Introduction

Clusters of galaxies host hot, diffuse, X-ray emitting gas, which we call the intracluster medium (ICM). Some clusters, especially disturbed and merging clusters, also host a rich variety of diffuse MHz-GHz radio emission in their ICM: radio synchrotron halos, bridges, relics, and phoenices powered by relativistic cosmic-ray electrons (CRe; van Weeren et al. 2019). These CRe cool via synchrotron radiation and inverse-Compton scattering off cosmic microwave background photons over megayears to gigayears, reaching 1-100 MeV energies. Because radiative power losses decrease at lower electron energies, and Coulomb collisions are weak in the ICM, MeV "fossil" CRe may persist in clusters for Gyr (Enßlin 1999;Petrosian 2001;Pinzke et al. 2013).

Fossil CRe energies are too low to emit detectable radio synchrotron emission. However, a reacceleration of 100× in energy can make fossil CRe shine again in radio synchrotron and permit them to contribute to the power budget of radio emission in the ICM (Brunetti et al. 2001;van Weeren et al. 2019;Brunetti & Vazza 2020). Many mechanisms can energize fossil CRe: large-scale adiabatic compression from subsonic sloshing or shocks (Enßlin & Gopal-Krishna 2001;Markevitch et al. 2005), diffusive shock acceleration in cluster merger shocks (Kang et al. 2012;Guo et al. 2014;Kang & Ryu 2016;van Weeren et al. 2017;Ha et al. 2022), and wave damping or reconnection within a turbulent scale-by-scale cascade (Brunetti & Lazarian 2007, 2011, 2016).

We consider another possibility for reaccelerating fossil CRe, wherein large-scale deformation-compression, dilation, or shear-drives small-scale plasma waves that might directly scatter and energize CRe. When the ICM deforms on timescales shorter than the Coulomb collision time and longer than the Larmor gyration time, the B-perpendicular temperature T ⊥ changes due to conservation of particle magnetic moment p B 2 ^, and the B-parallel temperature T ∥ changes due to conservation of particle bounce invariant ∮ p ∥ ds integrated along a field line (assuming periodicity in parallel motion). As T ⊥ and T ∥ evolve independently, the plasma becomes temperature and pressure anisotropic: Δ ≡ T ⊥ /T ∥ -1 ≠ 0. Because the ICM's thermal pressure dominates over magnetic pressure, i.e., its plasma beta β p = P thermal /P magnetic ? 1, Δ ≠ 0 easily triggers the growth of various Larmor-scale plasma waves (Kasper et al. 2002;Bale et al. 2009;Kunz et al. 2014Kunz et al. , 2019)). The strongest waves reside at proton Larmor scales; although they are triggered by and regulated by proton anisotropy, they may also interact with fossil CRe, which gyrate more slowly and have larger Larmor radii than typical ICM thermal electrons.

We focus on CRe interaction with ion cyclotron (IC) waves driven by thermal ICM proton (i.e., ion) anisotropy Δ > 0, with the anisotropy in turn driven by continuous compression. IC waves interact with electrons via the gyroresonance condition:

where ω is wave angular frequency, k = 2π/λ is wavenumber, λ is wavelength, v ∥ is electron velocity parallel to B, Ω e = -eB/(m e c) is the signed nonrelativistic electron cyclotron frequency, and γ is the electron's Lorentz factor. Equation (1) specifies an "anomalous" resonance, wherein an electron overtaking the wave (|v ∥ | > |ω/k|) sees the Doppler-shifted IC wave polarization as right-circular rather than left-circular, thus enabling gyroresonance (Tsurutani & Lakhina 1997;Terasawa & Matsukiyo 2012). The resonance condition is simplified in the low-frequency limit, appropriate for ICM plasma with Alfvén speed v A /c = 1 and ion-electron mass ratio m i /m e ? 1:

is the ion thermal velocity. The form of Equation (2) anticipates that k -1 is on the order of the ion Larmor radius ρ i for temperature-anisotropy-driven IC waves at marginal stability (Davidson & Ogden 1975;Yoon et al. 2010;Sironi & Narayan 2015). 5 For ICM temperatures T i ≈ T e ∼ 1-10 keV (Chen et al. 2007), IC waves with k ρ i ∼ 0.5, and m i /m e = 1836 for a proton-electron plasma, we anticipate resonant momenta p ∥ ∼ 7 to 21 m e c within the expected range for fossil CRe in the ICM, p ∼ 1 to 300 m e c (Pinzke et al. 2013). We thus expect that IC waves may efficiently scatter fossil CRe.

Gyroresonant IC wave scattering may energize CRe in at least two different ways. First, the nonzero phase velocity of IC waves will transfer energy from waves to CRe via second-order Fermi acceleration (Fermi 1949), but this is slow because the energy gain per cycle scales with the square of the scatterers' velocity, ( ) v c 1 A 2  for IC waves. Second, pitch-angle scattering couples parallel and perpendicular momenta p ∥ , p ⊥ and drives CRe toward isotropy. Pitch-angle scattering, in isolation, conserves particle energy. However, scattering during bulk deformation can heat particles via magnetic pumping if the scattering rate is comparable to the bulk deformation rate (Berger et al. 1958;Lichko et al. 2017).

Magnetic pumping in a compressing plasma works as follows. Because particle momenta p ⊥ and p ∥ have different adiabatic responses to compression, a scattering rate comparable to the bulk compression rate can cause a net transfer of energy from p ⊥ to p ∥ over one compression-decompression cycle; this energy transfer may be linked to a phase difference between pressure anisotropy and magnetic field compression (Lichko et al. 2017). Magnetic pumping has been previously studied in the contexts of plasma confinement, planetary magnetospheres, and the solar wind (Alfvén 1950;Schlüter 1957;Berger et al. 1958;Goertz 1978;Borovsky et al. 1981Borovsky et al. , 2017;;Borovsky 1986;Lichko et al. 2017;Fowler et al. 2020;Lichko & Egedal 2020).

In high-β p plasmas with Δ > 0, anisotropy-driven IC waves may not be the dominant fluctuations. Nonpropagating structures created by the mirror instability are thought to prevail over IC waves, based on theory (e.g., Shoji et al. 2009;Isenberg et al. 2013) and measurements in Earth's magnetosheath (Schwartz et al. 1996) and the solar wind (Bale et al. 2009). Nevertheless: IC waves may coexist with mirror structures; IC waves appear in 3D hybrid simulations of turbulent high-β p plasma (Markovskii et al. 2020;Arzamasskiy et al. 2023); there may be local regions of the ICM with reduced plasma β p or with reduced electron/ion temperature ratio T e /T i (Fox & Loeb 1997) more conducive for IC wave growth. Mirror modes also have k i 1 r ~-, so they may nonresonantly scatter fossil CRe and drive magnetic pumping as well. The same will likely hold for firehose modes excited when Δ < 0.

