The Energy Of The Tides

Tides as Waves

The movements of the ocean are traditionally classified into currents, waves, and tides. In one sense, however, tides are waves, albeit of a special kind. Unlike wind waves and swells, which are caused by the wind, and tsunamis, which are caused by earthquakes and similar disturbances, tides are caused by the gravitational pull of the moon and sun. They are waves nevertheless; to distinguish them from other waves, we shall call them tide waves.

A tide wave, as we noted in chapter 7, should not be confused with a "tidal wave"; the latter is a misnomer for a tsunami. Tide waves have some very special properties. They are controlled predominantly by the moon—the effect of the sun is only half as great—and because of this their period is equal to one-half a lunar day on average, or twelve hours and twenty-five minutes (a lunar day, twenty-four hours and fifty minutes, is the time it takes for the earth to rotate so that the moon makes one complete circuit, from a given compass direc-

tion on one day to the same compass direction on the following day). Because the lunar day is fifty minutes longer than the twenty-four-hour solar day, the tides come roughly fifty minutes later by the clock on each succeeding day. The wavelength of a tide wave is about 22,000 km, which is half the earth's circumference at the equator.

The reason these figures are not exact is that the tides are affected by the sun as well as by the moon, and the relative positions of sun and moon are always changing. Moreover, a tide wave is a shallow-water wave because its wavelength is so great relative to the depth of the ocean that its speed varies from place to place as it travels. It slows down if the water becomes shallow and speeds up again where it deepens. Because of this, high tide comes late wherever a shallow continental shelf extends a long way to seaward of the shoreline.

The range of the tide (the difference in water level between high and low tide) also varies from place to place. The range depends on the shape of the shoreline and the pattern of the depth contours: it is often especially great in deep, narrow inlets where an entering tide wave piles up and has nowhere to spread out.

Normally there are two high tides and two low tides per lunar day; that is, the tides are semidiurnal. The highs are not equally high, nor are the lows equally low. The relative heights vary from tide to tide and from day to day, depending on the distances of the moon and the sun to the north and to the south of the equator, which vary from day to day all through the year. As shore dwellers know well, at most places the sequence of tides in a lunar day is higher high, higher low, lower high, and lower low. But in a few places, for a few days in each lunar month, the tides are diurnal, with only one high tide and one low tide per lunar day. This is most surprising when you consider how the tides are caused, as we do in the next section. Diurnal tides happen when the low tide between two high tides is itself so high that the high tides before and after it seem to be one long, uninterrupted high tide; likewise, two succeeding low tides seem to be one long low tide. Two further complications deserve mention.

First, there are internal tide waves, in other words, internal tides. These are tides affecting an internal surface in the sea, the surface whose undulations are the internal waves described in chapter 7.

Second, there are tide currents or tide streams.1 These are the currents that flow back and forth as the tide rises and falls. Along a coastline with numerous offshore islands separated by winding channels, tide currents are forced to follow correspondingly winding courses. At each turn of the tide, the current reverses direction; its speed is greatest at midtide. The unique characteristic of tide currents is that, in contrast with all other currents, the speed of flow is the same through the whole depth of the water, from the surface right down to the level, close to the bottom, where drag slows it.2 In the deep ocean, the tide current is slow: the water shifts about 1 km during each half-period (of 6 h, 12.5 min) so the average speed is only 160 meters an hour. In shallow inshore waters, a tide current flows much farther between each reversal of direction, at speeds that may exceed 1 km/h.

The movement of the water molecules within a tide wave is an exaggerated form of the movement shown in figure 7.7 for a shallow-water swell. The wavelength of the tide wave is more than five thousand times the water's depth; consequently the elliptical paths of the molecules are so flattened as to be indistinguishable from horizontal straight lines; figure 7.7 can be modified to represent the tide wave by replacing the stack of ellipses with a stack of double-headed horizontal arrows all of the same length.

If you live near a gently sloping ocean beach, you can watch these currents any time the water is glassy calm: what you see is the water creeping slowly landward up the beach as the tide rises and then down the beach as it falls, back and forth twice a day, endlessly.

That each tide is a "wave" and causes "currents" should not obscure the fact that tide waves and tide currents are utterly different from other waves and currents, because their causes are extraterrestrial. The causes and consequences of the tides are always in step over the whole world; in contrast, the causes and consequences of all other ocean movements are always regional or local.

The Energy That Drives the Tides

It is often said (for example, in the preceding section) that the tides are caused by the gravitational pull of the moon and the sun on the oceans. This is true, but it is so oversimplified as to disguise what's really happening.

Disregard the sun for the moment—it is much less important than the moon in the context of tides—and visualize the earth-moon system. The earth has a mass of 6 x 1024 kg (or more impressively, 6 billion trillion metric tons), eighty-one times that of the moon (which, at more than 70 billion billion metric tons, isn't negligible). The two bodies are about 384,400 km apart.3 Each exerts a gravitational pull on the other, so why don't they fall in on each other and become one?

