Main

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. CRs, through their transport characteristics, not only influence their environment but also modulate their own (re)acceleration and confinement efficiency as well as the observable photon and neutrino 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-β plasma, that is, one in which the thermal pressure greatly 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-β plasma, two characteristic scales are relevant for describing the global transport of CRs: the macroscale of the magnetic turbulence, characterized by the correlation length lc or the ‘Alfvén scale’ lA, and the microscale lmm of the micromirrors created by the mirror instability1,2. The prefix ‘micro’ refers to scales much smaller than the ≳μpc gyroradii of ≳100 MeV CRs, and serves to distinguish the plasma-kinetic-scale ‘micromirrors’ (denoted by subscript mm) from the large-scale magnetic mirrors that also influence CR transport3,4,5. While there are other micro-instabilities, the magnetic fluctuations created by the mirror instability are stronger and thus more influential for CR transport6,7. The physics associated with these micro- and macroscales introduces three distinct transport regimes, which depend on the CR energy.

First, in the high-energy (subscript he) limit, CRs with gyroradii rg ≫ lc (energies E ≳ 100 EeV for typical turbulence-driving scales in galaxy clusters) undergo scattering by small angles of order δΘ ≈ lc/rg at characteristic times δt ≈ lc/c, leading to a scattering rate νhe ≈ δΘ2/δt ≈ clc/rg2 and, therefore, to a CR diffusion coefficient κhe ≈ c2/νhe ≈ crg2/lc∝ E2/lc (see, for example, ref. 8). This scaling is indeed observed in both numerical simulations (for example, refs. 9,10,11) and (scaled) laboratory experiments12 and serves as input for propagation models of ultrahigh-energy CRs in galaxy clusters13.

As the high-energy regime is, thus, believed to be understood, recent studies of CR transport have predominantly focused on the second, mesoscale regime, rg ≲ lc, in which CRs scatter resonantly off inertial-range turbulent structures14,15,16,17. This second regime is probably relevant for explaining CR spectra18: for example, the steepening of the CR spectrum in the Galactic Centre from gigaelectronvolt to petaelectronvolt energies11,19,20,21.

We argue, and confirm numerically, that in high-β 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 lmm playing the role of lc and the requirement that rg ≫ lmm (E ≫ 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-β plasma, and determine the transition between micro- and macrophysics-dominated transport to be at teraelectronvolt energies, only weakly influenced by mesoscale physics. We confirm this theory with a novel method (Methods) that incorporates the microscales (lmm ≈ 100 npc), the macroscales (lc ≈ 100 kpc) and the vast range in between.

Results

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 instability22. 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 plasma22,23,24,25. In high-β, weakly collisional plasmas, there also exist a variety of instabilities that are driven by pressure anisotropies and generate magnetic fluctuations on ion Larmor scales (see ref. 2, and references therein). The pressure anisotropies arise from the (approximate) conservation of particles’ adiabatic invariants during the local stretching and compression of magnetic fields26. In the present context, the mirror instability27,28 is of particular interest because its saturated amplitude δBmm ≈ B/3 is of the same order of magnitude as the ambient magnetic field B (refs. 6,29,30). 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: δBf ≈ (τΩi)−1/4B (refs. 6,7), where τ is the timescale over which a firehose-susceptible plasma evolves macroscopically, and Ωi ≈ 0.01(B/3 μG) s−1 is the non-relativistic thermal-ion gyrofrequency. In the ICM, τΩi ≈ 1011 (ref. 31), so δBf ≈ 10−3B. By analogy to equation (3), it follows that the scattering rate νf of CRs off firehoses in the ICM is much smaller than that off the mirrors: νf/νmm ≈ 10−7, with the firehose scale taken to be comparable to the thermal-ion gyroradius6,7.

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. Note that a previous assumption of weaker micromirror fluctuations led to a different, much larger diffusion coefficient based on calculations using quasilinear theory32. The velocity change δv of a relativistic CR with gamma factor γ = (1 − v2/c2)−1/2, charge q = Ze and mass m in a magnetic-field structure of scale lmm ≪ rg and vector amplitude δBmm is given in the small-angle limit ∣δv∣ ≪ v by integrating the equation of motion γm dv/dt = q(v/c) × δBmm along the CR path:

$$\updelta {\mathbf{v}} \approx \frac{q}{\gamma mcv}\mathop{\int}\nolimits_{0}^{{l}_{{\rm{mm}}}}{\rm{d}}l\,{\mathbf{v}}\times \updelta {{\mathbf{B}}}_{{\rm{mm}}}.$$
(1)

Assuming relativistic CRs with v ≈ c, E = γmc2 and rg = γmc2/qB determined by B ≳ δBmm, the scattering angle at δt ≈ lmm/c is

$$\updelta {\Theta} \approx \frac{| \updelta {\mathbf{v}}| }{c} \sim \frac{{l}_{{\rm{mm}}}}{{r}_{{\rm{g}}}}\frac{\updelta {B}_{{\rm{mm}}}}{B}.$$
(2)

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

$${\nu }_{{\rm{mm}}} \sim \frac{\updelta \varTheta ^{2}}{\updelta t} \approx \frac{c\,{l}_{{\rm{mm}}}}{{r}_{{\rm{g}}}^{2}}{\left(\frac{\updelta {B}_{{\rm{mm}}}}{B}\right)}^{2},$$
(3)

which implies a spatial diffusion coefficient of

$${\kappa }_{{\rm{mm}}} \approx \frac{{c}^{2}}{{\nu }_{{\rm{mm}}}} \approx \frac{c{r}_{{\rm{g}}}^{2}}{{l}_{{\rm{mm}}}}{\left(\frac{\updelta {B}_{{\rm{mm}}}}{B}\right)}^{-2}\propto {E}^{2}{l}_{{\rm{mm}}}^{-1}.$$
(4)

As usual, more energetic CRs diffuse much faster.

