Solution of Laplace’s equations for the M 2 tide in the world oceans

Abstract
Laplace’s tidal equations were solved for the M 2 tide in a realistic model of the world oceans based on the observed topography of the ocean bottom. When friction was neglected, the theoretical tides came out high, but of the right order of magnitude. The tidal heights proved to be sensitive to changes in the configuration of the coastline, indicating a state of near resonance. However, the positions of the amphidromic centres remained stable and were located close to the empirically deduced amphidromes. A fine computational grid was required to assure numerical convergence of the solution. A solution was also carried out with friction included, on the assumption of a bottom frictional force proportional to the first power of the velocity. For a coefficient of friction appropriate to tidal currents in shallow waters, the theoretical tidal heights came out in fair agreement with the observed values. With friction included, the solution proved insensitive to changes in the configuration of the coastline, and the numerical convergence was also improved. The resulting rate of tidal dissipation was higher than the observationally deduced values, but the excess is within the uncertainty of the latter. Our solution for the tide is given in figure 10, and the corresponding tidal current chart in figure 12. Figure 11 gives a comparison of the Middle South Atlantic amphidrome, which was derived from theory, with tidal observations in island stations.

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