The answer is that the system is rotating, like a barbell with very unequal weights, around an axis through its center of gravity (CG). Figure 8.1 illus-

Figure 8.1 (a) A lopsided barbell rotating counterclockwise around its center of gravity (CG). (b) Analogous rotation of the earth-moon system, as seen looking down on the North Pole. Not to scale: the moon is disproportionately large relative to the earth, which brings the system's CG outside the earth instead of inside. The whole earth-moon system rotates once in 27.32 days; the earth also rotates around its own axis once in twenty-four hours relative to the sun.

Figure 8.1 (a) A lopsided barbell rotating counterclockwise around its center of gravity (CG). (b) Analogous rotation of the earth-moon system, as seen looking down on the North Pole. Not to scale: the moon is disproportionately large relative to the earth, which brings the system's CG outside the earth instead of inside. The whole earth-moon system rotates once in 27.32 days; the earth also rotates around its own axis once in twenty-four hours relative to the sun.

trates the resemblance except that it shows both CGs located somewhere between the two masses—weights in the case of the barbell, heavenly bodies in the case of the earth-moon system. In reality, because the mass of the earth is so much greater than that of the moon, their CG is inside the earth; it is below the ground, about one-fourth of the way from the surface to the center. In the figure, the CG has been placed in the space between the two bodies merely for clarity and to emphasize that the moon's orbit is centered not on the earth's center, but rather on the CG of the earth-moon system. That it is inside the earth makes no difference to the dynamics.

Because the whole system is spinning, at the rather stately speed of one revolution each 27.32 days, the two bodies are kept apart by centrifugal force.4 The distance between them and the speed of rotation of the system are in equilibrium: that is, the gravitational attraction tending to make the bodies fall together exactly balances the centrifugal force tending to drive them apart. Which raises the next question: Why is the system spinning?

Think of a child's top. If you put it on the floor, it just lies there; it will spin only if you give it a sharp twist to impart rotational energy to it. Likewise with the earth-moon system; it has rotational energy. Where does the rotational energy come from? Undoubtedly it is part of the original rotational energy of the solar nebula—a vast rotating cloud of dust and gas—that was the precursor of the solar system.5 As the primordial dust gradually accreted into solid bodies and groups of bodies, they retained their rotational energy, and much of it (not all, as we shall see) persists. In a nutshell, the energy that drives the tides is a fragment of the rotational energy of the infant solar system, surviving in the rotation of the earth-moon system.

Figure 8.1 also shows what happens to the ocean; note that for the present we are still disregarding the presence of the sun, and also of the continents, which cover less than 30 percent of the earth's surface. Because the earth-moon system spins, the shell of ocean water encasing the earth is deformed: its surface takes on the shape of a football, with one end pointing toward the moon, the other end away from it. In the figure the football's length is greatly exaggerated, for clarity; the difference in depth between the deepest water and the shallowest is really only a meter or two. The deformation arises because the moon's gravitational pull raises the water nearest it up into a bulge, while at the same time the water on the side farthest from the moon, and therefore on the outside of the rotating system, bulges because of centrifugal force.

More precisely, gravity acts to pull both earth and ocean toward the moon, while centrifugal force drives both earth and ocean away from the moon. The effects balance: on the side of the earth nearest the moon, the moon's gravitational pull exceeds the centrifugal push, and vice versa on the side farthest from the moon; hence the symmetry of the football.

Now recall that the earth rotates on its axis, relative to the moon, once per lunar day of 24 h, 50 min. As it does so, every point on the surface passes, suc-

Third quarter

First quarter

Figure 8.2. The relative position of earth, sun, and moon; N is the North Pole. (a) and (b) The three bodies are aligned, giving spring tides. (c) and (d) The three bodies form a right angle, giving neap tides. Note that the outline of the ocean forms a more elongated ellipse at spring tides than at neap tides, giving a bigger tide range. (Both c and d are viewed from the side; the sunlit half of the moon is white. Seen from the earth, the first-quarter moon appears D-shaped, and the third-quarter moon appears C-shaped.)

cessively, first under an ocean bulge, then under a shallow part, then another bulge, and then another shallow part of the "football." This represents the familiar sequence of high tide, low tide, high tide, low tide.

Next we consider two details that complicate this simple picture. The first detail is the sun. In the mathematical theory of the tides the sun's effect is ex-

Moon

Earth

Figure 8.3. The axis through the tide waves (the ocean "bulges" raised by tidal forces) does not coincide with the line joining earth and moon; it lags behind because of drag.

ceedingly complex, but all we need to know here is that the sun reinforces the effect of the moon when earth, sun, and moon are all aligned (as they are at new moon and full moon) and partly negates its effect when the sun's gravitational pull is at right angles to the moon's (as it is when the moon is in its first and third quarters). Figure 8.2 shows what happens. In each month, the tide range reaches a maximum around the time of new moon and full moon, giving spring tides; tide range is at a minimum around the times of first quarter and third quarter, giving neap tides.

