Geophys. J . R . a m . SOC.(1981) 67,515-527
Tidal amphidrome movement and energy dissipation
in the Irish Sea
D.T. PU&
Institute of Oceanographic Sciences, Bidston Observatory,
Birkenhend, Merseyside L43 7RA
Received 1981 April 7 ; in original form 1980 December 11
Summary. The positions of points of zero tidal range are normally plotted as
the amphidromes of individual harmonic tidal constituents on a series of
cotidal and coamplitude charts. However, the amphidrome may also be considered as a time-dependent position of zero tidal range for a complete tidal
band. Daily movement of the semidiurnal tidal amphidrome in the southern
Irish Sea, though a spring-neap cycle, extends over 70 km in a direction
perpendicular to the entrance channel axis. At spring tides the amphidrome is
degenerate, being located inland, but at neap tides the position moves into
the Irish Sea. This movement is due to proportionally more energy being
absorbed from spring than from neap tides. The area-averaged power law of
energy dissipation is approximately cubic for small tidal ranges; however, it
reduces towards a square law for large ranges, because dissipation is limited
by the available inward energy flux. The model applied can be extended to
explain why anomalously weak spring tidal currents occur in parts of the
Celtic Sea.
Introduction
In his report on a series of coordinated sea-level measurements made at 22 sites around
Ireland in 1842, Airy (1845) noted that at Courtown, County Wexford, ‘The tide was sometimes apparent as a semidiurnal tide, but with considerable irregularity; at other times the
character of semidirunal tide was . . . completely lost, and in its stead there was a small tide
four times a day; in all cases the tide was small’. Recent observations (for example, Fig. 1)
confirm the unusual nature of the Courtown tides. During the period illustrated, the range
varies from near zero to more than 1.4 m. Overall, the mean range on spring tides (2 (M2 +
s2)) is 0.66 m, compared with the mean neap range (2 (M2 - S2)) of 0.02 m. By contrast,
the tides at Arklow, 17 km to the north, follow the usual pattern of spring-neap modulation
for the Irish Sea, with spring ranges twice those at neaps.
The semidiurnal tides of the Irish Sea have been described (Taylor 1919) in terms of the
superposition of two progressive waves, one entering through St Georges Channel, and the
other a reflected wave travelling out through the same Channel. These combine to give small
516
D.T.Pugh
2Imtrn
0
-2
-
1
215
I
220
I
225
I
2x)
I
235
M Y No 1979
Figure 1. Twenty days of sea-levels at Courtown, 1979 August. The range of the semidiurnal tide varies
f r o m 1.4 m t o near zero. Fourth-diurnal non-linear effects are particularly obvious when the range is
small.
tidal ranges north of St Georges Channel, characterized on M2 and S2 cotidal charts
(Robinson 1979) by degenerate amphidromes. For the M2 tide this degenerate amphidrome
is located some 40 k m inland from the Irish coast at 53" N. In this paper we investigate the
implications of the behaviour of the Courtown tides for the daily movement of the semidiurnal amphidrome, and consider the physical reasons why the amphidrome moves as the
tidal range changes.
Observation and analysis
Harmonic analyses were made of hourly sea-level data, measured by the Institute of Oceanographic Sciences at Wicklow, Arklow, Courtown, Rosslare and Aberystwyth (Fig. 2). Table 1
summarizes the positions of these stations, and the lengths of data analysed. These analyses,
including all the shallow-water constituents, were then used to generate a tidal synthesis for
the common period, days 255 1978 to 247 1979.
A least-squares daily harmonic analysis was then made at 25 hourly values from each midnight to t h e following midnight to determine the amplitude ( H ) and the phase ( G ) of the
semidiurnal tide at each of the five stations and for each day. It is convenient to call the
composite harmonic so determined D2, for consistency with the normal tidal nomenclature.
The values o f H and G are not constant from day t o day, because the range varies, and the
G values, which are related to midnight of each solar day, advance on average 24.4" per day
because of the dominance of the lunar M2 constituent.
The values of H and G at the five stations formed a set from which the amphidrome
position on each day could be estimated. This was done by fitting least-squares planes to the
surface H sin(G) and H c o s ( G ) , which themselves represent the sea-levels at a quarter tidal
period separation, and then solving the equations of the two lines of zero vertical displacement for their coordinates of intersection. Fitting of a simple plane may be justified
qualitatively by reference to the sea-surface contours plotted by Mungall & Matthews
(1978), and quantitatively by the low average residual standard deviation of 0.07 m obtained
from the fits.
