Impact of Cosmic Ray Distribution on the Growth and Saturation of Bell Instability

TL;DR

This paper examines how the momentum distribution of CRs influences the Bell (NRSI) instability in weakly magnetized plasmas using 1D kinetic simulations. It combines fluid and kinetic analytic insights with fully self-consistent PIC runs to show that linear growth is governed by the CR current and is largely insensitive to the CR distribution, while nonlinear saturation is controlled by CR isotropization and momentum-dependent current relaxation. Mono-energetic CRs saturate at larger magnetic-field amplitudes than wide power-law tails with the same anisotropy parameter ξ, because high-energy CRs isotropize less efficiently; for broad PL spectra, a generalized saturation prescription involves an effective cutoff p′_eff ≈ 9.3 p′_min. The authors propose a layered CR confinement picture upstream of shocks, in which lower-energy CRs trap closer to the shock and higher-energy CRs drive successive NRSI stages, with implications for magnetic-field amplification and PeV particle acceleration in supernova remnants.

Abstract

Cosmic rays (CRs) streaming in weakly magnetized plasmas can drive large-amplitude magnetic fluctuations via nonresonant streaming instability (NRSI), or Bell instability. Using one-dimensional kinetic simulations, we investigate how mono-energetic and power-law CR momentum distributions influence the growth and saturation of NRSI. The linear growth is governed solely by the CR current and is largely insensitive to the CR distribution. However, the saturation depends strongly on the CR distribution and is achieved through CR isotropization, which quenches the driving current. Mono-energetic CRs effectively amplify the magnetic field and isotropize. For power-law distributions, the lowest-energy CRs dominate current relaxation and magnetic growth, while the highest-energy CRs remain weakly scattered, limiting their contribution to saturation. In the absence of low-energy CRs, high-energy particles amplify magnetic fields effectively and isotropize. We provide a modified saturation prescription accounting for these effects and propose a layered CR-confinement scenario upstream of astrophysical shocks, relevant to particle acceleration to high energies.

Figures (11)

• Figure 1: Analytic growth rates of NRSI derived for mono-energetic (ME, dotted) and power-law (PL, dashed) momentum distributions of CRs. The blue curve represents the growth rate obtained from the fluid approach (Equation \ref{['eq:growth_rate_fluid']}), while all other curves are obtained from the kinetic approach (Equations \ref{['eq:dispersion_iso']} and \ref{['eq:dispersion_cone']}). The maximum growth rates and the corresponding wavenumbers remain almost the same; they depend on the bulk CR current.
• Figure 2: Diagnostics of the magnetic fields for a mono-energetic (upper panels) and a power-law (lower panels) isotropic CR distributions. [Left panels] The spatial profiles of the magnetic field components, normalized to the initial magnetic field $B_0$, at $t \approx 7\gamma_{\text{fast}}^{-1}$. The $x$-axes are normalized to the wavelengths of the fastest-growing modes $(\lambda_{\text{fast}}=2 \pi /k_{\text{fast}} \approx 90 d_i)$. From the shaded zoomed-in part, the dominant wavelengths, almost equal to $\lambda_{\text{fast}}$, are prominent. [Right panels] Time evolution of helicities $\Delta \phi (k)$. The positive and negative values signify R-handed NRSI and L-handed RSI, respectively.
• Figure 3: Similar to Figure \ref{['fig:MagFld_Snaps_pcr_5']}, except that the CRs initially have cone distributions. [Left panels] The spatial profiles of the magnetic field components at $t \approx 7\gamma_{\text{fast}}^{-1}$. The $x$-axes are normalized to $\lambda_{\text{fast}}=2 \pi /k_{\text{fast}} \approx 50 d_i$. In the shaded zoomed-in part, the dominant wavelengths $\sim \lambda_{\text{fast}}$ are prominent. [Right panels] Time evolution of helicities $\Delta \phi (k)$.
• Figure 4: Time evolution of the transverse magnetic fields $B_{\perp}$ for the runs in Table \ref{['tab:sims_data']}. Their linear growth shows good agreement with the analytical prediction (grey solid lines). However, the saturated magnetic fields are different.
• Figure 5: The saturated magnetic field $\left( B_{\perp}/B_0 \right)_{\rm sat}$ versus the anisotropy parameter $\xi$. The shapes and colors represent different simulations (see the table on the right; see Table \ref{['tab:sims_data']} for the nomenclature of the runs). Momenta separated by '/' represent different runs (e.g. PLI:5-30/80 implies PLI:5-30 and PLI:5-80). For the points with the same colors and shapes, larger $\xi$'s correspond to larger momenta (e.g. for PLI:5-30/80, $\xi$ for PLI:5-80 is larger than PLI:5-30). The grey dashed line represents the prediction, i.e. Equation \ref{['eq:xi_vs_saturation']}. The runs with mono-energetic CRs mostly follow this prediction, unlike those with power-law CRs. Inset: Modelling of $\left( B_{\perp}/B_0 \right)_{\rm sat}$ using Equation \ref{['eq:xi_vs_saturation_fitting']} (red dashed line).