Electrostatic instabilities in ExB discharges: comparison of the linear theory dispersion relation with the reconstructed power spectrum

In the last decades, the ExB community has shown a great interest in plasma instabilities, from theoretical, numerical and experimental point of view. Although it has been discussed for quite a long time how these instabilities determine the anomalous electron mobility coefficient, and consequently the efficiency of an Hall Thruster (HT), some work still needs to be accomplished to fully understand the origin of instabilities and their consequences. In our work, we show that it is possible to retrieve the dispersion relations (DRs) of the most known ExB instabilities (i.e. IAW, MTSI and ITTI) from the same 3D fluid electrostatic DR. This procedure allows us to consider not only the purely 1D DRs, that are commonly studied, but also the instabilities with a minor component developing along a secondary axis. The comparison of the analytic DRs with the results of PIC simulations, in conditions similar to the ones of a HT, gives important insights about the origin and the development of these instabilities. The Two-Points Power Spectral Density reconstruction method allows to investigate the evolution of the spectrum along the thruster axis, in particular, it permits to make and challenge some hypotheses about the instabilities wavefronts bending and about the temporal evolution of their amplitude. A complete discussion about the instabilities development and a comparison with the analytic DRs are carried out to unveil the bi-dimensional nature of these instabilities.