Tide Heights

I have already stated in the main body of this article that [for all practical purposes] the Earth experiences just two tide-generating influences: the Moon and the Sun. It would seem, therefore, that all that needs to be done to calculate the theoretical height of the tides is to solve the hydrodynamical equations representing the action of these tide-generating forces. As ever in matters astronomical however, it's not quite that simple!

There are three main complicating factors: the Moon is not a constant distance from the Earth (due to its eccentric orbit); the Moon does not orbit in the plane of the Earth's orbit round the Sun, and the Earth's spin axis is tilted with respect to its orbital plane. The first factor means that the tide-generating potential is not constant, the second means that the tides produced by the Moon and the Sun are rarely aligned and the third means that the highest tides are not always on the equator. One must also bear in mind the variable distance of the Earth from the Sun (which causes a small change in the tide-generating potential of the Sun) and the many other minor complications due to perturbations in the Moon's orbit. All these various effects produce changes in the level of the tides, each of which has its own magnitude and timescale. For example, the Moon passes through the plane of the Earth's orbit once every 13.606 days (half the draconic period); is at its nearest to the Earth every 27.555 days, and regains its same position relative to the Sun and Earth every 29.531 days. Many of these variables also interact, further complicating the issue. The end result is that the tidal record for even the simplistic case of an Earth with no land-masses is quite different from the "up-down-up-down" pattern one might imagine.

Fortunately, because the geometry of these complicating factors is well-known it is possible to derive (very complex!) equations describing the variation in the tide-generating force produced by each of them and then solve the equations to integrate their effect over the entire globe of the Earth. The end-result is another equation, consisting of a series of terms involving sines and cosines - that is, simple "up and down" waves. The first term is a constant, showing that the mere presence of the Moon causes a permanent change in the overall profile of the ocean surface. The next terms give us the contributions of the diurnal (24hr) and semi-diurnal (12hr) influences, and there are also higher-order terms giving monthly, yearly and longer influences. It is thus possible to separate out the various influences as a set of tidal components, each of which can be analysed individually. There are rather a lot of these components, but fortunately only one effect produces an absolute increase in tide height - variable distance (to both Moon and Sun). All other effects simply alter the relative sizes of successive high tides without making any of them greater than they would otherwise have been. For example, the tilt of the Earth's axis is the main cause of the diurnal variation - the phenomenon whereby high tides are alternately slightly higher than usual and slightly lower than usual. However, at no time do the "higher highs" exceed what the high tide [on the equator] would have been if there was no tilt: this is well shown by the diagrams of the diurnal variation on the main page. It is also the case that, while such things as the diurnal variation produce fluctuations on a timescale of a day, the tidal driving force is always twice a day, due to the presence of the two tidal bulges. The only "real" tide is thus the semi-diurnal tide. This makes the job of calculating the maximum tide height much easier!

All this is only true in the idealised case, of course. When land-masses and coastlines are added the equations are no longer accurate, as the tides are heavily influenced by the local topography. The various components will still be present, however, so we just need a way to sort them out. Faced with this type of problem, a mathematician has a powerful weapon in his armoury - Fourier analysis. This is a process by which a complex waveform can be broken down into a set of basic waveforms, each of which has the simple "up and down" form. Calculation will then give the amplitude, frequency and phase relationship of each component required in order to construct the original complex waveform. Fourier analysis of actual tidal records will thus tell us the magnitude and phase of each of the idealised tidal components present at the given location. Because the characteristics of the components are constant at a given place, once found they can be used to predict future tides for that location: this is the way such predictions were made until the advent of computational hydrodynamic models.

Before moving on, I must cover some issues of nomenclature. All tidal components have names, in order to identify them, which usually consist of a letter and a number. The letters seem to have almost no logic to them but diurnal tidal components usually have the number 1 and semi-diurnal components have the number 2. In the case of the semi-diurnal components, the "real" lunar and solar tides are thus called M2 and S2 respectively. For reasons which are not clear, the components which (taken together) define the variation due to the eccentricity of the Moon's orbit are called N2 and L2: the corresponding ones for the Earth's orbit are T2 and R2. Although we don't need to consider them in terms of the maximum tide (as stated above) I shall add that the semi-diurnal component caused by the obliquity of the Earth's axis (the declinational variation) is called K2 and the long-period component which corresponds to the tilt of the orbit of the Moon is called Mf. There are three such diurnal declinational components (K1, O1 and P1), and many other named components corresponding to minor variations.

Once identified, the relative magnitudes of the components can be found by solving the differential equations defining the various interactions, as tabulated in [1]. The values given in these tables are quoted from other references, but I have checked these out just to make sure everything ties up! The actual magnitudes are then found by substitution of actual values for the Earth/Moon system into these solutions, as in [2]. The results are given in the following table.

The entry for R2 is incomplete as I could not find a value for it. However, by analogy with the N2 and L2 pair I would expect it to be very small (as the eccentricity of the Earth's orbit is much less than that of the Moon) and so its absence will not cause a significant error. Although I will not be using them to calculate the maximum height, I have given the values for K2, Mf and the main diurnal components to show how significant they are, and thus illustrate the extent of the diurnal variation in tide height.

