OSOcean ScienceOSOcean Sci.1812-0792Copernicus PublicationsGöttingen, Germany10.5194/os-13-599-2017The double high tide at Port Ellen: Doodson's criterion revisitedByrneHannah A. M.https://orcid.org/0000-0001-6928-488XGreenJ. A. Mattiasm.green@bangor.ac.ukBowersDavid G.Bangor University, School of Ocean Sciences, Menai Bridge,
Anglesey, LL59 5AB, UKJ. A. Mattias Green (m.green@bangor.ac.uk)20July201713459960715March20177April20173June201721June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://os.copernicus.org/articles/13/599/2017/os-13-599-2017.htmlThe full text article is available as a PDF file from https://os.copernicus.org/articles/13/599/2017/os-13-599-2017.pdf
Doodson proposed a minimum criterion to predict the occurrence of double
high (or double low) waters when a higher-frequency tidal harmonic is added
to the semi-diurnal tide. If the phasing of the harmonic is optimal, the
condition for a double high water can be written bn2/a>1 where b is
the amplitude of the higher harmonic, a is the amplitude of the
semi-diurnal tide, and n is the ratio of their frequencies. Here we expand
this criterion to allow for (i) a phase difference ϕ between the
semi-diurnal tide and the harmonic and (ii) the fact that the double high
water will disappear in the event that b/a becomes large enough for the
higher harmonic to be the dominant component of the tide. This can happen,
for example, at places or times where the semi-diurnal tide is very small.
The revised parameter is br2/a, where r is a number generally less
than n, although equal to n when ϕ=0. The theory predicts that a
double high tide will form when this parameter exceeds 1 and then disappear
when it exceeds a value of order n2 and the higher harmonic becomes
dominant. We test these predictions against observations at Port Ellen in
the Inner Hebrides of Scotland. For most of the data set, the largest
harmonic of the semi-diurnal tide is the sixth diurnal component, for which
n=3. The principal lunar and solar semi-diurnal tides are about equal at
Port Ellen and so the semi-diurnal tide becomes very small twice a month at
neap tides (here defined as the smallest fortnightly tidal range). A double
high water forms when br2/a first exceeds a minimum value of about 1.5
as neap tides are approached and then disappears as br2/a then exceeds a
second limiting value of about 10 at neap tides in agreement with the
revised criterion.
Introduction
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.
The
semi-diurnal tide on a particular day, changing in amplitude from day to day
in this way is called the D2 tide . Similarly, the amplitude of
the higher-frequency oscillation, b, is also that observed on the day and
will be different on other days. If the higher frequency oscillation is a
harmonic of the semi-diurnal tide it is referred to as a D4 tide in the case
of a quarter diurnal oscillation, D6 if the period is about 4 h, and so
on. One way to increase the value of the ratio b/a in the Doodson criterion
is to reduce the amplitude, a, of the D2 tide. A good place to look for
double high waters, therefore, is near an amphidrome in the semi-diurnal
tide. In fact, Southampton lies close to the nodal line for the semi-diurnal
tide in the English Channel. Taking another example, in the Irish Sea, there
is a degenerate amphidrome for the semi-diurnal tide close to Courtown on the
Irish coast. Here, the semi-diurnal tide is small and the higher harmonics
can be the dominant components in the tide . Alternatively, in
places where the two principal semi-diurnal constituents – those of the sun
and moon – are about equal, the D2 tide will become small twice a month at
neap tides and any higher-frequency oscillations will assert themselves. In
South Australia, this phenomenon is called the dodge tide
. Similarly, double low waters will occur in locations
near areas with double high waters. This is because we expect double tides
near a semi-diurnal amphidrome (where the semi-diurnal amplitude is small).
Consequently, a short movement across the amphidrome, the phase of the
semi-diurnal tide will change by 180∘, and if the phase of the overtide
does not change, there will be a switch from a double high to a double low
tide.
