We study the underlying physics of cosmic-ray (CR) driven instabilities that
play a crucial role for CR transport across a wide range of scales, from
interstellar to galaxy cluster environments. By examining the linear dispersion
relation of CR-driven instabilities in a magnetised electron-ion background
plasma, we establish that both, the intermediate and gyroscale instabilities
have a resonant origin and show that these resonances can be understood via a
simple graphical interpretation. These instabilities destabilise wave modes
parallel to the large-scale background magnetic field at significantly distinct
scales and with very different phase speeds. Furthermore, we show that
approximating the electron-ion background plasma with either
magnetohydrodynamics (MHD) or Hall-MHD fails to capture the fastest growing
instability in the linear regime, namely the intermediate-scale instability.
This finding highlights the importance of accurately characterising the
background plasma for resolving the most unstable wave modes. Finally, we
discuss the implications of the different phase speeds of unstable modes on
particle-wave scattering. Further work is needed to investigate the relative
importance of these two instabilities in the non-linear, saturated regime and
to develop a physical understanding of the effective CR transport coefficients
in large-scale CR hydrodynamics theories.Comment: 14 pages, 3 figures, submitted to JPP Letters, comments welcom