10.3: Tides on an Ocean Covered Earth

We can understand many features of tides and their variability by examining the theoretical effect of the sun and moon on a hypothetical planet Earth entirely covered by deep oceans with no friction between water and the seafloor. This approach is the basis of equilibrium tide theory.

The Fundamental Equilibrium Tides

Earth spins on its axis once every 24 h. At the same time, the moon orbits the Earth in the same direction as the Earth’s spin, but much more slowly (Fig. 10-9). By the time the Earth completes one rotation (24 h), the moon has moved forward a little in its orbit. The Earth must turn for an additional 49 min before it “catches up” with the moon. Hence, 24 h and 49 min elapse between successive times at which the moon is directly overhead at a specific location on the Earth’s surface. This is why the moon rises and sets almost an hour later each night.

On a hypothetical planet covered by oceans and without friction, we can consider what would be the effect of the moon alone during a solar day (Fig. 10-9). The moon would create upward bulges on the oceans at points aligned directly toward and directly away from the moon. As the Earth spun, the tidal forces would continue to pull water toward these points. The tide bulges would remain aligned with the moon as the Earth rotated, so the bulges would appear to migrate around the Earth each day. However, the moon does not always orbit directly over the equator. Instead, the angle between the moon’s orbital plane and the equator, called the declination, varies with time during the lunar month and on longer timescales (Fig. 10-9a). For now, we need only consider the much simplified case of the moon at its maximum declination to see how diurnal, semidiurnal, and mixed tides might result (Fig. 10-9b).

Figure 10-9. A diagram with the moon and an ocean-covered Earth illustrates the principal features of the tides. In this figure the continents are shown only to make it easier to identify Earth’s orientation. (a) The angle of the moon’s orbit with respect to the equator (the declination) oscillates regularly. The maximum declination is about 25°. (b) The tide bulges caused by the moon’s tidal forces are shown here with the declination and size of the bulges greatly exaggerated, and the effects of the continents ignored. (c) Looking down from above the North Pole at the Earth and moon, we would see that the moon moves through a small segment of its orbit each time the Earth rotates. Hence, the moon is “overhead” at any point on Earth a little later each day so the tidal day is 24h 49min long. (d) In the simplified system depicted in (b), an observer at latitude A would see a diurnal tide, an observer at C would see a semidiurnal tide, and observers anywhere else on Earth, such as B and D, would see mixed tides. As discussed later in the chapter, the tides are more complex than shown in this figure.

Imagine an observer standing at a point along latitude B in Figure 10-9b and rotating with the Earth. When the observer is at point B1 there is a high tide. As the Earth rotates, the observer passes through the low tide at the back of the Earth (B₂) after the Earth has rotated a little more than 90°, or a little more than 6 h later. After 12 h and 24½ min, the observer passes a second high tide (B₃), but it is not as high as the original high tide. B₃ is further from AM than B₁ is from TM. Thus, as the bulge passes the longitude of this location (B₃), the highest point of the bulge is farther away from the observer than it was when the bulge passed the longitude of the observer’s original position (B₁). The observer rotates farther with the Earth, passing a second low tide after about 18 hr (B₄), and another high tide after 24 h and 49 min (B₁). The extra 49 min are due to the moon’s progression in its orbit during the day (Fig. 10-9c).

We can follow the tidal patterns observed at points A, C, and D in a similar way. A diurnal tide is observed at A, a semidiurnal tide at C, and mixed tides at B and D (Fig. 10-9d). Note that, in this simplified model of tides, pure semidiurnal tides occur only at the equator. Note also that, in mixed tides, low tides do not occur exactly midway between high tides.

In the simple model shown in Figure 10-9, low tide would always be the same height at any specific location. However, the tide records in Figure 10-6 and Figure 10-8 show that this is not true. The reason is that the simplified system in Figure 10-9 does not include solar tides that are added to lunar tides or the effects of continents, both of which create much greater complexity in the actual tides on the Earth.

The Origin of Spring and Neap Tides

The sun exerts tidal forces on the Earth that are about half as strong as the lunar tidal forces. We can see how the solar and lunar tides interact by again considering a simplified ocean-covered Earth. Tide bulges are created by both the sun and the moon (Fig. 10-10). The tidal height at any point on our model Earth is the sum of the tidal heights of the lunar and solar tides. The moon’s orbit around the Earth and the Earth’s orbit around the sun are not quite in the same plane, but they are nearly so. We can ignore this small angle for now.

Figure 10-10. A simplified depiction of the Earth-moon-sun system shows how the solar and lunar tides interact to create spring and neap tides. (a) At the new moon, spring tides with a greater tidal range (higher high and lower low tides) occur because the tide bulges caused by the moon and the sun (both greatly exaggerated here) are aligned so that their maxima are at the same locations. (b) At the first-quarter moon, neap tides occur because the moon’s bulge and the sun’s bulge are aligned at 90° to each other and the maxima of the moon’s tide coincide with the minima of the sun’s tide. (c) At the full moon, spring tides occur again as the moon’s and the sun’s bulges are again aligned with their maxima at the same locations. (d) At the fourth-quarter moon, neap tides occur again as the moon’s and the sun’s bulges are again aligned at 90° to each other and the maxima of the moon’s tide coincide with the minima of the sun’s tide. The observed spring and neap sequences on the Earth are more complicated for various reasons, including that the orbits of the moon and sun are not exactly in the same plane as this simplified figure depicts. These orbits are inclined to each other, and the inclination changes with time.

