International Hydrographic Review, Monaco, LXII (2), July 1985
A MODEL FOR PREDICTION
OF THE TIDAL CURRENTS
IN THE ENGLISH CHANNEL
by M. FORNERINO <*’ and C. LE PROVOST <**>
ABSTRACT
A model for prediction o f tidal currents in the English Channel is presented.
It is based on the classical harmonic description of tides deduced from the spectral
development of the luni-solar tidal potential. The spatial distribution o f the
characteristic parameters (intensity, phase, and direction of the maximum velocity
vector, ellipticity o f the hodograph) for the 26 harmonic constituents introduced in
the prediction procedure are deduced from a numerical simulation o f 1 month’s
duration for the entire English Channel.
Two kinds o f documents can be produced from this model : instantaneous
velocity fields over a given area, and time series of the intensity and the direction
of the velocity vector at a given location, over a given period. Four examples o f
prediction are presented, corresponding to specific areas and over periods where
tidal currents have actually been observed. The comparison between predictions
and observations is very satisfactory.
1. INTRODUCTION
On the European continental shelf, the tidal effects dominate the currents; in
some areas such as the English Channel they have such an amplitude that their
prediction becomes necessary, not only for the safety o f navigation but, in a more
general sense, for everything related to the marine environment (maritime works,
environmental studies, etc.). Usually, for these areas, the most complete and
authentic source o f information on tidal currents is provided by the nautical
documents published by the Hydrographic Offices o f the bordering countries.
(*) Perm anent address : Z ulia University, Venezuela.
(**) Institut de M écanique de G renoble, C.N.R.S., D om aine U niversitaire, B.P. n° 68, 38402
Saint-M artin-d'H ères Cedex, France.
These documents give the amplitude and the direction o f the currents for typical
tidal cycles, usually for neap and spring tides, for a given number o f points
distributed throughout these areas. This information is obtained from averaged in
situ observations representative of these typical tidal currents. This constitutes a
very precious documentation, yet it does not enable a continuous and detailed
prediction o f the currents over long periods.
A very efficient prediction method is the harmonic method, which is
commonly used to predict the variations in the sea level in ports. It allows for a
prediction to be made at any time provided one knows the totality o f the harmonic
constituents characteristic of the phenomenon in a given area. Usually, these
harmonic constituents are determined from in situ measurements. However,
because o f difficulties in analysis o f these singals (related to the very peculiar form
o f the tidal spectrum whose lines are rather numerous and very close), the
observations have to be made over a !on £ period, spreading over several months.
Because o f the difficulty in carrying out observations at sea, the observation points
available are very limited, even in the case of a well studied littoral sea.
L e P r o v o s t (1981) previously suggested a global model for the prediction of
tidal levels in the Channel. At minimal cost it enables the computation of the
variation in level o f the surface o f the open sea, at any time, with an accuracy of
about 15 cm. The implementation o f this model, based on the method o f the
harmonic analysis o f tides, was possible due to the existence o f a set o f cotidal and
isoamplitude charts o f the main tidal constituents in the Channel, established by
C h a b e r t d ’H iE R ES and L e P r o v o s t (1979), with the help o f a physical small-scale
model. Tested with traditional coastal and offshore observations as well as with
altimetric measurements by satellite (Seasat 78), this model has given excellent
results.
Recent numerical modelling work ( F o r n e r i n o , 1982; L e P r o v o s t and
1984) has resulted in a collection of harmonic constants o f currents for
the Channel equivalent to the collection o f tidal levels established by C h a b e r t
d ’HiERES and L e P r o v o s t . We propose in this article a tidal current prediction
model valid for the whole Channel area and developed according to the same
principle as that used by L e P r o v o s t (1981) for the computation o f levels and using
that collection of characteristic constants o f the currents.
F o r n e r in o ,
2. DESCRIPTION OF THE COMPONENTS OF THE MODEL
2.1. The harmonic method
Generally speaking, one may describe the tide by means of its harmonic
constituents; each o f the parameters characterizing the phenomenon (sea level
variation or constituents of the velocity field) can be approximated by a relation
o f the type :
Sap (M,
t)
= S0 (M) +
2
fi S, (M) cos
[C0it
+ (V0 + v), - g, (M) ]
(1)
where :
So (M)
mean value o f the signal at point (M)
N
number o f constituents taken into account
Si (M)
amplitude o f the constituent having i as a subscript
phase difference o f this constituent in relation to the transit o f the
g. (M)
constituent at Greenwich Meridian
coi
: pulsation o f this constituent
Voi
phase o f the constituent in relation to the Greenwich Meridian at
time t = 0
fi and v amplitude and phase nodal correction coefficients of this consti­
tuent
In order to validate this approximation (1), it is o f course necessary to take
into account all significant constituents : not only the ones resulting from
astronomical generating potential, but also the ones induced by the non-linear
hydrodynamic mechanism to be found in the littoral seas. The amplitude and phase
of each o f these harmonic constituents depend on the intensity o f the generating
forces as well as on the geometrical and hydrodynamical factors typical o f the
basins where they develop. The nodal correction coefficients result from the
harmonic development o f the generating potential; they allow for the long term
variation o f astronomical parameters (over one year) while limiting the number o f
significant constituents, N, to be retained (cf. S c h u r e m a n , 1958).
The parameters (Oi, v0i, fi and Vj are defined as soon as the choice of the N
constituents is made, and so the main problem related to the application o f (1) lies
in determining the quantities Si and gi, which depend on the position o f the
considered point M in the observed zone.