In arriving at equation (4), we effectively assumed that micromirrors are described by only one characteristic scale, lmm. In reality, micromirrors are anisotropic (see ellipsoid-like shapes in Fig. 1) with scales perpendicular (⊥) and parallel (∥) to the ambient magnetic field that satisfy l⊥,mm ≪ l∥,mm. While gyrating through this field, CRs with rg ≫ lmm mostly traverse micromirrors perpendicularly. Only low-energy CRs satisfying rgv⊥/c ≲ l⊥,mm ≪ l∥,mm are an exception, and should be treated analogously to thermal electrons with negligible rg being scattered29,33 and trapped34 in the micromirrors. This subpopulation makes a negligible contribution to the overall transport of CRs with rg ≫ l⊥,mm considered here.

Fig. 1: Micromirror field generated by the PIC simulation.
figure 1

Colour shows fluctuations δB∥ along the field B, which is aligned with the x axis. We show two snapshots of the 3D field during its secular evolution, characterized by different δB/B, as indicated in the plots (the right-hand snapshot is later in the evolution).

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For rg ≫ l⊥,mm, CRs will sample many different micromirrors, with deflections adding up to a correlated random walk. During one gyro-orbit, CRs will travel Δl∥ ≈ 2πrgv∥/c in the field-parallel direction. CRs with large pitch angles satisfying Δl∥ ≲ l∥,mm, that is, v⊥/v∥ ≈ c/v∥ ≳ 2πrg/l∥,mm, that sample the same micromirror repeatedly, may become relevant only at low energies rg ≲ l∥,mm/2π, not considered in this study. The scattering rate associated with the parallel micromirror perturbation δB∥ ≈ Bmm decreases with decreasing pitch angle, but this is overcome by scattering at the perpendicular micromirror component δB⊥ ≈ δB∥l⊥,mm/l∥,mm for v⊥/v∥ ≲ l⊥,mm/l∥,mm ≪ 1. Except for this cone containing CRs with small pitch angles, from which they escape quickly on the timescale tesc ≈ νmm−1l⟂,mm/l∥,mm, gyrating CRs cross micromirrors perpendicularly faster than they traverse them in the parallel direction, implying lmm ≈ l⊥,mm to be the relevant scale.

For application to the ICM, we estimate lmm using an asymptotic theory of the mirror instability’s nonlinear evolution35 supported by previous numerical studies6,36: lmm ≈ (τΩi)1/8rg,i, where τ is the timescale over which a micromirror-susceptible plasma evolves macroscopically. Applying the theory to an ICM2,37 with B ≈ 3 μG, the thermal-ion gyroradius rg,i ≈ (2T/mi)1/2/Ωi ≈ 1 npc, the mean galaxy cluster temperature T ≈ 5 keV, the thermal-ion mass mi and τ ≈ 1012 s (refs. 31,35) yields lmm ≈ l⊥,mm ≈ 100rg,i ≈ 100 npc (ref. 36), only a factor of a few smaller than the gyroradius of a gigaelectronvolt CR. In combination with equation (4), this gives us the estimate

$$\begin{array}{ll}{\kappa }_{{\rm{mm}}}& \approx 1{0}^{30}\,{Z}^{-2}\,{\left(\frac{{l}_{{\rm{mm}}}}{100\,{\rm{npc}}}\right)}^{-1}{\left(\frac{B}{3\,\upmu {\rm{G}}}\right)}^{-2}{\left(\frac{\updelta {B}_{{\rm{mm}}}/B}{1/3}\right)}^{-2}{\left(\frac{E}{{\rm{TeV}}}\right)}^{2}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1},\\ & \approx 1{0}^{30}\,{Z}^{-2}\,{\left(\frac{T}{5\,{\rm{keV}}}\right)}^{-1/2}{\left(\frac{B}{3\,\upmu {\rm{G}}}\right)}^{-1}{\left(\frac{\updelta {B}_{{\rm{mm}}}/B}{1/3}\right)}^{-2}{\left(\frac{E}{{\rm{TeV}}}\right)}^{2}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1}.\end{array}$$
(5)

This estimate is valid provided that lmm ≪ rg and δt νmm ≈ (lmm/rg)2(δBmm/B)2 ≪ 1 (E ≫ 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 δt ≈ lmm/c moves the gyrocentre by a distance Δr⊥ ≈ rg δΘ in the plane perpendicular to the local magnetic-field line. Using the estimate (2) for the scattering angle leads to the perpendicular diffusion coefficient

$${\kappa }_{\perp ,{\rm{mm}}} \approx \frac{{\rm{\Delta }}{r}_{\perp }^{2}}{\delta t} \approx \frac{{r}_{\mathrm{g}}^{2}\,\updelta \varTheta ^{2}}{{l}_{{\rm{mm}}}/c} \approx c{l}_{{\rm{mm}}}{\left(\frac{\updelta {B}_{{\rm{mm}}}}{B}\right)}^{2} \approx \frac{{l}_{{\rm{mm}}}^{\,2}}{{r}_{{\rm{g}}}^{2}}{\left(\frac{\updelta {B}_{{\rm{mm}}}}{B}\right)}^{4}{\kappa }_{{\rm{mm}}}.$$
(6)

Since κ⊥,mm ≪ κmm for rg ≫ lmm and δBmm ≲ B, it is the parallel diffusion κ∥,mm ≈ κmm along field lines that dominates. The smaller perpendicular diffusion coefficients arise from anisotropic scattering. The degree to which this anisotropy enhances the parallel diffusion coefficient depends on the specifics of pitch-angle scattering and the properties of B⊥, both of which warrant further investigation.

