Tide

Tides represent the periodic fluctuation in sea level, primarily caused by the differential gravitational forces exerted by the Moon and the Sun, in conjunction with inertial effects stemming from the Earth–Moon system’s orbital mechanics and the Earth's rotational movement.

Tides are the periodic rise and fall of sea level resulting from the differential gravitational forces exerted primarily by the Moon and the Sun, combined with inertial effects associated with the Earth–Moon system’s orbital motion and the Earth's rotation.

Although these astronomical forcings establish the fundamental tidal potential, the empirically observed tides undergo significant modification by terrestrial elements, such as the morphological configuration of ocean basins, continental margins, seafloor topography, the Coriolis effect, frictional energy dissipation in shallow marine environments, and the resonant characteristics of coastlines.

Tidal variations manifest across timescales from hours to years, influenced by multiple determinants of the lunitidal interval. For precise documentation, tide gauges positioned at fixed locations continuously monitor water levels. These instruments disregard fluctuations induced by waves with periods less than a minute. The collected data are subsequently referenced against a designated datum, typically referred to as mean sea level.

While tides typically constitute the predominant source of short-term sea-level variability, sea levels are additionally susceptible to alterations from thermal expansion, wind stress, and atmospheric pressure differentials, which can lead to storm surges, particularly in shallow marine areas and coastal regions.

Tidal phenomena extend beyond oceanic environments, manifesting in any system where a spatiotemporally variable gravitational field exists. For instance, the Earth's solid body experiences subtle deformations due to Earth tides, although these are less overtly observable than oceanic tidal movements.

Characteristics

Oceanic tides exhibit a cyclic pattern, typically rising and falling approximately semidiurnally. The tidal cycle comprises four distinct stages:

• The cessation of water recession, marking a local minimum, is termed low tide.
• The subsequent increase in sea level over several hours, inundating the intertidal zone, is referred to as flooding.
• The cessation of water ascent, signifying a local maximum, is designated as high tide.
• The subsequent decrease in sea level over several hours, exposing the intertidal zone, is known as ebbing.

Oscillatory currents generated by tidal forces are identified as tidal streams or tidal currents. The instant at which the tidal current ceases is termed slack water or slack tide, after which the tide reverses direction, indicating a turning phase. While slack water typically coincides with periods near high and low water, certain geographical locations exhibit notable discrepancies between the timing of slack tide and the occurrences of high and low water.

Tides are frequently categorized as either semi-diurnal, characterized by two high waters and two low waters daily, or diurnal, featuring a single tidal cycle per day. The two daily high waters typically exhibit unequal heights, a phenomenon known as the daily inequality; these are designated as the higher high water and the lower high water in tidal prediction tables. Correspondingly, the two daily low waters are distinguished as the higher low water and the lower low water. This daily inequality is not uniform and generally diminishes when the Moon is positioned above the Equator.

Reference Levels

The subsequent reference tidal levels are defined, ordered from the highest to the lowest:

• Highest astronomical tide (HAT) – Represents the maximum predicted tidal elevation. It is important to note that meteorological factors can augment the height of the HAT.
• Mean high water springs (MHWS) – Denotes the average elevation of the two high tides occurring during spring tide periods.
• Mean high water neaps (MHWN) – Represents the average elevation of the two high tides observed during neap tide periods.
• Mean sea level (MSL) – Constitutes the average sea level. The MSL remains constant for a given geographical location over extended temporal durations.
• Mean low water neaps (MLWN) – Indicates the average depression of the two low tides occurring during neap tide periods.
• Mean low water springs (MLWS) – Signifies the average depression of the two low tides observed during spring tide periods.
• Lowest astronomical tide (LAT) – Designates the minimum predicted tidal depression.

Range Variation: Spring and Neap Tides

The semi-diurnal range, which represents the height differential between high and low waters over approximately half a day, exhibits a bi-weekly fluctuation. Roughly twice monthly, during the new moon and full moon phases, the Sun, Moon, and Earth align in a configuration termed a syzygy. In this alignment, the solar tidal force augments the lunar tidal force, resulting in the maximum tidal range, a phenomenon known as the spring tide. This term originates not from the season but from the etymological sense of "jump, burst forth, or rise," akin to a natural spring. Syzygy tides is an alternative designation for spring tides.

Conversely, when the Moon is in its first or third quarter, the Sun and Moon are positioned 90° apart relative to Earth, a configuration known as quadrature. During these periods, the solar tidal force partially counteracts the lunar tidal force, leading to the minimum tidal range. This phenomenon is termed the neap tide or neaps. The word "neap" derives from an Anglo-Saxon term signifying "without power." Quadrature tides is an alternative designation for neap tides.

Spring tides are characterized by higher-than-average high waters, lower-than-average low waters, a reduced duration of "slack water," and intensified tidal currents. In contrast, neap tides produce milder tidal conditions. Approximately seven days separate the occurrences of spring and neap tides.

Tidal constituents

Tidal constituents represent the cumulative effect of various factors influencing tidal fluctuations across specific temporal scales. Key constituents encompass Earth's rotation, the relative positions of the Moon and Sun to Earth, the Moon's elevation above the Earth's Equator, and bathymetry. Variations with periods shorter than half a day are classified as harmonic constituents, whereas cycles spanning days, months, or years are designated as long period constituents.

Tidal forces exert influence across the entire Earth system. Within the Earth's crust, these forces induce periodic vertical displacements measurable in centimeters, a phenomenon termed Earth tide. Concurrently, in the atmosphere, the gravitational forces from the Moon and Sun, combined with solar heating, generate global-scale oscillations in pressure, density, and wind, referred to as Atmospheric tide. While Earth tides manifest as elastic deformation, atmospheric tides are predominantly observed as fluctuations in pressure gradients and wind patterns.

Principal lunar semi-diurnal constituent

The most prominent tidal constituent in the majority of locations is the principal lunar semi-diurnal, alternatively identified as the M2 tidal constituent or M§56§ tidal constituent. This constituent exhibits a period of approximately 12 hours and 25.2 minutes, precisely half of a tidal lunar day. A tidal lunar day represents the average interval between successive lunar zeniths, corresponding to the time required for Earth to complete one rotation relative to the Moon. Basic tide clocks are designed to monitor this constituent. The lunar day exceeds the terrestrial day in duration because the Moon's orbital direction aligns with Earth's rotational direction.

Due to the Moon's orbital direction around Earth coinciding with Earth's axial rotation, approximately 24 hours and 50 minutes are required for the Moon to reappear at the same celestial position. Within this interval, the Moon undergoes one culmination (overhead passage) and one passage underfoot (corresponding to hour angles of 00:00 and 12:00, respectively). Consequently, in numerous regions, the period of maximal tidal forcing aligns with the aforementioned duration of approximately 12 hours and 25 minutes. It is important to note that the precise moment of highest tide does not invariably coincide with the Moon's closest proximity to zenith or nadir; however, the forcing period remains the determinant of the interval between high tides.

The Moon's gravitational field attenuates with increasing distance, resulting in a differential force exerted upon Earth. Specifically, a slightly stronger-than-average force acts on the Earth's side facing the Moon, while a marginally weaker force affects the opposing side. This differential gravitation causes the Moon to induce a slight "stretching" of Earth along the axis connecting the two celestial bodies. Although the solid Earth undergoes some deformation, oceanic water, owing to its fluid nature, exhibits significantly greater mobility in response to this tidal force, particularly in horizontal displacement.

