53 ELEMENTARY TIDAL THEORY
202. THE TIDE-GENERATING FORCES
The study of the causes of the tides has engaged the attention of many of the ablest mathematicians of the last three centuries. Notable amongst these mathematicians was the illustrious Sir Issac Newton, who was the first to formulate the laws of motion and gravity, according to which every particle of matter in the universe attracts every other particle with a force which is proportional to the product of their masses and inversely proportional to the square of their distance apart, forms of the basis of a theory of tide known as the Equilibrium Theory. Newton’s universal gravitation law is expressed as:
when appropriate units are employed and where
m1 and m2 are the masses and d is the distance apart.
According to the equilibrium theory of the tide, every drop of water on the Earth’s surface is balanced mainly under the action of centrifugal force and the forces of gravitational attraction of the Earth, Moon and Sun. It has been pointed out that, although the equilibrium theory of the tide expresses the law of the vertical oscillation of the sea surface, it is not a true theory because the problem related to the tide is one of motion and not of balance. Many mathematicians have, therefore, attempted to deal with the problems associated with the tide using dynamic and not statistical principles. However the general agreement between the actual tide found from tide gauge observations, and the computed equilibrium tide, which is obtained by considering the several forces involved, is so close that the basis of the greater part of tidal computation and prediction is the equilibrium theory. Let us, therefore, consider briefly the more important tide-generating agents according to this theory.
One of the factors in relation to the tide is the effect of the spin of the Earth. Because the Earth rotates, every particle on its surface (except at the poles) experiences a centrifugal force which acts perpendicularly to the Earth’s axis. Thin force varies as the cosine of the Lat., being maximum at the equator, where the radius of rotation is greatest, and non-existent at the poles. It may be resolved into vertical and horizontal components; the vertical component of this force is neutralised by the Earth’s force of gravity which acts vertically downwards while the horizontal component of centrifugal force on the Earth’s surface acts towards the equator, such that every particle tends to move along its meridian towards the equator. If this tidal factor alone acted we should get, if the Earth were uniformly covered with water, a permanent high water at every point on the equator, and a permanent low water at both poles.
The Moon and the Sun are the most important tide-generating agents. The tide-raising force exerted by each of these bodies depends upon the distance of the body from the Earth and upon the direction of the body relative to the plane of rotation of the Earth.
The distances of both the Moon and the Sun from the Earth are variable because the Earth’s orbit around the Sun and the Moon’s orbit around the Earth is elliptical. The distance between the Earth and the Sun is the least when the Earth is at a point in her orbit known as perihelion, and the distance is greatest when the Earth is at a point called aphelion. The Moon is nearest to the Earth when it is at a point in its orbit known as perigee, and at its greatest distance from the Earth when it is at a point in its orbit called apogee.
The directions of the Moon and the Sun relative to the plane of the Earth’s rotation vary with the changing declinations of these bodies. Dec., it will be remembered, is the angular distance N. or S. of the equinoctial being a great circle on the celestial sphere which is co-planar with the plane of the Earth’s rotation, i.e., with the plane of the Earth’s equator.
The force exerted between the Sun and the Earth is about 200 times greater than the force exerted between the Moon and the Earth. Tides are caused by the attraction of the Moon and the Sun on the Earth and the water surrounding the Earth, and it is therefore surprising that the Moon’s influence on the tide is more than double the Sun’s influence.
The reason for this is that the tide-raising force is a function of the difference between the attractive forces on two drops of water one at each end of a diameter of the Earth on the extension of which the tide-raising body lies. In fig. 53-1, let m represent the mass of each of two drops of water at A and B at opposite ends of the diameter of the Earth. Let M represent the mass of a tide-raising body (sun or Moon). If the distance between the centres of the Earth and of the tide-raising body is D, and the Earth’s radius is denoted by R, then by Newton’s universal law of gravitation, the tide-raising force is given by the expression:
The Earth’s mass is about 1/324,000th of that of the Sun, and about 80 times that of the Moon. The distance of the Moon from the Earth is about 60 times the Earth’s radius, and the distance between the Earth and the Sun is about 23,000 times the Earth’s radius. Evaluating the expressions in respect of the Moon and the Sun, it may be verified that the ratio between the tide-raising forces of the Moon and the Sun is about 21:10. In other words, the Moon as a tide-raising agent is about twice as effective as the Sun. The Moon and the Sun, pull more forcibly on the side of the Earth facing them than on the opposite side, and as stated above, it is the difference between the forces exerted on these two sides of the Earth which produces the tide.
