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Nonlinear shallow-water oscillations in a parabolic channel: exact solutions and trajectory analyses

Published online by Cambridge University Press:  26 April 2006

Alan Shapiro
Alan Shapiro
Affiliation:
Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, OK 73019, USA
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Abstract

A new exact solution of the nonlinear shallow-water equations is presented. The solution corresponds to divergent and non-divergent free oscillations in an infinite straight channel of parabolic cross-section on the rotating Earth. It provides a description of the one-dimensional subclass of shallow-water flows in paraboloidal basins considered by Ball (1964), Thacker (1981), Cushman-Roisin (1987) and others in which the velocity field varies linearly and the free-surface displacement varies quadratically with the spatial coordinates. In contrast to the previous exact solutions describing divergent oscillations in circular and elliptic paraboloidal basins, the oscillation frequency of the divergent oscillation in the parabolic channel is found to depend, in part, on the amplitudes of the relative vorticity and free-surface curvature. This result is consistent with Thacker's (1981) numerical finding that when the free surface in parabolic channel flow is curved, the oscillation frequency depends on the amplitude of the motion. Solutions for parcel trajectories are also presented. The exact solution provides a rare description of a class of nonlinear flows and is potentially valuable as a validation test for numerical shallow-water models in Eulerian and Lagrangian frameworks.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 318 , 10 July 1996 , pp. 49 - 76
Copyright
© 1996 Cambridge University Press

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