The Earth's rotation continuously alters the magnitude and direction of tidal forces at any given point on its surface. Although the ocean never achieves equilibrium—meaning the fluid lacks sufficient time to fully respond to a constant tidal force—these dynamic tidal forces nonetheless induce rhythmic fluctuations in sea surface height.

A tidal pattern characterized by two high tides and two low tides of differing heights within a single day is classified as a mixed semi-diurnal tide.

Lunar Distance

Variations in the Moon-Earth distance also influence tidal amplitudes. At perigee, when the Moon is closest to Earth, the tidal range expands; conversely, at apogee, when the Moon is farthest, the range diminishes. Annually, perigee aligns with either a new or full moon approximately six to eight times, resulting in perigean spring tides, which exhibit the maximum tidal range. The height differential between a perigean spring tide and a spring tide occurring at lunar apogee is site-dependent but can reach up to one foot.

Other Constituents

Approximately 62 tidal constituents are substantial enough for inclusion in marine tide prediction models. These encompass solar gravitational influences, the obliquity (tilt) of Earth's equator and rotational axis, the inclination of the lunar orbital plane, and the elliptical trajectory of Earth's orbit around the Sun.

Shallow-water constituents comprise overtides, defined as higher harmonics of a singular dominant tidal frequency, and compound tides, which arise from interactions among the primary lunar and solar constituents. While generally possessing smaller amplitudes than the fundamental tidal potential, these constituents introduce asymmetries between the rising and falling phases of tides within estuaries and continental shelf environments.

Phase and Amplitude

Given the prevalence of the M₂ tidal constituent in most areas, the concept of tidal stage or phase, expressed as the time in hours following high water, proves highly valuable. Tidal stage can also be quantified in degrees, with 360° representing a complete tidal cycle. Lines connecting points of constant tidal phase are termed cotidal lines; these are comparable to contour lines indicating constant altitude on topographic maps and, when depicted, constitute a cotidal map or cotidal chart. High water occurs concurrently along cotidal lines extending from the coastline seaward, and tidal phases (represented by cotidal lines) progress along the coast. Semi-diurnal and long-period phase constituents are referenced from high water, while diurnal constituents are referenced from the maximum flood tide. It is important to note that this discussion and subsequent explanations are strictly accurate only when considering a single tidal constituent.

Within an ocean basin approximating a circular shape enclosed by a coastline, cotidal lines converge radially inward, ultimately meeting at a singular location known as an amphidromic point. An amphidromic point is characterized by simultaneous high and low waters, a condition fulfilled by zero tidal motion. (Rare exceptions exist where tides circulate around islands, such as New Zealand, Iceland, and Madagascar.) Tidal motion typically diminishes with increasing distance from continental coasts; consequently, contours of constant amplitude (defined as half the vertical distance between high and low water) intersect the cotidal lines and decrease to zero at the amphidromic point. For a semi-diurnal tide, an amphidromic point can be conceptualized as the center of a clock face, where the hour hand indicates the direction of the high water cotidal line, diametrically opposite the low water cotidal line. High water circulates around the amphidromic point every 12 hours, moving in the direction of advancing cotidal lines and away from receding ones. This rotation, induced by the Coriolis effect, is generally clockwise in the Southern Hemisphere and counterclockwise in the Northern Hemisphere. The deviation of a cotidal phase from a designated reference tide's phase is termed the epoch. The reference tide is defined as the hypothetical "equilibrium tide" constituent on a landless Earth, measured at 0° longitude (the Greenwich meridian).

In the North Atlantic, the counterclockwise circulation of cotidal lines around the amphidromic point results in high tide reaching New York Harbor approximately one hour earlier than Norfolk Harbor. South of Cape Hatteras, tidal dynamics become considerably more intricate, precluding reliable prediction solely based on North Atlantic cotidal line patterns.

History

History of Tidal Theory

In pre-scientific traditions, the periodic rise and fall of tides were often ascribed to mythological or animistic explanations. For instance, Indian and East Asian texts depicted the sea as breathing or pulsating like a living entity, while other narratives invoked the influence of a supernatural maritime force.

The study of tidal physics played a crucial role in the nascent stages of celestial mechanics. Initially, the occurrence of two daily tides was attributed to the Moon's gravitational pull. Subsequently, a more precise explanation emerged, detailing the interaction between the gravitational forces of both the Moon and the Sun.

Around 150 BC, Seleucus of Seleucia posited that the Moon was the cause of tides. Ptolemy's work, Tetrabiblos, also referenced the Moon's influence on aquatic bodies.

In 725, Bede, in his work De temporum ratione (The Reckoning of Time), established a connection between semidiurnal tides, fluctuating tidal heights, and the Moon's phases. He observed that tides consistently rose and fell 4/5 of an hour later each day, mirroring the Moon's delayed rise and set times. Bede further highlighted that over two lunar months (59 days), the Moon orbits the Earth 57 times, corresponding to 114 tides. He also noted monthly variations in tidal height, designating increasing tides as malinae and decreasing tides as ledones. The month, he explained, was segmented into four periods of seven or eight days, characterized by alternating malinae and ledones. Within the same text, Bede acknowledged the capacity of winds to impede tidal flow. Furthermore, he documented geographical variations in tidal timing, noting that tides occurred earlier north of his location (Monkwearmouth) and later to the south. He elucidated this phenomenon by stating that the tide "deserts these shores in order to be able all the more to be able to flood other [shores] when it arrives there," and that "the Moon which signals the rise of tide here, signals its retreat in other regions far from this quarter of the heavens."

Subsequent medieval comprehension of tides largely derived from the writings of Muslim astronomers, which became accessible in Latin translations beginning in the 12th century. Abu Ma'shar al-Balkhi (d. circa 886), in his Introductorium in astronomiam, posited that the Moon was responsible for ebb and flood tides. He also explored the influence of wind and the Moon's phases, relative to the Sun, on tidal patterns. During the 12th century, al-Bitruji (d. circa 1204) advanced the theory that tides resulted from the general circulation of the heavens.

In his 1608 treatise, De spiegheling der Ebbenvloet (The theory of ebb and flood), Simon Stevin refuted numerous prevailing misconceptions regarding ebb and flood tides. Stevin advocated for the concept that lunar attraction caused tides, articulating clear definitions for ebb, flood, spring tide, and neap tide, while emphasizing the necessity of further investigation.

In 1609, Johannes Kepler also accurately proposed that lunar gravitation was the cause of tides, grounding his hypothesis in ancient observations and correlations.

In his 1632 work, Dialogue Concerning the Two Chief World Systems, originally titled Dialogue on the Tides, Galileo Galilei integrated his tidal theory as a central component of his physical argument for Copernican heliocentrism. He posited that tides arose from the synergistic motions of the Earth's rotation and revolution. Galilei explicitly repudiated Kepler's assertion of lunar influence on the seas, characterizing such lunar attraction as an invocation of occult qualities rather than a mechanistically comprehensible cause.

Isaac Newton (1642–1727) pioneered the explanation of tides, attributing them to the gravitational attraction exerted by celestial bodies. His comprehensive theory, detailed in the Principia (1687), utilized universal gravitation to elucidate how lunar and solar attractions generate tidal forces. Prior to Pierre-Simon Laplace, Newton and his contemporaries approached this phenomenon through an equilibrium theory, a static system model that approximated tidal behavior in a hypothetical, non-inertial ocean uniformly covering the Earth. While the concept of tide-generating force (or its potential) remains crucial in modern tidal theory, it now serves as an intermediate forcing function rather than a definitive outcome. Contemporary theory additionally incorporates the Earth's dynamic tidal response to these applied forces, a response modulated by factors such as ocean depth, planetary rotation, and other environmental variables.