The amphdrome position moved regularly back and forth within the narrow area shaded
in Fig. 2. This movement, which is shown in more detail in Fig. 3 relative to the Eire coast,
reached its maximum westward displacement during spring tides in the Irish Sea, and its
517
Tidal amphidrome movement in the Irish Sea
550
5 3c
51'
Figure 2. Map of the Irish Sea showing the positions of the tidal stations, the coordinate system adopted,
and the region within which the amphidrome tracked.
Table 1. Locations and periods of data analysed to produce the constants for tidal synthesis and amphidrome movement.
Wicklow
Arklow
Courtown
Rosslare
Aberystwyth
Latitude
Longitude
Days
analysed
Observation
period
Total
constituents
5 2 59.5"
5248"
52 39' N
52 15"
52 25"
6 03'W
6 09'W
6 13'W
6 21'W
4 055'W
35 9
35 9
35 1
3 19
53
255178-248179
255 / I 8 -248119
255/18-240179
21 l/12-253/13
238172-290172
103
103
103
60
34
maximum eastward displacement during neap tides. The range of this east-west movement
exceeded 70 km, whereas the north-south movement was only 14km. Also marked on Fig. 3
are the positions of the M2, S2 and N2 amphidromes determined from the H and G values
for these individual semidiurnal tidal constituents from the long-period analyses at the five
sites. The S2 amphidrome is displaced inland from the M2 amphidrome by 40km. This
518
D.T.Pugh
I
IRISH
i
Km
60 -
EIRE
ROSSLARE
80-
i
Figure 3. Detailed movement of the semidiurnal amphidrome within the box shown in Fig. 2. The
positions of the M 2 , S2 and N2 individual amphidromes are also plotted.
displacement is also implicit in the best available cotidal charts (Robinson 1979). For neap
tides the daily amphidrome is real, that is, zero tidal range occurs at a point within the Irish
Sea; however, on spring tides this point is degenerate and lies up to 40km inland.
A simple theoretical model
In order to understand physically why the amphidrome should move in this way, it is useful
to return to Taylor’s idea of the tidal behaviour being represented as the sum of ingoing and
outgoing waves and to develop a simple physical model on this basis. Taylor (1921) showed
that for a semi-infinite rectangular channel on a rotating earth, in the absence of energy
losses, the elevation amphidromes due to ingoing and outgoing Kelvin waves lie along the
central axis, at half-wavelength intervals, with the first amphidrome a quarter wavelength
from the reflecting boundary. In the case of the Irish Sea, he also considered (Taylor 1919)
the amplitude ratio between the ingoing and outgoing waves in St Georges Channel, and
estimated, from the rate of tidal phase progression, that the reflected, outgoing wave at
spring tides had only half the amplitude of the ingoing wave. Further theoretical analyses of
this problem have been made by Brown (1973) and Rienecker & Teubner (1980).
For an ingoing Kelvin wave, using the coordinate system shown in Fig. 2, the elevations
are given b y :
b @,Y,0 = bo exp (-fY/&)
cos W ( f
-X
l V m
where f is the Coriolis parameter, g the gravitational acceleration, h the channel depth,
assumed uniform, and w is the angular speed of the tidal oscillation. If this wave is reflected
and has a returning amplitude attenuation factor a,that is all the energy loss is assumed to
take place at the reflecting boundary, then for the combined two wave system and for the
first amphidrome, the coordinates are:
Tidal amphidrome movement in the Irish Sea
519
This result is used by Hendershott & Speranza (1971) in their study of M2 tides in the
Adriatic Sea and the Gulf of California. They also consider the circumstances in which the
propagation of tidal energy in the form of Poincari waves is significant. Poincark waves are
the other form of wave motions, apart from Kelvin waves, which satisfy the hydrodynamic
equations for tidal propagation on a rotating earth. For a basin with the width and depth of
the Irish Sea, the Poincari modes decay exponentially away from the reflecting boundary
with an exp(- 1) decay distance of some 50 km. Therefore, they need not be considered for
the model in St Georges Channel, some 300km from the head of the Irish Sea, where the
imperfect reflection is assumed to occur.
Another potential error in the assumption that the energy in the outgoing wave consists
solely of the reflected part of the ingoing energy, is the possibility of additional energy flux
through the smaller North Channel entrance to the Irish Sea. However, Taylor (1919), and
more recently Robinson (1979), have shown from current and elevation observations, that
this flux is small compared with that through St Georges Channel.
