Ocean and shelf tides
In this article the term tide refers to the astronomical tide.
Contents
Introduction
Tidal motion is the oscillation of ocean waters under influence of the attractive gravitational forces of the moon and the sun. The response of the ocean to the gravitational forces follows a pattern of rotating ('amphidromic') systems, as a consequence of earth's rotation. The frequencies are determined by the relative periodic motions of moon, sun and earth surface. The amplitude of the tidal oscillation is very small compared to ocean depths. The ocean tidal oscillation in each point can therefore be represented by a linear superposition of sinusoidal tidal components with frequencies corresponding to the different cycles of the gravitational forces. The most important tidal components have a periodicity which is close to semidiurnal or diurnal, due to earth's rotation.
This article is an adaptation of the sections on ocean tides from the book Dynamics of Coastal Systems ^{[1]}.
Tide generation
Moon and sun
The regular daily upward and downward motion of the water surface along the coastline is the most visible expression of tidal forcing. In ancient Greece it was recognized that tides are in some way related to sun and moon, but this relationship could not be explained ^{[2]}. The explanation of tidal motion as a consequence of gravitational forces was given by Newton. The lunar and solar tide generating forces acting on the oceans are extremely small; they are about a factor [math]\, 10^{7} \,[/math] smaller than the gravitational force of the earth. The tidegenerating force of the moon is about twice as large as that of the sun.
Tidal wave and ocean basin resonance
Ocean tides owe their strength for a large part to resonance; tidal motion is amplified for tidal components with a frequency close to the frequency of free oscillations in the ocean basins. This is the case in particular for the semidiurnal tide in the North Atlantic and Pacific Oceans ^{[3]}, and also for the diurnal tide in the Pacific ocean ^{[4]}. Tidal waves in wide ocean basins (width typically larger than a few thousand km) rotate around points of zero amplitude, the amphidromic points; the rotation is counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere. This due to earth's rotation, which induces Coriolis acceleration of tidal currents. Fig.1 shows the semidiurnal tidal wave in the world's oceans; it comprises a large number of amphidromic systems, where the tide rotates in resonance with the semidiurnal component of the gravitational force. This ocean tidal wave pattern can be described in first approximation by a combination of Kelvin and Sverdrup waves (also called Poincaré waves, see Tidal motion in shelf seas).
Modelling ocean tides
Modelling ocean tides remains a challenge, even with modern fast computers. Because the tidal wavelength is much larger than ocean depth, these models are generally based on the hydrostatic shallowwater equations. Relatively simple models considering the response of a homogeneous ocean to the tidegenerating forces yield results in qualitative agreement with observations. However, for better quantitative agreement stratification and internal tide generation (at ocean ridges, in particular) should be included. Other factors that need to be taken into account are:
 Tideinduced seafloor deformation under the weight of the water column;
 Tideinduced seafloor deformation related to the solid earth tide;
 Gravitational attraction induced by the mass of the ocean on the ocean itself.
See Hendershott (2007) ^{[6]} for a discussion of these different factors.
Tidal components
Semidiurnal periodicity
In order to explain tidal motion, it is not sufficient to consider only the gravitational forces exerted by moon and sun on the earth's water masses. Tide generation results from the local imbalance at the earth's surface of two opposing factors: the gravitational forces acting between the earth and the moon (and between the earth and the sun), and the centrifugal acceleration related to the orbital motions of these celestial bodies. In fact, only the tangential component of the gravitational force is relevant  water is not accelerated away from the earth's surface. Because the gravitational force and the centrifugal acceleration are in balance at any location on the earth's surface twice during each diurnal rotation, the major periodicity is approximately semidiurnal (a little more than 12 hours because of moon's orbital motion).
Diurnal tide and springneap cycle
Because the solar and lunar orbits do not coincide with the equatorial plane (the angle between orbital plane and equatorial plane is called declination), a daily inequality arises in the semidiurnal cycle. The daily inequality is strongest in shelf seas that resonate at diurnal frequency and which are situated close to amphodromic points of the semidiurnal tide. In such regions the tide is mainly diurnal.
