Definitions

tide

tide

tide, alternate and regular rise and fall of sea level in oceans and other large bodies of water. These changes are caused by the gravitational attraction of the moon and, to a lesser extent, of the sun on the earth. More generally, tides are the deformations of celestial bodies from a perfectly spherical shape that result from stresses created by their mutual gravitational attraction (see gravitation). Another way of viewing the tide is as the longest possible ocean wave, one which stretches all the way around the earth. The tide regarded as a wave is sometimes referred to as a tidal wave, although this term has been commonly applied to the shock wave propagated by an underwater earthquake. (To avoid confusion, such shock waves are now called tsunamis, their Japanese name, or seismic sea waves.) Numerous schemes have been proposed to harness the earth's tides, especially in various estuaries, as a practical source of power, but none as yet have proved economically or technologically feasible.

Tidal Effect on the Earth

Tides are raised in the earth's solid crust and atmosphere as well as in the oceans. Every body in the universe has some tidal effect, however small, on every other body. This effect is directly proportional to the mass of the body causing the tide but inversely proportional to the cube of the distance between the bodies. The earth's nearby moon is about 2.17 times as effective as the more massive sun in raising tides on the earth, even though the sun exerts a much greater total force on the earth than does the moon. Thus, the moon's proximity explains its dominant role in creating tides.

Direct and Indirect Tides

At any given time, there are two high tides on the earth, the direct tide on the side facing the moon and the indirect tide on the opposite side. As the earth rotates on its axis, the location of the two diametrically opposed tidal bulges varies on the earth's surface. The earth's rotation and the moon's revolution, which have the same direction, bring each point on the earth opposite the moon once every 24 hr and 50 min. Therefore, the average interval between direct and indirect high tides is about 12 hr and 25 min. In many places along the Atlantic coasts of N America and Europe, the two daily low tides are of nearly equal duration and magnitude, called semidiurnal tides.

In certain shallow seas and narrow estuaries, the tides differ from this simple pattern. For example, in certain regions such as the Pacific coast of N America, one of the two daily tides is appreciably higher than the other or the interval between successive tides is unequal; these are called mixed tides. In other regions, such as the Gulf of Mexico, there is only one high tide per day called a diurnal tide, with a period of 24 hr and 50 min.

The Magnitude and Effects of Tidal Ranges

The range of the tides is the difference in sea level between high and low tides. Spring tide, having the maximum range, occurs during the full moon when the earth is between the moon and the sun, and new moon when the moon is between the earth and the sun. At these times in the lunar cycle when the moon, earth, and sun are aligned the condition is known as syzygy. The term king tide is used in some regions to describe the highest tides of the year. Neap tide, having the minimum range, occurs during the moon's first and last quarters, when the moon, earth, and sun form a right angle. The typical tidal range in the open ocean is 2 ft (0.61 m) but is much greater near the coast. Tidal ranges vary around the world and average about 6 to 10 ft (2 to 3 m). The world's widest tidal range occurs in the Bay of Fundy, in E Canada, where the sea level changes by 40 ft (12 m) during the day, while the Mediterranean, Baltic, and Caribbean Seas are relatively tideless.

As the tides change, currents must flow to redistribute the ocean's water. Near the coast, the direction of the current changes every 61/4 hr from toward the shore (flood current) to away from the shore (ebb current). In the open ocean, the tidal currents are rotary, shifting through all directions of the compass in a period matching that of the local tide. When tidal currents flow into the mouth of a river, they speed up. In extreme cases, the tidal rise advances up the river as a solid wall of water often several feet high, a rare phenomenon called a tidal bore. During times of high tide accompanied by high wind and low pressure, as during a hurricane, a tidal surge can occur, causing coastal erosion, flooding, and damage to coastal cities.

The Prediction of Tides

Detailed prediction of ocean tides from theories of classical mechanics and hydrodynamics has not been entirely successful, largely because of complications introduced by the irregular shape of the ocean basins and coastlines. Useful results are obtained empirically by analyzing records of previous tides at a particular location to predict future tides. The importance of tides for maritime activities has prompted the compilation of tide tables for harbors, which give the time and height of high water and low water based on past observations and corrected for the varying positions of celestial bodies.

Bibliography

See A. C. Redfield, Introduction to Tides (1982); D. Arnold, Tides and Currents (1987); G. Marchuk and B. A. Kagan, Dynamics of Ocean Tides (1989).

