Tides and their relationship to vertical datums.

With some exceptions, everybody in our country and Canada has seen an ocean. If you were to spend a few hours on the beach, you would see the water level rise and fall. Depending on your location, the rise and fall can vary from less than 3 feet at Key West, Fla., to more than 52 feet at Nova Scotia's Minas Basin, Canada. But what causes tides, and how are tides related to the vertical datums used to establish elevations on bench marks? The word "tide" is used to define the alternating rise and fall in sea level with respect to the land, produced by the gravitational attraction of the moon and sun. The most familiar evidence of the tides along our seashores is the recurrence of high and low water-usually, but not always, twice daily. Configuration of the coastline, local depth of water and ocean floor topography can play an important role in altering the range of and interval between high and low water, and the times of arrival of the tides.

Figure 1. Centrifugal force plus gravitation.

## Gravitation

Sir Isaac Newton discovered gravitation and reported it in Philosophiae Naturalis Principia Mathematica in 1687. Newton's Law of Universal Gravitation1 states: "Every particle of matter in the universe attracts every other particle of Matter with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them."

The equation for gravitational force is:

F = - G Mm/rÂ² (1)

where

G = 6.673 x 10^-11 mÂ³ kg^-1 s^-2,

M = mass of the Earth,

m = mass of any object,

r = distance from the center of mass of the Earth to the object.

G is called Newton's Gravitational Constant.

But what's the difference between gravitation and gravity? Gravity is the sum of gravitation and the centrifugal force of our rotating Earth. Referring to Figure 1 below, the gravitational force F is directed toward the center of mass of the Earth and centrifugal force C is directed outward from the Earth. The sum of these two forces (vectors) is called gravity. The symbol, g, is the acceleration due to gravity, and is approximately 32 ft/sÂ².

When the Earth rotates, the Earth's atmosphere rotates with it. An airplane flying at approximately 35,000 feet above the Earth is affected by gravity; if the engines were to fail the airplane would fall into the Earth. How about GPS satellites? GPS satellites, as well as other near-Earth satellites are outside the Earth's atmosphere, and for all practical purposes, are not subject to the Earth's centrifugal force. Satellites in orbit are falling less than the 32 ft/sÂ² for an object on Earth, but are moving fast enough that the change in Earth curvature matches the falling distance. If you look at a sky plot of a GPS satellite, you'll see that the Earth rotates under the satellites.

## Tides

At the surface of the Earth, the Earth's gravitational attraction is inward (that's the reason for the negative signs in the equation to the left), which holds the ocean waters confined. However, the gravitational forces of the moon and sun also act externally upon the Earth's ocean waters. These forces draw the ocean waters to positions on the Earth's surface directly beneath, i.e. toward, the "sublunar" and "subsolar" points. The tide raising force of the moon is insufficient to "lift" the waters of the Earth physically against the pull of the Earth's gravity. Instead, the tides are produced by that component of the tide-raising force of the moon, which acts to draw the waters of the Earth horizontally over its surface toward the sublunar point and the point on the Earth directly opposite called the antipodal point. Since the horizontal component is not opposed to gravity and can act to draw the particles of water freely over the Earth's surface, it becomes the effective force in generating tides. Because of the closeness of the moon to the Earth, as opposed to the sun, the tide-producing force of the moon is approximately 2.5 times that of the sun.

High tides are produced in the ocean waters by the "heaping" action resulting from the horizontal flow of water toward the two regions of the Earth representing positions of maximum attraction of combined lunar and solar gravitational forces. Low tides are created by a compensating maximum withdrawal of water from the regions around the world midway between these two humps. The alternation of high and low tides is caused by the daily rotation of the Earth with respect to these two tidal humps and two tidal depressions.

Tides are on average higher at latitudes over 45Â° north and south of the equator. The path of the sun, the ecliptic, is inclined 23.5Â° to the Earth's Equator. The plane of the moon's orbit is inclined only about 5Â° to the ecliptic, which means the moon's monthly revolution around the Earth remains close to the ecliptic. Sublunar points are never over 28.5Â° north and south of the equator, so the horizontal draw pulls the water away from higher latitudes. Tide tables for Anchorage, Alaska are in the neighborhood of 30 feet. However, it is not wise to generalize and say that all northern latitudes have high tides. Figure 2 to the right shows the Tide Table for Massacre Bay, on Attu Island, Alaska, where the high tide is only 2.56 feet.

Tides can be predicted at specific locations. Figures 3 and 4 to the right show a fishing boat at Parrsboro, Nova Scotia, at high and low tide. As I stated earlier, in the Minas Basin in Nova Scotia off the Bay of Fundy the water level at high tide can be as much as 16 meters (52.5 feet) higher than at low tide.

