Ceophys. J. R. astr. SOC.(1978) 53,541-552
The quarter-diurnal tide in the English Channel
James Hamilton*
Department of Oceanography, University of Southumpton
Received 1977 October 25;in original form 1977 August 9
Summary. The English Channel is modelled by a simplified geometry which
enables a quasi-analytic solution to be obtained for the semidiurnal tide.
This solution is then used to calculate the forced components of the quarterdiurnal tide from the non-linear terms in the equations of motion.
Introduction
The English Channel is noted for the large amplitudes of the so called shallow-water components of the tide, the quarterdiurnal (period approximately six hours) being the most
important.
Southampton in particular has both strong quarterdiurnal and sixthdiurnal components
and these latter have been attributed by Macmillan (1964) to ‘modifications brought about
by the existence of two entrances to the Solent’.
Although the two entrances to the Solent may produce unusual frictional effects which
may account to some extent for the large sixth-diurnal components (the origin of this
component is mainly frictional (Gallagher & Munk 1971)), the two entrances of themselves
will not produce quarterdiurnal components. Indeed these components are found not only
in the Solent, but also throughout the Channel. Macmillan’s (1964) explanation of the
presence of the quarter-diurnal constituents in terms of an ’unstable nodal area’ in the Isle
of Wight and the Cherbourg peninsular, is however not sufficiently precise to be meaningful.
The general mechanism by which for example the energy in the M4 (quarterdiurnal)
component which has no astronomical source may be generated from the equivalent
dominant semi-diumal component (the M2) was demonstrated by Lamb (1932) and may be
described as a simple firstader non-linear interaction of the M2 component with itself.
Thus Gallagher & Munk (1971) have been able to calculate the amplitudes of the shallowwater constituents generated in a onedimensional canal of fixed length by incident astronomical components.
However, their calculations are of little use in the strongly two-dimensional situation
of the English Channel, where the interplay of the Coriolis effect with the configuration
Present address: Department of Applied Mathematics, University of Waterloo, Ontario N2L 3G1,
Canada.
542
J. Hamilton
may be expected significantly to modify the amplitudes and phases of the M4 and M2
components vish-vis the twodimensional theory.
We could of course write a numerical model for the Enghsh Channel as a whole, solving
the full non-linear equations and performing a spectroscopic analysis of the result to separate
the semidiumal and quarter-diurnal constituents. This procedure however, would tend to
obscure both the mechanism by which the quarter-diurnal constituents are generated and
any simple explanation for the patterns of M4 and M2 obtained. The complicated geometry
might also make it difficult to separate local effects from effects due the the overall configuration.
We instead combine the method of Lamb (1932) for calculating the M4 constituent from
the M2 with the collocation method of Brown (1973) for obtaining a quasi-analytic solution
of the (linearized ,vertically-averaged) equations in a rectangular basin (avoiding discretization errors in solving the equations). A rectangular basin is evidently not a good
representation of the English Channel, however we are able to generalize Brown’s result (see
also Taylor 1920) to the case of a wedge-shaped channel with parabolic depth variation from
the apex. This model seems to possess the essential features necessary to explain the
observed M4 and M2 tidal patterns in the Englrsh Channel.
Equations
We measure time t in units of l / w where w = 0.0805 cycleslhr is the frequency of the M2
tide (thus 2n units = 12 hr), -and horizontal coordinates x, y (coordinates in a locally
conformal map of the Earth’s surface, where the English Channel is considered to be small
in comparison with the Earth’s radius), in units of c o l a , where co =ghoisthe phase speed of
long waves on depth ‘h,’ (h, is a typical depth). If the vertical coordinate is measured in
units of ho we obtain the nondimensional form of the vertically-averaged equations
qt + fk
x q +vq = - V % q 2 + qx v x q
(2)
where ~ ( xy,, t) is the wave amplitude (measured in units ho), q is the mean horizontalvelocity vector and f = f */a,
where f* is the Coriolis parameter. We assume that the tidegenerating forces have neghgible effect in the English Channel compared with the radiated
tide incident from the Atlantic Ocean, also friction is neglected since although it is of the
same order as the non-linear terms on the right of (l), (2) it is known (e.g. Gallagher &
Munk 1971) to generate odd harmonics of the fundamental oscillation (i.e. M6). Dissipation
will be provided for by the use of the radiative boundary condition suggested by Proudman
(1 941).