Regular, periodic rise and fall of the surface of the sea, occurring in most places twice a day. Tides result from differences in the gravitational forces exerted at different points on the Earth's surface by another body (such as the Moon). Although any celestial body (e.g., Jupiter) produces minute tidal effects, the majority of the tidal forces on the Earth are raised by the Sun (because of its enormous mass) and the Moon (because of its proximity to Earth). In fact, the tidal forces from the Moon are about twice as strong as those from the Sun. The largest tides (spring tides, exhibiting very large change in sea level between high and low tides) occur at the new and full moon, when the Earth, the Moon, and the Sun are aligned and the Sun's tidal forces are added to those of the Moon. The smallest tides (neap tides) occur when the Sun and Moon are at right angles (from Earth), when the tidal forces from the Sun partially cancel those from the Moon. The geometry of the coastline and of the water's basin also affects the range of the tides.

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Discoloration of seawater caused by dinoflagellates during periodic blooms (population increases). Toxic substances released by these organisms into the water may be lethal to fish and other marine life, and they irritate the human respiratory system. Coastal resorts sometimes close when breaking waves release the toxic substances into the air. The causes of red tide are uncertain; it may require the confluence of several natural phenomena, in which human influence may or may not play a part.

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Tides are the cyclic rising and falling of Earth's ocean surface caused by the tidal forces of the Moon and the Sun acting on the oceans. Tides cause changes in the depth of the marine and estuarine water bodies and produce oscillating currents known as tidal streams, making prediction of tides important for coastal navigation (see Navigation). The strip of seashore that is submerged at high tide and exposed at low tide, the intertidal zone, is an important ecological product of ocean tides (see Intertidal ecology).

The changing tide produced at a given location is the result of the changing positions of the Moon and Sun relative to the Earth coupled with the effects of Earth rotation and the bathymetry of oceans, seas and estuaries. Sea level measured by coastal tide gauges may also be strongly affected by wind. More generally, tidal phenomena can occur in other systems besides the ocean, whenever a gravitational field that varies in time and space is present (see Other tides).

Characteristics

A tide is a repeated cycle of sea level changes in the following stages:

  • Over several hours the water rises or advances up a beach in the flood
  • The water reaches its highest level and stops at high water. Because tidal currents cease this is also called slack water or slack tide. The tide reverses direction and is said to be turning.
  • The sea level recedes or falls over several hours during the ebb tide.
  • The level stops falling at low water. This point is also described as slack or turning.

Tides may be semidiurnal (two high waters and two low waters each day), or diurnal (one tidal cycle per day). In most locations, tides are semidiurnal. Because of the diurnal contribution, there is a difference in height (the daily inequality) between the two high waters on a given day; these are differentiated as the higher high water and the lower high water in tide tables. Similarly, the two low waters each day are referred to as the higher low water and the lower low water. The daily inequality changes with time and is generally small when the Moon is over the equator.

The various frequencies of orbital forcing which contribute to tidal variations are called constituents. In most locations, the largest is the "principal lunar semidiurnal" constituent, also known as the M2 (or M2) tidal constituent. Its period is about 12 hours and 25.2 minutes, exactly half a tidal lunar day, the average time separating one lunar zenith from the next, and thus the time required for the Earth to rotate once relative to the Moon. This is the constituent tracked by simple tide clocks.

Tides vary on timescales ranging from hours to years, so to make accurate records tide gauges measure the water level over time at fixed stations which are screened from variations caused by waves shorter than minutes in period. These data are compared to the reference (or datum) level usually called mean sea level.

Constituents other than M2 arise from factors such as the gravitational influence of the Sun, the tilt of the Earth's rotation axis, the inclination of the lunar orbit and the ellipticity of the orbits of the Moon about the Earth and the Earth about the Sun. Variations with periods of less than half a day are called harmonic constituents. Long period constituents have periods of days, months, or years.

Range variation: springs and neaps

The semidiurnal tidal range (the difference in height between high and low waters over about a half day) varies in a two-week or fortnightly cycle. Around new and full moon when the Sun, Moon and Earth form a line (a condition known as syzygy), the tidal forces due to the Sun reinforce those of the Moon. The tide's range is then maximum: this is called the spring tide, or just springs and is derived not from the season of spring but rather from the verb meaning "to jump" or "to leap up". 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 forces induced by the Sun partially cancel those of the Moon. At these points in the lunar cycle, the tide's range is minimum: this is called the neap tide, or neaps. 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.

The changing distance of the Moon from the Earth also affects tide heights. When the Moon is at perigee the range is increased, and when it is at apogee the range is reduced. Every 7½ lunations, perigee coincides with either a new or full moon causing perigean tides with the largest tidal range. If a storm happens to be moving onshore at this time, the consequences (in the form of property damage, etc.) can be especially severe.

Phase and amplitude

Because the M2 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. It is also measured in degrees, with 360° per tidal cycle. Lines of constant tidal phase are called cotidal lines. 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. 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 is at once cotidal with high and low waters, which is satisfied by zero tidal motion. (The rare exception occurs when the tide circles around an island, as it does around New Zealand.) Indeed tidal motion generally lessens moving away from the continental coasts, so that crossing the cotidal lines are contours of constant amplitude (half of the distance between high and low water) which decrease to zero at the amphidromic point. For a 12 hour semidiurnal tide the amphidromic point behaves roughly like 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 once every 12 hours in the direction of rising cotidal lines, and away from ebbing cotidal lines. The difference of cotidal phase from the phase of a reference tide is the epoch.

