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Theory of tides

The theory of tides refers to the body of scientific principles and models that explain the periodic rise and fall of ocean levels, known as tides, resulting from the gravitational interactions between Earth, the Moon, and the Sun, which produce differential forces that deform the oceans into two opposing bulges daily.^[1]
These tides manifest as two high waters and two low waters per lunar day (approximately 24 hours and 50 minutes), with the Moon's gravitational pull dominating due to its proximity to Earth—despite the Sun's greater mass, the Moon's tide-generating force is about twice as strong because tidal forces vary inversely with the cube of the distance to the attracting body.^[2]
The foundational explanation emerged in 1687 when Isaac Newton, in his Philosophiæ Naturalis Principia Mathematica, applied his law of universal gravitation to tides, describing them as the response of ocean waters to these celestial attractions, including the centrifugal force from Earth's rotation that contributes to the far-side bulge.^[3]
Newton's equilibrium theory assumes a frictionless, global ocean that instantly adjusts to form a static, ellipsoidal shape aligned with the Earth-Moon axis, predicting basic tidal cycles such as semidiurnal tides (twice daily) and variations like spring tides (higher highs and lower lows during full and new moons when Sun, Moon, and Earth align) and neap tides (weaker tides during quarter moons).^[4]
However, this idealized model overlooks Earth's rotation, continental barriers, ocean depth variations, and frictional delays, leading to inaccuracies in predicting actual tidal amplitudes and timings observed in coastal regions.^[3]
In the late 18th century, Pierre-Simon Laplace advanced the field with the dynamic theory of tides, incorporating hydrodynamic equations to account for wave propagation, Coriolis effects from Earth's rotation, basin geometries, and friction, which better explain real-world complexities like amphidromic systems (rotating tidal patterns around fixed points) and the generation of over 400 harmonic tidal constituents.^[3]
Subsequent refinements by scientists such as George Darwin and Arthur Doodson in the 19th and 20th centuries, along with modern integrations of satellite altimetry and numerical modeling, have enabled precise global tide predictions essential for navigation, coastal engineering, and environmental management.^[3]

Basic Principles

Gravitational Mechanisms

The tides on Earth are primarily driven by the differential gravitational forces exerted by the Moon and the Sun on the planet's oceans. The Moon's gravity pulls more strongly on the side of Earth facing it, drawing ocean water into a bulge toward the Moon, while the gravitational pull on the opposite side is weaker, allowing the water there to form a second bulge due to the centrifugal force arising from the Earth-Moon orbital motion.^[2] These two bulges, aligned with the Moon, result in high tides at those locations and low tides midway between them as Earth rotates.^[1] Although the Sun is vastly more massive than the Moon, its tidal influence is approximately half as strong because tidal forces scale inversely with the cube of the distance from the tide-generating body.^[5] The Moon, being much closer to Earth (about 390 times nearer than the Sun), exerts a dominant effect despite its smaller mass, with its tide-generating force roughly double that of the Sun's.^[2] This proximity amplifies the Moon's differential pull across Earth's diameter, making it the primary driver of tidal cycles.^[5] The Sun's gravitational contribution interacts with the Moon's to modulate tidal ranges through their relative alignments. When the Sun, Earth, and Moon are aligned during new or full moons, their gravitational pulls reinforce each other, producing spring tides with exceptionally high high tides and low low tides.^[1] Conversely, during first and third quarter moons, when the Sun and Moon are at right angles to each other relative to Earth, their pulls partially cancel, resulting in neap tides with moderate tidal ranges.^[6] Tidal ranges also vary due to the elliptical orbits of Earth and the Moon around their common center of mass and of Earth around the Sun. The Moon's perigee, when it is closest to Earth (occurring monthly and varying by about 31,000 miles in distance), intensifies its gravitational pull and increases tidal heights, while apogee, when farthest, diminishes them.^[7] Similarly, Earth's perihelion (around January 2) brings it closer to the Sun, enhancing solar tidal effects, whereas aphelion (around July 2) reduces them; the coincidence of these positions with lunar phases can lead to extreme tidal variations.^[7]