The second complication is illustrated in figure 8.3. It is crucial from the energy point of view. The figure shows, more precisely than was possible in small-scale diagrams, the orientation of the football-shaped shell of ocean encasing the earth. Notice that its long axis is not along the line joining earth and moon; it is deflected through a small angle. This is because the crests of the two tide waves (the bulges at the ends of the football) cannot keep up with the moon: they lag behind, held back by drag between the water and the ocean floor. The delay at any observing station on the coast depends on the depth of the water offshore and on the topography of the bottom. If the water is deep and the bottom flat, the lag will be short, but if the water is shallow and the bottom hilly—perhaps with islands breaking the surface—the lag may be several hours. The lag shows that drag is operating and, consequently, that energy is being dissipated.

The Dissipation of Tidal Energy

The energy of the tides is dissipated by the drag of water on the ocean floor and along the shorelines of the world; the drag is especially strong in shallow

Earth

Figure 8.3. The axis through the tide waves (the ocean "bulges" raised by tidal forces) does not coincide with the line joining earth and moon; it lags behind because of drag.

seas overlying continental shelves. As always, drag produces heat, increasing the entropy of the oceans. The rate at which heat is produced is believed to be about 2.5 x 1012 W—think of the energy from 25 billion 100 watt lightbulbs.6

Drag also causes the earth to lose rotational energy, a loss that is going on all the time. The drag of the tides slows the rotation of earth so that, imperceptibly, the days become longer and longer. Here we are considering the rotational energy not of the earth-moon system as a whole, but of the earth as a single body, spinning on its axis once every twenty-four hours relative to the sun.

Now for some numbers. What is the rotational energy of the earth (also called, more briefly, the spin energy), and how is it computed?

Recall (see chapter 3) that any object moving in a straight line has kinetic energy, KE. Further, if the mass of the object is M kg and it is moving with velocity v meters per second, then we can compute its KE from the formula KE = V2 Mv2 J. Likewise, a spinning object, such as the earth rotating on its axis, has energy by virtue of its spin, but a different formula is required to measure it, a formula using the object's moment of inertia, I, and its angular velocity, w. The formula is spin energy = V2 Iw2 J; it is of the same form as the formula for KE, so that if we know the relevant values of I and w, the spin energy is easy to compute. Before doing so, however, it is necessary to explain moment of inertia and angular velocity.

First, moment of inertia, I: in the same way that a nonrotating body's mass is a measure of its resistance to being "pushed" (accelerated or decelerated) in a straight line, a rotating body's moment of inertia is a measure of its resistance to having its rate of spin altered (either increased or decreased). The moment of inertia depends on the mass and also on the shape of the body and the location of its spin axis. For example, imagine two gates of the same weight, one wide and one narrow; it is obvious from experience that the wide gate will be harder than the narrow gate to swing on its hinges. This is equivalent to saying that it has a greater moment of inertia. Similarly, imagine two flywheels of the same weight, one made of aluminum and the other of lead, and suppose they are spinning at the same speed; it will take more effort to slow the aluminum wheel than the lead one, because its diameter is greater, and therefore its moment of inertia is greater. Without going into details on how the moment of inertia of a body is computed, it suffices to say that the earth's is7

Next consider angular velocity, w: this is the rate at which a body spins, measuring the angle in radians (1 radian = 57.3°). For the earth, w = 1 rotation per 24 hours = 7.27 x 10-5 radians per second.

Then, for the spin energy of the earth, we have

This is the spin energy at the present time. It is being gradually lost because of the braking effect of the tides, which slows the earth's rotation. The days are lengthening at the rate of 0.0024 seconds per century.8 If the rate continues, the day will be ten minutes longer than at present 25 million years hence, 20 minutes longer 50 million years hence, and so on.

This "lost " rotational energy is not truly lost, nor is it converted into entropy. On the contrary, it is conserved: as the earth slows down, energy is transferred outside the earth to the earth-moon system as a whole. The rotation rate of the system increases, causing the distance between earth and moon to increase too. In brief, the rotational energy that originally belonged to the earth alone is gradually being shared with the earth-moon system.

Other Tides

Tides affect the atmosphere and also the "solid" earth itself. As you would expect, the atmosphere is as strongly affected as the ocean by tidal forces, and at first thought it is surprising that atmospheric tides should be so elusive—nobody notices them. The atmosphere does indeed respond to tidal forces as it should, but the effect is almost completely masked by volume changes resulting from the diurnal heating and cooling of the atmosphere by the sun. As a result, the atmospheric tides are imperceptible without sensitive instruments. Their contribution to the earth's energy balance is negligible compared with that of ocean tides.

The great contrast between the atmosphere and the oceans in the prominence of their tides arises chiefly because air is compressible and water (almost) incompressible. Water's expansion and contraction in response to temperature changes are much too slight to mask the tides.

The solid earth is slightly affected by tidal forces too, because it is not rigid. The shape of the earth is deformed in the same way that the ocean is deformed, but only to a minuscule extent: the crustal bulges facing toward and away from the moon are less than a meter high on either side of the earth, a sphere 12,750 km in diameter.

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