Double high waters in the semi-diurnal tide are an intriguing and sometimes puzzling feature of coastal tidal observations. In a double high water, the tide rises to a first maximum followed by a short dip in water level before it rises again to a second maximum and then falls towards the subsequent low water. This is because the spatial scale of the higher harmonics of the tide is shorter than that of the dominating semi-diurnal constituent. Consequently, any location with a double high has a nearby location with a double low tide. The most famous and best studied example of a double high water is that at Southampton on the south coast of England, where the extended period of deep water associated with the double high tide gave the port a commercial advantage over its rivals. The Southampton double high has an associated double low in Weymouth, some 50 km away Other examples of double high waters, and associated double lows, can be found along the eastern side of the North Sea, e.g. a double in Den Helder in the Netherlands. On the US east coast, Woods Hole (among other locations) experiences a double high, whereas a double low water is seen at Providence in Narragansett Bay (; see also , for a discussion).

No explanation of double high (or low) waters can be found in the direct action on the ocean of the tidal forces of the moon and sun. To create a double high water in the semi-diurnal tide it is necessary to add a higher-frequency oscillation with an appropriate phase and sufficient amplitude. Higher harmonics of the semi-diurnal tide (that is oscillations with a frequency higher than twice per day) are created as the tidal wave enters shallow water, either through the reduction of water depth, through the enhanced effects of quadratic bottom friction, or through streamline curvature around coasts and sandbanks . Transient seiches created by the tide can also serve to provide the right conditions for double tides .

The amplitude and phase of the higher-frequency oscillation must meet certain conditions to produce a double high water. The simplest of these conditions was described by . If the frequency of the oscillation is n times that of the semi-diurnal tide and the oscillation has a trough exactly coinciding with the peak of high water, a double high water will form if bn2a>1, where b is the amplitude of the oscillation and a the amplitude of the semi-diurnal tide. For example, if the oscillation has the period of a quarter of a day, then n=2 and the amplitude must be at least one-fourth that of the semi-diurnal tide to produce a double high water. This is a minimum requirement – in most cases, the quarter diurnal component will not be big enough on its own to produce a double high water and additional higher harmonics will be required. As the frequency, n, of the higher harmonic increases, Eq. (1) shows that the required amplitude ratio, b/a, becomes smaller.

It is worth noting here that the amplitude, a, of the semi-diurnal tide in Eq. (1) is that observed on the day – i.e. it is the sum of the lunar and solar semi-diurnal tidal constituents on that day. The value of it therefore changes through the fortnightly spring-neap cycle.

In the following we use “neap” to describe the smallest fortnightly tidal range. Note that this is not necessarily at first or third lunar quadrature.

Figure 1 illustrates the formation and subsequent disappearance of a double high water for the sum of a semi-diurnal D2 tide (amplitude a) and its D6 harmonic (amplitude b). In Fig. 1a the ratio b/a=0.25, a=1 m, and the phase difference is 0.5 h (i.e. the minimum in D6 occurs half an hour after the maximum in D2). Adding a D6 curve to D2 creates a double high water and also a double low water, as the maximum in D6 also coincides with the low water in D2. In Fig. 1b the amplitude of D2 has been reduced to 0.1 m, keeping the D6 curve the same. There is now no evidence of a double high water – the tide is instead best described as sixth diurnal with a weak remnant semi-diurnal modulation in mean water level.

In general, the sum of a semi-diurnal tide and a single higher harmonic can be written as y=acos⁡[ωt]-bcos⁡[nω(t-ϕ)], where y is sea level and ω=2π/12 lunar hours is the angular frequency of the semi-diurnal tide. The phase ϕ represents the time difference between the maximum in the semi-diurnal tide and the nearest minimum in the higher harmonic, so that D2 has a maximum for t=0 and the n harmonic a minimum at t=ϕ. Time, t, is measured from zero at the high water in the semi-diurnal tide. For a small range of times around t=0 and t=ϕ the cosine curves can be represented by quadratic curves (using the expansion that cos⁡(x)=1-0.5x2 for small x): y=a[1-0.5ω2t2]-b[1-0.5n2ω2(t-ϕ)2].

When there is a double high water, there is a turning point (marked as D in Fig. 1a) which marks the centre of the dip in sea level between the two high waters either side. At this turning point, occurring at time t′, dy/dt=0 and so, for small values of ϕ, from Eq. (), t′=n2(b/a)n2(b/a)-1ϕ.

Since the term in brackets is always greater than 1, t′>ϕ. It is a feature of double high waters that the turning point lies further away from the (semidiurnal) high water than does the minimum in the harmonic that creates it.

