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Tides are the rise and fall of
sea level Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardise ...
s caused by the combined effects of the gravitational forces exerted by the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
(and to a much lesser extent, the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
) and are also caused by the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
and
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
orbiting one another.
Tide table Tide tables, sometimes called tide charts, are used for tidal prediction and show the daily times and levels of high and low tides, usually for a particular location. Tide heights at intermediate times (between high and low water) can be approxi ...
s can be used for any given locale to find the predicted times and
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
(or " tidal range"). The predictions are influenced by many factors including the alignment of the Sun and Moon, the phase and amplitude of the tide (pattern of tides in the deep ocean), the amphidromic systems of the oceans, and the shape of the coastline and near-shore
bathymetry Bathymetry (; ) is the study of underwater depth of ocean floors (''seabed topography''), lake floors, or river floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The first recorded evidence of water ...
(see '' Timing''). They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. Many shorelines experience semi-diurnal tides—two nearly equal high and low tides each day. Other locations have a diurnal tide—one high and low tide each day. A "mixed tide"—two uneven magnitude tides a day—is a third regular category. Tides vary on timescales ranging from hours to years due to a number of factors, which determine the
lunitidal interval The lunitidal interval measures the time lag from lunar culmination to the next high tide at a given location. It is also called the high water interval (HWI). Sometimes a term is not used for the time lag, but instead the terms ''age'' or ''est ...
. To make accurate records,
tide gauge A tide gauge is a device for measuring the change in sea level relative to a vertical datum. It its also known as mareograph, marigraph, sea-level recorder and limnimeter. When applied to freshwater continental water bodies, the instrument ma ...
s at fixed stations measure water level over time. Gauges ignore variations caused by waves with periods shorter than minutes. These data are compared to the reference (or datum) level usually called
mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
. While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also subject to change from
thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kinetic ...
, wind, and barometric pressure changes, resulting in storm surges, especially in shallow seas and near coasts. Tidal phenomena are not limited to the oceans, but can occur in other systems whenever a gravitational field that varies in time and space is present. For example, the shape of the solid part of the Earth is affected slightly by Earth tide, though this is not as easily seen as the water tidal movements.


Characteristics

Tide changes proceed via the two main stages: * The water stops falling, reaching a local minimum called low tide. * The water stops rising, reaching a local maximum called high tide. In some regions, there are additional two possible stages: * Sea level rises over several hours, covering the intertidal zone; flood tide. * Sea level falls over several hours, revealing the intertidal zone; ebb tide. Oscillating
currents Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
produced by tides are known as tidal streams or
tidal currents Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ca ...
. The moment that the tidal current ceases is called ''
slack water Slack water is a short period in a body of tidal water when the water is completely unstressed, and there is no movement either way in the tidal stream, and which occurs before the direction of the tidal stream reverses. Slack water can be esti ...
'' or ''slack tide''. The tide then reverses direction and is said to be turning. Slack water usually occurs near high water and low water, but there are locations where the moments of slack tide differ significantly from those of high and low water. Tides are commonly ''semi-diurnal'' (two high waters and two low waters each day), or ''diurnal'' (one tidal cycle per day). The two high waters on a given day are typically not the same height (the daily inequality); these are the ''higher high water'' and the ''lower high water'' in
tide table Tide tables, sometimes called tide charts, are used for tidal prediction and show the daily times and levels of high and low tides, usually for a particular location. Tide heights at intermediate times (between high and low water) can be approxi ...
s. Similarly, the two low waters each day are the ''higher low water'' and the ''lower low water''. The daily inequality is not consistent and is generally small when the Moon is over the Equator.


Reference levels

The following reference tide levels can be defined, from the highest level to the lowest: * ''Highest astronomical tide'' (HAT) – The highest tide which can be predicted to occur. Note that meteorological conditions may add extra height to the HAT. * '' Mean high water springs'' (MHWS) – The average of the two high tides on the days of spring tides. * ''Mean high water neaps'' (MHWN) – The average of the two high tides on the days of neap tides. * ''
Mean sea level There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
'' (MSL) – This is the average sea level. The MSL is constant for any location over a long period. * ''Mean low water neaps'' (MLWN) – The average of the two low tides on the days of neap tides. * '' Mean low water springs'' (MLWS) – The average of the two low tides on the days of spring tides. * ''
Lowest astronomical tide A chart datum is the water level surface serving as origin of depths displayed on a nautical chart. A chart datum is generally derived from some tidal phase, in which case it is also known as a tidal datum. Common chart datums are ''lowest astr ...
'' (LAT) – The lowest tide which can be predicted to occur.


