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We present an explicit analysis of wave-resonant instability of swirlingflows inside fast rotating cylindrical containers. The linear dynamics aredecomposed into the interaction between the horizontal inner centrifugal edgewaves, the outer vertical gravity waves with the aim of understanding thedynamics of the centrifugal waves. We show how the far field velocity inducedrespectively by the centrifugal and the gravity waves affect each other'spropagation rates and amplitude growth. We follow this with an analysis of theinstability in terms of a four wave interaction, two centrifugal and twogravity ones, and explain why the resonant instability can be obtained onlybetween a pair of two counter-propagating waves, one centrifugal and onegravity. Furthermore, a near resonant regime which does not yield instabilityis shown to result from a phase-locking configuration between a pair of acounter-propagating centrifugal wave and a pro-propagating gravity one, wherethe interaction affects the waves' propagation rates but not the amplitudegrowth.

On the physical mechanism of centrifugal-gravity wave resonant instability in swirling free surface rotating Polygons