Equilibrium Tide Theory

The equilibrium tide theory models the response of a hypothetical global ocean to the gravitational influences of the Moon and Sun, assuming the ocean surface adjusts instantaneously and frictionlessly to form a static equilibrium shape. This theory derives from Newton's law of universal gravitation, which states that the force F between two masses M and m separated by distance d is F = G \frac{M m}{d^2}, where G is the gravitational constant.^[8] The tidal effect arises not from the absolute gravitational pull but from its gradient across Earth's radius, creating a differential force that deforms the ocean surface into an ellipsoid. For the Moon, with mass M_m and average distance d_m \approx 384,400 km, this gradient produces a stronger tidal force than the Sun's, despite the Sun's greater mass M_s, due to the Moon's closer proximity; the lunar tidal force is approximately 2.2 times that of the solar.^[8]^[9] The tidal potential \Omega, representing the gravitational potential perturbation, is obtained by expanding Newton's law in a Taylor series around Earth's center and retaining the quadrupole term (second-order Legendre polynomial) relevant for tides: \Omega = -\frac{G M}{d} \sum_{n=2}^{\infty} \left( \frac{r}{d} \right)^n P_n (\cos \psi), where r is the distance from Earth's center, \psi is the angular separation between the point and the attracting body, and P_2 (\cos \psi) = \frac{1}{2} (3 \cos^2 \psi - 1). For the dominant n=2 term at Earth's surface (r = R, Earth's radius), the potential simplifies to \Omega \approx \frac{G M R^2}{2 d^3} (3 \cos^2 \theta - 1), with \theta the angle from the sub-lunar (or sub-solar) point. The equilibrium tide height h is then the negative of this potential divided by gravitational acceleration g \approx 9.81 m/s², yielding h(\theta) = \frac{3}{2} \frac{G M R^2}{g d^3} \cos^2 \theta - C, where C is a constant ensuring zero mean height (equivalent to shifting by \frac{1}{3} of the amplitude to account for the full P_2 form). For the Moon, this predicts a maximum amplitude of about 0.36 m, and for the Sun, 0.16 m.^[8]^[9] This model illustrates two high-tide bulges: one facing the Moon (or Sun) where gravitational pull is strongest, and an antipodal bulge on the opposite side where the pull is weakest, balanced by the centrifugal force from the Earth-Moon orbital motion. At \theta = 0^\circ or $180^\circ (sub-lunar and anti-sub-lunar points), h reaches its maximum; at \theta = 90^\circ, the tide height is zero, forming the low-tide nodal line. As Earth rotates beneath these fixed bulges (relative to the stars), an observer experiences two high and two low tides per lunar day (approximately 24 hours 50 minutes). Spring tides occur when lunar and solar bulges align (full or new moon), amplifying heights to about 0.52 m, while neap tides at right angles (quarter moons) reduce them to 0.20 m.^[8]^[9] Despite its foundational role, the equilibrium theory has significant limitations, as it assumes a rigid, spherical, non-rotating Earth covered entirely by a shallow, frictionless ocean that responds instantly without inertia or basin constraints. In reality, this leads to unrealistically small predicted amplitudes—around 50 cm globally—compared to observed tides of several meters in many coastal regions, due to unaccounted dynamic effects like ocean depth variations and continental boundaries. The model also neglects the Coriolis force from Earth's rotation, which distorts the equilibrium shape in practice.^[8]^[9]

Historical Development

Ancient and Classical Views

Early observations of tides date back to ancient civilizations, where they were often described phenomenologically without a unified explanatory framework. In ancient Greece, the explorer Pytheas of Massalia, around the 4th century BCE, provided one of the earliest recorded links between tides and lunar phases during his voyages to northern Europe. He noted that tides rose and fell twice per lunar day and varied in amplitude with the Moon's waxing and waning, attributing the phenomenon to the Moon's influence, though his accounts were preserved only through later quotations.^[10] In contrast, Aristotle, in his Meteorology (c. 350 BCE), rejected a direct lunar connection and instead explained tides as resulting from subterranean winds and occasional earthquakes that displaced water, particularly in confined seas like the Euripus Strait, where he observed irregular tidal currents.^[11] By the 2nd century BCE, Seleucus of Seleucia advanced a more celestial-oriented view, proposing that the Moon exerted a direct influence on ocean tides through what could be interpreted as an early notion of gravitational pull. Observing diurnal inequalities in the tides of the Erythraean Sea (modern Red Sea and Persian Gulf), he correlated tidal variations specifically with the Moon's position relative to the horizon and its declination, marking the first explicit attribution of tides to lunar attraction in surviving records.^[10] This idea gained some traction but remained controversial among Greek scholars. In the Roman era, Pliny the Elder compiled observational knowledge in his Natural History (c. 77 CE), describing tides as occurring twice daily and twice nightly in sync with the Moon's risings, with greater ranges during equinoxes due to the Sun's alignment and shortly after new moons. He acknowledged regional irregularities, such as in estuaries, but emphasized the Moon's dominant role in driving the cycles without delving into mechanisms.^[12] Similarly, ancient Indian texts reflected awareness of lunar correlations; for instance, the Mahabharata (c. 400 BCE–400 CE) attributes large sea waves and tides to the Moon's rising and falling, integrating this into broader cosmological descriptions of celestial influences on earthly waters.^[13] Despite these insights, ancient and classical views lacked consensus, with tides frequently regarded as local meteorological events driven by winds, earthquakes, or divine forces rather than systematic celestial mechanics. This fragmented understanding persisted until the Renaissance, when renewed interest bridged to medieval Islamic scholarship that further refined lunar observations.^[10]