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 (‘Micromirror field from PIC simulations’). For our numerical experiment, we selected two representative realizations of the three-dimensional (3D) field during its secular evolution, visualized in Fig. 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 Fig. 2. The diffusion coefficients in the micromirror fields show good agreement with equations (5) and (6).

Fig. 2: Diffusion coefficients of CRs in micromirror fields.
figure 2

Our predictions (5) (grey dashed line) and (6) (grey dash–dotted line), valid for lmm ≪ rg, agree well with the computed diffusion coefficients. κ is dominated by the parallel diffusion; the perpendicular diffusion coefficient κ⊥ is negligible. Note that κ decreases with δBmm/B (triangles versus stars). We include results for CRs with gyroradii smaller than lmm (vertical orange line) to highlight the change in transport regimes, which follows from our theory. The grey 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.

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The micro–macrophysics transition is at teraelectronvolt CR energies

In ‘Effect of micromirrors on large-scale CR transport’, we derived the diffusion coefficient κmm ∝ lmm−1E2 associated with CR scattering at micromirrors of scale lmm. To determine the upper bound for the energies at which the scattering off micromirrors dominates CR transport, we need a model of the competing contribution from the resonant scattering off mesoscale magnetic turbulence. In ‘Model of CR scattering at mesoscales’, we present models of CR diffusion based on resonant scattering in the (mesoscale) inertial range of magnetic turbulence stirred at the macroscale lc, leading to the diffusion coefficient

$${\kappa }_{{\rm{res}}} \sim c\,{l}_{{\rm{c}}}{\left(\frac{{r}_{{\rm{g}}}}{{l}_{{\rm{c}}}}\right)}^{\delta }\propto {E}^{\delta }{l}_{{\rm{c}}}^{-\delta +1}\,,$$
(7)

where the model-dependent exponent is 0 ≤ δ ≤ 1/2. The mechanism with the smallest diffusion coefficient dominates CR transport. The transition between the micromirror and resonant-scattering transport regimes occurs when κmm ≈ κres. Equating (4) and (7) determines the gyroradius corresponding to this transition:

$${r}_{{\rm{g}}} \approx {l}_{{\rm{c}}}{\left(\frac{\updelta {B}_{{\rm{mm}}}}{B}\right)}^{2/(2-\delta )}{\left(\frac{{l}_{{\rm{mm}}}}{{l}_{{\rm{c}}}}\right)}^{1/(2-\delta )}.$$
(8)

This translates into a δ-dependent estimate for the transition energy:

$$\begin{array}{l}E \approx Z\,\left(\frac{B}{3\,\upmu {\rm{G}}}\right)\left(\frac{{l}_{{\rm{c}}}}{100\,{\rm{kpc}}}\right)\\\times \left\{\begin{array}{ll}300\,{\left(\frac{\updelta {B}_{{\rm{mm}}}}{B}\right)}^{2/(2-\delta )}{\left(\frac{{l}_{{\rm{mm}}}}{{l}_{{\rm{c}}}}\right)}^{1/(2-\delta )}\,{\rm{EeV}}\,,\quad &\,\text{general}\,\,\delta ,\\ 5\,{\left(\frac{\updelta {B}_{{\rm{mm}}}/B}{1/3}\right)}^{6/5}{\left(\frac{{l}_{{\rm{mm}}}/{l}_{{\rm{c}}}}{1{0}^{-12}}\right)}^{3/5}\,{\rm{TeV}}\,,\quad &\delta =1/3\,,\\ 600\,{\left(\frac{\updelta {B}_{{\rm{mm}}}/B}{1/3}\right)}^{4/3}{\left(\frac{{l}_{{\rm{mm}}}/{l}_{{\rm{c}}}}{1{0}^{-12}}\right)}^{2/3}\,{\rm{GeV}}\,,\quad &\delta =1/2.\end{array}\right.\end{array}$$
(9)

Below this energy, magnetic micromirrors dominate CR diffusion. The factor involving the ratio lmm/lc accounts for the scale separation between micro- and macrophysics, which is ~10−12 in our fiducial ICM under the same assumptions as in ‘Effect of micromirrors on large-scale CR transport’.

To test this prediction, we performed numerical simulations of CR transport in the ICM (detailed in Methods), modelling the effects of both the micromirrors (‘Model of small-angle scattering in magnetic micromirrors’) and of the turbulent cascade up to lc ≈ 100 kpc (‘Model of CR scattering at mesoscales’). 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 δ ∈ [1/3, 1/2]. While this estimate of the micro–macrophysics transition at teraelectronvolt CR energies is numerically confirmed using synthetic turbulence, we also used magnetic fields from direct PIC and magnetohydrodynamic (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, detailed as points (1)–(3) in ‘Model of CR scattering at mesoscales’.

Fig. 3: Diffusion coefficients of CRs in the ICM as functions of CR energy.
figure 3

The black stars show the diffusion coefficients in the micromirror field generated by a PIC simulation (‘Micromirror field from PIC simulations’). The olive open crosses show the diffusion coefficients of CRs in MHD turbulence without a guide field (‘Turbulence from MHD simulations’). The other open markers show the diffusion coefficients computed in isotropic synthetic turbulence with a large inertial range (‘Model of synthetic magnetic turbulence’), together with our stochastic micromirror-scattering model (‘Model of small-angle scattering in magnetic micromirrors’), assuming the volume-filling fraction fmm of micromirrors indicated in the legend (‘The case of spatially intermittent micromirrors’). The grey dash–dotted lines represent theories for CR transport depending on the macroscale lc according to equation (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 equation (18) in ‘Model of CR streaming’. The grey dashed line represents our prediction of the diffusion due to micromirrors according to equation (5). The vertical light-blue bar indicates our estimate (9) for the micro–macro transition for the most likely range of δ between 1/3 and 1/2. Simulation results from refs. 10 and 106 illustrate the best resolution towards the limit rg ≪ lc achieved before the present results with synthetic turbulence on a grid (2,0483 grid points) and nested grids, respectively. The effects of field-line tangling are not considered, which is expected to reduce the global CR diffusion coefficients by a factor of three.

