  Long wavelength plasma non-uniformities rotating in the azimuthal direction
("rotating spokes") have been observed in a number of experiments on Hall
thrusters or magnetron discharges. We use a two-dimensional (2D),
axial-azimuthal Particle-In-Cell Monte Carlo Collisions (PIC-MCC) model to
study the formation of instabilities in a direct current (dc) magnetron
discharge under conditions close to recent experiments. In spite of the
simplified 2D geometry of the model, the simulations can reproduce the main
features of the experimental results. At a given position above the cathode,
corresponding to the spoke location, the simulations show large amplitude
oscillations of the plasma density and a very sharp increase of the plasma
potential and electron temperature at the leading edge of the spoke, as in time
resolved probe measurements. Moreover, the simulations show that the
instability evolves in time from a gradient-drift type of instability in the
linear phase, to an ionization wave in the non-linear phase, with rotation in
the +ExB direction in the first phase and in the -ExB direction in the second
phase. The number of spokes is found to increase with pressure, as in
experiments. The mechanisms of electron heating and the role of the Bx$\nabla$B
drift in electron heating and in the coherence and direction of spoke rotation
are discussed.
