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Models for the formation of a critical layer in water wave propagation

Lookup NU author(s): Professor Robin Johnson


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A theory is presented which provides a model for the appearance of critical layers within the flow below a water wave. The wave propagates over constant depth, with constant (non-zero) vorticity. The mechanism described here involves adjusting the surface-pressure boundary condition; two models are discussed. In the first, the pressure at the surface is controlled (mimicking the movement of a low-pressure region associated with a storm) so that the speed and development of the pressure region ensure the appearance of a critical layer. In the second, the pressure boundary condition is allowed to accommodate the reduction of pressure with altitude, although the effects have to be greatly enhanced for this mechanism to produce a critical layer. These two problems are analysed using formal parameter asymptotics. In the second problem, this leads to a Korteweg-de Vries equation for the surface wave, and then the evolution of appropriate solutions of this equation gives rise to the appearance of a critical layer near the bottom; the corresponding problem at the surface can be formulated but not completely resolved. The appearance of a stagnation point and then a critical layer, either at the surface or the bottom, are discussed; the nature of the flow, and the corresponding streamlines are obtained and some typical flow fields are depicted.

Publication metadata

Author(s): Johnson RS

Publication type: Article

Publication status: Published

Journal: Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences

Year: 2012

Volume: 370

Issue: 1964

Pages: 1638-1660

Print publication date: 13/04/2012

ISSN (print): 1364-503X

ISSN (electronic): 1471-2962

Publisher: The Royal Society Publishing


DOI: 10.1098/rsta.2011.0456


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