Skip to main content Accessibility help
×
×
Home

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

  • David Henry (a1)

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.

Copyright

Corresponding author

Email address for correspondence: d.henry@ucc.ie

References

Hide All
Bennett, A. 2006 Lagrangian Fluid Dynamics. Cambridge University Press.
Constantin, A. 2006 The trajectories of particles in Stokes waves. Invent. Math. 166, 523535.
Constantin, A. 2011 Nonlinear Water Waves with Applications to Wave–Current Interactions and Tsunamis. (CBMS-NSF Conference Series in Applied Mathematics, vol. 81) , SIAM.
Constantin, A. 2012 An exact solution for equatorially trapped waves. J. Geophys. Res. 117, C05029.
Constantin, A. 2013 Some three-dimensional nonlinear equatorial flows. J. Phys. Oceanogr. 43, 165175.
Constantin, A. 2014 Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves. J. Phys. Oceanogr. 44, 781789.
Constantin, A. & Germain, P. 2013 Instability of some equatorially trapped waves. J. Geophys. Res. 118, 28022810.
Constantin, A. & Johnson, R. S. 2015 The dynamics of waves interacting with the equatorial undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311358.
Constantin, A. & Johnson, R. S. 2016 An exact, steady, purely azimuthal equatorial flow with a free surface. J. Phys. Oceanogr. 46, 19351945.
Constantin, A. & Strauss, W. 2010 Pressure beneath a Stokes wave. Commun. Pure Appl. Math. 53, 533557.
Cushman-Roisin, B. & Beckers, J.-M. 2011 Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic.
Fedorov, A. V. & Brown, J. N. 2009 Equatorial waves. In Encyclopedia of Ocean Sciences (ed. Steele, J.), pp. 36793695. Academic.
Genoud, F. & Henry, D. 2014 Instability of equatorial water waves with an underlying current. J. Math. Fluid Mech. 16, 661667.
Henry, D. 2008 On the deep-water Stokes flow. Int. Math. Res. Not. 22, 071.
Henry, D. 2013 An exact solution for equatorial geophysical water waves with an underlying current. Eur. J. Mech. (B/Fluids) 38, 1821.
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.
Hsu, H.-C. 2014 Some nonlinear internal equatorial flows. Nonlinear Anal. Real World Appl. 18, 6974.
Ionescu-Kruse, D. 2015 An exact solution for geophysical edge waves in the f-plane approximation. Nonlinear Anal. Real World Appl. 24, 190195.
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, 110123.
Johnson, G. C., McPhaden, M. J. & Firing, E. 2001 Equatorial Pacific ocean horizontal velocity, divergence, and upwelling. J. Phys. Oceanogr. 31, 839849.
Matioc, A.-V. 2012 An exact solution for geophysical equatorial edge waves over a sloping beach. J. Phys. A 45, 365501.
Matioc, A.-V. 2013 Exact geophysical waves in stratified fluids. Appl. Anal. 92, 22542261.
Mollo-Christensen, E. 1978 Gravitational and geostrophic billows: some exact solutions. J. Atmos. Sci. 35, 13951398.
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, 397411.
Nachbin, A. & Ribeiro-Junior, R. 2014 A boundary integral formulation for particle trajectories in Stokes waves. J. Discrete Continuous Dyn. Syst. A 34, 31353153.
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, 16871702.
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed