Arthur Thomas Doodson - Doodson Numbers

Doodson Numbers

In order to specify the different harmonic components of the tide-generating potential, 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. (In the case that the biased multiple exceeds 9, the system adopts X for 10, and E for 11.)

The Doodson arguments are specified in the following way, in order of decreasing frequency:

is 'Mean Lunar Time', the Greenwich Hour Angle of the mean Moon plus 12 hours.

is the mean longitude of the Moon.

is the mean longitude of the Sun.

is the longitude of the Moon's mean perigee.

is the negative of the longitude of the Moon's mean ascending node on the ecliptic.

or is the longitude of the Sun's mean perigee.

In these expressions, the symbols, and refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:-

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

is the mean anomaly of the Sun (distance from its perigee).

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

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 due to the sun, 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. 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.

A number of further examples can be seen in Theory of tides - Tidal constituents.