Tidal constituents

''Tidal constituents'' are the net result of multiple influences impacting tidal changes over certain periods of time. Primary constituents include the Earth's rotation, the position of the Moon and Sun relative to the Earth, the Moon's altitude (elevation) above the Earth's Equator, and
bathymetry Bathymetry (; ) is the study of underwater depth of ocean floors (''seabed topography''), lake floors, or river floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The first recorded evidence of water ...
. Variations with periods of less than half a day are called ''harmonic constituents''. Conversely, cycles of days, months, or years are referred to as ''long period'' constituents. Tidal forces affect the entire earth, but the movement of solid Earth occurs by mere centimeters. In contrast, the atmosphere is much more fluid and compressible so its surface moves by kilometers, in the sense of the contour level of a particular low pressure in the outer atmosphere.


Principal lunar semi-diurnal constituent

In most locations, the largest constituent is the ''principal lunar semi-diurnal'', also known as the ''M2 tidal constituent'' or ''M2 tidal constituent''. Its period is about 12 hours and 25.2 minutes, exactly half a ''tidal lunar day'', which is the average time separating one lunar
zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "high ...
from the next, and thus is the time required for the Earth to rotate once relative to the Moon. Simple tide clocks track this constituent. The lunar day is longer than the Earth day because the Moon orbits in the same direction the Earth spins. This is analogous to the minute hand on a watch crossing the hour hand at 12:00 and then again at about 1:05½ (not at 1:00). The Moon orbits the Earth in the same direction as the Earth rotates on its axis, so it takes slightly more than a day—about 24 hours and 50 minutes—for the Moon to return to the same location in the sky. During this time, it has passed overhead ( culmination) once and underfoot once (at an
hour angle In astronomy and celestial navigation, the hour angle is the angle between two planes: one containing Earth's axis and the zenith (the '' meridian plane''), and the other containing Earth's axis and a given point of interest (the ''hour circle ...
of 00:00 and 12:00 respectively), so in many places the period of strongest tidal forcing is the above-mentioned, about 12 hours and 25 minutes. The moment of highest tide is not necessarily when the Moon is nearest to
zenith The zenith (, ) is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction ( plumb line) opposite to the gravity direction at that location ( nadir). The zenith is the "high ...
or
nadir The nadir (, ; ar, نظير, naẓīr, counterpart) is the direction pointing directly ''below'' a particular location; that is, it is one of two vertical directions at a specified location, orthogonal to a horizontal flat surface. The direc ...
, but the period of the forcing still determines the time between high tides. Because the gravitational field created by the Moon weakens with distance from the Moon, it exerts a slightly stronger than average force on the side of the Earth facing the Moon, and a slightly weaker force on the opposite side. The Moon thus tends to "stretch" the Earth slightly along the line connecting the two bodies. The solid Earth deforms a bit, but ocean water, being fluid, is free to move much more in response to the tidal force, particularly horizontally (see equilibrium tide). As the Earth rotates, the magnitude and direction of the tidal force at any particular point on the Earth's surface change constantly; although the ocean never reaches equilibrium—there is never time for the fluid to "catch up" to the state it would eventually reach if the tidal force were constant—the changing tidal force nonetheless causes rhythmic changes in sea surface height. When there are two high tides each day with different heights (and two low tides also of different heights), the pattern is called a ''mixed semi-diurnal tide''.


Range variation: springs and neaps

The semi-diurnal range (the difference in height between high and low waters over about half a day) varies in a two-week cycle. Approximately twice a month, around new moon and
full moon The full moon is the lunar phase when the Moon appears fully illuminated from Earth's perspective. This occurs when Earth is located between the Sun and the Moon (when the ecliptic longitudes of the Sun and Moon differ by 180°). This means ...
when the Sun, Moon, and Earth form a line (a configuration known as a syzygy), the
tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
due to the Sun reinforces that due to the Moon. The tide's range is then at its maximum; this is called the spring tide. It is not named after the
season A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of the axial parallelism of Earth's tilted orbit around the Sun. In temperate and ...
, but, like that word, derives from the meaning "jump, burst forth, rise", as in a natural spring. Spring tides are sometimes referred to as ''syzygy tides''. When the Moon is at first quarter or third quarter, the Sun and Moon are separated by 90° when viewed from the Earth, and the solar tidal force partially cancels the Moon's tidal force. At these points in the lunar cycle, the tide's range is at its minimum; this is called the neap tide, or neaps. "Neap" is an Anglo-Saxon word meaning "without the power", as in ''forðganges nip'' (forth-going without-the-power). Neap tides are sometimes referred to as ''quadrature tides''. Spring tides result in high waters that are higher than average, low waters that are lower than average, "
slack water Slack water is a short period in a body of tidal water when the water is completely unstressed, and there is no movement either way in the tidal stream, and which occurs before the direction of the tidal stream reverses. Slack water can be esti ...
" time that is shorter than average, and stronger tidal currents than average. Neaps result in less extreme tidal conditions. There is about a seven-day interval between springs and neaps. File:High tide sun moon same side beginning.png, ''Spring tide:'' Sun and Moon on the same side (0°) File:Low tide sun moon 90 degrees.png, ''Neap tide:'' Sun and Moon at 90° File:High tide sun moon opposite side.png, ''Spring tide:'' Sun and Moon at opposite sides (180°) File:Low tide sun moon 270 degrees.png, ''Neap tide:'' Sun and Moon at 270° File:High tide sun moon same side end.png, ''Spring tide:'' Sun and Moon at the same side (cycle restarts)