For the validity of (l), (2), it would appear to be sufficient to make the following
assumptions :
qr+ V-h(x,y)q=- V.qq
(i) The flow is sensibly horizontal.
(ii) Vertical accelerations in the fluid are much smaller than the acceleration due to gravity
multiplied by a typical ratio of amplitude to depth of the tidal wave. This condition enables
us to neglect frequency dispersion compared to the non-linear terms and is well justified
for tides in the English Channel.
(iii) The deficit in the mass flux caused by the presence of the bottom boundary layer is
small compared with the total vertically-integrated mass flux.
Conditions (i) and (ii) amount to the conventional long-wave assumption and the hydrostatic assumption respectively. The first implies the smallness of the bottom slope; however,
Quarterdiurnal tide in the English Channel
543
recent work (Hamilton 1977) has suggested that for sufficiently long-period oscillations (and
presumably in the absence of significant stratification) it may be possible to relax this
condition. The third assumption is necessary to ensure that the mean of the horizontal
advection terms are given for example, approximately by,
where (-) here denotes vertical averaging. Because the length scales of waves of tidal
frequency in the channel are typically considerably larger than the distances over which
particles may be advected in a tidal cycle, it is unlikely that even large errors in this
assumption will qualitatively affect the predictions obtained. However, a systematically large
over or under estimation of the advective terms in (l), would lead to a systematic over or
under estimation of the size of the quarter-diurnal constituents predicted.
Now, if h (x, y ) is constant then (l), (2) have constant coefficients and the linearized
equations possess elementary solutions in a parallel-sided channel. These elementary
solutions are the basis of the investigations of Taylor (1920); Defant (1961); Hendershott &
Speranza (1971) and Brown (1973). A constant depth parallel-sided channel is however
evidently not a suitable idealization of the English Channel. We must therefore generalize
(I), (2) to a case of varying depth and non-parallel sides, in such a way that we obtain
elementary analytic solutions.
We consider a conformal mapping
(x,
v)
+
(t,t)
(3)
with velocity components u, u in the directions i,f defined by
q = UVE t
uvg.
(4)
We may thus define an equivalent velocity in the (t,f) coordinates
(5)
<=uitut
and vector operator
(1) and (2) may then be rewritten as
-q t t f k x < + V v = J < x V x < - Q - ? h J < '
qttJV.h<=-JV.q$
where k = e x and J(5, {)= IVT 1' = IV{ Iz is the Jacobean of the transformation.
Evidently for constant coefficients on the left-hand sides we require
e
h(5, f) = 1/J(S 9'1) = exp ( 2 W )
(7)
(8)
(9)
say, where L determines the length scale of the bottom topography. It is easily shown that
is the appropriate mapping and hence
h = (x2 t y 2 ) / L z
where h = 1 at xz + y 2 = L2.
J. Hamilton
544
-51"N
Figure 1. Idealized depths, metres.
Thus we may obtain elementary first-order solutions for tidal waves in a bay of wedge
shape (bounded by lines { = constant), with the depth varying parabolically from the apex.
In Figs 1 and 2 may be found the idealization suggested for the English Channel, the reader
should consult the British Admiralty chart no. 1598 to convince himself that the depths
shown are not unreasonable values to take for the mean crosschannel depth at these points.
Generation of the M4 constituent
If the tidal amplitudes are small in comparison to the local depth we say write:
+ E 2 7 p ...
i=
eq(1) t E q(?). . .
77 = E
7 p
then ~
( lq(')
)
satisfy the equations
q y ) +fk x q(') t Q T p = 0
2
77p
+ fj. q(') t - q(1) . e = 0.
L
160
f
.
t
I2O
t
Distance fromI C.Griz Nez (degrees)
Figure 2. Averagecross-channel depths. Arrows indicate maximum depths at a section.