For the dip to be a minimum between the double high waters, a further condition is that d2y/dt2>0 at the dip. At this point it is necessary to include the next term (a function of (t-ϕ)4) in the expansion of the cosine curve for the higher harmonic. Applying the condition that d2y/dt2|t=t′>0 gives the condition for a double high water as bar2>1, where r2=n2[1-(0.5n2)ω2(t′-ϕ)2], which is the result obtained by . Note that when the phase difference ϕ=0 it follows that t′=0 and therefore that r2=n2; the condition for a double high water then becomes the Doodson criterion in Eq. (). For all other values of ϕ, r2<n2; then, according to Eq. (), b/a must be larger than the value required for zero ϕ in order to create a double high water. It is possible for r2, and hence (b/a)r2, to be negative. This will happen when either b/a is small or ϕ is large. When r2 is negative, it is impossible to satisfy Eq. () whatever the value of b/a, and no double high water can be formed (the actual physical interpretation of a negative r2 is intriguing, but left for a future paper).

As an illustration, we show in Fig. 2 the solution to the approximate parameterization in Eqs. ()–() for the case of n=3, that is the sum of D2 and D6 tides. The solution is symmetrical for negative and positive values of ϕ and so in Fig. 2 we have drawn the curves as a function of the absolute value of ϕ. It is necessary to iterate to reach the solution – the critical value of b/a must satisfy both Eqs. (5) and (6). For D2 and D6 tides, the critical value of b/a for zero phase is 1/9, or 0.111. As the phase difference increases from zero, the critical value of b/a also increases, so that for a phase difference of 1 h, it is necessary for the amplitude of D6 to be at least 0.27 times that of D2 to produce a double high tide. Also shown in Fig. 2 is the exact numerical solution to the problem, starting with the cosine curves in Eq. (3). The approximation of the cosine curves as quadratic curves close to their maxima and minima means that the analytical solution is not exact, but it does capture the essential features of the numerical solution, especially for ϕ less than about 1.5 h. In fact, at values of |ϕ| greater than about 1 h the dip becomes so detached from the high water that the tidal curve is no longer recognizable as a double high water. In practice, therefore, for a recognizable double high water produced by D2 and D6 tides, we can limit attention to the region in which ϕ is less than about 1 h in Fig.  2.

As the amplitude ratio b/a continues to increase above the critical value for the formation of a double high water, there is a gradual transition towards a tide dominated by the higher harmonic. As this happens, the level of water in the dip between the high waters falls towards the level of the low tide (Fig. 1). There comes a point when the level of water between the dips is virtually the same as that at the low tide, when we can say that the transition to the higher harmonic is complete and the double tide has disappeared. At the time of the minimum in the dip, t=t′ and sea level is given by y=a(1-0.5ω2t2)-b[1-(0.5n2)ω2(t-ϕ)2]. The first term in the right-hand side of this equation represents the fall in water level between t=0 and t=t′ due to the semi-diurnal tide; the second term is the fall in the same time due to the harmonic. The first term will occur in the absence of the harmonic, so the size of the dip due to the harmonic is equal to the second term. Referring to Eq. (), this second term can be written as b(r/n)2. The size of the dip relative to the amplitude of the semi-diurnal tide is therefore (b/a)(r/n)2. The higher harmonic will become dominant when this ratio exceeds a certain value, of order 1. That is the upper limit of b/a for a recognizable double high water is n2f>bar2, where the factor f represents the size of the dip relative to a. For example, if f=1 and n=3, the upper bound for b/a is 9/r2. In general, we can write the critical condition for the formation of a recognizable double high water in the form n2f>bar2>1, which is the condition we want to test. Note that the theory leads us to expect that the same parameter, namely (b/a)r2, is important in predicting the initial onset of the double high water and the disappearance of the high water as the harmonic becomes dominant.

Port Ellen lies on the south coast of Islay, an island which is part of the Inner Hebrides on the west coast of Scotland (Fig. 3). Tidal data sampled every 15 min at Port Ellen are available from the UK Tide Gauge Network as part of The National Tidal and Sea Level Facility (available from https://www.bodc.ac.uk/data/hosted_data_systems/sea_level/uk_tide_gauge_network/processed/), and Fig. 4 shows a record from Port Ellen for the second half of February, 2010. The 2-week observational period we use began on a new and ended on a full moon. The semi-diurnal tide is unusual because the two principal semi-diurnal tidal constituents (M2 and S2) are about equal in amplitude and, as a consequence, the semi-diurnal tide virtually disappears twice each month when these two constituents are 180∘ out of phase. This effect can be seen for 25 February in Fig. 4. A similar phenomenon is the dodge tide in South Australia , and the vanishing tide in Courtown, Ireland .

