We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas ( A Unified Approach to Boundary Value Problems , 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces a strong coupling between the carrier wave and the mean-surface displacement. The effects of the background shear on the modulational instability of plane waves is also studied, where it is shown that shear can suppress instability, although not for all carrier wavelengths in the presence of surface tension. These results expand upon the findings of Thomas et al. ( Phys. Fluids , vol. 24 (12), 2012, 127102). Using a modification of the generalized Lagrangian mean theory in Andrews & McIntyre ( J. Fluid Mech. , vol. 89, 1978, pp. 609–646) and approximate formulas for the velocity field in the fluid column, explicit, asymptotic approximations for the Lagrangian and Stokes drift velocities are obtained for plane-wave and Jacobi elliptic function solutions of the nonlinear Schrödinger equation. Numerical approximations to particle trajectories for these solutions are found and the Lagrangian and Stokes drift velocities corresponding to these numerical solutions corroborate the theoretical results. We show that background currents have significant effects on the mean transport properties of waves. In particular, certain combinations of background shear and carrier wave frequency lead to the disappearance of mean-surface mass transport. These results provide a possible explanation for the measurements reported in Smith ( J. Phys. Oceanogr. , vol. 36, 2006, pp. 1381–1402). Our results also provide further evidence of the viability of the modification of the Stokes drift velocity beyond the standard monochromatic approximation, such as recently proposed in Breivik et al. ( J. Phys. Oceanogr. , vol. 44, 2014, pp. 2433–2445) in order to obtain a closer match to a range of complex ocean wave spectra.
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Nonlinear travelling internal waves with piecewise-linear shear profiles
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.
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- Award ID(s):
- 1715039
- NSF-PAR ID:
- 10220371
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 856
- ISSN:
- 0022-1120
- Page Range / eLocation ID:
- 984 to 1013
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2018.679