IC-resonant scattering of relativistic MeV electrons also occurs in Earth's radiation belts and can precipitate electrons into the upper atmosphere (e.g., Thorne & Kennel 1971;Meredith et al. 2003;Zhang et al. 2016;Adair et al. 2022). In particular, Borovsky et al. (2017) studied the same mechanism as this article-compression-driven IC waves energizing relativistic electrons via magnetic pumping-applied to Earth's outer radiation belt.

Methods

We simulate continuously compressed ICM plasma using the relativistic particle-in-cell (PIC) code TRISTAN-MP (Buneman 1993;Spitkovsky 2005). The PIC equations are solved in comoving coordinates while subject to global compression or expansion, as implemented by Sironi & Narayan (2015), similar to hybrid expanding box simulations in the literature (Liewer et al. 2001;Hellinger et al. 2003;Hellinger & Trávníček 2005;Innocenti et al. 2019;Bott et al. 2021). To do this, Sironi & Narayan (2015) transform from the physical laboratory frame (t lab , x lab ) to a comoving coordinate frame ( ) x t , ¢ ¢ via a transformation law x Lx lab = ¢, where:

and the differential transformation law is:

The scale factors a x , a y , and a z are >1 for expansion and <1 for contraction. We report quantities (fields, particle positions, momenta, distribution function moments) in physical CGS units in the plasma's local rest frame; i.e., the unprimed coordinates Sironi & Narayan (2015). We use a 1D domain parallel to a background magnetic field B, which permits the growth of parallel-propagating IC waves and precludes the growth of the mirror instability. Our domain and magnetic field B are aligned along y; all wavenumbers k ≡ k y in this article. We compress along both x and z axes by choosing scale factors:

where q > 0 is a tunable constant controlling the compression rate. We fix a y (t) = 1. The background field evolves consistent with flux freezing as

 where B 0 is the initial field strength. The imposed B -perpendicular compression conserves two particle invariants, p B 2 ^and p ∥ , if there is no wave-particle interaction (Sironi & Narayan 2015, Appendix A.2). The ICM is modeled as a thermal ion-electron plasma with Maxwell-Jüttner distributions of initial temperature T 0 and density n 0 for each species. The fossil CRe are modeled as test particles, i.e., passive tracer particles, which advance according to the electromagnetic fields on the grid but do not contribute to the plasma dynamics-in the PIC algorithm, they have no weight and so deposit no current. The treatment of fossil CRe as passive tracers is motivated by their low kinetic energy density, ∼10 4 × smaller than the thermal ICM, in cluster outskirts as simulated by Pinzke et al. (2013, Figure 3). However, fossil CRe could become dynamically important in the recently shocked ICM responsible for radio relics; see, e.g., Böss et al. (2023, Figure 11) and Ha et al. (2022).

Standard length-and timescales are defined as follows for thermal plasma species s ä {i, e}. The signed, nonrelativistic particle cyclotron frequency Ω s = q s B/(m s c). = is a thermal velocity. Subscript 0 in Ω s0 , ω ps0 , ρ s0 , and other symbols hereafter means that the quantity is evaluated at t = 0. Subscripts ⊥ and ∥ indicate vector projections with respect to the background magnetic field direction ŷ.

Our results center on one "fiducial" simulation with ion-toelectron mass ratio m i /m e = 8, initial plasma beta In each time step, the electric current is smoothed with 32 passes of a three-point binomial ("1-2-1") filter, approximating a Gaussian filter with a standard deviation of four cells (Birdsall & Langdon 1991, Appendix C). Outputs are saved at 1 i0 1 ~W-intervals. We use two different initial test-particle CRe distributions f (p)dp depending on our analysis needs: f (p) constant (flat) or f (p) ∝ p -1 to uniformly sample p or p log , respectively. Both distributions are isotropic. The f (p) constant case uses 2,880,000 CRe in p = 0 to 70 m e c, and the f (p) ∝ p -1 case uses 14,400,000 CRe in p = 0.0014 to 1400 m e c. Neither case mimics nature, but the uniform p and p log sampling means that our results can be reweighted to describe any initially isotropic CRe distribution. The test-particle distributions span the momentum range of CRe that should be efficiently scattered by IC waves in our simulation: p ∥ ∼ 4 to 25 m e c based on Equation (2). 6Besides our fiducial simulation, we also run simulations with varying q, m i /m e , v A0 /c, and β p0 ; the detailed parameters are given in Appendix F and Table 1. The domain size is pinned to ∼80ρ i0 for all such simulations. The test-particle CRe spectrum is kept flat ( f (p) constant), but the upper bound is rescaled according to (m i /m e )(v th,i /c) per Equation (2) to capture the momentum range of the expected IC gyroresonance. The simulations with varying β p0 are not presented in the main text and appear only in Appendix B. In cases with slow compression, e.g., q 3200

or m i /m e = 32 in Table 1, we saw gyrophase-dependent numerical errors in particle momenta when using single-precision (32 bit) floats in the PIC algorithm. We therefore use double-precision (64 bit) floats for all simulations in the article, except for convergence checks in Appendix E.

Besides IC waves, whistlers (i.e., electron cyclotron waves) are excited by the thermal electrons in our simulations. To help separate the effects of whistler and IC waves upon fossil CRe energy gain, we perform simulations in which one particle species, ions or electrons, is compressed isotropically in order to suppress that species' cyclotron waves. The species may still participate in plasma dynamics by generating currents. To implement isotropic compression, we modify the comoving momentum equation (Boris particle push):



For the chosen species, we set the diagonal elements of LL 1 in the Boris pusher to a x = a y = a z = 1/(1 + q iso t). We choose q iso = 2q/3 to match the initial energy input rate from anisotropic compression; i.e., at t = 0, the determinant

has first derivative equal to that for the anisotropic ℓ = 1/(1 + qt) 2 . The rest of the code in the PIC algorithm retains the anisotropic compression. For electrons, isotropic forcing is only applied to regular particles (thermal ICM) and not test particles (fossil CRe).