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The 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 probably 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 (~1/β; see, e.g., refs. 36,38 and references therein). This gives rise to an effectively two-phase plasma (see Fig. 4 for an illustration), with two different effective scattering rates: νmm ≈ c2/κmm in micromirror patches and νres ≈ c2/κres elsewhere (instead of κres, one could also use κst associated with the streaming instability—‘Discussion’). For the purpose of modelling CR scattering in such a plasma, we introduce the effective micromirror fraction fmm to quantify the probability of CR being scattered by the micromirrors.

Fig. 4: Visualization of example CR trajectories through spatially intermittent micromirror patches.
figure 4

The numerical experiment to study the effective CR transport in a two-phase medium is described in ‘Model of a static two-phase inhomogeneous medium’. The grey surfaces are the isosurfaces of a threshold field strength B>. In our simplified numerical experiment, these isosurfaces are assumed to enclose the micromirror patches, inside which the diffusion coefficient is much smaller than it is outside. Therefore, a given choice of B> corresponds to a certain value of fmm (‘Model of a static two-phase inhomogeneous medium’). Example CR trajectories show increased deflections within the micromirror patches (see, for example, the 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 fmm as demonstrated in ‘Model of a static two-phase inhomogeneous medium’.

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By definition, the effective scattering rate in a two-phase medium is39,40

$${\nu }_{{\rm{eff}}}={f}_{{\rm{mm}}}\,{\nu }_{{\rm{mm}}}+(1-{f}_{{\rm{mm}}})\,{\nu }_{{\rm{res}}}.$$
(10)

The effective diffusion coefficient is then

$${\kappa }_{{\rm{eff}}} \approx \frac{{\kappa }_{{\rm{mm}}}}{{f}_{{\rm{mm}}}+(1-{f}_{{\rm{mm}}}){\kappa }_{{\rm{mm}}}/{\kappa }_{{\rm{res}}}}.$$
(11)

The transition at which micromirror transport takes over from resonant scattering is entirely independent of fmm: κeff ≈ κres when κmm ≈ κres. However, the asymptotic scaling κeff ≈ κmm/fmm is only reached at CR energies for which κmm/κres ≲ fmm/(1 − fmm), pulling the transition energy down by a factor of fmm1/(2−δ). This is not a very strong modification of our cruder (fmm = 1) estimate (9) unless fmm is extremely small.

The most intuitive interpretation of fmm is that it is the fraction of the plasma volume occupied by the micromirrors. This, however, requires at least two caveats. (1) The lifetime of micromirror patches, determined by the turbulent dynamics, can be shorter than the time for a CR to diffuse through the patch. (2) If the micromirror patches form solid macroscopically extended 3D blobs, it is possible to show that CRs typically do not penetrate much farther than the mean free path λmm = c/νmm into the patches. This leaves the patch volume largely uncharted (Appendix 3 of ref. 40). The second concern obviates the first (‘Model of a static two-phase inhomogeneous medium’). 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 them41. However, the scenario of micromirror-dominated transport is made more plausible as the diffusion is mostly along the field lines. In this one-dimensional (1D) problem, CRs bounce between mirror patches on the same field line until they have a lucky streak in diffusion and pass directly through a micromirror patch. This trapping effect leads to efficient confinement, if the influence of potential field-line separation and cross-field diffusion is found to be negligible, though this remains subject to further research.

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 (‘Model of a static two-phase inhomogeneous medium’). Its results are shown in Fig. 5. It is a matter for future work to determine the precise dependence of fmm on the morphology and dynamics of magnetic fields and micromirror-unstable patches in high-β turbulence—itself a system that has only recently become amenable to numerical modelling38,42,43. Here, we proceed to discuss the implications of dominant micromirror transport, assuming that fmm is not tiny, namely, fmm ≳ 0.1, as indeed observed in recent numerical simulations33,43,44,45. Studies of Faraday depolarization of radio emission from radio galaxies could be in principle used to constrain the volume-filling fraction of micromirrors, because depolarization increases proportionally to fmm. We tested the expected Faraday depolarization arising from micromirrors within galaxy clusters in a numerical experiment using our PIC simulations (‘Micromirror field from PIC simulations’). The expected depolarization angles due to micromirror fluctuations for wavelengths observed with the Very Large Telescope (VLT) and the Low-Frequency Array (LOFAR) are too small to constrain fmm. Details of the resolution element in radio telescope observations may affect this result, an issue reserved for future studies.

Fig. 5: Effective diffusion coefficient of CRs in a two-phase medium versus the effective micromirror fraction fmm.
figure 5

We model the CR transport through a two-phase medium in 3D and 1D (to reduce simulation costs), as explained in ‘Model of a static two-phase inhomogeneous medium’. The 3D case (red diamonds) shows the diffusion coefficient of 300 GeV CRs computed at their trajectory lengths of ~10lc. The blue and grey lines show the 1D results on 400 test trajectories using our recurrent neural network (RNN) and the classical analysis method. Error bars for the 3D data represent the s.d. across different realizations of patches for each fmm, while for the 1D cases the coloured contours show the s.d. across 400 CRs from a single realization. The black dash–dotted line represents expected values for averaged scattering frequencies of CRs in the two-phase plasma, formally expressed in equation (11). κmm and κres are recovered for fmm = 1 and fmm = 0, respectively.