Medieval to 18th Century Advances

In the medieval Islamic world, scholars advanced the understanding of tides through empirical observations and connections to celestial bodies. Abū al-Rayhān al-Bīrūnī, an 11th-century polymath, conducted detailed observations of tidal variations and linked their cycles to the phases of the Moon, noting higher tides during new and full moons. He further associated tidal heights with the combined influences of the Moon and Sun, observing that tides were more pronounced when these bodies aligned, based on reports from coastal regions including the Arabian Sea and Persian Gulf.^[14] Al-Bīrūnī's work, documented in his extensive treatises on astronomy and geography, represented an early quantitative approach, measuring tidal periods and amplitudes to correlate them with lunar positions.^[14] During the Renaissance, European thinkers began exploring gravitational hypotheses for tides, though with mixed success. In 1609, Johannes Kepler proposed that the Moon's gravitational attraction caused ocean tides, analogizing it to magnetic forces pulling on water, though he provided no mathematical quantification.^[15] Kepler's idea, outlined in Astronomia Nova, marked a shift toward celestial mechanics but remained qualitative.^[15] Conversely, Galileo Galilei rejected gravitational explanations involving the Moon as "occult" and unsupported, instead attributing tides to the sloshing motion of oceans induced by Earth's daily rotation and annual orbit around the Sun, as detailed in his 1616 Discourse on the Tides.^[16] Galileo's mechanical model aimed to support Copernican heliocentrism but failed to explain semidiurnal tidal patterns or lunar correlations.^[16] The foundational gravitational theory emerged with Isaac Newton's Philosophiæ Naturalis Principia Mathematica in 1687. Newton mathematically derived tidal forces from the inverse-square law of universal gravitation, showing how the Moon (and to a lesser extent, the Sun) creates differential pulls on Earth's oceans, resulting in two tidal bulges—one toward the attracting body and one on the opposite side due to reduced effective gravity.^[17] This equilibrium model predicted semidiurnal tides but underestimated their amplitude, partly because it neglected Earth's rotation and treated oceans as static.^[17] In the 18th century, mathematicians refined Newton's framework by explicitly incorporating centrifugal forces from the Earth-Moon orbital motion. Leonhard Euler and Daniel Bernoulli developed variational approaches to tidal dynamics, balancing gravitational attractions with centrifugal effects to better model bulge formation and tidal asymmetries.^[8] Their contributions, including Euler's work on fluid equilibria and Bernoulli's analyses of oscillatory motions, laid groundwork for more accurate predictions while addressing limitations in Newton's static assumptions.^[8]