In 1740, the Académie Royale des Sciences in Paris instituted a prize for the most compelling theoretical treatise on tides. This prestigious award was jointly conferred upon Daniel Bernoulli, Leonhard Euler, Colin Maclaurin, and Antoine Cavalleri.

Maclaurin applied Newtonian theory to demonstrate that a smooth, ocean-covered sphere, subjected to the tidal force of a singular celestial body, would assume the shape of a prolate spheroid—an elongated three-dimensional oval—with its major axis oriented towards the perturbing body. He was also the first to document the influence of Earth's rotation on motion. Euler subsequently identified the horizontal component of the tidal force as the primary driver of tidal phenomena, rather than its vertical counterpart. In 1744, Jean le Rond d'Alembert investigated atmospheric tidal equations, though his models did not incorporate rotational effects.

In 1770, James Cook's barque, HMS Endeavour, ran aground on the Great Barrier Reef. Initial attempts to refloat the vessel during the subsequent high tide proved unsuccessful; however, the tide following that successfully dislodged it. During the seven-week repair period at the mouth of the Endeavour River, Cook meticulously observed tidal patterns. He noted that during neap tides, the two daily tides exhibited similar heights, whereas during spring tides, the morning tide ascended 7 feet (2.1 m) while the evening tide reached 9 feet (2.7 m).

Pierre-Simon Laplace developed a system of partial differential equations that correlated the ocean's horizontal current with its surface elevation, establishing the foundational dynamic theory for water tides. These Laplace tidal equations remain pertinent in contemporary oceanography. William Thomson, 1st Baron Kelvin, later reformulated Laplace's equations using vorticity, thereby enabling the derivation of solutions that describe tidally induced, coastally trapped waves, now recognized as Kelvin waves.

Subsequent to Laplace, scholars such as Kelvin and Henri Poincaré advanced his theoretical framework. Building upon these advancements and E. W. Brown's lunar theory, which detailed the Moon's orbital mechanics, Arthur Thomas Doodson published in 1921 the inaugural modern formulation of the tide-generating potential in harmonic form. Doodson's work identified 388 distinct tidal frequencies, and several of his methodologies continue to be employed.

The historical trajectory of tidal observation.

Since antiquity, the sophistication of tidal observation and discourse has progressively evolved, initially focusing on daily recurrences and subsequently on the intricate relationship between tides, the Sun, and the Moon.

Ancient surviving records of tidal observation indicate that Pytheas, during his expedition to the British Isles circa 325 BC, noted tidal patterns that correlated with the lunar cycle. Fragments ascribed to his work, On the Ocean, suggest an early recognition of the connection between tidal range and lunar phases.

During the 2nd century BC, Seleucus of Seleucia, a Hellenistic astronomer, accurately characterized the tidal phenomenon to substantiate his heliocentric model. He correctly posited that the Moon caused tides, though he hypothesized that this interaction was facilitated by "pneuma." Seleucus also observed that tidal timing and magnitude differed across various global locations. As documented by Strabo (1.1.9), Seleucus was the first to establish a connection between tides and lunar gravitational attraction, further noting that tidal height is contingent upon the Moon's position relative to the Sun.

Pliny the Elder's Naturalis Historia documents numerous tidal observations; for instance, spring tides occur a few days after or before the new and full moon and reach their peak around the equinoxes, although Pliny also recorded several correlations now considered speculative. In his Geography, Strabo detailed that tides in the Persian Gulf exhibited their maximum range when the Moon was most distant from the equatorial plane. These observations were made despite the comparatively limited tidal amplitude within the Mediterranean basin. Notably, the vigorous currents observed in the Euripus Strait and the Strait of Messina presented a puzzle to Aristotle. In Book Five of The Life of Apollonius of Tyana, Philostratus explored tides, acknowledging the Moon's influence but attributing the phenomena to "spirits." Around 730 AD in Europe, the Venerable Bede documented the synchronous rise of the tide on one British Isles coast with its fall on another, further detailing the temporal progression of high water along the Northumbrian coastline.

The first tide table in China emerged in 1056 AD, primarily intended for visitors desiring to observe the renowned tidal bore in the Qiantang River. The first known British tide table is attributed to John Wallingford, who passed away as Abbot of St. Albans in 1213, which calculated high water occurring 48 minutes later daily and three hours earlier at the Thames estuary compared to its upstream location in London.

In 1614, Claude d'Abbeville published the treatise "Histoire de la mission de pères capucins en l'Isle de Maragnan et terres circonvoisines," in which he revealed that the Tupinambá people possessed an understanding of the relationship between the Moon and tides prior to European recognition.

William Thomson (Lord Kelvin) initiated the inaugural systematic harmonic analysis of tidal records in 1867. A primary outcome was the construction of a tide-predicting machine, which utilized a pulley system to synthesize six harmonic time functions. This device was "programmed" through the adjustment of gears and chains to modify phasing and amplitudes. Comparable machines remained in use until the 1960s.

The first known sea-level record of an entire spring–neap cycle was documented in 1831 at the Navy Dock within the Thames Estuary. By 1850, numerous major ports had established automatic tide gauge stations.

John Lubbock was among the pioneers in mapping co-tidal lines for Great Britain, Ireland, and their adjacent coasts in 1840. William Whewell subsequently expanded upon this research, culminating in a nearly global chart by 1836. To ensure the consistency of these maps, he posited the existence of a mid-oceanic region devoid of tidal rise or fall, where co-tidal lines converge. The presence of such an amphidromic point, a term now used to describe these locations, was corroborated in 1840 by Captain William Hewett, RN, through meticulous soundings conducted in the North Sea.

Subsequently, in the late 20th century, geologists identified tidal rhythmites, which serve as geological evidence for the presence of ancient tides, particularly during the Carboniferous period.

Physics

Equilibrium Theory

The foundational model, attributed to Isaac Newton, that elucidates the phenomenon of two daily tides is termed equilibrium theory. This theory incorporates three primary simplifications: 1) the omission of Earth's landmasses, 2) the assumption of water's negligible viscosity, allowing instantaneous gravitational response, and 3) the disregard for friction between the Earth and water. Within a coordinate system co-rotating with the Earth-Moon system, the Earth-Moon distance remains constant, signifying a state of equilibrium. This equilibrium is conceptualized as a balance between the Moon's gravitational force and the centrifugal force generated by rotation. At the Earth's core, these forces are precisely equal and opposite. However, at other locations, these forces do not perfectly counterbalance, and the resultant residual force is designated as the tide-generating force. On the Earth's surface, points nearest to the Moon experience a marginally stronger gravitational pull, while points most distant exhibit a slightly stronger centrifugal force. At the poles, situated away from the Earth-Moon axis, a minor net force is directed inward towards the Earth. Ocean water is minimally affected by these specific forces. Conversely, in regions between the poles and the equator, a component of this minor force acts horizontally across the Earth's surface, directed towards the equator. This horizontal force encounters no opposing resistance. Consequently, ocean water flows in response, receding from the poles and accumulating near the equator. This process culminates in the formation of a double tidal bulge aligned with the Earth-Moon axis, with the bulge on the side closer to the Moon being marginally larger. As the Earth rotates on its axis, various points on its surface traverse these bulges, thereby providing a general explanation for the occurrence of daily double tides. Both oceanic water and the solid Earth are subject to these differential gravitational pulls; however, the rigid Earth resists deformation, maintaining its approximately spherical shape, whereas the fluid redistributes itself to accommodate the imbalance, thus forming the characteristic bulges. The equilibrium tide represents this idealized tidal condition, predicated on the assumption of a landless Earth.

