The tides


Tide refers to the process of periodic sea level change, usually semi-diurnal (period close to 12 hours), but diurnal in some regions. The tide is due to the lunar attraction, and to a lesser extent to the attraction of the Sun which modulates its amplitude according to the phase of the Moon and different astronomical periods. Very violent local currents are associated with tides in coastal areas. In addition to these well-known local effects, the tide plays a global role in climate by contributing to the vertical mixing of the ocean. Tidal oscillation excites internal waves propagating along interfaces due to temperature differences within the ocean, producing mixing as they break. Finally, on a geological time scale, the tide slows down the Earth’s rotation and keeps the Moon away from the Earth.


The links between the tide and the movement of the Moon have been known empirically since antiquity (ref 1). The time of low water1 (low water) and high water (high water) delays by about 25 minutes per tide, or 1/60th day, corresponding to the displacement of the Moon on its orbit by 1/60th of a turn in 12 hours (1 turn in 30 days). Because of this delay the effective tidal period is 12 h 25 mn. It is also known that tides are more intense during the full and new Moon (bright waters) than during the first and last districts (dead waters). This indicates that the effect is governed by the Moon, with a more modest contribution from the Sun. The tide is particularly strong at equinoxes and also depends on the distance from the Moon, which varies by about 10% because of its elliptical orbit. The tidal amplitude in a given place is thus modulated by a tidal coefficient that varies from 20 to 120 according to the different astronomical periods.

Tidal range, the difference in height between low water and high water, also depends greatly on the location, with the highest values reaching nearly 20 m in Ungava Bay (Quebec), 16.5 m in the Severn Estuary (Great Britain) and 15 m at Mont Saint-Michel. However, it is limited to a few tens of centimetres in other parts of the ocean.

Due to its elasticity, the solid Earth is also subject to a tidal effect but with a smaller amplitude, in the order of 10 cm. What we observe at the seaside is the difference between the ocean tide and this land tide.

Newton’s static theory

Tides were very early interpreted as an attraction effect of the Moon and the Sun. However, these explanations came up against the fact that the sea is raised not only on the side of the Moon, but also on the opposite side, leading to the 12-hour period (rather than 24 hours). It was Isaac Newton (1643-1727) who first understood this paradox thanks to his theory of universal gravitation published in 1687 in his famous book Philosophiae Naturalis Principia (ref. 2). Tides play an important role because they were the most tangible effect of attraction by a body outside the Earth at that time.

Newton first understood that if the Earth keeps the Moon in orbit by its attraction force, the Moon must in turn exert an equal and opposite force on the Earth: this is the principle of action and reaction. Thus the Earth revolves a little around the Moon, more precisely around their common barycentre (point G in the figure in Box 2). Every body on Earth accompanies it in its movement around this barycentre, in the same way that an orbiting cosmonaut remains weightless near his spacecraft. The reason is that every body undergoes the same acceleration in a field of gravity regardless of its mass. What will move the ocean relative to the Earth is therefore not the attraction field of the Moon, but the difference between this field and the one acting at the centre of the Earth. An excess attraction is exerted as close as possible to the Moon and a lack of attraction on the opposite side further away, producing a bead on each side (see Box 2).

In the so-called static theory proposed by Newton, this bead is assumed to be fixed in relation to the Earth-Moon system. During its rotation around the Earth, a point thus passes successively through each bead leading to two high tides per day, hence the semi-day period.

The effect of the Sun is added to that of the Moon when it is in the same direction (new Moon), but also when it is in the opposite direction (full Moon), due to the double bead. This explains the alternation observed between spring tides and dead waters. The attraction force of the Sun is stronger than that of the Moon, but the difference between the two sides is smaller due to the long distance, producing a lower tidal effect (see Box 2).

Tidal amplitude variations

A complication is that these different rotations are carried out along different axes: the Earth’s axis of rotation is inclined by 23°26′ with respect to the plane of the Earth’s orbit, itself close to the plane of the Moon’s orbit. Thus a point of the Earth runs through the beads in an inclined circle. The result is an asymmetry between the two semi-diurnal tides, which can be interpreted as a small diurnal component added to the main semi-diurnal component. This diurnal component dominates the tide in some places, see section 4.

The Earth’s axis of rotation is oriented towards or away from the Sun at the solstices, while it is oriented transversely to the direction of the Sun at the equinox. We can convince ourselves that the latter case is more favourable to tides by considering the limit case of a 90° inclination: at the solstice the axis of the Earth would be oriented along the axis of the solar bulge, a rugby ball whose tip would be directed towards the Sun, then a point on Earth would turn without variation in height. At the equinox, on the other hand, the axis of rotation would be perpendicular to the axis of the bead, and the point would pass successively through the beads and hollows. It is therefore conceivable that the equinox tides of living waters are stronger than the solstice tides.

Finally, the Moon’s orbit is not circular, but elliptical, so that the distance D to the Earth varies by 10% between the perihelion (minimum) and the aphelion (maximum). As a result, the tidal effect is greater at perihelion by 30% (due to the dependence in 1 / D3). These different astronomical effects are taken into account very precisely to establish tide tables.