Lunar distance

The changing distance separating the Moon and Earth also affects tide heights. When the Moon is closest, at
perigee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...
, the range increases, and when it is at
apogee An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ell ...
, the range shrinks. Six or eight times a year perigee coincides with either a new or full moon causing
perigean spring tide A perigean spring tide is a tide that occurs three or four times per year when a perigee (the point nearest Earth reached by the Moon during its 27.3-day elliptic orbit) coincides with a spring tide (when the Sun, the Moon, and Earth are nearl ...
s with the largest '' tidal range''. The difference between the height of a tide at perigean spring tide and the spring tide when the moon is at apogee depends on location but can be large as a foot higher.


Other constituents

These include solar gravitational effects, the obliquity (tilt) of the Earth's Equator and rotational axis, the inclination of the plane of the lunar orbit and the elliptical shape of the Earth's orbit of the Sun. A compound tide (or overtide) results from the shallow-water interaction of its two parent waves.


Phase and amplitude

Because the ''M''2 tidal constituent dominates in most locations, the stage or ''phase'' of a tide, denoted by the time in hours after high water, is a useful concept. Tidal stage is also measured in degrees, with 360° per tidal cycle. Lines of constant tidal phase are called ''cotidal lines'', which are analogous to contour lines of constant altitude on topographical maps, and when plotted form a ''cotidal map'' or ''cotidal chart''. High water is reached simultaneously along the cotidal lines extending from the coast out into the ocean, and cotidal lines (and hence tidal phases) advance along the coast. Semi-diurnal and long phase constituents are measured from high water, diurnal from maximum flood tide. This and the discussion that follows is precisely true only for a single tidal constituent. For an ocean in the shape of a circular basin enclosed by a coastline, the cotidal lines point radially inward and must eventually meet at a common point, the
amphidromic point An amphidromic point, also called a tidal node, is a geographical location which has zero tidal amplitude for one harmonic constituent of the tide. The tidal range (the peak-to-peak amplitude, or the height difference between high tide and low ...
. The amphidromic point is at once cotidal with high and low waters, which is satisfied by ''zero'' tidal motion. (The rare exception occurs when the tide encircles an island, as it does around New Zealand,
Iceland Iceland ( is, Ísland; ) is a Nordic island country in the North Atlantic Ocean and in the Arctic Ocean. Iceland is the most sparsely populated country in Europe. Iceland's capital and largest city is Reykjavík, which (along with its s ...
and
Madagascar Madagascar (; mg, Madagasikara, ), officially the Republic of Madagascar ( mg, Repoblikan'i Madagasikara, links=no, ; french: République de Madagascar), is an island country in the Indian Ocean, approximately off the coast of East Africa ...
.) Tidal motion generally lessens moving away from continental coasts, so that crossing the cotidal lines are contours of constant ''amplitude'' (half the distance between high and low water) which decrease to zero at the amphidromic point. For a semi-diurnal tide the amphidromic point can be thought of roughly like the center of a clock face, with the hour hand pointing in the direction of the high water cotidal line, which is directly opposite the low water cotidal line. High water rotates about the amphidromic point once every 12 hours in the direction of rising cotidal lines, and away from ebbing cotidal lines. This rotation, caused by the
Coriolis effect In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
, is generally clockwise in the southern hemisphere and counterclockwise in the northern hemisphere. The difference of cotidal phase from the phase of a reference tide is the ''epoch''. The reference tide is the hypothetical constituent "equilibrium tide" on a landless Earth measured at 0° longitude, the Greenwich meridian. In the North Atlantic, because the cotidal lines circulate counterclockwise around the amphidromic point, the high tide passes New York Harbor approximately an hour ahead of Norfolk Harbor. South of Cape Hatteras the tidal forces are more complex, and cannot be predicted reliably based on the North Atlantic cotidal lines.