These considerations about small values of a lead us to the conclusion that
there must be a second criterion for double high waters. As b/a increases
from the critical value given by Eq. (), a double high water
will first form and then, as b/a continues to increase, the double high
water will disappear as the higher harmonic becomes dominant. The Doodson
criterion in Eq. () for the production of double high tides
is therefore a necessary requirement but it is not, on its own, a sufficient
requirement, and it explicitly only deals with harmonics smaller than M2. To
develop a more general criterion we need also to impose an upper limit on the
ratio b/a to allow for times and places where the fundamental harmonic (in
Doodson's case, M2) is small compared to its harmonics. We can imagine that a
more general condition for double high waters might take the form
x>bn2a>1,
where x is a function to be determined. Our first aim in this paper is thus
to explore the nature of x.
A further restriction on the Doodson criterion is that it applies only to the
case where the phase of the higher harmonic is optimal – that is, in the case
of double high waters the minimum in the harmonic coincides with the maximum
in the semi-diurnal tide. suggested a modification to the
Doodson criterion to allow for a phase difference between the main tide and
its harmonic. A second aim of this paper is to take the opportunity to test
that idea against observations.
Theoretical background
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].
(a) A double high (and double low) water formed by the addition of a
semi-diurnal curve (dash-dot line, amplitude 1 m) and a sixth-diurnal harmonic
(dashed line, amplitude 0.25 m). There is phase difference of 0.5 h
between the curves, and time is measured from the peak of the semi-diurnal
tide. “D” marks the centre of the dip between the high waters. At D, the
gradient of y with respect to time is zero. (b) As (a) but the amplitude of the
semi-diurnal curve is reduced to 0.1 m. The double high and low waters
have disappeared and the tide is effectively sixth diurnal.
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.
Theoretical condition for the formation of double high waters
illustrated for the addition of a sixth diurnal (D6) harmonic to a
semi-diurnal D2 tide. The y axis is the ratio of the amplitudes of the
harmonic to the principal tide. The x axis is the time difference between the
minimum in D6 and the maximum in D2 (plus D1 in the case of the observations).
The solid curve shows the critical value of b/a for the formation of double
high waters according to the analytical solution (Eq. in text).
The dotted curve is an exact numerical solution to the problem. The points
show values of b/a and ϕ at Port Ellen. Points have been plotted as
solid circles for tides where there is a double high water, grey filled
circles where there is a stand in the tide near high water and open circles
when neither a stand nor a double high water is observed.
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.
Location of Port Ellen in the Inner Hebrides, Scotland. The open
circle on the channel marks the M2 amphidrome. The coastline was drawn using
GEBCO2014 seeand www.gebco.net, available from
https://www.bodc.ac.uk/data/hosted_data_systems/gebco_gridded_bathymetry_data/.
Observations
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.
Water level record for Port Ellen in the second half of February,
2010. Double high waters are seen on the mornings of 18 and 19 February and
the afternoons of 27 and 28 February.
Harmonic analysis
We have analysed short portions of the water level record for the amplitude
and phases of key harmonics using harmonic analysis see e.g.for
details. The method was first described by to
establish the fact that higher harmonics of the semi-diurnal tide were
important at Southampton. Later, used the technique to
test their inequality in Eq. () against observations of the
tide at Southampton.
Because the diurnal tide is important at Port Ellen, we have analysed the
data for harmonics starting with the diurnal tide as the first harmonic. The
data set of Fig. 4 was broken up into 25 h segments, each starting at
midnight on the chosen day. The selected segments of data therefore run for
1 h into the next day. The harmonic analysis is then applied to each
segment of data to calculate the amplitude and phase of a diurnal constituent
(D1, period 25 h), a semi-diurnal constituent (D2, period 12.5 h), a
quarter-diurnal constituent (D4, period 6.25 h), and a sixth-diurnal
constituent (D6, period 4.17 h). Note that this analysis produces
amplitudes and phases of the harmonics applicable to that day only. The
harmonics are therefore not the same as the tidal constituents such as M2 and
M4 treated by a full harmonic tidal analysis and which have constant
amplitude and phase on all days. The amplitude and phase of these daily
harmonics changes from day to day. The usefulness of the “D” harmonics has
been discussed by . It enables the relative importance of the
principal tide and its harmonics to be established on each day. The
relationship between the relative amplitudes and phases of the harmonics and
the production (or non-production) of double high waters can then be
explored.
