Coastal Trapped Waves: Normal Modes, Evolution Equations, and Topographic Generation

Samuel M. Kelly, Sebastine Ogbuka

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Coastal trapped waves (CTWs) transport energy along coastlines and drive coastal currents and upwelling. CTW modes are nonorthogonal when frequency is treated as the eigenvalue, preventing the separation of modal energy fluxes and quantification of longshore topographic scattering. Here, CTW modes are shown to be orthogonal with respect to energy flux (but not energy) when the longshore wavenumber is the eigenvalue. The modal evolution equation is a simple harmonic oscillator forced by longshore bathymetric variability, where downstream distance is treated like time. The energy equation includes an expression for modal topographic scattering. The eigenvalue problem is carefully discretized to produce numerically orthogonal modes, allowing CTW amplitudes, energy fluxes, and generation to be precisely quantified in numerical simulations. First, a spatially uniform K1 longshore velocity is applied to a continental slope with a Gaussian bump in the coastline. Mode-1 CTW generation increases quadratically with the amplitude of the bump and is maximum when the bump’s length of coastline matches the natural wavelength of the CTW mode, as predicted by theory. Next, a realistic K1 barotropic tide is applied to the Oregon coast. The forcing generates mode-1 and mode-2 CTWs with energy fluxes of 6 and 2 MW, respectively, which are much smaller than the 80 MW of M2 internal-tide generation in this region. CTWs also produce 1-cm sea surface displacements along the coast, potentially complicating the interpretation of future satellite altimetry. Prospects and challenges for quantifying the global geography of CTWs are discussed.

Original languageEnglish (US)
Pages (from-to)1835-1848
Number of pages14
JournalJournal of Physical Oceanography
Issue number8
StatePublished - Aug 2022

Bibliographical note

Funding Information:
Acknowledgments. Kelly and Ogbuka were supported by the National Science Foundation Grant OCE-1635560. Ogbuka was also supported by the University of Minnesota Duluth Physics and Astronomy Department. This work benefited greatly from conversations with Ruth Musgrave, who also helped us setup the MITgcm simulations. Two anonymous reviewers improved the paper with insightful comments.

Publisher Copyright:
© 2022 American Meteorological Society.


  • Coastal flows
  • Inertia-gravity waves
  • Internal waves
  • Kelvin waves
  • Tides
  • Topographic effects
  • Waves, oceanic


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