mathrm{m}}=1024$ log-
spaced wavemodes are sufficient.

We compared our key results on CR transport obtained with the above method
with those obtained with an alternative method for synthetic turbulence, where
we followed the approach proposed, e.g., in [38, 72]. In this alternative
method, synthetic turbulence is precomputed and stored on many discrete nested
grids at different scales, with magnetic fluctuations between scales
$l_{\mathrm{min},i}$ and $l_{\mathrm{max},i}$ with individual magnetic-field
strengths $\delta B_{i}^{2}=\delta
B^{2}(l_{\mathrm{max},i}^{\xi-1}-l_{\mathrm{min},i}^{\xi-1})/(l_{\mathrm{max}}^{\xi-1}-l_{\mathrm{min}}^{\xi-1})$
and $l_{\mathrm{max}}\sim 5l_{\mathrm{c}}$. We found agreement between results
obtained using both methods.

Note that CRs below PeV energies diffuse on time scales smaller than the
typical timescale of large-scale turbulence motions, justifying our approach
of static turbulence. CRs are “frozen” within the ICM as the plasma evolves,
as discussed in Section 2.

###  3.2 CR trajectories

Computing CR trajectories in magnetic fields involves solving the equation of
motion for charged particles. These are then used to calculate the statistical
transport characteristics of CRs. We use the Boris-push method for this task,
as implemented in [6]. This captures the dynamics of charged particles in
magnetic fields while preserving key properties, such as CR energy.

###  3.3 Model of small-angle scattering in magnetic micromirrors

We model the effect of magnetic micromirrors as a change of propagation after
a distance $s$ by an angle $\delta\theta$ given the scattering rate
$\nu_{\mathrm{mm}}=\delta\theta^{2}\,c/s$. Particles traveling the mean free
path $\lambda_{\mathrm{mm}}=c/\nu_{\mathrm{mm}}$ will have lost information of
their original direction. At each step $s$, chosen to be smaller than
$\lambda_{\mathrm{mm}}$, we introduce a small deflection

$\delta\theta=X\,\sqrt{\frac{s\,\nu_{\mathrm{mm}}}{c}}\,,$ (17)

where the random Gaussian variable $X$ with mean 0 and standard deviation 1
represents the assumption that CRs random-walk their way through the magnetic
micromirrors. Alternatively, micromirrors could be directly modeled in the
magnetic field, which would, however, necessitate step sizes $s\lesssim
l_{\mathrm{mm}}\ll\lambda_{\mathrm{mm}}$, leading to significantly longer
simulation times.

###  3.4 Model of scattering at mesoscales

There is an ongoing debate about the dominant mechanism of CR scattering off
mesoscale fluctuations, with theories including “extrinsic” (cascading)
turbulence and “self-excitation” (by kinetic CR-driven instabilities)
scenarios (see [53] and [44] for recent overviews). Here, we focus on the
extrinsic scenario, with the self-excitation discussed in Section 2.

The CR diffusion coefficient due to resonant scattering is, by dimensional
analysis,

$\kappa_{\mathrm{res}}\sim\frac{c\,r_{\mathrm{g}}}{f(r_{\mathrm{g}})}\,,$ (18)

where $f(r_{\mathrm{g}})$ is a dimensionless model-dependent numerical factor
expressing the efficiency of the resonant scattering off turbulent magnetic
structures at the scale $l\sim r_{\mathrm{g}}$. Quasi-linear theory [50]
determines this factor for isotropic turbulence as the fraction of the
parallel turbulent power located at the gyroresonant scales
$l=2\pi/k_{\parallel}\sim r_{\mathrm{g}}$, viz.
$f(r_{\mathrm{g}})\sim\int_{2\pi/r_{\mathrm{g}}}^{\infty}\mathrm{d}k_{\parallel}\,P(k_{\parallel})/(B^{2}/8\pi)\leq
1$, where $k_{\parallel}$ is the parallel wavenumber and $P(k_{\parallel})$ is
the parallel magnetic-energy spectrum. Assuming an undamped turbulent cascade
with $P(k_{\parallel})\propto k_{\parallel}^{-\xi}$ gives
$f(r_{\mathrm{g}})\sim(r_{\mathrm{g}}/l_{\mathrm{c}})^{\xi-1}$, where
$l_{\mathrm{c}}$ is the energy-containing scale. Therefore,

$\kappa_{\mathrm{res}}\sim
c\,l_{\mathrm{c}}\left(\frac{r_{\mathrm{g}}}{l_{\mathrm{c}}}\right)^{2-\xi}\propto
E^{2-\xi}l_{\mathrm{c}}^{\xi-1}=E^{\delta}l_{\mathrm{c}}^{-\delta+1}\,,$ (19)

where $\delta=2-\xi$ is defined for convenience. In Section 1.2, we confirm
this scaling via numerical simulations with unprecedented spatial resolution
for a synthetic turbulence composed of plane waves.

An important qualitative result is that the cases $\delta=2$ ($\xi=0$)777While
there is no realistic turbulence model with $\xi=0$, this case formally
corresponds to the small-angle scattering limit of CRs with $r_{\mathrm{g}}\gg
l_{\mathrm{c}}$: equation (7) then yields $\kappa\sim
c\,r_{\mathrm{g}}^{2}/l_{\mathrm{c}}$, which is identical to (4) if one
replaces $l_{\mathrm{mm}}$ and $\delta B_{\mathrm{mm}}$ with $l_{\mathrm{c}}$
and $B$, respectively. This is the standard theory for the high-energy regime
referred to in the Introduction. and $\delta=1$ ($\xi=1$)888The scaling (7)
implies that Bohm diffusion ($\kappa_{\mathrm{res}}\sim
l_{\mathrm{c}}\,r_{\mathrm{g}}$) as the dominant process for $r_{g}\ll l_{c}$
is only possible for $\xi=1$, which has not been observed to occur in
realistic astrophysical environments over the whole turbulent cascade. Bohm
diffusion, however, does apply at $r_{g}\sim l_{c}$ in the limit $\delta B\gg
B_{0}$, where quasi-linear theory is expected to fail [90, 94, 76, 98]. apply
to $r_{\mathrm{g}}\gg l_{\mathrm{c}}$ and $r_{\mathrm{g}}\sim l_{\mathrm{c}}$,
respectively, leaving only scalings with weaker energy dependence for
$r_{\mathrm{g}}\ll l_{\mathrm{c}}$. In this limit, three popular choices for
this scaling have appeared in the literature: (i) $\delta=1/3$, corresponding
to isotropic turbulence with a Kolmogorov [56] spectrum ($\xi=5/3$); (ii)
$\delta=1/2$, corresponding to $\xi=3/2$, which in the past was associated
with the theory by Iroshnikov [47] and Kraichnan [60] for weak, isotropic
magnetohydrodynamic (MHD) turbulence (now known not to exist); and (iii)
$\delta=0$, corresponding to the $\xi=2$ Goldreich–Sridhar parallel spectrum
of critically balanced Alfvénic turbulence ([39]; see [83] for a review) by
adhering to (7) (note that this is not observed; see, e.g., [36]).
Historically, Alfvénic turbulence was favored until it was realized that
scale-dependent anisotropy [24, 70, 29], damping [89, 44], and intermittency
[87, 73] might lead to inefficient gyroresonant scattering. A putative
$\xi=3/2$ cascade of fast MHD modes, if isotropic and robust against
steepening [51, 53] and various damping mechanisms, may help by generating
fluctuations with large enough frequencies and amplitudes to scatter CRs
efficiently [99, 28]. More recently, the exponents $\delta=1/3$ and
$\delta=1/2$ were ascribed to CR scattering in intermittent distributions of
sharp magnetic-field bends in Goldreich–Sridhar turbulence [68] and in MHD
turbulent dynamo [55]. Scaling exponents in the range
$0.3\lesssim\delta\lesssim 0.5$ are in broad agreement with constraints from
Galactic observations (see [10] for a review).

###  3.5 Computation of CR diffusion coefficient

We use the CR propagation software CRPropa 3.2 [6] and extend the framework
with our custom modules for the generalized nested turbulence, different
turbulence geometries, and micromirror scattering. We choose sufficiently
small step sizes
$s_{\mathrm{step}}\sim\min\\{\lambda_{\mathrm{mm}}/10^{3},\lambda_{\mathrm{res}}/10^{3},\lambda_{\mathrm{he}}/10^{3}\\}$,
with mean free path $\lambda_{i}\sim\kappa_{i}/c$ to resolve the small-angle
scattering at micromirrors, the resonant scattering in the extrinsic turbulent
cascade, and the small-angle scattering in the high-energy limit,
respectively. The option $\lambda_{\mathrm{he}}/10^{3}$ is only included for
CR energies above $10\,$EeV, as the high-energy limit is not valid below that
energy. We compute sufficiently long CR trajectories
$d\sim\min\\{10^{3}\lambda_{\mathrm{mm}},10^{3}\lambda_{\mathrm{res}},l_{\mathrm{max}}\\}$
(min is used here to save computation resources as only the more efficient
scattering process needs to be resolved) for $E\lesssim 10\,$EeV and
$d\sim\max\\{10^{3}\lambda_{\mathrm{he}},l_{\mathrm{max}}\\}$ otherwise. The
time-dependent diffusion coefficient $\kappa(t)$ for CRs performing a
correlated random walk is [25]

