Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T20:31:45.329Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M., Fokas, A. & Musslimani, Z. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.Google Scholar
Akers, B. F., Ambrose, D., Pond, K. & Wright, J. 2016 Overturned internal capillary–gravity waves. Eur. J. Mech. (B/Fluids) 57, 143–151.Google Scholar
Appel, J. 2004 Oceanic internal wavs and solitons. In Synthetic Aperture Radar Marine User’s Manual, pp. 189–206. U.S. Department of Commerce.Google Scholar
Ashton, A. & Fokas, A. 2011 A non-local formulation of rotational water waves. J. Fluid Mech. 689, 129–148.Google Scholar
Choi, W. 2009 Nonlinear surface waves interacting with a linear shear current. Maths Comput. Simul. 80 (1), 29–36.Google Scholar
Constantin, A. 2011 Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. SIAM.Google Scholar
Craig, W., Guyenne, P. & Kalisch, K. 2005a Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Maths 58, 1587–1641.Google Scholar
Craig, W., Guyenne, P., Nicholls, D. P. & Sulem, C. 2005b Hamiltonian long–wave expansions for water waves over a rough bottom. Proc. R. Soc. Lond. A 461, 839–873.Google Scholar
Curtis, C. & Deconinck, B. 2010 On the convergence of Hill’s method. Maths Comput. 79, 169–187.Google Scholar
Curtis, C., Oliveras, K. & Morrison, T. 2016 Shallow waves in density stratified shear currents. Eur. J. Mech. (B/Fluids) (accepted).Google Scholar
Dalrymple, R. 1974 Water waves on a bilinear shear current. In Proc. 14th Conf. on Coastal Engng. (ed. Edge, B.), pp. 626–641. American Society of Civil Engineers.Google Scholar
Deconinck, B. & Kutz, J. 2006 Computing spectra of linear operators using the Floquet–Fourier–Hill method. J. Comput. Phys. 219, 296–321.Google Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.Google Scholar
Deconinck, B. & Trichtchenko, O.2016 High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. DCDS-B (to appear).Google Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary-gravity waves. Annu. Rev. Fluid Mech. 31, 301–346.Google Scholar
Farmer, D. & Armi, L. 1999 The generation and trapping of solitary waves over topography. Science 283, 188–191.Google Scholar
Fokas, A. S. 2008 A Unified Approach to Boundary Value Problems, vol. 78. SIAM.Google Scholar
Francius, M. & Kharif, C. 2006 Three-dimensional instabilities of periodic gravity waves in shallow water. J. Fluid Mech. 561, 417–437.Google Scholar
Grue, J., Jensen, A., Rusøas, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.Google Scholar
Haut, T. & Ablowitz, M. 2009 A reformulation and applications of interfacial fluids with a free surface. J. Fluid Mech. 631, 375–396.Google Scholar
Helfrich, K. & Melville, W. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.Google Scholar
Hur, V. & Johnson, M. A. 2015 Modulational instability in the whitham equation with surface tension and vorticity. Nonlinear Anal. Theory Meth. Applics. 129, 104–118.Google Scholar
Ko, J. & Strauss, W. 2008 Effect of vorticity on steady water waves. J. Fluid Mech. 608, 197–215.Google Scholar
Luke, J. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.Google Scholar
McLean, J. W. 1982a Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331–341.Google Scholar
McLean, J. W. 1982b Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
Moler, C. & Stewart, G. 1973 An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256.Google Scholar
Oliveras, K., Sprenger, P. & Vasan, V.2016 The instability of traveling waves with vorticity: Part I, infinite depth (in preparation).Google Scholar
Oliveras, K. & Vasan, V. 2013 A new equation describing travelling water waves. J. Fluid Mech. 717, 514–522.Google Scholar
Osborne, A. & Burch, T. 1980 Internal solitons in the Andaman Sea. Science 208, 451–460.Google Scholar
Pullin, D. & Grimshaw, R. 1983 Interfacial progressive gravity waves in a two-layer shear flow. Phys. Fluids 26, 1731–1739.Google Scholar
Pullin, D. & Grimshaw, R. 1986 Stability of finite-amplitude interfacial waves. Part 3. The effect of basic current shear for one-dimensional instabilities. J. Fluid Mech. 172, 277–306.Google Scholar
Pullin, D. & Grimshaw, R. 1988 Finite-amplitude solitary waves at the interface between two homogeneous fluids. Phys. Fluids 31, 3550–3559.Google Scholar
da Silva, A. T. & Peregrine, D. 1988 Steep, steady surface waves on wter of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302.Google Scholar
Simmen, J. & Saffman, P. 1985 Steady deep-water waves on a linear shear current. Stud. Appl. Maths 75, 35–57.Google Scholar
Swan, C., Cummins, I. & James, R. 2001 An experimental study of two-dimensional surface water waves propagating on depth-varying currents. Part 1. Regular waves. J. Fluid Mech. 428, 273–304.Google Scholar
Thomas, R., Kharif, C. & Manna, M. 2012 A nonlinear schrödinger equation for water waves on finite depth with constant vorticity. Phys. Fluids 24 (12), 127102.Google Scholar
Vanden-Broeck, J. 1994 Steep solitary waves in water of finite depth with constant vorticity. J. Fluid Mech. 274, 339–348.Google Scholar
Vasan, V. & Oliveras, K. 2014 Pressure beneath a traveling wave with constant vorticity. Discrete Continuous Dyn. Syst. 34, 3219–3239.Google Scholar