Tides 2

TIDES SOME CONCEPTS PART - II By Rakesh Sharma, Officer Surveyor Marine Geodesy Wing Geodetic & Research Branch

MOTIONS OF THE SUN AND MOON BEGINNING with the motion of the sun, it is well known that the apparent path of the sun through the heaven is in a plane cutting what is known as the celestial sphere in a path called the ecliptic. The plane of the ecliptic is inclined at a constant angle of 23° 27' to the plane of the equator. The point where the apparent path of the sun crosses the equator from south to north is called. by historical custom, “the first point of Aries” (denoted by 'γ '), or, alternatively, “the vernal equinox”

The position of any heavenly body is defined by: Its "longitude," which is the angular distance eastward along the ecliptic, measured from the vernal equinox, and its "latitude," measured positively to the north of the ecliptic along a great circle cutting the ecliptic at right angles.

Astronomical angular definitions.

Alternatively, its position can be expressed in terms of its angular distance along the equator from the vernal equinox (this angle being called the "right ascension"), together with its angular distance (called the "declination"), north or south of the equator, measured along a meridian.

Astronomical angular definitions.

The apparent motion of the sun is such that a complete revolution of the ecliptic is made in a mean solar year of 365.2422 mean solar days, and its movement in latitude is negligible.

Ecliptic

Equator

The apparent path of the moon oscillates somewhat about the ecliptic and observation shows that while the moon completes a revolution, measured along the ecliptic, in a period of 27.3216 mean solar days, the cycle of oscillation north and south of the ecliptic is completed in 27.2122 mean solar days.

INCLINATION OF SUN AND MOON’S PATH OVER EQUATOR

MOON’S PATH ECLIPTIC 5° 08′ 23° 27′

EQUATOR

The difference between the period of revolution in the orbit and the period of oscillation north and south of the equator is of considerable importance. Suppose that the moon commences a cycle at the vernal equinox. When the moon has completed its cycle north and south of the ecliptic, the revolution in orbit is not complete by 0.1094 day, on the average. 27.3216 – 27.2122 = 0.1094 days

The "ascending node" (that is, the point where the moon crosses from south to north of the ecliptic), thus travel westwards (i.e., back-wards, since the moon and sun travel eastwards), by 0.1094 day in every 27.2122 days,

INCLINATION OF SUN AND MOON’S PATH OVER EQUATOR

MOON’S PATH ECLIPTIC

EQUATOR

Therefore on the average this "regression of the nodes" will be completed in

27.3216 cycles in latitude 0.1094

or

27.3216 X 27.2122 years = 18.61 Julian years. 0.1094 X 365.25

A Julian year is equal to 365.25 mean solar days.

It is known from observations that the lunar orbit is in a plane which is inclined to the plane of the ecliptic by practically a constant angle of 5° 8', and the variations from this figure are negligible.

It is clear from the figure that the maximum declination of the moon will occur when the ascending node is at the vernal equinox ('γ '), when the north declination will rise during the following month to about 5° above the ecliptic (23° 27' + 5° 8' = 28° 35' in all) and a fortnight later the south declination will have an equal value.

Moon’s orbit Ecliptic

γ Equator

If, the descending node is at the vernal equinox, the maximum declination (north or south) will not be more than 23° 27' - 5° 8' =18° 19'. These values will recur at intervals of 18.61.years so that we have the important fact that any tidal variations associated with lunar declination will have a regular variation in a period of 18.61 years. (generally referred to briefly as the nineteenyearly variation).

γ

Ecliptic

Moon’s orbit Equator

The phenomena associated with the nineteen-yearly period are so interesting and so, important that it may be well to consider the matter a little more generally. For this purpose we shall refer to the periods of recurrence of lunar phases and of lunar distance, which are obtained from observations, so that in all we have four “months” as follows :(a)29.5306 mean solar days for the period of recurrence of lunar phases, this being the period generally referred to as a lunation; (b)27.5546 mean solar days for the period of oscillation in lunar distance; (c)27.2122 mean solar days for the period of oscillation of the moon in latitude; (d)27.3216 mean solar days for the period of revolution of the moon in longitude.

