Basic information on the tides may be found in How Deep is the Ocean, How High is the Sky?. This article describes what I have found in the excellent website UK Hydrographic Office, and is primarily an analysis of the tides actually observed at ports worldwide, with some attempt to explain the peculiarities, which are many.
One thing I shall do is relate the tides to the position and phases of the moon. The phases of the moon are shown on many calendars. Spring tides are associtated with new and full moons, while neap tides are associated with the quarters. These principal phases are about a week apart. Indeed, this is the origin of the week. The UKHO site predicts from the current date plus six days, covering the period between spring and neap tides quite nicely, so one access shows the whole picture. For the actual positions of the sun and moon, I use my planetarium program Sidera, but the reader may use a commercial equivalent.
It will probably be good to give a brief account of the physical basis of ocean tides here. The reader is referred to Butikov (Reference 3) for details. The theory is rather subtle, and it is easy to form incorrect notions. A tidal theory must explain the following principal observations: (1) tides are in synchronism with the Moon's motion, with a period of 24h 50m; (2) a double, or semidiurnal, tide is typically observed; (3) high and low water occur when the Moon is near the horizon; (4) spring and neap tides alternate as the relation between Sun and Moon changes; and (5) tidal signatures are quite various, as is the range of the tide.
The Chinese explained the tides as the breathing of the Earth. In classical times, the relation of the tides to the moon was clearly recognized and the reason assumed to have a physical cause, though this cause could not be explained. Kepler and Galileo tried to understand the formation of the tides, but the key was Newton's gravitational theory. Newton and Bernoulli described the tidal forces and the ellipsoidal ocean surface they would produce. This static theory explained the connection with the moon, the semidiurnal tide, and the alternation of spring and neap tides as the superposition of the lunar and solar tides, and gave a reasonable estimate of the tidal range. However, it failed badly on the phase of the tide and its great complexity.
The dynamical theory of Laplace and Airy considered the tides as the effect of the excitation of normal modes of oscillation of the ocean's surface. This explained the phase problem, as well as the extremely various nature of the tidal signatures. Although this was a complete and satisfactory explanation, it is very difficult to calculate tides a priori because of the complex nature of the oceans. Nevertheless, tides can be considered as the superposition of a large number of normal oscillations and quite reliably predicted from observations. We'll now sketch the dynamical theory of the tides.
The tidal forces per unit mass are shown in the figure at the right. They are symmetrical about the line joining the sun or moon and the earth, and have a quadrupole nature. They are due entirely to the differences in the gravitational attraction at different points of the body causing the tides. The gravitational acceleration due to the sun or moon at the centre of the earth is F = GM'/R2. The vertical component of the tidal force is Fv = (3/2)(r/R)F cos 2θ, and the horizontal component is Fh = (3/2)(r/R)F sin 2θ. These accelerations are very small (about 10-7 g). Constant terms have been omitted in these expressions, and only the terms that will vary as the earth rotates are included. The horizontal components will have the major effect, since they are perpendicular to the force of gravity.
The forces due to these accelerations would produce an ellipsoidal shape r(θ) = r + a cos 2θ in a static fluid. The constant a is given by 2a = (3/2)(M'/M)(r/R)3r. From this equation, the lunar tide should have an amplitude of 2a = 54 cm, and the solar tide 24 cm. These are quite reasonable values, but note that high and low tides come with the moon in the zenith or nadir, which is incorrect. Note especially the characteristic cubic dependence on r/R, which explains the predominance of the lunar tide, even though the forces due to the sun are greater on the earth.
The phase velocity of a wave of a wavelength long compared to the depth of water is v = √(gh). If the ocean is assumed to be 3.5 km deep on the average, v = 185.2 m/s. The period of a wave with a wavelength of half the circumference of the earth, or about 20,000 km, will be about 30 h. Such a wave will represent the semidiurnal tides, and is a normal mode of "sloshing" of the oceans on the earth. While thinking about these things, remember that the oceans are a very thin layer of water compared to the size of the earth, and their movement will be predominantly lateral, of which the observed vertical tides are only a reflection.
