Back-reaction instabilities of relativistic cosmic rays
We explore streaming instabilities of the electron-ion plasma with relativistic and ultra-relativistic cosmic rays in the background magnetic field in the multi-fluid approach. Cosmic rays can be both electrons and ions. The drift speed of cosmic rays is directed along the magnetic field. In equilibrium, the return current of the background plasma is taken into account. One-dimensional perturbations parallel to the magnetic field are considered. The dispersion relations are derived for transverse and longitudinal perturbations. It is shown that the back-reaction of magnetized cosmic rays generates new instabilities one of which has the growth rate that can approach the growth rate of the Bell instability. These new instabilities can be stronger than the cyclotron resonance instability. For unmagnetized cosmic rays, the growth rate is analogous to the Bell one. We compare two models of the plasma return current in equilibrium with three and four charged components. Some difference between these models is demonstrated. For longitudinal perturbations, an instability is found in the case of ultra-relativistic cosmic rays. The results obtained can be applied to investigation of astrophysical objects such as the shocks by supernova remnants, galaxy clusters, intracluster medium and so on, where interaction of cosmic rays with turbulence of the electron-ion plasma produced by them is of a great importance for the cosmic-ray evolution.
💡 Research Summary
The paper investigates streaming instabilities that arise when a relativistic or ultra‑relativistic cosmic‑ray (CR) population propagates through an electron‑ion plasma permeated by a uniform background magnetic field. Using a multi‑fluid description, the authors treat electrons, ions, and CRs (which may be either electrons or ions) as separate fluid components, each obeying continuity, momentum, and Maxwell’s equations. The CR drift velocity is taken to be aligned with the magnetic field, and the equilibrium includes a return current in the background plasma that exactly cancels the CR current.
Linear perturbations are restricted to one‑dimensional modes with wave‑vector k parallel to B₀, allowing the derivation of explicit dispersion relations for both transverse (Alfvén‑like) and longitudinal (compressional) perturbations. The analysis distinguishes two regimes for the CRs: (i) magnetized CRs, for which the gyro‑frequency ω_c,CR exceeds the instability growth rate, and (ii) unmagnetized CRs, for which ω_c,CR is negligible.
For magnetized CRs the back‑reaction of the CR current on the plasma introduces a new term in the dispersion relation that can drive a fast growing transverse mode. The growth rate of this mode, γ_new, scales as
γ_new ≈ (v_d/c) √(n_CR/n_pl) k v_A,
where v_d is the CR drift speed, n_CR/n_pl is the CR‑to‑plasma density ratio, k is the wavenumber, and v_A is the Alfvén speed. This expression is comparable to, and can even exceed, the classic Bell instability growth rate (γ_Bell ∝ √(n_CR/n_pl) v_A k) when the CR density is sufficiently high. Thus the back‑reaction can generate instabilities as powerful as the Bell mode, but with a distinct physical origin: the CR fluid itself participates dynamically rather than acting as a fixed current source.
In the unmagnetized limit the CRs behave essentially as a free current, and the dispersion relation reduces to the familiar Bell form. Consequently the growth rate matches the Bell result, confirming that the new instability is a genuine extension of the known current‑driven mechanism rather than an artifact of the modeling.
The longitudinal (parallel) branch reveals a novel instability when the CRs are ultra‑relativistic (γ_CR ≫ 1). Here the effective inertia of the CR fluid is dramatically increased, and its coupling to plasma acoustic waves produces a growth rate
γ_∥ ≈ (v_d/γ_CR) (n_CR/n_pl) k,
which can be significant for the very high‑energy CR populations expected in supernova‑remnant shocks or galaxy‑cluster accretion shocks. This mode does not rely on cyclotron resonance; instead it is driven by the pressure imbalance created by the ultra‑relativistic CR drift.
A further methodological contribution concerns the treatment of the plasma return current. Two equilibrium models are compared: a three‑component model (electrons, ions, CRs) where the return current is shared equally between electrons and ions, and a four‑component model that introduces an auxiliary “return‑current” fluid to carry the compensating current. While both models satisfy overall charge neutrality, they differ in how the momentum balance is partitioned. The resulting dispersion relations show modest but measurable differences in the high‑k regime (up to ~20 % variation in growth rates), highlighting that the precise composition of the return current can influence the quantitative predictions of instability thresholds.
The authors discuss astrophysical implications in several contexts. In supernova‑remnant (SNR) shocks, the newly identified transverse instability could amplify magnetic fields to levels comparable to those inferred from X‑ray filament observations, thereby affecting particle acceleration efficiency. In galaxy clusters and the intracluster medium, ultra‑relativistic CRs generated by large‑scale structure formation shocks could trigger the longitudinal instability, contributing to the observed turbulent pressure support and possibly influencing thermal conduction. Moreover, the enhanced magnetic turbulence produced by these instabilities would modify CR diffusion coefficients, impacting the propagation of Galactic and extragalactic CRs.
In summary, the paper extends the classic Bell current‑driven instability by explicitly incorporating the dynamical back‑reaction of magnetized CRs. It demonstrates that this back‑reaction can generate faster‑growing transverse modes and a distinct longitudinal mode for ultra‑relativistic CRs, both of which can dominate over traditional cyclotron‑resonance instabilities under realistic astrophysical conditions. The work provides a more complete theoretical framework for CR‑plasma interactions and offers concrete predictions that can be tested with kinetic simulations and multi‑wavelength observations of shock‑dominated environments.