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On Langmuir circulation in shallow waters

Published online by Cambridge University Press:  04 March 2014

W. R. C. Phillips  and
A. Dai
W. R. C. Phillips*
Affiliation:
Department of Mathematics, Swinburne University of Technology, Hawthorn 3122, Australia Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2935, USA
A. Dai
Affiliation:
Department of Water Resources and Environmental Engineering, Tamkang University, Taipei 25137, Taiwan Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2935, USA
*
†Email address for correspondence: wrphilli@illinois.edu
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Abstract

The instability of shallow-water waves on a moderate shear to Langmuir circulation is considered. In such instances, specifically at the shallow end of the inner coastal region, the shear can significantly affect the drift giving rise to profiles markedly different from the simple Stokes drift. Since drift and shear are instrumental in the instability to Langmuir circulation, of key interest is how that variation in turn affects onset to Langmuir circulation. Also of interest is the effect on onset of various boundary conditions. To that end the initial value problem describing the wave–mean flow interaction which accounts for the multiple time scales of the surface waves, evolving shear and evolving Langmuir circulation is crafted from scratch, and includes the wave-induced drift and a consistent set of free-surface boundary conditions. The problem necessitates that Navier–Stokes be employed side by side with a set of mean-field equations. Specifically, the former is used to evaluate events with the shortest time scale, that is the wave field, while the mean field set is averaged over that time scale. This averaged set, the CLg equations, follow from the generalized Lagrangian mean equations and for the case at hand take the same form as the well-known CL equations, albeit with different time and velocity scales. Results based upon the Stokes drift are used as a reference to which those based upon drift profiles corrected for shear are compared, noting that the latter are asymptotic to the former as the waves transition from shallow to deep. Two typical temporal flow fields are considered: shear-driven flow and pressure-driven flow. Relative to the reference case, shear-driven flow is found to be destabilizing while pressure driven are stabilizing to Langmuir circulation. In pressure-driven flows it is further found that multiple layers, as opposed to a single layer, of Langmuir circulation can form, with the most intense circulations at the ocean floor. Moreover, the layers can extend into a region of flow beyond that in which the instability applies, suggesting that Langmuir circulation excited by the instability can in turn drive, as a dynamic consequence, contiguous albeit less intense Langmuir circulation. Pressure-driven flows also admit two preferred spacings, one closely in accord with observation for small-aspect-ratio Langmuir circulation, the other well in excess of observed large-aspect-ratio Langmuir circulation.

JFM classification

Instability: Instability Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 743 , 25 March 2014 , pp. 141 - 169
Copyright
© 2014 Cambridge University Press 

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