Due to the [math]\approx[/math] 30day orbital motion of the moon, the moonearth and sunearth axes approximately coincide every 15 days (syzygy). This causes a 15day cycle of neap tide and spring tide; spring tide follows full moon and new moon and neap tide follows halfmoon (with a delay of one or two days due to frictional dissipation ^{[7]}). In fact, the 15day period corresponds to the frequency difference of the semidiurnal lunar component ([math]M_2[/math]) and the semidiurnal solar component ([math]S_2[/math]); the interference of these components produces the neapspring variation of the tidal amplitude.
Other tidal components
The different cycles in the relative motions of moon, sun and earth surface generate tidal waves with corresponding periods. The lunar semidiurnal tidal component ([math]M_2[/math], period [math]\approx[/math] 12 h 25 min) is generally the largest tidal constituent (see Fig.2), followed by the semidiurnal solar tide [math]S_2[/math]. Other important tidal constituents are:
 the diurnal component [math]K_1[/math], which is related to the declination of the lunar and solar orbits relative to the equatorial plane (period [math]\approx[/math] 24 h),
 the diurnal lunar component [math]O_1[/math] (period [math]\approx[/math] 26 h) and
 the diurnal solar component [math]P_1[/math] (period [math]\approx[/math] 24 h).
There are many other tidal components of smaller magnitude and lower frequency related to periodicity in the lunar and terrestrial orbits. A modulation of the mean tidal amplitude of the order of 5% is related to the 18.6 year oscillation in the declination of the lunar orbit.
Prediction of tides
Because the astronomical tide responds to a cyclic forcing by the gravitational motions of earth, moon and sun (including earth's rotation), tides are a cyclic phenomenon at any place on earth and can be predicted by analysing observations from the past. Therefore the amplitude and phase of all relevant tidal components need to be determined. The most usual method today is based on the leastsquares technique ^{[8]}. The tidal elevation is represented by a sum of the (sinusoidal) tidal components that are expected to yield a significant contribution. The amplitudes and phases of the tidal components are free parameters which are determined by a leastsquare fit of the representation to the observed tidal elevation record. The tidal record should be sufficiently long to eliminate meteorological influences. The observation record should also be longer (at least a few times) than the largest period appearing in the representation. For resolving two components with close periods [math]T[/math] and [math]T+\Delta T[/math] the length of the observation record should exceed a few times [math]\; T^2 / \Delta T [/math]. For many stations around the world tidal predictions are available (see for example https://maree.shom.fr/) .
Asymmetric ocean tides
Because of the minor role of friction in the propagation of ocean tides, individual astronomic tidal components are well described by sinusoidal functions. This implies that for each component the rising and falling branches are symmetric. This also holds on average for a superposition of tidal components, if the frequencies [math]\omega_i[/math] of the major tidal components are not linearly related with integer coefficients ([math]\omega_i[/math] is different from any combination of sums and differences of [math]\omega_j, \; j \neq i[/math]). Taken over a sufficiently long time interval the average period of rising tide then equals the average period of falling tide. Hence, at the continental shelf boundary there is symmetry between the astronomical forcing of flood currents and ebb currents.
However, certain combinations of astronomic components yield asymmetric tides, irrespective of the averaging period. The reason is that most tidal constituents result from a superposition of a limited number of basic cycles in the relative motions of earth, moon and sun. The frequencies of these tidal constituents correspond to sums and differences of the basic frequencies ^{[9]}. This implies that different tidal constituents may interfere in such a way that flood and ebb are modulated in a systematic, asymmetric way. Such asymmetries may become significant in the case of strong diurnal tides; for dominant semidiurnal tides this effect is small ^{[10]}. A world map of tidal asymmetry based on harmonic analysis of 335 tidal stations worldwide has been established by Song et al. (2011)^{[11]}.
A particular example is the combination of the [math]M_2, K_1[/math] and [math]O_1[/math] tides. The sum of the frequencies of the [math]K_1[/math] and [math]O_1[/math] tidal constituents equals the frequency of the [math]M_2[/math] tidal constituent. In regions where the amplitudes of the [math]K_1[/math] and [math]O_1[/math] tides are not small compared to the amplitude of the [math]M_2[/math] tide, the sum of these three tidal constituents yields an asymmetric tide, with an asymmetry depending on the respective phases of these tidal constituents. Along the Californian coast, for example, the combination of [math]M_2, K_1[/math] and [math]O_1[/math] yields a tidal asymmetry with longer duration of rising tide compared to falling tide ^{[12]}. Duration asymmetry of tidal rise and tidal fall induces an asymmetry in the strength of flood and ebb currents in tidal inlets. This has consequences for residual sediment transport and for the morphology of the inward tidal basins (see Morphology of estuaries for more details).