19th and 20th Century Progress

In the late 18th century, Pierre-Simon Laplace advanced the theory of tides beyond static models by developing a dynamic framework in his works from 1775 to 1778, incorporating the effects of ocean depth, friction, and Earth's rotation to explain tidal propagation as waves across the global ocean basins.^[18] This approach marked a pivotal shift from equilibrium theories, recognizing tides as responses to time-varying gravitational potentials influenced by celestial mechanics, though Laplace's equations provided the foundational mathematical structure for later refinements.^[19] During the 19th century, refinements to the tidal bulge concept emerged, with George Biddell Airy extending Newtonian ideas in his 1845 treatise Tides and Waves by quantifying the deformation of Earth's solid body under lunar and solar attractions, thus integrating geophysical responses into bulge theory.^[17] Thomas Young, in parallel efforts around the same period, contributed analytical insights into tidal mechanics that anticipated Airy's work, emphasizing wave propagation and equilibrium distortions without relying on overly simplistic assumptions.^[20] Meanwhile, William Ferrel introduced the role of Coriolis effects in his mid-19th-century studies, demonstrating how Earth's rotation deflects tidal currents and modifies flow patterns, thereby enhancing dynamic models with rotational dynamics and friction.^[21] In the early 20th century, Arthur Thomas Doodson standardized harmonic analysis through his 1921 expansion of the tide-generating potential, identifying over 400 distinct constituents derived from lunar and solar orbital parameters to enable precise tidal predictions worldwide.^[22] This development facilitated the decomposition of complex tidal signals into manageable sinusoidal components, laying the groundwork for systematic forecasting. A key technological milestone was the invention of tide-predicting machines by William Thomson (later Lord Kelvin) in 1872, which mechanically computed harmonic tide heights by summing multiple constituent waves via interconnected gears and pulleys, revolutionizing practical predictions for navigation and engineering.^[23] These analog devices, capable of handling up to 10 constituents initially, demonstrated the feasibility of automating tidal computations and influenced subsequent international efforts in tidal monitoring.^[24]

Dynamic Theory

Laplace's Tidal Equations

Laplace's tidal equations form the foundation of the dynamic theory of tides, extending beyond the static equilibrium model by incorporating the time-dependent response of ocean waters to gravitational forcing. These equations describe tides as shallow-water waves propagating across the Earth's oceans, driven by the periodic tidal potential arising from the Moon and Sun's gravitational influences. Unlike the equilibrium theory, which assumes instantaneous adjustment of the sea surface to the forcing potential, Laplace's framework accounts for inertia, Earth's rotation, and basin geometry, enabling predictions of wave propagation, reflection, and interference. The equations are derived from the linearized shallow-water approximations to the Navier-Stokes (or more precisely, Euler) equations for inviscid, incompressible flow under hydrostatic balance, assuming the ocean depth H is much smaller than the horizontal scales. In a local Cartesian coordinate system (suitable for mid-latitude approximations), the continuity equation expresses mass conservation:

\frac{\partial \eta}{\partial t} + H \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) = 0,

where \eta(x,y,t) is the sea surface elevation anomaly, u(x,y,t) is the velocity in the x-direction, and v(x,y,t) is the velocity in the y-direction. The momentum equation in the x-direction includes the pressure gradient, Coriolis force, and tidal forcing:

\frac{\partial u}{\partial t} - f v - g \frac{\partial \eta}{\partial x} = -\frac{\partial \Phi}{\partial x},

with the equation for the y-direction:

\frac{\partial v}{\partial t} + f u - g \frac{\partial \eta}{\partial y} = -\frac{\partial \Phi}{\partial y}.

Here, g is gravitational acceleration, f = 2 \Omega \sin \phi is the Coriolis parameter (\Omega Earth's rotation rate, \phi latitude), and \Phi(x,y,t) is the tidal potential, typically expanded in spherical harmonics reflecting the positions of the Moon and Sun. For one-dimensional channel flows, the y-derivatives and v may be neglected or set via geostrophy. These forms were first systematically derived by Pierre-Simon Laplace in the late 18th century, building on earlier hydrodynamic principles.^[25]^[26] Solutions to Laplace's equations represent forced oscillations at specific tidal frequencies determined by the relative orbital motions of the Earth, Moon, and Sun. The dominant semidiurnal M_2 constituent, for instance, has angular frequency \sigma = 2(\Omega - \omega_m), where \omega_m is the mean lunar orbital angular velocity, corresponding to twice the lunar frequency relative to Earth's rotation. The tidal potential \Phi thus acts as a body force with these periodicities (diurnal, semidiurnal, and long-period), exciting waves whose wavelengths for barotropic modes are on the order of basin scales (e.g., \lambda \approx 10,000 km for shallow-water gravity waves at tidal periods). In plane-wave form, solutions take the shape \eta = \Re \{ \hat{\eta} e^{i(kx - \sigma t)} \}, with dispersion relation \sigma^2 = f^2 + g H k^2 modified by forcing, allowing co-oscillating or progressive waves depending on basin dimensions.^[25]^[26] Boundary conditions are crucial for realistic solutions, with no normal flow imposed at coastal boundaries: \mathbf{u} \cdot \hat{\mathbf{n}} = 0, where \hat{\mathbf{n}} is the outward normal, ensuring conservation of volume in enclosed basins. Open-ocean boundaries may incorporate radiation conditions to allow outgoing waves. In semi-enclosed basins, these conditions lead to resonant amplification when the forcing frequency matches natural modes of the basin, resulting in standing wave patterns known as amphidromic systems. In such systems, tides rotate around a nodal point (amphidrome) of zero elevation, with amplitude increasing radially outward due to constructive interference; examples include the North Sea and Gulf of Mexico, where cotidal lines radiate from the amphidrome.^[25]^[26] A key insight from the dynamic theory is that tidal amplitudes can be significantly amplified beyond equilibrium predictions through wave interference and resonance, often by factors of up to 10 or more in coastal regions. For instance, while the equilibrium M_2 tide yields global elevations of about 0.5 m, dynamic effects in resonant basins like the Bay of Fundy produce ranges exceeding 15 m due to quarter-wave resonance in the funnel-shaped geometry. This amplification arises from the superposition of incident, reflected, and forced waves, highlighting the dynamic theory's superiority in explaining observed tidal ranges and phase lags.^[27]^[25]

