Non-Markovian Cosmic-Ray Pitch-Angle Transport from Mirror Interactions

Abstract

Cosmic-ray pitch-angle transport in magnetohydrodynamic (MHD) turbulence is governed by the interplay between magnetic mirroring and gyroresonant scattering. We develop a guiding-center (GC) Langevin model with explicit mirror drift and gyroresonant diffusion to describe the pitch angle evolution. This model accurately captures our test-particle simulation results in three-dimensional MHD turbulence, driven both solenoidally and compressively. We find that magnetic mirroring can drive anomalous pitch-angle diffusion at large pitch angles (including $90^\circ$) with non-Markovian memory effects, which arises from trapping of particles in magnetic wells. Gyroresonant scattering controls the escape rate from these wells. Across $M_{\rm A}$, large-pitch-angle particles are jointly regulated by mirror trapping and gyroresonant escape, exhibiting a transition from anomalous to normal diffusive pitch-angle transport as scattering strengthens, whereas small-pitch-angle particles remain gyroresonance-dominated and diffusive throughout. The pitch angle transport is found to be dominated by the compressible perturbations with marginal influence from Alfvén modes. In compressible turbulence with realistic damping accounted for, transit time damping (TTD) treatment fully recovers mirror interactions.

Figures (7)

• Figure 1: Time evolution of the mean pitch-angle increment $\langle \left| \Delta\mu(t) \right| \rangle$ for particles with different initial $\mu_0$ in compressively driven turbulence of $M_{\rm A}^{\rm C}=0.18, \ M_{\rm A}=0.24$, with $r_{\rm L}=5 \, l_{\rm grid}$. Solid lines show the test-particle results, while dotted lines show the GC Langevin results. (a) Initial pitch angle $\mu_0=0$, (b) $\mu_0=0.4$, (c) $\mu_0=0.8$, (d) $\mu_0$ is sampled from a uniform distibution $\mathcal{U}(-1,1)$. Colored dashed lines showcase different reference start time $t_0$.
• Figure 2: Statistics of particles in solenoidally driven turbulence of $M_{\rm A}=0.33, \ M_{\rm A}^{\rm C}=0.16$. (a) Time evolution of the mean pitch-angle increment $\langle \left| \Delta\mu(t) \right| \rangle$ of particles with $\mu_0=0$ and $r_{\rm L}=1 \, l_{\rm grid}$, so as to suppress gyroresonance and isolate the effects of magnetic mirroring and field-line curvature. Solid lines show the test-particle results, while dotted lines show the GC Langevin results. $\langle \Delta \mu \rangle$ rises faster initially in the C modes than in the whole datacube, because A modes increase $|B|$ without contributing to $\nabla_s B$, thus slightly reducing mirroring $\propto \nabla_s B / |B|$. (b) same as panel (a), but for $\mu_0=0.4$. (c) $\langle \left| \Delta\mu(t) \right| \rangle$ of particles with $\mu_0$ isotropically injected from a uniform distibution $\mathcal{U}(-1,1))$ and $r_{\rm L}=5 \, l_{\rm grid}$. Dash-dotted lines show results of particles injected at high-kurtosis regions and low-kurtosis regions separately. (d) Probability density distributions (PDF) of pitch angle increment $\Delta \theta$ of test-particle simulations for $\mu_0 \sim \mathcal{U}(-1,1)),\ \Delta t = 10 \, t_{\rm g}$. The dash-dotted curve shows the best-fit Gaussian distribution. (e) same as panel (d), but for particles injected at high-kurtosis regions. (f) same as panel (d), but for particles injected at low-kurtosis regions.
• Figure 3: Statistical properties of $\Delta\mu$ measured from test-particle simulations in a sub–Alfvénic compressive turbulence with $M_{\rm A}^{\rm C}=0.18$. (a) Probability density distributions (PDF) of pitch angle cosine increment $\Delta \mu$ of test-particle simulations for $\mu_0=0, \Delta t = 10 \, t_{\rm g}$. The dashed and dash-dotted curves show fits with a stretched Gaussian and a Gaussian distribution, respectively. The legend lists the reduced chi-square $\chi^2$ for each fit. (b) Same as (a), but for $\mu_0=0.8$. (c) Waiting time distribution for pitch angle cosine increment reaches a threshold $|\Delta \mu| = 0.05$ for $\mu_0 =0$. (d) Same as (b), but for $\mu_0=0.8$. (e) PDF of $\Delta\mu$ for $\mu_0=0$ and $\Delta t=10 \,t_{\rm g}$, computed after resetting $\mu(t_{\rm c})=\mu_0$ at different reference times $t_{\rm c}$, i.e. $\Delta\mu = \mu(t_{\rm c}+\Delta t)-\mu(t_{\rm c}).$ (f) Same as (e), but for $\mu_0 =0.8$.
• Figure 4: Time evolution of the mean pitch-angle increment $\langle \left| \Delta\mu(t) \right| \rangle$ for particles with initial $\mu_0=0$ in compressively driven turbulence of $M_{\rm A}^{\rm C}=0.41$. Solid lines show the test-particle results, while dotted lines show the GC Langevin results.
• Figure 5: Power spectrum density (PSD) for the (a) whole datacube (b) decomposed Alfvénic modes (c) decomposed compressible modes.