The shape of the shoreline and the ocean floor change the way that tides propagate, so there is no simple, general rule for predicting the time of high water from the position of the Moon in the sky. Coastal characteristics such as underwater topography and coastline shape mean that individual location characteristics need to be taken into consideration when forecasting tides; high water time may differ from that suggested by a model such as the one above due to the effects of coastal morphology on tidal flow.

Physics

Isaac Newton laid the foundations for the mathematical explanation of tides in the Philosophiae Naturalis Principia Mathematica (1687). In 1740, the Académie Royale des Sciences in Paris offered a prize for the best theoretical essay on tides. Daniel Bernoulli, Antoine Cavalleri, Leonhard Euler, and Colin Maclaurin 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 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 horizontal component of the tidal force (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. The first major theoretical formulation for water tides was made by Pierre-Simon Laplace, who formulated a system of partial differential equations relating the horizontal flow to the surface height of the ocean. 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, which are known as Kelvin waves.

Regarding the Earth–Moon system by itself (excluding the Sun for the moment) it is known that unless the spin axes of both partners are aligned and perpendicular to the orbital plane, oscillations are excited and these tidal deformations attribute somewhat to the tidal dissipation. This lack of alignment is the case for the Earth–Moon system. Thus, besides the tidal bulges, opposite to each other and comparable in size, that are associated with the so called equilibrium tide,additionally, a set of surface oscillations commonly known as the dynamical tide, characterized by a wide variety of harmonic frequencies, is established.

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 center of mass of the Earth. Thus, the tidal force depends not on the strength of the gravitational field of the Moon, but on its gradient (which falls off approximately as the inverse cube of the distance to the originating gravitational body; see NASA). The gravitational force exerted on the Earth by the Sun is on average 179 times stronger than that exerted on the Earth by the Moon, but because the Sun is on average 389 times farther from the Earth, the gradient of its field is weaker. The tidal force produced by the Sun is therefore only 46% as large as that produced by the Moon. (According to NASA the tidal force of the Moon is 2.21 times larger than that of the Sun. The effect of the other planets is much, much smaller, with the largest being Venus at 0.000113 times that of the Sun.)

Tidal forces can also be analyzed in the following way: besides the outward uniform centrifugal force acting on Earth each point of the Earth experiences the Moon's radially decreasing gravity differently; they are subject to the tidal forces of Figure 7, which dominate. Finally, most importantly, only the horizontal components of the tidal forces actually contribute tidal acceleration to the water particles since there is small resistance. The actual tidal force on a particle is only about a ten millionth of the force caused by the Earth's gravity.

The ocean's surface is closely approximated by an equipotential surface, (ignoring ocean currents) which is commonly referred to as the geoid. Since the gravitational force is equal to the gradient of the potential, there are no tangential forces on such a surface, and the ocean surface is thus in gravitational equilibrium. Now consider the effect of external, massive bodies such as the Moon and Sun. These bodies have strong gravitational fields that diminish with distance in space and which act to alter the shape of an equipotential surface on the Earth. Gravitational forces follow an inverse-square law (force is inversely proportional to the square of the distance), but tidal forces are inversely proportional to the cube of the distance. The ocean surface moves to adjust to changing tidal equipotential, tending to rise when the tidal potential is high, the part of the Earth nearest the Moon, and the farthest part. When the tidal equipotential changes, the ocean surface is no longer aligned with it, so that the apparent direction of the vertical shifts. The surface then experiences a down slope, in the direction that the equipotential has risen.

Laplace tidal equation

The depth of the oceans is 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:

  1. The vertical (or radial) velocity is negligible, and there is no vertical shear—this is a sheet flow.
  2. The forcing is only horizontal (tangential).
  3. The Coriolis effect appears as a fictitious lateral forcing proportional to velocity.
  4. The rate of change of the surface height 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 steers waves to the right in the northern hemisphere and to the left in the southern 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 54 cm 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 not rotating. The Sun similarly causes tides, of which the theoretical amplitude is about 25 cm (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 79 cm, while at neap tide the theoretical level is reduced to 29 cm. Since the orbits of the Earth about the Sun, and the Moon about the Earth, are elliptical, the amplitudes of the tides 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 93 cm.

Real amplitudes differ considerably, not only because of variations in ocean depth, and the obstacles to flow caused by the continents, but also because the natural period of wave propagation is of the same order of magnitude as the rotation period: about 30 hours. If there were no land masses, it would take about 30 hours for a long wavelength ocean surface wave to propagate along the equator halfway around the Earth (by comparison, the natural period of the Earth's lithosphere is about 57 minutes).

