Hostname: page-component-cb9f654ff-h4f6x Total loading time: 0 Render date: 2025-08-31T11:46:06.868Z Has data issue: false hasContentIssue false
  • English
  • Français

Equatorially trapped nonlinear water waves in a $\unicode[STIX]{x1D6FD}$ -plane approximation with centripetal forces

Published online by Cambridge University Press:  31 August 2016

David Henry
David Henry*
Affiliation:
Department of Applied Mathematics, University College Cork, Cork, Ireland
*
†Email address for correspondence: d.henry@ucc.ie
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we present an exact and explicit nonlinear solution of a $\unicode[STIX]{x1D6FD}$ -plane approximation to the governing equations which retains all Coriolis terms. The solution represents an equatorially trapped wave propagating in the presence of a constant underlying background current. In particular, we show that retention of the (relatively) small-scale centripetal forces in the governing equations enables us to admit currents of any physically plausible magnitude in the background flow.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Stratified flows Geophysical and Geological Flows: Waves in rotating fluids

Information

Type
Rapids
Information
Journal of Fluid Mechanics , Volume 804 , 10 October 2016 , R1
Check for updates
Copyright
© 2016 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.)

Article purchase

Temporarily unavailable

References

Bennett, A. 2006 Lagrangian Fluid Dynamics. Cambridge University Press.Google Scholar
Constantin, A. 2006 The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535.Google Scholar
Constantin, A. 2011 Nonlinear Water Waves with Applications to Wave–Current Interactions and Tsunamis. (CBMS-NSF Conference Series in Applied Mathematics, vol. 81) , SIAM.Google Scholar
Constantin, A. 2012 An exact solution for equatorially trapped waves. J. Geophys. Res. 117, C05029.Google Scholar
Constantin, A. 2013 Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43, 165–175.CrossRefGoogle Scholar
Constantin, A. 2014 Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr. 44, 781–789.CrossRefGoogle Scholar
Constantin, A. & Germain, P. 2013 Instability of some equatorially trapped waves. J. Geophys. Res. 118, 2802–2810.Google Scholar
Constantin, A. & Johnson, R. S. 2015 The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358.CrossRefGoogle Scholar
Constantin, A. & Johnson, R. S. 2016 An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr. 46, 1935–1945.Google Scholar
Constantin, A. & Strauss, W. 2010 Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 53, 533–557.Google Scholar
Cushman-Roisin, B. & Beckers, J.-M. 2011 Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic.Google Scholar
Fedorov, A. V. & Brown, J. N. 2009 Equatorial waves. In Encyclopedia of Ocean Sciences (ed. Steele, J.), pp. 3679–3695. Academic.Google Scholar
Genoud, F. & Henry, D. 2014 Instability of equatorial water waves with an underlying current. J. Math. Fluid Mech. 16, 661–667.Google Scholar
Henry, D. 2008 On the deep-water Stokes flow. Int. Math. Res. Not. 22, 071.Google Scholar
Henry, D. 2013 An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. (B/Fluids) 38, 18–21.Google Scholar
Henry, D. & Sastre-Gómez, S. 2016 Mean flow velocities and mass transport for equatorially-trapped water waves with an underlying current. J. Math. Fluid Mech.; doi:10.1007/s00021-016-0262-9.CrossRefGoogle Scholar
Hsu, H.-C. 2014 Some nonlinear internal equatorial flows. Nonlinear Anal. Real World Appl. 18, 69–74.CrossRefGoogle Scholar
Ionescu-Kruse, D. 2015 An exact solution for geophysical edge waves in the f-plane approximation. Nonlinear Anal. Real World Appl. 24, 190–195.Google Scholar
Izumo, T. 2005 The equatorial current, meridional overturning circulation, and their roles in mass and heat exchanges during the El Niño events in the tropical Pacific Ocean. Ocean Dyn. 55, 110–123.Google Scholar
Johnson, G. C., McPhaden, M. J. & Firing, E. 2001 Equatorial Pacific ocean horizontal velocity, divergence, and upwelling. J. Phys. Oceanogr. 31, 839–849.2.0.CO;2>CrossRefGoogle Scholar
Matioc, A.-V. 2012 An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A 45, 365501.Google Scholar
Matioc, A.-V. 2013 Exact geophysical waves in stratified fluids. Appl. Anal. 92, 2254–2261.Google Scholar
Mollo-Christensen, E. 1978 Gravitational and geostrophic billows: some exact solutions. J. Atmos. Sci. 35, 1395–1398.2.0.CO;2>CrossRefGoogle Scholar
Moum, J. N., Nash, J. D. & Smyth, W. D. 2011 Narrowband oscillations in the upper equatorial ocean. Part I. Interpretation as shear instability. J. Phys. Oceanogr. 41, 397–411.Google Scholar
Nachbin, A. & Ribeiro-Junior, R. 2014 A boundary integral formulation for particle trajectories in Stokes waves. J. Discrete Continuous Dyn. Syst. A 34, 3135–3153.Google Scholar
Umeyama, M. 2012 Eulerian/Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Phil. Trans. R. Soc. Lond. A 370, 1687–1702.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar