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Generation of surface waves by shear-flow instability

Published online by Cambridge University Press:  18 December 2013

W. R. Young  and
C. L. Wolfe
W. R. Young*
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
C. L. Wolfe
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
*
†Email address for correspondence: [email protected]
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Abstract

We consider the linear stability of an inviscid parallel shear flow of air over water with gravity and capillarity. The velocity profile in the air is monotonically increasing upwards from the sea surface and is convex, while the velocity in the water is monotonically decreasing from the surface and is concave. An archetypical example, the ‘double-exponential’ profile, is solved analytically and studied in detail. We show that there are two types of unstable mode which can, in some cases, co-exist. The first type is the ‘Miles mode’ resulting from a resonant interaction between a surface gravity wave and a critical level in the air. The second unstable mode is an interaction between surface gravity waves and a critical level in the water, resulting in the growth of ripples. The gravity–capillary waves participating in this second resonance have negative intrinsic phase speed, but are Doppler shifted so that their actual phase speed is positive, and matches the speed of the base-state current at the critical level. In both cases, the Reynolds stresses of an exponentially growing wave transfer momentum from the vicinity of the critical level to the zone between the crests and troughs of a surface wave.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Critical layers Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 739 , 25 January 2014 , pp. 276 - 307
Copyright
©2013 Cambridge University Press 

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