Dissipation

The tidal oscillations of the Earth introduce dissipation, at an average rate of about 3.75 terawatt. About 98% of this dissipation is by the tidal movement in the seas and oceans. The dissipation arises as the basin-scale tidal flow drives smaller-scale flows which experience turbulent dissipation. This tidal drag gives rise to a torque on the Moon that results in the gradual transfer of angular momentum to its orbit, and a gradual increase in the Earth–Moon separation. As a result of the equal and opposite torque on the Earth, the rotational velocity of the Earth is correspondingly slowed. Thus, over geologic time, the Moon recedes from the Earth, at about 3.8 cm/year, and the length of the terrestrial day increases, meaning that there is about 1 fewer day per 100 million years.

Observation and prediction

History

From ancient times, tides have been observed and discussed with increasing sophistication, first noting the daily recurrence, then its relationship to the Sun and Moon. Pytheas travelled to the British Isles and seems to be the first to have related spring tides to the phase of the Moon.

In the 2nd century BC, the Babylonian 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 the tides varied in time and strength in different parts of the world. According to Strabo (1.1.9), Seleucus was the first to state that the tides are due to the attraction of the Moon, 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 observations of detail: the spring tides being a few days after (or before) new and full moon, and that the spring tides around the time of the equinoxes were the highest, though there were also 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 feeble tides in the Mediterranean basin, though there are strong currents through the Strait of Messina and between Greece and the island of Euboea through the Euripus that puzzled Aristotle. In Europe around 730 AD, the Venerable Bede described how the rise of tide on one coast of the British Isles coincided with the fall on the other and described the progression in times of the same high water along the Northumbrian coast.

In the 9th century, the Arabian earth-scientist, Al-Kindi (Alkindus), wrote a treatise entitled Risala fi l-Illa al-Failali l-Madd wa l-Fazr (Treatise on the Efficient Cause of the Flow and Ebb), in which he presents an argument on tides which "depends on the changes which take place in bodies owing to the rise and fall of temperature." He describes a clear and precise laboratory experiment in order to prove his argument. The first tide table in China was recorded in 1056 A.D, primarily for the benefit of visitors wishing to see the famous tidal bore in the Qiantang River. The first known tide table is thought to be that of John, Abbott of Wallingford (d. 1213), based on high water occurring 48 minutes later each day, and three hours later upriver at London than at the mouth of the Thames.

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 functions of time. 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, and many large ports had automatic tide gage stations by 1850.

William Whewell first mapped co-tidal lines ending with a nearly global chart in 1836. In order to make these maps consistent, he hypothesized the existence of amphidromes where co-tidal lines meet in the mid-ocean. These points of no tide were confirmed by measurement in 1840 by Captain Hewett, RN, from careful soundings in the North Sea.

Timing

In most places there is a delay between the phases of the Moon and the effect on the tide. Springs and neaps in the North Sea, for example, are two days behind the new/full Moon and first/third quarter. This is called the age of the tide.

The exact time and height of the tide at a particular coastal point is also greatly influenced by the local bathymetry. There are some extreme cases: the Bay of Fundy, on the east coast of Canada, features the largest well-documented tidal ranges in the world, 16 meters (53 ft), because of the shape of the bay. Ungava Bay in northern Quebec, is believed by some experts to have higher tidal ranges than the Bay of Fundy, but it is free of pack ice for only about four months every year, whereas the Bay of Fundy rarely freezes.

Southampton in the United Kingdom has a double high water caused by the interaction between the different tidal harmonics within the region. This is contrary to the popular belief that the flow of water around the Isle of Wight creates two high waters. The Isle of Wight is important, however, since it is responsible for the 'Young Flood Stand', which describes the pause of the incoming tide about three hours after low water.

Because the oscillation modes of the Mediterranean Sea and the Baltic Sea do not coincide with any significant astronomical forcing period, the largest tides are close to their narrow connections with the Atlantic Ocean. Extremely small tides also occur for the same reason in the Gulf of Mexico and Sea of Japan. On the southern coast of Australia, because the coast is mainly straight (partly because of the tiny quantities of runoff flowing from rivers), tidal ranges are equally small.

Analysis

It was Isaac Newton's universal theory of gravitation that first enabled an explanation of why there were two tides a day, not one, and, via calculation of the forces, offered hope of detailed understanding. Although it may seem that tides could be predicted via a sufficiently detailed knowledge of the astronomical forcing terms, the actual tide at a given location is determined by the response of the oceans to the astronomical forces accumulated over a period of many days. To calculate this response requires a detailed knowledge of the shape of all the ocean basins — their bathymetry and coastline shape.