The period of the lunar tidal forces as the earth rotates is half of 24h 50m, or 12h 25m. Similarly, the period of the solar tide is 12h. Both these periods are less than the natural period of about 30 h. When an oscillator is driven by a force of period less than its natural period, the steady-state motion is opposite in phase to the driving force. Therefore, high and low water will occur for θ about equal to 90°, which explains the observed relation between the tides and the position of the moon. The exact relation, and the amplitude of the tide, depends sensitively on the decay constant of the normal mode, which is very difficult to determine. If we consider further modes of oscillation of the ocean surface, we can appreciate the possiblility of very complex tidal motions that can, however, be expressed as the superposition of modes.
It is interesting to note that the relation of the period of the natural motion and the period of the tidal forces is opposite for tides in the solid earth, so that earth tides are a maximum or minimum when the moon is at the zenith or nadir.
The first thing I want to do is to pick a definite port, and see what the tides are like there over a period of time. We can find the lunitidal interval, and the establishment of the port by these observations. The port I chose is Plymouth (Devonport), Devon, England, at about 4.5°W and 50.3°N, on the Atlantic Ocean. A tidal plot for Plymouth is shown below. (From UKHO)
This is a typical semidiurnal tide, but the small diurnal component is clearly visible in the alternating levels of the high tides. The tidal range for the tides of the 16th are 5.3 - 0.8 = 4.5 m and 5.5 - 0.7 = 4.8 m, taking what seem to be corresponding highs and lows. The average range is 4.65 m at the spring tide. On the 22nd, the ranges are 4.6 - 1.8 = 2.8 m and 4.3 - 2.0 = 2.3 m, with average 2.55 m.
In June 2003, 1st quarter was on the 7th, full moon on the 14th, last quarter on the 21st, and new moon on the 29th. The highest tides were on the 16th, and the lowest on the 23rd, in each case about 48 hours after the corresponding lunar phases. This is, then, roughly the "establishment of the port," which is the age of the tide at spring and neap.
The moon was on the meridian at 1.15 on the 16th, and the next high tide was at 6.57, so the lunitidal interval was 5h 42m. On the 22nd, the moon was on the meridian at 6.15, and the next high tide was at 11.53, for an interval of 5h 38m. The high tide does seem to be predictable from the position of the moon. At Plymouth, when you see the moon on the meridian, the high tide will occur about 5h 40m later.
This semidiurnal tidal behavior is found all around Britain, except for a few places where there are disturbing features, such as at Portland Bill. It is also found on the Atlantic coast of the United States, but not on the Gulf or Pacific coasts, where diurnal tides predominate.
Biloxi, Mississippi, USA is on the northern shore of the Gulf of Mexico, at about 89°W and 30.4°N. The tidal plot is quite different than the one at Plymouth: (From UKHO)
The tidal range is much smaller, less than 100 cm, and the tide is diurnal. There is little trace of any semidiurnal variation. The spring tide has a range of about 90 cm, but the neap range is only about 10 cm. The tidal plot looks very much like typical beats, except that the phase does not change at a node. It also looks like a diurnal wave amplitude modulated with a frequency of one month, which is probably closer to the truth.
Eleuthera Island is in the eastern part of the Bahamas, east of Nassau, facing the Atlantic Ocean at 76°W and 25°N. Its tidal plot is different from those of Plymouth and Biloxi: (From UKHO)
The semidiurnal and diurnal tides are mixed, with the typical Atlantic semidiurnal tide predominating. The diurnal tide is prominent in the spring tides, but decreases to a small amount at the neap tides. The tidal range is also small, not exceeding 1 m, but is still larger than that in the Gulf. This tidal plot is typical of all the Atlantic Islands, such as Bermuda and St. Helena, which have a principally semidiurnal tide of amplitude less than 1m with a more or less prominent diurnal tide making tidal heights alternate. Midocean tides normally have a small range, not amplified by the presence of continents or resonant basins.