(a) Observed and fitted tidal curves for 19 February. The double
high water on the morning is reproduced by the fitted curve. (b) D1 (blue), D2
(red), D4 (green), and D6 (purple) harmonics for 19 February. The high tide
in D1 (black line) occurs in the middle of the day and causes the diurnal
inequality seen in Fig. 4. The D6 harmonic (lightest grey) has a larger
amplitude than D4 and the minimum in D6 occurs closer to the high D2 than
does the minimum in D4. It is the D6 harmonic that is primarily responsible
for the double high water at Port Ellen.
Figure shows an example of the curve-fitting for 19 February,
whereas Fig. shows the daily amplitudes of the
harmonics throughout the observational period. Figure a shows
the observed and fitted curve for this day. The double high water on the
morning tide is clearly shown and the fitted curve reproduces this. There is
no double high water in the afternoon, in either the observed or fitted tide.
The fitted curve is not perfect, however, due to for instance meteorological
effects, seiches, and instrument noise. Most importantly, there is a double
low water in the fitted curve, which is not seen in the observations. Figure
5b shows the nature of the harmonics which sum to give the fitted curve on
this day. The semi-diurnal harmonic, D2, has the largest amplitude, followed
by D1, then D6, and finally D4. The low water in the D6 curve is close to the
high water in the D2 curve, a requirement for the formation of double high
waters. In contrast, the amplitude of D4 is smaller than D6 and its phase is
not optimal – the low in D4 occurs about 2 h after the semi-diurnal high
water in Fig. b (see also Fig. ).
The fact that there is a double high water in the morning of 19 February, but
not in the afternoon, can be explained by the effect of the D1 tide. The
diurnal tide is rising in the morning and falling in the afternoon (Figs. 5b
and ). This has the effect of pushing the time of high
water forwards in time in the morning, towards noon, and dragging it
backwards in time, also towards noon, in the afternoon. The time interval
between the two high waters produced by D2 and D1 together is therefore less
than 12.5 h. It is thus possible for the low tide in D6 to be close to
the high tide in D1 and D2 in the morning, but not so close in the afternoon.
In fact, the time interval between the D6 low and the (D1 + D2) high on
19 February is zero in the morning and 0.8 h in the afternoon. The close
coincidence of the low water in D6 and the high water in (D1 + D2) produces a
double high water in the morning but not in the afternoon. For this reason,
it is important to consider the time of the high water in D1 + D2.
The daily amplitudes for fits of D1 (blue), D2 (red), D4
(green), and D6 (purple).
The harmonic analysis illustrated in Fig. has been applied
to each of the days in the record and the results which we shall use in the
analysis are shown in Table 1. The phase of D6 is expressed as the absolute
value of the time interval between low water in this harmonic and the nearest
high water in D1 and D2 combined. The semi-diurnal tide virtually disappears
on 24 February, at neap tides, because of the equality in the amplitudes of the
solar and lunar tides.
The first column shows the day in February 2010 divided into two
tides, one in the morning (a.m.) and a second in the afternoon (p.m.). a is
the amplitude of the semi-diurnal tide in metres and is the same in the
morning and in the afternoon. b/a is the ratio of the amplitudes of the
sixth diurnal tide D6 to that of D2. |ϕ| is the absolute value of the
time difference in hours between the maximum in D1 and D2 combined and the
minimum in D6; t′ is the time interval calculated with Eq. (),
and r2 is the parameter in Eq. (). Finally, (b/a)r2 is the
discriminator for double high waters.
1 Tides in which a double high water is
observed. 2 Tides where there is a stand in the
tide, so that a double high water is close to being formed.