Tides on the continental shelf and in coastal basins
Tidal amplification
The tides generated in the ocean propagate to the continental shelf, where the tidal range may increase further. Different phenomena contribute to this amplification: resonant dimensions of the shelf sea, slowing down of waveenergy propagation ('shoaling') or concentration of the tidal energy flux in areas of reduced width ('funneling'). The maximum spring tidal range can exceed 14 m in some funnelshaped bays, such as the Bay of Fundy (Canada), Bristol Channel (UK) and Baie du Mont Saint Michel (France).
Tidal amplification in shelf seas is counteracted by frictional momentum dissipation. The strong tidal motion occurring in many shelf seas is thus not locally produced by tide generating forces, but results from cooscillation with ocean tides and from local topographic amplification. Numerical tidal studies for the northwest European shelf ^{[13]} and for the East China shelf ^{[14]} show that the local tide generating force influences the cooscillating semidiurnal tide in these shelf seas by no more than about 1%. This is the main reason why no substantial tidal motion is generated in shallow enclosed seas or lakes. Dissipation of ocean tidal energy is mainly due to frictional dissipation in shelf seas ^{[15]}; however, nonnegligible dissipation also takes place in the ocean ^{[16]}. Tidal resonance in shelf seas generally has a damping effect on ocean tides ^{[17]}.
Higher harmonic components
Tides on the continental shelf are due to cooscillation with the ocean tides at the continental shelf boundaries, as noted earlier. However, the nonlinearity of tidal propagation in shallow environments with strong topography (presence of large tidal flats, for example) alters the sinusoidal character of the ocean tide at the shelf boundary. The distortion of the tidal wave can be represented by additional tidal components corresponding to multiples of the dominant ocean tidal frequencies ^{[18]}^{[19]}^{[20]}. An important consequence of tidal wave distortion is the asymmetric character of tidal motion in shallow water: the periods of tidal rise and tidal fall are not equal. This asymmetry is represented in particular by a quarterdiurnal tidal component (indicated by M4), which induces different strengths of flood currents and ebb currents. In Fig. 3 coastal regions are indicated with an important M4 tidal component; they coincide largely with coastal zones where the semidiurnal tide is strong (Fig. 2). Tidal asymmetry in shallow coastal waters, estuaries and tidal rivers, plays a crucial role in tidetopography interaction ^{[1]}, because of the highly nonlinear dependence of sediment transport on tidal current strength. A further explanation is given below.
Ebbflood asymmetry and coastal basin morphodynamics
In bays and estuaries where the tidal amplitude is comparable to or larger than the average water depth the highwater crest of the tidal wave travels much faster than the lowwater trough. In this case the tidal wave distortion can become so strong that the highwater crest overtakes the lowwater trough. The tidal flood wave then propagates inland as a bore (see Fig. 4). Tidal bores are common in funnelshaped shallow bays and estuaries with strongly amplified tides (for example the funnelshaped bays mentioned above), especially for spring tidal conditions ^{[21]}. The flood current during the passage of the bore is very strong and largely exceeds the maximum currents occurring during ebb. The most famous tidal bore occurs in the Qiantang estuary (China) ^{[22]}. See also the article Tidal wave deformation and tidal bores.
More in general, one may distinguish two causes of tidal wave asymmetry ^{[23]}.
 Water depths in the tidal basin are substantially larger at high water than at low water and there is less frictional dissipation during the passage of the highwater wave crest than during the passage of the lowwater trough. The tidal wave crest then propagates faster into the basin than the tidal wave trough and the period of tidal rise (from low water to high water) will thus be shorter than the period of tidal fall (from high water to low water). This implies that flood currents will be on average stronger than ebb currents.
 The main tidal channel of the tidal basin is bordered by large intertidal flats. The propagation of the tidal wave crest is retarded, because of lateral expansion of the tidal flood wave over the intertidal flats. The highwater wave crest then propagates into the basin more slowly than the lowwater wave trough, which travels through the main tidal channel. In this case the period of tidal rise will be longer than the period of tidal fall and consequently ebb currents will be on average stronger than flood currents.
