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1. Modeling the Saturation of the Bell Instability Using Hybrid Simulations

   https://doi.org/10.3847/1538-4357/ad3960  

   Zacharegkas, Georgios; Caprioli, Damiano; Haggerty, Colby; Gupta, Siddhartha; Schroer, Benedikt (May 2024, The Astrophysical Journal)

   Abstract The nonresonant streaming instability (Bell instability) plays a pivotal role in the acceleration and confinement of cosmic rays (CRs), yet the exact mechanism responsible for its saturation and the magnitude of the final amplified magnetic field have not been assessed from first principles. Using a survey of hybrid simulations (with kinetic ions and fluid electrons), we study the evolution of the Bell instability as a function of the parameters of the CR population. We find that at saturation, the magnetic pressure in the amplified field is comparable with the initial CR anisotropic pressure, rather than with the CR energy flux, as previously argued. These results provide a predictive prescription for the total magnetic field amplification expected in the many astrophysical environments where the Bell instability is important. 

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2. Modeling the Saturation of the Bell Instability Using Hybrid Simulations

   Zacharegkas, G.; Caprioli, D.; Haggerty, C. (January 2019, ICRC2019)

   The nonresonant cosmic ray instability, predicted by Bell (2004), is thought to play an important role in the acceleration and confinement of cosmic rays (CRs) close to supernova remnants. Despite its importance, the exact mechanism responsible for the saturation of the instability has not been determined, and there is no first-principle prediction for the amplitude of the saturated magnetic field. Using a survey of self-consistent kinetic hybrid simulations (with kinetic ions and fluid electrons), we study the saturation of the non-resonant streaming instability as a function of the parameters of both the thermal background plasma and the CR population. The strength of the saturated magnetic field has important implications for both CR acceleration in supernova remnants and CR diffusion in the Galaxy. 

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3. Understanding Streaming Instabilities in the Limit of High Cosmic-Ray Current Density

   https://doi.org/10.3847/1538-4357/adadf5  

   Lichko, Emily; Caprioli, Damiano; Schroer, Benedikt; Gupta, Siddhartha (February 2025, The Astrophysical Journal)

   Abstract A critical component of particle acceleration in astrophysical shocks is the nonresonant (Bell) instability, where the streaming of cosmic rays (CRs) leads to the amplification of magnetic fields necessary to scatter particles. In this work we use kinetic particle-in-cell simulations to investigate the high-CR-current regime, where the typical assumptions underlying the Bell instability break down. Despite being more strongly driven, significantly less magnetic field amplification is observed than in low-current cases, an effect due to the anisotropic heating that occurs in this regime. We also find that electron-scale modes, despite being the fastest growing, mostly lead to moderate electron heating and do not affect the late evolution or saturation of the instability. 

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4. Hybrid Simulations of the Resonant and Non-Resonant Cosmic Ray Streaming Instability

   Haggerty, C.; Caprioli, D.; Zweibel, E. (July 2019, 36th International Cosmic Ray Conference (ICRC2019))

   Using hybrid simulations (kinetic ions--fluid electrons), we test the linear theory predictions of the cosmic ray (CR) streaming instability. We consider two types of CR distribution functions: a "hot" distribution where CRs are represented by a drifting power law in momentum and an anisotropic "beam" of monochromatic particles. Additionally, for each CR distribution we scan over different CR densities to transition from triggering the resonant to the non-resonant (Bell) streaming instability. We determine the growth rates of these instabilities in simulations by fitting an exponential curve during the linear stage, and we show that they agree well with the theoretical predictions as a function of wave number agree. We also examine the magnetic helicity as a function of time and wave number, finding a general good agreement with the predictions, as well as some unexpected non-linear features to the instability development. 

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5. Diffusive shock re-acceleration

   https://doi.org/10.1017/S0022377818000478  

   Caprioli, Damiano; Zhang, Horace; Spitkovsky, Anatoly (June 2018, Journal of Plasma Physics)

   We have performed two-dimensional hybrid simulations of non-relativistic collisionless shocks in the presence of pre-existing energetic particles (‘seeds’); such a study applies, for instance, to the re-acceleration of galactic cosmic rays (CRs) in supernova remnant (SNR) shocks and solar wind energetic particles in heliospheric shocks. Energetic particles can be effectively reflected and accelerated regardless of shock inclination via a process that we call diffusive shock re-acceleration. We find that re-accelerated seeds can drive the streaming instability in the shock upstream and produce effective magnetic field amplification. This can eventually trigger the injection of thermal protons even at oblique shocks that ordinarily cannot inject thermal particles. We characterize the current in reflected seeds, finding that it tends to a universal value $$J\simeq en_{\text{CR}}v_{\text{sh}}$$ , where $$en_{\text{CR}}$$ is the seed charge density and $$v_{\text{sh}}$$ is the shock velocity. When applying our results to SNRs, we find that the re-acceleration of galactic CRs can excite the Bell instability to nonlinear levels in less than $${\sim}10~\text{yr}$$ , thereby providing a minimum level of magnetic field amplification for any SNR shock. Finally, we discuss the relevance of diffusive shock re-acceleration also for other environments, such as heliospheric shocks, galactic superbubbles and clusters of galaxies. 

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