2.2. Aim of the model
Our aim is to develop a procedure for predicting tidal currents in the English
Channel. It is a vectorial quantity varying in space and time : three functions are
thus necessary to characterize it, such as, for instance, its projections on three axes
of reference Oxyz (further, in this article we take the plane Oxy horizontal with Ox
eastwards and Oy northwards). But the tide is typically a phenomenon o f long
waves and the velocity fields may, consequently, be considered as horizontal : so
the problem comes down to the definition of two functions, u and v, for instance,
projections of the speed vector on the Ox and Oy axes. On each point (x, y) the
velocity vector varies on the vertical, between bottom and surface. When there are
no surface constraints (i.e. when there is no wind) and if the density stratification
can be disregarded (which is indeed generally the case for the English Channel) this
vector is more or less constant, except near the bottom where a boundary layer
exists : in this study we will restrict ourselves to the definition o f this “mean” value
of the velocity, valid for practically the whole o f the fluid column. Therefore, using
the harmonic formula, this current vector will be expressed as follows :
Ü (x, y, t) = u T + v T
(2)
with :
N
U
= Uo +
X f, Ui ( x , y ) c o s [co.t + (V0 + v)i -
g ul ( x , y )]
1- 1
N
V
= Vo +
2
fi v, ( x , y ) c o s
[C0it + (V0 + v),
-
g v, (x , y )]
i- 1
where i and j are unit-vectors according to Ox and Oy.
2.3. Basic data
In order to apply this formula (2) on a field <X> we need to know for each
point the amplitude and the phase o f the various tidal constituents : this data base
is now available for the English Channel, following the work done by F o r n e r i n o
(1982). A group o f networks specifying the characteristics o f the significant
constituents of the tidal currents over the entire Channel was produced from a
numerical simulation based on a classical model with finite differences of the
“predictor-corrector” type, of which only the main characteristics are mentioned
here (cf. L e P r o v o s t and F o r n e r i n o , 1984, for more details). The zone under
consideration has two open boundaries (cf. figure 1) : the western one (Atlantic)
goes from Devonport on the English coast to Roscoff, or thereabouts, on the
French coast, and the northern one (North Sea) goes from Ramsgate on the English
coast to Ostend on the Belgian coast; the area is divided into a 10 km grid mesh.
The difference in level Ç(x, y, t) o f the open sea surface is calculated at the centre
o f each grid and the velocity consituents u and v, respectively, on the east-west and
north-south faces, as shown in figure 1. Along the solid boundaries the condition
F ig . 1. — Spatial gridding of the m odelled area, the English C hannel, with location of the points of
com parison betw een prediction and observation : Pi, P2, Pj, P4.
at the imposed limits is no normal flow; for the open boundaries on the Atlantic
and the North Sea, the variations o f level are prescribed at each moment and
calculated according to type 1 formula using a distribution of the harmonic
constituents obtained by C h a b e r t d ’HiERES and L e P r o v o s t (1979). This numerical
simulation reproduced the variations in level of the sea surface and the currents
over one month in real time. Characteristic networks o f the main harmonic
constituents of the current were established for the whole o f the gridded area
(English Channel, Dover Strait, cf. fig. 1) after a harmonic analysis o f the temporal
Series, computed at different points o f the grid, had been made.
The harmonic constituents presented by F o r n e r i n o (1982) number 22 : two
long-period non-linear constituents MS0 and MN„; three diurnal constituents : K,,
0 , and Qi, six astronomical semi-diurnal constituents : M2, S2, N 2, ji2, L2 and e2, five
non-linear semi-diurnal constituents : 2 MS2, 2 M N2, 2 SM2, M SN2 and MNS2,
three quarter-diurnal constituents : M4, MS4 and MN4, and three sixth-diurnal
constituents : M6, 2 MS6 and 2 M N 6. It should be noted that a certain number of
constituents found in the real spectrum are not included in this list; this is a direct
result o f the actual conditions o f the numerical experiment carried out in order to
get these results : to reduce the cost o f calculations, the numerical simulation was
only done for one month and it was decided consequently not to take into account
(for the definition of the boundary conditions) the waves close to the main
constituents S2, N 2,
and L2 which cannot be separated from the latter over a one
month period, i.e. K2 and T2, v2, 2 N 2 and Xi.
In order to define each current constituent, one has to use four parameters :
for instance, the amplitude and phase of each of the projections on Ox and Oy,
1.e. |j.i, gUi, Vi, gvi, if formula (2) is used (these are the quantities which will be used
later on in the prediction model). However, in order to visualize the current
characteristics o f these various constituents, it is easier to use another set of
parameters based on a polar representation of these currents. The hodograph of an
elementary harmonic constituent Ui o f (2) :
U; (x, y, t) = Ui (x, y) cos [coit - <pui (x, y) ] T + v; (x, y) cos [«it -
<pvi (x, y) ] f
is an ellipse (cf. figure 2); it can thus be characterized by three specific parameters :
— the amplitude o f its semi-major axis Ai : this is the amplitude o f the
maximum current associated with this constituent;
— its ellipticity R; = — x 100, expressed as a percentage o f the magnitude
Aj
of the minor axis a, to that o f the major axis Ai. For R = 0, the current
is rectilinear; for R = 100, the rose is circular;
— the direction o f the major axis compared with a direction o f reference
(usually the North is used).
It then suffices to specify the maximum phase of the current in relation to the
moment in time being used as a reference and the direction o f rotation o f the
velocity vector to completely define the constituent characteristics. By giving a
positive or a negative sign to the ellipticity according to the direct or reversed
direction o f rotation of the velocity vector, it is clear that the only fourth parameter
required is : the maximum current phase.
This is the presentation that F o r n e r i n o (1982) used for her results. To record
here all the networks obtained would take up too much space; yet, in order really
F ig. 2. — Param eters defining the velocity ellipse for any com ponent i.
— sem i-m ajor axis Ai (m odules of m axim um velocity vector)
— ellipticity : Ri = a /A ; x 100%
— m ajor axis orientation by reference to the North (Pi max) (direction of m axim um velocity
vector)
— m axim um velocity phase (ü>it)„„.
to understand the essential characteristics of the current fields in the prediction
zone ynder consideration, it is useful to be more specific about the salient features
o f the different groups o f constituents in the spectrum.