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Discussion

We have argued that CR diffusion in the ICM is determined by microscale (lmm) mirrors at CR energies GeV ≲ E ≲ TeV and by macroscale (lc) turbulence at E ≳ 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 equation (9) by the exponent δ, which depends on the details of the scattering mechanism and of the turbulent cascade (‘Model of CR scattering at mesoscales’). While our study is tailored to the ICM, it can be adapted to other nearly collisionless high-β plasmas, such as the hot interstellar medium and the Milky Way halo. In what follows, we discuss what this revised picture of CR diffusion in the ICM implies for our understanding of (Fermi/radio) bubbles and other similar large-scale morphologies.

The reduction of CR diffusion caused by micromirrors may act as a transport barrier in the ICM, reducing the escape of sub-teraelectronvolt CRs from their sources. For example, as radio bubbles rise through the ICM, their confinement of CRs may be dominated by micromirror confinement rather than by the conventional mechanism of magnetic draping; this depends on the details of diffusion coefficients both parallel and perpendicular to the mean magnetic field (see, for example, refs. 46,47,48). Reference 49 considered the CR transport in such systems, and showed that the sub-teraelectronvolt CRs form a thin layer on the surface of the bubble and acquire hardened spectral-density distributions. Such sharp boundaries and hardened CR spectra, which translate into hardened photon spectra, are indeed observed in popular morphologies such as the Fermi bubble50 and the radio bubble in the Ophiuchus galaxy cluster51. The decreased diffusion coefficients of sub-teraelectronvolt CRs could also be relevant in high-β regions within galaxies. One possible example is the Cygnus Cocoon, a Galactic PeVatron, located within a star-forming region, where the observationally constrained suppressed CR diffusion coefficients52,53 may be explained by the additional collisionality due to micromirrors.

We have shown that micromirror diffusion substantially reduces the CR mean free path, thus in principle making CR coupling to the ambient motions tighter. However, to assess what this does to the efficiency (or otherwise) of the (re)acceleration54,55,56,57,58,59,60, we must have a somewhat more detailed picture than we currently do of the nature of the ICM turbulence (that is, of turbulence in a weakly collisional, high-β plasma—a topic of active current investigations38,42,43) and of how the micromirror patches might be shaped and spatially distributed in this turbulence. This will require further study before the (re)acceleration question is settled.

A further example of how micromirror diffusion may impact the surrounding plasma arises from the observation that, as the scattering of sub-teraelectronvolt CRs at micromirrors increases the effective CR collisionality in high-β environments, the effective operation of the CR streaming instability within micromirror patches is put into doubt (‘CR streaming instability’). Note that a patchy distribution of the 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. Models designed to explain the thermal balance between heating and cooling of galaxy clusters based on collisionless, resonant mechanisms61,62,63 thus become less plausible.

Finally, the micromirror scattering matters for cosmological studies of the evolution of the ICM and galaxy clusters. The suppressed CR diffusion coefficients offer a compelling justification 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 surroundings59,64,65. This impacts the interplay between CRs and other astrophysical processes within these massive cosmic structures66,67.

While it is well established that macroscopic dynamics can trigger microscopic phenomena, the potentially transformative impact of micromirrors on CR diffusion provides a lesson that microphysics can reciprocally affect macroscopic dynamics and observable structures across a range of astrophysical scales.

Methods

Modelling CR transport across a wide energy spectrum from gigaelectronvolt to exaelectronvolt energies in a multiscale high-β plasma presents methodological and computational challenges. For our numerical simulations, we choose parameters from the ICM, a high-β plasma. Rotation-measure data indicate magnetic-field strengths of ~0.1–1 μG averaged over a cubic megaparsec ICM volume37,68, with typical field strengths of several microgauss in central regions (see, for example, ref. 69). Numerical simulations support these estimates (for example, ref. 70). We choose B ≈ 3 μG, δBmm/B ≈ 1/3, lmm ≈ 100 npc and turbulence lc ≈ 100 kpc (ref. 2). Simulations are performed using the publicly available tool CRPropa 3.2 (ref. 71) with additional extensions and modelling choices described below.

Model of synthetic magnetic turbulence

Modelling 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 ranges72. Even the current best grid resolutions of more than 1012 cells resolve less than four decades of scale separation. Synthetic turbulence, on the other hand, can be generated by summing over nm plane waves at an arbitrary particle position r as follows15,73:

$$\updelta {\mathbf{B}}({\mathbf{r}})={\rm{Re}}\left(\mathop{\sum }\limits_{n=1}^{{n}_{\mathrm{m}}}\updelta {{\mathbf{B}}}_{n}^{* }\,{{\rm{e}}}^{i{{\mathbf{k}}}_{n}\cdot {\mathbf{r}}}\right)=\sqrt{2}\,\updelta B\mathop{\sum }\limits_{n=1}^{{n}_{\mathrm{m}}}{{\mathbf{\xi }}}_{n}{A}_{n}\,\cos \left({k}_{n}\,{\hat{{\mathbf{k}}}}_{n}\cdot {\mathbf{r}}+{\phi }_{n}\right),$$
(12)

with normalized amplitudes An determined by the assumed turbulent energy spectrum, uniformly distributed phase factors ϕn ∈ [0, 2π], unit wavevectors \({\hat{{\mathbf{k}}}}_{n}\) and polarizations ξn, satisfying \({\hat{{\mathbf{k}}}}_{n}\cdot {{\mathbf{\upxi }}}_{n}=0\). We employ the performance-optimized method described in ref. 74. We investigated the number of wavemodes nm needed by analysing turbulence characteristics and diffusion coefficients of CRs and found that nm = 1,024 log-spaced wavemodes are sufficient for diffusion coefficients to converge.

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, for example, in refs. 75 and 72. In this alternative method, synthetic turbulence is precomputed and stored on many discrete nested grids at different scales, with magnetic fluctuations between scales lmin,i and lmax,i with individual magnetic-field strengths δBi2 = δB2(lmax,iξ−1 − lmin,iξ−1)/(lmaxξ−1 − lminξ−1) and lmax ≈ 5lc. We found agreement between results obtained using the two methods.

