Free-surface effects on spin-up

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Abstract
The effects of a free surface on the spin-up of a homogeneous fluid are studied, both analytically and experimentally. The analysis is carried out in cylindrical geometry and shows that the spin-up process is strongly modified as the rotational Froude number F = 4ω2L2/gH becomes large. The dynamic effect of the free surface causes delayed response outside a sidewall boundary layer of thickness LF−½. The timescale in the slowly decaying core is larger than the usual spin-up time by a factor of order F. A set of laboratory experiments using a cylinder with a parabolic bottom were carried out in order to test the theory. Reasonable agreement is found in all the experiments except close to the centre where an interesting deviation was observed, especially in cases corresponding to smaller Froude numbers. The deviation consisted of an anticyclonic vortex at the centre. It is shown that this phenomenon might be explained by Lagrangian mean motion resulting from inertial oscillations. In fact, the analysis shows that this motion produces a singular vortex at the centre.

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