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 the 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.
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 ^{[1]}. The explanation of tidal motion as resulting from 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 ^{[2]}, and also for the diurnal tide in the Pacific Ocean ^{[3]}. 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).
Tidal energy dissipation
The total power available from Moon and Sun on the global ocean is estimated at 3.7 TW ([math]10^{12}[/math] Watts); after allowing for a comparatively small dissipation in the atmosphere and the solid Earth, 3.5 TW remain to be dissipated in the ocean (Munk and Wunsch, 1998)^{[5]}. For comparison, the geothermal heat loss is 30 TW, and the equatortopole ocean heatflow is 2,000 TW. Solar radiation input (175,000 TW) is five orders of magnitude greater than the tidal power.
The tidal energy dissipation due to the [math]M_2[/math] tide can be exactly calculated from astronomic observation programs at 2.5 TW, of which 0.1 TW is dissipated in the atmosphere and the solid Earth (Munk, 1997)^{[7]}. It is estimated that 1.8 TW of [math]M_2[/math] energy (2.5 TW of all lunar and solar constituents) is lost in the turbulent bottom boundary layers of marginal seas due to the work against bottom friction which opposes tidal currents. This kind of energy dissipation is concentrated in a few shelf areas of strong tidal current, such as the northwest European Shelf, the Patagonian Shelf, the Yellow Sea, the Timor and Arafura Seas, and the Amazon Shelf (Fig. 2). The other 0.6 TW of [math]M_2[/math] tidal dissipation (1.0 TW of all lunar and solar constituents) occurs in the open ocean through the scattering by oceanbottom topography of surface tides into internal tidal waves (Egbert and Ray, 2000)^{[6]}. The breaking of internal waves may be concentrated in a few areas, generally near rugged bottom topography, for example seamounts and midocean ridges (Fig. 2), and is believed to be a principal contributor to drive the ocean interior mixing (Munk and Wunsch, 1998)^{[5]}.
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 (loading tide);
 Tideinduced seafloor deformation related to the solid earth tide;
 Gravitational attraction induced by the mass of the ocean on the ocean itself (selfattraction).
See Hendershott (2007) ^{[8]} 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 ^{[9]}). 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. 3), followed by the semidiurnal solar tide [math]S_2[/math] (period = 12 h). 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 ^{[10]}. 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 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 ^{[12]}. 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 ^{[13]}; for dominant semidiurnal tides this effect is small. Fig. 4 shows a world map of tidal asymmetry computed according to the tidal constants derived from TPXO7ATLAS 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 relative phase 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 ^{[14]}. 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 Tidal asymmetry and tidal basin morphodynamics and 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 ^{[15]} and for the East China shelf ^{[16]} show that the local tide generating force influences the cooscillating semidiurnal tide in these shelf seas by no more than about 1%. 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 (water depth no much greater than ten times the tidal amplitude, or less) 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 (overtides) or superposition (compound tides) 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. 5, coastal regions are indicated with an important M4 tidal component; they coincide largely with coastal zones where the semidiurnal tide is strong (Fig. 3).
Tides in estuaries and tidal lagoons
The tidal wave is further distorted when entering coastal inlets and estuaries. This is due the shallowness of these systems (the water depth is in many cases no greater than a few times the tidal amplitude or even less) and due to interaction with the basin topography (for example, expansion of the tidal flood wave over intertidal flats). As a consequence of tidal asymmetry, substantial differences may arise between the strength of flood and ebb currents. This results in a net import or export of sediment and corresponding sedimentation or erosion. The basin morphology will evolve and tidal asymmetry will change consequently. This morphodynamic feedback process is described in more detail in the article Tidal asymmetry and tidal basin morphodynamics. Tidal distortion in some estuaries and shallow tidal lagoons may become so strong that the tidal flood wave evolves into a tidal bore. This process is described in more detail in the article Tidal bore dynamics.
Related articles
 Tidal motion in shelf seas
 Coriolis acceleration
 Tidal asymmetry and tidal basin morphodynamics
 Morphology of estuaries
 Tidal bore dynamics
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
 ↑ 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
 ↑ Müller, M. 2007. The free oscillations of the world ocean in the period range 8 to 165 hours including the full loading effect. Geophys. Res. Letters 34, L05606, doi:10.1029/2006GL028870, 2007
 ↑ Bearman, G. (Ed.) 1991. Waves, tides and shallowwater processes. The Open University, Pergamon Press, Oxford
 ↑ ^{5.0} ^{5.1} Munk, W. H. and Wunsch C. 1998. Abyssal recipes II: Energetics of tidal and wind mixing, DeepSea Research 45: 19772010
 ↑ ^{6.0} ^{6.1} Egbert, G. D., and R. D. Ray (2000), Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data, Nature, 405, 775778
 ↑ Munk, W. H. 1997. Once again: once again–tidal friction, Progress in Oceanography 40: 735
 ↑ 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
 ↑ Garrett, C.J.R. and Munk, W.H. 1971. The age of the tide and the "Q" of the oceans. DeepSea Res. 18: 493503
 ↑ Parker, B.B. 2007. Tidal Analysis and Prediction. NOAA Special Publication NOS COOPS 3
 ↑ ^{11.0} ^{11.1} Song, D., X. H. Wang, A. E. Kiss, and Bao, X. 2011. The contribution to tidal asymmetry by different combinations of tidal constituents. J. Geophys. Res., 116, C12007
 ↑ Doodson, A.T. 1921. The harmonic development of the tidegenerating potential. Proc. R. Soc. London, Ser.A 100: 305329
 ↑ 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
 ↑ Nidzieko, J. 2010. Tidal asymmetry in estuaries with mixed semidiurnal/diurnal tides. J. Geophysical Research 115, C08006, doi:10.1029/2009JC005864
 ↑ Pingree, R.D. and Griffiths, D.K. 1987. Tidal friction for semidiurnal tides. Cont.Shelf Res. 7: 11811209
 ↑ Kang, S.K., Lee, S.R. and Lie, H.J. 1998. Finegrid tidal modelling of the Yellow and East China seas. Cont.Shelf Res. 18: 739772
 ↑ 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
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