Quarterdiurnal tide in the English Cltannel
Elementary solutions of these equations satisfying the boundary conditions
u ( ~ ) = o on
~=o,P
say, take the form:
Kelvin wave
utl) = 0
Poincard wave
where
n2n2
N2
=83-(U2-f2)(1
- l/u2L2)
and 'n' is an integer constant. With boundary conditions at [ = 0, -L2 (say), of the form
(Proudman 1941)
u-aOq=O
on
u + a L v = O on
C;=O,
O<{<p
f=-L2,
O<{<p
it is possible to construct solutions of (14), (15) by superimposing Kelvin waves (17) and
Poincard waves (1 8). The earlier method of Taylor (1920) in which the position of the end
of the channel is defined relative to an amphidromic point is inappropriate in our case and
moreover will not be applicable when travelling Poincard waves are present (N2< 0). We
instead use the method of collocation (satisfying 20, 21 at discrete values of 5) which has
been developed by Brown (1973) for a rectangular channel and seems to work well for all
values of N 2 .
Having obtained a representation of the first-order solution #I, q(') which we identify
with the semi-diurnal tide (taking u = I), we may substitute into the equation for the
second-order quantities qC2) q(') namely :
qi2) t fk x q(2) t pq(2) = exp {- 2 E / L )[q(*) x
0x
q(1) -
v.
-
q(1) q(1) t eq(1) . q(1)/~1
(22)
Thus the semi-diurnal tides will generate forced solutions O(e') through the non-linear
terms on the right-hand side of (22), (23). Since we have assumed that the first-order terms
have a time dependence of the form cos (t + t l ) say, the second-order solutions will consist
of terms having a typical time dependence of the form
cos ( t t t , ) cos ( t t f z ) = cos (2t
+- t3) t 113
(24)
546
J. Hamilton
say. Thus to order ez there will be an effect on the mean sea level (varying predominantly
on a spring-neaps cycle) and the generation of quarterdiurnal tidal constituents.
We will only consider the quarter-diurnal terms which may be obtained as follows:
(1) Evaluate the semidiurnal waves induced in the channel through the incoming tide from
the Atlantic. It tums out that we may ignore the tide radiated through the Strait of Dover
from the North Sea.
(2) Express the quarterdiurnal forcings in the form:
( 3 ) Hence obtain the forced components
N.B., there are no resonantly forced quarter-diurnal components.
( 4 ) Add in free solutions to satisfy the boundary conditions at 5 = 0,p.
( 5 ) Add in Kelvin and Poincard waves to satisfy the boundary conditions on k = 0, - L 2 .
Choice of parameters
A systematic investigation of the effect of varying the defining parameters (e, L , L 2 , p, cq,,
a,) will not be presented here although some cases (not shown) have been calculated. We
remark however, that they seem to show in common with the results of Brown (1973)
that the co-tide-co-range patterns are dominated by the existence or otherwise of
propagating Poincard modes. We restrict ourselves to obtaining results for values of the above
parameters in accord with semidiurnal and quarter-diurnal waves in the English Channel.
The fundamental parameter of the problem is L ; for a given depth h at a point r say,
L z = w2r2/gh
(27)
defines for a given base frequency w, both the location of the apex of the channel and the
depth at all points in it. It will be appreciated that attempting a fit to a given channel may
necessitate a compromise between the requirement that the idealized sides of the channel fit
the actual sides and the requirement that the depth be proportional to the square of the
distance from the apex of the channel.
In this case, average cross-channel depths were estimated by eye and their square roots
plotted against long-channel distance (see Fig. 2). A straight line was then fitted to these
points giving a position for the apex of the channel which, as can be seen, fits quite well
with the English Channel. This procedure gave a value for L of 4.83 based on a depth of
540 ft at a distance of 7 2 0 nm (nautical miles) from the apex. We use imperial units throughout for convenience in comparing co-range charts which at the time of writing were only
available in feet ( 1 nm = 1.8522 km,1 ft = 0.3048 m).
