Topographic waves and the evolution of coastal currents

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Abstract
An initial-value problem is considered for the oceanographically relevant case of slow flow over obstacles of small height and horizontal scale of order the fluid depth or larger. Previous work on starting flow over obstacles whose contours are closed (Johnson 1984) is extended to flow forced by a source–sink pair to cross a step change in depth bounded by a vertical sidewall. Bottom contours thus end abruptly and the near-periodic solutions of the earlier work are no longer possible. The relevant timescale for the motion is again the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the step height. This time is taken to be long compared with the inertial period but short compared with the advection time. It is shown that if shallow water lies to the right (looking away from the wall) a wavefront moves outwards exponentially fast leaving behind a flow equivalent to that obtained by replacing the step by a rigid wall. If shallow water lies to the left the wavefront approaches the wall, forming at the wall–step junction an unsteady, exponentially thinning, singular region that transports the whole flux. The relevance of these solutions to experiments and steady solutions for free-surface and two-layer flows in Davey, Gill, Johnson & Linden (1984, 1985) is discussed.

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