In most shallow tidal bays and estuaries both processes compete. This competition may lead to approximately equal strength of flood and ebb currents, in spite of strong nonlinearity of tidal wave propagation in these systems. When flood and ebb currents have approximately equal strength the tidally averaged residual sediment transport along the tidal basin will be small and the basin bathymetry will be in a state close to morphodynamic equilibrium. When flood sediment transport and ebb sediment transport are not in balance the tidal basin morphology will evolve (changing channel depth and intertidal area), until reaching a state of morphodynamic equilibrium ^{[24]}.
Related articles
 Tidal motion in shelf seas
 Coriolis acceleration
 Tidal wave deformation and tidal bores
 Morphology of estuaries
Further reading
Cartwright, D.E. 1999. Tides, a scientific history. Cambridge Univ.Press, UK, 292 pp.
Pedlosky, J. 1979. Geophysical Fluid Dynamics. Springer Verlag, 624 pp.
References
 ↑ ^{1.0} ^{1.1} Dronkers, J. 2017. Dynamics of Coastal Systems. World Scientific Publ. Co., Advanced Series on Ocean Engineering, 740 pp.
 ↑ Ekman, M. 1993. A concise history of the theories of tides, precessionnutation and polar motion. Surveys in Geophysics 14: 585617
 ↑ Heath, R.A. 1981. Estimates of the resonant period and Q in the semidiurnal tidal band in the North Atlantic and Pacific Oceans. DeepSea Research. Vol. 28A: 481 – 493
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 ↑ Bearman, G. (Ed.) 1991. Waves, tides and shallowwater processes. The Open University, Pergamon Press, Oxford
 ↑ Hendershott, M.C. 2007. Long Waves and Ocean Tides. In: Evolution of Physical Oceanography (Eds. B. Warren, and C. Wunsch) MIT OpenCourseWare, https://ocw.mit.edu
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 ↑ Hoitink, A.F.J., Hoekstra, P. and van Mare, D.S. 2003. Flow asymmetry associated with astronomical tides: Implications for residual transport of sediment. J.Geophys.Res. 108: 131  138
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 ↑ Pingree, R.D. and Griffiths, D.K. 1987. Tidal friction for semidiurnal tides. Cont.Shelf Res. 7: 11811209
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 ↑ Munk, W. 1997. Once again: once again  tidal friction. Prog. Oceanog. 40: 735
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 ↑ Arbic, B.K., Karsten, R.H. and Garrett, C. 2009. On tidal resonance in the global ocean and the back‐effect of coastal tides upon open‐ocean tides. AtmosphereOcean, 47: 239266
 ↑ Aubrey, D.G. and Speer, P.E. 1985. A study of nonlinear tidal propagation in shallow inlet/estuarine systems. Part I: Observations. Est. Coast. Shelf Sci. 21: 185205
 ↑ Friedrichs, C. T., and Aubrey, D. G. 1988. Nonlinear tidal distortion in shallow wellmixed estuaries: a synthesis. Estuarine, Coastal and Shelf Science, 27: 521545
 ↑ Boy, J.P., Llubes, M., Ray, R., Hinderer, J., Florsch, N., Rosat, S., Lyard, F. and Letellier, T. 2004. Nonlinear oceanic tides observed by superconducting gravimeters in Europe. J. Geodynamics 38: 391–405
 ↑ Bonneton, P., Bonneton, N., Parisot, J.P. and Castelle B. 2015. Tidal bore dynamics in funnelshaped estuaries, J. Geophys. Res. Oceans 120: 923–941, doi:10.1002/2014JC010267.
 ↑ Jiyu, C., Cangzi, L., Chongle, Z. and Walker, H.J. 1990. Geomorphological development and sedimentation in Qiantang estuary and Hangzou Bay. J.Coast.Res. 6: 559572
 ↑ Dronkers, J. 1986. Tidal asymmetry and estuarine morphology. Neth.J.Sea Res. 20: 117131
 ↑ Dronkers, J. 1998. Morphodynamics of the Dutch Delta. In: Physics of estuaries and coastal seas. Ed.: J.Dronkers and M.B.A.M. Scheffers, Balkema, Rotterdam: 297304
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