Wave Properties

Time Evolution

The simulation evolves as follows. The compression at first drives ( ) T B t T constant g  µ > = ^for all species while conserving the adiabatic invariants of magnetized particles (Northrop 1963), which can be recast according to the Chew-Goldberger-Low (CGL) fluid theory as pressure or temperature invariants (Chew et al. 1956). Instability is triggered, and waves grow, between t = 0.2q -1 and 0.5q -1 (Figures 1(a)-(c)). Right-circularly polarized (RCP) whistlers appear first and are the dominant mode at t = 0.2q -1 , followed by left-circularly polarized (LCP) ion cyclotron waves from t = 0.3 to 0.5q -1 . The wave polarizations are distinguished by Fourier transform of B z + iB x in Figure 1(a), which separates LCP and RCP waves into ω > 0 and <0, respectively, following Ley et al. (2019). The wave fluctuation power ( ) B B g 2 d ^saturates at a near-constant or slightly decreasing level by t ∼ 0.55q -1 (Figure 1(c)); while saturated, the IC wave power drifts toward lower ω and k (Figures 1(a)-(b)). We plot a manually chosen approximation to the k-space drift, The saturated waves drive the ion and electron temperature anisotropy Δ away from CGL-invariant conservation and toward a marginally stable state at late times t  1q -1 (Figure 1(d)). At marginal stability, we expect s 0.5  b D µ -for both ions (Gary & Lee 1994;Gary et al. 1994b;Hellinger et al. 2006) and electrons (Gary & Wang 1996;Gary & Karimabadi 2006), where

. We fit the relation

between t = 1q -1 and the simulation's end to obtain A i = 0.98 ± 0.02 and A e = 0.785 ± 0.012; the best-fit relations are the dotted lines in Figure 1(d). The uncertainty on A i and A e is one standard deviation estimated by assuming 1 reduced 2 c = , as no data uncertainty is used in fitting. We expect that the systematic uncertainty is larger.

When the IC waves saturate, we expect balance between compression increasing Δ and wave pitch-angle scattering decreasing Δ, as suggested by the marginal-stability scaling in Figure 1(d). This balance may be stated as:

In taking Δ = Δ i , we assume that only ions source and control the wave power ( ) B B 2 d ^at late times. In Figure 1(c), we show Equation (7) computed with arbitrary normalization and using Δ i = T i⊥ /T i∥ -1 measured from the simulation. Equation (7) does not explain the total late-time wave power in our simulation, but it better matches the power in currently unstable IC waves (Figures 1(a),(c)). We conjecture that waves in the unstable IC region may be most important for regulating Δ i , in contrast to the stronger IC wave power at lower k.

The total plasma beta decreases to half its initial value by the simulation's end, with ions hotter than electrons (Figure 1(e)). At early times t  0.1q -1 , β e⊥ deviates from the nonrelativistic CGL prediction because electrons are almost relativistic with k B T 0 = 0.2 m e c 2 .

Wave Identification

Let us now more closely study the wave properties and evolution. To predict the wave ω, k, and damping/growth as a function of time, we solve the nonrelativistic dispersion relation for B -parallel electromagnetic waves in a bi-Maxwellian ionelectron plasma: Davidson & Ogden (1975) and Stix (1992, Section 11-2), keeping only the n = 0, ± 1 resonant terms. The subscript s = i, e indexes component species,

is the plasma dispersion function (Fried & Conte 1961),

We approximate T s∥ and T s⊥ using the second moments of the ion and electron distributions in our simulations. In Equation (8), ω = ω R + iΓ is complex, but in all other text and figures, ω refers only to the real angular frequency ω R unless otherwise noted. The imaginary part Γ > 0 for instability and <0 for damping. We use Equation (8) to show the unstable ω range for LCP waves over time in Figure 1(b), and to show the expected ω and k for both LCP and RCP waves in the ω-k power spectra of Figure 2.

We note several features of interest in the B z + iB x spectrogram (Figure 1(b)). LCP and RCP modes both appear at t ∼ 0.2q -1 . The LCP mode is more monochromatic and has lower ω, while the RCP mode has broader bandwidth and higher ω. The LCP modes persist from t > 0.2q -1 through the rest of the simulation. The RCP modes appear in two transient bursts, at t = 0.2 and 0.4 q -1 , and the second RCP burst coincides with a growth in the LCP power and near-peak ion anisotropy Δ i . Some RCP power aliases from ω < 0 into ω > 0 at the top of Figure 1(b) and in each panel of Figure 2.

The LCP power splits into high-and low-frequency bands at t ≈ 0.8-1.0q -1 (Figure 1(a)); each band continues to, respectively, rise and fall in frequency over time. The high-frequency LCP power lies within the expected ω range of IC wave instability as predicted by Equation (8). The low-frequency LCP power resides in a frequency/wavenumber range that is not expected to spontaneously grow IC waves. We remain agnostic about why the low-frequency LCP power evolves toward low k, but we note that Ley et al. (2019, Figure 9) saw a similar drift of the IC wave power to low k in a shearing-box PIC simulation. In Appendix C, we show that wave power drifts to low frequencies even if compression halts at t = 0.5q -1 , so the low-frequency power drift is not caused by external compression or by a numerical artifact of the comoving PIC domain.

We verify that the LCP and RCP modes are IC waves and whistlers, respectively, by inspecting the ω-k power spectra in three time intervals (Figure 2). The LCP wave power agrees well with the predicted (ω, k) from Equation (8) in all time snapshots of Figure 2, and the previously noted highfrequency band in Figure 1(b) agrees well with the prediction for IC wave instability. The RCP wave power agrees with the bi-Maxwellian whistler dispersion in some respects. The phase speed ω/k agrees with Equation (8) at later times (Figures 2(b)-(c)). In simulations with higher m i /m e (Appendix B), the RCP phase speed ω/k increases with respect to the LCP phase speed and continues to agree with Equation (8). However, the RCP wave power disagrees with the bi-Maxwellian dispersion curve in some respects. At early times t = 0.2-0.3q -1 , the RCP mode is offset toward higher k than expected for the whistler mode; it does not appear to lie on a curve passing through (ω, k) = (0, 0). At later times, the RCP power shows better agreement with the whistler mode: the k offset disappears and RCP power connects continuously to

The later-time RCP power also has ω somewhat lower than that predicted by Equation (8) for k = 0.5-1.0ω pi0 /c (Figures 2(b)-(c)). Some more observations on the RCP mode are in Appendix B. All considered, despite the imperfect agreement with Equation (8), we attribute the RCP waves to thermal electron anisotropy and call them whistlers hereafter.

Equation (8) is approximate, as particles are not exactly bi-Maxwellian. Wave scattering alters distributions to quench instability, and the resulting anisotropic distributions can be stable to ion cyclotron waves (Isenberg et al. 2013). Appendix C checks the frequency of waves driven unstable by the actual particle distribution, and we find that the resulting waves do lie in a high-frequency LCP power band as predicted by Equation (8), validating our use of the bi-Maxwellian approximation in this context.

Equation (8) also does not account for the background plasma density and magnetic field varying during instability growth; the plasma properties are assumed to vary on a much longer timescale than is relevant to the linear dispersion calculation. The maximum IC growth rate predicted by Equation (8

, which is 80× faster than the compression rate q. The growth rate may be smaller in practice due to particles quenching their own instability; nevertheless, we expect that waves should grow on a short timescale that is well separated from the compression time.

Wave Scattering

Let us compare the CRe scattering directly measured in our simulations against the quasi-linear theory (QLT) description of resonant scattering as a diffusive process, in the limit of weak, uncorrelated, and broadband waves (Jokipii 1966;Kennel & Engelmann 1966;Kennel & Petschek 1966;Kulsrud & Pearce 1969). In particular, we wish to check the following.