Tidal waves

This static Newtonian scheme assumes that the oceanic bead, fixed with respect to the Moon, thus propagates with respect to the Earth at the opposite speed to its rotation, i.e. 450 m/s at the equator. This is not possible because a deformation of the ocean surface propagates at a finite speed, limited to about 200 m/s (box 3) and is therefore late in relation to the position of the Moon. This leads to a late offset of the bead as shown in Figure b in Box 2. The shape of the coasts also strongly constrains this propagation, with ocean basins behaving like large basins of water shaken by the tidal force. There are established specific modes of oscillation, somewhat similar to the modes of vibration of sound waves in a musical instrument.

Newton’s theory remains accurate as a driving force, but the resulting deformation therefore depends on these propagation phenomena and the resonances that occur when the excitation frequency coincides with the natural oscillation frequencies of ocean basins. The study of tides has generated a great deal of work throughout the 19th and 20th centuries (Box 1) because of its fundamental interest and importance for navigation and coastal development. The modulation of tides by different astronomical effects can be expressed as a sum of excitations at different periods, the main one being the semi-diurnal period (called M2). Excitement during the daytime period, about 20 times lower, results from a slight asymmetry in the passage through the two beads (see section 3). However, this daytime period dominates the tide in some regions (e. g. Vietnam) because it coincides with specific oscillation modes.

Tidal amplitude is now mapped to within one centimetre using altimetry satellites (see Figure 1). We see that the amplitude is very variable, depending on the bellies of oscillation (maximum in red) and the nodes of oscillation where the amplitude is cancelled (in blue). Lines of equal phase are also shown: they represent the delay of the maximum tide in relation to the passage of the Moon at its zenith. The tidal wave propagates perpendicular to these lines, i. e. by rotating around the knots. This rotation, due to the Coriolis force, is counter-clockwise in the Northern Hemisphere.

Current numerical models make it possible to reproduce and predict these tidal phenomena with an accuracy of around 1cm. The main difficulties are the consideration of friction on the ocean floor in turbulent conditions and energy losses due to internal tidal excitation (see section 5).

Figure 1. Amplitude and tidal phase M2 (semi-diurnal period) measured by the altimetric satellite Topex-Poseidon. The colours represent the amplitude (the tidal range is twice the amplitude), and the white lines the phase, i.e. the time separating the maximum from the passage from the Moon to the zenith.


Other influences on sea level

The tide is not the only effect influencing sea level. First of all, we can ask ourselves the question of the relevance of measurements to the nearest cm in a sea that is often rough with waves of several metres. But the average level over several square kilometres is very well defined even if the local level fluctuates very strongly. In addition, for tidal amplitudes such as those in Figure 1, the signal is filtered at a given time period (12h 25 min), similar to the frequency selection used to receive radio waves. Thus, effects acting at other frequencies are not taken into account.

Among these other effects, atmospheric pressure is a fairly immediate factor. A high pressure locally lowers the water level: an overpressure of 10 hPa = 103 N/m2 induces by simple hydrostatic equilibrium a decrease of the level by 10 cm (the thickness of a water column whose weight is 103 N/m√). On the contrary, low pressure raises the level. The overhang reaches a value of one metre at an atmospheric pressure of 913 hPa, occurring in the middle of extreme hurricanes. This rise in water levels amplifies the damage caused by waves and heavy rainfall in coastal areas.

A second effect, dynamic this time, is due to the friction force of the wind. When it is directed offshore, this force lowers the water level, and on the contrary pushes the water towards the shore if it is not. An elevation of about 1 m can be produced during heavy storms. The coincidence of these phenomena with high tides favours the breaking of protective dikes that cause flooding, such as the storm “Xynthia” that struck France in February 2010, or hurricane Katrina that flooded New Orleans in August 2005. These phenomena, depending on wind and pressure, are however much more predictable than the intense and very local precipitation that causes flash floods.

In the longer term, average sea level increases in the presence of global warming due to ocean expansion, which accounts for about 65%, and glacier melting for the remaining 35%. Recent measurements indicate an average elevation of about 2 mm/year.

Finally, the water level on the coast also depends on the evolution of the solid Earth. Sediment transport modifies the shoreline, through siltation or erosion. The latter effect currently tends to dominate due to dams on large rivers that reduce sediment input. The Louisiana coast is severely eroded due to the decrease in sediment brought by the Mississippi River. Deep geological movements also contribute, changing the shape of the coasts by continental drift over millions of years. In Canada and northern Europe, the most significant geological effect is the post-glacial rebound that raises Scandinavia by several mm per year following the relief due to the melting of the ice caps 10,000 years ago. This uplift induces by compensation a sinking of peripheral areas such as Brittany. Thus, menhirs erected on land 7000 years ago are found in the sea, after a sinking of the continent by about 7 m.

Harnessing the energy of the tide

Tidal mills have been used since the Middle Ages to harness tidal energy from favourable sites, with estuaries or coves sheltered from the waves and equipped with small dams. The principle was adopted for the Rance tidal power plant, which was commissioned in 1967. With an average capacity of 57 MW (installed capacity of 240 MW), it produces 3.5% of Brittany’s electricity consumption (and 45% of its electricity production). It remained the world’s largest tidal power plant for 45 years, until the slightly more powerful Sihwa Lake power plant in South Korea was commissioned in 2011 (254 MW installed). The facility uses a dam across the Rance estuary, with directional blade turbines that can operate in both directions, at rising or falling tide.