The formulae for x a n d y amphidrome displacement have the advantage of separating the
effects of the two physically variable quantities, the angular speed, w, and the amplitude
attenuation coefficient, a,in a way which means that the effects of each on the amphidromic
movement are orthogonal and so may be considered independently. To apply these
formulae, however, the line and direction of the central axis of the channel must be specified
in terms of the physical model. The coordinate system shown in Fig. 2 was determined by
adjusting the central axis until extrapolating the curve of amphidrome y-displacement
plotted against Heysham tidal range, gave zero y-displacement for zero tidal range - a
necessary condition giving no displacement when no energy is lost. The alignment of the axis
was less rigorously determined, by inspection of the general shape of the Irish Sea, as being
along a direction 187" T , as shown in Fig. 2.
Amphidrome-displacement related to angular speed
If the spring-neap tidal cycle is represented as the sum of two harmonics, M2 and S2,
denoted by the suffues 1 and 0 below, having angular speed 28.984" hr-' and 30" hr-', then
elevation changes may be written as a single modulated harmonic variation (Lamb 1932,
section 224) :
{ ( t )= c cos (w1t -6
-€)
where o1is the speed of the M2 constituent, e is a constant phase angle, C is a modulated
amplitude:
C=(~t2H1Hocosh+H~)"2
HI and Hoare the M2 and S2 amplitudes, h is an angular argument with speed (ao-w J and
hence with a period of 14.76 day, and 0 is a slowly changing phase angle:
which also varies over a 14.76 day period. Because of this modulated phase, the speed of the
composite semidiurnal tide:
D. T. Push
520
varies between:
Hi
WI
+Ho 00
HI - HO
For Irish Sea tides the ratioH,,/Hl is normally 0.33, giving a variation in the composite w
from 29.236" h f ' at spring tides to 28.246" hr-' at neap tides. For a mean depth of 60 m,
t h e corresponding spring-neap displacement of the first amphidrome was 7 km. The
computed absolute displacements were 269-276 km from the reflecting boundary. The
inclusion of other semidiurnal constituents, notably N2, would extend the range of w , and
hence of t h e amphidrome displacement, beyond these limits. The actual range of displacement in the coordinate system applied here, is nearly 20 km, and the maximum displacement
to the south occurs on spring rather than on neap tides. However, both the range and phase
o f the x-displacement are very sensitive to the direction chosen for the channel axis and this
could not be accurately identified with the physical model. The observations could be
reconciled with the theory by aligning the coordinate axis along a direction of 180"T,
implying that the effective reflecting boundary in the Irish Sea lies further west than
anticipated.
Amphidrome y-displacement related to attenuation factor a
The y-displacement, according to the simple model proposed, is proportional t o the
logarithm o f the attenuation factor. Fig. 4 shows the variation of this attenuation factor
computed from the y-displacement, during the range changes of the spring-neap tidal cycle,
as represented by the daily semidiurnal tidal range at Heysham. Heysham was chosen
KELVIN
WAVE
REFLECTION
COEFFICIENT
M2 4 2
M2
SEMlDlURNAL TIDAL AMPLITUDE
M2 +s2
AT HEYSWM
(m)
Figure 4. Variation of the Kelvin wave reflection coefficient ( 0 ) as a function of the semidiurnal tidal
range at Heysham, computed from the displacement of the amphidrome. (Y decreases rapidly as the tidal
range increases.
Tidal amphidrome movement in the Irish Sea
52 1
because it is near the reflecting boundary of the Irish Sea. This observed attenuation factor
varies between 0.60 and 0.72 on neap tides (M2 - S 2 , at Heysham) and between 0.43 and
0.47 on spring tides ( M 2 t S2, at Heysham). This spring tide attenuation factor compares
favourably with the 0.5 estimated by Taylor.
The reason for the observed amphidrome displacement lies in the variation of this factor
during the spring-neap cycle. This in turn, is due to the non-linear processes of energy
dissipation in the Irish Sea, so that proportionally more energy is removed from spring tides
than from neap tides. The implications of this variation in (Y for the laws of energy dissipation are discussed in the following section.
Energy dissipation related to tidal range
For progressive waves such as the Kelvin waves we consider here, the average energy flux at
position x through a channel of width 2a and depth h ( h s {), in our physical model is
(Taylor 19 19) :
p g h j r o r n r(x,Y,f)~(x,Y,f)~f~Y
0
where {(x, y , t ) is the Kelvin wave elevation as previously defined, and u ( x , y , t ) is the
corresponding current which, for a progressive wave, is related to the elevations:
u(x,.Y,t) = r ( X J , f ) ( g / V 2 .