(1) Do particles with a 90°pitch angle (i.e., p ⊥ ? p ∥ ) scatter efficiently in our simulations? As p ∥ → 0, the resonant wavenumber k res  ¥, and particles cannot scatter at exactly 90°pitch angle in QLT. (2) Does the resonant QLT description hold for our simulations? The saturated wave power δ B ⊥ /B ∼ 0.1 (Figure 1(a)) may be too strong to satisfy QLT (Liu et al. 2010). Strong waves may lead to, for example, momentum-space advection instead of diffusion (Albert & Bortnik 2009).

We compute the QLT diffusion coefficient D μ μ for the pitch-angle cosine p p cos

, assuming low-frequency (ω ≈ 0) waves, following Summers (2005):

 with + andsigns for CRe resonance with IC and whistler waves, respectively. We take W(k) to be the two-sided wave power spectrum of B 2 d ^measured directly from our simulation, with the sign of k specifying propagation direction. We decompose W(k) = W L (k) + W R (k) into LCP and RCP pieces by Fourier transforming B z + iB x over a time window of length 18.9 i0 1 W -, which is 4× larger than the time step used to measure particle scattering. The power at ω 0 is assigned to W L and the remainder to W R . We smooth W R (k) and W L (k) with a Hanning window of length 0.14ω pi0 /c (7 points) and then linearly interpolate to compute D μ μ for arbitrary (p, μ). Because our simulation has balanced forward-and backwardpropagating waves, we average D μ μ over μ < 0 and μ > 0 in Figure 3.

We directly measure 〈ΔμΔμ〉/(2Δt) by computing Δμ = μ (t + Δt) -μ(t) over an output time step t 4.7 i0 1 D = Wfor each test-particle CRe. The pitch angle α is defined with respect to the background field ( ) B t y g . Then, we compute the particleaveraged 〈ΔμΔμ〉 as a function of the phase-space coordinates (p, |μ|) using and 140 bins over p ä [0, 70]m e c and 50 bins over |μ| ä [0, 1]. The choice of Δt affects the shape and strength of the scattering regions in Figures 3(j)-(l). We find that time steps Δt = 4.7-18.8 i0 1 Wgive somewhat consistent scattering region shapes, but shorter time steps Δt = 0.9-1.9 i0 1 Wdo not resolve the scattering interaction, especially for the highest p CRe. Appendix D further shows and discusses the effect of varying Δt in our scattering measurement.

Figure 3 compares the measured pitch-angle scattering rates 〈ΔμΔμ〉/(2Δt) (Figures 3(j)-(l)) to the predicted rates D μ μ from LCP (Figures 3(d)-(f)) and RCP (Figures 3(g)-(i)) waves at t = 0.25, 0.45, and 1.05q -1 . The smoothed W L and W R used to compute D μ μ are shown in Figures 3(a)-(c); the one-sided spectra, as normalized, are averages of two-sided spectra over k > 0 and k < 0. The full QLT prediction for D μ μ is the sum of the middle two rows (d)-(i), which separate the ion cyclotron and whistler contributions to show their relative importance. White dotted lines mark all particles resonant with a wave of given k res according to Equation (10). At t = 0.25q -1 (left column), the whistler power is strong and the particles most efficiently scattered have small momenta p ∼ 1-5 m e c. At t = 0.45q -1 (middle column), the ion cyclotron power has overtaken whistlers in strength, with most resonant scattering predicted at the k = 0.3ω pi0 /c contour, though the measured scattering 〈ΔμΔμ〉 has a broader bandwidth in (p, |μ|) space and does not exactly follow the resonant contour shape of Equation (10). At t = 1.05q -1 (right column), the wave power is saturated (Figure 1(a)) and the IC spectrum has broadened to k = 0.1 ω pi0 /c, as seen in both the 1D wave spectrum (top row) and the QLT prediction (second row).

As time progresses, both the measured and modeled scattering extend toward larger p due to two effects. First, the increase in B g (t) leads to a rightward drift of the resonant contours p ∝ B g (t)/μ (Equation ( 10)) for fixed k res . Second, the saturated wave power drifts toward smaller k over time d ^ , and so we expect mirroring to be important at |μ|  0.12. We speculate that resonance broadening (e.g., Tonoian et al. 2022) or a nonmagnetostatic calculation with ω/ k ≠ 0 may also expand the scattering extent in (p, |μ|). In particular, the magnetostatic assumption is less valid for the higher v A /c in our simulations as compared to that of the real ICM. See also Holcomb & Spitkovsky (2019) for further recent discussion.

Particle Spectrum from Magnetic Pumping

We now seek a time-integrated view of the energy gain due to magnetic pumping from IC wave scattering during compression. Some particles scatter more efficiently and at different times than others, and it follows that some fossil CRe may gain more energy from magnetic pumping than others.

To frame the problem, we ask: given CRe of initial momentum p 0 at t = 0, what is their energy gain due to magnetic pumping during compression? We consider the following hypothetical scenario. After a compression to time t in our simulation, let the test-particle CRe decompress back to their initial volume, with no further wave scattering during decompression; i.e., map

 ^^and hold p ∥ constant for all particles. We call this adiabatic decompression a "reversion" of the particle distribution, and we say that the particles have undergone a "compress-revert" cycle. The decompressed particle energy is defined as

One cycle of compression to arbitrary time t, followed by a revert, yields an energy gain:

 where U(t) = 〈γ(t)〉, U 0 = U(t = 0), and angle brackets 〈L〉 are ensemble averages over particles in an initial momentum bin p 0 . Recall that our initial test-particle CRe distribution is isotropic; i.e., uniform on μ ä [-1, +1]. We use ΔU revert (t) as a proxy for magnetic pumping efficiency.

The "revert" is artificial; particles may scatter during decompression. However, the compress-revert cycle permits us to focus solely on magnetic pumping due to compressiondriven waves, without needing to also study and separate the effect of decompression-driven waves (e.g., firehose).

We shall now seek to understand how particles respond to a compress-revert cycle, before proceeding to use ΔU revert as a proxy for magnetic pumping efficiency. In Figures 45, we use a test-particle CRe spectrum ( ) dN dp f p p 1 = µthat uniformly samples p log with p ä [0.0014, 1400]m e c using 14,400,000 particles. However, we reiterate that our results can be reweighted to apply to any initial f (p), and Figure 5 shows one such reweighting to f (p) ∝ p -2 .