Effects of Earth's Rotation and Friction

The Coriolis effect, arising from Earth's rotation, significantly influences tidal propagation in the oceans by deflecting tidal currents to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. This deflection leads to the formation of amphidromic points, locations where the tidal range is zero and around which the tidal crest rotates counterclockwise in the Northern Hemisphere at a rate approximately equal to Earth's rotation period. Cotidal lines, which connect points experiencing high tide simultaneously, radiate outward from these amphidromic points, illustrating the progressive wave motion of the tide. In the dynamic theory, the Coriolis parameter f = 2 \Omega \sin \phi, where \Omega is Earth's angular velocity and \phi is latitude, is incorporated into the momentum equations to account for this rotational influence, modifying the idealized equilibrium tide patterns into more realistic co-oscillating systems across ocean basins.^[28]^[29] Friction and viscosity introduce dissipative processes that dampen tidal waves and introduce phase lags relative to the astronomical forcing. In the momentum equations of Laplace's dynamic theory, these effects are represented by damping terms, such as bottom friction parameterized as a quadratic drag \tau_b = -\rho C_d |\mathbf{u}| \mathbf{u} and viscous terms involving horizontal eddy viscosity, which extract energy from the barotropic tide and convert it into internal waves or turbulent mixing. This dissipation causes the tidal bulge to lag behind the equilibrium position, with the global average tidal energy dissipation rate estimated at approximately 3.7 terawatts (TW), primarily occurring in shallow marginal seas and through generation of internal tides in the deep ocean. The phase lag, typically on the order of 10–30 degrees for principal constituents like M2, results in a net torque that transfers angular momentum from Earth's rotation to the Moon's orbit.^[8]^[30]^[31] Tidal friction contributes to the secular slowing of Earth's rotation, lengthening the day by about 2.3 milliseconds per century. This deceleration arises from the gravitational interaction between the lagged tidal bulge and the Moon, producing a torque that reduces Earth's rotational angular momentum while increasing the Moon's orbital angular momentum, thereby causing the Moon to recede at approximately 3.8 cm per year. Observations from ancient eclipses and modern satellite laser ranging confirm this rate, with tidal friction accounting for roughly 90% of the observed lengthening of the day over geological timescales.^[32] Self-attraction and loading (SAL) effects further refine tidal height predictions by accounting for the gravitational attraction of the ocean's own water mass and the elastic deformation of Earth's crust under tidal loading. The self-attraction term modifies the tidal potential due to the redistributed ocean mass, while the loading term represents the vertical crustal displacement, which can reach several centimeters. In shallow seas, these combined SAL effects adjust computed tidal heights by 20–30%, with the dynamic response amplifying deviations in regions of complex bathymetry. Including SAL in models improves accuracy, particularly for the M2 constituent, by reducing phase errors in coastal areas.^[33]^[34]