2.3.1. The semi-diurnal astronom ical constituents
We show in figure 3 the M2 constituent (the most important one in the English
Channel tide). The distribution o f the current amplitude is typical of a Kelvin
amphidromic tide : the maximum intensity zone is located in the central part o f the
basin, between Cotentin and the Isle o f Wight, and coincides with the nodal zone
o f amplitude o f the differences in open sea surface levels. This maximum current
increases even more around the capes, mainly in La Hague, where the velocity
vector o f M2 exceeds 2 m /s. On the other hand, at the Channel entrance, and in
its eastern part, the velocity decreases to about 60 cm /s, and even less in certain
bays (St.-Brieuc, Mont St.-Michel, Seine, Somme). A band o f greater velocity is,
however, noted between Bréhat and Guernsey. It can be interpreted as a ventral
zone o f current, to be associated with a nodal zone of the sea surface elevations
at the entrance to the bay o ff Normandy and Brittany (Golfe normano-breton) in
which the M2 tide can exceed 4 m. Lastly, we should point out that the velocity
vector also increases locally in the Dover Strait, due to the narrowing o f the passage
available for the propagation o f the tidal wave in that area. The direction of the
F ig . 3. — Characteristics of the sem i-diurnal constituent M 2. Modules (cm/s) of maximum velocity
vector; direction (°) of maximum velocity vector; phase (°) of maximum velocity vector; ellipticity (%).
maximum velocity vector is almost everywhere in the order of 240° to 270°, that is
east-west, except in the Golfe normano-breton, where the network becomes very
complex due to the very strong gyratory aspect of the currents in this area, with
ellipticity values o f 40 to 60 %. The increase in ellipticity o f current roses in the
eastern part o f the Baie de la Seine is also to be noted.
The characteristic networks o f the other semi-diurnal group o f astronomical
constituents (S2, N 2, L2, |i2 •••) are very similar to those presented here for M2 : the
charts showing the direction and the ellipticity o f the current roses are exactly the
same; the phase networks are identical, given some phase shift; only the distribu­
tions o f the velocity amplitudes are slightly different, but their networks are similar
to that presented in figure 3 and may be deduced from it by simple affinity as a
first approximation.
As these semi-diurnal constituents are the most important of the spectrum,
their spatial characteristics, that we have just commented upon, determine the
salient features of the actual current fields.
2.3.2. The sem i-diurnal constituents o f non-linear origin
The constituents 2 MS2, 2 SM2, MSN2 and MNS2 are the main ones of this
group. It should be remembered that they result from the non-linear interaction
effects between the main constituents M2, S2 and N2 in shallow water areas. It is
not necessary to reproduce here a typical example o f their networks, since they
show maxima and minima of the current in the same areas as the astronomical
semi-diurnal constituents. The current directions are identical too; only the phases
and ellipticity networks are a little different in the western part o f the basin, mainly
in the Golfe normano-breton where shoal depths, which cause considerable
friction, induce energy transfers from the generating waves towards these semi­
diurnal non-linear constituents (cf. L e P r o v o s t , 1976).
2.3.3. The diurnal constituents
O f course, the networks related to these constituents are completely different
from the previous ones. The maxima o f the currents are located exclusively in the
Dover Strait area corresponding to their amphidromic points (cf. C h a b e r t
d ’H iERES and L e P r o v o s t , 1979). There, the K.i constituent amplitude reaches, for
instance, almost 13 cm /s (see figure 4); when we consider that in this area the M2
current amplitude does not exceed 120 cm /s, we might a priori forecast that the real
diurnal inequality in currents in the Dover Strait is appreciable. Also, on these
networks one notes a minimum amplitude zone between 0° and 1° o f longitude west
which goes along with an increase in the ellipticity o f the current rose; however,
these details are negligible, given the very low amplitude o f these constituents in
this area.
2.3.4. The quarter-diurnal constituents
The wave length o f these constituents (M4, MS4, M N4) is half that of the
semi-diurnal ones. As a result, the networks become more complex. To illustrate
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direction (°) of maximum velocity vector; phase (°) of maximum velocity vector; ellipticity (%).
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this p o in t, we show th e M 4 c o n stitu e n t in figure 5 : a m axim um cu rre n t a m p litu d e
can b e n o te d in th e centre o f th e ea ste rn C hannel, at the lo ngitude o f one o f the
a m p h id ro m ic p o in ts typical o f q u a rte r-d iu rn al cotidal charts, a n d a m inim um zone
o n eith er side w h ich c o rresp o n d s perfectly to th e stan d a rd schem e o f an a m p h i­
d ro m ic stru c tu re (o n e sh o u ld n o te th e m arked ellipticity o f th e cu rren t roses in
these a re a s o f m in im u m current). M oreover, the existence o f a n area o f m axim um
am p litu d e o f th e q u a rte r-d iu rn al cu rren ts betw een B réhat a n d G u ernsey, at the
en tran ce o f th e G o lfe n o rm a n o -b re to n a n d betw een Jersey and M o n t S aint-M ichel,
in th e very bay itself, sh o u ld also b e noted. T he am p litu d e o f these currents is
sign ifican t in th e D o v er Strait as w ell.
T h e sp a tia l d istrib u tio n o f th e se q uarter-d iu rn al co n stitu en ts is interesting
b ecause th is in tro d u ces an asy m m etry betw een flo o d a n d ebb in th e global signal.
An ex am p le o f th is is given later.
2.3.5. The sixth diurnal constituents
F ro m C h a b e r t d ’H iERES a n d L e P r o v o s t (1979) we know th a t th e C h an n el
is subject to sixth d iu rn a l o scillatio n s, m ainly betw een the Baie de la Seine a n d the
Isle o f W ight. T his ch aracteristic is also fo u n d o n the sixth d iu rn a l cu rren t fields
w hich are in th is a re a o f sig n ifican t am p litu d e : M 6 an d 2 M S 6 are o f a b o u t 3 to
6 c m /s . T h eir p resen ce in the global signal will th u s ap p reciab ly d isto rt the current
roses.
