Theory of tides

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High and low tide in the Bay of Fundy

The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans (especially Earth's oceans) under the gravitational loading of another astronomical body or bodies (especially the Moon and Sun).


Australian Aboriginal astronomy[edit]

The Yolngu people of north-eastern Arnhem Land in the Northern Territory of Australia identified a link between the moon and the tides.[1]


In 1609 Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides,[2] basing his argument upon ancient observations and correlations. The influence of the Moon on tides was mentioned in Ptolemy's Tetrabiblos as having derived from ancient observation.


In 1616, Galileo Galilei wrote Discourse on the Tides,[3]. He tried 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.[4] Therefore, as the Earth revolves, the force of the Earth's rotation causes the oceans to "alternately accelerate and retardate".[5] His view on the oscillation and "alternately accelerated and retardated" motion of the Earth's rotation is a "dynamic process" that deviated from the previous dogma, which proposed "a process of expansion and contraction of seawater."[6] However, Galileo's theory was erroneous.[3] In subsequent centuries, further analysis led to the current tidal physics. Galileo rejected Kepler's explanation of the tides. Galileo tried to use his tidal theory to prove the movement of the Earth around the Sun. Galileo theorized that because of the Earth's motion, borders of the oceans like the Atlantic and Pacific would show one high tide and one low tide per day. The Mediterranean had two high tides and low tides, though Galileo argued that this was a product of secondary effects and that his theory would hold in the Atlantic. However, Galileo's contemporaries noted that the Atlantic also had two high tides and low tides per day, which lead to Galileo omitting this claim from his 1632 Dialogue.[7]


Newton's three-body model

Newton, in the Principia, provided 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.[8]


The dynamic theory[edit]

The dynamic theory of tides describes and predicts the actual real behavior of ocean tides.[9]

While Newton explained the tides by describing the tide-generating forces and Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides, developed by Pierre-Simon Laplace in 1775,[10] describes the ocean's real reaction to tidal forces.[11] Laplace's theory of ocean tides took into account friction, resonance and natural periods of ocean basins. It predicted the large amphidromic systems in the world's ocean basins and explains the oceanic tides that are actually observed.[12] The equilibrium theory, based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, and other important effects, could not explain the real ocean tides.[13][14][15][16][17][18][19][20] Since measurements have confirmed the dynamic theory, many things have possible explanations now, like how the tides interact with deep sea ridges and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface.[21] The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters.[22] Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters.[23][24] Measurements from the CHAMP satellite closely match the models based on the TOPEX data.[25][26][27] Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels.[28]

Laplace's tidal equations[edit]

A. Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere.
B. This view shows same potential from 180° from view A. Viewed from above the Northern Hemisphere. Red up, blue down.

In 1776, Pierre-Simon Laplace formulated a single set of linear partial differential equations, for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamic equations, but they can also be derived from energy integrals via Lagrange's equation.

For a fluid sheet of average thickness D, the vertical tidal elevation ζ, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations:[29]

where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidal-forcing potential.

William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the curl to find an equation for vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity.

Tidal analysis and prediction[edit]

Harmonic analysis[edit]

Fourier transform of tides measured at Ft. Pulaski in 2012. Data downloaded from Fourier transform computed with

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 William Thomson's application of Fourier analysis to the tidal motions as harmonic analysis.

Thomson's work in this field was then 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.

Darwin's harmonic developments of the tide-generating forces were later improved when A T Doodson, applying the lunar theory of E W Brown,[30] developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies.[31] Doodson's work was carried out and published in 1921.[32]

Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson numbers, a system still in use.[33]

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 now superseded in this work by digital electronic computers that can be programmed to carry out the same computations.

Tidal constituents[edit]

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 of tidal constituents are given below for six example locations: Eastport, Maine (ME),[34] Biloxi, Mississippi (MS), San Juan, Puerto Rico (PR), Kodiak, Alaska (AK), San Francisco, California (CA), and Hilo, Hawaii (HI).