Note that CRs below petaelectronvolt energies diffuse on timescales smaller than the typical timescale of large-scale turbulence motions, justifying our modelling turbulence as static. CRs are frozen within the ICM as the plasma evolves (‘Discussion’).

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 ref. 71. This captures the dynamics of charged particles in magnetic fields while preserving key properties, such as CR energy.

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 δθ given νmm = δθ2 c/s. Particles travelling the mean free path λmm = c/νmm will have lost the information of their original direction. At each step s, chosen to be smaller than λmm, we introduce a small deflection

$$\updelta \theta =X\,\sqrt{\frac{s{\nu }_{{\rm{mm}}}}{c}},$$
(13)

where the random Gaussian variable X with mean 0 and s.d. 1 represents the assumption that CRs random-walk their way through the magnetic micromirrors. Alternatively, micromirrors could be directly modelled in the magnetic field, which would, however, necessitate step sizes s ≲ lmm ≪ λmm, leading to significantly longer simulation times.

Model of CR 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 refs. 76,77 for recent overviews). Here we focus on the extrinsic scenario, with the self-excitation described in ‘CR streaming instability’.

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

$${\kappa }_{{\rm{res}}} \approx \frac{c\,{r}_{{\rm{g}}}}{f({r}_{{\rm{g}}})},$$
(14)

where f(rg) is a dimensionless model-dependent numerical factor expressing the efficiency of the resonant scattering off turbulent magnetic structures at the scale l ≈ rg. We assume diffusion rather than superdiffusion, which would increase diffusion coefficients with time. This serves as a conservative estimate. Quasilinear theory14 determines f(rg) for isotropic turbulence as the fraction of the parallel turbulent power located at the gyroresonant scales l = 2π/k∥ ≈ rg, namely, \(f({r}_{{\rm{g}}}) \approx \mathop{\int}\nolimits_{2\uppi /{r}_{{\rm{g}}}}^{\infty }{\rm{d}}{k}_{\parallel }\,P({k}_{\parallel })/({B}^{2}/8\uppi )\le 1\), where k∥ is the parallel wavenumber and P(k∥) is the parallel magnetic-energy spectrum. Assuming an undamped turbulent cascade with P(k∥) ∝ k∥−ξ gives f(rg) ≈ (rg/lc)ξ−1, where lc is the energy-containing scale. Therefore,

$${\kappa }_{{\rm{res}}} \approx c\,{l}_{{\rm{c}}}{\left(\frac{{r}_{{\rm{g}}}}{{l}_{{\rm{c}}}}\right)}^{2-\xi }\propto {E}^{2-\xi }{l}_{{\rm{c}}}^{\xi -1}={E}^{\delta }{l}_{{\rm{c}}}^{-\delta +1}\,,$$
(15)

where δ = 2 − ξ is defined for convenience. In ‘The micro–macrophysics transition is at teraelectronvolt CR energies’, we confirmed 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 δ = 2 (ξ = 0) and δ = 1 (ξ = 1) apply only to rg ≫ lc and rg ≈ lc, respectively. While there is no realistic turbulence model with ξ = 0, this case formally corresponds to the small-angle scattering limit of CRs with rg ≫ lc: equation (7) then yields κ ≈ crg2/lc, which is identical to equation (4) if we replace lmm and δBmm with lc and B, respectively. This is the standard theory for the high-energy regime referred to in the introduction.

In fact, only scalings with weaker energy dependence for rg ≪ lc are typically considered. In this limit, three popular choices for this scaling have appeared in the literature: (1) δ = 1/3, corresponding to isotropic turbulence with a ref. 78 spectrum (ξ = 5/3); (2) δ = 1/2, corresponding to ξ = 3/2, which in the past was associated with the theory by refs. 79 and 80 for weak, isotropic Alfvénic MHD turbulence (now known not to exist); (3) δ = 0, corresponding to the ξ = 2 Goldreich–Sridhar parallel spectrum of critically balanced Alfvénic turbulence81 (see ref. 82 for a review) by adhering to equation (7) (note that this is not observed: see, for example, ref. 83); nevertheless, this spectrum provides a simple way to estimate the decreased CR-scattering efficiency expected for the anisotropic turbulent cascade). Historically, Alfvénic turbulence was favoured until it was realized that scale-dependent anisotropy84,85,86, damping77,87 and intermittency88,89 might lead to inefficient gyroresonant scattering. A putative ξ = 3/2 cascade of fast MHD modes, if isotropic and robust against steepening76,90 and various damping mechanisms, may help by generating fluctuations with large enough frequencies and amplitudes to scatter CRs efficiently91,92. More recently, the exponents δ = 1/3 and δ = 1/2 were ascribed to CR scattering in intermittent distributions of sharp magnetic-field bends in Goldreich–Sridhar turbulence16 and in an MHD turbulent dynamo17. Scaling exponents in the range 0.3 ≲ δ ≲ 0.5 are in broad agreement with constraints from Galactic observations (see ref. 93, for a review).

An additional process that may contribute to the diffusion of low-energy CRs arises from CRs following diffusing magnetic-field lines. The Alfvénic scale lA ≳ 1 kpc (ref. 94) approximates the mean free path of CRs following these field lines95. The associated CR diffusion coefficient κflrw ≈ clA ≳ 1032 cm2 s−1 does not fall significantly below κres for rg ≪ lc. Given this estimate, we neglect this transport process in our simulation set-up, but indicate the value of the diffusion coefficient corresponding to it as an upper boundary in Fig. 3.