T he review o f th e salien t fe a tu re s o f these d ifferent categories o f co n stitu en ts
alread y enab les us to p ro d u ce a ro u g h d escrip tio n o f w hat the tid al cu rren ts sh o u ld
b e in th e v ario u s zones o f th e C h a n n e l :
— m ax im a zones : central p a rt, C ap de la H ague, B arfleur, S t.-C atherine,
D o v er Strait, en tran ce to th e G olfe n o rm an o -b reto n ;
— stro n g ellipticity zones : e a ste rn Baie de la Seine, G olfe n o rm an o -b reto n ;
— sig n ifican t d iu rn al in e q u a lity zones : D over Strait;
— zone w here there is a stro n g asym m etry betw een flood and ebb : betw een
F écam p a n d N ew haven.
T he illu stra tio n s w hich fo llo w w ill confirm som e o f these points.
2.4. Characteristics o f the prediction model
W ith th is co llectio n o f c h a ra c teristic netw orks o f the different sig n ifican t tidal
co n stitu en ts w e th e n have a b ase w hich will enable us to c o n sid er m aking
p re d ic tio n s b ased o n th e h a rm o n ic fo rm u la ( 2).
T h e u sefu l h a rm o n ic p a ra m e te rs (ui, vi; gui, gvi) resulting from the n u m erical
sim u latio n are k n o w n at various in tersectio n p o in ts o f the grid : w ith a reso lu tio n
o f 10 km x 10 km , 716 d ata p o in ts p ro v ide accurate info rm atio n on the spatial
d istrib u tio n o f th ese p aram eters ex cep t, p erhaps, in th e coastal zones a n d in the
zones w ith sp atial v ariatio n s o f very im p o rta n t am p litu d e such as capes.
It is to be n o te d th a t F o r n e r i n o ’s sim ulation (1982) did n o t pro v id e fo r the
d e te rm in a tio n o f certain c o n stitu e n ts w hich w ere not included, b e c a u se the
sim u latio n p e rio d w as lim ited to o n e m o n th. T his applies m ainly to co n stitu en ts T 2
vector; direction (°) of maximum
velocity vector; phase (°) of maximum
velocity
velocity vector; ellipticity (%).
F i g . 5. — Characteristics of the quarter diurnal constituent M<. Modules (mm/s) of maximum
an d K 2, 2 N 2, v 2, k 2 w hose cycles o f sep a ra tio n are respectively one year with S2,
a n d six m onths w ith S2, p.2, v2, L2. H ow ever, we have alread y m en tio n ed in
p a ra g ra p h 2.3 the sim ilarity b etw een co n stitu en ts o f the sam e g ro u p a n d origin; this
also applies to th e above c o n stitu en ts. We can thus d ed u ce th e ir characteristic
n etw o rk s from th o se o f the n earest k n ow n constituents by using the coefficients o f
p ro p o rtio n a lity b etw een w aves a n d th e p h ase differences observed in the level
c o n stitu en ts (cf. Le P r o v o s t , 1981).
O n e p ro b lem is, how ever, th e cho ice o f the n u m b er o f co n stitu en ts to be tak en
in to acco u n t in fo rm u la (2). In th e co astal zones, it is unav o id ab ly high b ecause the
d isto rtio n s d u e to th e n o n -lin e a r p rocesses have to be rep ro d u ced . W e have given
in T ab le 1 th e m ax im u m a m p litu d e s o f the various constituents stu d ied in
F o r n e r in o ’s n u m erical sim u latio n , as well as o th er significant co n stitu en ts
d e d u c e d from them by sim ilitu d e. By lim iting ourselves to the co n stituents, w hose
m ax im u m am p litu d e exceeds 3 c m /s . we com e to a n um ber o f 26 : it is Dreciselv
this sam e n u m b er th a t L e P rovost chose fo r th e equivalent m odel o f level
p re d ic tio n w hich led to a m ean sq u are value rated at 20 cm in Le H avre, w here the
tid e is rep u ted fo r its co m p lex characteristics. F o r the tid al c u rren t p red ictio n
m odel we have in fact u sed 28 co n stituents, because the nu m erical sim ulation
allo w ed us to estab lish th e ch aracteristics o f tw o long-period constituents M S0
(sem i-m onthly) an d M N„ (m o n th ly ) co rresp o n d in g to the astronom ical waves M sf
a n d M m, b u t resulting, in fact, from th e n on linear interactions betw een M 2 an d S2,
o n th e one h a n d , an d M 2 an d N 2, o n th e o th e r : these tw o constitu en ts co n trib u te
in a sig n ifican t w ay to the c u rre n t fields a ro u n d th e capes (L a H ague, B arfleur,
P o in te S ain te-C ath erin e) an d at th e en d o f the M ont Saint-M ichel Bay w ith a
m ax im u m am p litu d e o f 6 c m /s an d 3 c m /s, respectively.
T he cu rren t p re d ic tio n m odel th u s fo rm ulated offers tw o types o f p ro d u cts in
p ractice :
— The p re d ic tio n o f a c u rre n t field at any desired tim e t, accurately defined
by its d a te (year, m o n th , day) a n d its tim e (hour, m inute, second) for a given group
o f p o in ts (x, y) w hich m ig h t very well cover the w hole C hannel : fo r each p o in t the
m o d el pred icts th e m o d u le a n d th e d irectio n o f the current. An exam ple o f this type
o f p ro d u c t is given in figure 6 : fo r th e w hole o f th e C hannel a c u rre n t p a tte rn is
esta b lish e d fo r d iffe re n t tim es d u rin g a tide o f average am plitude.