Testing the condition for double high waters
A full analysis of the conditions that produce a double high water would
include all of the relevant harmonics of the semi-diurnal tide. In the case
of the short record at Port Ellen, however, the mean amplitude of D4 is
0.026 m and that of D6 is 0.044 m. The fact that D6 is generally larger,
coupled with the fact that it is of higher frequency and so more potent at
producing double high tides, means that, if we are to consider just a single
harmonic, then that should be D6. The actual advantage for D6 is
(9/4)×(44/26)∼3.8, so we limit our analysis in this section to
the production of double high waters by a combination of D6 and D2 tides, and
use this combination to test the theory of Sect. 2. We bear in mind,
however, that the comparison between observations and theory may not be exact
because we are neglecting other harmonics, notably D4 and also D1. It can be
seen in Table 1 that on all days the ratio of amplitudes b/a for D6 and D2
tides is greater than the value of 0.111 required to satisfy the condition (1).
Moreover, the value of b/a is nearly as great on the afternoon of
24 February (0.223, when no double high water is seen) as it is on the afternoon of
the 28 February (0.224, when there is a double high water). The straightforward
Doodson criterion, neglecting the effect of phase difference between the
harmonic and the principal tide, is therefore not the best discriminator for
double high waters. The rows in Table 1 have been marked for tides where a
double high water is observed, or nearly so. Tides in which a double high
water is observed are shaded in the darker grey and those where there is a
stand in the tide, so that a double high water is close to being formed are
marked in lighter grey. On days when there is a clear double high water, that
is on the morning of 18 and 19 February and on the afternoons of 27 and
28 February, the value of (b/a)r2 is greater than or equal to 1.5. On days when
there is a stand in the tide close to high water, (b/a)r2 is greater than
about 0.5 and less than 1.5. On all other days, when there is no double high
water (excepting 25 and 26 February) the value of (b/a)r2 is less than 0.5.
The amended Doodson condition therefore performs well as a predictor of
double tides, although the transitions are not as clear-cut in the
observations as theory would predict. This is likely to be partly because of
difficulty in deciding when a double high water is there by visual inspection
alone and partly because the real Port Ellen tide has harmonics other than
the D2 and D6 components treated in the theory, as well as other sea-level
processes and sampling noise being present.
On 25 and 26 February, (b/a)r2 reaches its highest values,
over 11 on 25 February and a more modest level of over 2 on 26 February. There is
no clear double high water on these days, though. Instead the tide is
confused with no clear semi-diurnal pattern (Fig. 4). Closer inspection of
the tides on these days (using plots equivalent to Fig. )
shows that on 25 February the diurnal and quarter diurnal tides are dominant –
there is certainly no double high water in the semi-diurnal tide on this day.
The situation on 26 February is more complicated. The diurnal tide is the
largest component on this day. If the diurnal tide is removed from the
reconstruction and a tidal curve is drawn as the sum of the D2, D4, and D6
harmonics, then a double high water can be seen in both the morning and
afternoon. The predictor is therefore doing a reasonable job, within its
limitations, throughout the fortnight of the observations. When the parameter
(b/a)r2 has a value less than about 1, no double high water is formed
because the higher harmonic is too weak. When it has a value greater than
about 10, then again no double high water is formed because the higher
harmonic is too strong. Between these limiting values, double high waters, or
at least a tidal stand, are likely to be observed, provided they are not
obscured by the presence of other harmonics.
We have plotted in Fig. 2 values of b/a for the D2 and D6 tides (from
Table 1) against the phase difference ϕ (also from Table 1). The points
have been coded so that those on tides in which a double high tide is
observed are shown as filled circles and those on which no double high tide
is observed are shown as open circles. Tides where there is a point of
inflection, or a stand in the tide, have been marked as grey-filled circles.