$\kappa(t)=\frac{\left\langle\Delta
r^{2}\right\rangle}{2t}\left(1+\frac{2\left\langle\cos\Theta\right\rangle}{1-\left\langle\cos\Theta\right\rangle}\right)\stackrel{{\scriptstyle
t\gtrsim\lambda/c}}{{\approx}}\frac{\left\langle\Delta
r^{2}\right\rangle}{2t}\,,$ (20)

with $r$ representing the CR spatial displacement and the operation
$\left\langle...\right\rangle$ averaging over CRs. The approximation for
$t\gtrsim\lambda/c$ stems from the convergence to central-limit behavior,
assuming that the deflection angles $\Theta$ are uniformly distributed after
CRs travel a distance equivalent to their mean free path $\lambda$. We compute
the steady-state diffusion coefficient $\kappa$ by averaging over the value
$\kappa(t)$ of $10^{3}$ CRs. When using synthetic turbulence, we average the
quantity over ten different realizations.

To compute the CR diffusion coefficients efficiently, we developed a recurrent
neural network (RNN) that comprised a long short-term memory (LSTM) layer
followed by a fully connected layer, linking the output of the LSTM to our
output size, which is the predicted diffusion coefficient. This architecture
allows us to capture the temporal dependencies in our data, making the network
well-suited to use the CR trajectories as input for the model. For training
data, we used trajectories generated by Monte Carlo simulations using the
method described in Section 3.8, with various different effective diffusion
coefficients. These effective diffusion coefficients served as labels for the
training process. We chose low statistics of only 400 CRs for the 1D cases to
illustrate the superiority of our neural network (trained on only 1600 CR
trajectories in less than a minute on a conventional CPU) over the classical
computation of the running diffusion coefficient. We tested the convergence of
the latter method to the theoretical expectation in the limit of large times
and CR numbers and found good agreement. We demonstrate the capabilities of
the model in Figure 4. The good performance of the network indicates that it
could also be employed as a robust and transformative framework for assessing
$f_{\mathrm{mm}}$ in MHD turbulence by propagating charged particles through
the magnetic field. Reliable computation of the mean-squared diffusion
coefficient requires many trajectories to deduce $f_{\mathrm{mm}}$. Our use of
a neural network demonstrates a concept for efficient trajectory
classification and the computation of transport characteristics in
astrophysical systems.

###  3.6 Micromirrors field from PIC simulations

The micromirror field shown in Fig. 1 was self-consistently generated using
the hybrid-kinetic code Pegasus++ , which models the collisionless ions using
a PIC method and the electrons as an isotropic, isothermal fluid. The code can
simulate a plasma’s expansion or contraction using a coordinate transform
method (see [15] for further details), which via double-adiabatic conservation
laws produces an ion pressure anisotropy that becomes mirror unstable.

In our simulation, we initialized a uniformly magnetized plasma
($\boldsymbol{B}_{0}=B_{0}\hat{\boldsymbol{x}}$) with a Maxwellian population
(3000 ion macro-particles per cell) on a cubic domain. Its size is
$L_{0}^{3}=(76.8r_{\mathrm{g},i0})^{3}$, where $r_{\mathrm{g},i0}$ is the
initial gyroradius of thermal ions, and the grid resolution is $\Delta
x=0.1r_{\mathrm{g},i0}$, $\Delta y=\Delta z=0.3r_{\mathrm{g},i0}$. The initial
ion plasma beta is $\beta_{i0}=50$. The scale $L_{\perp}$ of the plasma in the
direction perpendicular to the background field then evolves as
$L_{\perp}=L_{0}(1+t/\tau_{\rm crt})^{-2}$, with a contraction timescale of
$\tau_{\rm crt}=5\times 10^{3}\Omega_{i0}^{-1}$, while the parallel scale
remains fixed. This gives rise to an ion pressure anisotropy $\Delta_{i}\equiv
T_{\perp,i}/T_{\parallel,i}-1=(1+t/\tau_{\rm crt})^{2}-1$ that increases with
time. The mirror instability is triggered at $t\approx\tau_{\rm
crt}/2\beta_{i0}\simeq 0.01\tau_{\rm crt}$, and then back-reacts at $t\approx
0.1\tau_{\rm crt}$, entering the secular (i.e., power-law) phase of growth
[64, 78]. The simulation is then run until $t_{\rm end}\approx 0.25\tau_{\rm
crt}$. The snapshots of the field shown in Fig. 1 are taken at $t\approx
0.15\tau_{\rm crt}$ and $t=t_{\rm end}$, respectively.