Forces

The tidal force generated by a massive celestial body (subsequently referred to as the Moon) upon a small particle situated on or within an extensive body (subsequently referred to as the Earth) is defined as the vector difference between the gravitational force exerted by the Moon on that particle and the gravitational force that would be exerted on the particle if it were positioned at the Earth's center of mass.

While the gravitational force exerted by a celestial body on Earth exhibits an inverse square relationship with its distance from Earth, the maximal tidal force demonstrates an approximate inverse cube proportionality to this distance. If the tidal force generated by each body were equivalent to its total gravitational force—a condition not met due to the free fall of the entire Earth, not merely its oceans, towards these bodies—a distinct pattern of tidal forces would manifest. For instance, the Sun's influence would appear considerably more potent than the Moon's. Although the solar gravitational force on Earth is, on average, 179 times greater than the lunar force, the Sun's average distance from Earth is 389 times greater, resulting in a weaker field gradient. The overarching proportionality can be expressed as:

${\displaystyle {\text{tidal force}}\propto {\frac {M}{d^{3}}}\propto \rho \left({\frac {r}{d}}\right)^{3},}$

Here, M denotes the mass of the celestial body, d represents its distance, ρ signifies its average density, and r indicates its radius. The ratio r/d correlates with the angular size of the object as observed from Earth. Despite the Sun and Moon exhibiting nearly identical apparent diameters in the terrestrial sky, the Sun's tidal force is considerably weaker than the Moon's. This disparity arises because the Sun's average density is significantly lower, and its tidal influence is merely 46% of the Moon's. Consequently, during a spring tide, the Moon accounts for 69% of the total tidal force, whereas the Sun contributes 31%. Specifically, the lunar tidal acceleration, measured along the Moon–Earth axis at the Earth's surface, approximates 1.1ק1516§−7 g. In contrast, the solar tidal acceleration, along the Sun–Earth axis at the Earth's surface, is approximately 0.52ק2122§−7 g, where g represents the gravitational acceleration at the Earth's surface. The tidal effects exerted by other planets fluctuate in accordance with their varying distances from Earth. At its perigee, Venus's tidal influence amounts to 0.000113 times that of the Sun. Conversely, at different orbital configurations, Jupiter or Mars may exert the predominant planetary tidal influence.

The ocean's surface is conventionally modeled by the geoid, a theoretical equipotential surface that accounts for both Earth's gravitational force and the centrifugal force arising from its rotation. Subsequently, the influence of substantial external celestial bodies, specifically the Moon and Sun, must be considered. These celestial entities possess potent gravitational fields that attenuate with increasing distance, inducing deviations in the ocean's surface from the geoid. This interaction establishes a dynamic equilibrium ocean surface, characterized by bulges oriented both towards and directly opposite the Moon. Earth's rotation relative to this evolving configuration generates the diurnal tidal cycle. The oceanic surface continuously strives to achieve this perpetually shifting equilibrium configuration, though it never fully reaches it. Disalignment between the actual ocean surface and this equilibrium state creates an apparent slope, prompting water to accelerate in the downslope direction.

Laplace's Tidal Equations

Oceanic depths are considerably less than their horizontal dimensions. Consequently, the oceanic response to tidal forcing can be accurately modeled using Laplace's tidal equations, which integrate the subsequent characteristics:

• Vertical (or radial) velocity is considered negligible, and the absence of vertical shear implies a sheet flow regime.
• The applied forcing is exclusively horizontal (tangential).
• The Coriolis effect manifests as a fictitious inertial force, acting perpendicular to the flow direction and directly proportional to velocity.
• The temporal rate of change of surface height is directly proportional to the negative divergence of velocity, scaled by the depth. When horizontal velocity induces stretching or compression of the oceanic sheet, the water volume consequently thins or thickens.

Boundary conditions stipulate zero flow perpendicular to coastlines and a free-slip condition at the seabed.

The Coriolis effect, an inertial force, deflects flows approaching the Equator westward and flows receding from the Equator eastward, thereby facilitating the formation of coastally trapped waves. Furthermore, a dissipation term, analogous to viscosity, can be incorporated into the equations.

Tidal Amplitude and Periodicity

The maximum theoretical amplitude of oceanic tides induced by the Moon is approximately 54 centimetres (21 in). This value represents the amplitude achievable under idealized conditions, specifically a uniform ocean depth, the absence of landmasses, and Earth's synchronous rotation with the Moon's orbit. Similarly, the Sun generates tides with a theoretical amplitude of approximately 25 centimetres (9.8 in), constituting 46% of the Moon's tidal amplitude, and exhibiting a 12-hour cycle. During spring tides, these two effects combine synergistically, resulting in a theoretical amplitude of 79 centimetres (31 in). Conversely, at neap tides, the theoretical amplitude diminishes to 29 centimetres (11 in). Given the elliptical nature of Earth's orbit around the Sun and the Moon's orbit around Earth, tidal amplitudes exhibit variations due to fluctuating Earth–Sun and Earth–Moon distances. This orbital eccentricity introduces a variation in tidal force and theoretical amplitude of approximately ±18% for lunar tides and ±5% for solar tides. Should both the Sun and Moon simultaneously occupy their closest orbital positions and align during a new moon phase, the theoretical tidal amplitude could attain 93 centimetres (37 in).

Actual tidal amplitudes exhibit substantial variation, due to factors such as bathymetric variations, continental landmasses, and the oceanic wave propagation's natural period, which approximates the Earth's rotational period. For instance, a long-wavelength surface wave would require approximately 30 hours to traverse half the Earth's circumference along the Equator in the absence of landmasses (In contrast, the Earth's lithosphere possesses a natural period of approximately 57 minutes). Furthermore, the ocean's response to tidal forces is significantly complicated by both Earth tides, which induce vertical displacement of the seafloor, and the gravitational self-attraction of the tides themselves.

Dissipation

Terrestrial tidal oscillations generate an average dissipation rate of approximately 3.75 terawatts. Marine tidal movements account for approximately 98% of this energy dissipation. This dissipation occurs when basin-scale tidal flows induce smaller-scale currents, leading to turbulent energy loss. The resulting tidal drag exerts torque on the Moon, progressively transferring angular momentum to its orbit and causing a gradual increase in the Earth–Moon separation. Concurrently, the equal and opposite torque exerted on Earth reduces its rotational velocity. Consequently, over geological timescales, the Moon recedes from Earth at an approximate rate of 3.8 centimeters (1.5 in) per year, thereby extending the terrestrial day.

Over the past 600 million years, the duration of a terrestrial day has increased by approximately 2 hours. Under the simplifying assumption of a constant deceleration rate, this suggests that 70 million years ago, the day length was approximately 1% shorter, resulting in roughly four additional days per year.