Tidal Analysis and Prediction

Harmonic Analysis Methods

Harmonic analysis methods in tidal theory involve the mathematical decomposition of observed sea level variations from tide gauges into a series of sinusoidal components, each corresponding to specific astronomical forcing frequencies. This approach allows for the isolation of periodic tidal signals driven by gravitational interactions between the Earth, Moon, and Sun, enabling both scientific understanding and practical predictions of tidal behavior.^[3] The core technique is Fourier-based harmonic analysis, which represents the tidal elevation η(t) as a sum of cosine functions:

\eta(t) = \sum_k A_k \cos(\omega_k t + \phi_k)

Here, A_k denotes the amplitude of the k-th constituent, ω_k its angular frequency, t the time, and φ_k the phase offset. This formulation captures the periodic nature of tides by fitting the model to observed data, where each term represents a distinct tidal constituent arising from combinations of lunar and solar motions.^[3] The analysis process employs least-squares fitting to determine the amplitudes and phases by minimizing the differences between observed tide gauge records and the harmonic model. To resolve closely spaced frequencies—such as those differing by mere cycles per year—data spanning at least one full 18.6-year lunar nodal cycle is typically required, as this period encompasses the regression of the Moon's orbital nodes and ensures separation of long-period effects like the 18.6-year variation in tidal range. Shorter records, such as one year, suffice for preliminary analyses of dominant constituents but may introduce errors in resolving subtle interactions.^[3] A key advantage of harmonic analysis lies in its ability to separate astronomical tidal components from geophysical influences, such as atmospheric pressure variations, wind-driven surges, ocean currents, and local bathymetric effects. After fitting the model, the residuals—differences between observed and predicted tides—reveal non-astronomical contributions, aiding in the study of weather-related sea level changes and shallow-water nonlinearities that generate additional harmonics.^[3] Historically, the foundations of these methods were laid in the 1880s by George Darwin, whose work on harmonic analysis of tidal observations established the framework for identifying and quantifying the principal tidal constituents, with standard predictions relying on 37 such components derived from his developments.^[22]^[3]

Classification of Tidal Constituents

Tidal constituents are periodic components of the tide resulting from gravitational interactions between Earth, the Moon, and the Sun, classified primarily by their periods and astronomical origins into semidiurnal, diurnal, long-period, and other categories such as shallow-water overtones.^[35] These classifications arise from the frequencies of celestial motions, with semidiurnal and diurnal types dominating daily tidal cycles in most coastal areas.^[3] Semidiurnal constituents, with periods near 12 hours, are the most prominent in global oceans, often comprising the majority of tidal energy. The principal lunar semidiurnal constituent M₂, driven by the Moon's direct gravitational pull, has a period of 12.42 hours and typically dominates, often accounting for the majority of the tidal amplitude in semidiurnal-dominated locations.^[35]^[3] The principal solar semidiurnal constituent S₂, resulting from solar gravitational forcing, has a period of 12.00 hours and contributes significantly where solar influence is amplified.^[35] Diurnal constituents feature periods around 24 hours and prevail in regions like the Gulf of Mexico and parts of the Pacific. The luni-solar diurnal constituent K₁, combining lunar and solar effects from Earth's declination, has a period of 23.93 hours.^[35] The principal lunar diurnal O₁, due to the Moon's declination, has a period of 25.82 hours.^[35] Long-period constituents exhibit cycles spanning days to months, modulating shorter tides through orbital variations. The lunar fortnightly constituent M_f, arising from the Moon's perigee-apogee cycle, has a period of 13.66 days.^[35] The lunar monthly constituent M_m, linked to the Moon's nodal cycle, has a period of 27.55 days.^[35] Additional categories include solar-related long-term constituents and shallow-water overtones generated by nonlinear interactions in coastal zones. The annual solar Sa has a period of 365.25 days, while the semiannual Ssa has 182.62 days, both influenced by Earth's orbital tilt.^[35] Shallow-water overtones, such as the quarter-diurnal M₄ (an overtide of M₂) with a period of 6.21 hours, arise from frictional distortions in shallow areas.^[35] In total, over 400 tidal constituents are theoretically possible from astronomical combinations, but approximately 60-70 are typically significant at a given site, with their relative amplitudes strongly shaped by local bathymetry and coastal geometry.^[3]^[36]