Doodson coefficients Doodson
Amplitude at example location (cm) NOAA
Species n1 (L) n2 (m) n3 (y) n4 (mp) ME MS PR AK CA HI
Principal lunar semidiurnal M2 12.4206012 28.9841042 2 255.555 268.7 3.9 15.9 97.3 58.0 23.0 1
Principal solar semidiurnal S2 12 30 2 2 −2 273.555 42.0 3.3 2.1 32.5 13.7 9.2 2
Larger lunar elliptic semidiurnal N2 12.65834751 28.4397295 2 −1 1 245.655 54.3 1.1 3.7 20.1 12.3 4.4 3
Larger lunar evectional ν2 12.62600509 28.5125831 2 −1 2 −1 247.455 12.6 0.2 0.8 3.9 2.6 0.9 11
Variational μ2 12.8717576 27.9682084 2 −2 2 237.555 2.0 0.1 0.5 2.2 0.7 0.8 13
Lunar elliptical semidiurnal second-order 2"N2 12.90537297 27.8953548 2 −2 2 235.755 6.5 0.1 0.5 2.4 1.4 0.6 14
Smaller lunar evectional λ2 12.22177348 29.4556253 2 1 −2 1 263.655 5.3 0.1 0.7 0.6 0.2 16
Larger solar elliptic T2 12.01644934 29.9589333 2 2 −3 272.555 3.7 0.2 0.1 1.9 0.9 0.6 27
Smaller solar elliptic R2 11.98359564 30.0410667 2 2 −1 274.555 0.9 0.2 0.1 0.1 28
Shallow water semidiurnal 2SM2 11.60695157 31.0158958 2 4 −4 291.555 0.5 31
Smaller lunar elliptic semidiurnal L2 12.19162085 29.5284789 2 1 −1 265.455 13.5 0.1 0.5 2.4 1.6 0.5 33
Lunisolar semidiurnal K2 11.96723606 30.0821373 2 2 275.555 11.6 0.9 0.6 9.0 4.0 2.8 35


Doodson coefficients Doodson
Amplitude at example location (cm) NOAA
Species n1 (L) n2 (m) n3 (y) n4 (mp) ME MS PR AK CA HI
Lunar diurnal K1 23.93447213 15.0410686 1 1 165.555 15.6 16.2 9.0 39.8 36.8 16.7 4
Lunar diurnal O1 25.81933871 13.9430356 1 −1 145.555 11.9 16.9 7.7 25.9 23.0 9.2 6
Lunar diurnal OO1 22.30608083 16.1391017 1 3 185.555 0.5 0.7 0.4 1.2 1.1 0.7 15
Solar diurnal S1 24 15 1 1 −1 164.555 1.0 0.5 1.2 0.7 0.3 17
Smaller lunar elliptic diurnal M1 24.84120241 14.4920521 1 155.555 0.6 1.2 0.5 1.4 1.1 0.5 18
Smaller lunar elliptic diurnal J1 23.09848146 15.5854433 1 2 −1 175.455 0.9 1.3 0.6 2.3 1.9 1.1 19
Larger lunar evectional diurnal ρ 26.72305326 13.4715145 1 −2 2 −1 137.455 0.3 0.6 0.3 0.9 0.9 0.3 25
Larger lunar elliptic diurnal Q1 26.868350 13.3986609 1 −2 1 135.655 2.0 3.3 1.4 4.7 4.0 1.6 26
Larger elliptic diurnal 2Q1 28.00621204 12.8542862 1 −3 2 125.755 0.3 0.4 0.2 0.7 0.4 0.2 29
Solar diurnal P1 24.06588766 14.9589314 1 1 −2 163.555 5.2 5.4 2.9 12.6 11.6 5.1 30

Long period[edit]

Doodson coefficients Doodson
Amplitude at example location (cm) NOAA
Species n1 (L) n2 (m) n3 (y) n4 (mp) ME MS PR AK CA HI
Lunar monthly Mm 27.554631896 661.3111655 0.5443747 0 1 −1 65.455 0.7 1.9 20
Solar semiannual Ssa 182.628180208 4383.076325 0.0821373 0 2 57.555 1.6 2.1 1.5 3.9 21
Solar annual Sa 365.256360417 8766.15265 0.0410686 0 1 56.555 5.5 7.8 3.8 4.3 22
Lunisolar synodic fortnightly MSf 14.765294442 354.3670666 1.0158958 0 2 −2 73.555 1.5 23
Lunisolar fortnightly Mf 13.660830779 327.8599387 1.0980331 0 2 75.555 1.4 2.0 0.7 24

