Does anyone know where there is a detailed definition for tidal harmonic constituents? There is loads of stuff on $M_2$, $S_2$, $N_2$, etc. (all the common ones), but there are a great many more, some (I believe) involved in the calculation of tidal current speeds and directions. I am interested in what their physical significance is and the relationship to earth/moon/sun positions.

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    $\begingroup$ I'll be watching this question with interest. There's a rather complete (?) list of constituents at iho.int/mtg_docs/com_wg/IHOTC/IHOTC_Misc/…, but I've always struggled to find an explanation of what astronomical period most of then represent. (although note that not all of the short-period ones are astronomical at sll) $\endgroup$ Jul 18, 2016 at 16:29

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The explanation of all the tidal constituents is pretty complex because of the interactions between the different frequencies involved. The harmonic analysis is an expansion of the Equilibrium Theory of Tides into a set of harmonic terms. The list of relevant periods includes:

  • the lunar day (period of lunar rotation), 24.84 mean solar hours.
  • the sidereal month (period of lunar declination), 27.32 mean solar days.
  • the tropical year (period of solar declination), 365.24 mean solar days.
  • the period of the lunar perigee, 8.85 years (1 year = 365.2421988 days).
  • the period of the lunar node, 18.61 years.
  • the period of the solar perihelion, 20940 years.

The frequency of any tidal constituent can be computed by a combination of small numbers that multiply each of the periods (usually they are expressed as speeds, e.g., degrees per mean solar hour) above. The most complete algebraic expansion of the constituents was produced by Doodson in 1921. The set of small numbers are called Doodson numbers and represent the importance of each of the period in the specific constituent.

For instance, the set of Doodson numbers for the $M_2$ is (2,0,0,0,0,0), with the ”2” referring to "twice per lunar day" and the rest of the terms are zero (no influence). This is expected as the $M_2$ represents the passage of the same point on Earth through the "tidal bulge", created by the tidal forces, twice in a lunar day. The set of Doodson numbers for the $O_1$ is (1,-1,0,0,0,0), which represents 1 cycle/lunar day minus 1 cycle/tropical month. This is the result of the inclination between the Earth's axis and the Moon's orbit that causes the "tidal bulge" to be offset from the Equator and thus a point on Earth will experience one single absolute maximum per day.

The explanation of the lesser tidal constituents become rather tricky as the combination of Doodson numbers increases. For instance, $MSN_2$ is a combination of lunar, solar, and nodal influences and has a set of Doodson numbers of (2,3,-2,-1,0,0) making the interpretation rather complex.

While a list of origins of the constituents can be found (e.g., Table 4.1 in Pugh, 1996), ultimately it is simpler to try to understand each constituent as the result of a harmonic (~ Fourier expansion) decomposition, rather than a complex interaction between the bodies and their periods.

A few good books dealing with the analysis of tides are:

  • Pugh, D. T. (1996). Tides, surges and mean sea-level. John Wiley & Sons Ltd. (check Table 4.1 for a list of constituents and their origin).
  • Godin, G. (1972). The analysis of tides, 264 pp. University of Toronto.
  • Kowalik, Z., & Luick, J. (2013). The Oceanography of Tides. University of Alaska Fairbanks.

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