The Role of Acoustic Instability in Cosmic-Ray Self-Confinement

TL;DR

The paper investigates acoustic (Drury) instability as a mechanism for cosmic-ray self-confinement near shocks by coupling CR pressure gradients to upstream plasma dynamics. A simple dispersion relation, ${omega^2 = c_s^2 k^2 + i (k \cdot \nabla P_CR)/\rho_0}$, predicts two growth regimes and a field-dependent maximum, with parallel fields reproducing hydrodynamic growth and perpendicular fields modifying the rate via ${c_ms^2 = c_s^2 + v_A^2}$. Magnetohydrodynamic simulations validate the linear growth and reveal dramatic nonlinear amplification of magnetic energy, driving small-scale turbulence and forming dense, shock-bearing structures that could suppress CR diffusion. The nonlinear evolution indicates robust magnetic-field amplification and potential CR self-confinement near shocks, though results are tempered by resolution limits and the absence of CR back-reaction; Bell instability is discussed as a broader context for precursor perturbations. Overall, the work demonstrates a viable pathway for CR self-confinement through Drury instability, with implications for accelerating particles to high energies in shock environments.

Abstract

Over the past decades, there has been growing observational and theoretical evidence that cosmic-ray-induced instabilities play an important role in both acceleration and transport of cosmic rays (CRs). For instance, the efficient acceleration of charged particles at supernova remnant shocks requires rapidly growing instabilities, so much so that none of the proposed processes seem sufficient to warrant acceleration to PeV energies. In this work, we investigate whether an acoustic instability triggered by the presence of a CR pressure gradient can lead to significant self-confinement of charged particles in the vicinity of shocks. We validate the expected growth rates and obtain the scale and energy of magnetic field perturbations induced by such system using magnetohydrodynamical simulations. Our results suggest a strong suppression of the diffusion coefficient for particles with Larmor radius around a thousandth of the precursor scale length.

Figures (5)

• Figure 1: Real and imaginary parts of $\omega$ (multiplied by $t_{\rm adv}=L/u_0$) from relations (\ref{['eq:dispersion']})/(\ref{['eq:kpar-dispersion']}) and (\ref{['eq:kperp-dispersion']}), assuming $|\nabla P_{\rm CR}|=\xi_{\rm CR}\rho_0u_0^2/L$ parallel to $\mathbf{k}$. Parameter values are chosen to be $\xi_{\rm CR}=0.1$, $M=100$, and $M_{ms}=100/\sqrt{5}$. Asymptotic behaviors are represented by the gray dashed and colored dotted lines.
• Figure 2: Left: Steady-state perturbation profiles for $\xi_{\rm CR}=0.1$, $M=100$, $\delta\rho_0/\rho_0=10^{-4}$, $B_0=0$, $k_xL=80\pi$, and $k_y=0$. Right: Constant-$y$ slice of profiles on the left, comparing the simulation growth of perturbations with the expected one in equation (\ref{['eq:growth-rate']}).
• Figure 3: Density, pressure and velocity steady-state profiles in simulations where perturbations become larger than their background values. Left: Injection of $k_x$ mode with $\delta\rho_0< \rho_0$, growing under the Drury instability until the nonlinear stage. Right: Several modes are injected in both directions, generating turbulence and short-lived stalled regions.
• Figure 4: Amplification of the average magnetic energy in perturbations at steady state. For comparison, dashed lines represent the total magnetic energy at $t=0$, which is entirely due to mean fields that remain mostly unchanged as the simulation progresses.
• Figure 5: Dimensionless power spectrum of magnetic field perturbations at steady-state, for the simulation with $\xi_{\rm CR}=0.6$ and $M=100$. We show the total power in the whole box (black line), as well as the power in each eighth of the box (colored lines).