Instead of a direct calculation, the procedure for analysing tides is pragmatic: At each place of interest, the tide heights are measured for at least a lunar cycle. The tide heights are compared to the known frequencies of the astronomical tide-raising forces. The behaviour of the tide heights is expected to follow the behaviour of the tide force, with the amplitude and delays of those responses remaining constant. Because astronomical frequencies and phases can be calculated with certainty, the tide height at other times can be predicted once the response to the astronomical states has been found.

The main patterns in the tides are

  • the twice-daily combined lunar and solar tide, caused by the rotation of the Earth,
  • the difference between the first and second tide of a day, due to the Moon or Sun being north or south of the equator,
  • the spring-neap cycle in amplitude (related to the relative positions of the Moon and Sun), and
  • the adjustment of spring tide heights due to the varying distances to the Moon and Sun.

The Highest Astronomical Tide is the perigean spring tide when both the Sun and the Moon are closest to the Earth.

When confronted by a periodically varying function, the standard approach is to employ Fourier series, a form of orthogonal analysis that uses sinusoidal functions as a basis set, having frequencies that are zero, one, two, three, etc. times the frequency of a particular fundamental cycle. These multiples are called harmonics of the fundamental frequency, and the process is termed harmonic analysis. If the basis set of sinusoidal functions are well-suited to the behaviour being modelled, relatively few harmonic terms need to be carried in the analysis. Fortunately orbital paths are very nearly circular, so sinusoidal variations are suitable for tides.

For the analysis of tide heights, the Fourier series approach is best made more elaborate. While the theorem remains true and the tidal height could be analysed in terms of a single frequency and its harmonics, a large number of significant terms would be required. A much more compact decomposition for the tides involves a combination of sinusoids having more than one fundamental frequency. Specifically, the incommensurable periods of one revolution of the Earth (equivalently, of the Sun around the Earth), and one orbit of the Moon about the Earth are used (for simplicity in phrasing, this discussion is entirely geocentric, but is informed by the heliocentric model).

To represent both the lunar and solar influences using one frequency would require many harmonic terms, but allowing two incommensurable frequencies requires only a few terms. That is, the sum of two sinusoids, one at the Sun's frequency and the second at the Moon's frequency, requires those two terms only, but their representation as a Fourier series having only one fundamental frequency and its (integer) multiples would require many terms. For tides then, although the process is still termed harmonic analysis, it is not limited to harmonics of a single frequency. To demonstrate this offers a tidal height pattern converted into an .mp3 sound file, and the rich sound is quite different from a pure tone. In other words, the harmonies are multiples of many fundamental frequencies, not just of the one fundamental frequency of the simpler Fourier series approach.

The study of tide height by harmonic analysis was begun by Laplace, William Thomson (Lord Kelvin), and George Darwin. Their work was extended by A.T. Doodson who introduced the Doodson Number notation to organise the hundreds of terms that result. This approach has been the international standard ever since, and the complications arise as follows: the tide-raising force is notionally given by sums of several terms. Each term is of the form

A·cos(w·t + p)
where A is the amplitude, w is the angular frequency usually given in degrees per hour corresponding to t measured in hours, and p is the phase offset with regard to the astronomical state at time t = 0 . There is one term for the Moon and a second term for the Sun. The phase p of the first harmonic for the Moon term is called the lunitidal interval or high water interval. If the orbits were circular, that would be the end of the matter, but of course they are not. Accordingly, the value of A is not a constant but also varying with time, slightly, about some average figure. Replace it then by A(t) , but what functional form? It turns out that another sinusoid gives an excellent approximation for the changing amplitude, similar to the cycles and epicycles of Ptolemaic theory. Accordingly,
A(t) = A·(1 + Aa·cos(wa·t + pa)) ,
which is to say an average value A with a sinusoidal variation about it of magnitude Aa , with frequency wa and phase pa . Thus the simple term is now a compound term, the product of two cosine factors:
A·[1 + Aa·cos(wa + pa)]·cos(w·t + p)

Now, given that for any x and y

cos(x)·cos(y) = ½·cos(x + y ) + ½·cos(xy ) ,
it is clear that a compound term involving the product of two cosine terms each with their own frequency is the same as three simple cosine terms that are to be added, at the original frequency and also at the sum and difference of the two frequencies of the product term. (Three, not two terms, since the whole expression is (1 + cos(x))·cos(y) .) Consider further that the tidal force on a location depends also on whether the Moon (or the Sun) is above or below the plane of the equator, and that these attributes have their own periods also incommensurable with a day and a month, and it is clear that many combinations result. With a careful choice of the basic astronomical frequencies, the Doodson Number annotates the particular additions and differences of them to form the frequency of each simple cosine term.

