Tides

Although acting at vastly greater distances than the other three forces of nature, we have discovered that gravity decreases with the square of the distance between the objects and proportionally to the mass of the objects. Think about three masses called A, B and C. If Mass A was four times as heavy as Mass B, Mass A would pull four times as hard on Mass C as Mass B does if it was the same distance away. If Mass A was the same size as Mass B , but half as far away, Mass A would pull four times harder on Mass C than Mass B would. To have the same effect on Mass C, Mass A would have to be four times as heavy as Mass B if it were twice as far away. In the figre below, Mass A is four times as heavy as Mass B but twice as far away. These two nodies exert equal magnitude forces on Mass C. With gravity, being closer is more important than being big.

Our solar system is our gravitational neighborhood. In it we have the sun, the planets, the planetary moons, asteroids and comets. The sun is the heavyweight in our little corner of the universe because it contains 98% of the entire mass of the solar system.

Our Earth is a planet some 8000 miles in diameter located about 93 million miles from the sun. It seems big to us, but the sun is 330,000 times bigger. ( The Earth's mass is about 6 septillion kilograms, a 6 followed by 24 zeros, expressed conveniently as 6x10²⁴ kilograms. This is equivalent to approximately one trillion metric tons per person. The mass of all the plants and animals on earth is about one trillion metric tons. The Sun has a mass of 2x10³⁰ kilograms. Our Moon's mass is 7x10 ²² kilograms.)

The moon is 1.2 % of the Earth's mass. No other planet in the solar system has a Moon as big compared to itself. Still, the Sun in 30 million times more massive than the Moon. The Moon is on average 240,000 miles from the Earth. Even though the Moon is much closer than the Sun, it is so much smaller than the Sun that the gravitational force the Sun exerts on the Earth is several thousand times stronger than the gravitation force the Moon exerts on the Earth.

The force we name gravity pulls the Earth directly toward the Sun. So why doesn't the Earth just fall into the Sun? It doesn't because it is moving. It is a mass moving through space. It has inertia. Unless acted upon by some force, it will continue to move in a straight line. But the force of gravity between the Earth and the Sun bends the straight line path of the moving Earth into an elliptical orbit around the sun. The force of gravity balances the inertial forces of the moving Earth in its yearly passage around the Sun. The Earth does not fall into the Sun but remains circling the Sun at a relatively constant distance. A balance is maintained between the forces of gravity and the forces of inertia as the Earth and Sun rotate around the center of mass of the two bodies. This is a good thing. ( The Earth also pulls on the Sun. Because the Earth's mass is so small compared to the Sun there is a minuscule difference between the spatial center of the Sun and the center of mass of the Sun and the Earth. The Earth's gravitational and inertial effects on the Sun are small. )

The same balance of forces exists between the Earth and the Moon. The force of gravity overcomes the inertia of the Moon and bends the motion of the Moon into an elliptical orbit around the Earth, keeping the Moon from flying away from the Earth. For amore this is a good thing too, no?

When we jump in the car to go to the grocery store to get milk, eggs and a loaf of bread, we aren't concerned about the strong solar gravity that keeps us slinging around the Sun at 65000 miles per hour, or the 1000 miles per hour that we spin in space as the Earth rotates on its own axis. These forces are largely balanced out. Earth's gravity keeps us firmly planted to the Earth. At least most of the time. We do not feel a net effect as we go about our daily lives. The motion of the car as it turns the corner, knocking over the bag of groceries, has a much more noticeable effect.

But over larger distances (from one side of the Earth to the other) and/or longer times, (thousands of years) the slight differences in the balance of these forces have many effects. The tides are one of them. Let's see how this comes about.

Remember we said that gravity gets weaker the farther apart the two masses are. Consider the exaggerated and simplified representation of the Moon's gravity upon the Earth, as shown below. Matter on the side of the Earth closest to the Moon is pulled more strongly toward the Moon than the center of the Earth and the center of the Earth is pulled more strongly than the matter on the far side of the Earth. Thus the near side of the Earth moves away from the center and the center moves away from the far side of the Earth. However, relative to the Earth's center that we all relate to on a day to day basis, the matter on the near side moves away from the center of the Earth. But the the matter on the far side of the Earth moves AWAY from the Earth's center also. This produces one bulge toward the Moon and another one away from it.

The Earth's surface is rigid, so it resists the differential pull of the Moon's gravity. Over its 8000 mile diameter, it bulges several centimeters as it rotates underneath the Moon. Not that we ever notice. The sea water, being more fluid, bulges more even though it is a very thin layer, less that 3 miles deep on average. (The bulge in the very much less dense air in negligible compared to the daily effects of temperature from the Sun. But the atmosphere does have tides.)