History


History of tidal theory

Investigation into tidal physics was important in the early development of
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, with the existence of two daily tides being explained by the Moon's gravity. Later the daily tides were explained more precisely by the interaction of the Moon's and the Sun's gravity. Seleucus of Seleucia theorized around 150 BC that tides were caused by the Moon. The influence of the Moon on bodies of water was also mentioned in
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
's '' Tetrabiblos''. In (''
The Reckoning of Time ''The Reckoning of Time'' ( la, De temporum ratione) is an Anglo-Saxon era treatise written in Medieval Latin by the Northumbrian monk Bede in 725. The treatise includes an introduction to the traditional ancient and medieval view of the cosm ...
'') of 725 Bede linked semidurnal tides and the phenomenon of varying tidal heights to the Moon and its phases. Bede starts by noting that the tides rise and fall 4/5 of an hour later each day, just as the Moon rises and sets 4/5 of an hour later. He goes on to emphasise that in two lunar months (59 days) the Moon circles the Earth 57 times and there are 114 tides. Bede then observes that the height of tides varies over the month. Increasing tides are called ''malinae'' and decreasing tides ''ledones'' and that the month is divided into four parts of seven or eight days with alternating ''malinae'' and ''ledones''. In the same passage he also notes the effect of winds to hold back tides. Bede also records that the time of tides varies from place to place. To the north of Bede's location (
Monkwearmouth Monkwearmouth is an area of Sunderland, Tyne and Wear in North East England. Monkwearmouth is located at the north side of the mouth of the River Wear. It was one of the three original settlements on the banks of the River Wear along with Bish ...
) the tides are earlier, to the south later. He explains that the tide "deserts these shores in order to be able all the more to be able to flood other horeswhen it arrives there" noting that "the Moon which signals the rise of tide here, signals its retreat in other regions far from this quarter of the heavens". Medieval understanding of the tides was primarily based on works of Muslim astronomers, which became available through Latin translation starting from the 12th century.
Abu Ma'shar al-Balkhi Abu Ma'shar al-Balkhi, Latinized as Albumasar (also ''Albusar'', ''Albuxar''; full name ''Abū Maʿshar Jaʿfar ibn Muḥammad ibn ʿUmar al-Balkhī'' ; , AH 171–272), was an early Persian Muslim astrologer, thought to be the greatest ast ...
(d. circa 886), in his , 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 Nur ad-Din al-Bitruji () (also spelled Nur al-Din Ibn Ishaq al-Betrugi and Abu Ishâk ibn al-Bitrogi) (known in the West by the Latinized name of Alpetragius) (died c. 1204) was an Iberian-Arab astronomer and a Qadi in al-Andalus. Al-Biṭrūjī ...
(d. circa 1204) contributed the notion that the tides were caused by the general circulation of the heavens.
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
, in his 1608 (''The theory of ebb and flood''), dismissed a large number of misconceptions that still existed about ebb and flood. Stevin pleaded for the idea that the attraction of the Moon was responsible for the tides and spoke in clear terms about ebb, flood,
spring tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables ca ...
and
neap tide Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the Moon (and to a much lesser extent, the Sun) and are also caused by the Earth and Moon orbiting one another. Tide tables c ...
, stressing that further research needed to be made. In 1609 Johannes Kepler also correctly suggested that the gravitation of the Moon caused the tides, which he based upon ancient observations and correlations.
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He wa ...
in his 1632 ''
Dialogue Concerning the Two Chief World Systems The ''Dialogue Concerning the Two Chief World Systems'' (''Dialogo sopra i due massimi sistemi del mondo'') is a 1632 Italian-language book by Galileo Galilei comparing the Copernican system with the traditional Ptolemaic system. It was tran ...
'', whose working title was ''Dialogue on the Tides'', gave an explanation of the tides. The resulting theory, however, was incorrect as he attributed the tides to the sloshing of water caused by the Earth's movement around the Sun. He hoped to provide mechanical proof of the Earth's movement. The value of his tidal theory is disputed. Galileo rejected Kepler's explanation of the tides.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