Remember that astronomical tides do not include the effect of weather, and changes to local conditions (sandbank movement, dredging harbour mouths, etc.) away from those prevailing at the time of measurement can affect the timing and magnitude of the actual tide. Organisations quoting a "highest astronomical tide" for some location can exaggerate the figure as a safety factor against uncertainties of analysis, extrapolation from the nearest point of measurement, changes since the time of observation, possible ground subsidence, etc., to protect the organisation against blame should an engineering work be overtopped. If the size of a "weather surge" is assessed by subtracting the astronomical tide from the observed tide at the time, care is needed.

Careful Fourier data analysis over a nineteen-year period (the National Tidal Datum Epoch in the U.S.) uses frequencies called the tidal harmonic constituents. Nineteen years is preferred because the relative positions of the Earth, Moon and Sun repeat almost exactly in the Metonic cycle of 18.6 years. This analysis can be done using only the knowledge of the period of forcing, but without detailed understanding of the mathematical derivation, which means that useful tidal tables have been constructed for centuries. The resulting amplitudes and phases can then be used to predict the expected tides. These are usually dominated by the constituents near 12 hours (the semidiurnal constituents), but there are major constituents near 24 hours (diurnal) as well. Longer term constituents are 14 day or fortnightly, monthly, and semiannual. Most of the coastline is dominated by semidiurnal tides, but some areas such as the South China Sea and the Gulf of Mexico are primarily diurnal. In the semidiurnal areas, the primary constituents M2 (lunar) and S2 (solar) periods differ slightly, so that the relative phases, and thus the amplitude of the combined tide, change fortnightly (14 day period).

In the M2 plot above each cotidal line differs by one hour from its neighbors, and the thicker lines show tides in phase with equilibrium at Greenwich. The lines rotate around the amphidromic points counterclockwise in the northern hemisphere so that from Baja California to Alaska and from France to Ireland the M2 tide propagates northward. In the southern hemisphere this direction is clockwise. On the other hand M2 tide propagates counterclockwise around New Zealand, but this because the islands act as a dam and permit the tides to have different heights on opposite sides of the islands. (But the tides do propagate northward on the east side and southward on the west coast, as predicted by theory.)

The exception is the Cook Strait where the tidal currents periodically link high to low water. This is because cotidal lines 180° around the amphidromes are in opposite phase, for example high water across from low water. Each tidal constituent has a different pattern of amplitudes, phases, and amphidromic points, so the M2 patterns cannot be used for other tide components.

Example calculation

Figure 10 shows the common pattern of two tidal peaks in a day, though remember that the repeat time is not exactly twelve hours but 12.4206 hours. The two peaks are not equal: the twin tidal bulges beneath the Moon and on the far side of the Earth are aligned with the Moon. Bridgeport is north of the equator, so when the Moon is north of the equator also and shining upon Bridgeport, Bridgeport is closer to its maximum effect than approximately twelve hours later when Bridgeport is on the far side of the Earth from the Moon and the high tide bulge at Bridgeport's longitude has its maximum south of the equator. Thus the two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again. Likewise for the low tides.

Figure 11 shows the spring tide / neap tide cycle in the amplitudes of the tides as the Moon orbits the Earth from being in line (Sun–Earth–Moon, or Sun–Moon–Earth) when the two main influences combine to give the spring tides, to when the two forces are opposing each other as when the angle Moon–Earth–Sun is close to ninety degrees producing the neap tides. Note also as the Moon moves around its orbit it also changes from north of the equator to south of the equator. The alternation in the heights of the high tides becomes smaller, until they are the same (the Moon is above the equator), then redevelops but with the other polarity, waxing to a maximum difference and then waning again.

Figure 12 shows just over a year's worth of tidal height calculations. The Sun also cycles between being north or south of the equator and as well the Earth–Sun and Earth–Moon distances change on their own cycles. None of the various cycle periods are commensurate.

Remember always that calculated tidal heights take no account of weather effects, nor include any changes to conditions since the coefficients were determined, such as movement of sandbanks or dredging, etc.

Current

The flow pattern caused by tidal influence is much more difficult to analyse, and also, data is much more difficult to collect. A tidal height is a simple number which applies to a wide region simultaneously — often as far as the eye can see. A flow has both a magnitude and a direction, which both can vary substantially over just a short distance due to local bathymetry, and also vary with depth below the water surface. Also, although the centre of a channel is the most useful measuring site, mariners will not accept a current measuring installation obstructing navigation, so a flexible approach is required. A flow proceeding up a curved channel is the same flow, even though its direction varies continuously along the channel. But contrary even to the obvious expectation, flood and ebb flows are often not in opposite directions. The direction of a flow is determined by the shape of the channel it is coming from, not the shape where it will shortly be. Likewise, eddies can form in one direction but not the other.

Nevertheless, analysis of currents proceeds on the same basis as tides: At a given location in the simple case, the great majority of the flood flow will be in one direction, and the ebb flow in another (not necessarily opposite) direction. The velocities measured along the flood direction are taken as positive, and along the ebb direction as negative, and analysis proceeds as if these were tide height figures.

