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.