Short period[edit]

Doodson coefficients Doodson
Amplitude at example location (cm) NOAA
Species n1 (L) n2 (m) n3 (y) n4 (mp) ME MS PR AK CA HI
Shallow water overtides of principal lunar M4 6.210300601 57.9682084 4 455.555 6.0 0.6 0.9 2.3 5
Shallow water overtides of principal lunar M6 4.140200401 86.9523127 6 655.555 5.1 0.1 1.0 7
Shallow water terdiurnal MK3 8.177140247 44.0251729 3 1 365.555 0.5 1.9 8
Shallow water overtides of principal solar S4 6 60 4 4 −4 491.555 0.1 9
Shallow water quarter diurnal MN4 6.269173724 57.4238337 4 −1 1 445.655 2.3 0.3 0.9 10
Shallow water overtides of principal solar S6 4 90 6 6 −6 * 0.1 12
Lunar terdiurnal M3 8.280400802 43.4761563 3 355.555 0.5 32
Shallow water terdiurnal 2"MK3 8.38630265 42.9271398 3 −1 345.555 0.5 0.5 1.4 34
Shallow water eighth diurnal M8 3.105150301 115.9364166 8 855.555 0.5 0.1 36
Shallow water quarter diurnal MS4 6.103339275 58.9841042 4 2 −2 473.555 1.8 0.6 1.0 37

Doodson numbers[edit]

In order to specify the different harmonic components of the tide-generating potential, Arthur Thomas Doodson devised a practical system which is still in use,[35] involving what are called the "Doodson numbers" based on the six "Doodson arguments" or Doodson variables.

The number of different tidal frequencies is large, but they can all be specified on the basis of combinations of small-integer multiples, positive or negative, of six basic angular arguments. In principle the basic arguments can possibly be specified in any of many 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 one 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. (In the case that the biased multiple exceeds 9, the system adopts X for 10, and E for 11.)[36]

The Doodson arguments are specified in the following way, in order of decreasing frequency:[36]

is 'Mean Lunar Time', the Greenwich Hour Angle of the mean Moon plus 12 hours.
is the mean longitude of the Moon.
is the mean longitude of the Sun.
is the longitude of the Moon's mean perigee.
is the negative of the longitude of the Moon's mean ascending node on the ecliptic.
or is the longitude of the Sun's mean perigee.

In these expressions, the symbols , , and refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:-

is the mean anomaly of the Moon (distance from its perigee).
is the mean anomaly of the Sun (distance from its perigee).
is the Moon's mean argument of latitude (distance from its node).
is the Moon's mean elongation (distance from the sun).

It is possible to define several auxiliary variables on the basis of combinations of these.

In terms of this system, each tidal constituent frequency can be identified by its Doodson numbers. The strongest tidal constituent "M2" has a frequency of 2 cycles per lunar day, its Doodson numbers are usually written 273.555, meaning that its frequency is composed of twice the first Doodson argument, +2 times the second, -2 times the third, and zero times each of the other three. The second strongest tidal constituent "S2" is due to the sun, its Doodson numbers are 255.555, meaning that its frequency is composed of twice the first Doodson argument, and zero times all of the others.[37] This aggregates to the angular equivalent of mean solar time + 12 hours. These two strongest component frequencies have simple arguments for which the Doodson system might appear needlessly complex, but each of the hundreds of other component frequencies can be briefly specified in a similar way, showing in the aggregate the usefulness of the encoding.