In more complex situations, the flow will not be dominated by the main ebb and flow directions, with the flow direction and magnitude tracing out an ellipse over a tidal cycle (on a polar plot) instead of along the two lines of ebb and flow direction. In this case, analysis might proceed along two pairs of directions, the primary flow directions and the secondary directions at right angles. Alternatively, the tidal flows can be treated as complex numbers, as each value has both a magnitude and a direction.

As with tide height predictions, tide flow predictions based only on astronomical factors do not take account of weather conditions, which can completely change the situation.

The tidal flow through Cook Strait between the two main islands of New Zealand is particularly interesting, as on each side of the strait the tide is almost exactly out of phase so that high water on one side meets low water on the other. Strong currents result, with almost zero tidal height change in the centre of the strait. Yet, although the tidal surge should flow in one direction for six hours and then the reverse direction for six hours, a particular surge might last eight or ten hours with the reverse surge enfeebled. In especially boisterous weather conditions, the reverse surge might be entirely overcome so that the flow remains in the same direction through three surge periods and longer. A further complication for Cook Strait's pattern of current flow is that the tides at the north end have the ordinary two cycles of spring-neap tides in a month (as found along the west side of the country), but the south end's tidal pattern has only one cycle of spring-neap tides a month, as found on the east side of the country. Tidal currents are much more complex than tidal heights!

Power generation

Power can be extracted by two means: inserting a water turbine into a tidal current, or building impoundment ponds so as to release or admit water through a turbine. In the first case, the generation is entirely determined by the timing and magnitude of the tidal currents, and the best currents may be unavailable because the turbines would obstruct navigation. In the second, the impoundment dams are expensive to construct, the natural water cycles are completely disrupted, as is navigation, but with multiple impoundment ponds power can be generated at chosen times. So far, there are few systems for tidal power generation (most famously, La Rance by Saint Malo, France) and many difficulties. Aside from environmental issues, simply withstanding sea-water corrosion and fouling by biological growths poses engineering challenges.

Proponents of tidal power systems point out that, unlike wind power systems, the generation pattern can be predicted years ahead. However, weather effects are still problematic. Another assertion is that some generation is possible for most of the tidal cycle. This may be true in principle since the time of still water is short, but in practice turbines lose efficiency at partial operating powers. Since the power available from a flow is proportional to the cube of the flow speed, the times during which high power generation is possible turn out to be rather brief. An obvious fallback then is to have several tidal power generation stations, at locations where the tide phase is different enough so that low power from one station is filled in by high power from another. Again, New Zealand has particularly interesting opportunities. Because the tidal pattern is such that a state of high water orbits the country once per cycle, there is always somewhere around the coast where the tide is at its peak, and somewhere else where it is at its lowest, so that via the electricity transmission network, there could always be supply from tidal generation somewhere. The most convenient situation is presented with Auckland city, which is between Manukau harbour and Waitemata harbour so that both power stations would be close to the load.

But, because the power available varies with the cube of the flow, even with the optimum phase difference of three hours between two stations, there are still significant amounts of time when neither tidal flow is rapid enough for significant generation, and worse, during the time of neap tides, the flow is weak all of the day, and there is no getting around this via multiple stations, because the neap tides apply to the whole Earth at once. The most feeble neap tides would be when the Sun's influence is maximum whilst the Moon's is weakest, and as far as the Sun is concerned, it is closest to the Earth during the time of the southern hemisphere's summer, which is when electricity demand is the least there, a small bonus.

As a result, interest must fall on the Kaipara harbour which not only is large, but also is two-lobed in shape, and thus almost pre-designed for a tidal impoundment scheme where one lobe could be filled by high water and the other emptied by a low water, and then via a canal from one to the other generation would be possible at a time of choice.

There is scant likelihood of any such scheme proceeding because of the disruption to natural conditions.

Navigation

Tidal flows are of importance in navigation, and significant errors in position will occur if they are not taken into account. Tidal heights are also important; for example many rivers and harbours have a shallow "bar" at the entrance which will prevent boats with significant draft from entering at certain states of the tide.

The timings and velocities of tidal flow can be found by looking at a tidal chart or tidal stream atlas for the area of interest. Tidal charts come in sets, with each diagram of the set covering a single hour between one high water and another (they ignore the extra 24 minutes) and give the average tidal flow for that one hour. An arrow on the tidal chart indicates the direction and the average flow speed (usually in knots) for spring and neap tides. If a tidal chart is not available, most nautical charts have "tidal diamonds" which relate specific points on the chart to a table of data giving direction and speed of tidal flow.