The Gulf of Mexico is a subcircular bay about 1000 miles from east to west, and 600-800 miles from north to south, with a surface area of about 715,000 mi2. Its centre is true sea floor, at a depth of 12,425 ft. in the Sigsbee Deep. Its formation began in the Mesozoic, when the shallow proto-Atlantic evaporated huge thicknesses of salt and anhydrite, mainly in the Jurassic. A constructive plate boundary then formed, opening the Gulf and separating the salt beds in the Cretaceous, while accreting terranes blocked of access to the Pacific. The structure and history of the Gulf are not yet well known, except sketchily. Water enters the Gulf through the Strait of Yucatan, about 140 miles wide, in the southeast, and leaves through the Straits of Florida, about 90 miles wide, in the northeast, to form the warm Gulf Stream.
The Caribbean Sea is wider than the Gulf, about 1900 miles from Central America to the Leeward Islands, but narrower, about 550 miles wide, and with a slightly larger area, 750,000 mi2. A tectonic plate occupies the area of the Sea, wedged between the subducting Pacific Plate on the west, and itself subducting beneath the Atlantic plate on the east. Its average depth probably does not exceed that of the Gulf by much, but on its northern side, in a line along the Sierra Maestra of Cuba and between the Caymans and Trinidad, is a subduction trench, the Bartlett Deep, that reaches 22,788 ft. in depth. The structure and history of the Caribbean are also not well known, but both the Gulf and the Caribbean are of great geological interest.
I have compared the tides at Biloxi, Galveston, Tampico and Progreso, Yucatan, which are spaced around the Gulf. If the tidal plots are superimposed in front of a strong light, it is found that low water occurs almost precisely at the same time at these widely-separated points. High water is almost as simultaneous, but there is some variation. It is clear that water is not sloshing from side to side of the Gulf, but the level at the centre is rising and falling with a period of one day. It would be impossible for all this water to enter and leave by the straits of Florida and Yucatan and distribute itself around the Gulf in the time available, so this simple drum-like oscillation is the only possibility.
Furthermore, the period of oscillation must be quite close to 24 hours, so that the diurnal stimulus can resonate to give the observed tidal ranges. The semidiurnal stimulus would not excite a significant amplitude, as is observed. The favoring of diurnal oscillations and the discouragement of semidiurnal oscillations explains the observed tides quite well, at least qualitatively.
Caribbean tides show more variations than Gulf tides, but are small in amplitude, less than 50 cm, and diurnal tides are prominent. In the eastern Caribbean, the tides at St. Croix, Virgin Islands, and at Willemstad, Nederlandse Antillen, are almost entirely diurnal, like Gulf tides. The amplitude of the spring tide is about 40 cm, and of the neap tide, 15 to 20 cm at both locations. High spring tide at St. Croix occurs about 3 hours earlier than at Willemstad, but low tide about 2h 20m later. As the amplitude of the diurnal tide decreases and the small semidiurnal tide increases at neap tides, the time sequence may vary irregularly. The tides are not simultaneous, as in the Gulf.
Further west, the tides at Colón, Panama and Port Royal, Jamaica have a similar pattern, the diurnal tide predominating, but with a significant semidiurnal component that is increasingly important towards neap tides. The phases at these two rather widely-separated ports are almost exactly the same, for both components. Most of the time, there is one high tide a day, with a small advance and recession between the low tide and the next high tide. The neap tide may, however be double, with two high tides about 10 hours apart. The Colón tide is about 2h later than the Port Royal tide.
At Belize City, in the far west of the Caribbean, the tide is mainly semidiurnal, but with a diurnal component that increases toward spring tides, as usual. Typically, there are two high tides with a shallow minimum between them, separated by deeper low tides once a day. The tidal range is small, from little more than 10 cm up to 25 cm. The phase of the semidiurnal tide is 180° out of phase with the semidiurnal component at Colón, so there seems to be some semidiurnal sloshing in this area.
The diurnal resonance does not appear to be as strong in the Caribbean as it is in the Gulf, nor is it as simple. There is good evidence for semidiurnal sloshing from phase relations in the western Caribbean, from north to south. The small tidal ranges show that the tidal oscillations are not part of the Atlantic tides, which have much larger amplitudes.