See also[edit]

References and notes[edit]

  1. ^ "Moon". Australian Indigenous Astronomy. Retrieved 8 October 2020.
  2. ^ Johannes Kepler, Astronomia nova … (1609), p. 5 of the Introductio in hoc opus
  3. ^ a b Rice University: Galileo's Theory of the Tides, by Rossella Gigli, retrieved 10 March 2010
  4. ^ Tyson, Peter. "Galileo's Big Mistake". NOVA. PBS. Retrieved 19 February 2014.
  5. ^ Palmieri, Paolo (1998). Re-examining Galileo's Theory of Tides. Springer-Verlag. p. 229.
  6. ^ Palmeri, Paolo (1998). Re-examining Galileo's Theory of Tides. Springer-Verlag. p. 227.
  7. ^ Naylor, Ron (2007). "Galileo's Tidal Theory". Isis. 98 (1): 1–22. Bibcode:2007Isis...98....1N. doi:10.1086/512829. PMID 17539198.
  8. ^ "Archived copy". Archived from the original on 10 April 2014. Retrieved 14 April 2014.CS1 maint: archived copy as title (link)
  9. ^ "Higher Education | Pearson" (PDF).
  10. ^ "Short notes on the Dynamical theory of Laplace". 20 November 2011.
  11. ^
  12. ^
  13. ^ Tidal theory website South African Navy Hydrographic Office
  14. ^ "Dynamic theory for tides". Retrieved 2 June 2012.
  15. ^ "Dynamic Theory of Tides".
  16. ^ "Dynamic Tides – In contrast to "static" theory, the dynamic theory of tides recognizes that water covers only three-quarters o". Archived from the original on 13 January 2013. Retrieved 2 June 2012.
  17. ^ "The Dynamic Theory of Tides". Archived from the original on 19 December 2013. Retrieved 2 June 2012.
  18. ^ [1]
  19. ^ "Tides – building, river, sea, depth, oceans, effects, important, largest, system, wave, effect, marine, Pacific". 27 June 2010. Retrieved 2 June 2012.
  20. ^ "TIDES". Retrieved 2 June 2012.
  21. ^ Floor Anthoni. "Tides". Retrieved 2 June 2012.
  22. ^ "The Cause & Nature of Tides".
  23. ^ "Scientific Visualization Studio TOPEX/Poseidon images". Retrieved 2 June 2012.
  24. ^ "TOPEX/Poseidon Western Hemisphere: Tide Height Model : NASA/Goddard Space Flight Center Scientific Visualization Studio : Free Download & Streaming : Internet Archive". 15 June 2000.
  25. ^ TOPEX data used to model actual tides for 15 days from the year 2000 |url=
  26. ^
  27. ^ "OSU Tidal Data Inversion". Retrieved 2 June 2012.
  28. ^ "Dynamic and residual ocean tide analysis for improved GRACE de-aliasing (DAROTA)". Archived from the original on 2 April 2015.
  29. ^ "The Laplace Tidal Equations and Atmospheric Tides" (PDF).
  30. ^ D E Cartwright, "Tides: a scientific history", Cambridge University Press 2001, at pages 163–4.
  31. ^ S Casotto, F Biscani, "A fully analytical approach to the harmonic development of the tide-generating potential accounting for precession, nutation, and perturbations due to figure and planetary terms", AAS Division on Dynamical Astronomy, April 2004, vol.36(2), 67.
  32. ^ A T Doodson (1921), "The Harmonic Development of the Tide-Generating Potential", Proceedings of the Royal Society of London. Series A, Vol. 100, No. 704 (1 December 1921), pp. 305–329.
  33. ^ See e.g. T D Moyer (2003), "Formulation for observed and computed values of Deep Space Network data types for navigation", vol.3 in Deep-space communications and navigation series, Wiley (2003), e.g. at pp.126–8.
  34. ^ NOAA. "Eastport, ME Tidal Constituents". NOAA. Retrieved 22 May 2012.
  35. ^ See e.g. T D Moyer (2003), "Formulation for observed and computed values of Deep Space Network data types for navigation", vol.3 in Deep-space communications and navigation series, Wiley (2003), e.g. at pp.126-8.
  36. ^ a b Melchior, P. (1971). "Precession-nutations and tidal potential". Celestial Mechanics. 4 (2): 190–212. Bibcode:1971CeMec...4..190M. doi:10.1007/BF01228823. and T D Moyer (2003) already cited.
  37. ^ See for example Melchior (1971), already cited, at p.191.

External links[edit]