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The linear stability of the Hall thruster discharge is analysed against axial-azimuthal perturbations in the low frequency range using a time-dependent 2D code of the discharge. This azimuthal stability analysis is spatially global, as opposed to the more common local stability analyses, already afforded previously (D. Escobar and E. Ahedo, Phys. Plasmas 21(4), 043505 (2014)). The study covers both axial and axial-azimuthal oscillations, known as breathing mode and spoke, respectively. The influence on the spoke instability of different operation parameters such as discharge voltage, mass flow, and thruster size is assessed by means of different parametric variations and compared against experimental results. Additionally, simplified models are used to unveil and characterize the mechanisms driving the spoke. The results indicate that the spoke is linked to azimuthal oscillations of the ionization process and to the Bohm condition in the transition to the anode sheath. Finally, results obtained from local and global stability analyses are compared in order to explain the discrepancies between both methods.