PREDICTION OF TIDES BY THE HARMONIC METHOD

The harmonic method is the practical application of the dynamic theory of tides and is the most usual and satisfactory method for the prediction of tidal heights. It makes use of the knowledge that the observed tide is the sum of a number of harmonic constituents or partial tides, each of whose periods precisely corresponds with the period of some component of the relative astronomical motions between Earth, Sun and Moon. For any coastal location, each partial tide has a particular amplitude and phase. In this context, phase means the fraction of the partial tidal cycle that has been completed at a given reference time. It depends upon the period of the tide- producing force concerned, and upon the lag (Section 2.3) of the partial tide for that particular location.

71 The basic concept is analogous to that illustrated in Figure 1.9, though with a great many more component wave motions (partial tides). The wave form that represents the actual tide at a particular place (e.g. Figure 2.1) is the resultant or sum of all of the partial tides at that place. An example using just two partial tides is illustrated in Figure 2.17: the combination of a

diurnal and a semi-diurnal component produces two unequal high tides (H and h) and two unequal low tides (L and 1) each day, and the time interval between the higher low tide (1) and the lower high tide (h) is significantly shorter than that between H and 1 or L and h. Tides like these, characterized by high and low tides of unequal height, are known as mixed tides, and are common, for example, along the Pacific coast of North America (see also Figure 2.18).

Figure 2.17 Mixed tide (purple) produced by the combination of a diurnal (red) and a semi- diurnal (light blue) partial tidal constituent.

H and h = high tides; L and I= low tides. For simplicity, the semi-diurnal period is shown as 12 hr, whereas the M2 period is 12 hr 25 rnin.

In order to make accurate tidal predictions for a location such as a seaport, the amplitude and phase for each partial tide that contributes to the actual tide must first be determined from analysis of the observed tides. This requires a record of measured tidal heights obtained over a time that is long compared with the periods of the partial tides concerned. As many as 390 harmonic constituents have been identified. Table 2.1 shows the nine most important of these: four semi-diurnal, three diurnal and two longer-period constituents.

Table 2.1 Some principal tidal constituents. The coefficient ratio (column 4) is the ratio of the amplitude of the tidal component to that of M2.

Name of tidal Symbol Period in Coefficient

component solar hours ratio

(1142= lO0) Semi-diurnal:

Principal lunar M2 12.42 100

Principal solar $2 12.00 46.6

Larger lunar elliptic N2 12.66 19.2

Luni-solar K2 11.97 12.7

Diurnal:

Luni-solar K1 23.93 58.4

Principal lunar O1 25.82 41.5

Principal solar P1 24.07 19.4

Longer period:

Lunar fortnightly Me 327.86 17.2

Lunar monthly M m 661.30 9.1

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Semi-diurnal partial tides result from tide-producing forces that are symmetrically distributed over the Earth's surface with respect to the Sun and Moon, as illustrated in Figures 2.3 and 2.4. M2 and $2 are the most important ones, because they control the spring-neap cycle (Figure 2.12).

The last column of Table 2.1 shows that $2 has only 46.6 per cent of the amplitude of M2, because although the Sun is much more massive than the Moon it is also much further away (Section 2.2).

Diurnal tides are principally a consequence of lunar and solar declination (Figures 2.8 and 2.11), and relate to the diurnal inequalities described on p.60. Thus, on Figure 2.9 there is a high diurnal tide at point Y and a low diurnal tide at point X. Half a day later, the diurnal tide will be high at X and low at Y, as the Earth rotates within the equilibrium tidal bulge. The tidal range at both X and Y will be lowest when the Moon is at zero declination. However, at locations where the semi-diurnal influence is minimal (p.73), only diurnal tides occur (see Figure 2.18(a), p.74) and tidal ranges are smallest when lunar declination is zero.*

Before moving on, it is worth noting some regularities among other constituents in Table 2.1. The luni-solar diurnal partial tide, K1, has twice the period of its semi-diurnal counterpart K2, but has much greater amplitude, while the average of K1 and P1 is exactly 24 hours. Small departures of the periods of some semi-diurnal and diurnal constituents (e.g. N2, P1) from a simple relationship with those of M2 and $2 result mainly from complications related to the orbits of Moon and Earth

(Figures 2.10 and 2.11). With regard to the longer cycles listed in Table 2.1, the lunar fortnightly period (Me) w o r k s o u t t o 13.66 days, almost exactly half the 27.3-day period of the Moon's rotation about the Earth-Moon centre of mass; while the lunar monthly period (Mm) is very close to the perigee-apogee cycle of 27.5 days mentioned in relation to Figure 2.10.

There are of course still longer cycles, an obvious example beingthe 18.6- year period related to precession of the lunar orbit (Figure 2.10); and there are shorter-period constituents as well (see Section 2.4).

Even using the few major constituents in Table 2.1, analysis of tidal records and production of fide-tables for a port for an entire year used to be a very time-consuming activity. In the early years of harmonic analysis, they were computed by hand. The first machine to do the job was invented by Lord Kelvin in 1872. Electronic computers are admirably suited to this repetitive procedure, and fide-tables for individual ports all over the world now take little time to prepare.

The precision achieved by radar altimeters (Section 1.7.1) is such that tidal ranges in the deep oceans can be determined using information on tidal amplitude and phase extracted from the satellite data. Results are in good agreement with predicted values, and are nowadays supplemented by tidal data from the deep-sea pressure gauges mentioned in Section 1.7, placed at strategic locations in the oceans, far from land.

*The period between high and low 'lunar' diurnal tides is much shorter than that between high and low 'solar' diurnal tides. That is because the Moon's declination changes from zero to maximum and back every two weeks or so (Figure 2.8), whereas the same change in solar declination takes about six months (Figure 2.11).

