Theory of tides


The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans under the gravitational loading of another astronomical body or bodies.

History

Kepler

In 1609 Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides, basing his argument upon ancient observations and correlations. The influence of the Moon on tides was mentioned in Ptolemy's Tetrabiblos as having derived from ancient observation.

Galileo

In 1616, Galileo Galilei wrote Discourse on the Tides,. He tried to explain the tides as the result of the Earth's rotation and revolution around the Sun, believing that the oceans moved like water in a large basin: as the basin moves, so does the water. Therefore, as the Earth revolves, the force of the Earth's rotation causes the oceans to "alternately accelerate and retardate". His view on the oscillation and "alternately accelerated and retardated" motion of the Earth's rotation is a "dynamic process" that deviated from the previous dogma, which proposed "a process of expansion and contraction of seawater." However, Galileo's theory was erroneous. In subsequent centuries, further analysis led to the current tidal physics. Galileo rejected Kepler's explanation of the tides. Galileo tried to use his tidal theory to prove the movement of the Earth around the Sun. Galileo theorized that because of the Earth's motion, borders of the oceans like the Atlantic and Pacific would show one high tide and one low tide per day. The Mediterranean had two high tides and low tides, though Galileo argued that this was a product of secondary effects and that his theory would hold in the Atlantic. However, Galileo's contemporaries noted that the Atlantic also had two high tides and low tides per day, which lead to Galileo omitting this claim from his 1632 Dialogue.

Newton

Newton, in the Principia, provided 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.

Laplace

The dynamic theory

The dynamic theory of tides describes and predicts the actual real behavior of ocean tides.
While Newton explained the tides by describing the tide-generating forces and 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 took into account friction, resonance and natural periods of ocean basins. It predicted 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, Pierre-Simon 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 dynamic 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 satisfy Laplace's tidal equations:
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 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

Harmonic analysis

Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by William Thomson's application of Fourier analysis to the tidal motions as harmonic analysis.
Thomson's work in this field was then further developed and extended by George Darwin, applying the lunar theory current in his time. Darwin's symbols for the tidal harmonic constituents are still used.
Darwin's harmonic developments of the tide-generating forces were later improved when A T Doodson, applying the lunar theory of E W Brown, developed the tide-generating potential in harmonic form, distinguishing 388 tidal frequencies. Doodson's work was carried out and published in 1921.
Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson numbers, a system still in use.
Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy. The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tide-predicting machine, a special-purpose form of analog computer now superseded in this work by digital electronic computers that can be programmed to carry out the same computations.

Tidal constituents

Tidal constituents combine to give an endlessly varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an illustrating the way in which the components used to be mechanically combined in the tide-predicting machine. Amplitudes of tidal constituents are given below for six example locations:
Eastport, Maine, Biloxi, Mississippi, San Juan, Puerto Rico, Kodiak, Alaska, San Francisco, California, and Hilo, Hawaii.

Semi-diurnal

Diurnal

Long period

Short period

Doodson numbers

In order to specify the different harmonic components of the tide-generating potential, Arthur Thomas Doodson devised a practical system which is still in use, involving what are called the "Doodson numbers" based on the six "Doodson arguments" or Doodson variables.
The number of different tidal frequencies is large, but they can all be specified on the basis of combinations of small-integer multiples, positive or negative, of six basic angular arguments. In principle the basic arguments can possibly be specified in any of many ways; Doodson's choice of his six "Doodson arguments" has been widely used in tidal work. In terms of these Doodson arguments, each tidal frequency can then be specified as a sum made up of a small integer multiple of each one of the six arguments. The resulting six small integer multipliers effectively encode the frequency of the tidal argument concerned, and these are the Doodson numbers: in practice all except the first are usually biased upwards by +5 to avoid negative numbers in the notation.
The Doodson arguments are specified in the following way, in order of decreasing frequency:
In these expressions, the symbols,, and refer to an alternative set of fundamental angular arguments, in which:-
It is possible to define several auxiliary variables on the basis of combinations of these.
In terms of this system, each tidal constituent frequency can be identified by its Doodson numbers. The strongest tidal constituent "M2" has a frequency of 2 cycles per lunar day, its Doodson numbers are usually written 273.555, meaning that its frequency is composed of twice the first Doodson argument, +2 times the second, -2 times the third, and zero times each of the other three. The second strongest tidal constituent "S2" is due to the sun, its Doodson numbers are 255.555, meaning that its frequency is composed of twice the first Doodson argument, and zero times all of the others. This aggregates to the angular equivalent of mean solar time + 12 hours. These two strongest component frequencies have simple arguments for which the Doodson system might appear needlessly complex, but each of the hundreds of other component frequencies can be briefly specified in a similar way, showing in the aggregate the usefulness of the encoding.