Like all PIC simulations, the finite number of macroparticles in Pegasus++
leads to grid-scale noise in the electromagnetic fields. To diagnose the
influence of this noise on our calculation of CR propagation, we performed an
experiment in which we removed the micromirrors from our Pegasus++ simulation
using a Fourier filter, and integrated CRs through the residual magnetic
field. While the resulting diffusion coefficients show that PIC noise also
leads to diffusive CR transport, their values are much larger than the ones
associated with the micromirrors and so can be safely ignored (see Fig. 2).

###  3.7 Turbulence from MHD simulations

At macroscales, we computed CR diffusion coefficients in forced incompressible
MHD turbulence from the John Hopkins Turbulence Databases999Johns Hopkins
Turbulence Databases, doi:10.7281/T1930RBS[35] to validate the consistency of
our numerical approach that relies on synthetic turbulence. The MHD turbulence
was generated in a direct numerical simulation of the incompressible MHD
system of equations without guide fluid using $1024^{3}$ nodes, employing a
pseudo-spectral method, with energy input from a Taylor-Green flow stirring
force.

Figure 5: Visualization of the numerical experiment described in Section 3.8
to study the effective CR transport in a two-phase medium. The grey surfaces
are the isosurfaces of a threshold $B_{\mathrm{>}}$. In our simplified
numerical experiment, these isosurfaces enclose the micromirror patches,
inside which the diffusion coefficient is much smaller than outside.
Therefore, a given choice of $B_{>}$ corresponds to a certain value of
$f_{\mathrm{mm}}$ (see Section 3.8). Example CR trajectories show increased
deflections within the micromirror patches (see, e.g., lower right corner of
the left panel and the zoom into a micromirror patch in the right panel).
Taking the patches to be static is suitable for small $f_{\mathrm{mm}}$ as
demonstrated in Section 3.8.

###  3.8 Model of a static two-phase inhomogeneous medium

We model the CR transport through a two-phase inhomogeneous medium in 1D and
in 3D (see a visualization in Figure 5). In 3D, we have modified the
computationally intensive numerical experiment described in the previous
subsections as follows: instead of imposing an additional effective scattering
rate $\nu_{\mathrm{mm}}$ on all CRs propagating through our turbulent field at
all times and places, we now turn this scattering on only if the CR is seeing
a magnetic field above a certain threshold $B_{>}$. This models qualitatively
the fact that micromirrors are likely to appear in regions of more vigorous
magnetic-field amplification. By varying $B_{>}$ between $0$ and $\infty$, we
effectively vary the micromirror volume-filling fraction between $1$ and $0$,
respectively. We assume the most intuitive interpretation that this volume-
filling fraction is the effective micromirror fraction $f_{\mathrm{mm}}$ in
(10). As our estimate of the effective diffusion coefficient works in all
dimensionalities, we also model the CR transport in a two-phase inhomogeneous
medium in 1D. This allows us to compute diffusion coefficients effectively for
many different values of $f_{\mathrm{mm}}$. In doing so, we employ a
simplified Monte Carlo model with $\nu(x)=\nu_{\mathrm{mm}}$ for $x$ mod
$1\leq f_{\mathrm{mm}}$ and $\nu(x)=\nu_{\mathrm{res}}$ otherwise.

We now justify modeling micromirror patches experienced by diffusing CRs as
static, based on the assumption of the short residence time of CRs in patches