Figure 4 shows one compress-revert cycle acting upon the simulated CRe, where the "Final" particle distribution is from the simulation's end, and the "Revert" particle distribution is taken after one compress-revert cycle. The "Final,CGL" distribution shows the same compression as for "Final," but without scattering. We call attention to four points. First, the "Revert" particle spectrum is skewed; although the mean "revert" particle momentum is ∼1.1-1.3 × p 0 , individual particles may be energized up to ∼2.4 × p 0 (Figures 4(a)-(c)). Second, scattering is strongest for low starting p 0 and weakens toward higher p 0 , as judged by the particles' deviation from the predictions for adiabatic compression and adiabatic decompression (Figures 4(d)-(l), black curves). Third, the final particle momentum correlates with the cosine of the particle's c) LCP (blue line) and RCP (orange line) magnetic power spectra W L and W R . The spectra are normalized using the number of grid points N and the Fourier spacing Δk so that ( )

The thick line is the spectrum smoothed with a Hanning window of length 0. initial pitch angle μ 0 , and that correlation strengthens for larger p 0 (Figures 4(d)-(f)). The energy gain for particles with large p 0 is nearly consistent with adiabatic compression, shown by comparing the "Final" particle distributions to the "Final,CGL" curve in Figures 4(a)-(c) and thick black curves in Figures 4(d) -(l). Fourth, the "Final" particle distribution extends rightward of the expected maximum momentum from adiabatic compression alone, ( ) p B t B 0 0 , from comparing the "Final" and "Final,CGL" distributions in Figures 4(a)-(c). We attribute the particles with ( ) p p B t B 0 0 > to momentum diffusion D pp ; the number of such particles decreases as we lower v A0 /c toward realistic values for the ICM and hence decrease D pp .

We can model the magnetic pumping upon any isotropic CRe spectrum f (p) by computing the response of a Dirac-delta distribution δ(p -p 0 ) to one compress-revert cycle, for multiple choices of constant p 0 , in the spirit of a Green's function. Let p and p¢ be momentum coordinates before and after a compressrevert cycle, respectively. Define ( ) G p p dp , 0 ¢ ¢ as the distribution obtained by applying one compress-revert cycle to an initial distribution f (p)dp = δ(p -p 0 )dp with p 0 being an arbitrary constant, similar to Figures 4(a)-(c). To construct G, we average over μ, even though the particle spectrum after a compress-revert cycle is not isotropic (Figures 4(j)-(l)). Then, the action of one revert cycle upon f (p) is:

for each of 300 logarithmically spaced bins over p ä [0.0014, 1400]m e c with 96,000 test-particle CRe per bin.

Figure 5 demonstrates the effect of magnetic pumping for an "Initial" spectrum f (p)dp ∝ p -2 dp with lower bound p = 10 -0.5 m e c. The "Revert" spectrum f revert has two distinct bumps compared to the Initial spectrum (Figure 5(a)). We attribute the higher-p bump at p ∼ 10-100 m e c to the IC wave resonance; hereafter, we call this the "IC bump." The lower-p bump with maximum at p ∼ 1 m e c has shape similar to a thermal Maxwell-Jüttner distribution. At high energies p  300 m e c, particle momenta remain nearly adiabatic through a compress-revert cycle, as previously seen in Figures 4(c), (f), (i), (l). We visualize the convolution of f (p) by plotting the kernels ( ) G p p , ¢ for various p (Figure 5(a)); these kernels are constructed using the same procedure as the 1D "Revert" spectra in Figures 4(a)-(c), up to details of numerical binning and normalization.

The IC bump in f revert (p) has an upper bound at p ∼ 100 m e c that is not exceeded by multiple pump cycles. What sets this p bound? We attribute this bound to the rightward skew of the convolution kernel ( ) G p p , ¢ , most visible for the kernels with p between 10 1 and 10 2 m e c in Figure 5(a). In contrast, the mean (μ-averaged) energy gain after one compress-revert cycle has a maximum of ∼30% for CRe with initial momenta p 0 ∼ 20-30 m e c, which we will shortly see in Figure 6; see also the mean energy gain (vertical orange lines) in Figures 4(a this thermal bump. The IC bump is not affected by the low-p boundary, which confirms that the thermal and fossil electrons are well separated in momentum space.

The IC bump extends toward higher momenta for longer compression duration. In Figure 5(c) we show f revert computed for three evenly spaced times t = 0.47, 0.94, 1.41q -1 in our fiducial simulation. The spectrum at t = 0.47q -1 shows a very weak IC bump, which we attribute to weaker IC scattering at early times when IC waves are not yet saturated. The IC bump becomes more prominent at t = 0.94q -1 and 1.41q -1 . We further explore the link between the compression duration and the onset of scattering at high p later in this article.

We also consider the effect of multiple compress-revert cycles by assuming that, at the end of each compress-revert cycle, f revert (p) instantly becomes isotropic in μ; the result is shown in Figure 5(d). Multiple cycles strengthen the IC energy gain between p = 10 to 100 m e c. The IC pumping does not extend to p ? 100 m e c; CRe with p ∼ 10 3 m e c remain adiabatic through multiple compress-revert cycles. The assumption of instant isotropization between each compressrevert cycle is questionable; we know from Figures 4(j)-(l) that the revert spectra are far from isotropic. The effect of scattering during decompression, which should bring electrons closer to isotropy, is left for future work.

Cumulative Energy Gain from Magnetic Pumping

Let us now focus on the efficiency metric ΔU revert , abstracting away details of the underlying μ-dependent particle spectra. Figure 6 shows ΔU revert /U 0 computed for all testparticle CRe in our simulation, binned by the initial CRe momentum with bin size Δp 0 = 0.5 m e c. We emphasize three main features. The lowest-energy CRe, p 0 ∼ 1-10 m e c, gain little energy from magnetic pumping. The medium-energy CRe, p 0 ∼ 10-30 m e c, pump the most efficiently by virtue of their having initial momenta at or above the expected resonant p ∥ ∼ 4-25 m e c (Equation ( 2)). The highest-energy CRe, p 0  30 m e c, gain energy at later times; as compression proceeds, CRe of progressively higher p 0 "turn on" their energy gain.

We also introduce U gain to represent the time-integrated energy gain from all mechanisms other than adiabatic compression, in particular momentum diffusion. To compute U gain , we decompose each particle's energy gain over a time step Δt into adiabatic and nonadiabatic pieces: In Figure 6 we draw three conclusions concerning U gain . First, both U gain and ΔU revert show the same qualitative features in (t, p 0 ) coordinates. We attribute this to the shared gyroresonant nature of both energy-gain processes: D pp for nonadiabatic diffusive energization U gain , and D μ μ for magnetic pumping ΔU revert . Second, the magnitude of U gain is ∼10% that of the initial particle energy by the end of the simulation; however, U gain (t) is small compared to the total particle energy U(t) arising from compression, which is 200% of the initial particle energy U 0 by the end of the simulation. Finally, U gain decreases as v A0 /c is lowered toward a more realistic value, whereas ΔU revert does not vary as strongly with v A0 /c; we show this decrease in U gain later in the article (Figure 12). On the basis of these observations, we view U gain and hence D pp as a minor player in CRe energization through our compressive cycle.