Bathymetry

The morphology of coastlines and the ocean floor significantly influences tidal propagation, precluding a simple, universal rule for predicting high water times based solely on the Moon's celestial position. Consequently, local coastal characteristics, including bathymetry and shoreline configuration, impact tide forecasting, leading to potential discrepancies between actual high water times and heights and those derived from predictive models, owing to the influence of coastal morphology on tidal currents. Nevertheless, for any specific location, the relationship between lunar altitude and the timing of high or low tide (known as the lunitidal interval) remains relatively consistent and predictable, as does the timing of tides relative to other points along the same coastline. For instance, in Norfolk, Virginia, U.S., high tide consistently occurs approximately two and a half hours prior to the Moon's zenith passage.

Continental landmasses and oceanic basins impede the free global movement of water, and their diverse geomorphologies influence tidal frequencies. Consequently, tidal patterns exhibit considerable regional variation. For instance, the U.S. East Coast and Europe's Atlantic coasts primarily experience semi-diurnal tides, whereas the U.S. West Coast is characterized by mixed tides. Anthropogenic modifications to the landscape can also substantially modify local tidal regimes.

Observation and Prediction

Timing

Lunar and solar tidal forces generate extensive long waves that propagate throughout the ocean, tracing paths depicted in co-tidal charts. The arrival time of a wave crest at a specific port determines the local high water time. The oceanic propagation time of these waves introduces a temporal lag between lunar phases and their corresponding tidal effects. For instance, spring and neap tides in the North Sea occur approximately two days after the new/full moon and first/third quarter moon, respectively. This phenomenon is termed the tide's age.

Oceanic bathymetry significantly dictates the precise timing and amplitude of tides at specific coastal locations. Notable extreme examples include the Bay of Fundy, situated on Canada's east coast, which is frequently cited for possessing the world's highest tides, a characteristic attributed to its unique geomorphology, bathymetry, and proximity to the continental shelf edge. In November 1998, measurements at Burntcoat Head within the Bay of Fundy documented a maximum tidal range of 16.3 meters (53 ft) and a highest predicted extreme of 17 meters (56 ft). Comparable measurements conducted in March 2002 at Leaf Basin, Ungava Bay, northern Quebec, yielded similar values (accounting for measurement uncertainties), with a maximum range of 16.2 meters (53 ft) and a highest predicted extreme of 16.8 meters (55 ft). While both Ungava Bay and the Bay of Fundy are similarly positioned relative to the continental shelf edge, Ungava Bay remains ice-free for only approximately four months annually, whereas the Bay of Fundy seldom freezes.

Southampton in the United Kingdom experiences a double high water phenomenon, which results from the interaction between the M₂ and M§67§ tidal constituents, identified as shallow water overtides of the principal lunar tide. Similarly, Portland exhibits double low waters due to the same underlying mechanism. The M§1011§ tide is prevalent along the entire southern coast of the United Kingdom; however, its impact is most pronounced in the area between the Isle of Wight and Portland, primarily because the M§14_{15§ tide reaches its minimum amplitude within this specific region.}

The Mediterranean Sea and the Baltic Sea exhibit their largest tidal amplitudes near their constricted connections with the Atlantic Ocean, a consequence of their oscillation modes not aligning with any substantial astronomical forcing periods. Analogously, the Gulf of Mexico and the Sea of Japan experience exceptionally minimal tides for identical reasons. In other geographical contexts, such as along the southern Australian coastline, reduced tidal ranges may be attributed to the proximity of an amphidromic point.

Analysis

Isaac Newton's gravitational theory provided the initial framework for explaining the typical occurrence of two daily tides, rather than a single one, thereby fostering prospects for a comprehensive understanding of tidal forces and their dynamics. While it might appear that tidal phenomena could be predicted solely through precise knowledge of instantaneous astronomical forcings, the actual tidal behavior at any specific location is influenced by astronomical forces that accumulate within the water body over several days. Furthermore, achieving accurate predictions necessitates detailed information regarding the morphology of all ocean basins, including their bathymetry and coastline configurations.

The contemporary methodology for tidal analysis employs the harmonic analysis technique, which William Thomson introduced in the 1860s. This approach posits that astronomical theories concerning the Sun's and Moon's motions define numerous component frequencies, each associated with a force component that induces tidal motion. Crucially, at every specific geographical location on Earth, tides respond to each of these frequencies with a unique amplitude and phase characteristic of that particular site. Consequently, at each location of interest, tidal heights are meticulously measured over an extended duration—typically exceeding one year for newly investigated ports—to facilitate the analytical differentiation of responses at each significant tide-generating frequency. This process allows for the extraction of tidal constants for a sufficient number of the most prominent known components of astronomical tidal forces, thereby enabling practical tide prediction. Tidal heights are anticipated to correspond to the tidal force, exhibiting a consistent amplitude and phase delay for each component. Given that astronomical frequencies and phases can be determined with precision, future tidal heights can be accurately predicted once the response to the harmonic components of the astronomical tide-generating forces has been established.

The primary patterns observed in tidal phenomena include:

• the semidiurnal variation;
• the diurnal inequality;
• the spring-neap cycle;
• the annual variation.

The Highest Astronomical Tide (HAT) is defined as the perigean spring tide, which occurs when both the Sun and the Moon are at their closest orbital proximity to Earth.

In the analysis of periodically varying functions, the conventional methodology involves the application of Fourier series. This analytical technique utilizes sinusoidal functions as a basis set, with frequencies that are integer multiples (zero, one, two, three, and so forth) of a specific fundamental cycle's frequency. These multiples are designated as harmonics of the fundamental frequency, and the entire procedure is referred to as harmonic analysis. When the chosen basis set of sinusoidal functions accurately represents the modeled behavior, a relatively small number of harmonic terms is sufficient. Given that orbital paths are approximately circular, sinusoidal variations are particularly well-suited for modeling tidal phenomena.

In the practical analysis of tidal heights, the Fourier series methodology necessitates a more sophisticated application than merely employing a single frequency and its associated harmonics. Tidal patterns are consequently decomposed into numerous sinusoids, each possessing distinct fundamental frequencies. These frequencies correspond, akin to lunar theory, to various combinations of the Earth's and Moon's motions, alongside the angular parameters that delineate the configuration and position of their respective orbits.

For tidal analysis, harmonic analysis extends beyond harmonics derived from a singular frequency. Consequently, the constituent harmonics are derived from multiple fundamental frequencies, diverging from the singular fundamental frequency characteristic of a basic Fourier series methodology. Representing these phenomena solely through a Fourier series with a single fundamental frequency and its integer multiples would necessitate an extensive number of terms and would exhibit significant temporal limitations regarding its applicability.

The investigation of tide height through harmonic analysis was initiated by Laplace, William Thomson (Lord Kelvin), and George Darwin. A.T. Doodson subsequently expanded upon this research, establishing the Doodson Number notation to systematically categorize the numerous resultant terms. This methodology has subsequently attained international standardization. The complexities inherent in this model stem from the conceptual representation of the tide-raising force as a summation of multiple constituent terms. Specifically, each term adheres to the following mathematical structure:

${\displaystyle A_{o}\cos(\omega t+p),}$

wherein

A_o represents the amplitude;

ω denotes the angular frequency, typically expressed in degrees per hour, corresponding to the variable t when measured in hours;

p signifies the phase offset relative to the astronomical conditions at time t = 0.

This formulation includes one term specifically for the Moon and a distinct second term for the Sun. The phase parameter p associated with the Moon term's primary harmonic is designated as the lunitidal interval or, alternatively, the high water interval.