(1642–1727) was the first person to explain tides as the product of the gravitational attraction of astronomical masses. His explanation of the tides (and many other phenomena) was published in the '' Principia'' (1687) and used his theory of universal gravitation to explain the lunar and solar attractions as the origin of the tide-generating forces. Newton and others before
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
worked the problem from the perspective of a static system (equilibrium theory), that provided an approximation that described the tides that would occur in a non-inertial ocean evenly covering the whole Earth. The tide-generating force (or its corresponding
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
) is still relevant to tidal theory, but as an intermediate quantity (forcing function) rather than as a final result; theory must also consider the Earth's accumulated dynamic tidal response to the applied forces, which response is influenced by ocean depth, the Earth's rotation, and other factors. In 1740, the
Académie Royale des Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the ...
in Paris offered a prize for the best theoretical essay on tides. Daniel Bernoulli,
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
,
Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for bei ...
and Antoine Cavalleri shared the prize. Maclaurin used Newton's theory to show that a smooth sphere covered by a sufficiently deep ocean under the tidal force of a single deforming body is a
prolate A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
spheroid (essentially a three-dimensional oval) with major axis directed toward the deforming body. Maclaurin was the first to write about the Earth's rotational effects on motion. Euler realized that the tidal force's ''horizontal'' component (more than the vertical) drives the tide. In 1744
Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
studied tidal equations for the atmosphere which did not include rotation. In 1770 James Cook's
barque A barque, barc, or bark is a type of sailing vessel with three or more masts having the fore- and mainmasts rigged square and only the mizzen (the aftmost mast) rigged fore and aft. Sometimes, the mizzen is only partly fore-and-aft rigged, b ...
HMS ''Endeavour'' grounded on the
Great Barrier Reef The Great Barrier Reef is the world's largest coral reef system composed of over 2,900 individual reefs and 900 islands stretching for over over an area of approximately . The reef is located in the Coral Sea, off the coast of Queensland, ...
. Attempts were made to refloat her on the following tide which failed, but the tide after that lifted her clear with ease. Whilst she was being repaired in the mouth of the
Endeavour River The Endeavour River ( Guugu Yimithirr: ''Wabalumbaal''), inclusive of the Endeavour River Right Branch, the Endeavour River South Branch, and the Endeavour River North Branch, is a river system located on Cape York Peninsula in Far North Queens ...
Cook observed the tides over a period of seven weeks. At neap tides both tides in a day were similar, but at springs the tides rose in the morning but in the evening. Pierre-Simon Laplace formulated a system of partial differential equations relating the ocean's horizontal flow to its surface height, the first major dynamic theory for water tides. The Laplace tidal equations are still in use today.
William Thomson, 1st Baron Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
, rewrote Laplace's equations in terms of
vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
which allowed for solutions describing tidally driven coastally trapped waves, known as
Kelvin wave A Kelvin wave is a wave in the ocean or atmosphere that balances the Earth's Coriolis force against a topographic boundary such as a coastline, or a waveguide such as the equator. A feature of a Kelvin wave is that it is non-dispersive, i.e., the ...
s. Others including Kelvin and Henri Poincaré further developed Laplace's theory. Based on these developments and the lunar theory of E W Brown describing the motions of the Moon,
Arthur Thomas Doodson Arthur Thomas Doodson (31 March 1890 – 10 January 1968) was a British oceanographer. Early life He was born at Boothstown, Salford, the son of cotton-mill manager Thomas Doodson. He was educated at Rochdale secondary school and then in 1908 en ...
developed and published in 1921 the first modern development of the tide-generating potential in harmonic form: Doodson distinguished 388 tidal frequencies. Some of his methods remain in use.


