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 ...
(and to a much lesser extent, the
Sun) 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 surf ...
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 ...
orbiting one another.
Tide tables 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 a ...
(or "
tidal range
Tidal range is the difference in height between high tide and low tide. Tides are the rise and fall of sea levels caused by gravitational forces exerted by the Moon and Sun and the rotation of Earth. Tidal range depends on time and location.
...
").
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 seabed, 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 w ...
(see ''
Timing
Timing is the tracking or planning of the spacing of events in time. It may refer to:
* Timekeeping, the process of measuring the passage of time
* Synchronization, controlling the timing of a process relative to another process
* Time metrolog ...
''). 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. To make accurate records,
tide gauges 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 surge
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 ...
s, 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
The intertidal zone, also known as the foreshore, is the area above water level at low tide and underwater at high tide (in other words, the area within the tidal range). This area can include several types of habitats with various species ...
; flood tide.
* Sea level falls over several hours, revealing the intertidal zone; ebb tide.
Oscillating
currents produced by tides are known as tidal streams or
tidal currents. The moment that the tidal current ceases is called ''
slack water'' 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 tables. 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
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can al ...
.
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
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 ...
'' (MLWS) – The average of the two low tides on the days of spring tides.
* ''
Lowest astronomical tide'' (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 seabed, 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 w ...
. 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 ''M
2 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 clock
A tide clock is a specially designed clock that keeps track of the Moon's diurnal motion, apparent motion around the Earth. Along many coastlines, the Moon contributes the major part (67%) of the combined lunar and solar tides. The exact interva ...
s 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 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, 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
In astronomy, the new moon is the first lunar phase, when the Moon and Sun have the same ecliptic longitude. At this phase, the lunar disk is not visible to the naked eye, except when it is silhouetted against the Sun during a solar eclip ...
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 mean ...
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 phenom ...
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 po ...
, 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
Concerning the lunar month of ~29.53 days as viewed from Earth, the lunar phase or Moon phase is the shape of the Moon's directly sunlit portion, which can be expressed quantitatively using areas or angles, or described qualitatively using the t ...
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" 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 el ...
, the range shrinks. Six or eight times a year perigee coincides with either a new or full moon causing
perigean spring tides with the largest ''
tidal range
Tidal range is the difference in height between high tide and low tide. Tides are the rise and fall of sea levels caused by gravitational forces exerted by the Moon and Sun and the rotation of Earth. Tidal range depends on time and location.
...
''. 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
In modern mapping, a topographic map or topographic sheet is a type of map characterized by large- scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but historic ...
, 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. 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 ...
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
Seleucus of Seleucia ( el, Σέλευκος ''Seleukos''; born c. 190 BC; fl. c. 150 BC) was a Hellenistic astronomer and philosopher. Coming from Seleucia on the Tigris, Mesopotamia, the capital of the Seleucid Empire, or, alternatively, Seleuk ...
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 import ...
'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
Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English monk at the monastery of St Peter and its companion monastery of St Paul in the Kingdom ...
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) 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 (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 (d. circa 1204) contributed the notion that the tides were caused by the general circulation of the heavens.
Simon Stevin, 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 and
neap tide, 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 w ...
in his 1632 ''
Dialogue Concerning the Two Chief World Systems'', 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 g ...
(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) 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 t ...
in Paris offered a prize for the best theoretical essay on tides.
Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mech ...
,
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 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 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 '' Encyclop ...
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 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 equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
s 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 physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did importan ...
, rewrote Laplace's equations in terms of
vorticity 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., t ...
s.
Others including Kelvin and
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
further developed Laplace's theory. Based on these developments and the
lunar theory Lunar theory attempts to account for the motions of the Moon. There are many small variations (or perturbations) in the Moon's motion, and many attempts have been made to account for them. After centuries of being problematic, lunar motion can now b ...
of
E W Brown describing the motions of the Moon,
Arthur Thomas Doodson 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
Pytheas of Massalia (; Ancient Greek: Πυθέας ὁ Μασσαλιώτης ''Pythéas ho Massaliōtēs''; Latin: ''Pytheas Massiliensis''; born 350 BC, 320–306 BC) was a Greek geographer, explorer and astronomer from the Greek colo ...
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 (O ...
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
Seleucus of Seleucia ( el, Σέλευκος ''Seleukos''; born c. 190 BC; fl. c. 150 BC) was a Hellenistic astronomer and philosopher. Coming from Seleucia on the Tigris, Mesopotamia, the capital of the Seleucid Empire, or, alternatively, Seleuk ...
correctly described the phenomenon of tides in order to support his
heliocentric
Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth ...
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 ...
, although he believed that the interaction was mediated by the
pneuma. 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 bo ...
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 and Southern Europe and Anatolia, on the south by North Africa, and on th ...
basin tides. (The strong currents through the
Euripus Strait and the
Strait of Messina puzzled
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
.)
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 prob ...
discussed tides in Book Five of ''The Life of
Apollonius of Tyana
Apollonius of Tyana ( grc, Ἀπολλώνιος ὁ Τυανεύς; c. 3 BC – c. 97 AD) was a Greek Neopythagorean philosopher from the town of Tyana in the Roman province of Cappadocia in Anatolia. He is the subject of ...
''. Philostratus mentions the moon, but attributes tides to "spirits". In Europe around 730 AD, the Venerable
Bede
Bede ( ; ang, Bǣda , ; 672/326 May 735), also known as Saint Bede, The Venerable Bede, and Bede the Venerable ( la, Beda Venerabilis), was an English monk at the monastery of St Peter and its companion monastery of St Paul in the Kingdom ...
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 in
China was recorded in 1056 AD primarily for visitors wishing to see the famous
tidal bore 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 ...
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
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...
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, 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 on the European continental shelf, it connects to the Atlantic Ocean through the English Channel in the south and the Norwegian S ...
.
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 stro ...
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 i ...
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
:
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 (physics), drag). This is the steady gain in speed caused exclusively by the force of gravitational attract ...
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 exten ...
, 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 parallel ...
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 a ...
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
A lithosphere () is the rigid, outermost rocky shell of a terrestrial planet or natural satellite. On Earth, it is composed of the crust and the portion of the upper mantle that behaves elastically on time scales of up to thousands of years ...
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 calibration and
VLBI measurements. Precise astronomical angular measurements require knowledge of the Earth's rotation rate and
polar motion, 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 tides'' 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 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
A rip tide, or riptide, is a strong offshore current that is caused by the tide pulling water through an inlet along a barrier beach, at a lagoon or inland marina where tide water flows steadily out to sea during ebb tide. It is a strong tidal flo ...
,
storm tide,
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 ha ...
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 phycotoxin, algae-produced toxins, mechanical damage to other organisms, or by other means. HABs are ...
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
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Notes
References
Further reading
150 Years of Tides on the Western Coast: The Longest Series of Tidal Observations in the AmericasNOAA (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 & AntarcticaTide and Current Predictor, for stations around the world
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