Theory of tides

 ( Tides ) 

Isaac 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.

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.Bryden, I.G. (2003). "Tidal Power Systems", in Meyers, R.A. (ed.) Encyclopedia of Physical Science and Technology. Aberdeen: Academic Press, p. 753. doi: 10.1016/b0-12-227410-5/00778-x 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.http://www.geomag.us/info/Ocean/m2_CHAMP+longwave_SSH.swf 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.

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}
     

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.

Darwin's harmonic developments of the tide-generating forces were later improved when A.T. Doodson, applying the lunar theory of E.W. Brown,

The Doodson arguments are specified in the following way, in order of decreasing frequency: and T D Moyer (2003) already cited.

$\backslash beta\_1\; =\; \backslash tau\; =\; (\; \backslash theta\_M\; +\; \backslash pi\; -\; s\; )$ is mean Lunar time, the Greenwich hour angle of the mean Moon plus 12 hours.

$\backslash beta\_2\; =\; s\; =\; (\; F\; +\; \backslash Omega\; )$ is the mean longitude of the Moon.

$\backslash beta\_3\; =\; h\; =\; (\; s\; -\; D\; )$ is the mean longitude of the Sun.

$\backslash beta\_4\; =\; p\; =\; (\; s\; -\; l\; )$ is the longitude of the Moon's mean perigee.

$\backslash beta\_5\; =\; N\text{'}\; =\; (\; -\backslash Omega\; )$ is the negative of the longitude of the Moon's mean ascending node on the ecliptic.

$\backslash beta\_6\; =\; p\_l$ or $p\_s\; =\; (\; s\; -\; D\; -\; l\text{'}\; )$ is the longitude of the Sun's mean perigee.

In these expressions, the symbols $l$, $l\text{'}$, $F$ and $D$ refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:-

$l$ is the mean anomaly of the Moon (distance from its perigee).

$l\text{'}$ is the mean anomaly of the Sun (distance from its perigee).

$F$ is the Moon's mean argument of latitude (distance from its node).

$D$ is the Moon's mean elongation (distance from the sun).

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 255.555, meaning that its frequency is composed of twice the first Doodson argument, and zero times all of the others. The second strongest tidal constituent "S₂" is influenced by the sun, and its Doodson numbers are 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.See for example Melchior (1971), already cited, at p.191. 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.

• Long-period tides
• Lunar node § Effect on tides
• Kelvin wave
• Tide table

• Contributions of satellite laser ranging to the studies of earth tides
• Dynamic Theory of Tides
• Tidal Observations
• Publications from NOAA's Center for Operational Oceanographic Products and Services
• Understanding Tides
• 150 Years of Tides on the Western Coast
• Our Relentless Tides
• GeoTide Tidal Analysis System