The Moon and the Earth revolve around their common centre of gravity, the barycentre (see fig 53-1). It is convenient to think of the Earth and the Moon continually falling towards each other. The Earth’s radius is about 4,000 miles, and the radius of the Moon’s orbit is about 240,000 miles. Thus, the point on the Earth’s surface which is nearest to the Moon is 3⅓ of the radius of the Moon’s orbit nearer to Moon than is the point on the Earth’s surface most remote from the Moon.
The Moon’s force of attraction on any point on the hemisphere of the Earth which faces the Moon is greater than its force at the Earth’s centre. The water, being mobile, falls towards the Moon more than does the Earth. The water, therefore, piles up at X in fig. 53-2. The water on the remote hemisphere falls towards the Moon to a less extent than does the Earth, thus the water on this hemisphere piles up at Y.
The differential attraction of the Moon on opposite sides of the Earth produces an ellipsoid of water surrounding the Earth, with its major diameter in line with the direction of the Moon. As the Earth rotates within this ellipsoid of water, the water level at any place would rise and fall. This rising and falling which is due to the Moon alone are known as the lunar tide.
The lunar tide has a period, i.e., the interval between the times of successive High Waters, equal to half a lunar day, the average length of which is 24h. 50m. of mean solar time. Hence, the period of the lunar tide is 12h. 25m. approximately, and is semidiurnal in character. The constituent of the total theoretical tide due to the Moon alone is called the Mean Lunar Semidiurnal Tide, abbreviated to M2, the letter M for Moon and the suffix 2 to indicate two high water per lunar day.
The Sun produces a similar effect to the Moon, but to a less marked degree. The differential solar attraction on opposite sides of the Earth is small compared with the differential lunar attraction. The effect of the Sun is the creation of a rising and falling of the water level which is known as the solar tide. The solar tide has a period of 12h. 00m. and is therefore also semidiurnal in character. The constituent of the total theoretical tide due to the Sun alone is called the Mean Solar Semi-diurnal Tide. abbreviated to S2, the letter S for Sun and the suffix 2 to indicate two high water per solar day.
203. The Luni-Solar Tide
The combination of the lunar and solar tides is known as the Luni-solar tide. At the times of the New Moon and Full Moon, when the Earth, the Sun and the Moon are in a straight line, the tide-raising forces of the Moon and the Sun are said to be in phase (M2 + S2), and this results in a big amplitude, or range, of the tide known as a Spring Tide (from the Saxon word spring, which means to swell, because spring tides are the highest tides during the lunation).
Fig. 53-3 illustrates the spring tide which occurs when the Sun and Moon are in conjunction, i.e., when the age of the Moon is 0 days, while fig. 53-4 illustrates the spring tide which occurs when the Moon and Sun are in opposition, i.e., when the age of the Moon is 14½ days.
As the Moon ages, the tide-raising forces of the Moon and the Sun become more and more out of the stop, or out of phase until the times of quadrature, i.e., at the first or third quarter, when the angle at the Earth between the Sun and the Moon is 90°. The range of the tide at these times is the least because the Moon’s force endeavours to create high waters on meridians on which the Sun’s force is endeavouring to create low waters (M2 – S2).
These tides are called Neap Tides (from the Saxon word neafte, meaning scarcity, because they have the lowest high waters during the lunation). Neap tides are illustrated in fig. 53-5.
Suppose the lunar tide had a range of 6 metres and the solar tide had a range of 3 metres, then:
At Springs (M2 + S2) = (6 + 3) = 9 meters – At Neaps (M2 – S2) = (6 – 3) = 3 meters
The difference in range is therefore 6 metres, and this difference is called the Phase Inequality of Heights as it is due to the phase of the Moon. The mean lunar day is about 50 minutes longer than the solar day, so that normally an interval of about 24 hrs. 50 mins. separates the high water of one day from the corresponding high water of the next day at a given place, and the average interval between HW and the next LW is about 6hrs. 12mins. where the factors of the luni-solar tide are the only ones taken into consideration.