The Sun has an very similar effect. Although the Sun's gravitational field is many thousands of times stronger than the Moon's field, because the Sun is so far away, the differences from one side of the Earth to the other are much smaller in relation to the strength on the field itself. In fact, the Sun's differential gravitational force from one side of the planet to the other is about half that of the Moon. So the Sun also creates a diurnal (twice a day) tide of about half the Moon's strength.

The real tides are much more complex than the frictionless, uniform depth model that was depicted above. Oceans are confined to basins between continental masses. Shoreline and bottom configuration dominate the size of the tidal fluctuations in the shallows of the continental margins. Atmospheric pressure raises and lowers the surface of the waters, temperature fluctuations in the water column affect sea surface heights, winds create stress that piles up water and ocean currents moving on the surface of the rotating Earth slant the surface of the ocean.

The gravitational forces of the sun and moon create very long-period waves that move through the ocean basins (These are not tsunamis!). Tides originate in the oceans and progress toward the coastlines where they appear as the regular rise and fall of the sea surface. When the highest part, or crest of the wave reaches a particular location, high tide occurs; low tide corresponds to the lowest part of the wave, or its trough. The difference in height between the high tide and the low tide is called the tidal range.

The Sun and the Moon both produce a tide. The Sun produces two bulges that "circle" the Earth in a day. But the Moon produces two bulges that circle the Earth about every 24 hours and 50 minutes. This is because the Moon revolves around the Earth every 27 days 7 hours 43 minutes in the same direction as the Earth rotates about its axis. Thus in that time, the same point on the Earth only rotates under the Moon 26 times. While the Moon rotates around the Earth, the Earth and the Moon are rotating around the Sun. During the 27+ days it takes for the Moon to rotate around the Earth, the Earth and Moon have moved a significant portion of the way around the Sun. In order for the Sun and Moon to align again it takes a little longer for the Moon to rotate another part of the way around the Earth to compensate for the angle difference due to the rotation of the Earth around the Sun.

The alignment of the Moon with the Earth and Sun is easily seen by how much of the Sun's light we can see reflected from the Moon's surface. This phase of the Moon has a period of 29.5 days, the lunar month. Each day the lunar tidal bulges lag about 50 minutes behind the previous days corresponding bulge..

A complication to this symmetric rhythmic pattern occurs because the Moon does not rotate around the Earth in the same plane as the Earth's daily rotation. Since the bulges of the lunar tides are formed toward and away from the Moon, in the highest latitudes there may be only one high and one low as the Earth rotates in the skewed elliptic gravitational envelope. Think about the tidal height that would be observed by someone in Antarctica or Greenland rotating around the North South Axis under the exaggerated ocean tidal bulge in the diagram below. At lower latitudes the there would be a higher high tide, then a low tide followed by a high tide not as high as the previous one and then a low tide. To complicate matters more, the angle of the Moon to the Earth's equator changes over time in a 19 year cycle.

Figure showing individual contributions of solar and lunar tides during early and late summer.

Figure showing individual contributions of solar and lunar tides during winter.

Here is a table of predicted tidal heights for high and low tides in feet for Tofino, British Columbia, Canada. All times are Mountain Standard Time. High tide heights are in bold type. Moon phase data from StarDate.org. See if you can note the effects of the waxing and waning Moon, spring and summer seasons in these real life tables.

	June 2004
	Night		Day				Night		Moon
1	05:05	1.0	11:25	9.8	1650	4.6	2300	12.3
2	05:55	0.3	12:20	10.2	1740	4.6	2340	12.8
3	06:40	-0.3	13:10	10.2	1830	4.9	0030	12.8
4	07:30	-0.3	14:05	10.2	1920	4.9
5	01:20	12.5	08:20	-0.3	1455	10.2	2015	4.9
6	02:10	11.8	09:15	0.3	1545	10.2	2115	5.2
7	03:05	11.2	10:05	1.0	1640	9.8	2220	5.2
8	04:05	10.5	1100	1.6	1735	9.8	2330	4.9
9	05:15	9.5	1155	2.3	1830	10.2
10	00:45	4.6	0625	8.9	1250	3.3	1925	10.2
11	01.55	4.3	0745	8.5	1345	3.9	2015	10.5
12	03:00	3.6	0900	8.5	1440	4.6	2100	10.5
13	03:50	3.0	1005	8.9	1535	4.9	2140	10.8
14	04:40	2.3	1100	8.9	1620	5.2	2220	10.8
15	05:25	2.0	1145	9.2	1700	5.2	2255	10.8
16	06:00	1.6	1230	9.2	1740	5.6	2330	10.8
17	06:40	1.3	1305	9.2	1815	5.6
18	00:05	10.8	0715	1.3	1345	9.2	1850	5.6
19	00:40	10.8	0745	1.3	1420	9.2	1930	5.6
20	01:15	10.8	0820	1.6	1455	9.2	2005	5.6
21	01:55	10.5	0855	1.6	1535	9.2	2050	5.6
22	02:35	10.2	0930	2.0	1610	9.2	2140	5.6
23	03:20	9.5	1010	2.3	1655	9.2	2235	5.2
24	04:15	9.2	1055	3.0	1735	9.5	2335	4.9
25	05:15	8.9	1140	3.3	1825	9.8
26	00:40	4.6	0630	8.2	1230	3.9	1910	10.2
27	01:50	3.6	0755	8.2	1325	4.3	2000	10.8
28	02:50	2.6	0910	8.5	1425	4.9	2050	11.2
29	03:50	1.6	1020	8.9	1520	5.2	2140	11.8
30	04:45	1.0	1120	9.5	1625	5.2	2230	12.1
1	05:40	0.0	1215	9.8	1725	5.2	2325	12.5
2	06:30	-0.3	1305	10.2	1820	4.9
3	00:15	12.5	0720	-0.3	1350	10.2	1915	4.6

