The motion of fluid contained between two concentric spherical surfaces is analysed in the limit of strong rotation appropriate to large scale flows and arbitrary gap width. To do so, the dynamical equations are written in the natural cylindrical co-ordinate system that gives a central role to the axis of rotation. The case of a homogeneous fluid allows us to give a general solution of the inviscid, steady flow when sources and sinks have prescribed boundary distributions. Fluid can cross the equatorial plane without breaking rotational constraints provided the source-sink forcing is antisymmetric. However the cylindrical surface tangent to the inner sphere at the equator is singular and calls for higher order inertial and/or viscous effects. No specific solution is obtained in the stratified case, instead a number of integral constraints along the axis of rotation are derived allowing us to relate the interior motion to the surface forcing distributions. The unsteady low frequency waves with Taylor column-like motions are obtained exactly and we extend the non dispersive limit of classical Rossby wave theory in concentric spheres of arbitrary gap width. In the stratified case, a new mode that has no counterpart in the classical, shallow fluid theory is found at the equator.
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