On detached shear layers and western boundary currents in a rotating homogeneous liquid

This paper reports a study of slow motions produced mechanically in a rapidly rotating homogeneous liquid when the side-wall of the container is a circular cylinder of radius b′, one of the two plane end-walls is inclined at an angle ½π – α to the axis of the cylinder (which is also the axis of basic rotation with angular speed Ω) and the other end-wall is perpendicular to the rotation axes and is split at a distance a from the axis so that the inner part of the end-wall can be rotated with relative angular speed εΩ. The investigation thus extends to cases when α ≠ 0 and a ≠ b′ extensive previous studies of detached shear layers in systems having α = 0 and a ≠ b′ and of western boundary currents in systems having α ≠ 0 and a = b′ (sliced cylinder). A full account of the experiments outlined in this paper (see Section 4)-which covered a wide range of Rossby number |ε| and Ekman number E ≅ v/Ωa2-where v is the coefficient of kinematic viscosity-will be deferred until a numerical study of the non-linear governing equations has been completed. The subsequent analytical study of the linear case (when |ε| ≪ E1/2) presented in Section 3 shows how the general flow pattern changes as α increases, revealing three flow regimes typified by the limiting cases tan α ≪ E1/2, E1/2 ≪ tan α ≪ E1/4 and E1/4 ≪ tan α ≪ 1. It also provides some information on nonlinear effects, including the magnitude of |sε| at which linear theory should break down in each flow regime and the relative importance of non-linear effects in different parts of the flow.