Starting flow for an obstacle moving transversely in a rapidly rotating fluid

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Abstract
The initial-value problem for Taylor columns is considered for the oceanographically relevant case of slow flow over obstacles of small slope, and horizontal scale of order of the fluid depth or larger. In such flows depth variations cause, in the neighbourhood of the obstacle, a gradient of background potential vorticity. Topographic Rossby waves, forced in the starting flow, propagate across the gradient, cycling the obstacle in times of order of the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the obstacle height. The linear inertialwave equation and the quasigeostrophic equations are related by considering the case where the inertial period is small compared with the topographic time, which is in turn small compared with the advection time.When viscosity is present the waves decay to a steady motion in a time of order of the viscous spin-up time. In inviscid flow the waves do not decay. The flow oscillates about steady, irrotational, zero-circulation flow. The drag and lift on the obstacle oscillate with almost constant frequency and amplitude equal to the Coriolis force on a body of fluid whose volume matches that of the obstacle. Particles above a right cylinder move in closed orbits of diameter 1/S, where S is the topographic parameter introduced by Hide (1961). Particles away from the cylinder move irrotationally, but asymmetrically, from upstream to downstream. A shear layer is present at the boundary of the cylinder throughout the motion. The initial vorticity distribution for flow over a paraboloid is everywhere finite. However, regions of positive and negative vorticity interlace ever more closely during the motion, causing at large time a similar shear layer at the column boundary. By this time the vorticity is increasing without bound above the obstacle and neglected nonlinear, horizontal viscous, or vertical effects are important.

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