On the Formation of Coherent Vortices beneath Nonbreaking Free-Propagating Surface Waves

Wu-ting Tsai Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan

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Guan-hung Lu Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan

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Jheng-rong Chen Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan

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Albert Dai Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan

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William R. C. Phillips Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria, Australia

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Abstract

Numerical simulations of monochromatic surface waves freely propagating over an initially quiescent flow field are conducted and found to reveal an array of quasi-streamwise vortices of alternating orientation in a manner akin to that of Langmuir circulation beneath wind-driven surface waves. A linear instability analysis of the wave-averaged Craik–Leibovich (CL) equation is then conducted to determine whether the structures in the simulations can be explained by the Craik–Leibovich type 2 (CL2) instability, which requires the presence of spanwise-independent drift and mean shear of the same sign. There is no imposed shear in the simulations, but they confirm the theoretical analysis of Longuet-Higgins that an Eulerian-mean shear with a magnitude comparable to that of Lagrangian Stokes drift occurs at the edge of the surface boundary layer in the otherwise irrotational oscillatory flow. The spanwise wavelength of the least stable disturbance is found to be close to the spacing between predominant vortex pairs, which likely are excited by the CL2 instability.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author address: Wu-ting Tsai, wttsai@ntu.edu.tw

Abstract

Numerical simulations of monochromatic surface waves freely propagating over an initially quiescent flow field are conducted and found to reveal an array of quasi-streamwise vortices of alternating orientation in a manner akin to that of Langmuir circulation beneath wind-driven surface waves. A linear instability analysis of the wave-averaged Craik–Leibovich (CL) equation is then conducted to determine whether the structures in the simulations can be explained by the Craik–Leibovich type 2 (CL2) instability, which requires the presence of spanwise-independent drift and mean shear of the same sign. There is no imposed shear in the simulations, but they confirm the theoretical analysis of Longuet-Higgins that an Eulerian-mean shear with a magnitude comparable to that of Lagrangian Stokes drift occurs at the edge of the surface boundary layer in the otherwise irrotational oscillatory flow. The spanwise wavelength of the least stable disturbance is found to be close to the spacing between predominant vortex pairs, which likely are excited by the CL2 instability.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author address: Wu-ting Tsai, wttsai@ntu.edu.tw
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