However, few high tide sites allow the construction of facilities of this size, and the need to preserve natural sites now makes it difficult to build them by the sea. A much more ambitious project consisted in damming the bay of Mont Saint-Michel, a particularly exceptional site in terms of tidal amplitude, but this project was abandoned in favour of the development of nuclear power plants in the 1970s.

The current trend is to directly use the currents produced by the tide through hydro turbines, the marine equivalent of wind turbines. These turbines do not require reservoirs and therefore have a lower environmental impact. However, the developments are only at the prototype stage of a few MW, with test sites in Scotland and Brittany (see Figure 2). In Scotland, the objective by 2020 is to build hydroelectric farms producing 1,000 MW. The total estimated resource in Europe is around 10,000 MW installed (5,000 MW average), 80% of which is in France and Great Britain. This energy would actually be taken from the Earth’s rotational energy (see section 8), which is dissipated as heat anyway in the absence of capture.

Figure 2. Map of the speed amplitudes of tidal currents, and prototype installation sites on the Breton coast: Raz de Sein, Ouessant and Raz Blanchard. In insert, hydrolienne model developed by EDF (diameter 10 m).

Internal tide

The density of the ocean increases with depth, with surface water being warmer (and therefore less dense) than deep water. Such density stratification can also result from salinity, for example in the Strait of Gibraltar where ocean water enters the Mediterranean Sea while remaining at the surface due to its lower density. This situation can be summarized by a two-layer model of different densities (Figure 3).

Internal wave oscillations, known as internal waves, can propagate along this interface in a similar way to surface waves. However, they are much slower, described by replacing gravity g with a reduced gravity g/, where / is the relative density difference between the two layers. In a surface layer of thickness H, the propagation velocity of the waves is therefore c=(H g/ i.e., for a typical value / 30 times less than the surface waves of a layer of the same thickness : c=1 m/s for a layer of thickness H=100 m.

As we have seen (see Figure 3), the tide carries a horizontal current over the entire water level. However, as a slope passes, this current acquires a vertical component that deforms the interface and thus generates an internal wave, called in this case the internal tide. These waves have a wavelength of about 100 km (distance travelled at 1 m /s during the 12-hour tidal period). In addition, they tend to be located in solitons, compact waves of high amplitude. Although these waves propagate at depth, the horizontal currents they generate are seen on the surface by changing wave shapes that change the brightness of the sea (Figure 3).

Internal tidal generation is observed in many parts of the ocean. One of the most active is the Strait of Luzon, separating Taiwan and the Philippines, where an underwater ridge generates internal waves in the China Sea whose vertical displacement exceeds 300 m. The dissipation of these waves by breaking contributes to the vertical mixing of the ocean, which in turn affects its general circulation and climate (see also section 6).

Astronomical effects and energy dissipation

Over astronomical times, tides increase the length of the day by 2 ms per century and push the Moon 3.8 cm/year away. The slowing down of the Earth’s rotation, corresponding to the increase in the duration of the day, is very well measured with today’s atomic clocks. Over 200 million years, this represents about an hour. As for the distance from the Moon, it is measured directly with an accuracy of 1cm by measuring the round trip time of laser pulses sent on reflectors deposited by Apollo lunar missions (ref 5).

This slowdown is confirmed by the observation of fossil corals (ref 6), whose daily growth circles make it possible to count the days in a year. Thus the year counted 410 days 400 million years ago, a 21.5-hour day. Monthly bands associated with the full moon also indicate that the year was 13 months long, i.e. the Moon was spinning faster (and was therefore closer to the Earth).

These effects are easily understood with the diagram in Figure b in Box 2. The Earth’s rotation tends to cause the bead to move, which is therefore out of phase with Newton’s static model. The lunar attraction thus exerts a torque that slows down the Earth and reciprocally brings energy to the Moon. Counterintuitively at first sight, such an energy supply tends to drive away the Moon, and therefore to slow down its rotation, whose speed decreases in 1/√r. However, the kinetic moment of the Moon, produced by the speed by the distance to the Earth r increases well (in √r), in accordance with the motor direction of the torque. The kinetic moment of the Earth decreases in the same proportion so that the total kinetic moment is preserved. The total mechanical energy decreases, converted into heat when the marine currents produced by the tide dissipate.

Astronomical measurements make it possible to accurately determine the decrease in rotational energy and thus to deduce the total power dissipated by the tides: 2.9 x1012 Watts. Oceanographers, for their part, have estimated a power dissipated by about half by studying tidal currents, most of which are active in coastal areas. It is now established that the’missing dissipation‘ is due to the excitation of the internal tide (see section 5), which spreads inside the ocean and eventually dissipates. This dissipation occurs by waves breaking, producing a slow vertical mixing of the ocean. The influence of these effects on thermo-haline circulation is currently under active research.

  1. low water is a term equivalent to low water

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