Ocean tides

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 can be used for any given locale to find the predicted times and amplitude (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 (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. 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.

While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also subject to change from thermal expansion, 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 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.

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'' (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'' (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. 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 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 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 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 and full moon when the Sun, Moon, and Earth form a line (a configuration known as a syzygy), the tidal force 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, 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" 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, the range increases, and when it is at apogee, 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''. 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. 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 and Madagascar.) 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, 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, 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's ''Tetrabiblos''.

In (''The Reckoning of Time'') 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) 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 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 (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 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 in Paris offered a prize for the best theoretical essay on tides. Daniel Bernoulli, Leonhard Euler, 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 studied tidal equations for the atmosphere which did not include rotation.

In 1770 James Cook's barque HMS ''Endeavour'' grounded on the Great Barrier Reef. 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 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, rewrote Laplace's equations in terms of vorticity which allowed for solutions describing tidally driven coastally trapped waves, known as Kelvin waves.

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 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 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, 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 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 having their greatest range when the moon was furthest from the plane of the Equator. All this despite the relatively small amplitude of Mediterranean basin tides. (The strong currents through the Euripus Strait and the Strait of Messina puzzled Aristotle.) Philostratus 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 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 mouth than upriver at London.

In 1614 Claude d'Abbeville published the work “”, where he exposed that the Tupinambá people 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. 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 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.

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 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 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 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, which takes into consideration the gravitational force exerted by the earth as well as centrifugal force 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 (tangential).
* The Coriolis effect 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 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 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 galaxy, satellite galaxies orbiting them. The galactic tide's effects on the Solar System's Oort cloud are believed to cause 90 percent of long-period comets.

Misnomers

Tsunamis, 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 current, rip tide, storm tide, hurricane tide, and oil spill, black or red tides. 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 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

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External links

NOAA Tides and Currents information and data







UK Admiralty Easytide


* [http://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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