Prediction Models and Doodson Numbers

Tidal predictions are generated through the harmonic method, which involves superimposing sinusoidal constituents derived from prior analysis of observed tidal data at a specific location. Each constituent is characterized by an amplitude and phase lag, adjusted by nodal factors to account for long-term astronomical variations such as the 18.6-year lunar nodal cycle. The predicted tidal height h(t) at time t is computed as the sum of these terms plus a mean level:

h(t) = H_0 + \sum_{i=1}^{n} f_i H_i \cos(a_i t + (V_0 + u)_i - \kappa_i)

where H_0 is the mean water level, f_i and u_i are nodal corrections, H_i and \kappa_i are the amplitude and phase lag from analysis, and a_i is the angular speed of the constituent.^[3] This approach relies on transfer functions to adapt equilibrium tide values to local conditions, ensuring predictions reflect site-specific hydrodynamic responses.^[8] The Doodson numbering system provides a standardized 6-digit code for identifying tidal constituents, facilitating their use in analysis and prediction. Developed by Arthur Thomas Doodson, the code represents the frequency of each constituent as multiples of six fundamental angular speeds derived from the orbital motions of the Earth, Moon, and Sun, expressed in multiples of the mean semi-diurnal lunar hour angle. The digits d_1 d_2.d_3 d_4 d_5 d_6 denote: d_1 for the species (e.g., 2 for semidiurnal), d_2 and d_3 for contributions from the mean longitudes of the Moon and Sun, d_4 for the lunar perigee, d_5 for the lunar node, and d_6 for the solar perigee argument. For example, the principal lunar semidiurnal constituent M₂ has the Doodson number 255.555, corresponding to a speed of 28.984 degrees per solar hour.^[8] These numbers enable precise computation of constituent speeds and grouping of related terms, such as satellite constituents sharing the first three digits, which simplifies handling perturbations in prediction algorithms.^[3] For sites where harmonic analysis is challenging due to short data records or non-stationary signals, the response method offers an alternative by modeling the tidal signal as the equilibrium tide filtered through a transfer function that captures local amplification and phase shifts. This admittance-based approach, often using cross-spectral techniques, infers amplitudes and phases for multiple constituents from a reference signal, adjusting for effects like ocean loading and self-attraction.^[3] To enhance accuracy in shallow coastal areas, predictions incorporate shallow-water terms arising from nonlinear interactions, such as overtides (e.g., M₄ as a quarter-diurnal multiple of M₂) and compound tides, which account for friction and continuity effects that distort the tidal curve. These terms, identified via Doodson numbers with higher species digits (e.g., 4 for quarter-diurnal), can contribute significantly to total variance in regions with large tidal ranges, like estuaries.^[8] Analytical harmonic predictions using up to 149 constituents, including shallow-water effects, typically achieve root-mean-square errors of 1-10 cm over a 24-hour period when validated against long-term observations, though residuals from nontidal influences like weather can increase this.^[3] Since the 1980s, such methods have increasingly been supplemented or replaced by numerical hydrodynamic models for complex domains, but harmonic approaches with Doodson-coded constituents remain foundational for global tide tables and real-time forecasting.^[8]

