Classical era The tides received relatively little attention in the civilizations around the
Mediterranean Sea, as the tides there are relatively small, and the areas that experience tides do so unreliably. A number of theories were advanced, however, from comparing the movements to breathing or blood flow to theories involving
whirlpools or river cycles. An ancient Indian
Purana text dated to 400-300 BC refers to the ocean rising and falling because of
heat expansion from the light of the Moon. The
Yolngu people of northeastern
Arnhem Land in the
Northern Territory of Australia identified a link between the Moon and the tides, which they mythically attributed to the Moon filling with water and emptying out again. Ultimately the link between the
Moon (and
Sun) and the tides became known to the
Greeks, although the exact date of discovery is unclear; references to it are made in sources such as
Pytheas of Massilia in 325 BC and
Pliny the Elder's
Natural History in 77 AD. Although the schedule of the tides and the link to lunar and solar movements was known, the exact mechanism that connected them was unclear. Classicists
Thomas Little Heath claimed that both Pytheas and
Posidonius connected the tides with the moon, "the former directly, the latter through the setting up of winds".
Eratosthenes (3rd century BC) and Posidonius (1st century BC) both produced detailed descriptions of the tides and their relationship to the
phases of the Moon, Posidonius in particular making lengthy observations of the sea on the Spanish coast, although little of their work survived. The influence of the Moon on tides was mentioned in
Ptolemy's
Tetrabiblos as evidence of the reality of
astrology.
Seleucus of Seleucia is thought to have theorized around 150 BC that tides were caused by the Moon as part of his
heliocentric model.
Aristotle, judging from discussions of his beliefs in other sources, is thought to have believed the tides were caused by winds driven by the Sun's heat, and he rejected the theory that the Moon caused the tides. An apocryphal legend claims that he committed suicide in frustration with his failure to fully understand the tides. However, he made no progress regarding the question of how exactly the Moon created the tides.
Dante references the Moon's influence on the tides in his
Divine Comedy.
Abu Ma'shar al-Balkhi, in his
Introductorium in astronomiam, taught that ebb and flood tides were caused by the Moon. In 1609,
Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides, which he compared to
magnetic attraction basing his argument upon ancient observations and correlations. In 1616,
Galileo Galilei wrote
Discourse on the Tides. He strongly and mockingly rejects the lunar theory of the tides, But his contemporaries noticed that this made predictions that did not fit observations.
René Descartes theorized that the tides (alongside the movement of planets, etc.) were caused by
aetheric vortices, without reference to Kepler's theories of gravitation by mutual attraction; this was extremely influential, with numerous followers of Descartes expounding on this theory throughout the 17th century, particularly in France. However, Descartes and his followers acknowledged the influence of the Moon, speculating that pressure waves from the Moon via the
aether were responsible for the correlation.
Equilibrium tidal theory Newton, in the
Principia, provides a correct explanation for the
tidal force, which can be used to explain tides on a planet covered by a uniform ocean but which takes no account of the distribution of the continents or ocean
bathymetry. This form of the theory is known as
equilibrium theory. Equilibrium theory makes three simplifications: 1) ignore Earth's land, 2) ignore the viscosity of water so it can respond to gravity instantly, 3) ignore friction between the Earth and water. In a coordinate system rotating with the Earth-Moon pair, the distance between the Earth and Moon is constant: they are in equilibrium. This equilibrium can be explained as a balance of the force of the Moon's gravity and
centrifugal force from rotation. At the center of the Earth, the forces are equal and opposite. For other points the forces do not exactly balance and the residual force is called the
tide-generating force. For points on Earth's surface but closest to the Moon, gravity is very slightly stronger; or points farthest away, centrifugal force is slightly stronger. On the poles away from the Earth-Moon line, the small net force points into the Earth. The ocean water is barely affected by these forces. In between the poles and the equator, a component of the small force points horizontal to the surface of the Earth and towards the equator. No force opposes this small force. The ocean water flows in response to this force, leaving the poles and accumulating near the equator. The result is a double tidal bulge along the Earth-Moon axis, somewhat larger on the side closer to the Moon. As the Earth rotates on its axis, different points on Earth move through these bulges, roughly explaining the daily double tides. Both the oceans' water and the solid Earth experience these differences in pull, but the rigid Earth resists deformation and keeps its roughly spherical shape, while the fluid redistributes to match the imbalance, forming the bulges. The
equilibrium tide is the idealized tide assuming a landless Earth.
Dynamic theory While Newton explained the tides by describing the tide-generating forces and
Daniel Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the
dynamic theory of tides, developed by
Pierre-Simon Laplace in 1775, describes the ocean's real reaction to tidal forces. Laplace's theory of ocean tides takes into account
friction,
resonance and natural periods of ocean basins. It predicts the large
amphidromic systems in the world's ocean basins and explains the oceanic tides that are actually observed. The equilibrium theory—based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, and other important effects—could not explain the real ocean tides. Since measurements have confirmed the dynamic theory, many things have possible explanations now, like how the tides interact with deep sea ridges, and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface. The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters. Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters. Measurements from the
CHAMP satellite closely match the models based on the
TOPEX data. Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels.
Laplace's tidal equations In 1776, Laplace formulated a single set of linear
partial differential equations for tidal flow described as a
barotropic two-dimensional sheet flow.
Coriolis effects are introduced as well as lateral forcing by
gravity. Laplace obtained these equations by simplifying the
fluid dynamics equations, but they can also be derived from energy integrals via
Lagrange's equation. For a fluid sheet of average thickness
D, the vertical tidal elevation
ζ, as well as the horizontal velocity components
u and
v (in the
latitude φ and
longitude λ directions, respectively) satisfy '''Laplace's tidal equations''': : \begin{align} \frac{\partial \zeta}{\partial t} &+ \frac{1}{a \cos( \varphi )} \left[ \frac{\partial}{\partial \lambda} (uD) + \frac{\partial}{\partial \varphi} \left(vD \cos( \varphi )\right) \right] = 0, \\[2ex] \frac{\partial u}{\partial t} &- v \, 2 \Omega \sin( \varphi ) + \frac{1}{a \cos( \varphi )} \frac{\partial}{\partial \lambda} \left( g \zeta + U \right) = 0, \quad \text{and} \\[2ex] \frac{\partial v}{\partial t} &+ u \, 2 \Omega \sin( \varphi ) + \frac{1}{a} \frac{\partial}{\partial \varphi} \left( g \zeta + U \right) = 0, \end{align} where Ω is the
angular frequency of the planet's rotation,
g is the planet's
gravitational acceleration at the mean ocean surface,
a is the planetary radius, and
U is the external gravitational tidal-forcing
potential.
William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the
curl to find an equation for
vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity. ==Tidal analysis and prediction==