A long-wavelength instability involving the stress tensor

Cosmic ray acceleration through first-order Fermi acceleration in a collisionless plasma relies on efficient scattering off magnetic field fluctuations. Scattering is most efficient for magnetic field fluctuations with wavelengths on the order of the gyroradius of the particles. In order to determine the highest energy to which cosmic rays can be accelerated, it is important to understand the growth of the magnetic field on these large scales. We derive the growth rate of the long-wavelength fluctuations in the linear regime, using the kinetic equation coupled to Maxwell’s equations for the background plasma. The instability, driven by the cosmic ray current, acts on large scales due to the stress tensor and efficient scattering on small scales, and operates for both left- and right circular polarisations. This long-wavelength instability is potentially important in determining the acceleration efficiency and maximum energy of cosmic rays around shock waves such as in supernova remnants.

💡 Research Summary

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The paper addresses a long‑standing problem in cosmic‑ray astrophysics: how magnetic fields can be amplified on scales larger than the gyroradius of the highest‑energy particles, thereby allowing efficient diffusive shock acceleration (DSA) up to the “knee” (~10¹⁵ eV). While the non‑resonant Bell instability (Bell 2004) explains rapid growth on scales much smaller than the gyroradius, it does not directly inform the evolution of magnetic fluctuations on the much larger, long‑wavelength side of the spectrum. Schure and Bell therefore develop a linear kinetic theory that explicitly includes the second‑order anisotropy (the stress tensor) of the cosmic‑ray distribution and the effect of small‑scale scattering, characterised by an effective collision frequency ν.

Starting from the Vlasov‑Fokker‑Planck equation, the authors expand the distribution function into isotropic (f₀), first‑order (f₁) and second‑order (f₂) moments. The first‑order moment represents the cosmic‑ray current that drives the instability; the second‑order moment (the stress tensor) describes the transport of perpendicular current and couples the current to the background plasma. Small‑scale magnetic turbulence scatters particles, providing an effective ν that appears in the momentum‑transport equation. By retaining f₂, the authors obtain a closed set of equations for the perturbed magnetic field components (B_y, B_z) and the first‑order moments (f_y, f_z).

Assuming a uniform background magnetic field B₀ aligned with the shock normal (x‑direction) and plane‑wave perturbations ∝ exp