## Two True Tide Stories

In 1977, while teaching at Iowa State University, my family and I drove from Iowa to New Brunswick, Canada, to attend a conference. On the way, we stayed overnight in Saint John, New Brunswick, which is on the coastline of the Bay of Fundy.

After arriving in Saint John, we found a restaurant with a beautiful view of a waterfall on a river that runs into the Bay. The waterfall was at least 20 feet high. The strange thing was that, just below the waterfall, were several large boats at anchor. The waiter at the restaurant told us the boats were waiting for the tide to come in so they could proceed up river to a paper mill a short distance beyond the waterfall. The tide would raise the water more than 30 feet so that the waterfall disappeared, and the boats would pull anchor and travel upstream to the paper mill. After the boats were loaded with the finished products from the paper mill, they had to wait until high tide to return to the Bay of Fundy.

In June of 2003 I flew into Halifax, Nova Scotia, Canada, to attend a conference. I saw the high tides and fishing boats on the river bottoms as shown in Figures 3 and 4. I also saw a tidal bore, the front of the water produced by the incoming tide to a river or stream. At a visitor's center in Nova Scotia I picked up a time schedule for the tidal bore at Truro, Nova Scotia. On the day I was there, the bore was due to arrive at 2:15 p.m. At 2:15 p.m. I was at the river, along with about 100 other tourists, and as predicted a large wave of water came rushing up the small river. The wave was only about a foot high, but it rushed passed us bringing in more water from the Bay of Fundy. The water in the river was now flowing upstream, and in 15 or so minutes the level of water had risen about 5 feet.

Figures 3 and 4. Tides can be predicted at specific locations like this fishing boat in Parrsboro, Nova Scotia, Canada, shows.

## Sea Level

With the tides rising and falling each day, how is average sea level determined? A series of tidal observations at one location is required over a period of time. This is accomplished with a tide gauge, a device that measures the rise and fall of the water. The time required to determine mean sea level at a tide gauge location is extremely long-18.6 years-which is one nutation cycle of the Earth's spin axis. Within this period, all significant astronomical modifications of tides will occur. The National Ocean Survey, a component of the National Oceanic and Atmospheric Administration (NOAA), maintains approximatly 140 tide gauges located along the coasts and within major embayments of the United States, its possessions, and the United Nations Trust Territories under its jurisdiction.

Sea level has been used as a surface for the vertical datum for at least 150 years. The following is a list of events relating sea level and vertical datums:

• In 1864, the International Geodetic Conference adopted a Swiss recommendation for the execution of a precise leveling network over a large part of Europe. The method of observation and the use of a mean sea level were prescribed in the resolution.
• The first recorded effort by the U.S. Coast Survey was a geodetic leveling project in 1856-57. To support detailed studies of the tides and currents in New York Bay and the Hudson River, a series of tide gages was established along the Hudson River that connected to a line of leveling established by G.B. Voss. In the 1857 Report of the Superintendent of the Coast Survey, Voss wrote, "As you directed, a double series of leveling were made throughout the whole route and every doubtful step was retraced"¦ It appears that the probable error for the entire distance from New York to Greenbush (near Albany) does not exceed two-tenths of a foot." A bench mark on this line provided the sea-level datum to which subsequent leveling by the U.S. Lake Survey were referred in determining elevations of the water surfaces of the Great Lakes.
• By 1900, geodetic leveling by the Coast and Geodetic Survey and other agencies became so extensive that a general adjustment of the results became necessary to obtain consistent and accurate elevations for all control points. Data from 21,095 km of leveling were obtained by the Coast and Geodetic Survey from the U.S. Geological Survey, the Corps of Engineers (U.S. Lake Survey, Mississippi River Commission, Missouri River Commission, Deep Waterways Commission and others) and the Pennsylvania Railroad.
• The adjustment of this, the first national network, produced elevations for about 4,200 control points that were referred to mean sea level as determined at the following tide gauges: Boston, Mass.; New York, N.Y.; Sandy Hook, N.J.; Washington, D.C.; and Biloxi, Miss. A connection to sea level on the Pacific coast had not yet been obtained.
• The General Adjustment of 1929 incorporated 75,159 km of leveling in the United States, and for the first time, 31,565 km of leveling in Canada-24 ties connected the U.S. and Canadian networks. A fixed elevation of zero was assigned to the points on mean sea level at 26 tide stations.
• The North American Vertical Datum of 1988 (NAVD 88) is not a sea level datum, but the origin for the datum is the tide station Father Point on the St. Lawrence River in Quebec, Canada.

Although not mentioned, there is a bench mark established at each tide gauge. The surveyors running levels from a tidal station started at that bench mark, which had an elevation referenced to the tide gauge.