From the last two we have already deduced the period of revolution of the moon's nodes, and it is clear that the moon's motions will tend to recur after this period in the sense that the moon will cross the ecliptic at the same point after 18.61 years, But the sun will not be in the same position relatively to the moon after that period, for the sun is only found in the same position in the ecliptic after an exact mean solar year. It is a very remarkable fact, discovered long ago by Meton, that 235 lunations occur almost exactly in 19 mean solar years. Again, it has been known for many centuries that eclipses tend to recur in a period of nearly 18 years and 11 days. Now the extent of an eclipse is largely governed by the distance of the moon as well as its nearness to the ecliptic, so that this period is related to the months (a) and (b), in that it includes 223 lunations and 239 oscillations in lunar distance, almost exactly.

We get three important periods :

18.61 years as the period of revolution of the moon's nodes.



19.00 years, the Metonic cycle, giving the recurrence of lunar phases.



18.03 years, the Saros, recurrence of eclipses.

giving

the

The question is often asked as to whether these periods have any direct use in avoiding the necessity of predicting tides. The answer is that if the Saros is used the lunar declination is not repeated exactly, and that the extra 11 days on the exact number of years affects the solar tides, while if the Metonic cycle is used the lunar distances are not the same. The necessary corrections to the observations of 19 years ago would be as troublesome as the direct prediction, and of course the meteorological influences on the old observations would not be easily corrected.

ORBITAL ELEMENTS AT ZERO HOUR, G.M.T. s = Mean Celestial Longitude of Moon. h = Mean Celestial Longitude of Sun. p = Mean Celestial Longitude of Lunar Perigee. N = Mean Celestial Longitude of Ascending Node. p’ = Mean Celestial Longitude of Solar Perigee.

s = 277°.02 + 129°.3848 ( Y – 1900) + 13°.1764 (D+i) h = 280°.19 -

0°.2387 ( Y – 1900) + 0°.9857 (D+i)

p = 334°.39 + 40°.6625 ( Y – 1900) + 0°.1114 (D+i) N = 259°.16 - 19°.3282 ( Y – 1900) - 0°.0530 (D+i) p’ = 282°.00 for the century 1900 to 2000. Y = the year. D = the number of days elapsed since january 1st in the year. i = the integral part of 0.25 (Y - 1901).

RESOLUTION OF TIDE The complicated oscillatory force can be resolved into sum of a number of ‘Constituents’. Each of which is a simple oscillation of the type F cos (E), which is quite regular and can be simply calculated. The Amplitude, F, is the constant maximum to which the constituent rises in either direction; its Phase, E, is an angle which increases at a constant rate with time for each constituent. The number of these constituents which it is necessary to take into account depends on the accuracy with which it is desired to reproduce the true variation of the tide-raising forces while the number of constituents it is possible to obtain by analysis depends on the length of period of the observations being analysed.

F

E

F

H=F Cos (E)

RESOLUTION OF TIDAL CURVE OBSERVED DATA

RESOLVED TIDE

SPEED OF THE CONSTITUENTS The rate of which the angle E increases with time for any constituent is known as the “speed” of that constituent; this speed (which is not to be confused with the actual speed of the tidal disturbance over the ground) is really an angular velocity and is usually expressed in degrees per hour. The “speeds” of over 400 such constituents have been calculated from the astronomical formulae for the apparent motions of the sun and moon.