Continuous Compression Controls the Efficiency of Magnetic Pumping

The 2D structure of ( ) U t p , revert 0 D encodes information regarding which particles scatter and when they scatter; i.e., it encodes the time-and k-dependent wave spectrum W(k, t), but we lack a mapping from W(k, t) and B g (t) to ( ) U t p , revert 0 D . To understand the 2D structure of ΔU revert , we perform Fokker-Planck (F-P) simulations of compression with time-dependent pitch-angle scattering:

We sample 280,000 CRe with momenta between p 0 between 0.25 to 69.75 m e c and an isotropic pitch-angle distribution (i.e., uniform μ ä [ -1, + 1]). Then, we subject the CRe to the same continuous compression as in our fiducial simulation, ( )

W -. At first, the compression is adiabatic to mimic the relatively weak wave power at early times in our fiducial simulation (Figures 1(a)-(c)). After t = 0.3q -1 , we begin scattering all particles that satisfy:

min is a user-chosen function. The scattering is implemented as a 1D random walk in pitch angle α. For each time t n , each particle satisfying Equation (12) takes a randomly signed step Δα = ± 0.04 prior to the compression step p ⊥ (t n ) → p ⊥ (t n+1 ). The variance of the total displacement after N steps is 〈ΔαΔα〉 N = N(Δα) 2 , so the effective diffusion coefficient D μ μ ∼ (1 -μ 2 )〈ΔαΔα〉/(2Δt) is approximately 8.5 × 10 -4 (1 -μ 2 ) Ω i0 . This D μ μ value is weaker than the scattering rate measured in our fiducial simulation (Figures 3(j) -(l)); nevertheless, the F-P model returns a comparable value of ΔU revert . Also, our F-P model deviates from quasi-linear theory in having no 90°barrier; particles with μ = 0 scatter efficiently in order to mimic the presence of scattering at μ = 0 in Figures 3(j)-(l). Varying the start time of scattering to either t = 0.0q -1 or 0.6q -1 has only a small effect on the F-P model energy gain; the time evolution of ( ) k t min is more important. Figure 7 shows the magnetic-pumping energy gain in our F-P model for four different choices of k min . We first consider constant k 0.3 min = , 0.15, and 0.09 ω pi0 /c in Figures 7(a)-(c). Then, we adopt a time-dependent ( ) k k t min IC = , using Equation (5) to mimic the decreasing-k drift of ion cyclotron wave power in our fiducial PIC simulation. We draw three conclusions. First, the magnetic-pumping energy gain has a self-similar geometric structure in (t, p 0 ) coordinates for k min constant in time; changing k min is the same as rescaling p 0 by a factor k 1 min (Equation ( 12)), so Figures 7(a)-(c) are identical up to linear rescaling along the y-axis. Second, the particles gaining the most energy from magnetic pumping have p 0 somewhat higher than the initial resonant p ∥ at t = 0. For example, choosing k c 0.09 = broadens the energy-gain "resonance" feature in ΔU revert toward higher p 0 (Figure 7(d)).

To understand how magnetic pumping interacts with continuously driven compression to "select" a range of p 0 with the highest magnetic pumping efficiency, Figure 8 shows how isotropic, monoenergetic particle distributions with p 0 = 4, 12, 36 m e c evolve over time while subjected to both compression and pitch-angle scattering (after t = 0.3q -1 ) for all particles with k c 0.15 min pi0 w = (Figure 7(b)). The lowestenergy particles, p 0 = 4 m e c (blue), scatter promptly at all pitch angles from t 0.3q -1 onward, so the magnetic pumping is less efficient. The medium-energyparticles, p 0 = 12 m e c (orange), only scatter near μ = 0 at early times t ∼ 0.3q -1 , but their scattering extends to most μ values by the simulation's end. The highest-energy particles, p 0 = 36 m e c (green), are mostly adiabatic; few such particles scatter until later times, so their energy gain from magnetic pumping is small. Preferential scattering near μ = 0, where compression gives the most energy (as compared to larger |μ|), causes mediumenergy particles to migrate to large |μ| and "lock in" their compressive energy gain; therefore, medium-energy particles participate most efficiently in magnetic pumping. We interpret orange particles accumulating at the scattering region boundaries in Figure 8, as well as the skewed particles at large |μ| in Figures 4(j)-(l), as evidence for energy locking. The highestenergy particles also scatter from μ ∼ 0 toward the scattering boundary (Figure 8), but (1) fewer particles are able to participate, and (2) the smaller μ of the scattering boundary causes more compressive energy gain to be removed in decompression. The lowest-energy particles, because they scatter at all μ, easily flow between μ ∼ 0 and |μ| ∼ 1; there is no region of (p, |μ|) space in which particles may lock energy gained from μ ∼ 0.

The drift of IC power toward low k further modifies particle energization. In Figure 8, the gray scattering region expands rightward as time progresses:

, and k res decreasing in time will hasten that expansion and therefore widen the band of medium-energy particles. Previously, Matsukiyo & Hada (2009, Section 4) have also noted how Alfvénic waves drifting to low k may help accelerate particles that can stay within the range of resonant momenta of the time-evolving waves.

Compression and the drift of IC power toward low k together can thus explain, qualitatively, the distinct low-, medium-, and high-energy CRe structure of ΔU revert as a function of t and p 0 (Figure 6).

Compression Rate Dependence

In our simulations, the compression timescale q 800

1 i0 1 = W --corresponds to q -1 ∼ 10 -3 yr if one assumes B 0 = 3 μG, which is much smaller than the actual soundcrossing time ∼10 8 yr for cluster-scale ICM bulk motion. How do the CRe energy gain and the IC wave spectrum change with q -1 in our simulations? For larger q -1 , linearly unstable IC waves grow earlier and attain smaller k at late times (Figure 9), so we expect the IC wave resonance to broaden toward higher p.

We also expect the wave power B 2 d ^to weaken for larger q -1 as per Equation (7), which may be rewritten more explicitly as

In Figure 10, we check if the linear scaling with q predicted by Equation (13) holds in our simulations. Both ( ) B B g 2 d ^and Δ i decrease when q decreases (Figures 10(a)-(b)). At t = 1.2q -1 , we sample and plot ( ) B B g 2 d ^as a function of q (Figure 10(c), solid markers). We similarly compute and plot the left-hand side (LHS) of Equation (13) (Figure 10(c), hollow markers). Both quantities appear to follow a power-law scaling q n with exponent n  0.5, which is a weaker proportionality than that predicted by Equation (13).