CR streaming instability

Let us now explain why the streaming instability can be ignored in our multiscale model of CR transport within micromirror patches. The self-confinement of CRs due to the streaming instability is believed to play an important role in the Galaxy24 and in galaxy clusters62. 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. At gyroscale22,

$${\gamma }_{{\rm{SI}}} \approx \varOmega _{i}\frac{{n}_{{\rm{CR}}}( {>}E\,)}{{n}_{i}}\left(\frac{{v}_{{\rm{st}}}}{{v}_{{\rm{A}}}}-1\right) \approx 1{0}^{-14}\,\left(\frac{B}{3\,\upmu {\rm{G}}}\right){\left(\frac{E}{{\rm{TeV}}}\right)}^{-1.6}\,{{\rm{s}}}^{-1},$$
(16)

where nCR(>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, ni is the ambient ion density, vA is the Alfvén speed and vst is the streaming speed, believed to be of the order of vA in saturation for the ~GeV CRs62,96. In the second estimate in equation (16), we employed the common assumptions (see, for example, refs. 97,98, and references therein) that (vst/vA − 1) ≈ 1 and nCR(>E)/ni ≈ 10−7(E/GeV)1−α in galaxy clusters, with α ≈ 2.6.

Scattering of sub-teraelectronvolt CRs at micromirrors increases the effective CR collisionality in high-β environments. A comparison of the gyroscale growth rate (16) with the scattering rate at micromirrors (3) gives

$$\frac{{\gamma }_{{\rm{SI}}}}{{\nu }_{{\rm{mm}}}} \approx 1{0}^{-5}\,{Z}^{-2}{\left(\frac{T}{5\,{\rm{keV}}}\right)}^{-1/2}{\left(\frac{\updelta {B}_{{\rm{mm}}}/B}{1/3}\right)}^{-2}{\left(\frac{E}{{\rm{TeV}}}\right)}^{0.4}.$$
(17)

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

Another way to gauge the importance of the streaming instability is to imagine that it is not suppressed and then check for self-consistency. In particular, for self-confined CRs, the CR-density scale height H is set by the properties of the ambient thermal gas and is of the order of the thermal-gas-density scale height Hρ. If scattering by micromirrors is present with diffusion coefficient κmm, the associated diffusive flux is smaller than the minimum flux required for the streaming instability to operate if κmm/H ≲ vA. This corresponds to scattering by micromirrors suppressing the anisotropy in the CR distribution function to levels below ~vA/c, which is the threshold anisotropy for the streaming instability to operate in the first place. Thus, because κmm/H ≲ vA for sub-teraelectronvolt CRs, where H ≈ Hρ ≳ 10 kpc (refs. 99,100), CRs may not self-confine in a self-consistent manner.

Model of CR streaming

Let us imagine that the instability is not suppressed, despite the arguments made in ‘CR streaming instability’, and estimate95

$${\kappa }_{{\rm{st}}} \approx {l}_{{\rm{A}}}{v}_{{\rm{st}}}\gtrsim {l}_{{\rm{A}}}{v}_{{\rm{A}}}\gtrsim 3\times 1{0}^{28}\,\left(\frac{{l}_{{\rm{A}}}}{1\,{\rm{kpc}}}\right)\left(\frac{{v}_{{\rm{A}}}}{100\,{\rm{km}}\,{{{\rm{s}}^{-1}}}}\right)\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1},$$
(18)

where we have used vA ≈ 100 km s−1 as the lower limit of the streaming speed. We also assumed that the magnetic-field lines stochastically tangled on the scale lA. In super-Alfvénic turbulence, this marks the transition towards fully MHD turbulence (see, for example, ref. 95), as it is the scale at which the turbulent velocity matches the Alfvén speed. lA ≳ 1 kpc under typical ICM conditions94.

Comparing this CR diffusivity with the one caused by micromirrors (5) gives

$$\begin{array}{l}\frac{{\kappa }_{{\rm{mm}}}}{{\kappa }_{{\rm{st}}}}\lesssim {Z}^{-2}{\left(\frac{E}{100\,{\rm{GeV}}}\right)}^{2}{\left(\frac{{l}_{{\rm{A}}}}{1{\,\rm{kpc}}}\right)}^{-1}\\{\left(\frac{T}{5\,{\rm{keV}}}\right)}^{-1/2}{\left(\frac{B}{3\,\upmu {\rm{G}}}\right)}^{-1}{\left(\frac{\updelta {B}_{{\rm{mm}}}/B}{1/3}\right)}^{-2}{\left(\frac{{v}_{{\rm{A}}}}{100\,{\rm{km}}\,{{{\rm{s}}^{-1}}}}\right)}^{-1},\end{array}$$
(19)

demonstrating that κmm dominates CR transport in the ICM up to almost teraelectronvolt energies. This comparison, together with the arguments for the suppression of the streaming instability (‘CR streaming instability’), justifies neglecting CR streaming in our numerical experiments. Nevertheless, we show this estimate in Fig. 3.

Computation of CR diffusion coefficient

We use the CR propagation software CRPropa 3.2 (ref. 71) 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 sstep ≈ min{λmm/103, λres/103, λhe/103}, with mean free path λi ≈ κi/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 λhe/103 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 ≈ min{103λmm, 103λres, lmax} (min is used here to save computation resources as only the more efficient scattering process needs to be resolved) for E ≲ 10 EeV and d ≈ max{103λhe, lmax} otherwise. The time-dependent diffusion coefficient κ(t) for CRs performing a correlated random walk is101

$$\kappa (t)=\frac{\left\langle {\rm{\Delta }}{r}^{2}\right\rangle }{2t}\left(1+\frac{2\left\langle \cos \varTheta \right\rangle }{1-\left\langle \cos \varTheta \right\rangle }\right)\mathop{\approx }\limits^{t\,\gg \lambda /c}\frac{\left\langle {\rm{\Delta }}{r}^{2}\right\rangle }{2t},$$
(20)

with r representing the CR spatial displacement and the operation 〈…〉 averaging over CRs. The approximation for t ≳ λ/c stems from the convergence to central-limit behaviour, assuming that the deflection angles Θ are uniformly distributed after CRs travel a distance equivalent to their mean free path λ. We compute the steady-state diffusion coefficient κ by averaging κ(t) for 103 CRs. When using synthetic turbulence, we average this quantity over ten different realizations.