	December 2004
	Night		Day				Night		Moon
1	0345	9.8	0850	6.6	1430	10.5	2140	3.0
2	0430	9.5	0940	6.6	1515	10.2	2220	3.3
3	0515	9.5	1040	6.6	1610	9.5	2310	3.9
4	0605	9.5	1150	6.6	1720	9.2
5	0000	4.3	0655	9.8	1305	5.9	1840	8.9
6	0055	4.6	0740	10.5	1410	5.2	2000	8.9
7	0150	4.9	0825	10.8	1505	3.9	2115	9.2
8	0245	5.2	0905	11.5	1555	3.0	2215	9.5
9	0335	5.2	0950	12.1	1645	2.0	2310	10.2
10	0425	5.6	1030	12.8	1730	1.0
11	0000	10.5	0515	5.6	1115	13.1	1820	0.3
12	0050	10.8	0605	5.6	1200	13.5	1905	0.0
13	0140	10.8	0655	5.6	1250	13.1	1955	0.3
14	0230	10.8	0745	5.6	1345	12.8	2045	0.7
15	0320	10.8	0845	5.6	1435	12.1	2135	1.3
16	0410	10.8	0945	5.6	1535	11.5	2225	2.0
17	0500	10.8	1055	5.6	1640	10.5	2315	3.0
18	0555	11.2	1210	5.2	1750	9.8
19	0010	3.6	0650	11.2	1325	4.6	1910	9.2
20	0110	4.6	0740	11.2	1430	3.9	2030	9.2
21	0210	5.2	0830	11.5	1530	3.3	2145	9.2
22	0305	5.9	0915	11.8	1620	3.0	2245	9.5
23	0400	5.9	1000	11.8	1710	2.3	2335	9.8
24	0445	6.2	1040	6.2	1750	2.0
25	0015	10.2	0530	6.2	1120	11.8	1825	2.0
26	0035	10.2	0605	6.2	1155	11.8	1600	2.0
27	0135	10.2	0640	6.2	1230	11.8	1935	2.0
28	0205	10.2	0715	6.2	1305	11.5	2005	2.0
29	0240	10.2	0755	6.2	1340	11.2	2040	2.3
30	0315	10.2	0835	6.2	1420	10.8	2110	2.6
31	0350	10.2	0920	5.9	1500	10.5	2145	3.0

Below is a graph of tidal heights for Friday Harbor, San Juan Islands, WA for June 1, 2004 to July 3, 2004. Notice the obvious change in tidal range between the neap and spring tides. The largest tidal range in on the full Moon.

This detailed plot of tidal heights during the full moon of early June shows the near absence of a tidal low at night and a big tidal range during the day.





Here is the tidal range plot for December 2003 at Friday Harbor. The neap and spring tide is still obvious. The largest tidal range is during the new moon.



In the winter the large tidal ranges occur at night and there is almost no low tide during the day.


In spring and fall the full moon tides show much more equal high and low tides and they are much smaller. Here is a plot of tidal heights during the full moon in March at Friday Harbor. Notice that the tidal ranges are much smaller.

In addition to the daily and seasonal fluctuations, there are some very long term and dramatic effects of the tides. There is friction between the more rigid Earth surface and the double bulge of water created by the Moon. As a result of this friction, the Earth's daily spin pulls the bulge of water slightly ahead of the Moon's rotation. Since the bulge of the water has an ever so slightly greater gravitational pull on the Moon (more mass ahead of the Moon than behind), the tides transfer energy from the rotation of the Earth to the rotation of the Moon. This causes the Earth to slow down ( 100 years from now the Earth's day will be 2 milliseconds longer) and the Moon to speed up and drift away from the Earth (3.8 centimeters per year). Given enough time, the Earth will be slowed down to rotate at exactly the same speed as the Moon. Thus the same side of the of the Earth will always point to the Moon. One side of the Earth will see the Moon all the time and the other side will never see it. This has already happened to the Moon. The Moon always presents nearly the same face to the Earth.