History of tidal observation

From ancient times, tidal observation and discussion has increased in sophistication, first marking the daily recurrence, then tides' relationship to the Sun and moon. Pytheas travelled to the
British Isles The British Isles are a group of islands in the North Atlantic Ocean off the north-western coast of continental Europe, consisting of the islands of Great Britain, Ireland, the Isle of Man, the Inner and Outer Hebrides, the Northern Isles, ...
about 325 BC and seems to be the first to have related spring tides to the phase of the moon. In the 2nd century BC, the Hellenistic astronomer Seleucus of Seleucia correctly described the phenomenon of tides in order to support his heliocentric theory. He correctly theorized that tides were caused by the
moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
, although he believed that the interaction was mediated by the
pneuma ''Pneuma'' () is an ancient Greek word for "breath", and in a religious context for " spirit" or "soul". It has various technical meanings for medical writers and philosophers of classical antiquity, particularly in regard to physiology, and is ...
. He noted that tides varied in time and strength in different parts of the world. According to Strabo (1.1.9), Seleucus was the first to link tides to the lunar attraction, and that the height of the tides depends on the moon's position relative to the Sun. The ''Naturalis Historia'' of
Pliny the Elder Gaius Plinius Secundus (AD 23/2479), called Pliny the Elder (), was a Roman author, naturalist and natural philosopher, and naval and army commander of the early Roman Empire, and a friend of the emperor Vespasian. He wrote the encyclopedic ' ...
collates many tidal observations, e.g., the spring tides are a few days after (or before) new and full moon and are highest around the equinoxes, though Pliny noted many relationships now regarded as fanciful. In his ''Geography'', Strabo described tides in the
Persian Gulf The Persian Gulf ( fa, خلیج فارس, translit=xalij-e fârs, lit=Gulf of Fars, ), sometimes called the ( ar, اَلْخَلِيْجُ ٱلْعَرَبِيُّ, Al-Khalīj al-ˁArabī), is a mediterranean sea in Western Asia. The bod ...
having their greatest range when the moon was furthest from the plane of the Equator. All this despite the relatively small amplitude of
Mediterranean The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Western Europe, Western and Southern Europe and Anatolia, on the south by North Africa ...
basin tides. (The strong currents through the
Euripus Strait The Euripus Strait ( el, Εύριπος ) is a narrow channel of water separating the Greek island of Euboea in the Aegean Sea from Boeotia in mainland Greece. The strait's principal port is Chalcis on Euboea, located at the strait's narrowest poi ...
and the
Strait of Messina The Strait of Messina ( it, Stretto di Messina, Sicilian: Strittu di Missina) is a narrow strait between the eastern tip of Sicily ( Punta del Faro) and the western tip of Calabria ( Punta Pezzo) in Southern Italy. It connects the Tyrrhenian S ...
puzzled
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
.)
Philostratus Philostratus or Lucius Flavius Philostratus (; grc-gre, Φιλόστρατος ; c. 170 – 247/250 AD), called "the Athenian", was a Greek sophist of the Roman imperial period. His father was a minor sophist of the same name. He was born probab ...
discussed tides in Book Five of ''The Life of Apollonius of Tyana''. Philostratus mentions the moon, but attributes tides to "spirits". In Europe around 730 AD, the Venerable Bede described how the rising tide on one coast of the British Isles coincided with the fall on the other and described the time progression of high water along the Northumbrian coast. The first
tide table Tide tables, sometimes called tide charts, are used for tidal prediction and show the daily times and levels of high and low tides, usually for a particular location. Tide heights at intermediate times (between high and low water) can be approxi ...
in China was recorded in 1056 AD primarily for visitors wishing to see the famous
tidal bore Tidal is the adjectival form of tide. Tidal may also refer to: * ''Tidal'' (album), a 1996 album by Fiona Apple * Tidal (king), a king involved in the Battle of the Vale of Siddim * TidalCycles, a live coding environment for music * Tidal (serv ...
in the Qiantang River. The first known British tide table is thought to be that of John Wallingford, who died Abbot of St. Albans in 1213, based on high water occurring 48 minutes later each day, and three hours earlier at the
Thames The River Thames ( ), known alternatively in parts as the River Isis, is a river that flows through southern England including London. At , it is the longest river entirely in England and the second-longest in the United Kingdom, after the R ...
mouth than upriver at
London London is the capital and List of urban areas in the United Kingdom, largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary dow ...
. In 1614 Claude d'Abbeville published the work “”, where he exposed that the
Tupinambá people The Tupinambá are one of the various Tupi ethnic groups that inhabited present-day Brazil since before the conquest of the region by Portuguese colonial settlers. In the first years of contact with the Portuguese, the Tupinambás lived in the wh ...
already had an understanding of the relation between the Moon and the tides before Europe. William Thomson (Lord Kelvin) led the first systematic harmonic analysis of tidal records starting in 1867. The main result was the building of a tide-predicting machine using a system of pulleys to add together six harmonic time functions. It was "programmed" by resetting gears and chains to adjust phasing and amplitudes. Similar machines were used until the 1960s. The first known sea-level record of an entire spring–neap cycle was made in 1831 on the Navy Dock in the
Thames Estuary The Thames Estuary is where the River Thames meets the waters of the North Sea, in the south-east of Great Britain. Limits An estuary can be defined according to different criteria (e.g. tidal, geographical, navigational or in terms of salini ...
. Many large ports had automatic tide gauge stations by 1850. John Lubbock was one of the first to map co-tidal lines, for Great Britain, Ireland and adjacent coasts, in 1840.
William Whewell William Whewell ( ; 24 May 17946 March 1866) was an English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science. He was Master of Trinity College, Cambridge. In his time as a student there, he achieved ...
expanded this work ending with a nearly global chart in 1836. In order to make these maps consistent, he hypothesized the existence of a region with no tidal rise or fall where co-tidal lines meet in the mid-ocean. The existence of such an
amphidromic point An amphidromic point, also called a tidal node, is a geographical location which has zero tidal amplitude for one harmonic constituent of the tide. The tidal range (the peak-to-peak amplitude, or the height difference between high tide and low ...
, as they are now known, was confirmed in 1840 by Captain William Hewett, RN, from careful soundings in the
North Sea The North Sea lies between Great Britain, Norway, Denmark, Germany, the Netherlands and Belgium. An epeiric sea, epeiric sea on the European continental shelf, it connects to the Atlantic Ocean through the English Channel in the south and the ...
.


