theory of tides

History

Classical era

The tides received relatively little attention in the civilizations around the Mediterranean Sea, as the tides there are relatively small, and the areas that experience tides do so unreliably. A number of theories were advanced, however, from comparing the movements to breathing or blood flow to theories involving whirlpools or river cycles. A similar "breathing earth" idea was considered by some Asian thinkers. Plato reportedly believed that the tides were caused by water flowing in and out of undersea caverns. Crates of Mallus attributed the tides to "the counter-movement (ἀντισπασμός) of the sea” and Apollodorus of Corcyra to "the refluxes from the Ocean". An ancient Indian Purana text dated to 400-300 BC refers to the ocean rising and falling because of heat expansion from the light of the Moon. The Yolngu people of northeastern Arnhem Land in the Northern Territory of Australia identified a link between the Moon and the tides, which they mythically attributed to the Moon filling with water and emptying out again.

Ultimately, the link between the Moon (and Sun) and the tides became known to the Greeks, although the exact date of discovery is unclear; references to it are made in sources such as Pytheas of Massilia in 325 BC and Pliny the Elder's Natural History in 77 AD. Although the schedule of the tides and the link to lunar and solar movements was known, the exact mechanism that connected them was unclear. Classicists Thomas Little Heath claimed that both Pytheas and Posidonius connected the tides with the moon, "the former directly, the latter through the setting up of winds". Seneca mentions in De Providentia the periodic motion of the tides controlled by the lunar sphere. Eratosthenes (3rd century BC) and Posidonius (1st century BC) both produced detailed descriptions of the tides and their relationship to the phases of the Moon, Posidonius in particular making lengthy observations of the sea on the Spanish coast, although little of their work survived. The influence of the Moon on tides was mentioned in Ptolemy's Tetrabiblos as evidence of the reality of astrology. Seleucus of Seleucia is thought to have theorized around 150 BC that tides were caused by the Moon as part of his heliocentric model.

Aristotle, judging from discussions of his beliefs in other sources, is thought to have believed the tides were caused by winds driven by the Sun's heat, and he rejected the theory that the Moon caused the tides. An apocryphal legend claims that he committed suicide in frustration with his failure to fully understand the tides. Heraclides also held "the sun sets up winds, and that these winds, when they blow, cause the high tide and, when they cease, the low tide". Dicaearchus also "put the tides down to the direct action of the sun according to its position". Philostratus discusses tides in Book Five of Life of Apollonius of Tyana (circa 217-238 AD); he was vaguely aware of a correlation of the tides with the phases of the Moon but attributed them to spirits moving water in and out of caverns, which he connected with the legend that spirits of the dead cannot move on at certain phases of the Moon.

Medieval period

The Venerable Bede discusses the tides in The Reckoning of Time and shows that the twice-daily timing of tides is related to the Moon and that the lunar monthly cycle of spring and neap tides is also related to the Moon's position. He goes on to note that the times of tides vary along the same coast and that the water movements cause low tide at one place when there is high tide elsewhere. However, he made no progress regarding the question of how exactly the Moon created the tides.

Medieval rule-of-thumb methods for predicting tides were said to allow one "to know what Moon makes high water" from the Moon's movements. Dante references the Moon's influence on the tides in his Divine Comedy.

Medieval European understanding of the tides was often based on works of Muslim astronomers that became available through Latin translation starting from the 12th century. Abu Ma'shar al-Balkhi, in his Introductorium in astronomiam, taught that ebb and flood tides were caused by the Moon. Abu Ma'shar discussed the effects of wind and Moon's phases relative to the Sun on the tides. In the 12th century, al-Bitruji contributed the notion that the tides were caused by the general circulation of the heavens. Medieval Arabic astrologers frequently referenced the Moon's influence on the tides as evidence for the reality of astrology; some of their treatises on the topic influenced western Europe. Some theorized that the influence was caused by lunar rays heating the ocean's floor.

Modern era

Simon Stevin in his 1608 De spiegheling der Ebbenvloet (The Theory of Ebb and Flood) dismisses a large number of misconceptions that still existed about ebb and flood. Stevin pleads for the idea that the attraction of the Moon was responsible for the tides and writes in clear terms about ebb, flood, spring tide and neap tide, stressing that further research needed to be made. In 1609, Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides, which he compared to magnetic attraction basing his argument upon ancient observations and correlations.

In 1616, Galileo Galilei wrote Discourse on the Tides. He strongly and mockingly rejects the lunar theory of the tides, and tries to explain the tides as the result of the Earth's rotation and revolution around the Sun, believing that the oceans moved like water in a large basin: as the basin moves, so does the water. But his contemporaries noticed that this made predictions that did not fit observations.