Waves at differing k may not contribute equally toward balancing the compression-driven anisotropy; recall how the strongest waves lie outside the unstable ω range in Figure 1(a), and how Equation (7) agrees better with the unstable wave power rather than the total wave power in Figure 1(c). We thus suspect that low-frequency wave power may participate less in regulating the ion anisotropy. Does the anisotropy-driven highfrequency wave power, rather than the total wave power, scale linearly with q as per Equation (13)? We select wave power with ω/Ω i0 > 0.9 by computing the average wave power spectral density (PSD) in the top-right white boxes of Figures 9(a)-(e);foot_2 the resulting PSD is plotted against q in Figure 10(d). The PSD multiplied by Δ i (2Δ i + 3)/(Δ i + 1) appears to follow a power-law scaling q n with exponent n between 0.5 and 1. Least-squares fits of form (

W , with free parameters A and n, are plotted as solid black lines in Figures 10(c)-(d). For Equation ( 13 = , as no data uncertainty is used in fitting. We expect that the systematic uncertainty is larger.

We warn that our ω/Ω i0 > 0.9 threshold does not cleanly separate low-and high-frequency wave power for every simulation because the ω range of the wave power varies with q (Figure 9). Altering the ω threshold will also alter the qscaling exponent in Figure 10(d). A multicomponent fit to the power spectrum may better separate the low-and highfrequency wave power and so provide a better test of Equation ( 7), but we omit such detailed modeling for now.

We also show how the CRe energy gain ΔU revert changes with q in Figure 11. As q -1 increases, the optimal p 0 range for magnetic pumping both widens and moves to higher momenta, which we ascribe to both the lower late-time k and earlier onset of waves with respect to the compression timescale q -1 . We suspect that wave evolution toward lower k is the dominant effect altering the shape of ΔU revert for varying q -1 . We do not observe, by eye, a trend in the peak magnitude of ΔU revert with respect to q.

Scaling to Realistic ICM Plasma Parameters

How do more realistic simulation parameters (higher m i /m e , lower v A /c) alter our results? Let us define a dimensionless CRe momentum

for our fiducial simulation parameters, motivated by the gyroresonance scaling (Equation ( 2)); recall that v th,i /c ∝ v A /c for fixed β p . As in Sections 5-6, p0 is the value of p for CRe particles at t = 0. If simulations of varying v A0 /c and m i /m e have a similar IC wave spectrum W(t, k) for fixed q/Ω i0 , then particle scattering and energization should also have a similar structure in p.

We vary v A0 /c of our fiducial simulation by factors of 2 and measure the particle scattering rates 〈ΔpΔp/p 2 〉/(2Δt) and 〈ΔμΔμ〉/(2Δt) in discrete (p, μ) bins. As in Section 4, the time step is t 5 i0 1 D » W -. The momentum bin width 0.5m e c is fixed Figure 9. Wave power spectrogram of B z + iB x for varying q, from small q -1 /fast compression (left) to large q -1 /slow compression (right). (a)-(e) Spectrogram in (t, ω) coordinates with a finite time binning for each pixel. The white curve on the top left of each panel is Ω i (t). Within the white boxes (t > 1q -1 and ω/Ω i0 > 0.9), we average the PSD to estimate the power due to unstable IC waves at high ω, omitting the linearly stable IC waves at low ω. (f)-(j) Wave power spectrum in (t, k) coordinates without time binning. The red-dash-framed panels (c), (h) mark the fiducial simulation; i.e., same data as Figure 1. (c) Wave power at t = 1.2 q -1 plotted as a function of q (solid circles). The same wave power is multiplied by Δ i (2Δ i + 3))(Δ i + 1), i.e., the left-hand side (LHS) of Equation (13) (hollow squares), to test the linear q scaling of Equation (13). The solid, dashed light-gray lines are ∝q, q scalings, respectively. The solid black lines are least-squares power-law fits. for all simulations, so the plotted p bin width varies between simulations in Figure 12.

The measured scattering rates indeed have similar shape in ( ˜| |) p , m coordinates for varying v A0 /c (Figure 12). At lower v A0 /c, a double-lobed scattering region appears along the resonant contours. Lower v A0 /c also alters the apparent edge of the scattering region at p 25 > toward possibly better agreement with the predicted resonant contours from Equation (10), although the scattering region edge still disagrees at low p 25 < . To explore how scattering scales with v A0 /c, we average the scattering rates over |μ| and ˜[ ] p mc 5, 25 e Î to sample the strongest IC wave signal in momentum space. The average rates are plotted as a function of time in Figures 13(a)-(b); the same rates sampled at three discrete times are then plotted as a function of v A0 /c in Figures 13(c)-(d). The pitch-angle and momentum scattering rates increase and decrease, respectively, as v A0 /c decreases. We interpret the data as showing a transition from mildly relativistic to nonrelativistic behavior as we lower v A0 /c. At lower v A0 /c than shown, we expect that the pitch-angle scattering should become independent of v A0 /c, while momentum scattering should scale as ( ) v c A0 2 . We also verify the expected QLT scaling: 13(e), which shows a power-law-like scaling consistent through the entire range of v A0 /c considered.

As previously claimed, momentum scattering is not important in a single compress-revert cycle for our simulation parameters. We see that 〈ΔpΔp/p 2 〉/(2Δt) is ∼10 -2 × smaller than 〈ΔμΔμ〉/(2Δt), and the QLT scaling assures us that momentum scattering is even less important in real ICM with v A /c  10 -3 . In Figure 13(e), the separation between data measured at different times in the same simulation may be partly attributed to time variation in v A (t)/c. We proceed to vary m i /m e and v A0 /c together, now focusing solely on the magnetic pumping efficiency ΔU revert , in Figure 14. Across all panels, we observe a similar three-band structure as in our fiducial simulation: low-energy CRe ( p 5 0  ) gain little energy, medium-energy CRe ( p 10 0 = -30) gain the most energy, and high-energy CRe ( p 30 0  ) progressively "turn on" their energy gain over time, later for higher-energy CRe. If we remove whistler waves by compressing electrons isotropically (Section 2), comparing Figures 14(e)-(h) against Figures 14(i)-(l): the region of the most efficient energy gain shifts to higher p0 , and the maximum value of ΔU revert /U 0 decreases in magnitude by ∼0.05. Otherwise, the overall shape of ΔU revert remains similar when comparing simulations with and without whistler waves.

Conclusions and Outlook

We have used 1D PIC simulations to show how ICM fossil CRe gain energy from bulk compression by scattering upon IC waves excited by anisotropic thermal ions. The energy gain comes from magnetic pumping, and we have measured the momentum-dependent pumping efficiency. Some summary points follow. First, high-β p plasma microinstabilities have a convenient wavelength-comparable to the Larmor radius of thermal protons-to interact with and scatter fossil CRe in the ICM of galaxy clusters. Second, continuous compression and wave-power drift toward low k both increase, over time, the CRe momentum p that can resonantly scatter on IC waves and hence gain energy via magnetic pumping. The increase in the resonant p may be viewed as a time-delayed scattering for high-p CRe, which can help increase the pumping energy gain compared to continuous scattering from beginning to end of the simulation. Third, IC wave pumping is robust with respect to m i /m e and v A0 /c and is not sensitive to the presence or absence of whistler waves driven by thermal electrons. Although the simulated m i /m e and v A0 /c are not realistic, the lower m i /m e and higher v A0 /c cancel such that the simulated resonant momenta are only 2-3× lower than those of real fossil CRe.