— The p re d ic tio n o f th e ev o lu tio n in tim e o f th e cu rren t at a given p o in t (x,
y), fo r a given p e rio d b etw een tw o d a te s ti a n d t2. T he m odel co m p u tes th e m odule
a n d th e d irectio n o f th e c u rre n t v ecto r at each m om ent t = ti 4- kAt betw een ti
an d t 2 w ith a tim e interval At sp ecified by the user. Such exam ples are show n in
figures 7, 8 , 9, 10.
T his m odel is easily a d a p ta b le to the m icro-com puters currently available on
th e m ark et (w ith a m em ory o f 16 K bytes). This allow s p red ictions to be m ade at
an extrem ely low cost.
2MK,
Maximum
4 (B)
amplitudes : constituents 1 to 19 established
27.8860712
1
from
a numerical sim ulation; constituents 20 to 26 derived by similituc
List of constituents included in the prediction.
TABLE
3. D ISC U S S IO N - C O M PA R ISO N OF THE RESULTS
WITH TH E OBSERVATIONS
A useful co llectio n o f cu rren t o bservations in the C h an n e l is availab le, in
p a rtic u la r in the archives o f th e “ B anque n atio n ale des données o c é a n o g ra p h iq u es”
(N atio n a l o ce a n o g ra p h ic d a ta center) (C O B -C N E X O Brest). In o rd e r to assess the
quality o f th e results o f th is m odel, we selected the m easu rem en t points giving the
longest co n tin u o u s series o f o bservations for two reasons : first, so as to be able
to assess th e b e h av io u r o f the m odel ov er a long p e rio d o f tim e, a n d second,
becau se these long observed series can , a p rio ri, be analyzed in term s o f h a rm o n ic
develo p m en ts a n d th u s enable us to u n d e rstan d any possible faults on som e
co m p o n e n t o f the p re d ic tio n m odel. W e have d ecided to give here fo u r typical
exam ples o f such com parisons.
3.1. Point Pi — Baie de la Seine
O bservations o f surface curren ts w ere m ade by the “ C en tre O céan o lo g iq u e de
B retagne” in 1979, over a p e rio d o f 20 days (geographical lo ca tio n :
49°41' N - 0°50' W) a n d were an aly sed by B e r t h e r a t , C a r c e l an d L e P r o v o s t
(1981). Figure 7 show s a plo t o f the m o d u le and the c u rre n t direction observed
(lim ited to the first 15 days due to th e page lay-out) an d in T able 2 a list o f som e
o f th e constitu en ts co m p u ted th ro u g h h arm o n ic analysis o f this signal. O v er the
m easu rem en t p erio d , th e m axim um c u rre n t is in the reg io n o f 1 m /s , its d ire ctio n
being altern ately so u th -east at flo o d a n d north-w est at ebb w ith a very low
ellipticity (the c u rre n t rose is alm ost rectilinear).
T h e p re d ic te d signal is p lo tted in d o tte d lines in th e sam e graph. It can be
seen th a t th e d ifferences betw een p re d ic te d a n d observed values are negligible; if
one carefully studies the curve w hich gives the m odule d ifference, one can see th at,
fo r low coefficients, b etw een th e 13th a n d th e 17th day, th e m axim um d ifferences
are n o t m ore th a n 10 c m /s a n d th a t, fo r th e significant coefficients, these m axim um
differences are at m ost ab o u t 20 c m /s . It can be seen th a t th e m ajo r d ev ia tio n s
observed at the b eg in n in g o f th e reco rd in g are d u e to th e m alfu n ctio n in g o f th e
cu rren t m eter; ex cep t fo r th e first three days, the average ab so lu te value o f th e
d ev iatio n is 4 c m /s a n d the m ean sq u are d eviation is 6.6 c m /s. T he p re d ic te d
cu rren ts are, m oreover, perfectly p h a se d w ith the observed currents.
O n the o th e r h a n d , th ere is a m ajo r a n d system atic d ifference in th e d ire ctio n
o f a b o u t 23°, p a rtly du e to a d ifference o f a b o u t 10° in th e m ain directio n o f th e
cu rren t rose a n d partly du e to a fa u lt in th e ellipticity. T his differen ce results in
a system atic erro r in th e p red ictio n s clearly illu strated in figure 7 e w hich in d ic ates
the a m p litu d e v ariatio n s o f th e v ectorial d ifferences betw een p red ic tio n a n d
observation.
T hese defects a n d qualities o f th e p red ictio n are also c lear if w e co m p are th e
m odel an d in-situ values o f the m ain h a rm o n ic co n stitu en ts, as show n in T ab le 2.
T he quality o f th e p h ase an d am p litu d e p aram eters can b e seen : th ere is only a
d ifference o f 4.2 c m /s fo r M 2, 0.1 c m /s fo r S 2 a n d 2.1 c m /s for N 2, a n d a few sm all
in accu racies w ith regard to phases. O n the o th er h an d , a discrepancy in the
ellipticity o f th e co n stitu en ts M 2 a n d N 2, as well as a dev iatio n o f ab o u t 10° in the
d irectio n o f th e m axim um cu rren ts o f M 2 and S2 will be noticed. Besides, it will be
n o te d th a t th e p aram eters o f th e h ig h e r constituents (fourth a n d sixth diurnal) are
qu ite well defin ed .
uj
a
_4
1
10
TEMPS
(J)
360
h
270
w
n
180
a.
ucr
f
S
90
0
TEMPS
10
(J)
TEMPS < J )
F i g . 7. — O bservation and prediction of tidal currents at point Pi (Baie de la Seine) since 1 9 /6 /7 9 at
19 h 20.
observation
prediction
TABLE 2
Harmonic constituents for tidal currents at point Pi.
V alues observed in situ a n d co m p u ted from a num erical sim u latio n o v er one
m onth. V elocity in c m /s , p h ase in degrees related to th e tra n sit o f the c o n stitu e n t
at G reen w ich m erid ian , d irectio n in degrees related to th e n o rth g eographical pole.
Cons­
tituent
o,
Maximum
velocity
Obs.