Figure 10-10a shows the relative positions of the sun, moon, and Earth at a new moon. Because the entire disk of the moon is in shade as seen from the Earth, the moon must be between the sun and the Earth. Lunar eclipses occur when the three bodies are precisely aligned. When the alignment is not exact, we see the moon as a thin sliver of light. This is a new moon. At a new moon, we can see that the high-tide bulges of the solar and lunar tides are located at the same point on the Earth. The lunar high-tide bulge directly under the moon and the solar high-tide bulge directly under the sun combine in the same location, and the lunar and solar high tide bulges on the side of Earth directly away from the sun and moon also coincide. Finally, the locations on Earth where the low tide areas of the solar and lunar tide bulge also coincide. Thus, at a new moon the solar and lunar high tides are added together and the solar and lunar low tides are added together, producing the maximum tidal range with the highest high tides and also the lowest low tides during the lunar month. These are spring tides.

Each day, the moon’s movement in its orbit is the equivalent of the angle through which the Earth rotates in about 49 minutes. Approximately 7½ (7.38) days after full moon, the moon, Earth, and sun are aligned at right angles (Fig. 10-10b). The right half of the moon (as seen from the Northern Hemisphere) is now lit by the sun. The other part of the moon that is lit is hidden from an observer on the Earth. This is the moon’s first quarter. (The lunar month is counted from a new moon.) At the first-quarter moon, the lunar tide bulge coincides with the low-tide region of the solar tide, and vice versa. Because the tides are additive, the low tide of the solar tide partially offsets and reduces the height of the lunar high tide, and the high tide of the solar tide partially offsets the lunar low tide. The tidal range is therefore smaller than at full moon. These tides are the neap tides.

Fifteen days after a new moon, the moon is directly between the Earth and the sun (Fig. 10-10c). Now the moon is entirely lit by the sun as seen from the Earth and there is a full moon. At full moon, the solar and lunar tide bulges again coincide and there is a second set of spring tides for the month. These tides are equal in height and range to the spring tides that occurred 15 days previously at new moon. Similarly, a little more than 22 days (22.14) after new moon, there is a third-quarter moon. The left side of the moon is lit for Northern Hemisphere observers (Fig. 10-10d). The solar and lunar tides again offset each other, and there is a second set of neap tides.

After about 29½ (29.53) days, the moon is new again, we have a new set of spring tides and begin a new lunar month. From this simple model, we can see why we have two sets of spring tides and two sets of neap tides in each 29½-day lunar month (Fig. 10-9).

Other Tidal Variations

From the simple model we have been discussing, we have seen that the height of the tide due to the moon varies on a cycle that is 24 h 49 min long. High tides actually occur every 12 h 24½ min because there are two bulges. Similarly, the solar tide varies on a 24-h cycle, and high tides occur every 12 h. The motions of the moon and the Earth in their orbits are much more complex than the simple model suggests, and tidal variations occur on many other timescales. These additional variations are generally smaller than the daily or monthly variations, but they must be taken into account when precise tidal calculations are made. They are due to periodic changes in the distances between the Earth and the sun and between the Earth and the moon, and the various declinations. Declinations are the angles between the Earth’s plane of orbit around the sun, the Earth’s axis of spin, and the moon’s orbital plane. Table 10-1 lists some of the more important variations of Earth–moon–sun orbits, along with the periodicity of the partial tides that they cause.

TABLE 10-1. Selected Tidal Components 

Tidal Component	Frequency	Period (h)	Relative Amplitude	Description
Principal lunar	Twice per day	12-42	100.0	Rotation of Earth relative to moon
Principal solar	Twice per day	12-00	46.6	Rotation of Earth relative to sun
Larger lunar ecliptic	Twice per day	12-66	19.2	Variation in moon-Earth distance
Lunisolar semidiurnal	Twice per day	11.97	10.7	Changes in declination of sun and moon
Lunisolar diurnal	Daily	23.93	58.4	Changes in declination of sun and moon
Principal lunar diurnal	Daily	25.82	41.5	Rotation of Earth relative to moon
Principal solar diurnal	Daily	24.07	19.4	Rotation of Earth relative to sun
Lunar fortnightly	Biweekly	327.9	17.2	Moon’s orbit declination variation from zero to maximum and back to zero
Lunar monthly	Monthly	661.3	9.1	Time for moon-Earth distance to change from minimum to maximum and back
Solar semiannual	Semiannually	4382.4 (182.6 days)	8.0	Time for sun’s declination to change from zero to maximum and back to zero
Anomalistic year	Annually	8766.2  (365.2 days)	1.3	Time for Earth-sun distance to change from minimum to maximum and back