A subsequent refinement involves incorporating harmonic terms that account for the elliptical configurations of the orbits. To achieve this, the amplitude is considered a time-variant quantity, oscillating around the mean amplitude A_o, rather than a constant. Specifically, A_o in the preceding equation is substituted with A(t), where A itself represents an additional sinusoidal function, conceptually analogous to the cycles and epicycles within Ptolemaic astronomical models. This substitution yields the following expression:

A ( t ) = A o ( §3031§ + A a cos ⁡ ( ω a t + p a ) ) , {\displaystyle A(t)=A_{o}{\bigl (}1+A_{a}\cos(\omega _{a}t+p_{a}){\bigr )},}

This expression indicates an average value A_o, modulated by a sinusoidal variation of magnitude A_a, characterized by a frequency ω_a and a phase p_a. The substitution of this expression for A_o into the initial equation results in a product comprising two cosine factors:

A o ( §2021§ + A a cos ⁡ ( ω a t + p a ) ) cos ⁡ ( ω t + p ) . {\displaystyle A_{o}{\bigl (}1+A_{a}\cos(\omega _{a}t+p_{a}){\bigr )}\cos(\omega t+p).}

It is established that for any variables x and y,

cos ⁡<!-- ⁡ --> x cos ⁡<!-- ⁡ --> y = §2526§ §2728§ cos ⁡ ( x + y ) + §5253§ §5455§ cos ⁡ ( x − y ) , {\displaystyle \cos x\cos y={\tfrac {1}{2}}\cos(x+y)+{\tfrac {1}{2}}\cos(x-y),}

It is evident that a compound term, formed by the product of two cosine terms each possessing a distinct frequency, is equivalent to three simple cosine terms. These terms are subsequently added at the original frequency, as well as at frequencies representing the sum and difference of the two constituent frequencies from the product term. This yields three terms, rather than two, because the complete expression is given by ${\displaystyle (1+\cos x)\cos y}$. Furthermore, the tidal force exerted on a specific location is also contingent upon the Moon's (or Sun's) position relative to the equatorial plane (i.e., above or below). These orbital characteristics introduce additional periods that are incommensurable with both a day and a month, leading to a multitude of resulting combinations. Through a meticulous selection of fundamental astronomical frequencies, the Doodson Number systematically quantifies the specific additions and differences required to derive the frequency for each individual simple cosine term.

It is crucial to note that astronomical tides explicitly do not incorporate meteorological influences. Furthermore, alterations in local environmental conditions, such as sandbank migration or the dredging of harbor entrances, can significantly impact the actual timing and magnitude of tides, especially when these conditions deviate from those present during measurement. Entities that publish a "highest astronomical tide" for a given location may inflate this value. This exaggeration often serves as a safety margin to account for analytical uncertainties, the distance from the closest measurement point, temporal changes since the last observation, ground subsidence, and other factors, thereby mitigating potential liability if an engineering structure were to be overtopped. Consequently, particular caution is warranted when determining the magnitude of a "weather surge" by deducting the astronomical tide from the empirically observed tide.

A meticulous Fourier data analysis, spanning a nineteen-year interval (designated as the National Tidal Datum Epoch in the United States), employs frequencies known as tidal harmonic constituents. This nineteen-year duration is favored due to the near-exact recurrence of the relative positions of the Earth, Moon, and Sun within the Metonic cycle, a period sufficient to encompass the 18.613-year lunar nodal tidal constituent. Such analysis can be performed solely with knowledge of the forcing period, obviating a detailed comprehension of the mathematical derivation; consequently, practical tidal tables have been compiled for centuries. The derived amplitudes and phases subsequently facilitate the prediction of anticipated tides. While constituents approximating 12 hours (termed semi-diurnal constituents) typically exert the primary influence, significant constituents also exist around 24 hours (diurnal). Additional longer-term constituents include those occurring every 14 days (or fortnightly), monthly, and semiannually. Although semi-diurnal tides prevail along most coastlines, certain regions, such as the South China Sea and the Gulf of Mexico, exhibit predominantly diurnal tidal patterns. Within semi-diurnal zones, the principal constituents, M§1415§ (lunar) and S§1819§ (solar), possess slightly divergent periods. This discrepancy leads to fortnightly (14-day period) variations in their relative phases and, consequently, in the amplitude of the resultant combined tide.

Within the aforementioned M₂ plot, each cotidal line represents a one-hour phase difference from adjacent lines, with thicker lines indicating tides in equilibrium phase at Greenwich. In the Northern Hemisphere, these lines exhibit a counterclockwise rotation around amphidromic points, causing the M§6_{7§ tide to propagate northward from the Baja California Peninsula to Alaska and from France to Ireland. Conversely, in the Southern Hemisphere, this propagation direction is clockwise. However, the M§1011§ tide propagates counterclockwise around New Zealand, a phenomenon attributed to the islands' barrier effect, which permits differential tidal heights on their opposing sides. (Notably, the tides propagate northward along the east coast and southward along the west coast, consistent with theoretical predictions.)}

An exception occurs at Cook Strait, where tidal currents periodically connect high water to low water. This phenomenon arises because cotidal lines situated 180° around amphidromes are in opposing phases, exemplified by high water occurring opposite low water at each extremity of Cook Strait. Given that each tidal constituent possesses a distinct configuration of amplitudes, phases, and amphidromic points, the patterns observed for M₂ are not transferable to other tidal components.

Tidal Tables

Tidal tables provide predicted times and amplitudes (or "tidal range") for specific geographical locations. These predictions are subject to numerous influences, such as the alignment of the Sun and Moon, the phase and amplitude of the tide (representing the deep ocean tidal pattern), the oceanic amphidromic systems, and the morphology of the coastline and near-shore bathymetry.

While tables offer predictions, the actual timing and height of tides are influenced by meteorological factors such as wind and atmospheric pressure. Numerous coastal regions exhibit semi-diurnal tides, characterized by two approximately equal high and low tides daily. Conversely, other locations experience diurnal tides, featuring a single high and low tide per day. A third common classification is the "mixed tide," which involves two tides of unequal magnitude occurring within a day.

Illustrative Calculation

Due to the Moon's orbital motion around the Earth in the same direction as Earth's rotation, a terrestrial point must rotate an additional distance to realign, resulting in a period between semi-diurnal tides of 12.4206 hours, approximately twenty-five minutes longer than twelve hours. The two daily tidal peaks are not equivalent in magnitude. The two daily high tides exhibit alternating maximum heights: a lower high (slightly under three feet), followed by a higher high (just over three feet), and then another lower high. A similar pattern is observed for the low tides.

When the Earth, Moon, and Sun are aligned (syzygy, i.e., Sun–Earth–Moon or Sun–Moon–Earth configurations), their combined gravitational influences generate spring tides; conversely, when these two forces oppose each other, such as when the Moon–Earth–Sun angle approaches ninety degrees, neap tides occur. During its orbital progression, the Moon traverses from positions north of the Equator to south of the Equator. This orbital shift causes the alternation in high tide heights to diminish, eventually becoming equal (at the lunar equinox, when the Moon is positioned above the Equator). Subsequently, the alternation re-emerges with the opposite polarity, increasing to a maximal difference before gradually decreasing again.

