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Nonlinear travelling internal waves with piecewise-linear shear profiles

Published online by Cambridge University Press:  12 October 2018

K. L. Oliveras Open the ORCID record for K. L. Oliveras [Opens in a new window]  and
C. W. Curtis
K. L. Oliveras*
Affiliation:
Mathematics Department, Seattle University, Seattle, WA 98122, USA
C. W. Curtis
Affiliation:
Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA
*
†Email address for correspondence: oliveras@seattleu.edu
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Abstract

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

JFM classification

Instability: Instability Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 856 , 10 December 2018 , pp. 984 - 1013
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Copyright
© 2018 Cambridge University Press 

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