This integral is readily evaluated in terms of the channel dimensions and other physical
constants, but for the following arguments it is sufficient to note that the average flux may
be written as:
average flux through a section at x = $c2(x, 0)
where (J is a constant in terms of the other physical variables. For our model, taking a crosschannel section through the first amphidrome, and assuming that the energy fluxes of the
ingoing and outgoing Kelvin waves may be added arithmetically (h s {, particularly in an
amphidromic region), the net energy flux into the Irish Sea is:
where tosignifies the elevation amplitude of the ingoing Kelvin wave at the channel axis.
The factor (1 -a2) represents the proportion of tidal energy removed from the ingoing
wave. It ranges from 0.78 to 0.82 at spring tides and from 0.50 to 0.65 at neap tides. When
the daily semidiurnal range is equivalent to that of the M2 tidal constituent the proportion
of energy absorbed is 0.65-0.75. For these comparisons, the value of C0 at the channel axis
is not available (measurements of currents and elevations would be necessary to separate
hgoing and outgoing waves), and so the range at Heysham has again been used. Heysham is
well removed from the amphidromic region under discussion, and to a first approximation
it probably has a tidal range which is linearly related to
On the basis of this assumption
the $ factor may be redefined to include this proportionality.
The total energy absorption by the Irish Sea may then be written:
c0.
4(1 -a’) {&
where {H is the amplitude of the semidiurnal tide at Heysham. The average value of the
energy absorption, related to the energy absorption for an M 2 tide, is of interest for relating
522
D. T. Pugh
I0
20
30
40
50
YMlOlURNAL TIDAL AMRITUOE AT W W A M (ml
Figure 5. Relative variation of the total energy absorption by the Irish Sea Sh (1 -a’) as a function of the
semidiurnal tidal range at Heysham. The units are not absolute.
the dissipation of energy within M2 tidal models to the total average dissipation. From the
determined values of a,the average value of Irish Sea tidal energy dissipation over the complete year o f computations was a factor of 1.33 greater than the dissipation on a mean M2
tide. Jeffries (1976) gives formulae which predict a factor of 1.24 for a combination of M2
and S2 alone, when the M2 amplitude is three times the S2 amplitude. Fig. 5 shows the
variation of the absorption factor with tidal range, and indicated a ratio of 19 between
extreme dissipation rates.
The enhanced absorption of energy at larger tidal ranges is again characteristic of a nonlinear physical system. Suppose that, averaged over a tidal cycle, this nonlinear absorption is
expressed as p{h, where 0 is a constant involving the current drag coefficients, the area over
which dissipation takes place, and the relationship between toand tH;n is the power law of
t h e loss process. The balance between the energy flux into the Irish Sea and the sum of the
absorbed and reflected energy gives:
1 -- a’ = (P/@) {h-
2,
Hence, t h e gradient of a plot of log (1 - a’) against log CH will give the value of (n -- 2 ) . Fig.
6 shows this relationship and indicates that (n -2) varies during the spring-neap cycle.
Turbulence theory suggests a quadratic law of bottom friction, which gives a cubic law for
energy dissipation, because of the distance moved against this force of bottom friction in
unit time. The corresponding gradient for n = 3 which is plotted in Fig. 6, shows that this law
is approximately true for the small tidal ranges of neap tides, but that the slope gradually
reduces towards n = 2 (horizontal in Fig. 6) at larger tidal ranges. This difference between a
macroscopic determination of n and the smaller scale local determinations of n , needs
further examination.
Consider the propagation of a Kelvin wave along a narrow channel of width 2a. Averaged
over a tidal cycle, the energy flux through a section o f x is:
Ywg (gh)”’ C:
j;aexp [-2fYl(gh)”* 1 dY =
4y
r:.
The difference between the flux throughx and x t Ax is:
Tidal amphidrome movement in the Irish Sea
03
I
523
04
MZ-Sz
M2
M2 'S2
LOG,* (DzI
-re
6. Logarithmic plot of the energy absorption factor (1 -a') against the semidiurnal tidal amplitude
at Heysham. A slope of 1 .O, as plotted, corresponds to a cubic law of energy dissipation for the Irish Sea.
However, for larger ranges the slope tends to zero, corresponding to a squared law of regional dissipation.
while the energy lost due to dissipation is: 03;" Ax, where /3 involves the channel width and
depth. m is the local power loss law.