Our 1D setup with an adiabatic "revert" is unrealistic in some ways. The compression factor ∼6 at the end of our simulation exceeds the expected density contrast of both weak ICM shocks and subsonic compressive ICM turbulence (e.g., Gaspari & Churazov 2013). More realistic, nonadiabatic decompression may excite firehose modes that should also resonantly scatter CRe and alter ΔU revert (Melville et al. 2016;Riquelme et al. 2018;Ley et al. 2023). In 2D or 3D simulations, the low-k drift of IC wave power may not persist, and mirror modes may weaken IC waves; both effects will weaken the energy gain from IC wave pumping. Nevertheless, h), but disabling whistler waves by compressing electrons isotropically at the rate q iso = 2q/3. The red-dash framed panels (b), (f) are the fiducial simulation, previously shown in Figure 6.

magnetic pumping via resonant scattering on firehose fluctuations or nonresonant scattering on mirror modes remains possible, as both firehose and mirror modes will also have a convenient wavelength to interact with fossil CRe. Varying | B | in solenoidal, shear-deforming flows will also excite the same high-β p plasma microinstabilities to scatter and magnetically pump CRe.

Our treatment of a collisionless ion-electron plasma has neglected (1) Coulomb collisions and (2) the presence of heavier ions. Regarding (1), the collision rate varies within a cluster. The ICM density decreases to ∼10 -5 -10 -4 cm -3 at large radii from cluster centers, and the proton collision time can reach 100 Myr, comparable to the sound-crossing time as discussed in Section 8. In denser gas closer to cluster centers, collisions may inhibit large-scale eddies from driving particle anisotropy. However, we expect that the turbulent cascade will eventually reach an eddy scale where the turnover rate is faster than the collision rate, so that particle anisotropy may be collisionlessly driven. Regarding (2), He and heavier ions are known to exist in the ICM (Abramopoulos et al. 1981;Peng & Nagai 2009;Berlok & Pessah 2015;Mernier et al. 2018). He+ + and other ions will modify the parallel plasma dispersion relation (Smith & Brice 1964) and proton cyclotron instability growth rate (Gary et al. 1993), and He++ cyclotron waves may themselves be excited (Gary et al. 1994a). Mirror and firehose linear instability thresholds will be altered as well (Hellinger 2007;Chen et al. 2016). The precise wave spectrum and hence CRe energy gain would thus change, but we expect that CRe may still gain energy by magnetic pumping in the presence of heavier ICM ions.

How does CRe energization by high-β p IC wave magnetic pumping fit into the broader context of large-scale ICM flows and turbulence? At ion Larmor scales, we expect power from high-β p plasma microinstabilities to be much larger than power from the direct turbulent cascade. Let us suppose that the ICM has a turbulent magnetic energy spectrum:



We suppose that the simulated B g 2 corresponds to the total magnetic energy 〈B 2 〉 in the ICM, because energy resides at the largest scales in the Kolmogorov spectrum.

Let us further consider IC waves driven by a compressive eddy at a galaxy cluster's outer scale, ∼1 Mpc. Using the estimate from the beginning of Section 8, the compression timescale q -1 will be 10 11 × larger than in our simulation.

Combined with the scaling B q 2 0.32 d µ ^from Figure 10(c), we should decrease our estimate of k i W PIC (k i ) in Equation (15) by a factor of 3 × 10 3 in order to extrapolate to realistic conditions. The IC wave power so extrapolated remains 10 5 times larger than the power expected from the turbulent direct cascade at ion Larmor scales.

The excess power at ion Larmor scales may also contribute to stochastic reacceleration via momentum scattering (D pp ), as explored for Alfvénic cascades by Blasi (2000) and Brunetti et al. (2004). Let us suppose that ( ) D v c q pp A 2 0.32 µ , from Figure 13 and its accompanying discussion. Again, we take the ICM outer scale q -1 ∼ 10 11 × larger than our simulation, and also take ICM v A /c ∼ 10 -3 and 10 yr i 1 6

W --. Our measured momentum scattering rate 10 -4 Ω i (Figure 13)then extrapolates to 〈ΔpΔp/p 2 〉/(2Δt) ∼ 10 -11 Ω i . The corresponding acceleration time ∼10 5 yr is short compared to cosmological timescales.

What is the efficiency of magnetic pumping, as well as stochastic reacceleration, upon IC waves in this slowly forced, turbulent setting? A quantitative answer is beyond the scope of this work, but we make a few remarks. For CRe momenta within the band of IC wave resonance, scattering will occur quickly and persist throughout the bulk compression. Both resonant magnetic pumping and stochastic reacceleration will be limited by the available IC wave bandwidth, so electrons will not reach arbitrarily high energies. If the IC wave drift rate toward low k scales with Ω i rather than q, owing to the smaller q in reality, wave energy may continue to cascade to smaller k than in our simulations and so help scatter and pump CRe at even higher momenta. At galaxy cluster merger shocks, nonthermal protons may also alter the growth and damping of IC waves and hence their resulting bandwidth (e.g., dos Santos et al. 2015). In a turbulent flow, the microinstabilities will not be volume filling; CRe streaming in and out of the scattering regions may also alter the energy gain from magnetic pumping (Egedal & Lichko 2021;Egedal et al. 2021).

Appendix F Simulation Parameters

Table 1 provides the input parameters for all simulations in this article: first the fiducial simulation, followed by parameter sweeps of q -1 , k B T 0 (equivalently v A0 /c), m i /m e , and β p0 . The simulations with varying β p0 are only used in Appendix B. Simulations with varying particle count (Appendix E) or with one species isotropic are not explicitly shown.

We define some input parameters in code units: my is the domain size in cells; intv is the number of time steps between output file dumps, relevant for wave-power spectra and particle scattering measurements; dur is the simulation duration in time steps. Other key parameters such as the grid cell size, particles per cell, current filtering, and numerical speed of light are identical across all simulations and are stated in Section 2. The black squares at t = 1q -1 correspond to the same symbols in (d)-(l). Right three columns: wave power and anisotropy, measured at t = 1q -1 for simulations with varying q -1 (d)-(f), v A0 /c (g)-(i), and m i /m e (j)-(l). Each marker set represents one simulation from the main article with varied particle sampling. The black squares represent fiducial simulation in all panels (d)-(l), and correspond to the data and markers in (a)-(c). The vertical light-gray bar indicates the fiducial particle sampling of 16,384 ions and electrons per cell (excluding test-particle CRe); all markers within the light-gray bar correspond to a simulation from the main article (see Table 1). The legends above each column report the ratio of q -1 , v A0 /c, and m i /m e with respect to the fiducial simulation (1×).