Modern Extensions

Numerical Modeling Approaches

Numerical modeling approaches in tidal theory build upon the foundational dynamic theory by solving Laplace's tidal equations computationally to simulate complex ocean responses that analytical methods cannot fully capture. These methods discretize the continuous ocean domain into grids or meshes, enabling simulations of tidal propagation, resonance, and interactions with coastal geometry on global or regional scales. Finite difference and finite element models are prominent, where the former approximates derivatives on structured grids and the latter uses unstructured meshes for irregular coastlines, both iterating solutions over time steps to predict sea surface elevations and currents.^[37] A key example is the TPXO (Tidal Prediction Using Cross-shelf Observational data) model series, developed at Oregon State University, which employs finite difference methods on a global grid to solve the shallow-water equations derived from Laplace's theory, assimilating satellite altimetry to estimate tidal harmonics with resolutions up to 1/30 degree. TPXO models, such as TPXO10, provide gridded outputs of tidal elevations and currents for over 100 constituents, achieving accuracies of 1-2 cm in open ocean comparisons with in-situ data, and have been instrumental in refining global tidal atlases since the 1990s. Finite element models like SELFE (Semi-implicit Eulerian-Lagrangian Finite Element) extend this by handling wetting-drying processes in estuaries, simulating tides with vertical resolution for three-dimensional flows.^[38]^[37]^[39] Global simulations integrate these models for operational forecasting, incorporating atmospheric forcing to predict tides alongside storm surges. The ADCIRC (Advanced Circulation) model, using finite element discretization on unstructured grids, supports real-time global storm tide predictions, as demonstrated in its v55 version which resolves tides and surges at 1/60 degree resolution with computation times under 24 hours on high-performance clusters. SELFE similarly enables coupled simulations of tides and surges in coastal regions, such as hindcasting Hurricane Sandy inundation with errors below 20 cm against tide gauges. These codes facilitate nowcasting by running on supercomputers, outputting forecasts every few hours for applications like port operations and flood warning.^[40]^[39]^[40] Data assimilation enhances model fidelity by merging observational data into simulations, iteratively adjusting parameters to minimize discrepancies. Satellite altimetry from missions like TOPEX/Poseidon (1992-2006) and the Jason series (Jason-1 to Jason-3, ongoing) provides global coverage of sea surface heights, with data assimilation techniques such as representer methods or Kalman filters incorporating these alongside tide gauge records to refine tidal potentials and dissipation rates. More recently, data from the Surface Water and Ocean Topography (SWOT) mission (launched 2022) has been assimilated to enhance resolution of internal and coastal tides, as demonstrated in 2025 studies. For instance, TPXO assimilates over 20 years of multi-mission altimetry to constrain open-ocean tides, reducing variances by up to 50% compared to gauge-only inversions, while regional models blend Jason data with coastal gauges for boundary conditions. This approach addresses observational gaps, improving predictions in data-sparse areas like the Southern Ocean.^[41]^[42]^[38]^[43] By 2025, machine learning enhancements have advanced sub-hourly tidal predictions, integrating neural networks with traditional models to bypass the limitations of over 400 harmonic constituents in classical analysis, particularly for non-stationary effects under climate change. Hybrid deep learning frameworks, such as convolutional neural networks combined with long short-term memory units, trained on altimetry and gauge archives, achieve sub-hourly forecasts with root mean square errors under 5 cm for lead times up to 6 hours, outperforming pure harmonic methods in dynamic coastal zones. These AI-driven models, like those using physics-guided transformers, simulate climate-induced tidal alterations—such as amplified resonances from sea level rise—by learning from reanalysis data, enabling projections of future inundation risks with reduced computational overhead compared to full hydrodynamic runs. This integration addresses traditional theory's incompleteness in capturing long-term modulations, supporting adaptive coastal management amid rising seas.^[44]^[45]^[46]

Applications to Planetary Tides

The theory of tides extends beyond Earth to other celestial bodies, where gravitational interactions drive significant geological and atmospheric processes. On Jupiter's moon Io, tidal heating arises from its orbital eccentricity, maintained by the Laplace resonance with siblings Europa and Ganymede, which pumps energy into Io's interior through periodic flexing by Jupiter's gravity.^[47] This dissipation generates immense heat, powering Io's extensive volcanism and making it the most geologically active body in the solar system.^[47] Saturn's moon Enceladus provides another striking example, where tidal flexing from its eccentric orbit around Saturn induces internal heating that sustains a global subsurface ocean beneath its icy crust.^[48] This ocean, kept liquid by frictional dissipation in the ice shell and rocky core, drives cryovolcanic geysers at the south pole, ejecting water plumes detectable from space.^[49] Observations confirm that tidal energy input balances conductive cooling, enabling potential habitability conditions within the ocean.^[50] For exoplanets, tidal theory informs the dynamics of worlds in habitable zones around other stars, particularly through tidal locking, where a planet's rotation synchronizes with its orbital period due to gravitational torques, resulting in permanent day and night sides.^[51] This locking is prevalent for close-in planets around low-mass stars, influencing atmospheric circulation and potential biosignatures.^[51] Tidal interactions also facilitate planetary migration, as protoplanetary disks exert torques that inward-migrate gas giants, explaining the prevalence of hot Jupiters near their stars.^[52] In these systems, disk-driven migration circularizes orbits before tidal decay dominates post-disk evolution.^[52] General relativity introduces minor corrections to tidal calculations in planetary systems, which remain negligible compared to Newtonian effects except in extreme environments like black hole vicinities.^[53] Recent James Webb Space Telescope observations in 2025 have confirmed tidal influences on hot Jupiter atmospheres, revealing enhanced heat redistribution and compositional asymmetries driven by orbital decay and stellar irradiation.^[54]

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