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Barotropic Tides
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Barotropic Tidal Dissipation

The barotropic tide is basically a large-scale wave with a wavelength of about 6000 km that sloshes around the ocean basins, while being forced by the gravitational forces of the moon and sun. The tidal elevation and current are defined by harmonic constants of amplitude and phase, e.g., η(t) = η cos(ωt - Gη) describes the time variation of sea-surface height at some position as a function of time given an amplitude (η) and phase (Gη). ω is one of the frequencies of the tide, e.g., the M2 frequency (period of about 12.4 hours). A global map of the M2 tidal elevation can be seen HERE. Note that the wave crests in the North Pacific are almost parallel to the Hawaiian Ridge. The waves carry an energy flux that is given by

(Fx, Fy)=(<p'u'>,<p'v'>)

where p is pressure (=ρgη) and <.> refers to a time average, so that

(Fx, Fy)= ½ρgη( u*cos(Gη - Gu), v*cos(Gη - Gv) )

where ρ is the density of seawater, g is gravitational acceleration, η is the amplitude of tidal elevation, u and v are amplitude of zonal and meridional current, respectively, and Gη,u,v are the phases of elevation, zonal current and meridional current.
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Barotropic tide energy flux derived from the TPXO.3 global tidal model. The HOME farfield array is shown. The triangle at the top of the figure is the RTE87 tomography experiment that first detected internal-tide radiation from the Hawaiian Ridge.

This equation was used to calculate the energy flux of the M2 tide for the region of the Hawaiian Ridge. Perhaps not surprisingly, the result (to the right) shows that much of the tidal energy is flowing around the Hawaiian Ridge. Some of this energy is lost at the Hawaiian Ridge to bottom friction, generation of internal waves, and other processes. The naive view of this energy loss is that a large energy flux approachs the Hawaiian Ridge and a small energy flux leaves the Ridge - with the difference in energy flux being the energy that is lost as the tidal wave (which is NOT a tsunami!) passes across the Hawaiian Ridge. The actual picture is more complicated than that, however, so that numerical models are actually used to calculate the divergence of energy flux at the Hawaiian Ridge.

The values for energy flux are rather large, and the divergence of energy flux is small and basically results from a difference of large flux values. Relatively small errors in energy flux can therefore result in large errors in the divergence of energy flux. One goal of the HOME farfield experiment is to measure the tidal elevations and currents to an accuracy sufficient to reduce the uncertainty in the divergence of energy flux. The accurate measurements of the harmonic constants for tidal elevation and currents by the HOME farfield instruments will be used to provide tight constraints on the tidal model, which will then give a more accurate estimate of energy lost from the barotropic tide at the Hawaiian Ridge.

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This material is based upon work supported by the National Science Foundation under Grant No. 9819527.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.