Comp.
1.3
Max. current
phase
Max. current
direction
Ellipticity
Obs.
Comp.
Obs.
1.5
42.8
176.
- 42.5
Comp.
Obs.
Comp.
18.
45.
293.
- 21.
180.
133.
K,
2.0
1.4
32.5
62.
n2
16.9
14.8
14.
15.
4.3
-
6.
298.
299.
m2
96.2
92.
33.
33.
4.
-
6.
311.6
300.
s2
28.9
29.
77.7
82.
8.2
-
8.
310.
301.
-
116.
m sn 2
0.
-
1.
1.5
12.5
79.
0.
mn4
2.15
1.7
34.
329.
28.
m4
6.25
5.
167.6
170.
m s4
4.04
9.
180.
48.
112.
148.
- 57.4
- 52.
340.
326.
- 20.
- 38.
171.
150.
3.5
32.
36.
2 MS6
7.2
5.
39.
24.
9.
326.
316.
m6
5.6
5.7
173.
161.
14.
12.
135.
139.
2 MN,
2.2
2.6
323.
14.
16.
296.
315.
2 SM2
1.
1.5
109.
0.
- 11.
90.
114.
5.6
105.
4.5
3.2. P oint P 2 — P aluel
A 40-day stu d y is av ailable fo r th e a re a o ff Paluel. It w as m ad e by the E D F
in 1976 (geo g rap h ical lo catio n 49°46' N, 1°40'W ) an d an analysis was m ade by
B e r t h e r a t (1981). This series is p a rtic u la rly interesting, since it show s a very
m ark ed asym m etry betw een flo o d an d ebb. Figure 8 show s a p lo t o f th e velocity
m o d u le am p litu d e an d its d irectio n from th e 20th to th e 35th day o f o b servation.
As for the p revious exam ple, th e p red icted values given by th e m o d el are
show n on the sam e diag ram as d o tte d lines. First, we can see how well th e
directio n s are d e te rm in e d : th e R M S e rro r betw een observed a n d co m p u ted values
is, over the 40 days available, only 9° !
T he c u rre n t m o d u le is also very accu rately defin ed : the asym m etry betw een
flo o d an d ebb is very well re p ro d u c e d b etw een the 8th a n d th e 31st d a y o f
o b se rv a tio n ; b etw een th e 2nd an d th e 8th , n o t show n h ere, a n d betw een th e 32nd
an d th e 38th day, th e ebbs are still w ell re p ro d u ce d , b u t a m ajo r deficiency in th e
flood leading to a d ifferen ce o f 25 c m /s is to be n o ted (see curve 8 b show ing th e
m o d u le difference). This is m ost certain ly due to m eteorological effects (the
ob serv atio n s w ere m ad e in N o v em b er a n d D ecem ber), since an y o th e r in te rp re ta ­
tion is not logical. A co m p ariso n b etw een th e in-situ a n d th e m o d el values o f the
m ajo r h a rm o n ic co n stitu en ts is in d eed very favourable (cf. T able 3). O ne can notice
th e im p o rta n c e o f th e q u a rte r-d iu rn al constituents, th e sum o f w hich co rresponds
to 19% o f th a t o f the m ajo r sem i-d iu rn al co n stituents, and w hich pro d u ces this
asym m etry betw een eb b a n d flood.
In sp ite o f these m ark ed deficiencies over certain periods, one sh o u ld keep
in m ind th a t the overall sta n d a rd dev iation over th e 40 available days is only
ui - 1.
30
TEMP S
C j)
360
270
L
100
a.
Œ
a
90
0
30
TEMPS
(J)
E
PU
U
LI
uz
_J b l
Z3 a.
aO
uj.
XL
L.
30
TEMPS ( J )
F i g . 8.
— O bservation and prediction of tidal currents at point P2 (Paluel) from day 20 to day 35 since
18/11/76 at 22 h 20.
T A B LE 3
Harmonic constituents for tidal currents at point P 2 .
V alues o bserved in situ a n d co m p u ted from a num erical sim u latio n over one
m on th . V elocity in c m /s , p h a se in degrees related to the tra n sit o f the co n stitu en t
at G reen w ich m erid ian , d irectio n in degrees re late d to the n o rth geo g rap h ical pole.
M axim um velocity
M axim um cu rren t phase
C o n s­
titu e n t
O bserved
C o m p u ted
O bserved
m2
81.6
83.0
59.1
58.0
s2
23.4
25.9
285.0
288.0
n2
13.3
13.6
218.0
216.0
m4
11.4
12.1
114.0
98.0
m s4
7.4
7.6
171.0
155.0
m n4
3.8
4.2
74.5
79.0
C o m p u te d
6.7 c m /s a n d the m ean overall dev iatio n 8.9 c m /s. T he agreem ent betw een the
p re d ic tio n at this p o in t an d th e reco rd ed v alues is all the m ore rem ark ab le as it is
situ a te d n e a r the coast w here th e grid is u n av o id ab ly coarse, given the 10 km grid
o f th e sim ulation m odel.
3.3. Point P } — Central Channel
T he b eh av io u r o f the p red ictio n m o d el in the central p a rt o f th e C h an n e l,
w here cu rren ts are very strong, is interesting. F o r this com p ariso n we have selected
p o in t P 3 (co o rd in ates : 50°01' N - 1°47' W) w here the F ren ch “ Service H y d ro g ra ­
p h iq u e d e la M arin e ” has co m p leted a 16-day observation w ith a cu rre n t-m e ter
su b m erg ed at 29 m in d ep th s o f a b o u t 58 m. SH O M also m a d e a h arm o n ic analysis
o f this reco rd in g ( B e s s e r o , 1980).