Physics


Forces

The tidal force produced by a massive object (Moon, hereafter) on a small particle located on or in an extensive body (Earth, hereafter) is the vector difference between the gravitational force exerted by the Moon on the particle, and the gravitational force that would be exerted on the particle if it were located at the Earth's center of mass. Whereas the
gravitational force In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
subjected by a celestial body on Earth varies inversely as the square of its distance to the Earth, the maximal tidal force varies inversely as, approximately, the cube of this distance. If the tidal force caused by each body were instead equal to its full gravitational force (which is not the case due to the
free fall In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...
of the whole Earth, not only the oceans, towards these bodies) a different pattern of tidal forces would be observed, e.g. with a much stronger influence from the Sun than from the Moon: The solar gravitational force on the Earth is on average 179 times stronger than the lunar, but because the Sun is on average 389 times farther from the Earth, its field gradient is weaker. The tidal force is proportional to :\text\propto\frac M\propto\rho\left(\frac rd\right)^3 where is the mass of the heavenly body, is its distance, ρ is its average density, and is its radius. The ratio is related to the angle subtended by the object in the sky. Since the sun and the moon have practically the same diameter in the sky, the tidal force of the sun is less than that of the moon because its average density is much less, and it is only 46% as large as the lunar, thus during a spring tide, the Moon contributes 69% while the Sun contributes 31%. More precisely, the lunar tidal acceleration (along the Moon–Earth axis, at the Earth's surface) is about 1.1 × 10−7 ''g'', while the solar tidal acceleration (along the Sun–Earth axis, at the Earth's surface) is about 0.52 × 10−7 ''g'', where ''g'' is the
gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
at the Earth's surface. The effects of the other planets vary as their distances from Earth vary. When Venus is closest to Earth, its effect is 0.000113 times the solar effect. At other times, Jupiter or Mars may have the most effect. The ocean's surface is approximated by a surface referred to as the
geoid The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
, which takes into consideration the gravitational force exerted by the earth as well as
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parall ...
due to rotation. Now consider the effect of massive external bodies such as the Moon and Sun. These bodies have strong gravitational fields that diminish with distance and cause the ocean's surface to deviate from the geoid. They establish a new equilibrium ocean surface which bulges toward the moon on one side and away from the moon on the other side. The earth's rotation relative to this shape causes the daily tidal cycle. The ocean surface tends toward this equilibrium shape, which is constantly changing, and never quite attains it. When the ocean surface is not aligned with it, it's as though the surface is sloping, and water accelerates in the down-slope direction.


Equilibrium

The equilibrium tide is the idealized tide assuming a landless Earth. It would produce a tidal bulge in the ocean, elongated towards the attracting body (Moon or Sun). It is ''not'' caused by the vertical pull nearest or farthest from the body, which is very weak; rather, it is caused by the tangent or "tractive" tidal force, which is strongest at about 45 degrees from the body, resulting in a horizontal tidal current.


Laplace's tidal equations

Ocean depths are much smaller than their horizontal extent. Thus, the response to tidal forcing can be modelled using the Laplace tidal equations which incorporate the following features: * The vertical (or radial) velocity is negligible, and there is no vertical shear—this is a sheet flow. * The forcing is only horizontal (
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
ial). * The
Coriolis effect In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
appears as an inertial force (fictitious) acting laterally to the direction of flow and proportional to velocity. * The surface height's rate of change is proportional to the negative divergence of velocity multiplied by the depth. As the horizontal velocity stretches or compresses the ocean as a sheet, the volume thins or thickens, respectively. The boundary conditions dictate no flow across the coastline and free slip at the bottom. The Coriolis effect (inertial force) steers flows moving towards the Equator to the west and flows moving away from the Equator toward the east, allowing coastally trapped waves. Finally, a dissipation term can be added which is an analog to viscosity.