Tidal Current

Analyzing the influence of tides on currents or flow presents significant challenges, as does the collection of relevant data. Tidal height, a scalar quantity, exhibits smooth variation across extensive geographical areas. Conversely, flow constitutes a vector quantity, possessing both magnitude and direction, which can fluctuate considerably with depth and across short distances, primarily influenced by local bathymetry. Furthermore, while the central axis of a water channel offers the most advantageous location for measurements, the deployment of current-measuring equipment often encounters opposition from mariners due to potential obstruction of waterways. A flow traversing a curved channel may maintain a consistent magnitude despite continuous directional changes along its path. Counterintuitively, flood and ebb flows frequently do not exhibit diametrically opposed directions. The morphology of the upstream channel dictates the direction of flow, rather than the downstream configuration. Similarly, the formation of eddies may be restricted to a single flow direction.

Despite these complexities, the analysis of tidal currents generally parallels that of tidal heights. In a simplified scenario, at a specific location, the flood flow predominantly moves in one direction, while the ebb flow proceeds in an alternative direction. Flood velocities are conventionally assigned a positive sign, whereas ebb velocities receive a negative sign. This analytical approach treats these velocities analogously to tidal heights.

In more intricate scenarios, the primary ebb and flood flows may not be the dominant factors. Rather, the flow's direction and magnitude delineate an elliptical path over a complete tidal cycle when plotted on a polar graph, diverging from a simple linear progression along ebb and flood axes. Consequently, analysis in such instances may involve pairs of orthogonal directions, designated as primary and secondary. An alternative methodological approach involves representing tidal flows as complex numbers, given that each value inherently possesses both magnitude and direction.

Information regarding tidal flow is typically disseminated via nautical charts, where it is presented in tabular format, detailing flow speeds and bearings at hourly increments, with distinct tables provided for spring and neap tides. The temporal referencing for this data is established relative to high water at a designated harbor exhibiting a comparable tidal pattern, irrespective of its geographical distance.

Analogous to tide height predictions, forecasts for tidal flow derived solely from astronomical parameters do not account for meteorological conditions, which possess the capacity to completely alter the predicted outcomes.

Cook Strait

The tidal flow traversing the Cook Strait, situated between New Zealand's two principal islands, presents a notable case study due to the nearly exact out-of-phase relationship of tides on either side, resulting in one side's high water coinciding with the other's low water. This phenomenon generates powerful currents, accompanied by an almost negligible change in tidal height at the strait's central point. However, while a tidal surge typically maintains a unidirectional flow for six hours before reversing for an equivalent duration, specific surges can extend for eight or ten hours, with the subsequent reverse surge significantly diminished. Under particularly severe meteorological conditions, the reverse surge may be entirely suppressed, causing the flow to persist in a single direction across three or more successive surge periods.

The characteristics of the Cook Strait's tidal cycle also exhibit coastal variations. Along the west coast and within Tasman/Golden Bay, tidal ranges typically adhere to a conventional fortnightly spring–neap cycle, characterized by amplified ranges during spring tides and reduced ranges during neap tides, a phenomenon primarily influenced by the syzygy of the Moon and Sun. Conversely, in certain eastern coastal regions, including areas near Wellington and Napier, the tidal pattern demonstrates a more pronounced monthly modulation, which correlates with perigean–apogean signals (variations in the Moon's orbital distance), and consequently does not display as distinct a fortnightly spring–neap signal.

A graphical representation of Cook Strait's tides, spanning through November 2007, delineates the heights and times of both high and low water. These data points are not direct measurements but rather computations derived from tidal parameters established through historical measurements spanning several years. Nautical charts pertaining to Cook Strait provide comprehensive tidal current information. For example, the January 1979 edition, covering the coordinates 41°13.9′S 174°29.6′E (situated northwest of Cape Terawhiti), referenced tidal timings to Westport, whereas the January 2004 edition utilized Wellington as its reference point.

Near Cape Terawhiti, situated in the central Cook Strait, tidal height variation is negligible, while the tidal current attains its peak velocity, particularly around the prominent Karori Rip. Beyond meteorological influences, the prevailing currents through Cook Strait are shaped by the tidal height differentials between the strait's two extremities. Significantly, only one of the two spring tides at the northwestern end of the strait near Nelson is mirrored by a corresponding spring tide at the southeastern end (Wellington); consequently, the tidal dynamics do not align with either reference port's characteristics.

Nantucket Shoals

In the Nantucket Shoals region of the Atlantic Ocean, tidal currents exhibit a rotary nature, wherein the flow direction progressively shifts across all compass points throughout a tidal cycle, as opposed to merely oscillating linearly. Across the shoals, these currents typically exhibit a clockwise rotation, with peak velocities commonly ranging from 1.5 to 2.5 knots and minimums approximating 0.5 knot; however, velocities are subject to considerable variation based on geographical position and tidal phase. Given the continuous rotational shift in current direction, rather than an abrupt reversal, a distinct slack water period—characteristic of simple reversing currents—is absent. Instead, current velocity fluctuates throughout the cycle, while its directional vector undergoes perpetual alteration.

Power Generation

Tidal energy can be harnessed through two primary methodologies: the deployment of hydrokinetic turbines within tidal currents, or the construction of impoundment ponds that regulate water flow through turbines. For the former, energy yield is solely contingent upon the temporal dynamics and magnitude of the tidal current. Nevertheless, optimal current locations may be inaccessible due to potential obstruction of maritime navigation by turbine installations. Conversely, the latter approach involves significant construction costs for impoundment dams, leads to substantial disruption of natural hydrological cycles, and impedes ship navigation. Nonetheless, the utilization of multiple impoundment basins offers the flexibility to generate power at predetermined intervals. To date, the implementation of tidal power generation systems remains limited (with the La Rance facility in Saint Malo, France, being a notable example), and these projects encounter numerous challenges. Beyond ecological considerations, the inherent difficulties of resisting corrosion and mitigating biofouling present considerable engineering hurdles.

Tidal power exhibits greater predictability compared to wind energy; however, turbine efficiency diminishes significantly at low flow velocities, and given that power output is proportional to the cube of velocity, peak generation periods are inherently short-lived. Such fluctuations can be ameliorated via energy storage solutions, sophisticated turbine control systems, distributed tidal energy arrays, or through hybridization with other renewable energy sources.

Navigation

Tidal flows are crucial for maritime navigation, and failure to account for these flows can result in substantial positional inaccuracies. Furthermore, tidal heights hold considerable significance; for instance, numerous rivers and harbors feature a shallow "bar" at their entrance, precluding vessels with substantial draft from ingress during low tide.

Prior to the widespread adoption of automated navigation systems, proficiency in computing tidal effects was an essential skill for naval officers. Historically, the Royal Navy's lieutenant examination certificate attested to a candidate's ability to "shift his tides," signifying their mastery of tidal calculations.

Information regarding tidal flow timings and velocities is typically presented in tide charts or a dedicated tidal stream atlas. These charts are commonly issued in sequential sets. Each individual chart depicts the average tidal flow for a specific one-hour interval between successive high waters, omitting the residual 24-minute period. A directional arrow on the tidal chart signifies both the direction and the average flow speed (conventionally expressed in knots) for both spring and neap tides. In the absence of a dedicated tide chart, the majority of nautical charts incorporate "tidal diamonds," which correlate specific chart locations with a tabular presentation of tidal flow direction and speed.

The conventional methodology for compensating for tidal influences during navigation involves three steps: (1) computing a "dead reckoning" (DR) position based on distance traveled and course steered, (2) marking this position on the chart with a vertical cross, and (3) projecting a line from the DR in the direction of the tidal current. The displacement of the vessel along this projected line, caused by the tidal current, is then calculated using the tidal speed, yielding an "estimated position" (EP), which is conventionally denoted by a dot enclosed within a triangle.