The diurnal tide is also important at Port Ellen and produces a significant diurnal inequality which can be seen at the beginning and end of the record in Fig. 4. In the figure, the diurnal inequality takes the form of the low tide in the middle of the day being higher than the low tide at the beginning and end of the day. The diurnal inequality is important in the selection of the morning or afternoon tide for the production of double high waters, as we shall see.

The double high water is intermittent and occurs most clearly in the morning high tide in the first part of the record (18 and 19 February) and the afternoon high tide towards the end of the record (27 and 28 February). There are also times when there is a stand in water level around the time of high water – it is sometimes difficult to tell if there is actually a double high tide present on these occasions or not. These “stand” tides are observed on all tidal cycles between the morning of 15 February and the morning of 17 February, and on the mornings of 20, 23, and 27 February.

The theoretical considerations presented in this paper, supported by a small data set, suggest that a single parameter can be used to predict the presence of a double high water when a higher harmonic is added to a semi-diurnal tide. As we might expect, this parameter – (b/a)r2 – depends on the amplitude and phase of the harmonic (compared to the semi-diurnal tide) and on the ratio of frequencies of harmonic and main tide. The data in Table 1 can be divided into four categories:

(b/a)r2≤0: no double high water is seen.

In the terms posed in the introduction, it is necessary to place both lower and upper bounds on the criterion for a double high water to allow for the higher harmonic being too small and too large.

A limitation of the theory, as presented here, is that it considers just a single higher harmonic added to the semi-diurnal tide. There will be places, and times, when this is appropriate. In the case of the data from Port Ellen we present here, the theory adequately represents the data for most of the time. The theory can be extended to include other harmonics in a straightforward way. The condition for the initial formation of a double high water by two harmonics added to the semi-diurnal tide, for example, can be written note that discussed a triple-harmonic case, but for amplitudes only: b1ar12+b2ar22>1.

Here, b1 and b2 are the amplitudes of the two harmonics and r12 and r22 are calculated from Eq. () with the appropriate phases. The time at the centre of the dip, t′, is calculated from a modified form of Eq. () which allows for both harmonics acting together. It can happen that the two harmonics support each other in producing a double high water or (because the r2 parameter can be negative) that one of the harmonics suppresses a double high water that would otherwise be formed by a single harmonic acting on its own.

Turning now to the observations, it is, in fact, difficult to tell, in marginal cases, when there is a double high water by visual inspection alone. This is because, when the double high first forms, the dip between the two high waters is small – perhaps just a centimetre or so – and this is often at the noise level of the measurements. In this paper, we have tried to avoid this problem by referring to a tidal stand – that is, there is clearly a portion of the tidal record where the water level remains flat for a time, but it is difficult to say for sure if there is a double high tide.

For most of the data set, the two most important harmonics at Port Ellen are the semi-diurnal (D2) and the sixth-diurnal D6. Other harmonics, in particular D1 and D4 play a role, however. The effect of the diurnal harmonic, D1, in allowing a double high water in one tide but not in the second on the same day has already been mentioned. The role of the D4 harmonic at Port Ellen is to suppress the formation of the double low tide, although it is quite likely that there will be a double low near Port Ellen cf. Southampton–Weymouth;. The sum of D2 and D6 harmonics tends to produce a double high tide and a double low tide (Fig. 1a). At Port Ellen, the minimum in D4 occurs at about the time of the minimum in D2 (see Fig. b). The D4 harmonic, though small, tends to flatten out the low tide and prevents the sixth diurnal harmonic (which has a maximum at this time) from producing a double low water. The opposite is true at high tide, when the minimum in D4 helps the minimum in D6 to produce a double high water.

The theory presented here has extended the original Doodson criterion and now includes the phase, ϕ, and it allows for b/a to be large. We have, however, not covered all situations, and we do not include phases larger than 2 h in Fig. 2. We also show that the phase can be the most important variable in producing double high waters. That is, b/a can be large enough but there is no double high because the phase prevents its formation. This has been overlooked in previous papers on the subject. Also, we have focused on double highs in the paper, but the theory is just as applicable for double lows, where the phase goes from ϕ to ϕ+180∘. There is most likely a double low in the vicinity of Port Ellen, but locating and quantifying it is left for a future study.

Overall, the new theory captures the double highs at Port Ellen very well. It shows that it is the D6 harmonic which is the dominant one at our location, rather than the usual D4. The formation of double high and double low waters in the semi-diurnal tide is a fascinating problem with important practical implications, e.g. to understand the differences between mean tide level and mean sea level see the discussion in.