Amplitude and cycle time

The theoretical
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of oceanic tides caused by the Moon is about at the highest point, which corresponds to the amplitude that would be reached if the ocean possessed a uniform depth, there were no landmasses, and the Earth were rotating in step with the Moon's orbit. The Sun similarly causes tides, of which the theoretical amplitude is about (46% of that of the Moon) with a cycle time of 12 hours. At spring tide the two effects add to each other to a theoretical level of , while at neap tide the theoretical level is reduced to . Since the orbits of the Earth about the Sun, and the Moon about the Earth, are elliptical, tidal amplitudes change somewhat as a result of the varying Earth–Sun and Earth–Moon distances. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. If both the Sun and Moon were at their closest positions and aligned at new moon, the theoretical amplitude would reach . Real amplitudes differ considerably, not only because of depth variations and continental obstacles, but also because wave propagation across the ocean has a natural period of the same order of magnitude as the rotation period: if there were no land masses, it would take about 30 hours for a long wavelength surface wave to propagate along the Equator halfway around the Earth (by comparison, the Earth's lithosphere has a natural period of about 57 minutes). Earth tides, which raise and lower the bottom of the ocean, and the tide's own gravitational self attraction are both significant and further complicate the ocean's response to tidal forces.


Dissipation

Atmospheric tides are both gravitational and thermal in origin and are the dominant dynamics from about , above which the molecular density becomes too low to support fluid behavior.


Earth tides

Earth tides or terrestrial tides affect the entire Earth's mass, which acts similarly to a liquid gyroscope with a very thin crust. The Earth's crust shifts (in/out, east/west, north/south) in response to lunar and solar gravitation, ocean tides, and atmospheric loading. While negligible for most human activities, terrestrial tides' semi-diurnal amplitude can reach about at the Equator— due to the Sun—which is important in
GPS The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
calibration and
VLBI Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth or in space. T ...
measurements. Precise astronomical angular measurements require knowledge of the Earth's rotation rate and
polar motion Polar motion of the Earth is the motion of the Earth's rotational axis relative to its crust. This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called ''Earth-centered, Earth-fixed'' or ECEF reference ...
, both of which are influenced by Earth tides. The semi-diurnal ''M''2 Earth tides are nearly in phase with the Moon with a lag of about two hours.


Galactic tides

''
Galactic tide A galactic tide is a tidal force experienced by objects subject to the gravitational field of a galaxy such as the Milky Way. Particular areas of interest concerning galactic tides include galactic collisions, the disruption of dwarf or satelli ...
s'' are the tidal forces exerted by galaxies on stars within them and satellite galaxies orbiting them. The galactic tide's effects on the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
's
Oort cloud The Oort cloud (), sometimes called the Öpik–Oort cloud, first described in 1950 by the Dutch astronomer Jan Oort, is a theoretical concept of a cloud of predominantly icy planetesimals proposed to surround the Sun at distances ranging from ...
are believed to cause 90 percent of long-period comets.


Misnomers

Tsunami A tsunami ( ; from ja, 津波, lit=harbour wave, ) is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater exp ...
s, the large waves that occur after earthquakes, are sometimes called ''tidal waves'', but this name is given by their ''resemblance'' to the tide, rather than any causal link to the tide. Other phenomena unrelated to tides but using the word ''tide'' are rip tide,
storm tide A storm surge, storm flood, tidal surge, or storm tide is a coastal flood or tsunami-like phenomenon of rising water commonly associated with low-pressure weather systems, such as cyclones. It is measured as the rise in water level above the n ...
, hurricane tide, and
black Black is a color which results from the absence or complete absorption of visible light. It is an achromatic color, without hue, like white and grey. It is often used symbolically or figuratively to represent darkness. Black and white ...
or
red tide A harmful algal bloom (HAB) (or excessive algae growth) is an algal bloom that causes negative impacts to other organisms by production of natural algae-produced toxins, mechanical damage to other organisms, or by other means. HABs are sometimes ...
s. Many of these usages are historic and refer to the earlier meaning of tide as "a portion of time, a season" and "a stream, current or flood".


See also

* * * * * * * * * * * * * * * * * * * * * * *


Notes


References


Further reading


150 Years of Tides on the Western Coast: The Longest Series of Tidal Observations in the Americas
NOAA (2004).
Eugene I. Butikov: ''A dynamical picture of the ocean tides''


Why the centrifugal force does not explain the tide's opposite lobe (with nice animations).
O. Toledano ''et al.'' (2008): ''Tides in asynchronous binary systems''
* Gaylord Johnso
"How Moon and Sun Generate the Tides"
''Popular Science'', April 1934 *


External links


NOAA Tides and Currents information and data





UK Admiralty Easytide


* ttp://www.bom.gov.au/oceanography/tides/index.shtml Tide Predictions for Australia, South Pacific & Antarctica
Tide and Current Predictor, for stations around the world
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