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A simple model of Rossby-wave hydraulic behaviour

Published online by Cambridge University Press:  26 April 2006

P. H. Haynes ,
E. R. Johnson  and
R. G. Hurst
P. H. Haynes
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
R. G. Hurst
Affiliation:
Department of Mathematics, University College, Gower Street, London WC1E 6BT, UK
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Abstract

This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique.

The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 359 - 384
Copyright
© 1993 Cambridge University Press

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