The surface barometric pressure, averaging 1013.25 mb at sea level, fluctuates slowly by a maximum of about 50 mb because of weather conditions. There is also a regular daily variation of up to a few mb showing diurnal and semidiurnal components, like the ocean tide. This atmospheric tide is completely solar, the lunar component being too small to observe. The effect is greatest at the equator and at continental locations, with a diurnal component of 0.3 to 0.5 mb, and a semidiurnal component of 1 - 2 mb, for a total range of 3 to 4 mb daily. This range is least at the solstices, and greatest at the equinoxes. In polar regions, the amplitude of the variation is only about 0.3 mb. At Washington, DC, the daily variation is 1.2 mb. The variation is similar at all stations, more similar than that of the ocean tides, but still with characteristic differences.
The variation is a maximum at 10.00 hours and a minimum at 16.00 hours, with a second maximum at 22.00 hours and a second minimum at 04.00 hours, that is generally of lower amplituded. At continental stations such as Kansas City and Grand Junction, the night-time maximum and minimum are largely suppressed, but at the equator the two amplitudes are about the same. The diurnal effect is attributed to the heating and cooling of the atmosphere, but the semidiurnal effect is a tidal variation. The diagram at right shows the relation of the tidal variation to the earth and sun. In this diagram, only the earth and atmosphere are rotating, while the sun and elliptical shape of the atmosphere (greatly accentuated) remain fixed. The pressure maximum is seen to lead the sun by about 30°. The atmospheric shape is very much like that usually drawn to illustrate the tides.
While the ocean is confined in the ocean basins, and by inlets, estuaries and straits, the atmosphere is unconfined, and can move much more freely and regularly. Like the ocean, the atmosphere is a very thin sheet; half of its mass is below 6 km altitude. Tidal motions are, therefore, mainly in the horizontal with only small vertical velocities. The wavelengths of wave motions are much greater than the thickness of the atmosphere. In addition to the atmospheric pressure waves, normal modes of oscillation have been investigated. The periods of these modes of oscillation are all less than about 24 hours. One of these oscillations was estimated to have a natural frequency of 11.94 hours, very close to half a day. The solar tide resonates with this mode, while the lunar tide does not. Since the losses are small, the resonance is sharp, and the amplitude of the oscillation excited by the regular stimulus of the sun's attraction can grow to the observed value. This is the reason for the predominance of the semidiurnal component, and the absence of the lunar tide. Without the resonance, atmospheric tides would be too small to observe.
The atmospheric tide makes observations of 3-hour variations in atmospheric pressure at equatorial locations meaningless, since the variations sought are of the same magnitude.
The tidal plots used as illustrations are from the EasyTide service of UK Hydrographic Office. I presume that this educational, noncommercial use of a few examples is permissible, and I hestitate to pester authorities with such small matters, especially since the copyright notices specifically refer to commercial uses, and the plots used as examples in this page are of past tides, which are of no commercial value whatsoever. I have, of course, given proper credit, and have maintained the copyright notices.
F. A. Berry, E. Bollay and N. R. Beers, eds., Handbook of Meteorology (New York: McGraw-Hill, 1945). pp. 483-498 and 746-749. Atmospheric tides.
E. I. Butikov, A dynamical picture of the oceanic tides, Am. J. Phys. 70, 1001-1011 (2002).
R. de Levie, Tidal Analysis on a Spreadsheet, Am. J. Phys. 72, 644-651 (2004). A short history of tidal studies, and the use of Excel macros to analyze tidal data. The observed tides at Eastport, Maine from 1 June to 31 July 2001 are the example. This is a primarily semidiurnal tide with a small diurnal component. The diurnal component shows up clearly in the power spectrum, but the author does not draw attention to it, instead looking for a solar semidiurnal tide (which is not evident) close to the strong lunar component. Long-period components are ignored.
Composed by J. B. Calvert
Created 17 June 2003
Last revised 27 April 2004