  Recent observations revealed that the solar atmosphere is highly structured
in density, temperature and magnetic field. The presence of these gradients may
lead to the appearance of currents in the plasma, which in the weakly
collisional corona can constitute sources of free energy for driving
micro-instabilities. Such instabilities are very important since they represent
a possible source of ion-cyclotron waves which have been conjectured to play a
prominent role in coronal heating, but whose solar origin remains unclear.
Considering a density stratification transverse to the magnetic field, this
paper aims at studying the possible occurrence of gradient-induced plasma
micro-instabilities under typical conditions of coronal holes. Taking into
account the WKB (Wentzel-Kramers-Brillouin) approximation, we perform a Fourier
plane waves analysis using the collisionless multi-fluid model. By neglecting
the electron inertia, this model allows us to take into account ion-cyclotron
wave effects that are absent from the one-fluid model of magnetohydrodynamics
(MHD). Realistic models of density and temperature, as well as a 2-D analytical
magnetic-field model, are used to define the background plasma in the
open-field funnel in a polar coronal hole. The ray-tracing theory is used to
compute the ray path of the unstable waves, as well as the evolution of their
growth rates during the propagation. We demonstrate that in typical coronal
hole conditions, and when assuming typical transverse density length scales
taken from radio observations, the current generated by a relative electron-ion
drift provides enough free energy for driving the mode unstable. This
instability results from a coupling between oppositely propagating slow-mode
waves.