$\Delta t_{\mathrm{p}}\sim\frac{l_{\mathrm{p}}}{c}\,,$ (21)

where $l_{\mathrm{p}}$ is the characteristic size of the patch. The residence
time of CRs in a patch can be understood intuitively: when CRs penetrate a
patch, they do so ballistically up to their mean free path
$\lambda_{\mathrm{mm}}\sim\kappa_{\mathrm{mm}}/c$ for a time $\Delta
t_{\mathrm{bal}}\sim\lambda_{\mathrm{mm}}/c$, followed by isotropic diffusion
over a time $\Delta t_{\mathrm{diff}}$. To get a rough estimate for $\Delta
t_{\mathrm{diff}}$, we can consider the simplified 1D case, where CRs exit
either at the point of entry or at the point on the opposite end of the patch,
with probabilities
$p_{1}\sim(l_{\mathrm{p}}-\lambda_{\mathrm{mm}})/l_{\mathrm{p}}$ and
$p_{2}\sim\lambda_{\mathrm{mm}}/l_{\mathrm{p}}$, respectively. The
corresponding times101010Note that in the limit $\lambda_{\mathrm{mm}}\ll
l_{\mathrm{p}}$, $\Delta t_{2}\sim l_{\mathrm{p}}^{2}/c\lambda_{\mathrm{mm}}$
can be larger than $\Delta t_{\mathrm{lifetime}}$, meaning that patches cannot
be treated as being static anymore. However, this only affects a small
fraction $p_{2}\sim\lambda_{\mathrm{mm}}/l_{\mathrm{p}}\ll 1$ of CRs. needed
to exit the patch via diffusive transport are $\Delta
t_{1}\sim\lambda_{\mathrm{mm}}^{2}/\kappa_{\mathrm{mm}}$ and $\Delta
t_{2}\sim(l_{\mathrm{p}}-\lambda_{\mathrm{mm}})^{2}/\kappa_{\mathrm{mm}}$. The
mean time duration of the diffusive transport is then given by

$\begin{split}\Delta t_{\mathrm{diff}}&\sim p_{1}\Delta t_{1}+p_{2}\Delta
t_{2}\\\
&\sim\frac{l_{\mathrm{p}}-\lambda_{\mathrm{mm}}}{l_{\mathrm{p}}}\frac{\lambda_{\mathrm{mm}}^{2}}{c\lambda_{\mathrm{mm}}}+\frac{\lambda_{\mathrm{mm}}}{l_{\mathrm{p}}}\frac{(l_{\mathrm{p}}-\lambda_{\mathrm{mm}})^{2}}{c\lambda_{\mathrm{mm}}}\\\
&=\frac{l_{\mathrm{p}}-\lambda_{\mathrm{mm}}}{c}\,,\end{split}$ (22)

resulting in $\Delta t_{\mathrm{p}}\sim\Delta t_{\mathrm{bal}}+\Delta
t_{\mathrm{diff}}\sim l_{\mathrm{p}}/c$. Ref. [59] obtains this result by a
more general method of solving the time-dependent diffusion equation. This
result means that CRs with small diffusion coefficients, corresponding to
short mean free paths, will exit the patch dominantly near their point of
entry. In contrast, CRs with large diffusion coefficients traverse the patch
(quasi)ballistically in time $\Delta t_{\mathrm{p}}\sim\Delta
t_{\mathrm{bal}}\sim l_{\mathrm{p}}/c$. As $\Delta t_{\mathrm{diff}}\ll\Delta
t_{\mathrm{lifetime}}$, micromirrors can be treated as being static for most
CRs.

Acknowledgments  PR thanks J. Becker Tjus, F. Effenberger, H. Fichtner, R.
Grauer, J. Lübke, and L. Schlegel for valuable discussions on the usage of
synthetic turbulence. AAS thanks E. Churazov and S. Komarov for useful
discussions of the ICM and CR physics. The authors thank P. Oh, E. Quataert,
L. Silva, L. Turica, D. Uzdensky, and especially W. Xu for insightful
discussions of CR transport in the ICM. We also acknowledge the generous
hospitality of the Wolfgang Pauli Institute, Vienna, where these ideas were
discussed during the 14th Plasma Kinetics Working Meeting.

The work of PR was supported by a Gateway Fellowship. AFAB’s work was
supported by a UKRI Future Leaders Fellowship (grant number MR/W006723/1).
RJE’s work was supported by a UK EPSRC studentship. MWK’s work was supported
in part by NSF CAREER Award No. 1944972. The work of AAS and GG was supported
in part by a grant from STFC (ST/W000903/1). The work of AAS was also
supported by a grant from EPSRC (EP/R034737/1) and by the Simons Foundation
via a Simons Investigator Award. PK was supported by the Lyman Spitzer, Jr.
Fellowship at Princeton University. UKRI is a Plan S funder, so for the
purpose of Open Access, the author has applied a CC BY public copyright
license to any Author Accepted Manuscript version arising from this
submission.
Data availability. The simulation data presented in this paper are available
upon reasonable request.

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