10.4: Tides in the Earth's Oceans

Tides in the Earth’s oceans behave somewhat differently from those on the hypothetical ocean-covered Earth. For example, on the model Earth, pure semidiurnal tides would occur only at the equator (Fig. 10-9), except when the moon’s declination was zero, and then all points on the Earth would have pure semidiurnal tides. However, on the real Earth semidiurnal tides occur at places other than the equator (Figs. 10-6, 10-7, 10-8).

Four major interrelated factors alter the Earth’s tides from the equilibrium model: the Earth’s landmasses, the shallow depth of the oceans relative to the wavelength of tides, the latitudinal variation of orbital velocity due to the Earth’s spin around its axis, and the Coriolis effect (CC12). When these factors are included in tide calculations, the resulting tides are called “dynamic tides.”

Effects of Continents and Ocean Depth

In discussing the factors that affect tidal motions, it is convenient to envision tide waves moving across the Earth’s surface from east to west, even though, in fact, it is the Earth that is spinning. The peaks of the tide waves remain fixed in their orientation toward and away from the moon or sun, moving only slowly as the moon, Earth, and sun move in their orbits.

The presence of landmasses prevents the tide wave from traveling around the world. The continents are generally oriented north–south and bisect the oceans. The equilibrium tide wave moves from east to west. When the tide wave encounters a continent, its energy is dissipated or reflected, and the wave must be “restarted” on the other side of the continent. Because continents, landmasses, and the ocean basins are complicated in shape, the tide wave is dispersed, refracted, and reflected in a complex and variable way within each ocean basin.

The average depth of the oceans is about 4 km, and the maximum depth is only 11 km. In contrast, the wavelength of the tide wave is one-half of the Earth’s circumference, or about 20,000 km at the equator. Hence, the water depth is always considerably less than 0.05 times the wavelength of the tide wave (that is, the water depth is considerably less than L/20), and the tide wave acts as a shallow-water wave (Chap. 9). The speed of a shallow-water wave is controlled only by the water depth. In the average depth of the oceans (4 km), the tide wave speed is approximately 700 km•h^–1. The equilibrium model tide wave always travels across the Earth’s surface aligned with the movements of the moon and sun. However, to travel across the Earth’s surface and remain always exactly lined up with the moon (or sun), the tide wave would have to travel at the same speed as the Earth’s spin. At the equator, the wave would have to travel at about 1620 km•h^–1. Because it is a shallow-water wave that can travel at only 700 km•h^–1, the tide wave at the equator must lag behind the moon as the moon moves across the Earth’s surface.

The interaction between the moving tide wave that lags behind the moon’s orbital movement and the tendency for a new tide wave to form ahead of the lagging wave is complex. However, the result is that the tide wave lags behind the moon’s orbital movement, but not by as much as it would if it were not continuously recreated. The tide wave is said to be a forced wave because it is forced to move faster than an ideal shallow-water wave.

Because the tide wave is a shallow-water wave and its speed depends on the water depth, the tide wave is refracted in the same way that shallow-water wind waves are refracted as they enter shallow water. Dynamic-tide theory therefore must include the refraction patterns created by the passage of tide waves over oceanic ridges, trenches, shallow continental shelves, and other large features of the ocean basins.

Latitudinal Variation of the Earth’s Spin Velocity

The tidal time lag changes with latitude because the orbital velocity (due to the Earth’s spin) of points on the Earth’s surface decreases with latitude (Fig. 10-11). At latitudes above about 65°, the shallow-water wave speed is equal to or greater than the orbital velocity. Hence, there is no tidal time lag at these latitudes, and the tide wave even tends to run ahead of the moon’s orbital movement. Another complication in calculating the tide wave time lag is the fact that the Earth’s axis is usually tilted relative to the moon’s orbit (and to the Earth’s orbit with the sun). Therefore, even on a planet with no continents, the tide waves would not normally travel exactly east–west.

Figure 10-11. The tide wave moves across the Earth’s surface as the Earth spins in relation to the moon and sun. If it were to do so exactly in time with the Earth’s spin, it would have to move at different speeds across the Earth’s surface at different latitudes. This is because the orbital velocity of points on the Earth’s surface due to the Earth’s spin is at a maximum at the equator and decreases with increasing latitude until it is zero at the poles.

Coriolis Effect and Amphidromic Systems

Tide waves are shallow-water waves in which water particles move in extremely flattened elliptical orbits. In fact, the orbits are so flattened that we can consider the water movements to be horizontal. Water moving within a tide wave is subject to the Coriolis effect (CC12). One important consequence of the Coriolis deflection of tide waves is that a unique form of standing wave can be created in an ocean basin of the correct dimensions. This type of standing wave is called an amphidromic system, in which the high- and low-tide points (the wave crest and trough, respectively) move around the basin in a rotary path—counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere.

Figure 10-12 shows how an amphidromic system is established in a wide Northern Hemisphere basin. In the Northern Hemisphere, a standing-wave crest (antinode) enters the basin at its east side. As water flows westward, it is deflected cum sole (meaning “with the sun”) to the north, causing a sea surface elevation on the north side of the basin as the wave height at the east boundary decreases behind the crest. Water now flows toward the south with the north-to-south pressure gradient created by the sea surface elevation on the north side of the basin. As water flows south, it is deflected cum sole to the west. The sea surface elevation continues to move around the basin counterclockwise in this way until it returns to the east side of the basin. If the standing-wave crest returns to the east side of the basin after exactly 12 h and 24½ min (or any multiple thereof), it will meet the crest of the next lunar tide wave and the oscillation is then said to be tuned.

Figure 10-12. Development of an amphidromic standing-wave oscillation in response to the Coriolis effect in the Northern Hemisphere. (a) An amphidromic system is created in the principal solar semidiurnal or diurnal component of the tide when the tide wave enters a basin and is deflected by the Coriolis effect so that it rotates around the basin and returns to the entrance exactly 12 or 24 h later. (b) In an idealized Northern Hemisphere amphidromic system, the tidal range is the same for all places on any circle centered on the amphidromic point. The line along which the low tide or high tide is located progresses around the basin counterclockwise (clockwise in the Southern Hemisphere). In this plan view, high tide or low tide occurs along each of the radial lines at the indicated number of hours it occurs along the line labeled “0 h.” (c) A cross-sectional view of the basin shows the up-and-down motion about the node upon which the rotation around the basin is superimposed.

One important characteristic of amphidromic systems is the amphidromic point (node) near the center of the basin, at which the tidal range is zero (Fig. 10-12). Amphidromic systems are established in all the ocean basins (Fig. 10-13). However, different amphidromic systems are set up for each tidal component (Table 10-1) because the components have different wavelengths. Hence, a location in an ocean basin that is an amphidromic point for one component of the tide will have a zero tidal range for that component, but will have tides generated by tidal components with other periods. In addition, some basins tune more easily with a particular tidal component. In parts of such basins, the tidal range due to that tidal component can be enhanced in relation to the ranges due to other components. This effect explains the presence of dominant diurnal and semidiurnal tides at many locations where they would not be predicted by equilibrium tide theory.

Figure 10-13. Amphidromic systems of the principal lunar semidiurnal tidal component in the Earth’s oceans. The lines show the location of the tidal maximum. The numbers indicate the number of hours that elapse as the maximum tide (crest) travels from the line labeled 0 to the indicated line. Notice that the crest of the tide wave moves in rotary amphidromic systems in most of the ocean basins, and that the rotation is counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. Because it is a shallow-water wave, the tide wave is also refracted as water depth changes, resulting in some of the complex bending of the wave in certain parts of the oceans.