Hydrographic charts indicate charted water depths at specific points via soundings and through bathymetric contour lines, which delineate the morphology of the submerged seafloor. These measurements are referenced to a specific "chart datum," typically defined as the water level at the lowest astronomical tide (LAT). However, alternative datums have been historically employed, and meteorological factors can cause tidal variations exceeding or falling below astronomical predictions. Consequently, charted depths represent the minimum water column available during a tidal cycle. Additionally, "drying heights" may be indicated, signifying the elevation of the exposed seabed at the lowest astronomical tide.

Tide tables provide daily data on high and low water heights and their corresponding times. Determining the actual water depth necessitates adding the charted depth to the predicted tide height. For intermediate times, water depths can be interpolated using tidal curves, which are typically available for significant ports. In the absence of precise tidal curves, the "rule of twelfths" offers a sufficient approximation. This method postulates that the increase in depth over the six-hour period between low and high water follows a specific distribution: 1/12 during the first hour, 2/12 during the second, 3/12 during the third, 3/12 during the fourth, 2/12 during the fifth, and 1/12 during the sixth.

Biological Considerations

Intertidal Ecology

Intertidal ecology focuses on the investigation of ecosystems situated between the low- and high-water marks of coastal environments. During periods of low tide, this zone becomes exposed (or emersed), while at high tide, it is submerged (or immersed). Consequently, intertidal ecologists analyze the complex interactions between organisms and their physical environment, alongside interspecific relationships. The predominant ecological interactions are contingent upon the specific characteristics of the intertidal community. Primary classifications of these communities are typically determined by substrate type, distinguishing between rocky shores and soft-bottom habitats.

Organisms inhabiting intertidal zones confront a highly fluctuating and frequently challenging environment, necessitating specialized adaptations to survive and even thrive under these conditions. A prominent characteristic is vertical zonation, where the biological community segregates into discrete horizontal bands of particular species, each occupying a specific elevation above the low-water mark. The upper distributional limit of a species is primarily dictated by its tolerance to desiccation, whereas interspecific competition typically establishes its lower boundary.

Human populations utilize intertidal regions for both sustenance and recreational activities. Direct damage to intertidal ecosystems can result from overexploitation. Furthermore, other anthropogenic impacts, including the introduction of invasive species and the effects of climate change, exert significant detrimental influences. The establishment of Marine Protected Areas represents a viable strategy for communities to safeguard these critical habitats and facilitate scientific investigation.

Biological Rhythms

The approximately 12-hour and fortnightly tidal cycles profoundly influence both intertidal and marine organisms. Consequently, their biological rhythms frequently manifest as approximate multiples of these tidal periodicities. Numerous other animal taxa, including vertebrates, exhibit comparable circatidal rhythms. Illustrative examples encompass gestation periods and egg hatching. In humans, the menstrual cycle approximates a lunar month, which is an even multiple of the tidal period. These observed parallels suggest a potential common evolutionary lineage for all animals, originating from a marine ancestor.

Other Tidal Phenomena

Internal Tides

Oscillating tidal currents within a stratified ocean, when interacting with irregular seafloor topography, induce the generation of internal waves characterized by tidal frequencies.

Lake Tides

Significant lakes, including Superior and Erie, may exhibit tidal ranges of 1 to 4 cm (0.39 to 1.6 in); however, these subtle fluctuations are often obscured by meteorologically driven phenomena like seiches. For instance, the tidal amplitude in Lake Michigan is reported to be between 1.3 and 3.8 cm (0.5 to 1.5 in), or alternatively, 4.4 cm (1+3⁄§56§ in). Such minimal tidal magnitudes are entirely overshadowed by other more substantial influences, leading to these lakes being categorized as effectively non-tidal.

Atmospheric Tides

At ground level and typical aviation altitudes, atmospheric tides are considered negligible, as their effects are obscured by more significant meteorological phenomena. These tides originate from both gravitational and thermal forces and constitute the primary dynamic processes within the mesosphere and lower thermosphere, approximately 80 to 120 kilometers (50 to 75 mi) above Earth's surface, beyond which the molecular density is insufficient to sustain fluid-like behavior.

Earth Tides

Terrestrial tides, also known as Earth tides, exert an influence across the entire mass of the planet, which behaves akin to a fluid gyroscope encased in a remarkably thin crust. The Earth's crust undergoes displacements—vertically, longitudinally, and latitudinally—as a consequence of gravitational forces from the Moon and Sun, oceanic tidal effects, and atmospheric pressure variations. Although generally insignificant for most human endeavors, the semi-diurnal amplitude of terrestrial tides can attain approximately 55 centimeters (22 in) at the Equator, with 15 centimeters (5.9 in) attributable to solar influence. This phenomenon holds considerable importance for the calibration of Global Positioning Systems (GPS) and Very Long Baseline Interferometry (VLBI) measurements. Accurate astronomical angular measurements necessitate precise data on Earth's rotational velocity and polar motion, both of which are modulated by Earth tides. The semi-diurnal M₂ Earth tides exhibit a near-synchronous phase with the Moon, displaying a lag in the Earth's primary semi-diurnal lunar tide of 0.204°±0.047°, which translates to an approximate time delay of 25 seconds. Consequently, this solid tide is estimated to dissipate a minimum of 110 GW (150,000,000 hp) of tidal energy, representing approximately 5% of the energy dissipated by ocean tides.

Galactic Tides

Galactic tides refer to the gravitational forces exerted by galaxies on their constituent stars and on satellite galaxies in orbit around them. Within the Solar System, the influence of galactic tides on the Oort cloud is hypothesized to be responsible for the genesis of 90 percent of long-period comets.

Misnomers

Tsunamis, which are substantial waves generated by seismic activity, are occasionally mislabeled as tidal waves. This nomenclature, however, stems from their resemblance to tidal phenomena rather than any direct causal relationship. Additional phenomena that employ the term tide but are unrelated to gravitational tides include rip tide, storm tide, hurricane tide, and black or red tides. Many of these applications are historically rooted, reflecting an older definition of "tide" as "a segment of time, a season" or "a current, stream, or flood."

Cultural Significance

Beyond their scientific implications, tides have historically possessed profound cultural significance, influencing mythological narratives, literary works, linguistic expressions, and human perspectives on the world. Scholars and historians observe that the ocean's rhythmic ebb and flow has served as a pervasive metaphor for concepts such as time, transformation, and the human condition across diverse cultural traditions. Concurrently, cultural historians have meticulously documented the intricate ways in which tidal phenomena intertwine with folklore and literary discourse.

Prior to the advent of scientific elucidation, numerous ancient societies frequently interpreted tidal movements through cosmological frameworks or attributed them to divine interventions. This approach underscored both a sense of reverence and an element of mystery in humanity's comprehension of oceanic rhythms.

Throughout history, artists and poets have consistently utilized tidal imagery to investigate themes of metamorphosis and introspection. Tides are frequently featured in folklore and literary metaphors, symbolizing life cycles, destiny, and the fluctuating nature of emotions. Furthermore, the Moon's gravitational influence is evident in coastal ceremonial practices and maritime traditions, highlighting the intricate relationship between natural periodicities and cultural narratives.

Notes

Notes

References

• History of tide prediction Archived 2015-05-09 at the Wayback Machine
• UK Admiralty Easytide
• Tide Predictions for Australia, South Pacific & Antarctica
• Tide and Current Predictor, for stations around the world