The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography

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The method of decomposing surface and internal tides determines the expression for internal tide energy, energy flux, and energy conversion. The de facto standard is to define surface tides as depth-averaged pressure and horizontal velocity and internal tides as the residuals. This decomposition, which is equivalent to projecting motion onto vertical modes that obey a rigid lid, is known to produce spurious energy conversion CS through movement of the free surface. Here, motion is instead projected onto modes that obey a linear, free-surface boundary condition. The free-surface modes are shown to obey a more complicated orthogonality condition than rigid-lid modes but are still straightforward to calculate numerically. The resulting decomposition (i) completely eliminates spurious energy conversion CS and (ii) leads to a more precise expression for topographic internal tide generation C, which now depends on horizontal gradients in the vertical structure of the surface tide. Numerical simulations and rough global estimates indicate that corrections to C are a maximum of a few percent. However, CS produces spurious energy flux divergences/convergences in the open ocean, which are the same order of magnitude [O(1-10) mW m-2] as open-ocean internal tide energy dissipation.

Original languageEnglish (US)
Pages (from-to)3777-3788
Number of pages12
JournalJournal of Physical Oceanography
Issue number12
StatePublished - 2016

Bibliographical note

Funding Information:
This work was motivated and guided by a discussion with Bill Young and was supported by NSF Grant OCE1434352 and NASA Grant NNX16AH75G. Eric Kunze provided helpful comments on a preliminary draft of the manuscript. MATLAB functions to determine modes and eigenspeeds are available online (at; smkelly/). Two anonymous reviewers provided comments that improved the manuscript.


  • Internal waves
  • Tides


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