Hence, for steady state energy balance conditions:
Integrating between the limits x1 and xz, where {, = f1 at x1 and tX= a f l at x, gives, for
m z 2:
For m > 2 this becomes smaller as (xz-xl) and t1increase. The energy loss betweenx, and
x2is:
E=4y,5:(1 - a').
Substituting for a,for small energy losses ((xz - xl) small or tl small), i.e. a = 1,
8=P
:!r
(X2
--x
1)
but for large energy losses ((x, -xl) large or
E = 4y
large), a tends to zero, and
r:.
For large energy losses the dissipation is governed by the available flux of energy, which
varies as {:, while for small losses the local (cubic) dissipation law applies.
In the Irish Sea, if the reflector is assumed perfect, and is located at $4 (xl +x,), then the
channel theory may be applied with b o t h x and x2 at the amphidrome position, but for the
524
D. T. Pugh
ingoing and outgoing waves respectively. At neap tides the relatively weak dissipation leads
to a cubic law, whereas the increase in t1on spring tides gives a tendency towards a squared
dissipation law, as t h e energy lost is being limited by the flux available. This is as observed in
Fig. 6.
Anomalous amplitude ratios for tidal constituents
One consequence of the variable reflection factor is that the amplitude ratio between the
individual solar and lunar semidiurnal constituents, S2 and M2, varies over a significant area
in the vicinity of t h e amphidrome. At Courtown the ratio is 1 .O, compared with the normal
ratio for t h e Irish and Celtic Seas of 0.33.
Heuristic argument suggests that along the line of amphidrome movements the ratio
should be close to unity around the extreme displacement positions, where the range falls to
zero once in each spring-neap cycle. Also, in seas where the spring amphidrome is not
degenerate, the smallest semidiurnal tides of the spring-neap cycle near that amphidrome
position will occur when the rest of the seas are experiencing spring tides. This will give an
anomalous tidal ‘age’ - the time between maximum gravitational tidal forcing and maximum
observed tidal range - to the local tides. At intermediate places there will be two periods of
zero semidiurnal tidal range, and two of maximum range during the spring-neap cycle. There
is a n analogy here with the annual variations of solar radiation in the tropics. Harmonically
this peculiar tidal behaviour will be represented by enhancement of the shallow-water nonlinear harmonic terms which beat with M2 and S2 over a period of seven days, notably 2SM2
and 2MS2 respectively. At Courtown the constituent 2MS2 has an amplitude 56 per cent of
t h e M2 amplitude; 2SM2 has an amplitude 7 per cent of that of M2.
The extent of the anomalous S2/M2 amplitude ratios along the axis of the channel has
proved difficult to determine from the analytical superposition of two modulated Kelvin
waves. As an alternative, a numerical addition of the two Kelvin waves was generated by
computer, using the observed 01 variations, and analysed harmonically. The computed values
of the S2/M2 ratio for elevations along a line displaced 60km to the west of the central
axis, and with depth and Coriolis parameters appropriate for the Irish Sea are plotted in
Fig. 7. This ratio exceeds 0.40 over a range of more than 70 km, showing that the influence
o f amphidrome movement extends far beyond the actual area of tracking, affecting tides
over a substantial part of the channel.
The effects of an elevation-dependent reflection coefficient also extend to the currents in
the channel. Fig. 7(b) shows that for currents, the S2/M2 ratio is normal in the region of the
elevation amphidrome, one quarter-wavelength from the reflecting boundary. However, at a
distance of a half-wavelength from the reflecting boundary, there is a region of weak
currents and large elevations: this is the current equivalent of the elevation amphidrome. As
a consequence of the variation of 01 with tidal range, enhanced S2/M2 ratios in the currents
are indicated for the region to the right of the channel axis. Fig. 7(b) shows theoretical ratios
in excess o f 0.4 extend over 100 km along the direction of the channel axis, at a displacement of 60 km. Application of this simple physical model to the Irish and Celtic Seas is
complicated by the additional influence of the Bristol Channel and the English Channel on
the behaviour of the Celtic Sea tides. However, anomalously high S2/M2 ratios have been
observed in the Celtic Sea, particularly, towards the Irish Coast around Cork, as would be
predicted b y the theory. Table 2 summarizes the amplitude of the M2 and S2 components
in the major axis of the current ellipses, at five stations (M. J. Howarth, private communication). Station E, which is closest to the expected ‘current amphidrome’ has both weak tidal
currents and an anomalous S2/M2 ratio.