However, the cycles of changes in the Dec. of the Sun and the Moon are regarded as being distinct tidal factors which must also be considered. The plane of the Earth’s orbit around the Sun is inclined to the plane of the Earth’s rotation at an angle of 23½° (the obliquity of the ecliptic explained in §s 1-4. The Sun, therefore, undergoes a change in its Dec. in a period of one year. The plane of the Moon’s orbit around the Earth is inclined to the plane of the Earth’s rotation by an angle of about 5¼° and the Moon completes a cycle of changes in her Dec. in a period of one month.
The maximum value of the Moon’s Dec. in any monthly period lies between (23½° – 5¼°) and (23½° + 5¼°), the actual value depending upon the relative positions of the points of intersection of the planes of the equinoctial and the Earth’s orbit – the equinoxes, and upon those of the planes of the Earth’s orbit and the Moon’s orbit – the nodes.
Dec. is mainly responsible for a difference in the heights of two successive high water. This phenomenon is known as diurnal inequality. In fig. 53-6, Lat. C equals the Moon’s Dec. The tide shown is higher high water at C, where the Moon is over the upper meridian, than at A on the lower meridian. At position A there is a lower high water, and this place, on being carried around by the Earth’s rotation, gets low water on reaching B, when it is on the low water trough BB1, and, later, gets the higher high water on reaching C. The same applies to the parallel C1B1A1. This gives a diurnal component of the tide due to the Moon’s Dec.. The Dec. of the Sun produces a similar diurnal force affecting the tide. Thus we get two further constituents of the theoretical tide: –
K1 – the constituent which allows for part of the Moon’s Dec. and all of the Sun’s Dec., or the combined average lunar and solar diurnal force.
01 – the constituent which allows for the remainder of the Moon’s Dec., or the principal variation in lunar diurnal force.
204. Tidal Predictions by Harmonic Analysis
It has been shown above that tile theoretical tide is composed of a number of distinct tidal factors. Each of these tidal factors is called a tidal constituent, and the four major tidal constituents have been introduced and described. These are the lunar and solar semidiurnal constituents M2 and S2 respectively, and the Dec. or diurnal constituents K1 and 01.
There are, of course, a great many more tidal constituents which take account of the Earth’s centrifugal force, the varying distance of the Moon and the Sun from the Earth, friction between water and land, the varying depths of the seas and the irregular shapes of land masses. These so-called minor constituents have a bewildering array of names such as 2N2, N2,v2, L2,T2, R2, K2, 2Q1, Q1, M1, P1, etc, etc.
Because a graph of a constituent against time is a cosine curve, the constituents are referred as being harmonic. A harmonic curve is a symmetrical curve which recurs periodically. The tidal constituents are periodic and they can be represented by harmonic curves. The sine and cosine curves in trigonometry are examples of harmonic curves and the cosine curve is illustrated in fig. 53-7.
The periods and effects of the various Astronomical tidal constituents are known; and their phase relationships are also known. It is therefore possible to compute the tide resulting from each constituent, the sum of the results being a prediction of the times and heights of the L.W.’s and H.W.’s at any place.
On this basis, the tidal curve of the actual tide (a curve which may be readily obtained from tide gauge observations) is regarded as being the summation of several cosine curves, each representing one of the several tidal constituents.
The method of analysing a tidal curve obtained from a tide gauge observation record, so breaking it down into its harmonic constituents, and the reverse process of combining a series of constituents to form a hypothetical tidal curve, is known as harmonic analysis. The Admiralty Method of Tidal Prediction, in which the four principal constituents M2, S2, K1 and 01 are combined, is simply a practical, albeit complex, example of harmonic analysis which the ocean navigator must, however, on occasion perform because of the limitations of the data in Volumes 2 and 3 of Admiralty Tide Tables.
This method requires the use of specially prepared forms and diagrams (N.P. 159) which are obtainable from Admiralty Chart Agents. Full instructions for plotting the tidal curve for the whole day for any place for which the tidal constants are available, are given in N.P. 159.