Standard procedure to counteract the effects of tides on navigation is to (1) calculate a "dead reckoning" position (or DR) from distance and direction of travel, (2) mark this on the chart (with a vertical cross like a plus sign) and (3) draw a line from the DR in the direction of the tide. The distance the tide will have moved the boat along this line is computed by the tidal speed, and this gives an "estimated position" or EP (traditionally marked with a dot in a triangle).

Nautical charts display the "charted depth" of the water at specific locations with "soundings" and the use of bathymetric contour lines to depict the shape of the submerged surface. These depths are relative to a "chart datum", which is typically the level of water at the lowest possible astronomical tide (tides may be lower or higher for meteorological reasons) and are therefore the minimum water depth possible during the tidal cycle. "Drying heights" may also be shown on the chart, which are the heights of the exposed seabed at the lowest astronomical tide.

Heights and times of low and high water on each day are published in tide tables. The actual depth of water at the given points at high or low water can easily be calculated by adding the charted depth to the published height of the tide. The water depth for times other than high or low water can be derived from tidal curves published for major ports. If an accurate curve is not available, the rule of twelfths can be used. This approximation works on the basis that the increase in depth in the six hours between low and high water will follow this simple rule: first hour - 1/12, second - 2/12, third - 3/12, fourth - 3/12, fifth - 2/12, sixth - 1/12.

Biological aspects

Intertidal ecology

Intertidal ecology is the study of intertidal ecosystems, where organisms live between the low and high water lines. At low water, the intertidal is exposed (or ‘emersed’) whereas at high water, the intertidal is underwater (or ‘immersed’). Intertidal ecologists therefore study the interactions between intertidal organisms and their environment, as well as between different species of intertidal organisms within a particular intertidal community. The most important environmental and species interactions may vary based on the type of intertidal community being studied, the broadest of classifications being based on substrates - rocky shore and soft bottom communities.

Organisms living in this zone have a highly variable and often hostile environment, and have evolved various adaptations to cope with and even exploit these conditions. One easily visible feature of intertidal communities is vertical zonation, where the community is divided into distinct vertical bands of specific species going up the shore. A species' ability to cope with desiccation determines their upper limits, while competition with other species sets their lower limits.

Intertidal regions are utilized by humans for food and recreation, but anthropogenic actions also have major impacts, with overexploitation, invasive species and climate change being among the problems faced by intertidal communities. In some places Marine Protected Areas have been established to protect these areas and aid in scientific research.

Biological rhythms

Intertidal organisms are greatly affected by the approximately fortnightly cycle of the tides, and hence their biological rhythms tend to occur in rough multiples of this period. This is seen not only in the intertidal organisms however, but also in many other terrestrial animals, such as the vertebrates. Examples include gestation and the hatching of eggs. In humans, for example, the menstrual cycle lasts roughly a month, an even multiple of the period of the tidal cycle. This may be evidence of the common descent of all animals from a marine ancestor.

Other tides

In addition to oceanic tides, there are atmospheric tides as well as earth tides. All of these are continuum mechanical phenomena, the first two being fluids and the third solid (with various modifications).

Atmospheric tides are negligible from ground level and aviation altitudes, drowned by the much more important effects of weather. Atmospheric tides are both gravitational and thermal in origin and are the dominant dynamics from about 80 km to 120 km where the molecular density becomes too small to behave as a fluid.

Earth tides or terrestrial tides affect the entire rocky mass of the Earth. The Earth's crust shifts (up/down, east/west, north/south) in response to the Moon's and Sun's gravitation, ocean tides, and atmospheric loading. While negligible for most human activities, the semidiurnal amplitude of terrestrial tides can reach about 55 cm at the equator (15 cm is due to the Sun) which is important in GPS calibration and VLBI measurements. Also to make precise astronomical angular measurements requires knowledge of the Earth's rate of rotation and nutation, both of which are influenced by earth tides. The semi-diurnal M2 Earth tides are nearly in phase with the Moon with tidal lag of about two hours. Terrestrial tides also need to be taken in account in the case of some particle physics experiments. For instance, at the CERN or SLAC, the very large particle accelerators were designed while taking terrestrial tides into account for proper operation. Among the effects that need to be taken into account are circumference deformation for circular accelerators and particle beam energy. Since tidal forces generate currents of conducting fluids within the interior of the Earth, they affect in turn the Earth's magnetic field itself.

When oscillating tidal currents in the stratified ocean flow over uneven bottom topography, they generate internal waves with tidal frequencies. Such waves are called internal tides.

The galactic tide is the tidal force exerted by galaxies on stars within them and satellite galaxies orbiting them. The effects of the galactic tide on the Solar System's Oort cloud are believed to be the cause of 90 percent of all observed long-period comets.

Misapplications

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 actual link to the tide. Other phenomena unrelated to tides but using the word tide are rip tide, storm tide, hurricane tide, and black or red tides.

See also

External links

Tide predictions

References and notes

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