However, these are still theoretical answers for the idealised situation where the system has reached equilibrium, and of course we have two largely separate systems to consider - the solid body of the Earth itself and the oceans sitting upon it. Because neither the solid body of Earth nor the oceans can respond to the tide-generating potential sufficiently quickly to achieve tidal equilibrium, to determine the actual tide height the numbers must be multiplied by a factor related to the rigidity of the substance being deformed (either the Earth itself or the oceans). This is given by the so-called Love Number h2 [the 2 again implying semi-diurnal], which is well-defined in the case of the solid Earth. I took my value, 0.609, from the Handbook of Physical Constants [3]. Applying this factor to the M2 and S2 components produces the following results:-

Mean lunar solid tide = 38.5cm
Mean solar solid tide = 17.9cm
Total = 56.4cm

If we now do the same for the N2 & L2 values and to T2, and add them to M2 and S2 respectively (i.e. take the case where both Moon and Sun are at their nearest to the Earth), we get:-

Max. lunar solid tide = 46.9cm
Max. solar solid tide = 19.0cm
Total = 65.9cm

Again simply for comparison, I shall do the same calculations for the diurnal components. In this case the relevant Love Number is of course h1, but it does not have the same magnitude as h2 as the Earth responds differently to the 24hr periodicity. In fact, it is slightly different for each of K1, O1 and P1 but averages out at around 0.568. Using the correct values for each component and adding them all together we find that the maximum diurnal solid tide is 42.1cm i.e. rather more than the mean semi-diurnal tide caused by the Moon alone.

The calculation of corresponding results for the deep-ocean tides is more problematic however, as the value of h2 is uncertain for this situation: I could not find it defined anywhere. Further research indicated that help might be at hand in the shape of a slightly different Love Number, k2. This refers to the extra gravitational potential produced by the deformation in shape of the tidally-affected body. Vitally though, it is involved in the calculation of the energy dissipated in the tides, as defined in [4]. The formula is D = 101.4 x k2sinP, where P is the tidal lag value and the result is in Terrawatts (10¹²W). It is known that the energy dissipated in the principal semi-diurnal lunar oceanic tides is 2.421TW [5] and for oceanic tides P=2.9deg [6]. k2 is thus 0.472, which is consistent with the only reference I could find (in a paper on Saturn's moon Titan, admittedly!), which gave 0.48 as typical for an ocean on the surface of a solid planet. [Note that while 2.421TW is a large amount by normal standards, the heat flow from the Earth's interior amounts to 30TW and the power in solar radiation is a massive 2 x 10⁵ TW. Tidal dissipation is thus, relatively speaking, very small.]

Now we've got k2, in principle we could calculate h2 by using the theoretical relationship that h2 = 5 x k2 / 3. However, the table in Ref.3 implies that for the solid Earth h2 is twice k2 not 5/3: this is supported by other papers which quote both values. I thus took an average between these two multipliers - 11/6. Using this factor, our derived value for h2 is 0.865 which produces the following results:-

Mean lunar ocean tide = 54.7cm
Mean solar ocean tide = 25.4cm
Total = 80.1cm

Max. lunar ocean tide = 66.7cm
Max. solar ocean tide = 26.9cm
Total = 93.6cm

If we assume that the ratio of Love Numbers h1 and h2 for oceanic tides is the same as for solid tides, the maximum diurnal oceanic tide is 59.8cm, again somewhat more than the mean lunar tide.

It should be noted that the values calculated for the semi-diurnal ocean tides are almost exactly those quoted in Wikipedia (without any citation or reference to say how they were derived) so maybe the author of this section took the same line as I have! Note also that the M2 tide is 46.6% of the S2 tide, which should be compared with the value of 45.9% derived from simple basic principles. Given the complexities involved in the creation of the tides, this close agreement is quite remarkable. Finally, we can see that the tidal dissipation equation used to derive our value of k2 for the ocean tides also gives consistent answers for the solid tides. If we take k2 to be 0.302 for the solid tides [as per [3]] and P=0.204deg [as per [4]] then D=109GW (10⁹ watts). Ref.5 tells us that the total semi-diurnal tidal dissipation is 2.536TW, so subtracting the 2.421TW oceanic dissipation leaves 115GW, of which ~10GW is estimated to be due to "air-tides" (i.e. dissipation in the atmosphere). This leaves 105GW for the solid-body tides, very close to the calculated figure.

My final comment is to remark that even though I took the oceanic and solid tides to be independently calculable, this is not quite true in practice, as each affects the other to a minor degree. Firstly, the oceans tend to depress the surface of the Earth upon which they are sitting. Any variations in the depth of the ocean will result in different amounts of depression of the surface, and so an oceanic tide will induce a tide in the solid body, called the "load tide". Secondly, deformations in the solid body will cause a change in the local gravitational potential and hence in the tide-generating force, thus affecting the magnitude of the tides - the solid body tide (and indeed the load tide) will therefore reflect back into the oceanic tide. These effects do have to be taken into account in the most accurate tide models, as the magnitude of the load tide is about 7% of that of the oceanic tide and in opposition to it.