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Time and Tides in the Gulf of Maine a Dockside Dialogue Between Two Old Friends

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Time and Tides in the Gulf of Maine a Dockside Dialogue Between Two Old Friends 1 Time and Tides in the Gulf of Maine A dockside dialogue between two old friends by David A. Brooks It's impossible to visit Maine's coast and not notice the tides. The twice-daily rise and fall of sea level never fails to impress, especially downeast, toward the Canadian border, where the tidal range can exceed twenty feet. Proceeding northeastward into the Bay of Fundy, the range grows steadily larger, until at the head of the bay, "moon" tides of greater than fifty feet can leave ships wallowing in the mud, awaiting the water's return. My dockside companion, nodding impatiently, interrupts: Yes, yes, but why is this so? Why are the tides so large along the Maine coast, and why does the tidal range increase so dramatically northeastward? Well, my friend, before we address these important questions, we should review some basic facts about the tides. Here, let me sketch a few things that will remind you about our place in the sky. A quiet rumble, as if a dark cloud had suddenly passed overhead. Didn’t expect a physics lesson on this beautiful day. 2 The only physics needed, my friend, you learned as a child, so not to worry. The sketch is a top view, looking down on the earth’s north pole. You see the moon in its monthly orbit, moving in the same direction as the earth’s rotation. And while this is going on, the earth and moon together orbit the distant sun once a year, in about twelve months, right? Got it skippah. And I see the new and full moon positions in the monthly orbit. But why do we get two tides each day, more or less, since the moon passes by only once each day as we rotate under it? And why does the time of high water get later every day? Let me remind you of our own experience. Growing up on the Maine coast, surrounded by big tides, we soon learn to regulate our lives by two clocks. We awaken mornings, go to school, and more or less carry out our expected duties like normal folks, following the solar clock, the one hanging on the wall. But for us there is another persistent ticking, an insistent lunar synchrony that drifts in and out of step with the wall clock. Sometimes the lunar beat gets the upper hand and at three in the morning, under a bright moon, we find ourselves chasing a run of smelt in a tidal creek choked with flashing fish. But back to the story. The moon's obvious westward motion on a starry night is due to the earth's rotation, which carries us eastward under the celestial sphere at a rate of 15 degrees of longitude each hour. This is simple enough by itself to explain the regular motion of the stars, but the moon's position in the sky is influenced by its own eastward movement in a roughly 29 day orbit. Because of this, each successive day the earth's rotation must carry us a bit farther around before the moon passes our longitude again. For the same reason, moonrise and moonset at a particular location are delayed each day by the time it takes for the earth to "catch up" with the moon. 3 And how long is this delay? Well, in the 24 hours required for the earth to turn once on its axis, you see that the moon advances about 1/29th of its orbit, so the earth must turn an extra 1/29 rotation to bring the moon back in line with a particular location. Thus the extra time required is 24/29 hours, or about 50 minutes, and this is the fundamental reason why the tides are later each day at a given port. Hold on, I can read a tide table! The time between successive highs right here at this dock is about 12 hours, 25 minutes, not 12 hours, 50 minutes, and there's usually two tides a day. And if you're going to tell me the moon's gravity is the reason, how come we get high tides when the moon is long since set, clear around on the other side of the earth – what am I missing? It’s the moon’s gravity all right, but to answer your question we must realize that each month the earth and the moon orbit each other around their common center of mass. The earth is about 80 times more massive than the moon, so this balance point actually is inside the earth, about three-fourths of the way out from the center. As a result, every month all points on the earth travel around that balance point, known as the barycenter, following circular paths with the same radius. This may be difficult to visualize, so let me offer a couple more sketches: This is a side view, showing the moon and the earth connected by an imaginary thin stick. Because the earth is much more massive than the moon, the balance point on the stick is inside the earth, about 4700 km out from the rotation axis (distance “r” in the sketch). Now flip the picture so you are looking down on the North Pole… 4 …and you see the earth and moon together orbiting the barycenter once each month. The thin stick is gone, replaced by the gravitational attraction between the two bodies. To see what happens next, it helps to turn off the earth’s daily rotation and focus on the monthly orbital motion. Choose any points you like on or in the earth, for example points a, b and c shown in the sketch. Follow them over the course of the month, and you see that they all make circles with the same radius, r. Now recall the first time you learned how to whirl a stone or an apple on a string. To keep it in a circular path, you must provide an inward pull known as centripetal force. If the string breaks (or you just let go) the object continues in the tangential direction it is moving when the inward force vanishes. Maybe you also learned about centripetal force on a spinning carousel in the park, where that inward pull is necessary to avoid being launched into the dirt. Right, my sister taught me that lesson at recess in grammar school. So here’s the basic story: The moon’s gravity provides the inward pull, the centripetal force, the invisible “string,” needed to keep the earth and moon in a stable orbit. But the gravitational attraction between objects in space sensitively depends on their separation distance (the inverse square, actually - thank you, Isaac), so the attractive force is stronger on the side of the earth under the moon, at zenith, and weaker on the opposite side, at nadir. In contrast, all points on the earth, including those in the ocean, experience the same centripetal force because in their monthly orbit they all travel in circles with the same radius, as illustrated in the sketch. Here’s the key: At the center of the earth, the moon’s gravitational attraction provides exactly the centripetal force necessary to maintain the stable orbit. Therefore, on the side of the earth closer to the moon, the gravity force is larger than the centripetal force, and the reverse is true on the side away from the moon. The result is a tidal pull of the water toward the zenith point under the moon, where the gravitational force is stronger, and a second pull directed away from the moon on the opposite side of the earth, where the centripetal force wins out. The actual forces are tiny, but it’s the difference between them that matters. In the simplest interpretation, the 5 combination produces an egg-shaped distortion of the ocean surface, with the long axis of the egg pointing at the moon and following it in its orbit. Maybe another sketch would help, this time a side view showing the moon’s orbital plane and the tidal “egg,” tilted relative to the earth’s equator: You can see that an observer carried around under the tidal egg by the earth's daily rotation will experience two highs, one under each lobe of the egg, and likewise two lows separating them. You will also understand that the time separating successive highs or lows is therefore 12 hours plus about one-half of the "catch-up" time mentioned earlier. This is close to 12 hours, 25 minutes, comfortingly in agreement with tables and experience. An apprehensive sideways glance, and I suspect my trapped companion silently wishes he were with the flashing fish. And I suppose this artful sketch also explains the unequal highs that we often see on a particular day, and the unequal lows as well? Ah, yes, as you point out, the afternoon high may be higher than the morning one, or vice- versa, and the evening low correspondingly lower. To explain this diurnal inequality, note that the plane of the moon's orbit is tilted relative to the the plane of the earth's equator. The tilt angle, known as declination, is about 28.5 degrees. Thus in a month the moon swings above and below the equator, and the amount above or below on a given day depends on where the moon is in its orbit. And keep in mind that the long axis of the tidal egg, shown greatly exaggerated by the dashed oval, always points to the moon in its orbit.
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