Having examined the theory, let us see how the actual tides behave in different places. Every partial tide has its own set of amphidromic systems, and their amphidromic points do not necessarily coincide.

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It would be predominantly diumal. The tidal range increases with distance from the amphidromic point (Figure 2.16), so in this case, the tidal range due to the semi-diumal constituents would be small relative to that due to the diurnal constituents.

Tides can in fact be classified according to the ratio (F) of the sum of the amplitudes of the two main diumal constituents (K1 and O1) to the sum of the amplitudes of the two main semi-diumal constituents (M2 and $2). Some examples are illustrated in Figure 2.18.

*With increasing values of F (signifying greater influence of diurnal constituents) the time interval between maximum (or minimum) tidal ranges decreases from about 15 days

(approximately half of 29.5 days) to about 13.5 days (approximately half of 27.3 days).

Figure 2.18 shows only a selection of the many possible types of tides that can occur. The actual tides at any particular location result principally from the combination of amplitude and phase of the diurnal and semi-diurnal constituents (Table 2.1) at that location. A high value of F (say above 3.0) implies a diurnal tidal cycle, i.e. only one high tide occurs daily, and fluctuations in tidal range are largely due to changes in the Moon's

declination (Figure 2.9). Low values of F (say less than 0.25) imply a semi- diurnal tide, and the fluctuations in tidal range are mainly due to the relative positions of Sun and Moon, giving the spring-neap variation (Figure 2.12), and variations in lunar declination have only a relatively small effect.*

Between these two extremes are the mixed tidal types, where daily inequalities are important, and there can be considerable variations in the amplitudes of, and time intervals between, successive high tides. The middle two tidal records in Figure 2.18(a) show diurnal inequalities where typical 'large tides' alternate with 'half-tides' (cf. Figure 2.17), and there is an additional contribution to the diurnal inequality resulting from the changing declination of the Moon, i.e. the change from tropic to equatorial tides and back again (Section 2.1.1). For example, the transition between tropic and equatorial tides can be seen at around days 6-9 and 19-22 in the record for San Francisco (Figure 2.18(a)), as lunar declination passes through zero. However, changes in the Moon's declination have less effect at higher latitudes, and diurnal inequalities are therefore not an obvious feature of tides around Britain, for example.

The configuration of an ocean basin determines its natural resonant period (Section 1.6.4), and along open ocean coasts the type of tide (Figure 2.18(a)) depends upon whether the adjacent ocean responds more readily to diurnal or semi-diurnal constituents of the tide-producing forces. In the Atlantic Ocean and most of the Indian Ocean, the response is mainly semi- diurnal, though the natural period of the Gulf of Mexico appears to be about 24 hours, and diurnal tides predominate there. In the Pacific Ocean, the diurnal response is more significant and tides are usually of the mixed type, though they are predominantly diurnal in northern and some western parts of the Pacific (cf. Figure 2.18(a)).

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Figure 2.18 (a) Examples of different types of tidal curves, in England, the USA, the

Philippines, and Vietnam. Note that the vertical scales are not the same in each record. For full explanation, see text.

(b) The tidal measuring station at Cascais harbour, Portugal, in use since the late 19th century, is one of the longest-serving stations in the European tidal gauge network. The recorder is at the top of a 'stilling' well which dampens the oscillations of swell and other wind waves entering the harbour.

75 It is worth mentioning here that, at any particular location, the highest and lowest spring tides will occur at the same times of day (~6 hr 25 min. apart).

That is because the alternation of spring and neap tides is determined by the Sun (Figure 2.12) and the period of the $2 constituent is 24 hr (Table 2.1).

As you might expect, a similar relationship applies to neap tides. The feeding and reproductive behaviour of many marine animals, especially those living in nearshore and shallow shelf waters, is 'tuned' to tidal cycles, notably the 29.5-day lunar or synodic month (the spring-neap period, Section 2 . 2 . 1 ) - see also Section 2.4.1.

In shallow water, local effects can modify tidal constituents such as Me, particularly by producing harmonics whose frequencies are simple multiples of the frequency of the constituent concemed. These harmonics result from frictional interactions between the sea-bed and the ebb and flow of the tide - especially in shallow waters. For example, the quarter-diurnal constituent M4 (twice the frequency of M2) and the one-sixth-diurnal constituent M 6 (three times the frequency of M2) are generated in addition to the semi-diurnal constituents. In most locations, the effect of these two harmonics is

insignificant compared with the principal constituents, but along the Dorset and Hampshire coasts of the English Channel each has a larger amplitude than usual. Moreover, the two harmonics are in phase, and their combined amplitude is significant when compared to that of M2. (Just west of the Isle of Wight, M2 is about 0.5 m, M4 about 0.15 m, and M6 about 0.2 m.) The additive effect of all three constituents causes the double high waters at Southampton and the double low waters at Portland. However, there is no truth in the popular myth that double high water at Southampton is caused by the tide flooding at different times around either end of the Isle of Wight.

The Mediterranean and other enclosed seas (e.g. Black Sea, Baltic Sea) have small tidal ranges of about 0.5 m or less, because they are connected to the ocean basins only by narrow straits. The tidal waves of the major amphidromic systems (Figure 2.15) cannot themselves freely propagate through these restricted openings. However, interaction between Atlantic tides and the shallow-water shelf region near Gibraltar for example, results in the generation of internal waves, which do propagate into the

Mediterranean (Figure 1.23(b)) - and the internal waves seen in Figure 1.23(a) in the South China Sea may have a similar cause. By contrast, it is unlikely that similar packets of internal waves would occur where the Bosphorus connects to the Black Sea because the tidal range in the adjacent Mediterranean is negligible.