T he reco rd ed cu rren t m o d u le an d d ire c tio n are show n in figure 9. O ne will
n o tice th e signal am p litu d e w hich exceeds 2.5 m /s d u rin g spring tid es as well as
th e a lte rn a te e a s t/w e s t d irectio n o f the velocity vector. The p re d ic tio n over this
o b serv atio n p erio d gives relatively satisfacto ry results. W e no te a slight dev iatio n
in th e d irectio n ( 10° average) b u t, above all, a system atic erro r in th e c u rren t
am p litu d e due to a p h ase d ifferen ce o f th e p red icted signal p ro d u c in g an e rro r o f
a b o u t 20 c m /s , to w hich is a d d e d a system atic deficiency o f the ebb in tro d u cin g
a fu rth e r 20 c m /s difference.
R eferring to th e observed a n d m odel values o f th e h arm o n ic constitu en ts
relative to this p o in t (cf. T able 4) we see th a t this phase d ifference is d u e to a fau lt
o f th e sem i-d iu rn al n u m erical solu tio n s : M 2 is 10° too early, S2 : 7°4 a n d N 2 : 11°7;
a system atic deviatio n o f 7° in th e d irectio n o f th e sem i-d iu rn al co n stitu e n ts is to
be fo u n d again in this ta b le ; th e d ifference b etw een flo o d a n d ebb is m o re difficult
M/lM
w
TEMPS
(j)
TEMPS
(J)
180
* 2
W ÜJ
u.3
i
•fc♦H
Q
a 0
e
TEMPS ( J )
F ig . 9. — Observation and prediction o f tidal currents at point P 3 (middle of the English Channel).
to in te rp re t an d no e x p la n a tio n can be found in the harm onic constituents given
in T a b le 4.
In spite o f these system atic faults, the m ean deviation betw een predicted an d
ob serv ed values over th e 16 av ailab le days is only ab o u t 14 c m /s fo r currents
ex ceeding 2.5 m /s at certain p eriods. One can thus be reasonably satisfied w ith
these results.
TA B L E 4
Harmonic constituents for tidal currents at point P3.
Values o b serv ed in situ a n d co m p u ted from a num erical sim ulation over one
m onth. V elocity in c m /s , p hase in degrees related to th e tra n sit o f th e co n stitu en t
at G reen w ich m eridian, d irectio n in degrees re la ted to the n o rth g eo graphical pole.
Cons­
tituent
Maximum
velocity
Max. current
phase
Ellipticity
Obs.
Comp.
Obs.
Comp.
Obs.
0,
2.5
2.9
17.8
2.0
4.0
K,
2.6
2.7
45.6
63.0
m ns2
1.6
1.9
154.8
91.0
n2
18.5
25.8
186.8
2
155.2
157.5
226.1
m
Comp.
Max. current
direction
Obs.
Comp.
2.0
69.5
84.0
7.7
1.0
78.2
83.0
0.0
1.0
75.4
83.0
198.5
2.2
1.5
77.3
84.0
216.0
3.2
1.5
78.7
84.0
S2
47.5
50.0
271.4
264.0
2.9
1.5
77.5
84.0
m4
2.8
1.9
279.5
329.5
35.7
4.0
112.4
52.0
m s4
1.1
1.5
253.7
198.5
18.2
9.0
170.3
238.0
2 MS6
222.0
0.0
11.0
125.5
102.0
2.6
1.6
189.4
m6
1.8
2.2
177.0
171.5
0.0
4.0
119.3
101.0
2 MN(
1.0
1.1
171.0
163.0
0.0
6.0
116.4
98.0
3.4. Point 4 — O ff Cherbourg
H ow ever, this p red ictio n m odel d o es n o t alw ays provide the degree o f
accuracy fo u n d in th e p revious exam ples. W e shall end o u r in-situ co m parisons
with th e stu d y o f a p o in t located o ff C h erb o u rg (location : 49°46' N - 1°40' W ; 15
d ays’ o b se rv a tio n carried out by S H O M an d a n aly sed by B e s s e r o , 1980). T he m a jo r
deviations a p p e a rin g in figure 10, in th e m o d u le as well as in th e directio n s, can
be easily u n d e rsto o d if one rem em bers th e extrem ely co arse g rid o f C o te n tin (cf.
figure 1).
As reg ard s th e direction, a serious defect can be observed w hich co rre sp o n d s
in th e p red ictio n s to a ro tatio n o f the c u rre n t rose o p p o se d to th a t o f th e
observ atio n s. T his results in a m ean sq u are e rro r o f 44° (w ith a m ean ab so lu te value
of 25°). T he sign ifican t deviations on the m o d u le seem m ainly due to a system atic
phase d ifferen ce in th e p red ictio n , in adv an ce, as fo r the p rev io u s p o in t P3, on th e
observation. A stu dy o f the h a rm o n ic co n stitu ents given in T able 5 show s th a t, in
fact, th ese differen ces are n o t only the re su lt o f a p h ase difference o f the
co n stitu en t M 2 (17° o r h a lf an h our) b u t o f a n am p litu d e fau lt o f 8.4 c m /s o n M 2
and o f 5 c m /s on S 2 a n d N 2 as well. O n th a t tab le one notices th e negative values
o f th e sem i-d iu rn al co n stitu en t rose ellipticity, in c o n trad ic tio n w ith the d a ta
inferred fro m o b servations. It is clear th a t th e g ridded division o f th e n um erical
m odel is n o t fine en o u g h a ro u n d C o ten tin to give results as satisfactory as in th e
o th er areas stu d ied .
u
D
TEMPS
TEMPS
90
0
-9 0
1) J j J
t
(J)
CJ)
d
j j
Jb r LlJw li' M UJUJ UJiliill
r illW
.^1
1
r
r
r jv
fVv r
V'
'S '
-160
TEMPS
CJ)
£
ip
TEMPS < J )
Fig .
10. — O bservation and prediction of tidal currents at point P4 (off Cherbourg).
TABLE 5
Harmonic constituents for tidal currents at point P4.