LIST OF SOME HARMONIC CONSTITUENTS Symbol

Argument

Speed number

Sa

h

0.0411

Ssa

2h

0.0821

Mm

s–p

0.5444

Msf

2s – 2h

1.0159

Mf

2s

1.0980

CALCULATION OF SPEED FOR Sa (SOLAR ANNUAL)

Apparently Sun takes 365.2422 day to complete one revolution around Earth. ( in other words one cycle of 3600 is completed in 365.2422 days). Change in angle per day = 3600/365.2422 Change in angle per mean solar hour = 3600/(365.2422 X 24) = 0.04110

CALCULATION OF SPEED FOR Mm (LUNAR MONTHLY)

Period of oscillation of Moon in lunar distance = 27.5546 mean solar day ( in other words one cycle of 3600 is completed in 27.5546 days). Change in angle per day = 3600/27.5546 Change in angle per mean solar hour = 3600/(27.5546 X 24) = 0.54440

Nodal Corrections Due to the fact that the plane of moon’s orbit rotates slowly, returning to its original position in space after about 19 years, the magnitude and phase of each constituent (F and E respectively) vary slowly on either side of the values they would have if the moon’s orbit were fixed. This variation could be allowed for by additional constituents, but it has been found more convenient to allow for it by introducing a factor “f” and a phase correction “u”, which vary slightly about their mean values of unity and zero degrees respectively with a period of about 19 years. In the majority of cases for any period less than a year “u” and “f” can safely be assumed to be constant at the values calculated for the mid-point of the period. These Nodal Corrections “f” and “u” differ for constituent, year and date but are the same for a particular constituent, year and date all over the world. Hence ‘F’ and ‘E’ must be replaced by fF and (E + u) so that a tide-raising force constituent becomes fF cosine (E + u).

The Phase (E + u) of a Constituent of the Tide-Raising Force The angle ‘E’ may be regarded as the hour angle of a fictitious satellite which moves round the earth at a known constant angular velocity (the ‘speed’ of the constituent). Hence the value of ‘E’, relative to any particular meridian on the earth, can be calculated at any time. The value of ‘u’ depend only on the year and date, hence its value for any particular day can be calculated; this value will be the same for all meridians. The phase of a constituent, (E + u ), relative to any particular meridian, is the sum of the two, and it can be calculated for any year, date and time.

INPUT FOR THE HARMINIC ANALYSIS •

Hourly heights for 1 month or 1 year.

•

Values of s, h, p, N calculated at 00 hour on 1st day of data to be analysed.

•

Speeds of 100 constituents in their serial number.

•

Mean Sea Level Value of the port.

AFTER ANALYSIS WE GET Fn = Amplitude of the nth constituent; gn = Phase lag of nth constituents; n = Number of constituents ( say 100)

Height of tide can be calculated by the formula H=Zo+∑ f n Fn cos [Ent + Vn + un - gn}. where H = height of the tide at any time (t). Zo = mean height of water level above the adopted datum of predictions. Fn = mean amplitude of nth component. f n= factor for reducing Fn to the year of prediction. En = speed of nth component. t = time reckoned from some initial epoch (such as the commencement of the year of predictions). Vn =Value of equilibrium argument of nth component when t = 0. un = Nodal angle in degrees of nth constituent and gn = Phase-lag of nth component.

CALCULATION OF f AND u Value of ‘f’ and ‘u’ for various constituents can be calculated using the formulae e.g. Mm:

f=1.000-0.130 cos N,

u=0°.0

M2:

f=1.000-0.037 cos N,

u= - 2°.1 sin N

………… ………….

Vn can be calculated using formula; V1=h V2=h + h V3=s - p ……………. ……………. En are the speed of constituents; e.g. E1 = 0.04107 E2 = 0.08214 ………………… ………………..

INPUT FOR THE PREDICTION OF TIDES •

Amplitude (Fn) and Phase lag (gn) of 100 constituents.

•

Desired species ( Hourly predictions / High –Low water predictions.

•

Year and period of predictions.

RESOLUTION OF TIDAL CURVE OBSERVED DATA

RESOLVED TIDES

RECONSTITUTION OF RESOLVED TIDES

PREDICTIONS

RESOLVED TIDES