To compute the CR diffusion coefficients efficiently, we have developed a recurrent neural network that comprises a long short-term memory layer followed by a fully connected layer, linking the output of the long short-term memory to our output size, which is the predicted diffusion coefficient. This architecture allows us to capture the temporal dependence 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 with the method described in ‘Model of a static two-phase inhomogeneous medium’, using 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 1,600 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 Fig. 5. The good performance of the network is primarily attributable to its capacity to learn efficiently that the diffusion coefficients can be predicted accurately by the frequency of small-angle scattering, which becomes evident in a relatively small number of steps. Additionally, the ability to distinguish signal (deflection caused by fluctuations) from noise (constant gyration) further contributes to the network’s superior performance. These abilities indicate that the network could also be employed as a robust framework to assess fmm in MHD turbulence by propagating charged particles through the magnetic field. Reliable computation of the mean-squared diffusion coefficient requires many trajectories, necessary to deduce fmm. Our use of a neural network demonstrates a concept for efficient trajectory classification and the computation of transport characteristics in astrophysical systems.

Micromirror 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 ref. 102 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 (\({{\mathbf{B}}}_{0}={B}_{0}\hat{{\mathbf{x}}}\)) with a Maxwellian population (3,000 ion macroparticles per cell) on a cubic domain. Its size is L03 = (76.0rg,i0)3, where rg,i0 is the initial gyroradius of thermal ions, and the grid resolution is Δx = 0.1rg,i0, Δy = Δz = 0.3rg,i0. The initial ion plasma beta is βi0 = 50. The scale L⊥ of the plasma in the direction perpendicular to the background field then evolves as L⟂ = L0(1 + t/τcrt)−2, with a contraction timescale of τcrt = 5 × 103Ωi0−1, while the parallel scale remains fixed. This gives rise to an ion pressure anisotropy Δi ≡ T⟂,i/T∥,i − 1 = (1 + t/τcrt)2 − 1 that increases with time. The mirror instability is triggered at t ≈ τcrt/2βi0 ≃ 0.01τcrt, and then back-reacts at t ≈ 0.1τcrt, entering the secular (that is, power-law) phase of growth6,35. The simulation is then run until tend ≈ 0.25τcrt. The snapshots of the field shown in Fig. 1 were taken at t ≈ 0.15τcrt and t = tend, respectively.

As in 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 those associated with the micromirrors and so can be safely ignored (Fig. 2).

Turbulence from MHD simulations

At macroscales, we computed CR diffusion coefficients in forced incompressible MHD turbulence from the John Hopkins Turbulence Databases103,104 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 1,0243 nodes, employing a pseudospectral method, with energy input from a Taylor–Green flow stirring force. CR diffusion coefficients in this field are shown in Fig. 3.

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 Fig. 4). In 3D, we modified the computationally intensive numerical experiment described in the previous subsections as follows: instead of imposing an additional effective νmm 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 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 ∞, 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 fmm in equation (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 fmm. In doing so, we employ a simplified Monte Carlo model with ν(x) = νmm for x mod 1 ≤ fmm and ν(x) = νres otherwise.

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

$${\rm{\Delta }}{t}_{{\rm{p}}} \approx \frac{{l}_{{\rm{p}}}}{c},$$
(21)

where lp 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 λmm ≈ κmm/c for a time Δtbal ≈ λmm/c, followed by isotropic diffusion over a time Δtdiff. To obtain a rough estimate for Δtdiff, 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 p1 ≈ (lp − λmm)/lp and p2 ≈ λmm/lp, respectively. The corresponding times needed to exit the patch via diffusive transport are Δt1 ≈ λmm2/κmm and Δt2 ≈ (lp − λmm)2/κmm. Note that in the limit λmm ≪ lp, Δt2 ≈ lp2/cλmm can be larger than Δtlifetime, meaning that patches cannot be treated as being static anymore. However, this only affects a small fraction p2 ≈ λmm/lp ≪ 1 of CRs. The mean time duration of the diffusive transport is then given by

$${\rm{\Delta }}{t}_{{\rm{diff}}} \approx {p}_{1}\,{\rm{\Delta }}{t}_{1}+{p}_{2}\,{\rm{\Delta }}{t}_{2} \approx \frac{{l}_{{\rm{p}}}-{\lambda }_{{\rm{mm}}}}{{l}_{{\rm{p}}}}\frac{{\lambda }_{{\rm{mm}}}^{2}}{c{\lambda }_{{\rm{mm}}}}+\frac{{\lambda }_{{\rm{mm}}}}{{l}_{{\rm{p}}}}\frac{{({l}_{{\rm{p}}}-{\lambda }_{{\rm{mm}}})}^{2}}{c{\lambda }_{{\rm{mm}}}}=\frac{{l}_{{\rm{p}}}-{\lambda }_{{\rm{mm}}}}{c},$$
(22)

resulting in Δtp ≈ Δtbal + Δtdiff ≈ lp/c. Reference 40 obtained this result by a more general method of solving the time-dependent diffusion equation. It means that CRs with small diffusion coefficients, corresponding to short mean free paths, will exit the patch predominantly near their point of entry. In contrast, CRs with large diffusion coefficients traverse the patch (quasi)ballistically in time Δtp ≈ Δtbal ≈ lp/c. As Δtdiff ≪ Δtlifetime, micromirrors can be treated as being static for most CRs.