René Descartes theorized that the tides (alongside the movement of planets, etc.) were caused by aetheric vortices, without reference to Kepler's theories of gravitation by mutual attraction; this was extremely influential, with numerous followers of Descartes expounding on this theory throughout the 17th century, particularly in France. However, Descartes and his followers acknowledged the influence of the Moon, speculating that pressure waves from the Moon via the aether were responsible for the correlation.

Newton, in the Principia, provides a correct explanation for the tidal force, which can be used to explain tides on a planet covered by a uniform ocean but which takes no account of the distribution of the continents or ocean bathymetry.

This form of the theory is known as equilibrium theory. Equilibrium theory makes three simplifications: 1) ignore Earth's land, 2) ignore the viscosity of water so it can respond to gravity instantly, 3) ignore friction between the Earth and water. In a coordinate system rotating with the Earth-Moon pair, the distance between the Earth and Moon is constant: they are in equilibrium. This equilibrium can be explained as a balance of the force of the Moon's gravity and centrifugal force from rotation. At the center of the Earth, the forces are equal and opposite. For other points, the forces do not exactly balance and the residual force is called the tide-generating force. For points on Earth's surface closest to the Moon, gravity is very slightly stronger; for points farthest away, centrifugal force is slightly stronger. On the poles away from the Earth-Moon line, the small net force points into the Earth. The ocean water is barely affected by these forces. In between the poles and the equator, a component of the small force points horizontal to the surface of the Earth and towards the equator. No force opposes this small force. The ocean water flows in response to this force, leaving the poles and accumulating near the equator. The result is a double tidal bulge along the Earth-Moon axis, somewhat larger on the side closer to the Moon. As the Earth rotates on its axis, different points on Earth move through these bulges, roughly explaining the daily double tides. Both the oceans' water and the solid Earth experience these differences in pull, but the rigid Earth resists deformation and keeps its roughly spherical shape, while the fluid redistributes to match the imbalance, forming the bulges.

Tidal analysis and prediction

Harmonic analysis

Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by Lord Kelvin's application of Fourier analysis to the tidal motions as harmonic analysis. Thomson's work in this field was further developed and extended by George Darwin, applying the lunar theory current in his time. Darwin's symbols for the tidal harmonic constituents are still used, for example: M: moon/lunar; S: sun/solar; K: moon-sun/lunisolar.

Darwin's harmonic developments of the tide-generating forces were later improved when A.T. Doodson, applying the lunar theory of E.W. Brown, developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies. Doodson's work was carried out and published in 1921. Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson numbers, a system still in use.

Since the mid-twentieth century, further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy. The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tide-predicting machine, a special-purpose form of analog computer. More recently digital computers, using the method of matrix inversion, are used to determine the tidal harmonic constituents directly from tide gauge records.

Tidal constituents

Tidal constituents combine to give an endlessly varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an animation of the American Mathematical Society illustrating the way in which the components used to be mechanically combined in the tide-predicting machine. Amplitudes (half of peak-to-peak amplitude) of tidal constituents are given below for six example locations:

Eastport, Maine (ME), Biloxi, Mississippi (MS), San Juan, Puerto Rico (PR), Kodiak, Alaska (AK), San Francisco, California (CA), and Hilo, Hawaii (HI).

Doodson numbers

In order to specify the different harmonic components of the tide-generating potential, Doodson devised a practical system which is still in use, involving what are called the Doodson numbers based on the six Doodson arguments or Doodson variables. The number of different tidal frequency components is large, but each corresponds to a specific linear combination of six frequencies using small-integer multiples, positive or negative. In principle, these basic angular arguments can be specified in numerous ways; Doodson's choice of his six "Doodson arguments" has been widely used in tidal work. In terms of these Doodson arguments, each tidal frequency can then be specified as a sum made up of a small integer multiple of each of the six arguments. The resulting six small integer multipliers effectively encode the frequency of the tidal argument concerned, and these are the Doodson numbers: in practice, all except the first are usually biased upwards by +5 to avoid negative numbers in the notation. (If the biased multiple exceeds 9, the system adopts X for 10, and E for 11.)

The Doodson arguments are specified in the following way, in order of decreasing frequency:

β

1

=

τ

=

(

θ

M

+

π

−

s

)

{\displaystyle \beta _{1}=\tau =(\theta _{M}+\pi -s)}

is mean Lunar time, the Greenwich hour angle of the mean Moon plus 12 hours.

β

2