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Lecture

The Dynamic Theory of Tides

Amphidromic Systems(Click for larger view). In this image of the amphidromic systems of the world, the white lines are similar to the co-tidal lines of figure 9.14 in your text. The colors indicate the approximate height of high tides in coastal and open ocean locations. The black arrows indicate the direction of movement of the high tide wave crest through the ocean basin. (R. Ray, TOPEX/Poseidon: Revealing Hidden Tidal Energy GSFC, NASA)

Tides in the actual Earth’s oceans behave a bit differently than in our hypothetical ocean-covered Earth due to the placement of landmasses, the shallow depth of water relative to wavelength of tides, the latitudinal variation of the rotational velocity of Earth, and the Coriolis Effect. When we take these factors into effect we discover the dynamic theory of tides.

Ocean Depth and Rotational Velocity. Because tides are such long wavelength waves, they behave as shallow water waves. This means that all of the water in the oceans are effected by tides – from the water at the surface to the water at the deepest depths. Recall that the speed of a shallow water wave is directly proportional to the water depth – because the seafloor acts to slow down waves.

Based on our equilibrium theory of tides, ocean water always stayed in direct line with the sun and moon, meaning that in theory the waves traveled at the speed of rotation of the Earth. However, if we calculate the maximum speed tides can reach (being shallow water waves) we find that they travel slightly slower. This means that the bulge created by the gravitational pulls and centripetal force actually lags somewhat behind the moon as the moon orbits the earth. So when the moon is directly overhead a certain location, that location is not experiencing its high tide at that moment, it comes later.

However, at higher latitudes we find that the tides do not lag behind the sun and the moon. This is because the rotational velocity of the Earth decreases with latitude and even though the tides still interact with the seafloor, they are able to “keep up”. This lag time is shown  in figure 9.14 shows a map of these systems using co-tidal lines – lines showing the delay in time between when the moon is directly overhead and the actual high tide occurs.

Continents and the Coriolis Effect. Landmasses on Earths surface prevent the Earth from simply rotating into and out of tidal bulges. When the tidal bulge “hits” the side of a continent some of its energy is dissipated, and some of the energy is reflected back into the ocean basin. This reflection, coupled with the Coriolis Effect causes water to be rotated around an ocean basin, much in the way water would rotate around a cup if you move the cup back and forth.

This oscillation of water around an ocean basin is called an amphidromic system and causes the high tide wave crests and low tide wave troughs to move around ocean basins in a clockwise (S. Hemisphere) or counterclockwise (N. Hemisphere) pattern. In the center of this rotating wave is a node where the tidal range is zero.  These systems have been found to occur in all the ocean basins except the Southern Ocean, where tide crests and troughs simply move east to west.

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