The stability properties of the rotating free surface flow in a cylindrical container is studied using a global stability approach, considering successively three models. For the case of solid body rotation (Newton's bucket), all eigenmodes are found to be stable, and are classified into three families: gravity waves, singular inertial modes, and Rossby waves. For the case of a potential flow, an instability is found. The mechanism is explained as a resonance between gravity waves and centrifugal waves, and is thought to be at the origin of the "rotating polygon instability" observed in experiments where the flow is driven by rotation of the bottom plate (see L. Tophoj, J. Mougel, T. Bohr, D. Fabre, The Rotating Polygon Instability of a Swirling Free Surface Flow, Phys. Rev. Lett. 110 (2013) 194502). Finally, in the case of the Rankine vortex which in fact consists in the combination of the two first cases, we report a new instability mechanism involving Rossby and gravity waves.
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