Tidal amphidrome movement in the Irkh Sea
525
i i LYITiOnS
/("TI,
DISThNCE
FROM RFKLECTING
C W S T IXm)
0,
pipre 7. Numerically computed variations in M2 amplitude (-) and S2/M2 amplitude ratios (---) for
elevations and currents as a function of distance from the reflecting coast ( X = 0.0 km); Y = 60.0 km;
channel depth= 60.0 m ; variable reflection coefficient =0.65 (springs), =0.45 (neaps); input S2/M2
amplitude ratio=0.33. Values are computed numerically for the sum of ingoing and outgoing Kelvin
waves. Note the region of anomalous ratios at quarter-wavelength (elevation) and half-wavelength
(currents) displacement.
Table 2. Amplitudes of the maximum M2 and S2 tidal currents at five stations in the Celtic Sea, measured
by 10s in 1978 May.
Maximum amplitude
(m s-')
Ratio
Station
Latitude
Longitude
M2
s2
S2/M2
B
51 45"
51 19"
51 27.5"
50 32' N
40 37' N
6 38'W
6 35'W
150'W
7 29'W
8 35.5'W
0.325
0.238
0.194
0.310
0.328
0.113
0.103
0.094
0.128
0.128
0.342
0.433
0.484
0.412
0.390
C
E
F
G
Discussion
Movements of the semidiurnal tidal amphidrome at the southern entrance to the Irish Sea
have been computed from coastal tidal measurements, and used in conjunction with a simple
physical model t o interpret other aspects of the tidal dynamics of the Irish and Celtic Seas.
The simple physical model cannot reproduce the fine details of their behaviour to the extent
that a detailed numerical model can, but it can help the physical understanding of the
D.T.Pugk
526
processes involved. Also, consideration of the daily changes in the behaviour of the composite semidiurnal tide, rather than the separate behaviour of the individual harmonic components, has enabled the identification of the physical reasons for the generation of some
non-linear terms (2MS2,2SM2) in the tides.
Future experimental investigations of amphidrome movement should include a more
regular network of coastal level and offshore pressure gauges in the vicinity of amphidrome
movement. In particular, there should be an offshore pressure gauge and current meter
station at t h e point where the amphidrome would intersect the channel axis, so that the
relative amplitudes o f the ingoing and outgoing waves could be determined directly. The use
of Heysham as an independent indicator of the amplitude modulations of the tides in the
Irish Sea was not ideal, as the Heysham amplitude must be affected by attenuation of the
ingoing wave during transit. A proper determination of {, would avoid this difficulty and
probably remove some of the hysteresis observed in the plots of a against f H . If the data set
is collected simultaneously, then the influence of meteorological surges on the amphidrome
displacement, evidence of the tide-surge interaction, could be investigated. In this paper,
only the movements generated by tidal syntheses have been considered because the observations were not simultaneous.
Application of the model to other regions is feasible. For example, in the English
Channel, where a degenerate semidiurnal amphidrome exists to the north, near 2" W,the S2
amphidrome should be located even further inland than the M2 amphidrome. In this way, as
in the case of the Irish Amphidrome, the daily movement of the amphidrome is such as to be
further inland when M2 and S2 are in phase. Also, a region of anomalous S2/M2 current
ratios is predicted to the south of Cornwall.
These computations have also shown that the proportion of incident tidal energy (1 -az)
which is absorbed in shallow seas varies considerably through the spring-neap cycle. The
proportion reflected back is greater for neap tides than for spring tides. This has important
implications for ocean tides and their numerical modelling. Cartwright (1977) has summarized
the need for an energy radiation condition at the ocean shelf boundaries:
ku,
= X{
where k is t h e ocean depth, u, the current component normal to the boundary, and { is the
oceanic tidal amplitude. h is a complex parameter which allows for only a proportion of the
incident energy being absorbed. From these considerations it appears that h is not constant,
but dependent on f , and that allowance should be made for this variation. Removal of
proportionally more energy at springs than at neaps also implies an enhancement of the
energy, and hence the amplitude, of the ocean tides at neaps, relative to those at springs,
which in turn would be represented harmonically as a reduction in the S2/M2 amplitude
ratio from the 0.46 ratio of the equilibrium tide.
Acknowledgments
The programs to compute the movements of the tidal amphidrome were developed and run
b y Mr C. Loller of Liverpool Polytechnic. I am grateful to colleagues at Bidston who contributed to the data collection and with helpful discussion. Dr J . Huthnance, in particular,
helped to clarify the relationship between local and regional laws of dissipation. This work
was funded in part b y the Department of Energy.
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Tidal amphidrome movement in the Irish Sea
527
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