Considerations of topography then gave p and L 2 respectively by the angle between the
coasts P/L = 10.7" and the location of the eastern boundary L2 = 4.175 (approximately 5"
of latitude = 300 nm from the apex). There then remained the choice of e and the end
boundary conditions characterized by a. and a ~The
. value of a. corresponding to the
continental shelf is determined without much variability by the depth in the ocean basin.
Quarterdiurnal tide in the English Channel
547
According to Proudman’s analogy
ffo =&&
where h A , hc are the depths on either side of the shelf. We used
u - 4.7n = semidiurnal forcing,
on g = 0
(29)
where the ‘semidiurnal forcing’ was chosen as that analogous to a unit incident Kelvin wave,
namely
semi-diurnal forcing = - (1
+ 4.7) exp [ f(C;
- P/2)
d
m + if/L(E
-
P/2)1
06 = 4.7 corresponding to a depth in the deep ocean of approximately 3700 m.
The boundary condition at the eastern end of the channel (the Strait of Dover) is
however, more indeterminate. An a priori choice of aL= 1/3 would be reasonable, based on
the actual widths of the Strait relative to the width of the theoretical channel (approximately 2 nm as against 6 nm). This choice tended to place the amphidromic point of the
M4 in the east of the channel too far to the north; however, it proved possible partially to
correct for this by replacing the boundary condition
u t o ~ ~ q = Oon
(=-L2
by one of the form
utp=O
O<(<P/3
u=o
P/3 < < (3.
r
This is nominally an attempt to represent the northern location of the Strait of Dover
with respect to the channel. A value for e of 1/200 has been used for the M2 co-amplitude
and co-phase lines (Fig. 3) as giving reasonable agreement with the observed amplitudes of
the M2 (see Fig. 4). The incident Kelvin wave then has an amplitude of 2.7 ft (0.82 m) on
the continental shelf at the centre of the channel. With the above choice of parameters the
M4 amplitudes and phases have been calculated, using five grid points at each end of the
channel for the M2, and nine grid points at each end for the M4. The result is shown in
Fig. 5, which should be compared with the partial M 4 chart, Fig. 6, and the values given in
Table 1 . There is also good agreement with the M4 values observed at the experimental
facility in Grenoble (Chabert d’Hieres & Le Provost 1970).
!
Figure 3. M,predicted: coamplitude - - - - feet, cophase -degrees.
548
J. Hamilton
I
I
I
I
I
Figure 4. M, ‘observed’:coamplitude - - - - feet, cophase -degrees.
Figure 5. M, predicted: coamplitude - - - - feet, cophase -degrees.
Discussion of results
It can be seen that the broad features of the M2-M4 system are reproduced; suggesting that
detailed topographic features need not be considered in looking for explanations for the
quarterdiurnal tide. In particular, the degenerate amphidromic point of the M2 is present in
roughly the correct place. However, the cross-channel placement of the M 4 amphidromies
is not very accurate although their longchannel placement is reasonable. As a consequence
Quarterdiurnal tide in the English mannel
Figure 6. M, ‘observed’:coamplitude - - - - feet, cophase
549
-degrees.
Table 1.
M*
Port
Amplitude
m
(1) Dover
(2) Spithead
(3) Portland
(4) Penzance
(5) St Malo
(6) Cherbourg
(7) Le Havre
2.16
1.34
0.64
1.71
3.78
1.89
2.65
Ma
Phase
degrees
ft
7.1
4A
2.1
5.6
12.4
6.2
8.7
29
31
166
2 24
182
132
74
Phase
degrees
Amplitude
m
ft
0.21
0.18
0.1 2
0.12
0.27
0.12
0.24
0.7
0.6
0.4
0.4
0.9
0.4
0.8
138
349
330
184
82
9
215
the local maximum in the M4 amplitudes near the Isle of Wight is reproduced, but not to
quite the extent of the tide at Southampton. The additional rise in amplitude between
Freshwater Bay (0.53 ft) on the south of the Isle of Wight, and Southampton (0.82ft) must
be attributed to the enhancing effect of the comparatively shallow Solent and Southampton
Water.