The tides of 25 and 26 February, when the semi-diurnal tide is very small,
have been omitted from this diagram. Because Fig. 2 shows a graphical
solution to the inequality b/ar2>1 the inequality is satisfied and we
expect a double high water to form if we are above the curves (incidentally,
the same is true for double low waters; ϕ then becomes the time
difference between the D2 low and the Dn high). This
is very much the case – all four clear double high waters plot above the
theoretical transition curve. Similarly, points representing the tides with
no double high water lie below the critical curve. The grey points,
representing tides with a stand, lie close to the critical lines. It is
illuminating to note that the main spread of the points in Fig. 2 is along
the x axis. Because the amplitude of harmonics generated by the
semi-diurnal tide tend to increase with that of D2, the ratio b/a remains
fairly constant, and Fig. confirms that D6 depends on D2 but
D4 does not. The critical factor in deciding whether a double high tide will
form is actually the phase difference. As the time of the dip in D6 moves
close to the high tide in D2, the critical condition for a double high tide
is met. Looking at Fig. 2, most of the spread of the points is along the
ϕ axis – the variation of b/a is relatively small (because b is
proportional to a for D6 and D2). Consequently, what brings a point across
the critical line, in practice, is a change in the phase of the constituents.
Although the theory has it that both phase and amplitude ratio are important,
in practice for this data set, the phase difference between the harmonics is
the most important parameter in controlling the formation of a double tide.
Discussion
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.
0.5≤(b/a)r2≤∼1.5: a stand in the tide is observed,
but no clear double high water.
1.5≤(b/a)r2≤∼10: the regime of double high waters.
(b/a)r2>10: the harmonic dominates and there is again no
clear double high tide.
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.
Conclusions
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.
The sea-level data were provided by the UK National Tide
Gauge Network through the National Tidal and Sea Level Facility
(http://www.ntslf.org/), sponsored by the UK Environment Agency. The
data for Port Ellen can be downloaded from the BODC archive at
https://www.bodc.ac.uk/data/hosted_data_systems/sea_level/uk_tide_gauge_network/processed/.
The authors declare that they have no conflict of
interest.
Acknowledgements
Funding was provided by the Natural Environmental Research Council (grant
NE/I030224/1 to JAMG), and from the BurningHam research support foundation.
Constructive comments from Phil Woodworth and two anonymous reviewers greatly
improved the manuscript. Edited by:
Neil Wells Reviewed by: Philip Woodworth and two anonymous
referees
References
Airy, G. B.: On the laws of individual tides at Southampton and at Ipswich,
Philos. T. R. Soc. Lond., 133, 45–54, 1843.
Bowers, D., Macdonald, R., McKee, D., Nimmo-Smith, W., and Graham, G.: On the
formation of tide-produced seiches and double high waters in coastal seas,
Estuarine, Coast. Shelf Sci., 134, 108–116, 2013.Doodson, A. T. and Warburg, H.: Admiralty manual of tides, Her Majesty's
Stationery Office, London, 1941.
Emery, W. J. and Thomson, R. E.: Data analysis methods in physical
oceanography, 2nd Edn., Pergamon Press, 650 pp., 1996.
Godin, G.: An investigation of the phenomenon of double high water or double
low water at some harbours, Deutsche Hydrographische Zeitschrift, 45,
87–106, 1993.
Nunes, R. and Lennon, G.: Physical property distributions and seasonal trends
in Spencer Gulf, South Australia, an inverse estuary, Mar. Freshwater Res., 37, 39–53, 1986.
Pugh, D.: A comparison of recent and historical tides and mean sea-levels off
Ireland, Geophys. J. Roy. Astr. S., 71, 809–815, 1982.
Pugh, D. T.: Tides, surges and mean sea level: a handbook for engineers and
scientists, Wiley, Chichester, UK, 1987.
Pugh, D. and Woodworth, P.: Sea-Level Science: Understanding tides, surges,
tsunamis and mean sea-level changes, Cambridge University Press, Cambridge,
UK, 2014.
Redfield, A. C.: The tides of New England and New York, Woods Hole
Oceanographic Institution, USA, 1980.
Weatherall P., Marks, K. M., Jakobsson, M.,
Schmitt, T., Tani, S., Arndt, J. E., Rovere, M., Chayes, D., Ferrini, V., and Wigley,
R.: A new digital bathymetric model
of the world's oceans, Earth Space
Sci., 2, 331–345, doi:10.1002/2015EA000107, 2015.
Woodworth, P. L.: Differences between mean tide level and mean sea level,
J. Geodesy, 91, 69–90, 2017.