V alues o b serv ed in situ an d co m p u ted from a n u m erical sim ulation over one
m onth. V elocity in c m /s , p hase in degrees related to th e tra n sit o f th e co n stitu en t
at G reen w ich m erid ian , directio n in d egrees related to th e n o rth g eo graphical pole.
Cons­
tituent
Maximum
velocity
Obs.
Max. current
phase
Comp.
Max. current
direction
Ellipticity
Obs.
Comp.
Obs.
331.6
348.0
31.8
Comp.
Obs.
Comp.
5.0
69.7
89.0
o,
2.2
2.1
K,
2.7
2.1
41.3
51.0
3.7
5.0
64.9
91.0
m ns2
1.5
1.9
202.9
265.0
13.3
- 13.0
74.0
91.0
n2
27.5
22.6
176.8
183.0
13.4
-
7.0
79.4
91.0
m2
150.4
142.0
218.0
201.0
3.2
-
7.0
88.7
91.0
s2
39.0
44.0
253.4
249.0
-
7.0
83.9
91.0
MN,
2.2
1.3
274.8
123.0
18.2
-
5.0
24.8
85.0
m4
10.2
3.5
349.2
323.0
3.9
-
2.0
42.8
85.0
8.46
MS„
2.6
2.6
2.6
12.0
19.2
-
2.0
49.1
85.0
2 MS6
2.1
2.9
87.9
173.0
52.4
- 10.0
97.9
90.0
m6
4.1
3.5
172.5
129.0
46.0
-
8.0
92.9
91.0
2 MN6
2.3
1.7
136.1
117.0
47.8
-
7.0
95.8
90.0
4. C O N C L U SIO N S
W e have p re se n te d here a m odel fo r th e p red ic tio n o f the tid al cu rren ts in th e
English C h an n el. This m odel uses th e resu lts o f a n u m erical sim u latio n w hich h a d
previo u sly led to th e p ro d u c tio n o f a set o f ch arts defining th e characteristics o f
the m a jo r co n stitu en ts o f th e tid al sp ectru m . W ith th ese h arm o n ic c o n stitu e n ts,
reaso n ab ly precise p red ictio n s c a n b e m ad e w hich re p ro d u ce th e signal m o d u la ­
tions (sem i-d iu rn al, d iu rn al, sem i-m onthly, m onthly, etc.) as w ell as o th er ty p ical
aspects such as asym m etries b etw een flo o d an d ebb. M oreover, this m eth o d h as an
obvious p ractical adv an tag e, b ecau se th e co m p u tatio n eq u ip m en t re q u ire d is
m odest.
T h e m o d el h as b een checked by co m p arin g th e p re d ictio n s w ith som e in -situ
recordings. It is p e rtin e n t to n o te th a t these co m p ariso n s have b een m ade u sin g raw
recordings, w ith o u t filtering o f som e m easu red co n trib u tio n s n o t m odelized, such
as th e effects o f m eteorological c o n stra in ts; it sh o u ld also be u n d e rlin e d th a t o u r
results relate to th e m ean values o f velocities on each vertical, a n d are th u s n o t
valid fo r th e surface a n d th e b o tto m . In sp ite o f these shortcom ings, th e c o m p a ri­
sons b etw een p re d ic tio n s a n d o b serv atio n s are, in m ost cases, very fav o u ra b le :
ab o u t 10 c m /s sh o u ld be n o te d as ty p ical o f th e d eviations on th e m odules a n d
ab o u t 15° fo r th e in stan tan eo u s value o f in accu racy in d irection. T hese resu lts are
ra th e r su rp risin g given th e low sp atial resolution o f the num erical m odel on w hich
th e calcu latio n o f th e h arm o n ic c o n stitu en ts used fo r the pred ictio n s is based.
P assing o n to regional m o d els w ith finer grids w ould certainly give a b etter
accu racy for th e d e ta il o f th e c u rre n t patterns, p articu larly in coastal areas.
REFEREN CES
C. (1981) : Sur l’analyse et la prédiction des marées à partir d ’enregistrements
de courte durée. Thèse 3' cycle USMG-INPG, Grenoble, France.
B e r t h e r a t , C., C a r c e l , R., L e P r o v o s t , C. (1981) : Analyse de courants et niveaux en baie
de Seine. Rapport, contrat CNEXO n° 79/6041.
B e s s e r o (1980) : A new technique for the harmonie analysis of tidal currents and its
applications for the English Channel. Meteor. Forsch. Ergebnisse, n° 22, pp. 43-51.
C h a b e r t d ’HiERES, G. & L e P r o v o s t , C. (1979) : Atlas des composantes harmoniques de la
marée dans la Manche. Annales Hydrographiques, série 5, 6, 3, pp. 5-36.
D a r w i n , G.H. (1883) : Report on harmonie analysis of tidal observations. Brit. Assoc. Adv.
Sci. Rep. pp. 48-118.
D o o d s o n , A.T. (1921) : The harmonic development of the tide generating potential. Proceed­
ings o f the Royal Society A, Vol. 100, pp. 305-328.
F o r n e r in o , M. (1982) : Modélisation des courants de marée dans la Manche. Thèse de
Docteur-Ingénieur, INPG, Grenoble, 267 p.
L e P r o v o s t , C. (1976) : Theoretical analysis of the structure of the tidal wave’s spectrum in
shallow water areas. Mémoires Société Royale des Sciences de Liège, 6' série, tome X,
pp. 97-111.
Bertherat,
C. (1981) : A model for prediction of tidal elevation over the English Channel.
Oceanologica Acta, Vol. 4, n° 3, pp. 279-285.
L e P r o v o s t , C. & F o r n e r in o M. (1984) : Tidal spectroscopy of the English Channel with
a numerical model. To be published in Journal o f Physical Oceanography.
S c h u r e m a n , P. (1958) : Manual of harm onic analysis and prediction of tides. Spec. Publ. 98,
US Dept, of Commerce, Coast and Geod. Survey, U.S.A.
L e P rovost,