From the results obtained it is possible to formulate a qualitative explanation of the large
ratio of M4 and M2 tidal amplitudes in the neighbourhood of the Isle of Wight. The large
ratio is not due to any local topographic effect or to any local resonance of M4 period.
Instead an explanation must be sought in terms of the partial reflection of the M4 and M2
550
J. Hamilton
Grid points
Figure 7. M, residuals ___ ,forced component - - - - at E = - L 2 .
Grid points
Figure 8. M, residuals __ ,forced component - - - - at = 0.0.
Quarterdiurnal tide in the English Channel
551
constituents at the Strait of Dover. The wavelength of the quarter-diurnal constituents are
half the wavelengths of the semidiurnal constituents and result in the partial standing-wave
pattern set up in the English Channel having a node for the semidiurnal and an antinode for
the quarterdiurnal, in the neighbourhood of the Isle of Wight. Hence the relatively large
ratio of their amplitudes.
Accuracy
The accuracy of the numerical results is determined largely by the number of grid points at
which the end-boundary conditions are satisfied. This in turn is limited fairly rapidly in the
case of the M2 by the size of the computer as the number of forced M4 solutions generated
becomes very large. However, the effect of changing the number of grid points appears to
be confined to the amplitudes of only the higher order Poincard modes. Figs 7 and 8 are
graphs of the boundarycondition residuals and forcings for the M2 component. The
anomalous rise in the middle of the eastern-boundary residuals is due to the change in
boundary condition there.
Conclusions
We have shown that the distribution of the quarterdiurnal tide in the English Channel may
be explained using only monochromatic theory (i.e. one semidiurnal constituent) and a
second-order modal expansion in which the quarter-diurnal tide is generated solely by the
non-linear-distortion of the semidiurnal tide. The semidiurnal tide in turn is largely
explained as an incoming component radiated from the Atlantic and partially reflected at
the Strait of Dover.
On technical aspects we have extended the studies of Taylor etc. of tides in rectangular
bays to the case of linearly varying width and parabolically varying depth. We have also
extended the one-dimensional result of Lamb (1932) for the generation of quarter-diurnal
constituents in a narrow channel to a wide channel where Coriolis effects must be taken into
account.
Acknowledgments
'Ihis work was carried out while the author was supported in a research studentship by the
National Environmental Research Council. I should like also to thank Commander N. C.
Glen of the Hydrographic Office for ready access to their data and Dr P. A. Taylor of
Southampton University for advice and encouragement.
References
Brown,P., 1973. Kelvin wave reflection in a semi-infinite canal,J. Mar. Res., 31,l-10.
Chabert d'Hieres, G. & le Provost, C., 1970. Wtermination des charact6ristiques des ondes harmonique
M4 et M6 dans la Manche sur mod&lereduit hydraulique, Comptes Rendu Acad. Sci. Pans, 270,
1703-1706.
Defant, A., 1961. Physical oceanography Vol. 2, Pergamon, Oxford.
Gallagher, B. S. & Munk, W. H., 1971. Tides in shallow water: spectroscopy, Tellus, 23, (4-5). 346-363.
Hamilton, J., 1977. Differential equations for long period gravity waves on fluid of rapidly varying depth,
J. HuidMech.. 83,289-310.
J. Hamilton
552
Hendershott, M. C. & Speranza, A., 1971. Co-oscillating tides in long narrow bays: the Taylor problem
revisited,Deep-Sea Res., 18,959-980.
Lamb, H., 1932. Hydrodynamics, Cambridge University Press.
Macmillan. Cmdr D. H., 1964. A survey of Southampton and its region, ed. Monkhouse, F. J.,
Southampton University Press.
Proudman, J., 1941. The effect of coastal friction on the tides,Mon. Not. R. astr. SOC.
Geophys. Suppl.,
1,23-26.
Taylor, G. I., 1920. Tidal oscillations in gulfs and rectangular bays,Proc. Lond. Math. SOC.,(2), 20,148181.